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+# A LATENT MORPHOLOGY MODEL FOR OPEN-VOCABULARY NEURAL MACHINE TRANSLATION
+
+Duygu Ataman*
+
+University of Zurich
+
+ataman@cl.uzh.ch
+
+Wilker Aziz
+
+University of Amsterdam w.aziz@uva.nl
+
+Alexandra Birch
+
+University of Edinburgh a.birch@ed.ac.uk
+
+# ABSTRACT
+
+Translation into morphologically-rich languages challenges neural machine translation (NMT) models with extremely sparse vocabularies where atomic treatment of surface forms is unrealistic. This problem is typically addressed by either pre-processing words into subword units or performing translation directly at the level of characters. The former is based on word segmentation algorithms optimized using corpus-level statistics with no regard to the translation task. The latter learns directly from translation data but requires rather deep architectures. In this paper, we propose to translate words by modeling word formation through a hierarchical latent variable model which mimics the process of morphological inflection. Our model generates words one character at a time by composing two latent representations: a continuous one, aimed at capturing the lexical semantics, and a set of (approximately) discrete features, aimed at capturing the morphosynthetic function, which are shared among different surface forms. Our model achieves better accuracy in translation into three morphologically-rich languages than conventional open-vocabulary NMT methods, while also demonstrating a better generalization capacity under low to mid-resource settings.
+
+# 1 INTRODUCTION
+
+Neural machine translation (NMT) models are conventionally trained by maximizing the likelihood of generating the target side of a bilingual parallel corpus of observations one word at a time conditioned on their full observed context. NMT models must therefore learn distributed representations that accurately predict word forms in very diverse contexts, a process that is highly demanding in terms of training data as well as the network capacity. Under conditions of lexical sparsity, which includes both the case of unknown words and the case of known words occurring in surprising contexts, the model is likely to struggle. Such adverse conditions are typical of translation involving morphologically-rich languages, where any single root may lead to exponentially many different surface realizations depending on its syntactic context. Such highly productive processes of word formation lead to many word forms being rarely or ever observed with a particular set of morphosyntactic attributes. The standard approach to overcome this limitation is to pre-process words into subword units that are shared among words, which are, in principle, more reliable as they are observed more frequently in varying context (Sennrich et al., 2016; Wu et al., 2016). One drawback related to this approach, however, is that the estimation of the subword vocabulary relies on word segmentation methods optimized using corpus-dependent statistics, disregarding any linguistic notion of morphology and the translation objective. This often produces subword units that are semantically ambiguous as they might be used in far too many lexical and syntactic contexts (Ataman et al., 2017). Moreover, in this approach, a word form is then generated by prediction of multiple subword units, which makes generalizing to unseen word forms more difficult due to the possibility that a subword unit necessary to reconstruct a given word form may be unlikely in a given context. To alleviate the sub-optimal effects of using explicit segmentation and generalize better to new morphological forms, recent studies explored the idea of extending NMT to model translation directly at
+
+the level of characters (Kreutzer & Sokolov, 2018; Cherry et al., 2018), which, in turn, have demonstrated the requirement of using comparably deeper networks, as the network would then need to learn longer distance grammatical dependencies (Sennrich, 2017).
+
+In this paper, we explore the benefits of explicitly modeling variation in surface forms of words using techniques from deep latent variable modeling in order to improve translation accuracy for low-resource and morphologically-rich languages. Latent variable models allow us to inject inductive biases relevant to the task, which, in our case, is word formation during translation. In order to formulate the process of morphological inflection, design a hierarchical latent variable model which translates words one character at a time based on word representations learned compositionally from sub-lexical components. In particular, for each word, our model generates two latent representations: i) a continuous-space dense vector aimed at capturing the lexical semantics of the word in a given context, and ii) a set of (approximately) discrete features aimed at capturing that word's morphosyntactic role in the sentence. We then see inflection as decoding a word form, one character at a time, from a learned composition of these two representations. By forcing the model to encode each word representation in terms of a more compact set of latent features, we encourage them to be shared across contexts and word forms, thus, facilitating generalization under sparse settings. We evaluate our method in translating English into three morphologically-rich languages each with a distinct morphological typology: Arabic, Czech and Turkish, and show that our model is able to obtain better translation accuracy and generalization capacity than conventional approaches to open-vocabulary NMT.
+
+# 2 NEURAL MACHINE TRANSLATION
+
+In this paper, we use recurrent NMT architectures based on the model developed by Bahdanau et al. (2014). The model essentially estimates the conditional probability of translating a source sequence  $x = \langle x_1, x_2, \ldots, x_m \rangle$  into a target sequence  $y = \langle y_1, y_2, \ldots, y_l \rangle$  via an exact factorization:
+
+$$
+p (y | x, \theta) = \prod_ {i = 1} ^ {l} p \left(y _ {j} \mid x, y _ {<   i}, \theta\right) \tag {1}
+$$
+
+where  $y_{<i}$  stands for the sequence preceding the  $i$ th target word. At each step of the sequence, a fixed neural network architecture maps its inputs, the source sentence and the target prefix, to the probability of the  $i$ th target word observation in context. In order to condition on the source sentence fully, this network employs an embedding layer and a bi-directional recurrent neural network (bi-RNN) based encoder. Conditioning on the target prefix  $y_{<i}$  is implemented using a recurrent neural network (RNN) based decoder, and an attention mechanism which summarises the source sentence into a context vector  $\mathbf{c}_i$  as a function of a given prefix (Luong et al., 2015). Given a parallel training set  $\mathcal{D}$ , the parameters  $\theta$  of the network are estimated to attain a local minimum of the negative log-likelihood function  $\mathcal{L}(\theta|\mathcal{D}) = -\sum_{x,y \sim \mathcal{D}} \log p(y|x,\theta)$  via stochastic gradient-based optimization (Bottou & Cun, 2004).
+
+Atomic parameterization estimates the probability of generating each target word  $y_{i}$  in a single shot:
+
+$$
+p \left(y _ {i} \mid x, y _ {<   i}, \theta\right) = \frac {\exp \left(\mathbf {E} _ {y _ {i}} ^ {\top} \mathbf {h} _ {i}\right)}{\sum_ {e = 1} ^ {v} \exp \left(\mathbf {E} _ {e} ^ {\top} \mathbf {h} _ {i}\right)}, \tag {2}
+$$
+
+where  $\mathbf{E} \in \mathbb{R}^{v \times d}$  is the target embedding matrix and the decoder output  $\mathbf{h}_i \in \mathbb{R}^d$  represents  $x$  and  $y_{<i}$ . Clearly, the size  $v$  of the target vocabulary plays an important role in determining the complexity of the model, which creates an important bottleneck when translating into low-resource and morphologically-rich languages due to the sparsity in the lexical distribution.
+
+Recent studies approached this problem by performing NMT with subword units, a popular one of which is based on the Byte-Pair Encoding algorithm (BPE; Sennrich et al., 2016), which finds the optimal description of a corpus vocabulary by iteratively merging the most frequent character sequences. Atomic parameterization could also be used to model translation at the level of characters, which is found to be advantageous in generalizing to morphological variations (Cherry et al., 2018).
+
+Hierarchical paramaterization further factorizes the probability of a target word in context:
+
+$$
+p \left(y _ {i} \mid x, y _ {<   i}, \theta\right) = \prod_ {j = 1} ^ {l _ {i}} p \left(y _ {i, j} \mid x, y _ {<   i}, y _ {i, <   j}, \theta\right) \tag {3}
+$$
+
+where the  $i$ th word  $y_{i} = \langle y_{i,1},\dots ,y_{i,l_{i}}\rangle$  is seen as a sequence of  $l_{i}$  characters. Generation follows one character at a time, each with probability computed by a fixed neural network architecture with varying inputs, namely, the source sentence  $x$ , the target prefix  $y_{< i}$ , and the prefix  $y_{i,< j}$  of characters already generated for that word. In this case there are two recurrent cells, one updated at the boundary of each token, much like in the standard case, and another updated at the character level. Luong & Manning (2016) propose hierarchical parameterization to compute the probability  $p(y_i|x,y_{< i},\theta)$  for unknown words, while for known words they use the atomic parameterization. In this paper, we use the hierarchical parameterization method for generating all target words, where we also augment the input embedding layer with a character-level bi-RNN, which computes each word representation  $\mathbf{y}_i$  as a composition of the embeddings of their characters (Ling et al., 2015).
+
+# 3 A LATENT MORPHOLOGY MODEL (LMM) FOR LEARNING WORD REPRESENTATIONS
+
+The application of a hierarchical structure for learning word representations in language modeling (Vania & Lopez, 2017) or semantic role labeling (Sahin & Steedman, 2018) have shown that such representations encode many cues about the morphological features of words by establishing a mapping between phonetic units and lexical context. Although it can provide an alternative solution to open-vocabulary NMT by potentially alleviating the need for subword segmentation, the quality of word representations learned by an hierarchical model is still highly dependant on the amount of observations (Sahin & Steedman, 2018; Ataman et al., 2019), since the training data is essential in properly modeling the lexical distribution. On the other hand, the process of word formation, particularly morphological inflection, has many properties that remain universal across languages, where a word is typically composed of a lemma, representing its lexical semantics, and a distinct combination of categorical inflectional features expressing the word's syntactic role in the phrase or sentence. In this paper, we propose to manipulate this universal structure in order to enforce an inductive bias on the prior distribution of words and allow the hierarchical parameterization model in properly learning lexical representations under conditions of data sparsity.
+
+# 3.1 GENERATIVE MODEL
+
+Our generative LMM for NMT formulates word formation in terms of a stochastic process, where each word is generated one character at a time by composing two latent representations: a continuous vector aimed at capturing the lemma, and a set of sparse features aimed at capturing the inflectional features. The motivation for using a stochastic model is twofold. First, deterministic models are by definition unimodal: when presented with the same input (the same context) they always produce the same output. When we model the word formation process, it is reasonable to expect a larger degree of ambiguity, that is, for the same context (e.g. a noun prefix), we may continue by inflecting the word differently depending on the (latent) mode of operation we are at (e.g. generating nominative, accusative or dative noun). Second, in stochastic models, the choice of distribution gives us a mechanism to favour a particular type of representation. In our case, we use sparse distributions for inflectional features to accommodate the fact that morphosyntactic features are discrete in nature. Our latent variable model is an instance of a variational auto-encoder (VAE; Kingma & Welling, 2013) inspired by the model of Zhou & Neubig (2017) for morphological reinflation.
+
+Generation of the  $i$ th word starts by sampling a Gaussian-distributed representation in context. This requires predicting the Gaussian location  $\mathbf{u}_i$  and scale  $\mathbf{s}_i$  vectors, $^1$
+
+$$
+Z _ {i} | x, y _ {<   i} \sim \mathcal {N} (\mathbf {u} _ {i}, \operatorname {d i a g} (\mathbf {s} _ {i} \odot \mathbf {s} _ {i}))
+$$
+
+$$
+\mathbf {u} _ {i} = \operatorname {d e n s e} \left(\mathbf {h} _ {i}; \theta_ {\mathrm {u}}\right) \tag {4}
+$$
+
+$$
+\mathbf {s} _ {i} = \zeta (\operatorname {d e n s e} (\mathbf {h} _ {i}; \theta_ {\mathrm {s}}))
+$$
+
+![](images/88048b9b2890461a9152485f9b37fc30627cdaca5ef267a59b2d5a252452bca8.jpg)  
+Figure 1: LMM for computing word representations while translating the sentence '... went home' into Turkish ('eve-(to)home gitti(he/she/it)went'). The character-level decoder is initialized with the attentional vector  $\mathbf{h}_i$  computed by the attention mechanism using current context  $\mathbf{c}_i$  and the word representation  $\mathbf{t}_i$  as in Luong & Manning (2016).
+
+where prediction of the location (in  $\mathbb{R}^d$ ) and scale (in  $\mathbb{R}_{>0}^d$ ) from the word-level decoder hidden state  $\mathbf{h}_i$  (which represents  $x$  and  $y_{<i}$ ) is performed by two dense layers, and the scale values are ensured to be positive with the softmax ( $\zeta$ ) activation.
+
+Generation proceeds by then sampling a  $K$ -dimensional vector  $f_{i}$  of sparse scalar features (see §3.2) conditioned on the source  $x$ , the target prefix  $y_{< i}$ , and the sampled lemma  $z_{i}$ . We model sampling of  $f_{i}$  conditioned on  $z_{i}$  in order to capture the insight that inflectional transformations typically depend on the category of a lemma. Having sampled  $f_{i}$  and  $z_{i}$ , the representation of the  $i$ th target word is computed by a transformation of  $z_{i}$  and  $f_{i}$ , i.e.  $\mathbf{t}_i = \mathrm{dense}([z_i,f_i];\theta_{\mathrm{comp}})$ .
+
+As shown in Figure 1, our model generates each word character by character auto-regressively by conditioning on the word representation  $\mathbf{t}_i$  predicted by the LMM, the current context  $\mathbf{c}_i$ , and the previously generated characters following the hierarchical parameterization. See Algorithm 1 for details on generation.
+
+Input: model parameters  $\theta$ , latent lemma  $z_{i}$ , latent morphological attributes  $f_{i}$ , observed character sequence  $\langle y_{i,1},\ldots ,y_{i,l_i}\rangle$  if training or placeholders if test, decoder state  $\mathbf{h}_i$ , and context vector  $\mathbf{c}_i$
+
+Result: updated decoder hidden state, prediction (a word), probability of prediction (for loss) initialization;
+
+$\mathbf{t}_i = \mathrm{dense}([z_i,f_i];\theta_{\mathrm{comp}})$
+
+initialize char-rnn with a projection of  $[\mathbf{t}_i,\mathbf{c}_i]$
+
+for  $j < l_{i}$  and  $j < \max$  do
+
+```txt
+compute output layer from char-rnn state ; if training then set prediction to observation  $y_{i,j}$  . else set prediction to arg max of output softmax layer; end assess log-probability of prediction ; update word-level RNN decoder with prediction;
+```
+
+# end
+
+Algorithm 1: Word generation: in training the word is observed, thus we only update the decoder and assess the probability of the observation, in test, we use mean values of the distributions to represent most likely values for  $z$  and  $f$  and populate predictions with beam-search.
+
+# 3.2 SPARSE FEATURES
+
+Since each target word  $y_{i}$  may have multiple inflectional features, ideally, we would like  $f_{i}$  to be  $K$  feature indicators, which could be achieved by sampling from  $K$  independent Bernoulli distributions parameterized in context. The problem with this approach is that sampling Bernoulli outcomes is non-differentiable, thus, their training requires gradient estimation via REINFORCE (Williams, 1992) and sophisticated variance reduction techniques. An alternative approach that has recently become popular is to use relaxations such as the Concrete distribution or Gumbel-Softmax (Maddison et al., 2017; Jang et al., 2017) in combination with the straight-through estimator (ST; Bengio et al., 2013). This is based on the idea of relaxing the discrete variable from taking on samples in the discrete set  $\{0,1\}$  to taking on samples in the continuous set  $(0,1)$  using a distribution for which a reparameterization exists (e.g. Gumbel). Then, a non-differentiable activation (e.g. a threshold function) maps continuous outcomes to discrete ones. ST simply ignores the discontinuous activation in the backward pass, i.e. it assumes the Jacobian is the identity matrix. This does lead to biased estimates of the gradient of the loss, which is in conflict with the requirements behind stochastic optimization (Robbins & Monro, 1951).
+
+An alternative presented by Louizos et al. (2018) achieves a different compromise, it gets rid of bias at the cost of mixing both sparse and dense outcomes. The idea is to obtain a continuous sample  $c \in (0, 1)$  from a distribution for which a reparameterization exists and stretch it to a continuous support  $(l, r) \supset (0, 1)$  using a simple linear transformation  $s = l + (r - l)c$ . A rectifier is then employed to map the negative outcomes to 0 and the positive outcomes larger than one to 1, i.e.  $f = \min(1, \max(0, s))$ . The rectifier is only non-differentiable at  $s = 0$  and at  $s = 1$ , however, because the stretched variable  $s$  is sampled from a continuous distribution, the chance of sampling  $s = 0$  and  $s = 1$  is essentially 0. This stretched-and-rectified distribution allows: i) the sampling procedure to become differentiable with respect to the parameters of the distribution, ii) to sample sparse outcomes with an unbiased estimator, and iii) to calculate the probability of sampling  $f = 0$  and  $f = 1$  in closed form as a function of the parameters of the underlying distribution, which corresponds to the probability of sampling  $s < 0$  and  $s > 1$ , respectively.
+
+In their paper, Louizos et al. (2018) used the BinaryConcrete (or Gumbel-Sigmoid) as the underlying continuous distribution, the sparsity of which is controlled via a temperature parameter. However, in our study, we found this parameter difficult to predict, since it is very hard to allow a neural network to control its value without unstable gradient updates. Instead, we opt for a slight variant by Bastings et al. (2019) based on the Kumaraswamy distribution (Kumaraswamy, 1980), a two-parameters distribution that closely resembles a Beta distribution and is sparse whenever its (strictly positive) parameters are between 0 and 1. In the context of text classification, Bastings et al. (2019) shows this stretch-and-rectify technique to work better than methods based on REINFORCE.
+
+For each token  $y_{i}$ , we sample  $K$  independent Kumaraswamy variables in context,
+
+$$
+C _ {i, k} | x, y <   i, z _ {i} \sim \operatorname {K u m a} \left(a _ {i, k}, b _ {i, k}\right) \quad k = 1, \dots , K \tag {5}
+$$
+
+$$
+[ \mathbf {a} _ {i}, \mathbf {b} _ {i} ] = \zeta (\operatorname {d e n s e} ([ z _ {i}, \mathbf {h} _ {i} ]; \theta_ {\mathrm {a b}}))
+$$
+
+which makes a continuous random vector  $c_{i}$  in the support  $(0,1)^{K.4}$ . We then stretch-and-rectify the samples via  $f_{i,k} = \min(1, \max(0, l - (r - l)c_{i,k}))$  making  $f_{i}$  a random vector in the support  $[0,1]^{K.5}$ . The probability that  $f_{i,k}$  is exactly 0 is
+
+$$
+\pi_ {i, k} ^ {\{0 \}} = \int_ {0} ^ {\frac {- l}{r - l}} \operatorname {K u m a} (c | a _ {i, k}, b _ {i, k}) \mathrm {d} c \tag {6a}
+$$
+
+and the probability that  $f_{i,k}$  is exactly 1 is
+
+$$
+\pi_ {i, k} ^ {\{1 \}} = 1 - \int_ {0} ^ {\frac {1 - l}{r - l}} \operatorname {K u m a} (c | a _ {i, k}, b _ {i, k}) \mathrm {d} c \tag {6b}
+$$
+
+and therefore the complement
+
+$$
+\pi_ {i, k} ^ {(0, 1)} = 1 - \pi_ {i, k} ^ {\{0 \}} - \pi_ {i, k} ^ {\{1 \}} \tag {6c}
+$$
+
+is the probability that  $f_{i,k}$  be any continuous value in the open set  $(0,1)$ . In §3.4, we will derive regularizers based on  $\pi_{i,k}^{(0,1)}$  to promote sparse outcomes to be sampled with large probability.
+
+# 3.3 PARAMETER ESTIMATION
+
+Parameter estimation of neural network models is typically done via maximum-likelihood estimation (MLE), where we approach a local minimum of the negative log-likelihood function via stochastic gradient descent with gradient computation automated by the back-propagation algorithm. Using the following shorthand notation:
+
+$$
+\alpha \left(z _ {i}\right) \triangleq p \left(z _ {i} \mid x, y _ {<   i}, z _ {<   i}, f _ {<   i}, \theta\right) \tag {7a}
+$$
+
+$$
+\beta \left(f _ {i}\right) \triangleq \prod_ {k = 1} ^ {K} p \left(f _ {i, k} \mid x, y _ {<   i}, z _ {<   i}, f _ {<   i}, z _ {i}, \theta\right) \tag {7b}
+$$
+
+$$
+\gamma \left(y _ {i}\right) \triangleq \prod_ {j = 1} ^ {l _ {i}} p \left(y _ {i, j} \mid x, y _ {<   i}, z _ {\leq i}, f _ {\leq i}, y _ {i, <   j}, \theta\right). \tag {7c}
+$$
+
+The log-likelihood for a single data point can be formulated as:
+
+$$
+\log p (y | x, \theta) = \log \int \prod_ {i = 1} ^ {l} \alpha \left(z _ {i}\right) \beta \left(f _ {i}\right) \gamma \left(y _ {i}\right) \mathrm {d} z \mathrm {d} f \tag {8}
+$$
+
+the computation of which is intractable. Instead, we resort to variational inference (VI; Jordan et al., 1999), where we optimize a lower-bound on the log-likelihood
+
+$$
+\mathbb {E} _ {q (z, f | x, y, \lambda)} \left[ \sum_ {i = 1} ^ {l} \log \frac {\alpha \left(z _ {i}\right) \beta \left(f _ {i}\right) \gamma \left(y _ {i}\right)}{q (z , f | x , \lambda)} \right] \tag {9}
+$$
+
+expressed with respect to an independently parameterized posterior approximation  $q(z, f|x, y, \lambda)$ . For as long as sampling from the posterior is tractable and can be performed via a reparameterization, we can rely on stochastic gradient-based optimization. In order to have a compact parameterization, we choose
+
+$$
+q (z, f | x, y, \lambda) := \prod_ {i = 1} ^ {l} \alpha \left(z _ {i}\right) \beta \left(f _ {i}\right). \tag {10}
+$$
+
+This simplifies the lowerbound, which then takes the form of  $l$  nested expectations, the  $i$ th of which is  $\mathbb{E}_{\alpha (z_i)\beta (f_i)}[\log \gamma (y_i)]$ . This is similar to the stochastic decoder of Schulz et al. (2018), though our approximate posterior is in fact, also our parameterized prior. Although this objective does not particularly promote sparsity, we employ sparsity-inducing regularization techniques that will be discussed in the next section.
+
+Concretely, for a given source sentence  $x$ , target prefix  $y_{<i}$ , and a latent sample  $z_{\leq i}, f_{\leq i}$ , we obtain a single-sample estimate of the loss by computing  $\mathcal{L}_i(\theta) = -\log \gamma(y_i)$ .
+
+# 3.4 REGULARIZATION
+
+In order to promote sparse distributions for the inflectional features, we apply a regularizer inspired by expected  $L_{0}$  regularization (Louizos et al., 2018). Whereas  $L_{0}$  is a penalty based on the number of non-zero outcomes, we design a penalty based on the expected number of continuous outcomes, which corresponds to  $\pi_{i,k}^{(0,1)}$  as shown in Equation (6). For a given source sentence  $x$ , target prefix  $y_{<i}$ , and a latent sample  $z_{<i}$ ,  $f_{<i}$ , we aggregate this penalty for each feature
+
+$$
+\mathcal {R} _ {i} (\theta) = \sum_ {k = 1} ^ {K} \pi_ {i, k} ^ {(0, 1)} \tag {11}
+$$
+
+and add it to the cost function with a positive weight  $\rho$ . The final loss of the NMT model is
+
+$$
+\mathcal {L} (\theta | \mathcal {D}) = \sum_ {x, y \sim \mathcal {D}} \sum_ {i = 1} ^ {| y |} \mathcal {L} _ {i} (\theta) + \rho \mathcal {R} _ {i} (\theta). \tag {12}
+$$
+
+# 3.5 PREDICTIONS
+
+In our model, obtaining the conditional likelihood for predicting the most likely hypothesis requires marginalisation of the latent variables, which is intractable. An alternative approach is to heuristically search through the joint distribution,
+
+$$
+\underset {y, z, f} {\arg \max } p (y, z, f | x), \tag {13}
+$$
+
+rather than the marginal, an approximation that has been referred to as Viterbi decoding (Smith, 2011). During beam search, we populate the beam with alternative target words, and for each prefix  $y_{<i}$  in the beam, we resort to deterministically choosing the latent variables based on a single sample which we deem representative of their distributions, which is a common heuristic in VAEs for translation (Zhang et al., 2016; Schulz et al., 2018). For unimodal distributions, such as the Gaussian  $p(z_i|x, y_{<i}, z_{<i}, f_{<i})$ , we use the analytic mean, whereas for multimodal distributions, such as the Hard Kumaraswamy  $p(f_i|x, y_{<i}, z_{\leq i}, f_{<i})$ , we use the argmax.
+
+# 4 EVALUATION
+
+# 4.1 MODELS
+
+We evaluate our model by comparing it in machine translation against three baselines which constitute the conventional open-vocabulary NMT methods, including architectures using atomic parameterization either with subword units segmented with BPE (Sennrich et al., 2016) or characters, and the hierarchical parameterization method employed for generating all words in the output. We implement all architectures using Pytorch (Paszke et al., 2017) within the OpenNMT-py framework (Klein et al., 2017)<sup>8</sup>.
+
+# 4.2 DATA AND LANGUAGES
+
+In order to evaluate our model we design two sets of experiments. The experiments in §4.4.1 aim to evaluate different methods under low-resource settings, for languages with different morphological typology. We model the machine translation task from English into three languages with distinct morphological characteristics: Arabic (templatic), Czech (fusional), and Turkish (agglutinative). We use the TED Talks corpora (Cettolo, 2012) for training the NMT models for these experiments. In §4.4.3, we conduct more experiments in Turkish to demonstrate the case of increased data sparsity using multi-domain training corpora, where we extend the training set using corpora from EU Bookshop (Skadin's et al., 2014), Global Voices, Gnome, Tatoeba, Ubuntu (Tiedemann, 2012), KDE4 (Tiedemann, 2009), Open Subtitles (Lison & Tiedemann, 2016) and SETIMES (Tyers & Alperen, 2010) $^{9}$ . The statistical characteristics of the training sets are given in Tables 4 and 5. We use the official evaluation sets of the IWSLT $^{10}$  for validating and testing the accuracy of the models. In order to increase the number of unknown and rare words in the evaluation sets we measure accuracy on large test sets combining evaluation sets from many years (Table 6 presents the evaluation sets used for development and testing). The accuracy of each model output is measured using BLEU (Papineni et al., 2002) and chrF3 (Popovic, 2015) metrics, whereas the significance of the improvements are computed using bootstrap hypothesis testing (Clark et al., 2011). In order to measure the accuracy in predicting the correct syntactic description of the references, we also compute BLEU
+
+scores over the output sentences segmented using a morphological analyzer. We use the AlKhalil Morphosys (Boudchiche et al., 2017) for segmenting Arabic, Morphidata (Straková et al., 2014) for segmenting Czech and the morphological lexicon model of Offlazer (Offlazer & Kuruöz, 1994) and disambiguation tool of Sak (Sak et al., 2007) for segmenting Turkish sentences into sequences of lemmas and morphological features.
+
+# 4.3 TRAINING SETTINGS
+
+All models are implemented using gated recurrent units (GRU) (Cho et al., 2014), and have a single-layer bi-RNN encoder. The source sides of the data used for training all NMT models, and the target sides of the data used in training the subword-level NMT models are segmented using BPE with 16,000 merge rules. We implement all decoders using a comparable number of GRU parameters, including 3-layer stacked-GRU subword and character-level decoders, where the attention is computed after the  $1^{st}$  layer (Barone et al., 2017) and a 3-layer hierarchical decoder which implements the attention mechanism after the  $2^{nd}$  layer. All models use an embedding dimension and GRU size of 512. LMM uses the same hierarchical GRU architecture, where the middle layer is augmented using 4 multi-layer perceptrons with 256 hidden units. We use a lemma vector dimension of 150, 10 inflectional features (See §A.3 for experiments conducted to tune the feature dimensions) and set the regularization constant to  $\rho = 0.4$ . All models are trained using the Adam optimizer (Kinga & Ba, 2014) with a batch size of 100, dropout rate of 0.2, learning rate of 0.0004 and learning rate decay of 0.8, applied when the perplexity does not decrease at a given epoch.[11] Translations are generated with beam search with a beam size of 5, where the hierarchical models implement the hierarchical beam search algorithm (Ataman et al., 2019).
+
+# 4.4 RESULTS
+
+# 4.4.1 THE EFFECT OF MORPHOLOGICAL TYPLOGY
+
+The experiment results given in Table 1 show the performance of each model in translating English into Arabic, Czech and Turkish. In Turkish, the most sparse target language in our benchmark with rich agglutinative morphology, using character-based decoding shows to be more advantageous compared to the subword-level and hierarchical models, suggesting that increased granularity in the vocabulary units might aid in better learning accurate representations under conditions of high data sparsity. In Arabic, on the other hand, using a hierarchical decoding model shows to be advantageous compared to the subword and character-level models, as it might be useful in better learning syntactic dependencies. LMM obtains improvements of 0.51 and 0.30 BLEU points in Arabic and Turkish over the best performing baselines, respectively. The fact that our model can efficiently work in both Arabic and Turkish confirms that it can handle the generation of both concatenative and non-concatenative morphological transformations. The results in the English-to-Czech translation direction do not indicate a specific advantage of using either method for generating fusional morphology, where morphemes are already optimized at the surface level, although our model is still able to achieve translation accuracy comparable to the character and subword-level models.
+
+# 4.4.2 PREDICTING UNSEEN WORDS
+
+In addition to the general machine translation evaluation using automatic metrics, we perform a more focused statistical analysis to illustrate the performance of different methods in predicting unseen words by computing the average perplexity per character on the input sentences which contain out-of-vocabulary (OOV) words as suggested by Cotterell et al. (2018). We also analyze the outputs generated by each decoder in terms of the frequency of unknown words in each model output and the Kullback-Leibler (KL) divergence between the character trigram distributions of the references and outputs, which represents the coherence between the statistical distribution learned by each model and the reference translations.
+
+Our analysis results generally confirm the advantage of increased granularity during the generation of unseen words, where the character-level decoder can generate a higher rate of unseen word forms and higher KL-divergence with the reference, suggesting superior ability in generalizing to new
+
+<table><tr><td rowspan="3">Model</td><td colspan="9">(only in-domain)</td></tr><tr><td rowspan="2">BLEU</td><td rowspan="2">AR t-BLEU</td><td rowspan="2">chrF3</td><td colspan="3">CS</td><td colspan="3">TR</td></tr><tr><td>BLEU</td><td>t-BLEU</td><td>chrF3</td><td>BLEU</td><td>t-BLEU</td><td>chrF3</td></tr><tr><td>Subwords</td><td>14.27</td><td>51.24</td><td>0.3927</td><td>16.60</td><td>54.22</td><td>0.4123</td><td>8.52</td><td>38.03</td><td>0.3763</td></tr><tr><td>Char.s</td><td>12.72</td><td>47.56</td><td>0.3804</td><td>16.94</td><td>52.80</td><td>0.4103</td><td>10.63</td><td>40.63</td><td>0.3810</td></tr><tr><td>Hierarch.</td><td>15.55</td><td>54.01</td><td>0.4154</td><td>16.79</td><td>48.27</td><td>0.4068</td><td>9.74</td><td>35.91</td><td>0.3771</td></tr><tr><td>LMM</td><td>16.06</td><td>55.97</td><td>0.4251</td><td>16.97</td><td>50.35</td><td>0.4095</td><td>10.93</td><td>45.47</td><td>0.3889</td></tr></table>
+
+<table><tr><td rowspan="3">Model</td><td colspan="3">(multi-domain)</td></tr><tr><td colspan="3">TR</td></tr><tr><td>BLEU</td><td>t-BLEU</td><td>chrF3</td></tr><tr><td>Subwords</td><td>10.42</td><td>42.65</td><td>0.3722</td></tr><tr><td>Char.s</td><td>8.94</td><td>37.12</td><td>0.3274</td></tr><tr><td>Hierarch.</td><td>10.35</td><td>40.54</td><td>0.3870</td></tr><tr><td>LMM</td><td>11.48</td><td>48.23</td><td>0.3939</td></tr></table>
+
+output and not necessarily copying previous observations as the subword-level model, however, this advantage is more visible in Turkish and less in Czech or Arabic. The hierarchical decoder which performs the search at the level of words, on the other hand, behaves with less uncertainty in terms of the perplexity values although it cannot demonstrate the ability to generalize to new forms and neither can closely capture the actual distribution in the target language.
+
+Due to its stochastic nature, our model yields higher perplexity values compared to the hierarchical model, whereas the values range between subword and character-based models, possibly finding an optimal level of granularity between the two solutions. The KL-divergence and OOV rates confirm that our model has the potential in better generalize to new word forms as well as different morphological typology.
+
+Table 1: Above: Machine translation accuracy in Arabic (AR), Czech (CS) and Turkish (TR) in terms of BLEU and ChrF3 metrics as well as BLEU scores computed on the output sentences tagged with the morphological analyzer (t-BLEU) using in-domain training data. Below: The performance of models trained with multi-domain data. Best scores are in bold. All improvements over the baselines are statistically significant (p-value  $< 0.05$ ).  
+
+<table><tr><td>Model</td><td>OOV%</td><td>AR Ppl</td><td>KL-Div</td><td>OOV%</td><td>CS Ppl</td><td>KL-Div</td><td>OOV%</td><td>TR Ppl</td><td>KL-Div</td></tr><tr><td>Subwords</td><td>1.75</td><td>2.84</td><td>12,871</td><td>2.39</td><td>2.62</td><td>8,954</td><td>3.54</td><td>2.78</td><td>17,342</td></tr><tr><td>Char.s</td><td>3.08</td><td>2.46</td><td>29,607</td><td>1.90</td><td>2.61</td><td>17,092</td><td>4.28</td><td>2.38</td><td>38,043</td></tr><tr><td>Hierarch.</td><td>1.96</td><td>2.59</td><td>15,064</td><td>0.87</td><td>2.65</td><td>29,022</td><td>1.53</td><td>2.46</td><td>68,743</td></tr><tr><td>LMM</td><td>3.78</td><td>2.68</td><td>9,892</td><td>2.4</td><td>2.71</td><td>14,296</td><td>4.89</td><td>2.59</td><td>38,930</td></tr></table>
+
+Table 2: Percentage of out-of-vocabulary (OOV) words in the output, normalized perplexity measures (PPl) per characters and the KL divergence between the reference and outputs of systems trained with in-domain data on different language directions.
+
+# 4.4.3 THE EFFECT OF DATA SIZE
+
+Repeating the experiments in the English-to-Turkish translation direction by increasing the amount of training data with multi-domain corpora demonstrates a more challenging case, where there is a greater possibility of observing new words in varying context, either in the form of morphological inflections due to previously unobserved syntactic conditions, or a larger vocabulary extended with terminology from different domains. In this experiment, the character-level model experiences a drop in performance and its accuracy is much lower than the subword-level one, suggesting that its capacity cannot cope with the increased amount of training data. Empirical results suggest that with increased capacity, character-level models carry the potential to reach comparable performance to subword-level models (Cherry et al., 2018). On the other hand, our model reaches a much larger improvement of 0.82 BLEU points over the subword-level and 2.54 BLEU points over the character-level decoders, suggesting that it could make use of the increased amount of observations for improving the translation performance, which possibly aid the morphology model in becoming more accurate.
+
+# 4.4.4 THE IMPACT OF INFLECTIONAL FEATURES
+
+In order to understand whether the latent inflectional features in fact capture information about variations related to morphological transformations, we first try generating different surface forms of the same lemma by sampling a lemma vector with LMM for the input word 'go' and generating outputs using the fixed lemma vector and assigning different values to the inflectional features. In the second experiment, we assess the impact of the inflectional features by setting all features  $f$  to 0 and translating a set of English sentences with varying inflected forms in Turkish. Table 3 presents different sets of feature values and the corresponding outputs generated by the decoder and the outputs generated with or without the inflectional component.
+
+<table><tr><td>Features</td><td>Output</td><td>English Translation</td></tr><tr><td>[1,1,1,1,1,1,1,1,1]</td><td>git</td><td>go (informal)</td></tr><tr><td>[0,1,1,1,1,1,1,1,1]</td><td>‘a git</td><td>to go</td></tr><tr><td>[0,1,0,1,1,1,1,1,1,1]</td><td>‘da git</td><td>at go</td></tr><tr><td>[0,0,0,1,1,0,0,1,1,0]</td><td>gidin</td><td>go (formal)</td></tr><tr><td>[1,1,0,0,0,0,1,0,1,1]</td><td>gitmek</td><td>to go (infinitive)</td></tr><tr><td>[0,0,1,0,0,0,0,0,0,1]</td><td>gidiyor</td><td>(he/she/it is) going</td></tr><tr><td>[0,0,0,0,0,0,0,0,1,0]</td><td>gidip</td><td>by going (gerund)</td></tr><tr><td>[0,0,1,1,0,0,1,0,1,0]</td><td>gidiyoruz</td><td>(we are) going</td></tr><tr><td>Input</td><td>Output with f</td><td>Output without f</td></tr><tr><td>he went home.</td><td>eve gitti.</td><td>eve gitti.</td></tr><tr><td>he came from home.</td><td>evden geldi.</td><td>eve geldi.</td></tr><tr><td>it is good to be home.</td><td>evde olmak iyi.</td><td>evde olmak iyi.</td></tr><tr><td>his home has red walls.</td><td>evinde kırkızı duvarlar var.</td><td>evde kırkızı duvar var.</td></tr></table>
+
+Table 3: Above: Outputs of LMM based on the lemma 'git' ('go') and different sets of inflectional features. Below: Examples of predicting inflections in context with or without using features.
+
+The model generates different surface forms for different sets of features, confirming that the latent variables represent morphological features related to the infinitive form of the verb, as well as its formality conditions, prepositions, person, number and tense. Decoding the set of sentences given in the second experiment LMM always generates the correct inflectional form, although when the feature values are set to 0 the model omits some inflectional features in the output, suggesting that despite partially relying on the source-side context, it still encodes important information for generating correct surface forms in the inflectional features.
+
+# 5 CONCLUSION
+
+In this paper we presented a novel decoding architecture for NMT employing a hierarchical latent variable model to promote sparsity in lexical representations, which demonstrated promising application for morphologically-rich and low-resource languages. Our model generates words one character at a time by composing two latent features representing their lemmas and inflectional features. We evaluate our model against conventional open-vocabulary NMT solutions such as subword and character-level decoding methods in translation English into three morphologically-rich languages with different morphological typologies under low to mid-resource settings. Our results show that our model can significantly outperform subword-level NMT models, whereas demonstrates better capacity than character-level models in coping with increased amounts of data sparsity. We also conduct ablation studies on the impact of feature variations to the predictions, which prove that despite being completely unsupervised, our model can in fact manage to learn morphosyntactic information and make use of it to generalize to different surface forms of words.
+
+# 6 ACKNOWLEDGMENTS
+
+The authors would like to thank Marcello Federico, Orhan First, Adam Lopez, Graham Neubig, Akash Srivastava and Clara Vania for their feedback and suggestions. This project received funding from the European Union's Horizon 2020 research and innovation programme under grant agreements 825299 (GoURMET) and 688139 (SUMMA).
+
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+
+# A APPENDIX
+
+A.1 THE STATISTICAL CHARACTERISTICS OF EXPERIMENTAL DATA  
+
+<table><tr><td rowspan="2">Language Pair</td><td rowspan="2"># sentences</td><td colspan="2"># tokens</td><td colspan="2"># types</td></tr><tr><td>Source</td><td>Target</td><td>Source</td><td>Target</td></tr><tr><td>English-Arabic</td><td>238K</td><td>5M</td><td>4M</td><td>120K</td><td>220K</td></tr><tr><td>English-Czech</td><td>118K</td><td>2M</td><td>2M</td><td>50K</td><td>118K</td></tr><tr><td>English-Turkish</td><td>136K</td><td>2M</td><td>3M</td><td>53K</td><td>171K</td></tr></table>
+
+Table 4: Training sets based on the TED Talks corpora (M: Million, K: Thousand).  
+
+<table><tr><td rowspan="2">Language Pair</td><td rowspan="2"># sentences</td><td colspan="2"># tokens</td><td colspan="2"># types</td></tr><tr><td>Source</td><td>Target</td><td>Source</td><td>Target</td></tr><tr><td>English-Turkish</td><td>434K</td><td>8M</td><td>6M</td><td>135K</td><td>373K</td></tr></table>
+
+Table 5: The multi-domain training set (M: Million, K: Thousand).  
+
+<table><tr><td>Language</td><td colspan="2">Data sets</td><td># sentences</td></tr><tr><td rowspan="3">English-Arabic</td><td>Development</td><td>dev2010, test2010</td><td>6K</td></tr><tr><td rowspan="2">Testing</td><td>test2011, test2012</td><td rowspan="2">4K</td></tr><tr><td>test2013, test2014</td></tr><tr><td rowspan="3">English-Czech</td><td>Development</td><td>dev2010, test2010,</td><td>3K</td></tr><tr><td rowspan="2">Testing</td><td>test2011</td><td rowspan="2">3K</td></tr><tr><td>test2012, test2013</td></tr><tr><td rowspan="2">English-Turkish</td><td>Development</td><td>dev2010, test2010</td><td>3K</td></tr><tr><td>Testing</td><td>test2011, test2012</td><td>3K</td></tr></table>
+
+Table 6: Development and testing sets (K: Thousand).
+
+# A.2 THE KUMARASWAMY DISTRIBUTION
+
+![](images/581355b878caa4c7b22c96e6f10c54da39032ea97d7fea880de0ac9c9a2ccd4d.jpg)  
+Figure 2: The top row shows the density function of the continuous base distribution over  $(0,1)$ . The middle row shows the result of stretching it to include 0 and 1 in its support. The bottom row shows the result of rectification: probability mass under  $(l,0)$  collapses to 0 and probability mass under  $(1,r)$  collapses to 1, which cause sparse outcomes to have non-zero mass. Varying the shape parameters  $(a,b)$  of the underlying continuous distribution changes how much mass concentrates outside the support  $(0,1)$  in the stretched density, and hence the probability of sampling sparse outcomes.
+
+# A.3 THE EFFECT OF FEATURE DIMENSIONS
+
+We investigate the optimal lemma and inflectional feature sizes by measuring the accuracy in English-to-Turkish translation using different feature vector dimensions. The results given in Figure 3 show that gradually compressing the word representations computed by recurrent hidden states, with an original dimension of 512, from 500 to 100, leads to increased output accuracy, suggesting that encoding more compact representations might provide the model with a better generalization capability. Our results also show that using a feature dimension of 10 is sufficient in reaching the best accuracy.
+
+![](images/24aa99ee8b4171825d54a0b650e1e58336ee980ab5adf8d2441e3a67347c39ce.jpg)  
+Figure 3: The effect of feature dimensions on translation accuracy in Turkish.
\ No newline at end of file
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+# ALBERT: A LITE BERT FOR SELF-SUPERVISED LEARNING OF LANGUAGE REPRESENTATIONS
+
+Zhenzhong Lan $^{1}$  Mingda Chen $^{2*}$  Sebastian Goodman $^{1}$  Kevin Gimpel $^{2}$
+
+Piyush Sharma<sup>1</sup> Radu Soricut<sup>1</sup>
+
+$^{1}$ Google Research  $^{2}$ Toyota Technological Institute at Chicago
+
+{lanzhzh, seabass, piyushsharma, rsoricut}@google.com {mchen, kgimpel}@ttic.edu
+
+# ABSTRACT
+
+Increasing model size when pretraining natural language representations often results in improved performance on downstream tasks. However, at some point further model increases become harder due to GPU/TPU memory limitations and longer training times. To address these problems, we present two parameter-reduction techniques to lower memory consumption and increase the training speed of BERT (Devlin et al., 2019). Comprehensive empirical evidence shows that our proposed methods lead to models that scale much better compared to the original BERT. We also use a self-supervised loss that focuses on modeling inter-sentence coherence, and show it consistently helps downstream tasks with multi-sentence inputs. As a result, our best model establishes new state-of-the-art results on the GLUE, RACE, and SQuAD benchmarks while having fewer parameters compared to BERT-large. The code and the pretrained models are available at https://github.com/google-research/ALBERT.
+
+# 1 INTRODUCTION
+
+Full network pre-training (Dai & Le, 2015; Radford et al., 2018; Devlin et al., 2019; Howard & Ruder, 2018) has led to a series of breakthroughs in language representation learning. Many nontrivial NLP tasks, including those that have limited training data, have greatly benefited from these pre-trained models. One of the most compelling signs of these breakthroughs is the evolution of machine performance on a reading comprehension task designed for middle and high-school English exams in China, the RACE test (Lai et al., 2017): the paper that originally describes the task and formulates the modeling challenge reports then state-of-the-art machine accuracy at  $44.1\%$ ; the latest published result reports their model performance at  $83.2\%$  (Liu et al., 2019); the work we present here pushes it even higher to  $89.4\%$ , a stunning  $45.3\%$  improvement that is mainly attributable to our current ability to build high-performance pretrained language representations.
+
+Evidence from these improvements reveals that a large network is of crucial importance for achieving state-of-the-art performance (Devlin et al., 2019; Radford et al., 2019). It has become common practice to pre-train large models and distill them down to smaller ones (Sun et al., 2019; Turc et al., 2019) for real applications. Given the importance of model size, we ask: Is having better NLP models as easy as having larger models?
+
+An obstacle to answering this question is the memory limitations of available hardware. Given that current state-of-the-art models often have hundreds of millions or even billions of parameters, it is easy to hit these limitations as we try to scale our models. Training speed can also be significantly hampered in distributed training, as the communication overhead is directly proportional to the number of parameters in the model.
+
+Existing solutions to the aforementioned problems include model parallelization (Shazeer et al., 2018; Shoeybi et al., 2019) and clever memory management (Chen et al., 2016; Gomez et al., 2017).
+
+These solutions address the memory limitation problem, but not the communication overhead. In this paper, we address all of the aforementioned problems, by designing A Lite BERT (ALBERT) architecture that has significantly fewer parameters than a traditional BERT architecture.
+
+ALBERT incorporates two parameter reduction techniques that lift the major obstacles in scaling pre-trained models. The first one is a factorized embedding parameterization. By decomposing the large vocabulary embedding matrix into two small matrices, we separate the size of the hidden layers from the size of vocabulary embedding. This separation makes it easier to grow the hidden size without significantly increasing the parameter size of the vocabulary embeddings. The second technique is cross-layer parameter sharing. This technique prevents the parameter from growing with the depth of the network. Both techniques significantly reduce the number of parameters for BERT without seriously hurting performance, thus improving parameter-efficiency. An ALBERT configuration similar to BERT-large has 18x fewer parameters and can be trained about 1.7x faster. The parameter reduction techniques also act as a form of regularization that stabilizes the training and helps with generalization.
+
+To further improve the performance of ALBERT, we also introduce a self-supervised loss for sentence-order prediction (SOP). SOP primary focuses on inter-sentence coherence and is designed to address the ineffectiveness (Yang et al., 2019; Liu et al., 2019) of the next sentence prediction (NSP) loss proposed in the original BERT.
+
+As a result of these design decisions, we are able to scale up to much larger ALBERT configurations that still have fewer parameters than BERT-large but achieve significantly better performance. We establish new state-of-the-art results on the well-known GLUE, SQuAD, and RACE benchmarks for natural language understanding. Specifically, we push the RACE accuracy to  $89.4\%$ , the GLUE benchmark to 89.4, and the F1 score of SQuAD 2.0 to 92.2.
+
+# 2 RELATED WORK
+
+# 2.1 SCALING UP REPRESENTATION LEARNING FOR NATURAL LANGUAGE
+
+Learning representations of natural language has been shown to be useful for a wide range of NLP tasks and has been widely adopted (Mikolov et al., 2013; Le & Mikolov, 2014; Dai & Le, 2015; Peters et al., 2018; Devlin et al., 2019; Radford et al., 2018; 2019). One of the most significant changes in the last two years is the shift from pre-training word embeddings, whether standard (Mikolov et al., 2013; Pennington et al., 2014) or contextualized (McCann et al., 2017; Peters et al., 2018), to full-network pre-training followed by task-specific fine-tuning (Dai & Le, 2015; Radford et al., 2018; Devlin et al., 2019). In this line of work, it is often shown that larger model size improves performance. For example, Devlin et al. (2019) show that across three selected natural language understanding tasks, using larger hidden size, more hidden layers, and more attention heads always leads to better performance. However, they stop at a hidden size of 1024, presumably because of the model size and computation cost problems.
+
+It is difficult to experiment with large models due to computational constraints, especially in terms of GPU/TPU memory limitations. Given that current state-of-the-art models often have hundreds of millions or even billions of parameters, we can easily hit memory limits. To address this issue, Chen et al. (2016) propose a method called gradient checkpointing to reduce the memory requirement to be sublinear at the cost of an extra forward pass. Gomez et al. (2017) propose a way to reconstruct each layer's activations from the next layer so that they do not need to store the intermediate activations. Both methods reduce the memory consumption at the cost of speed. Raffel et al. (2019) proposed to use model parallelization to train a giant model. In contrast, our parameter-reduction techniques reduce memory consumption and increase training speed.
+
+# 2.2 CROSS-LAYER PARAMETER SHARING
+
+The idea of sharing parameters across layers has been previously explored with the Transformer architecture (Vaswani et al., 2017), but this prior work has focused on training for standard encoder-decoder tasks rather than the pretraining/finetuning setting. Different from our observations, Dehghani et al. (2018) show that networks with cross-layer parameter sharing (Universal Transformer, UT) get better performance on language modeling and subject-verb agreement than the standard
+
+transformer. Very recently, Bai et al. (2019) propose a Deep Equilibrium Model (DQE) for transformer networks and show that DQE can reach an equilibrium point for which the input embedding and the output embedding of a certain layer stay the same. Our observations show that our embeddings are oscillating rather than converging. Hao et al. (2019) combine a parameter-sharing transformer with the standard one, which further increases the number of parameters of the standard transformer.
+
+# 2.3 SENTENCE ORDERING OBJECTIVES
+
+ALBERT uses a pretraining loss based on predicting the ordering of two consecutive segments of text. Several researchers have experimented with pretraining objectives that similarly relate to discourse coherence. Coherence and cohesion in discourse have been widely studied and many phenomena have been identified that connect neighboring text segments (Hobbs, 1979; Halliday & Hasan, 1976; Grosz et al., 1995). Most objectives found effective in practice are quite simple. Skipthought (Kiros et al., 2015) and FastSent (Hill et al., 2016) sentence embeddings are learned by using an encoding of a sentence to predict words in neighboring sentences. Other objectives for sentence embedding learning include predicting future sentences rather than only neighbors (Gan et al., 2017) and predicting explicit discourse markers (Jernite et al., 2017; Nie et al., 2019). Our loss is most similar to the sentence ordering objective of Jernite et al. (2017), where sentence embeddings are learned in order to determine the ordering of two consecutive sentences. Unlike most of the above work, however, our loss is defined on textual segments rather than sentences. BERT (Devlin et al., 2019) uses a loss based on predicting whether the second segment in a pair has been swapped with a segment from another document. We compare to this loss in our experiments and find that sentence ordering is a more challenging pretraining task and more useful for certain downstream tasks. Concurrently to our work, Wang et al. (2019) also try to predict the order of two consecutive segments of text, but they combine it with the original next sentence prediction in a three-way classification task rather than empirically comparing the two.
+
+# 3 THE ELEMENTS OF ALBERT
+
+In this section, we present the design decisions for ALBERT and provide quantified comparisons against corresponding configurations of the original BERT architecture (Devlin et al., 2019).
+
+# 3.1 MODEL ARCHITECTURE CHOICES
+
+The backbone of the ALBERT architecture is similar to BERT in that it uses a transformer encoder (Vaswani et al., 2017) with GELU nonlinearities (Hendrycks & Gimpel, 2016). We follow the BERT notation conventions and denote the vocabulary embedding size as  $E$ , the number of encoder layers as  $L$ , and the hidden size as  $H$ . Following Devlin et al. (2019), we set the feed-forward/filter size to be  $4H$  and the number of attention heads to be  $H / 64$ .
+
+There are three main contributions that ALBERT makes over the design choices of BERT.
+
+Factorized embedding parameterization. In BERT, as well as subsequent modeling improvements such as XLNet (Yang et al., 2019) and RoBERTa (Liu et al., 2019), the WordPiece embedding size  $E$  is tied with the hidden layer size  $H$ , i.e.,  $E \equiv H$ . This decision appears suboptimal for both modeling and practical reasons, as follows.
+
+From a modeling perspective, WordPiece embeddings are meant to learn context-independent representations, whereas hidden-layer embeddings are meant to learn context-dependent representations. As experiments with context length indicate (Liu et al., 2019), the power of BERT-like representations comes from the use of context to provide the signal for learning such context-dependent representations. As such, untying the WordPiece embedding size  $E$  from the hidden layer size  $H$  allows us to make a more efficient usage of the total model parameters as informed by modeling needs, which dictate that  $H \gg E$ .
+
+From a practical perspective, natural language processing usually require the vocabulary size  $V$  to be large. If  $E \equiv H$ , then increasing  $H$  increases the size of the embedding matrix, which has size
+
+$V \times E$ . This can easily result in a model with billions of parameters, most of which are only updated sparsely during training.
+
+Therefore, for ALBERT we use a factorization of the embedding parameters, decomposing them into two smaller matrices. Instead of projecting the one-hot vectors directly into the hidden space of size  $H$ , we first project them into a lower dimensional embedding space of size  $E$ , and then project it to the hidden space. By using this decomposition, we reduce the embedding parameters from  $O(V \times H)$  to  $O(V \times E + E \times H)$ . This parameter reduction is significant when  $H \gg E$ . We choose to use the same  $E$  for all word pieces because they are much more evenly distributed across documents compared to whole-word embedding, where having different embedding size (Grave et al. (2017); Baevski & Auli (2018); Dai et al. (2019)) for different words is important.
+
+Cross-layer parameter sharing. For ALBERT, we propose cross-layer parameter sharing as another way to improve parameter efficiency. There are multiple ways to share parameters, e.g., only sharing feed-forward network (FFN) parameters across layers, or only sharing attention parameters. The default decision for ALBERT is to share all parameters across layers. All our experiments use this default decision unless otherwise specified. We compare this design decision against other strategies in our experiments in Sec. 4.5.
+
+Similar strategies have been explored by Dehghani et al. (2018) (Universal Transformer, UT) and Bai et al. (2019) (Deep Equilibrium Models, DQE) for Transformer networks. Different from our observations, Dehghani et al. (2018) show that UT outperforms a vanilla Transformer. Bai et al. (2019) show that their DQEs reach an equilibrium point for which the input and output embedding of a certain layer stay the same. Our measurement on the L2 distances and cosine similarity show that our embeddings are oscillating rather than converging.
+
+![](images/e9c7e0aa628c7132cc4211bc24456f81ee1d8dddaf1d13879ac07e5295de9fdb.jpg)  
+Figure 1: The L2 distances and cosine similarity (in terms of degree) of the input and output embedding of each layer for BERT-large and ALBERT-large.
+
+![](images/f1d65322d22d936be5975593e3f3247b84b1132710d667d129294296a19f6825.jpg)
+
+Figure 1 shows the L2 distances and cosine similarity of the input and output embeddings for each layer, using BERT-large and ALBERT-large configurations (see Table 1). We observe that the transitions from layer to layer are much smoother for ALBERT than for BERT. These results show that weight-sharing has an effect on stabilizing network parameters. Although there is a drop for both metrics compared to BERT, they nevertheless do not converge to 0 even after 24 layers. This shows that the solution space for ALBERT parameters is very different from the one found by DQE.
+
+Inter-sentence coherence loss. In addition to the masked language modeling (MLM) loss (Devlin et al., 2019), BERT uses an additional loss called next-sentence prediction (NSP). NSP is a binary classification loss for predicting whether two segments appear consecutively in the original text, as follows: positive examples are created by taking consecutive segments from the training corpus; negative examples are created by pairing segments from different documents; positive and negative examples are sampled with equal probability. The NSP objective was designed to improve performance on downstream tasks, such as natural language inference, that require reasoning about the relationship between sentence pairs. However, subsequent studies (Yang et al., 2019; Liu et al., 2019) found NSP's impact unreliable and decided to eliminate it, a decision supported by an improvement in downstream task performance across several tasks.
+
+We conjecture that the main reason behind NSP's ineffectiveness is its lack of difficulty as a task, as compared to MLM. As formulated, NSP conflates topic prediction and coherence prediction in a
+
+<table><tr><td colspan="2">Model</td><td>Parameters</td><td>Layers</td><td>Hidden</td><td>Embedding</td><td>Parameter-sharing</td></tr><tr><td rowspan="2">BERT</td><td>base</td><td>108M</td><td>12</td><td>768</td><td>768</td><td>False</td></tr><tr><td>large</td><td>334M</td><td>24</td><td>1024</td><td>1024</td><td>False</td></tr><tr><td rowspan="4">ALBERT</td><td>base</td><td>12M</td><td>12</td><td>768</td><td>128</td><td>True</td></tr><tr><td>large</td><td>18M</td><td>24</td><td>1024</td><td>128</td><td>True</td></tr><tr><td>xlarge</td><td>60M</td><td>24</td><td>2048</td><td>128</td><td>True</td></tr><tr><td>xxlarge</td><td>235M</td><td>12</td><td>4096</td><td>128</td><td>True</td></tr></table>
+
+Table 1: The configurations of the main BERT and ALBERT models analyzed in this paper.
+
+single task $^2$ . However, topic prediction is easier to learn compared to coherence prediction, and also overlaps more with what is learned using the MLM loss.
+
+We maintain that inter-sentence modeling is an important aspect of language understanding, but we propose a loss based primarily on coherence. That is, for ALBERT, we use a sentence-order prediction (SOP) loss, which avoids topic prediction and instead focuses on modeling inter-sentence coherence. The SOP loss uses as positive examples the same technique as BERT (two consecutive segments from the same document), and as negative examples the same two consecutive segments but with their order swapped. This forces the model to learn finer-grained distinctions about discourse-level coherence properties. As we show in Sec. 4.6, it turns out that NSP cannot solve the SOP task at all (i.e., it ends up learning the easier topic-prediction signal, and performs at random-baseline level on the SOP task), while SOP can solve the NSP task to a reasonable degree, presumably based on analyzing misaligned coherence cues. As a result, ALBERT models consistently improve downstream task performance for multi-sentence encoding tasks.
+
+# 3.2 MODEL SETUP
+
+We present the differences between BERT and ALBERT models with comparable hyperparameter settings in Table 1. Due to the design choices discussed above, ALBERT models have much smaller parameter size compared to corresponding BERT models.
+
+For example, ALBERT-large has about 18x fewer parameters compared to BERT-large, 18M versus 334M. An ALBERT-xlarge configuration with  $H = 2048$  has only 60M parameters and an ALBERT-xxlarge configuration with  $H = 4096$  has 233M parameters, i.e., around 70% of BERT-large's parameters. Note that for ALBERT-xxlarge, we mainly report results on a 12-layer network because a 24-layer network (with the same configuration) obtains similar results but is computationally more expensive.
+
+This improvement in parameter efficiency is the most important advantage of ALBERT's design choices. Before we can quantify this advantage, we need to introduce our experimental setup in more detail.
+
+# 4 EXPERIMENTAL RESULTS
+
+# 4.1 EXPERIMENTAL SETUP
+
+To keep the comparison as meaningful as possible, we follow the BERT (Devlin et al., 2019) setup in using the BOOKCORPUS (Zhu et al., 2015) and English Wikipedia (Devlin et al., 2019) for pretraining baseline models. These two corpora consist of around 16GB of uncompressed text. We format our inputs as "[CLS]  $x_{1}$  [SEP]  $x_{2}$  [SEP]", where  $x_{1} = x_{1,1}, x_{1,2} \cdots$  and  $x_{2} = x_{1,1}, x_{1,2} \cdots$  are two segments. We always limit the maximum input length to 512, and randomly generate input sequences shorter than 512 with a probability of  $10\%$ . Like BERT, we use a vocabulary size of 30,000, tokenized using SentencePiece (Kudo & Richardson, 2018) as in XLNet (Yang et al., 2019).
+
+We generate masked inputs for the MLM targets using  $n$ -gram masking (Joshi et al., 2019), with the length of each  $n$ -gram mask selected randomly. The probability for the length  $n$  is given by
+
+$$
+p (n) = \frac {1 / n}{\sum_ {k = 1} ^ {N} 1 / k}
+$$
+
+We set the maximum length of  $n$ -gram (i.e.,  $n$ ) to be 3 (i.e., the MLM target can consist of up to a 3-gram of complete words, such as "White House correspondents").
+
+All the model updates use a batch size of 4096 and a LAMB optimizer with learning rate 0.00176 (You et al., 2019). We train all models for 125,000 steps unless otherwise specified. Training was done on Cloud TPU V3. The number of TPUs used for training ranged from 64 to 512, depending on model size.
+
+The experimental setup described in this section is used for all of our own versions of BERT as well as ALBERT models, unless otherwise specified.
+
+# 4.2 EVALUATION BENCHMARKS
+
+# 4.2.1 INTRINSIC EVALUATION
+
+To monitor the training progress, we create a development set based on the development sets from SQuAD and RACE using the same procedure as in Sec. 4.1. We report accuracies for both MLM and sentence classification tasks. Note that we only use this set to check how the model is converging; it has not been used in a way that would affect the performance of any downstream evaluation, such as via model selection.
+
+# 4.2.2 DOWNSSTREAM EVALUATION
+
+Following Yang et al. (2019) and Liu et al. (2019), we evaluate our models on three popular benchmarks: The General Language Understanding Evaluation (GLUE) benchmark (Wang et al., 2018), two versions of the Stanford Question Answering Dataset (SQuAD; Rajpurkar et al., 2016; 2018), and the ReAding Comprehension from Examinations (RACE) dataset (Lai et al., 2017). For completeness, we provide description of these benchmarks in Appendix A.3. As in (Liu et al., 2019), we perform early stopping on the development sets, on which we report all comparisons except for our final comparisons based on the task leaderboards, for which we also report test set results. For GLUE datasets that have large variances on the dev set, we report median over 5 runs.
+
+# 4.3 OVERALL COMPARISON BETWEEN BERT AND ALBERT
+
+We are now ready to quantify the impact of the design choices described in Sec. 3, specifically the ones around parameter efficiency. The improvement in parameter efficiency showcases the most important advantage of ALBERT's design choices, as shown in Table 2: with only around  $70\%$  of BERT-large's parameters, ALBERT-xxlarge achieves significant improvements over BERT-large, as measured by the difference on development set scores for several representative downstream tasks: SQuAD v1.1  $(+1.9\%)$ , SQuAD v2.0  $(+3.1\%)$ , MNLI  $(+1.4\%)$ , SST-2  $(+2.2\%)$ , and RACE  $(+8.4\%)$ .
+
+Another interesting observation is the speed of data throughput at training time under the same training configuration (same number of TPUs). Because of less communication and fewer computations, ALBERT models have higher data throughput compared to their corresponding BERT models. If we use BERT-large as the baseline, we observe that ALBERT-large is about 1.7 times faster in iterating through the data while ALBERT-xxlarge is about 3 times slower because of the larger structure.
+
+Next, we perform ablation experiments that quantify the individual contribution of each of the design choices for ALBERT.
+
+# 4.4 FACTORIZED EMBEDDING PARAMETERIZATION
+
+Table 3 shows the effect of changing the vocabulary embedding size  $E$  using an ALBERT-base configuration setting (see Table 1), using the same set of representative downstream tasks. Under the non-shared condition (BERT-style), larger embedding sizes give better performance, but not by
+
+<table><tr><td colspan="2">Model</td><td>Parameters</td><td>SQuAD1.1</td><td>SQuAD2.0</td><td>MNLI</td><td>SST-2</td><td>RACE</td><td>Avg</td><td>Speedup</td></tr><tr><td rowspan="2">BERT</td><td>base</td><td>108M</td><td>90.4/83.2</td><td>80.4/77.6</td><td>84.5</td><td>92.8</td><td>68.2</td><td>82.3</td><td>4.7x</td></tr><tr><td>large</td><td>334M</td><td>92.2/85.5</td><td>85.0/82.2</td><td>86.6</td><td>93.0</td><td>73.9</td><td>85.2</td><td>1.0</td></tr><tr><td rowspan="4">ALBERT</td><td>base</td><td>12M</td><td>89.3/82.3</td><td>80.0/77.1</td><td>81.6</td><td>90.3</td><td>64.0</td><td>80.1</td><td>5.6x</td></tr><tr><td>large</td><td>18M</td><td>90.6/83.9</td><td>82.3/79.4</td><td>83.5</td><td>91.7</td><td>68.5</td><td>82.4</td><td>1.7x</td></tr><tr><td>xlarge</td><td>60M</td><td>92.5/86.1</td><td>86.1/83.1</td><td>86.4</td><td>92.4</td><td>74.8</td><td>85.5</td><td>0.6x</td></tr><tr><td>xxlarge</td><td>235M</td><td>94.1/88.3</td><td>88.1/85.1</td><td>88.0</td><td>95.2</td><td>82.3</td><td>88.7</td><td>0.3x</td></tr></table>
+
+much. Under the all-shared condition (ALBERT-style), an embedding of size 128 appears to be the best. Based on these results, we use an embedding size  $E = 128$  in all future settings, as a necessary step to do further scaling.
+
+Table 2: Dev set results for models pretrained over BOOKCORPUS and Wikipedia for 125k steps. Here and everywhere else, the Avg column is computed by averaging the scores of the downstream tasks to its left (the two numbers of F1 and EM for each SQuAD are first averaged).  
+
+<table><tr><td>Model</td><td>E</td><td>Parameters</td><td>SQuAD1.1</td><td>SQuAD2.0</td><td>MNLI</td><td>SST-2</td><td>RACE</td><td>Avg</td></tr><tr><td>ALBERT</td><td>64</td><td>87M</td><td>89.9/82.9</td><td>80.1/77.8</td><td>82.9</td><td>91.5</td><td>66.7</td><td>81.3</td></tr><tr><td>base</td><td>128</td><td>89M</td><td>89.9/82.8</td><td>80.3/77.3</td><td>83.7</td><td>91.5</td><td>67.9</td><td>81.7</td></tr><tr><td>not-shared</td><td>256</td><td>93M</td><td>90.2/83.2</td><td>80.3/77.4</td><td>84.1</td><td>91.9</td><td>67.3</td><td>81.8</td></tr><tr><td></td><td>768</td><td>108M</td><td>90.4/83.2</td><td>80.4/77.6</td><td>84.5</td><td>92.8</td><td>68.2</td><td>82.3</td></tr><tr><td>ALBERT</td><td>64</td><td>10M</td><td>88.7/81.4</td><td>77.5/74.8</td><td>80.8</td><td>89.4</td><td>63.5</td><td>79.0</td></tr><tr><td>base</td><td>128</td><td>12M</td><td>89.3/82.3</td><td>80.0/77.1</td><td>81.6</td><td>90.3</td><td>64.0</td><td>80.1</td></tr><tr><td>all-shared</td><td>256</td><td>16M</td><td>88.8/81.5</td><td>79.1/76.3</td><td>81.5</td><td>90.3</td><td>63.4</td><td>79.6</td></tr><tr><td></td><td>768</td><td>31M</td><td>88.6/81.5</td><td>79.2/76.6</td><td>82.0</td><td>90.6</td><td>63.3</td><td>79.8</td></tr></table>
+
+# 4.5 CROSS-LAYER PARAMETER SHARING
+
+Table 4 presents experiments for various cross-layer parameter-sharing strategies, using an ALBERT-base configuration (Table 1) with two embedding sizes ( $E = 768$  and  $E = 128$ ). We compare the all-shared strategy (ALBERT-style), the not-shared strategy (BERT-style), and intermediate strategies in which only the attention parameters are shared (but not the FNN ones) or only the FFN parameters are shared (but not the attention ones).
+
+The all-shared strategy hurts performance under both conditions, but it is less severe for  $E = 128$  (-1.5 on Avg) compared to  $E = 768$  (-2.5 on Avg). In addition, most of the performance drop appears to come from sharing the FFN-layer parameters, while sharing the attention parameters results in no drop when  $E = 128$  (+0.1 on Avg), and a slight drop when  $E = 768$  (-0.7 on Avg).
+
+There are other strategies of sharing the parameters cross layers. For example, We can divide the  $L$  layers into  $N$  groups of size  $M$ , and each size- $M$  group shares parameters. Overall, our experimental results show that the smaller the group size  $M$  is, the better the performance we get. However, decreasing group size  $M$  also dramatically increase the number of overall parameters. We choose all-shared strategy as our default choice.
+
+Table 3: The effect of vocabulary embedding size on the performance of ALBERT-base.  
+
+<table><tr><td></td><td>Model</td><td>Parameters</td><td>SQuAD1.1</td><td>SQuAD2.0</td><td>MNLI</td><td>SST-2</td><td>RACE</td><td>Avg</td></tr><tr><td rowspan="4">ALBERT 
+base 
+E=768</td><td>all-shared</td><td>31M</td><td>88.6/81.5</td><td>79.2/76.6</td><td>82.0</td><td>90.6</td><td>63.3</td><td>79.8</td></tr><tr><td>shared-attention</td><td>83M</td><td>89.9/82.7</td><td>80.0/77.2</td><td>84.0</td><td>91.4</td><td>67.7</td><td>81.6</td></tr><tr><td>shared-FFN</td><td>57M</td><td>89.2/82.1</td><td>78.2/75.4</td><td>81.5</td><td>90.8</td><td>62.6</td><td>79.5</td></tr><tr><td>not-shared</td><td>108M</td><td>90.4/83.2</td><td>80.4/77.6</td><td>84.5</td><td>92.8</td><td>68.2</td><td>82.3</td></tr><tr><td rowspan="4">ALBERT 
+base 
+E=128</td><td>all-shared</td><td>12M</td><td>89.3/82.3</td><td>80.0/77.1</td><td>82.0</td><td>90.3</td><td>64.0</td><td>80.1</td></tr><tr><td>shared-attention</td><td>64M</td><td>89.9/82.8</td><td>80.7/77.9</td><td>83.4</td><td>91.9</td><td>67.6</td><td>81.7</td></tr><tr><td>shared-FFN</td><td>38M</td><td>88.9/81.6</td><td>78.6/75.6</td><td>82.3</td><td>91.7</td><td>64.4</td><td>80.2</td></tr><tr><td>not-shared</td><td>89M</td><td>89.9/82.8</td><td>80.3/77.3</td><td>83.2</td><td>91.5</td><td>67.9</td><td>81.6</td></tr></table>
+
+Table 4: The effect of cross-layer parameter-sharing strategies, ALBERT-base configuration.
+
+# 4.6 SENTENCE ORDER PREDICTION (SOP)
+
+We compare head-to-head three experimental conditions for the additional inter-sentence loss: none (XLNet- and RoBERTa-style), NSP (BERT-style), and SOP (ALBERT-style), using an ALBERT-base configuration. Results are shown in Table 5, both over intrinsic (accuracy for the MLM, NSP, and SOP tasks) and downstream tasks.
+
+<table><tr><td rowspan="2">SP tasks</td><td colspan="3">Intrinsic Tasks</td><td colspan="6">Downstream Tasks</td></tr><tr><td>MLM</td><td>NSP</td><td>SOP</td><td>SQuAD1.1</td><td>SQuAD2.0</td><td>MNLI</td><td>SST-2</td><td>RACE</td><td>Avg</td></tr><tr><td>None</td><td>54.9</td><td>52.4</td><td>53.3</td><td>88.6/81.5</td><td>78.1/75.3</td><td>81.5</td><td>89.9</td><td>61.7</td><td>79.0</td></tr><tr><td>NSP</td><td>54.5</td><td>90.5</td><td>52.0</td><td>88.4/81.5</td><td>77.2/74.6</td><td>81.6</td><td>91.1</td><td>62.3</td><td>79.2</td></tr><tr><td>SOP</td><td>54.0</td><td>78.9</td><td>86.5</td><td>89.3/82.3</td><td>80.0/77.1</td><td>82.0</td><td>90.3</td><td>64.0</td><td>80.1</td></tr></table>
+
+The results on the intrinsic tasks reveal that the NSP loss brings no discriminative power to the SOP task (52.0% accuracy, similar to the random-guess performance for the "None" condition). This allows us to conclude that NSP ends up modeling only topic shift. In contrast, the SOP loss does solve the NSP task relatively well (78.9% accuracy), and the SOP task even better (86.5% accuracy). Even more importantly, the SOP loss appears to consistently improve downstream task performance for multi-sentence encoding tasks (around +1% for SQuAD1.1, +2% for SQuAD2.0, +1.7% for RACE), for an Avg score improvement of around +1%.
+
+# 4.7 WHAT IF WE TRAIN FOR THE SAME AMOUNT OF TIME?
+
+The speed-up results in Table 2 indicate that data-throughput for BERT-large is about  $3.17\mathrm{x}$  higher compared to ALBERT-xxlarge. Since longer training usually leads to better performance, we perform a comparison in which, instead of controlling for data throughput (number of training steps), we control for the actual training time (i.e., let the models train for the same number of hours). In Table 6, we compare the performance of a BERT-large model after  $400\mathrm{k}$  training steps (after 34h of training), roughly equivalent with the amount of time needed to train an ALBERT-xxlarge model with 125k training steps (32h of training).
+
+Table 5: The effect of sentence-prediction loss, NSP vs. SOP, on intrinsic and downstream tasks.  
+
+<table><tr><td>Models</td><td>Steps</td><td>Time</td><td>SQuAD1.1</td><td>SQuAD2.0</td><td>MNLI</td><td>SST-2</td><td>RACE</td><td>Avg</td></tr><tr><td>BERT-large</td><td>400k</td><td>34h</td><td>93.5/87.4</td><td>86.9/84.3</td><td>87.8</td><td>94.6</td><td>77.3</td><td>87.2</td></tr><tr><td>ALBERT-xxlarge</td><td>125k</td><td>32h</td><td>94.0/88.1</td><td>88.3/85.3</td><td>87.8</td><td>95.4</td><td>82.5</td><td>88.7</td></tr></table>
+
+After training for roughly the same amount of time, ALBERT-xxlarge is significantly better than BERT-large:  $+1.5\%$  better on Avg, with the difference on RACE as high as  $+5.2\%$ .
+
+# 4.8 ADDITIONAL TRAINING DATA AND DROPOUT EFFECTS
+
+The experiments done up to this point use only the Wikipedia and BOOKCORPUS datasets, as in (Devlin et al., 2019). In this section, we report measurements on the impact of the additional data used by both XLNet (Yang et al., 2019) and RoBERTa (Liu et al., 2019).
+
+Fig. 2a plots the dev set MLM accuracy under two conditions, without and with additional data, with the latter condition giving a significant boost. We also observe performance improvements on the downstream tasks in Table 7, except for the SQuAD benchmarks (which are Wikipedia-based, and therefore are negatively affected by out-of-domain training material).
+
+Table 6: The effect of controlling for training time, BERT-large vs ALBERT-xxlarge configurations.  
+
+<table><tr><td></td><td>SQuAD1.1</td><td>SQuAD2.0</td><td>MNLI</td><td>SST-2</td><td>RACE</td><td>Avg</td></tr><tr><td>No additional data</td><td>89.3/82.3</td><td>80.0/77.1</td><td>81.6</td><td>90.3</td><td>64.0</td><td>80.1</td></tr><tr><td>With additional data</td><td>88.8/81.7</td><td>79.1/76.3</td><td>82.4</td><td>92.8</td><td>66.0</td><td>80.8</td></tr></table>
+
+Table 7: The effect of additional training data using the ALBERT-base configuration.
+
+We also note that, even after training for 1M steps, our largest models still do not overfit to their training data. As a result, we decide to remove dropout to further increase our model capacity. The
+
+![](images/73ebecfa97fa9e9248ea37c9627731c814f4ad1f93c3d754f3da7954f4820d46.jpg)  
+(a) Adding data
+
+![](images/b0714aa9fd3c6ba9746c9e2c9a76b911ef2950fbc270a810dd82b455ac7f4535.jpg)  
+(b) Removing dropout  
+Figure 2: The effects of adding data and removing dropout during training.
+
+plot in Fig. 2b shows that removing dropout significantly improves MLM accuracy. Intermediate evaluation on ALBERT-xxlarge at around 1M training steps (Table 8) also confirms that removing dropout helps the downstream tasks. There is empirical (Szegedy et al., 2017) and theoretical (Li et al., 2019) evidence showing that a combination of batch normalization and dropout in Convolutional Neural Networks may have harmful results. To the best of our knowledge, we are the first to show that dropout can hurt performance in large Transformer-based models. However, the underlying network structure of ALBERT is a special case of the transformer and further experimentation is needed to see if this phenomenon appears with other transformer-based architectures or not.
+
+<table><tr><td></td><td>SQuAD1.1</td><td>SQuAD2.0</td><td>MNLI</td><td>SST-2</td><td>RACE</td><td>Avg</td></tr><tr><td>With dropout</td><td>94.7/89.2</td><td>89.6/86.9</td><td>90.0</td><td>96.3</td><td>85.7</td><td>90.4</td></tr><tr><td>Without dropout</td><td>94.8/89.5</td><td>89.9/87.2</td><td>90.4</td><td>96.5</td><td>86.1</td><td>90.7</td></tr></table>
+
+# 4.9 CURRENT STATE-OF-THE-ART ON NLU TASKS
+
+The results we report in this section make use of the training data used by Devlin et al. (2019), as well as the additional data used by Liu et al. (2019) and Yang et al. (2019). We report state-of-the-art results under two settings for fine-tuning: single-model and ensembles. In both settings, we only do single-task fine-tuning<sup>4</sup>. Following Liu et al. (2019), on the development set we report the median result over five runs.
+
+Table 8: The effect of removing dropout, measured for an ALBERT-xxlarge configuration.  
+
+<table><tr><td>Models</td><td>MNLI</td><td>QNLI</td><td>QQP</td><td>RTE</td><td>SST</td><td>MRPC</td><td>CoLA</td><td>STS</td><td>WNLI</td><td>Avg</td></tr><tr><td colspan="11">Single-task single models on dev</td></tr><tr><td>BERT-large</td><td>86.6</td><td>92.3</td><td>91.3</td><td>70.4</td><td>93.2</td><td>88.0</td><td>60.6</td><td>90.0</td><td>-</td><td>-</td></tr><tr><td>XLNet-large</td><td>89.8</td><td>93.9</td><td>91.8</td><td>83.8</td><td>95.6</td><td>89.2</td><td>63.6</td><td>91.8</td><td>-</td><td>-</td></tr><tr><td>RoBERTa-large</td><td>90.2</td><td>94.7</td><td>92.2</td><td>86.6</td><td>96.4</td><td>90.9</td><td>68.0</td><td>92.4</td><td>-</td><td>-</td></tr><tr><td>ALBERT (1M)</td><td>90.4</td><td>95.2</td><td>92.0</td><td>88.1</td><td>96.8</td><td>90.2</td><td>68.7</td><td>92.7</td><td>-</td><td>-</td></tr><tr><td>ALBERT (1.5M)</td><td>90.8</td><td>95.3</td><td>92.2</td><td>89.2</td><td>96.9</td><td>90.9</td><td>71.4</td><td>93.0</td><td>-</td><td>-</td></tr><tr><td colspan="11">Ensembles on test (from leaderboard as of Sept. 16, 2019)</td></tr><tr><td>ALICE</td><td>88.2</td><td>95.7</td><td>90.7</td><td>83.5</td><td>95.2</td><td>92.6</td><td>69.2</td><td>91.1</td><td>80.8</td><td>87.0</td></tr><tr><td>MT-DNN</td><td>87.9</td><td>96.0</td><td>89.9</td><td>86.3</td><td>96.5</td><td>92.7</td><td>68.4</td><td>91.1</td><td>89.0</td><td>87.6</td></tr><tr><td>XLNet</td><td>90.2</td><td>98.6</td><td>90.3</td><td>86.3</td><td>96.8</td><td>93.0</td><td>67.8</td><td>91.6</td><td>90.4</td><td>88.4</td></tr><tr><td>RoBERTa</td><td>90.8</td><td>98.9</td><td>90.2</td><td>88.2</td><td>96.7</td><td>92.3</td><td>67.8</td><td>92.2</td><td>89.0</td><td>88.5</td></tr><tr><td>Adv-RoBERTa</td><td>91.1</td><td>98.8</td><td>90.3</td><td>88.7</td><td>96.8</td><td>93.1</td><td>68.0</td><td>92.4</td><td>89.0</td><td>88.8</td></tr><tr><td>ALBERT</td><td>91.3</td><td>99.2</td><td>90.5</td><td>89.2</td><td>97.1</td><td>93.4</td><td>69.1</td><td>92.5</td><td>91.8</td><td>89.4</td></tr></table>
+
+Table 9: State-of-the-art results on the GLUE benchmark. For single-task single-model results, we report ALBERT at 1M steps (comparable to RoBERTa) and at 1.5M steps. The ALBERT ensemble uses models trained with 1M, 1.5M, and other numbers of steps.
+
+The single-model ALBERT configuration incorporates the best-performing settings discussed: an ALBERT-xxlarge configuration (Table 1) using combined MLM and SOP losses, and no dropout.
+
+The checkpoints that contribute to the final ensemble model are selected based on development set performance; the number of checkpoints considered for this selection range from 6 to 17, depending on the task. For the GLUE (Table 9) and RACE (Table 10) benchmarks, we average the model predictions for the ensemble models, where the candidates are fine-tuned from different training steps using the 12-layer and 24-layer architectures. For SQuAD (Table 10), we average the prediction scores for those spans that have multiple probabilities; we also average the scores of the "unanswerable" decision.
+
+Both single-model and ensemble results indicate that ALBERT improves the state-of-the-art significantly for all three benchmarks, achieving a GLUE score of 89.4, a SQuAD 2.0 test F1 score of 92.2, and a RACE test accuracy of 89.4. The latter appears to be a particularly strong improvement, a jump of  $+17.4\%$  absolute points over BERT (Devlin et al., 2019; Clark et al., 2019),  $+7.6\%$  over XLNet (Yang et al., 2019),  $+6.2\%$  over RoBERTa (Liu et al., 2019), and  $5.3\%$  over DCMI+ (Zhang et al., 2019), an ensemble of multiple models specifically designed for reading comprehension tasks. Our single model achieves an accuracy of  $86.5\%$ , which is still  $2.4\%$  better than the state-of-the-art ensemble model.
+
+<table><tr><td>Models</td><td>SQuAD1.1 dev</td><td>SQuAD2.0 dev</td><td>SQuAD2.0 test</td><td>RACE test (Middle/High)</td></tr><tr><td colspan="5">Single model (from leaderboard as of Sept. 23, 2019)</td></tr><tr><td>BERT-large</td><td>90.9/84.1</td><td>81.8/79.0</td><td>89.1/86.3</td><td>72.0 (76.6/70.1)</td></tr><tr><td>XLNet</td><td>94.5/89.0</td><td>88.8/86.1</td><td>89.1/86.3</td><td>81.8 (85.5/80.2)</td></tr><tr><td>RoBERTa</td><td>94.6/88.9</td><td>89.4/86.5</td><td>89.8/86.8</td><td>83.2 (86.5/81.3)</td></tr><tr><td>UPM</td><td>-</td><td>-</td><td>89.9/87.2</td><td>-</td></tr><tr><td>XLNet + SG-Net Verifier++</td><td>-</td><td>-</td><td>90.1/87.2</td><td>-</td></tr><tr><td>ALBERT (1M)</td><td>94.8/89.2</td><td>89.9/87.2</td><td>-</td><td>86.0 (88.2/85.1)</td></tr><tr><td>ALBERT (1.5M)</td><td>94.8/89.3</td><td>90.2/87.4</td><td>90.9/88.1</td><td>86.5 (89.0/85.5)</td></tr><tr><td colspan="5">Ensembles (from leaderboard as of Sept. 23, 2019)</td></tr><tr><td>BERT-large</td><td>92.2/86.2</td><td>-</td><td>-</td><td>-</td></tr><tr><td>XLNet + SG-Net Verifier</td><td>-</td><td>-</td><td>90.7/88.2</td><td>-</td></tr><tr><td>UPM</td><td>-</td><td>-</td><td>90.7/88.2</td><td></td></tr><tr><td>XLNet + DAAF + Verifier</td><td>-</td><td>-</td><td>90.9/88.6</td><td>-</td></tr><tr><td>DCMN+</td><td>-</td><td>-</td><td>-</td><td>84.1 (88.5/82.3)</td></tr><tr><td>ALBERT</td><td>95.5/90.1</td><td>91.4/88.9</td><td>92.2/89.7</td><td>89.4 (91.2/88.6)</td></tr></table>
+
+Table 10: State-of-the-art results on the SQuAD and RACE benchmarks.
+
+# 5 DISCUSSION
+
+While ALBERT-xxlarge has less parameters than BERT-large and gets significantly better results, it is computationally more expensive due to its larger structure. An important next step is thus to speed up the training and inference speed of ALBERT through methods like sparse attention (Child et al., 2019) and block attention (Shen et al., 2018). An orthogonal line of research, which could provide additional representation power, includes hard example mining (Mikolov et al., 2013) and more efficient language modeling training (Yang et al., 2019). Additionally, although we have convincing evidence that sentence order prediction is a more consistently-useful learning task that leads to better language representations, we hypothesize that there could be more dimensions not yet captured by the current self-supervised training losses that could create additional representation power for the resulting representations.
+
+# ACKNOWLEDGEMENT
+
+The authors would like to thank Beer Changpinyo, Nan Ding, Noam Shazeer, and Tomer Levinboim for discussion and providing useful feedback on the project; Omer Levy and Naman Goyal for clarifying experimental setup for RoBERTa; Zihang Dai for clarifying XLNet; Brandon Norick, Emma Strubell, Shaojie Bai, Chas Leichner, and Sachin Mehta for providing useful feedback on the paper; Jacob Devlin for providing the English and multilingual version of training data; Liang Xu, Chenjie Cao and the CLUE community for providing the training data and evaluation benchmark of the Chinese version of ALBERT models.
+
+# REFERENCES
+
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+
+# A APPENDIX
+
+# A.1 EFFECT OF NETWORK DEPTH AND WIDTH
+
+In this section, we check how depth (number of layers) and width (hidden size) affect the performance of ALBERT. Table 11 shows the performance of an ALBERT-large configuration (see Table 1) using different numbers of layers. Networks with 3 or more layers are trained by fine-tuning using the parameters from the depth before (e.g., the 12-layer network parameters are fine-tuned from the checkpoint of the 6-layer network parameters). Similar technique has been used in Gong et al. (2019). If we compare a 3-layer ALBERT model with a 1-layer ALBERT model, although they have the same number of parameters, the performance increases significantly. However, there are diminishing returns when continuing to increase the number of layers: the results of a 12-layer network are relatively close to the results of a 24-layer network, and the performance of a 48-layer network appears to decline.
+
+<table><tr><td>Number of layers</td><td>Parameters</td><td>SQuAD1.1</td><td>SQuAD2.0</td><td>MNLI</td><td>SST-2</td><td>RACE</td><td>Avg</td></tr><tr><td>1</td><td>18M</td><td>31.1/22.9</td><td>50.1/50.1</td><td>66.4</td><td>80.8</td><td>40.1</td><td>52.9</td></tr><tr><td>3</td><td>18M</td><td>79.8/69.7</td><td>64.4/61.7</td><td>77.7</td><td>86.7</td><td>54.0</td><td>71.2</td></tr><tr><td>6</td><td>18M</td><td>86.4/78.4</td><td>73.8/71.1</td><td>81.2</td><td>88.9</td><td>60.9</td><td>77.2</td></tr><tr><td>12</td><td>18M</td><td>89.8/83.3</td><td>80.7/77.9</td><td>83.3</td><td>91.7</td><td>66.7</td><td>81.5</td></tr><tr><td>24</td><td>18M</td><td>90.3/83.3</td><td>81.8/79.0</td><td>83.3</td><td>91.5</td><td>68.7</td><td>82.1</td></tr><tr><td>48</td><td>18M</td><td>90.0/83.1</td><td>81.8/78.9</td><td>83.4</td><td>91.9</td><td>66.9</td><td>81.8</td></tr></table>
+
+Table 11: The effect of increasing the number of layers for an ALBERT-large configuration.
+
+A similar phenomenon, this time for width, can be seen in Table 12 for a 3-layer ALBERT-large configuration. As we increase the hidden size, we get an increase in performance with diminishing returns. At a hidden size of 6144, the performance appears to decline significantly. We note that none of these models appear to overfit the training data, and they all have higher training and development loss compared to the best-performing ALBERT configurations.
+
+<table><tr><td>Hidden size</td><td>Parameters</td><td>SQuAD1.1</td><td>SQuAD2.0</td><td>MNLI</td><td>SST-2</td><td>RACE</td><td>Avg</td></tr><tr><td>1024</td><td>18M</td><td>79.8/69.7</td><td>64.4/61.7</td><td>77.7</td><td>86.7</td><td>54.0</td><td>71.2</td></tr><tr><td>2048</td><td>60M</td><td>83.3/74.1</td><td>69.1/66.6</td><td>79.7</td><td>88.6</td><td>58.2</td><td>74.6</td></tr><tr><td>4096</td><td>225M</td><td>85.0/76.4</td><td>71.0/68.1</td><td>80.3</td><td>90.4</td><td>60.4</td><td>76.3</td></tr><tr><td>6144</td><td>499M</td><td>84.7/75.8</td><td>67.8/65.4</td><td>78.1</td><td>89.1</td><td>56.0</td><td>74.0</td></tr></table>
+
+# A.2 DO VERY WIDE ALBERT MODELS NEED TO BE DEEP(ER) TOO?
+
+In Section A.1, we show that for ALBERT-large ( $H = 1024$ ), the difference between a 12-layer and a 24-layer configuration is small. Does this result still hold for much wider ALBERT configurations, such as ALBERT-xxlarge ( $H = 4096$ )?
+
+Table 12: The effect of increasing the hidden-layer size for an ALBERT-large 3-layer configuration.  
+
+<table><tr><td>Number of layers</td><td>SQuAD1.1</td><td>SQuAD2.0</td><td>MNLI</td><td>SST-2</td><td>RACE</td><td>Avg</td></tr><tr><td>12</td><td>94.0/88.1</td><td>88.3/85.3</td><td>87.8</td><td>95.4</td><td>82.5</td><td>88.7</td></tr><tr><td>24</td><td>94.1/88.3</td><td>88.1/85.1</td><td>88.0</td><td>95.2</td><td>82.3</td><td>88.7</td></tr></table>
+
+Table 13: The effect of a deeper network using an ALBERT-xxlarge configuration.
+
+The answer is given by the results from Table 13. The difference between 12-layer and 24-layer ALBERT-xxlarge configurations in terms of downstream accuracy is negligible, with the Avg score being the same. We conclude that, when sharing all cross-layer parameters (ALBERT-style), there is no need for models deeper than a 12-layer configuration.
+
+# A.3 DOWNSTREAM EVALUATION TASKS
+
+GLUE GLUE is comprised of 9 tasks, namely Corpus of Linguistic Acceptability (CoLA; Warstadt et al., 2018), Stanford Sentiment Treebank (SST; Socher et al., 2013), Microsoft Research Paraphrase Corpus (MRPC; Dolan & Brockett, 2005), Semantic Textual Similarity Benchmark (STS; Cer et al., 2017), Quora Question Pairs (QQP; Iyer et al., 2017), Multi-Genre NLI (MNLI; Williams et al., 2018), Question NLI (QNLI; Rajpurkar et al., 2016), Recognizing Textual Entailment (RTE; Dagan et al., 2005; Bar-Haim et al., 2006; Giampiccolo et al., 2007; Bentivogli et al., 2009) and Winograd NLI (WNLI; Levesque et al., 2012). It focuses on evaluating model capabilities for natural language understanding. When reporting MNLI results, we only report the "match" condition (MNLI-m). We follow the finetuning procedures from prior work (Devlin et al., 2019; Liu et al., 2019; Yang et al., 2019) and report the held-out test set performance obtained from GLUE submissions. For test set submissions, we perform task-specific modifications for WNLI and QNLI as described by Liu et al. (2019) and Yang et al. (2019).
+
+SQuAD SQuAD is an extractive question answering dataset built from Wikipedia. The answers are segments from the context paragraphs and the task is to predict answer spans. We evaluate our models on two versions of SQuAD: v1.1 and v2.0. SQuAD v1.1 has 100,000 human-annotated question/answer pairs. SQuAD v2.0 additionally introduced 50,000 unanswerable questions. For SQuAD v1.1, we use the same training procedure as BERT, whereas for SQuAD v2.0, models are jointly trained with a span extraction loss and an additional classifier for predicting answerability (Yang et al., 2019; Liu et al., 2019). We report both development set and test set performance.
+
+RACE RACE is a large-scale dataset for multi-choice reading comprehension, collected from English examinations in China with nearly 100,000 questions. Each instance in RACE has 4 candidate answers. Following prior work (Yang et al., 2019; Liu et al., 2019), we use the concatenation of the passage, question, and each candidate answer as the input to models. Then, we use the representations from the “[CLS)” token for predicting the probability of each answer. The dataset consists of two domains: middle school and high school. We train our models on both domains and report accuracies on both the development set and test set.
+
+# A.4 HYPERPARAMETERS
+
+Hyperparameters for downstream tasks are shown in Table 14. We adapt these hyperparameters from Liu et al. (2019), Devlin et al. (2019), and Yang et al. (2019).
+
+<table><tr><td></td><td>LR</td><td>BSZ</td><td>ALBERT DR</td><td>Classifier DR</td><td>TS</td><td>WS</td><td>MSL</td></tr><tr><td>CoLA</td><td>1.00E-05</td><td>16</td><td>0</td><td>0.1</td><td>5336</td><td>320</td><td>512</td></tr><tr><td>STS</td><td>2.00E-05</td><td>16</td><td>0</td><td>0.1</td><td>3598</td><td>214</td><td>512</td></tr><tr><td>SST-2</td><td>1.00E-05</td><td>32</td><td>0</td><td>0.1</td><td>20935</td><td>1256</td><td>512</td></tr><tr><td>MNLI</td><td>3.00E-05</td><td>128</td><td>0</td><td>0.1</td><td>10000</td><td>1000</td><td>512</td></tr><tr><td>QNLI</td><td>1.00E-05</td><td>32</td><td>0</td><td>0.1</td><td>33112</td><td>1986</td><td>512</td></tr><tr><td>QQP</td><td>5.00E-05</td><td>128</td><td>0.1</td><td>0.1</td><td>14000</td><td>1000</td><td>512</td></tr><tr><td>RTE</td><td>3.00E-05</td><td>32</td><td>0.1</td><td>0.1</td><td>800</td><td>200</td><td>512</td></tr><tr><td>MRPC</td><td>2.00E-05</td><td>32</td><td>0</td><td>0.1</td><td>800</td><td>200</td><td>512</td></tr><tr><td>WNLI</td><td>2.00E-05</td><td>16</td><td>0.1</td><td>0.1</td><td>2000</td><td>250</td><td>512</td></tr><tr><td>SQuAD v1.1</td><td>5.00E-05</td><td>48</td><td>0</td><td>0.1</td><td>3649</td><td>365</td><td>384</td></tr><tr><td>SQuAD v2.0</td><td>3.00E-05</td><td>48</td><td>0</td><td>0.1</td><td>8144</td><td>814</td><td>512</td></tr><tr><td>RACE</td><td>2.00E-05</td><td>32</td><td>0.1</td><td>0.1</td><td>12000</td><td>1000</td><td>512</td></tr></table>
+
+Table 14: Hyperparameters for ALBERT in downstream tasks. LR: Learning Rate. BSZ: Batch Size. DR: Dropout Rate. TS: Training Steps. WS: Warmup Steps. MSL: Maximum Sequence Length.
\ No newline at end of file
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+# A MUTUAL INFORMATION MAXIMIZATION PERSPECTIVE OF LANGUAGE REPRESENTATION LEARNING
+
+Lingpeng Kong\*, Cyprien de Masson d'Autume\*, Wang Ling\*, Lei Yu\*, Zihang Dai\* Dani Yogatama\*
+
+DeepMind\*, Carnegie Mellon University, Google Brain
+
+London, United Kingdom
+
+{lingpenk, cyprien, lingwang, leiyu, zihangd, dyogatama}@google.com
+
+# ABSTRACT
+
+We show state-of-the-art word representation learning methods maximize an objective function that is a lower bound on the mutual information between different parts of a word sequence (i.e., a sentence). Our formulation provides an alternative perspective that unifies classical word embedding models (e.g., Skip-gram) and modern contextual embeddings (e.g., BERT, XLNet). In addition to enhancing our theoretical understanding of these methods, our derivation leads to a principled framework that can be used to construct new self-supervised tasks. We provide an example by drawing inspirations from related methods based on mutual information maximization that have been successful in computer vision, and introduce a simple self-supervised objective that maximizes the mutual information between a global sentence representation and  $n$ -grams in the sentence. Our analysis offers a holistic view of representation learning methods to transfer knowledge and translate progress across multiple domains (e.g., natural language processing, computer vision, audio processing).
+
+# 1 INTRODUCTION
+
+Advances in representation learning have driven progress in natural language processing. Performance on many downstream tasks have improved considerably, achieving parity with human baselines in benchmark leaderboards such as SQuAD (Rajpurkar et al., 2016; 2018) and GLUE (Wang et al., 2019). The main ingredient is the "pretrain and fine-tune" approach, where a large text encoder is trained on an unlabeled corpus with self-supervised training objectives and used to initialize a task-specific model. Such an approach has also been shown to reduce the number of training examples that is needed to achieve good performance on the task of interest (Yogatama et al., 2019).
+
+In contrast to first-generation models that learn word type embeddings (Mikolov et al., 2013; Pennington et al., 2014), recent methods have focused on contextual token representations—i.e., learning an encoder to represent words in context. Many of these encoders are trained with a language modeling objective, where the representation of a context is trained to be predictive of a target token by maximizing the log likelihood of predicting this token (Dai & Le, 2015; Howard & Ruder, 2018; Radford et al., 2018; 2019). In a vanilla language modeling objective, the target token is always the next token that follows the context. Peters et al. (2018) propose an improvement by adding a reverse objective that also predicts the word token that precedes the context. Following this trend, current state-of-the-art encoders such as BERT (Devlin et al., 2018) and XLNet (Yang et al., 2019) are also trained with variants of the language modeling objective: masked language modeling and permutation language modeling.
+
+In this paper, we provide an alternative view and show that these methods also maximize a lower bound on the mutual information between different parts of a word sequence. Such a framework is inspired by the InfoMax principle (Linsker, 1988) and has been the main driver of progress in self-supervised representation learning in other domains such as computer vision, audio processing, and reinforcement learning (Belghazi et al., 2018; van den Oord et al., 2019; Hjelm et al., 2019;
+
+Bachman et al., 2019; O'Connor & Veeling, 2019). Many of these methods are trained to maximize a particular lower bound called InfoNCE (van den Oord et al., 2019)—also known as contrastive learning (Arora et al., 2019). The main idea behind contrastive learning is to divide an input data into multiple (possibly overlapping) views and maximize the mutual information between encoded representations of these views, using views derived from other inputs as negative samples. In §2, we provide an overview of representation learning with mutual information maximization. We then show how the skip-gram objective (§3.1; Mikolov et al. 2013), masked language modeling (§3.2; Devlin et al. 2018), and permutation language modeling (§3.3; Yang et al. 2019), fit in this framework.
+
+In addition to providing a principled theoretical understanding that bridges progress in multiple areas, our proposed framework also gives rise to a general class of word representation learning models which serves as a basis for designing and combining self-supervised training objectives to create better language representations. As an example, we show how to use this framework to construct a simple self-supervised objective that maximizes the mutual information between a sentence and  $n$ -grams in the sentence ( $\S 4$ ). We combine it with a variant of the masked language modeling objective and show that the resulting representation performs better, particularly on tasks such as question answering and linguistics acceptability ( $\S 5$ ).
+
+# 2 MUTUAL INFORMATION MAXIMIZATION
+
+Mutual information measures dependencies between random variables. Given two random variables  $A$  and  $B$ , it can be understood as how much knowing  $A$  reduces the uncertainty in  $B$  or vice versa. Formally, the mutual information between  $A$  and  $B$  is:
+
+$$
+I (A, B) = H (A) - H (A \mid B) = H (B) - H (B \mid A).
+$$
+
+Consider  $A$  and  $B$  to be different views of an input data (e.g., a word and its context, two different partitions of a sentence). Consider a function  $f$  that takes  $A = a$  and  $B = b$  as its input. The goal of training is to learn parameters of the function  $f$  that maximizes  $I(A, B)$ .
+
+Maximizing mutual information directly is generally intractable when the function  $f$  consists of modern encoders such as neural networks (Paninski, 2003), so we need to resort to a lower bound on  $I(A,B)$ . One particular lower bound that has been shown to work well in practice is InfoNCE (Logeswaran & Lee, 2018; van den Oord et al., 2019), which is based on Noise Contrastive Estimation (NCE; Gutmann & Hyvarinen, 2012). InfoNCE is defined as:
+
+$$
+I (A, B) \geq \mathbb {E} _ {p (A, B)} \left[ f _ {\boldsymbol {\theta}} (a, b) - \mathbb {E} _ {q (\tilde {\mathcal {B}})} \left[ \log \sum_ {\tilde {b} \in \tilde {\mathcal {B}}} \exp f _ {\boldsymbol {\theta}} (a, \tilde {b}) \right] \right] + \log | \tilde {\mathcal {B}} |, \tag {1}
+$$
+
+where  $a$  and  $b$  are different views of an input sequence,  $f_{\theta} \in \mathbb{R}$  is a function parameterized by  $\pmb{\theta}$  (e.g., a dot product between encoded representations of a word and its context, a dot product between encoded representations of two partitions of a sentence), and  $\tilde{\mathcal{B}}$  is a set of samples drawn from a proposal distribution  $q(\tilde{\mathcal{B}})$ . The set  $\tilde{\mathcal{B}}$  contains the positive sample  $b$  and  $|\tilde{\mathcal{B}}| - 1$  negative samples.
+
+Learning representations based on this objective is also known as contrastive learning. Arora et al. (2019) show representations learned by such a method have provable performance guarantees and reduce sample complexity on downstream tasks.
+
+We note that InfoNCE is related to cross-entropy. When  $\tilde{\mathcal{B}}$  always includes all possible values of the random variable  $B$  (i.e.,  $\tilde{\mathcal{B}} = \mathcal{B}$ ) and they are uniformly distributed, maximizing InfoNCE is analogous to maximizing the standard cross-entropy loss:
+
+$$
+\mathbb {E} _ {p (A, B)} \left[ f _ {\boldsymbol {\theta}} (a, b) - \log \sum_ {\tilde {b} \in \mathcal {B}} \exp f _ {\boldsymbol {\theta}} (a, \tilde {b}) \right]. \tag {2}
+$$
+
+Eq. 2 above shows that InfoNCE is related to maximizing  $p_{\theta}(b \mid a)$ , and it approximates the summation over elements in  $\mathcal{B}$  (i.e., the partition function) by negative sampling. As a function of the negative samples, the InfoNCE bound is tighter when  $\widetilde{\mathcal{B}}$  contains more samples (as can be seen in Eq. 1 above by inspecting the log  $|\widetilde{\mathcal{B}}|$  term). Approximating a softmax over a large vocabulary with negative samples is a popular technique that has been widely used in natural language processing in the past. We discuss it here to make the connection under this framework clear.
+
+# 3 MODELS
+
+We describe how Skip-gram, BERT, and XLNet fit into the mutual information maximization framework as instances of InfoNCE. In the following, we assume that  $f_{\theta}(a,b) = g_{\psi}(b)^{\top}g_{\omega}(a)$ , where  $\pmb{\theta} = \{\pmb{\omega},\pmb{\psi}\}$ . Denote the vocabulary set by  $\mathcal{V}$  and the size of the vocabulary by  $V$ . For word representation learning, we seek to learn an encoder parameterized by  $\pmb{\omega}$  to represent each word in a sequence  $\pmb{x} = \{x_1,x_1,\dots ,x_T\}$  in  $d$  dimensions. For each of the models we consider in this paper,  $a$  and  $b$  are formed by taking different parts of  $x$  (e.g.,  $a := x_0$  and  $b := x_T$ ).
+
+# 3.1 SKIP-GRAM
+
+We first start with a simple word representation learning model Skip-gram (Mikolov et al., 2013). Skip-gram is a method for learning word representations that relies on the assumption that a good representation of a word should be predictive of its context. The objective function that is maximized in Skip-gram is:  $\mathbb{E}_{p(x_i, x_j^i)}[p(x_j^i \mid x_i)]$ , where  $x_i$  is a word token and  $x_j^i$  is a context word of  $x_i$ .
+
+Let  $b$  be the context word to be predicted  $x_{j}^{i}$  and  $a$  be the input word  $x_{i}$ . Recall that  $f_{\theta}(a,b)$  is  $g_{\psi}(b)^{\top}g_{\omega}(a)$ . The skip-gram objective function can be written as an instance of InfoNCE (Eq. 1) where  $g_{\psi}(b)$  and  $g_{\omega}(a)$  are embedding lookup functions that map each word type to  $\mathbb{R}^d$ . (i.e.,  $g_{\psi}(b), g_{\omega}(a): \mathcal{V} \to \mathbb{R}^d$ ).
+
+$p(x_{j}^{i}\mid x_{i})$  can either be computed using a standard softmax over the entire vocabulary or with negative sampling (when the vocabulary is very large). These two approaches correspond to different choices of  $\tilde{\mathcal{B}}$ . In the softmax approach,  $\tilde{\mathcal{B}}$  is the full vocabulary set  $\mathcal{V}$  and each word in  $\mathcal{V}$  is uniformly distributed. In negative sampling,  $\tilde{\mathcal{B}}$  is a set of negative samples drawn from e.g., a unigram distribution.
+
+While Skip-gram has been widely accepted as an instance contrastive learning (Mikolov et al., 2013; Mnih & Kavukcuoglu, 2013), we include it here to illustrate its connection with modern approaches such as BERT and XLNet described subsequently. We can see that the two views of an input sentence that are considered in Skip-gram are two words that appear in the same sentence, and they are encoded using simple lookup functions.
+
+# 3.2 BERT
+
+Devlin et al. (2018) introduce two self-supervised tasks for learning contextual word representations: masked language modeling and next sentence prediction. Previous work suggests that the next sentence prediction objective is not necessary to train a high quality BERT encoder and the masked language modeling appears to be the key to learn good representations (Liu et al., 2019; Joshi et al., 2019; Lample & Conneau, 2019), so we focus on masked language modeling here. However, we also show how next sentence prediction fits into our framework in Appendix A.
+
+In masked language modeling, given a sequence of word tokens of length  $T$ ,  $\pmb{x} = \{x_{1},\dots,x_{T}\}$ , BERT replaces  $15\%$  of the tokens in the sequence with (i) a mask symbol  $80\%$  of the time, (ii) a random word  $10\%$  of the time, or (iii) its original word. For each replaced token, it introduces a term in the masked language modeling training objective to predict the original word given the perturbed sequence  $\hat{\pmb{x}}_i = \{x_1,\dots,\hat{x}_i,\dots,x_T\}$  (i.e., the sequence  $\pmb{x}$  masked at  $x_{i}$ ). This training objective can be written as:  $\mathbb{E}_{p(x_i,\hat{x}_i)}[p(x_i\mid \hat{x}_i)]$ .
+
+Following our notation in §2, we have  $f_{\theta}(a,b) = g_{\psi}(b)^{\top}g_{\omega}(a)$ . Let  $b$  be a masked word  $x_{i}$  and  $a$  be the masked sequence  $\hat{\pmb{x}}_i$ . Consider a Transformer encoder parameterized by  $\omega$  and denote  $g_{\omega}(\hat{\pmb{x}}_i)\in \mathbb{R}^d$  as a function that returns the final hidden state corresponding to the  $i$ -th token (i.e., the masked token) after running  $\hat{\pmb{x}}_i$  through the Transformer. Let  $g_{\psi}:\mathcal{V}\to \mathbb{R}^{d}$  be a lookup function that maps each word type into a vector and  $\tilde{\mathcal{B}} = \mathcal{B}$  be the full vocabulary set  $\mathcal{V}$ . Under this formulation, the masked language modeling objective maximizes Eq. 1 and different choices of masking probabilities can be understood as manipulating the joint distributions  $p(a,b)$ . In BERT, the two views of a sentence correspond to a masked word in the sentence and its masked context.
+
+Contextual vs. non-contextual. It is generally understood that the main difference between Skip-gram and BERT is that Skip-gram learns representations of word types (i.e., the representation for a word is always the same regardless of the context it appears in) and BERT learns representations of word tokens. We note that under our formulation for either Skip-gram or BERT, the encoder that we want to learn appears in  $g_{\omega}$ , and  $g_{\psi}$  is not used after training. We show that Skip-gram and BERT maximizes a similar objective, and the main difference is in the choice of the encoder that forms  $g_{\omega}$  —a context dependent Transformer encoder that takes a sequence as its input for BERT and a simple word embedding lookup for Skip-gram.
+
+# 3.3 XLNET
+
+Yang et al. (2019) propose a permutation language modeling objective to learn contextual word representations. This objective considers all possible factorization permutations of a joint distribution of a sentence. Given a sentence  $\pmb{x} = \{x_{1},\dots,x_{T}\}$ , there are  $T!$  ways to factorize its joint distribution. Given a sentence  $\pmb{x}$ , denote a permutation by  $z\in \mathbb{Z}$ . XLNet optimizes the objective function:
+
+$$
+\mathbb {E} _ {p (\boldsymbol {x})} \left[ \mathbb {E} _ {p (\boldsymbol {z})} \left[ \sum_ {t = 1} ^ {T} \log p (x _ {t} ^ {\boldsymbol {z}} \mid \boldsymbol {x} _ {<   t} ^ {\boldsymbol {z}}) \right] \right].
+$$
+
+As a running example, consider a permutation order  $3, 1, 5, 2, 4$  for a sentence  $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}$ . Given the order, XLNet is only trained to predict the last  $S$  tokens in practice. For  $S = 1$ , the context sequence used for training is  $x_{1}, x_{2}, x_{3}, \ldots, x_{5}$ , with  $x_{4}$  being the target word.
+
+In addition to replacing the Transformer encoder with Transformer XL (Dai et al., 2019), a key architectural innovation of XLNet is the two-stream self-attention. In two-stream self-attention, a shared encoder is used to compute two sets of hidden representations from one original sequence. They are called the query stream and the content stream. In the query stream, the input sequence is masked at the target position, whereas the content stream sees the word at the target position. Words at future positions for the permutation order under consideration are also masked in both streams. These masks are implemented as two attention mask matrices. During training, the final hidden representation for a target position from the query stream is used to predict the target word.
+
+Since there is only one set of encoder parameters for both streams, we show that we can arrive at the permutation language modeling objective from the masked language modeling objective with an architectural change in the encoder. Denote a hidden representation by  $\mathbf{h}_t^k$ , where  $t$  indexes the position and  $k$  indexes the layer, and consider the training sequence  $x_{1}, x_{2}, x_{3}, \ldots, x_{5}$  and the permutation order  $3, 1, 5, 2, 4$ . In BERT, we compute attention scores to obtain  $\mathbf{h}_t^k$  from  $\mathbf{h}_t^{k-1}$  for every  $t$  (i.e.,  $t = 1, \ldots, T$ ), where  $\mathbf{h}_4^0$  is the embedding for the mask symbol. In XLNet, the attention scores for future words in the permutation order are masked to 0. For example, when we compute  $\mathbf{h}_1^k$ , only the attention score from  $\mathbf{h}_3^{k-1}$  is considered (since the permutation order is  $3, 1, 5, 2, 4$ ). For  $\mathbf{h}_5^k$ , we use  $\mathbf{h}_1^{k-1}$  and  $\mathbf{h}_3^{k-1}$ . XLNet does not require a mask symbol embedding since the attention score from a masked token is always zeroed out with an attention mask (implemented as a matrix). As a result, we can consider XLNet training as masked language modeling with stochastic attention masks in the encoder.
+
+It is now straightforward to see that the permutation language modeling objective is an instance of Eq.1, where  $b$  is a target token  $x_{i}$  and  $a$  is a masked sequence  $\hat{\pmb{x}}_i = \{x_1,\dots ,\hat{x}_i,\dots ,x_T\}$ . Similar to
+
+Table 1: Summary of methods as instances of contrastive learning. See text for details.  
+
+<table><tr><td>Objective</td><td>a</td><td>b</td><td>p(a,b)</td><td>gω</td><td>gψ</td></tr><tr><td>Skip-gram</td><td>word</td><td>word</td><td>word and its context</td><td>lookup</td><td>lookup</td></tr><tr><td>MLM</td><td>context</td><td>masked word</td><td>masked tokens probability</td><td>Transformer</td><td>lookup</td></tr><tr><td>NSP</td><td>sentence</td><td>sentence</td><td>(non-)consecutive sentences</td><td>Transformer</td><td>lookup</td></tr><tr><td>XLNet</td><td>context</td><td>masked word</td><td>factorization permutation</td><td>TXL++</td><td>lookup</td></tr><tr><td>DIM</td><td>context</td><td>masked n-grams</td><td>sentence and its n-grams</td><td>Transformer</td><td>not used</td></tr></table>
+
+BERT, we have a Transformer encoder parameterized by  $\omega$  and denote  $g_{\omega}(\hat{x}_i) \in \mathbb{R}^d$  as a function that returns the final hidden state corresponding to the  $i$ -th token (i.e., the masked token) after running  $\hat{x}_i$  through the Transformer. Let  $g_{\psi}: \mathcal{V} \to \mathbb{R}^d$  be a lookup function that maps each word type into a vector and  $\tilde{\mathcal{B}} = \mathcal{B}$  be the full vocabulary set  $\mathcal{V}$ . The main difference between BERT and XLNet is that the encoder that forms  $g_{\omega}$  used in XLNet implements attention masking based on a sampled permutation order when building its representations. In addition, XLNet and BERT also differ in the choice of  $p(a, b)$  since each of them has its own masking procedure. However, we can see that both XLNet and BERT maximize the same objective.
+
+# 4 INFOWORD
+
+Our analysis on Skip-Gram, BERT, and XLNet shows that their objective functions are different instances of InfoNCE in Eq.1, although they are typically trained using the entire vocabulary set for  $\tilde{\mathcal{B}}$  instead of negative sampling. These methods differ in how they choose which views of a sentence they use as  $a$  and  $b$ , the data distribution  $p(a,b)$ , and the architecture of the encoder for computing  $g_{\omega}$  which we summarize in Table 1. Seen under this unifying framework, we can observe that progress in the field has largely been driven by using a more powerful encoder to represent  $g_{\omega}$ . While we only provide derivations for Skip-gram, BERT, and XLNet, it is straightforward to show that other language-modeling-based pretraining-objectives such as those used in ELMo (Peters et al., 2018) and GPT-2 (Radford et al., 2019) can be formulated under this framework.
+
+Our framework also allows us to draw connections to other mutual information maximization representation learning methods that have been successful in other domains (e.g., computer vision, audio processing, reinforcement learning). In this section, we discuss an example derive insights to design a simple self-supervised objective for learning better language representations.
+
+Deep InfoMax (DIM; Hjelm et al., 2019) is a mutual information maximization based representation learning method for images. DIM shows that maximizing the mutual information between an image representation and local regions of the image improves the quality of the representation. The complete objective function that DIM maximizes consists of multiple terms. Here, we focus on a term in the objective that maximizes the mutual information between local features and global features. We describe the main idea of this objective for learning representations from a one-dimensional sequence, although it is originally proposed to learn from a two-dimensional object.
+
+Given a sequence  $\pmb{x} = \{x_{1}, x_{2}, \dots, x_{T}\}$ , we consider the "global" representation of the sequence to be the hidden state of the first token (assumed to be a special start of sentence symbol) after contextually encoding the sequence  $g_{\omega}(\pmb{x})$ , and the local representations to be the encoded representations of each word in the sequence  $g_{\psi}(x_{t})$ . We can use the contrastive learning framework to design a task that maximizes the mutual information between this global representation vector and its corresponding "local" representations using local representations from other sequences  $g_{\psi}(\hat{x}_{t})$  as negative samples. This is analogous to training the global representation vector of a sentence to choose which words appear in the sentence and which words are from other sentences. However, if we feed the original sequence  $\pmb{x}$  to the encoder and take the hidden state of the first token as the global
+
+representation, the task becomes trivial since the global representation is built using all the words in the sequence. We instead use a masked sequence  $a \coloneqq \hat{\pmb{x}}_t = \{x_1, \dots, \hat{x}_t, \dots, x_T\}$  and  $b \coloneqq x_t$ .
+
+State-of-the-art methods based on language modeling objectives consider all negative samples since the second view of the input data (i.e., the part denoted by  $b$  in Eq. 1) that are used is simple and it consists of only a target word—hence the size of the negative set is still manageable. A major benefit of the contrastive learning framework is that we only need to be able to take negative samples for training. Instead of individual words, we can use  $n$ -grams as the local representations.<sup>6</sup> Denote an  $n$ -gram by  $x_{i:j}$  and a masked sequence masked at position  $i$  to  $j$  by  $\hat{x}_{i:j}$ . We define  $\mathcal{I}_{\mathrm{DIM}}$  as:
+
+$$
+\mathbb {I} _ {\mathrm {D I M}} = \mathbb {E} _ {p (\hat {\boldsymbol {x}} _ {i: j}, \boldsymbol {x} _ {i: j})} \left[ g _ {\boldsymbol {\omega}} (\hat {\boldsymbol {x}} _ {i: j}) ^ {\top} g _ {\boldsymbol {\omega}} (\boldsymbol {x} _ {i: j}) - \log \sum_ {\tilde {\boldsymbol {x}} _ {i: j} \in \tilde {\mathcal {S}}} \exp (g _ {\boldsymbol {\omega}} (\hat {\boldsymbol {x}} _ {i: j}) ^ {\top} g _ {\boldsymbol {\omega}} (\tilde {\boldsymbol {x}} _ {i: j})) \right],
+$$
+
+where  $\hat{\pmb{x}}_{i:j}$  is a sentence masked at position  $i$  to  $j$ ,  $\pmb{x}_{i:j}$  is an  $n$ -gram spanning from  $i$  to  $j$ , and  $\tilde{\pmb{x}}_{i:j}$  is an  $n$ -gram from a set  $\tilde{\mathbb{S}}$  that consists of the positive sample  $\pmb{x}_{i:j}$  and negative  $n$ -grams from other sentences in the corpus. We use one Transformer to encode both views, so we do not need  $g_{\psi}$  here.
+
+Since the main goal of representation learning is to train an encoder parameterized by  $\omega$ , it is possible to combine multiple self-supervised tasks into an objective function in the contrastive learning framework. Our model, which we denote INFOWORD, combines the above objective—which is designed to improve sentence and span representations—with a masked language modeling objective  $\mathcal{I}_{\mathrm{MLM}}$  for learning word representations. The only difference between our masked language modeling objective and the standard masked language modeling objective is that we use negative sampling to construct  $\tilde{\mathcal{V}}$  by sampling from the unigram distribution. We have:
+
+$$
+\mathcal {I} _ {\mathrm {M L M}} = \mathbb {E} _ {p (\hat {\boldsymbol {x}} _ {i}, \boldsymbol {x} _ {i})} \left[ g _ {\boldsymbol {\omega}} (\hat {\boldsymbol {x}} _ {i}) ^ {\top} g _ {\boldsymbol {\psi}} (\boldsymbol {x} _ {i}) - \log \sum_ {\tilde {\boldsymbol {x}} _ {i} \in \tilde {\mathcal {V}}} \exp (g _ {\boldsymbol {\omega}} (\hat {\boldsymbol {x}} _ {i}) ^ {\top} g _ {\boldsymbol {\psi}} (\tilde {\boldsymbol {x}} _ {i})) \right],
+$$
+
+where  $\hat{x}_i$  a sentence masked at position  $i$  and  $x_i$  is the  $i$ -th token in the sentence.
+
+Our overall objective function is a weighted combination of the two terms above:
+
+$$
+\mathcal {I} _ {\text {I N F O W O R D}} = \lambda_ {\text {M L M}} \mathcal {I} _ {\text {M L M}} + \lambda_ {\text {D I M}} \mathcal {I} _ {\text {D I M}},
+$$
+
+where  $\lambda_{\mathrm{MLM}}$  and  $\lambda_{\mathrm{DIM}}$  are hyperparameters that balance the contribution of each term.
+
+# 5 EXPERIMENTS
+
+In this section, we evaluate the effects of training masked language modeling with negative sampling and adding  $\mathcal{I}_{\mathrm{DIM}}$  to the quality of learned representations.
+
+# 5.1 SETUP
+
+We largely follow the same experimental setup as the original BERT model (Devlin et al., 2018). We have two Transformer architectures similar to  $\mathrm{BERT}_{\mathrm{BASE}}$  and  $\mathrm{BERT}_{\mathrm{LARGE}}$ .  $\mathrm{BERT}_{\mathrm{BASE}}$  has 12 hidden layers, 768 hidden dimensions, and 12 attention heads (110 million parameters); whereas  $\mathrm{BERT}_{\mathrm{LARGE}}$  has 24 hidden layers, 1024 hidden dimensions, and 16 attention heads (340 million parameters).
+
+For each of the Transformer variant above, we compare three models in our experiments:
+
+- BERT: The original BERT model publicly available in https://github.com/google-research/bert.  
+- BERT-NCE: Our reimplementation of BERT. It differs from the original implementation in several ways: (1) we only use the masked language modeling objective and remove next sentence prediction, (2) we use negative sampling instead of softmax, and (3) we only use one sentence for each training example in a batch.
+
+- INFOWORD: Our model described in §4. The main difference between INFOWORD and BERT-NCE is the addition of  $\mathcal{I}_{\mathrm{DIM}}$  to the objective function. We discuss how we mask the data for  $\mathcal{I}_{\mathrm{DIM}}$  in §5.2.
+
+# 5.2 PRETRAINING
+
+We use the same training corpora and apply the same preprocessing and tokenization as BERT. We create masked sequences for training with  $\mathcal{I}_{\mathrm{DIM}}$  as follows. We iteratively sample  $n$ -grams from a sequence until the masking budget ( $15\%$  of the sequence length) has been spent. At each sampling iteration, we first sample the length of the  $n$ -gram (i.e.,  $n$  in  $n$ -grams) from a Gaussian distribution  $\mathcal{N}(5, 1)$  clipped at 1 (minimum length) and 10 (maximum length). Since BERT tokenizes words into subwords, we measure the  $n$ -gram length at the word level and compute the masking budget at the subword level. This procedure is inspired by the masking approach in Joshi et al. (2019).
+
+For negative sampling, we use words and  $n$ -grams from other sequences in the same batch as negative samples (for MLM and DIM respectively). There are approximately 70,000 subwords and 10,000  $n$ -grams (words and phrases) in a batch. We discuss hyperparameter details in Appendix B.
+
+# 5.3 FINE-TUNING
+
+We evaluate on two benchmarks: GLUE (Wang et al., 2019) and SQuAD(Rajpurkar et al., 2016). We train a task-specific decoder and fine-tune pretrained models for each dataset that we consider. We describe hyperparameter details in Appendix B.
+
+GLUE is a set of natural language understanding tasks that includes sentiment analysis, linguistic acceptability, paraphrasing, and natural language inference. Each task is formulated as a classification task. The tasks in GLUE are either a single-sentence classification task or a sentence pair classification task. We follow the same setup as the original BERT model and add a start of sentence symbol (i.e., the CLS symbol) to every example and use a separator symbol (i.e., the SEP symbol) to separate two concatenated sentences (for sentence pair classification tasks). We add a linear transformation and a softmax layer to predict the correct label (class) from the representation of the first token of the sequence.
+
+SQuAD is a reading comprehension dataset constructed from Wikipedia articles. We report results on SQuAD 1.1. Here, we also follow the same setup as the original BERT model and predict an answer span—the start and end indices of the correct answer in the context. We use a standard span predictor as the decoder, which we describe in details in Appendix C.
+
+# 5.4 RESULTS
+
+We show our main results in Table 2 and Table 3. Our BERT reimplementation with negative sampling underperforms the original BERT model on GLUE but is significantly better on SQuAD. However, we think that the main reasons for this performance discrepancy are the different masking procedures (we use span-based masking instead of whole-word masking) and the different ways training examples are presented to the model (we use one consecutive sequence instead of two sequences separated by the separator symbol). Comparing BERT-NCE and INFOWORD, we observe the benefit of the new self-supervised objective  $\mathcal{I}_{\mathrm{DIM}}$  (better overall GLUE and SQuAD results), particularly on tasks such as question answering and linguistics acceptability that seem to require understanding of longer phrases. In order to better understand our model, we investigate its performance with varying numbers of training examples and different values of  $\lambda_{\mathrm{DIM}}$  on the SQuAD development set and show the results in Figure 1 (for models with the BASE configuration). We can see that INFOWORD consistently outperforms BERT-NCE and the performance gap is biggest when the dataset is smallest, suggesting the benefit of having better pretrained representations when there are fewer training examples.
+
+Table 2: Summary of results on GLUE.  
+
+<table><tr><td></td><td>Model</td><td>CoLA</td><td>SST-2</td><td>MRPC</td><td>QQP</td><td>MNLI (M/MM)</td><td>QNLI</td><td>RTE</td><td>GLUE AVG</td></tr><tr><td rowspan="3">BASE</td><td>BERT</td><td>52.1</td><td>93.5</td><td>88.9</td><td>71.2</td><td>84.6/83.4</td><td>90.5</td><td>66.4</td><td>78.8</td></tr><tr><td>BERT-NCE</td><td>50.8</td><td>93.0</td><td>88.6</td><td>70.5</td><td>83.2/83.0</td><td>90.9</td><td>65.9</td><td>78.2</td></tr><tr><td>INFOWORD</td><td>53.3</td><td>92.5</td><td>88.7</td><td>71.0</td><td>83.7/82.4</td><td>91.4</td><td>68.3</td><td>78.9</td></tr><tr><td rowspan="3">LARGE</td><td>BERT</td><td>60.5</td><td>94.9</td><td>89.3</td><td>72.1</td><td>86.7/85.9</td><td>92.7</td><td>70.1</td><td>81.5</td></tr><tr><td>BERT-NCE</td><td>54.7</td><td>93.1</td><td>89.5</td><td>71.2</td><td>85.8/85.0</td><td>92.7</td><td>72.5</td><td>80.6</td></tr><tr><td>INFOWORD</td><td>57.5</td><td>94.2</td><td>90.2</td><td>71.3</td><td>85.8/84.8</td><td>92.6</td><td>72.0</td><td>81.1</td></tr></table>
+
+Table 3: Summary of results on SQuAD 1.1.  
+
+<table><tr><td rowspan="2" colspan="2">Model</td><td colspan="2">DEV</td><td colspan="2">TEST</td></tr><tr><td>F1</td><td>EM</td><td>F1</td><td>EM</td></tr><tr><td rowspan="3">BASE</td><td>BERT</td><td>88.5</td><td>80.8</td><td>-</td><td>-</td></tr><tr><td>BERT-NCE</td><td>90.2</td><td>83.3</td><td>90.9</td><td>84.4</td></tr><tr><td>INFOWORD</td><td>90.7</td><td>84.0</td><td>91.4</td><td>84.7</td></tr><tr><td rowspan="3">LARGE</td><td>BERT</td><td>90.9</td><td>84.1</td><td>91.3</td><td>84.3</td></tr><tr><td>BERT-NCE</td><td>92.0</td><td>85.9</td><td>92.7</td><td>86.6</td></tr><tr><td>INFOWORD</td><td>92.6</td><td>86.6</td><td>93.1</td><td>87.3</td></tr></table>
+
+# 5.5 DISCUSSION
+
+Span-based models. We show how to design a simple self-supervised task in the InfoNCE framework that improves downstream performance on several datasets. Learning language representations to predict contiguous masked tokens has been explored in other context, and the objective introduced in  $\mathcal{I}_{\mathrm{DIM}}$  is related to these span-based models such as SpanBERT (Joshi et al., 2019) and MASS (Song et al., 2019). While our experimental goal is to demonstrate the benefit of contrastive learning for constructing self-supervised tasks, we note that INFOWORD is simpler to train and exhibits similar trends to SpanBERT that outperforms baseline models. We leave exhaustive comparisons to these methods to future work.
+
+Mutual information maximization. A recent study has questioned whether the success of InfoNCE as an objective function is due to its property as a lower bound on mutual information and provides an alternative hypothesis based on metric learning (Tschannen et al., 2019). Regardless of the prevailing perspective, InfoNCE is widely accepted as a good representation learning objective, and formulating state-of-the-art language representation learning methods under this framework offers valuable insights that unifies many popular representation learning methods.
+
+Regularization. Image representation learning methods often incorporate a regularization term in its objective function to encourage learned representations to look like a prior distribution (Hjelm et al., 2019; Bachman et al., 2019). This is useful for incorporating prior knowledge into a representation learning model. For example, the DeepInfoMax model has a term in its objective that encourages the learned representation from the encoder to match a uniform prior. Regularization is not commonly used when learning language representations. Our analysis and the connection we draw to representation learning methods used in other domains provide an insight into possible ways to incorporate prior knowledge into language representation learning models.
+
+Future directions. The InfoNCE framework provides a holistic way to view progress in language representation learning. The framework is very flexible and suggests several directions that can be explored to improve existing methods. We show that progress in the field has been largely driven by innovations in the encoder which forms  $g_{\omega}$ . InfoNCE is based on maximizing the mutual information between different views of an input data, and it facilitates training on structured views as long as we can perform negative sampling (van den Oord et al., 2019; Bachman et al., 2019). Our analysis demonstrates that existing methods based on language modeling objectives only consider a single target word as one of the views. We think that incorporating more complex views (e.g., higher-order or skip  $n$ -grams, syntactic and semantic parses, etc.) and designing appropriate self-supervised tasks
+
+![](images/6ecc71334866f92d0e1238e0b57faedd1436b394a026ae99a3530085313f278a.jpg)  
+Figure 1: The left plot shows  $F_{1}$  scores of BERT-NCE and INFOWORD as we increase the percentage of training examples on SQuAD (dev). The right plot shows  $F_{1}$  scores of INFOWORD on SQuAD (dev) as a function of  $\lambda_{\mathrm{DIM}}$ .
+
+![](images/8bf4b5b9e553f8da2cd450a9f887d3753ac6c6b079e38a910313b659b1c57e7b.jpg)
+
+is a promising future direction. A related area that is also underexplored is designing methods to obtain better negative samples.
+
+# 6 CONCLUSION
+
+We analyzed state-of-the-art language representation learning methods from the perspective of mutual information maximization. We provided a unifying view of classical and modern word embedding models and showed how they relate to popular representation learning methods used in other domains. We used this framework to construct a new self-supervised task based on maximizing the mutual information between the global representation and local representations of a sentence. We demonstrated the benefit of this new task via experiments on GLUE and SQuAD.
+
+# REFERENCES
+
+Sanjeev Arora, Hrishikesh Khandeparkar, Mikhail Khodak, Orestis Plevrakis, and Nikunj Saunshi. A theoretical analysis of contrastive unsupervised representation learning. In Proc. of ICML, 2019.  
+Philip Bachman, R Devon Hjelm, and William Buchwalter. Learning representations by maximizing mutual information across views. arXiv preprint 1906.00910, 2019.  
+Mohamed Ishmael Belghazi, Aristide Baratin, Sai Rajeswar, Sherjil Ozair, Yoshua Bengio, Aaron Courville, and R Devon Hjelm. Mine: Mutual information neural estimation. In Proc. of ICML, 2018.  
+Andrew M. Dai and Quoc V. Le. Semi-supervised sequence learning. In Proc. of NIPS, 2015.  
+Zihang Dai, Zhilin Yang, Yiming Yang, Jaime Carbonell, Quoc V. Le, and Ruslan Salakhutdinov. Transformer-XL: Attentive language models beyond a fixed-length context. In Proc. of ACL, 2019.  
+Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. BERT: Pre-training of deep bidirectional transformers for language understanding. In Proc. of NAACL, 2018.  
+M. D. Donsker and S. R. S. Varadhan. Asymptotic evaluation of certain markov process expectations for large time. iv. Communications on Pure and Applied Mathematics, 36(2):183-212, 1983.  
+Michael U. Gutmann and Aapo Hyvarinen. Noise-contrastive estimation of unnormalized statistical models, with applications to natural image statistics. Journal of Machine Learning Research, 13: 307-361, 2012.  
+R Devon Hjelm, Alex Fedorov, Samuel Lavoie-Marchildon, Karan Grewal, Phil Bachman, Adam Trischler, and Yoshua Bengio. Learning deep representations by mutual information estimation and maximization. In Proc. of ICLR, 2019.  
+Jeremy Howard and Sebastian Ruder. Universal language model fine-tuning for text classification. In Proc. of ACL, 2018.
+
+Mandar Joshi, Danqi Chen, Yinhan Liu, Daniel S. Weld, Luke Zettlemoyer, and Omer Levy. Span-BERT: Improving pre-training by representing and predicting spans. arXiv preprint 1907.10529, 2019.  
+Diederik P. Kingma and Jimmy Lei Ba. Adam: a method for stochastic optimization. In Proc. of ICLR, 2015.  
+Guillaume Lample and Alexis Conneau. Cross-lingual language model pretraining. arXiv preprint 1901.07291, 2019.  
+Ralph Linsker. Self-organization in a perceptual network. Computer, 21(3):105-117, 1988.  
+Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Mandar Joshi, Danqi Chen, Omer Levy, Mike Lewis, Luke Zettlemoyer, and Veselin Stoyanov. RoBERTa: A robustly optimized bert pretraining approach. arXiv preprint 1907.11692, 2019.  
+Lajanugen Logeswaran and Honglak Lee. An efficient framework for learning sentence representations. In Proc. of ICLR, 2018.  
+Tomas Mikolov, Ilya Sutskever, Kai Chen, Greg Corrado, and Jeffrey Dean. Distributed representations of words and phrases and their compositionality. In Proc. of NIPS, 2013.  
+Andriy Mnih and Koray Kavukcuoglu. Learning word embeddings efficiently with noise-contrastive estimation. In Proc. of NIPS, 2013.  
+Sebastian Nowozin, Botond Cseke, , and Ryota Tomioka. f-gan: Training generative neural samplers using variational divergence minimization. In Proc. of NIPS, 2016.  
+Sindy Lowe Peter O'Connor and Bastiaan S. Veeling. Greedy infomax for biologically plausible self-supervised representation learning. In Proc. of NeurIPS, 2019.  
+Liam Paninski. Estimation of entropy and mutual information. Neural computation, 15(6):1191-1253, 2003.  
+Jeffrey Pennington, Richard Socher, and Christopher D. Manning. Glove: Global vectors for word representation. In Proc. of EMNLP, 2014.  
+Matthew E. Peters, Mark Neumann, Mohit Iyyer, Matt Gardner, Christopher Clark, Kenton Lee, and Luke Zettlemoyer. Deep contextualized word representations. In Proc. of NAACL, 2018.  
+Ben Poole, Sherjil Ozair, Aaron van den Oord, Alexander A. Alemi, and George Tucker. On variational lower bounds of mutual information. In Proc. of ICML, 2019.  
+Alec Radford, Karthik Narasimhan, Tim Salimans, and Ilya Sutskever. Improving language understanding by generative pre-training. Technical report, OpenAI, 2018.  
+Alec Radford, Jeffrey Wu, Rewon Child, David Luan, Dario Amodei, and Ilya Sutskever. Language models are unsupervised multitask learners. Technical report, OpenAI, 2019.  
+Pranav Rajpurkar, Jian Zhang, Konstantin Lopyrev, and Percy Liang. SQuAD: 100,000+ questions for machine comprehension of text. In Proc. of EMNLP, 2016.  
+Pranav Rajpurkar, Robin Jia, and Percy Liang. Know what you don't know: Unanswerable questions for squad. In Proc. of ACL, 2018.  
+Kaitao Song, Xu Tan, Tao Qin, Jianfeng Lu, and Tie-Yan Liu. MASS: Masked sequence to sequence pre-training for language generation. In Proc. of ICML, 2019.  
+Michael Tschannen, Josip Djolonga, Paul K. Rubenstein, and Sylvain Gelly. On mutual information maximization for representation learning. arXiv preprint 1907.13625, 2019.  
+Aaron van den Oord, Yazhe Li, and Oriol Vinyals. Representation learning with contrastive predictive coding. arXiv preprint 1807.03748, 2019.
+
+Alex Wang, Amanpreet Singh, Julian Michael, Felix Hill, Omer Levy, and Samuel R. Bowman. GLUE: A multi-task benchmark and analysis platform for natural language understanding. In Proc. of ICLR, 2019.  
+Zhilin Yang, Zihang Dai, Yiming Yang, Jaime Carbonell, Ruslan Salakhutdinov, and Quoc V. Le. XLNet: Generalized autoregressive pretraining for language understanding. arXiv preprint 1906.08237, 2019.  
+Dani Yogatama, Cyprien de Masson d'Autume, Jerome Connor, Tomas Kocisky, Mike Chrzanowski, Lingpeng Kong, Angeliki Lazaridou, Wang Ling, Lei Yu, Chris Dyer, and Phil Blunsom. Learning and evaluating general linguistic intelligence. arXiv preprint 1901.11373, 2019.
+
+# A NEXT SENTENCE PREDICTION
+
+We show that the next sentence prediction objective used in BERT is an instance of contrastive learning in this section. In next sentence prediction, given two sentences  $\pmb{x}^1$  and  $\pmb{x}^2$ , the task is to predict whether these are two consecutive sentences or not. Training data for this task is created by sampling a random second sentence  $\hat{\pmb{x}}^2$  from the corpus to be used as a negative example 50% of the time.
+
+Consider a discriminator (i.e., a classifier with parameters  $\phi$ ) that takes encoded representations of concatenated  $x^1$  and  $x^2$  and returns a score. We denote this discriminator by  $d_{\phi}(x^1, x^2)$ . The next sentence prediction objective function is:
+
+$$
+\mathbb {E} _ {p \left(\boldsymbol {x} ^ {1}, \boldsymbol {x} ^ {2}\right)} \left[ \log d _ {\phi} \left(g _ {\omega} \left(\left[ \boldsymbol {x} ^ {1}, \boldsymbol {x} ^ {2} \right]\right)\right) + \log \left(1 - d _ {\phi} \left(g _ {\omega} \left(\left[ \boldsymbol {x} ^ {1}, \tilde {\boldsymbol {x}} ^ {2} \right]\right)\right)\right) \right].
+$$
+
+This objective function—which is used for training BERT—is known in the literature as "local" Noise Contrastive Estimation (Gutmann & Hyvarinen, 2012). Since summing over all possible negative sentences is intractable, BERT approximates this by using a binary classifier to distinguish real samples and noisy samples.
+
+An alternative approximation to using a binary classifier is to use "global NCE", which is what InfoNCE is based on. Here, we have:
+
+$$
+\mathbb {E} _ {p \left(\boldsymbol {x} ^ {1}, \boldsymbol {x} ^ {2}\right)} \left[ \psi^ {\top} g _ {\boldsymbol {\omega}} \left([ \boldsymbol {x} ^ {1}, \boldsymbol {x} ^ {2}) ]\right) - \log \sum_ {\tilde {\boldsymbol {x}} ^ {2} \in \tilde {\mathcal {X}} ^ {2}} \exp \left(\psi^ {\top} \left(g _ {\boldsymbol {\omega}} \left([ \boldsymbol {x} ^ {1}, \tilde {\boldsymbol {x}} ^ {2} ]\right)\right)\right) \right],
+$$
+
+where we sample negative sentences from the corpus and combine it with the positive sentence to construct  $\tilde{\mathcal{X}}^2$ . To make the connection of this objective function with InfoNCE in Eq. 1 explicit, let  $a$  and  $b$  be two consecutive sentences  $\pmb{x}_1$  and  $\pmb{x}_2$ . Let  $f_{\pmb{\theta}}(a,b)$  be  $\psi^\top g_{\omega}([a,b])$ , where  $\psi \in \mathbb{R}^d$  is a trainable parameter,  $[a,b]$  denotes a concatenation of  $a$  and  $b$ . Consider a Transformer encoder parameterized by  $\omega$ , and let  $g_{\omega}([a,b]) \in \mathbb{R}^d$  be a function that returns the final hidden state of the first token after running the concatenated sequence to the Transformer. Note that the encoder that we want to learn only depends on  $g_{\omega}$ , so both of these approximations can be used for training next sentence prediction.
+
+# B HYPERPARAMETERS
+
+Pretraining. We use Adam (Kingma & Ba, 2015) with  $\beta_{1} = 0.9$ ,  $\beta_{2} = 0.98$  and  $\epsilon = 1e - 6$ . The batch size for training is 1024 with a maximum sequence length of 512. We train for 400,000 steps (including 18,000 warmup steps) with a weight decay rate of 0.01. We set the learning rate to  $4e^{-4}$  for all variants of the BASE models and  $1e^{-4}$  for the LARGE models. We set  $\lambda_{\mathrm{MLM}}$  to 1.0 and tune  $\lambda_{\mathrm{DIM}} \in \{0.4, 0.6, 0.8, 1.0\}$ .
+
+GLUE. We set the maximum sequence length to 128. For each GLUE task, we use the respective development set to choose the learning rate from  $\{5e^{-6}, 1e^{-5}, 2e^{-5}, 3e^{-5}, 5e^{-5}\}$ , and the batch size from  $\{16, 32\}$ . The number of training epochs is set to 4 for CoLA and 10 for other tasks, following Joshi et al. (2019). We run each hyperparameter configuration 5 times and evaluate the best model on the test set (once).
+
+SQuAD. We set the maximum sequence length to 512 and train for 4 epochs. We use the development set to choose the learning rate from  $\{5e^{-6}, 1e^{-5}, 2e^{-5}, 3e^{-5}, 5e^{-5}\}$  and the batch size from  $\{16, 32\}$ .
+
+# C QUESTION ANSWERING DECODER
+
+We use a standard span predictor as follows. Denote the length of the context paragraph by  $M$ , and  $\mathbf{x}^{\mathrm{context}} = \{x_1^{\mathrm{context}}, \ldots, x_M^{\mathrm{context}}\}$ . Denote the encoded representation of the  $m$ -th token in the context by  $\mathbf{x}_{t,m}^{\mathrm{context}}$ . The question answering decoder introduces two sets of parameters:  $\mathbf{w}_{\mathrm{start}}$  and  $\mathbf{w}_{\mathrm{end}}$ . The probability of each context token being the start of the answer is computed as:  $p(\mathrm{start} = x_{t,m}^{\mathrm{context}} \mid \mathbf{x}_t) = \frac{\exp(\mathbf{w}_{\mathrm{start}}^\top \mathbf{x}_{t,m}^{\mathrm{context}})}{\sum_{n=0}^{M} \exp(\mathbf{w}_{\mathrm{start}}^\top \mathbf{x}_{t,n}^{\mathrm{context}})}$ . The probability of the end index of the answer is computed analogously using  $\mathbf{w}_{\mathrm{end}}$ . The predicted answer is the span with the highest probability after multiplying the start and end probabilities.
\ No newline at end of file
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+# AND THE BIT GOES DOWN: REVISITING THE QUANTIZATION OF NEURAL NETWORKS
+
+Pierre Stock $^{1,2}$ , Armand Joulin $^{1}$ , Rémi Gribonval $^{2}$ , Benjamin Graham $^{1}$ , Hervé Jégou $^{1}$
+
+<sup>1</sup>Facebook AI Research, <sup>2</sup>Univ Rennes, Inria, CNRS, IRISA
+
+# ABSTRACT
+
+In this paper, we address the problem of reducing the memory footprint of convolutional network architectures. We introduce a vector quantization method that aims at preserving the quality of the reconstruction of the network outputs rather than its weights. The principle of our approach is that it minimizes the loss reconstruction error for in-domain inputs. Our method only requires a set of unlabelled data at quantization time and allows for efficient inference on CPU by using byte-aligned codebooks to store the compressed weights. We validate our approach by quantizing a high performing ResNet-50 model to a memory size of 5 MB ( $20 \times$  compression factor) while preserving a top-1 accuracy of  $76.1\%$  on ImageNet object classification and by compressing a Mask R-CNN with a  $26 \times$  factor. $^{1}$
+
+# 1 INTRODUCTION
+
+There is a growing need for compressing the best convolutional networks (or ConvNets) to support embedded devices for applications like robotics and virtual/augmented reality. Indeed, the performance of ConvNets on image classification has steadily improved since the introduction of AlexNet (Krizhevsky et al., 2012). This progress has been fueled by deeper and richer architectures such as the ResNets (He et al., 2015) and their variants ResNeXts (Xie et al., 2017) or DenseNets (Huang et al., 2017). Those models particularly benefit from the recent progress made with weak supervision (Mahajan et al., 2018; Yaliniz et al., 2019; Berthelot et al., 2019). Compression of ConvNets has been an active research topic in the recent years, leading to networks with a  $71\%$  top-1 accuracy on ImageNet object classification that fit in 1 MB (Wang et al., 2018b).
+
+In this work, we propose a compression method particularly adapted to ResNet-like architectures. Our approach takes advantage of the high correlation in the convolutions by the use of a structured quantization algorithm, Product Quantization (PQ) (Jégou et al., 2011). More precisely, we exploit the spatial redundancy of information inherent to standard convolution filters (Denton et al., 2014). Besides reducing the memory footprint, we also produce compressed networks allowing efficient inference on CPU by using byte-aligned indexes, as opposed to entropy decoders (Han et al., 2016).
+
+Our approach departs from traditional scalar quantizers (Han et al., 2016) and vector quantizers (Gong et al., 2014; Carreira-Perpinañán & Idelbayev, 2017) by focusing on the accuracy of the activations rather than the weights. This is achieved by leveraging a weighted  $k$ -means technique. To our knowledge this strategy (see Section 3) is novel in this context. The closest work we are aware of is the one by Choi et al. (2016), but the authors use a different objective (their weighted term is derived from second-order information) along with a different quantization technique (scalar quantization). Our method targets a better in-domain reconstruction, as depicted by Figure 1.
+
+Finally, we compress the network sequentially to account for the dependency of our method to the activations at each layer. To prevent the accumulation of errors across layers, we guide this compression with the activations of the uncompressed network on unlabelled data: training by distillation (Hinton et al., 2014) allows for both an efficient layer-by-layer compression procedure and a global fine-tuning of the codewords. Thus, we only need a set of unlabelled images to adjust the codewords. As opposed to recent works by Mishra & Marr (2017) or Lopes et al. (2017), our distillation scheme is sequential and the underlying compression method is different (PQ vs. scalar).
+
+![](images/2fb21d71c12f31b09eae125143f0c8720d067c5fbaf19dde5ed4626c79ed83f8.jpg)  
+Figure 1: Illustration of our method. We approximate a binary classifier  $\varphi$  that labels images as dogs or cats by quantizing its weights. Standard method: quantizing  $\varphi$  with the standard objective function (1) promotes a classifier  $\widehat{\varphi}_{\mathrm{standard}}$  that tries to approximate  $\varphi$  over the entire input space and can thus perform badly for in-domain inputs. Our method: quantizing  $\varphi$  with our objective function (2) promotes a classifier  $\widehat{\varphi}_{\mathrm{activations}}$  that performs well for in-domain inputs. Images lying in the hatched area of the input space are correctly classified by  $\varphi_{\mathrm{activations}}$  but incorrectly by  $\varphi_{\mathrm{standard}}$ .
+
+We show that applying our approach to the semi-supervised ResNet-50 of Yalniz et al. (Yalniz et al., 2019) leads to a 5 MB memory footprint and a  $76.1\%$  top-1 accuracy on ImageNet object classification (hence  $20\times$  compression vs. the original model). Moreover, our approach generalizes to other tasks such as image detection. As shown in Section 4.3, we compress a Mask R-CNN (He et al., 2017) with a size budget around 6 MB ( $26\times$  compression factor) while maintaining a competitive performance.
+
+# 2 RELATED WORK
+
+There is a large body of literature on network compression. We review the works closest to ours and refer the reader to two recent surveys (Guo, 2018; Cheng et al., 2017) for a comprehensive overview.
+
+Low-precision training. Since early works like those of Courbariaux et al. (2015), researchers have developed various approaches to train networks with low precision weights. Those approaches include training with binary or ternary weights (Shayer et al., 2017; Zhu et al., 2016; Li & Liu, 2016; Rastegari et al., 2016; McDonnell, 2018), learning a combination of binary bases (Lin et al., 2017) and quantizing the activations (Zhou et al., 2016; 2017; Mishra et al., 2017). Some of these methods assume the possibility to employ specialized hardware that speed up inference and improve power efficiency by replacing most arithmetic operations with bit-wise operations. However, the back-propagation has to be adapted to the case where the weights are discrete.
+
+Quantization. Vector Quantization (VQ) and Product Quantization (PQ) have been extensively studied in the context of nearest-neighbor search (Jegou et al., 2011; Ge et al., 2014; Norouzi & Fleet, 2013). The idea is to decompose the original high-dimensional space into a cartesian product of subspaces that are quantized separately with a joint codebook. To our knowledge, Gong et al. (2014) were the first to introduce these stronger quantizers for neural network quantization, followed by Carreira-Perpinañán & Idelbayev (2017). As we will see in the remainder of this paper, employing this discretization off-the-shelf does not optimize the right objective function, and leads to a catastrophic drift of performance for deep networks.
+
+Pruning. Network pruning amounts to removing connections according to an importance criteria (typically the magnitude of the weight associated with this connection) until the desired model size/accuracy tradeoff is reached (LeCun et al., 1990). A natural extension of this work is to prune structural components of the network, for instance by enforcing channel-level (Liu et al., 2017) or filter-level (Luo et al., 2017) sparsity. However, these methods alternate between pruning and re-training steps and thus typically require a long training time.
+
+Dedicated architectures. Architectures such as SqueezeNet (Iandola et al., 2016), NASNet (Zoph et al., 2017), ShuffleNet (Zhang et al., 2017; Ma et al., 2018), MobileNets (Sandler et al., 2018) and EfficientNets (Tan & Le, 2019) are designed to be memory efficient. As they typically rely on a combination of depth-wise and point-wise convolutional filters, sometimes along with channel shuffling, they are less prone than ResNets to structured quantization techniques such as PQ. These architectures are either designed by hand or using the framework of architecture search (Howard et al., 2019). For instance, the respective model size and test top-1 accuracy of ImageNet of a MobileNet are  $13.4\mathrm{MB}$  for  $71.9\%$  to be compared with a vanilla ResNet-50 with size  $97.5\mathrm{MB}$  for a top-1 of  $76.2\%$ . Moreover, larger models such as ResNets can benefit from large-scale weakly- or semi-supervised learning to reach better performance (Mahajan et al., 2018; Yaliniz et al., 2019).
+
+Combining some of the mentioned approaches yields high compression factors as demonstrated by Han et al. with Deep Compression (DC) (Han et al., 2016) or more recently by Tung & Mori (Tung & Mori, 2018). Moreover and from a practical point of view, the process of compressing networks depends on the type of hardware on which the networks will run. Recent work directly quantizes to optimize energy-efficiency and latency time on a specific hardware (Wang et al., 2018a). Finally, the memory overhead of storing the full activations is negligible compared to the storage of the weights for two reasons. First, in realistic real-time inference setups, the batch size is almost always equal to one. Second, a forward pass only requires to store the activations of the current layer –which are often smaller than the size of the input– and not the whole activations of the network.
+
+# 3 OUR APPROACH
+
+In this section, we describe our strategy for network compression and we show how to extend our approach to quantize a modern ConvNet architecture. The specificity of our approach is that it aims at a small reconstruction error for the outputs of the layer rather than the layer weights themselves. We first describe how we quantize a single fully connected and convolutional layer. Then we describe how we quantize a full pre-trained network and finetune it.
+
+# 3.1 QUANTIZATION OF A FULLY-CONNECTED LAYER
+
+We consider a fully-connected layer with weights  $\mathbf{W} \in \mathbf{R}^{C_{\mathrm{in}} \times C_{\mathrm{out}}}$  and, without loss of generality, we omit the bias since it does not impact reconstruction error.
+
+Product Quantization (PQ). Applying the PQ algorithm to the columns of  $\mathbf{W}$  consists in evenly splitting each column into  $m$  contiguous subvectors and learning a codebook on the resulting  $mC_{\mathrm{out}}$  subvectors. Then, a column of  $\mathbf{W}$  is quantized by mapping each of its subvector to its nearest codeword in the codebook. For simplicity, we assume that  $C_{\mathrm{in}}$  is a multiple of  $m$ , i.e., that all the subvectors have the same dimension  $d = C_{\mathrm{in}} / m$ .
+
+More formally, the codebook  $\mathcal{C} = \{\mathbf{c}_1,\dots ,\mathbf{c}_k\}$  contains  $k$  codewords of dimension  $d$ . Any column  $\mathbf{w}_j$  of  $\mathbf{W}$  is mapped to its quantized version  $\mathbf{q}(\mathbf{w}_j) = (\mathbf{c}_{i_1},\ldots ,\mathbf{c}_{i_m})$  where  $i_{1}$  denotes the index of the codeword assigned to the first subvector of  $\mathbf{w}_j$ , and so forth. The codebook is then learned by minimizing the following objective function:
+
+$$
+\left\| \mathbf {W} - \widehat {\mathbf {W}} \right\| _ {2} ^ {2} = \sum_ {j} \left\| \mathbf {w} _ {j} - \mathbf {q} \left(\mathbf {w} _ {j}\right) \right\| _ {2} ^ {2}, \tag {1}
+$$
+
+where  $\widehat{\mathbf{W}}$  denotes the quantized weights. This objective can be efficiently minimized with  $k$ -means. When  $m$  is set to 1, PQ is equivalent to vector quantization (VQ) and when  $m$  is equal to  $C_{\mathrm{in}}$ , it is the scalar  $k$ -means algorithm. The main benefit of PQ is its expressivity: each column  $\mathbf{w}_j$  is mapped to a vector in the product  $\mathcal{C} = \mathcal{C}\times \dots \times \mathcal{C}$ , thus PQ generates an implicit codebook of size  $k^m$ .
+
+Our algorithm. PQ quantizes the weight matrix of the fully-connected layer. However, in practice, we are interested in preserving the output of the layer, not its weights. This is illustrated in the case of a non-linear classifier in Figure 1: preserving the weights a layer does not necessarily guarantee preserving its output. In other words, the Frobenius approximation of the weights of a layer is not guaranteed to be the best approximation of the output over some arbitrary domain (in particular for in-domain inputs). We thus propose an alternative to PQ that directly minimizes the reconstruction error on the output activations obtained by applying the layer to in-domain inputs. More precisely, given a batch of  $B$  input activations  $\mathbf{x} \in \mathbf{R}^{B \times C_{\mathrm{in}}}$ , we are interested in learning a codebook  $\mathcal{C}$
+
+that minimizes the difference between the output activations and their reconstructions:
+
+$$
+\left\| \mathbf {y} - \widehat {\mathbf {y}} \right\| _ {2} ^ {2} = \sum_ {j} \left\| \mathbf {x} \left(\mathbf {w} _ {j} - \mathbf {q} \left(\mathbf {w} _ {j}\right)\right) \right\| _ {2} ^ {2}, \tag {2}
+$$
+
+where  $\mathbf{y} = \mathbf{x}\mathbf{W}$  is the output and  $\widehat{\mathbf{y}} = \mathbf{x}\widehat{\mathbf{W}}$  its reconstruction. Our objective is a re-weighting of the objective in Equation (1). We can thus learn our codebook with a weighted  $k$ -means algorithm. First, we unroll  $\mathbf{x}$  of size  $B\times C_{\mathrm{in}}$  into  $\widetilde{\mathbf{x}}$  of size  $(B\times m)\times d$  i.e. we split each row of  $\mathbf{x}$  into  $m$  subvectors of size  $d$  and stack these subvectors. Next, we adapt the EM algorithm as follows.
+
+(1) E-step (cluster assignment). Recall that every column  $\mathbf{w}_j$  is divided into  $m$  subvectors of dimension  $d$ . Each subvector  $\mathbf{v}$  is assigned to the codeword  $\mathbf{c}_j$  such that
+
+$$
+\mathbf {c} _ {j} = \underset {\mathbf {c} \in \mathcal {C}} {\operatorname {a r g m i n}} \| \widetilde {\mathbf {x}} (\mathbf {c} - \mathbf {v}) \| _ {2} ^ {2}. \tag {3}
+$$
+
+This step is performed by exhaustive exploration. Our implementation relies on broadcasting to be computationally efficient.
+
+(2) M-step (codeword update). Let us consider a codeword  $\mathbf{c} \in \mathcal{C}$ . We denote  $(\mathbf{v}_p)_{p \in I_{\mathbf{c}}}$  the subvectors that are currently assigned to  $\mathbf{c}$ . Then, we update  $\mathbf{c} \gets \mathbf{c}^{\star}$ , where
+
+$$
+\mathbf {c} ^ {\star} = \underset {\mathbf {c} \in \mathbf {R} ^ {d}} {\operatorname {a r g m i n}} \sum_ {p \in I _ {\mathbf {c}}} \| \widetilde {\mathbf {x}} (\mathbf {c} - \mathbf {v} _ {p}) \| _ {2} ^ {2}. \tag {4}
+$$
+
+This step explicitly computes the solution of the least-squares problem<sup>2</sup>. Our implementation performs the computation of the pseudo-inverse of  $\widetilde{\mathbf{x}}$  before alternating between the Expectation and Minimization steps as it does not depend on the learned codebook  $\mathcal{C}$ .
+
+We initialize the codebook  $\mathcal{C}$  by uniformly sampling  $k$  vectors among those we wish to quantize. After performing the E-step, some clusters may be empty. To resolve this issue, we iteratively perform the following additional steps for each empty cluster of index  $i$ . (1) Find codeword  $\mathbf{c}_0$  corresponding to the most populated cluster; (2) define new codewords  $\mathbf{c}_0' = \mathbf{c}_0 + \mathbf{e}$  and  $\mathbf{c}_i' = \mathbf{c}_0 - \mathbf{e}$ , where  $\mathbf{e} \sim \mathcal{N}(\mathbf{0}, \varepsilon \mathbf{I})$  and (3) perform again the E-step. We proceed to the M-step after all the empty clusters are resolved. We set  $\varepsilon = 1\mathrm{e} - 8$  and we observe that its generally takes less than 1 or 2 E-M iterations to resolve all the empty clusters. Note that the quality of the resulting compression is sensitive to the choice of  $\mathbf{x}$ .
+
+# 3.2 CONVOLUTIONAL LAYERS
+
+Despite being presented in the case of a fully-connected layer, our approach works on any set of vectors. As a consequence, our approoach can be applied to a convolutional layer if we split the associated 4D weight matrix into a set of vectors. There are many ways to split a 4D matrix in a set of vectors and we are aiming for one that maximizes the correlation between the vectors since vector quantization based methods work the best when the vectors are highly correlated.
+
+Given a convolutional layer, we have  $C_{\mathrm{out}}$  filters of size  $K\times K\times C_{\mathrm{in}}$ , leading to an overall 4D weight matrix  $\mathbf{W}\in \mathbf{R}^{C_{\mathrm{out}}\times C_{\mathrm{in}}\times K\times K}$ . The dimensions along the output and input coordinate have no particular reason to be correlated. On the other hand, the spatial dimensions related to the filter size are by nature very correlated: nearby patches or pixels likely share information. As depicted in Figure 2, we thus reshape the weight matrix in a way that lead to spatially coherent quantization. More precisely, we quantize  $\mathbf{W}$  spatially into subvectors of size  $d = K\times K$  using the following procedure. We first reshape  $\mathbf{W}$  into a 2D matrix of size  $(C_{\mathrm{in}}\times K\times K)\times C_{\mathrm{out}}$ . Column  $j$  of the reshaped matrix  $\mathbf{W}_{\mathrm{r}}$  corresponds to the  $j^{\mathrm{th}}$  filter of  $\mathbf{W}$  and is divided into  $C_{\mathrm{in}}$  subvectors of size  $K\times K$ . Similarly, we reshape the input activations  $\mathbf{x}$  accordingly to  $\mathbf{x}_{\mathrm{r}}$  so that reshaping back the matrix  $\mathbf{x}_{\mathrm{r}}\mathbf{W}_{\mathrm{r}}$  yields the same result as  $\mathbf{x}*\mathbf{W}$ . In other words, we adopt a dual approach to the one using bi-level Toeplitz matrices to represent the weights. Then, we apply our method exposed in Section 3.1 to quantize each column of  $\mathbf{W}_{\mathrm{r}}$  into  $m = C_{\mathrm{in}}$  subvectors of size  $d = K\times K$  with  $k$  codewords, using  $\mathbf{x}_{\mathrm{r}}$  as input activations in (2). As a natural extension, we also quantize with larger subvectors, for example subvectors of size  $d = 2\times K\times K$ , see Section 4 for details.
+
+![](images/eaf4edb4795996ba60ba907c24a18819f722eda59eb054566c42640a1f0d84d7.jpg)  
+Figure 2: We quantize  $C_{\mathrm{out}}$  filters of size  $C_{\mathrm{in}} \times K \times K$  using a subvector size of  $d = K \times K$ . In other words, we spatially quantize the convolutional filters to take advantage of the redundancy of information in the network. Similar colors denote subvectors assigned to the same codewords.
+
+![](images/773452dc0c965b129cbab8a4700d09d7ca2975c121974d07df03a4c6c15cbe3c.jpg)
+
+![](images/3209edcff003e01b0014615b9bcafd60f5d42ecda4eef060639c6be29e1ab80f.jpg)
+
+In our implementation, we adapt the reshaping of  $\mathbf{W}$  and  $\mathbf{x}$  to various types of convolutions. We account for the padding, the stride, the number of groups (for depthwise convolutions and in particular for pointwise convolutions) and the kernel size. We refer the reader to the code for more details.
+
+# 3.3 NETWORK QUANTIZATION
+
+In this section, we describe our approach for quantizing a neural network. We quantize the network sequentially starting from the lowest layer to the highest layer. We guide the compression of the student network by the non-compressed teacher network, as detailed below.
+
+Learning the codebook. We recover the current input activations of the layer, i.e. the input activations obtained by forwarding a batch of images through the quantized lower layers, and we quantize the current layer using those activations. Experimentally, we observed a drift in both the reconstruction and classification errors when using the activations of the non-compressed network rather than the current activations.
+
+Finetuning the codebook. We finetune the codewords by distillation (Hinton et al., 2014) using the non-compressed network as the teacher network and the compressed network (up to the current layer) as the student network. Denoting  $y_{\mathrm{t}}$  (resp.  $y_{\mathrm{s}}$ ) the output probabilities of the teacher (resp. student) network, the loss we optimize is the Kullback-Leibler divergence  $\mathcal{L} = \mathrm{KL}(\mathbf{y}_{\mathrm{s}},\mathbf{y}_{\mathrm{t}})$ . Finetuning on codewords is done by averaging the gradients of each subvector assigned to a given codeword. More formally, after the quantization step, we fix the assignments once for all. Then, denoting  $(\mathbf{b}_p)_{p\in I_{\mathbf{c}}}$  the subvectors that are assigned to codeword  $\mathbf{c}$ , we perform the SGD update with a learning rate  $\eta$
+
+$$
+\mathbf {c} \leftarrow \mathbf {c} - \eta \frac {1}{\left| I _ {\mathbf {c}} \right|} \sum_ {p \in I _ {\mathbf {c}}} \frac {\partial \mathcal {L}}{\partial \mathbf {b} _ {p}}. \tag {5}
+$$
+
+Experimentally, we find the approach to perform better than finetuning on the target of the images as demonstrated in Table 3. Moreover, this approach does not require any labelled data.
+
+# 3.4 GLOBAL FINETUNING
+
+In a final step, we globally finetune the codebooks of all the layers to reduce any residual drifts and we update the running statistics of the BatchNorm layers: We empirically find it beneficial to finetune all the centroids after the whole network is quantized. The finetuning procedure is exactly the same as described in Section 3.3, except that we additionally switch the BatchNorms to the training mode, meaning that the learnt coefficients are still fixed but that the batch statistics (running mean and variance) are still being updated with the standard moving average procedure.
+
+We perform the global finetuning using the standard ImageNet training set for 9 epochs with an initial learning rate of 0.01, a weight decay of  $10^{-4}$  and a momentum of 0.9. The learning rate is decayed by a factor 10 every 3 epochs. As demonstrated in the ablation study in Table 3, finetuning on the true labels performs worse than finetuning by distillation. A possible explanation is that the supervision signal coming from the teacher network is richer than the one-hot vector used as a traditional learning signal in supervised learning (Hinton et al., 2014).
+
+# 4 EXPERIMENTS
+
+# 4.1 EXPERIMENTAL SETUP
+
+We quantize vanilla ResNet-18 and ResNet-50 architectures pretrained on the ImageNet dataset (Deng et al., 2009). Unless explicit mention of the contrary, the pretrained models are taken from the PyTorch model zoo<sup>3</sup>. We run our method on a 16 GB Volta V100 GPU. Quantizing a ResNet-50 with our method (including all finetuning steps) takes about one day on 1 GPU. We detail our experimental setup below. Our code and the compressed models are open-sourced.
+
+Compression regimes. We explore a large block sizes (resp. small block sizes) compression regime by setting the subvector size of regular  $3 \times 3$  convolutions to  $d = 9$  (resp.  $d = 18$ ) and the subvector size of pointwise convolutions to  $d = 4$  (resp.  $d = 8$ ). For ResNet-18, the block size of pointwise convolutions is always equal to 4. The number of codewords or centroids is set to  $k \in \{256, 512, 1024, 2048\}$  for each compression regime. Note that we clamp the number of centroids to  $\min(k, C_{\mathrm{out}} \times m/4)$  for stability. For instance, the first layer of the first stage of the ResNet-50 has size  $64 \times 64 \times 1 \times 1$ , thus we always use  $k = 128$  centroids with a block size  $d = 8$ . For a given number of centroids  $k$ , small blocks lead to a lower compression ratio than large blocks.
+
+Sampling the input activations. Before quantizing each layer, we randomly sample a batch of 1024 training images to obtain the input activations of the current layer and reshape it as described in Section 3.2. Then, before each iteration (E+M step) of our method, we randomly sample 10,000 rows from those reshaped input activations.
+
+Hyperparameters. We quantize each layer while performing 100 steps of our method (sufficient for convergence in practice). We finetune the centroids of each layer on the standard ImageNet training set during 2,500 iterations with a batch size of 128 (resp 64) for the ResNet-18 (resp.ResNet-50) with a learning rate of 0.01, a weight decay of  $10^{-4}$  and a momentum of 0.9. For accuracy and memory reasons, the classifier is always quantized with a block size  $d = 4$  and  $k = 2048$  (resp.  $k = 1024$ ) centroids for the ResNet-18 (resp., ResNet-50). Moreover, the first convolutional layer of size  $7 \times 7$  is not quantized, as it represents less than  $0.1\%$  (resp.,  $0.05\%$  ) of the weights of a ResNet-18 (resp.ResNet-50).
+
+Metrics. We focus on the tradeoff between accuracy and memory. The accuracy is the top-1 error on the standard validation set of ImageNet. The memory footprint is calculated as the indexing cost (number of bits per weight) plus the overhead of storing the centroids in float16. As an example, quantizing a layer of size  $128 \times 128 \times 3 \times 3$  with  $k = 256$  centroids (1 byte per subvector) and a block size of  $d = 9$  leads to an indexing cost of  $16\mathrm{kB}$  for  $m = 16,384$  blocks plus the cost of storing the centroids of  $4.5\mathrm{kB}$ .
+
+# 4.2 IMAGE CLASSIFICATION RESULTS
+
+We report below the results of our method applied to various ResNet models. First, we compare our method with the state of the art on the standard ResNet-18 and ResNet-50 architecture. Next, we show the potential of our approach on a competitive ResNet-50. Finally, an ablation study validates the pertinence of our method.
+
+Vanilla ResNet-18 and ResNet-50. We evaluate our method on the ImageNet benchmark for ResNet-18 and ResNet-50 architectures and compare our results to the following methods: Trained Ternary Quantization (TTQ) (Zhu et al., 2016), LR-Net (Shayer et al., 2017), ABC-Net (Lin et al., 2017), Binary Weight Network (XNOR-Net or BWN) (Rastegari et al., 2016), Deep Compression (DC) (Han et al., 2016) and Hardware-Aware Automated Quantization (HAQ) (Wang et al., 2018a). We report the accuracies and compression factors in the original papers and/or in the two surveys (Guo, 2018; Cheng et al., 2017) for a given architecture when the result is available. We do not compare our method to DoReFa-Net (Zhou et al., 2016) and WRPN (Mishra et al., 2017) as those approaches also use low-precision activations and hence get lower accuracies, e.g.,  $51.2\%$  top-1 accuracy for a XNOR-Net with ResNet-18. The results are presented in Figure 4.2. For better readability, some results for our method are also displayed in Table 1. We report the average accuracy and standard deviation over 3 runs. Our method significantly outperforms state of the art papers for
+
+![](images/cbf756c3f7df7b8154c6ff54daf2111274e54972cc40107e8ecd27d7713f21ba.jpg)  
+Figure 3: Compression results for ResNet-18 and ResNet-50 architectures. We explore two compression regimes as defined in Section 4.1: small block sizes (block sizes of  $d = 4$  and 9) and large block sizes (block sizes  $d = 8$  and 18). The results of our method for  $k = 256$  centroids are of practical interest as they correspond to a byte-compatible compression scheme.
+
+![](images/b08501c4f9b543ab725be5bfae4571aa6f1e1b38db25a2908adc02bd099edb0a.jpg)
+
+Table 1: Results for vanilla ResNet-18 and ResNet-50 architectures for  $k = {256}$  centroids.  
+
+<table><tr><td>Model (original top-1)</td><td>Compression</td><td>Size ratio</td><td>Model size</td><td>Top-1 (%)</td></tr><tr><td rowspan="2">ResNet-18 (69.76%)</td><td>Small blocks</td><td>29x</td><td>1.54 MB</td><td>65.81 ±0.04</td></tr><tr><td>Large blocks</td><td>43x</td><td>1.03 MB</td><td>61.10 ±0.03</td></tr><tr><td rowspan="2">ResNet-50 (76.15%)</td><td>Small blocks</td><td>19x</td><td>5.09 MB</td><td>73.79 ±0.05</td></tr><tr><td>Large blocks</td><td>31x</td><td>3.19 MB</td><td>68.21 ±0.04</td></tr></table>
+
+various operating points. For instance, for a ResNet-18, our method with large blocks and  $k = 512$  centroids reaches a larger accuracy than ABC-Net ( $M = 2$ ) with a compression ratio that is 2x larger. Similarly, on the ResNet-50, our compressed model with  $k = 256$  centroids in the large blocks setup yields a comparable accuracy to DC (2 bits) with a compression ratio that is 2x larger.
+
+The work by Tung & Mori (Tung & Mori, 2018) is likely the only one that remains competitive with ours with a 6.8 MB network after compression, with a technique that prunes the network and therefore implicitly changes the architecture. The authors report the delta accuracy for which we have no direct comparable top-1 accuracy, but their method is arguably complementary to ours.
+
+Semi-supervised ResNet-50. Recent works (Mahajan et al., 2018; Yalniz et al., 2019) have demonstrated the possibility to leverage a large collection of unlabelled images to improve the performance of a given architecture. In particular, Yalniz et al. (Yalniz et al., 2019) use the publicly available YFCC-100M dataset (Thomee et al., 2015) to train a ResNet-50 that reaches  $79.1\%$  top-1 accuracy on the standard validation set of ImageNet. In the following, we use this particular model and refer to it as semi-supervised ResNet-50. In the low compression regime (block sizes of 4 and 9), with  $k = 256$  centroids (practical for implementation), our compressed semi-supervised ResNet-50 reaches  $76.12\%$  top-1 accuracy. In other words, the model compressed to  $5\mathrm{MB}$  attains the performance of a vanilla, non-compressed ResNet50 (vs.97.5MB for the non-compressed ResNet-50).
+
+Comparison for a given size budget. To ensure a fair comparison, we compare our method for a given model size budget against the reference methods in Table 2. It should be noted that our method can further benefit from advances in semi-supervised learning to boosts the performance of the non-compressed and hence of the compressed network.
+
+Ablation study. We perform an ablation study on the vanilla ResNet-18 to study the respective effects of quantizing using the activations and finetuning by distillation (here, finetuning refers both to the per-layer finetuning and to the global finetuning after the quantization described in Section 3). We refer to our method as Act + Distill. First, we still finetune by distillation but change the quantization: instead of quantizing using our method (see Equation (2)), we quantizing using the standard PQ algorithm and do not take the activations into account, see Equation (1). We refer to this method as No act + Distill. Second, we quantize using our method but perform a standard finetuning using
+
+Table 2: Best test top-1 accuracy on ImageNet for a given size budget (no architecture constraint).  
+
+<table><tr><td>Size budget</td><td>Best previous published method</td><td>Ours</td></tr><tr><td>~1 MB</td><td>70.90% (HAQ (Wang et al., 2018a), MobileNet v2)</td><td>64.01% (vanilla ResNet-18)</td></tr><tr><td>~5 MB</td><td>71.74% (HAQ (Wang et al., 2018a), MobileNet v1)</td><td>76.12% (semi-sup.ResNet-50)</td></tr><tr><td>~10 MB</td><td>75.30% (HAQ (Wang et al., 2018a), ResNet-50)</td><td>77.85% (semi-sup.ResNet-50)</td></tr></table>
+
+Table 3: Ablation study on ResNet-18 (test top-1 accuracy on ImageNet).  
+
+<table><tr><td>Compression</td><td>Centroids k</td><td>No act + Distill</td><td>Act + Labels</td><td>Act + Distill (ours)</td></tr><tr><td rowspan="4">Small blocks</td><td>256</td><td>64.76</td><td>65.55</td><td>65.81</td></tr><tr><td>512</td><td>66.31</td><td>66.82</td><td>67.15</td></tr><tr><td>1024</td><td>67.28</td><td>67.53</td><td>67.87</td></tr><tr><td>2048</td><td>67.88</td><td>67.99</td><td>68.26</td></tr><tr><td rowspan="4">Large blocks</td><td>256</td><td>60.46</td><td>61.01</td><td>61.18</td></tr><tr><td>512</td><td>63.21</td><td>63.67</td><td>63.99</td></tr><tr><td>1024</td><td>64.74</td><td>65.48</td><td>65.72</td></tr><tr><td>2048</td><td>65.94</td><td>66.21</td><td>66.50</td></tr></table>
+
+the image labels (Act + Labels). The results are displayed in Table 3. Our approach consistently yields significantly better results. As a side note, quantizing all the layers of a ResNet-18 with the standard PQ algorithm and without any finetuning leads to top-1 accuracies below  $25\%$  for all operating points, which illustrates the drift in accuracy occurring when compressing deep networks with standard methods (as opposed to our method).
+
+# 4.3 IMAGE DETECTION RESULTS
+
+To demonstrate the generality of our method, we compress the Mask R-CNN architecture used for image detection in many real-life applications (He et al., 2017). We compress the backbone (ResNet-50 FPN) in the small blocks compression regime and refer the reader to the open-sourced compressed model for the block sizes used in the various heads of the network. We use  $k = 256$  centroids for every layer. We perform the fine-tuning (layer-wise and global) using distributed training on 8 V100 GPUs. Results are displayed in Table 4. We argue that this provides an interesting point of comparison for future work aiming at compressing such architectures for various applications.
+
+# 5 CONCLUSION
+
+We presented a quantization method based on Product Quantization that gives state of the art results on ResNet architectures and that generalizes to other architectures such as Mask R-CNN. Our compression scheme does not require labeled data and the resulting models are byte-aligned, allowing for efficient inference on CPU. Further research directions include testing our method on a wider variety of architectures. In particular, our method can be readily adapted to simultaneously compress and transfer ResNets trained on ImageNet to other domains. Finally, we plan to take the non-linearity into account to improve our reconstruction error.
+
+Table 4: Compression results for Mask R-CNN (backbone ResNet-50 FPN) for  $k = 256$  centroids (compression factor  $26 \times$ ).  
+
+<table><tr><td>Model</td><td>Size</td><td>Box AP</td><td>Mask AP</td></tr><tr><td>Non-compressed</td><td>170 MB</td><td>37.9</td><td>34.6</td></tr><tr><td>Compressed</td><td>6.51 MB</td><td>33.9</td><td>30.8</td></tr></table>
+
+# REFERENCES
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+
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+# AN EXPONENTIAL LEARNING RATE SCHEDULE FOR DEEP LEARNING
+
+Zhiyuan Li
+
+Princeton University
+
+zhiyuanli@cs.princeton.edu
+
+Sanjeev Arora
+
+Princeton University and Institute for Advanced Study arora@cs.princeton.edu
+
+# ABSTRACT
+
+Intriguing empirical evidence exists that deep learning can work well with exotic schedules for varying the learning rate. This paper suggests that the phenomenon may be due to Batch Normalization or BN(Ioffe & Szegedy, 2015), which is ubiquitous and provides benefits in optimization and generalization across all standard architectures. The following new results are shown about BN with weight decay and momentum (in other words, the typical use case which was not considered in earlier theoretical analyses of stand-alone BN (Ioffe & Szegedy, 2015; Santurkar et al., 2018; Arora et al., 2018)
+
+- Training can be done using SGD with momentum and an exponentially increasing learning rate schedule, i.e., learning rate increases by some  $(1 + \alpha)$  factor in every epoch for some  $\alpha > 0$ . (Precise statement in the paper.) To the best of our knowledge this is the first time such a rate schedule has been successfully used, let alone for highly successful architectures. As expected, such training rapidly blows up network weights, but the network stays well-behaved due to normalization.  
+- Mathematical explanation of the success of the above rate schedule: a rigorous proof that it is equivalent to the standard setting of  $\mathrm{BN} + \mathrm{SGD} + \mathrm{Standard}$  Rate Tuning  $^+$  Weight Decay  $^+$  Momentum. This equivalence holds for other normalization layers as well, Group Normalization(Wu & He, 2018), Layer Normalization(Ba et al., 2016), Instance Norm(Ulyanov et al., 2016), etc.  
+- A worked-out toy example illustrating the above linkage of hyperparameters. Using either weight decay or BN alone reaches global minimum, but convergence fails when both are used.
+
+# 1 INTRODUCTION
+
+Batch Normalization (BN) offers significant benefits in optimization and generalization across architectures, and has become ubiquitous. Usually best performance is attained by adding weight decay and momentum in addition to BN.
+
+Usually weight decay is thought to improve generalization by controlling the norm of the parameters. However, it is fallacious to try to separately think of optimization and generalization because we are dealing with a nonconvex objective with multiple optima. Even slight changes to the training surely lead to a different trajectory in the loss landscape, potentially ending up at a different solution! One needs trajectory analysis to have a hope of reasoning about the effects of such changes.
+
+In the presence of BN and other normalization schemes, including GroupNorm, LayerNorm, and InstanceNorm, the optimization objective is scale invariant to the parameters, which means rescaling parameters would not change the prediction, except the parameters that compute the output which do not have BN. However, Hoffer et al. (2018b) shows that fixing the output layer randomly doesn't harm the performance of the network. So the trainable parameters satisfy scale invariance.(See more in Appendix E) The current paper introduces new modes of analysis for such settings. This rigorous analysis yields the surprising conclusion that the original learning rate (LR) schedule and weight decay(WD) can be folded into a new exponential schedule for learning rate: in each iteration multiplying it by  $(1 + \alpha)$  for some  $\alpha > 0$  that depends upon the momentum and weight decay rate.
+
+Theorem 1.1 (Main, Informal). SGD on a scale-invariant objective with initial learning rate  $\eta$ , weight decay factor  $\lambda$ , and momentum factor  $\gamma$  is equivalent to SGD with momentum factor  $\gamma$  where at iteration  $t$ , the learning rate  $\tilde{\eta}_t$  in the new exponential learning rate schedule is defined as  $\tilde{\eta}_t = \alpha^{-2t - 1}\eta$  without weight decay  $(\lambda = 0)$  where  $\alpha$  is a non-zero root of equation
+
+$$
+x ^ {2} - (1 + \gamma - \lambda \eta) x + \gamma = 0, \tag {1}
+$$
+
+Specifically, when momentum  $\gamma = 0$ , the above schedule can be simplified as  $\tilde{\eta}_t = (1 - \lambda \eta)^{-2t - 1}\eta$ .
+
+The above theorem requires that the product of learning rate and weight decay factor,  $\lambda \eta$ , is small than  $(1 - \sqrt{\gamma})^2$ , which is almost always satisfied in practice. The rigorous and most general version of above theorem is Theorem 2.12, which deals with multi-phase LR schedule, momentum and weight decay.
+
+There are other recently discovered exotic LR schedules, e.g. Triangular LR schedule(Smith, 2017) and Cosine LR schedule(Loshchilov & Hutter, 2016), and our exponential LR schedule is an extreme example of LR schedules that become possible in presence of BN. Such an exponential increase in learning rate seems absurd at first sight and to the best of our knowledge, no deep learning success has been reported using such an idea before. It does highlight the above-mentioned viewpoint that in deep learning, optimization and regularization are not easily separated. Of course, the exponent trumps the effect of initial lr very fast (See Figure 4), which explains why training with BN and WD is not sensitive to the scale of initialization, since with BN, tuning the scale of initialization is equivalent to tuning the initial LR  $\eta$  while fixing the product of LR and WD,  $\eta \lambda$  (See Lemma 2.7).
+
+Note that it is customary in BN to switch to a lower LR upon reaching a plateau in the validation loss. According to the analysis in the above theorem, this corresponds to an exponential growth with a smaller exponent, except for a transient effect when a correction term is needed for the two processes to be equivalent (see discussion around Theorem 2.12).
+
+Thus the final training algorithm is roughly as follows: Start from a convenient LR like 0.1, and grow it at an exponential rate with a suitable exponent. When validation loss plateaus, switch to an exponential growth of LR with a lower exponent. Repeat the procedure until the training loss saturates.
+
+In Section 3, we demonstrate on a toy example how weight decay and normalization are inseparably involved in the optimization process. With either weight decay or normalization alone, SGD will achieve zero training error. But with both turned on, SGD fails to converge to global minimum.
+
+In Section 4, we experimentally verify our theoretical findings on CNNs and ResNets. We also construct better exponential LR schedules by incorporating the Cosine LR schedule on CIFAR10, which opens the possibility of even more general theory of rate schedule tuning towards better performance.
+
+# 1.1 RELATED WORK
+
+There have been other theoretical analyses of training models with scale-invariance. (Cho & Lee, 2017) proposed to run Riemannian gradient descent on Grassmann manifold  $\mathcal{G}(1,n)$  since the weight matrix is scaling invariant to the loss function. observed that the effective stepsize is proportional to  $\frac{\eta_w}{\|\pmb{w}_t\|^2}$ . (Arora et al., 2019) show the gradient is always perpendicular to the current parameter vector which has the effect that norm of each scale invariant parameter group increases monotonically, which has an auto-tuning effect. (Wu et al., 2018) proposes a new adaptive learning rate schedule motivated by scale-invariance property of Weight Normalization.
+
+Previous work for understanding Batch Normalization. (Santurkar et al., 2018) suggested that the success of BNhas does not derive from reduction in Internal Covariate Shift, but by making landscape smoother. (Kohler et al., 2018) essentially shows linear model with BN could achieve exponential convergence rate assuming gaussian inputs, but their analysis is for a variant of GD with an inner optimization loop rather than GD itself. (Bjorck et al., 2018) observe that the higher learning rates enabled by BN empirically improves generalization. (Arora et al., 2019) prove that with certain mild assumption, (S)GD with BN finds approximate first order stationary point with any fixed learning rate. None of the above analyses incorporated weight decay, but (Zhang et al., 2019; Hoffer et al., 2018a; van Laarhoven, 2017) argued qualitatively that weight decay makes parameters
+
+have smaller norms, and thus the effective learning rate,  $\frac{\eta_w}{\|\pmb{w}_t\|^2}$  is larger. They described experiments showing this effect but didn't have a closed form theoretical analysis like ours. None of the above analyses deals with momentum rigorously.
+
+# 1.2 PRELIMINARIES AND NOTATIONS
+
+For batch  $\mathcal{B} = \{x_i\}_{i=1}^B$ , network parameter  $\pmb{\theta}$ , we denote the network by  $f_{\pmb{\theta}}$  and the loss function at iteration  $t$  by  $L_t(f_{\pmb{\theta}}) = L(f_{\pmb{\theta}}, \mathcal{B}_t)$ . When there's no ambiguity, we also use  $L_t(\pmb{\theta})$  for convenience.
+
+We say a loss function  $L(\theta)$  is scale invariant to its parameter  $\theta$  is for any  $c \in \mathbb{R}^+$ ,  $L(\theta) = L(c\theta)$ . In practice, the source of scale invariance is usually different types of normalization layers, including Batch Normalization (Ioffe & Szegedy, 2015), Group Normalization (Wu & He, 2018), Layer Normalization (Ba et al., 2016), Instance Norm (Ulyanov et al., 2016), etc.
+
+Implementations of SGD with Momentum/Nesterov comes with subtle variations in literature. We adopt the variant from Sutskever et al. (2013), also the default in PyTorch (Paszke et al., 2017). L2 regularization (a.k.a. Weight Decay) is another common trick used in deep learning. Combining them together, we get the one of the mostly used optimization algorithms below.
+
+Definition 1.2. [SGD with Momentum and Weight Decay] At iteration  $t$ , with randomly sampled batch  $\mathcal{B}_t$ , update the parameters  $\theta_t$  and momentum  $v_t$  as following:
+
+$$
+\boldsymbol {\theta} _ {t} = \boldsymbol {\theta} _ {t - 1} - \eta_ {t - 1} \boldsymbol {v} _ {t} \tag {2}
+$$
+
+$$
+\boldsymbol {v} _ {t} = \gamma \boldsymbol {v} _ {t - 1} + \nabla_ {\boldsymbol {\theta}} \left(L _ {t} \left(\boldsymbol {\theta} _ {t - 1}\right) + \frac {\lambda_ {t - 1}}{2} \| \boldsymbol {\theta} _ {t - 1} \| ^ {2}\right), \tag {3}
+$$
+
+where  $\eta_{t}$  is the learning rate at epoch  $t$ ,  $\gamma$  is the momentum coefficient, and  $\lambda$  is the factor of weight decay. Usually,  $\mathbf{v}_0$  is initialized to be 0.
+
+For ease of analysis, we will use the following equivalent of Definition 1.2.
+
+$$
+\frac {\boldsymbol {\theta} _ {t} - \boldsymbol {\theta} _ {t - 1}}{\eta_ {t - 1}} = \gamma \frac {\boldsymbol {\theta} _ {t - 1} - \boldsymbol {\theta} _ {t - 2}}{\eta_ {t - 2}} - \nabla_ {\boldsymbol {\theta}} \left(\left(L \left(\boldsymbol {\theta} _ {t - 1}\right) + \frac {\lambda_ {t - 1}}{2} \| \boldsymbol {\theta} _ {t - 1} \| _ {2} ^ {2}\right), \right. \tag {4}
+$$
+
+where  $\eta_{-1}$  and  $\theta_{-1}$  must be chosen in a way such that  $\pmb{v}_0 = \frac{\pmb{\theta}_0 - \pmb{\theta}_{-1}}{\eta_{-1}}$  is satisfied, e.g. when  $\pmb{v}_0 = \mathbf{0}$ ,  $\pmb{\theta}_{-1} = \pmb{\theta}_0$  and  $\eta_{-1}$  could be arbitrary.
+
+A key source of intuition is the following simple lemma about scale-invariant networks Arora et al. (2019). The first property ensures GD (with momentum) always increases the norm of the weight.(See Lemma C.1 in Appendix C) and the second property says that the gradients are smaller for parameteres with larger norm, thus stabilizing the trajectory from diverging to infinity.
+
+Lemma 1.3 (Scale Invariance). If for any  $c \in \mathbb{R}^+$ ,  $L(\pmb{\theta}) = L(c\pmb{\theta})$ , then
+
+(1).  $\langle \nabla_{\pmb{\theta}}L,\pmb {\theta}\rangle = 0;$  
+(2).  $\nabla_{\pmb{\theta}}L\big|_{\pmb{\theta} = \pmb{\theta}_0} = c\nabla_{\pmb{\theta}}L\big|_{\pmb{\theta} = c\pmb{\theta}_0},$  for any  $c > 0$
+
+# 2 DERIVING EXPONENTIAL LEARNING RATE SCHEDULE
+
+As a warm-up in Section 2.1 we show that if momentum is turned off then Fixed  $LR +$  Fixed WD can be translated to an equivalent Exponential LR. In Section 2.2 we give a more general analysis on the equivalence between Fixed  $LR +$  Fixed WD + Fixed Momentum Factor and Exponential  $LR +$  Fixed Momentum Factor. While interesting, this still does completely apply to real-life deep learning where reaching full accuracy usually requires multiple phases in training where LR is fixed within a phase and reduced by some factor from one phase to the next. Section 2.3 shows how to interpret such a multi-phase LR schedule + WD + Momentum as a certain multi-phase exponential LR schedule with Momentum.
+
+# 2.1 REPLACING WD BY EXPONENTIAL LR IN MOMENTUM-FREE SGD
+
+We use notation of Section 1.2 and assume LR is fixed over iterations, i.e.  $\eta_t = \eta_0$ , and  $\gamma$  (momentum factor) is set as 0. We also use  $\lambda$  to denote WD factor and  $\theta_0$  to denote the initial parameters.
+
+The intuition should be clear from Lemma 1.3, which says that shrinking parameter weights by factor  $\rho$  (where  $\rho < 1$ ) amounts to making the gradient  $\rho^{-1}$  times larger without changing its direction. Thus in order to restore the ratio between original parameter and its update (LR×Gradient), the easiest way would be scaling LR by  $\rho^2$ . This suggests that scaling the parameter  $\theta$  by  $\rho$  at each step is equivalent to scaling the LR  $\eta$  by  $\rho^{-2}$ .
+
+To prove this formally we use the following formalism. We'll refer to the vector  $(\theta, \eta)$  the state of a training algorithm and study how this evolves under various combinations of parameter changes. We will think of each step in training as a mapping from one state to another. Since mappings can be composed, any finite number of steps also correspond to a mapping. The following are some basic mappings used in the proof.
+
+1. Run GD with WD for a step:  $\mathbf{GD}_t^\rho (\pmb {\theta},\eta) = (\rho \pmb {\theta} - \eta \nabla L_t(\pmb {\theta}),\eta);$  
+2. Scale the parameter  $\pmb{\theta}$  ..  $\Pi_1^c (\pmb {\theta},\eta) = (c\pmb {\theta},\eta);$  
+3. Scale the LR  $\eta$  ..  $\Pi_2^c (\pmb {\theta},\eta) = (\pmb {\theta},c\eta)$
+
+For example, when  $\rho = 1$ ,  $\mathrm{GD}_t^1$  is vanilla GD update without WD, also abbreviated as  $\mathrm{GD}_t$ . When  $\rho = 1 - \lambda \eta_0$ ,  $\mathrm{GD}_t^{1 - \lambda \eta_0}$  is GD update with WD  $\lambda$  and LR  $\eta_0$ . Here  $L_{t}$  is the loss function at iteration  $t$ , which is decided by the batch of the training samples  $\mathcal{B}_t$  in  $t$ th iteration. Below is the main result of this subsection, showing our claim that  $\mathrm{GD} + \mathrm{WD} \Leftrightarrow \mathrm{GD} + \mathrm{ExpLR}$  (when Momentum is zero). It will be proved after a series of lemmas.
+
+Theorem 2.1 (WD  $\Leftrightarrow$  Exp LR). For every  $\rho < 1$  and positive integer  $t$  following holds:
+
+$$
+\mathbf {G D} _ {t - 1} ^ {\rho} \circ \dots \circ \mathbf {G D} _ {0} ^ {\rho} = \left[ \Pi_ {1} ^ {\rho^ {t}} \circ \Pi_ {2} ^ {\rho^ {2 t}} \right] \circ \Pi_ {2} ^ {\rho^ {- 1}} \circ \mathbf {G D} _ {t - 1} \circ \Pi_ {2} ^ {\rho^ {- 2}} \circ \dots \circ \mathbf {G D} _ {1} \circ \Pi_ {2} ^ {\rho^ {- 2}} \circ \mathbf {G D} _ {0} \circ \Pi_ {2} ^ {\rho^ {- 1}}.
+$$
+
+With WD being  $\lambda$ ,  $\rho$  is set as  $1 - \lambda \eta_0$  and thus the scaling factor of LR per iteration is  $\rho^{-2} = (1 - \lambda \eta_0)^{-2}$ , except for the first iteration it's  $\rho^{-1} = (1 - \lambda \eta_0)^{-1}$ .
+
+We first show how to write GD update with WD as a composition of above defined basic maps.
+
+Lemma 2.2.  $GD_{t}^{\rho} = \Pi_{2}^{\rho}\circ \Pi_{1}^{\rho}\circ GD_{t}\circ \Pi_{2}^{\rho^{-1}}$ .
+
+Below we will define the proper notion of equivalence such that (1).  $\Pi_1^\rho \sim \Pi_2^{\rho^{-2}}$ , which implies  $\mathrm{GD}_t^\rho \sim \Pi_2^{\rho^{-1}}\circ \mathrm{GD}_t\circ \Pi_2^{\rho^{-1}}$ ; (2) the equivalence is preserved under future GD updates.
+
+We first extend the equivalence between weights (same direction) to that between states, with additional requirement that the ratio between the size of GD update and that of parameter are the same among all equivalent states, which yields the notion of Equivalent Scaling.
+
+Definition 2.3 (Equivalent States).  $(\pmb {\theta},\eta)$  is equivalent to  $(\pmb {\theta}^{\prime},\eta^{\prime})$  iff  $\exists c > 0$ $(\widetilde{\theta},\widetilde{\eta}) = [\Pi_1^c\circ$ $\Pi_2^{c^2}](\pmb {\theta},\eta) = (c\pmb {\theta},c^2\eta)$ , which is also denoted by  $(\widetilde{\theta},\widetilde{\eta})\stackrel {c}{\sim}(\pmb {\theta},\eta)$ .  $\Pi_1^c\circ \Pi_2^{c^2}$  is called Equivalent Scaling for all  $c > 0$
+
+The following lemma shows that equivalent scaling commutes with GD update with WD, implying that equivalence is preserved under GD update (Lemma 2.4). This anchors the notion of equivalence — we could insert equivalent scaling anywhere in a sequence of basic maps(GD update, LR/parameter scaling), without changing the final network.
+
+Lemma 2.4. For any constant  $c, \rho > 0$  and  $t \geq 0$ ,  $GD_{t}^{\rho} \circ [\Pi_{1}^{c} \circ \Pi_{2}^{c^{2}}] = [\Pi_{1}^{c} \circ \Pi_{2}^{c^{2}}] \circ GD_{t}^{\rho}$ .
+
+In other words,  $(\pmb {\theta},\eta)\stackrel {c}{\sim}(\pmb{\theta}^{\prime},\eta^{\prime})\Longrightarrow GD_{t}^{p}(\pmb {\theta},\eta)\stackrel {c}{\sim}GD_{t}^{p}(\pmb{\theta}^{\prime},\eta^{\prime}).$
+
+Now we formally define equivalence relationship between maps using equivalent scalings.
+
+Definition 2.5 (Equivalent Maps). Two maps  $F, G$  are equivalent iff  $\exists c > 0$ ,  $F = \Pi_1^c \circ \Pi_2^{c^2} \circ G$ , which is also denoted by  $F \stackrel{c}{\sim} G$ .
+
+Proof of Theorem 2.1. By Lemma 2.2.,  $\mathrm{GD}_t^\rho \stackrel{\rho}{\sim} \Pi_2^{\rho^{-1}} \circ \mathrm{GD}_t \circ \Pi_2^{\rho^{-1}}$ . By Lemma 2.4, GD update preserves map equivalence, i.e.  $F \stackrel{c}{\sim} G \Rightarrow \mathrm{GD}_t^\rho \circ F \stackrel{c}{\sim} \mathrm{GD}_t^\rho \circ G, \forall c, \rho > 0$ . Thus,
+
+$$
+\mathbf {G D} _ {t - 1} ^ {\rho} \circ \dots \circ \mathbf {G D} _ {0} ^ {\rho} \stackrel {{\rho^ {t}}} {{\sim}} \Pi_ {2} ^ {\rho^ {- 1}} \circ \mathbf {G D} _ {t - 1} \circ \Pi_ {2} ^ {\rho^ {- 2}} \circ \dots \circ \mathbf {G D} _ {1} \circ \Pi_ {2} ^ {\rho^ {- 2}} \circ \mathbf {G D} _ {0} \circ \Pi_ {2} ^ {\rho^ {- 1}}.
+$$
+
+![](images/c5ba33072b0848101660b4a7a8b22d2e96770734e890b3702bde88ac79f31295.jpg)
+
+![](images/3c80e7a182c62d988306159ec82c93ba8a6829bf00da16e1f26caee4e10963e0.jpg)  
+Figure 1: Taking PreResNet32 with standard hyperparameters and replacing WD during first phase (Fixed LR) by exponential LR according to Theorem 2.9 to the schedule  $\widetilde{\eta}_t = 0.1 \times 1.481^t$ , momentum 0.9. Plot on right shows weight norm  $\boldsymbol{w}$  of the first convolutional layer in the second residual block grows exponentially, satisfying  $\frac{\|\boldsymbol{w}_t\|^2}{\widetilde{\eta}_t} = \text{constant}$ . Reason being that according to the proof it is essentially the norm square of the weights when trained with Fixed LR + WD + Momentum, and published hyperparameters kept this norm roughly constant during training.
+
+![](images/6906a52d3652308108348e92e778b1ce6ffde4479ba3b15423ae8b91b148288c.jpg)
+
+# 2.2 REPLACING WD BY EXPONENTIAL LR: CASE OF CONSTANT LR WITH MOMENTUM
+
+In this subsection the setting is the same to that in Subsection 2.1 except that the momentum factor is  $\gamma$  instead of 0. Suppose the initial momentum is  $v_{0}$ , we set  $\theta_{-1} = \theta_0 - v_0\eta$ . Presence of momentum requires representing the state of the algorithm with four coordinates,  $(\theta ,\eta ,\theta^{\prime},\eta^{\prime})$ , which stand respectively for the current parameters/LR and the buffered parameters/LR (from last iteration) respectively. Similarly, we define the following basic maps and equivalence relationships.
+
+1. Run GD with WD for a step:  $\mathrm{GD}_t^\rho (\pmb {\theta},\eta ,\pmb {\theta}',\eta ') = \Bigl (\rho \pmb {\theta} + \eta \left(\gamma \frac{\pmb{\theta} - \pmb{\theta}'}{\eta'} -\nabla L_t(\pmb {\theta})\right),\eta ,\pmb {\theta},\eta \Bigr);$  
+2. Scale Current parameter  $\pmb{\theta}$ $\Pi_1^c (\pmb {\theta},\pmb {\eta},\pmb {\theta}',\pmb {\eta}') = (c\pmb {\theta},\pmb {\eta},\pmb {\theta}',\pmb {\eta}')$  
+3. Scale Current LR  $\eta$ :  $\Pi_2^c (\pmb {\theta},\pmb {\eta},\pmb {\theta}',\eta ') = (\pmb {\theta},c\eta ,\pmb {\theta}',\eta ');$  
+4. Scale Buffered parameter  $\pmb{\theta}^{\prime}$  ..  $\Pi_3^c (\pmb {\theta},\pmb {\eta},\pmb {\theta}',\pmb {\eta}') = (\pmb {\theta},\pmb {\eta},c\pmb {\theta}',\pmb {\eta}')$  
+5. Scale Buffered parameter  $\eta^{\prime}$  ..  $\Pi_4^c (\pmb {\theta},\eta ,\pmb {\theta}^\prime ,\eta^\prime) = (\pmb {\theta},\eta ,\pmb {\theta}^\prime ,cn\eta^\prime).$
+
+Definition 2.6 (Equivalent States).  $(\pmb {\theta},\pmb {\eta},\pmb{\theta}^{\prime},\pmb{\eta}^{\prime})$  is equivalent to  $(\widetilde{\theta},\widetilde{\eta},\widetilde{\theta}^{\prime},\widetilde{\eta}^{\prime})$  iff  $\exists c > 0$ $(\pmb {\theta},\pmb {\eta},\pmb{\theta}^{\prime},\pmb{\eta}^{\prime}) = \left[\Pi_{1}^{c}\circ \Pi_{2}^{c^{2}}\circ \Pi_{3}^{c}\circ \Pi_{4}^{c^{2}}\right]$ $(\widetilde{\pmb{\theta}},\widetilde{\pmb{\eta}},\widetilde{\pmb{\theta}}^{\prime},\widetilde{\pmb{\eta}}^{\prime}) = (c\widetilde{\pmb{\theta}},c^{2}\widetilde{\pmb{\eta}},c\widetilde{\pmb{\theta}}^{\prime},c^{2}\widetilde{\pmb{\eta}}^{\prime})$  , which is also denoted by  $(\pmb {\theta},\pmb {\eta},\pmb {\theta}^{\prime},\pmb {\eta}^{\prime})\stackrel {c}{\sim}(\widetilde{\pmb{\theta}},\widetilde{\pmb{\eta}},\widetilde{\pmb{\theta}}^{\prime},\widetilde{\pmb{\eta}}^{\prime})$  . We call  $\Pi_1^c\circ \Pi_2^{c^2}\circ \Pi_3^c\circ \Pi_4^{c^2}$  Equivalent Scalings for all  $c > 0$
+
+Again by expanding the definition, we show equivalent scalings commute with GD update.
+
+Lemma 2.7.  $\forall c,\rho >0$  and  $t\geq 0$ $GD_{t}^{\rho}\circ \left[\Pi_{1}^{c}\circ \Pi_{2}^{c^{2}}\circ \Pi_{3}^{c}\circ \Pi_{4}^{c^{2}}\right] = \left[\Pi_{1}^{c}\circ \Pi_{2}^{c^{2}}\circ \Pi_{3}^{c}\circ \Pi_{4}^{c^{2}}\right]\circ GD_{t}^{\rho}$
+
+Similarly, we can rewrite  $\mathrm{GD}_t^\rho$  as a composition of vanilla GD update and other scalings by expanding the definition, when the current and buffered LR are the same in the input of  $\mathrm{GD}_t^\rho$ .
+
+Lemma 2.8. For any input  $(\pmb {\theta},\eta ,\pmb{\theta}^{\prime},\eta)$  , if  $\alpha >0$  is a root of  $\alpha +\gamma \alpha^{-1} = \rho +\gamma$  , then
+
+$GD_{t}^{\rho}(\pmb {\theta},\eta ,\pmb{\theta}^{\prime},\eta) = \left[\Pi_{4}^{\alpha}\circ \Pi_{2}^{\alpha}\circ \Pi_{1}^{\alpha}\circ GD_{t}\circ \Pi_{2}^{\alpha^{-1}}\circ \Pi_{3}^{\alpha}\circ \Pi_{4}^{\alpha}\right](\pmb {\theta},\eta ,\pmb{\theta}^{\prime},\eta).$  In other words,
+
+$$
+G D _ {t} ^ {\rho} \left(\boldsymbol {\theta}, \eta , \overline {{\boldsymbol {\theta} ^ {\prime}}}, \eta\right) \stackrel {\alpha} {\sim} \left[ \Pi_ {3} ^ {\alpha^ {- 1}} \circ \Pi_ {4} ^ {\alpha^ {- 1}} \circ \Pi_ {2} ^ {\alpha^ {- 1}} \circ G D _ {t} \circ \Pi_ {2} ^ {\alpha^ {- 1}} \circ \Pi_ {3} ^ {\alpha} \circ \Pi_ {4} ^ {\alpha} \right] (\boldsymbol {\theta}, \eta , \boldsymbol {\theta} ^ {\prime}, \eta). \tag {5}
+$$
+
+Though looking complicated, the RHS of Equation 5 is actually the desired  $\Pi_2^{\alpha^{-1}}\circ \mathrm{GD}_t\circ \Pi_2^{\alpha^{-1}}$  conjugated with some scaling on momentum part  $\Pi_3^\alpha \circ \Pi_4^\alpha$ , and  $\Pi_3^{\alpha^{-1}}\circ \Pi_4^{\alpha^{-1}}$  in the current update cancels with the  $\Pi_3^\alpha \circ \Pi_4^\alpha$  in the next update. Now we are ready to show the equivalence between WD and Exp LR schedule when momentum is turned on for both.
+
+Theorem 2.9 (GD + WD ⇌ GD+ Exp LR; With Momentum). The following defined two sequences of parameters,  $\{\pmb{\theta}_t\}_{t=0}^{\infty}$  and  $\{\widetilde{\pmb{\theta}}_t\}_{t=0}^{\infty}$ , satisfy  $\widetilde{\pmb{\theta}}_t = \alpha^t\pmb{\theta}_t$ , thus they correspond to the same networks in function space, i.e.  $f_{\pmb{\theta}_t} = f_{\widetilde{\pmb{\theta}}_t}$ ,  $\forall t \in \mathbb{N}$ , given  $\widetilde{\pmb{\theta}}_0 = \pmb{\theta}_0$ ,  $\widetilde{\pmb{\theta}}_{-1} = \pmb{\theta}_{-1}\alpha$ , and  $\widetilde{\eta}_t = \eta_0\alpha^{-2t - 1}$ .
+
+1.  $\frac{\pmb{\theta}_{t} - \pmb{\theta}_{t - 1}}{\eta_0} = \frac{\gamma(\pmb{\theta}_{t - 1} - \pmb{\theta}_{t - 2})}{\eta_0} -\nabla_{\pmb{\theta}}(L(\pmb{\theta}_{t - 1}) + \frac{\lambda}{2}\| \pmb{\theta}_{t - 1}\| _2^2)$  
+2.  $\frac{\widetilde{\boldsymbol{\theta}}_t - \widetilde{\boldsymbol{\theta}}_{t - 1}}{\widetilde{\eta}_t} = \frac{\gamma(\widetilde{\boldsymbol{\theta}}_{t - 1} - \widetilde{\boldsymbol{\theta}}_{t - 2})}{\widetilde{\eta}_{t - 1}} -\nabla_\boldsymbol {\theta}L(\widetilde{\boldsymbol{\theta}}_{t - 1})$
+
+![](images/fd653e70f2b0ef510f544f13b4bbcb08d4cb7658c7699163e7f769de70938c0f.jpg)  
+Figure 2: PreResNet32 trained with standard Step Decay and its corresponding Tapered-Exponential LR schedule. As predicted by Theorem 2.12, they have similar trajectories and performances.
+
+![](images/ac8636ea6a506175bdf3525fb3377cc8544492f95b2eedc32a723f96299d4d0b.jpg)
+
+where  $\alpha$  is a positive root of equation  $x^{2} - (1 + \gamma -\lambda \eta_{0})x + \gamma = 0$ , which is always smaller than 1(See Appendix B.1). When  $\gamma = 0$ ,  $\alpha = 1 - \lambda \eta_0$  is the unique non-zero solution.
+
+Remark 2.10. Above we implicitly assume that  $\lambda \eta_0 \leq (1 - \sqrt{\gamma})^2$  such that the roots are real and this is always true in practice. For instance of standard hyper-parameters where  $\gamma = 0.9$ ,  $\eta_0 = 0.1$ ,  $\lambda = 0.0005$ ,  $\frac{\lambda \eta_0}{(1 - \sqrt{\gamma})^2} \approx 0.019 \ll 1$ .
+
+Proof. Note that  $(\widetilde{\pmb{\theta}}_0,\widetilde{\eta}_0,\widetilde{\pmb{\theta}}_{-1},\widetilde{\eta}_{-1}) = \left[\Pi_2^{\alpha^{-1}}\circ \Pi_3^{\alpha}\circ \Pi_4^{\alpha}\right](\pmb {\theta}_0,\eta_0,\pmb {\theta}_0,\eta_0)$ , it suffices to show that
+
+$$
+\left[ \Pi_ {3} ^ {\alpha^ {- 1}} \circ \Pi_ {4} ^ {\alpha^ {- 1}} \circ \Pi_ {2} ^ {\alpha^ {- 1}} \circ \mathbf {G D} _ {t - 1} \circ \Pi_ {2} ^ {\alpha^ {- 2}} \circ \dots \circ \mathbf {G D} _ {1} \circ \Pi_ {2} ^ {\alpha^ {- 2}} \circ \mathbf {G D} _ {0} \circ \Pi_ {2} ^ {\alpha^ {- 1}} \circ \Pi_ {3} ^ {\alpha} \circ \Pi_ {4} ^ {\alpha} \right] (\boldsymbol {\theta} _ {0}, \eta_ {0}, \boldsymbol {\theta} _ {0}, \eta_ {0})
+$$
+
+$$
+\stackrel {\alpha^ {t}} {\sim} \mathbf {G D} _ {t - 1} ^ {1 - \lambda \eta_ {0}} \circ \dots \circ \mathbf {G D} _ {0} ^ {1 - \lambda \eta_ {0}} (\pmb {\theta} _ {0}, \eta_ {0}, \pmb {\theta} _ {0}, \eta_ {0}), \quad \forall t \geq 0.
+$$
+
+which follows immediately from Lemma 2.7 and Lemma 2.8 by induction.
+
+![](images/77a2c773821254a6f247a25986d34149425698fa59e6b0b46ed8cbcb1a0e3070.jpg)
+
+# 2.3 REPLACING WD BY EXPONENTIAL LR: CASE OF MULTIPLE LR PHASES
+
+Usual practice in deep learning shows that reaching full training accuracy requires reducing the learning rate a few times.
+
+Definition 2.11. Step Decay is the (standard) learning rate schedule, where training has  $K$  phases  $I = 0, 1, \dots, K - 1$ , where phase  $I$  starts at iteration  $T_{I}$  ( $T_{0} = 0$ ), and all iterations in phase  $I$  use a fixed learning rate of  $\eta_{I}^{*}$ .
+
+The algorithm state in Section 2.2, consists of 4 components including buffered and current LR. When LR changes, the buffered and current LR are not equal, and thus Lemma 2.8 cannot be applied any more. In this section we show how to fix this issue by adding extra momentum correction. In detail, we show the below defined Exp LR schedule leads the same trajectory of networks in function space, with one-time momentum correction at the start of each phase. We empirically find on CIFAR10 that ignoring the correction term does not change performance much.
+
+Theorem 2.12 (Tapered-Exponential LR Schedule). There exists a way to correct the momentum only at the first iteration of each phase, such that the following Tapered-Exponential LR schedule (TEXP)  $\{\widetilde{\eta}_{lt}\}$  with momentum factor  $\gamma$  and no WD, leads the same sequence networks in function space as that of Step Decay LR schedule(Definition 2.11) with momentum factor  $\gamma$  and WD  $\lambda$ .
+
+$$
+\widetilde {\eta} _ {t} = \left\{ \begin{array}{l l} \widetilde {\eta} _ {t - 1} \times \left(\alpha_ {I - 1} ^ {*}\right) ^ {- 2} & \text {i f} T _ {I - 1} + 1 \leq t \leq T _ {I} - 1, I \geq 1; \\ \widetilde {\eta} _ {t - 1} \times \frac {\eta_ {I} ^ {*}}{\eta_ {I - 1} ^ {*}} \times \left(\alpha_ {I} ^ {*}\right) ^ {- 1} \left(\alpha_ {I - 1} ^ {*}\right) ^ {- 1} & \text {i f} t = T _ {I}, I \geq 1, \end{array} \right. \tag {6}
+$$
+
+where  $\alpha_{I}^{*} = \frac{1 + \gamma - \lambda\eta_{I}^{*} + \sqrt{\left(1 + \gamma - \lambda\eta_{I}^{*}\right)^{2} - 4\gamma}}{2},\widetilde{\eta}_{0} = \eta_{0}\cdot (\alpha_{0}^{*})^{-1} = \eta_{0}^{*}\cdot (\alpha_{0}^{*})^{-1}.$
+
+The analysis in previous subsections give the equivalence within each phase, where the same LR is used throughout the phase. To deal with the difference between buffered LR and current LR when entering new phases, the idea is to pretend  $\eta_{t - 1} = \eta_t$  and  $\theta_{t - 1}$  becomes whatever it needs to maintain  $\frac{\bar{\theta}_t - \theta_{t - 1}}{\eta_{t - 1}}$  such that we can again apply Lemma 2.8, which requires the current LR of the input state is equal to its buffered LR. Because scaling  $\alpha$  in RHS of Equation 5 is different in different phases, so unlike what happens within each phase, they don't cancel with each other at phase transitions, thus remaining as a correction of the momentum. The proofs are delayed to Appendix B, where we prove a more general statement allowing phase-dependent WD,  $\{\lambda_I\}_{I = 0}^{K - 1}$ .
+
+Alternative interpretation of Step Decay to exponential LR schedule: Below we present a new LR schedule,  $TEXP++$ , which is exactly equivalent to Step Decay without the need of one-time correction of momentum when entering each phase. We further show in Appendix B.1 that when translating from Step Decay, the  $TEXP++$  we get is very close to the original  $TEXP(Equation 9)$ , i.e. the ratio between the LR growth per round,  $\frac{\widetilde{\eta}_{t+1}}{\widetilde{\eta}_t} / \frac{\widetilde{\eta}_{t+1}'}{\widetilde{\eta}_t'}$  converges to 1 exponentially each phase. For example, with WD 0.0005, max LR 0.1, momentum factor 0.9, the ratio is within  $1 \pm 0.0015 * 0.9^{t-T_I}$ , meaning  $TEXP$  and  $TEXP++$  are very close for Step Decay with standard hyperparameters.
+
+Theorem 2.13. The following two sequences of parameters,  $\{\pmb{\theta}_t\}_{t=0}^{\infty}$  and  $\{\widetilde{\pmb{\theta}}_t\}_{t=0}^{\infty}$ , define the same sequence of network functions, i.e.  $f_{\pmb{\theta}_t} = f_{\widetilde{\pmb{\theta}}_t}$ ,  $\forall t \in \mathbb{N}$ , given the initial conditions,  $\widetilde{\pmb{\theta}}_0 = P_0\pmb{\theta}_0$ ,  $\widetilde{\pmb{\theta}}_{-1} = P_{-1}\pmb{\theta}_{-1}$ .
+
+1.  $\frac{\pmb{\theta}_{t} - \pmb{\theta}_{t - 1}}{\eta_{t - 1}} = \gamma \frac{\pmb{\theta}_{t - 1} - \pmb{\theta}_{t - 2}}{\eta_{t - 2}} -\nabla_{\pmb{\theta}}\left((L(\pmb{\theta}_{t - 1}) + \frac{\lambda_{t - 1}}{2}\| \pmb{\theta}_{t - 1}\| _2^2\right),$  for  $t = 1,2,\ldots$  
+2.  $\frac{\widetilde{\boldsymbol{\theta}}_t - \widetilde{\boldsymbol{\theta}}_{t - 1}}{\widetilde{\eta}_{t - 1}} = \gamma \frac{\widetilde{\boldsymbol{\theta}}_{t - 1} - \widetilde{\boldsymbol{\theta}}_{t - 2}}{\widetilde{\eta}_{t - 2}} -\nabla_\boldsymbol {\theta}L(\widetilde{\boldsymbol{\theta}}_{t - 1})$  , for  $t = 1,2,\ldots$
+
+where  $\widetilde{\eta}_t = P_tP_{t + 1}\eta_t$ ,  $P_{t} = \prod_{i = -1}^{t}\alpha_{i}^{-1}$ ,  $\forall t\geq -1$  and  $\alpha_{t}$  recursively defined as
+
+$$
+\alpha_ {t} = - \eta_ {t - 1} \lambda_ {t - 1} + 1 + \frac {\eta_ {t - 1}}{\eta_ {t - 2}} \gamma \left(1 - \alpha_ {t - 1} ^ {- 1}\right), \forall t \geq 1. \tag {7}
+$$
+
+The LR schedule  $\{\widetilde{\eta}_t\}_{t=0}^{\infty}$  is called Tapered Exponential ++, or  $TEXP++$ .
+
+# 3 EXAMPLE ILLUSTRATING INTERPLAY OF WD AND BN
+
+The paper so far has shown that effects of different hyperparameters in training are not easily separated, since their combined effect on the trajectory is complicated. We give a simple example to illustrate this, where convergence is guaranteed if we use either BatchNorm or weight decay in isolation, but convergence fails if both are used. (Momentum is turned off for clarity of presentation)
+
+Setting: Suppose we are fine-tuning the last linear layer of the network, where the input of the last layer is assumed to follow a standard Gaussian distribution  $\mathcal{N}(0, I_m)$ , and  $m$  is the input dimension of last layer. We also assume this is a binary classification task with logistic loss,  $l(u, y) = \ln(1 + \exp(-uy))$ , where label  $y \in \{-1, 1\}$  and  $u \in \mathbb{R}$  is the output of the neural network. The training algorithm is SGD with constant LR and WD, and without momentum. For simplicity we assume the batch size  $B$  is very large so we could assume the covariance of each batch  $\mathcal{B}_t$  concentrates and is approximately equal to identity, namely  $\frac{1}{B} \sum_{i=1}^{B} \boldsymbol{x}_{t,b} \boldsymbol{x}_{t,b}^{\top} \approx I_m$ . We also assume the input of the last layer are already separable, and w.l.o.g. we assume the label is equal to the sign of the first coordinate of  $\boldsymbol{x} \in \mathbb{R}^m$ , namely sign  $(x_1)$ . Thus the training loss and training error are simply
+
+$$
+L(\boldsymbol {w}) = \underset {\boldsymbol {x}\sim \mathcal{N}(0,I_{m}),y = \operatorname {sign}(x_{1})}{\mathbb{E}}\left[\ln (1 + \exp (-\boldsymbol{x}^{\top}\boldsymbol {w}y))\right],\quad \underset {\boldsymbol {x}\sim \mathcal{N}(0,I_{m}),y = \operatorname {sign}(x_{1})}{\Pr}\left[\boldsymbol{x}^{\top}\boldsymbol {w}y\leq 0\right] = \frac{1}{\pi}\arccos \frac{w_{1}}{\|\boldsymbol{w}\|}
+$$
+
+Case 1: WD alone: Since both the above objective with L2 regularization is strongly convex and smooth in  $\boldsymbol{w}$ , vanilla GD with suitably small learning rate could get arbitrarily close to the global minimum for this regularized objective. In our case, large batch SGD behaves similarly to GD and can achieve  $O\left(\sqrt{\frac{\eta\lambda}{B}}\right)$  test error following the standard analysis of convex optimization.
+
+Case 2: BN alone: Add a BN layer after the linear layer, and fix scalar and bias term to 1 and 0. The objective becomes
+
+$$
+L _ {B N} (\boldsymbol {w}) = \underset {\boldsymbol {x} \sim \mathcal {N} (0, I _ {m}), y = \operatorname {s i g n} (x _ {1})} {\mathbb {E}} \left[ L _ {B N} (\boldsymbol {w}, \boldsymbol {x}) \right] = \underset {\boldsymbol {x} \sim \mathcal {N} (0, I _ {m}), y = \operatorname {s i g n} (x _ {1})} {\mathbb {E}} \left[ \ln (1 + \exp (- \boldsymbol {x} ^ {\top} \frac {\boldsymbol {w}}{\| \boldsymbol {w} \|} y)) \right].
+$$
+
+From Appendix B.6, there's some constant  $C$ , such that  $\forall \boldsymbol{w} \in \mathbb{R}^m$  with constant probability,  $\| \nabla_{\boldsymbol{w}} L_{BN}(\boldsymbol{w}, \boldsymbol{x}) \| \geq \frac{C}{\|\boldsymbol{w}\|}$ . By Pythagorean Theorem,  $\| \boldsymbol{w}_{t+1} \|^4 = (\| \boldsymbol{w}_t \|^2 + \eta^2 \| \nabla_{\boldsymbol{w}} L_{BN}(\boldsymbol{w}_t, \boldsymbol{x}) \|^2)^2 \geq \| \boldsymbol{w}_t \|^4 + 2 \eta^2 \| \boldsymbol{w}_t \|^2 \| \nabla_{\boldsymbol{w}} L_{BN}(\boldsymbol{w}_t, \boldsymbol{x}) \|^2$ . As a result, for any fixed learning rate,  $\| \boldsymbol{w}_{t+1} \|^4 \geq 2 \sum_{i=1}^{t} \eta^2 \| \boldsymbol{w} \|^2 \| \nabla_{\boldsymbol{w}} L_{BN}(\boldsymbol{w}_i, \boldsymbol{x}) \|^2$  grows at least linearly with high probability. Following the analysis of Arora et al. (2019), this is like reducing the effective learning
+
+rate, and when  $\| \pmb{w}_t\|$  is large enough, the effective learning rate is small enough, and thus SGD can find the local minimum, which is the unique global minimum.
+
+Case 3: Both BN and WD: Suppose WD factor is  $\lambda$ , LR is  $\eta$ , we have the following theorem:
+
+Theorem 3.1. [Nonconvergence] Starting from iteration any  $T_0$ , with probability  $1 - \delta$  over the randomness of samples, the training error will be larger than  $\frac{\varepsilon}{\pi}$  at least once for the following consecutive  $\frac{1}{2(\eta\lambda - 2\varepsilon^2)}\ln \frac{64\|w_{T_0}\|^2\varepsilon\sqrt{B}}{\eta\sqrt{m - 2}} +9\ln \frac{1}{\delta}$  iterations.
+
+Sketch. (See full proof in Appendix B.) The high level idea of this proof is that if the test error is low, the weight is restricted in a small cone around the global minimum, and thus the amount of the gradient update is bounded by the size of the cone. In this case, the growth of the norm of the weight by Pythagorean Theorem is not large enough to cancel the shrinkage brought by weight decay. As a result, the norm of the weight converges to 0 geometrically. Again we need to use the lower bound for size of the gradient, that  $\| \nabla_w L_t\| = \Theta (\frac{\eta\sqrt{m}}{\|\pmb{w}_t\|})$  holds with constant probability. Thus the size of the gradient will grow along with the shrinkage of  $\| \pmb {w}_t\|$  until they're comparable, forcing the weight to leave the cone in next iteration.
+
+# 4 EXPERIMENTS
+
+The translation to exponential LR schedule is exact except for one-time momentum correction term entering new phases. The experiments explore the effect of this correction term. The Tapered Exponential(TEXP) LR schedule contains two parts when entering a new phase I: an instant LR decay  $\left(\frac{\eta_I}{\eta_{I-1}}\right)$  and an adjustment of the growth factor  $(\alpha_{I-1}^* \to \alpha_I^*)$ . The first part is relative small compared to the huge exponential growing. Thus a natural question arises: Can we simplify TEXP LR schedule by dropping the part of instant LR decay?
+
+Also, previously we have only verified our equivalence theorem in Step Decay LR schedules. But it's not sure how would the Exponential LR schedule behave on more rapid time-varying LR schedules such as Cosine LR schedule.
+
+Settings: We train PreResNet32 on CIFAR10. The initial learning rate is 0.1 and the momentum is 0.9 in all settings. We fix all the scalar and bias of BN, because otherwise they together with the following conv layer grow exponentially, sometimes exceeding the range of Float32 when trained with large growth rate for a long time. We fix the parameters in the last fully connected layer for scale invariance of the objective.
+
+# 4.1 THE BENEFIT OF INSTANT LR DECAY
+
+We tried the following LR schedule (we call it  $TEXP--$ ). Interestingly, up to correction of momentum when entering a new phase, this schedule is equivalent to a constant LR schedule, but with the weight decay coefficient reduced correspondingly at the start of each phase. (See Theorem B.2 and Figure 6)
+
+$$
+\text {T E X P - - :} \quad \widetilde {\eta} _ {t + 1} = \left\{ \begin{array}{l l} \widetilde {\eta} _ {t} \times \left(\alpha_ {I - 1} ^ {*}\right) ^ {- 2} & \text {i f} T _ {I - 1} + 1 \leq t \leq T _ {I} - 1, I \geq 1; \\ \widetilde {\eta} _ {t} \times \left(\alpha_ {I} ^ {*}\right) ^ {- 1} \left(\alpha_ {I - 1} ^ {*}\right) ^ {- 1} & \text {i f} t = T _ {I}, I \geq 1, \end{array} \right. \tag {8}
+$$
+
+where  $\alpha_{I}^{*} = \frac{1 + \gamma - \lambda\eta_{I}^{*} + \sqrt{\left(1 + \gamma - \lambda\eta_{I}^{*}\right)^{2} - 4\gamma}}{2},\widetilde{\eta}_{0} = \eta_{0}\cdot (\alpha_{0}^{*})^{-1} = \eta_{0}^{*}\cdot (\alpha_{0}^{*})^{-1}.$
+
+# 4.2 BETTER EXPONENTIAL LR SCHEDULE WITH COSINE LR
+
+We applied the TEXP LR schedule (Theorem 2.12) on the Cosine LR schedule (Loshchilov & Hutter, 2016), where the learning rate changes every epoch, and thus correction terms cannot be ignored. The LR at epoch  $t \leq T$  is defined as:  $\eta_t = \eta_0 \frac{1 + \cos(\frac{t}{T} \pi)}{2}$ . Our experiments show this hybrid schedule with Cosine LR performs better on CIFAR10 than Step Decay, but this finding needs to be verified on other datasets.
+
+# 5 CONCLUSIONS
+
+The paper shows rigorously how BN allows a host of very exotic learning rate schedules in deep learning, and verifies these effects in experiments. The lr increases exponentially in almost every
+
+![](images/c1ae6d15d7e4aacb774a417f5ff2f5a6371d9bad7a91da94c3ea4353ec57bac7.jpg)  
+Figure 3: Instant LR decay is crucial when LR growth  $\widetilde{\eta}_t / \widetilde{\eta}_{t-1} - 1$  is very small. The original LR of Step Decay is decayed by 10 at epoch 80, 120 respectively. In the third phase, LR growth  $\widetilde{\eta}_t / \widetilde{\eta}_{t-1} - 1$  is approximately 100 times smaller than that in the third phase, it would take TEXP-- hundreds of epochs to reach its equilibrium. As a result, TEXP achieves better test accuracy than TEXP--. As a comparison, in the second phase,  $\widetilde{\eta}_t / \widetilde{\eta}_{t-1} - 1$  is only 10 times smaller than that in the first phase and it only takes 70 epochs to return to equilibrium.
+
+![](images/4ba8f3d0ab7e6bcf2acb239e249ea3913ab698d9c7e632659056c9e6afaf3c51.jpg)
+
+![](images/4972979154b572f8f6fe72589ec5a24399ee777821932807a5aa06c1fa50fd27.jpg)  
+Figure 4: Instant LR decay has only temporary effect when LR growth  $\widetilde{\eta}_t / \widetilde{\eta}_{t - 1} - 1$  is large. The blue line uses an exponential LR schedule with constant exponent. The orange line multiplies its LR by the same constant each iteration, but also divide LR by 10 at the start of epoch 80 and 120. The instant LR decay only allows the parameter to stay at good local minimum for 1 epoch and then diverges, behaving similarly to the trajectories without no instant LR decay.
+
+![](images/d24bf6534ac120a2b73a528de64c57593a50bbec184eaa6683e3df2291233032.jpg)
+
+![](images/30e4336c8a920e733acb50241027a1144bb809747c30204cb17b21a1554d499f.jpg)  
+Figure 5: Both Cosine and Step Decay schedule behaves almost the same as their exponential counterpart, as predicted by our equivalence theorem. The (exponential) Cosine LR schedule achieves better test accuracy, with a entirely different trajectory.
+
+![](images/cc35de13712ed9eb6bceebf3c1e324d16cea04e0621ec4d524c0f5b13148aa77.jpg)
+
+iteration during training. The exponential increase derives from use of weight decay, but the precise expression involves momentum as well. We suggest that the efficacy of this rule may be hard to explain with canonical frameworks in optimization.  
+Our analyses of BN is a substantial improvement over earlier theoretical analyses, since it accounts for weight decay and momentum, which are always combined in practice.  
+Our tantalising experiments with a hybrid of exponential and cosine rates suggest that more surprises may lie out there. Our theoretical analysis of interrelatedness of hyperparameters could also lead to faster hyperparameter search.
+
+# REFERENCES
+
+Sanjeev Arora, Nadav Cohen, and Elad Hazan. On the optimization of deep networks: Implicit acceleration by overparameterization. In International Conference on Machine Learning, pp. 244-253, 2018.  
+Sanjeev Arora, Zhiyuan Li, and Kaifeng Lyu. Theoretical analysis of auto rate-tuning by batch normalization. In International Conference on Learning Representations, 2019. URL https://openreview.net/forum?id=rkxQ-nA9FX.  
+Jimmy Lei Ba, Jamie Ryan Kiros, and Geoffrey E Hinton. Layer normalization. arXiv preprint arXiv:1607.06450, 2016.  
+Nils Bjorck, Carla P Gomes, Bart Selman, and Kilian Q Weinberger. Understanding batch normalization. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett (eds.), Advances in Neural Information Processing Systems 31, pp. 7705-7716. Curran Associates, Inc., 2018.  
+Minhyung Cho and Jaehyung Lee. Riemannian approach to batch normalization. In I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett (eds.), Advances in Neural Information Processing Systems 30, pp. 5225-5235. Curran Associates, Inc., 2017.  
+Sanjoy Dasgupta and Anupam Gupta. An elementary proof of a theorem of johnson and lindenstrauss. *Random Structures & Algorithms*, 22(1):60-65, 2003.  
+Robert Mansel Gower, Nicolas Loizou, Xun Qian, Alibek Sailanbayev, Egor Shulgin, and Peter Richtárik. Sgd: General analysis and improved rates. arXiv preprint arXiv:1901.09401, 2019.  
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 770-778, 2016a.  
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Identity mappings in deep residual networks. In European conference on computer vision, pp. 630-645. Springer, 2016b.  
+Elad Hoffer, Ron Banner, Itay Golan, and Daniel Soudry. Norm matters: efficient and accurate normalization schemes in deep networks. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett (eds.), Advances in Neural Information Processing Systems 31, pp. 2164-2174. Curran Associates, Inc., 2018a.  
+Elad Hoffer, Itay Hubara, and Daniel Soudry. Fix your classifier: the marginal value of training the last weight layer. In International Conference on Learning Representations, 2018b. URL https://openreview.net/forum?id=S1Dh8Tg0-.  
+Sergey Ioffe and Christian Szegedy. Batch normalization: Accelerating deep network training by reducing internal covariate shift. In International Conference on Machine Learning, pp. 448-456, 2015.  
+Jonas Kohler, Hadi Daneshmand, Aurelien Lucchi, Ming Zhou, Klaus Neymeyr, and Thomas Hofmann. Exponential convergence rates for batch normalization: The power of length-direction decoupling in non-convex optimization. arXiv preprint arXiv:1805.10694, 2018.
+
+Ilya Loshchilov and Frank Hutter. SGDR: Stochastic Gradient Descent with Warm Restarts. arXiv e-prints, art. arXiv:1608.03983, Aug 2016.  
+David Page. How to train your resnet 6: Weight decay? URL https://myrtle.ai/ how-to-train-your-resnet-6-weight-decay/.  
+Adam Paszke, Sam Gross, Soumith Chintala, Gregory Chanan, Edward Yang, Zachary DeVito, Zeming Lin, Alban Desmaison, Luca Antiga, and Adam Lerer. Automatic differentiation in pytorch. 2017.  
+Shibani Santurkar, Dimitris Tsipras, Andrew Ilyas, and Aleksander Madry. How does batch normalization help optimization? In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesabianchi, and R. Garnett (eds.), Advances in Neural Information Processing Systems 31, pp. 2488-2498. Curran Associates, Inc., 2018.  
+Leslie N Smith. Cyclical learning rates for training neural networks. In 2017 IEEE Winter Conference on Applications of Computer Vision (WACV), pp. 464-472. IEEE, 2017.  
+Ilya Sutskever, James Martens, George Dahl, and Geoffrey Hinton. On the importance of initialization and momentum in deep learning. In Proceedings of the 30th International Conference on International Conference on Machine Learning - Volume 28, ICML'13, pp. III-1139-III-1147. JMLR.org, 2013. URL http://dl.acm.org/citation.cfm?id=3042817.3043064.  
+Dmitry Ulyanov, Andrea Vedaldi, and Victor Lempitsky. Instance normalization: The missing ingredient for fast stylization. arXiv preprint arXiv:1607.08022, 2016.  
+Twan van Laarhoven. L2 regularization versus batch and weight normalization. arXiv preprint arXiv:1706.05350, 2017.  
+Xiaoxia Wu, Rachel Ward, and Léon Bottou. WNGrad: Learn the Learning Rate in Gradient Descent. arXiv preprint arXiv:1803.02865, 2018.  
+Yuxin Wu and Kaiming He. Group normalization. In The European Conference on Computer Vision (ECCV), September 2018.  
+Yang You, Igor Gitman, and Boris Ginsburg. Large Batch Training of Convolutional Networks. arXiv e-prints, art. arXiv:1708.03888, Aug 2017.  
+Guodong Zhang, Chaoqi Wang, Bowen Xu, and Roger Grosse. Three mechanisms of weight decay regularization. In International Conference on Learning Representations, 2019. URL https://openreview.net/forum?id=B1lz-3Rct7.
+
+# A VIEWING EXP LR VIA CANONICAL OPTIMIZATION FRAMEWORK
+
+This section tries to explain why the efficacy of exponential LR in deep learning is mysterious to us, at least as viewed in the canonical framework of optimization theory.
+
+Canonical framework for analysing 1st order methods This focuses on proving that each—or most—steps of GD noticeably reduce the objective, by relying on some assumption about the spectrum norm of the hessian of the loss, and most frequently, the smoothness, denoted by  $\beta$ . Specifically, for GD update  $\pmb{\theta}_{t + 1} = \pmb{\theta}_t - \eta \nabla L(\pmb{\theta}_t)$ , we have
+
+$$
+L (\boldsymbol {\theta} _ {t + 1}) - L (\boldsymbol {\theta} _ {t}) \leq (\boldsymbol {\theta} _ {t + 1} - \boldsymbol {\theta} _ {t}) ^ {\top} \nabla L (\boldsymbol {\theta} _ {t}) + \frac {\beta}{2} \| \boldsymbol {\theta} _ {t + 1} - \boldsymbol {\theta} _ {t} \| ^ {2} = - \eta (1 - \frac {\beta \eta}{2}) \| \nabla L (\boldsymbol {\theta} _ {t}) \| ^ {2}.
+$$
+
+When  $\beta < \frac{2}{\eta}$ , the first order term is larger than the second order one, guaranteeing the loss value decreases. Since the analysis framework treats the loss as a black box (apart from the assumed bounds on the derivative norms), and the loss is non-convex, the best one can hope for is to prove speedy convergence to a stationary point (where gradient is close to 0). An increasing body of work proves such results.
+
+Now we turn to difficulties in understanding the exponential LR in context of the above framework and with scale-invariance in the network.
+
+1. Since loss is same for  $\theta$  and  $c\cdot \theta$  for all  $c > 0$  a simple calculation shows that along any straight line through the origin, smoothness is a decreasing function of  $c$ , and is very high close to origin. (Note: it is also possible to one can show the following related fact: In any ball containing the origin, the loss is nonconvex.)
+
+Thus if one were trying to apply the canonical framework to argue convergence to a stationary point, the natural idea would be to try to grow the norm of the parameters until smoothness drops enough that the above-mentioned Canonical Framework starts to apply. Arora et al. (2019) showed this happens in GD with fixed LR (WD turned off), and furthermore the resulting convergence rate to stationary point is asymptotically similar to analyses of nonconvex optimization with learning rate set as in the Canonical framework. Santurkar et al. (2018) observed similar phenomenon in experiments, which they described as a smoothening of the objective due to BN.
+
+2. The Canonical Framework can be thought of as a discretization of continuous gradient descent (i.e., gradient flow): in principle it is possible to use arbitrarily small learning rate, but one uses finite learning rate merely to keep the number of iterations small. The discrete process approximates the continuous process due to smoothness being small.
+
+In case of gradient flow with weight decay (equivalently, with exponential LR schedule) the discrete process cannot track the continuous process for very long, which suggests that any explanation of the benefits of exponential LR may need to rely on discrete process being somehow better. The reason being that for gradient flow one can decouple the speed of the  $\theta_{t}$  into the tangential and the radial components, where the former one has no effect on the norm and the latter one has no effect on the objective but scales the tangential gradient exponentially. Thus the Gradient Flow with WD gives exactly the same trajectory as vanilla Gradient Flow does, excepting a exponential reparametrization with respect to time  $t$ .
+
+3. It can be shown that if the local smoothness is upperbounded by  $\frac{2}{\eta}$  (as stipulated in Canonical Framework) during a sequence  $\pmb{\theta}_t$  ( $t = 1, 2, \ldots$ ) of GD updates with WD and constant LR then such sequence satisfies  $\pmb{\theta}_t \to \mathbf{0}$ . This contrasts with the usual experimental observation that  $\pmb{\theta}_t$  stays bounded away from  $\mathbf{0}$ . One should thus conclude that in practice, with constant LR and WD, smoothness doesn't always stay small (unlike the above analyses where WD is turned off).
+
+# B OMITTED PROOFS
+
+# B.1 OMITTED PROOF IN SECTION 2
+
+Lemma B.1 (Some Facts about Equation 1). Suppose  $z^1, z^2 (z^1 \geq z^2)$  are the two real roots of the following equation, we have
+
+$$
+x ^ {2} - (1 + \gamma - \lambda \eta) x + \gamma = 0
+$$
+
+1.  $z^1 = \frac{1 + \gamma - \lambda\eta + \sqrt{(1 - \gamma)^2 - 2(1 + \gamma)\lambda\eta + \lambda^2\eta^2}}{2}, z^2 = \frac{1 + \gamma - \lambda\eta - \sqrt{(1 - \gamma)^2 - 2(1 + \gamma)\lambda\eta + \lambda^2\eta^2}}{2}$  
+2.  $z^1, z^2$  are real  $\Longleftrightarrow \lambda \eta \leq (1 - \sqrt{\gamma})^2$ ;  
+3.  $z^1 z^2 = \gamma, z^1 + z^2 = (1 + \gamma - \lambda \eta)$ ;  
+4.  $\gamma \leq z^2 \leq z^1 \leq 1$ ;  
+5. Let  $t = \frac{\lambda\eta}{1 - \gamma}$ , we have  $z^1 \geq \frac{1}{1 + t} \geq 1 - t = 1 - \frac{\lambda\eta}{1 - \gamma}$ .  
+6. if we view  $z^1(\lambda \eta)$ ,  $z^2(\lambda \eta)$  as functions of  $\lambda \eta$ , then  $z^1(\lambda \eta)$  is monotone decreasing,  $z^2(\eta)$  is monotone increasing.
+
+Proof.
+
+4. Let  $f(x) = x^{2} - (1 + \gamma - \lambda \eta)x + \gamma$ , we have  $f(1) = f(\gamma) = \lambda \eta \geq 0$ . Note the minimum of  $f$  is taken at  $x = \frac{1 + \gamma - \lambda\eta}{2} \in [0,1]$ , the both roots of  $f(x) = 0$  must lie between 0 and 1, if exists.
+
+5
+
+$$
+\begin{array}{l} 1 - z ^ {1} = \frac {1 - \gamma + \lambda \eta - \sqrt {(1 - \gamma) ^ {2} - 2 (1 + \gamma) \lambda \eta + \lambda^ {2} \eta^ {2}}}{2} \\ = (1 - \gamma) \frac {1 + t - \sqrt {1 - \frac {1 + \gamma}{1 - \gamma} t + t ^ {2}}}{2} \\ = (1 - \gamma) \frac {2 t + 2 \frac {1 + \gamma}{1 - \gamma} t}{2 (1 + t + \sqrt {1 - \frac {1 + \gamma}{1 - \gamma} t + t ^ {2}})} \\ \leq (1 - \gamma) \frac {\frac {4}{1 - \gamma} t}{4 (1 + t)} \\ = \frac {t}{(1 + t)} \\ \end{array}
+$$
+
+6. Note that  $(z^1 - z^2)^2 = (z^1 + z^2)^2 - 4z^1z^2 = (1 + \gamma - \lambda\eta)^2 - 4\gamma$  is monotone decreasing, since  $z^1(\lambda\eta) + z^2(\lambda\eta)$  is constant,  $z^1(\lambda\eta) \geq z^2(\lambda\eta)$ ,  $z^1(\lambda\eta)$  must be decreasing and  $z^2(\lambda\eta)$  must be increasing.
+
+![](images/f0daabe686891f1bd6a90df4cf9f2e60f7e23b35cf8221a2460d7eab7c96e500.jpg)
+
+# B.2 OMITTED PROOFS IN SECTION 2.1
+
+Proof of Lemma 2.2. For any  $(\theta, \eta)$ , we have
+
+$$
+\mathbf {G D} _ {t} ^ {\rho} (\pmb {\theta}, \eta) = (\rho \pmb {\theta} - \eta \nabla L _ {t} (\pmb {\theta}), \eta) = [ \Pi_ {1} ^ {\rho} \circ \Pi_ {2} ^ {\rho} \circ \mathbf {G D} _ {t} ] (\pmb {\theta}, \frac {\eta}{\rho}) = [ \Pi_ {1} ^ {\rho} \circ \Pi_ {2} ^ {\rho} \circ \mathbf {G D} _ {t} \circ \Pi_ {2} ^ {\rho^ {- 1}} ] (\pmb {\theta}, \eta).
+$$
+
+Proof of Lemma 2.4. For any  $(\theta, \eta)$ , we have
+
+$$
+\begin{array}{l} \left[ \mathbf {G D} _ {t} \circ \Pi_ {1} ^ {c} \circ \Pi_ {2} ^ {c ^ {2}} \right] (\pmb {\theta}, \eta) = \mathbf {G D} _ {t} (c \pmb {\theta}, c ^ {2} \eta) = (c \pmb {\theta} - c ^ {2} \pmb {\theta} \nabla L _ {t} (c \pmb {\theta}), c ^ {2} \eta) \stackrel {*} {=} (c (\pmb {\theta} - \nabla L _ {t} (\pmb {\theta})), c ^ {2} \eta) \\ = \left[ \Pi_ {1} ^ {c} \circ \Pi_ {2} ^ {c ^ {2}} \circ \mathbf {G D} _ {t} \right] (\boldsymbol {\theta}, \eta). \quad \left(\stackrel {*} {=} \text {: S c a l e I n v a r i a n c e , L e m m a 1 . 3}\right) \\ \end{array}
+$$
+
+# B.3 OMITTED PROOFS IN SECTION 2.2
+
+Proof of Lemma 2.7. For any input  $(\pmb{\theta}, \eta, \pmb{\theta}', \eta')$ , it's easy to check both composed maps have the same outputs on the 2,3,4th coordinates, namely  $(c^2 \eta, c \pmb{\theta}, c^2 \eta')$ . For the first coordinate, we have
+
+$$
+\begin{array}{l} \left[ \mathbf {G D} ^ {\rho} (c \boldsymbol {\theta}, c ^ {2} \eta , c \boldsymbol {\theta} ^ {\prime}, c ^ {2} \eta) \right] _ {1} = \rho c \boldsymbol {\theta} + c ^ {2} \eta \left(\gamma \frac {\boldsymbol {\theta} - \boldsymbol {\theta} ^ {\prime}}{\eta^ {\prime}} - \nabla L _ {t} (c \boldsymbol {\theta})\right) \stackrel {*} {=} c \left(\boldsymbol {\theta} + \eta \left(\gamma \frac {\boldsymbol {\theta} - \boldsymbol {\theta} ^ {\prime}}{\eta^ {\prime}} - \nabla L _ {t} (\boldsymbol {\theta})\right)\right) \\ = c \left[ \mathrm {G D} ^ {\rho} \left(\boldsymbol {\theta}, \eta , \boldsymbol {\theta} ^ {\prime}, \eta\right) \right] _ {1}. \quad \stackrel {*} {=} \text {S c a l e I n v a r i a n c e}, \text {L e m m a} 1. 3 \\ \end{array}
+$$
+
+Proof of Lemma 2.8. For any input  $(\pmb{\theta}, \eta, \pmb{\theta}', \eta')$ , it's easy to check both composed maps have the same outputs on the 2,3,4th coordinates, namely  $(\eta, \pmb{\theta}, \eta)$ . For the first coordinate, we have
+
+$$
+\begin{array}{l} \left[ \left[ \Pi_ {3} ^ {\alpha} \circ \Pi_ {4} ^ {\alpha} \circ \Pi_ {2} ^ {\alpha} \circ \mathbf {G D} _ {t} \circ \Pi_ {2} ^ {\alpha^ {- 1}} \circ \Pi_ {3} ^ {\alpha} \circ \Pi_ {4} ^ {\alpha} \right] (\pmb {\theta}, \eta , \pmb {\theta} ^ {\prime}, \eta) \right] _ {1} = \alpha \left[ \mathbf {G D} _ {t} (\pmb {\theta}, \alpha^ {- 1} \eta , \alpha \pmb {\theta} ^ {\prime}, \alpha \eta) \right] _ {1} \\ = \alpha (\boldsymbol {\theta} + \alpha^ {- 1} \eta \left(\gamma \frac {\boldsymbol {\theta} - \boldsymbol {\theta} ^ {\prime}}{\eta} - \nabla L _ {t} (\boldsymbol {\theta})\right)) \\ = (\alpha + \gamma \alpha^ {- 1}) \boldsymbol {\theta} - \eta \nabla L _ {t} (\boldsymbol {\theta}) - \eta \gamma \frac {\boldsymbol {\theta} ^ {\prime}}{\eta} \\ = (\rho + \gamma) \boldsymbol {\theta} - \eta \nabla L _ {t} (\boldsymbol {\theta}) - \gamma \boldsymbol {\theta} ^ {\prime} = [ \mathbf {G D} _ {t} ^ {\rho} (\boldsymbol {\theta}, \eta , \boldsymbol {\theta} ^ {\prime}, \eta) ] _ {1} \\ \end{array}
+$$
+
+# B.4 OMITTED PROOFS OF THEOREM 2.12
+
+In this subsection we will prove a stronger version of Theorem 2.12(restated below), allowing the  $\mathrm{WD},\lambda_I$  changing each phase.
+
+Theorem B.2 (A stronger version of Theorem 2.12). There exists a way to correct the momentum only at the first iteration of each phase, such that the following Tapered-Exponential LR schedule (TEXP)  $\{\widetilde{\eta}_t\}$  with momentum factor  $\gamma$  and no WD, leads the same sequence networks in function space compared to that of Step Decay LR schedule(Definition 2.11) with momentum factor  $\gamma$  and phase-dependent WD  $\lambda_I^*$  in phase  $I$ , where phase  $I$  lasts from iteration  $T_{I}$  to iteration  $T_{I + 1}, T_0 = 0$ .
+
+$$
+\widetilde {\eta} _ {t + 1} = \left\{ \begin{array}{l l} \widetilde {\eta} _ {t} \times \left(\alpha_ {I - 1} ^ {*}\right) ^ {- 2} & \text {i f} T _ {I - 1} + 1 \leq t \leq T _ {I} - 1, I \geq 1 \\ \widetilde {\eta} _ {t} \times \frac {\eta_ {I} ^ {*}}{\eta_ {I - 1} ^ {*}} \times \left(\alpha_ {I} ^ {*}\right) ^ {- 1} \left(\alpha_ {I - 1} ^ {*}\right) ^ {- 1} & \text {i f} t = T _ {I}, I \geq 1 \end{array} , \right. \tag {9}
+$$
+
+where  $\alpha_{I}^{*} = \frac{1 + \gamma - \lambda_{I}^{*}\eta_{I}^{*} + \sqrt{\left(1 + \gamma - \lambda_{I}^{*}\eta_{I}^{*}\right)^{2} - 4\gamma}}{2},\widetilde{\eta}_{0} = \eta_{0}(\alpha_{0}^{*})^{-1} = \eta_{0}^{*}(\alpha_{0}^{*})^{-1}.$
+
+Towards proving Theorem 2.12, we need the following lemma which holds by expanding the definition, and we omit its proof.
+
+Lemma B.3 (Canonicalization). We define the Canonicalization map as  $N(\pmb{\theta}, \eta, \pmb{\theta}', \eta') = (\pmb{\theta}, \eta, \pmb{\theta} - \frac{\eta}{\eta'} (\pmb{\theta} - \pmb{\theta}')', \eta)$ , and it holds that
+
+1.  $GD_{t}^{\rho}\circ N = GD_{t}^{\rho},\forall \rho >0,t\geq 0.$  
+2.  $N\circ \left[\Pi_1^c\circ \Pi_2^{c^2}\circ \Pi_3^c\circ \Pi_4^{c^2}\right] = \left[\Pi_1^c\circ \Pi_2^{c^2}\circ \Pi_3^c\circ \Pi_4^{c^2}\right]\circ N,\forall c > 0.$
+
+Similar to the case of momentum-free SGD, we define the notion of equivalent map below
+
+Definition B.4 (Equivalent Maps). For two maps  $F$  and  $G$ , we say  $F$  is equivalent to  $G$  iff  $\exists c > 0$ ,  $F = \left[ \Pi_1^c \circ \Pi_2^{c^2} \circ \Pi_3^c \circ \Pi_4^{c^2} \right] \circ G$ , which is also denoted by  $F \stackrel{c}{\sim} G$ .
+
+Note that for any  $(\pmb {\theta},\eta ,\pmb{\theta}^{\prime},\eta^{\prime})$ $[N(\pmb {\theta},\eta ,\pmb{\theta}^{\prime},\eta^{\prime})]_{2} = [N(\pmb {\theta},\eta ,\pmb{\theta}^{\prime},\eta^{\prime})]_{4}$  . Thus as a direct consequence of Lemma 2.8, the following lemma holds.
+
+Lemma B.5.  $\forall \rho, \alpha > 0$ ,  $GD_{t}^{\rho} \circ N \stackrel{\alpha}{\sim} \Pi_{3}^{\alpha^{-1}} \circ \Pi_{4}^{\alpha^{-1}} \circ \Pi_{2}^{\alpha^{-1}} \circ GD_{t} \circ \Pi_{2}^{\alpha^{-1}} \circ \Pi_{3}^{\alpha} \circ \Pi_{4}^{\alpha} \circ N$ .
+
+Proof of Theorem2.12. Starting with initial state  $(\pmb{\theta}_0, \eta_0, \pmb{\theta}_{-1}, \eta_{-1})$  where  $\eta_{-1} = \eta_0$  and a given LR schedule  $\{\eta_t\}_{t \geq 0}$ , the parameters generated by GD with WD and momentum satisfies the following relationship:
+
+$$
+\left(\boldsymbol {\theta} _ {t + 1}, \eta_ {t + 1}, \boldsymbol {\theta} _ {t}, \eta_ {t}\right) = \left[ \Pi_ {2} ^ {\frac {\eta_ {t + 1}}{\eta_ {t}}} \circ \mathrm {G D} _ {t} ^ {1 - \eta_ {t} \lambda_ {t}} \right] \left(\boldsymbol {\theta} _ {t}, \eta_ {t}, \boldsymbol {\theta} _ {t - 1}, \eta_ {t - 1}\right).
+$$
+
+Define  $\bigcirc_{t=a}^{b} F_{t}=F_{b} \circ F_{b-1} \circ \ldots \circ F_{a}$ , for  $a \leq b$ . By Lemma B.3 and Lemma B.5, letting  $\alpha_{t}$  be the root of  $x^{2} - (\gamma + 1 - \eta_{t-1} \lambda_{t-1}) x + \gamma = 0$ , we have
+
+$$
+\begin{array}{l} \bigoplus_ {t = 0} ^ {T - 1} \left[ \Pi_ {2} ^ {\frac {\eta_ {t + 1}}{\eta_ {t}}} \circ \mathbf {G D} _ {t} ^ {1 - \eta_ {t} \lambda_ {t}} \right] \\ = \bigoplus_ {t = 0} ^ {T - 1} \left[ \Pi_ {2} ^ {\frac {\eta_ {t + 1}}{\eta_ {t}}} \circ \mathbf {G D} _ {t} ^ {1 - \eta_ {t} \lambda_ {t}} \circ N \right] \\ \end{array}
+$$
+
+$$
+\begin{array}{l} \overset {\prod_ {i = 0} ^ {T - 1} \alpha_ {i}} {\underset {t = 0} {\sim}} \underset {t = 0} {T - 1} \left[ \Pi_ {2} ^ {\frac {\eta_ {t + 1}}{\eta_ {t}}} \circ \Pi_ {3} ^ {\alpha_ {t + 1} ^ {- 1}} \circ \Pi_ {4} ^ {\alpha_ {t + 1} ^ {- 1}} \circ \Pi_ {2} ^ {\alpha_ {t + 1} ^ {- 1}} \circ \mathbf {G D} _ {t} \circ \Pi_ {2} ^ {\alpha_ {t + 1} ^ {- 1}} \circ \Pi_ {3} ^ {\alpha_ {t + 1}} \circ \Pi_ {4} ^ {\alpha_ {t + 1}} \circ N \right] \\ = \Pi_ {2} ^ {\frac {\eta_ {T}}{\eta_ {T - 1}}} \circ \Pi_ {3} ^ {\alpha_ {T - 1} ^ {- 1}} \circ \Pi_ {4} ^ {\alpha_ {T - 1} ^ {- 1}} \circ \Pi_ {2} ^ {\alpha_ {T} ^ {- 1}} \circ \mathbf {G D} _ {T - 1} \circ \left(\bigotimes_ {t = 1} ^ {T - 1} \left[ \Pi_ {2} ^ {\alpha_ {t + 1} ^ {- 1} \alpha_ {t} ^ {- 1}} \circ H _ {t} \circ \mathbf {G D} _ {t - 1} \right]\right) \circ \Pi_ {2} ^ {\alpha_ {1} ^ {- 1}} \circ \Pi_ {3} ^ {\alpha_ {1}} \circ \Pi_ {4} ^ {\alpha_ {1}} \circ N, \tag {10} \\ \end{array}
+$$
+
+where  $\stackrel{T-1}{\underset{i=0}{\overset{\prod\alpha_i}{\sim}}}$  is because of Lemma B.5, and  $H_{t}$  is defined as
+
+$$
+H _ {t} = \Pi_ {2} ^ {\alpha_ {t}} \circ \Pi_ {2} ^ {\frac {\eta_ {t - 1}}{\eta_ {t}}} \circ \Pi_ {3} ^ {\alpha_ {t + 1}} \circ \Pi_ {4} ^ {\alpha_ {t + 1}} \circ N \circ \Pi_ {3} ^ {\alpha_ {t} ^ {- 1}} \circ \Pi_ {4} ^ {\alpha_ {t} ^ {- 1}} \circ \Pi_ {2} ^ {\alpha_ {t} ^ {- 1}} \circ \Pi_ {2} ^ {\frac {\eta_ {t}}{\eta_ {t - 1}}}.
+$$
+
+Since the canonicalization map  $N$  only changes the momentum part of the state, it's easy to check that  $H_{t}$  doesn't touch the current parameter  $\theta$  and the current LR  $\eta$ . Thus  $H_{t}$  only changes the momentum part of the input state. Now we claim that  $H_{t} \circ \mathrm{GD}_{t-1} = \mathrm{GD}_{t-1}$  whenever  $\eta_{t} = \eta_{t-1}$ . This is because when  $\eta_{t} = \eta_{t-1}$ ,  $\alpha_{t} = \alpha_{t+1}$ , thus  $H_{t} \circ \mathrm{GD}_{t-1} = \mathrm{GD}_{t-1}$ . In detail,
+
+$$
+\begin{array}{l} H _ {t} \circ \mathrm {G D} _ {t - 1} \\ = \Pi_ {2} ^ {\alpha_ {t}} \circ \Pi_ {3} ^ {\alpha_ {t}} \circ \Pi_ {4} ^ {\alpha_ {t}} \circ N \circ \Pi_ {3} ^ {\alpha_ {t} ^ {- 1}} \circ \Pi_ {4} ^ {\alpha_ {t} ^ {- 1}} \circ \Pi_ {2} ^ {\alpha_ {t} ^ {- 1}} \circ \mathbf {G D} _ {t - 1} \\ \stackrel {*} {=} \Pi_ {2} ^ {\alpha_ {t}} \circ \Pi_ {3} ^ {\alpha_ {t}} \circ \Pi_ {4} ^ {\alpha_ {t}} \circ \Pi_ {3} ^ {\alpha_ {t} ^ {- 1}} \circ \Pi_ {4} ^ {\alpha_ {t} ^ {- 1}} \circ \Pi_ {2} ^ {\alpha_ {t} ^ {- 1}} \circ \mathbf {G D} _ {t - 1} \\ = \mathrm {G D} _ {t - 1}, \\ \end{array}
+$$
+
+where  $\stackrel{*}{=}$  is because GD update  $\mathrm{GD}_t$  sets  $\eta'$  the same as  $\eta$ , and thus ensures the input of  $N$  has the same momentum factor in buffer as its current momentum factor, which makes  $N$  an identity map.
+
+Thus we could rewrite Equation 10 with a "sloppy" version of  $H_{t}$ ,  $H_{t}^{\prime} = \left\{ \begin{array}{ll} H_{t} & \eta_{t} \neq \eta_{t-1}; \\ I d & \text{o.w.} \end{array} \right.$ :
+
+$$
+\begin{array}{l} \begin{array}{l} T - 1 \\ \bigcirc \\ t = 0 \end{array} \left[ \Pi_ {2} ^ {\frac {\eta_ {t + 1}}{\eta_ {t}}} \circ \mathbf {G D} _ {t} ^ {1 - \eta_ {t} \lambda_ {t}} \right] \\ = \Pi_ {2} ^ {\overline {{\eta_ {T}}}} \circ \Pi_ {3} ^ {\alpha_ {T - 1} ^ {- 1}} \circ \Pi_ {4} ^ {\alpha_ {T - 1} ^ {- 1}} \circ \Pi_ {2} ^ {\alpha_ {T} ^ {- 1}} \circ \mathbf {G D} _ {T - 1} \circ \left(\bigoplus_ {t = 1} ^ {T - 1} \left[ \Pi_ {2} ^ {\alpha_ {t + 1} ^ {- 1} \alpha_ {t} ^ {- 1}} \circ H _ {t} ^ {\prime} \circ \mathbf {G D} _ {t - 1} \right]\right) \circ \Pi_ {2} ^ {\alpha_ {1} ^ {- 1}} \circ \Pi_ {3} ^ {\alpha_ {1}} \circ \Pi_ {4} ^ {\alpha_ {1}} \circ N \\ = \Pi_ {2} ^ {\frac {\eta_ {T}}{\eta_ {T - 1}}} \circ \Pi_ {3} ^ {\alpha_ {T - 1} ^ {- 1}} \circ \Pi_ {4} ^ {\alpha_ {T - 1} ^ {- 1}} \circ \Pi_ {2} ^ {\alpha_ {T} ^ {- 1}} \circ \left(\bigotimes_ {t = 1} ^ {T - 1} \left[ \mathbf {G D} _ {t} \circ \Pi_ {2} ^ {\alpha_ {t + 1} ^ {- 1} \alpha_ {t} ^ {- 1}} \circ H _ {t} ^ {\prime} \right]\right) \circ \mathbf {G D} _ {0} \circ \Pi_ {2} ^ {\alpha_ {1} ^ {- 1}} \circ \Pi_ {3} ^ {\alpha_ {1}} \circ \Pi_ {4} ^ {\alpha_ {1}} \circ N \tag {11} \\ \end{array}
+$$
+
+Now we construct the desired sequence of parameters achieved by using the Tapered Exp LR schedule 9 and the additional one-time momentum correction per phase. Let  $(\widetilde{\pmb{\theta}}_0,\widetilde{\eta}_0,\widetilde{\pmb{\theta}}_{-1},\widetilde{\eta}_{-1}) = (\pmb {\theta}_0,\pmb {\eta}_0,\pmb {\theta}_{-1},\pmb {\eta}_0)$ , and
+
+$$
+\begin{array}{l} \left(\widetilde {\boldsymbol {\theta}} _ {1}, \widetilde {\boldsymbol {\eta}} _ {1}, \widetilde {\boldsymbol {\theta}} _ {0}, \widetilde {\boldsymbol {\eta}} _ {0}\right) = \left[ \mathbf {G D} _ {0} \circ \Pi_ {2} ^ {\alpha_ {1} ^ {- 1}} \circ \Pi_ {3} ^ {\alpha_ {1}} \circ \Pi_ {4} ^ {\alpha_ {1}} \circ N \right] \left(\widetilde {\boldsymbol {\theta}} _ {0}, \widetilde {\boldsymbol {\eta}} _ {0}, \widetilde {\boldsymbol {\theta}} _ {- 1}, \widetilde {\boldsymbol {\eta}} _ {- 1}\right) \\ = \left[ \mathbf {G D} _ {0} \circ \Pi_ {2} ^ {\alpha_ {1} ^ {- 1}} \circ \Pi_ {3} ^ {\alpha_ {1}} \circ \Pi_ {4} ^ {\alpha_ {1}} \right] (\widetilde {\boldsymbol {\theta}} _ {0}, \widetilde {\eta} _ {0}, \widetilde {\boldsymbol {\theta}} _ {- 1}, \widetilde {\eta} _ {- 1}); \\ \end{array}
+$$
+
+$$
+\left(\widetilde {\boldsymbol {\theta}} _ {t + 1}, \widetilde {\eta} _ {t + 1}, \widetilde {\boldsymbol {\theta}} _ {t}, \widetilde {\eta} _ {t}\right) = \left[ \mathrm {G D} _ {t} \circ \Pi_ {2} ^ {\alpha_ {t + 1} ^ {- 1} \alpha_ {t} ^ {- 1}} \circ H _ {t} ^ {\prime} \right] \left(\widetilde {\boldsymbol {\theta}} _ {t}, \widetilde {\eta} _ {t}, \widetilde {\boldsymbol {\theta}} _ {t - 1}, \widetilde {\eta} _ {t - 1}\right).
+$$
+
+we claim  $\{\widetilde{\theta}_t\}_{t=0}$  is the desired sequence of parameters. We've already shown that  $\theta_t \sim \widetilde{\theta}_t$ ,  $\forall t$ . Clearly  $\{\widetilde{\theta}_t\}_{t=0}$  is generated using only vanilla GD, scaling LR and modifying the momentum part of the state. When  $t \neq T_I$  for any  $I$ ,  $\eta_t = \eta_{t-1}$  and thus  $H_t' = Id$ . Thus the modification on the momentum could only happen at  $T_I (I \geq 0)$ . Also it's easy to check that  $\alpha_t = \alpha_I^*$ , if  $T_I + 1 \leq t \leq T_{I+1}$ .
+
+# B.5 OMITTED PROOFS OF THEOREM 2.13
+
+Theorem B.6. The following two sequences of parameters,  $\{\pmb{\theta}_t\}_{t=0}^{\infty}$  and  $\{\widetilde{\pmb{\theta}}_t\}_{t=0}^{\infty}$ , define the same sequence of network functions, i.e.  $f_{\pmb{\theta}_t} = f_{\widetilde{\pmb{\theta}}_t}$ ,  $\forall t \in \mathbb{N}$ , given the initial conditions,  $\widetilde{\pmb{\theta}}_0 = P_0\pmb{\theta}_0$ ,  $\widetilde{\pmb{\theta}}_{-1} = P_{-1}\pmb{\theta}_{-1}$ .
+
+1.  $\frac{\pmb{\theta}_t - \pmb{\theta}_{t-1}}{\eta_{t-1}} = \gamma \frac{\pmb{\theta}_{t-1} - \pmb{\theta}_{t-2}}{\eta_{t-2}} - \nabla_{\pmb{\theta}} \left( (L(\pmb{\theta}_{t-1}) + \frac{\lambda_{t-1}}{2} \| \pmb{\theta}_{t-1} \|_2^2 \right)$ , for  $t = 1, 2, \ldots$ ;  
+2.  $\frac{\widetilde{\boldsymbol{\theta}}_t - \widetilde{\boldsymbol{\theta}}_{t - 1}}{\widetilde{\eta}_{t - 1}} = \gamma \frac{\widetilde{\boldsymbol{\theta}}_{t - 1} - \widetilde{\boldsymbol{\theta}}_{t - 2}}{\widetilde{\eta}_{t - 2}} -\nabla_\boldsymbol {\theta}L(\widetilde{\boldsymbol{\theta}}_{t - 1})$ , for  $t = 1,2,\ldots$
+
+where  $\widetilde{\eta}_t = P_tP_{t + 1}\eta_t$ ,  $P_{t} = \prod_{i = -1}^{t}\alpha_{i}^{-1}$ ,  $\forall t\geq -1$  and  $\alpha_{t}$  recursively defined as
+
+$$
+\alpha_ {t} = - \eta_ {t - 1} \lambda_ {t - 1} + 1 + \frac {\eta_ {t - 1}}{\eta_ {t - 2}} \gamma \left(1 - \alpha_ {t - 1} ^ {- 1}\right), \forall t \geq 1. \tag {12}
+$$
+
+needs to be always positive. Here  $\alpha_0, \alpha_{-1}$  are free parameters. Different choice of  $\alpha_0, \alpha_{-1}$  would lead to different trajectory for  $\{\widetilde{\pmb{\theta}}_t\}$ , but the equality that  $\widetilde{\pmb{\theta}}_t = P_t\pmb{\theta}_t$  is always satisfied. If the initial condition is given via  $\pmb{v}_0$ , then it's also free to choose  $\eta_{-1}, \pmb{\theta}_{-1}$ , as long as  $\frac{\pmb{\theta}_0 - \pmb{\theta}_{-1}}{\eta_{-1}} = \pmb{v}_0$ .
+
+Proof of Theorem 2.13. We will prove by induction. By assumption  $S(t): P_t \pmb{\theta}_t = \widetilde{\pmb{\theta}}_t$  for  $t = -1, 0$ . Now we will show that  $S(t) \Longrightarrow S(t + 1), \forall t \geq 0$ .
+
+$$
+\begin{array}{l} \frac {\boldsymbol {\theta} _ {t} - \boldsymbol {\theta} _ {t - 1}}{\eta_ {t - 1}} = \gamma \frac {\boldsymbol {\theta} _ {t - 1} - \boldsymbol {\theta} _ {t - 2}}{\eta_ {t - 2}} - \nabla_ {\boldsymbol {\theta}} \left(\left(L \left(\boldsymbol {\theta} _ {t - 1}\right) + \frac {\lambda_ {t - 1}}{2} \| \boldsymbol {\theta} _ {t - 1} \| _ {2} ^ {2}\right) \right. \\ \xrightarrow {\text {T a k e g r a d i e n t}} \frac {\boldsymbol {\theta} _ {t} - \boldsymbol {\theta} _ {t - 1}}{\eta_ {t - 1}} = \gamma \frac {\boldsymbol {\theta} _ {t - 1} - \boldsymbol {\theta} _ {t - 2}}{\eta_ {t - 2}} - \nabla_ {\boldsymbol {\theta}} L (\boldsymbol {\theta} _ {t - 1}) + \lambda_ {t - 1} \boldsymbol {\theta} _ {t - 1} \\ \xrightarrow {\text {S c a l e I n v a r i a n c e}} \frac {\boldsymbol {\theta} _ {t} - \boldsymbol {\theta} _ {t - 1}}{\eta_ {t - 1}} = \gamma \frac {\boldsymbol {\theta} _ {t - 1} - \boldsymbol {\theta} _ {t - 2}}{\eta_ {t - 2}} - P _ {t - 1} \nabla_ {\boldsymbol {\theta}} L (\widetilde {\boldsymbol {\theta}} _ {t - 1}) + \lambda_ {t - 1} \boldsymbol {\theta} _ {t - 1} \\ \xrightarrow {\text {R e s c a l i n g}} \frac {P _ {t} \left(\boldsymbol {\theta} _ {t} - \boldsymbol {\theta} _ {t - 1}\right)}{P _ {t} P _ {t - 1} \eta_ {t - 1}} = \gamma \frac {P _ {t - 2} \left(\boldsymbol {\theta} _ {t - 1} - \boldsymbol {\theta} _ {t - 2}\right)}{P _ {t - 1} P _ {t - 2} \eta_ {t - 2}} - \nabla_ {\boldsymbol {\theta}} L (\widetilde {\boldsymbol {\theta}} _ {t - 1}) - \lambda_ {t - 1} \frac {\boldsymbol {\theta} _ {t - 1}}{P _ {t - 1}} \\ \xlongequal {\text {S i m p l f y i n g}} \frac {P _ {t} \boldsymbol {\theta} _ {t} - \alpha_ {t} ^ {- 1} \widetilde {\boldsymbol {\theta}} _ {t - 1}}{\widetilde {\eta} _ {t - 1}} = \gamma \frac {\alpha_ {t - 1} \widetilde {\boldsymbol {\theta}} _ {t - 1} - \widetilde {\boldsymbol {\theta}} _ {t - 2}}{\widetilde {\eta} _ {t - 2}} - \nabla_ {\boldsymbol {\theta}} L (\widetilde {\boldsymbol {\theta}} _ {t - 1}) - \eta_ {t - 1} \lambda_ {t - 1} \frac {P _ {t} \boldsymbol {\theta} _ {t - 1}}{\eta_ {t - 1} P _ {t - 1} P _ {t}} \\ \xlongequal {\text {S i m p l f y i n g}} \frac {P _ {t} \boldsymbol {\theta} _ {t} - \alpha_ {t} ^ {- 1} \widetilde {\boldsymbol {\theta}} _ {t - 1}}{\widetilde {\eta} _ {t - 1}} = \gamma \frac {\alpha_ {t - 1} \widetilde {\boldsymbol {\theta}} _ {t - 1} - \widetilde {\boldsymbol {\theta}} _ {t - 2}}{\widetilde {\eta} _ {t - 2}} - \nabla_ {\boldsymbol {\theta}} L (\widetilde {\boldsymbol {\theta}} _ {t - 1}) - \eta_ {t - 1} \lambda_ {t - 1} \frac {\alpha_ {t} ^ {- 1} \widetilde {\boldsymbol {\theta}} _ {t - 1}}{\widetilde {\eta} _ {t - 1}} \\ \xlongequal {\text {S i m p l f y i n g}} \frac {P _ {t} \boldsymbol {\theta} _ {t} - \alpha_ {t} ^ {- 1} \left(1 - \eta_ {t - 1} \lambda_ {t - 1}\right) \widetilde {\boldsymbol {\theta}} _ {t - 1}}{\widetilde {\eta} _ {t - 1}} = \gamma \frac {\alpha_ {t - 1} \widetilde {\boldsymbol {\theta}} _ {t - 1} - \widetilde {\boldsymbol {\theta}} _ {t - 2}}{\widetilde {\eta} _ {t - 2}} - \nabla_ {\boldsymbol {\theta}} L (\widetilde {\boldsymbol {\theta}} _ {t - 1}) \\ \end{array}
+$$
+
+To conclude that  $P_{t}\pmb{\theta}_{t} = \widetilde{\pmb{\theta}}_{t}$ , it suffices to show that the coefficients before  $\widetilde{\pmb{\theta}}_{t - 1}$  is the same to that in (2). In other words, we need to show
+
+$$
+\frac {- 1 + \alpha_ {t} ^ {- 1} (1 - \eta_ {t - 1} \lambda_ {t - 1})}{\widetilde {\eta} _ {t - 1}} = \frac {\gamma (1 - \alpha_ {t - 1})}{\widetilde {\eta} _ {t - 2}},
+$$
+
+which is equivalent to the definition of  $\alpha_{t}$ , Equation 12.
+
+![](images/c7ae9b85d9adb1fef8ef8da3d023f9f04efe62435e9572b17dd8bd84edd2a838.jpg)
+
+Lemma B.7 (Sufficient Conditions for positivity of  $\alpha_{t}$ ). Let  $\lambda_{max} = \max_{t}\lambda_{t},\eta_{max} = \max_{t}\eta_{t}$ . Define  $z_{min}$  is the larger root of the equation  $x^{2} - (1 + \gamma -\lambda_{max}\eta_{max})x + \gamma = 0$ . To guarantee the existence of  $z_{max}$  we also assume  $\eta_{max}\lambda_{max}\leq (1 - \sqrt{\gamma})^2$ . Then we have
+
+$$
+\forall \alpha_ {- 1}, \quad \alpha_ {0} = 1 \Longrightarrow z _ {\min } \leq \alpha_ {t} \leq 1, \forall t \geq 0 \tag {13}
+$$
+
+Proof. We will prove the above theorem with a strengthened induction —
+
+$$
+S (t): \qquad \forall 0 \leq t ^ {\prime} \leq t,   z _ {m i n} \leq \alpha_ {t ^ {\prime}} \leq 1 \bigwedge \frac {\alpha_ {t ^ {\prime}} ^ {- 1} - 1}{\eta_ {t ^ {\prime} - 1}} \leq \frac {z _ {m i n} ^ {- 1} - 1}{\eta_ {m a x}}.
+$$
+
+Since  $\alpha_0 = 1$ ,  $S(0)$  is obviously true. Now suppose  $S(t)$  is true for some  $t \in \mathbb{N}$ , we will prove  $S(t + 1)$ .
+
+First, since  $0 < \alpha_{t} \leq 1$ ,  $\alpha_{t+1} = -\eta_{t}\lambda_{t} + 1 + \frac{\eta_{t}}{\eta_{t-1}}\gamma(1 - \alpha_{t}^{-1}) \leq 1$ .
+
+Again by Equation 12, we have
+
+$$
+1 - \alpha_ {t + 1} = \eta_ {t} \lambda_ {t} + \frac {\alpha_ {t} ^ {- 1} - 1}{\eta_ {t - 1}} \eta_ {t} \gamma = \eta_ {t} \lambda_ {t} + \frac {z _ {m i n} ^ {- 1} - 1}{\eta_ {m a x}} \eta_ {t} \gamma \leq \eta_ {t} \lambda_ {t} + (z _ {m i n} ^ {- 1} - 1) \gamma = 1 - z _ {m i n},
+$$
+
+which shows  $\alpha_{t + 1}\geq z_{min}$ . Here the last step is by definition of  $z_{min}$
+
+Because of  $\alpha_{t + 1}\geq z_{min}$  , we have
+
+$$
+\frac {\alpha_ {t + 1} ^ {- 1} - 1}{\eta_ {t}} \leq z _ {m i n} ^ {- 1} \frac {1 - \alpha_ {t + 1}}{\eta_ {t}} \leq z _ {m i n} ^ {- 1} (\lambda_ {t} + \frac {\alpha_ {t} ^ {- 1} - 1}{\eta_ {t - 1}} \gamma) \leq z _ {m i n} ^ {- 1} (\lambda_ {m a x} + \frac {z _ {m i n} ^ {- 1} - 1}{\eta_ {m a x}} \gamma) = z _ {m i n} ^ {- 1} \frac {1 - z _ {m i n}}{\eta_ {m a x}} = \frac {z _ {m i n} ^ {- 1} - 1}{\eta_ {m a x}}.
+$$
+
+![](images/59ee56980d120fbe8e7e6f41ea95a264a79e564183e9aca9a119d9407c722365.jpg)
+
+Now we are ready to give the formal statement about the closeness of Equation 9 and the reduced LR schedule by Theorem 2.13.
+
+Theorem B.8. Given a Step Decay LR schedule with  $\{T_I\}_{I = 0}^{K - 1},\{\eta_I^*\}_{I = 0}^{K - 1},\{\lambda_I^*\}_{I = 0}^{K - 1}$ , the TEXP++ LR schedule in Theorem 2.13 is the following  $(\alpha_0 = \alpha_{-1} = 1,T_0 = 0)$ :
+
+$$
+\alpha_ {t} = \left\{ \begin{array}{l} - \eta_ {I} ^ {*} \lambda_ {I} ^ {*} + 1 + \gamma (1 - \alpha_ {t - 1} ^ {- 1}), \forall T _ {I} + 2 \leq t \leq T _ {I + 1}, I \geq 0; \\ - \eta_ {I} ^ {*} \lambda_ {I} ^ {*} + 1 + \frac {\eta_ {I} ^ {*}}{\eta_ {I - 1} ^ {*}} \gamma (1 - \alpha_ {t - 1} ^ {- 1}), \forall t = T _ {I} + 1, I \geq 0; \end{array} \right.
+$$
+
+$$
+P _ {t} = \prod_ {i = - 1} ^ {t} \alpha_ {t} ^ {- 1};
+$$
+
+$$
+\hat {\eta} _ {t} = P _ {t} P _ {t + 1} \eta_ {t}.
+$$
+
+It's the same as the TEXP LR schedule( $\{\tilde{\eta}_t\}$ ) in Theorem 2.12 throughout each phase  $I$ , in the sense that
+
+$$
+\left| \frac {\hat {\eta} _ {t - 1}}{\hat {\eta} _ {t}} \Bigg / \frac {\widetilde {\eta} _ {t - 1}}{\widetilde {\eta} _ {t}} - 1 \right| <   3 \frac {\lambda_ {m a x} \eta_ {m a x}}{1 - \gamma} \left(\frac {\gamma}{z _ {m i n} ^ {2}}\right) ^ {t - T _ {I} - 1} \leq 3 \frac {\lambda_ {m a x} \eta_ {m a x}}{1 - \gamma} \left[ \gamma (1 + \frac {\lambda_ {m a x} \eta_ {m a x}}{1 - \gamma}) ^ {2} \right] ^ {(t - T _ {I} - 1)}, \quad \forall T _ {I} + 1 \leq t \leq T _ {I + 1}.
+$$
+
+where  $z_{min}$  is the larger root of  $x^{2} - (1 + \gamma - \lambda_{max}\eta_{max})x + \gamma = 0$ . In Appendix B, we show that  $z_{min}^{-1} \leq 1 + \frac{\eta_{max}\lambda_{max}}{1 - \gamma}$ . When  $\lambda_{max}\eta_{max}$  is small compared to  $1 - \gamma$ , which is usually the case in practice, one could approximate  $z_{min}$  by 1. For example, when  $\gamma = 0.9$ ,  $\lambda_{max} = 0.0005$ ,  $\eta_{max} = 0.1$ , the above upper bound becomes
+
+$$
+\left| \frac {\hat {\eta} _ {t - 1}}{\hat {\eta} _ {t}} \Bigg / \frac {\widetilde {\eta} _ {t - 1}}{\widetilde {\eta} _ {t}} - 1 \right| \leq 0. 0 0 1 5 \times 0. 9 0 0 9 ^ {t - T _ {I} - 1}.
+$$
+
+Proof of Theorem B.8. Assuming  $z_I^1$  and  $z_I^2 (z_I^1 \geq z_I^2)$  are the roots of Equation 1 with  $\eta = \eta_I$  and  $\lambda = \lambda_I$ , we have  $\gamma \leq z_{I'}^2 \leq \sqrt{\gamma} \leq z_{min} \leq z_I^1 \leq 1, \forall I, I' \in [K - 1]$  by Lemma B.1.
+
+We can rewrite the recursion in Theorem 2.13 as the following:
+
+$$
+\alpha_ {t} = - \eta_ {I} \lambda_ {I} + 1 + \gamma \left(1 - \alpha_ {t - 1} ^ {- 1}\right) = - \left(z _ {I} ^ {1} + z _ {I} ^ {2}\right) + z _ {I} ^ {1} z _ {I} ^ {2} \alpha_ {t - 1} ^ {- 1}. \tag {14}
+$$
+
+In other words, we have
+
+$$
+\alpha_ {t} - z _ {I} ^ {1} = \frac {z _ {I} ^ {2}}{\alpha_ {t - 1}} \left(\alpha_ {t - 1} - z _ {I} ^ {1}\right), t \geq 1. \tag {15}
+$$
+
+By Lemma B.7, we have  $\alpha_{t} \geq z_{min}, \forall t \geq 0$ . Thus  $|\frac{\alpha_{t}}{z_{I}^{1}} - 1| = \frac{z_{2}^{I}}{\alpha_{t-1}} |\frac{\alpha_{t-1}}{z_{I}^{1}} - 1| \leq \frac{\gamma}{z_{min}^{2}} |\frac{\alpha_{t-1}}{z_{I}^{1}} - 1| = \frac{\gamma}{z_{min}^{2}} |\frac{\alpha_{t-1}}{z_{I}^{1}} - 1| \leq \gamma (1 + \frac{\lambda \eta}{1-\gamma})^{2} |\frac{\alpha_{t-1}}{z_{I}^{1}}|$ , which means  $\alpha_{t}$  geometrically converges to its stable fixed point  $z_{I}^{1}$ . and  $\frac{\widetilde{\eta}_{t-1}}{\widetilde{\eta}_{t}} = (z_{I}^{1})^{2}$ . Since that  $z_{min} \leq \alpha_{t} \leq 1$ ,  $z_{min} \leq z_{I}^{1} \leq 1$ , we have  $|\frac{\alpha_{T_{I}}}{z_{I}^{1}} - 1| \leq \frac{1-z_{min}}{z_{min}} = \frac{\lambda_{max} \eta_{max}}{1-\gamma} \leq 1$ , and thus  $|\frac{\alpha_{t}}{z_{I}^{1}} - 1| \leq \frac{\lambda_{max} \eta_{max}}{1-\gamma} (\frac{\gamma}{z_{min}^{2}})^{t-T_{I}-1} \leq 1$ ,  $\forall T_{I} + 1 \leq t \leq T_{I+1}$ .
+
+Note that  $\alpha_{I}^{*} = z_{I}^{1}$ ,  $\frac{\hat{\eta}_{t - 1}}{\hat{\eta}_t} = \alpha_t\alpha_{t + 1}$  By definition of TEXP and TEXP++, we have
+
+$$
+\frac {\widetilde {\eta} _ {t - 1}}{\widetilde {\eta} _ {t}} = \left\{ \begin{array}{l l} \left(z _ {I - 1} ^ {1}\right) ^ {2} & \text {i f} T _ {I - 1} + 1 \leq t \leq T _ {I} - 1 \\ \frac {\eta_ {I - 1} ^ {*}}{\eta_ {I} ^ {*}} z _ {I} ^ {1} z _ {I - 1} ^ {1} & \text {i f} t = T _ {I}, I \geq 1 \end{array} \right. \tag {16}
+$$
+
+$$
+\frac {\hat {\eta} _ {t - 1}}{\hat {\eta} _ {t}} = \frac {\eta_ {t - 1}}{\eta_ {t}} \alpha_ {t + 1} \alpha_ {t} = \left\{ \begin{array}{l l} \alpha_ {t + 1} \alpha_ {t} & \text {i f} T _ {I - 1} + 1 \leq t \leq T _ {I} - 1 \\ \frac {\eta_ {I - 1} ^ {*}}{\eta_ {I} ^ {*}} \alpha_ {T _ {I} + 1} \alpha_ {T _ {I}} & \text {i f} t = T _ {I}, I \geq 1 \end{array} \right. \tag {17}
+$$
+
+Thus we have when  $t = T_I$
+
+$$
+\begin{array}{l} \left| \frac {\hat {\eta} _ {t - 1}}{\hat {\eta} _ {t}} \Bigg / \frac {\widetilde {\eta} _ {t - 1}}{\widetilde {\eta} _ {t}} - 1 \right| \leq \left| \frac {\alpha_ {T _ {I} + 1}}{z _ {I} ^ {1}} \frac {\alpha_ {T _ {I}}}{z _ {I - 1} ^ {1}} - 1 \right| \leq \left| \frac {\alpha_ {T _ {I} + 1}}{z _ {I} ^ {1}} - 1 \right| + \left| \frac {\alpha_ {T _ {I}}}{z _ {I - 1} ^ {1}} - 1 \right| + \left| \frac {\alpha_ {T _ {I} + 1}}{z _ {I} ^ {1}} - 1 \right| \left| \frac {\alpha_ {T _ {I}}}{z _ {I - 1} ^ {1}} - 1 \right| \\ \leq 3 \frac {\lambda_ {m a x} \eta_ {m a x}}{1 - \gamma}. \\ \end{array}
+$$
+
+When  $T_{I} + 1 \leq t \leq T_{I + 1}$ , we have
+
+$$
+\begin{array}{l} \left| \frac {\hat {\eta} _ {t - 1}}{\hat {\eta} _ {t}} \Bigg / \frac {\widetilde {\eta} _ {t - 1}}{\widetilde {\eta} _ {t}} - 1 \right| = \left| \frac {\alpha_ {t + 1}}{z _ {I - 1} ^ {1}} \frac {\alpha_ {t}}{z _ {I - 1} ^ {1}} - 1 \right| \leq \left| \frac {\alpha_ {t + 1}}{z _ {I - 1} ^ {1}} - 1 \right| + \left| \frac {\alpha_ {t}}{z _ {I - 1} ^ {1}} - 1 \right| + \left| \frac {\alpha_ {t + 1}}{z _ {I - 1} ^ {1}} - 1 \right| \left| \frac {\alpha_ {t}}{z _ {I - 1} ^ {1}} - 1 \right| \\ \leq 3 \frac {\lambda_ {m a x} \eta_ {m a x}}{1 - \gamma} \left(\frac {\gamma}{z _ {m i n} ^ {2}}\right) ^ {t - T _ {I} - 1}. \\ \end{array}
+$$
+
+Thus we conclude  $\forall I\in [K - 1],T_I + 1\leq t\leq T_{I + 1}$  , we have
+
+$$
+\left| \frac {\hat {\eta} _ {t - 1}}{\hat {\eta} _ {t}} \Bigg / \frac {\widetilde {\eta} _ {t - 1}}{\widetilde {\eta} _ {t}} - 1 \right| \leq 3 \frac {\lambda_ {m a x} \eta_ {m a x}}{1 - \gamma} \left(\frac {\gamma}{z _ {m i n} ^ {2}}\right) ^ {t - T _ {I} - 1} \leq 3 \frac {\lambda_ {m a x} \eta_ {m a x}}{1 - \gamma} \cdot \gamma^ {t - T _ {I} - 1} (1 + \frac {\lambda_ {m a x} \eta_ {m a x}}{1 - \gamma}) ^ {2 (t - T _ {I} - 1)}.
+$$
+
+![](images/551046e3eeaeb665bab698e7f53d2396b457e325ae3c115652ac91eb9bc66776.jpg)
+
+# B.6 OMITTED PROOFS IN SECTION 3
+
+We will use  $\hat{\boldsymbol{w}}$  to denote  $\frac{\boldsymbol{w}}{\|\boldsymbol{w}\|}$  and  $\angle u w$  to  $\arccos (\hat{u}^{\top}\hat{w})$ . Note that training error  $\leq \frac{\varepsilon}{\pi}$  is equivalent to  $\angle e_1\boldsymbol {w}_t < \varepsilon$ .
+
+Case 1: WD alone Since the objective is strongly convex, it has unique argmin  $w^{*}$ . By symmetry,  $w^{*} = \beta e_{1}$ , for some  $\beta > 0$ . By KKT condition, we have
+
+$$
+\lambda \beta = \underset {x _ {1} \sim \mathcal {N} (0, 1)} {\mathbb {E}} \left[ \frac {| x _ {1} |}{1 + \exp (\beta | x _ {1} |)} \right] \leq \underset {x _ {1} \sim \mathcal {N} (0, 1)} {\mathbb {E}} [ | x _ {1} | ] = \sqrt {\frac {2}{\pi}},
+$$
+
+which implies  $\| \pmb{w}^{*}\| = O\left(\frac{1}{\lambda}\right)$
+
+By Theorem 3.1 of Gower et al. (2019), for sufficiently large  $t$ , we have  $\mathbb{E}\left\| \boldsymbol{w}_t - \boldsymbol{w}^*\right\|^2 = O\left(\frac{\eta}{B\lambda}\right)$ . Note that  $\angle e_1\boldsymbol{w}_t = \angle \boldsymbol{w}^*\boldsymbol{w}_t \leq 2\sin \angle \boldsymbol{w}^*\boldsymbol{w}_t \leq 2\frac{\|\boldsymbol{w}^* - \boldsymbol{w}^t\|}{\|\boldsymbol{w}^*\|}$ , we have  $\mathbb{E}\left(\angle e_1\boldsymbol{w}_t\right)^2 = O\left(\frac{\eta\lambda}{B}\right)$ , so the expected error =  $\mathbb{E}\left(\angle e_1\boldsymbol{w}_t\right) / \pi \leq \sqrt{\mathbb{E}\left(\angle e_1\boldsymbol{w}_t\right)^2} / \pi = O\left(\sqrt{\frac{\eta\lambda}{B}}\right)$ .
+
+Case 3: Both BN and WD We will need the following lemma when lower bounding the norm of the stochastic gradient.
+
+Lemma B.9 (Concentration of Chi-Square). Suppose  $X_{1},\ldots ,X_{k}\stackrel {i.i.d}{\sim}\mathcal{N}(0,1)$  , then
+
+$$
+P r \left[ \sum_ {i = 1} ^ {k} X _ {i} ^ {2} <   k \beta \right] \leq \left(\beta e ^ {1 - \beta}\right) ^ {\frac {k}{2}}. \tag {18}
+$$
+
+Proof. This Chernoff-bound based proof is a special case of Dasgupta & Gupta (2003).
+
+$$
+\begin{array}{l} \Pr \left[ \sum_ {i = 1} ^ {k} X _ {i} ^ {2} <   k \beta \right] \leq \left(\beta e ^ {1 - \beta}\right) ^ {\frac {k}{2}} = \Pr \left[ \exp \left(k t \beta - t \sum_ {i = 1} ^ {k} X _ {i} ^ {2}\right) \geq 1 \right] \\ \leq \mathbb {E} \left[ \exp \left(k t \beta - t \sum_ {i = 1} ^ {k} X _ {i} ^ {2}\right) \right] (\text {M a r k o v I n e q u a l i t y}) \tag {19} \\ = e ^ {k t \beta} (1 + 2 t) ^ {- \frac {k}{2}}. \\ \end{array}
+$$
+
+The last equality uses the fact that  $\mathbb{E}\left[tX_i^2\right] = \frac{1}{\sqrt{1 - 2t}}$  for  $t < \frac{1}{2}$ . The proof is completed by taking  $t = \frac{1 - \beta}{2\beta}$ .
+
+Setting for Theorem B.6: Suppose WD factor is  $\lambda$ , LR is  $\eta$ , the width of the last layer is  $m \geq 3$ . Now the SGD updates have the form
+
+$$
+\begin{array}{l} \boldsymbol {w} _ {t + 1} = \boldsymbol {w} _ {t} - \frac {\eta}{B} \sum_ {b = 1} ^ {B} \nabla \left(\ln \left(1 + \exp \left(- \boldsymbol {x} _ {t, b} ^ {\top} \frac {\boldsymbol {w} _ {t}}{\| \boldsymbol {w} _ {t} \|} y _ {t, b}\right) + \frac {\lambda}{2} \| \boldsymbol {w} _ {t} \| ^ {2}\right) \right. \\ = (1 - \lambda \eta) \boldsymbol {w} _ {t} - \frac {\eta}{B} \sum_ {b = 1} ^ {B} \frac {y _ {t , b}}{1 + \exp (\boldsymbol {x} _ {t , b} \top \frac {\boldsymbol {w} _ {t}}{\| \boldsymbol {w} _ {t} \|} y _ {t , b})} \frac {\Pi_ {\boldsymbol {w} _ {t}} ^ {\perp} \boldsymbol {x} _ {t , b}}{\| \boldsymbol {w} _ {t} \|}, \\ \end{array}
+$$
+
+where  $\pmb{x}_{t,b}\stackrel{\mathrm{i.i.d.}}{\sim}\mathcal{N}(0,I_m),y_{t,b} = \mathrm{sign}\left([x_{t,b}]_1\right)$  and  $\Pi_{\pmb{w}_t}^{\perp} = I - \frac{\pmb{w}_t\pmb{w}_t^\top}{\|\pmb{w}_t\|^2}$ .
+
+Proof of Theorem B.6.
+
+Step 1: Let  $T_{1} = \frac{1}{2(\eta\lambda - 2\varepsilon^{2})}\ln \frac{64\|w_{T_{0}}\|^{2}\varepsilon\sqrt{B}}{\eta\sqrt{m - 2}}$ , and  $T_{2} = 9\ln \frac{1}{\delta}$ . Thus if we assume the training error is smaller than  $\varepsilon$  from iteration  $T_{0}$  to  $T_{0} + T_{1} + T_{2}$ , then by spherical triangle inequality,  $\angle w_{t}w_{t'} \leq \angle e_{1}w_{t'} + \angle e_{1}w_{t} = 2\varepsilon$ , for  $T_{0} \leq t, t' \leq T_{0} + T_{1} + T_{2}$ .
+
+Now let's define  $\pmb{w}_t' = (1 - \eta \lambda)\pmb{w}_t$  and for any vector  $\pmb{w}$ , and we have the following two relationships:
+
+1.  $\| \pmb{w}_t^{\prime}\| = (1 - \eta \lambda)\| \pmb {w}\|$  
+2.  $\| \pmb{w}_{t + 1}\| \leq \frac{\|\pmb{w}_t'\|}{\cos 2\varepsilon}.$
+
+The second property is because by Lemma 1.3,  $(\pmb{w}_{t + 1} - \pmb{w}_t') \perp \pmb{w}_t'$  and by assumption of small error,  $\angle \pmb{w}_{t + 1}\pmb{w}_t' \leq 2\varepsilon$ .
+
+Therefore
+
+$$
+\frac {\left\| \boldsymbol {w} _ {T _ {1} + T _ {0}} \right\| ^ {2}}{\left\| \boldsymbol {w} _ {T _ {0}} \right\| ^ {2}} \leq \left(\frac {1 - \eta \lambda}{\cos 2 \varepsilon}\right) ^ {2 T _ {1}} \leq \left(\frac {1 - \eta \lambda}{1 - 2 \varepsilon^ {2}}\right) ^ {2 T _ {1}} \leq \left(1 - \left(\eta \lambda - 2 \varepsilon^ {2}\right)\right) ^ {2 T _ {1}} \leq e ^ {- 2 T _ {1} \left(\eta \lambda - 2 \varepsilon^ {2}\right)} = \frac {\eta}{6 4 \| w _ {T _ {0}} \| ^ {2} \varepsilon} \sqrt {\frac {m - 2}{B}}.
+$$
+
+In other words,  $\| \pmb{w}_{T_0 + T_1} \|^2 \leq \frac{\eta}{64\varepsilon} \sqrt{\frac{m - 2}{B}}$ . Since  $\| \pmb{w}_{T_0 + t} \|$  is monotone decreasing,  $\| \pmb{w}_{T_0 + t} \|^2 \leq \frac{\eta}{64\varepsilon} \sqrt{\frac{m - 2}{B}}$  holds for any  $t = T_1, \ldots, T_1 + T_2$ .
+
+Step 2: We show that the norm of the stochastic gradient is lower bounded with constant probability. In other words, we want to show the norm of  $\pmb{\xi}_t = \sum_{b=1}^{B} \frac{y_{t,b}}{1 + \exp(\pmb{x}_{t,b}^\top \frac{\pmb{w}_t}{\|\pmb{w}_t\|} y_{t,b})} \frac{\Pi_{\pmb{w}_t}^\perp \pmb{x}_{t,b}}{\|\pmb{w}_t\|}$  is lower bounded with high probability.
+
+Let  $\Pi_{\boldsymbol{w}_t, \boldsymbol{e}_1}^{\perp}$  be the projection matrix for the orthogonal space spanned by  $\boldsymbol{w}_t$  and  $e_1$ . W.L.O.G, we can assume the rank of  $\Pi_{\boldsymbol{w}_t, \boldsymbol{e}_1}^{\perp}$  is 2. In case  $\boldsymbol{w}_t = e_1$ , we just exclude a random direction to make  $\Pi_{\boldsymbol{w}_t, \boldsymbol{e}_1}^{\perp}$  rank 2. Now we have  $\Pi_{\boldsymbol{w}_t, \boldsymbol{e}_1}^{\perp} \boldsymbol{x}_{t,b}$  are still i.i.d. multivariate gaussian random variables, for  $b = 1, \ldots, B$ , and moreover,  $\Pi_{\boldsymbol{w}_t, \boldsymbol{e}_1}^{\perp} \boldsymbol{x}_{t,b}$  is independent to  $\frac{y_{t,b}}{1 + \exp(\boldsymbol{x}_{t,b}^{\top} \frac{\boldsymbol{w}_t}{\|\boldsymbol{w}_t\|} y_{t,b})}$ . When  $m \geq 3$ , we can lower bound  $\|\xi_t\|$  by dealing with  $\|\Pi_{\boldsymbol{w}_t, \boldsymbol{e}_1}^{\perp} \boldsymbol{\xi}_t\|$ .
+
+It's not hard to show that conditioned on  $\{\pmb{x}_{t,b}^{\top}\frac{\pmb{w}_t}{\|\pmb{w}_t\|},[\pmb{x}_{t,b}]_1\}_{b = 1}^B$
+
+$$
+\sum_ {b = 1} ^ {B} \frac {y _ {t , b}}{1 + \exp \left(\boldsymbol {x} _ {t , b} ^ {\top} \frac {\boldsymbol {w} _ {t}}{\| \boldsymbol {w} _ {t} \|} y _ {t , b}\right)} \Pi_ {\boldsymbol {w} _ {t}} ^ {\perp} \boldsymbol {x} _ {t, b} \stackrel {{d}} {{=}} \sqrt {\sum_ {b = 1} ^ {B} \left(\frac {y _ {t , b}}{1 + \exp \left(\boldsymbol {x} _ {t , b} ^ {\top} \frac {\boldsymbol {w} _ {t}}{\| \boldsymbol {w} _ {t} \|} y _ {t , b}\right)}\right) ^ {2}} \Pi_ {\boldsymbol {w} _ {t}, \boldsymbol {e} _ {1}} ^ {\perp} \boldsymbol {x}, \tag {21}
+$$
+
+where  $\pmb{x} \sim \mathcal{N}(\mathbf{0}, I_m)$ . We further note that  $\|\Pi_{\pmb{w}_t, \pmb{e}_1}^\perp \pmb{x}\|^2 \sim \chi^2(m - 2)$ . By Lemma B.9,
+
+$$
+\Pr \left[ \| \Pi_ {\boldsymbol {w} _ {t}, \boldsymbol {e} _ {1}} ^ {\perp} \boldsymbol {x} _ {t} \| ^ {2} \geq \frac {m - 2}{8} \right] \geq 1 - \left(\frac {1}{8 e ^ {\frac {7}{8}}}\right) ^ {\frac {m - 2}{2}} \geq 1 - \left(\frac {1}{8 e ^ {\frac {7}{8}}}\right) ^ {\frac {1}{2}} \geq \frac {1}{3}. \tag {22}
+$$
+
+Now we will give a high probability lower bound for  $\sum_{b=1}^{B}\left(\frac{y_{t,b}}{1 + \exp(\boldsymbol{x}_{t,b}^{\top}\frac{\boldsymbol{w}_t}{\|\boldsymbol{w}_t\|}\boldsymbol{y}_{t,b})}\right)^2$ . Note that  $\boldsymbol{x}_t^\top \frac{\boldsymbol{w}_t}{\|\boldsymbol{w}_t\|} \sim \mathcal{N}(0,1)$ , we have
+
+$$
+\Pr \left[ \left| \boldsymbol {x} _ {t, b} ^ {\top} \frac {\boldsymbol {w} _ {t}}{\| \boldsymbol {w} _ {t} \|} \right| <   1 \right] \geq \frac {1}{2}, \tag {23}
+$$
+
+which implies the following, where  $A_{t,b}$  is defined as  $\mathbb{1}\left[|\pmb{x}_{t,b}^{\top}\frac{\pmb{w}_t}{\|\pmb{w}_t\|}| < 1\geq \frac{1}{2}\right]$ :
+
+$$
+\mathbb {E} A _ {t, b} = \Pr \left[ \| \frac {y _ {t , b}}{1 + \exp \left(\boldsymbol {x} _ {t , b} ^ {\top} \frac {\boldsymbol {w} _ {t}}{\| \boldsymbol {w} _ {t} \|} y _ {t}\right)} \| \geq \frac {1}{1 + e} \right] \geq \frac {1}{2}. \tag {24}
+$$
+
+Note that  $\sum_{b=1}^{B} A_{t,b} \leq B$ , and  $\mathbb{E} \sum_{b=1}^{B} A_{t,b} \geq \frac{B}{2}$ , we have  $\Pr \left[\sum_{b=1}^{B} A_{t,b} < \frac{B}{4}\right] \leq \frac{2}{3}$ . Thus,
+
+$$
+\Pr \left[ \sum_ {b = 1} ^ {B} \left(\frac {y _ {t , b}}{1 + \exp \left(\boldsymbol {x} _ {t , b} \top \frac {\boldsymbol {w} _ {t}}{\| \boldsymbol {w} _ {t} \|} y _ {t , b}\right)}\right) ^ {2} \geq \frac {B}{4 (1 + e) ^ {2}} \right] \geq \Pr \left[ \sum_ {b = 1} ^ {B} A _ {t, b} \geq \frac {B}{4} \right] \geq \frac {1}{3}. \tag {25}
+$$
+
+Thus w.p. at least  $\frac{1}{9}$ , equation 25 and equation 22 happen together, which implies
+
+$$
+\| \frac {\eta}{B} \sum_ {b = 1} ^ {B} \nabla \ln (1 + \exp (- \boldsymbol {x} _ {t, b} ^ {\top} \frac {\boldsymbol {w} _ {t}}{\| \boldsymbol {w} _ {t} \|} y _ {t, b})) \| = \| \frac {\eta}{B} \sum_ {b = 1} ^ {B} \frac {y _ {t , b}}{1 + \exp (\boldsymbol {x} _ {t , b} ^ {\top} \frac {\boldsymbol {w} _ {t}}{\| \boldsymbol {w} _ {t} \|} y _ {t})} \frac {\Pi_ {\boldsymbol {w} _ {t}} ^ {\perp} \boldsymbol {x} _ {t , b}}{\| \boldsymbol {w} _ {t} \|} \| \geq \frac {\eta}{1 + e} \frac {\sqrt {m - 2}}{8 \| \boldsymbol {w} _ {t} \|} \geq \frac {\eta}{3 2 \| \boldsymbol {w} _ {t} \|} \sqrt {\frac {m - 2}{B}}
+$$
+
+Step 3. To stay in the cone  $\{\pmb{w}|\angle \pmb{w}\pmb{e}_1\leq \varepsilon \}$ , the SGD update  $\| \pmb{w}_{t + 1} - \pmb{w}_t'\| = \| \frac{\eta}{B}\sum_{b = 1}^{B}\nabla \ln (1 + \exp (-\pmb{x}_{t,b}^\top \frac{\pmb{w}_t}{\|\pmb{w}_t\|} y_{t,b}))\|$  has to be smaller than  $\| \pmb{w}_t\|$  sin  $2\varepsilon$  for any  $t = T_0 + T_1,\ldots ,T_0 + T_1 + T_2$ . However, step 1 and 2 together show that  $\| \nabla \ln (1 + \exp (-\pmb{x}_t^\top \frac{\pmb{w}_t}{\|\pmb{w}_t\|} y_t))\| \geq 2\| \pmb{w}_t\| \varepsilon$  w.p.  $\frac{1}{9}$  per iteration. Thus the probability that  $\pmb{w}_t$  always stays in the cone for every  $t = T_0 + T_1,\dots ,T_0+$ $T_{1} + T_{2}$  is less than  $\left(\frac{8}{9}\right)^{T_2}\leq \delta$ .
+
+Remark B.10. It's interesting that the only property of the global minimum we use is that the if both  $\boldsymbol{w}_t, \boldsymbol{w}_{t+1}$  are  $\epsilon$  optimal, then the angle between  $\boldsymbol{w}_t$  and  $\boldsymbol{w}_{t+1}$  is at most  $2\epsilon$ . Thus we indeed have proved a stronger statement: At least once in every  $\frac{1}{2(\eta\lambda - 2\varepsilon^2)} \ln \frac{64\|w_{T_0}\|^2 \varepsilon \sqrt{B}}{\eta \sqrt{m - 2}} + 9 \ln \frac{1}{\delta}$  iterations, the angle between  $\boldsymbol{w}_t$  and  $\boldsymbol{w}_{t+1}$  will be larger than  $2\epsilon$ . In other words, if the amount of the update stabilizes in terms of angle, then this angle must be larger than  $\sqrt{2\eta\lambda}$  for this simple model.
+
+# B.7 OMITTED PROOFS IN SECTION A
+
+Lemma B.11. Suppose loss  $L$  is scale invariant, then  $L$  is non-convex in the following two sense:
+
+1. The domain is non-convex: scale invariant loss can't be defined at origin;  
+2. There exists no ball containing origin such that the loss is locally convex, unless the loss is constant function.
+
+Proof. Suppose  $L(\pmb{\theta}^{*}) = \sup_{\pmb{\theta} \in B} L(\pmb{\theta})$ . W.L.O.G, we assume  $\| \pmb{\theta}^{*} \| < 1$ . By convexity, every line segment passing  $\pmb{\theta}^{*}$  must have constant loss, which implies the loss is constant over set  $\mathcal{B} - \{c \frac{\pmb{\theta}^{*}}{\|\pmb{\theta}^{*}\|} \mid -1 \leq c \leq 0\}$ . Applying the above argument on any other maximum point  $\pmb{\theta}'$  implies the loss is constant over  $\mathcal{B} - \{0\}$ .
+
+Theorem B.12. Suppose the momentum factor  $\gamma = 0$ , LR  $\eta_t = \eta$  is constant, and the loss function  $L$  is lower bounded. If  $\exists c > 0$  and  $T \geq 0$  such that  $\forall t \geq T$ ,  $f(\pmb{\theta}_{t+1}) - f(\pmb{\theta}_t) \leq -c\eta \| \nabla L(\pmb{\theta}_t) \|^2$ , then  $\lim_{t \to \infty} \| \pmb{\theta}_t \| = 0$ .
+
+Proof in Item 3. By Lemma 1.3 and the update rule of GD with WD, we have
+
+$$
+\left\| \boldsymbol {\theta} _ {t} \right\| ^ {2} = \left\| (1 - \lambda \eta) \left\| \boldsymbol {\theta} \right\| _ {t - 1} + \eta \nabla L (\boldsymbol {\theta} _ {t - 1}) \right\| ^ {2} = (1 - \lambda \eta) ^ {2} \left\| \boldsymbol {\theta} _ {t - 1} \right\| ^ {2} + \eta^ {2} \left\| \nabla L (\boldsymbol {\theta} _ {t - 1}) \right\| ^ {2},
+$$
+
+which implies
+
+$$
+\left\| \boldsymbol {\theta} _ {t} \right\| ^ {2} = \sum_ {i = T} ^ {t - 1} (1 - \lambda \eta) ^ {2 (t - i - 1)} \eta^ {2} \left\| \nabla L \left(\boldsymbol {\theta} _ {t - 1}\right) \right\| ^ {2} + (1 - \lambda \eta) ^ {2 (t - T)} \left\| \boldsymbol {\theta} _ {T} \right\| ^ {2}.
+$$
+
+Thus for any  $T' > T$ ,
+
+$$
+\sum_ {t = T} ^ {T ^ {\prime}} \| \pmb {\theta} _ {t} \| ^ {2} \leq \frac {1}{1 - (1 - \lambda \eta) ^ {2}} \left(\sum_ {t = T} ^ {T ^ {\prime} - 1} \| \nabla L (\pmb {\theta} _ {t}) \| ^ {2} + \| \pmb {\theta} _ {T} \| ^ {2}\right) \leq \frac {1}{\lambda \eta} \left(\sum_ {t = T} ^ {T ^ {\prime} - 1} \| \nabla L (\pmb {\theta} _ {t}) \| ^ {2} + \| \pmb {\theta} _ {T} \| ^ {2}\right).
+$$
+
+Note that by assumption we have  $\sum_{t = T}^{T' - 1}\| \nabla L(\pmb{\theta}_t)\|^2 = \frac{1}{c\eta} f(\pmb{\theta}_T) - f(\pmb{\theta}_{T'})$
+
+As a conclusion, we have  $\sum_{t = T}^{\infty}\| \pmb {\theta}_t\| ^2\leq \frac{f(\pmb{\theta}_T) - \min_\pmb{\theta}f(\pmb{\theta})}{c\eta^2\lambda} +\frac{\|\pmb{\theta}_T\|^2}{\lambda\eta}$ , which implies  $\lim_{t\to \infty}\| \pmb {\theta}_t\| ^2 = 0$
+
+# C OTHER RESULTS
+
+Now we rigorously analyze norm growth in this algorithm. This greatly extends previous analyses of effect of normalization schemes (Wu et al., 2018; Arora et al., 2018) for vanilla SGD.
+
+Theorem C.1. Under the update rule 1.2 with  $\lambda_t = 0$ , the norm of scale invariant parameter  $\theta_t$  satisfies the following property:
+
+- Almost Monotone Increasing:  $\| \pmb{\theta}_{t + 1} \|^2 - \| \pmb{\theta}_t \|^2 \geq -\gamma^{t + 1} \frac{\eta_t}{\eta_0} (\| \pmb{\theta}_0 \|^2 - \| \pmb{\theta}_{-1} \|^2)$ .  
+Assuming  $\eta_t = \eta$  is a constant, then
+
+$$
+\| \pmb {\theta} _ {t + 1} \| ^ {2} = \sum_ {i = 0} ^ {t} \frac {1 - \gamma^ {t - i + 1}}{1 - \gamma} \left(\| \pmb {\theta} _ {i} - \pmb {\theta} _ {i + 1} \| ^ {2} + \gamma \| \pmb {\theta} _ {i - 1} - \pmb {\theta} _ {i} \| ^ {2}\right) - \gamma \frac {1 - \gamma^ {t + 1}}{1 - \gamma} (\| \pmb {\theta} _ {0} \| ^ {2} - \| \pmb {\theta} _ {- 1} \| ^ {2})
+$$
+
+Proof. Let's use  $R_{t},D_{t},C_{t}$  to denote  $\| \pmb {\theta}_t\| ^2,\| \pmb {\theta}_{t + 1} - \pmb {\theta}_t\| ^2,\pmb {\theta}_t^\top (\pmb{\theta}_{t + 1} - \pmb {\theta}_t)$  respectively.
+
+The only property we will use about loss is  $\nabla_{\pmb{\theta}}L_t^\top \pmb{\theta}_t = 0$
+
+Expanding the square of  $\| \pmb{\theta}_{t + 1} \|^2 = \| (\pmb{\theta}_{t + 1} - \pmb{\theta}_t) + \pmb{\theta}_t \|^2$ , we have
+
+$$
+\forall t \geq - 1 \quad S (t): \quad R _ {t + 1} - R _ {t} = D _ {t} + 2 C _ {t}.
+$$
+
+We also have
+
+$$
+\frac {C _ {t}}{\eta_ {t}} = \boldsymbol {\theta} _ {t} ^ {\top} \frac {\boldsymbol {\theta} _ {t + 1} - \boldsymbol {\theta} _ {t}}{\eta_ {t}} = \boldsymbol {\theta} _ {t} ^ {\top} \left(\gamma \frac {\boldsymbol {\theta} _ {t} - \boldsymbol {\theta} _ {t - 1}}{\eta_ {t - 1}} - \lambda_ {t} \boldsymbol {\theta} _ {t}\right) = \frac {\gamma}{\eta_ {t - 1}} \left(D _ {t} + C _ {t - 1}\right) - \lambda_ {t} R _ {t},
+$$
+
+namely,
+
+$$
+\forall t \geq 0 P (t): \frac {C _ {t}}{\eta_ {t}} - \frac {\gamma D _ {t}}{\eta_ {t - 1}} = \frac {\gamma}{\eta_ {t - 1}} C _ {t - 1} - \lambda_ {t} R _ {t}.
+$$
+
+Simplify  $\frac{S(t)}{\eta_t} -\frac{\gamma S(t - 1)}{\eta_{t - 1}} +P(t)$  , we have
+
+$$
+\frac {R _ {t + 1} - R _ {t}}{\eta_ {t}} - \gamma \frac {R _ {t} - R _ {t - 1}}{\eta_ {t - 1}} = \frac {D _ {t}}{\eta_ {t}} + \gamma \frac {D _ {t - 1}}{\eta_ {t - 1}} - 2 \lambda_ {t} R _ {t}. \tag {27}
+$$
+
+When  $\lambda_t = 0$ , we have
+
+$$
+\frac {R _ {t + 1} - R _ {t}}{\eta_ {t}} = \gamma^ {t + 1} \frac {R _ {0} - R _ {- 1}}{\eta_ {- 1}} + \sum_ {i = 0} ^ {t} \gamma^ {t - i} \left(\frac {D _ {i}}{\eta_ {i}} + \gamma \frac {D _ {i - 1}}{\eta_ {i - 1}}\right) \geq \gamma^ {t + 1} \frac {R _ {0} - R _ {- 1}}{\eta_ {0}}.
+$$
+
+Further if  $\eta_t = \eta$  is a constant, we have
+
+$$
+R _ {t + 1} = R _ {0} + \sum_ {i = 0} ^ {t} \frac {1 - \gamma^ {t - i + 1}}{1 - \gamma} (D _ {i} + \gamma D _ {i - 1}) - \gamma \frac {1 - \gamma^ {t + 1}}{1 - \gamma} (R _ {0} - R _ {- 1}),
+$$
+
+![](images/cb648864d84f166cf6f0de0bc9af0073c8554893beec6484bddad97d5c90d190.jpg)  
+Figure 6: The orange line corresponds to PreResNet32 trained with constant LR and WD divided by 10 at epoch 80 and 120. The blue line is TEXP-- corresponding to Step Decay schedule which divides LR by 10 at epoch 80 and 120. They have similar trajectories and performances by a similar argument to Theorem 2.12.(See Theorem B.2 and its proof in Appendix B)
+
+![](images/ca5d5598247135bbdad6f5bc7259ea8562fd870b431135a7082782dcd1e321fa.jpg)
+
+which covers the result without momentum in (Arora et al., 2019) as a special case:
+
+$$
+R _ {t + 1} = R _ {0} + \sum_ {i = 0} ^ {t} D _ {i}.
+$$
+
+![](images/ae470059582ac97e95b8e9f3c0c183fd502cd4cfa9e01b00ab617330a0af61c8.jpg)
+
+For general deep nets, we have the following result, suggesting that the mean square of the update are constant compared to the mean square of the norm. The constant is mainly determined by  $\eta \lambda$ , explaining why the usage of weight decay prevents the parameters to converge in direction.
+
+Theorem C.2. For SGD with constant LR  $\eta$ , weight decay  $\lambda$  and momentum  $\gamma$ , when the limits  $R_{\infty} = \lim_{T \to \infty} \frac{1}{T} \sum_{t=0}^{T-1} \| \boldsymbol{w}_t \|^2$ ,  $D_{\infty} = \lim_{T \to \infty} \frac{1}{T} \sum_{t=0}^{T-1} \| \boldsymbol{w}_{t+1} - \boldsymbol{w}_t \|^2$  exist, we have
+
+$$
+D _ {\infty} = \frac {2 \eta \lambda}{1 + \gamma} R _ {\infty}.
+$$
+
+Proof of Theorem C.2. Take average of Equation 27 over  $t$ , when the limits  $R_{\infty} = \lim_{T \to \infty} \frac{1}{T} \sum_{t=0}^{T-1} \| \boldsymbol{w}_t \|^2$ ,  $D_{\infty} = \lim_{T \to \infty} \frac{1}{T} \sum_{t=0}^{T-1} \| \boldsymbol{w}_{t+1} - \boldsymbol{w}_t \|^2$  exists, we have
+
+$$
+\frac {1 + \gamma}{\eta} D _ {\infty} = 2 \lambda R _ {\infty}.
+$$
+
+![](images/028b04f4354e371a5f5b69d82729161e69f68ef126001c18e7f5c4fd6c6dc545.jpg)
+
+# D ADDITIONAL EXPERIMENTAL FIGURES
+
+# E SCALE INVARIANCE IN MODERN NETWORK ARCHITECTURES
+
+In this section, we will discuss how Normalization layers make the output of the network scale-invariant to its parameters. Viewing a neural network as a DAG, we give a sufficient condition for the scale invariance which could be checked easily by topological order, and apply this on several standard network architectures such as Fully Connected(FC) Networks, Plain CNN, ResNet(He et al., 2016a), and PreResNet(He et al., 2016b). For simplicity, we restrict our discussions among networks with ReLU activation only. Throughout this section, we assume the linear layers and the bias after last normalization layer are fixed to its random initialization, which doesn't harm the performance of the network empirically(Hoffer et al., 2018b).
+
+# E.1 NOTATIONS
+
+Definition E.1 (Degree of Homogeneity). Suppose  $k$  is an integer and  $\theta$  is all the parameters of the network, then  $f$  is said to be homogeneous of degree  $k$ , or  $k$ -homogeneous, if  $\forall c > 0$ ,  $f(c\theta) = c^k f(\theta)$ . The output of  $f$  can be multi-dimensional. Specifically, scale invariance means degree of homogeneity is 0.
+
+Suppose the network only contains following modules, and we list the degree of homogeneity of these basic modules, given the degree of homogeneity of its input.
+
+(I) Input
+
+(L) Linear Layer, e.g. Convolutional Layer or Fully Connected Layer  
+(B) Bias Layer(Adding Trainable Bias to the output of the previous layer)  
+$(+)$  Addition Layer (adding the outputs of two layers with the same dimension $^2$ .)  
+(N) Normalization Layer without affine transformation(including BN, GN, LN, IN etc.)
+
+(NA) Normalization Layer with affine transformation
+
+<table><tr><td>Module</td><td>I</td><td>L</td><td>B</td><td>+</td><td>N</td><td>NA</td></tr><tr><td>Input</td><td>-</td><td>x</td><td>1</td><td>(x,x)</td><td>x</td><td>x</td></tr><tr><td>Output</td><td>0</td><td>x+1</td><td>1</td><td>x</td><td>0</td><td>1</td></tr></table>
+
+Table 1: Table showing how degree of homogeneity of the output of basic modules depends on the degree of homogeneity of the input. For the row of the Input, entry ' -' means the input of the network (I) doesn't have any extra input, entry ' 1' of Bias Layer means if the input is 1-homogeneous then the output is 1- homogeneous.  $\left( {x,x}\right)$  ’ for ‘+’ means if the inputs of Addition Layer have the same degree of homogeneity,the output has the same degree of homogeneity. ReLU, Pooling( and other fixed linear maps) are ignored because they keep the degree of homogeneity and can be omitted when creating the DAG in Theorem E.3.
+
+Remark E.2. For the purpose of deciding the degree of homogeneity of a network, there's no difference among convolutional layers, fully connected layer and the diagonal linear layer in the affine transformation of Normalization layer, since they're all linear and the degree of homogeneity is increased by 1 after applying them.
+
+On the other hand, BN and IN has some benefit which GN and LN doesn't have, namely the bias term (per channel) immediately before BN or IN has zero effect on the network output and thus can be removed. (See Figure 15)
+
+We also demonstrate the homogeneity of the output of the modules via the following figures, which will be reused to later to define network architectures.
+
+Theorem E.3. For a network only consisting of modules defined above and ReLU activation, we can view it as a Directed acyclic graph and check its scale invariance by the following algorithm.
+
+```txt
+Input : DAG  $G = (V,E)$  translated from a neural network; the module type of each node  $v_{i}\in V$    
+1 for  $\mathcal{V}$  in topological order of G do 2 Compute the degree of homogeneity of  $\mathcal{V}$  using Table 1; 3 if  $\mathcal{V}$  is not homogeneous then 4 return False; 5 if  $v_{\text{ouptut}}$  is O-homogeneous then 6 return True; 7 else 8 return False.
+```
+
+2 Addition Layer  $(+)$  is mainly used in ResNet and other similar architectures. In this section, we also use it as an alternative definition of Bias Layer(B). See Figure 7
+
+![](images/9067e61d5afa5c96baff81726ae81c72859879cfbd1076f014a2e5d75422a629.jpg)
+
+![](images/64e18b93742b25782544c8c637933ab17ff32a9a473f13669f757a6596df88b1.jpg)
+
+![](images/eb1f78f534e3b7c5a6faed41da3a7bd830a03ec6654db07bcc1b0faf59304374.jpg)  
+Figure 7: Degree of homogeneity of the output of basic modules given degree of homogeneity of the input.
+
+# E.2 NETWORKS WITHOUT AFFINE TRANSFORMATION AND BIAS
+
+We start with the simple cases where all bias term(including that of linear layer and normalization layer) and the scaling term of normalization layer are fixed to be 0 and 1 element-wise respectively, which means the bias and the scaling could be dropped from the network structure. We empirically find this doesn't affect the performance of network in a noticeable way. We will discuss the full case in the next subsection.
+
+Plain CNN/FC networks: See Figure 8.
+
+![](images/24a4623c3d29b0727252f0e885fad7eb1c9dc8c75db39d0d2dbc6b91c88ba33f.jpg)  
+Figure 8: Degree of homogeneity for all modules in vanilla CNNs/FC networks.
+
+![](images/c6a5d1a704610bc30d29714a6bb2030a71b3eb38fbe592991601f0801765c9c9.jpg)  
+Figure 9: An example of the full network structure of ResNet/PreResNet represented by composite modules defined in Figure 10,11,13,14, where 'S' denotes the starting part of the network, 'Block' denotes a normal block with residual link, 'D-Block' denotes the block with downsampling, and 'N' denotes the normalization layer defined previously. Integer  $x \in \{0,1,2\}$  depends on the type of network. See details in Figure 10,11,13,14.
+
+ResNet: See Figure 10. To ensure the scaling invariance, we add an additional normalization layer in the shortcut after downsampling. This implementation is sometimes used in practice and doesn't affect the performance in a noticeable way.
+
+![](images/ab1b108856c8db498c9df208bf5451e49a0b0090d8271ee480fd0f3052f710bb.jpg)
+
+![](images/6f86c27155a820fbad8e2079f80d0395f3ca55cc5c0019a245655660af002168.jpg)  
+(a) The starting part of ResNet  
+(b) A block of ResNet
+
+![](images/de19cda5fb2d407dcdb26bc01a4ab30750c44213218772cee2b118610e344351.jpg)  
+(c) A block of ResNet with downsampling  
+Figure 10: Degree of homogeneity for all modules in ResNet without affine transformation in normalization layer. The last normalization layer is omitted.
+
+Preactivation ResNet: See Figure 11. Preactivation means to change the order between convolutional layer and normalization layer. For similar reason, we add an additional normalizaiton layer in the shortcut before downsampling.
+
+![](images/fa909a086d3cd4647c53f58ebde166e3c671de758adfa229a8cbb5d4a19c714a.jpg)  
+(a) The starting part of PreResNet
+
+![](images/5d46b330c84b85a6847840595dd0be74903b4b2ed9e16f85ddff1a1a3d07f394.jpg)  
+(b) A block of PreResNet
+
+![](images/029c8fe7f24f89ddb14f4d59471eadff2b7e269e81d499b95a16616f63f9f541.jpg)  
+(c) A block of PreResNet with downsampling  
+Figure 11: Degree of homogeneity for all modules in ResNet without affine transformation in normalization layer. The last normalization layer is omitted.
+
+# E.3 NETWORKS WITH AFFINE TRANSFORMATION
+
+Now we discuss the full case where the affine transformation part of normalization layer is trainable. Due to the reason that the bias of linear layer (before BN) has 0 gradient as we mentioned in E.2, the bias term is usually dropped from network architecture in practice to save memory and accelerate training (even with other normalization methods). (See PyTorch Implementation (Paszke et al., 2017)). However, when LN or GN is used, and the bias term of linear layer is trainable, the network could be scale variant (See Figure 15).
+
+Plain CNN/FC networks: See Figure 12.
+
+![](images/ad7459ff964caf445a61709663bc1797e0bf03370051e082b95749f88c4b5b5b.jpg)  
+Figure 12: Degree of homogeneity for all modules in vanilla CNNs/FC networks.
+
+ResNet: See Figure 13. To ensure the scaling invariance, we add an additional normalization layer in the shortcut after downsampling. This implementation is sometimes used in practice and doesn't affect the performance in a noticeable way.
+
+![](images/602f389cfec5ed4c672bc3ec155187746541b28dcf2bf1a56e16e541b4fbc27d.jpg)  
+(a) The starting part of ResNet
+
+![](images/7259f7f9be5d882cb51cd8c7b5aab86697fabc15a0e4297d99e6fdc66e04993f.jpg)  
+(b) A block of ResNet
+
+![](images/a7cd42e7d46040b951e96d9fcbb7c958df9cbf5b2161e0b65b7db49be6bdb425.jpg)  
+(c) A block of ResNet with downsampling  
+Figure 13: Degree of homogeneity for all modules in ResNet with trainable affine transformation. The last normalization layer is omitted.
+
+Preactivation ResNet: See Figure 14. Preactivation means to change the order between convolutional layer and normalization layer. For similar reason, we add an additional normalization layer in the shortcut before downsampling.
+
+![](images/bbd17e8a374b63d135cc680f91f3faa5024672ac642a833fa5af621aba311a2f.jpg)  
+(a) The starting part of PreResNet
+
+![](images/493dba149eea57e4914232f0d74279639964464a87fafc9fd07b277dde5d724d.jpg)  
+(b) A block of PreResNet
+
+![](images/3893c841bed3607be14551b290f66395058b5d40a09c140fd83f29e3b06df9ed.jpg)  
+(c) A block of PreResNet with downsampling  
+Figure 14: Degree of homogeneity for all modules in PreResNet with trainable affine transformation. The last normalization layer is omitted.
+
+![](images/30bed673c7228994d585f799405758d1d82e079164a54ea2d01e98250f374e5c.jpg)  
+Figure 15: The network can be not scale variant if the GN or IN is used and the bias of linear layer is trainable. The red 'F' means the Algorithm 1 will return False here.
\ No newline at end of file
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+# A PROBABILISTIC FORMULATION OF UNSUPERVISED TEXT STYLE TRANSFER
+
+Junxian He*, Xinyi Wang*, Graham Neubig
+
+Carnegie Mellon University
+
+{junxianh,xinyiw1,gneubig}@cs.cmu.edu
+
+Taylor Berg-Kirkpatrick
+
+University of California San Diego
+
+tberg@eng.ucsd.edu
+
+# ABSTRACT
+
+We present a deep generative model for unsupervised text style transfer that unifies previously proposed non-generative techniques. Our probabilistic approach models non-parallel data from two domains as a partially observed parallel corpus. By hypothesizing a parallel latent sequence that generates each observed sequence, our model learns to transform sequences from one domain to another in a completely unsupervised fashion. In contrast with traditional generative sequence models (e.g. the HMM), our model makes few assumptions about the data it generates: it uses a recurrent language model as a prior and an encoder-decoder as a transduction distribution. While computation of marginal data likelihood is intractable in this model class, we show that amortized variational inference admits a practical surrogate. Further, by drawing connections between our variational objective and other recent unsupervised style transfer and machine translation techniques, we show how our probabilistic view can unify some known non-generative objectives such as backtranslation and adversarial loss. Finally, we demonstrate the effectiveness of our method on a wide range of unsupervised style transfer tasks, including sentiment transfer, formality transfer, word decipherment, author imitation, and related language translation. Across all style transfer tasks, our approach yields substantial gains over state-of-the-art non-generative baselines, including the state-of-the-art unsupervised machine translation techniques that our approach generalizes. Further, we conduct experiments on a standard unsupervised machine translation task and find that our unified approach matches the current state-of-the-art.<sup>1</sup>
+
+# 1 INTRODUCTION
+
+Text sequence transduction systems convert a given text sequence from one domain to another. These techniques can be applied to a wide range of natural language processing applications such as machine translation (Bahdanau et al., 2015), summarization (Rush et al., 2015), and dialogue response generation (Zhao et al., 2017). In many cases, however, parallel corpora for the task at hand are scarce. Therefore, unsupervised sequence transduction methods that require only non-parallel data are appealing and have been receiving growing attention (Bannard & Callison-Burch, 2005; Ravi & Knight, 2011; Mizukami et al., 2015; Shen et al., 2017; Lample et al., 2018; 2019). This trend is most pronounced in the space of text style transfer tasks where parallel data is particularly challenging to obtain (Hu et al., 2017; Shen et al., 2017; Yang et al., 2018). Style transfer has historically referred to sequence transduction problems that modify superficial properties of text - i.e. style rather than content. We focus on a standard suite of style transfer tasks, including formality transfer (Rao & Tetreault, 2018), author imitation (Xu et al., 2012), word decipherment (Shen et al., 2017), sentiment transfer (Shen et al., 2017), and related language translation (Pourdamghani & Knight, 2017). General unsupervised translation has not typically been considered style transfer, but for the purpose of comparison we also conduct evaluation on this task (Lample et al., 2017).
+
+Recent work on unsupervised text style transfer mostly employs non-generative or non-probabilistic modeling approaches. For example, Shen et al. (2017) and Yang et al. (2018) design adversarial discriminators to shape their unsupervised objective - an approach that can be effective, but often introduces training instability. Other work focuses on directly designing unsupervised training objectives by incorporating intuitive loss terms (e.g. backtranslation loss), and demonstrates state-of-the-art performance on unsupervised machine translation (Lample et al., 2018; Artetxe et al., 2019) and style transfer (Lample et al., 2019). However, the space of possible unsupervised objectives is extremely large and the underlying modeling assumptions defined by each objective can only be reasoned about indirectly. As a result, the process of designing such systems is often heuristic.
+
+In contrast, probabilistic models (e.g. the noisy channel model (Shannon, 1948)) define assumptions about data more explicitly and allow us to reason about these assumptions during system design. Further, the corresponding objectives are determined naturally by principles of probabilistic inference, reducing the need for empirical search directly in the space of possible objectives. That said, classical probabilistic models for unsupervised sequence transduction (e.g. the HMM or semi-HMM) typically enforce overly strong independence assumptions about data to make exact inference tractable (Knight et al., 2006; Ravi & Knight, 2011; Pourdamghani & Knight, 2017). This has restricted their development and caused their performance to lag behind unsupervised neural objectives on complex tasks. Luckily, in recent years, powerful variational approximation techniques have made it more practical to train probabilistic models without strong independence assumptions (Miao & Blunsom, 2016; Yin et al., 2018). Inspired by this, we take a new approach to unsupervised style transfer.
+
+We directly define a generative probabilistic model that treats a non-parallel corpus in two domains as a partially observed parallel corpus. Our model makes few independent assumptions and its true posterior is intractable. However, we show that by using amortized variational inference (Kingma & Welling, 2013), a principled probabilistic technique, a natural unsupervised objective falls out of our modeling approach that has many connections with past work, yet is different from all past work in specific ways. In experiments across a suite of unsupervised text style transfer tasks, we find that the natural objective of our model actually outperforms all manually defined unsupervised objectives from past work, supporting the notion that probabilistic principles can be a useful guide even in deep neural systems. Further, in the case of unsupervised machine translation, our model matches the current state-of-the-art non-generative approach.
+
+# 2 UNSUPERVISED TEXT STYLE TRANSFER
+
+We first overview text style transfer, which aims to transfer a text (typically a single sentence or a short paragraph – for simplicity we refer to simply “sentences” below) from one domain to another while preserving underlying content. For example, formality transfer (Rao & Tetreault, 2018) is the task of transforming the tone of text from informal to formal without changing its content. Other examples include sentiment transfer (Shen et al., 2017), word decipherment (Knight et al., 2006), and author imitation (Xu et al., 2012). If parallel examples were available from each domain (i.e. the training data is a bitext consisting of pairs of sentences from each domain), supervised techniques could be used to perform style transfer (e.g. attentional Seq2Seq (Bahdanau et al., 2015) and Transformer (Vaswani et al., 2017)). However, for most style transfer problems, only non-parallel corpora (one corpus from each domain) can be easily collected. Thus, work on style transfer typically focuses on the more difficult unsupervised setting where systems must learn from non-parallel data alone.
+
+The model we propose treats an observed non-parallel text corpus as a partially observed parallel corpus. Thus, we introduce notation for both observed text inputs and those that we will treat as latent variables. Specifically, we let  $X = \{x^{(1)}, x^{(2)}, \dots, x^{(m)}\}$  represent observed data from domain  $\mathcal{D}_1$ , while we let  $Y = \{y^{(m+1)}, y^{(m+2)}, \dots, y^{(n)}\}$  represent observed data from domain  $\mathcal{D}_2$ . Corresponding indices represent parallel sentences. Thus, none of the observed sentences share indices. In our model, we introduce latent sentences to complete the parallel corpus. Specifically,  $\bar{X} = \{\bar{x}^{(m+1)}, \bar{x}^{(m+2)}, \dots, \bar{x}^{(n)}\}$  represents the set of latent parallel sentences in  $\mathcal{D}_1$ , while  $\bar{Y} = \{\bar{y}^{(1)}, \bar{y}^{(2)}, \dots, \bar{y}^{(m)}\}$  represents the set of latent parallel sentences in  $\mathcal{D}_2$ . Then the goal of unsupervised text transduction is to infer these latent variables conditioned the observed non-parallel corpora; that is, to learn  $p(\bar{y} | x)$  and  $p(\bar{x} | y)$ .
+
+![](images/30ae366257b88b9fe9fd6b165524bb20124f9b225e1ca483ad168f2460e47f10.jpg)  
+Figure 1: Proposed graphical model for style transfer via bitext completion. Shaded circles denote the observed variables and unshaded circles denote the latents. The generator is parameterized as an encoder-decoder architecture and the prior on the latent variable is a pretrained language model.
+
+# 3 THE DEEP LATENT SEQUENCE MODEL
+
+First we present our generative model of bitext, which we refer to as a deep latent sequence model. We then describe unsupervised learning and inference techniques for this model class.
+
+# 3.1 MODEL STRUCTURE
+
+Directly modeling  $p(\bar{y} |x)$  and  $p(\bar{x} |y)$  in the unsupervised setting is difficult because we never directly observe parallel data. Instead, we propose a generative model of the complete data that defines a joint likelihood,  $p(X,\bar{X},Y,\bar{Y})$ . In order to perform text transduction, the unobserved halves can be treated as latent variables: they will be marginalized out during learning and inferred via posterior inference at test time.
+
+Our model assumes that each observed sentence is generated from an unobserved parallel sentence in the opposite domain, as depicted in Figure 1. Specifically, each sentence  $x^{(i)}$  in domain  $\mathcal{D}_1$  is generated as follows: First, a latent sentence  $\bar{y}^{(i)}$  in domain  $\mathcal{D}_2$  is sampled from a prior,  $p_{\mathcal{D}_2}(\bar{y}^{(i)})$ . Then,  $x^{(i)}$  is sampled conditioned on  $\bar{y}^{(i)}$  from a transduction model,  $p(x^{(i)}|\bar{y}^{(i)})$ . Similarly, each observed sentence  $y^{(j)}$  in domain  $\mathcal{D}_2$  is generated conditioned on a latent sentence,  $\bar{x}^{(j)}$ , in domain  $\mathcal{D}_1$  via the opposite transduction model,  $p(y^{(j)}|\bar{x}^{(j)})$ , and prior,  $p_{\mathcal{D}_1}(\bar{x}^{(j)})$ . We let  $\theta_{x| \bar{y}}$  and  $\theta_{y| \bar{x}}$  represent the parameters of the two transduction distributions respectively. We assume the prior distributions are pretrained on the observed data in their respective domains and therefore omit their parameters for simplicity of notation. Together, this gives the following joint likelihood:
+
+$$
+p \left(X, \bar {X}, Y, \bar {Y}; \theta_ {x | \bar {y}}, \theta_ {y | \bar {x}}\right) = \left(\prod_ {i = 1} ^ {m} p \left(x ^ {(i)} \mid \bar {y} ^ {(i)}; \theta_ {x | \bar {y}}\right) p _ {\mathcal {D} _ {2}} \left(\bar {y} ^ {(i)}\right)\right) \left(\prod_ {j = m + 1} ^ {n} p \left(y ^ {(j)} \mid \bar {x} ^ {(j)}; \theta_ {y | \bar {x}}\right) p _ {\mathcal {D} _ {1}} \left(\bar {x} ^ {(j)}\right)\right) \tag {1}
+$$
+
+The log marginal likelihood of the data, which we will approximate during training, is:
+
+$$
+\log p (X, Y; \theta_ {x | \bar {y}}, \theta_ {y | \bar {x}}) = \log \sum_ {\bar {X}} \sum_ {\bar {Y}} p (X, \bar {X}, Y, \bar {Y}; \theta_ {x | \bar {y}}, \theta_ {y | \bar {x}}) \tag {2}
+$$
+
+Note that if the two transduction models share no parameters, the training problems for each observed domain are independent. Critically, we introduce parameter sharing through our variational inference procedure, which we describe in more detail in Section 3.2.
+
+Architecture: Since we would like to be able to model a variety of transfer tasks, we choose a parameterization for our transduction distributions that makes no independence assumptions. Specifically, we employ an encoder-decoder architecture based on the standard attentional Seq2Seq model which has been shown to be successful across various tasks (Bahdanau et al., 2015; Rush et al., 2015). Similarly, our prior distributions for each domain are parameterized as recurrent language models which, again, make no independence assumptions. In contrast, traditional unsupervised generative sequence models typically make strong independence assumptions to enable exact inference (e.g. the HMM makes a Markov assumption on the latent sequence and emissions are one-to-one). Our model is more flexible, but exact inference via dynamic programming will be intractable. We address this problem in the next section.
+
+![](images/ea70c00dd49a21d384a7194db94fc3f2840641fefca3e4a40c32c3adb7c6055c.jpg)  
+Figure 2: Depiction of amortized variational approximation. Distributions  $q(\bar{y} |x)$  and  $q(\bar{x} |y)$  represent inference networks that approximate the model's true posterior. Critically, parameters are shared between the generative model and inference networks to tie the learning problems for both domains.
+
+# 3.2 LEARNING
+
+Ideally, learning should directly optimize the log data likelihood, which is the marginal of our model shown in Eq. 2. However, due to our model's neural parameterization which does not factorize, computing the data likelihood cannot be accomplished using dynamic programming as can be done with simpler models like the HMM. To overcome the intractability of computing the true data likelihood, we adopt amortized variational inference (Kingma & Welling, 2013) in order to derive a surrogate objective for learning, the evidence lower bound (ELBO) on log marginal likelihood<sup>3</sup>:
+
+$$
+\begin{array}{l} \log p (X, Y; \theta_ {x | \bar {y}}, \theta_ {y | \bar {x}}) \\ \geq \mathcal {L} _ {\mathrm {E L B O}} (X, Y; \theta_ {x | \bar {y}}, \theta_ {y | \bar {x}}, \phi_ {\bar {x}} [ y, \phi_ {\bar {y} | x}) \\ = \sum_ {i} \left[ \mathbb {E} _ {q (\bar {y} | x ^ {(i)}; \phi_ {\bar {y} | x})} [ \log p (x ^ {(i)} | \bar {y}; \theta_ {x | \bar {y}}) ] - D _ {\mathrm {K L}} \left(q (\bar {y} | x ^ {(i)}; \phi_ {\bar {y} | x}) | | p _ {\mathcal {D} _ {2}} (\bar {y})\right) \right] \tag {3} \\ + \sum_ {j} \underbrace {\left[ \mathbb {E} _ {q (\bar {x} | y ^ {(j)} ; \phi_ {\bar {x} | y})} [ \log p (y ^ {(j)} | \bar {x} ; \theta_ {y | \bar {x}}) ] - \underbrace {D _ {\mathrm {K L}} \left(q (\bar {x} | y ^ {(j)} ; \phi_ {\bar {x} | y}) | | p _ {\mathcal {D} _ {1}} (\bar {x})\right) \right]} _ {\text {R e c o n s t r u c t i o n l i k e l i h o o d}} \\ \end{array}
+$$
+
+The surrogate objective introduces  $q(\bar{y} | x^{(i)}; \phi_{\bar{y} | x})$  and  $q(\bar{x} | y^{(j)}; \phi_{\bar{x} | y})$ , which represent two separate inference network distributions that approximate the model's true posteriors,  $p(\bar{y} | x^{(i)}; \theta_{x | \bar{y}})$  and  $p(\bar{x} | y^{(j)}; \theta_{y | \bar{x}})$ , respectively. Learning operates by jointly optimizing the lower bound over both variational and model parameters. Once trained, the variational posterior distributions can be used directly for style transfer. The KL terms in Eq. 3, that appear naturally in the ELBO objective, can be intuitively viewed as regularizers that use the language model priors to bias the induced sentences towards the desired domains. Amortized variational techniques have been most commonly applied to continuous latent variables, as in the case of the variational autoencoder (VAE) (Kingma & Welling, 2013). Here, we use this approach for inference over discrete sequences, which has been shown to be effective in related work on a semi-supervised task (Miao & Blunsom, 2016).
+
+Inference Network and Parameter Sharing: Note that the approximate posterior on one domain aims to learn the reverse style transfer distribution, which is exactly the goal of the generative distribution in the opposite domain. For example, the inference network  $q(\bar{y} | x^{(i)}; \phi_{\bar{y} | x})$  and the generative distribution  $p(y | \bar{x}^{(i)}; \theta_{y | \bar{x}})$  both aim to transform  $D_1$  to  $D_2$ . Therefore, we use the same architecture for each inference network as used in the transduction models, and tie their parameters:  $\phi_{\bar{x} | y} = \theta_{x | \bar{y}}, \phi_{\bar{y} | x} = \theta_{y | \bar{x}}$ . This means we learn only two encoder-decoders overall – which are parameterized by  $\theta_{x | \bar{y}}$  and  $\theta_{y | \bar{x}}$  respectively – to represent two directions of transfer. In addition to reducing the number of learnable parameters, this parameter tying couples the learning problems for both domains and allows us to jointly learn from the full data. Moreover, inspired by recent work that
+
+builds a universal Seq2Seq model to translate between different language pairs (Johnson et al., 2017), we introduce further parameter tying between the two directions of transduction: the same encoder is employed for both  $x$  and  $y$ , and a domain embedding  $c$  is provided to the same decoder to specify the transfer direction, as shown in Figure 2. Ablation analysis in Section 5.3 suggests that parameter sharing is important to achieve good performance.
+
+Approximating Gradients of ELBO: The reconstruction and KL terms in Eq. 3 still involve intractable expectations due to the marginalization over the latent sequence, thus we need to approximate their gradients. Gumbel-softmax (Jang et al., 2017) and REINFORCE (Sutton et al., 2000) are often used as stochastic gradient estimators in the discrete case. Since the latent text variables have an extremely large domain, we find that REINFORCE-based gradient estimates result in high variance. Thus, we use the Gumbel-softmax straight-through estimator to backpropagate gradients from the KL terms. However, we find that approximating gradients of the reconstruction loss is much more challenging – both the Gumbel-softmax estimator and REINFORCE are unable to outperform a simple stop-gradient method that does not back-propagate the gradient of the latent sequence to the inference network. This confirms a similar observation in previous work on unsupervised machine translation (Lample et al., 2018). Therefore, we use greedy decoding without recording gradients to approximate the reconstruction term. Note that the inference networks still receive gradients from the prior through the KL term, and their parameters are shared with the decoders which do receive gradients from reconstruction. We consider this to be the best empirical compromise at the moment.
+
+Initialization. Good initialization is often necessary for successful optimization of unsupervised learning objectives. In preliminary experiments, we find that the encoder-decoder structure has difficulty generating realistic sentences during the initial stages of training, which usually results in a disastrous local optimum. This is mainly because the encoder-decoder is initialized randomly and there is no direct training signal to specify the desired latent sequence in the unsupervised setting. Therefore, we apply a self-reconstruction loss  $\mathcal{L}_{\mathrm{rec}}$  at the initial epochs of training. We denote the output the encoder as  $e(\cdot)$  and the decoder distribution as  $p_{\mathrm{dec}}$ , then
+
+$$
+\mathcal {L} _ {\mathrm {r e c}} = - \alpha \cdot \sum_ {i} [ p _ {\mathrm {d e c}} (e (x ^ {(i)}, c _ {x}) ] - \alpha \cdot \sum_ {j} [ p _ {\mathrm {d e c}} (e (y ^ {(j)}, c _ {y}) ], \tag {4}
+$$
+
+$\alpha$  decays from 1.0 to 0.0 linearly in the first  $k$  epochs.  $k$  is a tunable parameter and usually less than 3 in all our experiments.
+
+# 4 CONNECTION TO RELATED WORK
+
+Our probabilistic formulation can be connected with recent advances in unsupervised text transduction methods. For example, back translation loss (Sennrich et al., 2016) plays an important role in recent unsupervised machine translation (Artetxe et al., 2018; Lample et al., 2018; Artetxe et al., 2019) and unsupervised style transfer systems (Lample et al., 2019). In order to incorporate back translation loss the source language  $x$  is translated to the target language  $y$  to form a pseudo-parallel corpus, then a translation model from  $y$  to  $x$  can be learned on this pseudo bitext just as in supervised setting. While back translation was often explained as a data augmentation technique, in our probabilistic formulation it appears naturally with the ELBO objective as the reconstruction loss term.
+
+Some previous work has incorporated a pretrained language models into neural semi-supervised or unsupervised objectives. He et al. (2016) uses the log likelihood of a pretrained language model as the reward to update a supervised machine translation system with policy gradient. Artetxe et al. (2019) utilize a similar idea for unsupervised machine translation. Yang et al. (2018) employed a similar approach, but interpret the LM as an adversary, training the generator to fool the LM. We show how our ELBO objective is connected with these more heuristic LM regularizers by expanding the KL loss term (assume  $x$  is observed):
+
+$$
+D _ {\mathrm {K L}} \left(q (\bar {y} | x) \| p _ {\mathcal {D} _ {2}} (\bar {y})\right) = - H _ {q} - \mathbb {E} _ {q} \left[ \log p _ {\mathcal {D} _ {2}} (\bar {y}) \right], \tag {5}
+$$
+
+Note that the loss used in previous work does not include the negative entropy term,  $-H_{q}$ . Our objective results in this additional "regularizer", the negative entropy of the transduction distribution,  $-H_{q}$ . Intuitively,  $-H_{q}$  helps avoid a peaked transduction distribution, preventing the transduction
+
+from constantly generating similar sentences to satisfy the language model. In experiments we will show that this additional regularization is important and helps bypass bad local optima and improve performance. These important differences with past work suggest that a probabilistic view of the unsupervised sequence transduction may provide helpful guidance in determining effective training objectives.
+
+# 5 EXPERIMENTS
+
+We test our model on five style transfer tasks: sentiment transfer, word substitution decipherment, formality transfer, author imitation, and related language translation. For completeness, we also evaluate on the task of general unsupervised machine translation using standard benchmarks.
+
+We compare with the unsupervised machine translation model (UNMT) which recently demonstrated state-of-the-art performance on transfer tasks such as sentiment and gender transfer (Lample et al., 2019). To validate the effect of the negative entropy term in the KL loss term Eq. 5, we remove it and train the model with a back-translation loss plus a language model negative log likelihood loss (which we denote as  $\mathrm{BT + NLL}$ ) as an ablation baseline. For each task, we also include strong baseline numbers from related work if available. For our method we select the model with the best validation ELBO, and for UNMT or  $\mathrm{BT + NLL}$  we select the model with the best back-translation loss. Complete model configurations and hyperparameters can be found in Appendix A.1.
+
+# 5.1 DATASETS AND EXPERIMENT SETUP
+
+Word Substitution Decipherment. Word decipherment aims to uncover the plain text behind a corpus that was enciphered via word substitution where word in the vocabulary is mapped to a unique type in a cipher dictionary (Dou & Knight, 2012; Shen et al., 2017; Yang et al., 2018). In our formulation, the model is presented with a non-parallel corpus of English plaintext and the ciphertext. We use the data in (Yang et al., 2018) which provides 200K sentences from each domain. While previous work (Shen et al., 2017; Yang et al., 2018) controls the difficulty of this task by varying the percentage of words that are ciphered, we directly evaluate on the most difficult version of this task -  $100\%$  of the words are deciphered (i.e. no vocabulary sharing in the two domains). We select the model with the best unsupervised reconstruction loss, and evaluate with BLEU score on the test set which contains 100K parallel sentences. Results are shown in Table 2.
+
+Sentiment Transfer. Sentiment transfer is a task of paraphrasing a sentence with a different sentiment while preserving the original content. Evaluation of sentiment transfer is difficult and is still an open research problem (Mir et al., 2019). Evaluation focuses on three aspects: attribute control, content preservation, and fluency. A successful system needs to perform well with respect to all three aspects. We follow prior work by using three automatic metrics (Yang et al., 2018; Lample et al., 2019): classification accuracy, self-BLEU (BLEU of the output with the original sentence as the reference), and the perplexity (PPL) of each system's output under an external language model. We pretrain a convolutional classifier (Kim, 2014) to assess classification accuracy, and use an LSTM language model pretrained on each domain to compute the PPL of system outputs.
+
+We use the Yelp reviews dataset collected by Shen et al. (2017) which contains 250K negative sentences and 380K positive sentences. We also use a small test set that has 1000 human-annotated parallel sentences introduced in Li et al. (2018). We denote the positive sentiment as domain  $\mathcal{D}_1$  and the negative sentiment as domain  $\mathcal{D}_2$ . We use Self-BLEU and BLEU to represent the BLEU score of the output against the original sentence and the reference respectively. Results are shown in Table 1.
+
+Formality Transfer. Next, we consider a harder task of modifying the formality of a sequence. We use the GYAFC dataset (Rao & Tetreault, 2018), which contains formal and informal sentences from two different domains. In this paper, we use the Entertainment and Music domain, which has about  $52\mathrm{K}$  training sentences, 5K development sentences, and 2.5K test sentences. This dataset actually contains parallel data between formal and informal sentences, which we use only for evaluation. We follow the evaluation of sentiment transfer task and test models on three axes. Since the test set is
+
+Table 1: Results on the sentiment transfer, author imitation, and formality transfer. We list the PPL of pretrained LMs on the test sets of both domains. We only report Self-BLEU on the sentiment task to compare with existing work.  
+
+<table><tr><td>Task</td><td>Model</td><td>Acc.</td><td>BLEU</td><td>Self-BLEU</td><td>PPLD1</td><td>PPLD2</td></tr><tr><td rowspan="7">Sentiment</td><td>Test Set</td><td>-</td><td>-</td><td>-</td><td>31.97</td><td>21.87</td></tr><tr><td>Shen et al. (2017)</td><td>79.50</td><td>6.80</td><td>12.40</td><td>50.40</td><td>52.70</td></tr><tr><td>Hu et al. (2017)</td><td>87.70</td><td>-</td><td>65.60</td><td>115.60</td><td>239.80</td></tr><tr><td>Yang et al. (2018)</td><td>83.30</td><td>13.40</td><td>38.60</td><td>30.30</td><td>42.10</td></tr><tr><td>UNMT</td><td>87.17</td><td>16.99</td><td>44.88</td><td>26.53</td><td>35.72</td></tr><tr><td>BT+NLL</td><td>88.36</td><td>12.36</td><td>31.48</td><td>8.75</td><td>12.82</td></tr><tr><td>Ours</td><td>87.90</td><td>18.67</td><td>48.38</td><td>27.75</td><td>35.61</td></tr><tr><td rowspan="4">Author Imitation</td><td>Test Set</td><td>-</td><td>-</td><td>-</td><td>132.95</td><td>85.25</td></tr><tr><td>UNMT</td><td>80.23</td><td>7.13</td><td>-</td><td>40.11</td><td>39.38</td></tr><tr><td>BT+NLL</td><td>76.98</td><td>10.80</td><td>-</td><td>61.70</td><td>65.51</td></tr><tr><td>Ours</td><td>81.43</td><td>10.81</td><td>-</td><td>49.62</td><td>44.86</td></tr><tr><td rowspan="4">Formality</td><td>Test Set</td><td>-</td><td>-</td><td>-</td><td>71.30</td><td>135.50</td></tr><tr><td>UNMT</td><td>78.06</td><td>16.11</td><td>-</td><td>26.70</td><td>10.38</td></tr><tr><td>BT+NLL</td><td>82.43</td><td>8.57</td><td>-</td><td>6.57</td><td>8.21</td></tr><tr><td>Ours</td><td>80.46</td><td>18.54</td><td>-</td><td>22.65</td><td>17.23</td></tr></table>
+
+a parallel corpus, we only compute reference BLEU and ignore self-BLEU. We use  $\mathcal{D}_1$  to denote formal text, and  $\mathcal{D}_2$  to denote informal text. Results are shown in Table 1.
+
+Author Imitation. Author imitation is the task of paraphrasing a sentence to match another author's style. The dataset we use is a collection of Shakespeare's plays translated line by line into modern English. It was collected by Xu et al. (2012)<sup>7</sup> and used in prior work on supervised style transfer (Jhamtani et al., 2017). This is a parallel corpus and thus we follow the setting in the formality transfer task. We use  $\mathcal{D}_1$  to denote modern English, and  $\mathcal{D}_2$  to denote Shakespeare-style English. Results are shown in Table 1.
+
+Related Language Translation. Next, we test our method on a challenging related language translation task (Pourdamghani & Knight, 2017; Yang et al., 2018). This task is a natural test bed for unsupervised sequence transduction since the goal is to preserve the meaning of the source sentence while rewriting it into the target language. For our experiments, we choose Bosnian (bs) and Serbian (sr) as the related language pairs. We follow Yang et al. (2018) to report BLEU-1 score on this task since BLEU-4 score is close to zero. Results are shown in Table 2.
+
+Unsupervised MT. In order to draw connections with a related work on general unsupervised machine translation, we also evaluate on the WMT'16 German English translation task. This task is substantially more difficult than the style transfer tasks considered so far. We compare with the state-of-the-art UNMT system using the existing implementation from the XLM codebase, $^{8}$  and implement our approach in the same framework with XLM initialization for fair comparison. We train both systems on 5M non-parallel sentences from each language. Results are shown in Table 2.
+
+In Tables 1 we also list the PPL of the test set under the external LM for both the source and target domain. PPL of system outputs should be compared to PPL of the test set itself because extremely low PPL often indicates that the generated sentences are short or trivial.
+
+Table 2: BLEU for decipherment, related language translation (Sr-Bs), and general unsupervised translation (En-De).
+
+<table><tr><td>Model</td><td>Decipher</td><td>Sr-Bs</td><td>Bs-Sr</td><td>En-De</td><td>De-En</td></tr><tr><td>Shen et al. (2017)</td><td>50.8</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Yang et al. (2018)</td><td>49.3</td><td>31.0</td><td>33.4</td><td>-</td><td>-</td></tr><tr><td>UNMT</td><td>76.4</td><td>31.4</td><td>33.4</td><td>26.5</td><td>32.2</td></tr><tr><td>BT+NLL</td><td>78.0</td><td>29.6</td><td>31.4</td><td>-</td><td>-</td></tr><tr><td>Ours</td><td>78.4</td><td>36.2</td><td>38.3</td><td>26.9</td><td>32.0</td></tr></table>
+
+# 5.2 RESULTS
+
+Tables 1 and 2 demonstrate some general trends. First, UNMT is able to outperform other prior methods in unsupervised text style transfer, such as (Yang et al., 2018; Hu et al., 2017; Shen et al., 2017). The performance improvements of UNMT indicate that flexible and powerful
+
+architectures are crucial (prior methods generally do not have an attention mechanism). Second, our model achieves comparable classification accuracy to UNMT but outperforms it in all style transfer tasks in terms of the reference-BLEU, which is the most important metric since it directly measures the quality of the final generations against gold parallel data. This indicates that our method is both effective and consistent across many different tasks. Finally, the BT+NLL baseline is sometimes quite competitive, which indicates that the addition of a language model alone can be beneficial. However, our method consistently outperforms the simple BT+NLL method, which indicates the effectiveness of the additional entropy regularizer in Eq. 5 that is the byproduct of our probabilistic formulation.
+
+Next, we examine the PPL of the system outputs under pretrained domain LMs, which should be evaluated in comparison with the PPL of the test set itself. For both the sentiment transfer and the formality transfer tasks in Table 1, BT+NLL achieves extremely low PPL, lower than the PPL of the test corpus in the target domain. After a close examination of the output, we find that it contains many repeated and overly simple outputs. For example, the system generates many examples of "I love this place" when transferring negative to positive sentiment (see Appendix A.3 for examples). It is not surprising that such a trivial output has low perplexity, high accuracy, and low BLEU score. On the other hand, our system obtains reasonably competitive PPL, and our approach achieves the highest accuracy and higher BLEU score than the UNMT baseline.
+
+# 5.3 FURTHER ABLATIONS AND ANALYSIS
+
+Parameter Sharing. We also conducted an experiment on the word substitution decipherment task, where we remove parameter sharing (as explained in Section 3.2) between two directions of transduction distributions, and optimize two encoder-decoder instead. We found that the model only obtained an extremely low BLEU score and failed to generate any meaningful outputs.
+
+Performance vs. Domain Divergence. Figure 3 plots the relative improvement of our method over UNMT with respect to accuracy of a naive Bayes' classifier trained to predict the domain of test sentences. Tasks with high classification accuracy likely have more divergent domains. We can see that for decipherment and en-de translation, where the domains have
+
+![](images/d0702c63c86b3029cd11695155b137bb4664da782a79b9fb8c013f5e0f54c567.jpg)  
+Figure 3: Improvement over UNMT vs. classification accuracy.
+
+different vocabularies and thus are easily distinguished, our method yields a smaller gain over UNMT. This likely indicates that the (discrimination) regularization effect of the LM priors is less important or necessary when the two domains are very different.
+
+Why does the proposed model outperform UNMT? Finally, we examine in detail the output of our model and UNMT for the author imitation task. We pick this task because the reference outputs for the test set are provided, aiding analysis. Examples shown in Table 3 demonstrate that UNMT tends to make overly large changes to the source so that the original meaning is lost, while our method is better at preserving the content of the source sentence. Next, we quantitatively examine the outputs from UNMT and our method by comparing the F1 measure of words bucketed by their syntactic tags. We use the open-sourced compare-mt tool (Neubig et al., 2019), and the results are shown in Figure 4. Our system has outperforms UNMT in all word categories. In particular, our system is much better at generating nouns, which likely leads to better content preservation.
+
+![](images/de1a551fb93a71c1822b1b01cb59bf1c27babebf1a631a204589adc2d494d018.jpg)  
+Figure 4: Word F1 score by POS tag.
+
+Table 3: Examples for author imitation task  
+
+<table><tr><td>Methods</td><td>Shakespeare to Modern</td></tr><tr><td>Source</td><td>Not to his father&#x27;s .</td></tr><tr><td>Reference</td><td>Not to his father&#x27;s house .</td></tr><tr><td>UNMT</td><td>Not to his brother .</td></tr><tr><td>Ours</td><td>Not to his father&#x27;s house .</td></tr><tr><td>Source</td><td>Send thy man away .</td></tr><tr><td>Reference</td><td>Send your man away .</td></tr><tr><td>UNMT</td><td>Send an excellent word .</td></tr><tr><td>Ours</td><td>Send your man away .</td></tr><tr><td>Source</td><td>Why should you fall into so deep an O ?</td></tr><tr><td>Reference</td><td>Why should you fall into so deep a moan ?</td></tr><tr><td>UNMT</td><td>Why should you carry so nicely , but have your legs ?</td></tr><tr><td>Ours</td><td>Why should you fall into so deep a sin ?</td></tr></table>
+
+Table 4: Comparison of gradient approximation on the sentiment transfer task.  
+
+<table><tr><td>Method</td><td>train ELBO↑</td><td>test ELBO↑</td><td>Acc.</td><td>BLEUr</td><td>BLEUs</td><td>PPLD1</td><td>PPLD2</td></tr><tr><td>Sample-based</td><td>-3.51</td><td>-3.79</td><td>87.90</td><td>13.34</td><td>33.19</td><td>24.55</td><td>25.67</td></tr><tr><td>Greedy</td><td>-2.05</td><td>-2.07</td><td>87.90</td><td>18.67</td><td>48.38</td><td>27.75</td><td>35.61</td></tr></table>
+
+Table 5: Comparison of gradient propagation method on the sentiment transfer task.  
+
+<table><tr><td>Method</td><td>train ELBO↑</td><td>test ELBO↑</td><td>Acc.</td><td>BLEUr</td><td>BLEUs</td><td>PPLD1</td><td>PPLD2</td></tr><tr><td>Gumbel Softmax</td><td>-2.96</td><td>-2.98</td><td>81.30</td><td>16.17</td><td>40.47</td><td>22.70</td><td>23.88</td></tr><tr><td>REINFORCE</td><td>-6.07</td><td>-6.48</td><td>95.10</td><td>4.08</td><td>9.74</td><td>6.31</td><td>4.08</td></tr><tr><td>Stop Gradient</td><td>-2.05</td><td>-2.07</td><td>87.90</td><td>18.67</td><td>48.38</td><td>27.75</td><td>35.61</td></tr></table>
+
+Greedy vs. Sample-based Gradient Approximation. In our experiments, we use greedy decoding from the inference network to approximate the expectation required by ELBO, which is a biased estimator. The main purpose of this approach is to reduce the variance of the gradient estimator during training, especially in the early stages when the variance of sample-based approaches is quite high. As an ablation experiment on the sentiment transfer task we compare greedy and sample-based gradient approximations in terms of both train and test ELBO, as well as task performance corresponding to best test ELBO. After the model is fully trained, we find that the sample-based approximation has low variance. With a single sample, the standard deviation of the EBLO is less than 0.3 across 10 different test repetitions. All final reported ELBO values are all computed with this approach, regardless of whether the greedy approximation was used during training. The reported ELBO values are the evidence lower bound per word. Results are shown in Table 4, where the sampling-based training underperforms on both ELBO and task evaluations.
+
+# 5.4 COMPARISON OF GRADIENT PROPAGATION METHODS
+
+As noted above, to stabilize the training process, we stop gradients from propagating to the inference network from the reconstruction loss. Does this approach indeed better optimize the actual probabilistic objective (i.e. ELBO) or only indirectly lead to improved task evaluations? In this section we use sentiment transfer as an example task to compare different methods for propagating gradients and evaluate both ELBO and task evaluations.
+
+Specifically, we compare three different methods:
+
+- Stop Gradient: The gradients from reconstruction loss are not propagated to the inference network. This is the method we use in all previous experiments.  
+- Gumbel Softmax (Jang et al., 2017): Gradients from the reconstruction loss are propagated to the inference network with the straight-through Gumbel estimator.  
+- REINFORCE (Sutton et al., 2000): Gradients from reconstruction loss are propagated to the inference network with ELBO as a reward function. This method has been used in previous work for semi-supervised sequence generation (Miao & Blunsom, 2016; Yin et al., 2018), but often suffers from instability issues.
+
+We report the train and test ELBO along with task evaluations in Table 5, and plot the learning curves on validation set in Figure 5. While being much simpler, we show that the stop-gradient trick produces superior ELBO over Gumbel Softmax and REINFORCE. This result suggests that stopping gradient helps better optimize the likelihood objective under our probabilistic formulation in comparison with other optimization techniques that propagate gradients, which is counter-intuitive. A likely explanation is that as a gradient estimator, while clearly biased, stop-gradient has substantially reduced variance. In comparison with other techniques that offer reduced bias but extremely high variance when applied to our model class (which involves discrete sequences as latent variables), stop-gradient actually leads to better optimization of our objective because it achieves better balance of bias and variance overall.
+
+![](images/4121c3976c2d9cf2d697705b3f9c3ffd62ad96e0ae8798e53a28752be642a3d7.jpg)  
+Figure 5: ELBO on the validation set v.s. the number training steps.
+
+# 6 CONCLUSION
+
+We propose a probabilistic generative forumalation that unites past work on unsupervised text style transfer. We show that this probabilistic formulation provides a different way to reason about unsupervised objectives in this domain. Our model leads to substantial improvements on five text style transfer tasks, yielding bigger gains when the styles considered are more difficult to distinguish.
+
+# ACKNOWLEDGEMENT
+
+The work of Junxian He and Xinyi Wang is supported by the DARPA GAILA project (award HR00111990063) and the Tang Family Foundation respectively. The authors would like to thank Zichao Yang for helpful feedback about the project.
+
+# REFERENCES
+
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+Mikel Artetxe, Gorka Labaka, and Eneko Agirre. An effective approach to unsupervised machine translation. arXiv preprint arXiv:1902.01313, 2019.  
+Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. Neural machine translation by jointly learning to align and translate. In Proceedings of ICLR, 2015.  
+Colin Bannard and Chris Callison-Burch. Paraphrasing with bilingual parallel corpora. In Proceedings of ACL, 2005.  
+Samuel R Bowman, Luke Vilnis, Oriol Vinyals, Andrew Dai, Rafal Jozefowicz, and Samy Bengio. Generating sentences from a continuous space. In Proceedings of ConNLL, 2016.  
+Qing Dou and Kevin Knight. Dependency-based decipherment for resource-limited machine translation. Proceedings of EMNLP, 2012.  
+Di He, Yingce Xia, Tao Qin, Liwei Wang, Nenghai Yu, Tie-Yan Liu, and Wei-Ying Ma. Dual learning for machine translation. In Proceedings of NeurIPS, 2016.  
+Zhiting Hu, Zichao Yang, Xiaodan Liang, Ruslan Salakhutdinov, and Eric P Xing. Toward controlled generation of text. In Proceedings of ICML, 2017.  
+Eric Jang, Shixiang Gu, and Ben Poole. Categorical reparameterization with gumbel-softmax. In Proceedings of ICLR, 2017.  
+Harsh Jhamtani, Varun Gangal, Edward Hovy, and Eric Nyberg. Shakespearizing modern language using copy-enriched sequence-to-sequence models. Proceedings of EMNLP, 2017.
+
+Melvin Johnson, Mike Schuster, Quoc V Le, Maxim Krikun, Yonghui Wu, Zhifeng Chen, Nikhil Thorat, Fernanda Viégas, Martin Wattenberg, Greg Corrado, et al. Google's multilingual neural machine translation system: Enabling zero-shot translation. Transactions of the Association for Computational Linguistics, 2017.  
+Yoon Kim. Convolutional neural networks for sentence classification. In Proceedings of EMNLP, 2014.  
+Diederik P Kingma and Max Welling. Auto-encoding variational bayes. arXiv preprint arXiv:1312.6114, 2013.  
+Kevin Knight, Anish Nair, Nishit Rathod, and Kenji Yamada. Unsupervised analysis for decipherment problems. In Proceedings of the COLING/ACL 2006 Main Conference Poster Sessions, pp. 499-506, 2006.  
+Guillaume Lample, Alexis Conneau, Ludovic Denoyer, and Marc'Aurelio Ranzato. Unsupervised machine translation using monolingual corpora only. arXiv preprint arXiv:1711.00043, 2017.  
+Guillaume Lample, Myle Ott, Alexis Conneau, Ludovic Denoyer, and Marc'Aurelio Ranzato. Phrase-based & neural unsupervised machine translation. arXiv preprint arXiv:1804.07755, 2018.  
+Guillaume Lample, Sandeep Subramanian, Eric Smith, Ludovic Denoyer, Marc'Aurelio Ranzato, and Y-Lan Boureau. Multiple-attribute text rewriting. In Proceedings of ICLR, 2019.  
+Juncen Li, Robin Jia, He He, and Percy Liang. Delete, retrieve, generate: A simple approach to sentiment and style transfer. arXiv preprint arXiv:1804.06437, 2018.  
+Yishu Miao and Phil Blunsom. Language as a latent variable: Discrete generative models for sentence compression. In Proceedings of EMNLP, 2016.  
+Ronen Mir, Bjarke Felbo, Nick Obradovich, and Iyad Rahwan. Evaluating style transfer for text. In Proceedings of NAACL, 2019.  
+Masahiro Mizukami, Graham Neubig, Sakriani Sakti, Tomoki Toda, and Satoshi Nakamura. Linguistic individuality transformation for spoken language. In *Natural Language Dialog Systems and Intelligent Assistants*. 2015.  
+Graham Neubig, Zi-Yi Dou, Junjie Hu, Paul Michel, Danish Pruthi, and Xinyi Wang. compare-mt: A tool for holistic comparison of language generation systems. In *Meeting of the North American Chapter of the Association for Computational Linguistics (NAACL) Demo Track*, Minneapolis, USA, June 2019. URL http://arxiv.org/abs/1903.07926.  
+Nima Pourdamghani and Kevin Knight. Deciphering related languages. Proceedings of EMNLP, 2017.  
+Sudha Rao and Joel Tetreault. Dear sir or madam, may i introduce the gyafc dataset: Corpus, benchmarks and metrics for formality style transfer. arXiv preprint arXiv:1803.06535, 2018.  
+Sujith Ravi and Kevin Knight. Deciphering foreign language. In Proceedings of ACL, 2011.  
+Alexander M Rush, Sumit Chopra, and Jason Weston. A neural attention model for abstractive sentence summarization. In Proceedings of EMNLP, 2015.  
+Rico Sennrich, Barry Haddow, and Alexandra Birch. Improving neural machine translation models with monolingual data. In Proceedings of ACL, 2016.  
+Claude Elwood Shannon. A mathematical theory of communication. Bell system technical journal, 27(3):379-423, 1948.  
+Tianxiao Shen, Tao Lei, Regina Barzilay, and Tommi Jaakkola. Style transfer from non-parallel text by cross-alignment. In Proceedings of NIPS, 2017.  
+Richard S Sutton, David A McAllester, Satinder P Singh, and Yishay Mansour. Policy gradient methods for reinforcement learning with function approximation. In Proceedings of NeurIPS, 2000.
+
+Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Proceedings of NeurIPS, 2017.  
+Wei Xu, Alan Ritter, William B. Dolan, Ralph Grishman, and Cherry Colin. Paraphrasing for style. COLING, 2012.  
+Zichao Yang, Zhiting Hu, Chris Dyer, Eric P Xing, and Taylor Berg-Kirkpatrick. Unsupervised text style transfer using language models as discriminators. In Proceedings of NeurIPS, 2018.  
+Pengcheng Yin, Chunting Zhou, Junxian He, and Graham Neubig. Structvae: Tree-structured latent variable models for semi-supervised semantic parsing. In Proceedings of ACL, 2018.  
+Tiancheng Zhao, Ran Zhao, and Maxine Eskenazi. Learning discourse-level diversity for neural dialog models using conditional variational autoencoders. In Proceedings of ACL, 2017.
+
+# A APPENDIX
+
+# A.1 MODEL CONFIGURATIONS.
+
+We adopt the following attentional encoder-decoder architecture for UNMT, BT+NLL, and our method across all the experiments:
+
+- We use word embeddings of size 128.  
+- We use 1 layer LSTM with hidden size of 512 as both the encoder and decoder.  
+- We apply dropout to the readout states before softmax with a rate of 0.3.
+
+- Following Lample et al. (2019), we add a max pooling operation over the encoder hidden states before feeding it to the decoder. Intuitively the pooling window size would control how much information is preserved during transduction. A window size of 1 is equivalent to standard attention mechanism, and a large window size corresponds to no attention. See Appendix A.2 for how to select the window size.  
+- There is a noise function for UNMT baseline in its denoising autoencoder loss (Lample et al., 2017; 2019), which is critical for its success. We use the default noise function and noise hyperparameters in Lample et al. (2017) when running the UNMT model. For BT+NLL and our method we found that adding the extra noise into the self-reconstruction loss (Eq. 4) is only helpful when the two domains are relatively divergent (decipherment and related language translation tasks) where the language models play a less important role. Therefore, we add the default noise from UNMT to Eq. 4 for decipherment and related language translation tasks only, and do not use any noise for sentiment, author imitation, and formality tasks.
+
+# A.2 HYPERPARAMETER TUNING.
+
+We vary pooling windows size as  $\{1,5\}$ , the decaying patience hyperparameter  $k$  for self-reconstruction loss (Eq. 4) as  $\{1,2,3\}$ . For the baseliens UNMT and BT+NLL, we also try the option of not annealing the self-reconstruction loss at all as in the unsupervised machine translation task (Lample et al., 2018). We vary the weight  $\lambda$  for the NLL term (BT+NLL) or the KL term (our method) as  $\{0.001,0.01,0.03,0.05,0.1\}$ .
+
+# A.3 SENTIMENT TRANSFER EXAMPLE OUTPUTS
+
+We list some examples of the sentiment transfer task in Table 6. Notably, the BT+NLL method tends to produce extremely short and simple sentences.
+
+# A.4 REPETITIVE EXAMPLES OF BT+NLL
+
+In Section 5 we mentioned that the baseline  $\mathrm{BT + NLL}$  has a low perplexity for some tasks because it tends to generate overly simple and repetitive sentences. From Table 1 we see that two representative tasks are sentiment transfer and formatily transfer. In Appendix A.3 we have demonstrated some examples for sentiment transfer, next we show some repetitive samples of  $\mathrm{BT + NLL}$  in Table 7.
+
+Table 6: Random Sentiment Transfer Examples  
+
+<table><tr><td>Methods</td><td>negative to positive</td></tr><tr><td>Original</td><td>the cake portion was extremely light and a bit dry .</td></tr><tr><td>UNMT</td><td>the cake portion was extremely light and a bit spicy .</td></tr><tr><td>BT+NLL</td><td>the cake portion was extremely light and a bit dry .</td></tr><tr><td>Ours</td><td>the cake portion was extremely light and a bit fresh .</td></tr><tr><td>Original</td><td>the “ chicken ” strip were paper thin oddly flavored strips .</td></tr><tr><td>UNMT</td><td>the “ chicken ” were extra crispy noodles were fresh and incredible .</td></tr><tr><td>BT+NLL</td><td>the service was great .</td></tr><tr><td>Ours</td><td>the “ chicken ” strip were paper sweet &amp; juicy flavored .</td></tr><tr><td>Original</td><td>if i could give them a zero star review i would !</td></tr><tr><td>UNMT</td><td>if i could give them a zero star review i would !</td></tr><tr><td>BT+NLL</td><td>i love this place .</td></tr><tr><td>Ours</td><td>i love the restaurant and give a great review i would !</td></tr><tr><td></td><td>positive to negative</td></tr><tr><td>Original</td><td>great food , staff is unbelievably nice .</td></tr><tr><td>UNMT</td><td>no , food is n’t particularly friendly .</td></tr><tr><td>BT+NLL</td><td>i will not be back .</td></tr><tr><td>Ours</td><td>no apologies , staff is unbelievably poor .</td></tr><tr><td>Original</td><td>my wife and i love coming here !</td></tr><tr><td>UNMT</td><td>my wife and i do n’t come here !</td></tr><tr><td>BT+NLL</td><td>i will not be back .</td></tr><tr><td>Ours</td><td>my wife and i walked out the last time .</td></tr><tr><td>Original</td><td>my wife and i love coming here !</td></tr><tr><td>UNMT</td><td>my wife and i do n’t come here !</td></tr><tr><td>BT+NLL</td><td>i will not be back .</td></tr><tr><td>Ours</td><td>my wife and i walked out the last time .</td></tr><tr><td>Original</td><td>the premier hookah lounge of las vegas !</td></tr><tr><td>UNMT</td><td>the worst museum of las vegas !</td></tr><tr><td>BT+NLL</td><td>the worst frame shop of las vegas !</td></tr><tr><td>Ours</td><td>the hallways scam lounge of las vegas !</td></tr></table>
+
+Table 7: Repetitive examples of BT+NLL baseline on Formality transfer.  
+
+<table><tr><td>Original</td><td>Transferred</td></tr><tr><td colspan="2">formal to informal</td></tr><tr><td>I like Rhythm and Blue music .</td><td>I like her and I don’t know .</td></tr><tr><td>There’s nothing he needs to change .</td><td>I don’t know , but I don’t know .</td></tr><tr><td>I enjoy watching my companion attempt to role @-@ play with them .</td><td>I don’t know , but I don’t know .</td></tr><tr><td>I am watching it right now .</td><td>I don’t know , but I don’t know .</td></tr><tr><td>That is the key point , that you fell asleep .</td><td>I don’t know , but I don’t know .</td></tr><tr><td colspan="2">informal to formal</td></tr><tr><td>its a great source just download it .</td><td>I do not know , but I do not know .</td></tr><tr><td>Happy Days , it was the coolest !</td><td>I do not know , but I do not know .</td></tr><tr><td>I used to play flute but once I started sax , I got hooked .</td><td>I do not know , but I do not know .</td></tr><tr><td>The word you are looking for is ...... strengths</td><td>The word you are looking for is : )</td></tr><tr><td>Plus you can tell she really cared about her crew .</td><td>Plus you can tell she really cared about her crew .</td></tr></table>
\ No newline at end of file
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+# A SIGNAL PROPAGATION PERSPECTIVE FOR PRUNING NEURAL NETWORKS AT INITIALIZATION
+
+Namhoon Lee $^{1}$ , Thalaiyasingam Ajanthan $^{2}$ , Stephen Gould $^{2}$ , Philip H. S. Torr $^{1}$
+
+<sup>1</sup>University of Oxford
+
+$^{2}$ Australian National University
+
+1{namhoon,phst}@robots.ox.ac.uk
+
+2{thalaiyasingam.ajanthan,stephen.gould}@anu.edu.au
+
+# ABSTRACT
+
+Network pruning is a promising avenue for compressing deep neural networks. A typical approach to pruning starts by training a model and then removing redundant parameters while minimizing the impact on what is learned. Alternatively, a recent approach shows that pruning can be done at initialization prior to training, based on a saliency criterion called connection sensitivity. However, it remains unclear exactly why pruning an untrained, randomly initialized neural network is effective. In this work, by noting connection sensitivity as a form of gradient, we formally characterize initialization conditions to ensure reliable connection sensitivity measurements, which in turn yields effective pruning results. Moreover, we analyze the signal propagation properties of the resulting pruned networks and introduce a simple, data-free method to improve their trainability. Our modifications to the existing pruning at initialization method lead to improved results on all tested network models for image classification tasks. Furthermore, we empirically study the effect of supervision for pruning and demonstrate that our signal propagation perspective, combined with unsupervised pruning, can be useful in various scenarios where pruning is applied to non-standard arbitrarily-designed architectures.
+
+# 1 INTRODUCTION
+
+Deep learning has made great strides in machine learning and been applied to various fields from computer vision and natural language processing, to health care and playing games (LeCun et al., 2015). Despite the immense success, however, it remains challenging to deal with the excessive computational and memory requirements of large neural network models. To this end, lightweight models are often preferred, and network pruning, a technique to reduce parameters in a network, has been widely employed to compress deep neural networks (Han et al., 2016). Nonetheless, designing pruning algorithms has been often purely based on ad-hoc intuition lacking rigorous underpinning, partly because pruning was typically carried out after training the model as a post-processing step or interwoven with the training procedure, without adequate tools to analyze.
+
+Recently, Lee et al. (2019) have shown that pruning can be done on randomly initialized neural networks in a single-shot prior to training (i.e., pruning at initialization). They empirically showed that as long as the initial random weights are drawn from appropriately scaled Gaussians (e.g., Glorot & Bengio (2010)), their pruning criterion called connection sensitivity can be used to prune deep neural networks, often to an extreme level of sparsity while maintaining good accuracy once trained. However, it remains unclear as to why pruning at initialization is effective, how it should be understood theoretically and whether it can be extended further.
+
+In this work, we first look into the effect of initialization on pruning, and find that initial weights have critical impact on connection sensitivity, and therefore, pruning results. Deeper investigation shows that connection sensitivity is determined by an interplay between gradients and weights. Therefore when the initial weights are not chosen appropriately, the propagation of input signals into layers of
+
+these random weights can result in saturating error signals (i.e., gradients) under backpropagation, and hence unreliable connection sensitivity, potentially leading to a catastrophic pruning failure.
+
+This result leads us to develop a signal propagation perspective for pruning at initialization, and to provide a formal characterization of how a network needs to be initialized for reliable connection sensitivity measurements and in turn effective pruning. Precisely, we show that a sufficient condition to ensure faithful<sup>1</sup> connection sensitivity is layerwise dynamical isometry, which is defined as all singular values of the layerwise Jacobians being concentrated around 1. Our signal propagation perspective is inspired by the recent literature on dynamical isometry and mean field theory (Saxe et al., 2014; Poole et al., 2016; Schoenholz et al., 2017; Pennington et al., 2017), in which the general signal propagation in neural networks is studied. We extend this result to understanding and improving pruning at initialization.
+
+Moreover, we study signal propagation in the pruned sparse networks and its effect on trainability. We find that pruning neural networks can indeed break dynamical isometry, and hence, hinders signal propagation and degrades the training performance of the resulting sparse network. In order to address this issue, we propose a simple, yet effective data-free method to recover the layerwise orthogonality given the sparse topology, which in turn improves the training performance of the compressed network significantly. Our analysis further reveals that in addition to signal propagation, the choice of pruning method and sparsity level can influence trainability in sparse neural networks.
+
+Perfect layerwise dynamical isometry cannot always be ensured in the modern networks that have components such as ReLU nonlinearities (Pennington et al., 2017) and/or batch normalization (Yang et al., 2019). Even in such cases, however, our experiments on various modern architectures (including convolutional and residual neural networks) indicate that connection sensitivity computed based on layerwise dynamical isometry is robust and consistently outperforms pruning based on other initialization schemes. This indicates that the signal propagation perspective is not only important to theoretically understand pruning at initialization, but also it improves the results of pruning for a range of networks of practical interest.
+
+Furthermore, this signal propagation perspective for pruning poses another important question: how informative is the error signal computed on randomly initialized networks, or can we prune neural networks even without supervision? To understand this, we compute connection sensitivity scores with different unsupervised surrogate losses and evaluate the pruning results. Interestingly, our results indicate that we can in fact prune networks in an unsupervised manner to extreme sparsity levels without compromising accuracy, and it often compares competitively to pruning with supervision. Moreover, we test if pruning at initialization can be extended to obtain architectures that yield better performance than standard pre-designed architectures with the same number of parameters. In fact, this process, which we call neural architecture sculpting, compares favorably against hand-designed architectures, taking network pruning one step further towards neural architecture search.
+
+# 2 PRELIMINARIES
+
+Pruning at initialization. The principle behind conventional approaches for network pruning is to find unnecessary parameters, such that by eliminating them the complexity of the model is reduced while minimizing the impact on what is learned (Reed, 1993). Naturally, a typical pruning algorithm starts after convergence to a minimum or training to some degree. This pretraining requirement has been left unattended until Lee et al. (2019) recently showed that pruning can be performed on untrained networks at initialization prior to training. They proposed a method called SNIP which relies on a new saliency criterion, namely connection sensitivity, defined as follows:
+
+$$
+s _ {j} (\mathbf {w}; \mathcal {D}) = \frac {\left| g _ {j} (\mathbf {w} ; \mathcal {D}) \right|}{\sum_ {k = 1} ^ {m} \left| g _ {k} (\mathbf {w} ; \mathcal {D}) \right|}, \quad \text {w h e r e} \quad g _ {j} (\mathbf {w}; \mathcal {D}) = \left. \frac {\partial L (\mathbf {c} \odot \mathbf {w} ; \mathcal {D})}{\partial c _ {j}} \right| _ {\mathbf {c} = \mathbf {1}}. \tag {1}
+$$
+
+Here,  $s_j$  is the saliency of the parameter  $j$ ,  $\mathbf{w} \in \mathbb{R}^m$  is the network parameters,  $\mathbf{c} \in \{0,1\}^m$  is the auxiliary indicator variables representing the connectivity of network parameters,  $m$  is the total number of parameters in the network, and  $\mathcal{D}$  is a given dataset. Also,  $g_j$  is the derivative of the loss  $L$  with respect to  $c_j$ , which turns out to be an infinitesimal approximation of the change in the
+
+loss with respect to removing the parameter  $j$ . Designed to be computed at initialization, pruning is performed by keeping top-  $\kappa$  (where  $\kappa$  denotes a desired sparsity level) salient parameters based on the above sensitivity scores.
+
+Dynamical isometry and mean field theory. The success of training deep neural networks is due in large part to the initial weights (Hinton & Salakhutdinov, 2006; Glorot & Bengio, 2010; Pascanu et al., 2013). In essence, the principle behind these random weight initializations is to have the mean squared singular value of a network's input-output Jacobian close to 1, so that on average, an error vector will preserve its norm under backpropagation; however, this is not sufficient to prevent amplification or attenuation of an error vector on worst case. A stronger condition that having as many singular values as possible near 1 is called dynamical isometry (Saxe et al., 2014). Under this condition, error signals backpropagate isometrically through the network, approximately preserving its norm and all angles between error vectors. Alongside dynamical isometry, mean field theory is used to develop a theoretical understanding of signal propagation in neural networks with random parameters (Poole et al., 2016). Precisely, the mean field approximation states that preactivations of wide, untrained neural networks can be captured as a Gaussian distribution. Recent works revealed a maximum depth through which signals can propagate at initialization, and verified that networks are trainable when signals can travel all the way through them (Schoenholz et al., 2017; Yang & Schoenholz, 2017; Xiao et al., 2018).
+
+# 3 SIGNAL PROPAGATION PERSPECTIVE TO PRUNING RANDOM NETWORKS
+
+Problem setup. Consider a fully-connected, feed-forward neural network with weight matrices  $\mathbf{W}^l\in \mathbb{R}^{N\times N}$ , biases  $\mathbf{b}^l\in \mathbb{R}^N$ , pre-activations  $\mathbf{h}^l\in \mathbb{R}^N$ , and post-activations  $\mathbf{x}^l\in \mathbb{R}^N$ , for  $l\in \{1\dots K\}$  up to  $K$  layers. Now, the feed-forward dynamics of a network can be written as,
+
+$$
+\mathbf {x} ^ {l} = \phi \left(\mathbf {h} ^ {l}\right), \quad \mathbf {h} ^ {l} = \mathbf {W} ^ {l} \mathbf {x} ^ {l - 1} + \mathbf {b} ^ {l}, \tag {2}
+$$
+
+where  $\phi : \mathbb{R} \to \mathbb{R}$  is an elementwise nonlinearity, and the input is denoted by  $\mathbf{x}^0$ . Given the network configuration, the parameters are initialized by sampling from a probability distribution, typically a zero mean Gaussian with scaled variance (LeCun et al., 1998; Glorot & Bengio, 2010).
+
+# 3.1 EFFECT OF INITIALIZATION ON PRUNING
+
+It is observed in Lee et al. (2019) that pruning results tend to improve when initial weights are drawn from a scaled Gaussian, or so-called variance scaling initialization (LeCun et al., 1998; Glorot & Bengio, 2010; He et al., 2015). As we wish to better understand the role of these random initial weights in pruning, we will examine the effect of varying initialization on the pruning results.
+
+In essence, variance scaling schemes introduce normalization factors to adjust the variance  $\sigma$  of the weight sampling distribution, which can be summarized as  $\sigma \rightarrow \frac{\alpha}{\psi_l}\sigma$ , where  $\psi_l$  is a layerwise scalar that depends on an architecture specification such as the number of output neurons in the previous layer (e.g., fan-in), and  $\alpha$  is a global scalar throughout the network. Notice in case of a network with layers of the same width, the variance can be controlled by a single scalar  $\gamma = \frac{\alpha}{\psi}$  as  $\psi_l = \psi$  for all layers  $l$ . In particular, we take both linear and tanh multilayer perceptron networks (MLP) of layers  $K = 7$  and width  $N = 100$  on MNIST with  $\sigma = 1$  as the default, similar to Saxe et al. (2014). We initialize these networks with different  $\gamma$ , compute the connection sensitivity, prune it, and then visualize layerwise the resulting sparsity patterns  $\mathbf{c}$  as well as the corresponding connection sensitivity used for pruning in Figure 1.
+
+It is seen in the sparsity patterns that for the tanh network, unlike the linear case, more parameters tend to be pruned in the later layers than the earlier layers. As a result, this limits the learning capability of the subnetwork critically when a high sparsity level is requested; e.g., for  $\bar{\kappa} = 90\%$ , only a few parameters in later layers are retained after pruning. This is explained by the connection sensitivity plot. The sensitivity of parameters in the nonlinear network tends to decrease towards the later layers, and therefore, choosing the top- $\kappa$  parameters globally based on the sensitivity scores results in a subnetwork in which retained parameters are distributed highly non-uniformly and sparsely towards the end of the network. This result implies that the initial weights have a crucial effect on the connection sensitivity, and from there, the pruning results.
+
+![](images/2bd44a486af4c4a522d2676153835040a6683f21637587ccc242dda01ab6fdae.jpg)  
+Figure 1: (left) layerwise sparsity patterns  $c \in \{0,1\}^{100 \times 100}$  obtained as a result of pruning for the sparsity level  $\bar{\kappa} = \{10,..,90\} \%$ . Here, black(0)/white(1) pixels refer to pruned/retained parameters; (right) connection sensitivities (CS) measured for the parameters in each layer. All networks are initialized with  $\gamma = 1.0$ . Unlike the linear case, the sparsity pattern for the tanh network is nonuniform over different layers. When pruning for a high sparsity level (e.g.,  $\bar{\kappa} = 90\%$ ), this becomes critical and leads to poor learning capability as there are only a few parameters left in later layers. This is explained by the connection sensitivity plot which shows that for the nonlinear network parameters in later layers have saturating, lower connection sensitivities than those in earlier layers.
+
+![](images/e9ed295c32281b06f5e6c38c6b8406867b89660d25e675de00c4bb0c878c59a6.jpg)
+
+![](images/e2575c247dc4c089f6eac7c5cabde9a215558fb67fea4f0dd34d1ea4524d7ced.jpg)
+
+# 3.2 GRADIENT SIGNAL IN CONNECTION SENSITIVITY
+
+We posit that the unreliability of connection sensitivity observed in Figure 1 is due to poor signal propagation: an initialization that projects the input signal to be strongly amplified or attenuated in the forward pass will saturate the error signal under backpropagation (i.e., gradients), and hence will result in poorly calibrated connection sensitivity scores across layers, which will eventually lead to poor pruning results, potentially with complete disconnection of signal paths (e.g., entire layer).
+
+Precisely, we give the relationship between the connection sensitivity and the gradients as follows. From Equation 1, connection sensitivity is a normalized magnitude of gradients with respect to the connectivity parameters  $\mathbf{c}$ . Here, we use the vectorized notation where  $\mathbf{w}$  denotes all learnable parameters and  $\mathbf{c}$  denotes the corresponding connectivity parameters. From chain rule, we can write:
+
+$$
+\left. \frac {\partial L (\mathbf {c} \odot \mathbf {w} ; \mathcal {D})}{\partial \mathbf {c}} \right| _ {\mathbf {c} = 1} = \left. \frac {\partial L (\mathbf {c} \odot \mathbf {w} ; \mathcal {D})}{\partial (\mathbf {c} \odot \mathbf {w})} \right| _ {\mathbf {c} = 1} \odot \mathbf {w} = \frac {\partial L (\mathbf {w} ; \mathcal {D})}{\partial \mathbf {w}} \odot \mathbf {w}. \tag {3}
+$$
+
+Therefore,  $\partial L / \partial \mathbf{c}$  is the gradients  $\partial L / \partial \mathbf{w}$  amplified (or attenuated) by the corresponding weights  $\mathbf{w}$ , i.e.,  $\partial L / \partial c_{j} = \partial L / \partial w_{j}w_{j}$  for all  $j\in \{1\dots m\}$ . Considering  $\partial L / \partial c_{j}$  for a given  $j$ , since  $w_{j}$  does not depend on any other layers or signal propagation, the only term that depends on signal propagation in the network is the gradient term  $\partial L / \partial w_{j}$ . Hence, a necessary condition to ensure faithful  $\partial L / \partial \mathbf{c}$  (and connection sensitivity) is that the gradients  $\partial L / \partial \mathbf{w}$  need to be faithful. In the following section, we formalize this from a signal propagation perspective, and characterize an initial condition that ensures reliable connection sensitivity measurement.
+
+# 3.3 LAYERWISE DYNAMICAL ISOMETRY
+
+# 3.3.1 GRADIENTS IN TERMS OF JACOBIANS
+
+From the feed-forward dynamics of a network in Equation 2, the network's input-output Jacobian corresponding to a given input  $\mathbf{x}^0$  can be written, by the chain rule of differentiation, as:
+
+$$
+\mathbf {J} ^ {0, K} = \frac {\partial \mathbf {x} ^ {K}}{\partial \mathbf {x} ^ {0}} = \prod_ {l = 1} ^ {K} \mathbf {D} ^ {l} \mathbf {W} ^ {l}, \tag {4}
+$$
+
+where  $\mathbf{D}^l\in \mathbb{R}^{N\times N}$  is a diagonal matrix with entries  $\mathbf{D}_{ij}^{l} = \phi^{\prime}(h_{i}^{l})\delta_{ij}$ , with  $\phi^\prime$  denoting the derivative of nonlinearity  $\phi$ , and  $\delta_{ij} = \mathbb{1}[i = j]$  is the Kronecker delta. Here, we will use  $\mathbf{J}^{k,l}$  to denote the Jacobian from layer  $k$  to layer  $l$ . Now, we give the relationship between gradients and Jacobians:
+
+Proposition 1. Let  $\epsilon = \partial L / \partial \mathbf{x}^K$  denote the error signal and  $\mathbf{x}^0$  denote the input signal. Then,
+
+1. the gradients satisfy:
+
+$$
+\mathbf {g} _ {\mathbf {w} ^ {l}} ^ {T} = \epsilon \mathbf {J} ^ {l, K} \mathbf {D} ^ {l} \otimes \mathbf {x} ^ {l - 1}, \tag {5}
+$$
+
+where  $\mathbf{J}^{l,K} = \partial \mathbf{x}^K /\partial \mathbf{x}^l$  is the Jacobian from layer  $l$  to the output and  $\otimes$  is the Kronecker product. 2. additionally, for linear networks, i.e., when  $\phi$  is the identity:
+
+$$
+\mathbf {g} _ {\mathbf {w} ^ {l}} ^ {T} = \epsilon \mathbf {J} ^ {l, K} \otimes \left(\mathbf {J} ^ {0, l - 1} \mathbf {x} ^ {0} + \mathbf {a}\right), \tag {6}
+$$
+
+where  $\mathbf{J}^{0,l - 1} = \partial \mathbf{x}^{l - 1} / \partial \mathbf{x}^0$  is the Jacobian from the input to layer  $l - 1$  and  $\mathbf{a}\in \mathbb{R}^N$  is a constant term that does not depend on  $\mathbf{x}^0$ .
+
+Proof. This can be proved by an algebraic manipulation of the chain rule while using the feedforward dynamics in Equation 2. We provide the full derivation in Appendix A.  $\square$
+
+Notice that the gradient at layer  $l$  constitutes both the backward propagation of the error signal  $\epsilon$  up to layer  $l$  and the forward propagation of the input signal  $\mathbf{x}^0$  up to layer  $l - 1$ . Moreover, especially in the linear case, the signal propagation in both directions is governed by the corresponding Jacobians. We believe that this interpretation of gradients is useful as it sheds light on how signal propagation affects the gradients. To this end, we next analyze the conditions on the Jacobians, which would guarantee faithful signal propagation in the network, and consequently, faithful gradients.
+
+# 3.3.2 ENSURING FAITHFUL GRADIENTS
+
+Here, we first consider the layerwise signal propagation which would be useful to derive properties on the initialization to ensure faithful gradients. To this end, let us consider the layerwise Jacobian:
+
+$$
+\mathbf {J} ^ {l - 1, l} = \frac {\partial \mathbf {x} ^ {l}}{\partial \mathbf {x} ^ {l - 1}} = \mathbf {D} ^ {l} \mathbf {W} ^ {l}. \tag {7}
+$$
+
+Note that it is sufficient to have layerwise dynamical isometry in order to ensure faithful signal propagation in the network.
+
+Definition 1. (Layerwise dynamical isometry) Let  $\mathbf{J}^{l - 1,l} = \frac{\partial\mathbf{x}^l}{\partial\mathbf{x}^{l - 1}}\in \mathbb{R}^{N_l\times N_{l - 1}}$  be the Jacobian matrix of layer  $l$ . The network is said to satisfy layerwise dynamical isometry if the singular values of  $\mathbf{J}^{l - 1,l}$  are concentrated near 1 for all layers, i.e., for a given  $\epsilon >0$ , the singular value  $\sigma_{j}$  satisfies  $|1 - \sigma_{j}|\leq \epsilon$  for all  $j$ .
+
+This would guarantee that the signal from layer  $l$  to  $l - 1$  (or vice versa) is propagated without amplification or attenuation in any of its dimension. From Proposition 1 and Equation 7, by induction, it is easy to show that if the layerwise signal propagation is faithful, the error and input signals will faithfully propagate throughout the network, resulting in faithful gradients.
+
+For linear networks,  $\mathbf{J}^{l - 1,l} = W^l$ . Therefore, one can initialize the weight matrix to be orthogonal such that  $(\mathbf{W}^l)^T\mathbf{W}^l = \mathbf{I}$ , where  $\mathbf{I}$  is the identity matrix of dimension  $N$ . In this case, all singular values of  $\mathbf{W}^l$  are exactly 1 (i.e., exact dynamical isometry), and such an initialization guarantees faithful gradients. While a linear network is of little practical use, we note that it helps to develop theoretical analysis and provides intuition as to why dynamical isometry is a useful measure.
+
+For nonlinear networks, the diagonal matrix  $\mathbf{D}^l$  needs to be accounted for as it depends on the pre-activations  $\mathbf{h}^l$  at layer  $l$ . In this case, it is important to have the pre-activations  $\mathbf{h}^l$  fall into the linear region of the nonlinear function  $\phi$ . Precisely, mean-field theory assumes that for large- $N$  limit, the empirical distribution of the pre-activations  $\mathbf{h}^l$  converges to a Gaussian with zero mean and variance  $q^l$ , where the variance follows a recursion relation (Poole et al., 2016). Therefore, to achieve layerwise dynamical isometry, the idea becomes to find a fixed point  $q^*$  such that  $\mathbf{h}^l \sim \mathcal{N}(0, q^*)$  for all  $l \in \{1 \dots K\}$ . Such a fixed point makes  $\mathbf{D}^l = \mathbf{D}$  for all layers, and therefore, the pre-activations are placed in the linear region of the nonlinearity. Then, given the nonlinearity, one can find a rescaling such that  $(\mathbf{D}\mathbf{W}^l)^T (\mathbf{D}\mathbf{W}^l) = (\mathbf{W}^l)^T\mathbf{W}^l / \sigma_w^2 = \mathbf{I}$ . The procedure for finding the rescaling  $\sigma_w^2$  for various nonlinearities are discussed in Pennington et al. (2017; 2018). Also, this easily extends to convolutional neural networks using the initialization method in Xiao et al. (2018).
+
+Table 1: Jacobian singular values and resulting sparse networks for the 7-layer tanh MLP network considered in section 3.1. SG, CN, and Sparsity refer to Scaled Gaussian, Condition Number (i.e.,  $s_{\mathrm{max}} / s_{\mathrm{min}}$ , where  $s_{\mathrm{max}}$  and  $s_{\mathrm{min}}$  are the maximum and minimum Jacobian singular values), and a ratio of pruned prameters to the total number of parameters, respectively. SG ( $\gamma = 10^{-2}$ ) is equivalent to the variance scaling initialization as in LeCun et al. (1998); Glorot & Bengio (2010). The failure cases correspond to unreliable connection sensitivity resulted from poorly conditioned initial Jacobians.  
+
+<table><tr><td rowspan="2">Initialization</td><td colspan="3">Jacobian singular values</td><td colspan="8">Sparsity in pruned network (across layers)</td><td rowspan="2">Error</td></tr><tr><td>Mean</td><td>Std</td><td>CN</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td></td></tr><tr><td>SG (γ=10-4)</td><td>2.46e-07</td><td>9.90e-08</td><td>4.66e+00</td><td>0.97</td><td>0.80</td><td>0.80</td><td>0.80</td><td>0.80</td><td>0.81</td><td>0.48</td><td>2.66</td><td></td></tr><tr><td>SG (γ=10-3)</td><td>5.74e-04</td><td>2.45e-04</td><td>8.54e+00</td><td>0.97</td><td>0.80</td><td>0.80</td><td>0.80</td><td>0.80</td><td>0.81</td><td>0.48</td><td>2.67</td><td></td></tr><tr><td>SG (γ=10-2)</td><td>4.49e-01</td><td>2.51e-01</td><td>5.14e+01</td><td>0.96</td><td>0.80</td><td>0.80</td><td>0.80</td><td>0.81</td><td>0.81</td><td>0.49</td><td>2.67</td><td></td></tr><tr><td>SG (γ=10-1)</td><td>2.30e+01</td><td>2.56e+01</td><td>2.92e+04</td><td>0.96</td><td>0.81</td><td>0.82</td><td>0.82</td><td>0.82</td><td>0.80</td><td>0.45</td><td>2.61</td><td></td></tr><tr><td>SG (γ=100)</td><td>1.03e+03</td><td>2.61e+03</td><td>3.34e+11</td><td>0.85</td><td>0.88</td><td>0.99</td><td>1.00</td><td>1.00</td><td>1.00</td><td>0.91</td><td>90.2</td><td></td></tr><tr><td>SG (γ=101)</td><td>3.67e+04</td><td>2.64e+05</td><td>inf</td><td>0.84</td><td>0.95</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td><td>1.00</td><td>90.2</td><td></td></tr></table>
+
+We note that dynamical isometry is in fact a weaker condition than layerwise dynamical isometry. However, in practice, the initialization suggested in the existing works (Pennington et al., 2017; Xiao et al., 2018), i.e., orthogonal initialization for weight matrices in each layer with rescaling based on mean field theory, satisfy layerwise dynamical isometry, even though this term was not mentioned.
+
+Now, recall from Section 3.1 that a network is pruned with a global threshold based on connection sensitivity, and from Section 3.2 that the connection sensitivity is the gradients scaled by the weights. This in turn implies that the connection sensitivity scores across layers are required to be of the same scale. To this end, we require the gradients to be faithful and the weights to be in the same scale for all the layers. Notice, this condition is trivially satisfied when the layerwise dynamical isometry is ensured, as each layer is initialized identically (i.e., orthogonal initialization) and the gradients are guaranteed to be faithful.
+
+Finally, we verify the failure of pruning cases presented in Section 3.1 based on the signal propagation perspective. Specifically, we measure the singular value distribution of the input-output Jacobian  $(\mathbf{J}^{0,K})$  for the 7-layer tanh MLP network, and the results are reported in Table 1. Note that while connection sensitivity based pruning is robust to moderate changes in the Jacobian singular values, it failed catastrophically when the condition number of the Jacobian is very large ( $>1\mathrm{e} + 11$ ). In fact, these failure cases correspond to the completely disconnected networks, as a consequence of pruning with unreliable connection sensitivity resulted from poorly conditioned initial Jacobians. As we will show subsequently, these findings extend to modern architectures, and layerwise dynamical isometry yields well-conditioned Jacobians and in turn the best pruning results.
+
+# 4 SIGNAL PROPAGATION IN SPARSE NEURAL NETWORKS
+
+So far, we have shown empirically and theoretically that layerwise dynamical isometry can improve the process of pruning at initialization. One remaining question to address is the following: how well do signals propagate in the pruned sparse networks? In this section, we first examine the effect of sparsity on signal propagation after pruning. We find that indeed pruning can break dynamical isometry, degrading trainability of sparse networks. Then we follow up to present a simple, but effective data-free method to recover approximate dynamical isometry on sparse networks.
+
+Setup. The overall process is summarized as follows: Step 1. Initialize a network with a variance scaling (VS) or layerwise dynamical isometry (LDI) satisfying orthogonal initialization. Step 2. Prune at initialization for a sparsity level  $\bar{\kappa}$  based on connection sensitivity (CS); we also test random (Rand) and magnitude (Mag) based pruning for comparison. Step 3. (optional) Enforce approximate dynamical isometry, if specified. Step 4. Train the pruned sparse network using SGD. We measure signal propagation (e.g., Jacobian singular values) on the sparse network right before Step 4, and observe training behavior during Step 4. Different methods are named as {A}-{B}-{C}, where A, B, C stand for initialization scheme, pruning method, (optional) approximate isometry, respectively. We perform this on 7-layer linear and tanh MLP networks as before<sup>3</sup>.
+
+![](images/1a88362cfba48921f1628dcea4a321ab40a26ee5a14231ee9135156dc20d52b9.jpg)  
+(a) Signal propagation
+
+![](images/0d44134d62b4fefd0deda2e9cc93223bbbe65787857e956e595f527e2cc22844.jpg)  
+Figure 2: (a) Signal propagation (mean Jacobian singular values) in sparse networks pruned for varying sparsity levels  $\bar{\kappa}$ , and (b) training behavior of the sparse network at  $\bar{\kappa} = 90\%$ . Signal propagation, pruning scheme, and overparameterization affect trainability of sparse neural networks. We train using SGD with the initial learning rate of 0.1 decayed by 1/10 at every 20k iterations. All results are the average over 10 runs. We provide other singular value statistics (max, min, std), accuracy plot, and extended training results for random and magnitude pruning in Appendix C.
+
+![](images/8904355e48b4ea260fce72b24c4eba558a6c8de2d4f4e09b6ce32d7094a0963e.jpg)  
+(b) Training behavior
+
+![](images/c2f79b91a7e48bddc1a9efea659f3faeaea775520a31e78fad736e5b6f02d2c4.jpg)
+
+Effect of pruning on signal propagation and trainability. Let us first check signal propagation measurements in the pruned networks (see Figure 2a). In general, Jacobian singular values decrease continuously as the sparsity level increases (except for  $\{\cdot\}-\{\cdot\}$ -AI which we will explain later), indicating that the more parameters are removed, the less faithful a network is likely to be with regard to propagating signals. Also, notice that the singular values drop more rapidly with random pruning compared to connection sensitivity based pruning methods (i.e.,  $\{\cdot\}$ -Rand vs.  $\{\cdot\}$ -CS). This means that pruning using connection sensitivity is more robust to destruction of dynamical isometry and preserve better signal propagation in the sparse network than random pruning. We further note that, albeit marginal, layerwise dynamical isometry allows better signal propagation than variance scaling initialization, with relatively higher mean singular values and much lower standard deviations especially in the low sparsity regime (see Appendix C).
+
+Now, we look into the relation between signal propagation and trainability of the sparse networks. Figure 2b shows training behavior of the pruned networks  $(\bar{\kappa} = 90\%)$  obtained by different methods. We can see a clear correlation between signal propagation capability of a network and its training performance; i.e., the better a network propagates signals, the faster it converges during training. For instance, compare the trainability of a network before and after pruning. That is, compared to LDI-Dense  $(\bar{\kappa} = 0)$ , LDI-  $\{\mathrm{CS},\mathrm{Mag},\mathrm{Rand}\}$  decrease the loss much slowly; random pruning starts to decrease the loss around  $4\mathrm{k}$  iteration, and finally reaches close to zero loss around  $10\mathrm{k}$  iterations (see Appendix C), which is more than an order of magnitude slower than a network pruned by connection sensitivity. Recall that the pruned networks have much smaller singular values.
+
+Enforcing approximate dynamical isometry. The observation above indicates that the better signal propagation is ensured on sparse networks, the better their training performs. This motivates us to think of the following: what if we can repair the broken isometry, before we start training the pruned network, such that we can achieve trainability comparable to that of the dense network? Precisely, we consider the following:
+
+$$
+\min  _ {\mathbf {W} ^ {l}} \left\| \left(\mathbf {C} ^ {l} \odot \mathbf {W} ^ {l}\right) ^ {T} \left(\mathbf {C} ^ {l} \odot \mathbf {W} ^ {l}\right) - \mathbf {I} ^ {l} \right\| _ {F}, \tag {8}
+$$
+
+where  $\mathbf{C}^l$ ,  $\mathbf{W}^l$ ,  $\mathbf{I}^l$  are the sparse mask obtained by pruning, the corresponding weights, the identity matrix at layer  $l$ , respectively, and  $\| \cdot \|_F$  is the Frobenius norm. We optimize this for all layers identically using gradient descent. Given the sparsity topology  $\mathbf{C}^l$  and initial weights  $\mathbf{W}^l$ , this data-free method attempts to find an optimal  $\mathbf{W}^*$  such that the combination of the sparse topology and the weights to be layerwise orthogonal, potentially to the full rank capacity. This simple method (i.e.,  $\{\cdot\} - \{\cdot\}$ -AI, where AI is named for Approximate Isometry) turns out to be highly effective. The results are provided in Figure 2, and we summarize our key findings below:
+
+- Signal propagation (LDI-{CS, Rand} vs. LDI-{CS, Rand}-AI). The decreased singular values (by pruning  $\bar{\kappa} > 0$ ) bounce up dramatically and become close to the level before pruning. This means that orthogonality enforced by Equation 8 is achieved in the sparse topology of the pruned
+
+Table 2: Pruning results for various neural networks on different datasets. All networks are pruned at initialization for the sparsity  $\bar{\kappa} = 90\%$  based on connection sensitivity scores as in Lee et al. (2019). We report orthogonality scores (OS) and generalization errors (Error) on CIFAR-10 (VGG16, ResNets) and Tiny-ImageNet (WRN16); all results are the average over 5 runs. The first and second best results are highlighted in each column of errors. The orthogonal initialization with enforced approximate isometry method (i.e., LDI-AI) achieves the best results across all tested architectures.  
+
+<table><tr><td rowspan="2">Initialization</td><td colspan="2">VGG16</td><td colspan="2">ResNet32</td><td colspan="2">ResNet56</td><td colspan="2">ResNet110</td><td colspan="2">WRN16</td></tr><tr><td>OS</td><td>Error</td><td>OS</td><td>Error</td><td>OS</td><td>Error</td><td>OS</td><td>Error</td><td>OS</td><td>Error</td></tr><tr><td>VS-L</td><td>13.72</td><td>8.16</td><td>4.50</td><td>11.96</td><td>4.64</td><td>10.43</td><td>4.65</td><td>9.13</td><td>11.99</td><td>45.08</td></tr><tr><td>VS-G</td><td>13.60</td><td>8.18</td><td>4.55</td><td>11.89</td><td>4.67</td><td>10.60</td><td>4.67</td><td>9.17</td><td>11.50</td><td>44.56</td></tr><tr><td>VS-H</td><td>15.44</td><td>8.36</td><td>4.41</td><td>12.21</td><td>4.44</td><td>10.63</td><td>4.39</td><td>9.08</td><td>13.49</td><td>46.62</td></tr><tr><td>LDI</td><td>13.33</td><td>8.11</td><td>4.43</td><td>11.55</td><td>4.51</td><td>10.08</td><td>4.57</td><td>8.88</td><td>11.28</td><td>44.20</td></tr><tr><td>LDI-AI</td><td>6.43</td><td>7.99</td><td>2.62</td><td>11.47</td><td>2.79</td><td>9.85</td><td>2.92</td><td>8.78</td><td>6.62</td><td>44.12</td></tr></table>
+
+Table 3: Pruning results for VGG16 and ResNet32 with different activation functions on CIFAR-10. We report generalization errors (avg. over 5 runs), and the first and second best results are highlighted.  
+
+<table><tr><td rowspan="2">Initialization</td><td colspan="3">VGG16</td><td colspan="3">ResNet32</td></tr><tr><td>tanh</td><td>l-relu</td><td>selu</td><td>tanh</td><td>l-relu</td><td>selu</td></tr><tr><td>VS-L</td><td>9.07</td><td>7.78</td><td>8.70</td><td>13.41</td><td>12.04</td><td>12.26</td></tr><tr><td>VS-G</td><td>9.06</td><td>7.84</td><td>8.82</td><td>13.44</td><td>12.02</td><td>12.32</td></tr><tr><td>VS-H</td><td>9.99</td><td>8.43</td><td>9.09</td><td>13.12</td><td>11.66</td><td>12.21</td></tr><tr><td>LDI</td><td>8.76</td><td>7.53</td><td>8.21</td><td>13.22</td><td>11.58</td><td>11.98</td></tr><tr><td>LDI-AI</td><td>8.72</td><td>7.47</td><td>8.20</td><td>13.14</td><td>11.51</td><td>11.68</td></tr></table>
+
+Table 4: Unsupervised pruning results for  $K$ -layer MLP networks on MNIST. All networks are pruned for sparsity  $\bar{\kappa} = 90\%$  at orthogonal initialization. We report generalization errors (avg. over 10 runs).  
+
+<table><tr><td>Loss</td><td>Superv.</td><td>K=3</td><td>K=5</td><td>K=7</td></tr><tr><td>GT</td><td>✓</td><td>2.46</td><td>2.43</td><td>2.61</td></tr><tr><td>Pred. (raw)</td><td>✗</td><td>3.31</td><td>3.38</td><td>3.60</td></tr><tr><td>Pred. (softmax)</td><td>✗</td><td>3.11</td><td>3.37</td><td>3.56</td></tr><tr><td>Unif.</td><td>✗</td><td>2.77</td><td>2.77</td><td>2.94</td></tr></table>
+
+network (i.e., approximate dynamical isometry), and therefore, signal propagation on the sparse network is likely to behave similarly to the dense network. As expected, the training performance increased significantly (e.g., compare LDI-CS with LDI-CS-AI for trainability). This works more dramatically for random pruning; i.e., even for randomly pruned sparse networks, training speed increases significantly, implying the benefit of ensuring signal propagation.
+
+- Structure (LDI-Rand-AI vs. LDI-CS-AI). Even if the approximate dynamical isometry is enforced identically, the network pruned using connection sensitivity shows better trainability than the randomly pruned network. This potentially means that the sparse topology obtained by different pruning methods also matters, in addition to signal propagation characteristics.  
+- Overparameterization (LDI-Dense vs. LDI-{CS, Rand}-AI). Even though the singular values are restored to a level close to before pruning with approximate isometry, the non-pruned dense network converges faster than pruned networks. We hypothesize that in addition to signal propagation, overparameterization helps in optimization taking less time to find a minimum.
+
+While being simple and data free (thus fast), our signal propagation perspective not only can be used to improve trainability of sparse neural networks, but also to complement a common explanation for decreased trainability of compressed networks which is often attributed merely to a reduced capacity. Our results also extend to the case of convolutional neural network (see Figure 8 in Appendix C).
+
+# 5 VALIDATION AND EXTENSIONS
+
+In this section, we aim to demonstrate the efficacy of our signal propagation perspective on a wide variety of settings. We first evaluate the idea of employing layerwise dynamical isometry on various modern neural networks. In addition, we further study the role of supervision under the pruning at initialization regime, extending it to unsupervised pruning. Our results show that indeed, pruning can be approached from the signal propagation perspective at varying scale, bringing forth the notion of neural architecture sculpting. The experiment settings used to generate the presented results are detailed in Appendix B. The code can be found here: https://github.com/namhoonlee/spp-public.
+
+# 5.1 EVALUATION ON VARIOUS NEURAL NETWORKS AND DATASETS
+
+Here, we verify that our signal propagation perspective for pruning neural networks at initialization is indeed valid, by evaluating further on various modern neural networks and datasets. To this end, we provide orthogonality scores (OS) and generalization errors of the sparse networks obtained by different methods and show that layerwise dynamical isometry with enforced approximate isometry results in the best performance; here, we define OS as  $\frac{1}{l}\sum_{l}\| (\mathbf{C}^{l}\odot \mathbf{W}^{l})^{T}(\mathbf{C}^{l}\odot \mathbf{W}^{l}) - \mathbf{I}^{l}\|_{F}$ , which can be used to indicate how close are the weight matrices in each layer of the pruned network to being orthogonal. All results are the average of 5 runs, and we do not optimize anything specific for a particular case (see Appendix B for experiment settings). The results are presented in Table 2.
+
+The best pruning results are achieved when the approximate dynamical isometry is enforced on the pruned sparse network (i.e., LDI-AI), across all tested architectures. Also, the second best results are achieved with the orthogonal initialization that satisfies layerwise dynamical isometry (i.e., LDI). Looking closely, it is evident that there exists a high correlation between the orthogonality scores and the performance of pruned networks; i.e., the network initialized to have the lowest orthogonality scores achieves the best generalization errors after training. Note that the orthogonality scores being close to 0, by definition, states how faithful a network will be with regard to letting signals propagate without being amplified or attenuated. Therefore, the fact that a pruned network with the lowest orthogonality scores tends to yield good generalization errors further validates that our signal propagation perspective is indeed effective for pruning at initialization. Moreover, we test for other nonlinear activation functions (tanh, leaky-relu, selu), and found that the orthogonal initialization consistently outperforms variance scaling methods (see Table 3).
+
+# 5.2 PRUNING WITHOUT SUPERVISION
+
+So far, we have shown that pruning random networks can be approached from a signal propagation perspective by ensuring faithful connection sensitivity. Another factor that constitutes connection sensitivity is the loss term. At a glance, it is not obvious how informative the supervised loss measured on a random network will be for connection sensitivity. In this section, we look into the effect of supervision, by simply replacing the loss computed using ground-truth labels with different unsupervised surrogate losses as follows: replacing the target distribution using ground-truth labels with uniform distribution (Unif.), and using the averaged output prediction of the network (Pred.; softmax/raw). The results for MLP networks are in Table 4. Even though unsupervised pruning results are not as good as the supervised case, the results are still interesting, especially for the uniform case, in that there was no supervision given. We thus experiment further for the uniform case on other networks, and obtain the following results: 8.25, 11.69, 11.01, 8.82 errors (\%) for VGG16, ResNet32, ResNet56, ResNet110, respectively. Surprisingly, the results are often competitive to that of pruning with supervision (i.e., compare to LDI results in Table 2). Notably, previous pruning algorithms assume the existence of supervision a priori. Being the first demonstration, along with the signal propagation perspective, this unsupervised pruning strategy can be useful in scenarios where there are no labels or only weak supervision is available.
+
+To demonstrate further, we also conducted transfer of sparsity experiments such as transferring a pruned network from one task to another (MNIST  $\leftrightarrow$  Fashion-MNIST). Table 5 shows that, while pruning results may degrade if sparsity is transferred, or done without supervision, less impact is caused for unsupervised pruning when transferred to a different task (i.e., 0.52 to 0.14 on MNIST, and 1.11 to -0.78 on F
+
+Table 5: Transfer of sparsity experiment results for LeNet. We prune for  $\bar{\kappa} = 97\%$  at orthogonal initialization, and report gen. errors (average over 10 runs).  
+
+<table><tr><td rowspan="2">Category</td><td colspan="2">Dataset</td><td colspan="2">Error</td><td rowspan="2">Error rand</td></tr><tr><td>prune</td><td>train&amp;test</td><td>sup. → unsup.</td><td>(Δ)</td></tr><tr><td>Standard</td><td>MNIST</td><td>MNIST</td><td>2.42 → 2.94</td><td>+0.52</td><td>15.56</td></tr><tr><td>Transfer</td><td>F-MNIST</td><td>MNIST</td><td>2.66 → 2.80</td><td>+0.14</td><td>18.03</td></tr><tr><td>Standard</td><td>F-MNIST</td><td>F-MNIST</td><td>11.90 → 13.01</td><td>+1.11</td><td>24.72</td></tr><tr><td>Transfer</td><td>MNIST</td><td>F-MNIST</td><td>14.17 → 13.39</td><td>-0.78</td><td>24.89</td></tr></table>
+
+MNIST). This indicates that inductive bias exists in data, affecting transfer and unsupervised pruning, and potentially, that "universal" sparse topology might be obtainable if universal data distribution is known (e.g., extremely large dataset in practice). This may help in situations where different tasks from unknown data distribution are to be performed (e.g., continual learning). We also tested two other unsupervised losses, but none performed as well as uniform loss (e.g., Jacobian norms  $\| J\| _1$ : 5.03,  $\| J\| _2$ : 3.00 vs. Unif.: 2.94), implying that if pruning is to be unsupervised, the uniform loss would better be used, because other unsupervised losses depend on the input data (thus can suffer from inductive bias). Random pruning degrades significantly at high sparsity for all cases.
+
+# 5.3 NEURAL ARCHITECTURE SCULPTING
+
+We have shown that pruning at initialization, even when no supervision is provided, can be effective based on the signal propagation perspective. This begs the question of whether pruning needs to be limited to pre-shaped architectures or not. In other words, what if pruning is applied to an arbitrarily bulky network and is treated as sculpting an architecture? In order to find out, we conduct the following experiments: we take a popular pre-designed architecture (ResNet20 in He et al. (2016)) as a base network, and consider a range of variants that are originally bigger than the base model, but pruned to have the same number of parameters as the base dense network. Specifically, we consider the following equivalents: (1) the same number of residual blocks, but with larger widths; (2) a reduced number of residual blocks with larger widths; (3) a larger residual block and the same width (see Table 6 in Appendix B for details). The results are presented in Figure 3.
+
+Overall, the sparse equivalents record lower errors than the dense base model. Notice that some models are extremely sparse (e.g., Equivalent 1 pruned for  $\bar{\kappa} = 98.4\%$ ). While all networks have the same number of parameters, discovered sparse equivalents outperform the dense reference network. This result is well aligned with recent findings in Kalchbrenner et al. (2018): large sparse networks can outperform their small dense counterpart, while enjoying increased computational and memory efficiency via a dedicated implementation for sparsity in practice. Also, it seems that pruning wider networks tends to be more effective in producing a better model than pruning deeper ones (e.g., Equivalent 1 vs. Equivalent 3). We further note that unlike existing prior works, the sparse networks are discovered by sculpting arbitrarily-designed architecture, without pretraining nor supervision.
+
+![](images/5cb713bef823e0cde1435dcfedf9b2038e1b85c58a46bcda4d0c412596e6645f.jpg)  
+Figure 3: Neural architecture sculpting results on CIFAR-10. We report generalization errors (avg. over 5 runs). All networks have the same number of parameters (269k) and trained identically.
+
+# 6 DISCUSSION AND FUTURE WORK
+
+In this work, we have approached the problem of pruning neural networks at initialization from a signal propagation perspective. Based on observing the effect of varying the initialization, we found that initial weights have a critical impact on connection sensitivity measurements and hence pruning results. This led us to conduct theoretical analysis based on dynamical isometry and a mean field theory, and formally characterize a sufficient condition to ensure faithful signal propagation in a given network. Moreover, our analysis on compressed neural networks revealed that signal propagation characteristics of a sparse network highly correlates with its trainability, and also that pruning can break dynamical isometry ensured on a network at initialization, resulting in degradation of trainability of the compressed network. To address this, we introduced a simple, yet effective data-free method to recover the orthogonality and enhance trainability of the compressed network. Finally, throughout a range of validation and extension experiments, we verified that our signal propagation perspective is effective for understanding, improving, and extending the task of pruning at initialization across various settings. We believe that our results on the increased trainability of compressed networks can take us one step towards finding "winning lottery ticket" (i.e., a set of initial weights that given a sparse topology can quickly reach to a generalization performance that is comparable to the uncompressed network, once trained) suggested in Frankle & Carbin (2019).
+
+We point out, however, that there remains several aspects to consider. While pruning on enforced isometry produces trainable sparse networks, the two-stage orthogonalization process (i.e., prune first and enforce the orthogonality later) can be suboptimal especially at a high sparsity level. Also, network weights change during training, which can affect signal propagation characteristics, and therefore, dynamical isometry may not continue to hold over the course of training. We hypothesize that a potential key to successful neural network compression is to address the complex interplay between optimization and signal propagation, and it might be immensely beneficial if an optimization naturally takes place in the space of isometry. We believe that our signal propagation perspective provides a means to formulate this as an optimization problem by maximizing the trainability of sparse networks while pruning, and we intend to explore this direction as a future work.
+
+# ACKNOWLEDGMENTS
+
+This work was supported by the ERC grant ERC-2012-AdG 321162-HELIOS, EPSRC grant See-bibyte EP/M013774/1, EPSRC/MURI grant EP/N019474/1 and the Australian Research Council Centre of Excellence for Robotic Vision (project number CE140100016). We would also like to acknowledge the Royal Academy of Engineering and FiveAI, and thank Richard Hartley, Puneet Dokania and Amartya Sanyal for helpful discussions.
+
+# REFERENCES
+
+Jonathan Frankle and Michael Carbin. The lottery ticket hypothesis: Finding sparse, trainable neural networks. *ICLR*, 2019.  
+Xavier Glorot and Yoshua Bengio. Understanding the difficulty of training deep feedforward neural networks. AISTATS, 2010.  
+Song Han, Huizi Mao, and William J Dally. Deep compression: Compressing deep neural networks with pruning, trained quantization and huffman coding. *ICLR*, 2016.  
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Delving deep into rectifiers: Surpassing human-level performance on imagenet classification. ICCV, 2015.  
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. CVPR, 2016.  
+Geoffrey E Hinton and Ruslan R Salakhutdinov. Reducing the dimensionality of data with neural networks. Science, 2006.  
+Nal Kalchbrenner, Erich Elsen, Karen Simonyan, Seb Noury, Norman Casagrande, Edward Lockhart, Florian Stimberg, Aaron van den Oord, Sander Dieleman, and Koray Kavukcuoglu. Efficient neural audio synthesis. ICML, 2018.  
+Yann LeCun, Léon Bottou, Genevieve B. Orr, and Klaus-Robert Müller. Efficient backprop. Neural Networks: Tricks of the Trade, 1998.  
+Yann LeCun, Joshua Bengio, and Geoffrey Hinton. Deep learning. Nature, 2015.  
+Namhoon Lee, Thalaiyasingam Ajanthan, and Philip HS Torr. Snip: Single-shot network pruning based on connection sensitivity. *ICLR*, 2019.  
+Razvan Pascanu, Tomas Mikolov, and Yoshua Bengio. On the difficulty of training recurrent neural networks. 2013.  
+Jeffrey Pennington, Samuel Schoenholz, and Surya Ganguli. Resurrecting the sigmoid in deep learning through dynamical isometry: theory and practice. NeurIPS, 2017.  
+Jeffrey Pennington, Samuel S Schoenholz, and Surya Ganguli. The emergence of spectral universality in deep networks. AISTATS, 2018.  
+Ben Poole, Subhaneil Lahiri, Maithra Raghu, Jascha Sohl-Dickstein, and Surya Ganguli. Exponential expressivity in deep neural networks through transient chaos. NeurIPS, 2016.  
+Russell Reed. Pruning algorithms-a survey. Neural Networks, 1993.  
+Andrew M Saxe, James L McClelland, and Surya Ganguli. Exact solutions to the nonlinear dynamics of learning in deep linear neural networks. *ICLR*, 2014.  
+Samuel S Schoenholz, Justin Gilmer, Surya Ganguli, and Jascha Sohl-Dickstein. Deep information propagation. *ICLR*, 2017.  
+Wojciech Tarnowski, Piotr Warchol, Stanisław Jastrzebski, Jacek Tabor, and Maciej A Nowak. Dynamical isometry is achieved in residual networks in a universal way for any activation function. AISTATS, 2019.
+
+Lechao Xiao, Yasaman Bahri, Jascha Sohl-Dickstein, Samuel S Schoenholz, and Jeffrey Pennington. Dynamical isometry and a mean field theory of cnns: How to train 10,000-layer vanilla convolutional neural networks. ICML, 2018.
+
+Ge Yang and Samuel Schoenholz. Mean field residual networks: On the edge of chaos. NeurIPS, 2017.
+
+Greg Yang, Jeffrey Pennington, Vinay Rao, Jascha Sohl-Dickstein, and Samuel S Schoenholz. A mean field theory of batch normalization. *ICLR*, 2019.
+
+# A GRADIENTS IN TERMS OF JACOBIANS
+
+Proposition 1. Let  $\epsilon = \partial L / \partial \mathbf{x}^K$  denote the error signal and  $\mathbf{x}^0$  denote the input signal. Then,
+
+1. the gradients satisfy:
+
+$$
+\mathbf {g} _ {\mathbf {w} ^ {l}} ^ {T} = \epsilon \mathbf {J} ^ {l, K} \mathbf {D} ^ {l} \otimes \mathbf {x} ^ {l - 1}, \tag {9}
+$$
+
+where  $\mathbf{J}^{l,K} = \partial \mathbf{x}^K /\partial \mathbf{x}^l$  is the Jacobian from layer  $l$  to the output and  $\otimes$  is the Kronecker product. 2. additionally, for linear networks, i.e., when  $\phi$  is the identity:
+
+$$
+\mathbf {g} _ {\mathbf {w} ^ {l}} ^ {T} = \epsilon \mathbf {J} ^ {l, K} \otimes \left(\mathbf {J} ^ {0, l - 1} \mathbf {x} ^ {0} + \mathbf {a}\right), \tag {10}
+$$
+
+where  $\mathbf{J}^{0,l - 1} = \partial \mathbf{x}^{l - 1} / \partial \mathbf{x}^0$  is the Jacobian from the input to layer  $l - 1$  and  $\mathbf{a}\in \mathbb{R}^N$  is the constant term that does not depend on  $\mathbf{x}^0$ .
+
+Proof. The proof is based on a simple algebraic manipulation of the chain rule. The gradient of the loss with respect to the weight matrix  $\mathbf{W}^l$  can be written as:
+
+$$
+\mathbf {g} _ {\mathbf {w} ^ {l}} = \frac {\partial L}{\partial \mathbf {W} ^ {l}} = \frac {\partial L}{\partial \mathbf {x} ^ {K}} \frac {\partial \mathbf {x} ^ {K}}{\partial \mathbf {x} ^ {l}} \frac {\partial \mathbf {x} ^ {l}}{\partial \mathbf {W} ^ {l}}. \tag {11}
+$$
+
+Here, the gradient  $\partial \mathbf{y} / \partial \mathbf{x}$  is represented as a matrix of dimension  $\mathbf{y}$ -size  $\times$ $\mathbf{x}$ -size. For gradients with respect to matrices, their vectorized from is used. Notice,
+
+$$
+\frac {\partial \mathbf {x} ^ {l}}{\partial \mathbf {W} ^ {l}} = \frac {\partial \mathbf {x} ^ {l}}{\partial \mathbf {h} ^ {l}} \frac {\partial \mathbf {h} ^ {l}}{\partial \mathbf {W} ^ {l}} = \mathbf {D} ^ {l} \frac {\partial \mathbf {h} ^ {l}}{\partial \mathbf {W} ^ {l}}. \tag {12}
+$$
+
+Considering the feed-forward dynamics for a particular neuron  $i$ ,
+
+$$
+h _ {i} ^ {l} = \sum_ {j} W _ {i j} ^ {l} x _ {j} ^ {l - 1} + b _ {i} ^ {l}, \tag {13}
+$$
+
+$$
+\frac {\partial h _ {i} ^ {l}}{\partial W _ {i j} ^ {l}} = x _ {j} ^ {l - 1}.
+$$
+
+Therefore, using the Kronecker product, we can compactly write:
+
+$$
+\frac {\partial \mathbf {x} ^ {l}}{\partial \mathbf {W} ^ {l}} = (\mathbf {D} ^ {l}) ^ {T} \otimes (\mathbf {x} ^ {l - 1}) ^ {T}. \tag {14}
+$$
+
+Now, Equation 11 can be written as:
+
+$$
+\mathbf {g} _ {\mathbf {w} ^ {l}} = \left(\epsilon \mathbf {J} ^ {l, K} \mathbf {D} ^ {l}\right) ^ {T} \otimes \left(\mathbf {x} ^ {l - 1}\right) ^ {T}, \tag {15}
+$$
+
+$$
+\mathbf {g} _ {\mathbf {w} ^ {l}} ^ {T} = \epsilon \mathbf {J} ^ {l, K} \mathbf {D} ^ {l} \otimes \mathbf {x} ^ {l - 1}.
+$$
+
+Here,  $\mathbf{A}^T\otimes \mathbf{B}^T = (\mathbf{A}\otimes \mathbf{B})^T$  is used. Moreover, for linear networks  $\mathbf{D}^l = \mathbf{I}$  and  $\mathbf{x}^l = \mathbf{h}^l$  for all  $l\in \{1\dots K\}$ . Therefore,  $\mathbf{x}^{l - 1}$  can be written as:
+
+$$
+\begin{array}{l} \mathbf {x} ^ {l - 1} = \phi \left(\mathbf {W} ^ {l - 1} \phi \left(\mathbf {W} ^ {l - 2} \dots \phi \left(\mathbf {W} ^ {1} \mathbf {x} ^ {0} + \mathbf {b} ^ {1}\right) \dots + \mathbf {b} ^ {l - 2}\right) + \mathbf {b} ^ {l - 1}\right), \tag {16} \\ = \mathbf {W} ^ {l - 1} \left(\mathbf {W} ^ {l - 2} \dots \left(\mathbf {W} ^ {1} \mathbf {x} ^ {0} + \mathbf {b} ^ {1}\right) \dots + \mathbf {b} ^ {l - 2}\right) + \mathbf {b} ^ {l - 1}, \\ = \prod_ {k = 1} ^ {l - 1} \mathbf {W} ^ {k} \mathbf {x} ^ {0} + \prod_ {k = 2} ^ {l - 1} \mathbf {W} ^ {k} \mathbf {b} ^ {1} + \dots + \mathbf {b} ^ {l - 1}, \\ = \mathbf {J} ^ {0, l - 1} \mathbf {x} ^ {0} + \mathbf {a}, \\ \end{array}
+$$
+
+where  $\mathbf{a}$  is the constant term that does not depend on  $\mathbf{x}^0$ . Hence, the proof is complete.
+
+![](images/a15fd43979184f10a9b26a2d80130d63ecae216c8e1988bdca303df178bd4640.jpg)
+
+# B EXPERIMENT SETTINGS
+
+Pruning at initialization. By default, we perform pruning at initialization based on connection sensitivity scores as in Lee et al. (2019). When computing connection sensitivity, we always use all examples in the training set to prevent stochasticity by a particular mini-batch. Unless stated otherwise, we set the default sparsity level to be  $\bar{\kappa} = 90\%$  (i.e.,  $90\%$  of the entire parameters in a network is pruned away). For all tested architectures, pruning for such level of sparsity does not lead to a large accuracy drop. Additionally, we perform either random pruning (at initialization) or a magnitude based pruning (at pretrained) for comparison purposes. Random pruning refers to pruning parameters randomly and globally for a given sparsity level. For the magnitude based pruning, we first train a model and simply prune parameters globally in a single-shot based on the magnitude of the pretrained parameters (i.e., keep the large weights while pruning small ones). For initialization methods, we follow either variance scaling initialization schemes (i.e., VS-L, VS-G, VS-H, as in LeCun et al. (1998); Glorot & Bengio (2010); He et al. (2015), respectively) or (convolutional) orthogonal initialization schemes (Saxe et al., 2014; Xiao et al., 2018).
+
+Training and evaluation. Throughout experiments, we evaluate pruning results on MNIST, CIFAR-10, and Tiny-ImageNet image classification tasks. For training of the pruned sparse networks, we use SGD with momentum and train up to 80k (for MNIST) or 100k (for CIFAR-10 and Tiny-ImageNet) iterations. The initial learning rate is set to be 0.1 and is decayed by 1/10 at every 20k (MNIST) or 25k (CIFAR-10 and Tiny-ImageNet). The mini-batch size is set to be 100, 128, 200 for MNIST, CIFAR-10, Tiny-ImageNet, respectively. We do not optimize anything specific for a particular case, and follow the standard training procedure. For all experiments, we use  $10\%$  of training set for the validation set, which corresponds to 5400, 5000, 9000 images for MNIST, CIFAR-10, Tiny-IamgeNet, respectively. We evaluate at every 1k iteration, and record the lowest test error. All results are the average of either 10 (for MNIST) or 5 (for CIFAR-10 and Tiny-ImageNet) runs.
+
+Signal propagation and approximate dynamical isometry. We use the entire training set when computing Jacobian singular values of a network. In order to enforce approximate dynamical isometry when specified, given a pruned sparse network, we optimize for the objective in Equation 8 using gradient descent. The learning rate is set to be 0.1 and we perform 10k gradient update steps (although it usually reaches to convergence far before). This process is data-free and thus fast; e.g., depending on the size of the network and the number of update steps, it can take less than a few seconds on a modern computer.
+
+Neural architecture sculpting. We provide the model details in Table 6.
+
+Table 6: All models (Equivalents 1,2,3) are initially bigger than the base network (ResNet20), by either being wider or deeper, but pruned to have the same number of parameters as the base network (269k). The widening factor (k) refers to the filter multiplier; e.g., for the basic filter size of 16, the widening factor of  $\mathrm{k} = 2$  will result in 32 filters. The block size refers to the number of residual blocks in each block layer; all models have three block layers. More/less number of residual blocks means the network is deeper/shaller. The reported generalization errors are averages over 5 runs. We find that the technique of architecture sculpting, pruning randomly initialized neural networks based on our signal propagation perspective even in the absence of ground-truth supervision, can be used to find models of superior performance under the same parameter budget.
+
+<table><tr><td>Model category</td><td>Shape</td><td>Widening (k)</td><td>Block size</td><td>Init.</td><td>GT</td><td>Sparsity</td><td>Error</td></tr><tr><td>Base</td><td>ResNet20 (He et al., 2016)</td><td>1</td><td>3</td><td>VS-H</td><td>✓</td><td>0.0</td><td>8.046</td></tr><tr><td rowspan="4">Equivalent 1</td><td>wider</td><td>2</td><td>3</td><td>LDI</td><td>✗</td><td>74.8</td><td>7.618</td></tr><tr><td>wider</td><td>4</td><td>3</td><td>LDI</td><td>✗</td><td>93.7</td><td>7.630</td></tr><tr><td>wider</td><td>6</td><td>3</td><td>LDI</td><td>✗</td><td>97.2</td><td>7.708</td></tr><tr><td>wider</td><td>8</td><td>3</td><td>LDI</td><td>✗</td><td>98.4</td><td>7.836</td></tr><tr><td rowspan="3">Equivalent 2</td><td>wider &amp; shallower</td><td>2</td><td>2</td><td>LDI</td><td>✗</td><td>60.4</td><td>7.776</td></tr><tr><td>wider &amp; shallower</td><td>4</td><td>2</td><td>LDI</td><td>✗</td><td>90.1</td><td>7.876</td></tr><tr><td>wider &amp; shallower</td><td>6</td><td>2</td><td>LDI</td><td>✗</td><td>95.6</td><td>7.940</td></tr><tr><td>Equivalent 3</td><td>deeper</td><td>1</td><td>5</td><td>LDI</td><td>✗</td><td>42.0</td><td>7.912</td></tr></table>
+
+# C SIGNAL PROPAGATION IN SPARSE NETWORKS: ADDITIONAL RESULTS
+
+![](images/75d58dfbf1029ecf58e1d394ba0ceb404e083098d20e49ede0eaee48e6770fe3.jpg)  
+linear  $(K = 7)$
+
+![](images/c826844ef9561bbeecb978a08dddea69879e7e398860d6480717ecff9579f6cf.jpg)  
+tanh (K=7)
+
+![](images/4620c22575a6c5cc26ec300230b86acb65154fecd70deda71e257d66af03ee25.jpg)  
+linear  $(K = 7)$
+
+![](images/2e64de07f82287ae0f8abbbe5b2aeb502a1354457ca378e14bcb0ca3b79d362a.jpg)  
+linear  $(K = 7)$
+
+![](images/11559eca14468b37fdf6528356aeedfcbfe61e98fe92f9e39ac3d43d4ac494b1.jpg)  
+tanh  $(\mathrm{K} = 7)$
+
+![](images/530344968100a24bfb877a3c468734d47baf504f3a01577fdea6785424acfc44.jpg)  
+(a) Signal propagation (all statistics)
+
+![](images/9b46de0c83366488c03885239a32a61aa5d66321bf45d7438533d2ab3de13192.jpg)  
+tanh (K=7)
+
+![](images/914f53e16e1cd5809259789e5197bbca3565beb29f7b4a1f007d606f6a03a9fe.jpg)  
+tanh  $(\mathrm{K} = 7)$
+
+![](images/e8889add26d30b68a09151224307268850156c68a9cc192008eb5235e81bc4e1.jpg)  
+linear  $(K = 7)$
+
+![](images/55351ab648f6aac6d481dcd0ee5bea744a1d7b9ba2114f69363f06d65713401e.jpg)  
+linear  $(K = 7)$  
+(b) Training behavior (loss and accuracy)
+
+![](images/47f7060486220bd52b0edbc36b2951b4cc3f30074c47656ab2f8d7203394c4db.jpg)  
+tanh (K=7)
+
+![](images/467a8d6793d3c65aa7327ec01877866c22b247f2737acca3a73fb983a34632e6.jpg)  
+tanh (K=7)  
+Figure 4: Full results for (a) signal propagation (all signular value statistics), and (b) training behavior (including accuracy) for 7-layer linear and tanh MLP networks. We provide results of LDI-Rand, LDI-Rand-AI, VS-CS, LDI-CS, LDI-CS-AI on the linear case for both singular value statistics and training log. We also plot results of LDI-Mag and LDI-Dense on the tanh case for trainability; the training results of non-pruned (LDI-Dense) and magnitude (LDI-Mag) pruning are only reported for the tanh case, because the learning rate had to be lowered for the linear case (otherwise it explodes), which makes the comparison not entirely fair. We provide the singular value statistics for the magnitude pruning in Figure 6 to avoid clutter. Also, extended training logs for random and magnitude based pruning are provided separately in Figure 5 to illustrate the difference in convergence speed.
+
+![](images/5602bd04bce9a2cd4e84fbe41b46fc19e6dab63f386f670d33d8f9037bdb6d0b.jpg)  
+(a) Training behavior
+
+![](images/d42e0d172125fbb9829cbb36d0006ee8611ef86484e3edc05e68b7769fa08058.jpg)
+
+![](images/1407e418e128913380020fa5feaca4621e7e0478b063985c2460e4d17e684597.jpg)
+
+![](images/8a4603b34f36db4137a5608206e5edc97e39bc51eee90d5ce74236e03f03974f.jpg)  
+Figure 5: Extended training log (i.e., Loss and Accuracy) for random (Rand) and magnitude (Mag) pruning. The sparse networks obtained by random or magnitude pruning take a much longer time to train than that obtained by pruning based on connection sensitivity. All methods are pruned at the layerwise orthogonal initialization, and trained the same way as before.
+
+![](images/8649690c9dcce16e7c04dfd6292bf5b77c8a3b83c49035c132dfa5e38c6a84e8.jpg)
+
+![](images/dbf94f9e048ef22407b807ea8a14e1a0efe1fe734ba030f0e748016966cf2b5c.jpg)
+
+![](images/e28b95f17ab91445b5670399428d7f2e4e5d942dae07fe5419fc5bbe5bddff2d.jpg)  
+(a) Signal propagation (all statistics; magnitude based pruning)
+
+![](images/afd93c049b2584daa4035474d60e8bf66f2842320026607f4cf9da5dcba668e1.jpg)
+
+![](images/af60b59dc9e0de90666bb8c5ff3bc6fe4588c732d740c0b6d5affbc9ea2521d4.jpg)
+
+![](images/80ded1d01714665490d0fe2a552f6c2e6f547cd36e952ccc7ff46b4b8972520b.jpg)
+
+Figure 6: Signal propagation measurments (all signular value statistics) for the magnitude based pruning (Mag) on the 7-layer linear and tanh MLP networks. As described in the experiment settings in Appendix B, the magnitude based pruning is performed on a pretrained model. Notice that unlike other cases where pruning is done at initialization (i.e., using either random or connection sensitivity based pruning methods), the singular value distribution changes abruptly when pruned (i.e., note of the sharp change of singular values from 0 to  $10\%$  sparsity). Also, the singular values are not concentrated (note of high standard deviations), which explains rather inferior trainability compared to other methods. We conjecture that naively pruning based on the magnitude of parameters in a single-shot, without pruning gradually or employing some sophisticated tricks such as layerwise thresholding, can lead to a failure of training compressed networks.
+
+![](images/88d0cdc4eb3610675ae0a44158f8d9523fbed446a7ac0a126980d7aa54810c94.jpg)  
+Figure 7: Signal propagation and training behavior for ReLU and Leaky-ReLU activation functions. They resemble those of the tanh case as in Figure 2, and hence the conclusion holds about the same.
+
+![](images/2d35c1aa87cbbd9106e26ee9ef038495e2bb03188302af789a0b36bd9d378833.jpg)
+
+![](images/f96c9ee1c4c80d11a6169f8270290899291cd251c58ce7aabf43aa234d5a4e75.jpg)
+
+![](images/8e07c1178bafa5aa8f414c70f5260c01822073e76764c5b2cac529ee1596de32.jpg)
+
+![](images/68dfd445c93d88bdf3d3b0bcc3541af01991bc966c7e257a4f36470e0a60eb88.jpg)
+
+![](images/de42a2545879dab4e551becf2743ee0635fbda2a005cb1a744cdba8d3db7c6f5.jpg)  
+Figure 8: Training performance (loss and accuracy) by different methods for VGG16 on CIFAR-10. To examine the effect of initialization in isolation on the trainability of sparse neural networks, we remove batch normalization (BN) layers for this experiment, as BN tends to improve training speed as well as generalization performance. As a result, enforcing approximate isometry (LDI-CS-AIF) improves the training speed quite dramatically compared to the pruned network without isometry (LDI-CS). We also find that even compared to the non-pruned dense network (LDI-Dense) which is ensured layerwise dynamical isometry, LDI-CS-AIF trains faster in the early training phase. This result is quite promising and more encouraging than the previous case of MLP (see Figures 2 and 7), as it potentially indicates that an underparameterized network (by connection sensitivity pruning) can even outperform an overparameterized network, at least in the early phase of neural network training. Furthermore, we add results of using the spectral norm in enforcing approximate isometry in Equation 8 (LDI-CS-AIS), and find that it also trains faster than the case of broken isometry (LDI-CS), yet not as much as the case of using the Frobenius norm (LDI-CS-AIF).
\ No newline at end of file
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+# ASYMPTOTICS OF WIDE NETWORKS FROM FEYNMAN DIAGRAMS
+
+Ethan Dyer & Guy Gur-Ari*
+
+Google
+
+Mountain View, CA
+
+{edyer,guyga}@google.com
+
+# ABSTRACT
+
+Understanding the asymptotic behavior of wide networks is of considerable interest. In this work, we present a general method for analyzing this large width behavior. The method is an adaptation of Feynman diagrams, a standard tool for computing multivariate Gaussian integrals. We apply our method to study training dynamics, improving existing bounds and deriving new results on wide network evolution during stochastic gradient descent. Going beyond the strict large width limit, we present closed-form expressions for higher-order terms governing wide network training, and test these predictions empirically.
+
+# 1 INTRODUCTION
+
+Neural networks achieve remarkable performance on a wide array of machine learning tasks, yet a complete analytic understanding of deep networks remains elusive. One promising approach is to consider the large width limit, in which the number of neurons in one or several layers is taken to be large. In this limit one can use a mean-field approach to better understand the network's properties at initialization (Neal (1996); Lee et al. (2018)), as well as its training dynamics (Daniely (2017); Jacot et al. (2018)). Additional related works are cited below.
+
+Suppose that  $f(x)$  is the network function evaluated at an input  $x$ . Let us denote the vector of model parameters by  $\theta$ , whose elements are initially chosen to be i.i.d. Gaussian. In this work we consider a class of functions we call correlation functions, obtained by taking the ensemble averages of  $f$ , its products, and its derivatives with respect to the parameters  $\theta$ , evaluated on arbitrary inputs. Here are a few examples of correlation functions.
+
+$$
+\mathbb {E} _ {\theta} \left[ f \left(x _ {1}\right) f \left(x _ {2}\right) \right], \sum_ {\mu} \mathbb {E} _ {\theta} \left[ \frac {\partial f \left(x _ {1}\right)}{\partial \theta^ {\mu}} \frac {\partial f \left(x _ {2}\right)}{\partial \theta^ {\mu}} \right], \sum_ {\mu , \nu} \mathbb {E} _ {\theta} \left[ \frac {\partial f \left(x _ {1}\right)}{\partial \theta^ {\mu}} \frac {\partial f \left(x _ {2}\right)}{\partial \theta^ {\nu}} \frac {\partial^ {2} f \left(x _ {3}\right)}{\partial \theta^ {\mu} \partial \theta^ {\nu}} f \left(x _ {4}\right) \right]. \tag {1}
+$$
+
+Correlation functions often show up in the study of wide networks. For example, the first correlation function in (1) plays a central role in the Gaussian Process picture of wide networks (Lee et al. (2018)), and has been used to diagnose signal propagation in wide networks (Pennington et al. (2017)). The second example in (1) is the ensemble average of the Neural Tangent Kernel (NTK), which controls the evolution of wide networks under gradient flow (Jacot et al. (2018)), and the third example shows up when computing the time derivative of the NTK with MSE loss.
+
+While correlation functions can be computed analytically in some special cases (Cho & Saul (2009)), they are not analytically tractable in general. In this work, we present a method for bounding the asymptotic behavior of such functions at large width. In many cases of interest, the bound we obtain is tight. Derivation of the method relies on Feynman diagrams (Feynman (1949)), a technique for calculating multivariate Gaussian integrals, and specifically on the 't Hooft expansion ('t Hooft (1974)). However, applying the method is straightforward and does not require any knowledge of Feynman diagrams.
+
+Motivation. The method presented here is a general-purpose tool that can be added to a researcher's toolbox. It allows a researcher to quickly identify ways in which network behavior and training
+
+dynamics simplify at large width, for example by deriving which quantities vanish in this limit. As we demonstrate, using this method one is able to dramatically cut down on the amount of effort required to derive several key results related to wide networks, as well as to go beyond the infinite width limit and investigate ordinary neural networks.
+
+# Our contribution.
+
+1. We present a general method for bounding the asymptotic behavior of correlation functions. The method is an adaptation of Feynman diagrams to the case of wide neural networks. The adaptation involves a novel treatment of derivatives of the network function, an element that is not present in the original theoretical physics formulation.  
+2. We apply the method to the study of wide network evolution under gradient descent. We improve on existing results for gradient flow (Jacot et al. (2018)) by deriving tighter bounds, and extending the analysis to the case of stochastic gradient descent (SGD). Going beyond the infinite-width limit, we present a formalism for deriving finite-width corrections to network evolution, and present explicit formulas for the first order correction. To our knowledge, this is the first time this correction has been calculated.  
+3. As additional applications of our method, in Appendix E.2 we show that in the large width limit the SGD updates are linear in the learning rate, and in Appendix E.3 we discuss finite width corrections to the spectrum of the Hessian.
+
+Limitations of our approach. The main result of this paper is a conjecture. We test our predictions extensively using numerical experiments, and prove the conjecture in some cases, but we do not have a proof that applies to all the cases we tested, including for deep networks with general non-linearities. Furthermore, our method can only be used to derive asymptotic bounds at large width; it does not produce the width-independent coefficient, which is often of interest.
+
+Related work. For additional works on wide networks, including relating them to Gaussian processes, see de G. Matthews et al. (2018); Novak et al. (2019); Garriga-Alonso et al. (2019); Yang (2019); Pennington & Bahri (2017); Pennington & Worah (2018); Schoenholz et al. (2017); Xiao et al. (2018); Chen et al. (2018); Daniely et al. (2016); Lee et al. (2019); Du et al. (2018b;a); Allen-Zhu et al. (2018). For additional works discussing the training dynamics of wide networks see Geiger et al. (2019a); Arora et al. (2019). For a previous use of diagrams in this context, see Pennington & Worah (2017). The Neural Tangent Hierarchy presented in Huang & Yau (2019), published during the completion of this version, has significant overlap with the recursive differential equations (10) presented below.
+
+The rest of the paper is organized as follows. In Section 2 we present our main conjecture and supporting evidence. In Section 3 we apply the method to gradient descent evolution of wide networks, and in Section 4 we present details on Feynman diagrams, which is the basic technique used in our proofs. We conclude with a Discussion. Proofs, additional applications, and details can be found in the Appendices.
+
+Note: An earlier version of this work appeared in the ICML 2019 workshop, Theoretical Physics for Deep Learning (Dyer & Gur-Ari (2019)).
+
+# 2 CORRELATION FUNCTION ASYMPTOTICS
+
+In this section we present our main result: a method for computing asymptotic bounds on correlation functions of wide networks. We present the result as a conjecture, supported by analytic and empirical evidence.
+
+# 2.1 NOTATION
+
+Let  $f(x) \in \mathbb{R}$  be the network output of a deep network with  $d$  hidden layers and input  $x \in \mathbb{R}^{D_{\mathrm{in}}}$ , defined by
+
+$$
+f (x) = n ^ {- 1 / 2} V ^ {T} \sigma \left(n ^ {- 1 / 2} W ^ {d - 1} \dots \sigma \left(n ^ {- 1 / 2} W ^ {1} \sigma (U x)\right)\right). \tag {2}
+$$
+
+Here  $U \in \mathbb{R}^{n \times D_{\mathrm{in}}}$ ,  $V \in \mathbb{R}^n$ ,  $W^1, \ldots, W^{d-1}$  are weight matrices of dimension  $n$ , and  $\sigma: \mathbb{R} \to \mathbb{R}$  is the non-linearity. We denote the vector of all model parameters by  $\theta$ . At initialization, the elements of  $\theta$  are independent Gaussian variables, with each element  $\theta^\mu \sim \mathcal{N}(0, 1)$ . The corresponding distribution of  $\theta$  is denoted by  $\mathcal{P}_0$ .
+
+Let us now define correlation functions, the class of functions that is the focus of this work. These functions involve derivative tensors of the network function. We denote the rank-  $k$  derivative tensor by  $T_{\mu_1\dots \mu_k}(x;f)\coloneqq \partial^k f(x) / \partial \theta^{\mu_1}\dots \partial \theta^{\mu_k}$ . For  $k = 0$  we define  $T(x;f)\coloneqq f(x)$ , and still refer to this as a derivative tensor for consistency.
+
+Definition 1. A correlation function is the expectation value of a product of derivative tensors, evaluated at arbitrary inputs, where the tensor indices are summed in pairs over all the model parameters. A general correlation function  $C$  takes the form
+
+$$
+C \left(x _ {1}, \dots , x _ {m}\right) := \sum_ {\mu_ {1}, \dots , \mu_ {k _ {m}}} \Delta_ {\mu_ {1} \dots \mu_ {k _ {m}}} ^ {(\pi)} \mathbb {E} _ {\theta} \left[ T _ {\mu_ {1} \dots \mu_ {k _ {1}}} \left(x _ {1}\right) T _ {\mu_ {k _ {1} + 1} \dots \mu_ {k _ {2}}} \left(x _ {2}\right) \dots T _ {\mu_ {k _ {m - 1} + 1} \dots \mu_ {k _ {m}}} \left(x _ {m}\right) \right]. \tag {3}
+$$
+
+Here,  $0 \leq k_{1} \leq \dots \leq k_{m-1} \leq k_{m}$  are integers,  $m$  and  $k_{m}$  are even,  $\pi \in S_{k_{m}}$  is a permutation, and  $\Delta_{\mu_{1} \ldots \mu_{k_{m}}}^{(\pi)} = \delta_{\mu_{\pi(1)} \mu_{\pi(2)}} \cdots \delta_{\mu_{\pi(k_{m}-1)} \mu_{\pi(k_{m})}}$ . We use  $\delta$  to denote the Kronecker delta.
+
+If two derivative tensors in a correlation function have matching indices that are summed over, we say that they are contracted. For example, the correlation function  $\sum_{\mu} \mathbb{E}_{\theta} \left[ \frac{\partial f(x_1)}{\partial \theta^{\mu}} \cdot \frac{\partial f(x_2)}{\partial \theta^{\mu}} \right]$  has one pair of contracted tensors. See (1) for additional examples of correlation functions.
+
+# 2.2 ASYMPTOTIC BOUNDS ON WIDE NETWORKS
+
+We now present our main conjecture, which allows us to place asymptotic bounds on general correlation functions of wide networks.
+
+Conjecture 1. Let  $C(x_{1},\ldots ,x_{m})$  be a correlation function. The cluster graph  $G_{C}(V,E)$  of  $C$  is a graph with vertices  $V = \{v_{1},\dots,v_{m}\}$ , where  $v_{i} = T(x_{i})$  for all  $i$ , and edges  $E = \{(v_{i},v_{j}) \mid (T(x_{i}),T(x_{j}))$  contracted in  $C\}$ . Suppose that the cluster graph  $G_{C}$  has  $n_e$  connected components with an even size (even number of vertices), and  $n_o$  components of odd size. Then  $C(x_{1},\ldots ,x_{m}) = \mathcal{O}(n^{s_C})$ , where
+
+$$
+s _ {C} = n _ {e} + \frac {n _ {o}}{2} - \frac {m}{2}. \tag {4}
+$$
+
+We will refer to the connected components of a cluster graph  $G_{C}$  as the clusters of  $C$ . Table 1 lists examples of bounds derived using the Conjecture for several correlation functions. The intuition behind Conjecture 1 comes from the following result for deep linear networks.
+
+Theorem 1. Conjecture 1 holds for correlation functions of networks with linear activations, including for deep linear networks with biases (affine transformations).
+
+Let us discuss the intuition behind this theorem. Computing correlation functions of deep linear networks amounts to evaluating Gaussian integrals with polynomial integrands in  $\theta$ . One can evaluate such integrals using Isserlis' theorem, which tells us how to express moments of multivariate Gaussian variables in terms of their second moments. For example, given centered Gaussian variables  $z_{1},\ldots,z_{4}$ ,
+
+$$
+\mathbb {E} _ {z} \left[ z _ {1} z _ {2} z _ {3} z _ {4} \right] = \mathbb {E} _ {z} \left[ z _ {1} z _ {2} \right] \mathbb {E} _ {z} \left[ z _ {3} z _ {4} \right] + \mathbb {E} _ {z} \left[ z _ {1} z _ {3} \right] \mathbb {E} _ {z} \left[ z _ {2} z _ {4} \right] + \mathbb {E} _ {z} \left[ z _ {1} z _ {4} \right] \mathbb {E} _ {z} \left[ z _ {2} z _ {3} \right]. \tag {5}
+$$
+
+Therefore, correlation functions of deep linear networks can be expressed in terms of the covariances  $\mathbb{E}_{\theta}\left[U_{i\alpha}U_{j\beta}\right] = \delta_{ij}\delta_{\alpha \beta},\mathbb{E}_{\theta}\left[V_iV_j\right] = \delta_{ij}$ , and  $\mathbb{E}_{\theta}\left[W_{ij}^{(l)}W_{kl}^{(l)}\right] = \delta_{ik}\delta_{jl}$ . For example, for a deep linear network with 2 hidden layers, we have
+
+$$
+\mathbb {E} _ {\theta} \left[ f \left(x _ {1}\right) f \left(x _ {2}\right) \right] = \frac {1}{n ^ {2}} \mathbb {E} _ {\theta} \left[ V ^ {T} W U x _ {1} V ^ {T} W U x _ {2} \right] = \frac {x _ {1} ^ {T} x _ {2}}{n ^ {2}} \sum_ {i, k} ^ {n} \delta_ {i k} \delta_ {i k} \sum_ {j, l} ^ {n} \delta_ {j l} \delta_ {j l} = x _ {1} ^ {T} x _ {2}. \tag {6}
+$$
+
+Every correlation function of a deep linear network can be similarly reduced to sums over products of Kronecker delta functions and width-independent functions of the inputs. The asymptotic large width behavior is determined by these sums over delta functions, which are tedious to compute by hand. Feynman diagrams are a graphical tool for computing these sums, allowing us to obtain the asymptotic behavior with minimal effort. This tool, which is described in detail in Section 4, is used to prove Theorem 1.
+
+For networks with non-linear activations we further show the following
+
+Theorem 2. Conjecture 1 holds for (1) networks with ReLU activations, where all inputs are set to be equal, and for (2) networks with one hidden layer and smooth activation.
+
+For case (1), the idea behind the proof is to put an asymptotic bound on the ReLU network in terms of a corresponding deep linear network. For case (2), the basic idea is that each network function contains a single sum over the width, and by keeping track of these sums using Feynman diagrams we are able to bound the asymptotic behavior. We refer the reader to Appendix C for details.
+
+# 2.3 NUMERICAL EXPERIMENTS
+
+Table 1 lists asymptotic bounds on several correlation functions, derived using Conjecture 1. These are compared against the asymptotic behavior computed using numerical experiments. In addition to the results presented here, we performed experiments using the same correlation functions and experimental setup, but with weights sampled uniformly from  $\{\pm 1\}$  instead of from a Gaussian distribution. The results are shown in Appendix A.1.
+
+In all cases that were tested empirically, we found that Conjecture 1 holds. For networks with smooth activations, we found that Conjecture 1 always gives a tight bound. For networks with linear or ReLU activations, we always find that the Conjecture holds as an upper bound, but that sometimes the bound is not tight. One such case is highlighted in Table 1. In such cases, a tight bound can be obtained for deep linear networks using the complete Feynman diagram analysis presented below. $^4$
+
+<table><tr><td>Correlation function C</td><td>ne, no</td><td>sc</td><td>lin.</td><td>ReLU</td><td>tanh</td></tr><tr><td>Eθ[f(x1)f(x2)]</td><td>0,2</td><td>0</td><td>-0.02</td><td>0.003</td><td>-0.02</td></tr><tr><td>Eθ[f(x1)f(x2)f(x3)f(x4)]</td><td>0,4</td><td>0</td><td>-0.01</td><td>0.03</td><td>-0.03</td></tr><tr><td>∑μEθ[∂μf(x1)∂μf(x2)]</td><td>1,0</td><td>0</td><td>0.00</td><td>0.00</td><td>0.00</td></tr><tr><td>∑μ,νEθ[∂μf(x1)∂νf(x2)∂μ,νf(x3)f(x4)]</td><td>0,2</td><td>-1</td><td>-0.98</td><td>-1.03</td><td>-1.01</td></tr><tr><td>∑μ,ν,ρEθ[∂μf(x1)∂νf(x2)∂ρf(x3)∂μ,ν,ρf(x4)]</td><td>1,0</td><td>-1</td><td>-2.01*</td><td>-2.01*</td><td>-0.98</td></tr><tr><td>∑μ,ν,ρ,σEθ[∂μf(x1)∂νf(x2)∂μ,νf(x3)∂ρf(x4)∂σf(x5)∂ρ,σf(x6)]</td><td>0,2</td><td>-2</td><td>-2.05</td><td>-2.01</td><td>-1.99</td></tr></table>
+
+Table 1: Examples of bounds on correlation functions obtained from Conjecture 1. The 3 right-most columns list numerical results for fully-connected networks with 3 hidden layers and with linear, ReLU, and tanh activations. The values listed in these columns are the fitted exponents for networks with the corresponding activations. The entries marked with an asterisk are those for which the predicted bound is not tight. The numerical results are obtained by computing the correlation functions for networks with widths  $2^{7}, 2^{8}, \ldots, 2^{13}$ , each averaged over 1,000 initializations, and fitting the exponent. Inputs are chosen to be random vectors of dimension 4.
+
+# 3 APPLICATIONS TO TRAINING DYNAMICS
+
+In this section we apply Conjecture 1 to study the evolution of wide networks under gradient flow and gradient descent. We begin by briefly reviewing existing results. Let  $D_{\mathrm{tr}}$  be a training set of size  $M$ ,
+
+and let  $L = \sum_{(x,y) \in D_{\mathrm{tr}}} \ell(x,y)$  be the MSE loss, with single sample loss  $\ell(x,y) = \frac{1}{2} (f(x) - y)^2$ . The gradient flow equation is  $\frac{d\theta}{dt} = -\nabla_{\theta} L$ . The evolution of the network function under gradient flow is given by
+
+$$
+\frac {d f (x)}{d t} = - \sum_ {\left(x ^ {\prime}, y ^ {\prime}\right) \in D _ {\mathrm {t r}}} \Theta \left(x, x ^ {\prime}\right) \frac {\partial \ell \left(x ^ {\prime} , y ^ {\prime}\right)}{\partial f}. \tag {7}
+$$
+
+Here,  $\Theta$  is the Neural Tangent Kernel (NTK), defined by  $\Theta(x_1, x_2) \coloneqq \nabla_\theta f^T(x_1) \nabla_\theta f(x_2)$ . The authors of Jacot et al. (2018) showed that the kernel is constant during training up to  $\mathcal{O}(n^{-1/2})$  corrections. This leads to a dramatic simplification in training dynamics (Jacot et al. (2018); Lee et al. (2019)). In particular, for MSE loss the network map evaluated on the training data evolves as  $f(t) = y + e^{-t\Theta^{(0)}}(f^{(0)} - y)$ . We will use our technology to derive a tighter bound on finite-width corrections to the kernel during training and present explicit formulas for the leading correction.
+
+The following result is useful in analyzing the behavior of correlation functions under gradient flow.
+
+Lemma 1. Let  $C(\vec{x}) = \mathbb{E}_{\theta}\left[F(\vec{x})\right]$  be a correlation function, where  $\vec{x} = (x_{1},\dots,x_{m})$ , and suppose that  $C = \mathcal{O}(n^{s_C})$  for  $s_C$  as defined in Conjecture 1. Then  $\mathbb{E}_{\theta}\left[\frac{d^kF(\vec{x})}{dt^k}\right] = \mathcal{O}(n^{s_C})$  for all  $k$ .
+
+Here we prove the statement for  $k = 1$ . Appendix D.2 contains a proof for the general case.
+
+Proof  $(k = 1)$ . Let  $C$  have  $n_e$  even clusters and  $n_o$  odd clusters. Consider  $\mathbb{E}_{\theta}\left[\frac{dF(\vec{x})}{dt}\right] = -\sum_{\mu}\sum_{x'\in D_{\mathrm{tr}}}\mathbb{E}_{\theta}\left[\frac{\partial F(\vec{x})}{\partial\theta^{\mu}}\frac{\partial f(x')}{\partial\theta^{\mu}} f(x')\right]$ . Denote by  $n_e'(n_o')$  the number of even (odd) clusters in this correlation function, which has  $m' = m + 2$  derivative tensors. One can check that either  $(n_e', n_o') = (n_e + 1, n_o)$  or  $(n_e', n_o') = (n_e - 1, n_o + 2)$ , depending on whether the  $\partial_{\mu}F$  derivative is acting on an odd or even cluster in  $F$ . Therefore,  $n_e' + \frac{n_o'}{2} - \frac{m'}{2} \leq s_C$ .
+
+With this result, it is easy to understand the constancy of the NTK at large width. The first derivative of the NTK is given by
+
+$$
+\mathbb {E} _ {\theta} \left[ \frac {d \Theta \left(x _ {1} , x _ {2}\right)}{d t} \right] = - \sum_ {x ^ {\prime} \in D _ {\mathrm {t r}}} ^ {M} \sum_ {\mu , \nu} \mathbb {E} _ {\theta} \left[ \frac {\partial^ {2} f (x _ {1})}{\partial \theta^ {\mu} \partial \theta^ {\nu}} \frac {\partial f (x _ {2})}{\partial \theta^ {\mu}} \frac {\partial f \left(x ^ {\prime}\right)}{\partial \theta^ {\nu}} f \left(x ^ {\prime}\right) \right] + \left(x _ {1} \leftrightarrow x _ {2}\right). \tag {8}
+$$
+
+Here,  $(x_{1}\leftrightarrow x_{2})$  means "add the same term, but exchange  $x_{1}$  and  $x_{2}$ ". This correlation function has  $n_e = 0$ ,  $n_o = 2$ , and  $m = 4$ , and is therefore  $\mathcal{O}(n^{-1})$  by Conjecture 1. By Lemma 1, all higher-order time derivatives of the NTK are  $\mathcal{O}(n^{-1})$  as well. If we now assume that the time-evolved kernel  $\Theta (t)$  is analytic in training time  $t$ , and that we are free to exchange the Taylor expansion in time with the large width limit, then we find that  $\mathbb{E}_{\theta}\left[\Theta (t) - \Theta (0)\right] = \sum_{k = 1}^{\infty}\frac{t^{k}}{k!}\mathbb{E}_{\theta}\left[\frac{d^{k}\Theta(0)}{dt^{k}}\right] = \mathcal{O}(n^{-1})$  for any fixed  $t$ . This bound, which is tighter than that found in Jacot et al. (2018), was noticed empirically in Lee et al. (2019) as well as in our own experiments, see Figure 1a.
+
+This analysis can be extended to show that  $\mathbb{E}_{\theta}[\Theta (t) - \Theta (0)] = \mathcal{O}(n^{-1})$  for SGD as well. The technique is similar, and again relies on Conjecture 1. We refer the reader to Appendix E.1 for details. These results improve on existing ones in several ways. Our method applies to the case of SGD as well as to networks where all layer widths are increased simultaneously — a setup that has proven to be difficult to analyze. In addition, the  $\mathcal{O}(n^{-1})$  bound we derive on kernel corrections during SGD is empirically tight, and improves on the existing bound of  $\mathcal{O}(n^{-1 / 2})$  which was derived for gradient flow (Jacot et al. (2018)).
+
+# 3.1 FINITE WIDTH CORRECTIONS
+
+Next, we will compute the explicit time dependence of  $\Theta$  and  $f$  at order  $\mathcal{O}(n^{-1})$  under gradient flow. This is the leading correction to the infinite width result. We define the functions  $O_{1}(x) \coloneqq f(x)$  and
+
+$$
+O _ {s} \left(x _ {1}, \dots , x _ {s}\right) := \sum_ {\mu} \frac {\partial O _ {s - 1} \left(x _ {1} , \dots , x _ {s - 1}\right)}{\partial \theta_ {\mu}} \frac {\partial f \left(x _ {s}\right)}{\partial \theta_ {\mu}}, \quad s \geq 2. \tag {9}
+$$
+
+Notice that  $O_2 = \Theta$  is the kernel. It is easy to check that
+
+$$
+\frac {d O _ {s} \left(x _ {1} , \dots , x _ {s}\right)}{d t} = - \sum_ {\left(x ^ {\prime}, y ^ {\prime}\right) \in D _ {\mathrm {t r}}} O _ {s + 1} \left(x _ {1}, \dots , x _ {s}, x ^ {\prime}\right) \left(f \left(x ^ {\prime}\right) - y ^ {\prime}\right), \quad s \geq 1. \tag {10}
+$$
+
+We can use equations (9) and (10) to solve for the evolution of the kernel and network map. Notice that each function  $O_{s}$  has  $s$  derivative tensors and a single cluster. As a result, correlation functions involving operators with larger  $s$  are increasingly suppressed in width. In particular,  $\mathbb{E}_{\theta}\left[dO_4 / dt\right] = -\sum_{x\in D_{\mathrm{tr}}}\mathbb{E}_{\theta}\left[O_5(x)f(x)\right] = \mathcal{O}(n^{-2})$  at all times, using Conjecture 1 and Lemma 1. Thus to solve for  $f$  and  $\Theta$  at  $\mathcal{O}(n^{-1})$  we can set  $O_{s} = 0$  for all  $s\geq 5$ .
+
+Let us denote the time-evolved kernel by  $\Theta (t) = \Theta^{(0)} + \Theta_{1}(t) + \mathcal{O}(n^{-2})$ , where  $\Theta^{(0)}$  is the kernel at initialization, and  $\Theta_1(t)$  is the  $\mathcal{O}(n^{-1})$  correction we are seeking. Integrating equations (10) starting with  $s = 4$ , we find
+
+$$
+\begin{array}{l} \Theta_ {1} (x _ {1}, x _ {2}; t) = - \int_ {0} ^ {t} d t ^ {\prime} \sum_ {x \in D _ {\mathrm {t r}}} O _ {3} ^ {(0)} (x _ {1}, x _ {2}, x) \Delta f (x; t ^ {\prime}) \\ + \int_ {0} ^ {t} d t ^ {\prime} \int_ {0} ^ {t ^ {\prime}} d t ^ {\prime \prime} \sum_ {x, x ^ {\prime} \in D _ {\mathrm {t r}}} O _ {4} ^ {(0)} \left(x _ {1}, x _ {2}, x, x ^ {\prime}\right) \Delta f \left(x ^ {\prime}; t ^ {\prime \prime}\right) \Delta f \left(x; t ^ {\prime}\right). \tag {11} \\ \end{array}
+$$
+
+Here we have introduced the notation  $\Delta f(x; t) = e^{-t\Theta_0}(f^{(0)} - y)$ . A detailed derivation can be found in Appendix E.4. There we also evaluate the integrals in (11) in terms of the NTK spectrum.
+
+To obtain the  $\mathcal{O}(n^{-1})$  correction to the network map (evaluated for simplicity on the training data), we further integrate (10) for  $s = 1$  and find
+
+$$
+f (t) = f _ {0} (t) - e ^ {- t \Theta^ {(0)}} \int_ {0} ^ {t} d t ^ {\prime} e ^ {t ^ {\prime} \Theta^ {(0)}} \Theta_ {1} \left(t ^ {\prime}\right) e ^ {- t ^ {\prime} \Theta^ {(0)}} \left(f ^ {(0)} - y\right) + \mathcal {O} \left(n ^ {- 2}\right). \tag {12}
+$$
+
+Here we have denote the infinite width evolution by  $f_0(t) = y + e^{-t\Theta^{(0)}}(f^{(0)} - y)$ . Figures 1b and 1c compare these predictions against empirical results.
+
+![](images/aabb4aa5eeb087db6230c619ce913316a86ada5c669b70e4e9bc7da5254b7d00.jpg)  
+(a) Mean deviation from init.
+
+![](images/ad1311fa57e85e71be09d0b9866aede7cda36c045ba24649356e95d79862259c.jpg)  
+(b)  $f$  beyond leading order.  
+Figure 1: Empirical verification of predicted asymptotics. (a) The mean deviation of the NTK from its initial value for a variety of widths and activation functions. The fit (dashed) matches well with the predicted  $\mathcal{O}(n^{-1})$  asymptotics. (b-c) Comparison between the empirical evolution (solid) and the  $\mathcal{O}(n^{-1})$  predicted evolution (dashed) for the network function and the kernel. All experiments were performed on two-class MNIST, computing a single randomly-chosen component of  $\Theta$  or  $f$ . See Appendix A for additional experimental details.
+
+![](images/8e3bc880d80f6dad529b8de22862a22a57ac26019c3d9606076c145e45dbe916.jpg)  
+(c) Kernel evolution.
+
+![](images/540ee1f7b2e76ec657a89c804f21dd8bee6a1f1f3bbd986363f9636cc30835ba.jpg)  
+(a)
+
+![](images/d803878995ad85c83067892fe7802e38adcf333cab85a979c94e4a751728f5aa.jpg)  
+Figure 2: Feynman diagram examples. (a) The Feynman diagram of  $\mathbb{E}_{\theta}[f(x)f(x')]$ . (b)-(c) The Feynman diagrams of  $\mathbb{E}_{\theta}[f(x_1)f(x_2)f(x_3)f(x_4)]$ ; additional, equivalent diagrams are not shown.
+
+![](images/1a6847c98302b3147ba610d761346b74d599194c264c309b715ea72fcf46439d.jpg)  
+(b)
+
+![](images/b9bd1aa8f13762781622db8861ed38939f52e4ab0e09d24016a226bebf96b727.jpg)  
+(c)
+
+# 4 FEYNMAN DIAGRAMS FOR DEEP LINEAR NETWORKS
+
+In this section we present the Feynman diagram technique, and show how it allows us to compute the asymptotic behavior of correlation functions. We end this section with a proof of Theorem 1 for the case of networks with a single hidden layer and linear activations.
+
+Given a correlation function  $C$ , we map it to a family of graphs called Feynman diagrams. The graphs are independent of the inputs, and are defined as follows.
+
+Definition 2. Let  $C(x_{1},\ldots ,x_{m})$  be a correlation function for a network with  $d$  hidden layers. The family  $\Gamma (C)$  is the set of all graphs that have the following properties.
+
+1. There are  $m$  vertices  $v_{1},\ldots ,v_{m}$  , each of degree  $d + 1$  
+2. Each edge has a type  $t \in \{U, W^1, \ldots, W^{d-1}, V\}$ . Every vertex has one edge of each type.  
+3. If two derivative tensors  $T_{\mu_1,\dots,\mu_\ell}(x_i), T_{\nu_1,\dots,\nu_{\ell'}}(x_j)$  are contracted  $k$  times in  $C$ , the graph must have at least  $k$  edges (of any type) connecting the vertices  $v_i, v_j$ .
+
+The graphs in  $\Gamma(C)$  are called the Feynman diagrams of  $C$ .
+
+Single hidden layer. For the rest of this section we focus on networks with a single hidden layer. We refer the reader to Appendix B for a full treatment of deep linear networks. For networks with one hidden layer and linear activation, the network output is  $f(x) = n^{-1/2} V^T U x$ . Consider the correlation function  $C(x_1, x_2) = \mathbb{E}_{\theta} [f(x_1) f(x_2)]$ . We have
+
+$$
+C \left(x _ {1}, x _ {2}\right) = \frac {1}{n} \sum_ {i, j} ^ {n} \sum_ {\alpha , \beta} ^ {D _ {\text {i n}}} \mathbb {E} _ {V} \left[ V _ {i} V _ {j} \right] \mathbb {E} _ {U} \left[ U _ {i \alpha} U _ {j \beta} \right] x _ {1} ^ {\alpha} x _ {2} ^ {\beta} = \frac {x _ {1} ^ {T} x _ {2}}{n} \sum_ {i, j} ^ {n} \delta_ {i j} \delta_ {i j} = x _ {1} ^ {T} x _ {2}. \tag {13}
+$$
+
+As the factors of  $x_{1}$  and  $x_{2}$  are independent of  $n$ , we see that  $C(x_{1},x_{2}) = \mathcal{O}(n^{0})$ . Notice that there are two relevant contributions to this answer: each factor of the network function in the integrand contributes  $n^{-1 / 2}$ , and the summed-over product of Kronecker deltas contributes  $n$ . Other details, such as the input dependence, are irrelevant. Feynman diagrams allow us to encode only those details that affect the  $n$  scaling, ignoring the rest.
+
+The set  $\Gamma(C)$  for the correlation function (13) consists of a single Feynman diagram, shown in Figure 2a. The asymptotic bound on a correlation function is obtained by the following result, which is due to 't Hooft (1974).
+
+Theorem 3. Let  $C(x_{1},\ldots ,x_{m})$  be a correlation function with one hidden layer and linear activation. Then  $C = \mathcal{O}(n^{s})$  where  $s = \max_{\gamma \in \Gamma (C)}l_{\gamma} - \frac{m}{2}$ , and  $l_{\gamma}$  is the number of loops in  $\gamma$ .<sup>7</sup>
+
+Let us give some intuition for Theorem 3. Each Feynman diagram  $\gamma$  encodes a subset of the terms contributing to the correlation function. To get the asymptotic bound on the correlation function, we sum over the contributions of individual diagrams. We can compute the asymptotic behavior of a single diagram  $\gamma$  using the following Feynman rules: (1) each vertex contributes a factor of  $n^{-1/2}$ , and (2) each loop contributes a factor of  $n$ . Therefore, if a diagram has  $l_{\gamma}$  loops, its contribution to
+
+![](images/0ea584ba11eb4e5d86dbee5e766cd5a28169dd790df9972a487910b248f33e02.jpg)  
+(a)
+
+![](images/3aba3371af365c26296ebd8ece4a6e09904b316cfc41a9cceda047c6c1750053.jpg)  
+(b)  
+Figure 3: Feynman diagrams for  $\mathbb{E}_{\theta}[\Theta (x,x^{\prime})]$  with one hidden layer. The dashed vertical line represents vertices forced by contracted derivatives.
+
+the correlation function scales as  $n^{l_{\gamma} - m / 2}$ . Rule (1) is due to the explicit  $n^{-1 / 2}$  factor in the network definition. Rule (2) follows from applying Isserlis' theorem, as follows. Each covariance factor (such as the factor  $\mathbb{E}\left[V_iV_j\right] = \delta_{ij}$  in eq. (13)) corresponds to an edge in a Feynman diagram. A loop in the diagram corresponds to a sum over a product of Kronecker deltas, yielding a factor of  $n$ .
+
+Returning to our example, the graph in Figure 2a has two vertices and one closed loop, so we recover the asymptotic behavior  $\mathcal{O}(n^0)$  from Theorem 3. As another example, consider the correlation function  $C(x_{1},x_{2},x_{3},x_{4}) = \mathbb{E}_{\theta}\left[f(x_{1})f(x_{2})f(x_{3})f(x_{4})\right]$ . It has two Feynman diagrams, shown in Figures 2b and 2c. Using the Feynman rules, we find that the disconnected graph represents terms that scale as  $\mathcal{O}(n^0)$ , while the connected graph represents terms that scale as  $\mathcal{O}(n^{-1})$ . Therefore,  $C(x_{1},x_{2},x_{3},x_{4}) = \mathcal{O}(n^{0})$ . This is an example of a more general phenomenon, that connected graphs vanish faster at large  $n$  compared with disconnected graphs.
+
+Correlation functions with derivatives. We now extend the Feynman diagram technique to correlation functions that include derivatives of  $f$ . As a concrete example, consider the correlation function  $C(x,x^{\prime}) = \mathbb{E}_{\theta}[\Theta (x,x^{\prime})]$ , where  $\Theta$  is the kernel defined in Section 3. For a single hidden layer, we have
+
+$$
+C \left(x, x ^ {\prime}\right) = \sum_ {i = 1} ^ {n} \mathbb {E} _ {\theta} \left[ \frac {\partial f (x)}{\partial U _ {i}} \frac {\partial f \left(x ^ {\prime}\right)}{\partial U _ {i}} + \frac {\partial f (x)}{\partial V _ {i}} \frac {\partial f \left(x ^ {\prime}\right)}{\partial V _ {i}} \right]. \tag {14}
+$$
+
+The two derivative tensors in this correlation function are contracted: their indices are set to be equal and summed over. Therefore, according to Definition 2,  $\Gamma(C)$  includes all diagrams in which the corresponding vertices share at least one edge. The resulting diagrams are shown in Figure 3. The edges forced by the contraction are explicitly marked by dashed lines for clarity, but mathematically they are ordinary edges. We see that in fact there is only one diagram contributing to this correlation function — the same one shown in Figure 2a. Following the Feynman rules, we find that  $\mathbb{E}_{\theta}[\Theta(x,x')] = \mathcal{O}(n^0)$ .
+
+The fact that contracted derivatives should be mapped to forced edges in the Feynman diagrams is proved in Appendix B. The basic reason behind this rule is the relation  $\sum_{k} \frac{\partial V_i}{\partial V_k} \frac{\partial V_j}{\partial V_k} = \delta_{ij} = \mathbb{E}[V_i V_j]$ . Namely, when derivatives act in pairs they yield a Kronecker delta factor  $(\delta_{ij})$ , which is equal to the factor obtained from a covariance  $(\mathbb{E}[V_i V_j])$ . While Isserlis' theorem instructs us to sum over all possible covariance configurations (and therefore over all possible edge configurations), a pair of summed derivative leads to a particular covariance factor. Therefore, we should only consider graphs that include the edge corresponding to this covariance factor.
+
+We are now ready to prove Theorem 1 for the case of single hidden layer with linear activations. A proof for the general case can be found in Appendix B.
+
+Proof (Theorem 1, one hidden layer). Let  $C(x_{1},\ldots ,x_{m})$  be a correlation function for a network with a single hidden layer and linear activation. Let  $G_{C}(V,E)$  be the cluster graph of  $C$ , and let  $\gamma (V,E^{\prime})\in \Gamma (C)$  be a Feynman diagram. Notice that  $E\subset E^{\prime}$ . Indeed,  $E^{\prime}$  contains an edge corresponding to each pair of contracted derivative tensors in  $C$ , and  $E = \{(v_{i},v_{j})\mid (T(x_{i}),T(x_{j}))$  contracted in  $C\}$ . In addition, notice that  $\gamma$  only contains connected
+
+components (i.e. loops) with an even number of vertices, because every vertex has exactly one edge of each type. Therefore,  $n_{\gamma} \leq n_e + \frac{n_o}{2}$ , where  $n_{\gamma}$  is the number of loops in  $\gamma$ , and  $n_e(n_o)$  is the number of even (odd) components in  $G_C$ . The bound is saturated when every even component of  $G_C$  belongs to a different component of  $\gamma$ , and pairs of odd components in  $G_C$  belong to different components of  $\gamma$ . From Theorem 3, we have that  $C = \mathcal{O}(n^s)$  where  $s = \max_{\gamma \in \Gamma(C)} n_{\gamma} - \frac{m}{2} \leq n_e + \frac{n_o}{2} - \frac{m}{2}$ .
+
+# 5 DISCUSSION
+
+Ensemble averages of the network function and its derivatives are an important class of functions that often show up in the study of wide neural networks. Examples include the ensemble average of the train and test losses, the covariance of the network function, and the Neural Tangent Kernel (Jacot et al. (2018)). In this work we presented Conjecture 1, which allows one to derive the asymptotic behavior of such functions at large width.
+
+For the case of deep linear networks, we presented a complete analytic understanding of the Conjecture based on Feynman diagrams. In addition, we presented empirical and analytic evidence showing that the Conjecture also holds for deep networks with non-linear activations, as well as for networks with non-Gaussian initialization. We found that the Conjecture holds in all cases we tested.
+
+The basic tools presented in this work can be applied to many aspects of wide network research, greatly simplifying theoretical calculations. We presented several applications of our method to the asymptotic behavior of wide networks during stochastic gradient descent, and additional applications are presented in Appendix E. We were able to improve upon known results by tightening existing bounds, and by applying the technique to SGD as well as to gradient flow. In addition, we took a step beyond the infinite width limit, deriving closed-form expressions for the first finite-width correction to the network evolution. These novel results open the door to studying finite-width networks by systematically expanding around the infinite width limit.
+
+A central question in the study of wide networks is whether the infinite width limit is a good model for describing the behavior of realistic deep networks (Chizat et al. (2018); Ghorbani et al. (2019); Geiger et al. (2019b)). In this work we take a step toward answering this question, by working out the next order in a perturbative expansion around the infinite width limit, potentially bringing us closer to an analytic description of finite-width networks. We hope that the techniques presented here provide a basis to systematically answering these and other questions about the behavior of wide networks.
+
+# ACKNOWLEDGEMENTS
+
+The authors would like to thank Alex Alemi, Yasaman Bahri, Boris Hanin, Jared Kaplan, Jaehoon Lee, Behanm Neyshabur, Sam Schoenholz, Sylvia Smullin, and Jascha Sohl-Dickstein for useful discussion. The authors would especially like to thank Ying Xiao for extensive comments on early versions of this manuscript.
+
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+
+# A EXPERIMENTAL DETAILS AND ADDITIONAL RESULTS
+
+# A.1 NON-GAUSSIAN INITIALIZATION
+
+In Table 1 we listed asymptotic bounds on several correlation functions, where the model parameters were initialized from a Gaussian distribution. Table 2 shows additional results using the same experimental setup, but with weights sampled uniformly from  $\{\pm 1\}$ . We again find good agreement with the predictions of Conjecture 1.
+
+<table><tr><td>Correlation function C</td><td>ne, no</td><td>sc</td><td>lin.</td><td>ReLU</td><td>tanh</td></tr><tr><td>Eθ[f(x1)f(x2)]</td><td>0,2</td><td>0</td><td>0.00</td><td>0.00</td><td>0.02</td></tr><tr><td>Eθ[f(x1)f(x2)f(x3)f(x4)]</td><td>0,4</td><td>0</td><td>-0.07</td><td>0.06</td><td>-0.02</td></tr><tr><td>∑μEθ[∂μf(x1)∂μf(x2)]</td><td>1,0</td><td>0</td><td>0.00</td><td>0.00</td><td>0.00</td></tr><tr><td>∑μ,νEθ[∂μf(x1)∂νf(x2)∂μ,νf(x3)f(x4)]</td><td>0,2</td><td>-1</td><td>-1.02</td><td>-1.01</td><td>-0.97</td></tr><tr><td>∑μ,ν,ρEθ[∂μf(x1)∂νf(x2)∂ρf(x3)∂μ,ν,ρf(x4)]</td><td>1,0</td><td>-1</td><td>-2.00</td><td>-1.99</td><td>-2.02</td></tr><tr><td>∑μ,ν,ρ,σEθ[∂μf(x1)∂νf(x2)∂μ,νf(x3)∂ρf(x4)∂σf(x5)∂ρ,σf(x6)]</td><td>0,2</td><td>-2</td><td>-2.05</td><td>-2.01</td><td>-1.99</td></tr></table>
+
+Table 2: Examples of bounds on correlation functions obtained from Conjecture 1. The experimental setup is the same as in Table 1, but the model parameters are sampled uniformly from  $\{\pm 1\}$  instead of from a Gaussian distribution. We find good agreement with the theoretical predictions, and in many cases the bound is tight.
+
+# A.2 EXPERIMENTAL DETAILS
+
+The experiments in Figure 1 were performed on two-class MNIST, computing a single randomly-chosen component of the kernel  $\Theta$ . Sub-figure (a) uses networks trained for 1024 steps with learning rate 1.0 and 1000 samples per class, averaged over 100 initializations. Each curve in figure (b) represents a single instance of the network map evaluated on a random image over the corse of training. The models were trained with 10 samples per class and learning rate 0.1. The input to the network is normalized by the square root of the input dimension as in (Jacot et al. (2018))
+
+$$
+f (x) = n ^ {- 1 / 2} V ^ {T} \sigma \left(n ^ {- 1 / 2} W ^ {d - 1} \dots \sigma \left(n ^ {- 1 / 2} W ^ {1} \sigma \left(D _ {\text {i n}} ^ {- 1 / 2} U x\right)\right)\right). \tag {15}
+$$
+
+# B FEYNMAN DIAGRAMS FOR DEEP LINEAR NETWORKS
+
+Feynman diagrams can be used to derive asymptotic upper bounds on deep linear networks in the large width limit. In this section we describe the method in detail, and use it to prove Theorem 1.
+
+# B.1 FEYNMAN DIAGRAMS AND DOUBLE-LINE DIAGRAMS
+
+In this section we build on the results of Section 4 and consider correlation functions of deep linear networks with  $d$  hidden layers. The network function was defined in (2), and here we set the activation  $\sigma$  to be the identity. Definition 2 describes how to map a correlation function  $C$  to  $\Gamma(C)$ , a family of graphs called Feynman diagrams. The Feynman diagram method relies on Isserlis' theorem, which allows us to express arbitrary moments of multivariate Gaussian variables in terms of their covariance.
+
+Theorem 4 (Isserlis). Let  $z = (z_{1},\ldots ,z_{l})$  be a centered multivariate Gaussian variable. For any positive integer  $k$ ,
+
+$$
+\mathbb {E} _ {z} \left[ z _ {i _ {1}} \dots z _ {i _ {2 k}} \right] = \frac {1}{2 ^ {k} k !} \sum_ {\pi \in S _ {2 k}} \mathbb {E} \left[ z _ {i _ {\pi (1)}} z _ {i _ {\pi (2)}} \right] \mathbb {E} \left[ z _ {i _ {\pi (3)}} z _ {i _ {\pi (4)}} \right] \dots \mathbb {E} \left[ z _ {i _ {\pi (2 k - 1)}} z _ {i _ {\pi (2 k)}} \right], \tag {16}
+$$
+
+$$
+\mathbb {E} _ {z} \left[ z _ {i _ {1}} \dots z _ {i _ {2 k - 1}} \right] = 0. \tag {17}
+$$
+
+In particular, if the covariance matrix of  $z$  is the identity then
+
+$$
+\mathbb {E} _ {z} \left[ z _ {i _ {1}} \dots z _ {i _ {2 k}} \right] = \frac {1}{2 ^ {k} k !} \sum_ {\pi \in S _ {2 k}} \delta_ {i _ {\pi (1)} i _ {\pi (2)}} \delta_ {i _ {\pi (3)} i _ {\pi (4)}} \dots \delta_ {i _ {\pi (2 k - 1)} i _ {\pi (2 k)}}. \tag {18}
+$$
+
+Using this theorem, a correlation function  $C$  can be expressed as a sum over permutations as in (16). Each term in this sum maps to a Feynman diagram in  $\Gamma(C)$ .
+
+![](images/7c64193ff11a6748b1b5d1a2cd46d2275bf679de943e1c37f1c6f537770c23d2.jpg)  
+(a) Feynman diagram  
+Figure 4: Feynman diagrams for  $\mathbb{E}_{\theta}[f(x)f(x^{\prime})]$  with 2 hidden layers. Notice that the  $U, V$  edges in the Feynman diagram map to single edges in the double-line diagram, while the  $W$  edge maps to a double edge.
+
+![](images/0d77d2a6dd684ff045214d2ae29ab4cdd998c62d2fd32bb967e9df4f9cdb9153.jpg)  
+(b) Double-line diagram
+
+As an example, consider the correlation function  $C(x,x^{\prime}) = \mathbb{E}_{\theta}\left[f(x)f(x^{\prime})\right]$  for a network with 2 hidden layers. An explicit calculation gives
+
+$$
+\begin{array}{l} C \left(x, x ^ {\prime}\right) = \frac {1}{n ^ {2}} \sum_ {i, j, k, l} ^ {n} \sum_ {\alpha , \beta} ^ {D _ {\text {i n}}} \mathbb {E} _ {\theta} \left[ V _ {i} W _ {i j} ^ {1} U _ {j \alpha} V _ {k} W _ {k l} ^ {1} U _ {l \beta} \right] x _ {\alpha} x _ {\beta} ^ {\prime} (19) \\ = \frac {1}{n ^ {2}} \sum_ {i, j, k, l} ^ {n} \sum_ {\alpha , \beta} ^ {D _ {\mathrm {i n}}} \mathbb {E} _ {V} \left[ V _ {i} V _ {k} \right] \mathbb {E} _ {W} \left[ W _ {i j} ^ {1} W _ {k l} ^ {1} \right] \mathbb {E} _ {U} \left[ U _ {j \alpha} U _ {l \beta} \right] x _ {\alpha} x _ {\beta} ^ {\prime} (20) \\ = \frac {x ^ {T} x ^ {\prime}}{n ^ {2}} \sum_ {i, k = 1} ^ {n} \delta_ {i k} \delta_ {i k} \sum_ {j, l = 1} ^ {n} \delta_ {j l} \delta_ {j l} = x ^ {T} x ^ {\prime} = \mathcal {O} \left(n ^ {0}\right). (21) \\ \end{array}
+$$
+
+To get from (19) to (20), we applied Isserlis' theorem for every choice of indices  $i, j, k, l, \alpha, \beta$ . We find that there is at most one permutation  $\pi$  of the network parameters such that the covariances do not vanish. This is because the covariance of parameters across different layers vanishes identically (for example  $\mathbb{E}_{\theta}\left[V_i W_{jk}^1\right] = 0$ ). Correspondingly, this correlation function has a single Feynman diagram, shown in Figure 4a.
+
+For networks with one hidden layer, we saw in Section 4 that the asymptotic behavior is determined by the number of loops in a graph. This observation does not immediately generalize to networks with general depth, because it is not obvious how to count loops in diagrams such as the one in Figure 4a. The problem can be traced back to the fact that weight matrices have covariances of the form  $\mathbb{E}\left[W_{ij}W_{kl}\right] = \delta_{ik}\delta_{jl}$  involving two Kronecker deltas, but the procedure described so far assumes that each covariance (and each edge in the graph) corresponds to a single Kronecker delta.
+
+A similar question appeared in the context of theoretical physics, and the correct generalization is due to 't Hooft (1974). The idea is to treat each Feynman diagram as the triangulation of a Riemann surface, and to define the number of loops in a graph to be the number of faces of the triangulation. In practice, this involves mapping each Feynman diagram to a new double-line diagram: A graph in which each edge corresponds to a single Kronecker delta factor, and loops correspond to triangulation faces of the original diagram.
+
+Definition 3. Let  $\gamma \in \Gamma(C)$  be a Feynman diagram for a correlation function  $C$  involving  $k$  derivative tensors for a network of depth  $d$ . Its double-line graph,  $\mathrm{DL}(\gamma)$  is a graph with  $kd$  vertices of degree 2, defined by the following blow-up procedure.
+
+- Each vertex  $v^{(i)}$  in  $\gamma$  is mapped to  $d$  vertices  $v_{1}^{(i)}, \ldots, v_{d}^{(i)}$  in  $\mathrm{DL}(\gamma)$ .  
+- Each edge  $(v^{(i)}, v^{(j)})$  in  $\gamma$  of type  $U$  is mapped to a single edge  $(v_{1}^{(i)}, v_{1}^{(j)})$ .  
+- Each edge  $(v^{(i)}, v^{(j)})$  in  $\gamma$  of type  $W^l$  is mapped to two edges  $(v_{l}^{(i)}, v_{l}^{(j)}), (v_{l+1}^{(i)}, v_{l+1}^{(j)})$ .  
+- Each edge  $(v^{(i)}, v^{(j)})$  in  $\gamma$  of type  $V$  is mapped to a single edge  $(v_{d}^{(i)}, v_{d}^{(j)})$ .
+
+The number of faces in  $\gamma$  is given by the number of loops in the double-line graph  $\mathrm{DL}(\gamma)$ .
+
+Figure 4 shows the Feynman diagram and corresponding double-line diagram for  $\mathbb{E}_{\theta}[f(x)f(x^{\prime})]$  with 2 hidden layers. We can interpret this Feynman diagram as a triangulation of the disc: a 2-dimensional
+
+surface with a single boundary. The triangulation has 2 vertices, 3 edges corresponding to the edges of the Feynman diagram, and 2 faces corresponding to the loops of the double-line diagram. Figure 5 shows additional examples of double-line diagrams, and Figure 6 shows the double-line diagrams of a correlation function with derivatives. As explained in Section 4, contracted derivative tensors in a correlation function  $C$  map to forced edges in  $\Gamma(C)$ , and these are marked with dashed lines on the diagrams.
+
+![](images/cd0ffa90cfb010279d1ec83f22655c07a292461b4150beca26a6448eb37041ac.jpg)  
+(a)
+
+![](images/ceb22262aef073a92d3d90e056969169630370c91cbfbad7916eb09057af67ff.jpg)
+
+![](images/7bcd4c1759206c24b7551e1daedcc463d286ef7298fce061c92aa109167bbe6a.jpg)  
+(b)
+
+![](images/7fd563b2fc041f97b4776a6987c6913c4913da33554d2a2da0df3a7a9b5db0df.jpg)  
+Figure 5: Double-line diagrams for  $\mathbb{E}_{\theta}\left[f(x_{1})f(x_{2})f(x_{3})f(x_{4})\right]$  for a deep linear network with 2 hidden layers.
+
+![](images/e9bac44367caaa927d72916da184e4c815c5d7684c765e82812148ae3ec3cc12.jpg)  
+Figure 6: Double-line graphs describing the expectation value of the NTK,  $\mathbb{E}_{\theta}[\Theta]$ , for a deep linear network with two hidden layers. Crossed edges mark the edges that are forced by contracted derivatives. The derivative can act on either  $U$ ,  $V$ , or  $W^{1}$ , and therefore there are three diagrams. Each diagram has 2 vertices and 2 faces, and therefore the correlation function is  $\mathcal{O}(n^0)$  according to the Feynman rules.
+
+![](images/90f0244d7c06f7cce4364b3ddd99c2bd54b4f91c553d9d103c324b4860c102aa.jpg)
+
+Generally, every Feynman diagram of a deep linear network can be interpreted as the triangulation of a 2-dimensional manifold with at least one boundary. Intuitively, the presence of the boundary is due to the fact that the  $U, V$  weights at both ends of network have only one dimension that scales linearly with the width. As a result, in the double-line diagrams the  $U, V$  edges become single lines rather than double lines. These 'missing' lines translate into fewer faces in the triangulation, and the 'missing' faces can be geometrically interpreted as boundaries in the corresponding surface.[9]
+
+The discussion above is summarized by the following result, due to 't Hooft (1974), that describes how the asymptotic behavior of a general correlation function can be computed using the Feynman rules for deep linear networks.
+
+Theorem 5. Let  $C(x_{1},\ldots ,x_{m})$  be a correlation function of a deep linear network with  $d$  hidden layers, and let  $\gamma \in \Gamma (C)$  be a Feynman diagram. The diagram represents a subset of terms that contribute to  $C$ , and its asymptotic behavior is determined by the Feynman rules: the subset is  $\mathcal{O}(n^{s_{\gamma}})$  where  $s_{\gamma} = l_{\gamma} - \frac{dm}{2}$ , and  $l_{\gamma}$  is the number of loops in the double-line diagram  $\mathrm{DL}(\gamma)$ . Furthermore, the correlation function is  $C = \mathcal{O}(n^s)$ , where  $s = \max_{\gamma \in \Gamma (C)} s_{\gamma}$ .
+
+The intuition for the formula  $s_{\gamma} = l_{\gamma} - \frac{dm}{2}$  is similar to the single hidden layer case, Theorem 3. The term  $l_{\gamma}$  counts the number of factors of the form  $\sum_{i_1,\ldots ,i_k}\delta_{i_1i_2}\delta_{i_2i_3}\dots \delta_{i_ki_1} = n$  that appear in the Correlation function after applying Isserlis' theorem. The term  $\left(-\frac{dm}{2}\right)$  is due to the explicit  $n^{-d / 2}$  normalization of the network function.
+
+# B.2 ASYMPTOTICS OF DEEP LINEAR NETWORKS
+
+We now prove Theorem 1. The theorem follows from the following lemma, again due to 't Hooft (1974), that relates the asymptotic behavior to the number of connected components in a Feynman diagram.
+
+Lemma 2. Let  $C(x_{1},\ldots ,x_{m})$  be a correlation function for a deep linear network. Let  $c_{\gamma}$  be the number of connected components of a graph  $\gamma \in \Gamma (C)$ . Then  $C = \mathcal{O}(n^{s})$ , where
+
+$$
+s = \max  _ {\gamma \in \Gamma (C)} c _ {\gamma} - \frac {m}{2}. \tag {22}
+$$
+
+Proof. We prove the result for 1D inputs  $(D_{\mathrm{in}} = 1)$ , and it is easy to generalize to arbitrary input dimension. Let  $\gamma \in \Gamma(C)$  be a Feynman diagram and let  $\gamma'$  be a connected component of  $\gamma$  with  $v_{\gamma'}$  vertices. Notice that we can apply the Feynman rules of Theorem 5 separately to each component  $\gamma'$  and find a bound  $\mathcal{O}(n^{s_{\gamma'}})$ . Then,  $\gamma = \mathcal{O}(n^{s_{\gamma}})$  where  $s_{\gamma} = \sum_{\gamma'} s_{\gamma'}$ , and the sum runs over the connected components of  $s_{\gamma}$ . We will show below that  $s_{\gamma'} \leq 1 - \frac{v_{\gamma'}}{2}$ , and therefore  $s_{\gamma} \leq c_{\gamma} - \frac{m}{2}$ , which is sufficient to prove (22).
+
+Let us prove the remaining statement about  $s_{\gamma'}$ . The Euler character of the graph  $\gamma'$  with  $v = v_{\gamma'}$  vertices,  $e$  edges and  $f$  faces is  $\chi = v - e + f$ . The degree of each vertex in the graph is  $d + 1$ , and therefore  $e = \frac{(d + 1)v}{2}$ . Using Theorem 5 the graph is  $\mathcal{O}(n^{s_{\gamma'}})$  where  $s_{\gamma'} = f - \frac{dv}{2}$ . We therefore find that  $\chi = \frac{v}{2} + s_{\gamma'}$ . The graph  $\gamma'$  is a triangulation of some connected surface with at least one boundary. The Euler character for such a surface is bounded by  $\chi \leq 1$ , and therefore  $s_{\gamma'} \leq 1 - \frac{v}{2}$ .
+
+Let us now prove Theorem 1.
+
+Proof. Let  $C(x_1, \ldots, x_m)$  be a correlation function for a deep linear network. Suppose that the cluster graph  $G_C$  has  $n_e$  even size components and  $n_o$  odd size components. Let  $\gamma \in \Gamma(C)$  be a Feynman diagram with  $c_\gamma$  connected components. We will show that  $c_\gamma \leq n_e + \frac{n_o}{2}$ . It then follows immediately from Lemma 2 that  $C = \mathcal{O}(n^s)$  where  $s = n_e + \frac{n_o}{2} - \frac{m}{2}$ , concluding the proof.
+
+Let us derive the bound on  $c_{\gamma}$ . First, all vertices that belong to a given cluster (a component of  $G_C$ ) will also belong to the same connected component in  $\gamma$ . This is because every edge in  $G_C$  is also an edge in  $\gamma$  (note that  $G_C$  and  $\gamma$  have the same set of vertices). Therefore,  $c_{\gamma} \leq n_e + n_o$ . Second, note that every connected component of the graph  $\gamma$  has an even number of vertices. Indeed, each edge has a type  $t$ , and each vertex has exactly one edge of each type. Therefore, a connected component with  $v$  vertices has  $\frac{v}{2}$  edges of each type, and so  $v$  must be even. It follows that the vertices of even clusters can form their own connected components in a Feynman diagrams, while odd clusters must be connected in sets of 2 or more to form connected components. The bound on  $c_{\gamma}$  then follows.
+
+For deep linear networks with bias the proof is the same, except we use Lemma 3 below instead of Lemma 2.
+
+# B.3 DEEP LINEAR NETWORKS WITH BIAS
+
+In this section we argue that Conjecture 1 holds for deep linear networks with bias. Let us add bias to the definition (2) of a deep linear network (setting  $\sigma$  to be the identity). We choose  $x = 1$  without loss of generality. The activations  $z^l$  at layer  $l$  and the network function  $f$  are given by
+
+$$
+z ^ {1} = U, \tag {23}
+$$
+
+$$
+z ^ {l + 1} = n ^ {- 1 / 2} W ^ {l} z ^ {l} + b ^ {l}, \quad l = 1, \dots , d - 1, \tag {24}
+$$
+
+$$
+f = n ^ {- 1 / 2} V ^ {T} z ^ {d}. \tag {25}
+$$
+
+Here  $b^1, \ldots, b^{d-1} \in \mathbb{R}^n$  are biases whose elements are chosen independently from  $\mathcal{N}(0,1)$  at initialization. The network function can be written as
+
+$$
+f = \sum_ {k = 2} ^ {d + 1} n ^ {- (k - 1) / 2} V ^ {T} W ^ {d - 1} \dots W ^ {d + 2 - k} b ^ {d + 1 - k}, \tag {26}
+$$
+
+![](images/d3b1913057e424ba64e3a066ab132617c1e625c427a8a9df1665cf13fa9fae84.jpg)  
+(a) Feynman diagram
+
+![](images/79cedf2d723840eb4a386b7b6754290e2c3a661ab1591c12e17af97073340565.jpg)  
+(b) Double-line diagram  
+Figure 7: New Feynman diagrams for  $\mathbb{E}_{\theta}\left[f(x_{1})f(x_{2})f(x_{3})f(x_{4})\right]$  with 2 hidden layers and bias.
+
+where we defined  $b^0 = U$ . In the rest of this Section we extend our definitions of Feynman diagrams and double-line diagrams to deep linear networks with bias, and then extend several of our results to this case.
+
+Definition 4. Let  $C(x_{1},\ldots ,x_{m})$  be a correlation function for a network with  $d$  hidden layers and bias. The family  $\Gamma (C)$  is the set of all graphs that have the following properties.
+
+1. There are  $m$  vertices  $v_{1},\ldots ,v_{m}$  where each  $v_{i}$  has degree  $k_{i}\in \{2,\dots ,d + 1\}$  
+2. Each edge has a type  $t$ . Every vertex of degree  $k$  has one edge of each type  $t \in \{b^{d+1-k}, W^{d+2-k}, \ldots, W^{d-1}, V\}$ .  
+3. If two derivative tensors  $T_{\mu_1,\dots,\mu_\ell}(x_i), T_{\nu_1,\dots,\nu_{\ell'}}(x_j)$  are contracted  $s$  times in  $C$ , the graph must have at least  $s$  edges (of any type) connecting the vertices  $v_i, v_j$ .
+
+The intuition behind this definition is that term  $k$  in the sum in (26) is equal to the network function of a deep linear network with depth  $k - 1$ . Correlation functions of such a function can be represented diagrammatically by including vertices of degree  $k$ . To represent the entire sum we must consider all diagrams with vertices of varying degree.
+
+We now extend Definition 3 of double-line graphs to accommodate vertices with varying degree.
+
+Definition 5. Let  $\gamma \in \Gamma(C)$  be a Feynman diagram for a correlation function  $C$  involving  $m$  derivative tensors for a network of depth  $d$  with biases. Its double-line graph,  $\mathrm{DL}(\gamma)$  is a graph with vertices of degree 2, defined by the following blow-up procedure.
+
+- Each vertex  $v^{(i)}$  in  $\gamma$  of degree  $k_{i}$  is mapped to  $k_{i} - 1$  vertices  $v_{1}^{(i)}, \ldots, v_{k_{i} - 1}^{(i)}$  in  $\mathrm{DL}(\gamma)$ .  
+- Each edge  $(v^{(i)}, v^{(j)})$  in  $\gamma$  of type  $W^l$  is mapped to two edges,  $(v_{l + k_i - d - 1}^{(i)}, v_{l + k_j - d - 1}^{(j)})$  and  $(v_{l + k_i - d}^{(i)}, v_{l + k_j - d}^{(j)})$ .  
+- Each edge  $(v^{(i)}, v^{(j)})$  in  $\gamma$  of type  $b^{d+1-k_i}$  is mapped to a single edge  $(v_1^{(i)}, v_1^{(j)})$  (notice that  $k_i = k_j$ ).  
+- Each edge  $(v^{(i)}, v^{(j)})$  in  $\gamma$  of type  $V$  is mapped to a single edge  $(v_{k_i - 1}^{(i)}, v_{k_j - 1}^{(j)})$ .
+
+As in Definition 3, the number of faces in  $\gamma$  is given by the number of loops in the double-line graph  $\mathrm{DL}(\gamma)$ .
+
+Figure 7 is an example of the new single line and double line diagrams that appear in the correlation function  $\mathbb{E}_{\theta}\left[f(x_1)f(x_2)f(x_3)f(x_4)\right]$  for a two-hidden-layer network with biases.
+
+Let us now extend Theorem 5 (the Feynman rules) to accommodate bias. The only difference is in the explicit normalization. Previously, each vertex had degree  $d + 1$  and contributed a factor of  $n^{-d/2}$  due to the explicit normalization of the function map. With biases, as a result of (26), a vertex with degree  $k$  contributes a factor of  $n^{(k-1)/2}$ .
+
+Theorem 6. Let  $C(x_{1},\ldots ,x_{m})$  be a correlation function of a deep linear network with  $d$  hidden layers and with biases. Let  $\gamma \in \Gamma (C)$  be a Feynman diagram. The diagram represents a subset of
+
+terms that contribute to  $C$ , and its asymptotic behavior is determined by the Feynman rules: the subset is  $\mathcal{O}(n^{s_{\gamma}})$  where  $s_{\gamma} = l_{\gamma} - \sum_{i=1}^{m} \frac{k_i - 1}{2}$ , with  $k_i$  the degree of the  $i$ -th vertex, and  $l_{\gamma}$  the number of loops in the double-line diagram  $\mathrm{DL}(\gamma)$ . Furthermore, the correlation function is  $C = \mathcal{O}(n^s)$ , where  $s = \max_{\gamma \in \Gamma(C)} s_{\gamma}$ .
+
+We now introduce the following result, which generalizes Lemma 2 to the case of networks with bias. This Lemma is used above in the proof of Theorem 1.
+
+Lemma 3. Let  $C(x_{1},\ldots ,x_{m})$  be a correlation function for a deep linear network with biases. Let  $c_{\gamma}$  be the number of connected components of a graph  $\gamma \in \Gamma (C)$ . Then  $C = \mathcal{O}(n^{s})$ , where
+
+$$
+s = \max  _ {\gamma \in \Gamma (C)} c _ {\gamma} - \frac {m}{2}. \tag {27}
+$$
+
+Proof. It is enough to show that each connected component  $\gamma'$  in  $\gamma$  is bounded as  $\mathcal{O}(n^{s_{\gamma'}})$  where  $s_{\gamma'} \leq 1 - \frac{v_{\gamma'}}{2}$ . The graph  $\gamma'$  is a triangulation of a Riemann surface with  $f$  faces  $e$  edges, and  $v_{\gamma'}$  vertices of degrees  $k_1, \ldots, k_{v_{\gamma'}}$ . The Feynman rules give
+
+$$
+s _ {\gamma^ {\prime}} = f - \sum_ {i = 1} ^ {v _ {\gamma^ {\prime}}} \frac {k _ {i} - 1}{2}. \tag {28}
+$$
+
+Using the relation  $e = \sum_{i=1}^{v_{\gamma'}} \frac{k_i}{2}$  and the definition of the Euler character,  $\chi = v_{\gamma'} - e + f$  we have
+
+$$
+s _ {\gamma^ {\prime}} = \chi - \frac {v _ {\gamma^ {\prime}}}{2}. \tag {29}
+$$
+
+The diagram  $\gamma'$  is a triangulation of a Riemann surface with at least one boundary, thus  $\chi \leq 1$  and  $s_{\gamma'} \leq 1 - \frac{v_{\gamma'}}{2}$ .
+
+# C NON-LINEARITIES
+
+In previous sections we presented the Feynman diagram method for computing the large width asymptotics of correlation functions. In this section we show that the method applies as-is for deep networks with ReLU non-linearities and all-equal inputs, as well as to networks with a single hidden layer, a broader class of non-linearities, and arbitrary inputs. Theorem 2 follows immediately from the results presented in this section.
+
+# C.1 DEEP NETWORKS WITH RELU NON-LINEARITIES
+
+The following result guarantees that the presence of ReLU non-linearities does not change the asymptotic upper bound compared with linear activations, when all inputs are the same.
+
+Theorem 7. Let  $f_{\mathrm{nl}}$  be a network function of the form (2) with ReLU activation. Let  $f$  be a network with the same architecture but with linear activation. Let  $C(x_1, \ldots, x_m; f_{\mathrm{nl}})$  be a correlation function of the deep network  $f_{\mathrm{nl}}$  with width  $n$ , and let  $f$  be a deep linear network with the same width and depth. Suppose that all inputs are the same,  $x_1 = x_2 = \dots = x_m$ . Then
+
+$$
+C \left(x _ {1}, \dots , x _ {m}; f _ {\mathrm {n l}}\right) = \mathcal {O} \left(C \left(x _ {1}, \dots , x _ {m}; f\right)\right). \tag {30}
+$$
+
+Intuitively, we will rely on the fact that for ReLU networks we can, in some cases, treat the binary neuron activations as being statistically independent of the weights. This result is due to (Hanin & Nica (2018)). Given this result, we can bound the contribution of the binary activations, and the remaining Gaussian integral is equivalent to that found in a deep linear network. The proof does not work for correlation functions with non-equal inputs, because in that case the independence result of Hanin & Nica (2018) no longer holds.
+
+Proof. We can write the network function as
+
+$$
+f _ {\mathrm {n l}} (x) = n ^ {- d / 2} V ^ {T} D ^ {d} (x) W ^ {d - 1} D ^ {d - 1} (x) W ^ {d - 2} \dots D ^ {2} (x) W ^ {1} D ^ {1} (x) U x, \tag {31}
+$$
+
+$$
+D ^ {j} (x) := H \left(W ^ {j - 1} \sigma \left(W ^ {j - 2} \dots \sigma (U x)\right)\right), \quad j = 1, \dots , d. \tag {32}
+$$
+
+Here,  $H$  is the Heaviside step function acting elementwise on its vector argument. We now introduce the construction from Hanin & Nica (2018). Let  $\xi^j,\eta^j$ ,  $j = 1,\dots ,d$  be diagonal matrices of dimension  $n$ , whose diagonal elements are  $\pm 1$ -Bernoulli  $(p)$  variables with  $p = \frac{1}{2}$ . We define the new variables
+
+$$
+\hat {U} := \xi^ {1} U, \quad \hat {V} := \eta^ {d} V, \quad \hat {W} ^ {j} := \xi^ {j + 1} W ^ {j} \eta^ {j}, \quad j = 1, \dots , d. \tag {33}
+$$
+
+Let  $\hat{f}_{\mathrm{nl}}$  be a network function with the same architecture as  $f_{\mathrm{nl}}$  but using the re-defined weights. We define
+
+$$
+\begin{array}{l} \hat {f} _ {\mathrm {n l}} (x) = n ^ {- d / 2} \hat {V} ^ {T} \hat {D} ^ {d} (x) \hat {W} ^ {d - 1} \dots \hat {D} ^ {2} (x) \hat {W} ^ {1} \hat {D} ^ {1} (x) \hat {U} x (34) \\ = n ^ {- d / 2} V ^ {T} \rho^ {d} \hat {D} ^ {d} (x) W ^ {d - 1} \rho^ {d - 1} \dots \rho^ {2} \hat {D} ^ {2} (x) W ^ {1} \rho^ {1} \hat {D} ^ {1} (x) U x, (35) \\ \end{array}
+$$
+
+$$
+\hat {D} ^ {j} (x) := H \left(\hat {W} ^ {j - 1} \sigma \left(\hat {W} ^ {j - 2} \dots \sigma (\hat {U} x)\right)\right), \quad j = 1, \dots , d, \tag {36}
+$$
+
+$$
+\rho^ {j} := \eta^ {j} \xi^ {j}, \quad j = 1, \dots , d. \tag {37}
+$$
+
+The following was shown in Hanin & Nica (2018).
+
+1.  $f_{\mathrm{nl}}$  and  $\hat{f}_{\mathrm{nl}}$  are equal in distribution.  
+2.  $\{\hat{D}^j (x),\rho^j,j = 1,\ldots ,d\}$  are independent of  $\{U,V,W^1,\dots ,W^{d - 1}\}$  
+3.  $\{\hat{D}^j (x), j = 1,\ldots ,d\}$  are independent of each other for fixed  $x$ . The diagonal entries of each diagonal matrix  $\hat{D}^j (x)$  are independent, and take the values  $\{0,1\}$  with probability  $\frac{1}{2}$ .
+
+Now, the correlation function can be written as
+
+$$
+C \left(x _ {1}, \dots , x _ {m}; f _ {\mathrm {n l}}\right) = \mathbb {E} _ {\theta} \left[ F \left(x _ {1}, \dots , x _ {m}; f _ {\mathrm {n l}}\right) \right] = \mathbb {E} _ {\theta , \eta , \xi} \left[ F \left(x _ {1}, \dots , x _ {m}; \hat {f} _ {\mathrm {n l}}\right) \right]. \tag {38}
+$$
+
+The second equality follows from the fact that  $f_{\mathrm{nl}} \stackrel{d}{=} \hat{f}_{\mathrm{nl}}$ . Let us assume for now that the correlation function has no contracted derivatives. The we have
+
+$$
+\begin{array}{l} C \left(x _ {1}, \dots , x _ {m}; f _ {\mathrm {n l}}\right) = \mathbb {E} _ {\theta , \eta , \xi} \left[ \hat {f} _ {\mathrm {n l}} \left(x _ {1}\right) \dots \hat {f} _ {\mathrm {n l}} \left(x _ {m}\right) \right] (39) \\ = \frac {1}{n ^ {d m / 2}} \sum_ {\vec {i}} ^ {n} \sum_ {\vec {\alpha}} ^ {D _ {\mathrm {i n}}} \mathbb {E} \left[ \prod_ {l = 1} ^ {m} V _ {i _ {l, d}} \right] \left(\prod_ {l ^ {\prime} = 1} ^ {d - 1} \mathbb {E} \left[ \prod_ {l ^ {\prime \prime} = 1} ^ {m} W _ {i _ {l ^ {\prime \prime}, l ^ {\prime} + 1}, i _ {l ^ {\prime \prime}, l ^ {\prime}}} ^ {l ^ {\prime}} \right]\right) \mathbb {E} \left[ \prod_ {l = 1} ^ {m} U _ {i _ {l, 1}, \alpha_ {l}} \right] E _ {\vec {i}, \vec {\alpha}}. (40) \\ \end{array}
+$$
+
+Here,  $\vec{i} = \{i_{1,1},\dots ,i_{m,d}\}$  and  $\vec{\alpha} = \{\alpha_{1},\ldots ,\alpha_{m}\}$ , and
+
+$$
+E _ {\vec {i}, \vec {\alpha}} := \mathbb {E} \left[ \prod_ {l = 1} ^ {d} \prod_ {l ^ {\prime} = 1} ^ {m} \rho_ {i _ {l ^ {\prime}}, l} ^ {l} \hat {D} _ {i _ {l ^ {\prime}}, l} ^ {l} \left(x _ {l ^ {\prime}}\right) \right] \prod_ {l = 1} ^ {m} x _ {\alpha_ {l}} ^ {l}. \tag {41}
+$$
+
+In writing the equation (40) we used the facts that  $\hat{D}, \rho$  are independent of the parameters, and that parameters in different layers are independent. We now use Theorem 4 (Isserlis), which says that each of the expectation values over products of  $V$ ,  $W^j$ , and  $U$  elements is equal to a sum over permutations, where each term is a product over Kronecker delta functions — the covariance matrices of the parameters.
+
+$$
+C \left(x _ {1}, \dots , x _ {m}; f _ {\mathrm {n l}}\right) = \frac {1}{n ^ {d m / 2}} \sum_ {\sigma_ {1}, \dots , \sigma_ {d + 1} \in S _ {m}} \sum_ {\vec {i}} ^ {n} \sum_ {\vec {\alpha}} ^ {D _ {\mathrm {i n}}} \tilde {\Delta} _ {\vec {i}, \vec {\alpha}} ^ {\vec {\sigma}} E _ {\vec {i}, \vec {\alpha}}. \tag {42}
+$$
+
+Here,  $\tilde{\Delta}_{\vec{i},\vec{\alpha}}^{\vec{\sigma}}$  is a product of Kronecker delta functions. The precise form of this object will not be important for our purpose, though it can be easily derived using Theorem 4.
+
+We can now bound the correlation function as follows.
+
+$$
+\begin{array}{l} \left| C \left(x _ {1}, \dots , x _ {m}; f _ {\mathrm {n l}}\right) \right| \leq \frac {1}{n ^ {d m / 2}} \sum_ {\sigma_ {1}, \dots , \sigma_ {d + 1} \in S _ {m}} \sum_ {\vec {i}} ^ {n} \sum_ {\vec {\alpha}} ^ {D _ {\mathrm {i n}}} \tilde {\Delta} _ {\vec {i}, \vec {\alpha}} ^ {\vec {\sigma}} \left| E _ {\vec {i}, \vec {\alpha}} \right| (43) \\ \leq \frac {E _ {\max }}{n ^ {d m / 2}} \sum_ {\sigma_ {1}, \dots , \sigma_ {d + 1} \in S _ {m}} \sum_ {\vec {i}} ^ {n} \sum_ {\vec {\alpha}} ^ {D _ {\mathrm {i n}}} \tilde {\Delta} _ {\vec {i}, \vec {\alpha}} ^ {\vec {\sigma}} (44) \\ = E _ {\max } \left| C (v, \dots , v; f) \right|. (45) \\ \end{array}
+$$
+
+Here,  $E_{\mathrm{max}} = \max_{\vec{i},\vec{\alpha}}\left|E_{\vec{i},\vec{\alpha}}\right|$ , and  $v \in \mathbb{R}^{D_{\mathrm{in}}}$  is the all-ones vector. The diagonal elements of  $\rho^l, \hat{D}^l(x)$  are identical variables. Therefore,  $E_{\vec{i},\vec{\alpha}}$  only takes an  $\mathcal{O}(1)$  number of values in the large width limit. Note that each of these values are independent of  $n$ , and therefore  $E_{\mathrm{max}} = \mathcal{O}(1)$ . We managed to bound the correlation function for  $f_{\mathrm{nl}}$  in terms of the correlation function for the corresponding linear network  $f$  with fixed inputs. The asymptotics of the linear-network correlation function do not depend on the inputs, and therefore this concludes the proof.
+
+# C.2 SINGLE HIDDEN LAYER NETWORKS
+
+For networks with a single hidden layer, defined by
+
+$$
+f _ {\mathrm {n l}} (x) = \frac {1}{\sqrt {n}} \sum_ {i = 1} ^ {n} V _ {i} \sigma \left(U _ {i} ^ {T} x\right), \tag {46}
+$$
+
+we can extend our asymptotic analysis to smooth non-linearities. We will show in Theorem 8 that for any correlation function  $C$ , we have
+
+$$
+C \left(x _ {1}, \dots , x _ {m}; f _ {\mathrm {n l}}\right) = \mathcal {O} \left(C \left(x _ {1}, \dots , x _ {m}; f\right)\right), \tag {47}
+$$
+
+where  $f$  is a deep linear network of equal width and sufficient depth. Therefore, computing the asymptotics using Feynman diagrams for deep linear networks yields a bound on networks with a single hidden layer and smooth non-linearities.
+
+Before delving into the proof of this claim, we consider a few simple examples. Let us begin with the correlation function  $\mathbb{E}_{\theta}\left[f_{\mathrm{nl}}(x)f_{\mathrm{nl}}(x^{\prime})\right]$ .
+
+$$
+\begin{array}{l} \mathbb {E} _ {\theta} \left[ f _ {\mathrm {n l}} (x) f _ {\mathrm {n l}} \left(x ^ {\prime}\right) \right] = \frac {1}{n} \sum_ {i, j} ^ {n} \mathbb {E} _ {\theta} \left[ V _ {i} V _ {j} \sigma \left(U _ {i} ^ {T} x\right) \sigma \left(U _ {j} ^ {T} x ^ {\prime}\right) \right] (48) \\ = \frac {1}{n} \sum_ {i} ^ {n} \mathbb {E} _ {\theta} \left[ \sigma \left(U _ {i} ^ {T} x\right) \sigma \left(U _ {i} ^ {T} x ^ {\prime}\right) \right] = \mathcal {O} (1). (49) \\ \end{array}
+$$
+
+Here we used two facts. First, the weights  $V_{i}$  are unaffected by the non-linearity, so we can carry out the  $V$  integral. Second, the summand in the last equation,  $\mathbb{E}\left[\sigma (U_i^T x)\sigma (U_i^T x')\right]$ , is independent of both  $i$  and  $n$  because  $U_{i}$  are i.i.d. variables.
+
+Next, consider the following correlation function. For simplicity, here we set the input dimension to be  $D_{\mathrm{in}} = 1$ , and we set all inputs to 1; this does not change the asymptotics.
+
+$$
+\begin{array}{l} \mathbb {E} _ {\theta} \left[ \frac {d \Theta}{d t} \right] = \sum_ {j, k} ^ {n} \mathbb {E} _ {\theta} \left[ f _ {\mathrm {n l}} \frac {\partial^ {2} f _ {\mathrm {n l}}}{\partial U _ {j} \partial V _ {k}} \frac {\partial f _ {\mathrm {n l}}}{\partial U _ {j}} \frac {\partial f _ {\mathrm {n l}}}{\partial V _ {k}} \right] \\ = \frac {1}{n ^ {2}} \sum_ {i _ {1}, i _ {2}} ^ {n} \mathbb {E} _ {\theta} \left[ V _ {i _ {1}} V _ {i _ {2}} \sigma (U _ {i _ {1}}) \sigma^ {\prime} (U _ {i _ {2}}) \sigma^ {\prime} (U _ {i _ {2}}) \sigma (U _ {i _ {2}}) \right] \\ = \frac {1}{n ^ {2}} \sum_ {i _ {1}} ^ {n} \mathbb {E} _ {\theta} \left[ \sigma \left(U _ {i _ {1}}\right) ^ {2} \sigma^ {\prime} \left(U _ {i _ {1}}\right) ^ {2} \right] \\ = \mathcal {O} \left(n ^ {- 1}\right). \tag {50} \\ \end{array}
+$$
+
+In the last line, we again used the fact that the summands are equal and independent of  $n$ .
+
+In general, a correlation function  $C(x_{1},\ldots ,x_{m};f_{\mathrm{nl}})$  can be reduced to the form
+
+$$
+C = \frac {1}{n ^ {m / 2}} \sum_ {\alpha = 1} ^ {K} \sum_ {i _ {1}, \dots , i _ {r _ {\alpha}}} ^ {n} \mathcal {S} _ {i _ {1} \dots i _ {r _ {\alpha}}} ^ {(\alpha)}, \tag {51}
+$$
+
+$$
+\begin{array}{l} \mathcal {S} _ {i _ {1} \dots i _ {r _ {\alpha}}} ^ {(\alpha)} := \mathbb {E} _ {U} \left[ \left(\sigma^ {(\ell_ {1} ^ {\alpha})} \left(U _ {i _ {1}} ^ {T} x _ {\bar {\sigma} _ {\alpha} (1)}\right) \dots \sigma^ {(\ell_ {k _ {1}} ^ {\alpha})} \left(U _ {i _ {1}} ^ {T} x _ {\bar {\sigma} _ {\alpha} \left(k _ {1} ^ {\alpha}\right)}\right)\right) \times \dots \times \right. \\ \left. \left(\sigma^ {\left(\ell_ {m + 1} ^ {\alpha} - k _ {r _ {\alpha}} ^ {\alpha}\right)} \left(U _ {i _ {r _ {\alpha}}} ^ {T} x _ {\bar {\sigma} _ {\alpha} (m + 1 - k _ {r _ {\alpha}} ^ {\alpha})}\right) \dots \sigma^ {\left(\ell_ {m} ^ {\alpha}\right)} \left(U _ {i _ {r _ {\alpha}}} ^ {T} x _ {\bar {\sigma} _ {\alpha} (m)}\right)\right) F ^ {(\alpha)} \left(x _ {1}, \dots , x _ {m}\right) \right]. \tag {52} \\ \end{array}
+$$
+
+This form is obtained by carrying out the  $V_{i}$  integrals, as well as all the sums that can be trivially carried out due to the presence of Kronecker deltas. Here, the  $\alpha$  sum represents a sum over all  $K$  terms that appear from performing the  $V_{i}$  integrals using Isserlis' theorem;  $\sigma^{(\ell)}$  denotes the  $\ell$ -th derivative of the non-linearity;  $k_{s}^{\alpha}$  is the number of  $\sigma^{(\cdot)}(U_{i_s}^T x)$  factors sharing the  $i_{s}$  index;  $\bar{\sigma}_{\alpha}\in S_{m}$  is a permutation; and  $F^{(\alpha)}$  is a function whose exact form is not important for us. We will denote by  $r_{\mathrm{max}}$  the maximum number of index sums appearing in (51), namely  $r_{\mathrm{max}}\coloneqq \max_{\alpha}r_{\alpha}$ .
+
+Our approach to establishing the asymptotic scaling will be to first bound the maximum number of index sums appearing in any term in our correlation function, written in the form (51), and then to argue that the summands are bounded by an  $n$ -independent constant.
+
+Let us introduce a family of diagrams,  $\Gamma'(C)$ , which are different in general than the Feynman diagrams. A given diagram  $g \in \Gamma'(C)$  is constructed as follows.
+
+Definition 6. Let  $C(x_{1},\ldots ,x_{m};f_{\mathrm{nl}})$  be a correlation function, where  $f_{\mathrm{nl}}$  is the output of a network with one hidden layer, defined in (46). The family  $\Gamma^{\prime}(C)$  is the set of all graphs that have the following properties.
+
+- Each derivative tensor in  $C$  is mapped to a vertex in the graph.  
+Each edge has a type that corresponds to one of the weight vectors  $U$  or  $V$ .  
+Each vertex has exactly one edge of  $V$  type.  
+- If two derivative tensors are contracted in  $C$ , the graph must have at least one edge (of any type) connecting the corresponding vertices for each contraction.
+
+The reason for introducing this graphical structure is that it allows us to make two important statements. If we define  $\tilde{c}_g$  the number of connected components in a graph  $g\in \Gamma^{\prime}(C)$  and  $\tilde{c}_{\mathrm{max}}\coloneqq \max_{g\in \Gamma^{\prime}(C)}\tilde{c}_{g}$  then the following holds.
+
+1. The maximal number of sums appearing in a correlation function of the form (51) is  $\tilde{c}_{\mathrm{max}}$  
+2.  $\tilde{c}_{\mathrm{max}} = c_{\mathrm{max}}$ , where  $c_{\mathrm{max}}$  is the maximal number of connected components in the family of Feynman diagrams corresponding to a correlation function  $C(x_1, \ldots, x_m; f_d)$ , where  $f_d$  is a deep linear network with  $d$  hidden layers, and  $d$  is large enough that none of the derivative tensors vanish. $^{10}$
+
+These two results, combined with a bound on the summands occurring in  $C(x_{1},\ldots ,x_{m};f_{\mathrm{nl}})$  will establish the bound (47).
+
+Lemma 4. A correlation function  $C$  has  $r_{max} = \tilde{c}_{max}$ .
+
+Proof. Each vertex corresponds to a derivative tensor,  $T_{\mu_1 \ldots \mu_k}$ , which contains a single sum over paired  $U_i$  and  $V_i$  indices. If two vertices are connected by one or more edges, there is a Kronecker delta factor that sets the corresponding indices to be equal (due either to a derivative contraction, or to a  $V$  covariance), reducing the number of sums by one. The result is that any connected component of a graph  $g \in \Gamma'(C)$  corresponds to a single sum, and the total number of index sums corresponding to a particular graph is  $\tilde{c}_g$ , the number of connected components. As a result, the maximum number of sums corresponding to the collection of graphs  $\Gamma'(C)$  is  $\tilde{c}_{\max}$ .
+
+Lemma 5. The maximal number of connected components,  $\tilde{c}_{\text{max}}$ , over the collection of graphs  $\Gamma'(C)$  corresponding to  $C(x_1, \ldots, x_m; f_{\text{n1}})$  is bounded by the maximal number of components,  $c_{\text{max}}$ , over the collection  $\Gamma(C)$  corresponding to  $C(x_1, \ldots, x_m; f_d)$  of Feynman diagrams, where  $f_d$  is a deep linear network of sufficient depth  $d$ .
+
+Proof. Let  $n_e(n_o)$  be the number of even (odd) clusters in the cluster graph  $G_C$  of  $C(x_1, \ldots, x_m; f_d)$ . The cluster graph,  $G_C$  is a subgraph of any graph  $g \in \Gamma'(C)$ . We can thus think about the embedding of even and odd clusters into  $g$ . In any graph  $g \in \Gamma'(C)$ , an even cluster may belong to its own connected component, while for odd clusters there must be an even number of them in any connected
+
+component. This is because an even (odd) cluster contains an even (odd) number of factors of  $V$ , which must be paired up in any connected component. We find that
+
+$$
+\tilde {c} _ {\max } = n _ {e} + \frac {n _ {o}}{2} \leq c _ {\max } . \tag {53}
+$$
+
+The last inequality was used below Lemma 2 in the proof of Theorem 1.
+
+![](images/c71a5c04b2b8838d97a6c4d34d789a76f6fc72b386245bfd5aa293b1de738e53.jpg)
+
+Theorem 8. Let  $C(x_{1},\ldots ,x_{m};f_{\mathrm{nl}})$  be a correlation function for a one hidden layer network. Let  $c_{\mathrm{max}}$  be the maximal number of connected components over the collection of graphs  $\Gamma (C)$  corresponding to  $C(x_{1},\dots,x_{m};f_{d})$ , where  $f_{d}(x)$  is a deep linear network map, with width depth greater than or equal to the maximum number of derivatives appearing in any single derivative tensor in  $C$ . Then  $C = \mathcal{O}(n^{s})$  where  $s = c_{\mathrm{max}} - \frac{m}{2}$ . Furthermore,
+
+$$
+C \left(x _ {1}, \dots , x _ {m}; f _ {\mathrm {n l}}\right) = \mathcal {O} \left(C \left(x _ {1}, \dots , x _ {m}; f\right)\right). \tag {54}
+$$
+
+Proof. The correlation function in (51) can be bound as
+
+$$
+\left| C \left(x _ {1}, \dots , x _ {m}; f _ {\mathrm {n l}}\right) \right| \leq \frac {1}{n ^ {m / 2}} \sum_ {\alpha = 1} ^ {K} \sum_ {i _ {1}, \dots , i _ {r}} ^ {n} \left| \mathcal {S} _ {i _ {1}, \dots , i _ {r \alpha}} ^ {(\alpha)} \right|. \tag {55}
+$$
+
+For fixed inputs,  $S_{i_1,\dots,i_{r_\alpha}}^{(\alpha)}$  can take at most  $\mathcal{O}(1)$  different values as a function of its indices, and the values are independent of  $n$ . This is because the variables  $U_i$  are identical. We define  $s_{\max}$  as the maximum value of  $|S_{i_1,\dots,i_{r_\alpha}}^{(\alpha)}|$  as a function of  $\alpha$  and the indices. Combining this with the above lemmas we can then write.
+
+$$
+\left| C \left(x _ {1}, \dots , x _ {m}; f _ {\mathrm {n l}}\right) \right| \leq K s _ {\max } n ^ {c _ {\max } - m / 2}. \tag {56}
+$$
+
+The result of the theorem follows from Lemma 2.
+
+![](images/0e63d882156a449a003ee562540662b9bbedbdbe6120e7f9df50272adb7406c9.jpg)
+
+# D CORRELATION FUNCTION ASYMPTOTICS
+
+In this section we prove several general results about correlation function asymptotics in the large width limit. Throughout this section, we assume that Conjecture 1 holds.
+
+# D.1 VARIANCE ASYMPTOTICS
+
+Conjecture 1 can be used to bound the variance of the integrands that appear inside correlation functions.
+
+Lemma 6. Let  $\tilde{C}(x_1, \ldots, x_m) = \mathbb{E}_{\theta} [F_{\theta}(x_1, \ldots, x_m)]$  be a correlation function, and let  $G_C$  be the cluster graph of  $C$  with  $n_e$  even components and  $n_o$  odd components. Assume Conjecture 1 holds such that  $C = \mathcal{O}(n^{s_C})$ , where  $s_C = n_e + \frac{n_o}{2} - \frac{m}{2}$ . Then  $\mathrm{Var}_{\theta} [F_{\theta}(x_1, \ldots, x_m)] = \mathcal{O}(n^{2s_C})$ .
+
+Proof. To bound the variance, it is enough to bound the correlation function
+
+$$
+\tilde {C} \left(x _ {1}, \dots , x _ {2 m}\right) := \mathbb {E} _ {\theta} \left[ F _ {\theta} \left(x _ {1}, \dots , x _ {m}\right) F _ {\theta} \left(x _ {m + 1}, \dots , x _ {2 m}\right) \right], \tag {57}
+$$
+
+because  $\operatorname{Var}_{\theta}[F_{\theta}(x_1, \ldots, x_m)] \leq \tilde{C}(x_1, \ldots, x_m, x_1, \ldots, x_m)$ . The correlation function  $\tilde{C}$  has  $2n_e$  even clusters and  $2n_o$  odd clusters. It follows from Conjecture 1 that  $\tilde{C} = \mathcal{O}(n^{2s_C})$ .
+
+As a corollary, notice that if  $C = \mathcal{O}(n^s)$  according to Conjecture 1, then typical realizations of the integrand will also be  $\mathcal{O}(n^s)$ . In other words, the asymptotic bound of Conjecture 1 holds for typical initializations, not just in expectation.
+
+Table 3 shows empirical results for the variance of several functions. In all cases we find that the bound of Lemma 6 holds. For  $\sum_{\mu}\mathrm{Var}_{\theta}\left[\partial_{\mu}f(x_1)\partial_{\mu}f(x_2)\right]$  we prove a tight bound below in Appendix E.1 for deep linear networks.
+
+<table><tr><td>Function</td><td>ne, no</td><td>2sC</td><td>lin.</td><td>ReLU</td><td>tanh</td></tr><tr><td>Varθ[f(x1)f(x2)]</td><td>0,2</td><td>0</td><td>-0.08</td><td>-0.00</td><td>-0.02</td></tr><tr><td>Varθ[f(x1)f(x2)f(x3)f(x4)]</td><td>0,4</td><td>0</td><td>-0.03</td><td>0.02</td><td>-0.05</td></tr><tr><td>∑μ Varθ[∂μf(x1)∂μf(x2)]</td><td>1,0</td><td>0</td><td>-1.01</td><td>-1.02</td><td>-0.99</td></tr><tr><td>∑μ,ν Varθ[∂μf(x1)∂νf(x2)∂μ,νf(x3)f(x4)]</td><td>0,2</td><td>-2</td><td>-2.1</td><td>-2.13</td><td>-2.14</td></tr><tr><td>∑μ,ν,ρ Varθ[∂μf(x1)∂νf(x2)∂ρf(x3)∂μ,ν,ρf(x4)]</td><td>1,0</td><td>-2</td><td>-4.02</td><td>-4.1</td><td>-3.05</td></tr><tr><td>∑μ,ν,ρ,σ Varθ[∂μf(x1)∂νf(x2)∂μ,νf(x3)∂ρf(x4)∂σf(x5)∂ρ,σf(x6)]</td><td>0,2</td><td>-4</td><td>-4.09</td><td>-4.14</td><td>-4.01</td></tr></table>
+
+Table 3: Bounds on variances obtained from Lemma 6, where the predicted exponent is  $2s_{C}$ , compared with empirical results. The predicted exponent is  $2s_{C}$ , and the 3 right-most columns list the empirical exponents. The experimental setup is the same as that of Table 1.
+
+# D.2 GRADIENT FLOW
+
+The following results are used in the gradient flow calculations of Section 3.
+
+Lemma 7. Let  $G'$  be a graph with  $m'$  vertices,  $n_e'$  even components, and  $n_o'$  odd components. Let  $G$  be a subgraph of  $G'$  with  $m$  vertices,  $n_e$  even components, and  $n_o$  odd components. Then  $s(n_e, n_o, m) \geq s(n_e', n_o', m')$  where  $s(a, b, c) := a + \frac{b - c}{2}$ .
+
+Proof. It is enough to show that  $s(n_{e}, n_{o}, m)$  does not increase if we (1) add a vertex to  $G$ , or (2) add an edge to  $G$ , because  $G'$  can be obtained from  $G$  by performing such operations finitely many times. Adding a vertex to  $G$  changes  $n_{e} \mapsto n_{e}$ ,  $n_{o} \mapsto n_{o} + 1$ , and  $m \mapsto m + 1$ , leaving  $s(n_{e}, n_{o}, m)$  unchanged. Next, if we add an edge to  $G$  then  $m$  does not change, and there are 4 possibilities for how  $n_{e}$  and  $n_{o}$  change.
+
+1. The edge connects two even components. Then  $n_e \mapsto n_e - 1$ ,  $n_o \mapsto n_o$ , and  $s(n_e, n_o, m)$  decreases by 1.  
+2. The edge connects two odd components. Then  $n_e \mapsto n_e + 1$ ,  $n_o \mapsto n_o - 2$ , and  $s(n_e, n_o, m)$  does not change.  
+3. The edge connects an even component and an odd component. Then  $n_e \mapsto n_e - 1$ ,  $n_o \mapsto n_o$ , and  $s(n_e, n_o, m)$  decreases by 1.  
+4. The edge connects two vertices that belong to the same component. In this case  $n_e, n_o$ , and  $s(n_e, n_o, m)$  do not change.
+
+![](images/99064ab33ffc385b03eb8be363552dc666624715f63f8428336363b131534d5b.jpg)
+
+We now prove Lemma 1 giving the scaling of time derivatives of correlation functions at initialization. We prove the result for polynomial loss functions. Extension to more general loss functions requires interchanging the large width limit and the Taylor expansion of the loss, which we do not discuss.
+
+proof (Lemma 1). Notice that
+
+$$
+\mathbb {E} _ {\theta} \left[ \frac {d ^ {k} F \left(x _ {1} , \dots , x _ {m}\right)}{d t ^ {k}} \right] = \mathbb {E} _ {\theta} \left[ \left(\sum_ {\mu} \sum_ {\left(x ^ {\prime}, y ^ {\prime}\right) \in D _ {\mathrm {t r}}} \frac {\partial \ell \left(x ^ {\prime} , y ^ {\prime}\right)}{\partial f} \frac {\partial f \left(x ^ {\prime}\right)}{\partial \theta^ {\mu}} \frac {\partial}{\partial \theta^ {\mu}}\right) ^ {k} F \left(x _ {1}, \dots , x _ {m}\right) \right]. \tag {58}
+$$
+
+On the right-hand side we have a linear combination  $\sum_{A} \alpha_{A} C_{A}$  of correlation functions  $C_{A}$ , where the coefficients depend on the training set labels. Here we used the polynomial loss assumption. For each correlation function  $C_{A}$ , its integrand can be obtained from  $F$  by finitely many operations of the form (1) multiply the integrand by  $f(x)$  for some input  $x$ , and (2) act with a pair of contracted
+
+derivatives on two of the derivative tensors. In the cluster graph representation, these two operations correspond to (1) adding a vertex, and (2) adding an edge. Therefore, the cluster graph  $G_{C}$  of  $C = \mathbb{E}_{\theta}[F]$  is a subgraph of the cluster graph  $G_{C_A}$  of each one of the correlation functions  $C_A$ . By Lemma 7 we have that  $C_A = \mathcal{O}(n^{s_C})$  for all  $A$ , and therefore the bound applies to  $\mathbb{E}_{\theta}\left[d^{k}F / dt^{k}\right]$  as well.
+
+# D.3 STOCHASTIC GRADIENT DESCENT
+
+In this section we show that the asymptotics of a correlation function do not change under stochastic gradient descent (SGD) updates. Let  $C(x_1, \ldots, x_m) = \mathbb{E}_{\theta \sim \mathcal{P}_0} [F_\theta(x_1, \ldots, x_m)]$  be a correlation function, where  $F_\theta$  is the integrand which explicitly depends on the parameters  $\theta$ . Under a single SGD step, the parameters are updated as  $\theta_{t+1} = \theta_t - \eta \nabla L_t(\theta_t)$ , where  $L$  is the mini-batch loss at time  $t$ . Let  $\mathcal{P}_t$  denote the distribution of parameters at step  $t$ , where  $\mathcal{P}_0$  is the initial distribution. We define the evolved correlation function at step  $t$  to be
+
+$$
+C _ {t} \left(x _ {1}, \dots , x _ {m}\right) := \mathbb {E} _ {\theta \sim \mathcal {P} _ {t}} \left[ F _ {\theta} \left(x _ {1}, \dots , x _ {m}\right) \right]. \tag {59}
+$$
+
+We have the following
+
+Theorem 9. Let  $C$  be a correlation function for a network with linear activations, and assume that Conjecture 1 holds, namely  $C = \mathcal{O}(n^{sc})$  where  $s_C$  is defined in the Conjecture. If  $C_t$  is the evolved correlation function after  $t$  SGD steps, then  $C_t = \mathcal{O}(n^{sc})$ .
+
+Proof. Notice that
+
+$$
+C _ {t + 1} = \mathbb {E} _ {\theta \sim \mathcal {P} _ {t + 1}} \left[ F _ {\theta} \right] = \mathbb {E} _ {\theta \sim \mathcal {P} _ {t}} \left[ F _ {\theta - \eta \nabla L (\theta)} \right]. \tag {60}
+$$
+
+The integrand  $F_{\theta}$  can be written as a product of derivative tensors of the form  $T_{\mu_1\dots \mu_k}(x;\theta)$ , with contracted derivatives. Suppose that the network has  $d$  hidden layers. Then under an SGD step, we have
+
+$$
+\begin{array}{l} T _ {\mu_ {1} \dots \mu_ {l}} (x; \theta - \eta \nabla L _ {t} (\theta)) = \sum_ {k = 0} ^ {d + 1} \frac {(- \eta) ^ {k}}{k !} \sum_ {\nu_ {1} \dots \nu_ {k}} \frac {\partial L _ {t}}{\partial \theta^ {\nu_ {1}}} \dots \frac {\partial L _ {t}}{\partial \theta^ {\nu_ {k}}} T _ {\mu_ {1} \dots \mu_ {l} \nu_ {1} \dots \nu_ {k}} (x; \theta) (61) \\ = \sum_ {k = 0} ^ {d + 1} \frac {(- \eta) ^ {k}}{k !} \sum_ {x _ {1} ^ {\prime}, \dots , x _ {k} ^ {\prime} \in D _ {B}} \left[ \frac {\partial L _ {t}}{\partial f \left(x _ {1} ^ {\prime}\right)} \dots \frac {\partial L _ {t}}{\partial f \left(x _ {k} ^ {\prime}\right)} \right. \\ \left. \times \sum_ {\nu_ {1}, \dots , \nu_ {k}} T _ {\nu_ {1}} \left(x _ {1} ^ {\prime}\right) \dots T _ {\nu_ {k}} \left(x _ {k} ^ {\prime}\right) T _ {\mu_ {1} \dots \mu_ {l} \nu_ {1} \dots \nu_ {k}} (x) \right]. (62) \\ \end{array}
+$$
+
+Here  $D_B$  is the mini-batch, and  $x_{a}^{\prime}$  are mini-batch samples. The  $k$  sum is truncated because higher-order derivatives of  $f$  vanish.
+
+We can now see how taking a gradient descent step affects the cluster graph. After taking a step, each derivative tensor in  $C_t$  is replaced by a sum (over  $k, x_1', \ldots, x_k'$ ). Each term in the combination of these sums is a correlation function, whose cluster graph is a subgraph of  $C$ . Therefore, by Lemma 7,  $C_t = \mathcal{O}(n^{s_C})$ .
+
+We note that for general activation functions the sum in (62) may be infinite. In this case, to complete the proof we would need to show that the infinite sum obeys the same bound as each individual term in the sum. We leave this for future work.
+
+# E APPLICATIONS
+
+Here we present several applications of our Feynman diagram method for computing large width asymptotics. We assume throughout this section that Conjecture 1 holds.
+
+# E.1 NEURAL TANGENT KERNEL
+
+In this section we prove two results regarding the NTK at large width. We show that the kernel converges in probability, and that during gradient descent it is constant up to  $\mathcal{O}(n^{-1})$  corrections.
+
+Theorem 10. The Neural Tangent Kernel  $\Theta$  of a deep linear network converges in probability in the large width limit, and its variance is  $\mathcal{O}(n^{-1})$ .
+
+Conjecture 1 is not sufficient for proving this theorem, as we need to use a more detailed Feynman diagram argument. Therefore, we only prove the theorem for the case of deep linear networks.
+
+Proof. First, we will show that  $\operatorname{Var}[\Theta] = \mathcal{O}(n^{-1})$ . Given a model function  $f$ , the variance is given by
+
+$$
+\operatorname {V a r} \left[ \Theta \left(x, x ^ {\prime}\right) \right] = A \left(x, x ^ {\prime}\right) - B \left(x, x ^ {\prime}\right), \tag {63}
+$$
+
+$$
+A \left(x, x ^ {\prime}\right) = \sum_ {\mu , \nu} \mathbb {E} \left[ \frac {\partial f ^ {(1)} (x)}{\partial \theta_ {\mu}} \frac {\partial f ^ {(2)} \left(x ^ {\prime}\right)}{\partial \theta_ {\mu}} \frac {\partial f ^ {(3)} (x)}{\partial \theta_ {\nu}} \frac {\partial f ^ {(4)} \left(x ^ {\prime}\right)}{\partial \theta_ {\nu}} \right], \tag {64}
+$$
+
+$$
+B \left(x, x ^ {\prime}\right) = \sum_ {\mu , \nu} \mathbb {E} \left[ \frac {\partial f ^ {(1)} (x)}{\partial \theta_ {\mu}} \frac {\partial f ^ {(2)} \left(x ^ {\prime}\right)}{\partial \theta_ {\mu}} \right] \mathbb {E} \left[ \frac {\partial f ^ {(3)} (x)}{\partial \theta_ {\nu}} \frac {\partial f ^ {(4)} \left(x ^ {\prime}\right)}{\partial \theta_ {\nu}} \right]. \tag {65}
+$$
+
+Here  $f^{(1)} = \dots = f^{(4)} = f$ ; we introduced the superscripts so we can easily refer to individual factors. The crux of the argument is that the set of Feynman diagrams representing the expression (63) includes only connected graphs. Assuming this is the case, according to the bound (22) these diagrams will all scale as  $\mathcal{O}(n^{(2 - 4) / 2}) = \mathcal{O}(n^{-1})$  and so will the variance, completing the proof. To see why only connected graphs contribute, notice that
+
+-  $A(x, x')$  includes all diagrams in which the vertices corresponding to  $f^{(1)}, f^{(2)}$  share an edge, and also the vertices  $f^{(3)}, f^{(4)}$  share an edge (due to the explicit derivatives);  
+-  $B(x, x')$  includes all diagrams in which all edges are either between  $f^{(1)}, f^{(2)}$  or between  $f^{(3)}, f^{(4)}$ .
+
+Therefore, in the full expression (63), the only remaining diagrams (i.e. the diagrams that do not cancel between the two terms) are those that include
+
+- an edge connecting  $f^{(1)}, f^{(2)}$ ,  
+- an edge connecting  $f^{(3)}, f^{(4)}$ , and  
+- an edge connecting one of  $f^{(1)}, f^{(2)}$  to one of  $f^{(3)}, f^{(4)}$ .
+
+These diagrams are all connected graphs, and this completes the proof that  $\mathrm{Var}[\Theta] = \mathcal{O}(n^{-1})$ .
+
+Here and above we have established that  $\operatorname{Var}[\Theta] = \mathcal{O}(n^{-1})$  and  $\mathbb{E}[\Theta] = \mathcal{O}(n^0)$ . More generally, let us consider a random variable  $O$  equal to the product of  $f$  and its derivatives, where the derivatives indices are fully summed in pairs. As we will now show, if  $\mathbb{E}[O] = \mathcal{O}(n^0)$  and  $\operatorname{Var}[O] = \mathcal{O}(n^{-1})$  then (1) the limit  $\lim_{n\to \infty}\mathbb{E}[O]$  exists, and (2)  $O$  converges in probability to this limit. In particular, the NTK is an example of such a random variable, and so this will conclude the proof that the NTK converges in probability.
+
+First, consider the mean. There is a finite number of diagrams contributing to  $\mathbb{E}[O]$ , and each has a well-defined  $n$  scaling. Therefore, we can write
+
+$$
+\mathbb {E} [ O ] = \sum_ {k = 0} ^ {k _ {\max }} \frac {c _ {k}}{n ^ {k}} \tag {66}
+$$
+
+for some values of  $k_{\mathrm{max}} \geq 0$  (integer) and  $c_0, c_1, \ldots \in \mathbb{R}$ . We see that the mean has a well-defined large  $n$  limit,
+
+$$
+\lim  _ {n \rightarrow \infty} \mathbb {E} [ O ] = c _ {0}. \tag {67}
+$$
+
+Next, let us show that  $O - \mathbb{E}[O]$  converges in probability using Chebyshev's inequality. Indeed, by the variance assumption there exists  $\tilde{c} > 0$  such that
+
+$$
+P \left(\left| O - \mathbb {E} [ O ] \right| > \epsilon\right) \leq \frac {\operatorname {V a r} [ O ]}{\epsilon^ {2}} \leq \frac {\tilde {c}}{n \epsilon^ {2}} \rightarrow 0. \tag {68}
+$$
+
+Combining the facts that  $O - \mathbb{E}[O] \to_{p} 0$  and  $\mathbb{E}[O] \to c_0$ , we find that  $O$  converges in probability to  $c_0$ .
+
+Next, we show that the large width NTK is constant during training, and compute the asymptotics of the higher-order terms. The following argument is phrased for deep linear networks. More generally, the same argument holds for deep networks with smooth non-linear activations under the additional assumption that the large width limit and Taylor series can be exchanged (note that for such networks, the network function is analytic in the weights).
+
+Theorem 11. Let  $f(x)$  be the network output of a deep linear network with MSE loss  $L$ . Let  $\Theta_t(x, x')$  be the Neural Tangent Kernel at SGD step  $t$ , for some inputs  $x, x'$ . Then in the large width limit, the kernel is constant in  $t$  in expectation, and
+
+$$
+\mathbb {E} _ {\theta \sim \mathcal {P} _ {0}} \left[ \Theta_ {t} \left(x, x ^ {\prime}\right) - \Theta_ {0} \left(x, x ^ {\prime}\right) \right] = \mathcal {O} \left(n ^ {- 1}\right), \tag {69}
+$$
+
+$$
+\left. \operatorname {V a r} _ {\theta \sim \mathcal {P} _ {0}} \left[ \Theta_ {t} \left(x, x ^ {\prime}\right) - \Theta_ {0} \left(x, x ^ {\prime}\right) \right] = \mathcal {O} \left(n ^ {- 2}\right). \right. \tag {70}
+$$
+
+Proof. It is enough to show that  $\mathbb{E}_{\theta \sim \mathcal{P}_0}[\Theta_1(x,x') - \Theta_0(x,x')] = \mathcal{O}(n^{-1})$ . It then follows from Theorem 9 that  $\mathbb{E}_{\theta \sim \mathcal{P}_0}[\Theta_{t + 1}(x,x') - \Theta_t(x,x')] = \mathcal{O}(n^{-1})$  for all  $t$ , and therefore
+
+$$
+\mathbb {E} _ {\theta \sim \mathcal {P} _ {0}} \left[ \Theta_ {t} \left(x, x ^ {\prime}\right) - \Theta_ {0} \left(x, x ^ {\prime}\right) \right] = \sum_ {t ^ {\prime} = 0} ^ {t - 1} \mathbb {E} _ {\theta \sim \mathcal {P} _ {0}} \left[ \Theta_ {t ^ {\prime} + 1} \left(x, x ^ {\prime}\right) - \Theta_ {t ^ {\prime}} \left(x, x ^ {\prime}\right) \right] = \mathcal {O} \left(n ^ {- 1}\right), \tag {71}
+$$
+
+concluding the proof. We have
+
+$$
+\begin{array}{l} \mathbb{E}_{\theta \sim \mathcal{P}_{0}}\left[\Theta_{1}(x,x^{\prime}) - \Theta_{0}(x,x^{\prime})\right] = \sum_{\substack{k,l = 0\\ k + l\geq 1}}^{d + 1}\frac{(-\eta)^{k + l}}{k!l!}\sum_{\mu}\mathbb{E}_{\theta \sim \mathcal{P}_{0}}\left[\sum_{\mu_{1},\ldots ,\mu_{k}}\frac{\partial L}{\partial\theta^{\mu_{1}}}\dots \frac{\partial L}{\partial\theta^{\mu_{k}}} T_{\mu \mu_{1}\ldots \mu_{k}}(x)\right. \\ \left. \sum_ {\nu_ {1}, \dots , \nu_ {l}} \frac {\partial L}{\partial \theta^ {\nu_ {1}}} \dots \frac {\partial L}{\partial \theta^ {\nu_ {l}}} T _ {\mu \nu_ {1} \dots \nu_ {l}} \left(x ^ {\prime}\right) \right]. \tag {72} \\ \end{array}
+$$
+
+The fact that the  $k,l$  sums are truncated at  $d + 1$  follows from using linear activations, as higher-order derivatives of the network function vanish in this case. All terms in the sum over  $k,l$  include a tensor product of the form  $\sum_{\mu,\vec{\mu},\vec{\nu}}T_{\mu\mu_1\ldots\mu_k}(x)T_{\mu\nu_1\ldots\nu_l}(x')(\cdots)$  with either  $k \geq 1$  or  $l \geq 1$ , where  $(\cdots)$  stands for additional derivative tensor factors. Therefore, all terms in the  $k,l$  sum are correlation functions that have a cluster of size at least 3, including  $T(x)$ ,  $T(x')$ , and at least one other tensor contracted through the  $\mu_1$  or  $\nu_1$  index. It follows from the Conjecture that each term in the sum is  $\mathcal{O}(n^{-1})$ . Lemma 6 then implies that the variance of these updates is  $\mathcal{O}(n^{-2})$ .
+
+# E.2 WIDE NETWORK EVOLUTION IS LINEAR IN THE LEARNING RATE
+
+In this section we prove that, at large width, the NTK determines the evolution of the network function not just for continuous-time gradient descent but also for discrete-time gradient descent. A similar result holds for stochastic gradient descent, using a stochastic kernel. $^{11}$  Again we prove the deep linear case explicitly, but the result holds for deep networks with smooth non-linear activations under the additional assumption that the large width limit and Taylor series can be exchanged.
+
+Theorem 12. Let  $f(x)$  be the network output of a deep linear network, and let  $f_{t}(x)$  be the evolved function after  $t$  gradient descent steps, defined by  $\theta_{t + 1} = \theta_t - \eta \nabla L(\theta_t)$ . In the large width limit, each gradient descent step update of  $f_{t}$  is linear in the learning rate  $\eta$ . Furthermore,
+
+$$
+\mathbb {E} _ {\theta} \left[ f _ {t + 1} (x) - f _ {t} (x) \right] = - \eta \sum_ {\left(x ^ {\prime}, y ^ {\prime}\right) \in D _ {\mathrm {t r}}} \mathbb {E} _ {\theta} \left[ \Theta_ {0} \left(x, x ^ {\prime}\right) \frac {\partial \ell_ {t} \left(x ^ {\prime} , y ^ {\prime}\right)}{\partial f} \right] + \mathcal {O} \left(n ^ {- 1}\right). \tag {73}
+$$
+
+Here,  $\Theta_0$  is the NTK at initialization and  $\ell_t$  is the single-sample loss at time  $t$ .
+
+Proof. For a deep linear network, under a single gradient descent step we have
+
+$$
+\begin{array}{l} \mathbb {E} _ {\theta} \left[ f _ {t + 1} (x) - f _ {t} (x) \right] = \sum_ {k = 1} ^ {d + 1} \frac {(- \eta) ^ {k}}{k !} \sum_ {\mu_ {1}, \dots , \mu_ {k}} \mathbb {E} _ {\theta} \left[ \frac {\partial^ {k} f _ {t} (x)}{\partial \theta^ {\mu_ {1}} \cdots \partial \theta^ {\mu_ {k}}} \frac {\partial L _ {t}}{\partial \theta^ {\mu_ {1}}} \dots \frac {\partial L _ {t}}{\partial \theta^ {\mu_ {k}}} \right] (74) \\ = - \eta \sum_ {\left(x ^ {\prime}, y ^ {\prime}\right) \in D _ {\mathrm {t r}}} \mathbb {E} _ {\theta} \left[ \Theta_ {t} \left(x, x ^ {\prime}\right) \frac {\partial \ell_ {t} \left(x ^ {\prime} , y ^ {\prime}\right)}{\partial f} \right] \\ + \sum_ {k = 2} ^ {d + 1} \frac {(- \eta) ^ {k}}{k !} \sum_ {\mu_ {1}, \dots , \mu_ {k}} \mathbb {E} _ {\theta} \left[ \frac {\partial^ {k} f _ {t} (x)}{\partial \theta^ {\mu_ {1}} \cdots \partial \theta^ {\mu_ {k}}} \frac {\partial L _ {t}}{\partial \theta^ {\mu_ {1}}} \dots \frac {\partial L _ {t}}{\partial \theta^ {\mu_ {k}}} \right]. (75) \\ \end{array}
+$$
+
+First, consider the sum over  $k$  in (73). Each term in the sum is a correlation function for which the cluster graph contains a connected subgraph of size at least 3, and is therefore  $\mathcal{O}(n^{-1})$  by Lemma 7 and Theorem 1. In the remaining  $\mathcal{O}(\eta)$  term, by the same argument as Theorem 11 we can replace  $\Theta_t = \Theta_0 + \mathcal{O}(n^{-1})$  in the correlation function. The result is equation (73).
+
+As mentioned above, we note that the proof goes through when using stochastic gradient descent updates, with the difference that in (73) we should sum over mini-batch samples instead of over the entire training set.
+
+# E.3 SPECTRAL PROPERTIES OF THE NTK AND THE HESSIAN
+
+With an eye towards understanding the structure of the loss landscape at large width and as another example use case of our approach, we investigate the relation between the spectra of the Hessian, and the NTK.
+
+Among other observations, we present an argument that for a network with  $d$  hidden layers and MSE loss, the top  $D_{\mathrm{in}}$  ordered eigenvalues of the Hessian,  $\lambda_i^{(H)}$ , are related to those of the kernel  $\lambda_i^{(\Theta)}$  by
+
+$$
+\mathbb {E} _ {\theta} \left[ \lambda_ {i} ^ {(H)} - \lambda_ {i} ^ {(\Theta)} \right] = \left\{ \begin{array}{l l} \mathcal {O} \left(n ^ {- 1 / 2}\right) & , d = 1 \\ \mathcal {O} \left(n ^ {- 1}\right) & , d > 1 \end{array} \right.. \tag {76}
+$$
+
+The remaining Hessian eigenvalues vanish at large width as  $\mathcal{O}(n^{-1/2})$  for one hidden layer networks and  $\mathcal{O}(n^{-1})$  for deeper networks. This is experimentally corroborated in Figure 9.
+
+The Hessian of a general loss takes the form,
+
+$$
+H _ {\mu \nu} = \sum_ {(x, y) \in D _ {\mathrm {t r}}} \left[ \underbrace {\frac {\partial^ {2} \ell (x , y)}{\partial f ^ {2}} \frac {\partial f (x)}{\partial \theta^ {\mu}} \frac {\partial f (x)}{\partial \theta^ {\nu}}} _ {\mathcal {A} _ {\mu \nu}} + \underbrace {\frac {\partial \ell (x , y)}{\partial f} \frac {\partial^ {2} f (x)}{\partial \theta^ {\mu} \partial \theta^ {\nu}}} _ {\mathcal {B} _ {\mu \nu}} \right]. \tag {77}
+$$
+
+The moments of  $H$  are a useful way to understand its spectrum. As in other examples that we have seen, traces of powers of  $\mathcal{B}$  lead to connected diagrams with higher and higher numbers of vertices, and so scale as increasingly negative powers of  $n$ . More explicitly, traces involving powers of  $\mathcal{B}$  contain multiple contracted derivative tensors. As a result, taking the average of these traces over the
+
+![](images/7b18a799e4450f694779a7aa32ae8a02afeda2642678a5d73e74f09e1333b6e2.jpg)  
+Figure 8: A prototypical diagram corresponding to  $\mathbb{E}_{\theta}\left[\mathrm{Tr}\left(\mathcal{B}^{2}\right)\right]$
+
+weight distribution leads to correlation functions where the associated cluster graph,  $G_{C}$ , contains subgraphs with at least three connected vertices. By Lemma 7 and Conjecture 1 these correlation functions vanish at infinite width. For example  $\mathbb{E}_{\theta}\left[\mathrm{Tr}\left(\mathcal{AB}\right)\right] = \mathcal{O}(1 / n)$ . There is a single exception to this, as can be easily seen in figure 8,  $\mathbb{E}_{\theta}\left[\mathrm{Tr}(\mathcal{B}^{2})\right]$  is  $\mathcal{O}(1)$ . Thus most moments of the Hessian are equal to moments of  $\mathcal{A}$  in expectation.[13]
+
+$$
+\mathbb {E} _ {\theta} \left[ \operatorname {T r} \left(H ^ {m}\right) \right] = \left\{ \begin{array}{l l} \mathbb {E} _ {\theta} \left[ \operatorname {T r} \left(\mathcal {A} ^ {m}\right) \right] & , m \neq 2 \\ \mathbb {E} _ {\theta} \left[ \operatorname {T r} \left(\mathcal {A} ^ {2}\right) \right] + \mathbb {E} _ {\theta} \left[ \operatorname {T r} \left(\mathcal {B} ^ {2}\right) \right] & , m = 2 \end{array} + \mathcal {O} (1 / n). \right. \tag {78}
+$$
+
+What's more, we can relate moments of  $\mathcal{A}$  to those of the kernel, as both are built out of two logit derivatives. Explicitly,
+
+$$
+\operatorname {T r} \left(\mathcal {A} ^ {m}\right) = \operatorname {T r} \left(\left(M \Theta\right) ^ {m}\right), \tag {79}
+$$
+
+and so the moments of the Hessian are also related to those of the kernel. $^{14}$
+
+$$
+\mathbb {E} _ {\theta} \left[ \operatorname {T r} \left(H ^ {m}\right) \right] = \left\{ \begin{array}{l l} \mathbb {E} _ {\theta} \left[ \operatorname {T r} \left((M \Theta) ^ {m}\right) \right] & , m \neq 2 \\ \mathbb {E} _ {\theta} \left[ \operatorname {T r} \left((M \Theta) ^ {2}\right) \right] + \mathbb {E} _ {\theta} \left[ \operatorname {T r} \left(\mathcal {B} ^ {2}\right) \right] & , m = 2 \end{array} + \mathcal {O} (1 / n). \right. \tag {80}
+$$
+
+Here, we have defined,
+
+$$
+M \left(x _ {a}, x _ {b}\right) = \delta_ {a b} \frac {\partial^ {2} \ell \left(x _ {a} , y _ {a}\right)}{\partial f ^ {2}}: \left(x _ {a}, y _ {a}\right) \in D _ {\mathrm {t r}}. \tag {81}
+$$
+
+For the case of MSE loss, we can go even further. In that case,  $M(x_{a}, x_{b}) = \delta_{ab}$  and  $\mathcal{B}$  decays to zero during training. We thus have,
+
+$$
+\text {I n i t i a l l y :} \quad \mathbb {E} _ {\theta} \left[ \operatorname {T r} \left(H ^ {m}\right) \right] = \left\{ \begin{array}{l l} \mathbb {E} _ {\theta} \left[ \operatorname {T r} \left(\Theta^ {m}\right) \right] & , m \neq 2 \\ \mathbb {E} _ {\theta} \left[ \operatorname {T r} \left(\Theta^ {2}\right) \right] + \mathbb {E} _ {\theta} \left[ \operatorname {T r} \left(\mathcal {B} ^ {2}\right) \right] & , m = 2 \end{array} + \mathcal {O} (1 / n) \right. \tag {82}
+$$
+
+At late times:  $\mathbb{E}_{\theta}\left[\mathrm{Tr}(H^{m})\right] \to \mathbb{E}_{\theta}\left[\mathrm{Tr}(\Theta^{m})\right] \quad \forall m$
+
+These results indicate that the only difference between the spectra of the Hessian and the NTK comes from  $\mathcal{B}$  and that  $\mathcal{B}$  must have eigenvalues which scale as  $1 / \sqrt{\dim(\mathcal{B})}$ . As  $\dim(B) = \mathcal{O}(n)$  for one hidden layer networks and  $\mathcal{O}(n^2)$  for deep networks we are left with the relation (76) between the eigenvalues of  $\Theta$  and  $H$ . As the network trains, the difference between these eigenvalues gets even smaller. These results are confirmed experimentally in Figure 9 and Figure 10.
+
+Multi-class. This story can be generalized to multi-class neural network maps. In this case the logit is a map  $f^A: \mathbb{R}^{D_{\mathrm{in}}} \to \mathbb{R}^{N_{\mathrm{class}}}$ , and the NTK has two class indices.
+
+$$
+\Theta^ {A B} \left(x, x ^ {\prime}\right) = \frac {\partial f ^ {A} (x)}{\partial \theta^ {\mu}} \frac {\partial f ^ {B} \left(x ^ {\prime}\right)}{\partial \theta^ {\mu}}. \tag {83}
+$$
+
+Expression (80) relating the moments of the Hessian to the NTK still holds in this context provided we take  $\mathrm{Tr}\left((M\Theta)^m\right)\to \mathrm{Tr}\left((M_{AB}\Theta^{AB})^m\right)$ , with the more general matrix,
+
+$$
+M _ {A B} \left(x _ {a}, x _ {b}\right) = \delta_ {a b} \frac {\partial \ell \left(x _ {a} , y _ {a}\right)}{\partial f ^ {A} \partial f ^ {B}}: \left(x _ {a}, y _ {a}\right) \in D _ {\mathrm {t r}}. \tag {84}
+$$
+
+![](images/9d6b5a2298f535b785c1d4c0b1fb84841695b3089c09dbf814b560fbdb458e11.jpg)  
+(a) Hessian and NTK e.v. difference.
+
+![](images/7ad30d92242abc0c27b20f5af4fd2adda9eab0807775a0bc27aad6e498d2dd52.jpg)  
+(b) Degenerate spectrum.  
+Figure 9: (a) The mean difference between the top eigenvalues of the NTK and the Hessian at initializations for a two hidden layer ReLU network of varying width match well with the predicted  $\mathcal{O}(n^{-1})$  behavior. The mean is taken over 100 instances per width on two-class MNIST with 10 images per class. (b) The top 10 eigenvalues of both the Hessian and the NTK for a two-layer ReLU network of width 2048 trained on a three-class subset of MNIST. The top eigenvalues match well and show aspects of the repeated  $N_{\mathrm{class}}$  structure predicted at large width. The eigenvalues of  $H$  are computed using the lanczos algorithm.
+
+![](images/a84f1f2bf349c3d0eab22ba221987085b85bb04f584b96f0285ea7de38519652.jpg)  
+(a) Relative moment differences  
+Figure 10: Relation between the spectrum of the Hessian and NTK. (a) The first 4 moments of  $H$  and  $\Theta$  for two hidden layer ReLU networks. At initialization the first third and fourth moments are numerically similar, while the relative difference of the second moment is  $\mathcal{O}(1)$ . After training all moments are numerically close. (b) The root mean square eigenvalue of the  $\mathcal{B}$  matrix shows good agreement with the predicted  $1 / \sqrt{n}$  fall off for single hidden layer ReLU networks. Experiments were performed on ReLU networks trained on two-class MNIST with 100 images per class. Moments of the NTK were computed exactly, while moments of the Hessian and  $\mathcal{B}$  were computed by randomly sampling 1000 vectors and using Hutchinson's trick.
+
+![](images/ca6094d09418e465857aefe33789236ad94bd1673572e4f26514d42e9ba069e0.jpg)  
+(b) Second moment of  $\mathcal{B}$
+
+At large width, the NTK approaches the identity matrix in class space,  $\Theta^{AB} \to N_{\mathrm{class}}^{-1}\delta^{AB}\operatorname{Tr}(\Theta)$  (Jacot et al. (2018)). This implies that the NTK spectrum consists of  $N_{\mathrm{class}}$  repeated copies. This has consequences for the spectrum of the Hessian at large width. For instance, for the case of MSE error it also implies  $N_{\mathrm{class}}$  repeated copies of the Hessian spectrum (See Figure 9b). It is intriguing to think this could serve as a path towards understanding the emergence of the  $N_{\mathrm{class}}$  (or  $N_{\mathrm{class}} - 1$ ) large eigenvalues and corresponding subspace observed in the Hessian spectrum (Sagun et al. (2016; 2017); Gur-Ari et al. (2018); Ghorbani et al. (2019); Papyan (2019)).
+
+# E.4 HIGHER-ORDER NETWORK EVOLUTION
+
+In this section we will explain how to compute higher-order corrections to training dynamics. In principle, this prescription allows one to compute model dynamics as an expansion in  $1 / n$  to arbitrary order. We apply this explicitly to compute the  $\mathcal{O}(1 / n)$  training dynamics. In Figure 11, we present experimental confirmation of our predictions for the evolution of the NTK. In the main text, we presented these results for the special case of gradient flow with MSE loss.
+
+So far we have mostly been using diagrammatic techniques to understanding the leading order scaling of correlation functions. It is interesting to try and understand finite width networks by asking how training dynamics are modified beyond the leading order asymptotics. There are three clear sources of corrections to the leading behavior.
+
+1. The initial kernel,  $\Theta_0$ , receives finite width corrections.  
+2. The linear (in learning rate) update for the network function, equation (73), is modified away from infinite width.  
+3. The kernel is not constant at finite width.
+
+The first source of corrections is automatically taken into account in typical empirical settings, as the finite-width  $\Theta_0$  can be computed explicitly. The other sources of corrections are non-trivial, and will be the focus of this section. We begin by explaining how to take into account the non-constancy of the kernel order by order in  $1/n$ , while maintaining the continuous time approximation. Next, we back away from the continuous time limit and write down the full discrete evolution. We introduce a method to compute arbitrary order corrections, and explicitly work out the corrections at order  $1/n$  for MSE loss.
+
+# E.4.1 CONTINUOUS TIME.
+
+Continuous time evolution of the model function in neural networks is governed by the differential equation
+
+$$
+\frac {d f (x ; t)}{d t} = - \sum_ {\left(x ^ {\prime}, y ^ {\prime}\right) \in D _ {\mathrm {t r}}} \Theta \left(x, x ^ {\prime}; t\right) \frac {\partial \ell \left(x ^ {\prime} , y ^ {\prime}\right)}{\partial f}. \tag {85}
+$$
+
+As we have discussed at length, in the large width limit, this equation simplifies and  $\Theta(t)$  is asymptotically constant (Jacot et al. (2018)). Our goal is to move beyond this leading behavior at large width and solve (85) order-by-order in a  $1/n$  expansion. In Section 3 we described how to compute the  $\mathcal{O}(n^{-1})$  corrections for the case of MSE loss. Here we explain how to handle corrections more generally as well as giving a more detailed discussion of the MSE case.
+
+The network map  $f$  and kernel,  $\Theta$  are members of the following family of operators.
+
+$$
+O _ {s} \left(x _ {1}, \dots , x _ {s}; t\right) := \sum_ {\mu} \frac {\partial O _ {s - 1} \left(x _ {1} , \dots , x _ {s - 1} ; t\right)}{\partial \theta_ {\mu}} \frac {\partial f \left(x _ {s} ; t\right)}{\partial \theta_ {\mu}}, \quad s \geq 2, \tag {86}
+$$
+
+with  $O_1 \coloneqq f$ . Here, as above,  $O_2 = \Theta$  is the kernel. With a general loss function, these operators satisfy.
+
+$$
+\frac {d O _ {s} \left(x _ {1} , \dots , x _ {s} ; t\right)}{d t} = - \sum_ {\left(x ^ {\prime}, y ^ {\prime}\right) \in D _ {\mathrm {t r}}} O _ {s + 1} \left(x _ {1}, \dots , x _ {s}, x ^ {\prime}; t\right) \frac {\partial \ell \left(x ^ {\prime} , y ^ {\prime} ; t\right)}{\partial f}, \quad s \geq 1. \tag {87}
+$$
+
+Equations (86) and (87). Give an infinite tower of first order ODEs, the solution of which gives the time evolution of the network map and the kernel.
+
+$$
+\frac {d f (x _ {1} ; t)}{d t} = - \sum_ {(x, y) \in D _ {\mathrm {t r}}} \Theta (x _ {1}, x; t) \frac {\partial \ell (x , y)}{\partial f}
+$$
+
+$$
+\frac {d \Theta (x _ {1} , x _ {2} ; t)}{d t} = - \sum_ {(x, y) \in D _ {\mathrm {t r}}} O _ {3} (x _ {1}, x _ {2}, x; t) \frac {\partial \ell (x , y)}{\partial f}
+$$
+
+$$
+\frac {d O ^ {(1)} \left(x , x ^ {\prime} , x ^ {\prime \prime} ; t\right)}{d t} = - \sum_ {(x, y) \in D _ {\mathrm {t r}}} O _ {4} \left(x _ {1}, x _ {2}, x _ {3}, x; t\right) \frac {\partial \ell (x , y)}{\partial f} \tag {88}
+$$
+
+![](images/c10c97a46c2c41ac16292886d888bf0cb7a0d2241366c0e390ee730b29b133a4.jpg)
+
+Solving this infinite tower is not feasible. If we wish to work to a given order in an expansion in  $1 / n$ , however, there is a dramatic simplification, which makes a solution possible. Firstly, we can truncate these equations. To see this note that the operators  $O_{s}$  contain  $s$  contracted derivative tensors. As a result, by Lemma 7 and Conjecture 1, correlation functions involving the operators,  $O_{s}$  satisfy
+
+$$
+\mathbb {E} _ {\theta} \left[ O _ {s} \left(x _ {1}, \dots , x _ {s}; t\right) F (\vec {x}; t) \right] = \left\{ \begin{array}{l l} \mathcal {O} \left(n ^ {\frac {2 - s}{2}}\right) & , s \text {e v e n} \\ \mathcal {O} \left(n ^ {\frac {1 - s}{2}}\right) & , s \text {o d d} \end{array} \right. \tag {89}
+$$
+
+where  $F(\vec{x};t)$  is arbitrary additional contribution to the integrand.
+
+Thus, if we wish to work to solve for the time evolution up to corrections which scale as  $\mathcal{O}(n^{-r})$  we can truncate the tower at  $s = 2r$  and set  $O_{2r}(t)$  to be equal to its initial value. Note that the leading order solution, (85), is the result of this procedure with  $r = 1$  and the results presented in the main text for the  $\mathcal{O}(n^{-1})$  evolution correspond to  $r = 2$ .
+
+The truncation provides a dramatic simplification, however it is not immediately clear how to solve even the truncated differential equations in (87). We now describe how to organize the perturbative expansion of the operators  $O_{s}(t)$  (including  $f$  and  $\Theta$ ) in such a way that the differential equations become tractable.
+
+The central idea is to write each operator,  $O_{s}(t)$ , as an expansion.
+
+$$
+O _ {s} \left(x _ {1}, \dots , x _ {s}; t\right) = \sum_ {r = \lfloor \frac {s - 1}{2} \rfloor} ^ {\infty} O _ {s} ^ {(r)} \left(x _ {1}, \dots , x _ {s}; t\right) \tag {90}
+$$
+
+where each order  $O_s^{(r)}$  captures the  $\mathcal{O}(n^{-r})$  evolution of  $O_{s}$ . For example,
+
+$$
+f (x _ {1}; t) = f ^ {(0)} (x _ {1}; t) + f ^ {(1)} (x _ {1}; t) + \dots
+$$
+
+$$
+\Theta (x _ {1}, x _ {2}; t) = \Theta^ {(0)} (x _ {1}, x _ {2}; t) + \Theta^ {(1)} (x _ {1}, x _ {2}; t) + \dots
+$$
+
+$$
+O _ {3} \left(x _ {1}, x _ {2}, x _ {3}; t\right) = O _ {3} ^ {(1)} \left(x _ {1}, x _ {2}, x _ {3}; t\right) + O _ {3} ^ {(2)} \left(x _ {1}, x _ {2}, x _ {3}; t\right) + \dots \tag {91}
+$$
+
+![](images/eb8e21dab66d95807809ab11b3719b5facdf510d80f578f6be97877e7ef412db.jpg)
+
+The notation  $O_{s}^{(r)}$  means both that any correlation function containing  $O_{s}^{(r)}$  is  $\mathcal{O}(n^{-r})$  and, by Lemma 6, that typical realizations of the operators scale as  $\mathcal{O}(n^{-r})$ .<sup>15</sup>
+
+Once the operators  $O_{s}$  are organized in this way, solving the differential equations (85) is tractable. As the differential equations describing the evolution of  $O_{s}^{(r)}$  for  $r > 0$  only depend on the time dependent solutions of  $O_{s}^{(r - 1)}$ , we can iteratively solve for the  $O_{s}^{(r)}$  order by order. For example, in Section 3, we used the leading order solution for  $f$  and  $\Theta$  to solve for  $\Theta^{(1)}$ .
+
+In principle this procedure can be extended to arbitrary order in  $1 / n$ . Before going onto explain the finite step corrections to this procedure, we reproduce the results of section 3 in more detail.
+
+MSE continuous time example. The MSE loss is,
+
+$$
+L ^ {\mathrm {M S E}} = \frac {1}{2} \sum_ {(x, y) \in D _ {\mathrm {t r}}} (f (x) - y _ {a}) ^ {2} \tag {92}
+$$
+
+As such the update equation simplifies to,
+
+$$
+\frac {d f (x ; t)}{d t} = - \sum_ {\left(x ^ {\prime}, y ^ {\prime}\right) \in D _ {\mathrm {t r}}} \Theta \left(x, x ^ {\prime}; t\right) \left(f \left(x ^ {\prime}; t\right) - y ^ {\prime}\right), \tag {93}
+$$
+
+The leading order solution to equation (93) is exponential, kernel evolution,
+
+$$
+f ^ {(0)} (t) = y + e ^ {- t \Theta_ {0}} \left(f _ {0} - y\right) \tag {94}
+$$
+
+$$
+\Theta^ {(0)} (t) = \Theta_ {0}. \tag {95}
+$$
+
+Here we are using a condensed notation where  $f_0 \coloneqq f^{(0)}(0)$ ,  $\Theta_0 \coloneqq \Theta^{(0)}(0)$ . Equation (94) is a vector equation with  $f_0, f^{(0)}(t)$ , and  $y$  are vectors over the training set and the leading order kernel  $\Theta_0$  is a square matrix over the same space.
+
+As discussed above to study the  $\mathcal{O}(1/n)$  evolution, we can truncate the set of equations in (87) at  $s = 4$ , and set  $O_4(t) = O_4^{(1)}(t) = O_4(0)$  up to corrections which scale as  $\mathcal{O}(n^{-2})$ .
+
+Moving up a rung in (87), we have
+
+$$
+\frac {d O _ {3} \left(x _ {1} , x _ {2} , x _ {3} ; t\right)}{d t} = - \sum_ {(x, y) \in D _ {\mathrm {t r}}} O _ {4} \left(x _ {1}, x _ {2}, x _ {3}, x; t\right) \left(f (x; t) - y\right). \tag {96}
+$$
+
+To solve for  $O_3^{(1)}(t)$ , i.e. neglecting  $\mathcal{O}(n^{-2})$  corrections, we can plug in the leading order approximations, to  $O_4(t)$  and  $f(t)$ .
+
+$$
+\frac {d O _ {3} ^ {(1)} \left(x _ {1} , x _ {2} , x _ {3} ; t\right)}{d t} = - \sum_ {(x, y) \in D _ {\mathrm {t r}}} O _ {4} \left(x _ {1}, x _ {2}, x _ {3}, x; 0\right) \left(f ^ {(0)} (x; t) - y\right). \tag {97}
+$$
+
+This gives,
+
+$$
+O _ {3} ^ {(1)} \left(x _ {1}, x _ {2}, x _ {3}; t\right) = O _ {3} \left(x _ {1}, x _ {2}, x _ {3}; 0\right) - \int_ {0} ^ {t} d t ^ {\prime} \sum_ {(x, y) \in D _ {\mathrm {t r}}} O _ {4} \left(x _ {1}, x _ {2}, x _ {3}, x; 0\right) \left(f ^ {(0)} (x; t) - y\right). \tag {98}
+$$
+
+Using the explicit form of  $f^{(0)}(t)$  we can write,
+
+$$
+O _ {3} ^ {(1)} (\vec {x}; t) = O _ {3} (\vec {x}; 0) - \sum_ {a} O _ {4} (\vec {x}; 0) \cdot \left(\Theta_ {0} ^ {- 1} \left(1 - e ^ {- t \Theta_ {0}}\right) \left(f _ {0} - y\right)\right). \tag {99}
+$$
+
+Here we are again using a condensed notation.  $\vec{x} = (x_{1},x_{2},x_{3})$ .  $f_{0}$  and  $y$  are vectors over the training set while  $\Theta_0$  is a square matrix, and  $O_4(\vec{x};0)$  is also a vector over the training set, with the value  $\bar{O}_4(\vec{x},x';0)$  on the point  $x^{\prime}\in D_{\mathrm{tr}}$ .
+
+Moving one more step up the ladder gives
+
+$$
+\Theta (x _ {1}, x _ {2}; t) = \Theta_ {x _ {1}, x _ {2}; 0} + \Theta^ {(1)} (x _ {1}, x _ {2}; t) + \mathcal {O} (n ^ {- 2}),
+$$
+
+$$
+\Theta^ {(1)} \left(x _ {1}, x _ {2}; t\right) = - \int_ {0} ^ {t} d t ^ {\prime} \sum_ {(x, y) \in D _ {\mathrm {t r}}} O _ {3} ^ {(1)} \left(x _ {1}, x _ {2}, x; t ^ {\prime}\right) \left(f ^ {(0)} \left(x; t ^ {\prime}\right) - y\right). \tag {100}
+$$
+
+Plugging in  $O_3^{(1)}$ , and using the eigen-decomposition of  $\Theta_0$  to perform the integrals gives,
+
+$$
+\begin{array}{l} \Theta^ {(1)} (\vec {x}; t) = - \sum_ {i} \frac {1}{\lambda_ {i}} (O _ {3} (\vec {x}; 0) \cdot \hat {e} _ {i}) (\Delta f _ {0} \cdot \hat {e} _ {i}) [ 1 - e ^ {- t \lambda_ {i}} ] \\ + \sum_ {i j} \frac {1}{\lambda_ {j}} \left(\hat {e} _ {i} ^ {T} O _ {4} (\vec {x}; 0) \hat {e} _ {j}\right) \left(\Delta f _ {0} \cdot \hat {e} _ {i}\right) \left(\Delta f _ {0} \cdot \hat {e} _ {j}\right) \left[ \frac {1 - e ^ {- t \lambda_ {i}}}{\lambda_ {i}} - \frac {1 - e ^ {- t \left(\lambda_ {i} + \lambda_ {j}\right)}}{\lambda_ {i} + \lambda_ {j}} \right]. \tag {101} \\ \end{array}
+$$
+
+![](images/7c5455461bdb633227bf1d7a265f882600e20c7a2f4298d9ebd7ca0521ec9bd0.jpg)  
+(a) Evolution of network map.
+
+![](images/d9dc0571e0a6f42d97951bceec16d0c9d2d23d110eab7a2356ea935c23ab1e9b.jpg)  
+(b) Evolution of NTK.
+
+![](images/50b5f53ac439a99ea4780e695cc972e636306a10f1d54f1ed846cdc401fed65c.jpg)  
+(c) Evolution of network map for a narrower network.
+
+![](images/cace4131089bb9d21ca3fcb3ce38cdec76eb1b29403f5eba567d93caf852cecb.jpg)  
+(d) Evolution of NTK for a narrower network.  
+Figure 11: Single instance evolution for a randomly selected element of the NTK and network map. Theoretical predictions at  $\mathcal{O}(n^{-1})$  are represented by dashed lines, while the true network evolution is shown in solid. (a-b) Wide networks show excellent agreement with predictions. (c-d) Predictions match reasonably well even down to modest widths. All plots correspond to single training runs on 2-class MNIST with 10 samples per class. The learning rate is 0.1 for all runs. All runs reach training accuracy 1.
+
+Here, we have introduced the eigenvectors,  $\hat{e}_i$ , which are vectors over the training dataset and eigenvalues  $\lambda_i$  of the initial kernel,  $\Theta_0$ . The vector  $\vec{x} = (x_1, x_2)$  as  $\Theta$  depends on two inputs.  $O_3(\vec{x}; 0)$  is a vector over the dataset while  $O_4(\vec{x}; 0)$  is a square matrix and  $\Delta f_0 := f_0 - y$  is a vector over the training data.
+
+Finally, we can plug in the expression (100) into (93) and give the sub-leading behavior for  $f(t)$
+
+$$
+\frac {d f (x ; t)}{d t} = - \sum_ {(x, y) \in D _ {\mathrm {t r}}} \left(\Theta \left(x, x ^ {\prime}; 0\right) + \Theta^ {(1)} \left(x, x ^ {\prime}; t\right)\right) \left(f \left(x ^ {\prime}; t\right) - y\right) + \mathcal {O} \left(n ^ {- 2}\right) \tag {102}
+$$
+
+$$
+f (t) = y + e ^ {- \Theta_ {0} t} \left(1 - \int_ {0} ^ {t} d t ^ {\prime} e ^ {t ^ {\prime} \Theta_ {0}} \Theta^ {(1)} \left(t ^ {\prime}\right) e ^ {- t ^ {\prime} \Theta_ {0}}\right) \left(f _ {0} - y\right). \tag {103}
+$$
+
+This completes the full  $\mathcal{O}(n^{-1})$  time dependence. This prediction for the  $\mathcal{O}(1/n)$  time dependence is confirmed experimentally in Figure 11.
+
+Note, one consequence of (101) is that the  $\mathcal{O}(1 / n)$  corrections to  $\Theta$  go to a constant at late times.
+
+$$
+\Theta_ {\infty} ^ {(1)} := \lim  _ {t \rightarrow \infty} \Theta^ {(1)} (t) = - \sum_ {i} \frac {1}{\lambda_ {i}} \left(O _ {3} (\vec {x}; 0) \cdot \hat {e} _ {i}\right)\left(\Delta f _ {0} \cdot \hat {e} _ {i}\right) + \sum_ {i j} \frac {1}{\lambda_ {i} \left(\lambda_ {i} + \lambda_ {j}\right)} \left(\hat {e} _ {i} ^ {T} O _ {4} (\vec {x}; 0) \hat {e} _ {j}\right)\left(\Delta f _ {0} \cdot \hat {e} _ {i}\right)\left(\Delta f _ {0} \cdot \hat {e} _ {j}\right). \tag {104}
+$$
+
+And the asymptotic evolution of  $\Delta f(t)$  is again constant kernel evolution, but with a corrected kernel.
+
+$$
+f (t) \rightarrow y + e ^ {- t \left(\Theta_ {0} + \Theta_ {\infty} ^ {(1)}\right)} \left(f _ {0} - y\right). \tag {105}
+$$
+
+It is worth noting that these predictions are quite detailed, giving the full time dependence, rather than just the overall scaling with width. In deriving these results and in the discrete time analysis below, we implicitly make an assumption that we can control the discrete time corrections or in the continuous time case that  $f(t)$ ,  $\Theta (t)$  and all the  $O_{s}(t)$  are differentiable. This relies on being able to differentiate through the activation function. For non-smooth activations such as ReLU, this assumption is suspect, and indeed the evolution is more subtle in the ReLU case and a full analysis is left to future work. For this reason we present experimental results for networks with smooth activation functions.
+
+# E.4.2 DISCRETE TIME.
+
+With the  $1 / n$  corrections to continuous time evolution under our belts, it is possible to also keep track of corrections coming from the discrete update step. The essential point, that it is possible to solve iteratively, order by order in  $1 / n$  is not changed. To see this we can look at how the update equations are modified. Beginning with the network output update equation, we have,
+
+$$
+\begin{array}{l} f _ {t + 1} (x) - f _ {t} (x) = - \eta \sum_ {\left(x ^ {\prime}, y ^ {\prime}\right) \in D _ {\mathrm {t r}}} \Theta_ {t} \left(x, x ^ {\prime}\right) \frac {\partial \ell_ {t} \left(x ^ {\prime} , y ^ {\prime}\right)}{\partial f} \\ + \frac {\eta^ {2}}{2} \sum_ {\mu \nu} \frac {\partial^ {2} f _ {t} (x)}{\partial \theta^ {\mu} \partial \theta^ {\nu}} g _ {t} ^ {\mu} g _ {t} ^ {\nu} - \frac {\eta^ {3}}{3 !} \sum_ {\mu \nu \rho} \frac {\partial^ {3} f _ {t} (x)}{\partial \theta^ {\mu} \partial \theta^ {\nu} \partial \theta^ {\rho}} g _ {t} ^ {\mu} g _ {t} ^ {\nu} g _ {t} ^ {\rho} + \dots . \tag {106} \\ \end{array}
+$$
+
+In Section E.2 we argued that the  $\mathcal{O}(\eta^2)$  terms vanish at large width. In more detail, each factor of the gradient,  $g^{\mu}$ , contains a contracted derivative tensor. Thus, the cluster graph for correlation functions involving the  $\eta^2$  term will always contain three contracted vertices and thus the correlation functions will scale as  $\mathcal{O}(n^{-1})$ . The  $\eta^3$  term will give four vertices, and thus also scale as  $\mathcal{O}(1/n)$ , the  $\eta^4$  contribution will be  $\mathcal{O}(1/n^2)$  and so on.
+
+Just as in the continuous time case this equation is still solvable order by order in  $1 / n$ . What we mean by solvable is the following. All terms on the RHS appearing with  $f_{t}$  are already higher order as a result of their derivative structure. This means, just as in the continuous case, we can use the lower order solution of  $f_{t}$  in the gradient terms on the RHS to solve for  $f_{t}$  on the LHS.
+
+Similarly, the update equation for  $\Theta$  receives discrete time modifications.
+
+$$
+\Theta_ {t + 1} \left(x _ {1}, x _ {2}\right) - \Theta_ {t} \left(x _ {1}, x _ {2}\right) = - \eta \sum_ {a} O _ {3; t} \left(x _ {1}, x _ {2}, x\right) \frac {\partial \ell_ {t} (x , y)}{\partial f} + \frac {\eta^ {2}}{2} \sum_ {\mu \nu} \frac {\partial^ {2} \Theta_ {t} \left(x _ {1} , x _ {2}\right)}{\partial \theta^ {\mu} \partial \theta^ {\nu}} g _ {t} ^ {\mu} g _ {t} ^ {\nu} + \dots . \tag {107}
+$$
+
+Here we have adopted a notation where,  $\vec{x} = (x_{1},\dots ,x_{s})$ . Just as for  $f$ , increased powers in  $\eta$  are increasingly suppressed in  $n$  and we can iteratively solve this equation using solutions at leading orders in  $n$  to solve for sub-leading behavior.
+
+More generally the tower of differential equations defined recursively in (87) becomes
+
+$$
+\begin{array}{l} O _ {s; t + 1} (\vec {x}) - O _ {s; t} (\vec {x}) = - \sum_ {\left(x ^ {\prime}, y ^ {\prime}\right) \in D _ {\mathrm {t r}}} O _ {s + 1; t} \left(\vec {x}, x ^ {\prime}\right) \frac {\partial \ell \left(x ^ {\prime} , y ^ {\prime} ; t\right)}{\partial f} \\ + \sum_ {k = 2} ^ {\infty} \frac {\eta^ {k}}{k !} \sum_ {\mu_ {1}, \mu_ {2}, \dots , \mu_ {k}} \frac {\partial^ {k} O _ {s ; t + 1} (\vec {x})}{\partial \theta_ {\mu_ {1}} \partial \theta_ {\mu_ {2}} \dots \partial \theta_ {\mu_ {k}}} g _ {\mu_ {1}} g _ {\mu_ {2}} \dots g _ {\mu_ {k}}, \quad s \geq 1. \tag {108} \\ \end{array}
+$$
+
+As in the continuous time case, at a given order in  $n$ , we can truncate this tower and use lower order solutions to solve for the time evolution up to the desired order. To ground this discussion in a concrete use case, we again walk through the  $1/n$  corrections for MSE loss, now in discrete time.
+
+# MSE discrete time example.
+
+For discrete time evolution, the leading order behavior of the model map is
+
+$$
+f _ {t} ^ {(0)} = \left(1 - \eta \Theta_ {0}\right) ^ {t} (f _ {0} - y) \tag {109}
+$$
+
+$$
+\Theta_ {t} ^ {(0)} = \Theta_ {0}. \tag {110}
+$$
+
+At next order we can proceed by solving the truncated set of equations in (108). The first equation, for  $O_{4;t}$  still gives a constant solution,
+
+$$
+O _ {4; t} ^ {(1)} = O _ {4; 0}. \tag {111}
+$$
+
+Here, both the discrete and continuous time terms that would appear on the RHS are  $\mathcal{O}(1 / n^2)$ .
+
+Next, we must solve for  $O_{3;t}$ . The discrete time update is,
+
+$$
+O _ {3; t + 1} (\vec {x}) - O _ {3; t} (\vec {x}) = - \eta \sum_ {\left(x ^ {\prime}, y ^ {\prime}\right) \in D _ {\mathrm {t r}}} O _ {4; t} \left(\vec {x}, x ^ {\prime}\right) \left(f _ {t} \left(x ^ {\prime}\right) - y ^ {\prime}\right) + \frac {\eta^ {2}}{2} \sum_ {\mu \nu} \frac {\partial^ {2} O _ {3 ; t} (\vec {x})}{\partial \theta^ {\mu} \partial \theta^ {\nu}} g _ {t} ^ {\mu} g _ {t} ^ {\nu} + \dots . \tag {112}
+$$
+
+To order  $1 / n$  we can drop the order  $\eta^2$  and higher terms, as they contain expressions with 5 or more contracted derivative tensors. Thus at this order, we are left with the discrete analogue of (96).
+
+$$
+O _ {3; t + 1} (\vec {x}) - O _ {3; t} (\vec {x}) = - \sum_ {\left(x ^ {\prime}, y ^ {\prime}\right) \in D _ {\mathrm {t r}}} O _ {4; t} \left(\vec {x}, x ^ {\prime}\right) \left(f _ {t} \left(x ^ {\prime}\right) - y ^ {\prime}\right). \tag {113}
+$$
+
+which we can sum to get the discrete version of (98),
+
+$$
+O _ {3; t} ^ {(1)} (\vec {x}) = O _ {3; 0} (\vec {x}) - \eta \sum_ {t ^ {\prime} = 1} ^ {t} \sum_ {\left(x ^ {\prime}, y ^ {\prime}\right) \in D _ {\mathrm {t r}}} O _ {4; 0} \left(\vec {x}, x ^ {\prime}\right) \left(f _ {0; t} \left(x ^ {\prime}\right) - y ^ {\prime}\right). \tag {114}
+$$
+
+Explicitly, this takes the form
+
+$$
+O _ {3; t} ^ {(1)} (\vec {x}) = O _ {3; 0} (\vec {x}) - O _ {4; 0} (\vec {x}) \cdot \left(\Theta_ {0} ^ {- 1} (1 - \eta \Theta_ {0}) \left(1 - \left(1 - \eta \Theta_ {0}\right) ^ {t}\right) \left(f _ {0} - y\right)\right). \tag {115}
+$$
+
+Here we have adopted notation similar to above, where  $O_{4;0}(\vec{x})$  is a vector over the training dataset.
+
+So far, this procedure has been a discrete analogue of what we have done in the continuous time case, however as we move on to compute  $\Theta$  and  $f$  we will have to keep track of the novel corrections, which vanish in continuous time. Explicitly, at order  $1 / n$  the discrete update for  $\Theta_{t}$  is given by,
+
+$$
+\begin{array}{l} \Theta_ {t + 1} ^ {(1)} (\vec {x}) - \Theta_ {t} ^ {(1)} (\vec {x}) = - \eta \sum_ {a} O _ {3; t} ^ {(1)} (\vec {x}, x ^ {\prime}) \left(f _ {t} ^ {(0)} (x ^ {\prime}) - y ^ {\prime}\right) \\ + \frac {\eta^ {2}}{2} \sum_ {x ^ {\prime}, x ^ {\prime \prime}, y ^ {\prime}, y ^ {\prime \prime} \in D _ {\mathrm {t r}}} \sum_ {\mu \nu} \frac {\partial^ {2} \Theta_ {0} (\vec {x})}{\partial \theta^ {\mu} \partial \theta^ {\nu}} \frac {\partial f _ {0} \left(x ^ {\prime}\right)}{\partial \theta^ {\mu}} \frac {\partial f _ {0} \left(x ^ {\prime \prime}\right)}{\partial \theta^ {\nu}} \left(f _ {t} ^ {(0)} \left(x ^ {\prime}\right) - y ^ {\prime}\right) \left(f _ {t} ^ {(0)} \left(x ^ {\prime \prime}\right) - y ^ {\prime \prime}\right). \tag {116} \\ \end{array}
+$$
+
+This can be summed to give  $\Theta_t^{(1)}$ .
+
+To move up to the neural network map itself there is one additional complication. The  $\mathcal{O}(\eta^2 /n)$  term,  $\sum_{\mu \nu}\frac{\partial^2f_t(x)}{\partial\theta^\mu\partial\theta^\nu}\frac{\partial f_t(x')}{\partial\theta^\mu}\frac{\partial f_t(x'')}{\partial\theta^\nu}$ , in (106) has non trivial time dependence. We can deal with this just as we have been doing, by taking an extra time derivative and integrating. Defining,
+
+$$
+O _ {1, 1; t} \left(x _ {1}, x _ {2}, x _ {2}\right) := \sum_ {\mu \nu} \frac {\partial^ {2} f _ {t} \left(x _ {1}\right)}{\partial \theta^ {\mu} \partial \theta^ {\nu}} \frac {\partial f _ {t} \left(x _ {2}\right)}{\partial \theta^ {\mu}} \frac {\partial f _ {t} \left(x _ {3}\right)}{\partial \theta^ {\nu}}. \tag {117}
+$$
+
+We have
+
+$$
+O _ {1, 1; t + 1} ^ {(1)} (\vec {x}) - O _ {1, 1; t} ^ {(1)} (\vec {x}) = - \eta \sum_ {\left(x ^ {\prime}, y ^ {\prime}\right) \in D _ {\mathrm {t r}}} \sum_ {\mu} \frac {\partial O _ {1 , 1 ; 0} (\vec {x})}{\partial \theta^ {\mu}} \frac {\partial f _ {t} ^ {(0)} \left(x ^ {\prime}\right)}{\partial \theta^ {\mu}} \left(f _ {t} ^ {(0)} \left(x ^ {\prime}\right) - y ^ {\prime}\right), \tag {118}
+$$
+
+with the solution,
+
+$$
+O _ {1, 1; t} ^ {(1)} (\vec {x}) = O _ {1, 1; 0} (\vec {x}) - \sum_ {\mu} \frac {\partial O _ {1 , 1 ; 0} (\vec {x})}{\partial \theta^ {\mu}} \frac {\partial f _ {0}}{\partial \theta^ {\mu}} \cdot \left(\Theta_ {0} ^ {- 1} (1 - \eta \Theta_ {0}) (1 - (1 - \eta \Theta_ {0}) ^ {t}) \Delta f _ {0}\right) _ {a}. \tag {119}
+$$
+
+This can in turn be plugged into the update equation for  $f_{t}$ .
+
+$$
+\begin{array}{l} f _ {t + 1} (x) - f _ {t} (x) = - \eta \sum_ {a} \left(\Theta_ {0} (x, x _ {a}) + \Theta_ {t} ^ {(1)} (x, x _ {a})\right) \left(f _ {t} (x _ {a}) - y _ {a}\right) \\ + \frac {\eta^ {2}}{2} \sum_ {a, b} O _ {1, 1; t} ^ {(1)} (x, x _ {a}, x _ {b}) \left(f _ {t} ^ {(0)} (x _ {a}) - y _ {a}\right) \left(f _ {t} ^ {(0)} (x _ {b}) - y _ {b}\right) \\ - \frac {\eta^ {3}}{3 !} \sum_ {a, b, c} \frac {\partial^ {3} f _ {0} (x)}{\partial \theta^ {\mu} \partial \theta^ {\nu} \partial \theta^ {\rho}} \frac {\partial f _ {0} \left(x _ {a}\right)}{\partial \theta^ {\mu}} \frac {\partial f _ {0} \left(x _ {b}\right)}{\partial \theta^ {\nu}} \frac {\partial f _ {0} \left(x _ {c}\right)}{\partial \theta^ {\rho}} \left(f _ {t} \left(x _ {a}\right) - y _ {a}\right) \left(f _ {t} \left(x _ {b}\right) - y _ {b}\right) \left(f _ {t} \left(x _ {c}\right) - y _ {c}\right). \tag {120} \\ \end{array}
+$$
+
+Here  $(x_{a},y_{a})$  are elements of the training set  $D_{\mathrm{tr}}$  and summed over. This equation can be solved to give  $f_{t}^{(1)}$ .
+
+$$
+f _ {t} ^ {(1)} = \left(1 - \eta \Theta_ {0}\right) ^ {t} \sum_ {t ^ {\prime} = 1} ^ {t} \left(1 - \eta \Theta_ {0}\right) ^ {- \left(t ^ {\prime} + 1\right)} \left[ - \eta \Theta_ {\mathrm {N L O}, t ^ {\prime}} \left(1 - \eta \Theta_ {0}\right) ^ {t ^ {\prime}} \Delta f _ {0} + \operatorname {D i s c} _ {t ^ {\prime}} \right]. \tag {121}
+$$
+
+Where  $\mathrm{Disc}_t$  contains the discrete derivative updates at  $\mathcal{O}(1 / n)$ .
+
+$$
+\begin{array}{l} \operatorname {D i s c} _ {t} := \frac {\eta^ {2}}{2} \sum_ {a, b} O _ {1, 1; t} ^ {(1)} \left(x _ {a}, x _ {b}\right) \left(f _ {t} \left(x _ {a}\right) - y _ {a}\right) \left(f _ {t} \left(x _ {b}\right) - y _ {b}\right) \\ - \frac {\eta^ {3}}{3 !} \sum_ {a, b, c} \frac {\partial^ {3} f _ {0}}{\partial \theta^ {\mu} \partial \theta^ {\nu} \partial \theta^ {\rho}} \frac {\partial f _ {0} \left(x _ {a}\right)}{\partial \theta^ {\mu}} \frac {\partial f _ {0} \left(x _ {b}\right)}{\partial \theta^ {\nu}} \frac {\partial f _ {0} \left(x _ {c}\right)}{\partial \theta^ {\rho}} \left(f _ {t} \left(x _ {a}\right) - y _ {a}\right) \left(f _ {t} \left(x _ {b}\right) - y _ {b}\right) \left(f _ {t} \left(x _ {c}\right) - y _ {c}\right). \tag {122} \\ \end{array}
+$$
+
+These expressions may look fairly intimidating. The key point is that all terms in the summand in (116) and thus (121) are known functions of time and initial data, just as in the continuous time setting.
\ No newline at end of file
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+# AT STABILITY'S EDGE: HOW TO ADJUST HYPERPARAMETERS TO PRESERVE MINIMA SELECTION IN ASYNCHRONOUS TRAINING OF NEURAL NETWORKS?
+
+Niv Giladi*
+
+Mor Shpigel Nacson\*
+
+Elad Hoffer1,2
+
+Daniel Soudry<sup>1</sup>
+
+$^{1}$ Technion - Israel Institute of Technology, Haifa, Israel
+
+$^{2}$ Habana-Labs, Caesarea, Israel
+
+{giladiniv, mor.shpigel, elad.hoffer, daniel.soudry}@gmail.com
+
+# ABSTRACT
+
+Background: Recent developments have made it possible to accelerate neural networks training significantly using large batch sizes and data parallelism. Training in an asynchronous fashion, where delay occurs, can make training even more scalable. However, asynchronous training has its pitfalls, mainly a degradation in generalization, even after convergence of the algorithm. This gap remains not well understood, as theoretical analysis so far mainly focused on the convergence rate of asynchronous methods.
+
+Contributions: We examine asynchronous training from the perspective of dynamical stability. We find that the degree of delay interacts with the learning rate, to change the set of minima accessible by an asynchronous stochastic gradient descent algorithm. We derive closed-form rules on how the learning rate could be changed, while keeping the accessible set the same. Specifically, for high delay values, we find that the learning rate should be kept inversely proportional to the delay. We then extend this analysis to include momentum. We find momentum should be either turned off, or modified to improve training stability. We provide empirical experiments to validate our theoretical findings.
+
+# 1 INTRODUCTION
+
+Training deep neural networks (DNNs) requires large amounts of computational resources, often using many devices operating over days and weeks. Furthermore, with ever-increasing model size and available data, the amount of compute used was noted to increase exponentially over the past few years (Amodei & Hernandez, 2018). Fortunately, the operations made in DNN training are highly suitable for parallelization over many devices. Most commonly, the parallelization is done by "data-parallelism" in which the data is divided into separate batches which are distributed between different devices during training. By using many devices, with each device highly utilized, one could train on a large number of samples without paying in run-time. However, this training is done synchronously, with synchronization between the different devices done on every iteration (S-SGD). With the growth in the number of devices participating in the training, the synchronization becomes a prominent bottleneck.
+
+To overcome the synchronization bottleneck, asynchronous training (A-SGD) has been proposed (Dean et al., 2012; Recht et al., 2011). The basic idea behind such training is that when a device finishes calculating its batch gradients, an update to the DNN parameters is immediately performed, without waiting for other devices. This approach is known as asynchronous centralized training with a parameter server (PS) that updates the parameters (Li, 2014). The main problem in such training is that, since the PS updates the parameters whenever a device communicates with him, the parameters that are being used in other devices calculation are no longer up-to-date. This phenomena is called gradient staleness or delay as we will refer to it, and it causes a deterioration in generalization performance when using A-SGD.
+
+The generalization deterioration can be seen in Fig. 1. Unless the learning rate is significantly decreased, we can see a generalization gap: a deterioration in the generalization of A-SGD (with delay  $\tau = 32$ ) from the baseline (S-SGD,  $\tau = 0$ ) value near the steady state — i.e., after we are near full convergence, following many training epochs (2000). Due to the generalization gap demonstrated in the figure, A-SGD with a parameter server and large delays is not commonly used — despite it is relatively simple to implement and despite its vast potential to accelerate training. Therefore, our main goal in this work is to shed some theoretical light on the generalization gap problem and use it to improve A-SGD training.
+
+![](images/46143b7e0ee0d8b45955961cd49b5cbf60c20be2f733273f91f417ab4c1f981e.jpg)  
+Figure 1: Impact of learning rate and delay on generalization error. Validation error with delay  $(\tau = 32)$  and different learning rates  $\eta$ . We observe that unless we decrease the learning rate (proportionally to  $1 / \tau$ ) there is a generalization gap from the equivalent large batch training: (left) training curve. (right) validation error as a function of learning rate after reaching steady state. Horizontal dashed line is the validation error acquired by the equivalent large batch training (S-SGD,  $\tau = 0$ ). Learning rates larger than 0.05 did not converge with  $(\tau = 32)$  delay. CNN trained with MNIST. More details in section E.
+
+![](images/a9b7bba38b184edd55606ee84e4fc28b42872b199a007aa1c80065ba537c76dd.jpg)
+
+Previous theoretical analysis of A-SGD (Liu et al., 2018; Lian et al., 2016; 2015; Dai et al., 2018; Dutta et al., 2018; Arjevani et al., 2018) focused on analyzing the convergence rate, i.e., the time it takes to reach steady state, and not on the properties of the obtained solution. In contrast, in our paper we focus on understanding how the delay affects the selection of the solution we converge to, and changes in this selection can impact generalization.
+
+We tackle these questions from the perspective of dynamical stability. The dynamical stability approach was used in (Nar & Sastry, 2018; Wu et al., 2018), to study which minima are accessible under a specific choice of optimization algorithm and hyperparameters. In (Nar & Sastry, 2018), the authors analyzed gradient descent (GD) algorithm as a discrete-time nonlinear dynamical system. By analyzing the Lyapunov stability of this system for different minima, they showed a relation between the learning rate choice and the accessible minima set, i.e., the subset of the local optima that GD can converge to. In (Wu et al., 2018), the authors focused on stochastic gradient descent (SGD), defined a criterion to evaluate the stability of a specific minimum and used this criterion to show how the learning rate and batch-size play a role in SGD minima selection process.
+
+Here, we use dynamical stability to analyze the dynamics of A-SGD. Using this approach, the main question we try to tackle is:
+
+How do learning rate, delay and momentum interact and affect the minima selection process?
+
+# Contributions
+
+We start by modelling A-SGD as a dynamical system with delay. By analyzing the stability properties of that system, we find:
+
+- There exist an inverse linear relation between the delay of A-SGD, and the threshold learning rate i.e., the critical learning rate in which a minimum loses stability.  
+- This implies that in order to keep a specific minimum stability as we change the delay, we need to change the learning rate inversely proportional to the delay.  
+- Momentum has a crucial role in determining stability properties for A-SGD. Depending on the specific training algorithm, it can either hurt the stability or improve it. We show how to modify momentum to stabilize A-SGD training.
+
+With our theoretical findings, we derive closed form rules when training asynchronously. We suggest to scale the learning rate inversely to the delay in order to reach a better generalization performance. In section 3 we provide experiments on popular classification tasks to support our theoretical analysis. It is interesting to contrast this simple linear relation with the more complicated picture observed for the scaling of the learning rate with the batch size. Specifically, it has been observed by Shallue et al. (2018) that the optimal learning rate scales differently with the batch size when changing the models and datasets, i.e., there is no general "rule of thumb" that applies in all cases.
+
+A similar scaling rule for the learning rate was proposed before (Zhang et al., 2015), however, there it lacked a theoretical basis or a demonstration of its effectiveness with large effective batch sizes, i.e., the sum of all the minibatch sizes of all the workers, as we do here.
+
+# 2 THEORETICAL ANALYSIS
+
+# 2.1 PRELIMINARIES AND PROBLEM SETUP
+
+Consider the problem of minimizing the empirical loss
+
+$$
+f (\mathbf {x}) = \sum_ {i = 1} ^ {N} f _ {i} (\mathbf {x}), \tag {1}
+$$
+
+where  $N$  is the number of samples,  $\mathbf{x} \in \mathbb{R}^d$  and  $\forall i = 1, \dots, N$ :  $f_i: \mathbb{R}^d \to \mathbb{R}$  is twice continuously differentiable function. The update rule for minimizing  $f(\mathbf{x})$  in eq. 1 using asynchronous stochastic gradient descent (A-SGD) is given by
+
+$$
+\mathbf {x} _ {t + 1} = \mathbf {x} _ {t} - \eta \nabla f _ {n (t - \tau)} \left(\mathbf {x} _ {t - \tau}\right), \tag {2}
+$$
+
+where  $\eta$  is the learning rate,  $n(t)$  is a random selection process of a sample from  $\{1,2,\dots ,N\}$  at iteration  $t$ , and  $\tau$  is some delay due the asynchronous nature of our training. For simplicity, we focus on a fixed delay  $\tau$ . The extension to stochastic delay is discussed in section 2.2.1.
+
+Our main goal is understanding how different factors such as the gradient staleness  $\tau$  and the learning rate  $\eta$  affect the minima selection process in neural networks. In other words, we would like to understand how the different hyperparameters interact to divide the minima of the loss to two sets: those we can converge to and those we cannot converge to.
+
+To analyze this we suppose we are in a vicinity of some minimum point  $\mathbf{x}^*$ , which implies  $\nabla f(\mathbf{x}^*) = 0$ . Given some values of  $\eta$  and  $\tau$ , we ask whether or not we can converge to  $\mathbf{x}^*$ , using the theory of dynamical stability. As we shall see, if the learning rate  $\eta$  or delay  $\tau$  are too high, then this minimum loses stability, i.e., A-SGD cannot converge to  $\mathbf{x}^*$ , in expectation.
+
+Specifically, we examine the expectation of the linearized dynamics around  $\mathbf{x}^*$ . We show in appendix A that, to ensure stability, it is sufficient that the following one-dimensional equation is stable:
+
+$$
+x _ {t + 1} - x _ {t} + \eta a x _ {t - \tau} = 0, \tag {3}
+$$
+
+where we defined the Hessian  $\mathbf{H} = \frac{1}{N}\sum_{i = 1}^{N}\nabla^{2}f_{i}(\mathbf{x}^{*})$ , the sharpness term  $a = \lambda_{\max}(\mathbf{H})$  as the maximal eigenvalue of the Hessian, and  $x_{t}$  is the projection of the expectation of the perturbation  $\mathbf{x}_t - \mathbf{x}^*$  on the eigenvector which corresponds to the maximal eigenvalue  $a$ . Additionally, we show in appendix A that the characteristic equation of eq. 3 is
+
+$$
+z ^ {\tau + 1} - z ^ {\tau} + \eta a = 0. \tag {4}
+$$
+
+# 2.2 STABILITY ANALYSIS
+
+For given  $(\eta, \tau)$ , we would like to determine if the minimum point  $\mathbf{x}^*$  is asymptotically stable (Oppenheim, 1999) in expectation. This will enable us to divide the minima into two sets: those we might converge to since they are stable, and those we cannot possibly converge (except for a measure zero set of initialization) since they are unstable. We use the characteristic equation (eq. 4) to define the stability criterion:
+
+Definition 1 (Stability criterion) For given  $(\eta, \tau)$ , we say that the dynamics in eq. 3 are stable if all the roots of the corresponding characteristic equation (eq. 4) are inside the unit circle. This condition is equivalent to  $x_{t} \to 0$  for any initialization.
+
+We would like to find conditions on when and how can we change the hyperparameters  $(\eta, \tau)$  to maintain the same stability criterion. This, for example, will enable us to understand how to compensate for the delay introduced by asynchrony using the learning rate. In order to analyze stability we examine the characteristic equation in eq. 4 and ask: when does the maximal root of this polynomial have unit magnitude?
+
+# 2.2.1 THE INTERACTION BETWEEN LEARNING RATE AND DELAY
+
+Recall the characteristic equation
+
+$$
+z ^ {\tau + 1} - z ^ {\tau} + a \eta = 0. \tag {5}
+$$
+
+This polynomial has  $\tau + 1$  roots. In order to ensure stability we require that the root with the maximal magnitude, i.e., the maximal root, will be inside the unit circle. We want to find the threshold learning rate, i.e., the learning rate in which the maximal root is exactly on the unit circle, so we are at the threshold of stability. In appendix B, we show that this requires that
+
+$$
+a \eta = 2 \sin \left(\frac {\pi}{4 \tau + 2}\right). \tag {6}
+$$
+
+Using Taylor approximation we get
+
+$$
+a \eta = 2 \left(\frac {\pi}{4 \tau + 2} + O \left(\frac {1}{\tau^ {3}}\right)\right) \Rightarrow \frac {1}{a \eta} = \frac {1}{\pi} (2 \tau + 1) + O \left(\frac {1}{\tau}\right). \tag {7}
+$$
+
+We observe numerically that the error between the exact solution (eq. 6) and the linear approximation (eq. 7) is smaller than 0.05 for  $\tau \geq 1$ . Additionally, in the appendix Fig. 6 we demonstrate the high accuracy of the analytic approximation of the relation between  $\frac{1}{a\eta}$  and  $\tau$ , compared to the numerical evaluation of the value of  $\frac{1}{a\eta}$  for which the maximal root of eq. 5 is on the unit circle.
+
+Implications: We can see that to maintain stability for a given minimum point (or a given sharpness value) the learning rate should be kept inversely proportional to the delay. This implies that given some minimum  $\mathbf{x}^*$  for  $\tau = 0$  and a corresponding learning rate  $\eta_0$  for which this minimum is stable, we can evaluate the learning rate which will ensure this minimum remains stable for larger delay values. To do this, we first estimate  $a$  using eq. 6:  $a\eta_0 = 2\sin \left(\frac{\pi}{4\cdot 0 + 2}\right) = 2\Rightarrow a = \frac{2}{\eta_0}$ . Note that we do not use eq. 7 to evaluate  $a$  since we are interested in  $\tau = 0$ , while the relation in eq. 7 is only a good approximation for  $\tau \geq 1$ . Next, given some delay  $\tau$ , we substitute  $a$  into eq. 7 in order to evaluate a learning rate value for which the stability of all minima is maintained as in  $\tau = 0$
+
+$$
+\eta \approx \frac {\pi}{a (2 \tau + 1)} = \frac {\pi \eta_ {0}}{4 (\tau + 0 . 5)}. \tag {8}
+$$
+
+Related empirical results: In section 3.1 we show empirically the existence of a stability threshold for different values of delay and compare it with our theoretical threshold.
+
+Stochastic delay: In appendix C we extend our theoretical analysis to derive the characteristic equation for A-SGD with stochastic delay, i.e., when  $\tau$  is drawn from some discrete distribution. We analyze two cases:
+
+- Discrete uniform distribution:  $\tau \sim$  unif  $\{a,b\}$  
+- Gaussian distribution discrete approximation:  $\forall \mu, \forall k = 1, \dots, 2\mu : \Pr (\tau = k) = \Pr (k - 0.5 \leq \mathbf{z} \leq k + 0.5)$  where  $\mathbf{z} \sim \mathbb{N}(\mu, 1)$ . In Zhang et al. (2015), the authors showed empirically a delay distribution that resembles to Gaussian distribution.
+
+For both cases, we show numerically that the inverse relation between the learning rate  $\eta$  and the expected delay  $\mathbb{E}\tau$  still applies.
+
+# 2.2.2 THE EFFECTS OF MOMENTUM ON STABILITY
+
+The optimization step of A-SGD with momentum is defined as follows:
+
+$$
+\mathbf {v} _ {t + 1} = m \mathbf {v} _ {t} - \eta (1 - m) \nabla f _ {n (t - \tau)} (\mathbf {x} _ {t - \tau})
+$$
+
+$$
+\mathbf {x} _ {t + 1} = \mathbf {x} _ {t} + \mathbf {v} _ {t + 1}, \tag {9}
+$$
+
+where  $\mathbf{x}_t$  are the weights,  $\mathbf{v}_t$  and  $\nabla f_{n(t)}(\mathbf{x}_t)$  are the velocity and gradients at time  $t$ , respectively,  $m$  is the momentum parameter,  $\eta$  is the step size and  $\tau$  is the delay. The constant  $(1 - m)$  term which multiplies the gradients is called "dampening".
+
+In appendix D we extend our theoretical analysis to derive the characteristic equation for A-SGD with momentum. In addition, in section D.2.1 we show numerically that an inverse relation between the learning rate  $\eta$  and the delay  $\tau$  also applies when using momentum. Particularly,
+
+1. We need to keep the learning rate inversely proportional to the delay.  
+2. For a given delay, larger momentum values require smaller learning rate for stability.
+
+Therefore, the range of stable learning rates decreases when the momentum parameter  $(m^{\prime})$  increases which can make tuning the learning rate difficult, especially with large delays. These results suggest that, to ensure stability when training with large delay, it is recommended to work without momentum.
+
+Relation to previous results: The conclusion that when using A-SGD with high delay values it's recommended to turn off momentum is consistent with the results of Mitliagkas et al. (2017); Liu et al. (2018). In Liu et al. (2018), for streaming PCA, the authors showed it is necessary to reduce momentum in order to ensure convergence and acceleration through asynchrony. In Mitliagkas et al. (2017) the authors suggested that asynchrony adds implicit momentum to the training process and therefore it is necessary to reduce momentum when working asynchronously.
+
+Next, we discuss a new method for introducing momentum into asynchronous training. This method enables asynchronous training where increasing the momentum parameter improves the stability.
+
+Shifted momentum As discussed, training asynchronously is less stable when momentum is used. We propose to utilize momentum in an alternative method, called shifted momentum, for training asynchronously where the momentum benefits the training process and particularly its stability. We observe this method tends to improve the convergence stability compared to training without momentum (or with the original momentum), as is demonstrated in appendix Fig. 11. We suggest that instead of exchanging the gradients terms, as in eq. 9, we exchange the entire velocity term. This way, each worker calculates the velocity term based in its own gradients and momentum buffer. The formulation of this method is given in the following equation:
+
+$$
+\mathbf {v} _ {t + 1} = m \mathbf {v} _ {t - \tau} - \eta (1 - m) \nabla f _ {n (t - \tau)} (\mathbf {x} _ {t - \tau})
+$$
+
+$$
+\mathbf {x} _ {t + 1} = \mathbf {x} _ {t} + \mathbf {v} _ {t + 1}
+$$
+
+or
+
+$$
+\mathbf {x} _ {t + 1} = \mathbf {x} _ {t} + m (\mathbf {x} _ {t - \tau} - \mathbf {x} _ {t - \tau - 1}) - \eta (1 - m) \nabla f _ {n (t - \tau)} (\mathbf {x} _ {t - \tau}) \tag {10}
+$$
+
+In appendix D we show that the characteristic equation of eq. 10 is
+
+$$
+z ^ {\tau + 2} - z ^ {\tau + 1} + (\eta (1 - m) a - m) z + m = 0, \tag {11}
+$$
+
+where we use the sharpness term  $a = \lambda_{\max}(\mathbf{H})$  as before.
+
+We would like to characterize how the learning rate, delay and momentum affect minima stability. Thus, for each momentum value, we evaluate numerically the value of  $\frac{1}{a\tau}$  for which the maximal root of eq. 11 is on the unit circle, i.e., when we are on the threshold of stability. In Fig. 4 we see that the inverse relation between the learning rate  $\eta$  and the delay  $\tau$  still applies when using shifted momentum. Moreover, we can see that in contrast to regular momentum, in shifted momentum, a larger momentum value  $m$  enables us to work with larger step-size. Therefore, increasing the momentum improves the stability.
+
+Relation to previous results: The analysis of delayed gradients and velocity might also shed light on the stability of recent asynchronous training approaches, where the workers communicating through weight averaging (Lian et al., 2017; Assran et al., 2019). This method is similar to ours in the sense that each worker holds its own momentum buffer. In parallel to us, (Hakimi et al., 2019) used a variation of shifted momentum with Nesterov momentum (denoted as Multi-ASGD), and demonstrated empirically its generalization benefits, but without a theoretical analysis of this benefit.
+
+![](images/4da17758711cfae19b51a8e303897b2032ada2dc0c0204bc944b146136c74691.jpg)  
+Figure 2: Stability threshold is maintained when  $\eta \propto 1 / \tau$ : In the left figure we show the number of epochs it takes to diverge from a minimum as a function of the learning rate  $\eta$ . The black circles are the stability thresholds — below which, we do not escape the minimum. In the right figure for each value of delay  $\tau$  we show  $1 / \eta$  where  $\eta$  is the maximal learning rate in which we did not diverge. Due to sampling resolution, there might be up to  $8\%$  deviation from the maximal learning rate found. This deviation is represented in the error bars. VGG-11 trained with CIFAR10.
+
+![](images/9c37fe1119e8962d571bf06b7fcb59dca2b78d08eda843f8932c175f07823498.jpg)
+
+# 3 EXPERIMENTS
+
+In this section, we provide experiments to support our theoretical findings. In the following subsections we: (1) demonstrate the relationship between hyperparameters and minima selection, and (2) show our findings can help improve asynchronous training. Details about the experiments and the implementation can be found in appendix E.
+
+# 3.1 HOW HYPERPARAMETERS AFFECT MINIMA SELECTION
+
+In this section, we demonstrate how the delay and learning rate affect the accessible minima set. We train with a fixed learning rate.
+
+Minima stability. To demonstrate how delay and learning rate change the stability of a minimum, we start from a model that converged to some minimum and train with different values of delay and learning rate to see if the algorithm leaves that minimum, i.e., the minimum becomes unstable. We do so by training a VGG-11 on CIFAR10 for 10,000 epochs – until we reach a steady state. This training is done without delay, momentum and weight decay. Next, introduce a delay  $\tau$ , change the learning rate, and continue to train the model. Fig. 2 shows the number of epochs it takes to leave the steady state for a given delay  $\tau$  and learning rate  $\eta$ . We observe that for certain  $(\tau, \eta)$  pairs, the algorithm stays in the minimum (below the black circle) while for others it leaves that minimum after some number of epochs (above the black circle). Importantly, we can see in the right panel in Fig. 2 that, as predicted by the theory, the inverse learning rate,  $1 / \eta$ , where  $\eta$  is the maximal learning rate in which we did not diverge, scales linearly with the delay  $\tau$ . Additional details are given in appendix F.
+
+Generalization with delay. With delay, we need to adjust the hyperparameters in order to maintain the accessible minima set. However, even if a certain pair of  $(\tau, \eta)$  changes the accessible set, it is possible that there are still accessible minima that generalize well. We perform experiments to investigate what type of generalization we can expect from training with different  $(\tau, \eta)$  pairs. We examine this by training a VGG-11 on CIFAR10 with different  $(\tau, \eta)$  pairs until we reach a steady state (around 6,000 epochs). We use plain SGD without momentum or weight decay. We compare the generalization performance achieved by each pair of learning rate and delay. We repeat the experiment for each pair four times and report mean and standard deviation values. The results are presented in Fig. 3. We can see a linear scaling between the delay values and the learning rate empirical stability threshold, i.e., the learning rate in which the error diverge. In addition, we see that the smallest error (with small variance) is obtained at a learning rate which is somewhat smaller than the learning rate at the empirical threshold of stability, as expected (LeCun et al., 2012).
+
+![](images/62b35962ccf12faaf583fb21535e6f806f3a5fe9da7afb73c5a104f01583d5ff.jpg)
+
+![](images/5898466aa4933b697fc6766fd77f35e15a6c5e28d26eb1afb7c728b55f3507a7.jpg)
+
+![](images/ca645c08139ce4be52bcd15327210426a06b962e1b2b08fc292b5eb2fde79326.jpg)  
+Figure 3: Better generalization obtained near the stability threshold. Validation error Vs. learning rate  $\eta$  of VGG-11 trained with CIFAR10 for different delay values. Solid lines represent mean validation error. The margins represent one standard deviation. The vertical dashed line is the learning rate according to eq. 8. The horizontal dashed line is the accuracy obtained with  $\tau = 0$  (S-SGD).
+
+![](images/c7a1557b46c59f2328a0e477c369106c8d5970fe0777ffbdf953b835ef5a31d1.jpg)
+
+Shifted momentum stability. In subsection 2.2.2 we introduced shifted momentum as a modification of the standard momentum update, in which increasing the momentum value  $(m')$  improves stability. We validate the stability properties of this method by training a fully connected model with MNIST. The model has 3 layers, 1024 neurons in each layer with ReLU activations. Again, training is done synchronously with a large batch and no weight decay until it reaches a steady state. Then, after reaching a minimum, we continued to train the model, each time with a different triplet of  $(\tau, \eta, m)$ . We found the maximum learning rate for a given  $m$  and  $\tau$  for which training does not diverge, i.e., this maximum learning rate is at the edge of stability. Fig. 4 depicts the empirical results. The value of  $a = \lambda_{\max}(\mathbf{H})$  is estimated at the minimum point using power iteration. Two important results emerge from the graph. First, in order to maintain the stability threshold, the learning rate has to be inversely proportional to the delay, as described in subsection 2.2. Second, with increased momentum values, training becomes more stable. Meaning, we can use larger learning rates with larger momentum value. This stands in contrast to the analysis of the original momentum: as we show numerically in appendix D, increasing the momentum value  $m$  in the standard momentum algorithm decreases the stability and hence we need to decrease the learning rate as the momentum value grows larger.
+
+# 3.2 IMPROVING ASYNCHRONOUS TRAINING
+
+So far we focused on the steady state behavior of A-SGD, i.e. after many iterations. However, one of the prominent motivations for training asynchronously is to speed-up training. In this subsection, we examine whether our findings are also relevant for improving the accuracy of models trained asynchronously with little to no excess budget of epochs, compared to equivalent large batch regime. In large batch training, the learning rate is scaled as a function of the ratio between the large batch size and the small one. As Shallue et al. (2018) pointed out, we can expect a different learning rate scaling for different models/datasets. We start with a set of models and datasets that follow a square root scaling of the learning rate with the batch size (Hoffer et al., 2017). Recall that our inversely linear scaling rule (i.e.,  $\eta \propto 1 / \tau$ ) applies to the large batch learning rate. We demonstrate the validity of our inversely linear learning rate scaling by comparing our method with one that does not account for the delay  $\tau$ . The results are presented in Table 1. We can see that using the small batch regime with delay (ASGD) does not converge at all. With our inversely linear learning rate scaling (+LR), there is a small gap from the equivalent large batch. If we also increase the budget of epochs by  $30\%$ , our optimization regime (+30%) generalizes better than the large batch (LB). Because A-SGD has better run-time performance compared to S-SGD (Dutta et al., 2018; Assran et al., 2019), such excess in epochs may still result in improved overall wall clock time (this depends on the hardware details).
+
+Figure 4: Stability threshold is improved when in increasing the momentum parameter, when using shifted momentum: (left) For different values of momentum  $m$ , we examine the relation between  $\frac{1}{a\eta}$  and  $\tau$  obtained by numerically evaluating the value of  $\frac{1}{a\eta}$  for which the maximal roots of eq. 11 is on the unit circle. With shifted momentum, we maintain the stability threshold. (right) we evaluate empirically the relation depicted in the left panel by training, with shifted momentum, a fully connected model with MNIST. The markers are empirically evaluated, and the solid lines are qclinear fit to the markers.  
+![](images/7dad96e32b81febf3607b92e63094832228b00aa9c1bcd620c975cf98bf7a62a.jpg)  
+Table 1: Validation accuracy results, SB/LB represent small and large batch respectively with  $\tau = 0$ . ASGD is asynchronous training with  $\tau = 32$ , +LR stands for ASGD with our linear scaling, and +30% is also with additional 30% epochs budget.
+
+![](images/8687698cadbc6b4e9a51ef3a867ee0f482b78de74e8e18a0c9fcfabdbb7117b4.jpg)
+
+<table><tr><td>Network</td><td>Dataset</td><td>SB</td><td>LB</td><td>ASGD</td><td>+LR</td><td>+30%</td></tr><tr><td>Resnet44 (He et al., 2016)</td><td>Cifar10</td><td>92.87%</td><td>90.42%</td><td>10.0%</td><td>88.57%</td><td>91.65%</td></tr><tr><td>VGG11 (Simonyan &amp; Zisserman, 2014)</td><td>Cifar10</td><td>89.8%</td><td>84.55%</td><td>10.0%</td><td>61.41%</td><td>84.74%</td></tr><tr><td>C3 (Keskar et al., 2016)</td><td>Cifar100</td><td>60.0%</td><td>57.89%</td><td>1.0%</td><td>54.02%</td><td>58.94%</td></tr><tr><td>WResnet16-4 (Zagoruyko &amp; Komodakis, 2016)</td><td>Cifar100</td><td>76.78%</td><td>72.85%</td><td>17.8%</td><td>70.35%</td><td>73.81%</td></tr></table>
+
+ImageNet. We next experiment with ResNet50 and ImageNet, which is a popular benchmark for large batch and asynchronous training because of its complexity and scale. Our focus is asynchronous centralized training with high delay. To the best of our knowledge, the generalization gap is not closed yet in this setting. This is an important setting because of its simplicity and its scalability potential. Optimization details can be found in appendix E.1
+
+In Fig. 5, we compare three A-SGD optimizations trained with ImageNet. As can be seen in the figure, the generalization gap decreases from  $4.33\%$  to  $1.69\%$ , just by turning off momentum. This corresponds with our analysis and previous results about the relation between momentum and delay. Shifting momentum similarly improves generalization, as can be seen in the right panel. This validates our analysis on the beneficial role the momentum parameter has, using shifted momentum. The learning rate is first scaled linearly with the total batch size of all the workers, as in Goyal et al. (2017). Our inverse linear scaling with delay cancels this increase, and we get a similar learning rate as in regular small batch training. In contrast, when we scaled the learning rate with  $1 / \sqrt{\tau}$ , training did not converge at all.
+
+# 4 DISCUSSION
+
+Motivation. A fundamental problem with asynchronous training is that it seems to cause a degradation in generalization, even after training has converged to a steady state. Identifying the root of this problem and resolving it is essential in order to scale parallelism. When approaching this problem, we need to address two issues. First, what is the origin of this generalization gap? Second, how can we decrease this generalization gap?
+
+Conclusions. Our work tackles both questions by studying the behaviour of A-SGD and the relation between the delay and the hyperparameters, e.g., the learning rate and momentum. We analyze A-SGD from the theoretical perspective of dynamical stability. We find that a linear connection between the inverse learning rate and delay is necessary in order to keep the set of accessible minima
+
+![](images/df07cb43afd8d7d29a2563ecc7e7bf577eb9e7dcd9ff42f7fdc1ff708cab9895.jpg)  
+Figure 5: ResNet50 ImageNet asynchronous training. The baseline (in blue, S-SGD) is a large batch synchronous training, following Goyal et al. (2017), where  $\eta \propto$  batch size. For A-SGD, based on our analysis, we additionally change  $\eta \propto 1 / \tau$  (in contrast,  $\eta \propto 1 / \sqrt{\tau}$  does not converge at all). Thus, in this case, we get the same learning rate as in a small batch regime. We compare three A-SGD options (in orange): (1) with standard momentum (2) with momentum value  $m = 0$ ; and (3) with shifted momentum. S-SGD validation final error is  $23.8\%$ . A-SGD achieves  $28.13\%$  when training with standard momentum (left). The error drops to  $25.49\%$  when turning off momentum (middle). With shifted momentum, the error is similar,  $25.4\%$  (right). This matches the analysis in section 2.2.2. Solid lines are validation error and dashed lines are train error.
+
+![](images/5a333ed6e3c9c5b35c5408b87d780cefb8b2bf966b2323f76b4bdcb826f64ce9.jpg)
+
+![](images/5acb06d7869a0202c7612aaf5a94cc4fe4efd1c1c58f3b720d57645845edf65b.jpg)
+
+the same. In addition, we find how momentum affects the accessible minima set. Our analysis suggests that, when increasing delay, momentum degrades stability. Therefore, it should be turned off in order to maintain performance, as was previously observed empirically (Mitliagkas et al., 2017; Dai et al., 2018; Liu et al., 2018). However, we observe that momentum can have some benefits in improving convergence stability without delay. To maintain these benefits, we propose to modify momentum into a method called shifted momentum. We supply a theoretical analysis of A-SGD with shifted momentum and conclude that, in contrast to regular momentum, this method improves stability when delay is increased. In addition, we supply empirical evidence to support these findings. Lastly, we demonstrate, these findings can improve the generalization performance of distributed training algorithms and bring us one step closer to closing the generalization gap.
+
+Future Directions. It is interesting to examine how the dynamical stability analysis holds during training. Specifically, what implications can we derive from the analysis, before we reach the steady state (e.g., a minimum of the loss), given a tight epochs budget. Our empirical findings in this case (Fig. 13) suggest that the optimal learning rate chosen for the steady state may remain optimal during the convergence process. This might be because the hyperparameters at the stability threshold are closely related to the hyperparameters that achieve the optimal convergence speed to that minimum. For example, in GD it is easy to show that  $\eta_{\mathrm{threshold}} = 2\eta_{\mathrm{optimal}}$ . Therefore, both stability and convergence rate may be optimally maintained using similar scaling relations in the hyperparameters. This is an interesting question for future study.
+
+Another interesting question for future study is to understand how the stability conditions change when some synchronization is introduced. Several practical A-SGD methods use synchronization to some extent, e.g. (Assran et al., 2019; Chen et al., 2016). For synchronization methods that reduce the delay in expectation, we can intuitively expect an improvement in dynamical stability, from our analysis of stochastic delay (appendix C). However, further analysis of the stability method for each method is required for an exact answer.
+
+Answering these questions may lead to a more precise understanding of the trade-offs between S-SGD and the various A-SGD methods.
+
+# REFERENCES
+
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+
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+
+# Appendix
+
+# A STABILITY ANALYSIS FOR EQ. 2
+
+In this section, we would like to analyze the stability of eq. 2. In particular, we would like to examine if a given minimum point  $\mathbf{x}^*$  is stable. To analyze this we suppose we are in a vicinity of some minimum point  $\mathbf{x}^*$ , which implies  $\nabla f(\mathbf{x}^*) = 0$ , and ask whether or not we can converge to  $\mathbf{x}^*$  using the theory of dynamical stability. Formally, consider the linearized dynamics around  $\mathbf{x}^*$ :
+
+$$
+\mathbf {x} _ {t + 1} = \mathbf {x} _ {t} - \eta \left(G \left(\mathbf {x} ^ {*}; n _ {t - \tau}\right) + \nabla G \left(\mathbf {x} ^ {*}; n _ {t - \tau}\right) \left(\mathbf {x} _ {t - \tau} - \mathbf {x} ^ {*}\right)\right),
+$$
+
+where we denoted  $G(\mathbf{x}_t; n_t) = \nabla f_{n_t}(\mathbf{x}_t)$  and changed the notation of the time index in  $n(t)$  to the subscript, i.e.,  $n_t$ .
+
+Examining the expectation of this equation we obtain
+
+$$
+\begin{array}{l} \mathbb {E} \mathbf {x} _ {t + 1} \stackrel {(1)} {=} \mathbb {E} \mathbf {x} _ {t} - \eta \left(\frac {1}{N} \sum_ {i = 1} ^ {N} G (\mathbf {x} ^ {*}; i) + \frac {1}{N} \sum_ {i = 1} ^ {N} \nabla G (\mathbf {x} ^ {*}; i) \left(\mathbb {E} \mathbf {x} _ {t - \tau} - \mathbf {x} ^ {*}\right)\right) \\ \stackrel {(2)} {=} \mathbb {E} \mathbf {x} _ {t} - \eta \mathbf {H} \left(\mathbb {E} \mathbf {x} _ {t - \tau} - \mathbf {x} ^ {*}\right), \\ \end{array}
+$$
+
+where in (1) we used  $\forall k = 1,\dots,N:\operatorname *{Pr}(n_t = k) = \frac{1}{N}$ , and in (2) we used the fact that  $\mathbf{x}^*$  is a minimum point and thus  $\sum_{i = 1}^{N}G(\mathbf{x}^{*};i) = \nabla f(\mathbf{x}^{*}) = 0$  and denoted  $\mathbf{H} = \frac{1}{N}\sum_{i = 1}^{N}\nabla G(\mathbf{x}^{*};i) = \frac{1}{N}\sum_{i = 1}^{N}\nabla f_i(\mathbf{x}^*)$ .
+
+Using  $\mathbf{s}_t = \mathbb{E}\mathbf{x}_t - \mathbf{x}^*$  variable change we obtain
+
+$$
+\mathbf {s} _ {t + 1} = \mathbf {s} _ {t} - \eta \mathbf {H} \mathbf {s} _ {t - \tau}. \tag {12}
+$$
+
+From the Spectral Factorization Theorem we can write  $\mathbf{H} = \mathbf{U}\mathbf{A}\mathbf{U}^{\top}$  where  $\mathbf{U}$  is an orthogonal matrix and  $\mathbf{A} = \mathrm{diag}(a_1, \dots, a_n)$  is a diagonal matrix. Substituting this into eq. 12 we obtain:
+
+$$
+\mathbf {s} _ {t + 1} = \mathbf {s} _ {t} - \eta \mathbf {U A U} ^ {\top} \mathbf {s} _ {t - \tau}.
+$$
+
+Multiplying the last equation with  $\mathbf{U}^{\top}$  from the left, denoting  $\tilde{\mathbf{s}}_t = \mathbf{U}^\top \mathbf{s}_t$ , and using the fact that  $\mathbf{U}$  is an orthogonal matrix, i.e.,  $\mathbf{U}^{\top}\mathbf{U} = \mathbf{I}$  we get:
+
+$$
+\tilde {\mathbf {s}} _ {t + 1} = \tilde {\mathbf {s}} _ {t} - \eta \mathbf {A} \tilde {\mathbf {s}} _ {t - \tau}.
+$$
+
+Recall that  $\mathbf{A} = \mathrm{diag}(a_1, \dots, a_N)$  is a diagonal matrix. Therefore, analyzing the dynamics of the last equation is equivalent to analyzing the dynamics of the following  $N$  one-dimensional equations:
+
+$$
+\forall i = 1, \dots N: \tilde {\mathbf {s}} _ {t + 1} (i) = \tilde {\mathbf {s}} _ {t} (i) - \eta a _ {i} \tilde {\mathbf {s}} _ {t - \tau} (i),
+$$
+
+where  $\mathbf{s}_t(i)$  denotes the  $i^{\mathrm{th}}$  component of the vector  $\mathbf{s}_t$ .
+
+In order to ensure stability of the  $N$ -dimensional dynamical system, it's sufficient to require that
+
+$$
+\tilde {\mathbf {s}} _ {t + 1} (1) = \tilde {\mathbf {s}} _ {t} (1) - \eta a _ {1} \tilde {\mathbf {s}} _ {t - \tau} (1), \tag {13}
+$$
+
+where we assume without loss of generality  $a_1 \geq a_2 \geq \dots \geq a_N$ . This is true since the dynamics for  $i = 1$  is the first one that loses stability.
+
+To simplify notations we define  $x_{t} = \tilde{\mathbf{s}}_{t}(1)$  and the sharpness term  $a = a_{1} = \lambda_{\max}(\mathbf{H})$ , the maximal singular value of  $\mathbf{H}$ . Using these notations, we can re-write eq. 13 as:
+
+$$
+x _ {t + 1} = x _ {t} - \eta a x _ {t - \tau} \Leftrightarrow x _ {t + 1} - x _ {t} + \eta a x _ {t - \tau} = 0. \tag {14}
+$$
+
+Since this is a linear time invariant difference equation, of order  $\tau + 1$ , its solution is, generically:
+
+$$
+x _ {t} = \sum_ {i = 1} ^ {\tau + 1} c _ {i} z _ {i} ^ {t},
+$$
+
+where  $c_{i}$  are determined by the initial conditions, and  $z_{i}$  are the roots of the characteristic equation of eq. 14. To find this characteristic equation, we substitute  $x_{t} = z^{t}$  into eq. 14, and obtain
+
+$$
+z ^ {t + 1} - z ^ {t} + \eta a z ^ {t - \tau} = 0 \Leftrightarrow z ^ {\tau + 1} - z ^ {\tau} + \eta a = 0
+$$
+
+Note that for  $x_{t} \to 0$ , it is required that  $\forall i, |z_i| < 1$ .
+
+# B FINDING THE CONDITION FOR THE MAXIMAL ROOT OF THE CHARACTERISTIC EQUATION (EQ. 4) TO BE ON THE UNIT CIRCLE
+
+To simplify the notation, we denote  $\alpha = a\eta$ . Recall the characteristic eq. (eq. 5)
+
+$$
+z ^ {\tau + 1} - z ^ {\tau} + \alpha = 0.
+$$
+
+Since we are looking for solution on the unit circle we substitute  $z = e^{i\theta}$  into the equation where  $i$  is the unit imaginary number and  $\theta$  is some angle. We obtain:
+
+$$
+e ^ {i \theta (\tau + 1)} - e ^ {i \theta \tau} + \alpha = 0.
+$$
+
+Using  $e^{i\theta} = \cos (\theta) + i\sin (\theta)$  we get
+
+$$
+\cos (\theta (\tau + 1)) + i \sin (\theta (\tau + 1)) - (\cos (\theta \tau) + i \sin (\theta \tau)) + \alpha = 0 \Rightarrow
+$$
+
+$$
+\left\{ \begin{array}{l} \cos \left(\theta (\tau + 1)\right) - \cos \left(\theta \tau\right) + \alpha = 0 \\ \sin \left(\theta (\tau + 1)\right) - \sin \left(\theta \tau\right) = 0 \end{array} \right.
+$$
+
+Using trigonometric product-sum identities we obtain
+
+$$
+\left\{ \begin{array}{l} - 2 \sin \left(\frac {\theta (2 \tau + 1)}{2}\right) \sin \left(\frac {\theta}{2}\right) + \alpha = 0 \\ 2 \cos \left(\frac {\theta (2 \tau + 1)}{2}\right) \sin \left(\frac {\theta}{2}\right) = 0 \end{array} \right. \tag {15}
+$$
+
+If  $\sin \left(\frac{\theta}{2}\right) = 0$  then we get from the first equation  $\alpha = 0$ . This is a contradiction since we assume that  $\alpha > 0$ . Thus,  $\sin \left(\frac{\theta}{2}\right) \neq 0$ . This implies from the second equation in eq. 15 that
+
+$$
+\cos \left(\frac {\theta (2 \tau + 1)}{2}\right) = 0 \Rightarrow \frac {\theta (2 \tau + 1)}{2} = \frac {\pi}{2} + \pi k, \text {f o r} k = 0, 1, \dots , 2 \tau .
+$$
+
+Substituting this result into the first equation in eq. 15 and using  $\alpha >0$  we obtain that the possible solutions must satisfy
+
+$$
+\alpha = 2 \sin \left(\frac {(1 + 4 k) \pi}{4 \tau + 2}\right), \text {f o r} k = 0, 1, \dots , \tau .
+$$
+
+Also, since  $\sin (x) = \sin (\pi - x)$  we can eliminate symmetric solution and get
+
+$$
+\alpha = 2 \sin \left(\frac {(1 + 4 k) \pi}{4 \tau + 2}\right), \text {f o r} k = 0, 1, \dots , \lfloor \frac {\tau + 1}{2} \rfloor .
+$$
+
+Note that for  $k = 0,1,\dots,\lfloor \frac{\tau + 1}{2}\rfloor \colon 0 < \frac{(1 + 4k)\pi}{4\tau + 2} < \frac{\pi}{2}$  and thus  $\sin \left(\frac{(1 + 4k)\pi}{4\tau + 2}\right) > 0$  is monotonically increasing with  $k$ . Since the magnitude of the characteristic equation increases with  $\alpha$  and we are interested in the maximal root we are looking for the minimal  $\alpha$  value. This value corresponds to  $k = 0$ . Thus, the maximal root satisfies:
+
+$$
+\alpha = \sin \left(\frac {\pi}{4 \tau + 2}\right).
+$$
+
+# C STABILITY ANALYSIS FOR EQ. 2 WITH STOCHASTIC DELAY
+
+We assume that  $\tau(t)$  is drawn from some distribution. In this section, we would like to analyze the stability of eq. 2. In particular, we would like to examine if a given minimum point  $\mathbf{x}^*$  is stable. To analyze this we suppose we are in a vicinity of some minimum point  $\mathbf{x}^*$ , which implies  $\nabla f(\mathbf{x}^*) = 0$ , and ask whether or not we can converge to  $\mathbf{x}^*$  using the theory of dynamical stability. Formally, consider the linearized dynamics around  $\mathbf{x}^*$ :
+
+$$
+\mathbf {x} _ {t + 1} = \mathbf {x} _ {t} - \eta \left(G (\mathbf {x} ^ {*}; n _ {t - \tau (t)}) + \nabla G (\mathbf {x} ^ {*}; n _ {t - \tau (t)}) (\mathbf {x} _ {t - \tau (t)} - \mathbf {x} ^ {*})\right),
+$$
+
+where we denoted  $G(\mathbf{x}_t; n_t) = \nabla f_{n_t}(\mathbf{x}_t)$  and changed the notation of the time index in  $n(t)$  to the subscript, i.e.,  $n_t$ .
+
+![](images/3aef049eec7fbf8925575b109dc7f5789513baaf892c925271e382771b57d595.jpg)  
+Figure 6: It is necessary to keep the learning rate inversely proportional to the delay to maintain stability. In blue we see the relation between  $\frac{1}{a\eta}$  and  $\tau$  obtained by numerically evaluating the value of  $\frac{1}{a\eta}$  for which the maximal root of eq. 5 is on the unit circle. In orange we see the analytic approximation  $\frac{1}{a\eta} = \frac{1}{\pi} (2\tau + 1)$ . We can see that in order to maintain stability for a given minimum point, as the delay  $\tau$  increases we need to decrease the learning rate  $\eta$  to keep  $\eta$  inversely proportional to  $\tau$ . In appendix (Fig. 10) we show that with momentum, we also get similar linear relation, only with a higher slope.
+
+Examining the expectation of this equation we obtain
+
+$$
+\begin{array}{l} \mathbb {E} \mathbf {x} _ {t + 1} \stackrel {(1)} {=} \mathbb {E} \mathbf {x} _ {t} - \eta \left(\frac {1}{N} \sum_ {i = 1} ^ {N} G (\mathbf {x} ^ {*}; i) + \frac {1}{N} \sum_ {i = 1} ^ {N} \nabla G (\mathbf {x} ^ {*}; i) \left(\sum_ {k} \mathbb {E} \mathbf {x} _ {t - k} \Pr (\tau (t) = k) - \mathbf {x} ^ {*}\right)\right) \\ \stackrel {(2)} {=} \mathbb {E} \mathbf {x} _ {t} - \eta \mathbf {H} \sum_ {k} \Pr (\tau (t) = k) \left(\mathbb {E} \mathbf {x} _ {t - k} - \mathbf {x} ^ {*}\right), \\ \end{array}
+$$
+
+where in (1) we used  $\forall t$  and  $\forall k = 1,\dots,N:\operatorname *{Pr}(n_t = k) = \frac{1}{N}$ , and in (2) we used the fact that  $\mathbf{x}^*$  is a minimum point and thus  $\sum_{i = 1}^{N}G(\mathbf{x}^{*};i) = \nabla f(\mathbf{x}^{*}) = 0$  and denoted  $\mathbf{H} = \frac{1}{N}\sum_{i = 1}^{N}\nabla G(\mathbf{x}^{*};i) = \frac{1}{N}\sum_{i = 1}^{N}\nabla^{2}f_{i}(\mathbf{x}^{*})$ .
+
+Using  $\mathbf{s}_t = \mathbb{E}\mathbf{x}_t - \mathbf{x}^*$  variable change we obtain
+
+$$
+\mathbf {s} _ {t + 1} = \mathbf {s} _ {t} - \eta \mathbf {H} \sum_ {k} \Pr (\tau (t) = k) \mathbf {s} _ {t - k}. \tag {16}
+$$
+
+From the Spectral Factorization Theorem we can write  $\mathbf{H} = \mathbf{U}\mathbf{A}\mathbf{U}^{\top}$  where  $\mathbf{U}$  is an orthogonal matrix and  $\mathbf{A} = \mathrm{diag}(a_1, \ldots, a_n)$  is a diagonal matrix. Substituting this into eq. 16 we obtain:
+
+$$
+\mathbf {s} _ {t + 1} = \mathbf {s} _ {t} - \eta \mathbf {U A} \sum_ {k} \Pr (\tau (t) = k) \mathbf {U} ^ {\top} \mathbf {s} _ {t - k}.
+$$
+
+Multiplying the last equation with  $\mathbf{U}^{\top}$  from the left, denoting  $\tilde{\mathbf{s}}_t = \mathbf{U}^\top \mathbf{s}_t$ , and using the fact that  $\mathbf{U}$  is an orthogonal matrix, i.e.,  $\mathbf{U}^{\top}\mathbf{U} = \mathbf{I}$  we get:
+
+$$
+\tilde {\mathbf {s}} _ {t + 1} = \tilde {\mathbf {s}} _ {t} - \eta \mathbf {A} \sum_ {k} \Pr (\tau (t) = k) \tilde {\mathbf {s}} _ {t - k}.
+$$
+
+Recall that  $\mathbf{A} = \mathrm{diag}(a_1, \dots, a_N)$  is a diagonal matrix. Therefore, analyzing the dynamics of the last equation is equivalent to analyzing the dynamics of the following  $N$  one-dimensional equations:
+
+$$
+\forall i = 1, \dots N: \tilde {\mathbf {s}} _ {t + 1} (i) = \tilde {\mathbf {s}} _ {t} (i) - \eta a _ {i} \sum_ {k} \Pr (\tau (t) = k) \tilde {\mathbf {s}} _ {t - k} (i),
+$$
+
+where  $\mathbf{s}_t(i)$  denotes the  $i^{\mathrm{th}}$  component of the vector  $\mathbf{s}_t$ .
+
+In order to ensure stability of the  $N$ -dimensional dynamical system, it's sufficient to require that
+
+$$
+\tilde {\mathbf {s}} _ {t + 1} (1) = \tilde {\mathbf {s}} _ {t} (1) - \eta a _ {1} \sum_ {k} \Pr (\tau (t) = k) \tilde {\mathbf {s}} _ {t - k} (1), \tag {17}
+$$
+
+where we assume without loss of generality  $a_1 \geq a_2 \geq \dots \geq a_N$ . This is true since the dynamics for  $i = 1$  is the first one that loses stability.
+
+To simplify notations we define  $x_{t} = \tilde{\mathbf{s}}_{t}(1)$  and the sharpness term  $a = a_{1} = \lambda_{\max}(\mathbf{H})$ , the maximal singular value of  $\mathbf{H}$ . Using these notations, we can re-write eq. 13 as:
+
+$$
+x _ {t + 1} = x _ {t} - \eta a \sum_ {k} \Pr (\tau (t) = k) x _ {t - k} \Leftrightarrow x _ {t + 1} - x _ {t} + \eta a \sum_ {k} \Pr (\tau (t) = k) x _ {t - k} = 0. \tag {18}
+$$
+
+The characteristic equation of eq. 18 is
+
+$$
+z - 1 + \eta a \sum_ {k} \Pr (\tau (t) = k) z ^ {- k} = 0 \tag {19}
+$$
+
+# C.1 STABILIY ANALYSIS
+
+We would like to find conditions on when and how can we change the hyperparameters  $(\eta, \mathbb{E}\tau)$  to maintain the same stability criterion (definition 1). In order to analyze stability we examine the characteristic equation in eq. 18 and ask: when does the maximal root of this polynomial have unit magnitude?
+
+1.  $\tau \sim \mathrm{unif}\{a,b\}$
+
+$$
+z - 1 + \eta a \sum_ {k = a} ^ {b} \frac {1}{n} z ^ {- k} = 0 \tag {20}
+$$
+
+![](images/5a28c411fb511b6b92c334d0c2b8f82b92e9b8c8a65b3f54bb6e03cb9b01e023.jpg)  
+Figure 7:  $\tau \sim$  unif  $\{a,b\}$  where  $a = 1$  and  $b$  is an integer in the range [1, 64]. In this case,  $\mathbb{E}\tau = \frac{a + b}{2}$ . We see the relation between  $\frac{1}{a\eta}$  and  $\mathbb{E}\tau$  obtained by numerically evaluating the value of  $\frac{1}{a\eta}$  for which the maximal root of eq. 20 is on the unit circle. We can see that in order to maintain stability for a given minimum point, as the delay expectation  $\mathbb{E}\tau$  increases we need to decrease the learning rate  $\eta$  to keep  $\eta$  inversely proportional to  $\tau$ .
+
+2. Gaussian distribution discrete approximation:  $\forall \mu, \forall k = 1, \dots, 2\mu : \operatorname{Pr}(\tau = k) = \operatorname{Pr}(k - 0.5 \leq \mathbf{z} \leq k + 0.5)$  where  $\mathbf{z} \sim \mathbb{N}(\mu, 1)$ :
+
+![](images/ba4aed2e3945775b87b936ada9557af3b4f1bc654bcea6c11afe205232c301a5.jpg)  
+Figure 8:  $\forall \mu, \forall k = 1, \dots, 2\mu : \operatorname{Pr}(\tau = k) = \operatorname{Pr}(k - 0.5 \leq \mathbf{z} \leq k + 0.5)$  where  $\mathbf{z} \sim \mathbb{N}(\mu, 1)$  and  $\mu$  is an integer in the range [1, 30]. In this case,  $\mathbb{E}\tau = \mu$ . We see the relation between  $\frac{1}{a\eta}$  and  $\mathbb{E}\tau$  obtained by numerically evaluating the value of  $\frac{1}{a\eta}$  for which the maximal root of eq. 19 is on the unit circle. We can see that in order to maintain stability for a given minimum point, as the delay expectation  $\mathbb{E}\tau$  increases we need to decrease the learning rate  $\eta$  to keep  $\eta$  inversely proportional to  $\tau$ .
+
+![](images/f3fd2958b65c006ee78c0c1bdb71edb9470a532a476e6c2e55c7ff2bc3e84fff.jpg)  
+Figure 9: Gaussian distribution of delay. Comparison of ResNet50 trained asynchronously on ImageNet with two types of delay distributions: constant i.e., round robin (orange) and discrete Gaussian distribution (blue). Constant distribution reaches  $25.4\%$  validation error while Gaussian distribution reaches a similar error of  $25.3\%$ .
+
+# D THEORETICAL ANALYSIS WITH MOMENTUM
+
+In this section, we repeat the steps of the analysis in section 2 for A-SGD with momentum.
+
+# D.1 PRELIMINARIES AND PROBLEM SETUP
+
+Consider the problem of minimizing the empirical loss
+
+$$
+f (\mathbf {x}) = \sum_ {i = 1} ^ {N} f _ {i} (\mathbf {x}), \tag {21}
+$$
+
+where  $N$  is the number of samples,  $\mathbf{x} \in \mathbb{R}^d$  and  $\forall i = 1, \dots, N$ :  $f_i : \mathbb{R}^d \to \mathbb{R}$  is twice continuously differentiable function. The update rule for minimizing  $f(\mathbf{x})$  in eq. 21 using asynchronous stochastic gradient descent with momentum (A-MSGD) is given by
+
+$$
+\mathbf {v} _ {t + 1} = m \mathbf {v} _ {t} - \eta (1 - m) G (\mathbf {x} _ {t - \tau}; n _ {t - \tau})
+$$
+
+$$
+\mathbf {x} _ {t + 1} = \mathbf {x} _ {t} + \mathbf {v} _ {t + 1}
+$$
+
+or
+
+$$
+\mathbf {x} _ {t + 1} = \mathbf {x} _ {t} + m \left(\mathbf {x} _ {t} - \mathbf {x} _ {t - 1}\right) - \eta (1 - m) G \left(\mathbf {x} _ {t - \tau}; n _ {t - \tau}\right), \tag {22}
+$$
+
+where  $m$  is the momentum,  $\eta$  is the learning rate,  $G(\mathbf{x}_t; n_t) = \nabla f_{n_t}(\mathbf{x}_t)$ ,  $n_t$  is a random selection process of a sample from  $\{1, 2, \dots, N\}$  at iteration  $t$ , and  $\tau$  is some delay due the asynchronous nature of our training.
+
+Our main goal is understanding how different factors such as the gradient staleness  $\tau$ , and hyperparameters such as the momentum  $m$  and learning rate  $\eta$  affect the minima selection process in neural networks. In other words, we would like to understand how the different hyperparameters interact to divide the global minima of our network to two sets: those we can converge to and those we cannot converge to.
+
+To analyze this we suppose we are in a vicinity of a minimum point  $\mathbf{x}^*$ , which implies  $\nabla f(\mathbf{x}^*) = 0$ , and ask whether or not we can converge to  $\mathbf{x}^*$  using the theory of dynamical stability. Formally, consider the linearized dynamics around some minimum point  $\mathbf{x}^*$ :
+
+$$
+\mathbf {x} _ {t + 1} = \mathbf {x} _ {t} + m (\mathbf {x} _ {t} - \mathbf {x} _ {t - 1}) - \eta (1 - m) (G (\mathbf {x} ^ {*}; n _ {t - \tau}) + \nabla G (\mathbf {x} ^ {*}; n _ {t - \tau}) (\mathbf {x} _ {t - \tau} - \mathbf {x} ^ {*})).
+$$
+
+Examining the first moment of this equation we obtain
+
+$$
+\begin{array}{l} \mathbb {E} \mathbf {x} _ {t + 1} = \mathbb {E} \mathbf {x} _ {t} + m \left(\mathbb {E} \mathbf {x} _ {t} - \mathbb {E} \mathbf {x} _ {t - 1}\right) - \eta (1 - m) \left(\frac {1}{N} \sum_ {i = 1} ^ {N} G (\mathbf {x} ^ {*}; i) + \frac {1}{N} \sum_ {i = 1} ^ {N} \nabla G (\mathbf {x} ^ {*}; i) \left(\mathbb {E} \mathbf {x} _ {t - \tau} - \mathbf {x} ^ {*}\right)\right) \\ = \mathbb {E} \mathbf {x} _ {t} + m \left(\mathbb {E} \mathbf {x} _ {t} - \mathbb {E} \mathbf {x} _ {t - 1}\right) - \eta (1 - m) \mathbf {H} \left(\mathbb {E} \mathbf {x} _ {t - \tau} - \mathbf {x} ^ {*}\right), \\ \end{array}
+$$
+
+where  $\frac{1}{N}\sum_{i=1}^{N}G(\mathbf{x}^*;i) = \frac{1}{N}\nabla f(\mathbf{x}^*) = 0$  since  $\mathbf{x}^*$  is a minimum point and  $\mathbf{H} = \frac{1}{N}\sum_{i=1}^{N}\nabla G(\mathbf{x}^*;i) = \frac{1}{N}\sum_{i=1}^{N}\nabla^2 f_i(\mathbf{x}^*)$ .
+
+Using  $\mathbf{s}_t = \mathbb{E}\mathbf{x}_t - \mathbf{x}^*$  variable change we obtain
+
+$$
+\mathbf {s} _ {t + 1} = \mathbf {s} _ {t} + m \left(\mathbf {s} _ {t} - \mathbf {s} _ {t - 1}\right) - \eta (1 - m) \mathbf {H} \mathbf {s} _ {t - \tau}. \tag {23}
+$$
+
+From the Spectral Factorization Theorem we can write  $\mathbf{H} = \mathbf{U}\mathbf{D}\mathbf{U}^{\top}$  where  $\mathbf{U}$  is an orthogonal matrix and  $\mathbf{A} = \mathrm{diag}(a_1,\dots,a_n)$  is a diagonal matrix. Substituting this into eq. 23 we obtain:
+
+$$
+\mathbf {s} _ {t + 1} = \mathbf {s} _ {t} + m (\mathbf {s} _ {t} - \mathbf {s} _ {t - 1}) - \eta (1 - m) \mathbf {U A U} ^ {\top} \mathbf {s} _ {t - \tau}.
+$$
+
+Multiplying the last equation with  $\mathbf{U}^{\top}$  from the left, denoting  $\tilde{\mathbf{s}}_t = \mathbf{U}^\top \mathbf{s}_t$ , and using the fact that  $\mathbf{U}$  is an orthogonal matrix, i.e.,  $\mathbf{U}^{\top}\mathbf{U} = \mathbf{I}$  we get:
+
+$$
+\tilde {\mathbf {s}} _ {t + 1} = \tilde {\mathbf {s}} _ {t} + m (\tilde {\mathbf {s}} _ {t} - \tilde {\mathbf {s}} _ {t - 1}) - \eta (1 - m) \mathbf {A} \tilde {\mathbf {s}} _ {t - \tau}.
+$$
+
+Recall that  $\mathbf{A} = \mathrm{diag}(a_1, \dots, a_N)$  is a diagonal matrix. Therefore, analyzing the dynamics of the last equation is equivalent to analyzing the dynamics of the following  $N$  one-dimensional equations:
+
+$$
+\forall i = 1, \dots N: \tilde {\mathbf {s}} _ {t + 1} (i) = \tilde {\mathbf {s}} _ {t} (i) + m (\tilde {\mathbf {s}} _ {t} (i) - \tilde {\mathbf {s}} _ {t - 1} (i)) - \eta (1 - m) a _ {i} \tilde {\mathbf {s}} _ {t - \tau} (i),
+$$
+
+where  $\mathbf{s}_t(i)$  denotes the  $i^{\mathrm{th}}$  component of the vector  $\mathbf{s}_t$ .
+
+In order to ensure stability of the  $N$ -dimensional dynamical system, it's sufficient to require that
+
+$$
+\tilde {\mathbf {s}} _ {t + 1} (1) = \tilde {\mathbf {s}} _ {t} (1) + m \left(\tilde {\mathbf {s}} _ {t} (1) - \tilde {\mathbf {s}} _ {t - 1} (1)\right) - \eta (1 - m) a _ {1} \tilde {\mathbf {s}} _ {t - \tau} (1), \tag {24}
+$$
+
+where we assume without loss of generality  $a_1 \geq a_2 \geq \dots \geq a_N$ . This is true since the dynamics for  $a_1$  is the first one that loses stability.
+
+To simplify notations we define  $x_{t} = \tilde{\mathbf{s}}_{t}(1)$  and the sharpness term  $a = a_{1} = \lambda_{\max}(\mathbf{H})$ , the maximal singular value of  $\mathbf{H}$ . Using these notations, we can re-write eq. 24 as:
+
+$$
+x _ {t + 1} = x _ {t} + m \left(x _ {t} - x _ {t - 1}\right) - \eta (1 - m) a x _ {t - \tau} \Leftrightarrow
+$$
+
+$$
+x _ {t + 1} - (m + 1) x _ {t} + m x _ {t - 1} + \eta (1 - m) a x _ {t - \tau} = 0. \tag {25}
+$$
+
+The characteristic equation of this difference equation is
+
+$$
+z ^ {\tau + 1} - (1 + m) z ^ {\tau} + m z ^ {\tau - 1} + \eta (1 - m) a = 0. \tag {26}
+$$
+
+# D.2 STABILITY ANALYSIS
+
+We would like to find conditions on when and how can we change the hyperparameters  $(m, \eta, \tau)$  to maintain the same stability criterion (definition 1). In order to analyze stability we examine the characteristic equation in eq. 26 and ask: when does the maximal root of this polynomial have unit magnitude?
+
+# D.2.1 THE INTERACTION BETWEEN LEARNING RATE, DELAY, AND MOMENTUM
+
+For the general case and different values of momentum, we evaluate numerically the value of  $\frac{1}{a\tau}$  for which the maximal root of the general characteristic equation (eq. 26) is on the unit circle, i.e., when we are on the threshold of stability. We show the results in Fig. 10. We observe that there is still an inverse relation between the learning rate and delay when using momentum. Specifically, we observe that:
+
+1. We need to decrease the learning rate as the delay increases.  
+2. The lines become more steep for larger values of momentum. This implies that, for a given delay, maintaining stability for larger values of momentum requires smaller learning rate. Therefore, it is easier to maintain stability with  $m = 0$ .
+
+# D.3 SHIFTED MOMENTUM
+
+In this section, we repeat the steps of the analysis in section D.1 for A-SGD with shifted momentum. The update rule for minimizing  $f(\mathbf{x})$  in eq. 21 using asynchronous stochastic gradient descent with shifted momentum is given by
+
+$$
+\mathbf {v} _ {t + 1} = m \mathbf {v} _ {t - \tau} - \eta (1 - m) \nabla f _ {n (t - \tau)} (\mathbf {x} _ {t - \tau})
+$$
+
+$$
+\mathbf {x} _ {t + 1} = \mathbf {x} _ {t} + \mathbf {v} _ {t + 1}
+$$
+
+or
+
+$$
+\begin{array}{l} \mathbf {x} _ {t + 1} = \mathbf {x} _ {t} + m \mathbf {v} _ {t - \tau} - \eta (1 - m) \nabla f _ {n (t - \tau)} (\mathbf {x} _ {t - \tau}) \\ = \mathbf {x} _ {t} + m \left(\mathbf {x} _ {t - \tau} - \mathbf {x} _ {t - \tau - 1}\right) - \eta (1 - m) \nabla f _ {n (t - \tau)} (\mathbf {x} _ {t - \tau}) \\ \end{array}
+$$
+
+Following the same steps as in section D.1 we obtain the following one-dimensional equation:
+
+$$
+x _ {t + 1} = x _ {t} + m \left(x _ {t - \tau} - x _ {t - \tau - 1}\right) - \eta (1 - m) a x _ {t - \tau} \Leftrightarrow
+$$
+
+$$
+x _ {t + 1} - x _ {t} + (\eta (1 - m) a - m) x _ {t - \tau} + m x _ {t - \tau - 1} = 0.
+$$
+
+The last equation characteristic equation is:
+
+$$
+z ^ {\tau + 2} - z ^ {\tau + 1} + (\eta (1 - m) a - m) z + m = 0.
+$$
+
+![](images/53022377761721c89a05448cfb4820b8633aa88f497b8ca42d0fb90d9b12e61c.jpg)  
+Figure 10: Using (standard) momentum, it is still necessary to keep the learning rate inversely proportional to the delay to maintain stability. For different values of momentum  $m$ , we examine the relation between  $\frac{1}{a\eta}$  and  $\tau$  obtained by numerically evaluating the value of  $\frac{1}{a\eta}$  for which the maximal root of eq. 25 is on the unit circle. Similar to the results with  $m = 0$ , we can see that maintaining stability requires keeping the learning rate  $\eta$  inversely proportional to the delay  $\tau$ .
+
+![](images/4ea88dfa52f972ce9dc91afdf6947cf4416e84a84b76d938b162fcde6fa197f9.jpg)  
+Figure 11: Shifting momentum improves stability. ResNet44 trained with CIFAR10 with same hyperparameters and three training algorithms: A-SGD with momentum (red), A-SGD without momentum (Orange) and A-SGD with shifted momentum (Blue). We observed "spikes" in the training error that appear when training with large batch size or with delay for large number of epochs, without decreasing the learning rate. While momentum negates these "spikes" in large batch training, Shifting momentum improves the convergence stability of the model and negates these "spikes" when training with delay.
+
+![](images/dc9f022b0a1795caaf3593b0d69d6c7f2766f645301b750c0dd6908e25cd5fb8.jpg)
+
+# EXPERIMENTS DETAILS
+
+We experimented with a set of popular classification tasks: MNIST (Lecun et al., 1998), CIFAR10, CIFAR100 (Krizhevsky, 2009) and ImageNet (Deng et al., 2009). In order to incorporate delay in the experiments, we keep replicas of the model parameters and perform SGD step with different replica at a time according to a round robin scheduling. The implementation is done with PyTorch framework. Section C in the appendix shows that constant delay, i.e., round robin, acts similarly in terms of stability to several other delay distributions. In addition, Zhang et al. (2015) shows empirically with realistic distribution that constant delay distribution is a good model for delay distribution. Fig. 9 shows empirically that round robin scheduling works in a similar way to discrete Gaussian distribution.
+
+Although stochastic gradient descent with momentum was originally introduced with the dampening term, many drop the latter when using momentum SGD. This is true for most DNN frameworks like Caffe, PyTorch, and TensorFlow where the dampening term is set to zero by default. As Shallue et al.
+
+(2018); Yan et al. (2018) pointed out, without dampening, the momentum scales the learning rate so that the effective learning rate  $\eta_{eff}$  is equal to  $\eta_{eff} = \frac{\eta}{1 - m}$  after sufficiently many iterations  $T$  (i.e., when  $T\to \infty$ ). Because we alter the momentum value in our experiments, we use dampening and scale the learning rate accordingly to keep the effective learning rate the same as without dampening.
+
+# E.1 IMAGENET OPTIMIZATION DETAILS
+
+Our baseline is Goyal et al. (2017) large batch training. They trained ResNet50 for 90 epochs with a large batch of 8192 samples and reached the same accuracy level of  $23.8\%$  as the small batch training. They used linear scaling with the batch size (scaling the learning rate by the large batch size divided by the baseline batch size) and a warm-up of the learning rate at the first 5 epochs. We use 32 workers, each with a minibatch size of 256.
+
+# F MINIMA STABILITY EXPERIMENTS
+
+In this section, we provide additional details about the minima stability experiment presented in section 3. As discussed in section 3, we are interested to examine for which pairs of  $(\tau, \eta)$  the minima stability remains the same.
+
+In Fig. 12 we show the validation error of such  $(\tau, \eta)$  pairs as a function of epochs. We note that these graphs are the same experiment as in Fig. 2. As can be seen, for larger learning rates, it takes less epochs to leave the minimum. It is interesting to see that after leaving the minimum, the A-SGD algorithm converges again to a minimum with generalization as good as the baseline (at  $\tau = 0$ ). This suggests that the minima selection process of A-SGD is affected by the whole optimization path. In other words, suppose we start the optimization from a minimum with good generalization (since it was selected using optimization with  $\tau = 0$ ), and then it becomes unstable due to a change in the values of  $(\eta, \tau)$ , as in this experiment. These results in Fig. 12 suggest we typically converge to a stable minimum with similar generalization properties, possibly nearby the original minimum. In contrast, if we start to train from scratch using the same  $(\eta, \tau)$  pair which lost stability in our experiment, we typically get a generalization gap (as observed in our experiments), which suggests the optimization path might have taken a very different path from the start, leading to other regions with worse generalization than the original minimum.
+
+# G ADDITIONAL EXPERIMENTS
+
+We examine what generalization performance we can expect during training, before reaching a minimum. We do so by using the same experiment in Fig. 1. We sample the validation error of each learning rate at different epochs. Results are presented in Fig. 13. Using our suggested learning rate modification (dividing the baseline large batch learning rate by the delay) gives the near optimal learning rate that can be seen in the figure.
+
+![](images/1a65fd4f51cc799cb2a3e0d7c7424fa0e10f04052d2fe12a67bf41825b3c83ed.jpg)
+
+![](images/8dfed44ae598de9db88345fa83bce93a7ce384e18c921a0f078bde7053cd6952.jpg)
+
+![](images/dfb6a265fb5ae2e3f1775922ace3d9c073c8352abafd79077a8c498de94f2cec.jpg)  
+Figure 12: Learning rates larger than  $0.8 / \tau$  diverge from the minimum. We show the validation error Vs. epochs for different delay and learning rate values. The baseline learning rate used is 0.8 according to large batch training Hoffer et al. (2017). The minimum learning rate for each  $\tau$  is the stability threshold.
+
+![](images/1592482482a7d76e53a89eb0460abd46b6bf5c19afb92accdcea1fa91778dbe3.jpg)
+
+![](images/0d4a229df1ff4ed063ac472350a65ee5a46d9a74db58b5839ef6fbd2f646f044.jpg)  
+Figure 13: Impact of learning rate and delay on generalization error during training. Validation error with delay  $(\tau = 32)$  as a function of learning rate at different epochs. The optimal learning rate of  $\eta = 0.01$  chosen found in the scan (which matches our proposed learning rate scaling), stays the same throughout the training. This might suggest why it's beneficial to use the optimal hyperparameters for the minimum, even when the algorithm hasn't converged to the steady state yet. CNN trained with MNIST, same settings as in Fig 1.
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+# BEHAVIOUR SUITE FOR REINFORCEMENT LEARNING
+
+Ian Osband*, Yotam Doron, Matteo Hessel, John Aslanides  
+Eren Sezener, Andre Saraiva, Katrina McKinney, Tor Lattimore, Csaba Szepesvari  
+Satinder Singh, Benjamin Van Roy, Richard Sutton, David Silver, Hado Van Hasselt
+
+DeepMind
+
+# ABSTRACT
+
+This paper introduces the Behaviour Suite for Reinforcement Learning, or bsuite for short. bsuite is a collection of carefully-designed experiments that investigate core capabilities of reinforcement learning (RL) agents with two objectives. First, to collect clear, informative and scalable problems that capture key issues in the design of general and efficient learning algorithms. Second, to study agent behaviour through their performance on these shared benchmarks. To complement this effort, we open source github.com/deepmind/bsuite, which automates evaluation and analysis of any agent on bsuite. This library facilitates reproducible and accessible research on the core issues in RL, and ultimately the design of superior learning algorithms. Our code is Python, and easy to use within existing projects. We include examples with OpenAI Baselines, Dopamine as well as new reference implementations. Going forward, we hope to incorporate more excellent experiments from the research community, and commit to a periodic review of bsuite from a committee of prominent researchers.
+
+# 1 INTRODUCTION
+
+The reinforcement learning (RL) problem describes an agent interacting with an environment with the goal of maximizing cumulative reward through time (Sutton & Barto, 2017). Unlike other branches of control, the dynamics of the environment are not fully known to the agent, but can be learned through experience. Unlike other branches of statistics and machine learning, an RL agent must consider the effects of its actions upon future experience. An efficient RL agent must address three challenges simultaneously:
+
+1. Generalization: be able to learn efficiently from data it collects.  
+2. Exploration: prioritize the right experience to learn from.  
+3. Long-term consequences: consider effects beyond a single timestep.
+
+The great promise of reinforcement learning are agents that can learn to solve a wide range of important problems. According to some definitions, an agent that can learn to perform at or above human level across a wide variety of tasks is an artificial general intelligence (AGI) (Minsky, 1961; Legg et al., 2007).
+
+Interest in artificial intelligence has undergone a resurgence in recent years. Part of this interest is driven by the constant stream of innovation and success on high profile challenges previously deemed impossible for computer systems. Improvements in image recognition are a clear example of these accomplishments, progressing from individual digit recognition (LeCun et al., 1998), to mastering ImageNet in only a few years (Deng et al., 2009; Krizhevsky et al., 2012). The advances in RL systems have been similarly impressive: from checkers (Samuel, 1959), to Backgammon (Tesauro, 1995), to Atari games (Mnih et al., 2015a), to competing with professional players at DOTA (Pachocki et al., 2019) or StarCraft (Vinyals et al., 2019) and beating world champions at Go (Silver et al., 2016). Outside of playing games, decision systems are increasingly guided by AI systems (Evans & Gao, 2016).
+
+As we look towards the next great challenges for RL and AI, we need to understand our systems better (Henderson et al., 2017). This includes the scalability of our RL algorithms, the environments where we expect them to perform well, and the key issues outstanding in the design of a general intelligence system. We have the existence proof that a single self-learning RL agent can master the game of Go purely from self-play (Silver et al., 2018). We do not have a clear picture of whether such a learning algorithm will perform well at driving a car, or managing a power plant. If we want to take the next leaps forward, we need to continue to enhance our understanding.
+
+# 1.1 PRACTICAL THEORY OFTEN LAGS PRACTICAL ALGORITHMS
+
+The practical success of RL algorithms has built upon a base of theory including gradient descent (Bottou, 2010), temporal difference learning (Sutton, 1988) and other foundational algorithms. Good theory provides insight into our algorithms beyond the particular, and a route towards general improvements beyond ad-hoc tinkering. As the psychologist Kurt Lewin said, 'there is nothing as practical as good theory' (Lewin, 1943). If we hope to use RL to tackle important problems, then we must continue to solidify these foundations. This need is particularly clear for RL with nonlinear function approximation, or 'deep RL'. At the same time, theory often lags practice, particularly in difficult problems. We should not avoid practical progress that can be made before we reach a full theoretical understanding. The successful development of algorithms and theory typically moves in tandem, with each side enriched by the insights of the other.
+
+The evolution of neural network research, or deep learning, provides a poignant illustration of how theory and practice can develop together (LeCun et al., 2015). Many of the key ideas for deep learning have been around, and with successful demonstrations, for many years before the modern deep learning explosion (Rosenblatt, 1958; Ivakhnenko, 1968; Fukushima, 1979). However, most of these techniques remained outside the scope of developed learning theory, partly due to their complex and non-convex loss functions. Much of the field turned away from these techniques in a 'neural network winter', focusing instead of function approximation under convex loss (Cortes & Vapnik, 1995). These convex methods were almost completely dominant until the emergence of benchmark problems, mostly for image recognition, where deep learning methods were able to clearly and objectively demonstrate their superiority (LeCun et al., 1998; Krizhevsky et al., 2012). It is only now, several years after these high profile successes, that learning theory has begun to turn its attention back to deep learning (Kawaguchi, 2016; Bartlett et al., 2017; Belkin et al., 2018). The current theory of deep RL is still in its infancy. In the absence of a comprehensive theory, the community needs principled benchmarks that help to develop an understanding of the strengths and weaknesses of our algorithms.
+
+# 1.2 AN 'MNIST' FOR REINFORCEMENT LEARNING
+
+In this paper we introduce the Behaviour Suite for Reinforcement Learning (or bsuite for short): a collection of experiments designed to highlight key aspects of agent scalability. Our aim is that these experiments can help provide a bridge between theory and practice, with benefits to both sides. These experiments embody fundamental issues, such as 'exploration' or 'memory' in a way that can be easily tested and iterated. For the development of theory, they force us to instantiate measurable and falsifiable hypotheses that we might later formalize into provable guarantees. While a full theory of RL may remain out of reach, the development of clear experiments that instantiate outstanding challenges for the field is a powerful driver for progress. We provide a description of the current suite of experiments and the key issues they identify in Section 2.
+
+Our work on bsuite is part of a research process, rather than a final offering. We do not claim to capture all, or even most, of the important issues in RL. Instead, we hope to provide a simple library that collects the best available experiments, and makes them easily accessible to the community. As part of an ongoing commitment, we are forming a bsuite committee that will periodically review the experiments included in the official bsuite release. We provide more details on what makes an 'excellent' experiment in Section 2, and on how to engage in their construction for future iterations in Section 5.
+
+The Behaviour Suite for Reinforcement Learning is a not a replacement for 'grand challenge' undertakings in artificial intelligence, or a leaderboard to climb. Instead it is a collection of diagnostic experiments designed to provide insight into key aspects of agent behaviour. Just as the MNIST dataset offers a clean, sanitised, test of image recognition as a stepping stone to advanced computer vision; so too bsuite aims to instantiate targeted experiments for the development of key RL capabilities.
+
+The successful use of illustrative benchmark problems is not unique to machine learning, and our work is similar in spirit to the Mixed Integer Programming Library (MIPLIB) (miplib2017). In mixed integer programming, and unlike linear programming, the majority of algorithmic advances have (so far) eluded theoretical analysis. In this field, MIPLIB serves to instantiate key properties of problems (or types of problems), and evaluation on MIPLIB is a typical component of any new algorithm. We hope that bsuite can grow to perform a similar role in RL research, at least for those parts that continue to elude a unified theory of artificial intelligence. We provide guidelines for how researchers can use bsuite effectively in Section 3.
+
+# 1.3 OPEN SOURCE CODE, REPRODUCIBLE RESEARCH
+
+As part of this project we open source github.com/deepmind/bsuite, which instantiates all experiments in code and automates the evaluation and analysis of any RL agent on bsuite. This library serves to facilitate reproducible and accessible research on the core issues in reinforcement learning. It includes:
+
+- Canonical implementations of all experiments, as described in Section 2.  
+- Reference implementations of several reinforcement learning algorithms.  
+- Example usage of bsuite with alternative codebases, including 'OpenAI Gym'.  
+- Launch scripts for Google cloud that automate large scale compute at low cost.  
+- A ready-made bsuite Jupyter notebook with analyses for all experiments.  
+- Automated  $\mathrm{LATEX}$  appendix, suitable for inclusion in conference submission.
+
+We provide more details on code and usage in Section 4.
+
+We hope the Behaviour Suite for Reinforcement Learning, and its open source code, will provide significant value to the RL research community, and help to make key conceptual issues concrete and precise. bsuite can highlight bottlenecks in general algorithms that are not amenable to hacks, and reveal properties and scalings of algorithms outside the scope of current analytical techniques. We believe this offers an avenue towards great leaps on key issues, separate to the challenges of large-scale engineering (Nair et al., 2015). Further, bsuite facilitates clear, targeted and unified experiments across different code frameworks, something that can help to remedy issues of reproducibility in RL research (Tanner & White, 2009; Henderson et al., 2017).
+
+# 1.4 RELATED WORK
+
+The Behaviour Suite for Reinforcement Learning fits into a long history of RL benchmarks. From the beginning, research into general learning algorithms has been grounded by the performance on specific environments (Sutton & Barto, 2017). At first, these environments were typically motivated by small MDPs that instantiate the general learning problem. 'CartPole' (Barto et al., 1983) and 'MountainCar' (Moore, 1990) are examples of classic benchmarks that has provided a testing ground for RL algorithm development. Similarly, when studying specific capabilities of learning algorithms, it has often been helpful to design diagnostic environments with that capability in mind. Examples of this include 'RiverSwim' for exploration (Strehl & Littman, 2008) or 'Taxi' for temporal abstraction (Dietterich, 2000). Performance in these environments provide a targeted signal for particular aspects of algorithm development.
+
+As the capabilities or RL algorithms have advanced, so has the complexity of the benchmark problems. The Arcade Learning Environment (ALE) has been instrumental in driving
+
+progress in deep RL through surfacing dozens of Atari 2600 games as learning environments (Bellemare et al., 2013). Similar projects have been crucial to progress in continuous control (Duan et al., 2016; Tassa et al., 2018), model-based RL (Wang et al., 2019) and even rich 3D games (Beattie et al., 2016). Performing well in these complex environments requires the integration of many core agent capabilities. We might think of these benchmarks as natural successors to 'CartPole' or 'MountainCar'.
+
+The Behaviour Suite for Reinforcement Learning offers a complementary approach to existing benchmarks in RL, with several novel components:
+
+1. bsuite experiments enforce a specific methodology for agent evaluation beyond just the environment definition. This is crucial for scientific comparisons and something that has become a major problem for many benchmark suites (Machado et al., 2017) (Section 2).  
+2. bsuite aims to isolate core capabilities with targeted 'unit tests', rather than integrate general learning ability. Other benchmarks evolve by increasing complexity, bsuite aims to remove all confounds from the core agent capabilities of interest (Section 3).  
+3. bsuite experiments are designed with an emphasis on scalability rather than final performance. Previous 'unit tests' (such as 'Taxi' or 'RiverSwim') are of fixed size, bsuite experiments are specifically designed to vary the complexity smoothly (Section 2).  
+4. github.com/deepmind/bsuite has an extraordinary emphasis on the ease of use, and compatibility with RL agents not specifically designed for bsuite. Evaluating an agent on bsuite is practical even for agents designed for a different benchmark (Section 4).
+
+# 2 EXPERIMENTS
+
+This section outlines the experiments included in the Behaviour Suite for Reinforcement Learning 2019 release. In the context of bsuite, an experiment consists of three parts:
+
+1. Environments: a fixed set of environments determined by some parameters.  
+2. Interaction: a fixed regime of agent/environment interaction (e.g. 100 episodes).  
+3. Analysis: a fixed procedure that maps agent behaviour to results and plots.
+
+One crucial part of each bsuite analysis defines a 'score' that maps agent performance on the task to [0, 1]. This score allows for agent comparison 'at a glance', the Jupyter notebook includes further detailed analysis for each experiment. All experiments in bsuite only measure behavioural aspects of RL agents. This means that they only measure properties that can be observed in the environment, and are not internal to the agent. It is this choice that allows bsuite to easily generate and compare results across different algorithms and codebases. Researchers may still find it useful to investigate internal aspects of their agents on bsuite environments, but it is not part of the standard analysis.
+
+Every current and future bsuite experiment should target some key issue in RL. We aim for simple behavioural experiments, where agents that implement some concept well score better than those that don't. For an experiment to be included in bsuite it should embody five key qualities:
+
+- Targeted: performance in this task corresponds to a key issue in RL.  
+- Simple: strips away confounding/confusing factors in research.  
+- Challenging: pushes agents beyond the normal range.  
+- Scalable: provides insight on scalability, not performance on one environment.  
+- Fast: iteration from launch to results in under  $30\mathrm{min}$  on standard CPU.
+
+Where our current experiments fall short, we see this as an opportunity to improve the Behaviour Suite for Reinforcement Learning in future iterations. We can do this both through replacing experiments with improved variants, and through broadening the scope of issues that we consider.
+
+We maintain the full description of each of our experiments through the code and accompanying documentation at github.com/deepmind/bsuite. In the following subsections, we pick two bsuite experiments to review in detail: 'memory length' and 'deep sea', and review these examples in detail. By presenting these experiments as examples, we can emphasize what we think makes bsuite a valuable tool for investigating core RL issues. We do provide a high level summary of all other current experiments in Appendix A.
+
+To accompany our experiment descriptions, we present results and analysis comparing three baseline algorithms on bsuite: DQN (Mnih et al., 2015a), A2C (Mnih et al., 2016) and Bootstrapped DQN (Osband et al., 2016). As part of our open source effort, we include full code for these agents and more at bsuite/baselines. All plots and analysis are generated through the automated bsuite Jupyter notebook, and give a flavour for the sort of agent comparisons that are made easy by bsuite.
+
+# 2.1 EXAMPLE EXPERIMENT: MEMORY LENGTH
+
+Almost everyone agrees that a competent learning system requires memory, and almost everyone finds the concept of memory intuitive. Nevertheless, it can be difficult to provide a rigorous definition for memory. Even in human minds, there is evidence for distinct types of 'memory' handled by distinct regions of the brain (Milner et al., 1998). The assessment of memory only becomes more difficult to analyse in the context of general learning algorithms, which may differ greatly from human models of cognition. Which types of memory should we analyse? How can we inspect belief models for arbitrary learning systems? Our approach in bsuite is to sidestep these debates through simple behavioural experiments.
+
+We refer to this experiment as memory length; it is designed to test the number of sequential steps an agent can remember a single bit. The underlying environment is based on a stylized T-maze (O'Keefe & Dostrovsky, 1971), parameterized by a length  $N \in \mathbb{N}$ . Each episode lasts  $N$  steps with observation  $o_{t} = (c_{t}, t / N)$  for  $t = 1,.., N$  and action space  $\mathcal{A} = \{-1, +1\}$ . The context  $c_{1} \sim \mathrm{Unif}(\mathcal{A})$  and  $c_{t} = 0$  for all  $t \geq 2$ . The reward  $r_{t} = 0$  for all  $t < N$ , but  $r_{N} = \mathrm{Sign}(a_{N} = c_{1})$ . For the bsuite experiment we run the agent on sizes  $N = 1,.., 100$  exponentially spaced and look at the average regret compared to optimal after 10k episodes. The summary 'score' is the percentage of runs for which the average regret is less than 75% of that achieved by a uniformly random policy.
+
+![](images/54e69246908ee28d33db2724fd0e34d13fdb251229ec4068a38fb569a20b78f8.jpg)  
+Figure 1: Illustration of the 'memory length' environment
+
+Memory length is a good bsuite experiment because it is targeted, simple, challenging, scalable and fast. By construction, an agent that performs well on this task has mastered some use of memory over multiple timesteps. Our summary 'score' provides a quick and dirty way to compare agent performance at a high level. Our sweep over different lengths  $N$  provides empirical evidence about the scaling properties of the algorithm beyond a simple pass/fail. Figure 2a gives a quick snapshot of the performance of baseline algorithms. Unsurprisingly, actor-critic with a recurrent neural network greatly outperforms the feedforward DQN and Bootstrapped DQN. Figure 2b gives us a more detailed analysis of the same underlying data. Both DQN and Bootstrapped DQN are unable to learn anything for length  $> 1$ , they lack functioning memory. A2C performs well for all  $N \leq 30$  and essentially random for all  $N > 30$ , with quite a sharp cutoff. While it is not surprising that the recurrent agent outperforms feedforward architectures on a memory task, Figure 2b gives an excellent insight into the scaling properties of this architecture. In this case, we have a clear explanation for the observed performance: the RNN agent was trained via backprop-through-time with length 30. bsuite recovers an empirical evaluation of the scaling properties we would expect from theory.
+
+# 2.2 EXAMPLE EXPERIMENT: DEEP SEA
+
+Reinforcement learning calls for a sophisticated form of exploration called deep exploration (Osband et al., 2017). Just as an agent seeking to 'exploit' must consider the long term
+
+![](images/be32332d06e632ba851805649666109f36a74d8243a3c1e5360565c7e45cd441.jpg)  
+(a) Summary score  
+Figure 2: Selected output from bsuite evaluation on 'memory length'.
+
+![](images/9c4bbbac01ca3b37e220a4e4e56a039c27c86b48c2b571129658ebb94d3ab582.jpg)  
+(b) Examining learning scaling.
+
+consequences of its actions towards cumulative rewards, an agent seeking to 'explore' must consider how its actions can position it to learn more effectively in future timesteps. The literature on efficient exploration broadly states that only agents that perform deep exploration can expect polynomial sample complexity in learning (Kearns & Singh, 2002). This literature has focused, for the most part, on uncovering possible strategies for deep exploration through studying the tabular setting analytically (Jaksch et al., 2010; Azar et al., 2017). Our approach in bsuite is to complement this understanding through a series of behavioural experiments that highlight the need for efficient exploration.
+
+The deep sea problem is implemented as an  $N \times N$  grid with a one-hot encoding for state. The agent begins each episode in the top left corner of the grid and descends one row per timestep. Each episode terminates after  $N$  steps, when the agent reaches the bottom row. In each state there is a random but fixed mapping between actions  $A = \{0,1\}$  and the transitions 'left' and 'right'. At each timestep there is a small cost  $r = -0.01 / N$  of moving right, and  $r = 0$  for moving left. However, should the agent transition right at every timestep of the episode it will be rewarded with an additional reward of +1. This presents a particularly challenging exploration problem for two reasons. First, following the 'gradient' of small intermediate rewards leads the agent away from the optimal policy. Second, a policy that explores with actions uniformly at random has probability  $2^{-N}$  of reaching the rewarding state in any episode. For the bsuite experiment we run the agent on sizes  $N = 10,12,\ldots,50$  and look at the average regret compared to optimal after 10k episodes. The summary 'score' computes the percentage of runs for which the average regret drops below 0.9 faster than the  $2^{N}$  episodes expected by dithering.
+
+![](images/4616c0208ab91381816dcbd28abffe7424e5601e339c9f094775bdd3f0f89a4b.jpg)  
+Figure 3: Deep-sea exploration: a simple example where deep exploration is critical.
+
+Deep Sea is a good bsuite experiment because it is targeted, simple, challenging, scalable and fast. By construction, an agent that performs well on this task has mastered some key properties of deep exploration. Our summary score provides a 'quick and dirty' way to compare agent performance at a high level. Our sweep over different sizes  $N$  can help to provide empirical evidence of the scaling properties of an algorithm beyond a simple pass/fail. Figure 3 presents example output comparing A2C, DQN and Bootstrapped DQN on this
+
+task. Figure 4a gives a quick snapshot of performance. As expected, only Bootstrapped DQN, which was developed for efficient exploration, scores well. Figure 4b gives a more detailed analysis of the same underlying data. When we compare the scaling of learning with problem size  $N$  it is clear that only Bootstrapped DQN scales gracefully to large problem sizes. Although our experiment was only run to size 50, the regular progression of learning times suggest we might expect this algorithm to scale towards  $N > 50$ .
+
+![](images/ab1c75e09e8f8ac2feb8411be1f78834bd77f5027c231c9d7ec43ac036038e55.jpg)  
+(a) Summary score
+
+![](images/0f25fbcb5ff1baef4bbee6b760b45d692a52002889340b03ce7b2075a6867994.jpg)  
+(b) Examining learning scaling. Dashed line at  $2^{N}$  for reference.
+
+![](images/0f91e6f2b46951a55eaf36cdd75d83f3c3fe41410d7c766cb269aac5a1a8ed69.jpg)  
+Figure 4: Selected output from bsuite evaluation on 'deep sea'.
+
+![](images/bd5366ed5bbb88de459b73bd490a557d6cdefb4ed894d39a34820f16a84ce9ef.jpg)
+
+# 3 HOW TO USE BSUITE
+
+This section describes some of the ways you can use bsuite in your research and development of RL algorithms. Our aim is to present a high-level description of some research and engineering use cases, rather than a tutorial for the code installation and use. We provide examples of specific investigations using bsuite in Appendixes C, D and E. Section 4 provides an outline of our code and implementation. Full details and tutorials are available at github.com/deepmind/bsuite.
+
+A bsuite experiment is defined by a set of environments and number of episodes of interaction. Since loading the environment via bsuite handles the logging automatically, any agent interacting with that environment will generate the data required for required for analysis through the Jupyter notebook we provide (Pérez & Granger, 2007). Generating plots and analysis via the notebook only requires users to provide the path to the logged data. The 'radar plot' (Figure 5) at the start of the notebook provides a snapshot of agent behaviour, based on summary scores. The notebook also contains a complete description of every experiment, summary scoring and in-depth analysis of each experiment. You can interact with the full report at bit.ly/bsuite-agents.
+
+![](images/c5a7b7598fccd11d3018842096402c160c2aecb2f0c92c9687483c3b4841ba93.jpg)  
+Figure 5: We aggregate experiment performance with a snapshot of 7 core capabilities.
+
+If you are developing an algorithm to make progress on fundamental issues in RL, running on `bsuite` provides a simple way to replicate benchmark experiments in the field. Although
+
+many of these problems are 'small', in the sense that their solution does not necessarily require large neural architecture, they are designed to highlight key challenges in RL. Further, although these experiments do offer a summary 'score', the plots and analysis are designed to provide much more information than just a leaderboard ranking. By using this common code and analysis, it is easy to benchmark your agents and provide reproducible and verifiable research.
+
+If you are using RL as a tool to crack a 'grand challenge' in AI, such as beating a world champion at Go, then taking on bsuite gridworlds might seem like small fry. We argue that one of the most valuable uses of bsuite is as a diagnostic 'unit-test' for large-scale algorithm development. Imagine you believe that 'better exploration' is key to improving your performance on some challenge, but when you try your 'improved' agent, the performance does not improve. Does this mean your agent does not do good exploration? Or maybe that exploration is not the bottleneck in this problem? Worse still, these experiments might take days and thousands of dollars of compute to run, and even then the information you get might not be targeted to the key RL issues. Running on bsuite, you can test key capabilities of your agent and diagnose potential improvements much faster, and more cheaply. For example, you might see that your algorithm completely fails at credit assignment beyond  $n = 20$  steps. If this is the case, maybe this lack of credit-assignment over long horizons is the bottleneck and not necessarily exploration. This can allow for much faster, and much better informed agent development - just like a good suite of tests for software development.
+
+Another benefit of bsuite is to disseminate your results more easily and engage with the research community. For example, if you write a conference paper targeting some improvement to hierarchical reinforcement learning, you will likely provide some justification for your results in terms of theorems or experiments targeted to this setting. However, it is typically a large amount of work to evaluate your algorithm according to alternative metrics, such as exploration. This means that some fields may evolve without realising the connections and distinctions between related concepts. If you run on bsuite, you can automatically generate a one-page Appendix, with a link to a notebook report hosted online. This can help provide a scientific evaluation of your algorithmic changes, and help to share your results in an easily-digestible format, compatible with ICML, ICLR and NeurIPS formatting. We provide examples of these experiment reports in Appendices B, C, D and E.
+
+# 4 CODE STRUCTURE
+
+To avoid discrepancies between this paper and the source code, we suggest that you take practical tutorials directly from github.com/deepmind/bsuite. A good starting point is bit.ly/bsuite-tutorial: a Jupyter notebook where you can play the code right from your browser, without installing anything. The purpose of this section is to provide a high-level overview of the code that we open source. In particular, we want to stress is that bsuite is designed to be a library for RL research, not a framework. We provide implementations for all the environments, analysis, run loop and even baseline agents. However, it is not necessary that you make use of them all in order to make use of bsuite.
+
+The recommended method is to implement your RL agent as a class that implements a policy method for action selection, and an update method for learning from transitions and rewards. Then, simply pass your agent to our run loop, which enumerates all the necessary bsuite experiments and logs all the data automatically. If you do this, then all the experiments and analysis will be handled automatically and generate your results via the included Jupyter notebook. We provide examples of running these scripts locally, and via Google cloud through our tutorials.
+
+If you have an existing codebase, you can still use bsuite without migrating to our run loop or agent structure. Simply replace your environment with environment = bsuite.load_and_record(bsuite_id) and add the flag bsuite_id to your code. You can then complete a full bsuite evaluation by iterating over the bsuite_ids defined in
+
+sweep.SWEEP. Since the environments handle the logging themselves, your don't need any additional logging for the standard analysis. Although full bsuite includes many separate evaluations, no single bsuite environment takes more than 30 minutes to run and the sweep is naturally parallel. As such, we recommend launching in parallel using multiple processes or multiple machines. Our examples include a simple approach using Python's multiprocessing module with Google cloud compute. We also provide examples of running bsuite from OpenAI baselines (Dhariwal et al., 2017) and Dopamine (Castro et al., 2018).
+
+Designing a single RL agent compatible with diverse environments can cause problems, particularly for specialized neural networks. bsuite alleviates this problem by specifying an observation_spec that surfaces the necessary information for adaptive network creation. By default, bsuite environments implement the dm_env standards (Muldal et al., 2017), but we also include a wrapper for use through Openai gym (Brockman et al., 2016). However, if your agent is hardcoded for a format, bsuite offers the option to output each environment with the observation_spec of your choosing via linear interpolation. This means that, if you are developing a network suitable for Atari and particular observation_spec, you can choose to swap in bsuite without any changes to your agent.
+
+# 5 FUTURE ITERATIONS
+
+This paper introduces the Behaviour Suite for Reinforcement Learning, and marks the start of its ongoing development. With our opensource effort, we chose a specific collection of experiments as the bsuite2019 release, but expect this collection to evolve in future iterations. We are reaching out to researchers and practitioners to help collate the most informative, targeted, scalable and clear experiments possible for reinforcement learning. To do this, submissions should implement a sweep that determines the selection of environments to include and logs the necessary data, together with an analysis that parses this data.
+
+In order to review and collate these submissions we will be forming a bsuite committee. The committee will meet annually during the NeurIPS conference to decide which experiments will be included in the bsuite release. We are reaching out to a select group of researchers, and hope to build a strong core formed across industry and academia. If you would like to submit an experiment to bsuite or propose a committee member, you can do this via github pull request, or via email to bsuitecommittee@gmail.com.
+
+We believe that bsuite can be a valuable tool for the RL community, and particularly for research in deep RL. So far, the great success of deep RL has been to leverage large amounts of computation to improve performance. With bsuite, we hope to leverage largescale computation for improved understanding. By collecting clear, informative and scalable experiments; and providing accessible tools for reproducible evaluation we hope to facilitate progress in reinforcement learning research.
+
+# REFERENCES
+
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+
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+Pablo Samuel Castro, Subhodeep Moitra, Carles Gelada, Saurabh Kumar, and Marc G. Bellemare. *Dopamine: A Research Framework for Deep Reinforcement Learning*. 2018. URL http://arxiv.org/abs/1812.06110.  
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+
+# A EXPERIMENT SUMMARY
+
+This appendix outlines the experiments that make up the bsuite 2019 release. In the interests of brevity, we provide only an outline of each experiment here. Full documentation for the environments, interaction and analysis are kept with code at github.com/deepmind/bsuite.
+
+# A.1 BASIC LEARNING
+
+We begin with a collection of very simple decision problems, and standard analysis that confirms an agent's competence at learning a rewarding policy within them. We call these experiments 'basic', since they are not particularly targeted at specific core issues in RL, but instead test a general base level of competence we expect all general agents to attain.
+
+# A.1.1 SIMPLE BANDIT
+
+![](images/ef97416bdbde16221bd45e95e4a9613584874df1212509fb6f68b91a237227e9.jpg)
+
+<table><tr><td>component</td><td>description</td></tr><tr><td>environments</td><td>Finite-armed bandit with deterministic rewards [0,0.1,.1] (Gittins, 1979). 20 seeds.</td></tr><tr><td>interaction</td><td>10k episodes, record regret vs optimal.</td></tr><tr><td>score</td><td>regret normalized [random, optimal] → [0,1]</td></tr><tr><td>issues</td><td>basic</td></tr></table>
+
+# A.1.2 MNIST
+
+![](images/a92674e5270923d4c96ecb0a25d54d68a0c0484cb26334715d3e48c4fb453fd9.jpg)
+
+<table><tr><td>component</td><td>description</td></tr><tr><td>environments</td><td>Contextual bandit classification of MNIST with ±1 rewards (LeCun et al., 1998). 20 seeds.</td></tr><tr><td>interaction</td><td>10k episodes, record average regret.</td></tr><tr><td>score</td><td>regret normalized [random, optimal] → [0,1]</td></tr><tr><td>issues</td><td>basic, generalization</td></tr></table>
+
+# A.1.3 CATCH
+
+![](images/b25e6205f2f78485e001f5c5c36baeb3e1fba141e2644e82e8e14fa05b57f42d.jpg)
+
+<table><tr><td>component</td><td>description</td></tr><tr><td>environments</td><td>A 10x5 Tetris-grid with single block falling per column. The agent can move left/right in the bottom row to ‘catch’ the block. 20 seeds.</td></tr><tr><td>interaction</td><td>10k episodes, record average regret.</td></tr><tr><td>score</td><td>regret normalized [random, optimal] → [0,1]</td></tr><tr><td>issues</td><td>basic, credit assignment</td></tr></table>
+
+# A.1.4 CARTPOLE
+
+![](images/0ad6d49f644dcbaafaca749f6b2a9acad51f49f24e48d5f459d747bed861c35f.jpg)
+
+<table><tr><td>component</td><td>description</td></tr><tr><td>environments</td><td>Agent can move a cart left/right on a plane to keep a balanced pole upright (Barto et al., 1983), 20 seeds.</td></tr><tr><td>interaction</td><td>10k episodes, record average regret.</td></tr><tr><td>score</td><td>regret normalized [random, optimal] → [0,1]</td></tr><tr><td>issues</td><td>basic, credit assignment, generalization</td></tr></table>
+
+# A.1.5 MOUNTAIN CAR
+
+![](images/bfc8502b5231d12a201f654a8f7f7dddf4d116f472f0fffcd3a67b500b379513.jpg)
+
+<table><tr><td>component</td><td>description</td></tr><tr><td>environments</td><td>Agent drives an underpowered car up a hill (Moore, 1990), 20 seeds.</td></tr><tr><td>interaction</td><td>10k episodes, record average regret.</td></tr><tr><td>score</td><td>regret normalized [random, optimal] → [0,1]</td></tr><tr><td>issues</td><td>basic, credit assignment, generalization</td></tr></table>
+
+# A.2 STOCHASTICITY
+
+To investigate the robustness of RL agents to noisy rewards, we repeat the experiments from Section A.1 under differing levels of Gaussian noise. This time we allocate the 20 different seeds across 5 levels of Gaussian noise  $N(0,\sigma^2)$  for  $\sigma = [0.1,0.3,1,3,10]$  with 4 seeds each.
+
+# A.3 PROBLEM SCALE
+
+To investigate the robustness of RL agents to problem scale, we repeat the experiments from Section A.1 under differing reward scales. This time we allocate the 20 different seeds across 5 levels of reward scaling, where we multiply the observed rewards by  $\lambda = [0.01, 0.1, 1, 10, 100]$  with 4 seeds each.
+
+# A.4 EXPLORATION
+
+As an agent interacts with its environment, it observes the outcomes that result from previous states and actions, and learns about the system dynamics. This leads to a fundamental tradeoff: by exploring poorly-understood states and actions the agent can learn to improve future performance, but it may attain better short-run performance by exploiting its existing knowledge. Exploration is the challenge of prioritizing useful information for learning, and the experiments in this section are designed to necessitate efficient exploration for good performance.
+
+# A.4.1 DEEP SEA
+
+![](images/8d32cd8417d692e09928a379064b1c577aaee4f3c62c86835383dadfd7c3742c.jpg)
+
+<table><tr><td>component</td><td>description</td></tr><tr><td>environments</td><td>Deep sea chain environments size N=[5..50].</td></tr><tr><td>interaction</td><td>10k episodes, record average regret.</td></tr><tr><td>score</td><td>% of runs with ave regret &lt; 90% random</td></tr><tr><td>issues</td><td>exploration</td></tr></table>
+
+# A.4.2 STOCHASTIC DEEP SEA
+
+![](images/ece4b2c4d65f58778bcf900b7b9dcb99b9ec8351883ead2f823fa7da2c0e9349.jpg)
+
+<table><tr><td>component</td><td>description</td></tr><tr><td>environments</td><td>Deep sea chain environments with stochastic transitions, N(0,1) reward noise, size N=[5..50].</td></tr><tr><td>interaction</td><td>10k episodes, record average regret.</td></tr><tr><td>score</td><td>% of runs with ave regret &lt; 90% random</td></tr><tr><td>issues</td><td>exploration, stochasticity</td></tr></table>
+
+# A.4.3 CARTPOLE SWINGUP
+
+![](images/2754b41f736c88863ac2596986c838411e3d08af82da2d6dad69fc1e3f1a11db.jpg)
+
+<table><tr><td>component</td><td>description</td></tr><tr><td>environments</td><td>Cartpole ‘swing up’ problem with sparse re- ward (Barto et al., 1983), weigh limit x=[0, 0.5, ..., 0.95].</td></tr><tr><td>interaction</td><td>1k episodes, record average regret.</td></tr><tr><td>score</td><td>% of runs with average return &gt;0</td></tr><tr><td>issues</td><td>exploration, generalization</td></tr></table>
+
+# A.5 CREDIT ASSIGNMENT
+
+Reinforcement learning extends contextual bandit decision problem to allow long term consequences in decision problems. This means that actions in one timestep can effect dynamics in future timesteps. One of the challenges of this setting is that of credit assignment, and the experiments in this section are designed to highlight these issues.
+
+# A.5.1 UMBRELLA LENGTH
+
+![](images/741855d8c05ae475d451aab69767117d6cb9f71d0b39a2251a3d9d98b3cb6925.jpg)
+
+<table><tr><td>component</td><td>description</td></tr><tr><td>environments</td><td>Stylized ‘umbrella problem’, where only the first decision matters and long chain of confounding variables. Vary length 1..100 loga-rithmically.</td></tr><tr><td>interaction</td><td>1k episodes, record average regret.</td></tr><tr><td>score</td><td>regret normalized [random, optimal] → [0,1]</td></tr><tr><td>issues</td><td>credit assignment, stochasticity</td></tr></table>
+
+# A.5.2 UMBRELLA FEATURES
+
+![](images/35bf6e2f2a8f72df00ebee68abf0ae017b13f639036ec4c61879f1b341234794.jpg)
+
+<table><tr><td>component</td><td>description</td></tr><tr><td>environments</td><td>Stylized ‘umbrella problem’, where only the first decision matters and long chain of confounding variables. Vary features 1..100 loga-rithmically.</td></tr><tr><td>interaction</td><td>1k episodes, record average regret.</td></tr><tr><td>score</td><td>regret normalized [random, optimal] → [0,1]</td></tr><tr><td>issues</td><td>credit assignment, stochasticity</td></tr></table>
+
+# A.5.3 DISCOUNTING CHAIN
+
+![](images/cb95a3e2d6572a8b20e2bab5c124169d7e7da6a8c0cb2ebae122cceb5273af2f.jpg)
+
+<table><tr><td>component</td><td>description</td></tr><tr><td>environments</td><td>Experiment designed to highlight issues of dis-counting horizon.</td></tr><tr><td>interaction</td><td>1k episodes, record average regret.</td></tr><tr><td>score</td><td>regret normalized [random, optimal] → [0,1]</td></tr><tr><td>issues</td><td>credit assignment</td></tr></table>
+
+# A.6 MEMORY
+
+Memory is the challenge that an agent should be able to curate an effective state representation from a series of observations. In this section we review a series of experiments in which agents with memory can perform much better than those that only have access to the immediate observation.
+
+# A.6.1 MEMORY LENGTH
+
+![](images/8bce187e18ecf7ed4db53d78c2c2afaabf881f2431f3475553e9a2a24b89ad6d.jpg)
+
+<table><tr><td>component</td><td>description</td></tr><tr><td>environments</td><td>T-maze with a single binary context, grow length 1..100 logarithmically.</td></tr><tr><td>interaction</td><td>1k episodes, record average regret.</td></tr><tr><td>score</td><td>regret normalized [random, optimal] → [0,1]</td></tr><tr><td>issues</td><td>credit assignment</td></tr></table>
+
+# A.6.2 MEMORY BITS
+
+![](images/3d8649b8283a98102aa9b782d039f544476e95099d7fc87ca5e7ce1a2b6b02fc.jpg)
+
+<table><tr><td>component</td><td>description</td></tr><tr><td>environments</td><td>T-maze with length 2, vary number of bits to remember 1..100 logarithmically.</td></tr><tr><td>interaction</td><td>1k episodes, record average regret.</td></tr><tr><td>score</td><td>regret normalized [random, optimal] → [0,1]</td></tr><tr><td>issues</td><td>credit assignment</td></tr></table>
+
+# B BSuite REPORT AS CONFERENCE APPENDIX
+
+If you run an agent on bsuite, and you want to share these results as part of a conference submission, we make it easy to share a single-page 'bsuite report' as part of your appendix. We provide a simple LATEXfile that you can copy/paste into your paper, and is compatible out-the-box with ICLR, ICML and NeurIPS style files. This single page summary displays the summary scores for experiment evaluations for one or more agents, with plots generated automatically from the included ipython notebook. In each report, two sections are left for the authors to fill in: one describing the variants of the agents examined and another to give some brief commentary on the results. We suggest that authors promote more in-depth analysis to their main papers, or simply link to a hosted version of the full bsuite analysis online. You can find more details on our automated reports at github.com/deepmind/bsuite.
+
+The sections that follow are example bsuite reports, that give some example of how these report appendixes might be used. We believe that these simple reports can be a good complement to conference submissions in RL research, that 'sanity check' the elementary properties of algorithmic implementations. An added bonus of bsuite is that it is easy to set up a like for like experiment between agents from different 'frameworks' in a way that would be extremely laborious for an individual researcher. If you are writing a conference paper on a new RL algorithm, we believe that it makes sense for you to include a bsuite report in the appendix by default.
+
+# C bsuite report: benchmarking baseline agents
+
+The Behaviour Suite for Reinforcement Learning, or bsuite for short, is a collection of carefully-designed experiments that investigate core capabilities of a reinforcement learning (RL) agent. The aim of the bsuite project is to collect clear, informative and scalable problems that capture key issues in the design of efficient and general learning algorithms and study agent behaviour through their performance on these shared benchmarks. This report provides a snapshot of agent performance on bsuite2019, obtained by running the experiments from github.com/deepmind/bsuite (Osband et al., 2019).
+
+# C.1 AGENT DEFINITION
+
+In this experiment all implementations are taken from bsuite/baselines with default configurations. We provide a brief summary of the agents run on bsuite2019:
+
+- random: selects action uniformly at random each timestep.  
+- dqn: Deep Q-networks (Mnih et al., 2015b).  
+- boot_dqn: bootstrapped DQN with prior networks (Osband et al., 2016; 2018).  
+- actor_critic_rnn: an actor critic with recurrent neural network (Mnih et al., 2016).
+
+# C.2 SUMMARY SCORES
+
+Each bsuite experiment outputs a summary score in [0,1]. We aggregate these scores by according to key experiment type, according to the standard analysis notebook. A detailed analysis of each of these experiments may be found in a notebook hosted on Colaboratory bit.ly/bsuite-agents.
+
+![](images/abb416a3311b629895c5211833f068c9ea1d1564bc32d0fba41d4533f7379477.jpg)  
+Figure 6: A snapshot of agent behaviour. Figure 7: Score for each bsuite experiment.
+
+# C.3 RESULTS COMMENTARY
+
+- random performs uniformly poorly, confirming the scores are working as intended.  
+- dqn performs well on basic tasks, and quite well on credit assignment, generalization, noise and scale. DQN performs extremely poorly across memory and exploration tasks. The feedforward MLP has no mechanism for memory, and  $\epsilon = 5\%$  -greedy action selection is inefficient exploration.  
+- boot_dqn is mostly identically to DQN, except for exploration where it greatly outperforms. This result matches our understanding of Bootstrapped DQN as a variant of DQN designed to estimate uncertainty and use this to guide deep exploration.  
+- actor_critic_rnn typically performs worse than either DQN or Bootstrapped DQN on all tasks apart from memory. This agent is the only one able to perform better than random due to its recurrent network architecture.
+
+# D bsuite report: optimization algorithm in DQN
+
+The Behaviour Suite for Reinforcement Learning, or bsuite for short, is a collection of carefully-designed experiments that investigate core capabilities of a reinforcement learning (RL) agent. The aim of the bsuite project is to collect clear, informative and scalable problems that capture key issues in the design of efficient and general learning algorithms and study agent behaviour through their performance on these shared benchmarks. This report provides a snapshot of agent performance on bsuite2019, obtained by running the experiments from github.com/deepmind/bsuite (Osband et al., 2019).
+
+# D.1 AGENT DEFINITION
+
+All agents correspond to different instantiations of the DQN agent (Mnih et al., 2015b), as implemented in bsuite/baselines but with differnet optimizers from Tensorflow (Abadi et al., 2015). In each case we tune a learning rate to optimize performance on 'basic' tasks from {1e-1, 1e-2, 1e-3}, keeping all other parameters constant at default value.
+
+- sgd: vanilla stochastic gradient descent with learning rate 1e-2 (Kiefer & Wolfowitz, 1952).  
+- rmsprop: RMSProp with learning rate 1e-3 (Tieleman & Hinton, 2012).  
+- adam: Adam with learning rate 1e-3 (Kingma & Ba, 2015).
+
+# D.2 SUMMARY SCORES
+
+Each bsuite experiment outputs a summary score in [0,1]. We aggregate these scores by according to key experiment type, according to the standard analysis notebook. A detailed analysis of each of these experiments may be found in a notebook hosted on Colaboratory: bit.ly/bsuite-optim.
+
+![](images/7db648b683618ee262b16ef2d891c9568b63ba85b9de301e2c891d684d1b43bf.jpg)  
+Figure 8: A snapshot of agent behaviour.
+
+![](images/57e68b046973b4933ca8bd604ca55f6f57e7c1794b8703335c80f960902be5e7.jpg)  
+Figure 9: Score for each bsuite experiment.
+
+# D.3 RESULTS COMMENTARY
+
+Both RMSProp and Adam perform better than SGD in every category. In most categories, Adam slightly outperforms RMSprop, although this difference is much more minor. SGD performs particularly badly on environments that require generalization and/or scale. This is not particularly surprising, since we expect the non-adaptive SGD may be more sensitive to learning rate optimization or annealing.
+
+In Figure 11 we can see that the differences are particularly pronounced on the cartpole domains. We hypothesize that this task requires more efficient neural network optimization, and the non-adaptive SGD is prone to numerical issues.
+
+# Ebsuite report: ensemble size in Bootstrapped DQN
+
+The Behaviour Suite for Reinforcement Learning, or bsuite for short, is a collection of carefully-designed experiments that investigate core capabilities of a reinforcement learning (RL) agent. The aim of the bsuite project is to collect clear, informative and scalable problems that capture key issues in the design of efficient and general learning algorithms and study agent behaviour through their performance on these shared benchmarks. This report provides a snapshot of agent performance on bsuite2019, obtained by running the experiments from github.com/deepmind/bsuite (Osband et al., 2019).
+
+# E.1 AGENT DEFINITION
+
+In this experiment, all agents correspond to different instantiations of a Bootstrapped DQN with prior networks (Osband et al., 2016; 2018). We take the default implementation from bsuite/baselines. We investigate the effect of the number of models used in the ensemble, sweeping over  $\{1,3,10,30\}$ .
+
+# E.2 SUMMARY SCORES
+
+Each bsuite experiment outputs a summary score in [0,1]. We aggregate these scores by according to key experiment type, according to the standard analysis notebook. A detailed analysis of each of these experiments may be found in a notebook hosted on Colaboratory: bit.ly/bsuite-ensemble.
+
+![](images/0dc4d4b9183c1116e38457f09d32aba2ea7f89b477fd0b54af4573832c76f3a6.jpg)  
+Figure 10: A snapshot of agent behaviour. Figure 11: Score for each bsuite experiment.
+
+# E.3 RESULTS COMMENTARY
+
+Generally, increasing the size of the ensemble improves bsuite performance across the board. However, we do see significantly decreasing returns to ensemble size, so that ensemble 30 does not perform much better than size 10. These results are not predicted by the theoretical scaling of proven bounds (Lu & Van Roy, 2017), but are consistent with previous empirical findings (Osband et al., 2017; Russo et al., 2017). The gains are most extreme in the exploration tasks, where ensemble sizes less than 10 are not able to solve large 'deep sea' tasks, but larger ensembles solve them reliably.
+
+Even for large ensemble sizes, our implementation does not completely solve every cartpole swingup instance. Further examination learning curves suggests this may be due to some instability issues, which might be helped by using Double DQN to combat value overestimation (van Hasselt et al., 2016).
\ No newline at end of file
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+# CLEVRR: COLLISION EVENTS FOR VIDEO REPRESENTATION AND REASONING
+
+Kexin Yi*
+
+Harvard University
+
+Chuang Gan*
+
+MIT-IBM Watson AI Lab
+
+Yunzhu Li
+
+MIT CSAIL
+
+Pushmeet Kohli
+
+DeepMind
+
+Jiajun Wu
+
+MIT CSAIL
+
+Antonio Torralba
+
+MITCSAIL
+
+Joshua B. Tenenbaum
+
+MIT BCS, CBMM, CSAIL
+
+# ABSTRACT
+
+The ability to reason about temporal and causal events from videos lies at the core of human intelligence. Most video reasoning benchmarks, however, focus on pattern recognition from complex visual and language input, instead of on causal structure. We study the complementary problem, exploring the temporal and causal structures behind videos of objects with simple visual appearance. To this end, we introduce the CoLlision Events for Video REpresentation and Reasoning (CLEVRER) dataset, a diagnostic video dataset for systematic evaluation of computational models on a wide range of reasoning tasks. Motivated by the theory of human causal judgment, CLEVRER includes four types of question: descriptive (e.g., 'what color'), explanatory ('what's responsible for'), predictive ('what will happen next'), and counterfactual ('what if'). We evaluate various state-of-the-art models for visual reasoning on our benchmark. While these models thrive on the perception-based task (descriptive), they perform poorly on the causal tasks (explanatory, predictive and counterfactual), suggesting that a principled approach for causal reasoning should incorporate the capability of both perceiving complex visual and language inputs, and understanding the underlying dynamics and causal relations. We also study an oracle model that explicitly combines these components via symbolic representations.
+
+# 1 INTRODUCTION
+
+The ability to recognize objects and reason about their behaviors in physical events from videos lies at the core of human cognitive development (Spelke, 2000). Humans, even young infants, group segments into objects based on motion, and use concepts of object permanence, solidity, and continuity to explain what has happened, infer what is about to happen, and imagine what would happen in counterfactual situations. The problem of complex visual reasoning has been widely studied in artificial intelligence and computer vision, driven by the introduction of various datasets on both static images (Antol et al., 2015; Zhu et al., 2016; Hudson & Manning, 2019) and videos (Jang et al., 2017; Tapaswi et al., 2016; Zadeh et al., 2019). However, despite the complexity and variety of the visual context covered by these datasets, the underlying logic, temporal and causal structure behind the reasoning process is less explored.
+
+In this paper, we study the problem of temporal and causal reasoning in videos from a complementary perspective: inspired by a recent visual reasoning dataset, CLEVR (Johnson et al., 2017a), we simplify the problem of visual recognition, but emphasize the complex temporal and causal structure behind the interacting objects. We introduce a video reasoning benchmark for this problem, drawing inspirations from developmental psychology (Gerstenberg et al., 2015; Ullman, 2015). We also evaluate and assess limitations of various current visual reasoning models on the benchmark.
+
+Our benchmark, named CoLlision Events for Video Representation and Reasoning (CLEVRER), is a diagnostic video dataset for temporal and causal reasoning under a fully controlled environment. The design of CLEVRER follows two guidelines: first, the posted tasks should focus on logic reasoning in the temporal and causal domain while staying simple and exhibiting minimal biases on visual scenes and language; second, the dataset should be fully controlled and well-annotated in order to host the
+
+![](images/8a495a021bd6565062d8509f9549ec23ea51cd368361dba08a872bad9daec76d.jpg)  
+(a) First collision
+
+![](images/afa615bf00b1a4726e366e5ce998398a5fc5d32a77a2e91dc64fb1d34ed595e3.jpg)  
+(b) Cyan cube enters
+
+![](images/45e5b07764f7a0365abb4283ed5694fde1d816986da6be49f41aa2e10f546b07.jpg)  
+(c) Second collision
+
+![](images/152a69a907858ffdf62a0fef581aa7820bb606fa6fcfe29c11af78323b57f95a.jpg)  
+(d) Video ends
+
+# I. Descriptive
+
+Q: What shape is the object that collides with the cyan cylinder?
+
+Q: How many metal objects are moving when the video ends?
+
+A: cylinder
+
+A:3
+
+# III. Predictive
+
+Q: Which event will happen next
+
+a) The cube collides with the red object
+
+b) The cyan cylinder collides with the red object
+
+A: a)
+
+# IV. Counterfactual
+
+Q: Without the gray object, which event will not happen?
+
+a) The cyan cylinder collides with the sphere
+
+b) The red object and the sphere collide
+
+A: a), b)
+
+# II. Explanatory
+
+Q: Which of the following is responsible for the gray cylinder's colliding with the cube? a) The presence of the sphere
+
+b) The collision between the gray cylinder and the cyan cylinder
+
+A: b)
+
+Figure 1: Sample video, questions, and answers from our CoLision Events for Video REpresentation and Reasoning (CLEVRER) dataset. CLEVRER is designed to evaluate whether computational models can understand what is in the video (I, descriptive questions), explain the cause of events (II, explanatory), predict what will happen in the future (III, predictive), and imagine counterfactual scenarios (IV, counterfactual). In the four images (a-d), only for visualization purposes, we apply stroboscopic imaging to reveal object motion. The captions (e.g., 'First collision') are for the readers to better understand the frames, not part of the dataset.
+
+complex reasoning tasks and provide effective diagnostics for models on those tasks. CLEVRR includes 20,000 synthetic videos of colliding objects and more than 300,000 questions and answers (Figure 1). We focus on four specific elements of complex logical reasoning on videos: descriptive (e.g., 'what color'), explanatory ('what's responsible for'), predictive ('what will happen next'), and counterfactual ('what if'). CLEVRR comes with ground-truth motion traces and event histories of each object in the videos. Each question is paired with a functional program representing its underlying logic. As summarized in table 1, CLEVRR complements existing visual reasoning benchmarks on various aspects and introduces several novel tasks.
+
+We also present analysis of various state-of-the-art visual reasoning models on CLEVRER. While these models perform well on descriptive questions, they lack the ability to perform causal reasoning and struggle on the explanatory, predictive, and counterfactual questions. We therefore identify three key elements that are essential to the task: recognition of the objects and events in the videos; modeling the dynamics and causal relations between the objects and events; and understanding of the symbolic logic behind the questions. As a first-step exploration of this principle, we study an oracle model, Neuro-Symbolic Dynamic Reasoning (NS-DR), that explicitly joins these components via a symbolic video representation, and assess its performance and limitations.
+
+# 2 RELATED WORK
+
+Our work can be uniquely positioned in the context of three recent research directions: video understanding, visual question answering, and physical and causal reasoning.
+
+Video understanding. With the availability of large-scale video datasets (Caba Heilbron et al., 2015; Kay et al., 2017), joint video and language understanding tasks have received much interest. This includes video captioning (Guadarrama et al., 2013; Venugopalan et al., 2015; Gan et al., 2017), localizing video segments from natural language queries (Gao et al., 2017; Hendricks et al., 2017), and video question answering. In particular, recent papers have explored different approaches to acquire and ground various reasoning tasks to videos. Among those, MovieQA (Tapaswi et al., 2016), TGIF-QA (Jang et al., 2017), TVQA (Lei et al., 2018) are based on real-world videos and human-generated questions. Social-IQ (Zadeh et al., 2019) discusses causal relations in human social interactions based on real videos. COG (Yang et al., 2018) and MarioQA (Mun et al., 2017) use simulated environments to generate synthetic data and controllable reasoning tasks. Compared to them, CLEVRER focuses on the causal relations grounded in object dynamics and physical interactions, and introduces a wide range of tasks including description, explanation, prediction and counterfactuals. CLEVRER also emphasizes compositionality in the visual and logic context.
+
+Visual question answering. Many benchmark tasks have been introduced in the domain of visual question answering. The Visual Question Answering (VQA) dataset (Antol et al., 2015) marks an important milestone towards top-down visual reasoning, based on large-scale cloud-sourced real images and human-generated questions. The CLEVR dataset (Johnson et al., 2017a) follows a bottom-up approach by defining the tasks under a controlled close-domain setup of synthetic images and questions with compositional attributes and logic traces. More recently, the GQA dataset (Hudson & Manning, 2019) applies synthetic compositional questions to real images. The VCR dataset (Zellers
+
+<table><tr><td>Dataset</td><td>Video</td><td>Diagnostic Annotations</td><td>Temporal Relation</td><td>Explanation</td><td>Prediction</td><td>Counterfactual</td></tr><tr><td>VQA (Antol et al., 2015)</td><td>×</td><td>×</td><td>×</td><td>×</td><td>×</td><td>×</td></tr><tr><td>CLEVR (Johnson et al., 2017a)</td><td>×</td><td>✓</td><td>×</td><td>×</td><td>×</td><td>×</td></tr><tr><td>COG (Yang et al., 2018)</td><td>×</td><td>✓</td><td>✓</td><td>×</td><td>×</td><td>×</td></tr><tr><td>VCR (Zellers et al., 2019)</td><td>×</td><td>✓</td><td>×</td><td>✓</td><td>×</td><td>✓</td></tr><tr><td>GQA (Johnson et al., 2017a)</td><td>×</td><td>✓</td><td>×</td><td>×</td><td>×</td><td>×</td></tr><tr><td>TGIF-QA (Jang et al., 2017)</td><td>✓</td><td>×</td><td>✓</td><td>×</td><td>×</td><td>×</td></tr><tr><td>MovieQA (Tapaswi et al., 2016)</td><td>✓</td><td>×</td><td>✓</td><td>✓</td><td>×</td><td>×</td></tr><tr><td>MarioQA (Mun et al., 2017)</td><td>✓</td><td>×</td><td>✓</td><td>✓</td><td>×</td><td>×</td></tr><tr><td>TVQA (Lei et al., 2018)</td><td>✓</td><td>×</td><td>×</td><td>✓</td><td>×</td><td>×</td></tr><tr><td>Social-IQ (Zadeh et al., 2019)</td><td>✓</td><td>×</td><td>×</td><td>✓</td><td>×</td><td>×</td></tr><tr><td>CLEVRER (ours)</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td></tr></table>
+
+Table 1: Comparison between CLEVRER and other visual reasoning benchmarks on images and videos. CLEVRER is a well-annotated video reasoning dataset created under a controlled environment. It introduces a wide range of reasoning tasks including description, explanation, prediction and counterfactuals
+
+et al., 2019) discusses explanations and hypothesis judgements based on common sense. There have also been numerous visual reasoning models (Hudson & Manning, 2018; Santoro et al., 2017; Hu et al., 2017; Perez et al., 2018; Zhu et al., 2017; Mascharka et al., 2018; Suarez et al., 2018; Cao et al., 2018; Bisk et al., 2018; Misra et al., 2018; Aditya et al., 2018). Here we briefly review a few. The stacked attention networks (SAN) (Yang et al., 2016) introduce a hierarchical attention mechanism for end-to-end VQA models. The MAC network (Hudson & Manning, 2018) combines visual and language attention for compositional visual reasoning. The IEP model (Johnson et al., 2017b) proposes to answer questions via neural program execution. The NS-VQA model (Yi et al., 2018) disentangles perception and logic reasoning by combining an object-based abstract representation of the image with symbolic program execution. In this work we study a complementary problem of causal reasoning and assess the strengths and limitations of these baseline methods.
+
+Physical and causal reasoning. Our work is also related to research on learning scene dynamics for physical and causal reasoning (Lerer et al., 2016; Battaglia et al., 2013; Mottaghi et al., 2016; Fragkiadaki et al., 2016; Battaglia et al., 2016; Chang et al., 2017; Agrawal et al., 2016; Finn et al., 2016; Shao et al., 2014; Fire & Zhu, 2016; Pearl, 2009; Ye et al., 2018), either directly from images (Finn et al., 2016; Ebert et al., 2017; Watters et al., 2017; Lerer et al., 2016; Mottaghi et al., 2016; Fragkiadaki et al., 2016), or from a symbolic, abstract representation of the environment (Battaglia et al., 2016; Chang et al., 2017). Concurrent to our work, CoPhy (Baradel et al., 2020) studies physical dynamics prediction in a counterfactual setting. CATER (Girdhar & Ramanan, 2020) introduces a synthetic video dataset for temporal reasoning associated with compositional actions. CLEVRR complements these works by incorporating dynamics modeling with compositional causal reasoning, and grounding the reasoning tasks to the language domain.
+
+# 3 THE CLEVRR DATASET
+
+The CLEVRER dataset studies temporal and causal reasoning on videos. It is carefully designed in a fully-controlled synthetic environment, enabling complex reasoning tasks, providing effective diagnostics for models while simplifying video recognition and language understanding. The videos describe motion and collisions of objects on a flat tabletop (as shown in Figure 1) simulated by a physics engine, and are associated with the ground-truth motion traces and histories of all objects and events. Each video comes with four types of questions generated by machine, including descriptive ('what color', 'how many'), explanatory ('What is responsible for'), predictive ('What will happen next'), and counterfactual ('what if'). Each question is paired with a functional program.
+
+# 3.1 VIDEOS
+
+CLEVRER includes 10,000 videos for training, 5,000 for validation, and 5,000 for testing. All videos last for 5 seconds. The videos are generated by a physics engine that simulates object motion plus a graphs engine that renders the frames. Extra examples from the dataset can be found in supplementary material C.
+
+Objects and events. Objects in CLEVRR videos adopt similar compositional intrinsic attributes as in CLEVRR (Johnson et al., 2017a), including three shapes (cube, sphere, and cylinder), two materials (metal and rubber), and eight colors (gray, red, blue, green, brown, cyan, purple, and
+
+![](images/c988ea15553b8353f963d5f8ed62a88a3e61d8895154a30a16ed1e3221804491.jpg)  
+How many metal objects are moving when the video ends?
+
+![](images/b182bf032411e721abb1b4b65aa88f13c7a8b0b8a76b143c1d42c6ea2b8198f9.jpg)  
+Figure 2: Sample questions and programs from CLEVRR. Left: Descriptive question. Middle and right: multiple-choice question and choice. Each choice can pair with the question to form a joint logic trace.
+
+![](images/0c15ae6d9086495d314ba7866308bce4cb6b1268c9761ddf642467c3fa03390d.jpg)  
+Without the gray object, which event will happen?  
+The cube collides with the red object.
+
+yellow). All objects have the same size so no vertical bouncing occurs during collision. In each video, we prohibit identical objects, such that each combination of the three attributes uniquely identifies one object. Under this constraint, all intrinsic attributes for each object are sampled randomly. We further introduce three types of events: enter, exit and collision, each of which contains a fixed number of object participants: 2 for collision and 1 for enter and exit. The objects and events form an abstract representation of the video. These ground-truth annotations, together with the object motion traces, enable model diagnostics, one of the key advantages offered by a fully controlled environment.
+
+Causal structure. Objects and events in CLEVRER videos exhibit rich causal structures. An event can be either caused by an object if the event is the first one participated by the object, or another event if the cause event happens right before the outcome event on the same object. For example, if a sphere collides with a cube and then a cylinder, then the first collision and the cube jointly "cause" the second collision. The object motion traces with complex causal structures are generated by the following recursive process. We start with one randomly initialized moving object, and then add another object whose initial position and velocity are set such that it will collide with the first object. The same process is then repeated to add more objects and collisions to the scene. All collision pairs and motion trajectories are randomly chosen. We discard simulations with repetitive collisions between the same pair of objects.
+
+Video generation. CLEVRER videos are generated from the simulated motion traces, including each object's position and pose at each time step. We use the Bullet (Coumans, 2010) physics engine for motion simulation. Each simulation lasts for seven seconds. The motion traces are first down-sampled to fit the frame rate of the output video (25 frames per second). Then the motion of the first five seconds are sent to Blender (Blender Online Community, 2016) to render realistic video frames of object motion and collision. The remaining two seconds are held-out for predictive tasks. We further note CLEVRER adopts the same software and parameters for rendering as CLEVR.
+
+# 3.2 QUESTIONS
+
+We pair each video with machine-generated questions for descriptive, explanatory, predictive, and counterfactual reasoning. Sample questions of the four types can be found in Figure 1. Each question is paired with a functional program executable on the video's dynamical scene. Unlike CLEVR (Johnson et al., 2017a), our questions focus on the temporal and causal aspects of the objects and events in the videos. We exclude all questions on static object properties, which can be answered by looking at a
+
+![](images/54f4ce71aed154ace783387b465883573ce047b4bc283809253ed081486578ab.jpg)  
+Figure 3: Distribution of CLEVRER question types. Left: distribution of four main questions types. Right: distribution of descriptive sub-types.
+
+single frame. CLEVRR consists of 219,918 descriptive questions, 33,811 explanatory questions, 14,298 predictive questions and 37,253 counterfactual questions. Detailed distribution and split of the questions can be found in Figure 3 and supplementary material A.
+
+Descriptive. Descriptive questions evaluate a model's capability to understand and reason about a video's dynamical content and temporal relation. The reasoning tasks are grounded to the compositional space of both object and event properties, including intrinsic attributes (color, material, shape), motion, collision, and temporal order. All descriptive questions are 'open-ended' and can be answered by a single word. Descriptive questions contain multiple sub-types including count, exist, query color, query material, and query shape. Distribution of the sub-types is shown in Figure 3. We evenly sample the answers within each sub-type to reduce answer bias.
+
+Explanatory. Explanatory questions query the causal structure of a video by asking whether an object or event is responsible for another event. Event  $A$  is responsible for event  $B$  if  $A$  is among
+
+$B$ 's ancestors in the causal graph. Similarly, object  $O$  is responsible for event  $A$  if  $O$  participates in  $A$  or any other event responsible for  $A$ . Explanatory questions are multiple choice questions with at most four options, each representing an event or object in the video. Models need to select all options that match the question's description. There can be multiple correct options for each question. We sample the options to balance the number of correct and wrong ones, and minimize text-only biases.
+
+Predictive. Predictive questions test a model's capability of predicting possible occurrences of future events after the video ends. Similar to explanatory questions, predictive questions are multiple-choice, whose options represent candidate events that will or will not happen. Because post-video events are sparse, we provide two options for each predictive question to reduce bias.
+
+Counterfactual. Counterfactual questions query the outcome of the video under certain hypothetical conditions (e.g. removing one of the objects). Models need to select the events that would or would not happen under the designated condition. There are at most four options for each question. The numbers of correct and incorrect options are balanced. Both predictive and counterfactual questions require knowledge of object dynamics underlying the videos and the ability to imagine and reason about unobserved events.
+
+Program representation. In CLEVRR, each question is represented by a tree-structured functional program, as shown in Figure 2. A program begins with a list of objects or events from the video. The list is then passed through a sequence of filter modules, which select entries from the list and join the tree branches to output a set of target objects and events. Finally, an output module is called to query a designated property of the target outputs. For multiple choice questions, each question and option correspond to separate programs, which can be jointly executed to output a yes/no token that indicates if the choice is correct for the question. A list of all program modules can be found in the supplementary material B.
+
+Question generation. Questions in CLEVRR are generated by a multi-step procedure. For each question type, a pre-defined logic template is chosen. The logic template can be further populated by attributes that are associated with the context of the video (i.e. the color, material, shape that identifies the object to be queried) and then turned into natural language. We first generate an exhaustive list of all possible questions for each video. To avoid language bias, we sample from that list to maintain a balanced answer distribution for each question type and minimize the correlation between the questions and answers over the entire dataset.
+
+# 4 BASELINE EVALUATION
+
+In this section, we evaluate and analyse the performances of a wide range of baseline models for video reasoning on CLEVRER. For descriptive questions, the models treat each question as a multi-class classification problem over all possible answers. For multiple choice questions, each question-choice pair is treated as a binary classification problem indicating the correctness of the choice.
+
+# 4.1 MODEL DETAILS
+
+The baseline models we evaluate fall into three families: language-only models, models for video question answering, and models for compositional visual reasoning.
+
+Language-only models. This model family includes weak baselines that only relies on question input to assess language biases in CLEVRER. Q-type (random) uniformly samples an answer from the answer space or randomly select each choice for multiple-choice questions. Q-type (frequent) chooses the most frequent answer in the training set for each question type. LSTM uses a pretrained word embedding trained on the Google News corpus (Mikolov et al., 2013) to encode the input question and processes the sequence with a LSTM (Hochreiter & Schmidhuber, 1997). A MLP is then applied to the final hidden state to predict a distribution over the answers.
+
+Video question answering. We also evaluate the following models that relies on both video and language inputs. CNN+MLP extracts features from the input video via a convolutional neural network (CNN) and encodes the question by taking the average of the pretrained word embeddings (Mikolov et al., 2013). The video and language features are then jointly sent to a MLP for answer prediction. CNN+LSTM relies on the same architecture for video feature extraction but uses the final state of a LSTM for answer prediction. TVQA (Lei et al., 2018) introduces a multi-stream end-to-end neural model that sets the state of the art for video question answering. We apply attribute-aware object-centric features acquired by a video frame parser (TVQA+). We also include a recent model that incorporates heterogeneous memory with multimodal attention work (Memory) (Fan et al., 2019) that achieves superior performance on several datasets.
+
+<table><tr><td rowspan="2">Methods</td><td rowspan="2">Descriptive</td><td colspan="2">Explanatory</td><td colspan="2">Predictive</td><td colspan="2">Counterfactual</td></tr><tr><td>per opt.</td><td>per ques.</td><td>per opt.</td><td>per ques.</td><td>per opt.</td><td>per ques.</td></tr><tr><td>Q-type (random)</td><td>29.2</td><td>50.1</td><td>8.1</td><td>50.7</td><td>25.5</td><td>50.1</td><td>10.3</td></tr><tr><td>Q-type (frequent)</td><td>33.0</td><td>50.2</td><td>16.5</td><td>50.0</td><td>0.0</td><td>50.2</td><td>1.0</td></tr><tr><td>LSTM</td><td>34.7</td><td>59.7</td><td>13.6</td><td>50.6</td><td>23.2</td><td>53.8</td><td>3.1</td></tr><tr><td>CNN+MLP</td><td>48.4</td><td>54.9</td><td>18.3</td><td>50.5</td><td>13.2</td><td>55.2</td><td>9.0</td></tr><tr><td>CNN+LSTM</td><td>51.8</td><td>62.0</td><td>17.5</td><td>57.9</td><td>31.6</td><td>61.2</td><td>14.7</td></tr><tr><td>TVQA+</td><td>72.0</td><td>63.3</td><td>23.7</td><td>70.3</td><td>48.9</td><td>53.9</td><td>4.1</td></tr><tr><td>Memory</td><td>54.7</td><td>53.7</td><td>13.9</td><td>50.0</td><td>33.1</td><td>54.2</td><td>7.0</td></tr><tr><td>IEP (V)</td><td>52.8</td><td>52.6</td><td>14.5</td><td>50.0</td><td>9.7</td><td>53.4</td><td>3.8</td></tr><tr><td>TbD-net (V)</td><td>79.5</td><td>61.6</td><td>3.8</td><td>50.3</td><td>6.5</td><td>56.1</td><td>4.4</td></tr><tr><td>MAC (V)</td><td>85.6</td><td>59.5</td><td>12.5</td><td>51.0</td><td>16.5</td><td>54.6</td><td>13.7</td></tr><tr><td>MAC (V+)</td><td>86.4</td><td>70.5</td><td>22.3</td><td>59.7</td><td>42.9</td><td>63.5</td><td>25.1</td></tr></table>
+
+Table 2: Question-answering accuracy of visual reasoning baselines on CLEVRER. All models are trained on the full training set. The IEP (V) model and TbD-net (V) use 1000 programs to train the program generator.
+
+Compositional visual reasoning. The CLEVR dataset (Johnson et al., 2017a) opened up a new direction of compositional visual reasoning, which emphasizes complexity and compositionality in the logic and visual context. We modify several best-performing models and apply them to our video benchmark. The IEP model (Johnson et al., 2017b) applies neural program execution for visual reasoning on images. We apply the same approach to our program-based video reasoning task (IEP (V)) by substituting the program primitives by the ones from CLEVRER, and applying the execution modules on the video features extracted by a convolutional LSTM (Shi et al., 2015). TbD-net (Mascharka et al., 2018) follows a similar approach by parsing the input question into a program, which is then assembled into a neural network that acts on the attention map over the image features. The final attended image feature is then sent to an output layer for classification. We adopt the same approach through spatial-temporal attention over the video feature space (TbD-net (V)). MAC (Hudson & Manning, 2018) incorporates a joint attention mechanism on both the image feature map and the question, which leads to strong performance on CLEVR without program supervision. We modify the model by applying a temporal attention unit across the video frames to generate a latent encoding for the video (MAC (V)). The video feature is then input to the MAC network to output an answer distribution. We study an augmented approach: we construct object-aware video features by adding the segmentation masks of all objects in the frames and labeling them by the values of their intrinsic attributes (MAC  $(\mathbf{V} + )$  ).
+
+Implementation details. We use a pre-trained ResNet-50 (He et al., 2016) to extract features from the video frames. We use the 2,048-dimensional pool5 layer output for CNN-based methods, and the  $14 \times 14$  feature maps for MAC, IEP and TbD-net. The program generators of IEP and TbD-net are trained on 1000 programs. We uniformly sample 25 frames for each video as input. Object segmentation masks and attributes are obtained by a video parser consisted of an object detector and an attribute network.
+
+# 4.2 RESULTS
+
+We summarize the performances of all baseline models in Table 2. The fact that these models achieve different performances over the wide spectrum of tasks suggest that CLEVRR offers powerful assessment to the models' strength and limitations on various domains. All models are trained on the training set until convergence, tuned on the validation set and evaluated on the test set.
+
+Evaluation metrics. For descriptive questions, we calculate the accuracy by comparing the predicted answer token to the ground-truth. For multiple choice questions, we adopt two metrics: per-option accuracy measures the model's overall correctness on single options across all questions; per-question accuracy measures the correctness of the full question, requiring all choices to be selected correctly.
+
+Descriptive reasoning. Descriptive questions query the content of the video from various aspects. In order to do well on this question type, a model needs to both accurately recognize the objects and events that happen in the video, as well as understanding the compositional logic pattern behind the questions. In other words, descriptive questions require strong perception and logic operations on both visual and language signals. As shown in Table 2, the LSTM baseline that relies only on question input performs poorly on the descriptive questions, only outperforming the random baselines
+
+![](images/5d58729110141baac47dd86efc7abd5ec294b840f18b2e24406fa754cc1853c9.jpg)  
+Figure 4: Our model includes four components: a video frame parser that generates an object-based representation of the video frames; a question parser that turns a question into a functional program; a dynamics predictor that extracts and predicts the dynamic scene of the video; and a symbolic program executor that runs the program on the dynamic scene to obtain an answer.
+
+by a small margin. This suggests that CLEVRR has very small bias on the questions. Video QA models, including the state of the art model TVQA+ (Lei et al., 2018) achieve better performances. But because of their limited capability of handling the compositionality in the question logic and visual context, these models are still unable to thrive on the task. In contrast, models designed for compositional reasoning, including TbD-net that operates on neural program execution and MAC that introduces a joint attention mechanism, are able to achieve more competitive performances.
+
+Causal reasoning. Results on the descriptive questions demonstrate the power of models that combine visual and language perception with compositional logic operations. However, the causal reasoning tasks (explanatory, predictive, counterfactual) require further understanding beyond perception. Our evaluation results (Table 2) show poor performance of most baseline models on these questions. The compositional reasoning models that performs well on the descriptive questions (MAC (V) and TbD-net (V)) only achieve marginal improvements over the random and language-only baselines on the causal tasks. However, we do notice a reasonable gain in performance on models that inputs object-aware representations: TVQA+ achieves high accuracy on the predictive questions, and MAC  $(\mathrm{V}+)$  improves upon MAC (V) on all tasks.
+
+We identify the following messages suggested by our evaluation results. First, object-centric representations are essential for the causal reasoning tasks. This is supported by the large improvement on MAC  $(\mathrm{V}+)$  over MAC (V) after using features that are aware of object instances and attributes, as well as the strong performance of TVQA+ on the predictive questions. Second, all baseline models lack a component to explicitly model the dynamics of the objects and the causal relations between the collision events. As a result, they struggle on the tasks involving unobserved scenes and in particular performs poorly on the counterfactual tasks. The combination of object-centric representation and dynamics modeling therefore suggests a promising direction for approaching the causal tasks.
+
+# 5 NEURO-SYMBOLIC DYNAMIC REASONING
+
+Baseline evaluations on CLEVRER have revealed two key elements that are essential to causal reasoning: an object-centric video representation that is aware of the temporal and causal relations between the objects and events; and a dynamics model able to predict the object dynamics under unobserved or counterfactual scenarios. However, unifying these elements with video and language understanding posts the following challenges: first, all the disjoint model components should operate on a common set of representations of the video, question, dynamics and causal relations; second, the representation should be aware of the compositional relations between the objects and events. We draw inspirations from Yi et al. (2018) and study an oracle model that operates on a symbolic representation to join video perception, language understanding with dynamics modeling. Our model Neuro-Symbolic Dynamic Reasoning (NS-DR) combines neural nets for pattern recognition and dynamics prediction, and symbolic logic for causal reasoning. As shown in Figure 4, NS-DR consists of a video frame parser (Figure 4-I), a neural dynamics predictor (Figure 4-II), a question parser (Figure 4-III), and a program executor (Figure 4-IV). We present details of the model components below and in supplementary material B.
+
+Video frame parser. The video frame parser serves as a perception module that is disentangled from other components of the model, from which we obtain an object-centric representation of the video frames. The parser is a Mask R-CNN (He et al., 2017) that performs object detection and scene de-rendering on each video frame (Wu et al., 2017). We use a ResNet-50 FPN (Lin et al., 2017; He et al., 2016) as the backbone. For each input video frame, the network outputs the object mask proposals, a label that corresponds to the intrinsic attributes (color, material, shape) and a score representing the confidence level of the proposal used for filtering. Please refer to He et al. (He et al., 2017) for more details. The video parser is trained on 4000 video frames randomly sampled from the training set with object masks and attribute annotations.
+
+Neural dynamics predictor. We apply the Propagation Network (PropNet) (Li et al., 2019) for dynamics modeling, which is a learnable physics engine that, extending from the interaction networks (Battaglia et al., 2016), performs object- and relation-centric updates on a dynamic scene. The model inputs the object proposals from the video parser, and learns the dynamics of the objects across the frames for predicting motion traces and collision events. PropNet represents the dynamical system as a directed graph,  $G = \langle O, R \rangle$ , where the vertices  $O = \{o_i\}$  represent objects and edges  $R = \{r_k\}$  represent relations. Each object (vertex)  $o_i$  and relation (edge)  $r_k$  can be further written as  $o_i = \langle s_i, a_i^o \rangle$ ,  $r_k = \langle u_k, v_k, a_k^r \rangle$ , where  $s_i$  is the state of object  $i$ ;  $a_i^o$  denotes its intrinsic attributes;  $u_k, v_k$  are integers denoting the index of the receiver and sender vertices joined by edge  $r_k$ ;  $a_k^r$  represents the state of edge  $r_k$ , indicating whether there is collision between the two objects. In our case,  $s_i$  is a concatenation of tuple  $\langle c_i, m_i, p_i \rangle$  over a small history window to encode motion history, where  $c_i$  and  $m_i$  are the corresponding image and mask patches cropped at  $p_i$ , which is the  $x-y$  position of the mask in the original image (please see the cyan metal cube in Figure 4 for an example). PropNet handles the instantaneous propagation of effects via multi-step message passing. Please refer to supplementary material B for more details.
+
+Question parser. We use an attention-based seq2seq model (Bahdanau et al., 2015) to parse the input questions into their corresponding functional programs. The model consists of a bidirectional LSTM encoder plus an LSTM decoder with attention, similar to the question parser in (Yi et al., 2018). For multiple choice questions, we use two networks to parse the questions and choices separately. Given an input word sequence  $(x_{1}, x_{2}, \ldots, x_{I})$ , the encoder first generates a bi-directional latent encoding at each step:
+
+$$
+e _ {i} ^ {f}, h _ {i} ^ {f} = \operatorname {L S T M} \left(\Phi_ {I} \left(x _ {i}\right), h _ {i - 1} ^ {f}\right), \tag {1}
+$$
+
+$$
+e _ {i} ^ {b}, h _ {i} ^ {b} = \operatorname {L S T M} \left(\Phi_ {I} \left(x _ {i}\right), h _ {i + 1} ^ {b}\right), \tag {2}
+$$
+
+$$
+e _ {i} = \left[ e _ {i} ^ {f}, e _ {i} ^ {b} \right]. \tag {3}
+$$
+
+The decoder then generates a sequence of program tokens  $(y_{1},y_{2},\dots,y_{J})$  from the latent encodings using attention:
+
+$$
+v _ {j} = \operatorname {L S T M} \left(\Phi_ {O} \left(y _ {j - 1}\right)\right), \tag {4}
+$$
+
+$$
+\alpha_ {j i} \propto \exp \left(v _ {j} ^ {T} e _ {i}\right), \quad c _ {j} = \sum_ {i} \alpha_ {j i} e _ {i}, \tag {5}
+$$
+
+$$
+\hat {y} _ {j} \sim \operatorname {s o f t m a x} \left(W \cdot \left[ q _ {j}, c _ {j} \right]\right). \tag {6}
+$$
+
+At training time, the input program tokens  $\{y_{j}\}$  are used for generating the predicted labels  $y_{j-1} \rightarrow \hat{y}_{j}$  at each step  $j = 1,2,3,\ldots,J$ . The generated label  $\hat{y}_{j}$  is then compared with  $y_{j}$  to compute a loss for training. At test time, the decoder rolls out by feeding the sampled prediction at the previous step to the input of the current step  $\hat{y}_{j} = y_{j}$ . The roll-out stops when the end token appears or when the sequence reaches certain maximal length. For both the encoder and decoder LSTM, we use the same parameter setup of two hidden layers with 256 units, and a 300-dimensional word embedding vector.
+
+Program executor. The program executor explicitly executes the program on the motion and event traces extracted by the dynamics predictor and output an answer to the question. It consists of a collection of functional modules implemented in Python. Given an input program, the executor first assembles the modules and then iterates through the program tree. The output of the final module is the answer to the target question. There are three types of program modules: input module, filter module, and output module. The input modules are the entry points of the program trees and can directly query the output of the dynamics predictor. For example, the Events and Objects modules that commonly appear in the programs of descriptive questions indicate inputting all the observed events and objects from a video. The filter modules perform logic operations on the input objects/events
+
+<table><tr><td rowspan="2">Methods</td><td rowspan="2">Descriptive</td><td colspan="2">Explanatory</td><td colspan="2">Predictive</td><td colspan="2">Counterfactual</td></tr><tr><td>per opt.</td><td>per ques.</td><td>per opt.</td><td>per ques.</td><td>per opt.</td><td>per ques.</td></tr><tr><td>NS-DR</td><td>88.1</td><td>87.6</td><td>79.6</td><td>82.9</td><td>68.7</td><td>74.1</td><td>42.2</td></tr><tr><td>NS-DR (NE)</td><td>85.8</td><td>85.9</td><td>74.3</td><td>75.4</td><td>54.1</td><td>76.1</td><td>42.0</td></tr></table>
+
+Table 3: Quantitative results of NS-DR on CLEVRER. We evaluate our model on all four question types. We also study a variation of our model NS-DR (NE) with "no events" but only motion traces from the dynamics predictor. Our question parser is trained with 1000 programs.
+
+based on a designated intrinsic attribute, motion state, temporal order, or causal relation. The output modules return the answer label. For open-ended descriptive questions, the executor will return a token from the answer space. For multiple choice questions, the executor will first execute the choice program and then send the result to the question program to output a yes/no token. A comprehensive list of all modules in the NS-DR program executor is presented in supplementary material B.
+
+Results. We summarize the performance of NS-DR on CLEVRER in Table 3. On descriptive questions, our model achieves an  $88.1\%$  accuracy when the question parser is trained under 1,000 programs, outperforming other baseline methods. On explanatory, predictive, and counterfactual questions, our model achieves a more significant gain. We also study a variation of our model, NS-DR (No Events/NE), which uses the dynamics predictor to generate only the predictive and counterfactual motion traces, and collision events are identified by the velocity change of the colliding objects. NS-DR (NE) performs comparably to the full model, showing that our disentangled system is adaptable to alternative methods for dynamics modeling. We also conduct ablation study on the number of programs for training the question parser. As shown in Figure 5, NS-DR reaches full capability at 1,000 programs for all question types.
+
+We highlight the following contributions of the model. First, NS-DR incorporates a dynamics planner into the visual reasoning task, which directly enables predictions of unobserved motion and events, and enables the model for the predictive and counterfactual tasks. This suggests that dynamics planning has great potential for language-grounded visual reasoning tasks and NS-DR takes a preliminary step towards this direction. Second, symbolic representation provides a powerful common ground for vision, language, dynamics and causality. By design, it empowers the model to explicitly capture the compositionality behind the video's causal structure and the question logic.
+
+We further discuss limitations of NS-DR and suggest possible directions for future research. First, training of the video and question parser relies on extra supervisions such as object masks, attributes, and question programs. Even though the amount of data required for training is minimal compared to end-to-end approaches (i.e. thousands of annotated frames and programs), these data is hard to acquire in real-world applications. This constraint could be relaxed by applying unsupervised/weakly-supervised methods for scene decomposition and concept discovery (Burgess et al., 2019; Mao et al., 2019). Second, our model performance decreases on tasks that require long-term dynamics prediction such as the
+
+counterfactual questions. This suggests that we need a better dynamics model capable of generating more stable and accurate trajectories. CLEVRR provides a benchmark for assessing the predictive power of such dynamics models.
+
+![](images/aafd27e83313ef0839272219e52bab7629bbdc3baa6e54ea05788e921cbf6d26.jpg)  
+Figure 5: Performance of our model under different number of programs used for training the question parser.
+
+![](images/7deb95d6908389c2a2bc044e6f625ca4655735c282171ae52b42d37bca2a422a.jpg)
+
+# 6 CONCLUSION
+
+We present a systematic study of temporal and causal reasoning in videos. This profound and challenging problem deeply rooted to the fundamentals of human intelligence has just begun to be studied with 'modern' AI tools. We introduce a set of benchmark tasks to better facilitate the research in this area. We also believe video understanding and reasoning should go beyond passive knowledge extraction, and focus on building an internal understanding of the dynamics and causal relations,
+
+which is essential for practical applications such as dynamic robot manipulation under complex causal conditions. Our newly introduced CLEVRER dataset and the NS-DR model are preliminary steps toward this direction. We hope that with recent advances in graph networks, visual predictive models, and neuro-symbolic algorithms, the deep learning community can now revisit this classic problem in more realistic setups in the future, capturing true intelligence beyond pattern recognition.
+
+Acknowledgment. We thank Jiayuan Mao for helpful discussions and suggestions. This work is in part supported by ONR MURI N00014-16-1-2007, the Center for Brain, Minds, and Machines (CBMM, funded by NSF STC award CCF-1231216), and IBM Research.
+
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+Xingjian Shi, Zhourong Chen, Hao Wang, Dit-Yan Yeung, Wai-Kin Wong, and Wang-chun Woo. Convolutional LSTM network: A machine learning approach for precipitation nowcasting. In NeurIPS, 2015. 6  
+Elizabeth S. Spelke. Core Knowledge. Am. Psychol., 55(11):1233, 2000. 1  
+Joseph Suarez, Justin Johnson, and Fei-Fei Li. DDRprog: A Clevr Differentiable Dynamic Reasoning Programmer. arXiv:1803.11361, 2018. 3  
+Makarand Tapaswi, Yukun Zhu, Rainer Stiefelhagen, Antonio Torralba, Raquel Urtasun, and Sanja Fidler. MovieQA: Understanding Stories in Movies through Question-Answering. In CVPR, 2016. 1, 2, 3  
+Tomer David Ullman. On the Nature and Origin of Intuitive Theories: Learning, Physics and Psychology. PhD thesis, Massachusetts Institute of Technology, 2015. 1  
+Subhashini Venugopalan, Marcus Rohrbach, Jeffrey Donahue, Raymond Mooney, Trevor Darrell, and Kate Saenko. Sequence to sequence-video to text. In ICCV, 2015. 2  
+Nicholas Watters, Andrea Tacchetti, Theophane Weber, Razvan Pascanu, Peter Battaglia, and Daniel Zoran. Visual Interaction Networks: Learning a Physics Simulator from Video. In NeurIPS, 2017. 3  
+Jiajun Wu, Joshua B. Tenenbaum, and Pushmeet Kohli. Neural Scene De-rendering. In CVPR, 2017. 8
+
+Guangyu Robert Yang, Igor Ganichev, Xiao-Jing Wang, Jonathon Shlens, and David Sussillo. A Dataset and Architecture for Visual Reasoning with a Working Memory. In ECCV, 2018. 2, 3  
+Zichao Yang, Xiaodong He, Jianfeng Gao, Li Deng, and Alex Smola. Stacked Attention Networks for Image Question Answering. In CVPR, 2016. 3  
+Tian Ye, Xiaolong Wang, James Davidson, and Abhinav Gupta. Interpretable intuitive physics model. In ECCV, 2018. 3  
+Kexin Yi, Jiajun Wu, Chuang Gan, Antonio Torralba, Pushmeet Kohli, and Joshua B. Tenenbaum. Neural-Symbolic VQA: Disentangling Reasoning from Vision and Language Understanding. In NeurIPS, 2018. 3, 7, 8  
+Amir Zadeh, Michael Chan, Paul Pu Liang, Edmund Tong, and Louis-Philippe Morency. Social-IQ: A question answering benchmark for artificial social intelligence. In CVPR, 2019. 1, 2, 3  
+Rowan Zellers, Yonatan Bisk, Ali Farhadi, and Yejin Choi. From recognition to cognition: Visual commonsense reasoning. In CVPR, 2019. 2, 3  
+Chen Zhu, Yanpeng Zhao, Shuaiyi Huang, Kewei Tu, and Yi Ma. Structured Attentions for Visual Question Answering. In ICCV, 2017. 3  
+Yuke Zhu, Oliver Groth, Michael Bernstein, and Li Fei-Fei. Visual7w: Grounded question answering in images. In CVPR, 2016. 1
+
+# SUPPLEMENTARY MATERIALS
+
+# A DESCRIPTIVE QUESTION SUB-TYPES AND STATISTICS
+
+Descriptive questions in CLEVRER consists of five sub-types. When generating the questions, we balance the frequency of the answers within each question sub-type to reduce bias. The sub-types and their answer space distribution are shown in Figure
+
+<table><tr><td>Sub-type</td><td>Answers</td></tr><tr><td>Count</td><td>0, 1, 2, 3, 4, 5</td></tr><tr><td>Exist</td><td>yes, no</td></tr><tr><td>Query_material</td><td>rubber, metal</td></tr><tr><td>Query_shape</td><td>sphere, cube, cylinder</td></tr><tr><td>Query_color</td><td>gray, brown, green, red, blue, purple, yellow, cyan</td></tr></table>
+
+![](images/45d757998b5657900e6b1a56ad0e325c3bcc32180fd9c27d2d8897b518c030d1.jpg)  
+Figure 6: Descriptive question sub-type and answer space statistics.
+
+# B NS-DR MODEL DETAILS AND TRAINING PARADIGM
+
+Here we present further details of the NS-DR model and the training paradigm. A running example of the model can be found in Figure 7.
+
+Video frame parser. Our frame parser is trained on 4,000 frames randomly selected from the training videos plus ground-truth masks and attribute annotations of each object in the frames. We train the model for 30,000 iterations with stochastic gradient decent, using a batch size of 6 and learning rate 0.001. At test time, we keep the object proposals with a confidence score of  $>0.9$  and use those to form the object-level abstract representation of the video.
+
+Neural dynamics predictor. The object encoder  $f_{O}^{\mathrm{enc}}$  and the relation encoder  $f_{R}^{\mathrm{enc}}$  in PropNet are instantiated as convolutional neural networks and output a  $D_{\mathrm{enc}}$ -dim vector as the representation. We add skip connections between the object encoder  $f_{O}^{\mathrm{enc}}$  and the predictor  $f_{O}^{\mathrm{pred}}$  for each object to generate higher quality images. We also include a relation predictor  $f_{R}^{\mathrm{pred}}$  to determine whether two objects will collide or not in the next time step.
+
+At time  $t$ , we first encodes the objects and relations  $c_{i,t}^{o} = f_{O}^{\mathrm{enc}}(o_{i,t}), c_{k,t}^{r} = f_{R}^{\mathrm{enc}}(o_{u_{k},t}, o_{v_{k},t}, a_{k}^{r})$  where  $o_{i,t}$  denotes object  $i$  at time  $t$ . We then denote the propagating effect from relation  $k$  at propagation step  $l$  as  $e_{k,t}^{l}$ , and the effect from object  $i$  as  $h_{i,t}^{l}$ . For step  $1 \leq l \leq L$ , the propagation procedure can be described as
+
+Step 0:
+
+$$
+h _ {i, t} ^ {0} = \mathbf {0}, \quad i = 1 \dots | O |, \tag {7}
+$$
+
+Step  $l = 1,\dots ,L$
+
+$$
+e _ {k, t} ^ {l} = f _ {R} \left(c _ {k, t} ^ {r}, h _ {u _ {k}, t} ^ {l - 1}, h _ {v _ {k}, t} ^ {l - 1}\right), \quad k = 1 \dots | R |,
+$$
+
+$$
+h _ {i, t} ^ {l} = f _ {O} \left(c _ {i, t} ^ {o}, \sum_ {k \in \mathcal {N} _ {i}} e _ {k, t} ^ {l}, h _ {i, t} ^ {l - 1}\right), \quad i = 1 \dots | O |, \tag {8}
+$$
+
+Output:
+
+$$
+\hat {r} _ {k, t + 1} = f _ {R} ^ {\text {p r e d}} \left(c _ {k, t} ^ {r}, e _ {k, t} ^ {L}\right), \quad k = 1 \dots | R |, \tag {9}
+$$
+
+$$
+\hat {o} _ {i, t + 1} = f _ {O} ^ {\mathrm {p r e d}} \left(c _ {i, t} ^ {o}, h _ {i, t} ^ {L}\right), \quad i = 1 \dots | O |
+$$
+
+where  $f_{O}$  denotes the object propagator,  $f_{R}$  denotes the relation propagator, and  $\mathcal{N}_i$  denotes the relation set where object  $i$  is the receiver. The neural dynamics predictor is trained by minimizing the  $\mathcal{L}_2$  distance between the predicted  $\hat{r}_{k,t+1}, \hat{o}_{i,t+1}$  and the real future observation  $r_{k,t+1}, o_{i,t+1}$  using stochastic gradient descent.
+
+![](images/2150838fa07b1dcdd961c50506b71b02981ec4bc57076e6b05aa8bc63fa5c4e1.jpg)  
+Figure 7: Sample results of NS-DR on CLEVRER. 'Pred.' and 'CF.' indicate predictive and counterfactual events extracted by the model's dynamics predictor. The counterfactual condition shown in this example is to remove the cyan cylinder.
+
+![](images/e2e566258a6a52979fd1b0d13afa829d78c1ac20989aa4785fd867e4595e5f36.jpg)
+
+![](images/0b68f520dde41ecdf460819570bc7e55f9b5fd5d6d1caf66ed90a5b303b652fe.jpg)
+
+![](images/87c21b47099fe90162af566b17816f8cc76b41cf2f758c978cfdb28877163165.jpg)
+
+<table><tr><td colspan="6">Objects</td></tr><tr><td>ID</td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td></tr><tr><td>Color</td><td>Cyan</td><td>Gray</td><td>Yellow</td><td>Red</td><td>Red</td></tr><tr><td>Material</td><td>Rubber</td><td>Metal</td><td>Rubber</td><td>Rubber</td><td>Metal</td></tr><tr><td>Shape</td><td>Cylinder</td><td>Cylinder</td><td>Sphere</td><td>Sphere</td><td>Sphere</td></tr></table>
+
+<table><tr><td colspan="6">Events</td></tr><tr><td>Mode</td><td colspan="3">Observation</td><td>Pred.</td><td>CF.</td></tr><tr><td>Frame</td><td>50</td><td>65</td><td>70</td><td>155</td><td>70</td></tr><tr><td>Type</td><td>Collision</td><td>Enter</td><td>Collision</td><td>Collision</td><td>Collision</td></tr><tr><td>Object ID</td><td>1,4</td><td>5</td><td>1,2</td><td>4,5</td><td>2,4</td></tr></table>
+
+Question: What shape is the first object to collide with the cyan object?
+
+Program: query_shape(get_colpartner过滤器_order过滤器_collision(Events, filter_color(Objects,Cyan)), First),filter_color(Objects,Cyan)))
+
+Answer: Sphere
+
+Question: Which event will happen next? Choice: The red rubber sphere collides with the metal sphere
+
+Program: belong_to滤errollision滤errollion(Events, filter_shape过滤材料过滤color( Objects, Red), Rubber), Sphere)), filter_shape过滤材料(Objects, Metal), Sphere)), UnseenEvents)
+
+Answer:
+
+Question: Which event will happen without the cyan cylinder? Choice: The red rubber sphere collides with the yellow sphere
+
+Program: belong_to.filter cjssion.filter_coiis(Events, filter_shape.filter_material(filters color(Objects, Red), Rubber, Sphere)), filter_shape.filter_color(Objects, Yellow), Sphere)), get counterfact( filter_shape-filter_color(Objects, Cyan), Cylinder)))
+
+Answer: X
+
+The output of the neural dynamics predictor is  $\langle \{\hat{O}_t\}_{t = 1\dots T},\{\hat{R}_t\}_{t = 1\dots T}\rangle$ , the collection of object states and relations across all the observed and rollout frames. From this representation, one can recover the full motion and event traces of the objects under different rollout conditions and generate a dynamic scene representation of the video. For example, if we want to know what will happen if an object is removed from the scene, we just need to erase the corresponding vertex and associated edges from the graph and rollout using the learned dynamics to obtain the traces.
+
+Our neural dynamics predictor is trained on the proposed object masks and attributes from the frame parser. We filter out inconsistent object proposals by matching the object intrinsic attributes across different frames and keeping the ones that appear in more than 10 frames. For a video of length 125, we will sample 5 rollouts, where each rollout contains 25 frames that are uniformly sampled from the original video. We normalize the input data to the range of  $[-1,1]$  and concatenate them over a time window of size 3. We set the propagation step  $L$  to 2, and the dimension of the propagating effects  $D_{\mathrm{enc}}$  to 512. We use the Adam optimizer (Kingma & Ba, 2015) with an initial learning rate of  $10^{-4}$ , and a decay factor of 0.3 per 3 epochs. The model is trained for 9 epochs with batch size 2.
+
+Question parser. For open-ended descriptive questions, the question parser is trained on various numbers of randomly selected question-program pairs. For multiple choice questions, we separately train the question parser and the choice parser. Training with  $n$  samples means training the question parser with  $n$  questions randomly sampled from the multiple choice questions (all three types together), and then training the choice parser with  $4n$  choices sampled from the same pool. All models are trained using Adam (Kingma & Ba, 2015) for 30,000 iterations with batch size 64 and learning rate  $7 \times 10^{-4}$ .
+
+Program executor. A comprehensive list of all modules in the NS-DR program executor is summarized in Table 4 and Table 5. The input and output data types of each module are summarized in Table 6.
+
+# C EXTRA EXAMPLES FROM CLEVRRER
+
+We show extra examples from CLEVRER in Figure 8. We also present visualizations of question/choice programs of different question types in Figure 9 through Figure 11. The full dataset will be made available for download.
+
+![](images/099262afcc6a38ac41b3cb8e4973d47cc4ea751e4b176bd08931aeb34d67fc79.jpg)
+
+![](images/078efad96fe77c4beb86c935243675630c27d3db4c05309e6244ab299ad0af63.jpg)
+
+![](images/c6230ee6c1e1ee8fc3cf38991feb3dae8923e38cb047a3fceceb7f35d38473c9.jpg)
+
+![](images/5dea76bae722ac5104654a877b273dd3c2020d947a44bafa1d6aab51356baf4a.jpg)
+
+# Descriptive
+
+Q1: What shape is the first object to collide with the gray sphere?
+
+Q2: How many collisions happen?
+
+Q3: Are there any metal spheres that enter the scene before the brown object enters the scene?
+
+A1: sphere
+
+A2:2
+
+A3: no
+
+# Explanatory
+
+Q: Which of the following is not responsible for the collision between the gray sphere and the brown object?
+
+a) The collision between the gray sphere and the green sphere
+
+b) The presence of the green rubber sphere
+
+c) The presence of the cyan metal sphere
+
+A: c)
+
+# Predictive
+
+Q: Which event will happen next  
+a) The gray sphere and the cube collide  
+b) The cube collides with the cylinder
+
+# Counterfactual
+
+Q: What will happen without the gray sphere?  
+a) The cube collides with the cyan sphere  
+b) The green sphere and the cyan sphere collide  
+c) The brown object and the cyan sphere collide  
+d) The brown object collides with the cube
+
+A: a)
+
+A: d)
+
+![](images/aea0d959136002533e2d9c0e278a08960112787add1b07b1014b48166c302434.jpg)  
+Figure 8: Sample videos and questions from CLEVRR. Stroboscopic imaging is applied for motion visualization.
+
+![](images/c24d78d8562083e27ebe063817e9b04fa5eb4855b4290b96cd8dc78778a3f3d1.jpg)
+
+![](images/2acb1f44873e36a1defc9c45ac2f9fa837ba264e2a6af40f48d429a6a5d36c0e.jpg)
+
+![](images/9fa6d7eab4f138ed5c08afd98863f5396e8a695b6608675c952f3a93542e4fe3.jpg)
+
+# Descriptive
+
+Q1: Are there any moving red objects when the video ends?  
+Q2: What color is the object that enters the scene?  
+Q3: What is the color of the first object to collide with the rubber?
+
+sphere?
+
+A1: no
+
+A2:purple
+
+A3:green
+
+# Predictive
+
+Q: Which event will happen next  
+a) The green cylinder and the sphere collide  
+b) The green cylinder collides with the cube
+
+A: b)
+
+# Explanatory
+
+Q: Which of the following is responsible for the sphere's exiting the scene?
+
+a) The presence of the cyan object
+
+b) The presence of the purple rubber cube
+
+c) The presence of the green metal cylinder
+
+d) The collision between the sphere and the cyan cylinder
+
+A:a),c),d)
+
+# Counterfactual
+
+Q: Without the green cylinder, what will not happen?  
+a) The sphere and the cube collide  
+b) The sphere and the cyan cylinder collide  
+c) The cube and the cyan cylinder collide
+
+A: b), c)
+
+![](images/555ac7c53edede024ed800de9888a99798ab80a1dcfe5d23fce7c993af101b59.jpg)
+
+![](images/9415a4c8a8f09aee882fab027b43cbd1e2a4c80e72fe489f9dcd1109d52d3ea8.jpg)
+
+![](images/4a15b8923b46eb75e8d5a41005eb8533c74b481b2d24bb24db55b9361dd42580.jpg)
+
+![](images/aa773cabeb8cf4467e27265b507224b9ee2b07fbd2d987be48e3a545015b43f2.jpg)
+
+# Descriptive
+
+Q1: How many blue objects enter the scene?
+
+A1:2
+
+Q2: What shape is the last object to collide with the cylinder?
+
+A2: sphere
+
+Q3: What color is the object that is stationary when the video ends?
+
+A3: blue
+
+# Explanatory
+
+Q: Which of the following is responsible for the collision between the cylinder
+
+and the metal object?
+
+a) The cube's entering the scene
+
+b) The presence of the blue rubber cube
+
+c) The cube's colliding with the  $c$
+
+d) The presence of the blue rubber sphere
+
+A:a),b),c)
+
+# Predictive
+
+Q: Which event will happen next  
+a) The cube and the blue sphere collide  
+b) The metal object collides with the blue sphere
+
+A: a)
+
+# Counterfactual
+
+Q: What will happen if the metal object is removed?  
+a) The cylinder and the blue sphere collide  
+b) The cube collides with the blue sphere  
+c) The cube and the cylinder collide A: b), c)
+
+<table><tr><td>Module Type</td><td>Name / Description</td><td>Input Type</td><td>Output Type</td></tr><tr><td rowspan="12">Input Modules</td><td>Objects</td><td>-</td><td>objects</td></tr><tr><td>Returns all objects in the video</td><td></td><td></td></tr><tr><td>Events</td><td>-</td><td>events</td></tr><tr><td>Returns all events that happen in the video</td><td></td><td></td></tr><tr><td>UnseenEvents</td><td>-</td><td>events</td></tr><tr><td>Returns all future events after the video ends</td><td></td><td></td></tr><tr><td>AllEvents</td><td>-</td><td>events</td></tr><tr><td>Returns all possible events on / between any objects</td><td></td><td></td></tr><tr><td>Start</td><td>-</td><td>event</td></tr><tr><td>Returns the special “start” event</td><td></td><td></td></tr><tr><td>End</td><td>-</td><td>event</td></tr><tr><td>Returns the special “end” event</td><td></td><td></td></tr><tr><td rowspan="12">Object Filter Modules</td><td>Filter_color</td><td>(objects, color)</td><td>objects</td></tr><tr><td>Selects objects from the input list with the input color</td><td></td><td></td></tr><tr><td>Filter_material</td><td>(objects, material)</td><td>objects</td></tr><tr><td>Selects objects from the input list with the input material</td><td></td><td></td></tr><tr><td>Filter_shape</td><td>(objects, shape)</td><td>objects</td></tr><tr><td>Selects objects from the input list with the input shape</td><td></td><td></td></tr><tr><td>Filter-moving</td><td>(objects, frame)</td><td>objects</td></tr><tr><td>Selects all moving objects in the input frame</td><td></td><td></td></tr><tr><td>or the entire video (when input frame is “null”)</td><td></td><td></td></tr><tr><td>Filter stationed</td><td>(objects, frame)</td><td>objects</td></tr><tr><td>Selects all stationary objects in the input frame</td><td></td><td></td></tr><tr><td>or the entire video (when input frame is “null”)</td><td></td><td></td></tr><tr><td rowspan="24">Event Filter Modules</td><td>Filter_in</td><td>(event, objects)</td><td>events</td></tr><tr><td>Selects all incoming events of the input objects</td><td></td><td></td></tr><tr><td>Filter_out</td><td>(event, objects)</td><td>events</td></tr><tr><td>Selects all exiting events of the input objects</td><td></td><td></td></tr><tr><td>Filter_collision</td><td>(event, objects)</td><td>events</td></tr><tr><td>Selects all collisions that involve any of the input objects</td><td></td><td></td></tr><tr><td>Filter_before</td><td>(event, event)</td><td>events</td></tr><tr><td>Selects all events before the target event</td><td></td><td></td></tr><tr><td>Filter_after</td><td>(event, event)</td><td>events</td></tr><tr><td>Selects all events after the target event</td><td></td><td></td></tr><tr><td>Filter_order</td><td>(event, order)</td><td>event</td></tr><tr><td>Selects the event at the specific time order</td><td></td><td></td></tr><tr><td>Filter_ancestor</td><td>(event, event)</td><td>events</td></tr><tr><td>Selects all causal ancestors of the input event</td><td></td><td></td></tr><tr><td>Get_frame</td><td>event</td><td>frame</td></tr><tr><td>Returns the frame of the input event in the video</td><td></td><td></td></tr><tr><td>Get(counterfact</td><td>(event, object)</td><td>events</td></tr><tr><td>Selects all events that would happen if the object is removed</td><td></td><td></td></tr><tr><td>Get_colpartner</td><td>(event, object)</td><td>object</td></tr><tr><td>Returns the collision partner of the input object</td><td></td><td></td></tr><tr><td>(the input event must be a collision)</td><td></td><td></td></tr><tr><td>Get_object</td><td>event</td><td>object</td></tr><tr><td>Returns the object that participates in the input event</td><td></td><td></td></tr><tr><td>(the input event must be a incoming / outgoing event)</td><td></td><td></td></tr><tr><td></td><td>Unique</td><td>events /</td><td>event /</td></tr><tr><td></td><td>Returns the only event / object in the input list</td><td>objects</td><td>object</td></tr></table>
+
+Table 4: Functional modules of NS-DR's program executor.
+
+<table><tr><td>Module Type</td><td>Name / Description</td><td>Input Type</td><td>Output Type</td></tr><tr><td rowspan="14">Output Modules</td><td>Query_color</td><td>object</td><td>color</td></tr><tr><td>Returns the color of the input object</td><td></td><td></td></tr><tr><td>Query_material</td><td>object</td><td>material</td></tr><tr><td>Returns the material of the input object</td><td></td><td></td></tr><tr><td>Query_shape</td><td>object</td><td>shape</td></tr><tr><td>Returns the shape of the input object</td><td></td><td></td></tr><tr><td>Count</td><td>objects</td><td>int</td></tr><tr><td>Returns the number of the input objects</td><td></td><td></td></tr><tr><td>Exist</td><td>objects</td><td>bool</td></tr><tr><td>Returns “yes” if the input objects is not empty</td><td></td><td></td></tr><tr><td>Belong_to</td><td>(event, events)</td><td>bool</td></tr><tr><td>Returns “yes” if the input event belongs to the input set of events</td><td></td><td></td></tr><tr><td>Negate</td><td>bool</td><td>bool</td></tr><tr><td>Returns the negation of the input boolean</td><td></td><td></td></tr></table>
+
+Table 5: Functional modules of NS-DR's program executor (continued).  
+
+<table><tr><td>Data type</td><td>Description</td></tr><tr><td>object</td><td>A dictionary storing the intrinsic attributes of an object in a video</td></tr><tr><td>objects</td><td>A list of objects</td></tr><tr><td>event</td><td>A dictionary storing the type, frame, and participating objects of an event</td></tr><tr><td>events</td><td>A list of events</td></tr><tr><td>order</td><td>A string indicating the chronological order of an event out of “first”, “second”, “last”</td></tr><tr><td>color</td><td>A string indicating a color out of “gray”, “brown”, “green”, “red”, “blue”, “purple”, “yellow”, “cyan”</td></tr><tr><td>material</td><td>A string indicating a material out of “metal”, “rubber”</td></tr><tr><td>shape</td><td>A string indicating a shape out of “cube”, “cylinder”, “sphere”</td></tr><tr><td>frame</td><td>An integer representing the frame number of an event</td></tr></table>
+
+Table 6: Input/output data types of modules in the program executor.
+
+```txt
+Question:"How many spheres are moving when the video ends?" Question:"What shape is the second object to collide with the green object?"   
+Count( Filter_move( Filter_shape(Objects, "Sphere"), Get_frame End) )   
+Question:"Are there any stationary spheres when the metal sphere enters the scene?" 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1) 1)
+```
+
+Figure 9: Example of descriptive question programs.
+
+Question: "Which of the following is not responsible for the metal sphere's colliding with the rubber sphere?"  
+```txt
+Negate(   
+Belong_to(   
+Event, // Output of the Choice program   
+Filter_ancestor(   
+Filter_collision(   
+Filter_collision(   
+Events,   
+Filter_shape( Filter_material(Objects, "Metal"), "Sphere" ) ),   
+Filter_shape(   
+Filter_material(Objects, "Rubber"), "Sphere" )   
+)   
+)
+```
+
+Choice: "The collision between the green cube and the rubber sphere"  
+```txt
+Filter cjssion(   
+Filter_cossion( Events,   
+Filter_shape( Filter_color(Objects，"Cube"), "Cube") 1   
+Filter_shape(   
+Filter_material(Objects，"Rubber"), "Sphere"   
+）
+```
+
+Choice: "The green cube's entering the scene"  
+```txt
+Filter_in( Filter_shape( Filter_color(Objects，"Green")， "Cube" ）   
+）
+```
+
+Figure 10: Example of explanatory question and choice programs.
+
+Counterfactual question: "Which event will happen if the rubber sphere is removed?"  
+```cmake
+Belong_to(
+Event, // Output of the choice program
+Get(counterfact(
+Filter_shape(
+Filter_material(Objects, "Rubber"),
+"Sphere"
+)
+)
+```
+
+Predictive question: "What will happen next?"  
+```txt
+Belong_to(
+Event, // Output of the choice program
+UnseenEvents
+```
+
+Choice: "The cube collides with the metal sphere"  
+```autoit
+Filter cjss( Filter_cssion( Filter_cssion( AllEvents, Filter_shape(Objects, "Cube")) Filter_shape( Filter_material(Objects, "Metal"), ) Sphere" Filter_shape(Objects, "Cylinder")
+```
+
+Figure 11: Example of predictive / counterfactual question and choice programs.
\ No newline at end of file
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+# COMPRESSION BASED BOUND FOR NON-COMPRESSED NETWORK: UNIFIED GENERALIZATION ERROR ANALYSIS OF LARGE COMPRESSIBLE DEEP NEURAL NETWORK
+
+Taiji Suzuki
+
+Graduate School of Information Science and Technology, The University of Tokyo, Japan  
+Center for Advanced Intelligence Project, RIKEN, Japan  
+Japan Digital Design  
+taiji@mist.i.u-tokyo.ac.jp
+
+Hiroshi Abe
+
+iPride Co., Ltd., Japan, abe@ipride.co.jp,
+
+Tomoaki Nishimura
+
+NTT Data Corporation, Japan, Tomoaki.Nishimura@nttdata.com
+
+# ABSTRACT
+
+One of the biggest issues in deep learning theory is the generalization ability of networks with huge model size. The classical learning theory suggests that overparameterized models cause overfitting. However, practically used large deep models avoid overfitting, which is not well explained by the classical approaches. To resolve this issue, several attempts have been made. Among them, the compression based bound is one of the promising approaches. However, the compression based bound can be applied only to a compressed network, and it is not applicable to the non-compressed original network. In this paper, we give a unified framework that can convert compression based bounds to those for non-compressed original networks. The bound gives even better rate than the one for the compressed network by improving the bias term. By establishing the unified framework, we can obtain a data dependent generalization error bound which gives a tighter evaluation than the data independent ones.
+
+# 1 INTRODUCTION
+
+Deep learning has shown quite successful results in wide range of machine learning applications. such as image recognition (Krizhevsky et al., 2012), natural language processing (Devlin et al., 2018) and image synthesis tasks (Radford et al., 2015). The success of deep learning methods is mainly due to its flexibility, expression power and computational efficiency for large dataset training. Due to its significant importance in wide range of application areas, its theoretical analysis is also getting much important. For example, it has been known that the deep neural network has universal approximation capability (Cybenko, 1989; Hornik, 1991; Sonoda & Murata, 2015) and its expressive power grows up in an exponential order against the number of layers (Montufar et al., 2014; Bianchini & Scarselli, 2014; Cohen et al., 2016; Cohen & Shashua, 2016; Poole et al., 2016; Suzuki, 2019). However, theoretical understandings are still lacking in several important issues.
+
+Among several topics of deep learning theories, a generalization error analysis is one of the biggest issues in the machine learning literature. An important property of deep learning is that it generalizes well even though its parameter size is quite large compared with the sample size (Neyshabur et al., 2019). This can not be well explained by a classical VC-dimension type theory (Harvey et al., 2017) which suggests that overparameterized models cause overfitting and thus result in poor generalization ability.
+
+For this purpose, norm based bounds have been extensively studied so far (Neyshabur et al., 2015; Bartlett et al., 2017b; Neyshabur et al., 2017; Golowich et al., 2018). These bounds are beneficial
+
+because the bounds are not explicitly dependent on the number of parameters and thus are useful to explain the generalization error of overparameterized network (Bartlett, 1998; Neyshabur et al., 2015; 2019). However, these bounds are typically exponentially dependent on the number of layers and thus tends to be loose for deep network situations (Dziugaite & Roy, 2017; Arora et al., 2018; Nagarajan & Kolter, 2019). As a result, Arora et al. (2018) reported that a simple VC-dimension bound (Li et al., 2018; Harvey et al., 2017) can still give sharper evaluations than these norm based bounds in some practically used deep networks. Wei & Ma (2019) improved this issue by involving a data dependent Lipschitz constant as performed in Arora et al. (2018); Nagarajan & Kolter (2019).
+
+On the other hand, compression based bound is another promising approach for tight generalization error evaluation which can avoid the exponential dependence on the depth. The complexity of deep neural network model is regulated from several aspects. For example, we usually impose explicit regularization such as weight decay (Krogh & Hertz, 1992), dropout (Srivastava et al., 2014; Wager et al., 2013), batch-normalization (Ioffe & Szegedy, 2015), and mix-up (Zhang et al., 2018; Verma et al., 2018). Zhang et al. (2016) reported that such explicit regularization does not have much effect but implicit regularization induced by SGD (Hardt et al., 2016; Gunasekar et al., 2018; Ji & Telgarsky, 2019) is important. Through these explicit and implicit regularizations, deep learning tends to produce a simpler model than its full expression ability (Valle-Perez et al., 2019; Verma et al., 2018). To measure how "simple" the trained model is, one of the most promising approaches currently investigated is the compression bounds (Arora et al., 2018; Baykal et al., 2019; Suzuki et al., 2018). These bounds measure how much the network can be compressed and characterize the size of the compressed network as the implicit effective dimensionality. Arora et al. (2018) characterized the implicit dimensionality based on so called layer-cushion quantity and suggested to perform random projection to obtain a compressed network. Along with a similar direction, Baykal et al. (2019) proposed a pruning scheme called Corenet and derived a bound of the size of the compressed network. Suzuki et al. (2018) has developed a spectrum based bound for their compression scheme. Unfortunately, all of these bounds guarantee the generalization error of only the compressed network, not the original network. Hence, it does not give precise explanations about why large network can avoid overfitting.
+
+In this paper, we derive a unified framework to obtain a compression based bound for a non-compressed network. Unlike the existing researches, our bound is valid to evaluate the original network before compression, and thus gives a direct explanation about why deep learning generalizes despite its large network size. The difficulty to apply the compression bound to the original network lies in evaluation of the population  $L_{2}$ -bound between the compression network and the original network. A naive evaluation results in the VC-bound which is not preferable. This difficulty is overcome by developing novel data dependent capacity control technique using local Rademacher complexity bounds (Mendelson, 2002; Bartlett et al., 2005; Koltchinskii, 2006; Giné & Koltchinskii, 2006). Then, the bound is applied to some typical situations where the network is well compressed. Our analysis stands on the implicit bias hypothesis (Gunasekar et al., 2018; Ji & Telgarsky, 2019) that claims deep learning tends to produce rather simple models. Actually, Gunasekar et al. (2018); Ji & Telgarsky (2019) showed gradient descent results in (near) low rank parameter matrices in each layer in linear network settings. Martin & Mahoney (2018) evaluated the eigenvalue decays of the weight matrix through random matrix theories and several numerical experiments. These observations are also supported by the flat minimum analysis (Hochreiter & Schmidhuber, 1997; Wu et al., 2017; Langford & Caruana, 2002), that is, the product of the eigenvalues of the Hessian around the SGD solution tends to be small, which means SGD converges to a flat minimum and possess stability against small perturbations leading to good generalization. Based on these observations, we make use of the eigenvalue decay of the weight matrix and the covariance matrix among the nodes in each layer (this assumption is actually verified by numerical experiments in Appendix D). The eigenvalue decay speed characterizes the redundancy in each layer and thus is directly relevant to compression ability. Our contributions in this paper are summarized as follows:
+
+- We give a unified framework to obtain a compression based bound for non-compressed network which properly explains that a compressible network can generalize well. The bound can convert several existing compression based bounds to that for non-compressed one in a unifying manner. The bound is applied to near low rank models as concrete examples.  
+- We develop a data dependent capacity control technique to bound the discrepancy between the original network and compressed network. As a result, we obtain a sharp generalization
+
+Table 1: Comparison of each generalization error to our bound.  $R_{\mathrm{F}}$  is the Frobenius norm of the weight matrix,  $R_{2}$  is the operator norm of the weight matrix,  $R_{p\rightarrow q}$  is the  $(p,q)$  matrix norm,  $L$  is the depth,  $m$  is the maximum of the width,  $n$  is the sample size.  $\bar{R}_n$  and  $\dot{R}_r$  represent the Rademacher complexity and local Rademacher complexity respectively.  $\kappa$  is a Lipschitz constant between layers.  $\alpha$  represents the eigenvalue drop rate of the weight matrix, and  $\beta$  represents that of the covariance matrix among the nodes in each internal layer.  $\hat{r}$  is the bias induced by compression. "Original" indicates whether the bound is about the original network or not.  
+
+<table><tr><td>Authors</td><td>Rate</td><td>Bound type</td><td>Original</td></tr><tr><td>Neyshabur et al. (2015)</td><td>\(\frac{2^L R_{\mathrm{F}}^L}{\sqrt{n}}\)</td><td>Norm base</td><td>Yes</td></tr><tr><td>Bartlett et al. (2017b)</td><td>\(\frac{R_2^L}{\sqrt{n}} \left( L \frac{R_{2 \rightarrow 1}^{2/3}}{R_{2}^{2/3}} \right)^{3/2}\)</td><td>Norm base</td><td>Yes</td></tr><tr><td>Wei &amp; Ma (2019)</td><td>\(\frac{\left(1 + L\kappa^{\frac{4}{3}} R_{2 \rightarrow 1}^{2/3} + L\kappa^{\frac{2}{3}} R_{1 \rightarrow 1}^{2/3}\right)^{3/2}}{\sqrt{n}}\)</td><td>Norm base</td><td>Yes</td></tr><tr><td>Neyshabur et al. (2017)</td><td>\(\frac{R_2^L}{\sqrt{n}} \sqrt{L^3 m \frac{R_{\mathrm{F}}^2}{R_2^2}}\)</td><td>Norm base</td><td>Yes</td></tr><tr><td>Golowich et al. (2018)</td><td>\(R_{\mathrm{F}}^L \min \left\{ \frac{1}{n^{1/4}}, \sqrt{\frac{L}{n}} \right\}\)</td><td>Norm base</td><td>Yes</td></tr><tr><td>Li et al. (2018)</td><td>\(\frac{R_2^L \sqrt{L^2 m^2}}{\sqrt{n}}\)</td><td>VC-dim.</td><td>Yes</td></tr><tr><td>Harvey et al. (2017)</td><td></td><td></td><td></td></tr><tr><td>Arora et al. (2018)</td><td>\(\hat{r} + \sqrt{L^2 \max_{1 \leq i \leq n} |\hat{f}(x_i)|^2 \sum_{\ell=1}^{L} \frac{1}{\mu_\ell^2 \mu_{\ell \rightarrow \ell}^2}}\)</td><td>Compression</td><td>No</td></tr><tr><td>Suzuki et al. (2018)</td><td>\(\hat{r} + \sqrt{\frac{\sum_{\ell=1}^{L} m_{\ell+1}^2 m_{\ell}^2}{n}}\)</td><td>Compression</td><td>No</td></tr><tr><td>Ours (Thm. 1)</td><td>\(\hat{r} \sqrt{\frac{1}{n}} + \dot{R}_{\hat{r}}(\widehat{\mathcal{F}} - \widehat{\mathcal{G}}) + \bar{R}_n(\widehat{\mathcal{G}})\)</td><td>General</td><td>Yes</td></tr><tr><td>Ours (Cor. 1)</td><td>\(\sqrt{L(L\kappa^2)^{1/\alpha} \frac{Lm}{n}}\)</td><td>Low rank weight</td><td>Yes</td></tr><tr><td>Ours (Thm. 4)</td><td>\(\sqrt{L^{1+\frac{\beta}{4\alpha}(2\alpha-1)+\beta} \frac{(Lm)^{4/\beta+2(1-1/2\alpha)}}{n}}\)</td><td>Low rank weight</td><td>Yes</td></tr><tr><td></td><td></td><td>Low rank cov.</td><td></td></tr></table>
+
+error bound which is even better than that of the compressed network. All derived bounds are characterized by data dependent quantities.
+
+Other related work Recently, the role of over-parameterization for two layer networks has been extensively studied (Neyshabur et al., 2019; Arora et al., 2019). These are for the shallow network and the generalization error is essentially given by the norm based bounds. It is not obvious that these bounds also give sharp bounds for deep models.
+
+PAC-Bayes bound is also applied to obtain a non-vacuous compression based bound (Zhou et al., 2019). However, the bound is still for the compressed (quantized) models and it is not obvious that that bound can be converted to that for the original network.
+
+Relation between compression and learnability was traditionally studied in a different framework as in Littlestone & Warmuth (1986) and minimum description code length (Hinton & Van Camp, 1993). Our bound would share the same spirits with these studies but give a new analysis by incorporating recent observations in deep learning researches.
+
+# 2 PRELIMINARIES: PROBLEM FORMULATION AND NOTATIONS
+
+In this section, we give the problem setting and notations that will be used in the theoretical analysis. We consider the standard supervised leaning formulation where data consists of input  $x \in \mathbb{R}^d$  and output (or label)  $y \in \mathbb{R}$ . We consider a single output setting, i.e., the output  $y$  is a 1-dimensional real value, but it is straight forward to generalize the result to a multiple output case. Suppose that we are given  $n$  i.i.d. observations  $D_{n} = (x_{i},y_{i})_{i = 1}^{n}$  distributed from a probability distribution  $P$ . To measure the performance of a trained function  $f$ , we use a loss function  $\psi : \mathbb{R} \times \mathbb{R} \to \mathbb{R}$  and define
+
+a training error and its expected one as
+
+$$
+\widehat {\Psi} (f) := \frac {1}{n} \sum_ {i = 1} ^ {n} \psi \left(y _ {i}, f \left(x _ {i}\right)\right), \Psi (f) := \operatorname {E} \left[ \psi \left(Y, f (X)\right) \right],
+$$
+
+where the expectation is taken with respect to  $(X,Y)\sim P$ . Basically, we are interested in the generalization error  $\Psi (\widehat{f}) - \widehat{\Psi} (\widehat{f})$  for an estimator  $\widehat{f}$ . We denote the empirical  $L_{2}$ -norm by  $\| f\| _n\coloneqq \sqrt{\sum_{i = 1}^n f(z_i)^2 / n}$  for an empirical observation  $z_{i} = (x_{i},y_{i})$ $(i = 1,\ldots ,n)$ . The population  $L_{2}$ -norm is denoted by  $\| f\|_{L_2}\coloneqq \sqrt{\operatorname{E}_{Z\sim P}[f(Z)^2]}$ .
+
+This paper deals with deep neural networks as a model. The activation function is denoted by  $\eta$  which will be assumed to be 1-Lipschitz as satisfied by ReLU (Assumption 1). Let the depth of the network be  $L$  and the width of the  $\ell$ -th internal layer be  $m_{\ell}$  ( $\ell = 1, \ldots, L + 1$ ) where we set  $m_1 = d$  (dimension of input) and  $m_{L + 1} = 1$  (dimension of output) for convention. Then, the set of networks having depth  $L$  and width  $\mathbf{m} = (m_1, \ldots, m_L)$  with norm constraint as
+
+$$
+\begin{array}{l} \mathrm {N N} (\mathbf {m}, R _ {2} ^ {\prime}, R _ {\mathrm {F}} ^ {\prime}) := \left\{f (x) = G \circ \left(W ^ {(L)} \eta (\cdot)\right) \circ \left(W ^ {(L - 1)} \eta (\cdot)\right) \circ \dots \circ \left(W ^ {(1)} x\right) \mid \right. \\ W ^ {(\ell)} \in \mathbb {R} ^ {m _ {\ell} \times m _ {\ell + 1}}, \| W ^ {(\ell)} \| _ {2} \leq R _ {2} ^ {\prime}, \| W ^ {(\ell)} \| _ {\mathrm {F}} \leq R _ {\mathrm {F}} ^ {\prime} \Big \}. \\ \end{array}
+$$
+
+where  $\| W\| _2\coloneqq \sup_{u:\| W u\| \neq 0}\| W u\| /\| u\| ^1$  is the operator norm (the maximum singular value),  $\| W\|_{\mathrm{F}}\coloneqq \sqrt{\sum_{i,j}W_{i,j}^{2}}$  is the Frobenius norm, and  $G$  is the "clipping" operator that is defined by  $G(x) = \max \{-M,\min \{x,M\} \}$  for a constant  $M$ . The reason why we put the clipping operator  $G$  on top of the last layer is because the clipping operator restricts the  $L_{\infty}$ -norm by a constant  $M$  and then we can avoid unrealistically loose generalization error. Note that the clipping operator does not change the classification error for binary classification. We express  $\mathcal{F}$  to represent the "full model":  $\mathcal{F} = \mathrm{NN}(\mathbf{m},R_2,R_{\mathrm{F}})$  for a given  $R_{2},R_{\mathrm{F}} > 0$ . Here, we implicitly suppose that  $R_{2}$  is close to 1 so that the norm of the output from internal layers is not too much amplified, while  $R_{\mathrm{F}}$  could be moderately large.
+
+The Rademacher complexity is the typical tool to evaluate the generalization error on a function class  $\mathcal{F}'$ , which is denoted by  $\hat{R}_n(\mathcal{F}') \coloneqq \operatorname{E}_{\epsilon} \left[ \sup_{f \in \mathcal{F}'} \frac{1}{n} \sum_{i=1}^{n} \epsilon_i f(z_i) \mid D_n \right]$  where  $D_n = (z_i)_{i=1}^n = (x_i, y_i)_{i=1}^n$ , and  $\epsilon_i$  ( $i = 1, \dots, n$ ) is an i.i.d. Rademacher sequence ( $P(\epsilon_i = 1) = P(\epsilon_i = -1) = 1/2$ ). This is also called conditional Rademacher complexity because the expectation is taken conditioned on fixed  $D_n$ . Its expectation with respect to  $D_n$  is denoted by  $\bar{R}_n(\mathcal{F}') \coloneqq \operatorname{E}_{D_n}[\hat{R}_n(\mathcal{F}')]$ . Roughly speaking the Rademacher complexity measures the size of the model and it gives an upper bound of the generalization error (Vapnik, 1998; Mohri et al., 2012).
+
+The main difficulty in generalization error analysis of deep learning is that the Rademacher complexity of the full model  $\mathcal{F}$  is quite large. One of the successful approaches to avoid this difficulty is the compression based bound (Arora et al., 2018; Baykal et al., 2019; Suzuki et al., 2018) which measures how much the trained network  $\widehat{f}$  can be compressed. If the network can be compressed to much smaller one, then its intrinsic dimensionality can be regarded as small. To describe it more precisely, suppose that the trained network  $\widehat{f}$  is included in a subset of the neural network model:  $\widehat{f} \in \widehat{\mathcal{F}} \subset \mathcal{F}$ . For example,  $\widehat{\mathcal{F}}$  can be a set of networks with weight matrices that have bounded norms and are near low rank (Sec. 3.1 or Sec. 3.2). We do not assume a specific type of training procedure, but we give a uniform bound valid for any estimator  $\widehat{f}$  that falls into  $\widehat{\mathcal{F}}$  and satisfies the following compressibility condition. We suppose that the network  $\widehat{f}$  is easy to compress, that is,  $\widehat{f}$  can be compressed to a smaller network  $\widehat{g}$  which is included in a submodel:  $\widehat{g} \in \widehat{\mathcal{G}}$ . For example,  $\widehat{\mathcal{G}}$  can be a set of networks with a smaller size than  $\widehat{f}$ . How small the trained network  $\widehat{f}$  can be compressed has been characterized by several notions such as "layer-cushion" (Arora et al., 2018). Typical compression based bounds give generalization errors of the compressed model  $\widehat{g}$ , not the original network  $\widehat{f}$ . Our approach converts an error bound of  $\widehat{g}$  to that of  $\widehat{f}$  and eventually obtains a tighter evaluation.
+
+The biggest difficulty for transforming the compression bound to that of  $\widehat{f}$  lies in evaluation of the population  $L_{2}$ -norm between  $\widehat{f}$  and  $\widehat{g}$ . Basically, the compression based bounds are given as
+
+$$
+\Psi (\widehat {g}) \leq \widehat {\Psi} (\widehat {f}) + \| \widehat {f} - \widehat {g} \| _ {n} + C \bar {R} _ {n} (\widehat {\mathcal {G}}), \tag {1}
+$$
+
+for a constant  $C > 0$  under some assumptions (Table 1). The term  $\| \widehat{f} - \widehat{g} \|_n$  appears to adapt the empirical error of  $\widehat{f}$  to that of  $\widehat{g}$ , that is called "compression error" which can be seen as a bias term. We see that, in the right hand side, there appears the complexity of  $\widehat{\mathcal{G}}$  which is assumed to be much smaller than that of the full model  $\mathcal{F}$ . However, the left hand side is not the expected error of  $\widehat{f}$  but that of  $\widehat{g}$ . One way to transfer this bound to that of  $\widehat{f}$  is that we have  $|\Psi(\widehat{g}) - \Psi(\widehat{f})| \leq \| \widehat{g} - \widehat{f} \|_{L_2}$  by assuming Lipschitz continuity of the loss function and then convert the bound (1) to
+
+$$
+\Psi (\widehat {f}) \leq \widehat {\Psi} (\widehat {f}) + \left(\| \widehat {f} - \widehat {g} \| _ {n} + \| \widehat {f} - \widehat {g} \| _ {L _ {2}}\right) + \bar {R} _ {n} (\widehat {\mathcal {G}}).
+$$
+
+However, to bound the term  $\| \widehat{f} - \widehat{g} \|_n + \| \widehat{f} - \widehat{g} \|_{L_2}$ , there typically appears the complexity of the model  $\widehat{\mathcal{F}}$  which is larger than the compressed model  $\widehat{\mathcal{G}}$  like  $\| \widehat{f} - \widehat{g} \|_n \leq \sqrt{\| \widehat{f} - \widehat{g} \|_{L_2}^2 + O_p(\bar{R}(\widehat{\mathcal{F}}))}$ , which results in slow convergence rate. To overcome this difficulty, we need to carefully control the difference between the training and test error of  $\widehat{f}$  and  $\widehat{g}$  by utilizing the local Rademacher complexity technique (Mendelson, 2002; Bartlett et al., 2005; Koltchinskii, 2006; Giné & Koltchinskii, 2006). The local Rademacher complexity of a model  $\mathcal{F}'$  with radius  $r > 0$  is defined as
+
+$$
+\dot {R} _ {r} \left(\mathcal {F} ^ {\prime}\right) := \bar {R} _ {n} \left(\left\{f \in \mathcal {F} ^ {\prime} \mid \| f \| _ {L _ {2}} \leq r \right\}\right).
+$$
+
+The main difference from the standard Rademacher complexity is that the model is localized to a set of functions satisfying  $\| f\|_{L_2}\leq r$ . As a result, we obtain a tighter error bound.
+
+Throughout this paper, we always assume the following assumptions. Let  $P_{\mathcal{X}}$  and  $P_{\mathcal{Y}}$  denote the marginal distribution of  $x$  and that of  $y$  respectively.
+
+Assumption 1 (Lipschitz continuity of loss and activation functions). The loss function  $\psi$  is 1-Lipschitz continuous with respect to the function output:
+
+$$
+\left| \psi (y, u) - \psi (y, u ^ {\prime}) \right| \leq \left| u - u ^ {\prime} \right| (\forall y \in \operatorname {s u p p} (P _ {\mathcal {Y}}), u, u ^ {\prime} \in \mathbb {R}).
+$$
+
+The activation function  $\eta$  is also 1-Lipschitz continuous:  $\| \eta (u) - \eta (u^{\prime})\| \leq \| u - u^{\prime}\|$  ( $\forall u\in \mathbb{R}^{d'}$ ) where  $d^{\prime}$  is any positive integer.
+
+Assumption 2. The norm of input is bounded by  $B_{x} > 0$ :  $\| x\| \leq B_x$  ( $\forall x\in \mathrm{supp}(P_{\mathcal{X}})$ ).
+
+Assumption 3. The  $L_{\infty}$ -norms of all elements in  $\widehat{\mathcal{F}}$  and  $\widehat{\mathcal{G}}$  are bounded by  $M \geq 1$ :  $\|f\|_{\infty}, \|g\|_{\infty} \leq M$  for all  $f \in \widehat{\mathcal{F}}$  and  $g \in \widehat{\mathcal{G}}$ .
+
+This assumption can be ensured by applying the clipping operator  $G$  to the output of the functions. In this paper, all the variables  $L, m_{\ell}, R_{2}, R_{\mathrm{F}}, M, B_{x}$  are supposed to be  $o(n)$ . What we will derive in the following is a bound which has mild dependency on the depth  $L$  and depends on the width  $(m_{\ell})_{\ell = 1}^{L}$  in a sub-linear order by using the compression based approach.
+
+Existing bounds for no-compressed network Here we give a brief review of the generalization error bound for non-compressed models. (i) VC-bound: The Rademacher complexity of the full model  $\mathcal{F}$  can be bounded by a naive VC-dimension bound (Harvey et al., 2017) which is  $\bar{R}_n(\mathcal{F}) = O\left(\sqrt{\frac{L\sum_{\ell=1}^n m_\ell m_{\ell+1}}{n}\log(n)}\right)$ . In this bound, there appears the number of parameters  $\sum_{\ell=1}^n m_\ell m_{\ell+1}$  in the numerator. However, the number of parameters is often larger than the sample size  $n$  in practical use. Hence, this bound is not appropriate to evaluate generalization ability of overparameterized networks. (ii) Norm-based bound: Golowich et al. (2018) showed the norm based bound which is given as  $\bar{R}_n(\mathcal{F}) = O\left(\sqrt{\frac{LR_{\mathrm{F}}^L}{n}}\right)$ . However, this is exponentially dependent on the depth as  $R_{\mathrm{F}}^L$  resulting in quite loose bound. Neyshabur et al. (2017) showed a norm based bound of  $\bar{R}_n(\mathcal{F}) = O\left(\sqrt{\frac{L^3(\max_\ell m_\ell)R_{\mathrm{F}}^2 / R_2^2}{n}}\right)$  which avoids the exponential dependency. However, there is still dependency on the width,  $\sum_{\ell} m_\ell R_{\mathrm{F}}^2$ , which is larger than the
+
+linear order of the width since  $R_{\mathrm{F}}$  could be moderately large. Bartlett et al. (2017b) showed  $\bar{R}_n(\mathcal{F}) = O\left(\frac{R_2^L}{\sqrt{n}}\left(L\frac{R_{2\rightarrow 1}^{2/3}}{R_2^{2/3}}\right)^{3/2}\right)$ . The norm constraint on  $R_{2\rightarrow 1}$  implicitly assumes sparsity on the weight matrix and  $R_{2\rightarrow 1}$  typically depends on the width linearly. Wei & Ma (2019) improved the exponential dependency  $R_2^L$  appearing in this bound (Bartlett et al., 2017b) to obtain a bound  $O\left(\frac{1}{\sqrt{n}}\left(1 + L\kappa^{\frac{4}{3}}R_{2\rightarrow 1}^{2/3} + L\kappa^{\frac{2}{3}}R_{1\rightarrow 1}^{2/3}\right)^{3/2}\right)$  where  $\kappa$  is the Lipschitz continuity between layers. We can see that  $R_{2\rightarrow 1}^2$  and  $R_{1\rightarrow 1}^2$  can depend on the width linearly and quadratically respectively even though  $R_2$  is bounded.
+
+# 3 COMPRESSION BOUND FOR NONCOMPRESSED NETWORK
+
+Here, we give a general theoretical tool that converts a compression based bound to that for the original network  $\widehat{f}$ . We suppose the model classes  $\widehat{\mathcal{F}}$  and  $\widehat{\mathcal{G}}$  are fixed independently on each data observation $^2$ . For sets of functions,  $\mathcal{F}'$  and  $\mathcal{G}'$ , we denote the Minkowski difference of them by  $\mathcal{F}' - \mathcal{G}' := \{f - g \mid f \in \mathcal{F}', g \in \mathcal{G}'\}$ . We assume that the local Rademacher complexity of  $\widehat{\mathcal{F}} - \widehat{\mathcal{G}}$  has a concave shape with respect to  $r > 0$ : Suppose that there exists a function  $\phi : [0, \infty) \to [0, \infty)$  such that
+
+$$
+\dot {R} _ {r} (\widehat {\mathcal {F}} - \widehat {\mathcal {G}}) \leq \phi (r) \text {a n d} \phi (2 r) \leq 2 \phi (r) (\forall r > 0).
+$$
+
+This condition is not restrictive, and usual bounds for the local Rademacher complexity satisfy this condition (Mendelson, 2002; Bartlett et al., 2005). Using this notation, we define  $r_* = r_*(t)$  as
+
+$$
+r _ {*} (t) := \inf  \left\{r > 0 \left| 8 \frac {\phi (r)}{r ^ {2}} + M \sqrt {\frac {4 t}{r ^ {2} n}} + M ^ {2} \frac {2 t}{r ^ {2} n} \leq \frac {1}{2} \right. \right\}. \tag {2}
+$$
+
+This is roughly given by the fixed point of a function  $r^2 \mapsto \phi(r)$ , and it is useful to bound the ratio of the empirical  $L_2$ -norm and the population  $L_2$ -norm of an element  $h$  in  $\widehat{\mathcal{F}} - \widehat{\mathcal{G}}$ :  $\| h \|_{L_2}^2 / (\| h \|_n^2 + r_*^2) \leq 1/2$  with high probability. Finally, we denote  $\psi(\mathcal{F}') \coloneqq \{ \psi(\cdot, f(\cdot)) \mid f \in \mathcal{F}' \}$  for a set  $\mathcal{F}'$  of functions. Then, we obtain the following theorem that gives the compression based bound for non-compressed networks.
+
+Theorem 1. Suppose that the empirical  $L_{2}$ -distance between  $\widehat{f}$  and  $\widehat{g}$  is bounded by  $\| \widehat{f} - \widehat{g} \|_n \leq \widehat{r}^2$  for a fixed  $\widehat{r} > 0$  almost surely. Let  $\dot{r} \coloneqq \sqrt{2(\widehat{r}^2 + r_*^2)}$ , then, under Assumptions 1, 2, 3, there exists a universal constant  $C > 0$  such that
+
+$$
+\Psi (\widehat {f}) \leq \hat {\Psi} (\widehat {f}) + \underbrace {2 \bar {R} _ {n} (\widehat {\mathcal {G}}) + \sqrt {M \frac {2 t}{n}}} _ {\text {m a i n t e r m}} + C \underbrace {\left[ \dot {R} _ {\dot {r}} (\psi (\widehat {\mathcal {F}}) - \psi (\widehat {\mathcal {G}})) + \dot {r} \sqrt {\frac {t}{n}} + \frac {1 + t M}{n} \right]} _ {\text {b i a s t e r m}}.
+$$
+
+with probability at least  $1 - 3e^{-t}$  for all  $t \geq 1$ .
+
+The proof is given in Appendix A. The bound consists of two terms: "main term" and "bias term." The main term represents the complexity of the compressed model  $\widehat{\mathcal{G}}$  which could be much smaller than  $\widehat{\mathcal{F}}$ . The bias term represents a sample complexity to bridge the original model and the compressed model. Typically we have  $r_*^2 = o(1 / \sqrt{n})$ , and if we set  $\hat{r} = o_p(1)$ , then the bias term can be faster than the main term which is  $O(1 / \sqrt{n})$ . The term  $\dot{R}_{\hat{r}}(\psi(\widehat{\mathcal{F}}) - \psi(\widehat{\mathcal{G}}))$  can be refined a little bit and the refined term can be evaluated by using the covering number of the model. The refined version is given in Appendix A (Theorem 5). This bound is general, and can be combined with the compression bounds derived so far such as Arora et al. (2018); Baykal et al. (2019); Suzuki et al. (2018) where the complexity of  $\widehat{\mathcal{G}}$  and the bias  $\hat{r}$  are analyzed for their generalization error bounds.
+
+The main difference from the compression bound (1) for  $\widehat{g}$  is that the bias term  $\hat{r} = \| \widehat{f} -\widehat{g}\| _n$  is replaced by  $\frac{1}{\sqrt{n}}\| \widehat{f} -\widehat{g}\| _n$  which is  $\sqrt{n}$  times smaller. Since  $r_*^2$  is typically  $o(1 / \sqrt{n})$  and  $\dot{R}_{\dot{r}}(\psi (\widehat{\mathcal{F}}) -$
+
+$\psi (\widehat{\mathcal{G}})$  can be made in the same order as the main term or even faster by setting  $\hat{r}$  appropriately, we may neglect these terms. Then, the bound is informally written as
+
+$$
+\Psi (\widehat {f}) \leq \hat {\Psi} (\widehat {f}) + O _ {p} \left(\bar {R} _ {n} (\widehat {\mathcal {G}}) + \frac {1}{\sqrt {n}} \| \widehat {f} - \widehat {g} \| _ {n} + \sqrt {1 / n}\right).
+$$
+
+This allows us to obtain tighter bound than the compression bound for  $\widehat{g}$  because the bias term  $\hat{r} / \sqrt{n}$  is much smaller than  $\hat{r}$  and eventually we can let the variance term  $\bar{R}_n(\widehat{\mathcal{G}})$  much smaller by taking small compressed model  $\widehat{\mathcal{G}}$  when we balance the bias and variance trade-off. This is an advantageous point of directly bounding the generalization error of  $\widehat{f}$  instead of  $\widehat{g}$ .
+
+Finally, we note that some existing bounds such as Arora et al. (2018); Bartlett et al. (2017b); Wei & Ma (2019) assumes a constant margin so that the bias term can be a sufficiently small constant (which does not need to converge to 0). On the other hand, our bound does not assume it and the bias term should converge to 0 so that the bias is balanced with the variance term, which is a more difficult problem setting.
+
+Example 1. In practice, a trained network can be usually compressed to one with sparse weight matrix via pruning techniques (Denil et al., 2013; Denton et al., 2014). Based on this observation, Baykal et al. (2019) derived a compression based bound based on a pruning procedure. In this situation, we may suppose that  $\widehat{\mathcal{G}}$  is the set of networks with  $S$  non-zero parameters where  $S$  is much smaller than the total number of parameters:  $\widehat{\mathcal{G}} = \{f\in \mathrm{NN}(\mathbf{m},R_2,R_{\mathrm{F}})\mid \sum_{\ell = 1}^{L}\| W^{(\ell)}\| _0\leq S\}$  where  $\| W^{(\ell)}\| _0$  is the number of nonzero parameters of the weight matrix  $W^{(\ell)}$ . In this situation, its Rademacher complexity is bounded by  $\bar{R} (\widehat{\mathcal{G}})\leq CM\sqrt{L\frac{S}{n}\log(n)}$  (see Appendix B.2 for the proof). This is much smaller than the VC-dimension bound  $\sqrt{\frac{L\sum_{\ell = 1}^{n}m_{\ell}m_{\ell + 1}}{n}\log(n)}$  if  $S\ll \sum_{\ell = 1}^{n}m_{\ell}m_{\ell +1}$ .
+
+Although our bound can be adopted to several compression based bounds, we are going to demonstrate how small the obtained bound can be through some typical situations in the following.
+
+# 3.1 COMPRESSION BOUND WITH NEAR LOW RANK WEIGHT MATRIX
+
+Here, we analyze the situation where the trained network has near low rank weight matrices  $(W^{(\ell)})_{\ell = 1}^{L}$ . It has been reported that the trained network tends to have near low rank weight matrices experimentally (Gunasekar et al., 2018; Ji & Telgarsky, 2019) (see Appendix D for the empirical verification). This situation has been analyzed in Arora et al. (2018) where the low rank property is characterized by their original quantities such as layer cushion. However, we employ a much simpler and intuitive condition to highlight how the low rank property affects the generalization.
+
+Assumption 4. Assume that each of weight matrices  $W^{(\ell)}$  ( $\ell = 1, \dots, L$ ) of any  $f \in \widehat{\mathcal{F}}$  is near low rank, that is, there exists  $\alpha > 1/2$  and  $V_0 > 0$  such that
+
+$$
+\sigma_ {j} (W ^ {(\ell)}) \leq V _ {0} j ^ {- \alpha},
+$$
+
+where  $\sigma_{j}(W)$  is the  $j$ -th largest singular value of a matrix  $W$  ( $\sigma_{1}(W) \geq \sigma_{2}(W) \geq \dots \geq 0$ ).
+
+In this situation, we can see that for any  $1 \leq s \leq \min\{m_{\ell}, m_{\ell+1}\}$ , we can approximate  $W^{(\ell)}$  by a rank  $s$  matrix  $W'$  as  $\| W^{(\ell)} - W' \|_2 \leq V_0 s^{-\alpha}$ . Let the set of networks with exactly low rank weight matrices be  $\mathrm{NN}(\mathbf{m}, \mathbf{s}, R_2, R_{\mathrm{F}}) := \{f \in \mathrm{NN}(\mathbf{m}, R_2, R_{\mathrm{F}}) | \text{the weight matrix } W^{(\ell)} \text{ of } f \text{ has rank } s_{\ell}\}$  for  $\mathbf{s} = (s_1, \ldots, s_L)$ . If we set  $\widehat{\mathcal{G}} = \mathrm{NN}(\mathbf{m}, \mathbf{s}, R_2, R_{\mathrm{F}})$ , then we have the following theorem.
+
+Theorem 2. The compressed model  $\widehat{\mathcal{G}} = \mathrm{NN}(\mathbf{m},\mathbf{s},R_2,R_{\mathrm{F}})$  has the following complexity:
+
+$$
+\bar {R} _ {n} (\widehat {\mathcal {G}}) \leq C M \sqrt {L \frac {\sum_ {\ell = 1} ^ {L} s _ {\ell} (m _ {\ell} + m _ {\ell + 1})}{n} \log (n)}.
+$$
+
+If  $\widehat{\mathcal{F}}$  satisfies Assumption 4, we can set  $\hat{r} = V_0R_2^{L - 1}B_x\sum_{\ell = 1}^L s_\ell^{-\alpha}$ : for any  $\widehat{f}\in \widehat{\mathcal{F}}$ , there exists  $\widehat{g}\in \widehat{\mathcal{G}}$  such that  $\| \widehat{f} -\widehat{g}\| _n\leq \hat{r}$ . Then, letting  $A_{1} = L\frac{\sum_{\ell = 1}^{L}s_{\ell}(m_{\ell} + m_{\ell + 1})}{n}\log (n)$  and
+
+$A_{2} = L\frac{(\sum_{\ell = 1}^{L}m_{\ell})(2LV_{0}R_{2}^{L - 1}B_{x})^{1 / \alpha}}{n}$ , the overall generalization error is bounded by
+
+$$
+\Psi (\widehat {f}) \leq \widehat {\Psi} (\widehat {f}) + C \left[ M A _ {1} + M ^ {\frac {2 \alpha - 1}{2 \alpha + 1}} A _ {2} ^ {\frac {2 \alpha}{1 + 2 \alpha}} + \sqrt {\hat {r} ^ {2 (1 - 2 \alpha)} A _ {2}} + (\hat {r} + M) \sqrt {A _ {1}} + \frac {1 + t M}{n} \right],
+$$
+
+with probability  $1 - 3e^{-t}$  for any  $t > 1$  where  $C > 0$  is a constant depending on  $\alpha$ .
+
+See Appendix B.3 for the proof. This indicates that, if  $\alpha > 1/2$  is large (in other words, each weight matrix is close to rank 1), then we have a better generalization error bound. Note that the rank  $s_{\ell}$  can be arbitrary chosen and  $\hat{r}$  and  $A_{1}$  are in a trade-off relation. Hence, by selecting the rank appropriately so that this trade-off is balanced, then we obtain the optimal upper bound as in the following corollary.
+
+Corollary 1. Under Assumption 4, using the same notation as Theorem 2, it holds that
+
+$$
+\Psi (\widehat {f}) \leq \hat {\Psi} (\widehat {f}) + C \left[ M ^ {1 - 1 / 2 \alpha} \sqrt {L ^ {\frac {(\sum_ {\ell = 1} ^ {L} m _ {\ell}) (2 L V _ {0} R _ {2} ^ {L - 1} B _ {x}) ^ {1 / \alpha}}{n} \log (n)}} + M ^ {\frac {2 \alpha - 1}{2 \alpha + 1}} A _ {2} ^ {\frac {2 \alpha}{2 \alpha + 1}} + \frac {1 + t M}{n} \right]
+$$
+
+with probability  $1 - 3e^{-t}$  for any  $t > 1$  where  $C$  is a constant depending on  $\alpha$ .
+
+An important point here is that the bound is  $O\left(\sqrt{L \frac{\sum_{\ell=1}^{L} m_{\ell}}{n}}\right)$  which has linear dependency on the width  $m_{\ell}$  in the square root, but the naive VC-dimension bound has quadratic dependency  $O\left(\sqrt{L \frac{\sum_{\ell=1}^{L} m_{\ell} m_{\ell}}{n}}\right)$ . In other words, the term in the square root has linear dependency to the number of nodes instead of the number of parameters. This is huge gap because the width can be quite large in practice. This result implies that a compressible model achieves much better generalization than the naive VC-bound.
+
+In the generalization error bound, there appears  $R_2^L$ . Even though  $R_2$  can be much smaller than  $R_{\mathrm{F}}$ , the exponential dependency  $R_2^L$  can give loose bound as pointed out in Arora et al. (2018). This is due to a rough evaluation of the Lipschitz continuity between layers, but the practically observed Lipschitz constant is usually much smaller. To fix this issue, we give a refined version of Corollary 1 in Appendix B.4 by using data dependent Lipschitz constants such as interlayer cushion and interlayer smoothness introduced by Arora et al. (2018). The refined bound does not involve the exponential term  $R_2^L$ , but instead  $\kappa^2$  ( $\kappa$ : Lipschitz continuity) appears.
+
+# 3.2 COMPRESSION BOUND WITH NEAR LOW RANK COVARIANCE MATRIX
+
+Strictly speaking, the near low rank condition on the weight matrix in the previous section can be dealt with a standard Rademacher complexity argument. Here, we consider more data dependent bound: We assume the near low rank property of the covariance matrix among the nodes in an internal layer (see Appendix D for the empirical verification). A compression based bound for  $\widehat{g}$  using the low rank property of the covariance has been studied by Suzuki et al. (2018), but their analysis requires a bit strong condition on the weight matrix. In this paper, we employ a weaker assumption.
+
+Let  $\widehat{\Sigma}_{(\ell)} = \frac{1}{n}\sum_{i=1}^{n}\phi_{\ell}(x_i)\phi_{\ell}(x_i)^\top$  be the covariance matrix of the nodes in the  $\ell$ -th layer where  $\phi_{\ell}(x) = \eta \circ (W^{(\ell-1)}\eta(\cdot)) \circ \dots \circ (W^{(1)}x)$ .
+
+Assumption 5. Suppose that the trained network  $\widehat{f}$  satisfies the following conditions:
+
+$$
+\sigma_ {j} \left(\widehat {\Sigma} _ {(\ell)}\right) \leq \dot {\mu} _ {j} ^ {(\ell)} =: U _ {0} j ^ {- \beta}, \tag {3}
+$$
+
+for a fixed  $\beta > 1$  and  $U_0 > 0$ .
+
+If  $\widehat{f}$  satisfies this assumption, then we can show that  $\widehat{f}$  can be compressed to a smaller one  $f^{\sharp}$  that has width  $(m_{\ell}^{\sharp})_{\ell = 2}^{L}$  with compression error roughly evaluated as  $\| \widehat{f} - f^{\sharp} \|_n \lesssim \sum_{\ell} (m_{\ell}^{\sharp})^{-\beta / 2}$ . More precisely, for given  $\tilde{r}_{\ell} > 0$  ( $\ell = 1, \ldots, L$ ) which corresponds to the compression error in the  $\ell$ -th layer, let  $\dot{m}_{\ell} := \max \{1 \leq j \leq m_{\ell} \mid \dot{\mu}_j^{(\ell)} \geq \tilde{r}_{\ell}^2 / 4\}$ . Then, we define  $N_{\ell}(\tilde{\mathbf{r}}) = \frac{\beta + 1}{\beta - 1} \dot{m}_{\ell} + 8 \frac{(\sum_{k=1}^{\ell-1} R_2^{(\ell-1-k)} R_{\mathrm{F}} \tilde{r}_k)^2}{\tilde{r}_{\ell}^2}$ , for  $\tilde{\mathbf{r}} = (\tilde{r}_1, \ldots, \tilde{r}_L)$ . Correspondingly we set
+
+$$
+m _ {\ell} ^ {\sharp} := 5 N _ {\ell} (\tilde {\mathbf {r}}) \log (8 0 N _ {\ell} (\tilde {\mathbf {r}})).
+$$
+
+Then, we obtain the following theorem.
+
+Theorem 3. Let  $\hat{r} \coloneqq \sum_{k=1}^{L} R_2^{(L-k)} R_{\mathrm{F}} \tilde{r}_k$ . Then, under Assumption 5, there exists  $\widehat{g}$  with width  $\mathbf{m}^{\sharp} = (m_1, m_2^{\sharp}, \ldots, m_L^{\sharp})$  that satisfies  $\widehat{g} \in \mathrm{NN}(\mathbf{m}^{\sharp}, \sqrt{\frac{20}{3} \max_{\ell} m_{\ell}} R_2 \sqrt{\frac{20}{3} \max_{\ell} m_{\ell}} R_{\mathrm{F}})$  and
+
+$$
+\| \widehat {f} - \widehat {g} \| _ {n} \leq \widehat {r}.
+$$
+
+In particular, we may set  $\widehat{\mathcal{G}} = \mathrm{NN}(\mathbf{m}^{\sharp},\sqrt{\frac{20}{3}\max_{\ell}m_{\ell}} R_2,\sqrt{\frac{20}{3}\max_{\ell}m_{\ell}} R_{\mathrm{F}})$ , and then it holds that  $\bar{R}_n(\widehat{\mathcal{G}})\leq C\sqrt{L\sum_{\ell = 1}^{L}\frac{m_{\ell + 1}^{\sharp}m_{\ell}^{\sharp}}{n}\log(n)}$  for a constant  $C > 0$ .
+
+See Appendix B.5 for the proof. Here, we again observe that there appears a trade-off between  $\hat{r}$  and  $m_{\ell}^{\sharp}$  because as  $\tilde{r}_{\ell}$  becomes small, then  $\dot{m}_{\ell}$  becomes large and thus  $m_{\ell}^{\sharp}$  becomes large. The evaluation given in Theorem 3 can be substituted to the general bound (Theorem 1). If  $\widehat{\mathcal{F}}$  is the full model  $\mathcal{F}$ , then there appears the number  $\sum_{\ell = 1}^{L}m_{\ell}m_{\ell +1}$  of parameters which could be larger than  $n$ , which is unavoidable. This dependency on the number of parameters becomes much milder if both of Assumptions 4 and 5 are satisfied.
+
+Theorem 4. Under Assumptions 4 and 5, it holds that
+
+$$
+\begin{array}{l} \Psi (\widehat {f}) \leq \widehat {\Psi} (\widehat {f}) + C \left[ M \sqrt {\frac {[ P _ {L} \vee Q _ {L} ] L ^ {1 + \frac {\beta}{(2 \alpha - 1) ^ {+ \beta}}}}{n} \left(\sum_ {\ell = 1} ^ {L} m _ {\ell}\right) ^ {\frac {4 / \beta}{4 / \beta + 2 (1 - 1 / 2 \alpha)}} \log (n) ^ {3}} \right. \\ + M ^ {\frac {2 \alpha - 1}{2 \alpha + 1}} \left(L P _ {L} \frac {\sum_ {\ell = 1} ^ {L} m _ {\ell}}{n} \log (n)\right) ^ {\frac {2 \alpha}{2 \alpha + 1}} + M \frac {R _ {\mathrm {F}} ^ {2} L ^ {2}}{R _ {2} ^ {2}} \sqrt {\frac {\log (n) ^ {3}}{n}} + \frac {1 + M t}{n} \Biggr ], \\ \end{array}
+$$
+
+for  $P_{L} = (2LV_{0}R_{2}^{L - 1}B_{x})^{1 / \alpha}$  and  $Q_{L} = \left[\frac{4U_{0}R_{\mathrm{F}}^{2}(1\vee R_{2})^{L}\exp\left(\frac{1}{4}(2\sqrt{L} -1)\right)}{(0.25)^{4}(1\wedge R_{2})^{2L}}\right]^{2 / \beta}$  with probability  $1 - 3e^{-t}$  ( $t > 1$ ), where  $C$  is a constant depending on  $\alpha, \beta$ .
+
+If we omit  $L$  and  $\log (n)$  terms for simplicity of presentation, then the bound can be written as
+
+$$
+\tilde {O} \left(\sqrt {\frac {(\sum_ {\ell = 1} ^ {L} m _ {\ell}) ^ {\frac {4 / \beta}{4 / \beta + 2 (1 - 1 / 2 \alpha)}}}{n}} + \left(\frac {\sum_ {\ell = 1} ^ {L} m _ {\ell}}{n}\right) ^ {\frac {2 \alpha}{2 \alpha + 1}}\right),
+$$
+
+where the  $\tilde{O}(\cdot)$  symbol hides the poly-log order. This is tighter than that of Corollary 1. We can see that as  $\beta$  and  $\alpha$  get large, the bound becomes tighter. Actually, by taking the limit of  $\alpha, \beta \to \infty$ , then the bound goes to  $L^2 \sqrt{\frac{\log(n)}{n}} + L \frac{\sum_{\ell=1}^{L} m_\ell}{n}$ . Moreover, the term dependent on the width is  $O(1/n)$  with respect to the sample size  $n$  which is faster than the rate  $O\left(\sqrt{\frac{\sum_{\ell=1}^{L} m_\ell}{n}}\right)$  which was presented in Corollary 1. Hence, the low rank property of both the covariance matrix and the weight matrix helps to obtain better generalization. Although the bound contains the exponential term  $R_2^L$ , we can give a refined version that does not contain the exponential term by assuming interlayer cushion (Arora et al., 2018). See Appendix B.6 for the refined version.
+
+There appears  $\exp \left(\frac{1}{4} (2\sqrt{L} - 1)\right)$  which is exponentially dependent on  $L$ . However, this term is moderately small for realistic settings of the depth  $L$ . Actually, it is 7.27 for  $L = 20$  and 26.7 for  $L = 50$  (we can replace this term in exchange for larger polynomial dependency on  $L$ ). The bound is not optimized with respect to the dependency on the depth  $L$ . In particular, the term  $L^2 \sqrt{\log(n)/n}$  could be an artifact of the proof technique and the  $L^2$  term would be improved.
+
+Finally, we compare our bound with the following norm based bounds;  $O\left(\frac{R_2^L}{\sqrt{n}}\left(L\frac{R_{2\rightarrow 1}^{2/3}}{R_2^{2/3}}\right)^{3/2}\right)$  by Bartlett et al. (2017b) and  $O\left(\frac{1}{\sqrt{n}}\left(1 + L\kappa^{\frac{4}{3}}R_{2\rightarrow 1}^{2/3} + \kappa^{\frac{2}{3}}LR_{1\rightarrow 1}^{2/3}\right)^{3/2}\right)$  by Wei & Ma (2019). Since our bound and their bounds are derived from different conditions, we cannot tell which is better. Here, we consider a special case where  $m_\ell = m$  ( $\forall \ell$ ) and  $W^{(\ell)} = \frac{1}{m}\mathbf{11}^\top \in \mathbb{R}^{m\times m}$  which is an extreme case of low rank settings (note that  $W^{(\ell)}$  has rank 1). Then,  $R_2 = 1$ ,  $R_{\mathrm{F}} = 1$ ,
+
+$R_{2\rightarrow 1} = \sqrt{m}$  and  $R_{1\rightarrow 1} = m$  , and thus their bounds are  $O(\sqrt{m / n})$  and  $O(\sqrt{(m + m^2) / n})$  respectively. However,  $\beta$  and  $\alpha$  in our bound  $\sqrt{m^{\frac{4 / \beta}{4 / \beta + 2(1 - 1 / 2\alpha)}} / n} = \sqrt{m^{\frac{1}{1 + \beta(1 - 1 / 2\alpha) / 2}} / n}$  can be arbitrary large in this situation, so that our bound has much milder dependency on the width  $m$  . On the other hand, if the weight matrix has small norm and has no spectral decay (corresponding to small  $\alpha$  and  $\beta$  ), then our bound can be looser than theirs. Combining compression based bounds and norm based bounds would be interesting future work.
+
+# 4 CONCLUSION
+
+In this paper, we derived a compression based error bound for non-compressed network. The bound is general and it can be adopted to several compression based bound derived so far. The main difficulty lies in evaluating the population  $L_{2}$ -norm between the original network and the compressed network, but it can be overcome by utilizing the data dependent bound by the local Rademacher complexity technique. We have applied the derived bound to a situation where low rank properties of the weight matrices and the covariance matrices are assumed. The obtained bound gives much better dependency on the parameter size than ever obtained compression based ones.
+
+Acknowledgment We thank the anonymous reviewers for their valuable comments. TS was partially supported by MEXT Kakenhi (15H05707, 18K19793 and 18H03201), Japan Digital Design, and JST-CREST, Japan.
+
+# 5 PROOF OUTLINE OF THEOREM 1
+
+Remember that for a trained network  $\widehat{f},\widehat{g}$  is its compressed version that is included in a submodel  $\widehat{\mathcal{G}}$ . First, we decompose the generalization gap as
+
+$$
+\Psi (\widehat {f}) - \hat {\Psi} (\widehat {f}) = [ (\Psi (\widehat {f}) - \Psi (\widehat {g})) - (\hat {\Psi} (\widehat {f}) - \hat {\Psi} (\widehat {g})) ] + (\Psi (\widehat {g}) - \hat {\Psi} (\widehat {g})). \tag {4}
+$$
+
+The second block in the right hand side is easy to bound, i.e., by applying the standard Rademacher complexity bound (Theorem 3.1 of Mohri et al. (2012)) with the contraction inequality (Theorem 11.6 of Boucheron et al. (2013) or Theorem 4.12 of Ledoux & Talagrand (1991)), it holds that
+
+$$
+\Psi (\widehat {g}) - \hat {\Psi} (\widehat {g}) \leq 2 \bar {R} _ {n} (\widehat {\mathcal {G}}) + \sqrt {M \frac {2 t}{n}},
+$$
+
+with probability  $1 - e^{-t}$ . Since  $\widehat{\mathcal{G}}$  is a small model, this bound could be much smaller than a naive VC-dimension bound. The first block "bridges" the generalization gap of  $\widehat{g}$  to that of  $\widehat{f}$ , but bounding the first block is more involved. We make use of the local Rademacher complexity to bound the term. Suppose that the event in which  $\| \widehat{f} - \widehat{g} \|_{L_2} \leq \dot{r}$  holds has high probability (which should be proven later), then it is also expected that  $\psi(y, f) - \psi(y, g)$  has small  $L_2$ -norm. This is true because of the Lipschitz continuity assumption (Assumption 1). Actually, the Talagrand's concentration inequality yields that
+
+$$
+\Psi (\widehat {f}) - \Psi (\widehat {g}) - (\hat {\Psi} (\widehat {f}) - \hat {\Psi} (\widehat {g})) \leq C \left[ \Phi (\dot {r}) + \sqrt {\dot {r} \frac {t}{n}} + \frac {1 + t M}{n} \right],
+$$
+
+for  $\Phi (r)\coloneqq \bar{R}_n(\{\psi (f) - \psi (g)\mid f\in \widehat{\mathcal{F}},\widehat{g}\in \widehat{\mathcal{G}},\| f - g\|_{L_2}\leq \dot{r}\})$  with probability  $1 - e^{-t}$ . Since  $\Phi (\dot{r})$  requires the restriction  $\| f - g\|_{L_2}\leq \dot{r}$ , this quantity is much smaller than the standard Rademacher complexity  $\bar{R}_n(\psi (\widehat{\mathcal{F}}) - \psi (\widehat{\mathcal{G}}))$ , which yields fast convergence rate.
+
+Finally, we should bound the probability of  $\| \widehat{f} -\widehat{g}\|_{L_2}\leq \dot{r}$ . This can be bounded by utilizing the ratio type empirical process. Actually, we can show that
+
+$$
+P \left(\sup  _ {h \in \hat {\mathcal {F}} - \hat {\mathcal {G}}} \frac {(P - P _ {n}) (h ^ {2})}{P h ^ {2} + r _ {*} ^ {2}} \geq \frac {1}{2}\right) \leq e ^ {- t},
+$$
+
+for  $r_*$  defined in Eq. (2), where  $P_n f \coloneqq \frac{1}{n} \sum_{i=1}^{n} f(z_i)$  and  $P_f = \operatorname{E}[f(Z)]$ . This yields that  $\| \widehat{f} - \widehat{g} \|_{L_2}^2 - \| \widehat{f} - \widehat{g} \|_n^2 \leq \frac{1}{2} (\| \widehat{f} - \widehat{g} \|_{L_2}^2 + r_*^2)$  with probability  $1 - e^{-t}$  and equivalently  $\| \widehat{f} - \widehat{g} \|_{L_2}^2 \leq 2(\| \widehat{f} - \widehat{g} \|_n^2 + r_*^2)$ . Since  $\| \widehat{f} - \widehat{g} \|_n^2 \leq \hat{r}^2$  a.s., we have  $\| \widehat{f} - \widehat{g} \|_{L_2}^2 \leq 2(\hat{r}^2 + r_*^2) = \dot{r}^2$ .
+
+We can show that  $\Phi (\dot{r})\leq \dot{R}_r(\psi (\widehat{\mathcal{F}}) - \psi (\widehat{\mathcal{G}}))$  by using the Lipschitz continuity assumption (Assumption 1). Then, we obtain the assertion.
+
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+
+# NOTATION LISTS
+
+Since we use plenty of notations, we give the notation list in Table 2.
+
+# APPENDIX
+
+In the appendix, we give the proofs of the main text. We use the following notation throughout the appendix:
+
+$$
+P _ {n} f := \frac {1}{n} \sum_ {i = 1} ^ {n} f (z _ {i}), P f = \operatorname {E} [ f (Z) ].
+$$
+
+To evaluate it, the covering number is useful (van der Vaart & Wellner, 1996).
+
+Definition 1 (Covering number). For a metric space  $\tilde{\mathcal{F}}$  equipped with a metric  $\tilde{d}$ , the  $\epsilon$ -covering number  $\mathcal{N}(\tilde{\mathcal{F}}, \tilde{d}, \epsilon)$  is defined as the minimum number of balls with radius  $\epsilon$  (measured by the metric  $\tilde{d}$ ) to cover the metric space  $\tilde{\mathcal{F}}$ .
+
+Hereafter,  $C$  denotes a constant which will be dependent on the context. We let  $a \wedge b \coloneqq \min \{a, b\}$  and  $a \vee b \coloneqq \max \{a, b\}$  for  $a, b \in \mathbb{R}$ .
+
+# A PROOF OF THEOREM 1
+
+Denote the local Rademacher complexity of  $\{\psi(f) - \psi(g) \mid f \in \widehat{\mathcal{F}}, \widehat{g} \in \widehat{\mathcal{G}}, \|f - g\|_{L_2} \leq r\}$  by
+
+$$
+\Phi (r) := \bar {R} _ {n} (\{\psi (f) - \psi (g) \mid f \in \widehat {\mathcal {F}}, \widehat {g} \in \widehat {\mathcal {G}}, \| f - g \| _ {L _ {2}} \leq r \}).
+$$
+
+Here, we restate Theorem 1 in the following in more complete form. Remember that  $\widehat{f} \in \widehat{\mathcal{F}}$  and  $\widehat{g} \in \widehat{\mathcal{G}}$  are the trained original network and the compressed network respectively.
+
+Table 2: Notation list  
+
+<table><tr><td>notation</td><td>definition</td></tr><tr><td>n</td><td>sample size</td></tr><tr><td>zi=(xi,yi)</td><td>i-th observation (xi: input, yi: output)</td></tr><tr><td>Dn=(zi)i=1</td><td>training data</td></tr><tr><td>ψ(y,f(x))</td><td>loss function</td></tr><tr><td>M</td><td>L∞-norm bound of models</td></tr><tr><td>Bx</td><td>norm bound of input</td></tr><tr><td>||·||n</td><td>empirical L2-norm (||f||n := √∑i=1n f(zi)2/n)</td></tr><tr><td>||·||L2</td><td>population L2-norm (||f||L2 := √EZ~P[f(Z)2])</td></tr><tr><td>Ψ(f)</td><td>training error (empirical risk)</td></tr><tr><td>Ψ(f)</td><td>generalization error (expected risk)</td></tr><tr><td>(εi)i=1</td><td>Rademacher random variable</td></tr><tr><td>Rn(F&#x27;)</td><td>conditional Rademacher complexity</td></tr><tr><td>Rn(F&#x27;)</td><td>Rademacher complexity</td></tr><tr><td>Rr(F&#x27;)</td><td>local Rademacher complexity</td></tr><tr><td>r̂</td><td>upper bound of ||f̂ - g||n</td></tr><tr><td>r*</td><td>fixed point of the local Rademacher complexity (Eq. (2))</td></tr><tr><td>r̂</td><td>√2(r2+r*)</td></tr><tr><td>L</td><td>depth of networks</td></tr><tr><td>W(ℓ)</td><td>weight matrix of the ℓ-th layer</td></tr><tr><td>Σ(ℓ)</td><td>covariance matrix of the ℓ-th layer</td></tr><tr><td>R2</td><td>operator norm bound of W(ℓ)</td></tr><tr><td>RF</td><td>Frobenius norm bound of W(ℓ)</td></tr><tr><td>m=(m1,...,mL)</td><td>list of widths of networks</td></tr><tr><td>m#=(m1,m2,...,mL#)</td><td>list of widths of compressed networks</td></tr><tr><td>s=(s1,...,sL)</td><td>list of ranks of the weight matrices of compressed networks</td></tr><tr><td>F=NN(m,R2,RF)</td><td>the whole set of networks with width m</td></tr><tr><td>F̂</td><td>set of trained networks</td></tr><tr><td>Ĝ</td><td>set of compressed networks</td></tr><tr><td>f̂ ∈ F̂</td><td>trained network</td></tr><tr><td>ĝ ∈ Ĝ</td><td>compressed network</td></tr><tr><td>μj(ℓ)</td><td>an upper bound of the j-th largest eigenvalue of the covariance ma-trix in the ℓ-th layer of f̂</td></tr><tr><td>α</td><td>decreasing rate of the eigenvalues of W(ℓ)</td></tr><tr><td>β</td><td>decreasing rate of the eigenvalues of Σ(ℓ)</td></tr></table>
+
+Theorem 5. Suppose that the empirical  $L_{2}$ -distance between  $\widehat{f}$  and  $\widehat{g}$  is bounded by  $\| \widehat{f} - \widehat{g} \|_n \leq \widehat{r}^2$  for a fixed  $\widehat{r} > 0$  almost surely. Let  $\dot{r} \coloneqq \sqrt{2(\widehat{r}^2 + r_*^2)}$ , then, under Assumptions 1, 2, 3, there exists a universal constant  $C > 0$  such that
+
+$$
+\Psi (\widehat {f}) \leq \hat {\Psi} (\widehat {f}) + 2 \bar {R} _ {n} (\widehat {\mathcal {G}}) + \sqrt {\frac {2 M t}{n}} + C \left[ \Phi (\dot {r}) + \dot {r} \sqrt {\frac {t}{n}} + \frac {1 + t M}{n} \right].
+$$
+
+with probability at least  $1 - 3e^{-t}$  for all  $t \geq 1$ .
+
+Note that  $\dot{R} (\psi (\widehat{\mathcal{F}}) - \psi (\widehat{\mathcal{G}}))$  in the statement of Theorem 1 of the main body is replaced by refined quantity  $\Phi (\dot{r})$  (we can show  $\Phi (\dot{r})\leq \dot{R} (\psi (\widehat{\mathcal{F}}) - \psi (\widehat{\mathcal{G}}))$  from the Lipschitz continuity of  $\psi$ ).
+
+Proof of Theorems 1 and 5. First, by the standard Rademacher complexity analysis, we have that
+
+$$
+\Psi (\widehat {g}) - \hat {\Psi} (\widehat {g}) = \frac {1}{n} \sum_ {i = 1} ^ {n} \left(\psi \left(z _ {i}, \widehat {g} \left(x _ {i}\right)\right) - \operatorname {E} [ \psi (Z, \widehat {g} (X)) ]\right)
+$$
+
+$$
+\begin{array}{l} \leq \sup  _ {g \in \widehat {\mathcal {G}}} \frac {1}{n} \sum_ {i = 1} ^ {n} (\psi (z _ {i}, g (x _ {i})) - \operatorname {E} [ \psi (Z, g (X)) ]) \\ \leq 2 \mathrm {E} _ {D _ {n, \epsilon}} \left[ \sup  _ {g \in \widehat {\mathcal {G}}} \frac {1}{n} \sum_ {i = 1} ^ {n} \epsilon_ {i} \psi \left(z _ {i}, g \left(x _ {i}\right)\right) \right] + \sqrt {M \frac {2 t}{n}} \leq 2 \bar {R} _ {n} (\widehat {\mathcal {G}}) + \sqrt {M \frac {2 t}{n}} \tag {5} \\ \end{array}
+$$
+
+with probability  $1 - e^{-t}$ , where we used the Rademacher concentration inequality (Theorem 3.1 of Mohri et al. (2012)) in the third line and the contraction inequality (Theorem 11.6 of Boucheron et al. (2013) or Theorem 4.12 of Ledoux & Talagrand (1991) and its proof) in the last line. We let this event be  $\mathcal{E}_0(t)$ .
+
+Next, we observe that
+
+$$
+\begin{array}{l} \Psi (\hat {f}) - \hat {\Psi} (\hat {f}) = \Psi (\hat {f}) - \Psi (\hat {g}) + \Psi (\hat {g}) - \hat {\Psi} (\hat {g}) + \hat {\Psi} (\hat {g}) - \hat {\Psi} (\hat {f}) \\ = (\Psi (\widehat {f}) - \Psi (\widehat {g}) - (\hat {\Psi} (\widehat {f}) - \hat {\Psi} (\widehat {g}))) + \Psi (\widehat {g}) - \hat {\Psi} (\widehat {g}) \\ \leq \left[ \Psi (\widehat {f}) - \Psi (\widehat {g}) - \left(\hat {\Psi} (\widehat {f}) - \hat {\Psi} (\widehat {g})\right) \right] + 2 \bar {R} _ {n} (\widehat {\mathcal {G}}) + \sqrt {M \frac {2 t}{n}} \tag {6} \\ \end{array}
+$$
+
+where we used Eq. (5) in the last line. Here, it should be noticed that it is not a good strategy to bound the first term  $\Psi (\widehat{f}) - \Psi (\widehat{g}) - (\hat{\Psi} (\widehat{f}) - \hat{\Psi} (\widehat{g}))$  by bounding  $\Psi (\widehat{f}) - \hat{\Psi} (\widehat{f})$  and  $\Psi (\widehat{g}) - \hat{\Psi} (\widehat{g})$  independently. Instead, we should bound them simultaneously to obtain tighter bound. This can be accomplished by using the local Rademacher complexity technique.
+
+Note that  $\widehat{f}$  and  $\widehat{g}$  are date dependent and we can only bound the empirical  $L_{2}$ -distance between them. On the other hand, the local Rademacher complexity is characterized by the population  $L_{2}$ -norm. To bridge this gap, we need to bound the population  $L_{2}$ -distance  $\| \widehat{f} - \widehat{g} \|_{L_2}$  between  $\widehat{f}$  and  $\widehat{g}$  in terms of the empirical  $L_{2}$ -norm bound  $\| \widehat{f} - \widehat{g} \|_n \leq \hat{r}$ . To do so, we also bound the local Rademacher complexity of  $\widehat{\mathcal{F}} - \widehat{\mathcal{G}}$ :  $\dot{R}_r(\widehat{\mathcal{F}} - \widehat{\mathcal{G}})$ . Suppose that there exists a function  $\phi : [0, \infty) \to [0, \infty)$  such that the following conditions are satisfied:
+
+$$
+\dot {R} _ {r} (\widehat {\mathcal {F}} - \widehat {\mathcal {G}}) \leq \phi (r)
+$$
+
+and
+
+$$
+\phi (2 r) \leq 2 \phi (r).
+$$
+
+Note that Eq. (11) gives one example of  $\phi(r)$ . Then, by the so-called peeling device, we can show that for any  $r > 0$ ,
+
+$$
+P \left(\sup _ {h \in \widehat {\mathcal {F}} - \widehat {\mathcal {G}}} \frac {(P - P _ {n}) (h ^ {2})}{P h ^ {2} + r ^ {2}} \geq 8 \frac {\phi (r)}{r ^ {2}} + M \sqrt {\frac {4 t}{r ^ {2} n}} + M ^ {2} \frac {2 t}{r ^ {2} n}\right) \leq e ^ {- t}
+$$
+
+for all  $t > 0$  (Theorem 7.7 and Eq. (7.17) of Steinwart & Christmann (2008)). Hence, if we choose  $r_* = r_*(t)$  so that
+
+$$
+8 \frac {\phi (r _ {*})}{r _ {*} ^ {2}} + M \sqrt {\frac {4 t}{r _ {*} ^ {2} n}} + M ^ {2} \frac {2 t}{r _ {*} ^ {2} n} \leq \frac {1}{2},
+$$
+
+then it holds that
+
+$$
+P \left(h ^ {2}\right) \leq 2 P _ {n} \left(h ^ {2}\right) + 2 r _ {*} ^ {2}
+$$
+
+uniformly over all  $h \in \widehat{\mathcal{F}} - \widehat{\mathcal{G}}$  with probability greater than  $1 - e^{-t}$ . We let this event as  $\mathcal{E}_1(t)$ . In this event, if  $\| \widehat{f} - \widehat{g} \|_n^2 \leq \hat{r}^2$ , then it holds that
+
+$$
+\left\| \widehat {f} - \widehat {g} \right\| _ {L _ {2}} ^ {2} \leq 2 (\widehat {r} ^ {2} + r _ {*} ^ {2}) = \dot {r} ^ {2}.
+$$
+
+Next, we bound  $\Psi (\widehat{f}) - \Psi (\widehat{g}) - (\hat{\Psi} (\widehat{f}) - \hat{\Psi} (\widehat{g}))$ . To bound this term, we apply the Talagrand's concentration inequality (Proposition 2 and Talagrand (1996); Bousquet (2002)). To apply it, we should bound the variance and  $L_{\infty}$ -norm of  $\psi (y,f(x)) - \psi (y,g(x)) - (\operatorname {E}[\psi (Y,f(X))] - \operatorname {E}[\psi (Y,g(X)])$  for any  $f\in \widehat{\mathcal{F}},g\in \widehat{\mathcal{G}}$  with  $\| f - g\|_{L_2}\leq r$  (where  $r$  will be set  $2(\dot{r}^{2} + r_{*}^{2}))$ ). Due to the Lipschitz continuity of  $\psi$ , we have that
+
+$$
+\operatorname {V a r} [ \psi (Y, f (X)) - \psi (Y, g (X)) ] \leq \operatorname {V a r} [ f (X) - g (X) ] \leq \| f - g \| _ {L _ {2}} ^ {2} \leq r ^ {2}.
+$$
+
+Similarly, it holds that
+
+$$
+\begin{array}{l} \left. \left| \psi (y, f (x)) - \psi (y, g (x)) - (\operatorname {E} [ \psi (Y, f (X)) ] - \operatorname {E} [ \psi (Y, g (X)) ]) \right| \right. \\ \leq | \psi (y, f (x)) - \mathrm {E} [ \psi (Y, f (X)) ] | + | \psi (y, g (x)) - \mathrm {E} [ \psi (Y, g (X)) ]) | \leq 2 M. \\ \end{array}
+$$
+
+Hence, by the Talagrand's concentration inequality (Proposition 2 and Talagrand (1996); Bousquet (2002)), it holds that
+
+$$
+\begin{array}{l} \sup  \quad (P - P _ {n}) (\psi (Y, f (X)) - \psi (Y, g (X))) \\ f, g: \| f - g \| _ {L _ {2}} \leq r \\ \leq 2 \mathrm {E} \left[ \sup  _ {f, g: \| f - g \| _ {L _ {2}} \leq r} (P - P _ {n}) (\psi (Y, f (X)) - \psi (Y, g (X))) \right] + r \sqrt {\frac {2 t}{n}} + \frac {4 t M}{n}, \\ \end{array}
+$$
+
+with probability at least  $1 - e^{-t}$  for any  $t > 0$ . The first term in the right hand side can be bounded as
+
+$$
+\begin{array}{l} \operatorname {E} \left[ \sup  _ {f, g: \| f - g \| _ {L _ {2}} \leq r} (P - P _ {n}) (\psi (Y, f (X)) - \psi (Y, g (X))) \right] \\ \leq 2 \mathrm {E} _ {D _ {n, \epsilon}} \left[ \sup  _ {f, g: \| f - g \| _ {L _ {2}} \leq r} \frac {1}{n} \sum_ {i = 1} ^ {n} \epsilon_ {i} [ \psi (y _ {i}, f (x _ {i})) - \psi (y _ {i}, g (x _ {i})) ] \right] \\ = 2 \Phi (n), \\ \end{array}
+$$
+
+where we used the standard symmetrization argument (see Lemma 11.4 of Boucheron et al. (2013) for example).
+
+Combining these inequalities, it holds that
+
+$$
+\sup  _ {f, g: \| f - g \| _ {L _ {2}} \leq r} (P - P _ {n}) (\psi (Y, f (X)) - \psi (Y, g (X))) \leq C \left[ \Phi (r) + r \sqrt {\frac {t}{n}} + \frac {1 + M t}{n} \right].
+$$
+
+for a universal constant  $C > 0$  with probability at least  $1 - e^{-t}$  for all  $t > 0$ . We denote by this event as  $\mathcal{E}_2(t,r)$ .
+
+We define an event  $\mathcal{E}_3(t) = \mathcal{E}_1(t)\cap \mathcal{E}_2(t,\dot{r})$ . Then  $P(\mathcal{E}_3(t))\geq 1 - 2e^{-t}$  for all  $t > 0$ . In this event,  $\| \widehat{f} -\widehat{g}\|_{L_2}^2\leq \dot{r}^2$  and thus it holds that
+
+$$
+\Psi (\widehat {f}) - \Psi (\widehat {g}) - (\hat {\Psi} (\widehat {f}) - \hat {\Psi} (\widehat {g})) \leq C \left[ \Phi (\dot {r}) + \dot {r} \sqrt {\frac {t}{n}} + \frac {1 + t M}{n} \right],
+$$
+
+for a universal constant  $C > 0$ . Combining this and Eq. (6), we obtain the assertion on the event  $\mathcal{E}_0(t) \cap \mathcal{E}_3(t)$ . This gives the proof of Theorem 5.
+
+To show Theorem 1, note that  $\| \psi (f) - \psi (g)\|_{L_2}\leq \| f - g\|_{L_2}$  by the Lipschitz continuity of  $\psi$  and this yields  $\{\psi (f) - \psi (g)\mid f\in \widehat{\mathcal{F}},g\in \widehat{\mathcal{G}},\| f - g\|_{L_2}\leq r\} \subset \{\psi (f) - \psi (g)\mid f\in \widehat{\mathcal{F}},g\in \widehat{\mathcal{G}},\| \psi (f) - \psi (g)\|_{L_2}\leq r\}$ . Therefore, we have
+
+$$
+\Phi (\dot {r}) \leq \dot {R} _ {\dot {r}} (\psi (\widehat {\mathcal {F}}) - \psi (\widehat {\mathcal {G}})).
+$$
+
+Hereafter, we derive some upper bounds of the (local) Rademacher complexities under some covering number conditions.
+
+Lemma 1. For a given  $r > 0$ , let  $\hat{\gamma}_n = \hat{\gamma}_n(D_n) \coloneqq \sup \{\| f - g \|_n \mid \| f - g \|_{L_2} \leq r, f \in \widehat{\mathcal{F}}, g \in \widehat{\mathcal{G}}\}$ . Then, it holds that
+
+$$
+\begin{array}{l} \max  \{\Phi (r), \dot {R} _ {r} (\widehat {\mathcal {F}} - \widehat {\mathcal {G}}) \} \\ \leq C \left\{\frac {1}{n} + \mathrm {E} _ {D _ {n}} \left[ \int_ {1 / n} ^ {\hat {\gamma} _ {n}} \sqrt {\frac {\log \left(\mathcal {N} \left(\widehat {\mathcal {F}} , \| \cdot \| _ {n} , \epsilon / 2\right)\right)}{n}} \mathrm {d} \epsilon + \int_ {1 / n} ^ {\hat {\gamma} _ {n}} \sqrt {\frac {\log \left(\mathcal {N} \left(\widehat {\mathcal {G}} , \| \cdot \| _ {n} , \epsilon / 2\right)\right)}{n}} \mathrm {d} \epsilon \right] \right\}, \tag {7} \\ \end{array}
+$$
+
+and
+
+$$
+\Phi (r) \leq C ^ {\prime} \dot {R} _ {r} (\psi (\widehat {\mathcal {F}}) - \psi (\widehat {\mathcal {G}})) \sqrt {\log (n)} \log (2 n M), \tag {8}
+$$
+
+where  $C, C'$  are universal constants.
+
+Proof. The conditional Rademacher complexity of the set  $\{\psi(y, f(x)) - \psi(y, g(x)) \mid f \in \widehat{\mathcal{F}}, g \in \widehat{\mathcal{G}}, \|f - g\|_{L_2} \leq r\}$  can be bounded by a constant times the following Dudley integral (see Theorem 5.22 of Wainwright (2019) or Lemma A.5 of Bartlett et al. (2017a) for example):
+
+$$
+\begin{array}{l} \inf _ {\alpha > 0} \left[ \alpha + \int_ {\alpha} ^ {\hat {\gamma} _ {n}} \sqrt {\frac {\log (\mathcal {N} (\{\psi (f) - \psi (g) \mid f \in \widehat {\mathcal {F}} , g \in \widehat {\mathcal {G}} , \| f - g \| _ {L _ {2}} \leq r \} , \| \cdot \| _ {n} , \epsilon))}{n}} \mathrm {d} \epsilon \right] \\ \leq \frac {1}{n} + \int_ {1 / n} ^ {\hat {\gamma} _ {n}} \sqrt {\frac {\log (\mathcal {N} \left(\{\psi (f) - \psi (g) \mid f \in \widehat {\mathcal {F}} , g \in \widehat {\mathcal {G}} , \| f - g \| _ {L _ {2}} \leq r \} , \| \cdot \| _ {n} , \epsilon\right))}{n}} d \epsilon (9) \\ \leq \frac {1}{n} + \int_ {1 / n} ^ {\hat {\gamma} _ {n}} \sqrt {\frac {\log (\mathcal {N} (\psi (\widehat {\mathcal {F}}) , \| \cdot \| _ {n} , \epsilon / 2)) + \log (\mathcal {N} (\psi (\widehat {\mathcal {G}}) , \| \cdot \| _ {n} , \epsilon / 2))}{n}} d \epsilon \\ \leq \frac {1}{n} + \int_ {1 / n} ^ {\hat {\gamma} _ {n}} \sqrt {\frac {\log (\mathcal {N} (\widehat {\mathcal {F}} , \| \cdot \| _ {n} , \epsilon / 2)) + \log (\mathcal {N} (\widehat {\mathcal {G}} , \| \cdot \| _ {n} , \epsilon / 2))}{n}} d \epsilon \\ \leq \frac {1}{n} + \int_ {1 / n} ^ {\hat {\gamma} _ {n}} \sqrt {\frac {\log \left(\mathcal {N} \left(\widehat {\mathcal {F}} , \| \cdot \| _ {n} , \epsilon / 2\right)\right)}{n}} d \epsilon + \int_ {1 / n} ^ {\hat {\gamma} _ {n}} \sqrt {\frac {\log \left(\mathcal {N} \left(\widehat {\mathcal {G}} , \| \cdot \| _ {n} , \epsilon / 2\right)\right)}{n}} d \epsilon , (10) \\ \end{array}
+$$
+
+where we used  $\| \psi (f) - \psi (g) - (\psi (f^{\prime}) - \psi (g^{\prime}))\|_{n}\leq \| \psi (f) - \psi (f^{\prime})\|_{n} + \| \psi (g) - \psi (g^{\prime})\|_{n}\leq \epsilon$  for  $f,f^{\prime}\in \widehat{\mathcal{F}}$  and  $g,g^{\prime}\in \widehat{\mathcal{G}}$  with  $\| f - f^{\prime}\| _n\leq \epsilon /2$  and  $\| g - g^{\prime}\| _n\leq \epsilon /2$  in the third line, and 1-Lipschitz continuity of the loss function  $\psi$  in the fourth line (i.e.,  $|\psi (y,f(x)) - \psi (y,g(x))|\leq |f(x) - g(x)|$  which yields  $\| \psi (f) - \psi (g)\| _n\leq \| f - g\| _n$
+
+In the same way, we can see that  $\dot{R}_r(\widehat{\mathcal{F}} -\widehat{\mathcal{G}})$  is bounded by the Dudley integral as
+
+$$
+\begin{array}{l} \dot {R} _ {r} (\widehat {\mathcal {F}} - \widehat {\mathcal {G}}) \\ \leq \frac {C}{n} + C \mathrm {E} _ {D _ {n}} \left[ \int_ {1 / n} ^ {\hat {\gamma} _ {n}} \sqrt {\frac {\log (\mathcal {N} (\{\hat {f} - \hat {g} \mid \hat {f} \in \hat {\mathcal {F}} , \hat {g} \in \hat {\mathcal {G}} , \| \hat {f} - \hat {g} \| _ {L _ {2}} \leq r \} , \| \cdot \| _ {n} , \epsilon))}{n}} \mathrm {d} \epsilon \right] \\ \leq \frac {C}{n} + C \mathrm {E} _ {D _ {n}} \left[ \int_ {1 / n} ^ {\hat {\gamma} _ {n}} \sqrt {\frac {\log \left(\mathcal {N} \left(\widehat {\mathcal {F}} , \| \cdot \| _ {n} , \epsilon / 2\right)\right)}{n}} \mathrm {d} \epsilon + \int_ {1 / n} ^ {\hat {\gamma} _ {n}} \sqrt {\frac {\log \left(\mathcal {N} \left(\widehat {\mathcal {G}} , \| \cdot \| _ {n} , \epsilon / 2\right)\right)}{n}} \mathrm {d} \epsilon \right], \tag {11} \\ \end{array}
+$$
+
+where we used the same argument as Eq. (10) and  $C > 0$  is a universal constant. Then, we conclude Eq. (7).
+
+Next, we show Eq. (8). The term Eq. (9) can be evaluated by using the local Rademacher complexity of  $\psi (\widehat{\mathcal{F}}) - \psi (\widehat{\mathcal{G}})$ . Note that  $\| \psi (f) - \psi (g)\|_{L_2}\leq \| f - g\|_{L_2}$  by the Lipschitz continuity of  $\psi$  and this yields  $\{\psi (f) - \psi (g)\mid f\in \widehat{\mathcal{F}},g\in \widehat{\mathcal{G}},\| f - g\|_{L_2}\leq r\} \subset \{\psi (f) - \psi (g)\mid f\in \widehat{\mathcal{F}},g\in \widehat{\mathcal{G}},\| \psi (f) - \psi (g)\|_{L_2}\leq r\}$ . Then, the Sudakov's minoration (Corollary 4.14 of Ledoux & Talagrand (1991)) gives an upper bound of the right hand side of Eq. (9):
+
+$$
+\begin{array}{l} \int_ {1 / n} ^ {\hat {\gamma} _ {n}} \sqrt {\frac {\log (\mathcal {N} (\{\psi (f) - \psi (g) \mid f \in \widehat {\mathcal {F}} , g \in \widehat {\mathcal {G}} , \| f - g \| _ {L _ {2}} \leq r \} , \| \cdot \| _ {n} , \epsilon))}{n}} d \epsilon \\ \leq \int_ {1 / n} ^ {\hat {\gamma} _ {n}} \sqrt {\frac {\log (\mathcal {N} (\{\psi (f) - \psi (g) \mid f \in \widehat {\mathcal {F}} , g \in \widehat {\mathcal {G}} , \| \psi (f) - \psi (g) \| _ {L _ {2}} \leq r \} , \| \cdot \| _ {n} , \epsilon))}{n}} d \epsilon \\ \leq \int_ {1 / n} ^ {\hat {\gamma} _ {n}} \hat {R} _ {n, r} (\psi (\widehat {\mathcal {F}}) - \psi (\widehat {\mathcal {G}})) \sqrt {\log (n)} \frac {1}{\epsilon} \mathrm {d} \epsilon \leq \hat {R} _ {n, r} (\psi (\widehat {\mathcal {F}}) - \psi (\widehat {\mathcal {G}})) \sqrt {\log (n)} \log (n \hat {\gamma} _ {n}), \\ \end{array}
+$$
+
+where  $\hat{R}_{n,r}(\psi (\widehat{\mathcal{F}}) - \psi (\widehat{\mathcal{G}}))\coloneqq \operatorname{E}_{\epsilon}\left[\sup \{\frac{1}{n}\sum_{i = 1}^{n}\epsilon_{i}(\psi (y_{i},f(x_{i})) - \psi (y_{i},g(x_{i})))\mid f\in \widehat{\mathcal{F}},g\in \widehat{\mathcal{G}},\| \psi (f) - \psi (g)\|_{L_{2}}\leq r\}\right].$
+
+Since  $\hat{\gamma}_n \leq 2M$ , the expectation of the right hand side with respect to  $D_n$  is  $\dot{R}_r(\psi(\widehat{\mathcal{F}}) - \psi(\widehat{\mathcal{G}}))\sqrt{\log(n)}\log(2nM)$ . This gives an upper bound of the right hand side of Eq. (9) and yields Eq. (8).
+
+Lemma 2.
+
+$$
+\operatorname {E} \left[ \sup  \left\{P _ {n} h ^ {2} \mid h \in \widehat {\mathcal {F}} - \widehat {\mathcal {G}}: \| h \| _ {L _ {2}} \leq r \right\} \right] \leq r ^ {2} + 2 M \phi (r).
+$$
+
+Proof. By the contraction inequality of the Rademacher complexity (Theorem 4.12 of Ledoux & Talagrand (1991) and its proof), we have
+
+$$
+\begin{array}{l} \left. \operatorname {E} \left[ \sup  \left\{P _ {n} h ^ {2} \mid h \in \widehat {\mathcal {F}} - \widehat {\mathcal {G}}: \| h \| _ {L _ {2}} \leq r \right\} \right] \right. \\ \leq \operatorname {E} \left[ \sup  \left\{\left(P _ {n} - P\right) h ^ {2} \mid h \in \widehat {\mathcal {F}} - \widehat {\mathcal {G}}: \| h \| _ {L _ {2}} \leq r \right\} \right] + r ^ {2} \\ \leq 2 \mathrm {E} \left[ \sup  \left\{\frac {1}{n} \sum_ {i = 1} ^ {n} \epsilon_ {i} h \left(x _ {i}\right) ^ {2} \mid h \in \widehat {\mathcal {F}} - \widehat {\mathcal {G}}: \| h \| _ {L _ {2}} \leq r \right\} \right] + r ^ {2} \\ \end{array}
+$$
+
+(symmetrization; Lemma 11.4 of Boucheron et al. (2013))
+
+$$
+\begin{array}{l} \leq 2 M E \left[ \sup  \left\{\frac {1}{n} \sum_ {i = 1} ^ {n} \epsilon_ {i} h \left(x _ {i}\right) \mid h \in \widehat {\mathcal {F}} - \widehat {\mathcal {G}}: \| h \| _ {L _ {2}} \leq r \right\} \right] + r ^ {2} \quad (\because \text {c o n t r a c t i o n i n e q u a l i t y}) \\ = 2 M \dot {R} _ {r} (\widehat {\mathcal {F}} - \widehat {\mathcal {G}}) + r ^ {2} \leq 2 M \phi (r) + r ^ {2}. \\ \end{array}
+$$
+
+![](images/c97b4e9841e7b219277a63c246143c75005dd0108a66dd2c680ea1908fb18857.jpg)
+
+Lemma 3. Suppose that
+
+$$
+\sup  _ {D _ {n}} \log (\mathcal {N} (\widehat {\mathcal {F}}, \| \cdot \| _ {n}, \epsilon / 2)) + \sup  _ {D _ {n}} \log (\mathcal {N} (\widehat {\mathcal {G}}, \| \cdot \| _ {n}, \epsilon / 2)) \leq S _ {1} + S _ {2} \log (1 / \epsilon) + S _ {3} \epsilon^ {- 2 q}
+$$
+
+for  $q < 1$ . Then, for a universal constant  $C > 0$  and a constant  $C_q > 0$  which depends on  $q < 1$ , it holds that
+
+$$
+\begin{array}{l} \Phi (r) \leq C \max  \left\{\frac {1}{n} + M \frac {S _ {1} + S _ {2} \log (n)}{n} + r \sqrt {\frac {S _ {1} + S _ {2} \log (n)}{n}}, \right. \\ C _ {q} \left[ \frac {1}{n} + \left(\frac {M ^ {1 - q} S _ {3}}{n}\right) ^ {\frac {1}{1 + q}} + r ^ {1 - q} \sqrt {\frac {S _ {3}}{n}} \right] \}. \\ \end{array}
+$$
+
+In particular,
+
+$$
+r _ {*} ^ {2} \leq C \left[ M \frac {S _ {1} + S _ {2} \log (n)}{n} + \left(\frac {M ^ {1 - q} S _ {3}}{n}\right) ^ {\frac {1}{1 + q}} + \frac {1 + M t}{n} \right].
+$$
+
+Proof. Remember that  $\hat{\gamma}_n\coloneqq \sup \{\| f - g\| _n:\| f - g\|_{L_2}\leq r,f\in \widehat{\mathcal{F}},g\in \widehat{\mathcal{G}}\}$  for a given  $r > 0$  Under the assumption, we may take  $\phi (r)$  (defined below) as an upper bound of  $\Phi (r)$  by Lemma 1:
+
+$$
+\phi (r) = C \left(\frac {1}{n} + \frac {1}{\sqrt {n}} \mathrm {E} \left[ \int_ {1 / n} ^ {\hat {\gamma} _ {n}} \sqrt {S _ {1} + S _ {2} \log (\epsilon^ {- 1}) + S _ {3} \epsilon^ {- 2 q}} \mathrm {d} \epsilon \right]\right),
+$$
+
+where  $C > 0$  is a universal constant. The right hand side can be evaluated as
+
+$$
+\begin{array}{l} \mathrm {E} \left[ \int_ {1 / n} ^ {\hat {\gamma} _ {n}} \sqrt {S _ {1} + S _ {2} \log (1 / \epsilon) + S _ {3} \epsilon^ {- 2 q}} \mathrm {d} \epsilon \right] \leq \mathrm {E} \left[ \hat {\gamma} _ {n} \sqrt {S _ {1} + S _ {2} \log (n)} \right] + \frac {1}{1 - q} \sqrt {S _ {3}} \mathrm {E} [ \hat {\gamma} _ {n} ^ {1 - q} ] \\ \leq \sqrt {\operatorname {E} \left[ \left\{P _ {n} h ^ {2} \mid h \in \widehat {\mathcal {F}} - \widehat {\mathcal {G}} , \| h \| _ {L _ {2}} \leq r \right\} \right]} \sqrt {S _ {1} + S _ {2} \log (n)} \\ + \frac {\sqrt {S _ {3}}}{1 - q} \mathrm {E} [ \{P _ {n} h ^ {2} \mid h \in \widehat {\mathcal {F}} - \widehat {\mathcal {G}}, \| h \| _ {L _ {2}} \leq r \} ] ^ {\frac {1 - q}{2}} \\ \end{array}
+$$
+
+$$
+\leq \sqrt {r ^ {2} + 2 M \phi (r)} \sqrt {S _ {1} + S _ {2} \log (n)} + \frac {\sqrt {S _ {3}}}{1 - q} \left(r ^ {2} + 2 M \phi (r)\right) ^ {\frac {1 - q}{2}}, \tag {12}
+$$
+
+where we used Lemma 2. Hence, if the first term is larger than the second term, we have that
+
+$$
+\begin{array}{l} \phi (r) \leq C \left(\frac {1}{n} + \sqrt {\frac {S _ {1} + S _ {2} \log (n)}{n}} \sqrt {r ^ {2} + 2 M \phi (r)}\right) \\ \leq \frac {C}{n} + C ^ {2} M \frac {S _ {1} + S _ {2} \log (n)}{n} + C r \sqrt {\frac {S _ {1} + S _ {2} \log (n)}{n}} + \frac {\phi (r)}{2}. \\ \end{array}
+$$
+
+Therefore, we obtain that
+
+$$
+\phi (r) \leq \frac {2 C}{n} + 2 C ^ {2} M \frac {S _ {1} + S _ {2} \log (n)}{n} + 2 C r \sqrt {\frac {S _ {1} + S _ {2} \log (n)}{n}}. \tag {13}
+$$
+
+On the other hand, if the second term in Eq. (12) is larger than the first one, then Young's inequality gives that
+
+$$
+\begin{array}{l} \phi (r) \leq C \left(\frac {1}{n} + \sqrt {\frac {S _ {3}}{n (1 - q) ^ {2}}} \left(r ^ {2} + 2 M \phi (r)\right) ^ {\frac {1 - q}{2}}\right) \\ \leq \frac {C}{n} + C \left[ q \left(\frac {c ^ {\prime 1 - q} C ^ {2} S _ {3}}{n (1 - q) ^ {2}}\right) ^ {\frac {1}{1 + q}} + (1 - q) \frac {2 M \phi (r)}{c ^ {\prime}} + \sqrt {\frac {S _ {3}}{n (1 - q) ^ {2}} r ^ {2 (1 - q)}} \right]. \\ \end{array}
+$$
+
+where  $c' > 0$  is any positive real. Thus taking  $c'$  sufficiently large (which depends on  $M, q$ ), we conclude that
+
+$$
+\phi (r) \leq C _ {q} \left[ \frac {1}{n} + \left(\frac {M ^ {1 - q} S _ {3}}{n}\right) ^ {\frac {1}{1 + q}} + \sqrt {\frac {S _ {3}}{n} r ^ {2 (1 - q)}} \right], \tag {14}
+$$
+
+where  $C_q$  is a constant depending only on  $q < 1$ . These two inequalities (Eq. (13) and Eq. (14)) give the first assertion. By noticing the assumption  $M \geq 1$ ,  $r_*$  can be derived from a simple calculation.
+
+![](images/73850f20058b67790abb5a5d321dcb522abfb5ea30d48a0acacc37a8b7885100.jpg)
+
+# B DERIVATION OF COMPRESSION BASED BOUND FOR NON-COMPRESSSED NETWORKS
+
+# B.1 FULL MODEL BOUND
+
+Here, we assume that the model  $\widehat{\mathcal{F}}$  of the trained network is the full model  $\widehat{\mathcal{F}} = \mathcal{F} = \mathrm{NN}(\mathbf{m}, R_2, R_{\mathrm{F}})$  and  $\widehat{\mathcal{G}}$  is included in  $\mathrm{NN}(\mathbf{m}, \hat{R}_2, \hat{R}_{\mathrm{F}})$ . Then, their covering entropy is bounded by
+
+$$
+\log (\mathcal {N} (\widehat {\mathcal {F}}, \| \cdot \| _ {\infty}, \epsilon)) \leq (\sum_ {\ell = 1} ^ {L} m _ {\ell} m _ {\ell + 1}) \log (\epsilon^ {- 1}) + L (\sum_ {\ell = 1} ^ {L} m _ {\ell} m _ {\ell + 1}) \log (L (R _ {2} \vee 1) (\max _ {\ell} m _ {\ell} + 1)),
+$$
+
+$$
+\log (\mathcal {N} (\widehat {\mathcal {G}}, \| \cdot \| _ {\infty}, \epsilon)) \leq (\sum_ {\ell = 1} ^ {L} m _ {\ell} m _ {\ell + 1}) \log (\epsilon^ {- 1}) + L (\sum_ {\ell = 1} ^ {L} m _ {\ell} m _ {\ell + 1}) \log (L (\hat {R} _ {2} \lor 1) (\max _ {\ell} m _ {\ell} + 1)).
+$$
+
+Hence, the condition in Lemma 3 holds for  $S_{1} = \sum_{\ell = 1}^{L}m_{\ell}m_{\ell +1}$ ,  $S_{2} = LS_{1}\log (L(R_{2}\lor \hat{R}_{2}\lor 1)(\max_{\ell}m_{\ell} + 1))$  and  $S_{3} = 0$ . In this case, we can set
+
+$$
+r _ {*} ^ {2} = C \frac {(M + 1) (S _ {1} + 1 + S _ {2} \log (n)) + M t}{n}
+$$
+
+for a constant  $C > 0$
+
+# B.2 COMPLEXITY OF A SPARSE MODEL (PROOF OF EXAMPLE 1)
+
+Suppose that  $\widehat{\mathcal{G}}$  is the model with sparse weight matrices given in Example 1. Let  $m = \max_{\ell} m_{\ell}$  and  $B = R_2$ , then we can see that
+
+$$
+\hat {\mathcal {G}} \subset \Phi (L, m, S, B)
+$$
+
+where the definition of  $\Phi(L, m, S, B)$  is given in Appendix C.1. Therefore, its covering number is bounded by
+
+$$
+\begin{array}{l} \log (\mathcal {N} (\widehat {\mathcal {G}}, \| \cdot \| _ {\infty}, \epsilon)) \\ \leq S \log (\epsilon^ {- 1}) + L S \log (L (R _ {2} \vee 1) (\max _ {\ell} m _ {\ell} + 1)) \leq O (S L \log (n) + S \log (\epsilon^ {- 1})), \\ \end{array}
+$$
+
+by Lemma 4. Hence, the Rademacher complexity is bounded as
+
+$$
+\widehat {\mathcal {G}} \leq C M \sqrt {L \frac {S}{n} \log (n L (R _ {2} \vee 1) (\max _ {\ell} m _ {\ell} + 1))} = O \left(M \sqrt {L \frac {S}{n} \log (n)}\right),
+$$
+
+by the Dudley integral,  $\bar{R} (\widehat{\mathcal{G}})\lesssim \int_0^M\sqrt{\frac{\log(\mathcal{N}(\widehat{\mathcal{G}},\|.\cdot\|_{\infty},\epsilon))}{n}}\mathrm{d}\epsilon$  , where  $C$  is a universal constant.
+
+# B.3 NEAR LOW RANK CONDITION ON THE WEIGHT MATRIX (PROOF OF THEOREM 2 AND COROLLARY 1)
+
+Here, we give proofs of Theorem 2 and Corollary 1 which give a generalization error bound when the trained network has near low rank weight matrices  $(W^{(\ell)})_{\ell = 1}^{L}$  (Assumption 4).
+
+Under Assumption 4, we can see that for any  $1 \leq s \leq \min\{m_{\ell}, m_{\ell+1}\}$ , we can approximate  $W^{(\ell)}$  by a rank  $s$  matrix  $W'$  as
+
+$$
+\left\| W ^ {(\ell)} - W ^ {\prime} \right\| _ {2} \leq V _ {0} s ^ {- \alpha}, \left\| W ^ {\prime} \right\| _ {2} \leq \left\| W ^ {(\ell)} \right\| _ {2} \tag {15}
+$$
+
+$$
+\left\| W ^ {(\ell)} - W ^ {\prime} \right\| _ {\mathrm {F}} \leq \frac {1}{\sqrt {2 \alpha - 1}} V _ {0} (s - 1) ^ {(1 - \alpha) / 2}. \tag {16}
+$$
+
+This can be checked by discarding the singular vectors corresponding to the singular values smaller than the  $s$ -th largest one. This ensures that, for any  $f \in \widehat{\mathcal{F}}$ , there exists  $f' \in \mathcal{F}$  such that it has width  $\mathbf{s} = (s_1, \ldots, s_L)$ , weight matrix  $W^{\sharp(\ell)}$  with  $\| W^{\sharp(\ell)} \|_2 \leq R_2$ ,  $\| W^{\sharp(\ell)} \|_{\mathrm{F}} \leq R_{\mathrm{F}}$  and
+
+$$
+\left\| f - f ^ {\prime} \right\| _ {\infty} \leq \sum_ {\ell = 1} ^ {L} V _ {0} R _ {2} ^ {L - 1} s _ {\ell} ^ {- \alpha} B _ {x}. \tag {17}
+$$
+
+This can be proved as follows. Let  $f'(x) = G \circ (W^{\sharp(L)}\eta(\cdot)) \circ \dots \circ (W^{\sharp(1)}x)$  where  $W^{\sharp(\ell)}$  is a rank  $s_\ell$  matrix that satisfies Eqs. (15) and (16) for  $W' = W^{\sharp(\ell)}$ . Let  $f_\ell(x) = G \circ (W^{\sharp(L)}\eta(\cdot)) \circ \dots \circ (W^{\sharp(\ell+1)}\eta(\cdot)) \circ (W^{(\ell)}\eta(\cdot)) \circ \dots \circ (W^{(1)}x)$  and  $f_0(x) = f'$ . Then,  $\|f - f'\|_{\infty} \leq \sum_{\ell=1}^{L} \|f_\ell - f_{\ell-1}\|_{\infty}$ . We can see that  $\|(W^{(\ell)}\eta(\cdot)) \circ \dots \circ (W^{(1)}x)\| \leq \prod_{k=1}^{\ell} \|W^{(k)}\|_2 \|x\| \leq R_2^\ell B_x$ ,  $\|(W^{\sharp(\ell)}\eta(\cdot)) \circ (W^{(\ell-1)}\eta(\cdot)) \circ \dots \circ (W^{(1)}x) - (W^{(\ell)}\eta(\cdot)) \circ (W^{(\ell-1)}\eta(\cdot)) \circ \dots \circ (W^{(1)}x)\| \leq \|W^{(\ell)} - W^{\sharp(\ell)}\|_2 \|(W^{(\ell-1)}\eta(\cdot)) \circ \dots \circ (W^{(1)}x)\| \leq V_0s_\ell^{-\alpha}R_2^{\ell-1}B_x$ . This gives  $\|f_\ell - f_\ell\|_{\infty} \leq \prod_{k=\ell+1}^{L} \|W^{(k)}\|_2V_0s_\ell^{-\alpha}R_2^{\ell-1}B_x \leq R_2^{L-1}V_0s_\ell^{-\alpha}B_x$ . Finally, by summing up this from  $\ell = 1$  to  $\ell = L$ , we obtain Eq. (17).
+
+In particular, for any  $\epsilon > 0$ , by setting  $s_{\ell}^{\prime} = s_{\ell}^{\prime}(\epsilon) = \min \left\{\left[\left(\frac{1}{2}\right)\left(LV_{0}R_{2}^{L - 1}B_{x}\right)\right]^{-1 / \alpha}\right\}$ ,  $m_{\ell} \wedge m_{\ell + 1}$  for all  $\ell$ , then  $\|f - f^{\prime}\|_{\infty} \leq \epsilon$ . This indicates that, by Lemma 5, the covering entropy of  $\widehat{\mathcal{F}}$  is bounded by
+
+$$
+\begin{array}{l} \log \left(\mathcal {N} \left(\widehat {\mathcal {F}}, \| \cdot \| _ {\infty}, \epsilon\right)\right) \leq \left(\sum_ {\ell = 1} ^ {L} \left(m _ {\ell} + m _ {\ell + 1}\right) s _ {\ell} ^ {\prime} (\epsilon / 2)\right) \left[ \log \left(\epsilon^ {- 1}\right) + 2 L \log \left(2 L \left(R _ {2} \vee 1\right) \left(\max  _ {\ell} m _ {\ell} + 1\right)\right) \right] \tag {18} \\ \leq 2 \left(\sum_ {\ell = 1} ^ {L} m _ {\ell}\right) \left(2 L V _ {0} R _ {2} ^ {L - 1} B _ {x}\right) ^ {1 / \alpha} \epsilon^ {- 1 / \alpha} \left[ \log \left(\epsilon^ {- 1}\right) + 2 L \log \left(2 L \left(R _ {2} \vee 1\right) \left(\max _ {\ell} m _ {\ell} + 1\right)\right) \right]. \\ \end{array}
+$$
+
+As the compressed network  $\widehat{\mathcal{G}}$ , we may choose  $\widehat{\mathcal{G}} = \mathrm{NN}(\mathbf{m},\mathbf{s},R_2,R_{\mathrm{F}})$  for  $\mathbf{s} = (s_1,\ldots ,s_L)$  so that, for all  $f\in \widehat{\mathcal{F}}$ , there exists  $g\in \widehat{\mathcal{G}}$  satisfying
+
+$$
+\left\| f - g \right\| _ {\infty} \leq \left(V _ {0} R _ {2} ^ {L - 1} B _ {x}\right) \sum_ {\ell = 1} ^ {L} s _ {\ell} ^ {- \alpha}.
+$$
+
+Hence, we may set  $\hat{r}^2 = [(V_0R_2^{L - 1}B_x)\sum_{\ell = 1}^L s_\ell^{-\alpha}]^2$ . In this case, the covering number of  $\widehat{\mathcal{G}}$  is bounded as (18) by replacing  $s_\ell^\prime$  with  $s_\ell$ .
+
+Therefore, Lemma 3 gives that
+
+$$
+r _ {*} ^ {2} \leq C \frac {(M + 1) (S _ {1} + 1 + S _ {2} \log (n)) + M t}{n} \vee M ^ {\frac {2 \alpha - 1}{2 \alpha + 1}} \left(\frac {S _ {3}}{n}\right) ^ {\frac {2 \alpha}{1 + 2 \alpha}},
+$$
+
+where
+
+$$
+S _ {1} = \sum_ {\ell = 1} ^ {L} s _ {\ell} (m _ {\ell} + m _ {\ell + 1}),
+$$
+
+$$
+S _ {2} = L S _ {1} \log (L (R _ {2} \lor 1) (\max _ {\ell} m _ {\ell} + 1)),
+$$
+
+$$
+S _ {3} = \left(\sum_ {\ell = 1} ^ {L} m _ {\ell}\right) \left(2 L V _ {0} R _ {2} ^ {L - 1} B _ {x}\right) ^ {1 / \alpha} \left[ \log (n) + 2 L \log \left(2 L \left(R _ {2} \vee 1\right) \left(\max  _ {\ell} m _ {\ell} + 1\right)\right) \right],
+$$
+
+where  $q = 1 / 2\alpha$  was used. This indicates that, if  $\alpha > 1 / 2$  is large (in other words, each weight matrix is close to rank 1), then the local Rademacher complexity can be small. Actually, the bound is smaller than  $\frac{\sum_{\ell=1}^{L} m_{\ell} m_{\ell+1}}{n}$  because each rank  $s_{\ell}$  must satisfy  $s_{\ell} \leq \min \{m_{\ell}, m_{\ell+1}\}$ .
+
+Finally, we observe that
+
+$$
+\bar {R} _ {n} (\widehat {\mathcal {G}}) \leq C M \sqrt {L \frac {\sum_ {\ell = 1} ^ {L} s _ {\ell} (m _ {\ell} + m _ {\ell + 1})}{n} \log (n L (R _ {2} \vee 1) (\max  _ {\ell} m _ {\ell} + 1))},
+$$
+
+by Lemma 5 for  $\widehat{\mathcal{G}} = \mathrm{NN}(\mathbf{m},\mathbf{s},R_2,R_{\mathrm{F}})$  and the Dudley integral (van der Vaart & Wellner, 1996):  $\bar{R} (\widehat{\mathcal{G}})\lesssim \int_0^M\sqrt{\frac{\log(\mathcal{N}(\widehat{\mathcal{G}},\|.\cdot\|_{\infty},\epsilon))}{n}}\mathrm{d}\epsilon$  . This gives Theorem 2.
+
+Corollary 1 can be obtained by substituting  $s_{\ell} = \min \{m_{\ell}, m_{\ell + 1}, \lceil LV_0 R_2^{L - 1} B_x \rceil^{1 / \alpha}\}$ .
+
+# B.4 IMPROVED BOUND WITH LIPSCHITZ CONTINUITY CONSTRAINT
+
+In the generalization error bound of Theorem 2 and Corollary 1, there appears  $R_2^L$ . Even though  $R_2$  can be much smaller than  $R_{\mathrm{F}}$ , the exponential dependency  $R_2^L$  could give lose bound as pointed out in Arora et al. (2018). We improve this exponential dependency by assuming the following condition.
+
+Assumption 6 (Lipschitz continuity between layers: Interlayer cushion, interlayer smoothness (Arora et al., 2018)). For the trained network  $\widehat{f} = G\circ (W^{(L)}\eta (\cdot))\circ \dots \circ (W^{(1)}x)$ , let  $\phi_{\ell}(x) = \eta \circ (W^{(\ell -1)}\eta (\cdot))\circ \dots \circ (W^{(1)}x)$  be the input to the  $\ell$ -th layer and  $M_{\ell,\ell'}(x) = (W^{(\ell')}\eta (\cdot))\circ \dots \circ (W^{(\ell)}x)$  be the transformation from the  $\ell$ -th layer to  $\ell'$ -th layer. Then, we assume that there exists  $\kappa, \tau > 0$  such that  $\tau \leq 1/(2\kappa^2 L)$  and for any  $\ell, \ell' \in [L]$ ,
+
+$$
+\sum_ {i = 1} ^ {n} \left[ M _ {\ell , \ell^ {\prime}} \left(\phi_ {\ell} \left(x _ {i}\right) + \xi_ {i} ^ {(1)}\right) - M _ {\ell , \ell^ {\prime}} \left(\phi_ {\ell} \left(x _ {i}\right) + \xi_ {i} ^ {(1)} + \xi_ {i} ^ {(2)}\right) \right] ^ {2} \leq \kappa^ {2} \left(\| \xi^ {(1)} \| + \tau \| \xi^ {(2)} \|\right) ^ {2},
+$$
+
+for all  $\xi^{(1)} = (\xi_1^{(1)},\dots ,\xi_n^{(1)})^\top \in \mathbb{R}^n$  and  $\xi^{(2)} = (\xi_1^{(2)},\dots ,\xi_n^{(2)})^\top \in \mathbb{R}^n$
+
+This assumption is a simplified version of the interlayer cushion and the interlayer smoothness introduced in Arora et al. (2018). Although a trivial bound of  $\kappa$  is  $\kappa \leq R_2^L$ , the practically observed Lipschitz constant is usually much smaller. Assumption 6 captures this point and gives better dependency on the depth  $L$ . Actually, we can remove the exponential dependency on  $R_{2}$  as in the following corollary.
+
+Corollary 2. Under Assumptions 4 and 6, it holds that
+
+$$
+\Psi (\widehat {f}) \leq \hat {\Psi} (\widehat {f}) + C \left[ M ^ {1 - 1 / 2 \alpha} \sqrt {L \frac {(\sum_ {\ell = 1} ^ {L} m _ {\ell}) (2 L V _ {0} \kappa^ {2} B _ {x}) ^ {1 / \alpha}}{n} \log (n)} + M ^ {\frac {2 \alpha - 1}{2 \alpha + 1}} A _ {2} ^ {\prime} ^ {\frac {2 \alpha}{2 \alpha + 1}} + \frac {1 + t M}{n} \right]
+$$
+
+for  $A_2' = L \frac{(\sum_{\ell=1}^L m_\ell)(2LV_0\kappa^2 B_x)^{1/\alpha}}{n}$  with probability  $1 - 3e^{-t}$  for any  $t > 1$  where  $C$  is a constant depending on  $\alpha$ .
+
+This is almost same as Corollary 1, but the exponential dependency on  $R_2^L$  is replaced by the Lipschitz continuity  $\kappa^2$ .
+
+Proof of Corollary 2. Suppose that
+
+$$
+s _ {\ell} \geq \min  \left\{m _ {\ell}, m _ {\ell + 1}, \lceil (4 \kappa V _ {0} L) ^ {1 / \alpha} \rceil \right\}, \tag {19}
+$$
+
+then we show that Eq. (17) can be replaced by
+
+$$
+\left\| f - f ^ {\prime} \right\| _ {n} \leq 4 \sum_ {\ell = 1} ^ {L} V _ {0} \kappa^ {2} s _ {\ell} ^ {- \alpha} B _ {x}, \tag {20}
+$$
+
+where if  $s_{\ell} = \min \{m_{\ell}, m_{\ell + 1}\}$ , then  $s_{\ell}^{-\alpha}$  term can be replaced by 0 (which means no-compression in the layer  $\ell$ ). Once we obtain this evaluations, then the following argument is same as the proof of Theorem 2 and Corollary 1 (Sec. B.3).
+
+Let  $\phi_{\ell}(x) = \eta \circ (W^{(\ell)}\eta (\cdot))\circ \dots \circ (W^{(1)}x)$  and  $\phi_{\ell}^{\sharp}(x) = \eta \circ (W^{\sharp (\ell)}\eta (\cdot))\circ \dots \circ (W^{\sharp (1)}x)$  for  $\ell = 1,\ldots ,L - 1$  , and let  $\phi_L(x) = G\circ (W^{(L)}\eta (\cdot))\circ \dots \circ (W^{(1)}x)$  and  $\phi_L^\sharp (x) = G\circ (W^{\sharp (L)}\eta (\cdot))\circ$ $\dots \circ (W^{\sharp (1)}x)$  . Let  $C_B\coloneqq 2\kappa B_x$  . We will show that
+
+$$
+\left\| \phi_ {k} - \phi_ {k} ^ {\sharp} \right\| _ {n} \leq 2 \kappa V _ {0} C _ {B} \left(\sum_ {j = 1} ^ {k} s _ {j} ^ {- \alpha}\right), \left\| \phi_ {k} ^ {\sharp} \right\| _ {n} \leq C _ {B},
+$$
+
+for all  $k = 1, \ldots, L$ . We show this by inductive reasoning. To do so, we assume that, for  $k = 1, \ldots, \ell - 1$ , this is satisfied, and then we show this for  $k = \ell$ . Note that, for all  $k$  with  $k < \ell$ , it holds that, for any  $\ell' > k$ ,
+
+$$
+\begin{array}{l} \| M _ {k, \ell^ {\prime}} \circ \phi_ {k} ^ {\sharp} - M _ {k - 1, \ell^ {\prime}} \phi_ {k - 1} ^ {\sharp} \| _ {n} \leq \kappa (\| \phi_ {k} ^ {\sharp} - \eta (W ^ {(k)} \phi_ {k - 1} ^ {\sharp}) \| _ {n} + \tau \| \eta (W ^ {(k)} \phi_ {k - 1} ^ {\sharp}) - \phi_ {k} \| _ {n}) \\ \leq \kappa \big (V _ {0} s _ {k} ^ {- \alpha} \| \phi_ {k - 1} ^ {\sharp} \| _ {n} + \tau \kappa \| \phi_ {k - 1} ^ {\sharp} - \phi_ {k - 1} \| _ {n} \big) \\ \leq \kappa \left(V _ {0} s _ {k} ^ {- \alpha} C _ {B} + \tau \kappa 2 \kappa V _ {0} C _ {B} \sum_ {j \leq k - 1} s _ {j} ^ {- \alpha}\right) \quad (\text {b y i n d u c t i o n}) \\ \leq \kappa V _ {0} C _ {B} \left(s _ {k} ^ {- \alpha} + \frac {1}{L} \sum_ {j = 1} ^ {k - 1} s _ {j} ^ {- \alpha}\right) \quad (\text {b y}) \\ \end{array}
+$$
+
+Note that the term  $s_k^{-\alpha}$  can be replaced by 0 if  $s_k = \min \{m_k, m_{k+1}\}$  which corresponds to the full rank setting  $(W^{\sharp(k)} = W^{(k)})$ . Therefore, we have that
+
+$$
+\| \phi_ {\ell} - \phi_ {\ell} ^ {\sharp} \| _ {n} \leq \sum_ {j = 1} ^ {\ell} \| M _ {j, \ell} \circ \phi_ {j} ^ {\sharp} - M _ {j - 1, \ell} \circ \phi_ {j - 1} ^ {\sharp} \| _ {n} \leq 2 \kappa V _ {0} C _ {B} \left(\sum_ {k = 1} ^ {\ell} s _ {k} ^ {- \alpha}\right).
+$$
+
+Under the setting (19), this gives that
+
+$$
+\left\| \phi_ {\ell} - \phi_ {\ell} ^ {\sharp} \right\| _ {n} \leq C _ {B} / 2.
+$$
+
+Finally, noting that  $\| \phi_{\ell}\|_{n}\leq C_{B} / 2$  , we have
+
+$$
+\| \phi_ {\ell} ^ {\sharp} \| _ {n} \leq C _ {B}.
+$$
+
+This concludes the inductive reasoning.
+
+Finally, noting that  $f = \phi_L$  and  $f^{\prime} = \phi_{L}^{\sharp}$ , we have Eq. (20).
+
+![](images/a637afa5176f75ecf9d021c571548eb037f46dfbf843ca23c832c3c0ec753e8f.jpg)
+
+# B.5 NEAR LOW RANK CONDITION ON THE COVARIANCE MATRIX (PROOF OF THEOREM 3 AND THEOREM 4)
+
+Under Assumption 5,  $\widehat{f}$  can be compressed as follows. Suppose that the network is compressed to smaller one upto the  $\ell -1$ -th layer and the weight matrix of the compressed one is denoted by  $(W^{\sharp (k)})_{k = 1}^{\ell -1}$  where each  $W^{\sharp (k)}$  has size  $m_{k + 1}^{\sharp}\times m_k^{\sharp}$  (here,  $m_{k}^{\sharp}\leq m_{k}$  is assumed), and, in the  $\ell -1$ -th layer,  $W^{\sharp (\ell -1)}$  has size  $m_{\ell}\times m_{\ell -1}^{\sharp}$ . The input to the  $\ell$ -th layer of the compressed network is denoted by  $\phi_{\ell}^{\sharp}(x) = \eta (W^{\sharp (\ell -1)}\eta (\dots W^{\sharp (1)}x)\dots)$ . Let  $r_\ell^2 = \| |\phi_\ell -\phi_\ell^\sharp ||||_n^2$  and  $\Sigma_{(\ell)}^{\sharp}:= \frac{1}{n}\sum_{i = 1}^{n}\phi_{\ell}^{\sharp}(x_i)(\phi_{\ell}^{\sharp}(x_i))^{\top}$ .
+
+For a given matrix  $\Sigma$  and a precision  $r^2 > 0$ , the degrees of freedom<sup>3</sup> are defined as
+
+$$
+N _ {\ell} \left(r ^ {2}, \Sigma\right) := \sum_ {j = 1} ^ {m _ {\ell}} \frac {\sigma_ {j} (\Sigma)}{\sigma_ {j} (\Sigma) + r ^ {2}}. \tag {21}
+$$
+
+Since the degrees of freedom are monotonically increasing with respect to each  $\sigma_{j}(\Sigma)$ , we can see that  $N_{\ell}(r^{2},\Sigma) \geq N_{\ell}(r^{2},\Sigma^{\prime})$  if  $\Sigma \succeq \Sigma^{\prime}$ . Let
+
+$$
+m _ {\ell} ^ {\sharp} = \lceil 5 N _ {\ell} (r ^ {2}, \Sigma_ {(\ell)} ^ {\sharp}) \log (8 0 N _ {\ell} (r ^ {2}, \Sigma_ {(\ell)} ^ {\sharp})) \rceil ,
+$$
+
+then Proposition 1 tells that there exists a matrix  $\hat{A}_{\ell} \in m_{\ell} \times m_{\ell}^{\sharp}$  and  $J_{\ell} \subset \{1, \ldots, m_{\ell}\}^{m_{\ell}^{\sharp}}$  such that
+
+$$
+\left\| w ^ {\top} \phi_ {\ell} ^ {\sharp} - w ^ {\top} \hat {A} _ {\ell} \phi_ {\ell , J _ {\ell}} ^ {\sharp} \right\| _ {n} ^ {2} \leq 4 r ^ {2} w ^ {\top} \Sigma_ {(\ell)} ^ {\sharp} \left(\Sigma_ {(\ell)} ^ {\sharp} + r ^ {2} \mathrm {I}\right) ^ {- 1} w \leq 4 r ^ {2} \| w \| ^ {2}, \tag {22}
+$$
+
+for any  $w\in \mathbb{R}^{m_{\ell}5}$ , and the norm of  $\hat{A}_{\ell}$  is bounded as
+
+$$
+\| \hat {A} _ {\ell} \| _ {2} \leq \sqrt {\frac {2 0}{3} m _ {\ell}}.
+$$
+
+Next, we evaluate the degrees of freedom of  $\Sigma_{(\ell)}^{\sharp}$ . We bound this by using the degrees of freedom of  $\widehat{\Sigma}_{(\ell)}$ . First note that  $r_{\ell}^{2} = ||\phi_{\ell} - \phi_{\ell}^{\sharp}||_{n}^{2}$ . Let  $s \leq m$ . For any matrix  $U \in \mathbb{R}^{m_{\ell} \times s}$  such that  $U^{\top}U = I_s$ ,  $\mathrm{Tr}[U^{\top}\Sigma_{(\ell)}^{\sharp}U] = P_n[\phi_{\ell}^{\sharp}\top U U^{\top}\phi_{\ell}^{\sharp}] \leq 2\{P_n[\phi_{\ell}^{\top}U U^{\top}\phi_{\ell}] + P_n[(\phi_{\ell} - \phi_{\ell}^{\sharp})^{\top}U U^{\top}(\phi_{\ell} - \phi_{\ell}^{\sharp})]\}$  by the Cauchy-Schwartz inequality. Here, let  $U$  be the matrix that gives  $P_n[\phi_{\ell}^{\top}U U^{\top}\phi_{\ell}] = \sum_{j=m_{\ell}-s+1}^{m_{\ell}} \sigma_j(\widehat{\Sigma}_{(\ell)}) = \inf_{U:U^{\top}U = I_s} P_n[\phi_{\ell}^{\top}U U^{\top}\phi_{\ell}]$ , then by noticing  $P_n[(\phi_{\ell} - \phi_{\ell}^{\sharp})^{\top}U U^{\top}(\phi_{\ell} - \phi_{\ell}^{\sharp})] \leq P_n\|\phi_{\ell}-\phi_{\ell}^{\sharp}\|^{2} \leq r_{\ell}^{2}$ , we obtain that  $P_n[\phi_{\ell}^{\sharp}\top U U^{\top}\phi_{\ell}^{\sharp}] \leq 2[\sum_{j=m_{\ell}-s+1}^{m_{\ell}} \sigma_j(\widehat{\Sigma}_{(\ell)}) + r_{\ell}^{2}]$ . Finally, by minimizing the left hand side with respect to  $U$ , we obtain that
+
+$$
+\sum_ {j = m _ {\ell} - s + 1} ^ {m _ {\ell}} \sigma_ {j} \left(\Sigma_ {(\ell)} ^ {\sharp}\right) \leq 2 \left(\sum_ {j = m _ {\ell} - s + 1} ^ {m _ {\ell}} \sigma_ {j} \left(\widehat {\Sigma} _ {(\ell)}\right) + r _ {\ell} ^ {2}\right).
+$$
+
+By setting  $s = m_{\ell} - m + 1$  for  $1 \leq m \leq m_{\ell}$ , this indicates that
+
+$$
+\sum_ {j = m} ^ {m _ {\ell}} \sigma_ {j} \left(\Sigma_ {(\ell)} ^ {\sharp}\right) \leq 2 \left(\sum_ {j = m} ^ {m _ {\ell}} \sigma_ {j} \left(\widehat {\Sigma} _ {(\ell)}\right) + r _ {\ell} ^ {2}\right) \leq 2 \left(\sum_ {j = m} ^ {m _ {\ell}} \dot {\mu} _ {j} ^ {(\ell)} + r _ {\ell} ^ {2}\right).
+$$
+
+Now, let  $\dot{m}_{\ell} := \min \{j \in \{1, \ldots, m_{\ell}\} \mid \dot{\mu}_j^{(\ell)} \leq r^2\}$  (if  $\dot{\mu}_{m_{\ell}}^{(\ell)} > r^2$ , then we set  $\dot{m}_{\ell} = m_{\ell}$ ). Then,
+
+$$
+\begin{array}{l} N _ {\ell} (r ^ {2}, \Sigma_ {(\ell)} ^ {\sharp}) = \sum_ {j = 1} ^ {m _ {\ell}} \frac {\sigma_ {j} (\Sigma_ {(\ell)} ^ {\sharp})}{\sigma_ {j} (\Sigma_ {(\ell)} ^ {\sharp}) + r ^ {2}} \leq \dot {m} _ {\ell} + \sum_ {j > \dot {m} _ {\ell}} \frac {\sigma_ {j} (\Sigma_ {(\ell)} ^ {\sharp})}{\sigma_ {j} (\Sigma_ {(\ell)} ^ {\sharp}) + r ^ {2}} \\ \leq \dot {m} _ {\ell} + \sum_ {j > \dot {m} _ {\ell}} \frac {\sigma_ {j} \left(\Sigma_ {(\ell)} ^ {\sharp}\right)}{r ^ {2}} \leq \dot {m} _ {\ell} + \frac {2}{r ^ {2}} \left(r _ {\ell} ^ {2} + \sum_ {j > \dot {m} _ {\ell}} \sigma_ {j} (\widehat {\Sigma} _ {(\ell)})\right) \\ \leq \dot {m} _ {\ell} + \frac {2}{r ^ {2}} \left(r _ {\ell} ^ {2} + U _ {0} \frac {\dot {m} _ {\ell} ^ {1 - \beta}}{\beta - 1}\right) \leq \dot {m} _ {\ell} + \frac {2}{r ^ {2}} \left(r _ {\ell} ^ {2} + \frac {\dot {m} _ {\ell} r ^ {2}}{\beta - 1}\right) \\ \leq \dot {m} _ {\ell} + \frac {2}{r ^ {2}} \left(r _ {\ell} ^ {2} + \frac {\dot {m} _ {\ell} r ^ {2}}{\beta - 1}\right) = \frac {\beta + 1}{\beta - 1} \dot {m} _ {\ell} + 2 \frac {r _ {\ell} ^ {2}}{r ^ {2}}. \\ \end{array}
+$$
+
+Now, let
+
+$$
+r ^ {2} = \frac {1}{4} \tilde {r} _ {\ell} ^ {2},
+$$
+
+then
+
+$$
+N _ {\ell} (r ^ {2}, \Sigma_ {(\ell)} ^ {\sharp}) \leq \frac {\beta + 1}{\beta - 1} \dot {m} _ {\ell} + 8 \frac {r _ {\ell} ^ {2}}{\tilde {r} _ {\ell} ^ {2}}.
+$$
+
+We define the right hand side as  $m_{\ell}^{\sharp}$ :
+
+$$
+m _ {\ell} ^ {\sharp} := \frac {\beta + 1}{\beta - 1} \dot {m} _ {\ell} + 8 \frac {r _ {\ell} ^ {2}}{\check {r} _ {\ell} ^ {2}}.
+$$
+
+We have, by Eq. (22),
+
+$$
+\begin{array}{l} \sum_ {j = 1} ^ {m _ {\ell + 1}} \left\| \eta \left(W _ {j,:} ^ {(\ell)} \phi_ {\ell} ^ {\sharp}\right) - \eta \left(W _ {j,:} ^ {(\ell)} \hat {A} _ {\ell} \phi_ {\ell , J _ {\ell}} ^ {\sharp}\right) \right\| _ {n} ^ {2} \leq 4 \sum_ {j = 1} ^ {m _ {\ell + 1}} \left\| W _ {j,:} ^ {(\ell)} \right\| ^ {2} r ^ {2} \\ \leq 4 R _ {\mathrm {F}} ^ {2} \times \frac {1}{4} \tilde {r} _ {\ell} ^ {2} = R _ {\mathrm {F}} ^ {2} \tilde {r} _ {\ell} ^ {2}. \\ \end{array}
+$$
+
+By the induction assumption, we also have that
+
+$$
+\| \| \eta (W ^ {(\ell)} \phi_ {\ell} ^ {\sharp}) - \phi_ {\ell + 1} \| \| _ {n} ^ {2} = \| \| \eta (W ^ {(\ell)} \phi_ {\ell} ^ {\sharp}) - \eta (W ^ {(\ell)} \phi_ {\ell}) \| \| _ {n} ^ {2} \leq \| W ^ {(\ell)} \| _ {2} ^ {2} r _ {\ell} ^ {2} \leq R _ {2} ^ {2} r _ {\ell} ^ {2}.
+$$
+
+Combining these inequalities, if we define
+
+$$
+\phi_ {\ell + 1} ^ {\sharp} = \eta (W ^ {(\ell)} \hat {A} _ {\ell} \phi_ {\ell , J _ {\ell}} ^ {\sharp} (x)),
+$$
+
+and set  $W^{\sharp(\ell)} = W^{(\ell)}\hat{A}_{\ell}$  and reset  $W^{\sharp(\ell-1)} \gets W_{J_{\ell;}}^{\sharp(\ell-1)}$ , then it holds that
+
+$$
+\| \| \phi_ {\ell + 1} - \phi_ {\ell + 1} ^ {\sharp} \| \| _ {n} \leq r _ {\ell + 1}
+$$
+
+where we let
+
+$$
+r _ {\ell + 1} = R _ {2} r _ {\ell} + R _ {\mathrm {F}} \tilde {r} _ {\ell}.
+$$
+
+Letting  $r_0 = 0$ , by an induction argument, we obtain
+
+$$
+r _ {\ell + 1} \leq \sum_ {k = 1} ^ {\ell} R _ {2} ^ {(\ell - k)} R _ {\mathrm {F}} \tilde {r} _ {k}.
+$$
+
+Finally, we obtain
+
+$$
+\| \widehat {f} - f ^ {\sharp} \| _ {n} ^ {2} \leq r _ {L} ^ {2} \leq \left[ \sum_ {k = 1} ^ {L} R _ {2} ^ {(L - k)} R _ {\mathrm {F}} \tilde {r} _ {\ell} \right] ^ {2},
+$$
+
+for a compressed network  $f^{\sharp}$  that has width  $\mathbf{m}^{\sharp} = (m_1^{\sharp},\dots,m_L^{\sharp})$  with parameters  $W^{\sharp (\ell)} = W_{J_{\ell +1},\cdot}\hat{A}_{\ell}$ . Note that
+
+$$
+\| W ^ {\sharp (\ell)} \| _ {2} \leq \| W _ {J _ {\ell + 1,:}} ^ {(\ell)} \| _ {2} \| \hat {A} _ {\ell} \| _ {2} \leq R _ {2} \sqrt {\frac {2 0}{3} m _ {\ell}}, \| W ^ {\sharp (\ell)} \| _ {\mathrm {F}} \leq \| W _ {J _ {\ell + 1,:}} ^ {(\ell)} \| _ {\mathrm {F}} \| \hat {A} _ {\ell} \| _ {2} \leq R _ {\mathrm {F}} \sqrt {\frac {2 0}{3} m _ {\ell}}.
+$$
+
+Therefore, if we set  $\widehat{\mathcal{G}} = \mathrm{NN}(\mathbf{m}^{\sharp},\sqrt{\frac{20}{3}\max_{\ell}m_{\ell}} R_2,\sqrt{\frac{20}{3}\max_{\ell}m_{\ell}} R_F)$ , then there exists  $\widehat{g}\in \widehat{\mathcal{G}}$  such that
+
+$$
+\| \widehat {f} - \widehat {g} \| _ {n} \leq \hat {r}
+$$
+
+where
+
+$$
+\hat {r} ^ {2} = r _ {L} ^ {2}.
+$$
+
+Moreover, applying Lemma 5 to  $\widehat{\mathcal{G}}$  and the Dudley integral yields
+
+$$
+\bar {R} (\widehat {\mathcal {G}}) \leq C M \sqrt {L \frac {\sum_ {\ell = 1} ^ {L} m _ {\ell} ^ {\sharp} m _ {\ell + 1} ^ {\sharp}}{n} \log (n L (R _ {2} \vee 1) (\max _ {\ell} m _ {\ell} + 1) ^ {2})}.
+$$
+
+This gives the assertion of Theorem 3.
+
+Here, we consider a situation where  $R_{\mathrm{F}}^{2}\tilde{r}_{\ell}^{2} = c_{0}^{2}\frac{r_{\ell}^{2}R_{2}^{2}}{\ell}$  for some constant  $c_{0} > 0$ . Then it holds that
+
+$$
+r _ {\ell + 1} = R _ {2} \left(1 + \sqrt {\frac {c _ {0} ^ {2}}{\ell}}\right) r _ {\ell} = R _ {2} ^ {\ell} \prod_ {k = 1} ^ {\ell} \left(1 + \sqrt {\frac {c _ {0} ^ {2}}{k}}\right) r _ {1} \leq R _ {2} ^ {\ell} \exp \left(c _ {0} (2 \sqrt {\ell} - 1)\right) r _ {1}.
+$$
+
+Therefore, by setting  $C_L \coloneqq (1 \vee R_2)^L \exp \left(c_0(2\sqrt{L} - 1)\right)$ , it holds that  $r_\ell \leq C_L r_1$  for  $\ell = 1, \ldots, L$ , in particular, we have
+
+$$
+\hat {r} \leq C _ {L} r _ {1}.
+$$
+
+In this situation, the degrees of freedom are bounded by
+
+$$
+N _ {\ell} (r ^ {2}, \Sigma_ {(\ell)} ^ {\sharp}) \leq m _ {\ell} ^ {\sharp} = \frac {\beta + 1}{\beta - 1} \dot {m} _ {\ell} + 8 \ell \frac {R _ {\mathrm {F}} ^ {2}}{c _ {0} ^ {2} R _ {2} ^ {2}}.
+$$
+
+Next, we bound  $\dot{m}_{\ell}$ . To do so, we should bound  $\tilde{r}_{\ell}$  from below. Note that
+
+$$
+\begin{array}{l} \tilde {r} _ {\ell} = \frac {c _ {0} R _ {2}}{R _ {\mathrm {F}}} \frac {1}{\sqrt {\ell}} r _ {\ell} = \frac {c _ {0} R _ {2}}{R _ {\mathrm {F}}} \frac {1}{\sqrt {\ell}} R _ {2} ^ {\ell - 1} \prod_ {k = 1} ^ {\ell - 1} \left(1 + \sqrt {\frac {c _ {0} ^ {2}}{k}}\right) r _ {1} \geq \frac {c _ {0} R _ {2} ^ {\ell}}{R _ {\mathrm {F}}} \frac {1}{\sqrt {\ell}} \prod_ {k = 1} ^ {\ell - 1} \left(1 + \sqrt {\frac {c _ {0} ^ {2}}{k}}\right) r _ {1} \\ \geq \frac {c _ {0} R _ {2} ^ {\ell}}{R _ {\mathrm {F}}} \frac {1}{\sqrt {\ell}} \left(1 + \sum_ {k = 1} ^ {\ell - 1} \sqrt {\frac {c _ {0} ^ {2}}{k}}\right) r _ {1} \geq \frac {c _ {0} R _ {2} ^ {\ell}}{R _ {\mathrm {F}}} \frac {1}{\sqrt {\ell}} \left(1 + 2 c _ {0} (\sqrt {\ell} - 1)\right) r _ {1} \\ \geq \frac {c _ {0} R _ {2} ^ {\ell}}{R _ {\mathrm {F}}} \left(\frac {1}{\sqrt {\ell}} + 2 c _ {0} (1 - \frac {1}{\sqrt {\ell}})\right) r _ {1}. \\ \end{array}
+$$
+
+Hence,
+
+$$
+\begin{array}{l} \dot {m} _ {\ell} \leq (\tilde {r} _ {\ell} ^ {2} / (4 U _ {0})) ^ {- 1 / \beta} \leq (4 U _ {0}) ^ {1 / \beta} \left\{\frac {c _ {0} R _ {2} ^ {\ell}}{R _ {\mathrm {F}}} \left[ \frac {1}{\sqrt {\ell}} + 2 c _ {0} \left(1 - \frac {1}{\sqrt {\ell}}\right) \right] r _ {1} \right\} ^ {- 2 / \beta} \\ \leq \left(4 U _ {0}\right) ^ {1 / \beta} \left[ \frac {R _ {\mathrm {F}}}{c _ {0} R _ {2} ^ {\ell}} \right] ^ {2 / \beta} \left(\frac {1}{2} \wedge c _ {0}\right) ^ {- 2 / \beta} r _ {1} ^ {- 2 / \beta} \leq \left[ \frac {4 U _ {0} R _ {\mathrm {F}} ^ {2}}{\left(0 . 5 \wedge c _ {0}\right) ^ {2} c _ {0} ^ {2} R _ {2} ^ {2 \ell}} \right] ^ {1 / \beta} r _ {1} ^ {- 2 / \beta}. \\ \end{array}
+$$
+
+By Lemma 3, we can evaluate  $r_*^2$  for  $\widehat{\mathcal{F}}$  satisfying Assumption 4 as
+
+$$
+r _ {*} ^ {2} \leq C \frac {(M + 1) (S _ {1} + 1 + S _ {2} \log (n)) + M t}{n} \vee M ^ {\frac {2 \alpha - 1}{2 \alpha + 1}} \left(\frac {S _ {3}}{n}\right) ^ {\frac {2 \alpha}{1 + 2 \alpha}},
+$$
+
+where
+
+$$
+S _ {1} = \sum_ {\ell = 1} ^ {L} m _ {\ell} ^ {\sharp} m _ {\ell + 1} ^ {\sharp},
+$$
+
+$$
+S _ {2} = L S _ {1} \log (L (R _ {2} \lor 1) (\max _ {\ell} m _ {\ell} + 1) ^ {2}) = O \left(L \sum_ {\ell = 1} ^ {L} m _ {\ell} ^ {\sharp} m _ {\ell + 1} ^ {\sharp} \log (n)\right),
+$$
+
+$$
+\begin{array}{l} S _ {3} = \left(\sum_ {\ell = 1} ^ {L} m _ {\ell}\right) \left(2 L V _ {0} R _ {2} ^ {L - 1} B _ {x}\right) ^ {1 / \alpha} \left[ \log (n) + 2 L \log \left(2 L \left(R _ {2} \vee 1\right) \left(\max  _ {\ell} m _ {\ell} + 1\right) ^ {2}\right) \right] \\ = O \left(L (\sum_ {\ell = 1} ^ {L} m _ {\ell}) (2 L V _ {0} R _ {2} ^ {L - 1} B _ {x}) ^ {1 / \alpha} \log (n)\right), \\ \end{array}
+$$
+
+Then, the overall generalization error is upper bounded by
+
+$$
+\Psi (\widehat {f}) \leq \widehat {\Psi} (\widehat {f}) + C \left[ r _ {*} ^ {2} + \sqrt {\frac {S _ {3}}{n} \widehat {r} ^ {2 (1 - 1 / 2 \alpha)}} + \sqrt {(M ^ {2} + \widehat {r} ^ {2}) L \frac {\sum_ {\ell = 1} ^ {L} m _ {\ell} ^ {\sharp} m _ {\ell + 1} ^ {\sharp}}{n} \log (n)} + \frac {1 + M t}{n} \right],
+$$
+
+with probability  $1 - 3e^{-t}$  for all  $t \geq 1$ . By letting  $Q_{L,\alpha,n}^{\prime} \coloneqq L \frac{(2LV_0R_2^{L-1}B_x)^{1/\alpha} \log(n)}{n}$  and assuming  $\hat{r} \leq 1$ , the second and third terms in  $C[\cdot]$  is bounded by
+
+$$
+\begin{array}{l} \sqrt {Q _ {L , \alpha , n} ^ {\prime} (\sum_ {\ell = 1} ^ {L} m _ {\ell}) (C _ {L} r _ {1}) ^ {2 (1 - 1 / 2 \alpha)}} + C ^ {\prime} M \sqrt {L \frac {\sum_ {\ell = 1} ^ {L} [ \dot {m} _ {\ell} + \ell R _ {\mathrm {F}} ^ {2} / (c _ {0} ^ {2} R _ {2} ^ {2}) ] ^ {2}}{n} \log (n) ^ {3}} \\ \leq \sqrt {Q _ {L , \alpha , n} ^ {\prime} (\sum_ {\ell = 1} ^ {L} m _ {\ell}) (C _ {L} r _ {1}) ^ {2 (1 - 1 / 2 \alpha)}} + 2 C ^ {\prime} M \sqrt {L \frac {L \left[ \frac {4 U _ {0} R _ {\mathrm {F}} ^ {2}}{(0 . 5 \wedge c _ {0}) ^ {2} c _ {0} ^ {2} (1 \wedge R _ {2}) ^ {2 L}} \right] ^ {2 / \beta} r _ {1} ^ {- 4 / \beta} + L ^ {3} R _ {\mathrm {F}} ^ {4} / R _ {2} ^ {4}}{n} \log (n) ^ {3}}. \\ \end{array}
+$$
+
+Hence, by setting  $C_L r_1 = \left(\frac{\sum_{\ell=1}^L m_\ell}{L}\right)^{-\frac{1}{4/\beta + 2(1 - 1/2\alpha)}}$  which balances the first and the second terms, then  $\hat{r} \leq C_L r_1 \leq 1$  and the right hand side is bounded by
+
+$$
+\begin{array}{l} \sqrt {Q _ {L , \alpha , n} ^ {\prime} (\sum_ {\ell = 1} ^ {L} m _ {\ell}) ^ {\frac {4 / \beta}{4 / \beta + 2 (1 - 1 / 2 \alpha)}} L ^ {\frac {2 (1 - 1 / 2 \alpha)}{4 / \beta + 2 (1 - 1 / 2 \alpha)}}} \\ + C M \sqrt {L \frac {L ^ {\frac {2 (1 - 1 / 2 \alpha)}{4 / \beta + 2 (1 - 1 / 2 \alpha)}} \left(\sum_ {\ell = 1} ^ {L} m _ {\ell}\right) ^ {\frac {4 / \beta}{4 / \beta + 2 (1 - 1 / 2 \alpha)}} \left(C _ {L}\right) ^ {4 / \beta} \left[ \frac {4 U _ {0} R _ {\mathrm {F}} ^ {2}}{(0 . 5 \wedge c _ {0}) ^ {2} c _ {0} ^ {2} (1 \wedge R _ {2}) ^ {2 L}} \right] ^ {2 / \beta}}{n} \log (n) ^ {3}} \\ + C M \frac {R _ {\mathrm {F}} ^ {2}}{R _ {2} ^ {2}} \sqrt {\frac {L ^ {4}}{n} \log (n) ^ {3}}. \\ \end{array}
+$$
+
+Finally, by setting  $c_{0} = 1 / 4$ , we obtain the assertion for
+
+$$
+Q _ {L} = \left[ \frac {4 U _ {0} R _ {\mathrm {F}} ^ {2} (1 \vee R _ {2}) ^ {L} \exp \left(c _ {0} (2 \sqrt {L} - 1)\right)}{(0 . 5 \wedge c _ {0}) ^ {2} c _ {0} ^ {2} (1 \wedge R _ {2}) ^ {2 L}} \right] ^ {2 / \beta} \leq \left[ \frac {4 U _ {0} R _ {\mathrm {F}} ^ {2} (1 \vee R _ {2}) ^ {L} \exp \left(\frac {1}{4} (2 \sqrt {L} - 1)\right)}{(0 . 2 5) ^ {4} (1 \wedge R _ {2}) ^ {2 L}} \right] ^ {2 / \beta}.
+$$
+
+This gives Theorem 4.
+
+# B.6 IMPROVED BOUND OF THEOREM 4 WITH LIPSCHITZ CONTINUITY CONSTRAINT
+
+Here, we again note that there appears  $R_2^L$  in  $P_L$  and  $Q_L$  in the bound of Theorem 4. This is due to a rough evaluation of the interlayer Lipschitz continuity. We can reduce this exponential dependency under Assumption 6.
+
+Corollary 3. Assume Assumption 6 in addition to Assumptions 4 and 5, then the bound in Theorem 4 holds for the following redefined  $P_L$  and  $Q_L$ :
+
+$$
+P _ {L} = (2 L V _ {0} \kappa^ {2} B _ {x}) ^ {1 / \alpha}, Q _ {L} = \left[ \frac {4 U _ {0} R _ {\mathrm {F}} ^ {2} \exp \left(\frac {1}{4} (2 \sqrt {L} - 1)\right)}{(0 . 2 5) ^ {4}} \right] ^ {2 / \beta},
+$$
+
+except that the term  $M \frac{R_{\mathrm{F}}^2 L^2}{R_2^2} \sqrt{\frac{\log(n)^3}{n}}$  is replaced by  $M \kappa^2 R_{\mathrm{F}}^2 L^2 \sqrt{\frac{\log(n)^3}{n}}$ :
+
+$$
+\Psi (\widehat {f}) \leq \widehat {\Psi} (\widehat {f}) + C \sqrt {\frac {[ P _ {L} \vee Q _ {L} ] L ^ {1 + \frac {\beta}{(2 \alpha - 1) ^ {+ \beta}}}}{n} \left(\sum_ {\ell = 1} ^ {L} m _ {\ell}\right) ^ {\frac {4 / \beta}{4 / \beta + 2 (1 - 1 / 2 \alpha)}} \log (n) ^ {3}}
+$$
+
+$$
++ M ^ {\frac {2 \alpha - 1}{2 \alpha + 1}} \left(L P _ {L} \frac {\sum_ {\ell = 1} ^ {L} m _ {\ell}}{n} \log (n)\right) ^ {\frac {2 \alpha}{2 \alpha + 1}} + M \kappa^ {2} R _ {\mathrm {F}} ^ {2} L ^ {2} \sqrt {\frac {\log (n) ^ {3}}{n}} + \frac {1 + M t}{n} \Biggr ].
+$$
+
+Proof of Corollary 3. To show Corollary 3, we set
+
+$$
+\tilde {r} _ {\ell} = \frac {1}{\sqrt {\ell}} \prod_ {k = 1} ^ {\ell - 1} \left(1 + \sqrt {\frac {c _ {0} ^ {2}}{k}}\right) \frac {r _ {1}}{R _ {\mathrm {F}}},
+$$
+
+where  $c_{0}$  is a constant, and by the same argument as in the proof of Corollary 2, we can show that
+
+$$
+r _ {\ell} \leq 2 \kappa \sum_ {k = 1} ^ {\ell} \frac {1}{\sqrt {k}} \prod_ {j = 1} ^ {k - 1} \left(1 + \sqrt {\frac {c _ {0} ^ {2}}{j}}\right) r _ {1}.
+$$
+
+Then, through a cumbersome calculation, we have that
+
+$$
+\frac {r _ {\ell}}{\tilde {r} _ {\ell}} \leq C \frac {\kappa R _ {\mathrm {F}}}{c _ {0}} \sqrt {\ell},
+$$
+
+for a universal constant  $C$ . Moreover, we can show that  $r_{\ell}$  can be bounded as
+
+$$
+r _ {\ell} \leq 2 \kappa \frac {e ^ {c _ {0} (2 \sqrt {\ell + 1} - 1)} - e ^ {c _ {0}}}{c _ {0}} r _ {1}.
+$$
+
+This also gives
+
+$$
+r _ {L} \leq C ^ {\prime} \frac {\kappa}{c _ {0}} \exp (c _ {0} (2 \sqrt {L} - 1)) r _ {1},
+$$
+
+for a universal constant  $C'$ . Then, redefining  $C_L = \kappa \exp(c_0(2\sqrt{L} - 1))$ , we can apply the same argument as in the proof of Corollary 2. Indeed, we can show
+
+$$
+\dot {m} _ {\ell} \lesssim \left[ \frac {4 U _ {0} R _ {\mathrm {F}} ^ {2}}{(0 . 5 \wedge c _ {0}) ^ {2} c _ {0} ^ {2} \kappa^ {2}} \right] ^ {1 / \beta}, m _ {\ell} ^ {\sharp} = \frac {\beta + 1}{\beta - 1} \dot {m} _ {\ell} + C ^ {2} \ell \frac {R _ {\mathrm {F}} ^ {2}}{c _ {0} ^ {2}} r _ {1} ^ {- 2 / \beta}.
+$$
+
+From the above argument, if we set  $\widehat{\mathcal{G}} = \mathrm{NN}(\mathbf{m}^{\sharp},\sqrt{\frac{20}{3}\max_{\ell}m_{\ell}} R_2,\sqrt{\frac{20}{3}\max_{\ell}m_{\ell}} R_F)$ , then there exists  $\widehat{g}\in \widehat{\mathcal{G}}$  such that
+
+$$
+\| \widehat {f} - \widehat {g} \| _ {n} \leq \widehat {r}
+$$
+
+where
+
+$$
+\hat {r} ^ {2} = r _ {L} ^ {2}.
+$$
+
+Moreover, we can show
+
+$$
+r _ {*} ^ {2} \leq C \frac {(M + 1) (S _ {1} + 1 + S _ {2} \log (n)) + M t}{n} \vee M ^ {\frac {2 \alpha - 1}{2 \alpha + 1}} \left(\frac {S _ {3}}{n}\right) ^ {\frac {2 \alpha}{1 + 2 \alpha}},
+$$
+
+where
+
+$$
+S _ {1} = \sum_ {\ell = 1} ^ {L} m _ {\ell} ^ {\sharp} m _ {\ell + 1} ^ {\sharp},
+$$
+
+$$
+S _ {2} = L S _ {1} \log (L (R _ {2} \vee 1) (\max  _ {\ell} m _ {\ell} + 1) ^ {2}) = O \left(L \sum_ {\ell = 1} ^ {L} m _ {\ell} ^ {\sharp} m _ {\ell + 1} ^ {\sharp} \log (n)\right),
+$$
+
+$$
+\begin{array}{l} S _ {3} = \left(\sum_ {\ell = 1} ^ {L} m _ {\ell}\right) \left(2 L V _ {0} \kappa^ {2} B _ {x}\right) ^ {1 / \alpha} \left[ \log (n) + 2 L \log \left(2 L \left(R _ {2} \vee 1\right) \left(\max _ {\ell} m _ {\ell} + 1\right) ^ {2}\right) \right] \\ = O \left(L \left(\sum_ {\ell = 1} ^ {L} m _ {\ell}\right) \left(2 L V _ {0} \kappa^ {2} B _ {x}\right) ^ {1 / \alpha} \log (n)\right). \\ \end{array}
+$$
+
+Here, to evaluate  $S_{3}$ , we used the argument in Sec. B.4 (proof of Corollary 2).
+
+The remaining argument is the same as the proof of Theorem 3 and Theorem 4 (Sec. B.5).
+
+![](images/9b49b464512f1a346d2e105ebc583a76b84e8b403ac96bf756ef99ce08f963be.jpg)
+
+# C AUXILIARY LEMMAS
+
+In this section, we give several auxiliary lemmas that are used in the proof of the theorems. These results are not new at all, but we explicitly present them for completeness.
+
+# C.1 COVERING NUMBER OF DEEP NETWORK MODELS
+
+Define the neural network with height  $L$ , width  $m$ , sparsity constraint  $S$  and norm constraint  $B$  as
+
+$$
+\Phi (L, m, S, B) := \left\{G \circ \left(W ^ {(L)} \eta (\cdot) + b ^ {(L)}\right) \circ \dots \circ \left(W ^ {(1)} x + b ^ {(1)}\right) \mid W ^ {(L)} \in \mathbb {R} ^ {1 \times m}, b ^ {(L)} \in \mathbb {R}, \right.
+$$
+
+$$
+W ^ {(1)} \in \mathbb {R} ^ {m \times d}, b ^ {(1)} \in \mathbb {R} ^ {m}, W ^ {(\ell)} \in \mathbb {R} ^ {m \times m}, b ^ {(\ell)} \in \mathbb {R} ^ {m} (1 <   \ell <   L),
+$$
+
+$$
+\sum_ {\ell = 1} ^ {L} (\| W ^ {(\ell)} \| _ {0} + \| b ^ {(\ell)} \| _ {0}) \leq S, \max _ {\ell} \| W ^ {(\ell)} \| _ {\infty} \vee \| b ^ {(\ell)} \| _ {\infty} \leq B \},
+$$
+
+where  $\| \cdot \| _0$  is the  $\ell_0$ -norm of the matrix (the number of non-zero elements of the matrix) and  $\| \cdot \|_{\infty}$  is the  $\ell_{\infty}$ -norm of the matrix (maximum of the absolute values of the elements).
+
+The following evaluation of the covering number of the model  $\Phi(L, m, S, B)$  is shown by Schmidt-Hieber (2019); Suzuki (2019).
+
+Lemma 4 (Covering number evaluation (Schmidt-Hieber, 2019; Suzuki, 2019)). The covering number of  $\Phi(L, m, S, B)$  can be bounded by
+
+$$
+\begin{array}{l} \log \mathcal {N} (\Phi (L, m, S, B), \| \cdot \| _ {\infty}, \delta) \leq S \log (\delta^ {- 1} L (B \vee 1) ^ {L - 1} (m + 1) ^ {2 L}) \\ \leq 2 S L \log ((B \vee 1) (m + 1)) + S \log (\delta^ {- 1} L). \\ \end{array}
+$$
+
+Proof of Lemma 4. Given a network  $f \in \Phi(L, m, S, B)$  expressed as
+
+$$
+f (x) = G \circ \left(W ^ {(L)} \eta (\cdot) + b ^ {(L)}\right) \circ \dots \circ \left(W ^ {(1)} x + b ^ {(1)}\right),
+$$
+
+let
+
+$$
+\mathcal {A} _ {k} (f) (x) = \eta \circ \left(W ^ {(k - 1)} \eta (\cdot) + b ^ {(k - 1)}\right) \circ \dots \circ \left(W ^ {(1)} x + b ^ {(1)}\right),
+$$
+
+and
+
+$$
+\mathcal {B} _ {k} (f) (x) = G \circ \left(W ^ {(L)} \eta (\cdot) + b ^ {(L)}\right) \circ \dots \circ \left(W ^ {(k)} \eta (x) + b ^ {(k)}\right),
+$$
+
+for  $k = 2, \ldots, L$ . Corresponding to the last and first layer, we define  $\mathcal{B}_{L+1}(f)(x) = x$  and  $\mathcal{A}_1(f)(x) = x$ . Then, it is easy to see that  $f(x) = \mathcal{B}_{k+1}(f) \circ (W^{(k)} \cdot +b^{(k)}) \circ \mathcal{A}_k(f)(x)$ . Now, suppose that a pair of different two networks  $f, g \in \Phi(L, m, S, B)$  given by
+
+$$
+f (x) = G \circ (W ^ {(L)} \eta (\cdot) + b ^ {(L)}) \circ \dots \circ (W ^ {(1)} x + b ^ {(1)}), g (x) = G \circ (W ^ {(L) ^ {\prime}} \eta (\cdot) + b ^ {(L) ^ {\prime}}) \circ \dots \circ (W ^ {(1) ^ {\prime}} x + b ^ {(1) ^ {\prime}}),
+$$
+
+has a parameters with distance  $\delta$ :  $\| W^{(\ell)} - W^{(\ell)'}\|_{\infty} \leq \delta$  and  $\| b^{(\ell)} - b^{(\ell)'}\|_{\infty} \leq \delta$ . Now, not that  $\| \mathcal{A}_k(f)\|_{\infty} \leq \max_j \| W_{j,:}^{(k-1)}\|_1 \| \mathcal{A}_{k-1}(f)\|_{\infty} + \| b^{(k-1)}\|_{\infty} \leq mB \| \mathcal{A}_{k-1}(f)\|_{\infty} + B \leq (B \vee 1)(m+1)\| \mathcal{A}_{k-1}(f)\|_{\infty} \leq (B \vee 1)^{k-1}(m+1)^{k-1}$ , and similarly the Lipsshitz continuity of  $\mathcal{B}_k(f)$  with respect to  $\|\cdot\|_{\infty}$ -norm is bounded as  $(Bm)^{L-k+1}$ . Then, it holds that
+
+$$
+\begin{array}{l} \left| f (x) - g (x) \right| \\ = \left| \sum_ {k = 1} ^ {L} \mathcal {B} _ {k + 1} (g) \circ \left(W ^ {(k)} \cdot + b ^ {(k)}\right) \circ \mathcal {A} _ {k} (f) (x) - \mathcal {B} _ {k + 1} (g) \circ \left(W ^ {(k) ^ {\prime}} \cdot + b ^ {(k) ^ {\prime}}\right) \circ \mathcal {A} _ {k} (f) (x) \right| \\ \leq \sum_ {k = 1} ^ {L} (B m) ^ {L - k} \| (W ^ {(k)} \cdot + b ^ {(k)}) \circ \mathcal {A} _ {k} (f) (x) - (W ^ {(k) ^ {\prime}} \cdot + b ^ {(k) ^ {\prime}}) \circ \mathcal {A} _ {k} (f) (x) \| _ {\infty} \\ \leq \sum_ {k = 1} ^ {L} (B m) ^ {L - k} \delta [ m (B \vee 1) ^ {k - 1} m ^ {k - 1} + 1 ] \\ \leq \sum_ {k = 1} ^ {L} (B m) ^ {L - k} \delta (B \vee 1) ^ {k - 1} m ^ {k} \leq \delta L (B \vee 1) ^ {L - 1} (m + 1) ^ {L}. \\ \end{array}
+$$
+
+Thus, for a fixed sparsity pattern (the locations of non-zero parameters), the covering number is bounded by  $\left(\delta / [L(B \vee 1)^{L-1}(m + 1)^L]\right)^{-S}$ . There are the number of configurations of the sparsity pattern is bounded by  $\binom{(m+1)L}{S} \leq (m + 1)^{LS}$ . Thus, the covering number of the whole space  $\Phi$  is bounded as
+
+$$
+(m + 1) ^ {L S} \left\{\delta / [ L (B \lor 1) ^ {L - 1} (m + 1) ^ {L} ] \right\} ^ {- S} = [ \delta^ {- 1} L (B \lor 1) ^ {L - 1} (m + 1) ^ {2 L} ] ^ {S},
+$$
+
+which gives the assertion.
+
+![](images/fb7dc2027cd2c514ff902f5ce21771650227d401a9823d601fe49dc2197b9850.jpg)
+
+Lemma 5 (Covering number evaluation). Let  $\mathrm{NN}(\mathbf{m}, R_2, R_{\mathrm{F}})$  be the set of neural networks with depth  $\mathbf{m}(m_1, \ldots, m_L)$ ,  $\|W^{(\ell)}\|_{\infty} \leq R_2$  and  $\|W^{(\ell)}\|_{\mathrm{F}} \leq R_F$ . The covering number of  $\mathrm{NN}(\mathbf{m}, R_2, R_{\mathrm{F}})$  can be bounded by
+
+$$
+\begin{array}{l} \log \mathcal {N} (\mathrm {N N} (\mathbf {m}, R _ {2}, R _ {\mathrm {F}}), \| \cdot \| _ {\infty}, \delta) \\ \leq \left(\sum_ {\ell = 1} ^ {L} m _ {\ell} m _ {\ell + 1}\right) \log \left(\delta^ {- 1} L \left(R _ {2} \vee 1\right) ^ {L - 1} \left(\max  _ {\ell} m _ {\ell} + 1\right) ^ {L}\right) \\ \leq \left(\sum_ {\ell = 1} ^ {L} m _ {\ell} m _ {\ell + 1}\right) \log \left(\delta^ {- 1}\right) + L \left(\sum_ {\ell = 1} ^ {L} m _ {\ell} m _ {\ell + 1}\right) \log \left(L \left(R _ {2} \vee 1\right) \left(\max  _ {\ell} m _ {\ell} + 1\right)\right). \\ \end{array}
+$$
+
+Moreover, the set of networks with low rank weight matrices,  $\mathrm{NN}(\mathbf{m},\mathbf{s},R_2,R_{\mathrm{F}})$ , has the following covering number bound:
+
+$$
+\begin{array}{l} \log \mathcal {N} (\mathrm {N N} (\mathbf {m}, \mathbf {s}, R _ {2}, R _ {\mathrm {F}}), \| \cdot \| _ {\infty}, \delta) \\ \leq \sum_ {\ell = 1} ^ {L} s _ {\ell} \left(m _ {\ell} + m _ {\ell + 1}\right) \log \left(\delta^ {- 1} L \left(R _ {2} \vee 1\right) ^ {2 L - 1} \left(\max  _ {\ell} m _ {\ell} + 1\right) ^ {2 L}\right). \\ \end{array}
+$$
+
+Proof of Lemma 5. Let  $B = R_{2}$ ,  $m = \max_{\ell} m_{\ell}$ , and  $S = \sum_{\ell=1}^{L} m_{\ell} m_{\ell+1}$ , then we can see that  $\mathrm{NN}(\mathbf{m}, R_{2}, R_{\mathrm{F}})$  is a subset of  $\Phi(L, m, S, B)$  because  $\|W\|_{\infty} \leq \|W\|_{2}$ . Hence Lemma 4 gives the first assertion. As for the second one, we can easily check that the covering number of  $\mathrm{NN}(\mathbf{m}, \mathbf{s}, R_{2}, R_{\mathrm{F}})$  can be bounded by the one given in Lemma 5 for  $\Phi(2L, m, S, B)$  with  $S = \sum_{\ell=1}^{L} s_{\ell}(m_{\ell} + m_{\ell+1})$ . Then, we obtain the second assertion.
+
+![](images/f37a8e141a0a77dcd7db3716678d439c026ae315f1a946ce3f6b3faee5b1448c.jpg)
+
+# C.2 COMPRESSION ERROR BOUND FOR ONE LAYER
+
+The following proposition was shown by Bach (2017); Suzuki et al. (2018). Let  $\widehat{\Sigma}_{I,I'} \in \mathbb{R}^{K \times H}$  for integers  $K, H \in \mathbb{N}$  and a matrix  $\widehat{\Sigma}_{I,I'} \in \mathbb{R}^{K \times H}$  be a matrix  $(\widehat{\Sigma}_{i,j})_{i \in I, j \in I'}$  for the index sets  $I \in [m_{\ell}]^K$  and  $I' \in [m_{\ell}]^H$ . Let  $F = \{1, \dots, m_{\ell}\}$  be the full index set. Let the degrees of freedom corresponding to  $\widehat{\Sigma}$  be  $\hat{N}(\lambda) := N_{\ell}(\lambda, \widehat{\Sigma})$  (see Eq. (21)) for  $\lambda > 0$ .
+
+Proposition 1. Let  $U = (U_{j,l})_{j,l}$  is the orthogonal matrix that diagonalizes  $\widehat{\Sigma}$ , that is,  $\widehat{\Sigma} = U\mathrm{diag}\left(\widehat{\mu}_1,\ldots ,\widehat{\mu}_{m_\ell}\right)U^\top$  for  $\widehat{\mu}_j\geq 0$  ( $j = 1,\dots,m_{\ell}$ ). Define
+
+$$
+\tau_ {j} ^ {\prime} = \frac {1}{\hat {N} (\lambda)} \sum_ {l = 1} ^ {m _ {\ell}} U _ {j, l} ^ {2} \frac {\hat {\mu} _ {l} ^ {(\ell)}}{\hat {\mu} _ {l} ^ {(\ell)} + \lambda} = \frac {1}{\hat {N} (\lambda)} [ \widehat {\Sigma} (\widehat {\Sigma} + \lambda I) ^ {- 1} ] _ {j, j} (j \in \{1, \dots , m _ {\ell} \}). \tag {23}
+$$
+
+For  $\lambda > 0$ , if
+
+$$
+m \geq 5 \hat {N} (\lambda) \log (8 0 \hat {N} (\lambda)),
+$$
+
+then there exist  $v_{1},\ldots ,v_{m}\in \{1,\ldots ,m_{\ell}\}$  such that, for every  $\alpha \in \mathbb{R}^{m_{\ell}}$
+
+$$
+\inf  _ {\beta \in \mathbb {R} ^ {m}} \left\{\left\| \alpha^ {\top} \eta (\hat {F} _ {\ell - 1} (\cdot)) - \sum_ {j = 1} ^ {m} \beta_ {j} \eta (\hat {F} _ {\ell - 1} (\cdot)) _ {v _ {j}} \right\| _ {n} ^ {2} + m \lambda \| \beta \| _ {\tau^ {\prime}} ^ {2} \right\} \leq 4 \lambda \alpha^ {\top} \widehat {\Sigma} (\widehat {\Sigma} + \lambda I) ^ {- 1} \alpha , \tag {24}
+$$
+
+and  $\sum_{j=1}^{m} \tau_j'^{-1} \leq \frac{5}{3} m \times m_\ell$ , where  $\|\beta\|_{\tau'}^2 \coloneqq \sum_{j=1}^{m} \beta_j^2 \tau_j'$ . Let  $\tau \coloneqq m \lambda \tau'$  and  $\mathrm{I}_{\tau} = \mathrm{diag}(\tau)$ . Then,  $\hat{A} \coloneqq \widehat{\Sigma}_{F,J} (\widehat{\Sigma}_{J,J} + \mathrm{I}_\tau)^{-1}$  for  $J = \{v_1, \ldots, v_m\}$  satisfies
+
+$$
+\| \hat {A} \| _ {2} \leq \sqrt {\frac {2 0}{3} m _ {\ell}},
+$$
+
+and the optimal  $\beta$  that achieves the infimum is given by  $\hat{\beta} = \hat{A}^{\top}\alpha$  for any  $\alpha \in \mathbb{R}^{m_{\ell}}$ .
+
+# C.3 CONCENTRATION INEQUALITY
+
+Proposition 2 (Talagrand's Concentration Inequality (Talagrand, 1996; Bousquet, 2002)). Let  $\mathcal{G}$  be a function class on  $\mathcal{X}$  that is separable with respect to  $\infty$ -norm, and  $\{x_i\}_{i=1}^n$  be i.i.d. random variables with values in  $\mathcal{X}$ . Furthermore, let  $B \geq 0$  and  $U \geq 0$  be  $B := \sup_{g \in \mathcal{G}} \operatorname{E}[(g - \operatorname{E}[g])^2]$  and  $U := \sup_{g \in \mathcal{G}} \|g\|_{\infty}$ , then for  $Z := \sup_{g \in \mathcal{G}} \left| \frac{1}{n} \sum_{i=1}^{n} g(x_i) - \operatorname{E}[g] \right|$ , we have
+
+$$
+P \left(Z \geq 2 \mathrm {E} [ Z ] + \sqrt {\frac {2 B t}{n}} + \frac {2 U t}{n}\right) \leq e ^ {- t}, \tag {25}
+$$
+
+for all  $t > 0$
+
+# D NUMERICAL EXPERIMENTS
+
+In this section, we experimentally validate the assumptions we made in the theoretical analysis and investigate how large the intrinsic dimensionality becomes. We use VGG-19 network trained on CIFAR-10. The VGG-19 network have 16 convolution layers (named c0, ..., c15) and 3 fully connected layers (named 116, ..., 118). The size of each filter in each convolution layer is  $3 \times 3$ . Our theory does not support a convolution layer in a strict sense, but we adopt it as follows. If the convolution layer in the  $\ell$ -th layer has the input channel size  $m_{\ell}$  and the output channel size  $m_{\ell+1}$  with the filter size  $k \times k$  (in our case  $k = 3$ ), then the weight matrix is given as a 4-way tensor with the size  $m_{\ell+1} \times m_{\ell} \times k \times k$ :  $W^{(\ell)} \in \mathbb{R}^{m_{\ell+1} \times m_{\ell} \times k \times k}$ . Although a singular value of a 4-way tensor is not well-defined, we can perform a low rank approximation of the weight matrix by folding out the tensor to a large matrix. Actually, considering "similarity" between the filters as
+
+$$
+K _ {(\ell)} := \left(\sum_ {c = 1} ^ {m _ {\ell}} \sum_ {\kappa_ {1}, \kappa_ {2} = 1} ^ {k, k} W _ {i, c, \kappa_ {1}, \kappa_ {2}} ^ {(\ell)} W _ {j, c, \kappa_ {1}, \kappa_ {2}} ^ {(\ell)}\right) _ {i, j = 1} ^ {m _ {\ell + 1}} \in \mathbb {R} ^ {m _ {\ell + 1} \times m _ {\ell + 1}},
+$$
+
+then we can easily see that, for  $s_{\ell} \in [m_{\ell + 1}]$ , it holds that
+
+$$
+\| W ^ {(\ell)} - P ^ {\top} W ^ {\prime} \| _ {\mathrm {F}} = \sqrt {\sum_ {j ^ {\prime} = \dot {m}} ^ {m _ {\ell + 1}} \sigma_ {j ^ {\prime}} (K _ {(\ell)})},
+$$
+
+where  $P \in \mathbb{R}^{s_{\ell} \times m_{\ell + 1}}$  is a projection matrix to the eigen-space corresponding to the  $s_{\ell}$  largest singular values of  $K_{(\ell)}$ ,  $W' := PW^{(\ell)} = (\sum_{i' = 1}^{m_{\ell + 1}} P_{i, i'} W_{i', j, k, k'}^{(\ell)})_{i, j, k, k' \in [s_{\ell}] \times [m_{\ell}] \times [k] \times [k]}$  and the Frobenius norm  $\| \cdot \|_{\mathrm{F}}$  of a tensor is the Euclidean norm as a vector. Therefore, we can use the eigenvalues of  $K_{(\ell)}$  to evaluate the redundancy of parameters among filters.
+
+As for the covariance matrix in a convolution layer, we also apply the same argument. That is, the input to the  $\ell$ -th layer (which is a convolution layer) is given by  $\phi_{\ell}(x) \in \mathbb{R}^{m_{\ell} \times I \times J}$  where  $I$  and  $J$  are the width and height of the input, and we define the following "covariance" matrix as a similarity measure between the channels:
+
+$$
+\widehat {\Sigma} _ {(\ell)} = \left(\frac {1}{n} \sum_ {i ^ {\prime} = 1} ^ {n} \sum_ {1 \leq c _ {1} \leq I} \sum_ {1 \leq c _ {2} \leq J} \phi_ {\ell , (i, c _ {1}, c _ {2})} (x _ {i ^ {\prime}}) \phi_ {\ell , (j, c _ {1}, c _ {2})} (x _ {i ^ {\prime}})\right) _ {i, j = 1} ^ {m _ {\ell}, m _ {\ell}} \in \mathbb {R} ^ {m _ {\ell} \times m _ {\ell}}.
+$$
+
+This also serve the redundancy measure and analogous argument to the main text can be applied.
+
+Near low rank properties of the covariance matrix and weight matrix Here, we see plausibility of the near low rank assumptions we made in the analysis. Figure 1 presents the eigenvalues of covariance matrix  $\widehat{\Sigma}_{(\ell)}$  in each of layer c2, c7, c12 and 116. The eigenvalues are sorted in decreasing order. We can see that the eigenvalue distributions are highly concentrated around 0 and the eigenvalues decrease quickly, which indicates the near low rank property of  $\widehat{\Sigma}_{(\ell)}$ .
+
+![](images/85a48ad0a7960b428b86e801018db039e1035228cd2031ac93818c28d570e213.jpg)  
+(a) c2
+
+![](images/481445e29a8431eb5662689f1e71cf41fbbf99e5e26a157a53e6734bb45e922b.jpg)  
+(b) c7
+
+![](images/6152ab68394db6048196f3c88519e5496eadab32b56b40b2736d63aa7d8c8ac2.jpg)  
+(c) c12
+
+![](images/eeec648a6d2b66eca594f20a73108394e7a94a0dacf256b1eaff62a2569a74d9.jpg)  
+(d) 116  
+Figure 1: Eigenvalue distribution of the covariance matrices
+
+Next, we plot the eigenvalues of  $K_{(\ell)}$  in Figure 2 for layer c2, c7, c12 and 116. We again observe a rapid decrease of the eigenvalues. These results justify our theoretical assumptions.
+
+![](images/45b0606912acc30ce809acb5d61c931d929bd6feaaf3f57c7df98059fe0bd584.jpg)  
+(a) c2
+
+![](images/dd57a26b94b9a524840f00eabacf956625d1e75c6e1b484366372a9436301a7a.jpg)  
+(b) c7
+
+![](images/e2db5efeafdd1388ceb8e5608e78a4ee6e63052496355a7498b7ea02fef7b432.jpg)  
+(c) c12  
+Figure 2: Singular-value distribution of the weight matrices
+
+![](images/90f1d74fa30dc27d44fe1891221be5e5447280ad96269105789d7e5b051bb176.jpg)  
+(d) 116
+
+Intrinsic dimensionality Here, we calculate the intrinsic dimensionalities of the VGG-19 network. For that purpose, we set a threshold parameter  $\nu \in \{10^{-1}, 10^{-2}, 10^{-3}\}$  and compute
+
+Table 3: The effective ranks  $\dot{m}_{\ell}$  and  $s_{\ell}$  in each layer for each threshold  $\nu$ . "In/Out" indicates the channel sizes of the input and output.  
+
+<table><tr><td rowspan="2">layer</td><td rowspan="2">In/Out</td><td colspan="3">Cov (m)</td><td colspan="3">Weight (s)</td></tr><tr><td>ν:10-1</td><td>10-2</td><td>10-3</td><td>ν:10-1</td><td>10-2</td><td>10-3</td></tr><tr><td>c0</td><td>3/64</td><td>3</td><td>3</td><td>3</td><td>9</td><td>16</td><td>19</td></tr><tr><td>c1</td><td>64/64</td><td>5</td><td>21</td><td>51</td><td>26</td><td>62</td><td>64</td></tr><tr><td>c2</td><td>64/128</td><td>5</td><td>33</td><td>64</td><td>37</td><td>110</td><td>128</td></tr><tr><td>c3</td><td>128/128</td><td>9</td><td>76</td><td>128</td><td>50</td><td>128</td><td>128</td></tr><tr><td>c4</td><td>128/256</td><td>11</td><td>110</td><td>128</td><td>102</td><td>255</td><td>256</td></tr><tr><td>c5</td><td>256/256</td><td>13</td><td>200</td><td>256</td><td>93</td><td>256</td><td>256</td></tr><tr><td>c6</td><td>256/256</td><td>17</td><td>219</td><td>256</td><td>55</td><td>230</td><td>256</td></tr><tr><td>c7</td><td>256/256</td><td>11</td><td>110</td><td>253</td><td>37</td><td>175</td><td>252</td></tr><tr><td>c8</td><td>256/512</td><td>10</td><td>35</td><td>129</td><td>30</td><td>122</td><td>349</td></tr><tr><td>c9</td><td>512/512</td><td>6</td><td>33</td><td>79</td><td>16</td><td>60</td><td>217</td></tr><tr><td>c10</td><td>512/512</td><td>3</td><td>15</td><td>41</td><td>16</td><td>42</td><td>127</td></tr><tr><td>c11</td><td>512/512</td><td>2</td><td>12</td><td>23</td><td>11</td><td>39</td><td>129</td></tr><tr><td>c12</td><td>512/512</td><td>2</td><td>7</td><td>16</td><td>7</td><td>18</td><td>46</td></tr><tr><td>c13</td><td>512/512</td><td>3</td><td>6</td><td>16</td><td>9</td><td>22</td><td>46</td></tr><tr><td>c14</td><td>512/512</td><td>3</td><td>7</td><td>18</td><td>16</td><td>38</td><td>59</td></tr><tr><td>c15</td><td>512/512</td><td>4</td><td>14</td><td>28</td><td>36</td><td>51</td><td>92</td></tr><tr><td>l16</td><td>512/4096</td><td>5</td><td>10</td><td>11</td><td>11</td><td>21</td><td>49</td></tr><tr><td>l17</td><td>4096/4096</td><td>7</td><td>10</td><td>11</td><td>10</td><td>49</td><td>990</td></tr><tr><td>l18</td><td>4096/10</td><td>6</td><td>10</td><td>10</td><td>9</td><td>9</td><td>9</td></tr></table>
+
+$s_{\ell} := \# \{j \in [m_{\ell + 1}] \mid \sigma_j(K_{(\ell)}) \geq \nu \times \max_{j'} \sigma_{j'}(K_{(\ell)})\}$  and  $\dot{m}_{\ell} := \# \{j \in [m_{\ell}] \mid \sigma_j(\widehat{\Sigma}_{(\ell)}) \geq \nu \times \max_{j'} \sigma_{j'}(\widehat{\Sigma}_{(\ell)})\}$  (which corresponds to setting  $\tilde{r}_{\ell}^{2} = 4\nu \times \max_{j'} \sigma_{j'}(\widehat{\Sigma}_{(\ell)})$ ). Table 4 summarizes the effective ranks  $\dot{m}_{\ell}, s_{\ell}$  of all layers. We can see that the effective ranks can be much smaller than the channel sizes in several layers (especially layers from c8 to 118.), which indicates the network has high redundancy and its intrinsic dimensionality could be much smaller than the actual number of parameters.
+
+Next, we compute the intrinsic dimensionality of each layer based on the effective ranks calculated above. Basically, the main term of our bound (Theorem 4) is given by  $\sqrt{\sum_{\ell=1}^{L} \frac{\dot{m}_{\ell+1} \dot{m}_{\ell}}{n}}$  (note that  $m_{\ell}^{\sharp}$  is essentially controlled by  $\dot{m}_{\ell}$ ). Hence, we employ  $\dot{m}_{\ell+1} \times \dot{m}_{\ell}$  as the intrinsic dimensionality of each fully connected layer. As for a convolution layer, we employ  $\dot{m}_{\ell+1} \dot{m}_{\ell} k^2$  as the intrinsic dimensionality which is the number of parameters of compressed network. We also calculate the intrinsic dimensionality obtained by compressing only the weight matrix. That is given by  $s_{\ell} m_{\ell} k^2 + m_{\ell+1} s_{\ell}$ . Both of them are summarized in Table 4. We can see that the intrinsic dimensionality is smaller than the actual number of parameters. In particular, it is much smaller for higher layers such as c8 to 118. This indicates that the information required for classification is almost distilled in the first few layers and the contribution of the subsequent layers would be much smaller than the earlier layers. Moreover, we see that compressing the network using the covariance matrix gives smaller intrinsic dimensionalities than the weight matrix. This is because the improvement induced by decreasing  $\dot{m}_{\ell}$  is a quadratic order but that by  $s_{\ell}$  is just a linear order. Another reason is that the effective rank of the covariance matrix is more data dependent in a sense that it is strongly dependent on the distribution of the data, and thus it can capture data dependent redundancy more efficiently (see also Figures 1 and 2). Since the intrinsic dimensionality is much smaller than the actual number of parameters, the VC-dimension bound is too pessimistic and a compression based bound like ours gives a better generalization error bound.
+
+Finally, we give a comparison of intrinsic dimensionalities calculated by Arora et al. (2018) and ours. We borrowed the values presented in the paper (Arora et al., 2018). We would like to note that the comparison is not completely fair because the intrinsic dimensionality of both our analysis and that of Arora et al. (2018) neglect constant functors (such as depth), and thus the final generalization error is not merely determined by the raw values. However, the comparison offers better understanding of our analysis by observing difference and similarity between them. We can see
+
+Table 4: The intrinsic dimensionality in each layer for each threshold  $\nu$ . Here again, "In/Out" indicates the channel sizes of the input and output. "Orig" indicates the number of parameters  $(m_{\ell +1}m_{\ell}\times$  filter size) in each layer.  
+
+<table><tr><td rowspan="2">layer</td><td rowspan="2">In/Out</td><td rowspan="2">Orig</td><td colspan="3">Cov</td><td colspan="3">Weight</td></tr><tr><td>ν: 10-1</td><td>10-2</td><td>10-3</td><td>ν: 10-1</td><td>10-2</td><td>10-3</td></tr><tr><td>c0</td><td>3/64</td><td>1,728</td><td>135</td><td>567</td><td>1,377</td><td>910</td><td>910</td><td>910</td></tr><tr><td>c1</td><td>64/64</td><td>36,864</td><td>225</td><td>6,237</td><td>29,376</td><td>1,920</td><td>1,920</td><td>1,920</td></tr><tr><td>c2</td><td>64/128</td><td>73,728</td><td>405</td><td>22,572</td><td>73,728</td><td>6,336</td><td>11,264</td><td>13,376</td></tr><tr><td>c3</td><td>128/128</td><td>147,456</td><td>891</td><td>75,240</td><td>147,456</td><td>33,280</td><td>79,360</td><td>81,920</td></tr><tr><td>c4</td><td>128/256</td><td>294,912</td><td>1,287</td><td>198,000</td><td>294,912</td><td>52,096</td><td>154,880</td><td>180,224</td></tr><tr><td>c5</td><td>256/256</td><td>589,824</td><td>1,989</td><td>394,200</td><td>589,824</td><td>128,000</td><td>327,680</td><td>327,680</td></tr><tr><td>c6</td><td>256/256</td><td>589,824</td><td>1,683</td><td>216,810</td><td>582,912</td><td>261,120</td><td>652,800</td><td>655,360</td></tr><tr><td>c7</td><td>256/256</td><td>589,824</td><td>990</td><td>34,650</td><td>293,733</td><td>238,080</td><td>655,360</td><td>655,360</td></tr><tr><td>c8</td><td>256/512</td><td>1,179,648</td><td>540</td><td>10,395</td><td>91,719</td><td>154,880</td><td>647,680</td><td>720,896</td></tr><tr><td>c9</td><td>512/512</td><td>2,359,296</td><td>162</td><td>4,455</td><td>29,151</td><td>189,440</td><td>896,000</td><td>1,290,240</td></tr><tr><td>c10</td><td>512/512</td><td>2,359,296</td><td>54</td><td>1,620</td><td>8,487</td><td>153,600</td><td>624,640</td><td>1,786,880</td></tr><tr><td>c11</td><td>512/512</td><td>2,359,296</td><td>36</td><td>756</td><td>3,312</td><td>81,920</td><td>307,200</td><td>1,111,040</td></tr><tr><td>c12</td><td>512/512</td><td>2,359,296</td><td>54</td><td>378</td><td>2,304</td><td>81,920</td><td>215,040</td><td>650,240</td></tr><tr><td>c13</td><td>512/512</td><td>2,359,296</td><td>81</td><td>378</td><td>2,592</td><td>56,320</td><td>199,680</td><td>660,480</td></tr><tr><td>c14</td><td>512/512</td><td>2,359,296</td><td>108</td><td>882</td><td>4,536</td><td>35,840</td><td>92,160</td><td>235,520</td></tr><tr><td>c15</td><td>512/512</td><td>2,359,296</td><td>180</td><td>1,260</td><td>2,772</td><td>46,080</td><td>112,640</td><td>235,520</td></tr><tr><td>l16</td><td>512/4096</td><td>2,097,152</td><td>35</td><td>100</td><td>121</td><td>73,728</td><td>175,104</td><td>271,872</td></tr><tr><td>l17</td><td>4096/4096</td><td>16,777,216</td><td>42</td><td>100</td><td>110</td><td>294,912</td><td>417,792</td><td>753,664</td></tr><tr><td>l18</td><td>4096/10</td><td>40,960</td><td>42</td><td>90</td><td>90</td><td>45,166</td><td>86,226</td><td>201,194</td></tr></table>
+
+Table 5: Comparison of the intrinsic dimensionality of our analysis and that in Arora et al. (2018).  
+
+<table><tr><td rowspan="2">layer</td><td rowspan="2">In/Out</td><td rowspan="2">Orig</td><td rowspan="2">Arora et al. (2018)</td><td colspan="3">Cov</td></tr><tr><td>ν: 10-1</td><td>10-2</td><td>10-3</td></tr><tr><td>c0</td><td>3/64</td><td>1,728</td><td>1,645</td><td>135</td><td>567</td><td>1,377</td></tr><tr><td>c3</td><td>128/128</td><td>147,456</td><td>644,654</td><td>891</td><td>75,240</td><td>147,456</td></tr><tr><td>c5</td><td>256/256</td><td>589,824</td><td>3,457,882</td><td>1,989</td><td>394,200</td><td>589,824</td></tr><tr><td>c8</td><td>256/512</td><td>1,179,648</td><td>36,920</td><td>540</td><td>10,395</td><td>91,719</td></tr><tr><td>c11</td><td>512/512</td><td>2,359,296</td><td>22,735</td><td>36</td><td>756</td><td>3,312</td></tr><tr><td>c14</td><td>512/512</td><td>2,359,296</td><td>26,584</td><td>108</td><td>882</td><td>4,536</td></tr></table>
+
+that our intrinsic dimensionality gives a smaller number than theirs. This is because compression through the covariance matrix gives quadratic factor improvement while compression through low rank property of the weight matrices gives linear order improvement. This indicates considering near low rank properties of both of weight matrices and covariance matrices yields sharper bounds. It can be realized by our unified theoretical frame-work.
\ No newline at end of file
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+# CONDITIONAL LEARNING OF FAIR REPRESENTATIONS
+
+# Han Zhao* & Amanda Coston
+
+Machine Learning Department
+
+Carnegie Mellon University
+
+han.zhao@cs.cmu.edu
+
+acoston@andrew.cmu.edu
+
+# Tameem Adel
+
+Department of Engineering
+
+University of Cambridge
+
+tah47@cam.ac.uk
+
+# Geoffrey J. Gordon
+
+Microsoft Research, Montreal
+
+Machine Learning Department
+
+Carnegie Mellon University
+
+geoff.gordon@microsoft.com
+
+# ABSTRACT
+
+We propose a novel algorithm for learning fair representations that can simultaneously mitigate two notions of disparity among different demographic subgroups in the classification setting. Two key components underpinning the design of our algorithm are balanced error rate and conditional alignment of representations. We show how these two components contribute to ensuring accuracy parity and equalized false-positive and false-negative rates across groups without impacting demographic parity. Furthermore, we also demonstrate both in theory and on two real-world experiments that the proposed algorithm leads to a better utility-fairness trade-off on balanced datasets compared with existing algorithms on learning fair representations for classification.
+
+# 1 INTRODUCTION
+
+High-stakes settings, such as loan approvals, criminal justice, and hiring processes, use machine learning tools to help make decisions. A key question in these settings is whether the algorithm makes fair decisions. In settings that have historically had discrimination, we are interested in defining fairness with respect to a protected group, the group which has historically been disadvantaged. The rapidly growing field of algorithmic fairness has a vast literature that proposes various fairness metrics, characterizes the relationship between fairness metrics, and describes methods to build classifiers that satisfy these metrics (Chouldechova & Roth, 2018; Corbett-Davies & Goel, 2018). Among many recent attempts to achieve algorithmic fairness (Dwork et al., 2012; Hardt et al., 2016; Zemel et al., 2013; Zafar et al., 2015), learning fair representations has attracted increasing attention due to its flexibility in learning rich representations based on advances in deep learning (Edwards & Storkey, 2015; Louizos et al., 2015; Beutel et al., 2017; Madras et al., 2018). The backbone idea underpinning this line of work is very intuitive: if the representations of data from different groups are similar to each other, then any classifier acting on such representations will also be agnostic to the group membership.
+
+However, it has long been empirically observed (Calders et al., 2009) and recently been proved (Zhao & Gordon, 2019) that fairness is often at odds with utility. For example, consider demographic parity, which requires the classifier to be independent of the group membership attribute. It is clear that demographic parity will cripple the utility if the demographic group membership and the target variable are indeed correlated. To escape such inherent trade-off, other notions of fairness, such as equalized odds (Hardt et al., 2016), which asks for equal false positive and negative rates across groups, and accuracy parity (Zafar et al., 2017), which seeks equalized error rates across groups, have been proposed. It is a well-known result that equalized odds is incompatible with demographic parity (Barocas et al., 2017) except in degenerate cases where group membership is independent
+
+of the target variable. Accuracy parity and the so-called predictive rate parity (c.f. Definition 2.4) could be simultaneously achieved, e.g., the COMPAS tool (Dieterich et al., 2016). Furthermore, under some conditions, it is also known that demographic parity can lead to accuracy parity (Zhao & Gordon, 2019). However, whether it is possible to simultaneously guarantee equalized odds and accuracy parity remains an open question.
+
+In this work, we provide an affirmative answer to the above question by proposing an algorithm to align the conditional distributions (on the target variable) of representations across different demographic subgroups. The proposed formulation is a minimax problem that admits a simple reduction to cost-sensitive learning. The key component underpinning the design of our algorithm is the balanced error rate (BER, c.f. Section 2) (Feldman et al., 2015; Menon & Williamson, 2018), over the target variable and protected attributes. We demonstrate both in theory and on two real-world experiments that together with the conditional alignment, BER helps our algorithm to simultaneously ensure accuracy parity and equalized odds across groups. Our key contributions are summarized as follows:
+
+- We prove that BER plays a fundamental role in ensuring accuracy parity and a small joint error across groups. Together with the conditional alignment of representations, this implies that we can simultaneously achieve equalized odds and accuracy parity. Furthermore, we also show that when equalized odds is satisfied, BER serves as an upper bound on the error of each subgroup. These results help to justify the design of our algorithm in using BER instead of the marginal error as our loss functions.  
+- We provide theoretical results that our method achieves equalized odds without impacting demographic parity. This result shows that we can preserve the demographic parity gap for free while simultaneously achieving equalized odds.  
+- Empirically, among existing fair representation learning methods, we demonstrate that our algorithm is able to achieve a better utility on balanced datasets. On an imbalanced dataset, our algorithm is the only method that achieves accuracy parity; however it does so at the cost of decreased utility.
+
+We believe our theoretical results contribute to the understanding on the relationship between equalized odds and accuracy parity, and the proposed algorithm provides an alternative in real-world scenarios where accuracy parity and equalized odds are desired.
+
+# 2 PRELIMINARY
+
+We first introduce the notations used throughout the paper and formally describe the problem setup and various definitions of fairness explored in the literature.
+
+Notation We use  $\mathcal{X} \subseteq \mathbb{R}^d$  and  $\mathcal{Y} = \{0,1\}$  to denote the input and output space. Accordingly, we use  $X$  and  $Y$  to denote the random variables which take values in  $\mathcal{X}$  and  $\mathcal{Y}$ , respectively. Lower case letters  $\mathbf{x}$  and  $y$  are used to denote the instantiation of  $X$  and  $Y$ . To simplify the presentation, we use  $A \in \{0,1\}$  as the sensitive attribute, e.g., race, gender, etc. Let  $\mathcal{H}$  be the hypothesis class of classifiers. In other words, for  $h \in \mathcal{H}$ ,  $h: \mathcal{X} \to \mathcal{Y}$  is the predictor that outputs a prediction. Note that even if the predictor does not explicitly take the sensitive attribute  $A$  as input, this fairness through blindness mechanism can still be biased due to the potential correlations between  $X$  and  $A$ . In this work we study the stochastic setting where there is a joint distribution  $\mathcal{D}$  over  $X, Y$  and  $A$  from which the data are sampled. To keep the notation consistent, for  $a, y \in \{0,1\}$ , we use  $\mathcal{D}_a$  to mean the conditional distribution of  $\mathcal{D}$  given  $A = a$  and  $\mathcal{D}^y$  to mean the conditional distribution of  $\mathcal{D}$  given  $Y = y$ . For an event  $E$ ,  $\mathcal{D}(E)$  denotes the probability of  $E$  under  $\mathcal{D}$ . In particular, in the literature of fair machine learning, we call  $\mathcal{D}(Y = 1)$  the base rate of distribution  $\mathcal{D}$  and we use  $\Delta_{\mathrm{BR}}(\mathcal{D},\mathcal{D}') := |\mathcal{D}(Y = 1) - \mathcal{D}'(Y = 1)|$  to denote the difference of the base rates between two distributions  $\mathcal{D}$  and  $\mathcal{D}'$  over the same sample space.
+
+Given a feature transformation function  $g: \mathcal{X} \to \mathcal{Z}$  that maps instances from the input space  $\mathcal{X}$  to feature space  $\mathcal{Z}$ , we define  $g_{\sharp}\mathcal{D} := \mathcal{D} \circ g^{-1}$  to be the induced (pushforward) distribution of  $\mathcal{D}$  under  $g$ , i.e., for any event  $E' \subseteq \mathcal{Z}$ ,  $g_{\sharp}\mathcal{D}(E') := \mathcal{D}(g^{-1}(E')) = \mathcal{D}(\{x \in \mathcal{X} \mid g(x) \in E'\})$ . To measure the discrepancy between distributions, we use  $d_{\mathrm{TV}}(\mathcal{D},\mathcal{D}')$  to denote the total variation between
+
+them:  $d_{\mathrm{TV}}(\mathcal{D}, \mathcal{D}') \coloneqq \sup_E |\mathcal{D}(E) - \mathcal{D}'(E)|$ . In particular, for binary random variable  $Y$ , it can be readily verified that the total variation between the marginal distributions  $\mathcal{D}(Y)$  and  $\mathcal{D}'(Y)$  reduces to the difference of their base rates,  $\Delta_{\mathrm{BR}}(\mathcal{D}, \mathcal{D}')$ . To see this, realize that  $d_{\mathrm{TV}}(\mathcal{D}(Y), \mathcal{D}'(Y)) = \max \{| \mathcal{D}(Y = 1) - \mathcal{D}'(Y = 1) |, |\mathcal{D}(Y = 0) - \mathcal{D}'(Y = 0) | \} = |\mathcal{D}(Y = 1) - \mathcal{D}'(Y = 1)| = \Delta_{\mathrm{BR}}(\mathcal{D}, \mathcal{D}')$  by definition.
+
+Given a joint distribution  $\mathcal{D}$ , the error of a predictor  $h$  under  $\mathcal{D}$  is defined as  $\mathrm{Err}_{\mathcal{D}}(h) \coloneqq \mathbb{E}_{\mathcal{D}}[|Y - h(X)|]$ . Note that for binary classification problems, when  $h(X) \in \{0,1\}$ ,  $\mathrm{Err}_{\mathcal{D}}(h)$  reduces to the true error rate of binary classification. To make the notation more compact, we may drop the subscript  $\mathcal{D}$  when it is clear from the context. Furthermore, we use  $\mathrm{CE}_{\mathcal{D}}(\widehat{Y} \parallel Y)$  to denote the cross-entropy loss function between the predicted variable  $\widehat{Y}$  and the true label distribution  $Y$  over the joint distribution  $\mathcal{D}$ . For binary random variables  $Y$ , we define  $\mathrm{BER}_{\mathcal{D}}(\widehat{Y} \parallel Y)$  to be the balanced error rate of predicting  $Y$  using  $\widehat{Y}$ , e.g.,  $\mathrm{BER}_{\mathcal{D}}(\widehat{Y} \parallel Y) \coloneqq \mathcal{D}(\widehat{Y} = 0 \mid Y = 1) + \mathcal{D}(\widehat{Y} = 1 \mid Y = 0)$ . Realize that  $\mathcal{D}(\widehat{Y} = 0 \mid Y = 1)$  is the false negative rate (FNR) of  $\widehat{Y}$  and  $\mathcal{D}(\widehat{Y} = 1 \mid Y = 0)$  corresponds to the false positive rate (FPR). So the balanced error rate can also be understood as the sum of FPR and FNR using the predictor  $\widehat{Y}$ . We can similarly define  $\mathrm{BER}_{\mathcal{D}}(\widehat{A} \parallel A)$  as well.
+
+Problem Setup In this work we focus on group fairness where the group membership is given by the sensitive attribute  $A$ . We assume that the sensitive attribute  $A$  is available to the learner during training phase, but not inference phase. As a result, post-processing techniques to ensure fairness are not feasible under our setting. In the literature, there are many possible definitions of fairness (Narayanan, 2018), and in what follows we provide a brief review of the ones that are mostly relevant to this work.
+
+Definition 2.1 (Demographic Parity (DP)). Given a joint distribution  $\mathcal{D}$ , a classifier  $\widehat{Y}$  satisfies demographic parity if  $\widehat{Y}$  is independent of  $A$ .
+
+When  $\widehat{Y}$  is a deterministic binary classifier, demographic parity reduces to the requirement that  $\mathcal{D}_0(\widehat{Y} = 1) = \mathcal{D}_1(\widehat{Y} = 1)$ , i.e., positive outcome is given to the two groups at the same rate. Demographic parity is also known as statistical parity, and it has been adopted as the default definition of fairness in a series of work (Calders & Verwer, 2010; Calders et al., 2009; Edwards & Storkey, 2015; Johndrow et al., 2019; Kamiran & Calders, 2009; Kamishima et al., 2011; Louizos et al., 2015; Zemel et al., 2013; Madras et al., 2018; Adel et al., 2019). It is not surprising that demographic parity may cripple the utility that we hope to achieve, especially in the common scenario where the base rates differ between two groups (Hardt et al., 2016). Formally, the following theorem characterizes the trade-off in terms of the joint error across different groups:
+
+Theorem 2.1. (Zhao & Gordon, 2019) Let  $\widehat{Y} = h(g(X))$  be the classifier. Then  $\mathrm{Err}_{\mathcal{D}_0}(h \circ g) + \mathrm{Err}_{\mathcal{D}_1}(h \circ g) \geq \Delta_{\mathrm{BR}}(\mathcal{D}_0, \mathcal{D}_1) - d_{\mathrm{TV}}(g_\sharp \mathcal{D}_0, g_\sharp \mathcal{D}_1)$ .
+
+In this case of representations that are independent of the sensitive attribute  $A$ , then the second term  $d_{\mathrm{TV}}(g_{\sharp}\mathcal{D}_0,g_{\sharp}\mathcal{D}_1)$  becomes 0, and this implies:
+
+$$
+\operatorname {E r r} _ {\mathcal {D} _ {0}} (h \circ g) + \operatorname {E r r} _ {\mathcal {D} _ {1}} (h \circ g) \geq \Delta_ {\mathrm {B R}} (\mathcal {D} _ {0}, \mathcal {D} _ {1}).
+$$
+
+At a colloquial level, the above inequality could be understood as an uncertainty principle which says:
+
+For fair representations, it is not possible to construct a predictor that simultaneously minimizes the errors on both demographic subgroups.
+
+More precisely, by the pigeonhole principle, the following corollary holds:
+
+Corollary 2.1. If  $d_{\mathrm{TV}}(g_{\sharp}\mathcal{D}_0,g_{\sharp}\mathcal{D}_1) = 0$ , then for any hypothesis  $h$ ,  $\max \{\mathrm{Err}_{\mathcal{D}_0}(h\circ g),\mathrm{Err}_{\mathcal{D}_1}(h\circ g)\} \geq \Delta_{\mathrm{BR}}(\mathcal{D}_0,\mathcal{D}_1) / 2$ .
+
+In words, this means that for fair representations in the demographic parity sense, at least one of the subgroups has to incur a prediction error of at least  $\Delta_{\mathrm{BR}}(\mathcal{D}_0,\mathcal{D}_1) / 2$  which could be large in settings like criminal justice where  $\Delta_{\mathrm{BR}}(\mathcal{D}_0,\mathcal{D}_1)$  is large. In light of such inherent trade-off, an alternative definition is accuracy parity, which asks for equalized error rates across different groups:
+
+Definition 2.2 (Accuracy Parity). Given a joint distribution  $\mathcal{D}$ , a classifier  $\widehat{Y}$  satisfies accuracy parity if  $\mathcal{D}_0(\widehat{Y} \neq Y) = \mathcal{D}_1(\widehat{Y} \neq Y)$ .
+
+A violation of accuracy parity is also known as disparate mistreatment (Zafar et al., 2017). Different from the definition of demographic parity, the definition of accuracy parity does not eliminate the perfect predictor even when the base rates differ across groups. Of course, other more refined definitions of fairness also exist in the literature, such as equalized odds (Hardt et al., 2016).
+
+Definition 2.3 (Equalized Odds, a.k.a. Positive Rate Parity). Given a joint distribution  $\mathcal{D}$ , a classifier  $\widehat{Y}$  satisfies equalized odds if  $\widehat{Y}$  is independent of  $A$  conditioned on  $Y$ .
+
+The definition of equalized odds essentially requires equal true positive and false positive rates between different groups, hence it is also known as positive rate parity. Analogous to accuracy parity, equalized odds does not eliminate the perfect classifier (Hardt et al., 2016), and we will also justify this observation by formal analysis shortly. Last but not least, we have the following definition for predictive rate parity:
+
+Definition 2.4 (Predictive Rate Parity). Given a joint distribution  $\mathcal{D}$ , a classifier  $\widehat{Y}$  satisfies predictive rate parity if  $\mathcal{D}_0(Y = 1 \mid \widehat{Y} = c) = \mathcal{D}_1(Y = 1 \mid \widehat{Y} = c)$ ,  $\forall c \in [0,1]$ .
+
+Note that in the above definition we allow the classifier  $\widehat{Y}$  to be probabilistic, meaning that the output of  $\widehat{Y}$  could be any value between 0 and 1. For the case where  $\widehat{Y}$  is deterministic, Chouldechova (2017) showed that no deterministic classifier can simultaneously satisfy equalized odds and predictive rate parity when the base rates differ across subgroups and the classifier is not perfect.
+
+# 3 ALGORITHM AND ANALYSIS
+
+In this section we first give the proposed optimization formulation and then discuss through formal analysis the motivation of our algorithmic design. Specifically, we show in Section 3.1 why our formulation helps to escape the utility-fairness trade-off. We then in Section 3.2 formally prove that the BERs in the objective function could be used to guarantee a small joint error across different demographic subgroups. In Section 3.3 we establish the relationship between equalized odds and accuracy parity by providing an upper bound of the error gap in terms of both BER and the equalized odds gap. We conclude this section with a brief discussion on the practical implementation of the proposed optimization formulation in Section 3.4. Due to the space limit, we defer all the proofs to the appendix and focus on explaining the high-level intuition and implications of our results.
+
+As briefly discussed in Section 5, a dominant approach in learning fair representations is via adversarial training. Specifically, the following objective function is optimized:
+
+$$
+\min  _ {h, g} \max  _ {h ^ {\prime}} \mathrm {C E} _ {\mathcal {D}} (h (g (X)) \mid | Y) - \lambda \mathrm {C E} _ {\mathcal {D}} \left(h ^ {\prime} (g (X)) \mid | A\right) \tag {1}
+$$
+
+In the above formulation, the first term corresponds to minimization of prediction loss of the target variable and the second term represents the loss incurred by the adversary  $h'$ . Overall this minimax optimization problem expresses a trade-off (controlled by the hyperparameter  $\lambda > 0$ ) between utility and fairness through the representation learning function  $g$ : on one hand  $g$  needs to preserve sufficient information related to  $Y$  in order to minimize the first term, but on the other hand  $g$  also needs to filter out information related to  $A$  in order to maximize the second term.
+
+# 3.1 CONDITIONAL LEARNING OF FAIR REPRESENTATIONS
+
+However, as we introduced in Section 2, the above framework is still subjective to the inherent trade-off between utility and fairness. To escape such a trade-off, we advocate for the following optimization formulation instead:
+
+$$
+\min  _ {h, g} \max  _ {h ^ {\prime}, h ^ {\prime \prime}} \quad \operatorname {B E R} _ {\mathcal {D}} \left(h (g (X)) \mid | Y\right) - \lambda \left(\operatorname {B E R} _ {\mathcal {D} ^ {0}} \left(h ^ {\prime} (g (X)) \mid | A\right) + \operatorname {B E R} _ {\mathcal {D} ^ {1}} \left(h ^ {\prime \prime} (g (X)) \mid | A\right)\right) \tag {2}
+$$
+
+Note that here we optimize over two distinct adversaries, one for each conditional distribution  $\mathcal{D}^y$ ,  $y \in \{0,1\}$ . Intuitively, the main difference between (2) and (1) is that we use BER as our objective function in both terms. By definition, since BER corresponds to the sum of Type-I and Type-II errors in classification, the proposed objective function essentially minimizes the conditional errors instead of the original marginal error:
+
+$$
+\begin{array}{l} \mathcal {D} (\widehat {Y} \neq Y) = \mathcal {D} (Y = 0) \mathcal {D} (\widehat {Y} \neq Y \mid Y = 0) + \mathcal {D} (Y = 1) \mathcal {D} (\widehat {Y} \neq Y \mid Y = 1) \\ \operatorname {B E R} _ {\mathcal {D}} (\widehat {Y} \mid Y) \propto \frac {1}{2} \mathcal {D} (\widehat {Y} \neq Y \mid Y = 0) + \frac {1}{2} \mathcal {D} (\widehat {Y} \neq Y \mid Y = 1), \tag {3} \\ \end{array}
+$$
+
+which means that the loss function gives equal importance to the classification error from both groups. Note that the BERs in the second term of (2) are over  $\mathcal{D}^y$ ,  $y \in \{0,1\}$ . Roughly speaking, the second term encourages alignment of the conditional distributions  $g_{\sharp} \mathcal{D}_0^y$  and  $g_{\sharp} \mathcal{D}_1^y$  for  $y \in \{0,1\}$ . The following proposition shows that a perfect conditional alignment of the representations also implies that any classifier based on the representations naturally satisfies the equalized odds criterion:
+
+Proposition 3.1. For  $g: \mathcal{X} \to \mathcal{Z}$ , if  $d_{\mathrm{TV}}(g_{\sharp}\mathcal{D}_0^y, g_{\sharp}\mathcal{D}_1^y) = 0, \forall y \in \{0,1\}$ , then for any classifier  $h: \mathcal{Z} \to \{0,1\}$ ,  $h \circ g$  satisfies equalized odds.
+
+To understand why we aim for conditional alignment of distributions instead of aligning the marginal feature distributions, the following proposition characterizes why such alignment will help us to escape the previous trade-off:
+
+Proposition 3.2. For  $g: \mathcal{X} \to \mathcal{Z}$ , if  $d_{\mathrm{TV}}(g_{\sharp}\mathcal{D}_0^y, g_{\sharp}\mathcal{D}_1^y) = 0, \forall y \in \{0,1\}$ , then for any classifier  $h: \mathcal{Z} \to \{0,1\}$ ,  $d_{\mathrm{TV}}((h \circ g)_{\sharp}\mathcal{D}_0, (h \circ g)_{\sharp}\mathcal{D}_1) \leq \Delta_{\mathrm{BR}}(\mathcal{D}_0, \mathcal{D}_1)$ .
+
+As a corollary, this implies that the lower bound given in Theorem 2.1 now vanishes if we instead align the conditional distributions of representations:
+
+$$
+\operatorname {E r r} _ {\mathcal {D} _ {0}} (h \circ g) + \operatorname {E r r} _ {\mathcal {D} _ {1}} (h \circ g) \geq \Delta_ {\mathrm {B R}} (\mathcal {D} _ {0}, \mathcal {D} _ {1}) - d _ {\mathrm {T V}} ((h \circ g) _ {\sharp} \mathcal {D} _ {0}, (h \circ g) _ {\sharp} \mathcal {D} _ {1}) = 0,
+$$
+
+where the first inequality is due to Lemma 3.1 (Zhao & Gordon, 2019) and the triangle inequality by the  $d_{\mathrm{TV}}(\cdot ,\cdot)$  distance. Of course, the above lower bound can only serve as a necessary condition but not sufficient to ensure a small joint error across groups. Later (c.f. Theorem 3.2) we will show that together with a small BER on the target variable, it becomes a sufficient condition as well.
+
+# 3.2 THE PRESERVATION OF DEMOGRAPHIC PARITY GAP AND SMALL JOINT ERROR
+
+In this section we show that learning representations by aligning the conditional distributions across groups cannot increase the DP gap as compared to the DP gap of  $Y$ . Before we proceed, we first introduce a metric to measure the deviation of a predictor from satisfying demographic parity:
+
+Definition 3.1 (DP Gap). Given a joint distribution  $\mathcal{D}$ , the demographic parity gap of a classifier  $\widehat{Y}$  is  $\Delta_{\mathrm{DP}}(\widehat{Y}) \coloneqq |\mathcal{D}_0(\widehat{Y} = 1) - \mathcal{D}_1(\widehat{Y} = 1)|$ .
+
+Clearly, if  $\Delta_{\mathrm{DP}}(\widehat{Y}) = 0$ , then  $\widehat{Y}$  satisfies demographic parity. To simplify the exposition, let  $\gamma_{a} := \mathcal{D}_{a}(Y = 0), \forall a \in \{0,1\}$ . We first prove the following lemma:
+
+Lemma 3.1. Assume the conditions in Proposition 3.1 hold and let  $\widehat{Y} = h(g(X))$  be the classifier, then  $|\mathcal{D}_0(\widehat{Y} = y) - \mathcal{D}_1(\widehat{Y} = y)| \leq |\gamma_0 - \gamma_1| \cdot \left(\mathcal{D}^0 (\widehat{Y} = y) + \mathcal{D}^1 (\widehat{Y} = y)\right), \forall y \in \{0,1\}$ .
+
+Lemma 3.1 gives an upper bound on the difference of the prediction probabilities across different subgroups. Applying Lemma 3.1 twice for  $y = 0$  and  $y = 1$ , we can prove the following theorem:
+
+Theorem 3.1. Assume the conditions in Proposition 3.1 hold and let  $\widehat{Y} = h(g(X))$  be the classifier, then  $\Delta_{\mathrm{DP}}(\widehat{Y}) \leq \Delta_{\mathrm{BR}}(\mathcal{D}_0, \mathcal{D}_1) = \Delta_{\mathrm{DP}}(Y)$ .
+
+Remark Theorem 3.1 shows that aligning the conditional distributions of representations between groups will not add more bias in terms of the demographic parity gap. In particular, the DP gap of any classifier that satisfies equalized odds will be at most the DP gap of the perfect classifier. This is particularly interesting as it is well-known in the literature (Barocas et al., 2017) that demographic parity is not compatible with equalized odds except in degenerate cases. Despite this result, Theorem 3.1 says that we can still achieve equalized odds and simultaneously preserve the DP gap.
+
+In Section 3.1 we show that aligning the conditional distributions of representations between groups helps reduce the lower bound of the joint error, but nevertheless that is only a necessary condition. In the next theorem we show that together with a small Type-I and Type-II error in inferring the target variable  $Y$ , these two properties are also sufficient to ensure a small joint error across different demographic subgroups.
+
+Theorem 3.2. Assume the conditions in Proposition 3.1 hold and let  $\widehat{Y} = h(g(X))$  be the classifier, then  $\mathrm{Err}_{\mathcal{D}_0}(\widehat{Y}) + \mathrm{Err}_{\mathcal{D}_1}(\widehat{Y}) \leq 2\mathrm{BER}_{\mathcal{D}}(\widehat{Y} \parallel Y)$ .
+
+The above bound means that in order to achieve small joint error across groups, it suffices for us to minimize the BER if a classifier satisfies equalized odds. Note that by definition, the BER in the bound equals to the sum of Type-I and Type-II classification errors using  $\widehat{Y}$  as a classifier. Theorem 3.2 gives an upper bound of the joint error across groups and it also serves as a motivation for us to design the optimization formulation (2) that simultaneously minimizes the BER and aligns the conditional distributions.
+
+# 3.3 CONDITIONAL ALIGNMENT AND BALANCED ERROR RATES LEAD TO SMALL ERROR
+
+In this section we will see that a small BER and equalized odds together not only serve as a guarantee of a small joint error, but they also lead to a small error gap between different demographic subgroups. Recall that we define  $\gamma_{a} \coloneqq \mathcal{D}_{a}(Y = 0), \forall a \in \{0,1\}$ . Before we proceed, we first formally define the accuracy gap and equalized odds gap of a classifier  $\widehat{Y}$ :
+
+Definition 3.2 (Error Gap). Given a joint distribution  $\mathcal{D}$ , the error gap of a classifier  $\widehat{Y}$  is  $\Delta_{\mathrm{Err}}(\widehat{Y}) := |\mathcal{D}_0(\widehat{Y} \neq Y) - \mathcal{D}_1(\widehat{Y} \neq Y)|$ .
+
+Definition 3.3 (Equalized Odds Gap). Given a joint distribution  $\mathcal{D}$ , the equalized odds gap of a classifier  $\widehat{Y}$  is  $\Delta_{\mathrm{EO}}(\widehat{Y}) \coloneqq \max_{y \in \{0,1\}} |\mathcal{D}_0^y(\widehat{Y} = 1) - \mathcal{D}_1^y(\widehat{Y} = 1)|$ .
+
+By definition the error gap could also be understood as the accuracy parity gap between different subgroups. The following theorem characterizes the relationship between error gap, equalized odds gap and the difference of base rates across subgroups:
+
+Theorem 3.3. For any classifier  $\widehat{Y}$ ,  $\Delta_{\mathrm{Err}}(\widehat{Y}) \leq \Delta_{\mathrm{BR}}(\mathcal{D}_0, \mathcal{D}_1) \cdot \mathrm{BER}_{\mathcal{D}}(\widehat{Y} \parallel Y) + 2\Delta_{\mathrm{EO}}(\widehat{Y})$ .
+
+As a direct corollary of Theorem 3.3, if the classifier  $\widehat{Y}$  satisfies equalized odds, then  $\Delta_{\mathrm{EO}}(\widehat{Y}) = 0$ . In this case since  $\Delta_{\mathrm{BR}}(\mathcal{D}_0,\mathcal{D}_1)$  is a constant, minimizing the balanced error rate  $\mathrm{BER}_{\mathcal{D}}(\widehat{Y} || Y)$  also leads to minimizing the error gap. Furthermore, if we combine Theorem 3.2 and Theorem 3.3 together, we can guarantee that each of the errors cannot be too large:
+
+Corollary 3.1. For any joint distribution  $\mathcal{D}$  and classifier  $\widehat{Y}$ , if  $\widehat{Y}$  satisfies equalized odds, then  $\max \{\mathrm{Err}_{\mathcal{D}_0}(\widehat{Y}), \mathrm{Err}_{\mathcal{D}_1}(\widehat{Y})\} \leq \Delta_{\mathrm{BR}}(\mathcal{D}_0, \mathcal{D}_1) \cdot \mathrm{BER}_{\mathcal{D}}(\widehat{Y} \parallel Y) / 2 + \mathrm{BER}_{\mathcal{D}}(\widehat{Y} \parallel Y)$ .
+
+Remark It is a well-known fact that out of the three fairness criteria, i.e., demographic parity, equalized odds, and predictive rate parity, any two of them cannot hold simultaneously (Barocas et al., 2017) except in degenerate cases. By contrast, Theorem 3.3 suggests it is possible to achieve both equalized odds and accuracy parity. In particular, among all classifiers that satisfy equalize odds, it suffices to minimize the sum of Type-I and Type-II error  $\mathrm{BER}_{\mathcal{D}}(\widehat{Y} || Y)$  in order to achieve accuracy parity. It is also worth pointing out that Theorem 3.3 provides only an upper bound, but not necessarily the tightest one. In particular, the error gap could still be 0 while  $\mathrm{BER}_{\mathcal{D}}(\widehat{Y} || Y)$  is greater than 0. To see this, we have
+
+$$
+\left\{ \begin{array}{l} \operatorname {E r r} _ {\mathcal {D} _ {0}} (\widehat {Y}) = \mathcal {D} _ {0} (Y = 0) \cdot \mathcal {D} _ {0} (\widehat {Y} = 1 \mid Y = 0) + \mathcal {D} _ {0} (Y = 1) \cdot \mathcal {D} _ {0} (\widehat {Y} = 0 \mid Y = 1) \\ \operatorname {E r r} _ {\mathcal {D} _ {1}} (\widehat {Y}) = \mathcal {D} _ {1} (Y = 0) \cdot \mathcal {D} _ {1} (\widehat {Y} = 1 \mid Y = 0) + \mathcal {D} _ {1} (Y = 1) \cdot \mathcal {D} _ {1} (\widehat {Y} = 0 \mid Y = 1). \end{array} \right.
+$$
+
+Now if the predictor  $\widehat{Y}$  satisfies equalized odds, then
+
+$$
+\mathcal {D} _ {0} (\widehat {Y} = 1 \mid Y = 0) = \mathcal {D} _ {1} (\widehat {Y} = 1 \mid Y = 0) = \mathcal {D} (\widehat {Y} = 1 \mid Y = 0),
+$$
+
+$$
+\mathcal {D} _ {0} (\widehat {Y} = 0 \mid Y = 1) = \mathcal {D} _ {1} (\widehat {Y} = 0 \mid Y = 1) = \mathcal {D} (\widehat {Y} = 0 \mid Y = 1).
+$$
+
+Hence the error gap  $\Delta_{\mathrm{Err}}(\hat{Y})$  admits the following identity:
+
+$$
+\begin{array}{l} \Delta_ {\mathrm {E r r}} (\widehat {Y}) = \left| \mathcal {D} (\widehat {Y} = 1 \mid Y = 0) \big (\mathcal {D} _ {0} (Y = 0) - \mathcal {D} _ {1} (Y = 0) \big) + \mathcal {D} (\widehat {Y} = 0 \mid Y = 1) \big (\mathcal {D} _ {0} (Y = 1) - \mathcal {D} _ {1} (Y = 1) \big) \right| \\ = \Delta_ {\mathrm {B R}} (\mathcal {D} _ {0}, \mathcal {D} _ {1}) \cdot \left| \mathcal {D} (\widehat {Y} = 1 \mid Y = 0) - \mathcal {D} (\widehat {Y} = 0 \mid Y = 1) \right| \\ = \Delta_ {\mathrm {B R}} (\mathcal {D} _ {0}, \mathcal {D} _ {1}) \cdot \left| \operatorname {F P R} (\widehat {Y}) - \operatorname {F N R} (\widehat {Y}) \right|. \\ \end{array}
+$$
+
+In other words, if the predictor  $\widehat{Y}$  satisfies equalized odds, then in order to have equalized accuracy,  $\widehat{Y}$  only needs to have equalized FPR and FNR globally when the base rates differ across groups. This is a much weaker condition to ask for than the one asking  $\mathrm{BER}_{\mathcal{D}}(\widehat{Y} || Y) = 0$
+
+# 3.4 PRACTICAL IMPLEMENTATION
+
+We cannot directly optimize the proposed optimization formulation (2) since the binary  $0/1$  loss is NP-hard to optimize, or even approximately optimize over a wide range of hypothesis classes (Ben-David et al., 2003). However, observe that for any classifier  $\widehat{Y}$  and  $y \in \{0,1\}$ , the log-loss (cross-entropy loss)  $\mathrm{CE}_{\mathcal{D}^y}(\widehat{Y} \parallel Y)$  is a convex relaxation of the binary loss:
+
+$$
+\mathcal {D} (\widehat {Y} \neq y \mid Y = y) = \frac {\mathcal {D} (\widehat {Y} \neq y , Y = y)}{\mathcal {D} (Y = y)} \leq \frac {\mathrm {C E} _ {\mathcal {D} ^ {y}} (\widehat {Y} \mid \mid Y)}{\mathcal {D} (Y = y)}. \tag {4}
+$$
+
+Hence in practice we can relax the optimization problem (2) to a cost-sensitive cross-entropy loss minimization problem, where the weight for each class is given by the inverse marginal probability of the corresponding class. This allows us to equivalently optimize the objective function without explicitly computing the conditional distributions.
+
+# 4 EMPIRICAL STUDIES
+
+In light of our theoretic findings, in this section we verify the effectiveness of the proposed algorithm in simultaneously ensuring equalized odds and accuracy parity using real-world datasets. We also analyze the impact of imposing such parity constraints on the utility of the target classifier, as well as its relationship to the intrinsic structure of the binary classification problem, e.g., the difference of base rates across groups, the global imbalance of the target variable, etc. We analyze how this imbalance affects the utility-fairness trade-off. As we shall see shortly, we will empirically demonstrate that, in many cases, especially the ones where the dataset is imbalanced in terms of the target variable, this will inevitably compromise the target utility. While for balanced datasets, this trend is less obvious: the proposed algorithm achieves a better fairness-utility trade-off when compared with existing fair representation learning methods and we can hope to achieve fairness without sacrificing too much on utility.
+
+Table 1: Statistics about the Adult and COMPAS datasets.  
+
+<table><tr><td></td><td>Train / Test</td><td>D0(Y=1)</td><td>D1(Y=1)</td><td>ΔBR(D0, D1)</td><td>D(Y=1)</td><td>D(A=1)</td></tr><tr><td>Adult</td><td>30,162/15,060</td><td>0.310</td><td>0.113</td><td>0.196</td><td>0.246</td><td>0.673</td></tr><tr><td>COMPAS</td><td>4,320/1,852</td><td>0.400</td><td>0.529</td><td>0.129</td><td>0.467</td><td>0.514</td></tr></table>
+
+# 4.1 EXPERIMENTAL SETUP
+
+To this end, we perform experiments on two popular real-world datasets in the literature of algorithmic fairness, including an income-prediction dataset, known as the Adult dataset, from the UCI Machine Learning Repository (Dua & Graff, 2017), and the Propublica COMPAS dataset (Dieterich et al., 2016). The basic statistics of these two datasets are listed in Table 1.
+
+Adult Each instance in the Adult dataset describes an adult, e.g., gender, education level, age, etc, from the 1994 US Census. In this dataset we use gender as the sensitive attribute, and the processed data contains 114 attributes. The target variable (income) is also binary: 1 if  $\geq 50\mathrm{K}$ /year otherwise 0. For the sensitive attribute  $A$ ,  $A = 0$  means male otherwise female. From Table 1 we can see that the base rates are quite different (0.310 vs. 0.113) across groups in the Adult dataset. The dataset is also imbalanced in the sense that only around  $24.6\%$  of the instances have target label 1. Furthermore, the group ratio is also imbalanced: roughly  $67.3\%$  of the data are male.
+
+COMPAS The goal of the COMPAS dataset is binary classification on whether a criminal defendant will recidivate within two years or not. Each instance contains 12 attributes, including age, race, gender, number of prior crimes, etc. For this dataset, we use the race (white  $A = 0$  vs. black  $A = 1$ ) as our sensitive attribute and target variable is 1 iff recidivism. The base rates are different across groups, but the COMPAS dataset is balanced in both the target variable and the sensitive attribute.
+
+To validate the effect of ensuring equalized odds and accuracy parity, for each dataset, we perform controlled experiments by fixing the baseline network architecture so that it is shared among all the fair representation learning methods. We term the proposed method CFAIR (for conditional fair representations) that minimizes conditional errors both the target variable loss function and adversary loss function. To demonstrate the importance of using BER in the loss function of target variable, we compare with a variant of CFAIR that only uses BER in the loss of adversaries, denoted as CFAIR-EO. To see the relative effect of using cross-entropy loss vs  $L_{1}$  loss, we also show the results of LAFTR (Madras et al., 2018), a state-of-the-art method for learning fair representations. Note that LAFTR is closely related to CFAIR-EO but slightly different: LAFTR uses global cross-entropy loss for target variable, but conditional  $L_{1}$  loss for the adversary. Also, there is only one adversary in LAFTR, while there are two adversaries, one for  $\mathcal{D}^0$  and one for  $\mathcal{D}^1$ , in both CFAIR and CFAIR-EO. Lastly, we also present baseline results of FAIR (Edwards & Storkey, 2015), which aims for demographic parity representations, and NODEBIAS, the baseline network without any fairness constraint. For all the fair representation learning methods, we use the gradient reversal layer (Ganin et al., 2016) to implement the gradient descent ascent (GDA) algorithm to optimize the minimax problem. All the experimental details, including network architectures, learning rates, batch sizes, etc. are provided in the appendix.
+
+# 4.2 RESULTS AND ANALYSIS
+
+In Figure 1 and Figure 2 we show the error gap  $\Delta_{\mathrm{Err}}$ , equalized odds gap  $\Delta_{\mathrm{EO}}$ , demographic parity gap  $\Delta_{\mathrm{DP}}$  and the joint error across groups  $\mathrm{Err}_0 + \mathrm{Err}_1$  of the aforementioned fair representation learning algorithms on both the Adult and the COMPAS datasets. For each algorithm and dataset, we also gradually increase the value of the trade-off parameter  $\lambda$  and compute the corresponding metrics.
+
+![](images/d013b326ef263ac7b0b354d01db68e2a1c77ddc2dc9a64c4c2e48769a05de847.jpg)  
+Figure 1: The error gap  $\Delta_{\mathrm{Err}}$ , equalized odds gap  $\Delta_{\mathrm{EO}}$ , demographic parity gap  $\Delta_{\mathrm{DP}}$  and joint error  $\mathrm{Err}_0 + \mathrm{Err}_1$  on the Adult dataset with  $\lambda \in \{0.1, 1.0, 10.0, 100.0, 1000.0\}$ .
+
+![](images/a104ad558f7f42ed8eea26e775f8b59e952efd5ff8ed97608e1158347ce0d42d.jpg)
+
+![](images/c53c955f5ddc3dcf16ea4add0f71ac78232223d2cd05e67a8c080fa3fa9aacec.jpg)
+
+![](images/be91a01ee752c20eda43875051179766d4603955245e470531f56109801e5234.jpg)
+
+![](images/fc839eb8604c560c8fcd0bbb49b5b315495a979346fc121c53725f8ad60cac06.jpg)  
+Figure 2: The error gap  $\Delta_{\mathrm{Err}}$ , equalized odds gap  $\Delta_{\mathrm{EO}}$ , demographic parity gap  $\Delta_{\mathrm{DP}}$  and joint error  $\mathrm{Err}_0 + \mathrm{Err}_1$  on the COMPAS dataset with  $\lambda \in \{0.1, 1.0, 10.0\}$ .
+
+![](images/12a60135611964615546ab0b6fcff07b9ac4a321e2f911d77eb383a11cf59f18.jpg)
+
+![](images/85a74a6d068e46afe21800106a14c9dde0f5056a8fdc6ee5f41622fa3246a652.jpg)
+
+![](images/93f8a7ea610046ff54a075dff8e55b550a153877885fd87f262c317ec843a29e.jpg)
+
+Adult Due to the imbalance of  $A$  in the Adult dataset, in the first plot of Figure 1 we can see that all the algorithms except CFAIR have a large error gap of around 0.12. As a comparison, observe that the error gap of CFAIR when  $\lambda = 1e3$  almost reduces to 0, confirming the effectiveness of our algorithm in ensuring accuracy parity. From the second plot, we can verify that all the three methods, including CFAIR, CFAIR-EO and LAFTR successfully ensure a small equalized odds gap, and they also decrease demographic parity gaps (the third plot). FAIR is the most effective one in mitigating  $\Delta_{\mathrm{DP}}$  since its objective function directly optimizes for that goal. Note that from the second plot we can also confirm that CFAIR-EO is more effective than LAFTR in reducing  $\Delta_{\mathrm{EO}}$ . The reason is two-fold. First, CFAIR-EO uses two distinct adversaries and hence it effectively competes with
+
+stronger adversaries than LAFTR. Second, CFAIR-EO uses the cross-entropy loss instead of the  $L_{1}$  loss for the adversary, and it is well-known that the maximum-likelihood estimator (equivalent to using the cross-entropy loss) is asymptotically efficient and optimal. On the other hand, since the Adult dataset is imbalanced (in terms of  $Y$ ), using BER in the loss function of the target variable can thus to a large extent hurt the utility, and this is also confirmed from the last plot, where we show the joint error.
+
+COMPAS The first three plots of Figure 2 once again verify that CFAIR successfully leads to reduced error gap, equalized odds gap and also demographic parity gap. These experimental results are consistent with our theoretical findings where we show that if the representations satisfy equalized odds, then its  $\Delta_{\mathrm{DP}}$  cannot exceed that of the optimal classifier, as shown by the horizontal dashed line in the third plot. In the fourth plot of Figure 2, we can see that as we increase  $\lambda$ , all the fair representation learning algorithms sacrifice utility. However, in contrast to Figure 1, here the proposed algorithm CFAIR has the smallest trade-off: this shows that CFAIR is particularly suited in the cases when the dataset is balanced and we would like to simultaneously ensure accuracy parity and equalized odds. As a comparison, while CFAIR-EO is still effective, it is slightly worse than CFAIR in terms of both ensuring parity and achieving small joint error.
+
+# 5 RELATED WORK
+
+Algorithmic Fairness In the literature of algorithmic fairness, two key notions of fairness have been extensively proposed and explored, i.e., group fairness, including various variants defined in Section 2, and individual fairness, which means that similar individuals should be treated similarly. Due to the complexity in defining a distance metric to measure the similarity between individuals (Dwork et al., 2012), most recent research focuses on designing efficient algorithms to achieve group fairness (Zemel et al., 2013; Zafar et al., 2015; Hardt et al., 2016; Zafar et al., 2017; Madras et al., 2018; Creager et al., 2019; Madras et al., 2019). In particular, Hardt et al. (2016) proposed a post-processing technique to achieve equalized odds by taking as input the model's prediction and the sensitive attribute. However, the post-processing technique requires access to the sensitive attribute during the inference phase, which is often not available in many real-world scenarios. Another line of work uses causal inference to define notions of causal fairness and to formulate procedures for achieving these notions (Zhang et al., 2018; Wang et al., 2019; Madras et al., 2019; Kilbertus et al., 2017; Kusner et al., 2017; Nabi & Shpitser, 2018). These approaches require making untestable assumptions. Of particular note is the observation in Coston et al. (2019) that fairness-adjustment procedures based on  $Y$  in settings with treatment effects may lead to adverse outcomes. To apply our method in such settings, we would need to match conditional counterfactual distributions, which could be a direction of future research.
+
+Theoretical Results on Fairness Theoretical work studying the relationship between different kinds of fairness notions are abundant. Motivated by the controversy of the potential discriminatory bias in recidivism prediction instruments, Chouldechova (2017) showed an intrinsic incompatibility between equalized odds and predictive rate parity. In the seminal work of Kleinberg et al. (2016), the authors demonstrated that when the base rates differ between different groups, then a non-perfect classifier cannot simultaneously be statistically calibrated and satisfy equalized odds. In the context of cost-sensitive learning, Menon & Williamson (2018) show that if the optimal decision function is dissimilar to a fair decision, then the fairness constraint will not significantly harm the target utility. The idea of reducing fair classification to cost-sensitive learning is not new. Agarwal et al. (2018) explored the connection between fair classification and a sequence of cost-sensitive learning problems where each stage corresponds to solving a linear minimax saddle point problem. In a recent work (Zhao & Gordon, 2019), the authors proved a lower bound on the joint error across different groups when a classifier satisfies demographic parity. They also showed that when the decision functions are close between groups, demographic parity also implies accuracy parity. The theoretical results in this work establish a relationship between accuracy parity and equalized odds; these two fairness notions are fundamentally related by the base rate gap and the balanced error rate. Furthermore, we also show that for any predictor that satisfies equalized odds, the balanced error rate also serves as an upper bound on the joint error across demographic subgroups.
+
+Fair Representations Through the lens of representation learning, recent advances in building fair algorithmic decision making systems focus on using adversarial methods to learn fair representations
+
+that also preserve sufficient information for the prediction task (Edwards & Storkey, 2015; Beutel et al., 2017; Zhang et al., 2018; Madras et al., 2018; Adel et al., 2019). In a nutshell, the key idea is to frame the problem of learning fair representations as a two-player game, where the data owner is competing against an adversary. The goal of the adversary is to infer the group attribute as much as possible while the goal of the data owner is to remove information related to the group attribute and simultaneously to preserve utility-related information for accurate prediction. Apart from using adversarial classifiers to enforce group fairness, other distance metrics have also been used to learn fair representations, e.g., the maximum mean discrepancy (Louizos et al., 2015), and the Wasserstein-1 distance (Jiang et al., 2019). In contrast to these methods, in this paper we advocate for optimizing BER on both the target loss and adversary loss in order to simultaneously achieve accuracy parity and equalized odds. We also show that this leads to better utility-fairness trade-off for balanced datasets.
+
+# 6 CONCLUSION AND FUTURE WORK
+
+In this paper we propose a novel representation learning algorithm that aims to simultaneously ensure accuracy parity and equalized odds. The main idea underlying the design of our algorithm is to align the conditional distributions of representations (rather than marginal distributions) and use balanced error rate (i.e., the conditional error) on both the target variable and the sensitive attribute. Theoretically, we prove how these two concepts together help to ensure accuracy parity and equalized odds without impacting demographic parity, and we also show how these two can be used to give a guarantee on the joint error across different demographic subgroups. Empirically, we demonstrate on two real-world experiments that the proposed algorithm effectively leads to the desired notions of fairness, and it also leads to better utility-fairness trade-off on balanced datasets.
+
+Calibration and Utility Our work takes a step towards better understanding the relationships between different notions of fairness and their corresponding trade-off with utility. In some scenarios, e.g., the COMPAS tool, it is desired to have a decision making system that is also well calibrated. While it is well-known that statistical calibration is not compatible with demographic parity or equalized odds, from a theoretical standpoint it is still not clear whether calibration will harm utility and if so, what is the fundamental limit of a calibrated tool on utility.
+
+Fairness and Privacy Future work could also investigate how to make use of the close relationship between privacy and group fairness. At a colloquial level, fairness constraints require a predictor to be (to some extent) agnostic about the group membership attribute. The membership query attack in privacy asks the same question – is it possible to guarantee that even an optimal adversary cannot steal personal information through inference attacks. Prior work (Dwork et al., 2012) has described the connection between the notion of individual fairness and differential privacy. Hence it would be interesting to exploit techniques developed in the literature of privacy to develop more efficient fairness-aware learning algorithms. On the other hand, results obtained in the algorithmic fairness literature could also potentially lead to better privacy-preserving machine learning algorithms (Zhao et al., 2019).
+
+# ACKNOWLEDGMENTS
+
+HZ and GG would like to acknowledge support from the DARPA XAI project, contract #FA87501720152 and an Nvidia GPU grant. HZ would also like to thank Richard Zemel, Toniann Pitassi, David Madras and Elliot Creager for helpful discussions during HZ's visit to the Vector Institute.
+
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+Matt J Kusner, Joshua Loftus, Chris Russell, and Ricardo Silva. Counterfactual fairness. In Advances in Neural Information Processing Systems, pp. 4066-4076, 2017.  
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+Aditya Krishna Menon and Robert C Williamson. The cost of fairness in binary classification. In Conference on Fairness, Accountability and Transparency, pp. 107-118, 2018.  
+Razieh Nabi and Ilya Shpitser. Fair inference on outcomes. In Thirty-Second AAAI Conference on Artificial Intelligence, 2018.  
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+Muhammad Bilal Zafar, Isabel Valera, Manuel Gomez Rodriguez, and Krishna P Gummadi. Fairness constraints: Mechanisms for fair classification. arXiv preprint arXiv:1507.05259, 2015.  
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+Han Zhao, Jianfeng Chi, Yuan Tian, and Geoffrey J Gordon. Adversarial privacy preservation under attribute inference attack. arXiv preprint arXiv:1906.07902, 2019.
+
+# A PROOFS
+
+# A.1 PROOF OF PROPOSITION 3.1
+
+Proposition 3.1. For  $g: \mathcal{X} \to \mathcal{Z}$ , if  $d_{\mathrm{TV}}(g_{\sharp}\mathcal{D}_0^y, g_{\sharp}\mathcal{D}_1^y) = 0, \forall y \in \{0,1\}$ , then for any classifier  $h: \mathcal{Z} \to \{0,1\}$ ,  $h \circ g$  satisfies equalized odds.
+
+Proof. To prove this proposition, we first show that for any pair of distributions  $\mathcal{D},\mathcal{D}'$  over  $\mathcal{Z}$  and any hypothesis  $h:\mathcal{Z}\to \{0,1\}$ ,  $d_{\mathrm{TV}}(h_{\sharp}\mathcal{D},h_{\sharp}\mathcal{D}^{\prime})\leq d_{\mathrm{TV}}(\mathcal{D},\mathcal{D}^{\prime})$ . Note that since  $h$  is a hypothesis, there are only two events in the induced probability space, i.e.,  $h(\cdot) = 0$  or  $h(\cdot) = 1$ . Hence by definition of the induced (pushforward) distribution, we have:
+
+$$
+\begin{array}{l} d _ {\mathrm {T V}} (h _ {\sharp} \mathcal {D}, h _ {\sharp} \mathcal {D} ^ {\prime}) = \max  _ {E = h ^ {- 1} (0), \mathrm {o r} E = h ^ {- 1} (1)} | \mathcal {D} (E) - \mathcal {D} ^ {\prime} (E) | \\ \leq \sup  _ {E \text {i s m e a s u r a b l e s u b s e t o f} \mathcal {Z}} | \mathcal {D} (E) - \mathcal {D} ^ {\prime} (E) | \\ = d _ {\mathrm {T V}} (\mathcal {D}, \mathcal {D} ^ {\prime}). \\ \end{array}
+$$
+
+Apply the above inequality twice for  $y \in \{0,1\}$ :
+
+$$
+0 \leq d _ {\mathrm {T V}} ((h \circ g) _ {\sharp} \mathcal {D} _ {0} ^ {y}, (h \circ g) _ {\sharp} \mathcal {D} _ {1} ^ {y}) \leq d _ {\mathrm {T V}} (g _ {\sharp} \mathcal {D} _ {0} ^ {y}, g _ {\sharp} \mathcal {D} _ {1} ^ {y}) = 0,
+$$
+
+meaning
+
+$$
+d _ {\mathrm {T V}} \left(\left(h \circ g\right) _ {\sharp} \mathcal {D} _ {0} ^ {y}, \left(h \circ g\right) _ {\sharp} \mathcal {D} _ {1} ^ {y}\right) = 0,
+$$
+
+which further implies that  $h(g(X))$  is independent of  $A$  given  $Y = y$  since  $h(g(X))$  is binary.
+
+# A.2 PROOF OF PROPOSITION 3.2
+
+Proposition 3.2. For  $g: \mathcal{X} \to \mathcal{Z}$ , if  $d_{\mathrm{TV}}(g_{\sharp}\mathcal{D}_0^y, g_{\sharp}\mathcal{D}_1^y) = 0, \forall y \in \{0,1\}$ , then for any classifier  $h: \mathcal{Z} \to \{0,1\}$ ,  $d_{\mathrm{TV}}((h \circ g)_{\sharp}\mathcal{D}_0, (h \circ g)_{\sharp}\mathcal{D}_1) \leq \Delta_{\mathrm{BR}}(\mathcal{D}_0, \mathcal{D}_1)$ .
+
+Proof. Let  $\widehat{Y} = (h\circ g)(X)$  and note that  $\widehat{Y}$  is binary, we have
+
+$$
+d _ {\mathrm {T V}} \big ((h \circ g) _ {\sharp} \mathcal {D} _ {0}, (h \circ g) _ {\sharp} \mathcal {D} _ {1} \big) = \frac {1}{2} \left(\left| \mathcal {D} _ {0} (\widehat {Y} = 0) - \mathcal {D} _ {1} (\widehat {Y} = 0) \right| + \left| \mathcal {D} _ {0} (\widehat {Y} = 1) - \mathcal {D} _ {1} (\widehat {Y} = 1) \right|\right).
+$$
+
+Now, by Proposition 3.1, if  $d_{\mathrm{TV}}(g_{\sharp}\mathcal{D}_0^y,g_{\sharp}\mathcal{D}_1^y) = 0,\forall y\in \{0,1\}$ , it follows that  $d_{\mathrm{TV}}((h\circ g)_{\sharp}\mathcal{D}_0^y,(h\circ g)_{\sharp}\mathcal{D}_1^y) = 0,\forall y\in \{0,1\}$  as well. Applying Lemma 3.1, we know that  $\forall y\in \{0,1\}$
+
+$$
+\left| \mathcal {D} _ {0} (\widehat {Y} = y) - \mathcal {D} _ {1} (\widehat {Y} = y) \right| \leq \left| \mathcal {D} _ {0} (Y = 0) - \mathcal {D} _ {1} (Y = 0) \right| \cdot \left(\mathcal {D} ^ {0} (\widehat {Y} = y) + \mathcal {D} ^ {1} (\widehat {Y} = y)\right).
+$$
+
+Hence,
+
+$$
+\begin{array}{l} d _ {\mathrm {T V}} \left(\left(h \circ g\right) _ {\sharp} \mathcal {D} _ {0}, \left(h \circ g\right) _ {\sharp} \mathcal {D} _ {1}\right) = \frac {1}{2} \left(\left| \mathcal {D} _ {0} (\widehat {Y} = 0) - \mathcal {D} _ {1} (\widehat {Y} = 0) \right| + \left| \mathcal {D} _ {0} (\widehat {Y} = 1) - \mathcal {D} _ {1} (\widehat {Y} = 1) \right|\right) \\ \leq \frac {\left| \mathcal {D} _ {0} (Y = 0) - \mathcal {D} _ {1} (Y = 0) \right|}{2} \left(\left(\mathcal {D} ^ {0} (\widehat {Y} = 0) + \mathcal {D} ^ {1} (\widehat {Y} = 0)\right) + \left(\mathcal {D} ^ {0} (\widehat {Y} = 1) + \mathcal {D} ^ {1} (\widehat {Y} = 1)\right)\right) \\ = \frac {\left| \mathcal {D} _ {0} (Y = 0) - \mathcal {D} _ {1} (Y = 0) \right|}{2} \left(\left(\mathcal {D} ^ {0} (\widehat {Y} = 0) + \mathcal {D} ^ {0} (\widehat {Y} = 1)\right) + \left(\mathcal {D} ^ {1} (\widehat {Y} = 0) + \mathcal {D} ^ {1} (\widehat {Y} = 1)\right)\right) \\ = \frac {\left| \mathcal {D} _ {0} (Y = 0) - \mathcal {D} _ {1} (Y = 0) \right|}{2}. 2 \\ = \left| \mathcal {D} _ {0} (Y = 0) - \mathcal {D} _ {1} (Y = 0) \right| \\ = \Delta_ {\mathrm {B R}} (\mathcal {D} _ {0}, \mathcal {D} _ {1}). \\ \end{array}
+$$
+
+# A.3 PROOF OF LEMMA 3.1
+
+Recall that we define  $\gamma_{a} := \mathcal{D}_{a}(Y = 0), \forall a \in \{0,1\}$ .
+
+Lemma 3.1. Assume the conditions in Proposition 3.1 hold and let  $\widehat{Y} = h(g(X))$  be the classifier, then  $|\mathcal{D}_0(\widehat{Y} = y) - \mathcal{D}_1(\widehat{Y} = y)| \leq |\gamma_0 - \gamma_1| \cdot \left(\mathcal{D}^0 (\widehat{Y} = y) + \mathcal{D}^1 (\widehat{Y} = y)\right), \forall y \in \{0,1\}$ .
+
+Proof. To bound  $|\mathcal{D}_0(\widehat{Y} = y) - \mathcal{D}_1(\widehat{Y} = y)|$ , for  $y \in \{0, 1\}$ , by the law of total probability, we have:
+
+$$
+\begin{array}{l} \left| \mathcal {D} _ {0} (\widehat {Y} = y) - \mathcal {D} _ {1} (\widehat {Y} = y) \right| = \\ = \left| \left(\mathcal {D} _ {0} ^ {0} (\widehat {Y} = y) \mathcal {D} _ {0} (Y = 0) + \mathcal {D} _ {0} ^ {1} (\widehat {Y} = y) \mathcal {D} _ {0} (Y = 1)\right) - \left(\mathcal {D} _ {1} ^ {0} (\widehat {Y} = y) \mathcal {D} _ {1} (Y = 0) + \mathcal {D} _ {1} ^ {1} (\widehat {Y} = y) \mathcal {D} _ {1} (Y = 1)\right) \right| \\ \leq \left| \gamma_ {0} \mathcal {D} _ {0} ^ {0} (\widehat {Y} = y) - \gamma_ {1} \mathcal {D} _ {1} ^ {0} (\widehat {Y} = y) \right| + \left| (1 - \gamma_ {0}) \mathcal {D} _ {0} ^ {1} (\widehat {Y} = y) - (1 - \gamma_ {1}) \mathcal {D} _ {1} ^ {1} (\widehat {Y} = y) \right|, \\ \end{array}
+$$
+
+where the above inequality is due to the triangular inequality. Now apply Proposition 3.1, we know that  $\widehat{Y}$  satisfies equalized odds, so we have  $\mathcal{D}_0^0 (\widehat{Y} = y) = \mathcal{D}_1^0 (\widehat{Y} = y) = \mathcal{D}^0 (\widehat{Y} = y)$  and  $\mathcal{D}_0^1 (\widehat{Y} = y) = \mathcal{D}_1^1 (\widehat{Y} = y) = \mathcal{D}^1 (\widehat{Y} = y)$ , leading to:
+
+$$
+\begin{array}{l} = \left| \gamma_ {0} - \gamma_ {1} \right| \cdot \mathcal {D} ^ {0} (\widehat {Y} = y) + \left| (1 - \gamma_ {0}) - (1 - \gamma_ {1}) \right| \cdot \mathcal {D} ^ {1} (\widehat {Y} = y) \\ = \left| \gamma_ {0} - \gamma_ {1} \right| \cdot \left(\mathcal {D} ^ {0} (\widehat {Y} = y) + \mathcal {D} ^ {1} (\widehat {Y} = y)\right), \\ \end{array}
+$$
+
+which completes the proof.
+
+# A.4 PROOF OF THEOREM 3.1
+
+Theorem 3.1. Assume the conditions in Proposition 3.1 hold and let  $\widehat{Y} = h(g(X))$  be the classifier, then  $\Delta_{\mathrm{DP}}(\widehat{Y}) \leq \Delta_{\mathrm{BR}}(\mathcal{D}_0, \mathcal{D}_1) = \Delta_{\mathrm{DP}}(Y)$ .
+
+Proof. To bound  $\Delta_{\mathrm{DP}}(\widehat{Y})$ , realize that  $|\mathcal{D}_0(\widehat{Y} = 0) - \mathcal{D}_1(\widehat{Y} = 0)| = |\mathcal{D}_0(\widehat{Y} = 1) - \mathcal{D}_1(\widehat{Y} = 1)|$ , so we can rewrite the DP gap as follows:
+
+$$
+\Delta_ {\mathrm {D P}} (\widehat {Y}) = \frac {1}{2} \left(\left| \mathcal {D} _ {0} (\widehat {Y} = 0) - \mathcal {D} _ {1} (\widehat {Y} = 0) \right| + \left| \mathcal {D} _ {0} (\widehat {Y} = 1) - \mathcal {D} _ {1} (\widehat {Y} = 1) \right|\right).
+$$
+
+Now apply Lemma 3.1 twice for  $y = 0$  and  $y = 1$ , we have:
+
+$$
+\begin{array}{l} \left| \mathcal {D} _ {0} (\widehat {Y} = 0) - \mathcal {D} _ {1} (\widehat {Y} = 0) \right| \leq \left| \gamma_ {0} - \gamma_ {1} \right| \cdot \left(\mathcal {D} ^ {0} (\widehat {Y} = 0) + \mathcal {D} ^ {1} (\widehat {Y} = 0)\right) \\ \left| \mathcal {D} _ {0} (\widehat {Y} = 1) - \mathcal {D} _ {1} (\widehat {Y} = 1) \right| \leq \left| \gamma_ {0} - \gamma_ {1} \right| \cdot \left(\mathcal {D} ^ {0} (\widehat {Y} = 1) + \mathcal {D} ^ {1} (\widehat {Y} = 1)\right). \\ \end{array}
+$$
+
+Taking sum of the above two inequalities yields
+
+$$
+\begin{array}{l} \left| \mathcal {D} _ {0} (\widehat {Y} = 0) - \mathcal {D} _ {1} (\widehat {Y} = 0) \right| + \left| \mathcal {D} _ {0} (\widehat {Y} = 1) - \mathcal {D} _ {1} (\widehat {Y} = 1) \right| \\ \leq \left| \gamma_ {0} - \gamma_ {1} \right| \left(\left(\mathcal {D} ^ {0} (\widehat {Y} = 0) + \mathcal {D} ^ {1} (\widehat {Y} = 0)\right) + \left(\mathcal {D} ^ {0} (\widehat {Y} = 1) + \mathcal {D} ^ {1} (\widehat {Y} = 1)\right)\right) \\ = \left| \gamma_ {0} - \gamma_ {1} \right| \left(\left(\mathcal {D} ^ {0} (\widehat {Y} = 0) + \mathcal {D} ^ {0} (\widehat {Y} = 1)\right) + \left(\mathcal {D} ^ {1} (\widehat {Y} = 0) + \mathcal {D} ^ {1} (\widehat {Y} = 1)\right)\right) \\ = 2 \big | \gamma_ {0} - \gamma_ {1} \big |. \\ \end{array}
+$$
+
+Combining all the inequalities above, we know that
+
+$$
+\begin{array}{l} \Delta_ {\mathrm {D P}} (\widehat {Y}) = \frac {1}{2} \left(\left| \mathcal {D} _ {0} (\widehat {Y} = 0) - \mathcal {D} _ {1} (\widehat {Y} = 0) \right| + \left| \mathcal {D} _ {0} (\widehat {Y} = 1) - \mathcal {D} _ {1} (\widehat {Y} = 1) \right|\right) \\ \leq \left| \gamma_ {0} - \gamma_ {1} \right| \\ = \left| \mathcal {D} _ {0} (Y = 0) - \mathcal {D} _ {1} (Y = 0) \right| \\ = \left| \mathcal {D} _ {0} (Y = 1) - \mathcal {D} _ {1} (Y = 1) \right| \\ = \Delta_ {\mathrm {B R}} (\mathcal {D} _ {0}, \mathcal {D} _ {1}) = \Delta_ {\mathrm {D P}} (Y), \\ \end{array}
+$$
+
+completing the proof.
+
+# A.5 PROOF OF THEOREM 3.2
+
+Theorem 3.2. Assume the conditions in Proposition 3.1 hold and let  $\widehat{Y} = h(g(X))$  be the classifier, then  $\mathrm{Err}_{\mathcal{D}_0}(\widehat{Y}) + \mathrm{Err}_{\mathcal{D}_1}(\widehat{Y}) \leq 2\mathrm{BER}_{\mathcal{D}}(\widehat{Y} \parallel Y)$ .
+
+Proof. First, by the law of total probability, we have:
+
+$$
+\begin{array}{l} \operatorname {E r r} _ {\mathcal {D} _ {0}} (\widehat {Y}) + \operatorname {E r r} _ {\mathcal {D} _ {1}} (\widehat {Y}) = \mathcal {D} _ {0} (Y \neq \widehat {Y}) + \mathcal {D} _ {1} (Y \neq \widehat {Y}) \\ = \mathcal {D} _ {0} ^ {1} (\widehat {Y} = 0) \mathcal {D} _ {0} (Y = 1) + \mathcal {D} _ {0} ^ {0} (\widehat {Y} = 1) \mathcal {D} _ {0} (Y = 0) + \mathcal {D} _ {1} ^ {1} (\widehat {Y} = 0) \mathcal {D} _ {1} (Y = 1) + \mathcal {D} _ {1} ^ {0} (\widehat {Y} = 1) \mathcal {D} _ {1} (Y = 0) \\ \end{array}
+$$
+
+Again, by Proposition 3.1, the classifier  $\widehat{Y} = (h\circ g)(X)$  satisfies equalized odds, so we have
+
+$$
+\mathcal {D} _ {0} ^ {1} (\widehat {Y} = 0) = \mathcal {D} ^ {1} (\widehat {Y} = 0), \mathcal {D} _ {0} ^ {0} (\widehat {Y} = 1) = \mathcal {D} ^ {0} (\widehat {Y} = 1), \mathcal {D} _ {1} ^ {1} (\widehat {Y} = 0) = \mathcal {D} ^ {1} (\widehat {Y} = 0) \text {a n d}
+$$
+
+$$
+\begin{array}{l} \mathcal {D} _ {1} ^ {0} (\widehat {Y} = 1) = \mathcal {D} ^ {0} (\widehat {Y} = 1): \\ = \mathcal {D} ^ {1} (\widehat {Y} = 0) \mathcal {D} _ {0} (Y = 1) + \mathcal {D} ^ {0} (\widehat {Y} = 1) \mathcal {D} _ {0} (Y = 0) + \mathcal {D} ^ {1} (\widehat {Y} = 0) \mathcal {D} _ {1} (Y = 1) + \mathcal {D} ^ {0} (\widehat {Y} = 1) \mathcal {D} _ {1} (Y = 0) \\ = \mathcal {D} ^ {1} (\widehat {Y} = 0) \cdot \left(\mathcal {D} _ {0} (Y = 1) + \mathcal {D} _ {1} (Y = 1)\right) + \mathcal {D} ^ {0} (\widehat {Y} = 1) \cdot \left(\mathcal {D} _ {0} (Y = 0) + \mathcal {D} _ {1} (Y = 0)\right) \\ \leq 2 \mathcal {D} ^ {1} (\widehat {Y} = 0) + 2 \mathcal {D} ^ {0} (\widehat {Y} = 1) \\ = 2 \operatorname {B E R} _ {\mathcal {D}} (\widehat {Y} | | Y), \\ \end{array}
+$$
+
+which completes the proof.
+
+# A.6 PROOF OF THEOREM 3.3
+
+Theorem 3.3. For any classifier  $\widehat{Y}$ ,  $\Delta_{\mathrm{Err}}(\widehat{Y}) \leq \Delta_{\mathrm{BR}}(\mathcal{D}_0, \mathcal{D}_1) \cdot \mathrm{BER}_{\mathcal{D}}(\widehat{Y} \parallel Y) + 2\Delta_{\mathrm{EO}}(\widehat{Y})$ .
+
+Before we give the proof of Theorem 3.3, we first prove the following two lemmas that will be used in the following proof.
+
+Lemma A.1. Define  $\gamma_{a} := \mathcal{D}_{a}(Y = 0), \forall a \in \{0,1\}$ , then  $|\gamma_{0}\mathcal{D}_{0}^{0}(\widehat{Y} = 1) - \gamma_{1}\mathcal{D}_{1}^{0}(\widehat{Y} = 1)| \leq |\gamma_{0} - \gamma_{1}| \cdot \mathcal{D}^{0}(\widehat{Y} = 1) + \gamma_{0}\mathcal{D}^{0}(A = 1)\Delta_{\mathrm{EO}}(\widehat{Y}) + \gamma_{1}\mathcal{D}^{0}(A = 0)\Delta_{\mathrm{EO}}(\widehat{Y})$ .
+
+Proof. In order to prove the upper bound in the lemma, it suffices if we could give the desired upper bound for the following term
+
+$$
+\begin{array}{l} \left| \left| \gamma_ {0} \mathcal {D} _ {0} ^ {0} (\widehat {Y} = 1) - \gamma_ {1} \mathcal {D} _ {1} ^ {0} (\widehat {Y} = 1) \right| - \left| (\gamma_ {0} - \gamma_ {1}) \mathcal {D} ^ {0} (\widehat {Y} = 1) \right| \right| \\ \leq \left| \left(\gamma_ {0} \mathcal {D} _ {0} ^ {0} (\widehat {Y} = 1) - \gamma_ {1} \mathcal {D} _ {1} ^ {0} (\widehat {Y} = 1)\right) - (\gamma_ {0} - \gamma_ {1}) \mathcal {D} ^ {0} (\widehat {Y} = 1) \right| \\ = \left| \gamma_ {0} \left(\mathcal {D} _ {0} ^ {0} (\widehat {Y} = 1) - \mathcal {D} ^ {0} (\widehat {Y} = 1)\right) - \gamma_ {1} \left(\mathcal {D} _ {1} ^ {0} (\widehat {Y} = 1) - \mathcal {D} ^ {0} (\widehat {Y} = 1)\right) \right|, \\ \end{array}
+$$
+
+following which we will have:
+
+$$
+\begin{array}{l} \left| \gamma_ {0} \mathcal {D} _ {0} ^ {0} (\widehat {Y} = 1) - \gamma_ {1} \mathcal {D} _ {1} ^ {0} (\widehat {Y} = 1) \right| \leq \left| \left(\gamma_ {0} - \gamma_ {1}\right) \mathcal {D} ^ {0} (\widehat {Y} = 1) \right| \\ + \left| \gamma_ {0} \left(\mathcal {D} _ {0} ^ {0} (\widehat {Y} = 1) - \mathcal {D} ^ {0} (\widehat {Y} = 1)\right) - \gamma_ {1} \left(\mathcal {D} _ {1} ^ {0} (\widehat {Y} = 1) - \mathcal {D} ^ {0} (\widehat {Y} = 1)\right) \right|, \\ \end{array}
+$$
+
+and an application of the Bayes formula could finish the proof. To do so, let us first simplify  $\mathcal{D}_0^0 (\widehat{Y} = 1) - \mathcal{D}^0 (\widehat{Y} = 1)$ . Applying the Bayes's formula, we know that:
+
+$$
+\begin{array}{l} \mathcal {D} _ {0} ^ {0} (\widehat {Y} = 1) - \mathcal {D} ^ {0} (\widehat {Y} = 1) = \mathcal {D} _ {0} ^ {0} (\widehat {Y} = 1) - \left(\mathcal {D} _ {0} ^ {0} (\widehat {Y} = 1) \mathcal {D} ^ {0} (A = 0) + \mathcal {D} _ {1} ^ {0} (\widehat {Y} = 1) \mathcal {D} ^ {0} (A = 1)\right) \\ = \left(\mathcal {D} _ {0} ^ {0} (\widehat {Y} = 1) - \mathcal {D} _ {0} ^ {0} (\widehat {Y} = 1) \mathcal {D} ^ {0} (A = 0)\right) - \mathcal {D} _ {1} ^ {0} (\widehat {Y} = 1) \mathcal {D} ^ {0} (A = 1) \\ = \mathcal {D} ^ {0} (A = 1) \left(\mathcal {D} _ {0} ^ {0} (\widehat {Y} = 1) - \mathcal {D} _ {1} ^ {0} (\widehat {Y} = 1)\right). \\ \end{array}
+$$
+
+Similarly, for the second term  $\mathcal{D}_1^0 (\widehat{Y} = 1) - \mathcal{D}^0 (\widehat{Y} = 1)$ , we can show that:
+
+$$
+\mathcal {D} _ {1} ^ {0} (\widehat {Y} = 1) - \mathcal {D} ^ {0} (\widehat {Y} = 1) = \mathcal {D} ^ {0} (A = 0) \big (\mathcal {D} _ {1} ^ {0} (\widehat {Y} = 1) - \mathcal {D} _ {0} ^ {0} (\widehat {Y} = 1) \big).
+$$
+
+Plug these two identities into above, we can continue the analysis with
+
+$$
+\begin{array}{l} \left| \gamma_ {0} \left(\mathcal {D} _ {0} ^ {0} (\widehat {Y} = 1) - \mathcal {D} ^ {0} (\widehat {Y} = 1)\right) - \gamma_ {1} \left(\mathcal {D} _ {1} ^ {0} (\widehat {Y} = 1) - \mathcal {D} ^ {0} (\widehat {Y} = 1)\right) \right| \\ = \left| \gamma_ {0} \mathcal {D} ^ {0} (A = 1) (\mathcal {D} _ {0} ^ {0} (\widehat {Y} = 1) - \mathcal {D} _ {1} ^ {0} (\widehat {Y} = 1)) - \gamma_ {1} \mathcal {D} ^ {0} (A = 0) (\mathcal {D} _ {1} ^ {0} (\widehat {Y} = 1) - \mathcal {D} _ {0} ^ {0} (\widehat {Y} = 1)) \right| \\ \leq \left| \gamma_ {0} \mathcal {D} ^ {0} (A = 1) \left(\mathcal {D} _ {0} ^ {0} (\widehat {Y} = 1) - \mathcal {D} _ {1} ^ {0} (\widehat {Y} = 1)\right) \right| + \left| \gamma_ {1} \mathcal {D} ^ {0} (A = 0) \left(\mathcal {D} _ {1} ^ {0} (\widehat {Y} = 1) - \mathcal {D} _ {0} ^ {0} (\widehat {Y} = 1)\right) \right| \\ \leq \gamma_ {0} \mathcal {D} ^ {0} (A = 1) \Delta_ {\mathrm {E O}} (\widehat {Y}) + \gamma_ {1} \mathcal {D} ^ {0} (A = 0) \Delta_ {\mathrm {E O}} (\widehat {Y}). \\ \end{array}
+$$
+
+The first inequality holds by triangular inequality and the second one holds by the definition of equalized odds gap.
+
+Lemma A.2. Define  $\gamma_{a} \coloneqq \mathcal{D}_{a}(Y = 0), \forall a \in \{0,1\}$ , then  $|(1 - \gamma_0)\mathcal{D}_0^1 (\widehat{Y} = 0) - (1 - \gamma_1)\mathcal{D}_1^1 (\widehat{Y} = 0)| \leq |\gamma_0 - \gamma_1| \cdot \mathcal{D}^1 (\widehat{Y} = 0) + (1 - \gamma_0)\mathcal{D}^1 (A = 1)\Delta_{\mathrm{EO}}(\widehat{Y}) + (1 - \gamma_1)\mathcal{D}^1 (A = 0)\Delta_{\mathrm{EO}}(\widehat{Y})$ .
+
+Proof. The proof of this lemma is symmetric to the previous one, so we omit it here.
+
+Now we are ready to prove Theorem 3.3:
+
+Proof of Theorem 3.3. First, by the law of total probability, it is easy to verify that following identity holds for  $a \in \{0,1\}$ :
+
+$$
+\begin{array}{l} \mathcal {D} _ {a} (\widehat {Y} \neq Y) = \mathcal {D} _ {a} (Y = 1, \widehat {Y} = 0) + \mathcal {D} _ {a} (Y = 0, \widehat {Y} = 1) \\ = (1 - \gamma_ {a}) \mathcal {D} _ {a} ^ {1} (\widehat {Y} = 0) + \gamma_ {a} \mathcal {D} _ {a} ^ {0} (\widehat {Y} = 1). \\ \end{array}
+$$
+
+Using this identity, to bound the error gap, we have:
+
+$$
+\begin{array}{l} \left| \mathcal {D} _ {0} (Y \neq \widehat {Y}) - \mathcal {D} _ {1} (Y \neq \widehat {Y}) \right| = \left| \left((1 - \gamma_ {0}) \mathcal {D} _ {0} ^ {1} (\widehat {Y} = 0) + \gamma_ {0} \mathcal {D} _ {0} ^ {0} (\widehat {Y} = 1)\right) - \left((1 - \gamma_ {1}) \mathcal {D} _ {1} ^ {1} (\widehat {Y} = 0) + \gamma_ {1} \mathcal {D} _ {1} ^ {0} (\widehat {Y} = 1)\right) \right| \\ \leq \left| \gamma_ {0} \mathcal {D} _ {0} ^ {0} (\widehat {Y} = 1) - \gamma_ {1} \mathcal {D} _ {1} ^ {0} (\widehat {Y} = 1) \right| + \left| (1 - \gamma_ {0}) \mathcal {D} _ {0} ^ {1} (\widehat {Y} = 0) - (1 - \gamma_ {1}) \mathcal {D} _ {1} ^ {1} (\widehat {Y} = 0) \right|. \\ \end{array}
+$$
+
+Invoke Lemma A.1 and Lemma A.2 to bound the above two terms:
+
+$$
+\begin{array}{l} | \mathcal {D} _ {0} (Y \neq \widehat {Y}) - \mathcal {D} _ {1} (Y \neq \widehat {Y}) | \\ \leq \left| \gamma_ {0} \mathcal {D} _ {0} ^ {0} (\widehat {Y} = 1) - \gamma_ {1} \mathcal {D} _ {1} ^ {0} (\widehat {Y} = 1) \right| + \left| (1 - \gamma_ {0}) \mathcal {D} _ {0} ^ {1} (\widehat {Y} = 0) - (1 - \gamma_ {1}) \mathcal {D} _ {1} ^ {1} (\widehat {Y} = 0) \right| \\ \leq \gamma_ {0} \mathcal {D} ^ {0} (A = 1) \Delta_ {\mathrm {E O}} (\widehat {Y}) + \gamma_ {1} \mathcal {D} ^ {0} (A = 0) \Delta_ {\mathrm {E O}} (\widehat {Y}) \\ + (1 - \gamma_ {0}) \mathcal {D} ^ {1} (A = 1) \Delta_ {\mathrm {E O}} (\widehat {Y}) + (1 - \gamma_ {1}) \mathcal {D} ^ {1} (A = 0) \Delta_ {\mathrm {E O}} (\widehat {Y}) \\ + \left| \gamma_ {0} - \gamma_ {1} \right| \mathcal {D} ^ {0} (\widehat {Y} = 1) + \left| \gamma_ {0} - \gamma_ {1} \right| \mathcal {D} ^ {1} (\widehat {Y} = 0), \\ \end{array}
+$$
+
+Realize that both  $\gamma_0,\gamma_1\in [0,1]$  , we have:
+
+$$
+\begin{array}{l} \leq \mathcal {D} ^ {0} (A = 1) \Delta_ {\mathrm {E O}} (\widehat {Y}) + \mathcal {D} ^ {0} (A = 0) \Delta_ {\mathrm {E O}} (\widehat {Y}) + \mathcal {D} ^ {1} (A = 1) \Delta_ {\mathrm {E O}} (\widehat {Y}) + \mathcal {D} ^ {1} (A = 0) \Delta_ {\mathrm {E O}} (\widehat {Y}) \\ + \left| \gamma_ {0} - \gamma_ {1} \right| \mathcal {D} ^ {0} (\widehat {Y} = 1) + \left| \gamma_ {0} - \gamma_ {1} \right| \mathcal {D} ^ {1} (\widehat {Y} = 0) \\ = 2 \Delta_ {\mathrm {E O}} (\widehat {Y}) + | \gamma_ {0} - \gamma_ {1} | \mathcal {D} ^ {0} (\widehat {Y} = 1) + | \gamma_ {0} - \gamma_ {1} | \mathcal {D} ^ {1} (\widehat {Y} = 0) \\ = 2 \Delta_ {\mathrm {E O}} (\widehat {Y}) + \Delta_ {\mathrm {B R}} \left(\mathcal {D} _ {0}, \mathcal {D} _ {1}\right) \cdot \operatorname {B E R} _ {\mathcal {D}} (\widehat {Y} | | Y), \\ \end{array}
+$$
+
+which completes the proof.
+
+We also provide the proof of Corollary 3.1:
+
+Corollary 3.1. For any joint distribution  $\mathcal{D}$  and classifier  $\widehat{Y}$ , if  $\widehat{Y}$  satisfies equalized odds, then  $\max \{\mathrm{Err}_{\mathcal{D}_0}(\widehat{Y}), \mathrm{Err}_{\mathcal{D}_1}(\widehat{Y})\} \leq \Delta_{\mathrm{BR}}(\mathcal{D}_0, \mathcal{D}_1) \cdot \mathrm{BER}_{\mathcal{D}}(\widehat{Y} \parallel Y) / 2 + \mathrm{BER}_{\mathcal{D}}(\widehat{Y} \parallel Y)$ .
+
+Proof. We first invoke Theorem 3.3, if  $\widehat{Y}$  satisfies equalized odds, then  $\Delta_{\mathrm{EO}}(\widehat{Y}) = 0$ , which implies:
+
+$$
+\Delta_ {\operatorname {E r r}} (\widehat {Y}) = \left| \operatorname {E r r} _ {\mathcal {D} _ {0}} (\widehat {Y}) - \operatorname {E r r} _ {\mathcal {D} _ {1}} (\widehat {Y}) \right| \leq \Delta_ {\operatorname {B R}} (\mathcal {D} _ {0}, \mathcal {D} _ {1}) \cdot \operatorname {B E R} _ {\mathcal {D}} (\widehat {Y} | | Y).
+$$
+
+On the other hand, by Theorem 3.2, we know that
+
+$$
+\operatorname {E r r} _ {\mathcal {D} _ {0}} (\widehat {Y}) + \operatorname {E r r} _ {\mathcal {D} _ {1}} (\widehat {Y}) \leq 2 \operatorname {B E R} _ {\mathcal {D}} (\widehat {Y} \mid | Y).
+$$
+
+Combine the above two inequalities and recall that  $\max \{a,b\} = (|a + b| + |a - b|) / 2, \forall a,b \in \mathbb{R}$ , yielding:
+
+$$
+\begin{array}{l} \max  \{\operatorname {E r r} _ {\mathcal {D} _ {0}} (\widehat {Y}), \operatorname {E r r} _ {\mathcal {D} _ {1}} (\widehat {Y}) \} = \frac {| \operatorname {E r r} _ {\mathcal {D} _ {0}} (\widehat {Y}) - \operatorname {E r r} _ {\mathcal {D} _ {1}} (\widehat {Y}) | + | \operatorname {E r r} _ {\mathcal {D} _ {0}} (\widehat {Y}) + \operatorname {E r r} _ {\mathcal {D} _ {1}} (\widehat {Y}) |}{2} \\ \leq \frac {\Delta_ {\mathrm {B R}} (\mathcal {D} _ {0} , \mathcal {D} _ {1}) \cdot \operatorname {B E R} _ {\mathcal {D}} (\widehat {Y} \mid | Y) + 2 \operatorname {B E R} _ {\mathcal {D}} (\widehat {Y} \mid | Y)}{2} \\ = \Delta_ {\mathrm {B R}} \left(\mathcal {D} _ {0}, \mathcal {D} _ {1}\right) \cdot \operatorname {B E R} _ {\mathcal {D}} (\widehat {Y} | | Y) / 2 + \operatorname {B E R} _ {\mathcal {D}} (\widehat {Y} | | Y), \\ \end{array}
+$$
+
+completing the proof.
+
+# B EXPERIMENTAL DETAILS
+
+# B.1 THE ADULT EXPERIMENT
+
+For the baseline network NODEBIAS, we implement a three-layer neural network with ReLU as the hidden activation function and logistic regression as the target output function. The input layer contains 114 units, and the hidden layer contains 60 hidden units. The output layer only contain one unit, whose output is interpreted as the probability of  $\mathcal{D}(\widehat{Y} = 1\mid X = x)$ .
+
+For the adversary in FAIR and LAFTR, again, we use a three-layer feed-forward network. Specifically, the input layer of the adversary is the hidden representations of the baseline network that contains 60 units. The hidden layer of the adversary network contains 50 units, with ReLU activation. Finally, the output of the adversary also contains one unit, representing the adversary's inference probability  $\mathcal{D}(\widehat{A} = 1\mid Z = z)$ . The network structure of the adversaries in both CFAIR and CFAIR-EO are exactly the same as the one used in FAIR and LAFTR, except that there are two adversaries, one for  $\mathcal{D}^0 (\widehat{A} = 1\mid Z = z)$  and one for  $\mathcal{D}^1 (\widehat{A} = 1\mid Z = z)$ .
+
+The hyperparameters used in the experiment are listed in Table 2.
+
+Table 2: Hyperparameters used in the Adult experiment.  
+
+<table><tr><td>Optimization Algorithm</td><td>AdaDelta</td></tr><tr><td>Learning Rate</td><td>1.0</td></tr><tr><td>Batch Size</td><td>512</td></tr><tr><td>Training Epochs λ ∈ {0.1, 1.0, 10.0, 100.0, 1000.0}</td><td>100</td></tr></table>
+
+# B.2 THE COMPAS EXPERIMENT
+
+Again, for the baseline network NODEBIAS, we implement a three-layer neural network with ReLU as the hidden activation function and logistic regression as the target output function. The input layer contains 11 units, and the hidden layer contains 10 hidden units. The output layer only contain one unit, whose output is interpreted as the probability of  $\mathcal{D}(\widehat{Y} = 1\mid X = x)$ .
+
+For the adversary in FAIR and LAFTR, again, we use a three-layer feed-forward network. Specifically, the input layer of the adversary is the hidden representations of the baseline network that contains 60 units. The hidden layer of the adversary network contains 10 units, with ReLU activation. Finally, the output of the adversary also contains one unit, representing the adversary's inference probability  $\mathcal{D}(\widehat{A} = 1\mid Z = z)$ . The network structure of the adversaries in both CFAIR and CFAIR-EO are exactly the same as the one used in FAIR and LAFTR, except that there are two adversaries, one for  $\mathcal{D}^0 (\widehat{A} = 1\mid Z = z)$  and one for  $\mathcal{D}^1 (\widehat{A} = 1\mid Z = z)$ .
+
+The hyperparameters used in the experiment are listed in Table 3.
+
+Table 3: Hyperparameters used in the COMPAS experiment.  
+
+<table><tr><td>Optimization Algorithm</td><td>AdaDelta</td></tr><tr><td>Learning Rate</td><td>1.0</td></tr><tr><td>Batch Size</td><td>512</td></tr><tr><td>Training Epochs λ ∈ {0.1, 1.0}</td><td>20</td></tr><tr><td>Training Epochs λ = 10.0</td><td>15</td></tr></table>
\ No newline at end of file
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+# CONTINUAL LEARNING WITH HYPERNETWORKS
+
+Johannes von Oswald*, Christian Henning*, Benjamin F. Grewe, João Sacramento *Equal contribution
+
+Institute of Neuroinformatics  
+University of Zurich and ETH Zurich  
+Zürich, Switzerland  
+{voswaldj, Henningc, bgrewe, rjoao}@ethz.ch
+
+# ABSTRACT
+
+Artificial neural networks suffer from catastrophic forgetting when they are sequentially trained on multiple tasks. To overcome this problem, we present a novel approach based on task-conditioned hypernetworks, i.e., networks that generate the weights of a target model based on task identity. Continual learning (CL) is less difficult for this class of models thanks to a simple key feature: instead of recalling the input-output relations of all previously seen data, task-conditioned hypernetworks only require rehearsing task-specific weight realizations, which can be maintained in memory using a simple regularizer. Besides achieving state-of-the-art performance on standard CL benchmarks, additional experiments on long task sequences reveal that task-conditioned hypernetworks display a very large capacity to retain previous memories. Notably, such long memory lifetimes are achieved in a compressive regime, when the number of trainable hypernetwork weights is comparable or smaller than target network size. We provide insight into the structure of low-dimensional task embedding spaces (the input space of the hypernetwork) and show that task-conditioned hypernetworks demonstrate transfer learning. Finally, forward information transfer is further supported by empirical results on a challenging CL benchmark based on the CIFAR-10/100 image datasets.
+
+# 1 INTRODUCTION
+
+We assume that a neural network  $f(\mathbf{x}, \Theta)$  with trainable weights  $\Theta$  is given data from a set of tasks  $\{(\mathbf{X}^{(1)}, \mathbf{Y}^{(1)}), \ldots, (\mathbf{X}^{(T)}, \mathbf{Y}^{(T)})\}$ , with input samples  $\mathbf{X}^{(t)} = \{\mathbf{x}^{(t,i)}\}_{i=1}^{n_t}$  and output samples  $\mathbf{Y}^{(t)} = \{\mathbf{y}^{(t,i)}\}_{i=1}^{n_t}$ , where  $n_t \equiv |\mathbf{X}^{(t)}|$ . A standard training approach learns the model using data from all tasks at once. However, this is not always possible in real-world problems, nor desirable in an online learning setting. Continual learning (CL) refers to an online learning setup in which tasks are presented sequentially (see van de Ven & Tolias, 2019, for a recent review on CL). In CL, when learning a new task  $t$ , starting with weights  $\Theta^{(t-1)}$  and observing only  $(\mathbf{X}^{(t)}, \mathbf{Y}^{(t)})$ , the goal is to find a new set of parameters  $\Theta^{(t)}$  that (1) retains (no catastrophic forgetting) or (2) improves (positive backward transfer) performance on previous tasks compared to  $\Theta^{(t-1)}$  and (3) solves the new task  $t$  potentially utilizing previously acquired knowledge (positive forward transfer). Achieving these goals is non-trivial, and a longstanding issue in neural networks research.
+
+Here, we propose addressing catastrophic forgetting at the meta level: instead of directly attempting to retain  $f(\mathbf{x},\Theta)$  for previous tasks, we fix the outputs of a metamodel  $f_{\mathrm{h}}(\mathbf{e},\Theta_{\mathrm{h}})$  termed task-conditioned hypernetwork which maps a task embedding  $\mathbf{e}$  to weights  $\Theta$ . Now, a single point has to be memorized per task. To motivate such approach, we perform a thought experiment: we assume that we are allowed to store all inputs  $\{\mathbf{X}^{(1)},\ldots ,\mathbf{X}^{(T)}\}$  seen so far, and to use these data to compute model outputs corresponding to  $\Theta^{(T - 1)}$ . In this idealized setting, one can avoid forgetting by simply mixing data from the current task with data from the past,  $\{(\mathbf{X}^{(1)},\hat{\mathbf{Y}}^{(1)}),\dots ,(\mathbf{X}^{(T - 1)},\hat{\mathbf{Y}}^{(T - 1)}),(\mathbf{X}^{(T)},\mathbf{Y}^{(T)})\}$ , where  $\hat{\mathbf{Y}}^{(t)}$  refers to a set of synthetic targets generated using the model itself  $f(\cdot ,\Theta^{(t - 1)})$ . Hence, by training to retain previously acquired input-output mappings, one can obtain a sequential algorithm in principle as powerful as multi-task learning. Multi-task learning, where all tasks are learned
+
+simultaneously, can be seen as a CL upper-bound. The strategy described above has been termed rehearsal (Robins, 1995). However, storing previous task data violates our CL desiderata.
+
+Therefore, we introduce a change in perspective and move from the challenge of maintaining individual input-output data points to the problem of maintaining sets of parameters  $\{\Theta^{(t)}\}$ , without explicitly storing them. To achieve this, we train the metamodel parameters  $\Theta_{\mathrm{h}}$  analogous to the above outlined learning scheme, where synthetic targets now correspond to weight configurations that are suitable for previous tasks. This exchanges the storage of an entire dataset by a single low-dimensional task descriptor, yielding a massive memory saving in all but the simplest of tasks. Despite relying on regularization, our approach is a conceptual departure from previous algorithms based on regularization in weight (e.g., Kirkpatrick et al., 2017; Zenke et al., 2017) or activation space (e.g., He & Jaeger, 2018).
+
+Our experimental results show that task-conditioned hypernetworks do not suffer from catastrophic forgetting on a set of standard CL benchmarks. Remarkably, they are capable of retaining memories with practically no decrease in performance, when presented with very long sequences of tasks. Thanks to the expressive power of neural networks, task-conditioned hypernetworks exploit task-to-task similarities and transfer information forward in time to future tasks. Finally, the task-conditional metamodelling perspective that we put forth is generic, as it does not depend on the specifics of the target network architecture. We exploit this key principle and show that the very same metamodelling framework extends to, and can improve, an important class of CL methods known as generative replay methods, which are current state-of-the-art performers in many practical problems (Shin et al., 2017; Wu et al., 2018; van de Ven & Tolias, 2018).
+
+# 2 MODEL
+
+# 2.1 TASK-CONDITIONED HYPERNETWORKS
+
+Hypernetworks parameterize target models. The centerpiece of our approach to continual learning is the hypernetwork, Fig. 1a. Instead of learning the parameters  $\Theta_{\mathrm{trgt}}$  of a particular function  $f_{\mathrm{trgt}}$  directly (the target model), we learn the parameters  $\Theta_{\mathrm{h}}$  of a metamodel. The output of such metamodel, the hypernetwork, is  $\Theta_{\mathrm{trgt}}$ . Hypernetworks can therefore be thought of as weight generators, which were originally introduced to dynamically parameterize models in a compressed form (Ha et al., 2017; Schmidhuber, 1992; Bertinetto et al., 2016; Jia et al., 2016).
+
+![](images/f71272a1b6fe08e698a1044916a513fdf4927765104ca63e48b8dd0d2b388ca7.jpg)  
+a
+
+![](images/916fe741fd63f72bbfb2e5602d8e7d755e4e45b6f72e71579a14a6449b21c172.jpg)  
+b  
+Figure 1: Task-conditioned hypernetworks for continual learning. (a) Commonly, the parameters of a neural network are directly adjusted from data to solve a task. Here, a weight generator termed hypernetwork is learned instead. Hypernetworks map embedding vectors to weights, which parameterize a target neural network. In a continual learning scenario, a set of task-specific embeddings is learned via backpropagation. Embedding vectors provide task-dependent context and bias the hypernetwork to particular solutions. (b) A smaller, chunked hypernetwork can be used iteratively, producing a chunk of target network weights at a time (e.g., one layer at a time). Chunked hypernetworks can achieve model compression: the effective number of trainable parameters can be smaller than the number of target network weights.
+
+Continual learning with hypernetwork output regularization. One approach to avoid catastrophic forgetting is to store data from previous tasks and corresponding model outputs, and then fix such outputs. This can be achieved using an output regularizer of the following form, where past outputs play the role of pseudo-targets (Robins, 1995; Li & Hoiem, 2018; Benjamin et al., 2018):
+
+$$
+\mathcal {L} _ {\text {o u t p u t}} = \sum_ {t = 1} ^ {T - 1} \sum_ {i = 1} ^ {| \mathbf {X} ^ {(t)} |} \| f \left(\mathbf {x} ^ {(t, i)}, \Theta^ {*}\right) - f \left(\mathbf {x} ^ {(t, i)}, \Theta\right) \| ^ {2}, \tag {1}
+$$
+
+In the equation above,  $\Theta^{*}$  is the set of parameters before attempting to learn task  $T$ , and  $f$  is the learner. This approach, however, requires storing and iterating over previous data, a process that is known as rehearsing. This is potentially expensive memory-wise and not strictly online learning. A possible workaround is to generate the pseudo-targets by evaluating  $f$  on random patterns (Robins, 1995) or on the current task dataset (Li & Hoiem, 2018). However, this does not necessarily fix the behavior of the function  $f$  in the regions of interest.
+
+Hypernetworks sidestep this problem naturally. In target network weight space, a single point (i.e., one set of weights) has to be fixed per task. This can be efficiently achieved with task-conditioned hypernetworks, by fixing the hypernetwork output on the appropriate task embedding.
+
+Similar to Benjamin et al. (2018), we use a two-step optimization procedure to introduce memory-preserving hypernetwork output constraints. First, we compute a candidate change  $\Delta \Theta_{\mathrm{h}}$  which minimizes the current task loss  $\mathcal{L}_{\mathrm{task}}^{(T)} = \mathcal{L}_{\mathrm{task}}(\Theta_{\mathrm{h}},\mathbf{e}^{(T)},\mathbf{X}^{(T)},\mathbf{Y}^{(T)})$  with respect to  $\Theta$ . The candidate  $\Delta \Theta_{\mathrm{h}}$  is obtained with an optimizer of choice (we use Adam throughout; Kingma & Ba, 2015). The actual parameter change is then computed by minimizing the following total loss:
+
+$$
+\begin{array}{l} \mathcal {L} _ {\text {t o t a l}} = \mathcal {L} _ {\text {t a s k}} \left(\Theta_ {\mathrm {h}}, \mathbf {e} ^ {(T)}, \mathbf {X} ^ {(T)}, \mathbf {Y} ^ {(T)}\right) + \mathcal {L} _ {\text {o u t p u t}} \left(\Theta_ {\mathrm {h}} ^ {*}, \Theta_ {\mathrm {h}}, \Delta \Theta_ {\mathrm {h}}, \left\{\mathbf {e} ^ {(t)} \right\}\right) \\ = \mathcal {L} _ {\text {t a s k}} \left(\Theta_ {\mathrm {h}}, \mathbf {e} ^ {(T)}, \mathbf {X} ^ {(T)}, \mathbf {Y} ^ {(T)}\right) + \frac {\beta_ {\text {o u t p u t}}}{T - 1} \sum_ {t = 1} ^ {T - 1} \| f _ {\mathrm {h}} \left(\mathbf {e} ^ {(t)}, \Theta_ {\mathrm {h}} ^ {*}\right) - f _ {\mathrm {h}} \left(\mathbf {e} ^ {(t)}, \Theta_ {\mathrm {h}} + \Delta \Theta_ {\mathrm {h}}\right)) \| ^ {2}, \tag {2} \\ \end{array}
+$$
+
+where  $\Theta_{\mathrm{h}}^{*}$  is the set of hypernetwork parameters before attempting to learn task  $T$ ,  $\Delta \Theta_{\mathrm{h}}$  is considered fixed and  $\beta_{\mathrm{output}}$  is a hyperparameter that controls the strength of the regularizer. On Appendix D, we run a sensitivity analysis on  $\beta_{\mathrm{output}}$  and experiment with a more efficient stochastic regularizer where the averaging is performed over a random subset of past tasks.
+
+More computationally-intensive algorithms that involve a full inner-loop refinement, or use second-order gradient information by backpropagating through  $\Delta \Theta_{\mathrm{h}}$  could be applied. However, we found empirically that our one-step correction worked well. Exploratory hyperparameter scans revealed that the inclusion of the lookahead  $\Delta \Theta_{h}$  in (2) brought a minor increase in performance, even when computed with a cheap one-step procedure. Note that unlike in Eq. 1, the memory-preserving term  $\mathcal{L}_{\mathrm{output}}$  does not depend on past data. Memory of previous tasks enters only through the collection of task embeddings  $\{\mathbf{e}^{(t)}\}_{t=1}^{T-1}$ .
+
+Learned task embeddings. Task embeddings are differentiable deterministic parameters that can be learned, just like  $\Theta_{\mathrm{h}}$ . At every learning step of our algorithm, we also update the current task embedding  $\mathbf{e}^{(T)}$  to minimize the task loss  $\mathcal{L}_{\mathrm{task}}^{(T)}$ . After learning the task, the final embedding is saved and added to the collection  $\{\mathbf{e}^{(t)}\}$ .
+
+# 2.2 MODEL COMPRESSION WITH CHUNKED HYPERNETWORKS
+
+Chunking. In a straightforward implementation, a hypernetwork produces the entire set of weights of a target neural network. For modern deep neural networks, this is a very high-dimensional output. However, hypernetworks can be invoked iteratively, filling in only part of the target model at each step, in chunks (Ha et al., 2017; Pawlowski et al., 2017). This strategy allows applying smaller hypernetworks that are reusable. Interestingly, with chunked hypernetworks it is possible to solve tasks in a compressive regime, where the number of learned parameters (those of the hypernetwork) is effectively smaller than the number of target network parameters.
+
+Chunk embeddings and network partitioning. Reapplying the same hypernetwork multiple times introduces weight sharing across partitions of the target network, which is usually not desirable.
+
+To allow for a flexible parameterization of the target network, we introduce a set  $\mathcal{C} = \{\mathbf{c}_i\}_{i=1}^{N_{\mathrm{c}}}$  of chunk embeddings, which are used as an additional input to the hypernetwork, Fig. 1b. Thus, the full set of target network parameters  $\Theta_{\mathrm{trgt}} = [f_{\mathrm{h}}(\mathbf{e}, \mathbf{c}_1), \dots, f_{\mathrm{h}}(\mathbf{e}, \mathbf{c}_{N_{\mathrm{C}}})]$  is produced by iteration over  $\mathcal{C}$ , keeping the task embedding  $\mathbf{e}$  fixed. This way, the hypernetwork can produce distinct weights for each chunk. Furthermore, chunk embeddings, just like task embeddings, are ordinary deterministic parameters that we learn via backpropagation. For simplicity, we use a shared set of chunk embeddings for all tasks and we do not explore special target network partitioning strategies.
+
+How flexible is our approach? Chunked neural networks can in principle approximate any target weight configuration arbitrarily well. For completeness, we state this formally in Appendix E.
+
+# 2.3 CONTEXT-FREE INFERENCE: UNKNOWN TASK IDENTITY
+
+Determining which task to solve from input data. Our hypernetwork requires a task embedding input to generate target model weights. In certain CL applications, an appropriate embedding can be immediately selected as task identity is unambiguous, or can be readily inferred from contextual clues. In other cases, knowledge of the task at hand is not explicitly available during inference. In the following, we show that our metamodelelling framework generalizes to such situations. In particular, we consider the problem of inferring which task to solve from a given input pattern, a noted benchmark challenge (Farquhar & Gal, 2018; van de Ven & Tolias, 2019). Below, we explore two different strategies that leverage task-conditioned hypernetworks in this CL setting.
+
+Task-dependent predictive uncertainty. Neural network models are increasingly reliable in signalling novelty and appropriately handling out-of-distribution data. For categorical target distributions, the network ideally produces a flat, high entropy output for unseen data and, conversely, a peaked, low-entropy response for in-distribution data (Hendrycks & Gimpel, 2016; Liang et al., 2017). This suggests a first, simple method for task inference (HNET+ENT). Given an input pattern for which task identity is unknown, we pick the task embedding which yields lowest predictive uncertainty, as quantified by output distribution entropy. While this method relies on accurate novelty detection, which is in itself a far from solved research problem, it is otherwise straightforward to implement and no additional learning or model is required to infer task identity.
+
+Hypernetwork-protected synthetic replay. When a generative model is available, catastrophic forgetting can be circumvented by mixing current task data with replayed past synthetic data (for recent work see Shin et al., 2017; Wu et al., 2018). Besides protecting the generative model itself, synthetic data can protect another model of interest, for example, another discriminative model. This conceptually simple strategy is in practice often the state-of-the-art solution to CL (van de Ven & Tolias, 2019). Inspired by these successes, we explore augmenting our system with a replay network, here a standard variational autoencoder (VAE; Kingma & Welling, 2014) (but see Appendix F for experiments with a generative adversarial network, Goodfellow et al., 2014).
+
+Synthetic replay is a strong, but not perfect, CL mechanism as the generative model is subject to drift, and errors tend to accumulate and amplify with time. Here, we build upon the following key observation: just like the target network, the generator of the replay model can be specified by a hypernetwork. This allows protecting it with the output regularizer, Eq. 2, rather than with the model's own replay data, as done in related work. Thus, in this combined approach, both synthetic replay and task-conditional metamodelelling act in tandem to reduce forgetting.
+
+We explore hypernetwork-protected replay in two distinct setups. First, we consider a minimalist architecture (HNET+R), where only the replay model, and not the target classifier, is parameterized by a hypernetwork. Here, forgetting in the target network is obviated by mixing current data with synthetic data. Synthetic target output values for previous tasks are generated using a soft targets method, i.e., by simply evaluating the target function before learning the new task on synthetic input data. Second (HNET+TIR), we introduce an auxiliary task inference classifier, protected using synthetic replay data and trained to predict task identity from input patterns. This architecture requires additional modelling, but it is likely to work well when tasks are strongly dissimilar. Furthermore, the task inference subsystem can be readily applied to process more general forms of contextual information, beyond the current input pattern. We provide additional details, including network architectures and the loss functions that are optimized, in Appendices B and C.
+
+# 3 RESULTS
+
+We evaluate our method on a set of standard image classification benchmarks on the MNIST, CIFAR-10 and CIFAR-100 public datasets<sup>1</sup>. Our main aims are to (1) study the memory retention capabilities of task-conditioned hypernetworks across three continual learning settings, and (2) investigate information transfer across tasks that are learned sequentially.
+
+Continual learning scenarios. In our experiments we consider three different CL scenarios (van de Ven & Tolias, 2019). In CL1, the task identity is given to the system. This is arguably the standard sequential learning scenario, and the one we consider unless noted otherwise. In CL2, task identity is unknown to the system, but it does not need to be explicitly determined. A target network with a fixed head is required to solve multiple tasks. In CL3, task identity has to be explicitly inferred. It has been argued that this scenario is the most natural, and the one that tends to be harder for neural networks (Farquhar & Gal, 2018; van de Ven & Tolias, 2019).
+
+Experimental details. Aiming at comparability, for the experiments on the MNIST dataset we model the target network as a fully-connected network and set all hyperparameters after van de Ven & Tolias (2019), who recently reviewed and compared a large set of CL algorithms. For our CIFAR experiments, we opt for a ResNet-32 target neural network (He et al., 2016) to assess the scalability of our method. A summary description of the architectures and particular hyperparameter choices, as well as additional experimental details, is provided in Appendix C. We emphasize that, on all our experiments, the number of hypernetwork parameters is always smaller or equal than the number of parameters of the models we compare with.
+
+![](images/4e0c81aa130281a31d691870d59cee7c57f265f8f9088f1c992a3fad5e8122ff.jpg)  
+a  
+b
+
+![](images/7c197acced7ea2a62305cfa828a2566897d9f4ab98f47915352a0c93878fab78.jpg)  
+C  
+Figure 2: 1D nonlinear regression. (a) Task-conditioned hypernetworks with output regularization can easily model a sequence of polynomials of increasing degree, while learning in a continual fashion. (b) The solution found by a target network which is trained directly on all tasks simultaneously is similar. (c) Fine-tuning, i.e., learning sequentially, leads to forgetting of past tasks. Dashed lines depict ground truth, markers show model predictions.
+
+![](images/bdee14e297487a275ed6119ddb0abd07ce3418815b211ed5d38c1ee52e3e54f8.jpg)
+
+Nonlinear regression toy problem. To illustrate our approach, we first consider a simple nonlinear regression problem, where the function to be approximated is scalar-valued, Fig. 2. Here, a sequence of polynomial functions of increasing degree has to be inferred from noisy data. This motivates the continual learning problem: when learning each task in succession by modifying  $\Theta_{\mathrm{h}}$  with the memory-preserving regularizer turned off ( $\beta_{\mathrm{output}} = 0$ , see Eq. 2) the network learns the last task but forgets previous ones, Fig. 2c. The regularizer protects old solutions, Fig. 2a, and performance is comparable to an offline non-continual learner, Fig. 2b.
+
+Permuted MNIST benchmark. Next, we study the permuted MNIST benchmark. This problem is set as follows. First, the learner is presented with the full MNIST dataset. Subsequently, novel tasks are obtained by applying a random permutation to the input image pixels. This process can be repeated to yield a long task sequence, with a typical length of  $T = 10$  tasks. Given the low similarity of the generated tasks, permuted MNIST is well suited to study the memory capacity of a continual learner. For  $T = 10$ , we find that task-conditioned hypernetworks are state-of-the-art on CL1, Table 1. Interestingly, inferring tasks through the predictive distribution entropy (HNET+ENT) works well on the permuted MNIST benchmark. Despite the simplicity of the method, both synaptic intelligence (SI; Zenke et al., 2017) and online elastic weight consolidation (EWC; Schwarz et al., 2018) are overperformed on CL3 by a large margin. When complemented with generative replay
+
+![](images/5497c77ac1318743b178e4c38fbbe34ba20824cffe50e937a2f7baec1987a0e3.jpg)  
+a
+
+![](images/cb19f34c3472fcbc5f5391dbc77dcf2225393702443ac05d3a42dcb3b307e4f9.jpg)  
+b  
+Figure 3: Experiments on the permuted MNIST benchmark. (a) Final test set classification accuracy on the  $t$ -th task after learning one hundred permutations (PermutedMNIST-100). Task-conditioned hypernetworks (hnet, in red) achieve very large memory lifetimes on the permuted MNIST benchmark. Synaptic intelligence (SI, in blue; Zenke et al., 2017), online EWC (in orange; Schwarz et al., 2018) and deep generative replay (DGR+distill, in green; Shin et al., 2017) methods are shown for comparison. Memory retention in SI and DGR+distill degrade gracefully, whereas EWC suffers from rigidity and can never reach very high accuracy, even though memories persist for the entire experiment duration. (b) Compression ratio  $\frac{|\Theta_{\mathrm{h}} \cup \{\mathbf{e}^{(t)}\}|}{|\Theta_{\mathrm{tgt}}|}$  versus task-averaged test set accuracy after learning all tasks (labelled 'final', in red) and immediately after learning a task (labelled 'during', in purple) for the PermutedMNIST-10 benchmark. Hypernetworks allow for model compression and perform well even when the number of target model parameters exceeds their own. Performance decays nonlinearly: accuracies stay approximately constant for a wide range of compression ratios below unity. Hyperparameters were tuned once for compression ratio  $\approx 1$  and were then used for all compression ratios. Shaded areas denote STD (a) resp. SEM (b) across 5 random seeds.
+
+methods, task-conditioned hypernetworks (HNET+TIR and HNET+R) are the best performers on all three CL scenarios.
+
+Performance differences become larger in the long sequence limit, Fig. 3a. For longer task sequences  $(T = 100)$ , SI and DGR+distill (Shin et al., 2017; van de Ven & Tolias, 2018) degrade gracefully, while the regularization strength of online EWC prevents the method from achieving high accuracy (see Fig. A6 for a hyperparameter search on related work). Notably, task-conditioned hypernetworks show minimal memory decay and find high performance solutions. Because the hypernetwork operates in a compressive regime (see Fig. 3b and Fig. A7 for an exploration of compression ratios), our results do not naively rely on an increase in the number of parameters. Rather, they suggest that previous methods are not yet capable of making full use of target model capacity in a CL setting. We report a set of extended results on this benchmark on Appendix D, including a study of CL2/3  $(T = 100)$ , where HNET+TIR strongly outperforms the related work.
+
+Split MNIST benchmark. Split MNIST is another popular CL benchmark, designed to introduce task overlap. In this problem, the various digits are sequentially paired and used to form five binary classification tasks. Here, we find that task-conditioned hypernetworks are the best overall performers. In particular, HNET+R improves the previous state-of-the-art method DGR+distill on both CL2 and CL3, almost saturating the CL2 upper bound for replay models (Appendix D). Since HNET+R is essentially hypernetwork-protected DGR, these results demonstrate the generality of task-conditioned hypernetworks as effective memory protectors. To further support this, in Appendix F we show that our replay models (we experiment with both a VAE and a GAN) can learn in a class-incremental manner the full MNIST dataset. Finally, HNET+ENT again outperforms both EWC and SI, without any generative modelling.
+
+On the split MNIST problem, tasks overlap and therefore continual learners can transfer information across tasks. To analyze such effects, we study task-conditioned hypernetworks with two-dimensional task embedding spaces, which can be easily visualized. Despite learning happening continually, we
+
+Table 1: Task-averaged test accuracy (± SEM,  $n = 20$ ) on the permuted ('P10') and split ('S') MNIST experiments. In the table, EWC refers to online EWC and DGR refers to DGR+distill (results reproduced from van de Ven & Tolias, 2019). We tested three hypernetwork-based models: for HNET+ENT (HNET alone for CL1), we inferred task identity based on the entropy of the predictive distribution; for HNET+TIR, we trained a hypernetwork-protected recognition-replay network (based on a VAE, cf. Fig. A1) to infer the task from input patterns; for HNET+R the main classifier was trained by mixing current task data with synthetic data generated from a hypernetwork-protected VAE.  
+
+<table><tr><td></td><td>EWC</td><td>SI</td><td>DGR</td><td>HNET+ENT</td><td>HNET+TIR</td><td>HNET+R</td></tr><tr><td>P10-CL1</td><td>95.96 ± 0.06</td><td>94.75 ± 0.14</td><td>97.51 ± 0.01</td><td>97.57 ± 0.02</td><td>97.57 ± 0.02</td><td>97.87 ± 0.01</td></tr><tr><td>P10-CL2</td><td>94.42 ± 0.13</td><td>95.33 ± 0.11</td><td>97.35 ± 0.02</td><td>92.80 ± 0.15</td><td>97.58 ± 0.02</td><td>97.60 ± 0.01</td></tr><tr><td>P10-CL3</td><td>33.88 ± 0.49</td><td>29.31 ± 0.62</td><td>96.38 ± 0.03</td><td>91.75 ± 0.21</td><td>97.59 ± 0.01</td><td>97.76 ± 0.01</td></tr><tr><td>S-CL1</td><td>99.12 ± 0.11</td><td>99.09 ± 0.15</td><td>99.61 ± 0.02</td><td>99.79 ± 0.01</td><td>99.79 ± 0.01</td><td>99.83 ± 0.01</td></tr><tr><td>S-CL2</td><td>64.32 ± 1.90</td><td>65.36 ± 1.57</td><td>96.83 ± 0.20</td><td>87.01 ± 0.47</td><td>94.43 ± 0.28</td><td>98.00 ± 0.03</td></tr><tr><td>S-CL3</td><td>19.96 ± 0.07</td><td>19.99 ± 0.06</td><td>91.79 ± 0.32</td><td>69.48 ± 0.80</td><td>89.59 ± 0.59</td><td>95.30 ± 0.13</td></tr></table>
+
+![](images/a538df833166d6a897a1cb5161839f8b7cf65fc0a7a941a5faab77faaaebf87b.jpg)  
+Figure 4: Two-dimensional task embedding space for the split MNIST benchmark. Color-coded test set classification accuracies after learning the five splits, shown as the embedding vector components are varied. Markers denote the position of final task embeddings. (a) High classification performance with virtually no forgetting is achieved even when e-space is low-dimensional. The model shows information transfer in embedding space: the first task is solved in a large volume that includes embeddings for subsequently learned tasks. (b) Competition in embedding space: the last task occupies a finite high performance region, with graceful degradation away from the embedding vector. Previously learned task embeddings still lead to moderate, above-chance performance.
+
+![](images/3152cbd714ad35a51401f33d411e283ca470a3fb0dabc44b93fdef3bd51d09fc.jpg)
+
+find that the algorithm converges to a hypernetwork configuration that can produce target model parameters that simultaneously solve old and new tasks, Fig. 4, given the appropriate task embedding.
+
+Split CIFAR-10/100 benchmark. Finally, we study a more challenging benchmark, where the learner is first asked to solve the full CIFAR-10 classification task and is then presented with sets of ten classes from the CIFAR-100 dataset. We perform experiments both with a high-performance ResNet-32 target network architecture (Fig. 5) and with a shallower model (Fig. A3) that we exactly reproduced from previous work (Zenke et al., 2017). Remarkably, on the ResNet-32 model, we find that task-conditioned hypernetworks essentially eliminate altogether forgetting. Furthermore, forward information transfer takes place; knowledge from previous tasks allows the network to find better solutions than when learning each task individually from initial conditions. Interestingly, forward transfer is stronger on the shallow model experiments (Fig. A3), where we otherwise find that our method performs comparably to SI.
+
+# 4 DISCUSSION
+
+Bayesian accounts of continual learning. According to the standard Bayesian CL perspective, a posterior parameter distribution is recursively updated using Bayes' rule as tasks arrive
+
+![](images/00cb40b9d9d0529a9775598590e22c5c121be5e404331b5feb5e41b8a640e33a.jpg)  
+Figure 5: Split CIFAR-10/100 CL benchmark. Test set accuracies (mean  $\pm$  STD,  $n = 5$ ) on the entire CIFAR-10 dataset and subsequent CIFAR-100 splits of ten classes. Our hypernetwork-protected ResNet-32 displays virtually no forgetting; final averaged performance (hnet, in red) matches the immediate one (hnet-during, in blue). Furthermore, information is transferred across tasks, as performance is higher than when training each task from scratch (purple). Disabling our regularizer leads to strong forgetting (in yellow).
+
+(Kirkpatrick et al., 2017; Huszar, 2018; Nguyen et al., 2018). While this approach is theoretically sound, in practice, the approximate inference methods that are typically preferred can lead to stiff models, as a compromise solution that suits all tasks has to be found within the mode determined by the first task. Such restriction does not apply to hypernetworks, which can in principle model complex multimodal distributions (Louizos & Welling, 2017; Pawlowski et al., 2017; Henning et al., 2018). Thus, rich, hypernetwork-modelled priors are one avenue of improvement for Bayesian CL methods. Interestingly, task-conditioning offers an alternative possibility: instead of consolidating every task onto a single distribution, a shared task-conditioned hypernetwork could be leveraged to model a set of parameter posterior distributions. This conditional metamodel naturally extends our framework to the Bayesian learning setting. Such approach will likely benefit from additional flexibility, compared to conventional recursive Bayesian updating.
+
+Related approaches that rely on task-conditioning. Our model fits within, and in certain ways generalizes, previous CL methods that condition network computation on task descriptors. Task-conditioning is commonly implemented using multiplicative masks at the level of modules (Rusu et al., 2016; Fernando et al., 2017), neurons (Serra et al., 2018; Masse et al., 2018) or weights (Mallya & Lazebnik, 2018). Such methods work best with large networks and come with a significant storage overhead, which typically scales with the number of tasks. Our approach differs by explicitly modelling the full parameter space using a metamodel, the hypernetwork. Thanks to this metamodel, generalization in parameter and task space is possible, and task-to-task dependencies can be exploited to efficiently represent solutions and transfer present knowledge to future problems. Interestingly, similar arguments have been drawn in work developed concurrently to ours (Lampinen & McClelland, 2019), where task embedding spaces are further explored in the context of few-shot learning. In the same vein, and like the approach developed here, recent work in CL generates last-layer network parameters as part of a pipeline to avoid catastrophic forgetting (Hu et al., 2019) or distills parameters onto a contractive auto-encoding model (Camp et al., 2018).
+
+Positive backwards transfer. In its current form, the hypernetwork output regularizer protects previously learned solutions from changing, such that only weak backwards transfer of information can occur. Given the role of selective forgetting and refinement of past memories in achieving intelligent behavior (Brea et al., 2014; Richards & Frankland, 2017), investigating and improving backwards transfer stands as an important direction for future research.
+
+Relevance to systems neuroscience. Uncovering the mechanisms that support continual learning in both brains and artificial neural networks is a long-standing question (McCloskey & Cohen, 1989; French, 1999; Parisi et al., 2019). We close with a speculative systems interpretation (Kumaran et al., 2016; Hassabis et al., 2017) of our work as a model for modulatory top-down signals in cortex. Task embeddings can be seen as low-dimensional context switches, which determine the behavior of a modulatory system, the hypernetwork in our case. According to our model, the hypernetwork would in turn regulate the activity of a target cortical network.
+
+As it stands, implementing a hypernetwork would entail dynamically changing the entire connectivity of a target network, or cortical area. Such a process seems difficult to conceive in the brain. However, this strict literal interpretation can be relaxed. For example, a hypernetwork can output lower-dimensional modulatory signals (Marder, 2012), instead of a full set of weights. This interpretation
+
+is consistent with a growing body of work which suggests the involvement of modulatory inputs in implementing context- or task-dependent network mode-switching (Mante et al., 2013; Jaeger, 2014; Stroud et al., 2018; Masse et al., 2018).
+
+# 5 CONCLUSION
+
+We introduced a novel neural network model, the task-conditioned hypernetwork, that is well-suited for CL problems. A task-conditioned hypernetwork is a metamodel that learns to parameterize target functions, that are specified and identified in a compressed form using a task embedding vector. Past tasks are kept in memory using a hypernetwork output regularizer, which penalizes changes in previously found target weight configurations. This approach is scalable and generic, being applicable as a standalone CL method or in combination with generative replay. Our results are state-of-the-art on standard benchmarks and suggest that task-conditioned hypernetworks can achieve long memory lifetimes, as well as transfer information to future tasks, two essential properties of a continual learner.
+
+# ACKNOWLEDGMENTS
+
+This work was supported by the Swiss National Science Foundation (B.F.G. CRSII5-173721), ETH project funding (B.F.G. ETH-20 19-01) and funding from the Swiss Data Science Center (B.F.G, C17-18, J. v. O. P18-03). Special thanks to Simone Carlo Surace, Adrian Huber, Xu He, Markus Marks, Maria R. Cervera and Jannes Jegminat for discussions, helpful pointers to the CL literature and for feedback on our paper draft.
+
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+David Rolnick, Arun Ahuja, Jonathan Schwarz, Timothy P Lillicrap, and Greg Wayne. Experience replay for continual learning. arXiv preprint arXiv:1811.11682, 2018.  
+Andrei A. Rusu, Neil C. Rabinowitz, Guillaume Desjardins, Hubert Soyer, James Kirkpatrick, Koray Kavukcuoglu, Razvan Pascanu, and Raia Hadsell. Progressive Neural Networks. arXiv preprint arXiv:1606.04671, 2016.  
+Jürgen Schmidhuber. Learning to control fast-weight memories: An alternative to dynamic recurrent networks. Neural Computation, 4(1):131-139, 1992.  
+Jonathan Schwarz, Wojciech Czarnecki, Jelena Luketina, Agnieszka Grabska-Barwinska, Yee Whye Teh, Razvan Pascanu, and Raia Hadsell. Progress & compress: A scalable framework for continual learning. In Jennifer Dy and Andreas Krause (eds.), Proceedings of the 35th International Conference on Machine Learning, volume 80 of Proceedings of Machine Learning Research, pp. 4528-4537, Stockholm Männssan, Stockholm Sweden, 10-15 Jul 2018. PMLR.  
+Joan Serra, Didac Suris, Marius Miron, and Alexandros Karatzoglou. Overcoming catastrophic forgetting with hard attention to the task. In Jennifer Dy and Andreas Krause (eds.), Proceedings of the 35th International Conference on Machine Learning, volume 80 of Proceedings of Machine Learning Research, pp. 4548-4557, Stockholm, Sweden, 10-15 Jul 2018. PMLR.  
+Hanul Shin, Jung Kwon Lee, Jaehong Kim, and Jiwon Kim. Continual Learning with Deep Generative Replay. In I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett (eds.), Advances in Neural Information Processing Systems 30, pp. 2990-2999. Curran Associates, Inc., 2017.  
+Jake P. Stroud, Mason A. Porter, Guillaume Hennequin, and Tim P. Vogels. Motor primitives in space and time via targeted gain modulation in cortical networks. Nature Neuroscience, 21(12):1774, December 2018.  
+Siddharth Swaroop, Cuong V Nguyen, Thang D Bui, and Richard E Turner. Improving and understanding variational continual learning. *Continual Learning Workshop at NeurIPS*, 2018.  
+Gido M. van de Ven and Andreas S. Tolias. Generative replay with feedback connections as a general strategy for continual learning. arXiv preprint arXiv:1809.10635, 2018.  
+Gido M. van de Ven and Andreas S. Tolias. Three scenarios for continual learning. arXiv preprint arXiv:1904.07734, 2019.  
+Chenshen Wu, Luis Herranz, Xialei Liu, yaxing wang, Joost van de Weijer, and Bogdan Raducanu. Memory Replay GANs: Learning to Generate New Categories without Forgetting. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett (eds.), Advances in Neural Information Processing Systems 31, pp. 5962-5972. Curran Associates, Inc., 2018.  
+Friedemann Zenke, Ben Poole, and Surya Ganguli. Continual Learning Through Synaptic Intelligence. In Proceedings of the 34th International Conference on Machine Learning - Volume 70, ICML'17, pp. 3987-3995. JMLR.org, 2017.
+
+# A TASK-CONDITIONED HYPERNETWORKS: MODEL SUMMARY
+
+In our model, a task-conditioned hypernetwork produces the parameters  $\Theta_{\mathrm{trgt}} = f_{\mathrm{h}}(\mathbf{e},\Theta_{\mathrm{h}})$  of a target neural network. Given one such parameterization, the target model then computes predictions  $\hat{\mathbf{y}} = f_{\mathrm{trgt}}(\mathbf{x},\Theta_{\mathrm{trgt}})$  based on input data. Learning amounts to adapting the parameters  $\Theta_{\mathrm{h}}$  of the hypernetwork, including a set of task embeddings  $\{\mathbf{e}^{(t)}\}_{t = 1}^{T}$ , as well as a set of chunk embeddings  $\{\mathbf{c}_i\}_{i = 1}^{N_{\mathrm{C}}}$  in case compression is sought or if the full hypernetwork is too large to be handled directly. To avoid catastrophic forgetting, we introduce an output regularizer which fixes the behavior of the hypernetwork by penalizing changes in target model parameters that are produced for previously learned tasks.
+
+Variables that need to be stored while learning new tasks. What are the storage requirements of our model, when learning continually?
+
+1. Memory retention relies on saving one embedding per task. This collection  $\{\mathbf{e}^{(t)}\}_{t = 1}^{T}$  therefore grows linearly with  $T$ . Such linear scaling is undesirable asymptotically, but it turns out to be essentially negligible in practice, as each embedding is a single low-dimensional vector (e.g., see Fig. 4 for a run with 2D embeddings).  
+2. A frozen snapshot of the hypernetwork parameters  $\Theta_{\mathrm{h}}^{*}$ , taken before learning a new task, needs to be kept, to evaluate the output regularizer in Eq. 2.
+
+# B ADDITIONAL DETAILS ON HYPERNETWORK-PROTECTED REPLAY MODELS
+
+Variational autoencoders. For all HNET+TIR and HNET+R experiments reported on the main text we use VAEs as our replay models (Fig. A1a, Kingma & Welling, 2014). Briefly, a VAE consists of an encoder-decoder network pair, where the encoder network processes some input pattern  $\mathbf{x}$  and its outputs  $f_{\mathrm{enc}}(\mathbf{x}) = (\boldsymbol{\mu}, \boldsymbol{\sigma}^2)$  comprise the parameters  $\boldsymbol{\mu}$  and  $\boldsymbol{\sigma}^2$  (coded in log domain, to enforce nonnegativity) of a diagonal multivariate Gaussian  $p_Z(\mathbf{z}; \boldsymbol{\mu}, \boldsymbol{\sigma}^2)$ , which governs the distribution of latent samples  $\mathbf{z}$ . On the other side of the circuit, the decoder network processes a latent sample  $\mathbf{z}$  and a one-hot-encoded task identity vector and returns an input pattern reconstruction,  $f_{\mathrm{dec}}(\mathbf{z}, \mathbf{1}_t) = \hat{\mathbf{x}}$ .
+
+VAEs can preserve memories using a technique called generative replay: when training task  $T$ , input samples are generated from the current replay network for old tasks  $t < T$ , by varying  $\mathbf{1}_t$  and drawing latent space samples  $\mathbf{z}$ . Generated data can be mixed with the current dataset, yielding an augmented dataset  $\tilde{\mathcal{X}}$  used to relearn model parameters. When protecting a discriminative model, synthetic 'soft' targets can be generated by evaluating the network on  $\tilde{\mathcal{X}}$ . We use this strategy to protect an auxiliary task inference classifier in HNET+TIR, and to protect the main target model in HNET+R.
+
+Hypernetwork-protected replay. In our HNET+TIR and HNET+R experiments, we parameterize the decoder network through a task-conditioned hypernetwork,  $f_{\mathrm{h,dec}}(\mathbf{e}, \Theta_{\mathrm{h,dec}})$ . In combination with our output regularizer, this allows us to take advantage of the memory retention capacity of hypernetworks, now on a generative model.
+
+The replay model (encoder, decoder and decoder hypernetwork) is a separate subsystem that is optimized independently from the target network. Its parameters  $\Theta_{\mathrm{enc}}$  and  $\Theta_{\mathrm{h,dec}}$  are learned by minimizing our regularized loss function, Eq. 2, here with the task-specific term set to the standard VAE objective function,
+
+$$
+\mathcal {L} _ {\mathrm {V A E}} \left(\mathbf {X}, \Theta_ {\mathrm {e n c}}, \Theta_ {\mathrm {h , d e c}}\right) = \mathcal {L} _ {\mathrm {r e c}} \left(\mathbf {X}, \Theta_ {\mathrm {e n c}}, \Theta_ {\mathrm {d e c}}\right) + \mathcal {L} _ {\text {p r i o r}} \left(\mathbf {X}, \Theta_ {\mathrm {e n c}}, \Theta_ {\mathrm {d e c}}\right), \tag {3}
+$$
+
+with  $\Theta_{\mathrm{dec}} = f_{\mathrm{h,dec}}(\mathbf{e},\Theta_{\mathrm{h,dec}})$  introducing the dependence on  $\Theta_{\mathrm{h,dec}}$ .  $\mathcal{L}_{\mathrm{VAE}}$  balances a reconstruction  $\mathcal{L}_{\mathrm{rec}}$  and a prior-matching  $\mathcal{L}_{\mathrm{prior}}$  penalties. For our MNIST experiments, we choose binary cross-entropy (in pixel space) as the reconstruction loss, that we write below for a single example  $\mathbf{x}$
+
+$$
+\mathcal {L} _ {\text {r e c}} (\mathbf {x}, \Theta_ {\text {e n c}}, \Theta_ {\text {d e c}}) = \mathcal {L} _ {\text {x e n t}} (\mathbf {x}, f _ {\text {d e c}} (\mathbf {z}, \mathbf {1} _ {t (\mathbf {x})}, \Theta_ {\text {d e c}})), \tag {4}
+$$
+
+where  $\mathcal{L}_{\mathrm{xent}}(t,y) = -\sum_k t_k \log y_k$  is the cross entropy. For a diagonal Gaussian  $p_Z$ , the prior-matching term can be evaluated analytically,
+
+$$
+\mathcal {L} _ {\text {p r i o r}} = - \frac {1}{2} \sum_ {i = 1} ^ {| \mathbf {z} |} \left(1 + \log \sigma_ {i} ^ {2} - \sigma_ {i} ^ {2} - \mu_ {i} ^ {2}\right). \tag {5}
+$$
+
+Above,  $\mathbf{z}$  is a sample from  $p_Z(\mathbf{z};\boldsymbol {\mu}(\tilde{\mathbf{x}}),\sigma^2 (\tilde{\mathbf{x}}))$  obtained via the reparameterization trick (Kingma & Welling, 2014; Rezende et al., 2014). This introduces the dependency of  $\mathcal{L}_{\mathrm{rec}}$  on  $\Theta_{\mathrm{enc}}$ .
+
+Task inference network (HNET+TIR). In the HNET+TIR setup, we extend our system to include a task inference neural network classifier  $\alpha(\mathbf{x})$  parameterized by  $\Theta_{\mathrm{TI}}$ , where tasks are encoded with a  $T$ -dimensional softmax output layer. In both CL2 and CL3 scenarios we use a growing single-head setup for  $\alpha$ , and increase the dimensionality of the softmax layer as tasks arrive.
+
+This network is prone to catastrophic forgetting when tasks are learned continually. To prevent this from happening we resort to replay data generated from a hypernetwork-protected VAE, described above. More specifically, we introduce a task inference loss,
+
+$$
+\mathcal {L} _ {\mathrm {T I}} (\tilde {\mathbf {x}}, \Theta_ {\mathrm {T I}}) = \mathcal {L} _ {\text {x e n t}} \left(\mathbf {1} _ {t (\tilde {\mathbf {x}})}, \boldsymbol {\alpha} (\tilde {\mathbf {x}}, \Theta_ {\mathrm {e n c}})\right), \tag {6}
+$$
+
+where  $t(\tilde{\mathbf{x}})$  denotes the correct task identity for a sample  $\tilde{\mathbf{x}}$  from the augmented dataset  $\tilde{\mathcal{X}} = \{\tilde{\mathbf{X}}^{(1)},\dots \tilde{\mathbf{X}}^{(T - 1)},\tilde{\mathbf{X}}^{(T)}\}$  with  $\tilde{\mathbf{X}}^{(t)}$  being synthetic data  $f_{\mathrm{dec}}(\mathbf{z},\mathbf{1}_t,\Theta_{\mathrm{dec}})$  for  $t = 1\dots T - 1$  and  $\tilde{\mathbf{X}}^{(T)} = \mathbf{X}^{(T)}$  is the current task data. Importantly, synthetic data is essential to obtain a well defined objective function for task inference; the cross-entropy loss  $\mathcal{L}_{\mathrm{TI}}$  requires at least two groundtruth classes to be optimized. Note that replayed data can be generated online by drawing samples  $z$  from the prior.
+
+![](images/da2fd766eed71c0f88baffe3aaae4eb116e28fea424674e8986a5df1c38cba51.jpg)  
+Figure A1: Hypernetwork-protected replay model setups. (a) A hypernetwork-protected VAE, that we used for HNET+R and HNET+TIR main text experiments. (b) A hypernetwork-protected GAN, that we used for our class-incremental learning Appendix F experiments. (c) A task inference classifier protected with synthetic replay data, used on HNET+TIR experiments.
+
+![](images/32241445b3ecd661258f8471b757adf539ff3c1262ebed5d951b063b07b5709f.jpg)
+
+![](images/3277193531da0896f5ba745425f8c943b6accd8131fbba5af8a748ba2efcad18.jpg)
+
+Hypernetwork-protected GANs. Generative adversarial networks (Goodfellow et al., 2014) have become an established method for generative modelling and tend to produce higher quality images compared to VAEs, even at the scale of datasets as complex as ImageNet (Brock et al., 2019; Lučić et al., 2019; Donahue & Simonyan, 2019). This makes GANs perfect candidates for powerful replay models. A suitable GAN instantiation for CL is the conditional GAN (Mirza & Osindero, 2014) as studied by Wu et al. (2018). Recent developments in the GAN literature already allude towards the potential of using hypernetwork-like structures, e.g., when injecting the latent noise (Karras et al., 2019) or when using class-conditional batch-normalization as in (Brock et al., 2019). We propose to go one step further and use a hypernetwork that maps the condition to the full set of generator parameters  $\Theta_{\mathrm{gen}}^{*}$ . Our framework allows training a conditional GAN one condition at the time. This is potentially of general interest, and goes beyond the scope of replay models, since conditional GANs trained in a multi-task fashion as in Brock et al. (2019) require very large computational resources.
+
+For our showcase experiment on class-incremental MNIST learning, Fig. A8, we did not aim to compare to related work and therefore did not tune to have less weights in the hypernetwork than on the target network (for the VAE experiments, we use the same compressive setup as in the main text, see Appendix C). The GAN hypernetwork is a fully-connected chunked hypernetwork with 2 hidden layers of size 25 and 25 followed by an output size of 75,000. We used learning rates for both discriminator and the generator hypernetwork of 0.0001, as well as dropout of 0.4 in the discriminator and the system is trained for 10000 iterations per task. We use the Pearson Chi² Least-Squares GAN loss from Mao et al. (2017) in our experiments.
+
+# C ADDITIONAL EXPERIMENTAL DETAILS
+
+All experiments are conducted using 16 NVIDIA GeForce RTX 2080 TI graphics cards.
+
+For simplicity, we decided to always keep the previous task embeddings  $\mathbf{e}^{(t)}, t = 1,\dots,T - 1$ , fixed and only learn the current task embedding  $\mathbf{e}^{(T)}$ . In general, performance should be improved if the
+
+regularizer in Eq. 2 has a separate copy of the task embeddings  $\mathbf{e}^{(t,*)}$  from before learning the current task, such that  $\mathbf{e}^{(t)}$  can be adapted. Hence, the targets become  $f_{\mathrm{h}}(\mathbf{e}^{(t,*)}, \Theta_{\mathrm{h}}^*)$  and remain constant while learning task  $T$ . This would give the hypernetwork the flexibility to adjust the embeddings i.e. the preimage of the targets and therefore represent any function that includes all desired targets in its image.
+
+Nonlinear regression toy problem. The nonlinear toy regression from Fig. 2 is an illustrative example for a continual learning problem where a set of ground-truth functions  $\{g^{(1)},\ldots ,g^{(T)}\}$  is given from which we collect 100 noisy training samples per task  $\{(\mathbf{x},\mathbf{y})\mid \mathbf{y} = g^{(t)}(\mathbf{x}) + \epsilon$  with  $\epsilon \sim \mathcal{N}(0,\sigma^2 I),\mathbf{x}\sim \mathcal{U}(\mathcal{X}^{(t)})\}$ , where  $\mathcal{X}^{(t)}$  denotes the input domain of task  $t$ . We set  $\sigma = 0.05$  in this experiment.
+
+We perform 1D regression and choose the following set of tasks:
+
+$$
+g ^ {(1)} (x) = x + 3 \quad \mathcal {X} ^ {(1)} = [ - 4, - 2 ] \tag {7}
+$$
+
+$$
+g ^ {(2)} (x) = 2 x ^ {2} - 1 \quad \mathcal {X} ^ {(2)} = [ - 1, 1 ] \tag {8}
+$$
+
+$$
+g ^ {(3)} (x) = (x - 3) ^ {3} \quad \mathcal {X} ^ {(3)} = [ 2, 4 ] \tag {9}
+$$
+
+The target network  $f_{\mathrm{trgt}}$  consists of two fully-connected hidden layers using 10 neurons each. For illustrative purposes we use a full hypernetwork  $f_{\mathrm{h}}$  that generates all 141 weights of  $f_{\mathrm{trgt}}$  at once, also being a fully-connected network with two hidden-layers of size 10. Hence, this is the only setup where we did not explore the possibility of a chunked hypernetwork. We use sigmoid activation functions in both networks. The task embedding dimension was set to 2.
+
+We train each task for 4000 iterations using the Adam optimizer with a learning rate of 0.01 (and otherwise default PyTorch options) and a batch size of 32.
+
+To test our regularizer in Fig. 2a we set  $\beta_{\mathrm{output}}$  to 0.005, while it is set to 0 for the fine-tuning experiment in Fig. 2c.
+
+For the multi-task learner in Fig. 2b we trained only the target network (no hypernetwork) for 12000 iterations with a learning rate of 0.05. Comparable performance could be obtained when training the task-conditioned hypernetwork in this multi-task regime (data not shown).
+
+It is worth noting that the multi-task learner from Fig. 2b that uses no hypernetwork is only able to learn the task since we choose the input domains to be non-overlapping.
+
+Permuted MNIST benchmark. For our experiments conducted on MNIST we replicated the experimental setup proposed by van de Ven & Tolias (2019) whenever applicable. We therefore use the same number of training iterations, the same or a lower number of weights in the hypernetwork than in the target network, the same learning rates and the same optimizer. For the replay model, i.e., the hypernetwork-empowered VAE, as well as for the standard classifier we used 5000 training iterations per task and learning rate is set to 0.0001 for the Adam optimizer (otherwise PyTorch default values). The batchsize is set to 128 for the VAE whereas the classifier is simultaneously trained on a batch of 128 samples of replayed data (evenly distributed over all past tasks) and a batch of 128 images from the currently available dataset. MNIST images are padded with zeros, which results in network inputs of size  $32 \times 32$ , again strictly following the implementation of the compared work. We experienced better performance when we condition our replay model on a specific task input. We therefore construct for every task a specific input namely a sample from a standard multivariate normal of dimension 100. In practice we found the dimension to be not important. This input stays constant throughout the experiment and is not learned. Note that we use the same hyperparameters for all learning scenarios, which is not true for the reported related work since they have tuned special hyperparameters for all scenarios and all methods.
+
+- Details of hypernetwork for the VAE. We use one hypernetwork configuration to generate weights for all variational autoencoders used for our PermutedMNIST-10 experiments namely a fully-connected chunked hypernetwork with 2 hidden layers of size 25 and 25 followed by an output size of 85,000. We use ELU nonlinearities in the hidden layers
+
+![](images/d90b08b7d9cd5f8e529c4e98dc233c1f632fa8a9fc688f5b180d68ccccf30e07.jpg)  
+a
+
+<table><tr><td>βoutput = 1</td><td>βoutput = 0.05</td><td>βoutput = 0.001</td></tr><tr><td>βoutput = 0.5</td><td>βoutput = 0.01</td><td>βoutput = 0.0005</td></tr><tr><td>βoutput = 0.1</td><td>βoutput = 0.005</td><td></td></tr></table>
+
+![](images/b32369cc06b3c4ae0931ec25fc9b012a871c2e726252f882f01b7cf886265d14.jpg)  
+b
+
+![](images/15e0fe595c9592eed213184d8d30a7f789f006b2763b272c4a62e45d20e333f6.jpg)  
+C  
+Figure A2: Additional experiments on the PermutedMNIST-100 benchmark. (a) Final test set classification accuracy on the  $t$ -th task after learning one hundred permutations (PermutedMNIST-100). All runs use exactly the same hyperparameter configuration except for varying values of  $\beta_{\mathrm{output}}$ . The final accuracies are robust for a wide range of regularization strengths. If  $\beta_{\mathrm{output}}$  is too weak, forgetting will occur. However, there is no severe disadvantage of choosing  $\beta_{\mathrm{output}}$  too high (cmp. (c)). A too high  $\beta_{\mathrm{output}}$  simply shifts the attention of the optimizer away from the current task, leading to lower baseline accuracies when the training time is not increased. (b) Due to an increased number of output neurons, the target network for PermutedMNIST-100 has more weights than for PermutedMNIST-10 (this is only the case for CL1 and CL3). This plot shows that the performance drop is minor when choosing a hypernetwork with a comparable number of weights as the target network in CL2 (orange) compared to one that has a similar number of weights as the target network for CL1 in PermutedMNIST-100 (red). (c) Task-averaged test set accuracy after learning all tasks (labelled 'final', in red) and immediately after learning a task (labelled 'during', in purple) for the runs depicted in (a). For low values of  $\beta_{\mathrm{output}}$  final accuracies are worse than immediate once (forgetting occurs). If  $\beta_{\mathrm{output}}$  is too high, baseline accuracies decrease since the optimizer puts less emphasis on the current task (note that the training time per task is not increased). Shaded areas in (a) and (b) denote STD, whereas error bars in (c) denote SEM (always across 5 random seeds).
+
+of the hypernetwork. The size of task embeddings  $\mathbf{e}$  has been set to 24 and the size of chunk embeddings  $\mathbf{c}$  to 8. The parameter  $\beta_{\mathrm{output}}$  is 0.05. The number of weights in this hypernetwork is 2,211,907 (2,211,691 network weights + 216 task embedding weights). The corresponding target network (and therefore output of the chunked hypernetwork), as taken from related work, has 2,227,024 weights.
+
+- Details of the VAE for HNET+TIR. For this variational autoencoder, we use two fully-connected neural networks with layers of size 1000, 1000 for the encoder and 1000, 1000 for the decoder and a latent space of 100. This setup is again copied from work we compare against.  
+- Details of the VAE for HNET+R. For this variational autoencoder, we use two fully-connected neural networks with layers of size 400, 400 for the encoder and 400, 400 for the decoder (both 1000, 1000 in the related work) and a latent space of dimension 100. Here, we departure from related work by choosing a smaller architecture for the autoencoder. Note that we still use a hypernetwork with less trainable parameters than the target network (in this case the decoder) that is used in related work.  
+- Details of the hypernetwork for the target classifier in PermutedMNIST-10 (HNET+TIR & HNET+ENT). We use the same setup for the hypernetwork as used for the VAEs above, but since the target network is smaller we reduce the output of the hypernetwork to 78,000. We also adjust the parameter  $\beta_{\mathrm{output}}$  to 0.01, consistent with our PermutedMNIST-100 experiments. The number of weights in this hypernetwork is therefore 2,029,931 parameters (2,029,691 network weights + 240 task embedding weights). The corresponding target network (from related work) would have 2,126,100 weights for CL1 and CL3 and 2,036,010 for CL2 (only one output head).  
+- Details of the hypernetwork for the target classifier for PermutedMNIST-100. For these experiments we chose an architecture that worked well on the PermutedMNIST-10 benchmark and did not conduct any more search for new architectures. For PermutedMNIST-100, the reported results were obtained by using a chunked hypernetwork with 3 hidden layers of size 200, 250 and 350 (300 for CL2) and an output size of 7500 (6000 for CL2) (such that we approximately match the corresponding target network size for CL1/CL2/CL3). Interestingly, Fig. A2b shows that even if we don't adjust the number of hypernetwork weights to the increased number of target network weights, the superiority of our method is evident. Aside from this, the plots in Fig. 3 have been generated using the PermutedMNIST-10 HNET+TIR setup (note that this includes the conditions set by related work for PermutedMNIST-10, e.g., target network sizes, the number of training iterations, learning rates, etc.).  
+- Details of the VAE and the hypernetwork for the VAE in PermutedMNIST-100 for CL2/CL3. We use a very similar setup for the VAE and it's hypernetwork used in HNET+TIR for PermutedMNIST-10 as described above. We only applied the following changes: Fully-connected hypernetwork with one hidden layer of size 100; chunk embedding sizes are set to 12; task embedding sizes are set two 128 and the hidden layer sizes of the VAE its generator are 400, 600. Also we increased the regularisation strength  $\beta_{\mathrm{output}} = 0.1$  for the VAE its generator hypernetwork.  
+- Details of the target classifier for HNET+TIR & HNET+ENT. For this classifier, we use the same setup as in the study we compare to (van de Ven & Tolias, 2019), i.e., a fully-connected network with layers of size 1000, 1000. Note that if the classifier is used as a task inference model, it is trained on replay data and the corresponding hard targets, i.e., the argmax of the soft targets.
+
+Below, we report the specifications for our automatic hyperparameter search (if not noted otherwise, these specifications apply for the split MNIST and split CIFAR experiments as well):
+
+- Hidden layer sizes of the hypernetwork: (no hidden layer), "5,5" "10,10", "25,25", "50,50", "100,100", "10", "50", "100"  
+- Output size of the hypernetwork: fitted such that we obtain less parameters than the target network which we compare against  
+- Embedding sizes (for e and c): 8, 12, 24, 36, 62, 96, 128  
+-  $\beta_{\mathrm{output}}$ : 0.0005, 0.001, 0.005, 0.01, 0.005, 0.1, 0.5, 1.0
+
+- Hypernetwork transfer functions: linear, ReLU, ELU, Leaky-ReLU
+
+Note that only a random subset of all possible combinations of hyperparameters has been explored.
+
+After we found a configuration with promising accuracies and a similar number of weights compared to the original target network, we manually fine-tuned the architecture to increase/decrease the number of hypernetwork weights to approximately match the number of target network weights.
+
+The choice of hypernetwork architecture seems to have a strong influence on the performance. It might be worth exploring alternatives, e.g., an architecture inspired by those used in typical generative models. We note that in addition to the above specifications we explored manually some hyperparameter configurations to gain a better understanding of our method.
+
+Split MNIST benchmark. Again, whenever applicable we reproduce the setup from van de Ven & Tolias (2019). Differences to the PermutedMNIST-10 experiments are just the learning rate (0.001) and the number of training iterations (set to 2000).
+
+- Details of hypernetwork for the VAE. We use one hypernetwork configuration to generate weights for all variational autoencoders used for our split MNIST experiments, namely a fully-connected chunked hypernetwork with 2 hidden layers of size 10, 10 followed by an output size of 50,000. We use ELU nonlinearities in the hidden layers of the hypernetwork. The size of task embeddings  $\mathbf{e}$  has been set to 96 and the size of chunk embeddings  $\mathbf{c}$  to 96. The parameter  $\beta_{\mathrm{output}}$  is 0.01 for HNET+R and 0.05 for HNET+TIR. The number of weights in this hypernetwork is 553,576 (553,192 network weights + 384 task embedding weights). The corresponding target network (and therefore output of the chunked hypernetwork), as taken from related work, has 555,184 weights. For a qualitative analyses of the replay data of this VAE (class incrementally learned), see A8.  
+- Details of the VAE for HNET+TIR. For this variational autoencoder, we use two fully-connected neural networks with layers of size 400, 400 for the encoder and 50, 150 for the decoder (both 400, 400 in the related work) and a latent space of dimension 100.  
+- Details of the VAE for HNET+R. For this variational autoencoder, we use two fully-connected neural networks with layers of size 400, 400 for the encoder and 250, 350 for the decoder (both 400, 400 in the related work) and a latent space of dimension 100.  
+- Details of the hypernetwork for the target classifier in split MNIST (HNET+TIR & HNET+ENT). We use the same setup for the hypernetwork as used for the VAE above, but since the target network is smaller we reduce the output of the hypernetwork to 42,000. We also adjust the  $\beta_{\mathrm{output}}$  to 0.01 although this parameter seems to not have a strong effect on the performance. The number of weights in this hypernetwork is therefore 465,672 parameters (465,192 network weights + 480 task embedding weights). The corresponding target network (from related work) would have 478,410 weights for CL1 and CL3 and 475,202 for CL2 (only one output head).  
+- Details of the target classifier for HNET+TIR & HNET+ENT. For this classifier, we again use the same setup as in the study we compare to (van de Ven & Tolias, 2019), i.e., a fully-connected neural networks with layers of size 400, 400. Note that if the classifier is used as a task inference model, it is trained on replay data and the corresponding hard targets, i.e., the argmax the soft targets.
+
+Split CIFAR-10/100 benchmark. For these experiments, we used as a target network a ResNet-32 network (He et al. (2016)) and again produce the weights of this target network by a hypernetwork in a compressive manner. The hypernetwork in this experiment directly maps from the joint task and chunk embedding space (both dimension 32) to the output space of the hypernetwork, which is of dimension 7,000. This hypernetwork has 457,336 parameters (457,144 network weights + 192 task embedding weights). The corresponding target network, the ResNet-32, has 468,540 weights (including batch-norm weights). We train for 200 epochs per task using the Adam optimizer with an initial learning rate of 0.001 (and otherwise default PyTorch values) and a batch size of 32. In addition, we apply the two learning rate schedules suggested in the Keras CIFAR-10 example<sup>2</sup>.
+
+Due to the use of batch normalization, we have to find an appropriate way to handle the running statistics which are estimated during training. Note, these are not parameters which are trained through backpropagation. There are different ways how the running statistics could be treated:
+
+1. One could ignore the running statistics altogether and simply compute statistics based on the current batch during evaluation.  
+2. The statistics could be part of the hypernetwork output. Therefore, one would have to manipulate the target hypernetwork output of the previous task, such that the estimated running statistics of the previous task will be distilled into the hypernetwork.  
+3. The running statistics can simply be checkpointed and stored after every task. Note, this method would lead to a linear memory growth in the number of tasks that scales with the number of units in the target network.
+
+For simplicity, we chose the last option and simply checkpointed the running statistics after every task.
+
+For the fine-tuning results in Fig. 5 we just continually updated the running statistics (thus, we applied no checkpointing).
+
+# D ADDITIONAL EXPERIMENTS AND NOTES
+
+Split CIFAR-10/100 benchmark using the model of Zenke et al. (2017). We re-run the split CIFAR-10/100 experiment reported on the main text while reproducing the setup from Zenke et al. (2017). Our overall classification performance is comparable to synaptic intelligence, which achieves  $73.85\%$  task-averaged test set accuracy, while our method reaches  $71.29\% \pm 0.32\%$ , with initial baseline performance being slightly worse in our approach, Fig. A3.
+
+![](images/b29acba1fb2eac4f433e6fd2a1eae238dcc1d9f5c55ec48d53a60079329f00c1.jpg)  
+Figure A3: Replication of the split CIFAR-10/100 experiment of Zenke et al. (2017). Test set accuracies on the entire CIFAR-10 dataset and subsequent CIFAR-100 splits. Both task-conditioned hypernetworks (hnet, in red) and synaptic intelligence (SI, in green) transfer information forward and are protected from catastrophic forgetting. The performance of the two methods is comparable. For completeness, we report our test set accuracies achieved immediately after training (hnet-during, in blue), when training from scratch (purple), and with our regularizer turned off (fine-tuning, yellow).
+
+To obtain our results, we use a hypernetwork with 3 hidden-layers of sizes 100, 150, 200 and output size 5500. The size of task embeddings  $\mathbf{e}$  has been set to 48 and the size of chunk embeddings  $\mathbf{c}$  to 80. The parameter  $\beta_{\mathrm{output}}$  is 0.01 and the learning rate is set to 0.0001.
+
+The number of weights in this hypernetwork is 1,182,678 (1,182,390 network weights + 288 task embedding weights). The corresponding target network would have 1,276,508 weights.
+
+In addition to the above specified hyperparameter search configuration we also included the following learning rates: 0.0001, 0.0005, 0.001 and manually tuned some architectural parameters.
+
+![](images/88143fe82b6fd0bc4916b66c3de05bed7b344d1946b4cbfc75ad45306f3e8b30.jpg)  
+a
+
+![](images/15e63ac803a46328547dac4a2633c9baf535453f79fdd75883c4dbdb8a26dd79.jpg)  
+b  
+Figure A4: Context-free inference using hypernetwork-protected replay (HNET+TIR) on long task sequences. Final test set classification accuracy on the  $t$ -th task after learning one hundred permutations of the MNIST dataset (PermutedMNIST-100) for the CL2 (a) and CL3 (b) scenarios, where task identity is not explicitly provided to the system. As before, the number of hypernetwork parameters is not larger than that of the related work we compare to. (a) HNET+TIR displays almost perfect memory retention. We used a stochastic regularizer (cf. Appendix D note below) which evaluates the output regularizer in Eq. 2 only for a random subset of previous tasks (here, twenty). (b) HNET+TIR is the only method that is capable of learning PermutedMNIST-100 in this learning scenario. For this benchmark, the input data domains are easily separable and the task inference system achieves virtually perfect ( $\sim 100\%$ ) task inference accuracy throughout, even for this long experiment. HNET+TIR uses a divide-and-conquer strategy: if task inference is done right, CL3 becomes just CL1. Furthermore, once task identity is predicted, the final softmax computation only needs to consider the corresponding task outputs in isolation (here, of size 10). Curiously, for HNET+TIR, CL2 can be harder than CL3 as the single output layer (of size 10, shared by all tasks) introduces a capacity bottleneck. The related methods, on the other hand, have to consider the entire output layer (here, of size  $10^{*}100$ ) at once, which is known to be harder to train sequentially. This leads to overwhelming error rates on long problems such as PermutedMNIST-100. Shaded areas in (a) and (b) denote STD ( $n = 5$ ).
+
+Upper bound for replay models. We obtain an upper bound for the replay-based experiments (Table 2) by sequentially training a classifier, in the same way as for HNET+R and DGR, now using true input data from past tasks and a synthetic, self-generated target. This corresponds to the rehearsal thought experiment delineated in Sect. 1.
+
+Table 2: Task-averaged test accuracy ( $\pm$  SEM,  $n = 20$ ) on the permuted ('P10') and split ('S') MNIST experiments. For HNET+R and DGR+distill (van de Ven & Tolias, 2019) the classification network is trained sequentially on data from the current task and replayed data from all previous tasks. Our HNET+R comes close to saturating the corresponding replay upper bound RPL-UB.
+
+<table><tr><td></td><td>DGR</td><td>HNET+R</td><td>RPL-UB</td></tr><tr><td>P10-CL1</td><td>97.51 ± 0.01</td><td>97.85 ± 0.02</td><td>97.89 ± 0.02</td></tr><tr><td>P10-CL2</td><td>97.35 ± 0.02</td><td>97.60 ± 0.02</td><td>97.72 ± 0.01</td></tr><tr><td>P10-CL3</td><td>96.38 ± 0.03</td><td>97.71 ± 0.06</td><td>97.91 ± 0.01</td></tr><tr><td>S-CL1</td><td>99.61 ± 0.02</td><td>99.81 ± 0.01</td><td>99.83 ± 0.01</td></tr><tr><td>S-CL2</td><td>96.83 ± 0.20</td><td>97.88 ± 0.05</td><td>98.96 ± 0.03</td></tr><tr><td>S-CL3</td><td>91.79 ± 0.32</td><td>94.97 ± 0.18</td><td>98.38 ± 0.02</td></tr></table>
+
+Quantification of forgetting in our continual learning experiments. In order to quantify forgetting of our approach, we compare test set accuracies of every single task directly after training with it's test set accuracy after training on all tasks.
+
+Only CL1 is shown since other scenarios i.e. CL2 and CL3 depend on task inference which only is measurable after training on all tasks.
+
+Table 3: Task-averaged test accuracy (± SEM,  $n = 20$ ) on the permutedMNIST-10 ('P10') and splitMNIST ('S') experiments during and after training.
+
+<table><tr><td></td><td>HNET+TIR during</td><td>HNET+TIR after</td><td>HNET+R during</td><td>HNET+R after</td></tr><tr><td>S-CL1</td><td>99.79 ± 0.01</td><td>99.79 ± 0.01</td><td>99.82 ± 0.01</td><td>99.83 ± 0.01</td></tr><tr><td>P10-CL1</td><td>97.58 ± 0.02</td><td>97.57 ± 0.02</td><td>98.03 ± 0.01</td><td>97.87 ± 0.01</td></tr></table>
+
+Table 4: Task-averaged test accuracy (± SEM,  $n = 5$ ) on the permutedMNIST-100 ('P100') experiments during and after training.  
+
+<table><tr><td></td><td>HNET+TIR during</td><td>HNET+TIR after</td></tr><tr><td>P100-CL1</td><td>96.12 ± 0.08</td><td>96.18 ± 0.09</td></tr><tr><td>P100-CL2</td><td>-</td><td>95.97 ± 0.05</td></tr><tr><td>P100-CL3</td><td>-</td><td>96.00 ± 0.03</td></tr></table>
+
+Table 5: Task-averaged test accuracy (± SEM,  $n = 5$ ) on split CIFAR-10/100 on CL1 on two different target network architectures.  
+
+<table><tr><td></td><td>during</td><td>after</td></tr><tr><td>ZenkeNet</td><td>74.75 ± 0.09</td><td>71.29 ± 0.32</td></tr><tr><td>ResNet-32</td><td>82.36 ± 0.44</td><td>82.34 ± 0.44</td></tr></table>
+
+Robustness of  $\beta_{\mathrm{output}}$ -choice. In Fig. A2a and Fig. A2c we provide additional experiments for our method on PermutedMNIST-100. We show that our method performs comparable for a wide range of  $\beta_{\mathrm{output}}$ -values (including the one depicted in Fig. 3a).
+
+![](images/01ad2e8b3edc94135877ee0e1a900a3688e25c1be87a0878f2e3120a2d33f7e1.jpg)  
+a
+
+![](images/13f44cdc06a6eca4e0b8d7b012ca21cfdb0ee397effb2afc35b4872bfafc2f9e.jpg)  
+b  
+d
+
+![](images/69ad89d60136c76f18d1939ab96be7b2bb94e2a8292087f33ded773a033b611b.jpg)  
+c
+
+![](images/0da1529c46d0dae1971675fcb6edc206d2c1cb9aef98959d452ad292364fb53d.jpg)  
+Figure A5: Additional experiments with online EWC and fine-tuning on the PermutedMNIST-100 benchmark. (a) Final test set classification accuracy on the  $t$ -th task after learning one hundred permutations (PermutedMNIST-100) using the online EWC algorithm (Schwarz et al., 2018) to prevent forgetting. All runs use exactly the same hyperparameter configuration except for varying values of the regularization strength  $\lambda$ . Our method (hnet, in red) and the online EWC run ( $\lambda = 100$ , in orange) from Fig. 3a are shown for comparison. It can be seen that even when tuning the regularization strength one cannot attain similar performance as with our approach (cmp. Fig. A2a). Too strong regularization prevents the learning of new tasks whereas too weak regularization doesn't prevent forgetting. However, a middle ground (e.g., using  $\lambda = 100$ ) does not reach acceptable per-task performances. (b) Task-averaged test set accuracy after learning all tasks (labelled 'final', in red) and immediately after learning a task (labelled 'during', in purple) for a range of regularization strengths  $\lambda$  when using the online EWC algorithm. Results are complementary to those shown in (a). (c) Final test set classification accuracy on the  $t$ -th task after learning one hundred permutations (PermutedMNIST-100) when applying fine-tuning to the hypernetwork (labelled 'hnet fine-tuning', in blue) or target network (labelled 'fine-tuning', in green). Our method (hnet, in red) from Fig. 3a is shown for comparison. It can be seen that without protection the hypernetwork suffers much more severely from catastrophic forgetting as when training a target network only. (d) This plot is complementary to (c). See description of (b) for an explanation of the labels. Shaded areas in (a) and (c) denote STD, whereas error bars in (b) and (d) denote SEM (always across 5 random seeds).
+
+![](images/cce3cb3050fa5bce496b01a9a5441fd9443765a9f691c40a78d2a2d8e0b0dc19.jpg)  
+a
+
+![](images/9aee16a47d4ff3cc9b4d44c454cd66da97f82ca5ca06e6add444cfb92db3ca8a.jpg)  
+b  
+Figure A6: Hyperparameter search for online EWC and SI on the PermutedMNIST-100 benchmark. We conduct the same hyperparameter search as performed in van de Ven & Tolias (2018). We did not compute different random seeds for this search. (a) Hyperparameter search on the regularisation strength  $c$  for the SI algorithm. Accuracies during and after the experiment are shown. (b) Hyperparameter search for parameters  $\lambda$  and  $\gamma$  of the online EWC algorithm. Only accuracies after the experiment are shown.
+
+Varying the regularization strength for online EWC. The performance of online EWC in Fig. 3a is closest to our method (labelled hnet, in red) compared to the other methods. Therefore, we take a closer look at this method and show that further adjustments of the regularization strength  $\lambda$  do not lead to better performance. Results for a wide range of regularization strengths can be seen in Fig. A5a and Fig. A5b. As shown, online EWC cannot attain a performance comparable to our method when tuning the regularization strength only.
+
+The impact of catastrophic forgetting on the hypernetwork and target network. We have successfully shown that by shifting the continual learning problem from the target network to the hypernetwork we can successfully overcome forgetting due to the introduction of our regularizer in Eq. 2. We motivated this success by claiming that it is an inherently simpler task to remember a few input-output mappings in the hypernetwork (namely the weight realizations of each task) rather than the massive number of input-output mappings  $\{(\mathbf{x}^{(t,i)},\mathbf{y}^{(t,i)})\}_{i = 1}^{n_t}$  associated with the remembering of each task  $t$  by the target network.
+
+Further evidence of this claim is provided by fine-tuning experiments in Fig. A5c and Fig. A5d. Fine-tuning refers to sequentially learning a neural network on a set of tasks without any mechanism in place to prevent forgetting. It is shown that fine-tuning a target network (no hypernetwork in this setup) has no catastrophic influence on the performance of previous tasks. Instead there is a graceful decline in performance. On the contrary, catastrophic forgetting has an almost immediate affect when training a hypernetwork without protection (i.e., training our method with  $\beta_{\mathrm{output}} = 0$ . The performance quickly drops to chance level, suggesting that if we weren't solving a simpler task then preventing forgetting in the hypernetwork rather than in the target network might not be beneficial.
+
+Chunking and hypernetwork architecture sensitivity. In this note we investigate the performance sensitivity for different (fully-connected) hypernetwork architectures on split MNIST and PermutedMNIST-10, Fig. A7. We trained thousands of randomly drawn architectures from the following grid (the same training hyperparameters as reported for for CL1, see Appendix C, were used throughout): possible number of hidden layers 1,2, possible layer size 5,10,20,...,90,100, possible chunk embedding size 8,12,24,56,96 and hypernetwork output size in  $\{10,50,100,200,300,400,500,750,1k,2k,\ldots ,9k,10k,20k,30k,40k\}$ . Since we realize compression through chunking, we sort our hypernetwork architectures by compression ratio, and consider only architectures with small compression ratios.
+
+Performance of split MNIST stays in the high 90 percentages even when reaching compression ratios close to  $1\%$  whereas for PermutedMNIST-10 accuracies decline in a non-linear fashion. For both experiments, the choice of the chunked hypernetwork architecture is robust and high performing even in
+
+![](images/aac6c116638dacefc928bd0621948642707dbf2dc84a601b78dad9228be3795e.jpg)  
+a
+
+![](images/00a9caf44e91c597bc22a6b2b13ed7504a69844dfc8df562a9a4d1368d962125.jpg)  
+b  
+Figure A7: Robustness to hypernetwork architecture choice for a large range of compression ratios. Performance vs. compression for random hypernetwork architecture choices, for split MNIST and PermutedMNIST-10 (mean  $\pm$  STD,  $n = 500$  architectures per bin). Every model was trained with the same setup (including all hyperparameters) used to obtain results reported in Table 1 (CL1). We considered architectures yielding compression ratios  $|\Theta_{\mathrm{h}} \cup \{\mathbf{e}^{(t)}\}| / |\Theta_{\mathrm{trgt}}| \in [0.01, 2.0]$  (a) split MNIST performance for CL1 stays high even for compression ratios  $\approx 1\%$ . (b) PermutedMNIST-10 accuracies degrade gracefully when compression ratios decline to  $1\%$ . Notably, for both benchmarks, performance remained stable across a large pool of hypernetwork configurations.
+
+the compressive regime. Note that the discussed compression ratio compares the amount of trainable parameters in the hypernetwork to its output size, i.e. the parameters of the target network.
+
+Small capacity target networks for the permuted MNIST benchmark. Swaroop et al. (2018) argue for using only small capacity target networks for this benchmark. Specifically, they propose to use hidden layer sizes [100, 100]. Again, we replicated the setup of van de Ven & Tolias (2019) wherever applicable, except for the now smaller hidden layer sizes of [100, 100] in the target network. We use a fully-connected chunked hypernetwork with chunk embeddings c having size 12, hidden layers having size 100, 75, 50 and an output size of 2000, resulting in a total number of hypernetwork weights of 122,459 (including  $10 \times 64$  task embedding weights) compared to 122,700 weights that are generated for the target network.  $\beta_{\mathrm{output}}$  is set to 0.05. The experiments performed here correspond to CL1.
+
+We achieve an average accuracy of  $93.91 \pm 0.04$  for PermutedMNIST-10 after having trained on all tasks. In general, we saw that the hypernetwork training can benefit from noise injection. For instance, when training with soft-targets (i.e., we modified the 1-hot target to be 0.95 for the correct class and  $\frac{1 - 0.95}{\# \text{classes} - 1}$  for the remaining classes), we could improve the average accuracy to  $94.24 \pm 0.03$ .
+
+We also checked the challenging PermutedMNIST-50 benchmark with this small target network as previously investigated by Ritter et al. (2018). Therefore, we slightly adapted the above setup by using a hypernetwork with hidden layer sizes [100, 100] and a regularization strength of  $\beta_{\mathrm{output}} = 0.1$ . This hypernetwork is slightly bigger than the corresponding target network  $\frac{|\Theta_{\mathrm{h}}| \cup \{\mathbf{e}^{(t)}\}|}{|\Theta_{\mathrm{trgt}}|} = 1.37$ . With this configuration, we obtain an average accuracy of  $90.91 \pm 0.07$ .
+
+Comparison to HAT. Serra et al. (2018) proposed the hard attention to the task (HAT) algorithm, a strong CL1 method which relies on learning a per-task, per-neuron mask. Since the masks are pushed to become binary, HAT can be viewed as an algorithm for allocating subnetworks (or modules) within the target network, which become specialized to solve a given task. Thus, the method is similar to ours in the sense that the computation of the target network is task-dependent, but different in spirit, as it relies on network modularity.
+
+In HAT, task identity is assumed to be provided, so that the appropriate mask can be picked during inference (scenario CL1). HAT requires explicitly storing a neural mask for each task, whose size
+
+scales with the number of neurons in the target network. In contrast, our method allows solving tasks in a compressive regime. Thanks to the hypernetwork, whose input dimension can be freely chosen, only a low-dimensional embedding needs to be stored per task (cf. Fig. 4), and through chunking it is possible to learn to parameterize large target models with a small number of plastic weights (cf. Fig. 3b).
+
+Here, we compare our task-conditioned hypernetworks to HAT on the permuted MNIST benchmarks  $(T = 10$  and  $T = 100)$ , cf. Table 6. For large target networks, both methods perform strongly, reaching comparable final task-averaged accuracies. For small target network sizes, task-conditioned hypernetworks perform better, the difference becoming more apparent on PermutedMNIST-100.
+
+We note that the two algorithms use different training setups. In particular, HAT uses 200 epochs (batch size set to 64) and applies a learning rate scheduler that acts on a held out validation set. Furthermore, HAT uses differently tuned forgetting hyperparameters when target network sizes change. This is important to control for the target network capacity used per task and assumes knowledge of the (number of) tasks at hand. Using the code freely made available by the authors, we were able to rerun HAT for our target network size and longer task sequences. Here, we used the setup provided by the author's code for HAT-Large for PermutedMNIST-10 and PermutedMNIST-100. To draw a fairer comparison, when changing our usual target network size to match the ones reported in Serra et al. (2018), we trained for 50 epochs per task (no training loss improvements afterwards observed) and also changed the batch size to 64 but did not changed our training scheme otherwise; in particular, we did not use a learning rate scheduler.
+
+Table 6: Comparison of HNET and HAT, Serra et al. (2018). Task-averaged test accuracy on the PermutedMNIST experiment with  $T = 10$  and  $T = 100$  tasks ('P10', 'P100') with three different target network sizes, i.e., three fully connected neural networks with hidden layer sizes of (100, 100) or (500, 500) or (2000, 2000) are shown. For these architectures, a single accuracy was reported by Serra et al. (2018) without statistics provided. We reran HAT for PermutedMNIST-100 with code provided at https://github.com/joansj/hat, and for PermutedMNIST-10 with hidden layer size (1000, 1000) to match our setup. HAT and HNET perform similarly on large target networks for PermutedMNIST-10, while HNET is able to achieve larger performances with smaller target networks as well as for long task sequences.
+
+<table><tr><td></td><td>HAT</td><td>HNET</td></tr><tr><td>P10-100, 100</td><td>91.6</td><td>95.92 ± 0.02</td></tr><tr><td>P10-500, 500</td><td>97.4</td><td>97.35 ± 0.02</td></tr><tr><td>P10-2000, 2000</td><td>98.6</td><td>98.06 ± 0.02</td></tr><tr><td>P10-1000, 1000</td><td>97.67 ± 0.02</td><td>97.56 ± 0.02</td></tr><tr><td>P100-1000, 1000</td><td>86.04 ± 0.26</td><td>94.98 ± 0.07</td></tr></table>
+
+Efficient PermutedMNIST-250 experiments with a stochastic regularizer on subsets of previous tasks. An apparent drawback of Eq. 2 is that the runtime complexity of the regularizer grows linearly with the number of tasks. To overcome this obstacle, we show here that it is sufficient to consider a small random subset of previous tasks.
+
+In particular, we consider the PermutedMNIST-250 benchmark (250 tasks) on CL1 using the hyperparameter setup from our PermutedMNIST-100 experiments except for a hypernetwork output size of 12000 (to adjust to the bigger multi-head target network) and a regularization strength  $\beta_{\mathrm{output}} = 0.1$ . Per training iteration, we choose maximally 32 random previous tasks to estimate the regularizer from Eq. 2. With this setup, we achieve a final average accuracy of  $94.19 \pm 0.16$  (compared to an average during accuracy (i.e., the accuracies achieved right after training on the corresponding task) of  $95.54 \pm 0.05$ ). All results are across 5 random seeds. These results indicate that a full evaluation of the regularizer at every training iteration is not necessary such that the linear runtime complexity can be cropped to a constant one.
+
+Combining hypernetwork output regularizers with weight importance. Our hypernetwork regularizer pulls uniformly in every direction, but it is possible to introduce anisotropy using an EWC-like approach (Kirkpatrick et al., 2017). Instead of weighting parameters, hypernetwork outputs can be weighted. This would allow for a more flexible regularizer, at the expense of additional storage.
+
+Task inference through predictive entropy (HNET+ENT). In this setup, we rely on the capability of neural networks to separate in- from out-of-distribution data. Although this is a difficult research problem on its own, for continual learning, we face a potentially simpler problem, namely to detect and distinguish between the tasks our network was trained on. We here take the first minimal step exploiting this insight and compare the predictive uncertainty, as quantified by output distribution entropy, of the different models given an input. Hence, during test time we iterate over all embeddings and therefore the models our metamodel can generate and compare the predictive entropies which results in making a prediction with the model of lowest entropy. For future work, we wish to explore the possibility of improving our predictive uncertainty by taking parameter uncertainty into account through the generation of approximate, task-specific weight posterior distributions.
+
+Learning without task boundaries with hypernetworks. An interesting problem we did not address in this paper is that of learning without task boundaries. For most CL methods, it is crucial to know when learning one task ends and training of a new tasks begins. This is no exception for the methods introduced in this paper. However, this is not necessarily a realistic or desirable assumption; often, one desires to learn in an online fashion without task boundary supervision, which is particularly relevant for reinforcement learning scenarios where incoming data distributions are frequently subject to change (Rolnick et al., 2018). At least for discrete changes, with our hypernetwork setup, this boils down to a detection mechanism that activates the saving of the current model, i.e., the embedding  $\mathbf{e}^{(T)}$ , and its storage to the collection of embeddings  $\{\mathbf{e}^{(t)}\}$ . We leave the integration of our model with such a hypernetwork-specific switching detection mechanism for future work. Interestingly, our task-conditioned hypernetworks would fit very well with methods that rely on fast remembering (a recently proposed approach which appeared in parallel to our paper, He et al., 2019).
+
+# E UNIVERSAL FUNCTION APPROXIMATION WITH CHUNKED NEURAL NETWORKS
+
+Proposition 1. Given a compact subset  $K \subset \mathbb{R}^m$  and a continuous function on  $K$  i.e.  $f \in C(K)$ , more specifically,  $f: K \to \mathbb{R}^n$  with  $n = r \cdot N_C$ . Now  $\forall \epsilon > 0$ , there exists a chunked neural network  $f_{\mathrm{h}}^{\mathbf{c}}: \mathbb{R}^m \times \mathcal{C} \to \mathbb{R}^r$  with parameters  $\Theta_{\mathrm{h}}$ , discrete set  $\mathcal{C} = \{\mathbf{c}_1, \dots, \mathbf{c}_{N_C}\}$  and  $\mathbf{c}_i \in \mathbb{R}^s$  such that  $|\tilde{f}_{\mathrm{h}}^{\mathbf{c}}(\mathbf{x}) - f(\mathbf{x})| < \epsilon$ ,  $\forall \mathbf{x} \in K$  and with  $\tilde{f}_{\mathrm{h}}^{\mathbf{c}}(\mathbf{x}) = [f_{\mathrm{h}}^{\mathbf{c}}(\mathbf{x}, \mathbf{c}_1), \dots, f_{\mathrm{h}}^{\mathbf{c}}(\mathbf{x}, \mathbf{c}_{N_C})]$ .
+
+For the following proof, we assume the existence of one form of the universal approximation theorem (UAT) for neural networks (Leshno & Schocken, 1993; Hanin, 2017). Note that we will not restrict ourselves to a specific architecture, nonlinearity, input or output dimension. Any neural network that is proven to be a universal function approximator is sufficient.
+
+Proof. Given any  $\epsilon > 0$ , we assume the existence of a neural network  $f_{\mathrm{h}}: \mathbb{R}^{m} \to \mathbb{R}^{n}$  that approximates function  $f$  on  $K$ :
+
+$$
+\left| f _ {\mathrm {h}} (\mathbf {x}) - f (\mathbf {x}) \right| <   \frac {\epsilon}{2}, \quad \forall x \in K. \tag {10}
+$$
+
+We will in the following show that we can always find a chunked neural network  $f_{\mathrm{h}}^{\mathbf{c}}: \mathbb{R}^{m} \times \mathcal{C} \rightarrow \mathbb{R}^{r}$  approximating the neural network  $f_{\mathrm{h}}$  on  $K$  and conclude with the triangle inequality
+
+$$
+\left| \tilde {f} _ {\mathrm {h}} ^ {\mathrm {c}} (\mathbf {x}) - f (\mathbf {x}) \right| \leq \left| \tilde {f} _ {\mathrm {h}} ^ {\mathrm {c}} (\mathbf {x}) - f _ {\mathrm {h}} (\mathbf {x}) \right| + \left| f _ {\mathrm {h}} (\mathbf {x}) - f (\mathbf {x}) \right| <   \epsilon , \quad \forall x \in K. \tag {11}
+$$
+
+Indeed, given the neural network  $f_{\mathrm{h}}$  such that (10) holds true, we construct
+
+$$
+\hat {f} _ {\mathrm {h}} (\mathbf {x}, \mathbf {c}) = \left\{ \begin{array}{l l} f _ {\mathrm {h}} ^ {\mathbf {c} _ {i}} (\mathbf {x}) & \mathbf {c} = \mathbf {c} _ {i} \\ 0 & \text {e l s e} \end{array} \right. \tag {12}
+$$
+
+by splitting the full neural network  $f_{\mathrm{h}}(\mathbf{x}) = [f_{\mathrm{h}}^{\mathbf{c}_1}(\mathbf{x}),f_{\mathrm{h}}^{\mathbf{c}_2}(\mathbf{x}),\dots ,f_{\mathrm{h}}^{\mathbf{c}_{N_C}}(\mathbf{x})]$  with  $\hat{f}_{\mathrm{h}}:\mathbb{R}^{m}\times \mathcal{C}\to \mathbb{R}^{r}$ .
+
+Note that  $\hat{f}_{\mathrm{h}}$  is continuous on  $\mathbb{R}^m\times \mathcal{C}$  with the product topology composed of the topology on  $\mathbb{R}^m$  induced by the metric  $|\cdot -\cdot |:\mathbb{R}^m\times \mathbb{R}^m\to \mathbb{R}$  and the discrete topology on  $\mathcal{C}$ . Now we can make use
+
+of the UAT again: Given the compact  $K\subset \mathbb{R}^n$ , the discrete set  $\mathcal{C} = \{\mathbf{c}_1,\dots ,\mathbf{c}_{N_{\mathrm{c}}}\}$  and any  $\frac{\epsilon}{2N_{\mathrm{c}}} >0$  there exists a neural network function  $f_{\mathrm{h}}^{\mathbf{c}}:\mathbb{R}^{m}\times \mathbb{R}^{s}\to \mathbb{R}^{r}$  such that
+
+$$
+\left| f _ {\mathrm {h}} ^ {\mathbf {c}} (\mathbf {x}, \mathbf {c}) - \hat {f} _ {\mathrm {h}} (\mathbf {x}, \mathbf {c}) \right| <   \frac {\epsilon}{2 N _ {\mathrm {C}}}, \quad \forall \mathbf {x} \in K, \forall \mathbf {c} \in \mathcal {C}. \tag {13}
+$$
+
+It follows that
+
+$$
+\sum_ {i} \left| f _ {\mathrm {h}} ^ {\mathbf {c}} (\mathbf {x}, \mathbf {c} _ {i}) - \hat {f} _ {\mathrm {h}} (\mathbf {x}, \mathbf {c} _ {i}) \right| <   \sum_ {i} \frac {\epsilon}{2 N _ {\mathrm {C}}} = \frac {\epsilon}{2}, \quad \forall \mathbf {x} \in K, \tag {14}
+$$
+
+which is equivalent to
+
+$$
+\left| \left[ \begin{array}{c} f _ {\mathrm {h}} ^ {\mathbf {c}} (\mathbf {x}, \mathbf {c} _ {1}) \\ \vdots \\ f _ {\mathrm {h}} ^ {c} (\mathbf {x}, \mathbf {c} _ {\mathbf {N} _ {\mathrm {C}}}) \end{array} \right] - \left[ \begin{array}{c} \hat {f} _ {\mathrm {h}} (\mathbf {x}, \mathbf {c} _ {1}) \\ \vdots \\ \hat {f} _ {\mathrm {h}} (\mathbf {x}, \mathbf {c} _ {\mathbf {N} _ {\mathrm {C}}}) \end{array} \right] \right| = \left| \bar {f} _ {\mathrm {h}} ^ {\mathbf {c}} (\mathbf {x}) - f _ {\mathrm {h}} (\mathbf {x}) \right| <   \frac {\epsilon}{2}, \quad \forall \mathbf {x} \in K. \tag {15}
+$$
+
+We have shown (11) which concludes the proof.
+
+![](images/a71e7f9ae2974bfec015f46ce1ebe38c732378c214835d2395d3900fad30030a.jpg)
+
+Note that we did not specify the number of chunks  $N_{\mathrm{C}}$ ,  $r$  or the dimension  $s$  of the embeddings  $\mathbf{c}_i$ . Despite this theoretical result, we emphasize that we are not aware of a constructive procedure to define a chunked hypernetwork that comes with a useful bound on the achievable performance and/or compression rate. We evaluate such aspects empirically in our experimental section.
+
+# F QUALITATIVE ANALYSES OF HYPERNETWORK-PROTECTED REPLAY MODELS
+
+![](images/1d247c0524bde5ae557879472f989ca811a44d3171ffb243a1d9f1164a30c212.jpg)  
+a  
+Figure A8: Image samples from hypernetwork-protected replay models. The left column of both of the subfigures display images directly after training the replay model on the corresponding class, compared to the right column(s) where samples are obtained after training on eights and nines i.e. all classes. (a) Image samples from a class-incrementally trained VAE. Here the exact same training configuration to obtain results for split MNIST with the HNET+R setup are used, see Appendix C. (b) Image samples from a class-incrementally trained GAN. For the training configurations, see Appendix B. In both cases the weights of the generative part, i.e., the decoder or the generator, are produced and protected by a hypernetwork.
+
+![](images/77cfba879f676b9dd58033d6304ee0f58b7fe3fe7a018e923f277d466be9abe5.jpg)  
+b
\ No newline at end of file
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+# COPHY: COUNTERFACTUAL LEARNING OF PHYSICAL DYNAMICS
+
+Fabien Baradel<sup>1</sup> Natalia Neverova<sup>2</sup> Julien Mille<sup>3</sup> Greg Mori<sup>4</sup> Christian Wolf<sup>1,5</sup>
+
+<sup>1</sup>Université Lyon, INSA Lyon, CNRS, LIRIS, Villeurbanne, France  
+$^{2}$ Facebook AI Research, Paris, France  
+$^{3}$ Laboratoire d'Informatique de l'Univ. de Tours, INSA Centre Val de Loire, Blois, France  
+<sup>4</sup>Simon Fraser University and Borealis AI, Vancouver, Canada  
+<sup>5</sup>Inria, Chroma group, CITI Laboratory, Villeurbanne, France
+
+# ABSTRACT
+
+Understanding causes and effects in mechanical systems is an essential component of reasoning in the physical world. This work poses a new problem of counterfactual learning of object mechanics from visual input. We develop the CoPhy benchmark to assess the capacity of the state-of-the-art models for causal physical reasoning in a synthetic 3D environment and propose a model for learning the physical dynamics in a counterfactual setting. Having observed a mechanical experiment that involves, for example, a falling tower of blocks, a set of bouncing balls or colliding objects, we learn to predict how its outcome is affected by an arbitrary intervention on its initial conditions, such as displacing one of the objects in the scene. The alternative future is predicted given the altered past and a latent representation of the confounders learned by the model in an end-to-end fashion with no supervision of confounders. We compare against feedforward video prediction baselines and show how observing alternative experiences allows the network to capture latent physical properties of the environment, which results in significantly more accurate predictions at the level of super human performance.
+
+# 1 INTRODUCTION
+
+Reasoning is an essential ability of intelligent agents that enables them to understand complex relationships between observations, detect affordances, interpret knowledge and beliefs, and to leverage this understanding to anticipate future events and act accordingly. The capacity for observational discovery of causal effects in physical reality and making sense of fundamental physical concepts, such as mass, velocity, friction, etc., may be one of differentiating properties of human intelligence that ensures our ability to robustly generalize to new scenarios (Martin-Ordas et al., 2008).
+
+One way to express causality is based on the concept of counterfactual reasoning, that deals with a problem containing an if statement, which is untrue or unrealized. Predicting the effect of the interventions based on the given observations without explicitly observing the effect of the intervention on data is a hard task and requires modeling of the causal relationships between the variable on which the intervention is performed and the variable whose alternative future should be predicted (Balke & Pearl, 1994). Using counterfactuals has been shown to be a way to perform reasoning over causal relationships between the variables of low dimensional spaces and has been an unexplored direction for high dimensional signals such as videos.
+
+In this work, we develop the Counterfactual Physics benchmark (CoPhy) and propose a framework for causal learning of dynamics in mechanical systems with multiple degrees of freedom, as illustrated in Fig. 1. For a number of scenarios, such as tower of blocks falling, balls bouncing against walls or objects colliding, we are given the starting frame  $\mathbf{A} = X_0$  and a sequence of following frames  $\mathbf{B} = X_{1:\tau}$ , where  $\tau$  covers the range of 6 sec. The observed sequences  $\mathbf{B}$ , conditioned on the initial state  $\mathbf{A}$ , are direct effects of the physical principles (such as inertia, gravity or friction) applied to the closed system, that cause the objects change their positions and 3D poses over time.
+
+![](images/935c1c851b14e37352c90db9b87f40c776b431315548f30f1d5872740195e740.jpg)  
+BallsCF
+
+![](images/b3457eed088d2efd44b36c0f300fb6caeebc714f73f7c4c0aeb51c6eda62ecb9.jpg)  
+CollisionCF
+
+![](images/f3e7e28e32ae424e0db294e903f76dd1de7ad551adcfc1fb816edc654f324bdd.jpg)  
+Figure 1: We train a model for performing counterfactual learning of physical dynamics. Given an observed frame  $\mathbf{A} = X_0$  and a sequence of future frames  $\mathbf{B} = X_{1:\tau}$ , we ask how the outcome  $\mathbf{B}$  would have changed if we changed  $X_0$  to  $\bar{X}_0$  by performing a do-intervention (e.g. changing the initial positions of objects in the scene).
+
+![](images/7247a6b8e72c34778040c5f48d8a9f28dd054f511c9e0dc71a00e30724f2881f.jpg)  
+BlocktowerCF
+
+![](images/601611bc860a29dbdcf9d6cccd6d21b9ca82d2fb70872b0a84565ab5f9f1aa8d.jpg)
+
+The task is formulated as follows: having observed the tuple  $(\mathbf{A},\mathbf{B})$ , we wish to predict positions and poses of all objects in the scene at time  $t = \tau$ , if we had changed the initial frame  $X_0$  by performing an intervention. The intervention is formalized by the do-operator introduced by Pearl et al. (Pearl, 2009; Pearl & McKenzie, 2018) for dealing with causal inference (Spirtes, 2010). In our case, it implies modification of the variable  $\mathbf{A}$  to  $\mathbf{C}$ , defined as  $\mathbf{C} = \mathbf{do}(X_0 = \bar{X}_0)$ . Accordingly, for each experiment in the CoPhy benchmark, we provide pairs of original sequences  $X_{0:\tau}$  and their modified counterparts  $\bar{X}_{0:\tau}$  sharing the same values of all confounders.
+
+We note the fundamental difference between this problem of counterfactual future forecasting and the conventional setup of feedforward future forecasting, like video prediction (Mathieu et al., 2016). The latter involves learning spatio-temporal regularities and thereby predicting future frames  $X_{1:\tau}$  from one or several past frame(s)  $X_0$  (the causal chain of this problem is shown in Fig. 2a). On the other hand, counterfactual forecasting benefits from additional observations in the form of the original outcome  $X_{1:\tau}$  before the do-operator. This adds a confounder variable  $U$  into the causal chain (Fig. 2b), which provides information not observable in frame  $X_0$ . For instance, in the case of the CoPhy benchmark, observing the pair  $(\mathbf{A},\mathbf{B})$  might give us information on the masses, velocities or friction coefficients of the objects in the scene, which otherwise cannot be inferred from frame  $\bar{X}_0$  alone. Therefore, predicting the alternative outcome after performing counterfactual intervention then involves using the estimate of the confounder  $U$  together with the modified past  $\mathbf{d}\mathbf{o}(X_0 = \bar{X}_0)$ .
+
+Overall, we employ the idea of counterfactual intervention in predictive models and argue that counterfactual reasoning is an important step towards human-like reasoning and general intelligence. More specifically, key contributions of this work include:
+
+- a new task of counterfactual prediction of physical dynamics from high-dimensional visual input, as a way to access capacity of intelligent agents for causal discovery;  
+- a large-scale CoPhy benchmark with three physical scenarios and 300k synthetic experiments including rendered sequences of frames, metadata (object positions, angles, sizes) and values of confounders (masses, frictions, gravity). This benchmark was specifically designed in bias-free fashion to make the counter-factual reasoning task challenging by optimizing the impact of the confounders on the outcome of the experiment. The dataset is publicly available<sup>1</sup>.  
+- a counterfactual neural model predicting an alternative outcome of a physical experiment given an intervention, by estimating the latent representation of the confounders. The model outperforms state-of-the-art solutions implementing feedforward video prediction, successfully generalizes to unseen initial states and does not require supervision on the confounders. We provide extensive ablations on the effects of key design choices and compare results with human performance, that show that the task is hard for humans to solve. The code will be made publicly available.
+
+# 2 RELATED WORK
+
+This work is inspired by a significant number of prior studies from several subfields, including visual reasoning, learning intuitive physics and perceived causality.
+
+![](images/c943d50a1c3d51e98caf5536efcf13b58e198758ba433b3db7363633d0572a94.jpg)  
+(a) feedforward future forecasting
+
+![](images/f29623a099acfe72c328f2adeb1e405af47f953a941047f7e24c70916c5c7d61.jpg)  
+Figure 2: The difference between conventional video prediction (a) and counterfactual video prediction (b). The causal graph of the latter includes a confounder variable, which passes information from the original outcome to the outcome after do-intervention. The initially observed sequence  $(\mathbf{A},\mathbf{B})$  (on the left) and the counterfactual sequence after the do-intervention (on the right).
+
+![](images/ccbdcae08dced96ecbed7f13e378fb3ffef7c8b97734385c3704f1e8e4efd56e.jpg)  
+(b) counterfactual future forecasting
+
+Visual reasoning. The most recent works on visual reasoning approach the problem in the setting of visual question answering (Hu et al., 2017; Hudson & Manning, 2018; Johnson et al., 2017; Mao et al., 2019; Perez et al., 2018; Santoro et al., 2017), embodied AI (Wijmans et al., 2019), as well as learning intuitive physics (Lerer et al., 2016; Riochet et al., 2018). (Santoro et al., 2017) introduced Relation Networks (RN), a fully-differentiable trainable layer for reasoning in deep networks. Following the same trend, (Baradel et al., 2018) estimate object relations from semantically well defined entities using instance segmentation predictions for video understanding. (Santoro et al., 2018) build a challenging dataset for solving the problem of abstract reasoning on the visual domain with some tasks such as interpolation or extrapolation. External memory (Graves et al., 2016; Jaeger, 2016) extends known recurrent neural mechanisms by decoupling the size of a representation from the controller capacity and introduces the separation between long-term and short-term reasoning. (Reed et al., 2015) propose to learn analogies in a fully supervised way. Our work builds upon this literature and extends the idea of visual reasoning to the counterfactual setting.
+
+Intuitive physics. Fundamental studies on cognitive psychology have shown that humans perform poorly when asked to reason about expected outcomes of a physics based event, demonstrating striking deviations from Newtonian physics in their intuitions (McCloskey & Kohl, 1983; McClooskey et al., 1980; 1983; Kubricht et al., 2017). The questions of approximating these mechanisms, learning from noisy observed and non-observed physical quantities (such as sizes or velocities vs masses or gravity), as well as justifying importance of explicit physical concepts vs cognitive constructs in intelligent agents have been raised and explored in recent works on deep learning (Wu et al., 2015). (Lerer et al., 2016; Groth et al., 2018) follow this direction by training networks to predict stability of block towers. (Ye et al., 2018) build an interpretable intuitive physical model from visual signals using full supervision on the physical properties of each object. On similar tasks, (Wu et al., 2017) propose to learn physics by interpreting and reconstructing the visual information stream leading to inverting physics or a graphical engine. (Zheng et al., 2018) propose to solve this task by first extracting a visual perception of the world state and then predict the future. (Battaglia et al., 2016) introduce a fully-differentiable network physics engine called Interaction Network (IN), which learns to predict physical systems such as gravitational ones, rigid body dynamics, and mass-spring systems. Similarly, (van Steenkiste et al., 2018) discover objects and their interactions in a unsupervised manner from a virtual environment. In (Velicovic et al., 2018), attention and relational modules are combined on a graph structure. Recent approaches (Chang et al., 2017; Janner et al., 2019; Battaglia et al., 2018) based on Graph Convolution Networks (Kipf & Welling, 2017) have shown promising results on learning physics but are restricted to setups where physical properties need to be fully observable, which is not the case of our approach. The most similar to ours is work by (Ehrhardt et al., 2019) on unsupervised learning of intuitive physics from unpaired past experiences.
+
+Other physics benchmarks and simulators. The main objective for the creation of our benchmark is (a) to focus specifically on evaluating capabilities of state of the art models for performing counterfactual reasoning, (b) to be unbiased in terms of distributions of parameters to be estimated and balanced with respect to possible outcomes, and (c) to have sufficient variety in terms of scenarios and latent physical characteristics of the scene that are not visually observed and therefore can act as confounders. To the best of our knowledge, none of existing intuitive physics benchmarks have these properties. IntPhys (Riochet et al., 2018) focuses on a high level task of estimating physical plausibility in a black box fashion and modeling out of distribution events at test time. CATER (Girdhar & Ramanan, 2019) introduces a video dataset requiring spatiotemporal understanding in
+
+order to solve the tasks such as action recognition, compositional action recognition, and adversarial target tracking. Phyre (Bakhtin et al., 2019) is an environment for solving physics based puzzles, where achieving sample efficiency may implicitly require counterfactual reasoning, but this component is not explicitly evaluated, construction of parallel data with several alternative outcomes is not straightforward, and the trivial baseline performance levels are not easy to estimate. Adapting these benchmarks to counterfactual reasoning would require significant refactoring and changing the logic of the data sampling. CLEVRR (Yi et al., 2019) is a diagnostic video dataset for systematic evaluation of models on a wide range of reasoning tasks including counterfactual questions. They cast the task as a classification problem where the model has to choose between a set of possible answers, whereas our benchmark requires the predicting the dynamics of each object.
+
+Perceptual causality. In the ML community, causal reasoning gained mainstream attention relatively recently (Lopez-Paz et al., 2017; Lopez-Paz & Oquab, 2017; Kocaoglu et al., 2018; Rojas-Carulla et al., 2018; Mooij et al., 2016; Scholkopf et al., 2012), due to limitations of statistical learning becoming increasingly apparent (Pearl, 2018; Lake et al., 2017). The concept of perceived causality has been however explored in cognitive psychology (Michotte, 1963), where human subjects have been shown to consistently report causal impressions not aligned with underlying physical principles of the events (Gerstenberg et al., 2015; Kubricht et al., 2017). Exploiting the colliding objects scenario as a standard testbed for these studies led to discovery of a number of cognitive biases, e.g. Motor Object Bias (i.e. false perceived association of object's velocity with its mass).
+
+In this work, we bring the domains of visual reasoning, intuitive physics and perceived causality together in a single framework to tackle the new problem of counterfactual learning of physical dynamics. Following prior literature (Battaglia et al., 2013), we also compare counterfactual learning with human performance and expect that, similarly to learning intuitive vs Newtonian physics, modeling perceived vs true causality will get more attention from the ML community in the future.
+
+# 3 COPHY: COUNTERFACTUAL PHYSICS BENCHMARK SUITE
+
+In this paper we investigate visual reasoning problems involving a set of  $K$  physical objects and their interactions, while considering a specific setting of learning counterfactual prediction with the objective of estimating objects' alternative 3D positions from images after do-intervention. We introduce the Counterfactual Physics benchmark suite (CoPhy) for counterfactual reasoning of physical dynamics from raw visual input. It is composed of three tasks based on three physical scenarios: BlocktowerCF, BallsCF and CollisionCF, defined similarly to existing state-of-the-art environments for learning intuitive physics: Shape Stack (Groth et al., 2018), Bouncing balls environment (Chang et al., 2017) and Collision (Ye et al., 2018) respectively. This was done to ensure natural continuity between the prior art in the field and the proposed counterfactual formulation.
+
+Each scenario includes training and test samples, that we call experiments. Each experiment is represented by two sequences of  $\tau$  synthetic RGB images (covering the time span of 6 sec at 5 fps):
+
+- an observed sequence  $X = \{X_0, \dots, X_\tau\}$  demonstrates evolution of the dynamic system under the influence of laws of physics (gravity, friction, etc.), from its initial state  $X_0$  to its final state  $X_\tau$ . For simplicity, we denote  $\mathbf{A}$  the initial state  $X_0$  and  $\mathbf{B}$  the observed outcome  $X_0, \dots, X$ ;  
+- a counterfactual sequence  $\bar{X} = \{\bar{X}_0, \dots, \bar{X}_{\tau}\}$ , where  $\bar{X}_0$  (C) corresponds to the initial state  $X_0$  after the do-intervention, and  $\bar{X}_1, \dots, \bar{X}_{\tau}$  (D) correspond to the counterfactual outcome.
+
+A do-intervention is a visually observable change introduced to the initial physical setup  $x_0$  (such as, for instance, object displacement or removal).
+
+Finally, the physical world in each experiment is parameterized by a set of visually unobservable quantities, or confounders (such as object masses, friction coefficients, direction and magnitude of gravitational forces), that cannot be uniquely estimated from a single time step. Our dataset provides ground truth values of all confounders for evaluation purposes. However, we do not assume access to this information during training or inference, and do not encourage it.
+
+Each of the three scenarios in the CoPhy benchmark is defined as follows (see Fig. 1 for illustrations).
+
+BlocktowerCF — Each experiment involves  $K = 3$  or  $K = 4$  stacked cubes, which are initially at resting (but potentially unstable) positions. We define three different confounder variables:
+
+![](images/7354ea49b5a4cf94ede18f1b403f8cfd38c672a39bdf660ec48bba5bc9bc7b96.jpg)  
+Figure 3: Stability distribution for each confounder variable for heights  $K = 3$  and  $K = 4$  of the BlockTowerCF task. Masses, friction coefficients: 2 configurations per block,  $2^{K}$  total; gravity: 3 configurations for each axis  $\in \{x, y\}$ , 9 total.
+
+masses,  $m \in \{1, 10\}$  and friction coefficients,  $\mu \in \{0.5, 1\}$ , for each block, as well as gravity components in  $X$  and  $Y$  direction,  $g_{x,y} \in \{-1, 0, 1\}$ . The do-interventions include block displacement or removal. This set contains 146k sample experiments corresponding to 73k different geometric block configurations.
+
+BallsCF — Experiments show  $K$  bouncing balls ( $K = 2\dots 6$ ). Each ball has an initial random velocity. The confounder variables are the masses,  $m\in \{1,10\}$ , and the friction coefficients,  $\mu \in \{0.5,1\}$ , of each ball. There are two do-operators: block displacement or removal. There are in total 100k experiments corresponding to 50k different initial geometric configurations.
+
+CollisionCF — This set is about moving objects colliding with static objects (balls or cylinders). The confounder variables are the masses,  $m \in \{1, 10\}$ , and the friction coefficients,  $\mu \in \{0.5, 1\}$ , of each object. The do-interventions are limited to object displacement. This scenario includes 40k experiments with 20k unique geometric object configurations.
+
+Given this data, the problem can be formalized as follows. During training, we are given the quadruplets of visual observations  $\mathbf{A},\mathbf{B},\mathbf{C},\mathbf{D}$  (through sequences  $X$  and  $\bar{X}$ , including GT object positions for supervision), but do not have access to the values of the confounders. During testing, the objective is to reason on new visual data unobserved at training time and to predict the counterfactual outcome  $\mathbf{D}$ , having observed the first sequence  $(\mathbf{A},\mathbf{B})$  and the modified initial state  $\mathbf{C}$  after the do-intervention, which is known.
+
+The CoPhy benchmark is by construction balanced and bias free w.r.t. (1) global statistics of all confounder values within each scenario, (2) distribution of possible outcomes of each experiment over the whole set of possible confounder values (for a given do-intervention). We make sure that the data does not degenerate to simple regularities which are solvable by conventional methods predicting the future from the past. In particular, for each experimental setup, we enforce existence of at least two different confounder configurations resulting in significantly different object trajectories. This guarantees that estimating the confounder variable is necessary for visual reasoning on this dataset.
+
+More specifically, we ensure that for each experiment the set of possible counterfactual outcomes is balanced w.r.t. (1) tower stability for BlocktowerCF and (2) distribution of object trajectories for BallsCF and CollisionCF. As a result, the BlocktowerCF set, for example, has  $50 \pm 5\%$  of stable and unstable counterfactual configurations. The exact distribution of stable/unstable examples for each confounder in this scenario is shown in Fig. 3.
+
+All images for this benchmark have been rendered into the visual space (RGB, depth and instance segmentation) at a resolution of  $448 \times 448$  px with PyBullet (only RGB images are used in this work). We ensure diversity in visual appearance between experiments by rendering the pairs of sequences over a set of randomized backgrounds. The ground truth physical properties of each object (3D pose, 4D quaternion angles, velocities) are sampled at a higher frame rate (20 fps) and also stored. The training / validation / test split is defined as  $0.7:0.2:0.1$  for each of the three scenarios.
+
+![](images/5488791e2aefd244acdef15d7553922e45f3e28330eb985f8e7aac307e758a5f.jpg)  
+Figure 4: Our model learns counterfactual reasoning in a weakly supervised way: while we supervise the do-operator, we do not supervise the confounder variables (masses, frictions, gravity). Input images of the original past (A) and the original outcome (B) are de-rendered into latent representations which are converted into fully-connected attributed graphs. A Graph Network updates node features to augment them with contextual information, which is integrated temporally with a set of RNNs, one for each object, running over time. The last hidden RNN state is taken as an estimate of the confounder  $U$ . A second set of GCN+RNN predicts residual object positions (D) using the modified past (C) and the confounder representation  $U$ . For clarity we draw arrows for the red object only. Not shown: stability prediction and gating.
+
+# 4 COUNTERFACTUAL LEARNING OF PHYSICS
+
+The task as described in Section 3 requires reasoning from visual inputs. We propose a single neural model which can be trained end-to-end, as shown in Fig. 4. We address this problem by adding strong inductive biases to a deep neural network, structuring it in a way to favor counterfactual reasoning. More precisely, we add structure for (i) estimation of physical properties from images, (ii) modelling interactions between objects through graph convolutions (GCN), (iii) estimating latent representations of the confounder variables, and (iv) exploiting these representations for predictions of the output object positions. At this point we would like to stress again, that the representation of the confounders  $U$  is latent and discovered from data without supervision.
+
+# 4.1 UNSUPERVISED ESTIMATION OF THE CONFOUNDERS
+
+While our method is capable of handling raw RGB frames as input, its internal reasoning is done on estimated representations in object-centric viewpoints. We train a convolutional neural network to detect the  $K$  objects and their 3D position in the scene, denoted as  $O = \{o^k\}$ ,  $k = 0\dots K - 1$  where  $o^k$  corresponds to the 3D position of object  $k$ . The de-rendering module is explained in the appendix.
+
+Predicting the future of a given block  $k$  requires modelling its interactions (through friction and collisions) with the other blocks in the scene, which we do with Graph Convolution Networks (GCN) (Kipf & Welling, 2017; Battaglia et al., 2018). The set of  $K$  objects in the scene is represented as a graph  $\mathcal{G} = (\mathcal{V}, \mathcal{E})$  where the nodes  $V$  are associated to objects  $\{o^k\}$ , and the object interactions to edges  $(o^k, o^j) \in \mathcal{E}$  in the fully-connected graph. Object embeddings  $\{o^k\}$  are updated classically and as follows, resulting in new embeddings  $\{\tilde{o}^k\}$ :
+
+$$
+e ^ {k} = \frac {1}{\left| \Omega_ {k} \right|} \sum_ {o ^ {j} \in \Omega_ {k}} f \left(o ^ {k}, o ^ {j}\right), \quad e = \frac {1}{K} \sum_ {k} e ^ {k}, \quad \delta^ {k} = g \left(o ^ {k}, e ^ {k}, e\right), \tag {1}
+$$
+
+where  $\Omega_{k}$  is the set of neighboring objects of  $o^k$ .  $f(.)$  and  $g(.)$  are non-linear mappings (MLPs), and their inputs are by default concatenated. For simplicity, in what follows we will denote the update of an object  $o^k$  with GCN given a graph with set of nodes and embeddings  $O$  by  $\tilde{o}^{k} = \mathbf{GCN}(o^{k},O)$ .
+
+As mentioned above, we want to infer a latent representation  $U$  of the confounding quantities for each object  $k$  given the input sequences  $X_{1:\tau}$  (the original past  $\mathbf{A}$  and the original outcome  $\mathbf{B}$ ), without any supervision. This latent representation  $U$  is trained end-to-end by optimizing the counterfactual prediction loss. To this end, we pass the updated object states  $\tilde{o}^k$  through a recurrent network to model the temporal evolution of this representation. In particular, we run a dedicated RNN for each object, each object maintaining its own hidden state  $h^k$ :
+
+$$
+h _ {t} ^ {k} = \phi \left(\tilde {\sigma} _ {t} ^ {k}, h _ {t - 1} ^ {k}\right) \tag {2}
+$$
+
+where we index objects and states with subscript  $t$  indicating time, and  $\phi$  is a gated recurrent unit (GRU) (gate equations have been omitted for simplicity). The recurrent network parameters are shared over objects  $k$ , which results in a model which is invariant to the number of objects present in the set. This allows us to use do-operators which change the number of objects in the scene (removal). We set the latent representation of the confounders to be the set  $U = \{u^k\}$ , where  $u^k \triangleq h_\tau^k$  is the temporally last hidden state of the recurrent network.
+
+# 4.2 TRAJECTORY PREDICTION GATED BY STABILITY
+
+We predict the counterfactual outcome  $\mathbf{D}$ , i.e. the 3D positions of all objects of the sequence  $\bar{X}_{1:\tau}$  with a recurrent network, which takes into account the confounders  $U$ . We cast this problem as a sequential prediction task, at each time step  $t$  predicting the residual position  $\Delta_t^k$  w.r.t. to position  $t - 1$ , i.e. the velocity vector. As in the rest of the model, this prediction is obtained object-wise, albeit with explicit modelling of the inter-object relationships through a graph network. More precisely,
+
+$$
+\tilde {\bar {o}} _ {t} ^ {k} = \mathbf {G C N} \left(\bar {o} _ {t} ^ {k}, \left\{\left[ \bar {o} _ {t} ^ {k}: u ^ {k} \right] \right\}\right), \quad r _ {t} ^ {k} = \psi \left(\tilde {\bar {o}} _ {t} ^ {k}, r _ {t - 1} ^ {k}\right), \quad \Delta_ {t} ^ {k} = \boldsymbol {W} r _ {t} ^ {k}, \tag {3}
+$$
+
+where  $r_t^k$  is the hidden state of the GRU network denoted by  $\psi$ , and  $\mathbf{W}$  is the weight matrix of a linear output layer. GCN is a graph convolutional network as described in eq. (1) and thereafter.
+
+At each moment of time, each object can either remain stationary or move under the influence of external physical forces or by inertia. The first task for the model is therefore to detect which objects are moving (i.e. affected by the environment) and then estimate parameters of the motion if it occurs. This is aligned well with the concepts of whether-causation and how-causation defined in the field of perceived causality (Gerstenberg et al., 2015). In our work, the whether-cause is estimated in the form of a binary stability indicator  $s_t^k$  described below (for each object, updated at each time step) that is then leveraged to gate the object position predictor (how-cause estimator):
+
+$$
+\bar {o} _ {t + 1} ^ {k} = \bar {o} _ {t} ^ {k} + \sigma \left(\frac {1 - s _ {t} ^ {k}}{\lambda}\right) \Delta_ {t} ^ {k}, \tag {4}
+$$
+
+where  $\sigma(.)$  is the sigmoid function and  $\lambda$  is a sparsifying temperature term.
+
+Counterfactual estimation of stability — estimation of object stability  $s_t^k$  is a counterfactual problem, as stability depends on the physical properties, and therefore on the latent confounder representation  $u_k$ . We combine the confounders  $U = \{u^k\}$  with the past after do-intervention (C), encoded in object states denoted as  $\bar{O}_t = \{\bar{o}_t^k\}$  at time step  $t = t$ . In particular, for each node we concatenate its object features with its confounder representation and we update the resulting object state with a graph network to take into account inter-object relationships:
+
+$$
+s _ {t} ^ {\prime k} = \mathbf {G C N} \left(\left[ \bar {o} _ {t} ^ {k}: u ^ {k} \right], \left\{\left[ \bar {o} _ {t} ^ {k}: u ^ {k} \right] \right\}\right), \quad s _ {t} ^ {k} = \boldsymbol {V} s _ {t} ^ {\prime k}, \tag {5}
+$$
+
+where  $s_t^k$  corresponds to the logits of stability of object  $k$  at time  $t$  and  $V$  is the weight matrix of a linear layer (for simplifying notations we omit bias here and in the rest of the paper).
+
+Training — The full counterfactual prediction model is trained end-to-end in graph space only (i.e. not including the de-rendering engine) with the following losses:
+
+$$
+\mathcal {L} _ {e 2 e} = \sum_ {k = 1} ^ {K} \mathcal {L} _ {c e} (s _ {t} ^ {k}, s _ {t} ^ {k *}) + \sum_ {t = 0} ^ {\tau} \left[ \sum_ {k = 1} ^ {K} \mathcal {L} _ {m s e} (\bar {o} _ {t} ^ {k}, \bar {o} _ {t} ^ {k *}) \right]
+$$
+
+where  $\mathcal{L}_{ce}$  is the binary cross entropy loss between the stability prediction of object  $k$  at time  $t$  and its ground truth value (calculated by thresholding applied to ground truth speed vectors), and  $\mathcal{L}_{mse}$  is mean squared error between the predicted positions  $\bar{\sigma}_t^k$  and GT positions  $\bar{\sigma}_t^{k*}$  in the 3D space. A detailed description of the overall architecture is given in the Appendix.
+
+<table><tr><td></td><td>Scenario</td><td>Top</td><td>Middle</td><td>Bottom</td><td>Mean</td><td>σ</td></tr><tr><td rowspan="2">Human</td><td>Non-CF</td><td>108.89</td><td>53.15</td><td>13.67</td><td>58.57</td><td>33.61</td></tr><tr><td>CF</td><td>99.98</td><td>49.65</td><td>13.99</td><td>54.54</td><td>34.15</td></tr><tr><td rowspan="2">Copying</td><td>C→D</td><td>65.41</td><td>25.51</td><td>7.36</td><td>32.76</td><td>N/A</td></tr><tr><td>B→D</td><td>92.60</td><td>35.85</td><td>18.54</td><td>49.00</td><td>N/A</td></tr><tr><td rowspan="2">CoPhyNet</td><td>Non-CF</td><td>77.97</td><td>33.10</td><td>2.39</td><td>37.81</td><td>N/A</td></tr><tr><td>CF</td><td>57.10</td><td>23.26</td><td>4.40</td><td>28.25</td><td>N/A</td></tr></table>
+
+Table 1: Comparison with human performance in the BlockTowerCF scenario obtained with AMT studies. We report 2D pixel error for each block, as well as global mean and variance  $\sigma$  (reference resolution  $448\times 448$ ) on the test set with  $K = 3$  blocks.  
+
+<table><tr><td>Train→Test</td><td>Copy C</td><td>Copy B</td><td>IN</td><td>NPE</td><td>CoPhyNet</td><td>IN sup.</td></tr><tr><td>3 → 3</td><td>0.470</td><td>0.601</td><td>0.318</td><td>0.331</td><td>0.294</td><td>0.296</td></tr><tr><td>3 → 3 †</td><td>0.365</td><td>0.592</td><td>0.298</td><td>0.319</td><td>0.289</td><td>0.282</td></tr><tr><td>3 → 4</td><td>0.754</td><td>0.846</td><td>0.524</td><td>0.523</td><td>0.482</td><td>0.467</td></tr><tr><td>4 → 4</td><td>0.735</td><td>0.852</td><td>0.521</td><td>0.528</td><td>0.453</td><td>0.481</td></tr><tr><td>4 → 4 †</td><td>0.597</td><td>0.861</td><td>0.480</td><td>0.476</td><td>0.423</td><td>0.464</td></tr><tr><td>4 → 3</td><td>0.480</td><td>0.618</td><td>0.342</td><td>0.350</td><td>0.301</td><td>0.297</td></tr></table>
+
+Table 2: BlocktowerCF: MSE on 3D pose average over time. IN sup. methods in the last column exploit the ground truth confounder quantities as input and thus represent a soft upper bound (are not comparable). Test confounder configurations not seen during training (50/50 split).
+
+![](images/9622feb2e3889f76c7ee31f823b18aa80f75d6e00b1cb2d354a7065f975d6855.jpg)  
+Figure 5: Visual examples of human performance on the ill-posed task of feedforward, i.e. noncounterfactual, dynamic prediction from a single image (in the BlockTower scenario). The image shows the initial state C. Small dots correspond to human estimates of the objects' final positions. Larger circles indicate ground truth final positions of each block. We note that this task is ill-posed by construction, as the dynamics of each experiment is defined by physical properties of each block (e.g. masses) which cannot be observed from a single image.
+
+# 5 EXPERIMENTS
+
+Training details. All models were implemented in PyTorch. We used the Adam optimizer (Kingma & Ba, 2015) and a learning rate of 0.001. For training the de-rendering pipeline 200k frames were sampled for each of the three scenarios (see the appendix for more details).
+
+Human performance. We empirically measured human performance in the BlockTowerCF scenario with crowdsourcing (Amazon Mechanical Turk/AMT). For this study, we have collected predictions from 100 participants, where each subject was given 20 assignments in both non-counterfactual (Fig. 5) and counterfactual (Fig. 6) settings. The human subjects were given 10 sec to click on the final positions of each block in the image C after the tower has fallen (or remained stable). The obtained quantitative results for both settings are reported in Table 1. We compare against copying baselines (i.e. predicting block positions in the frame D by either copying them from C or from B).
+
+In conclusion, we observe that humans perform slightly better in the counterfactual setup after having observed the first dynamic sequence (A, B) together with C compared to the classical prediction where only C is shown. This behavior has also been previously observed in experiments on intuitive physics in cognitive psychology (Kubricht et al., 2017) that revealed poor human abilities to extrapolate physical dynamics from a single image. Similar human studies have also been conducted in (Battaglia et al., 2013) in a more simplistic setup of predicting the direction of falling, where the authors also reported that the task appeared to be challenging for human subjects.
+
+The empirical results indicate that the participants decisions may have been however driven by simple inductive biases, e.g. "observed (in)stability in  $(\mathbf{A},\mathbf{B})^{\prime \prime}\rightarrow$  predict (in)stability in  $(\mathbf{C},\mathbf{D})$ ". The evidence for this approach is demonstrated qualitatively in Fig. 6: the variance in predictions after having observed a stable sequence is decreased (first row), after having observed a falling case - increased (second row). In all cases, human performance remains inferior w.r.t. the copying baselines.
+
+The last part of Table 1 shows results of our model (denoted by CoPhyNet in the rest of the discussion) after projecting the estimated 3D positions of all objects back into the 2D image space. CoPhyNet significantly outperforms both human subjects and copying baselines.
+
+Performance and comparisons. We evaluate the counterfactual prediction performance of the proposed CoPhyNet model against various baselines (shown in Tables 2-4 separately for each of the three scenarios of the CoPhy benchmark). The evaluated Network Physics Engine (NPE) (Chang et al., 2017) and Interaction Network (IN) (Battaglia et al., 2016), are both non-counterfactual baselines,
+
+![](images/0f197578d37622d6663b2d391a15e328677bd8da30fbeeaa99550ab7df8a6857.jpg)  
+experiment I: confounders  $\{\mu_0,\dots ,\mu_m\}_0$
+
+![](images/0e6bd76e2012ceea364a5db4a084df8e38c17648f6ff0f7bd3f2523c9107cef9.jpg)  
+Feedforward:  $\mathbf{C} \to \mathbf{D}$  
+Counterfactual:  $(\mathbf{A},\mathbf{B}),\mathbf{C}\rightarrow \mathbf{D}$  
+,B),C→D image C
+
+Feedforward:  $\mathbf{C}\rightarrow \mathbf{D}$  
+![](images/70698c82c09cd0a22bc61b0631a958d18df4b9f65d111b0fc8f0d235063197f9.jpg)  
+experiment II: confounders  $\{\mu_0,\dots ,\mu_m\} _1$
+
+![](images/130be9066804848861a0bdde286863a1d9c3b89471492fb9028247945bfb0295.jpg)  
+Counterfactual:  $(\mathbf{A},\mathbf{B}),\mathbf{C}\rightarrow \mathbf{D}$
+
+![](images/42766193fc325cc758f4dafe228cfc6c60481b8cbcb28f5480f0add01340c217.jpg)
+
+![](images/573757634ed40bc7564f99d35fcb91f9fa2725a731f42e8117afe0a34d1d3817.jpg)  
+Figure 6: Visual examples of human performance on the task of counterfactual dynamic prediction (in the BlockTowerCF scenario). Each participant has been shown both (A, B) and C. Small dots correspond to human estimates of the objects' resting positions (outcome after do-intervention). Larger circles indicate the ground truth final positions. The images show state C.  
+Figure 7: Visual examples of the counterfactual predictions produced by CoPhyNet (in the BlocktowerCF scenario). Circles denote GT position and crosses correspond to predictions.
+
+that predict future block coordinates from past coordinates after do-intervention without taking the confounders into account. More details for IN and NPE are given in the appendix A.3. Our method consistently outperforms NPE and IN by a large margin in all scenarios. The CoPhyNet model also usually (but not always) outperforms the augmented variants of these methods that include the GT confounder quantities as input (a not comparable setting).
+
+Fig. 7 illustrates several randomly sampled experimental setups and corresponding counterfactual predictions by the CoPhyNet model in the BlocktowerCF scenario.
+
+Generalization. We evaluate the ability of the CoPhyNet model to generalize to new physical setups which were not observed in the training data. In Table 2 we show model performance on unseen confounder combinations and on unseen number of blocks in the BlocktowerCF scenario (lines marked with  $\dagger$ ). Our proposed solution generalizes well under unseen settings compared to other methods. In Table 3 we also demonstrate that our method outperforms the baselines by a large margin on unseen numbers of balls in the BallsCF setup. Finally, in the CollisionCF scenario (Table 4) we train on one type of moving objects and test on another type (spheres vs cylinders). In this case we also show that our method is able to generalize to the new object types even when it has not seen such a combination of <moving-object>, static-object> before. Our method is able to estimate the object properties when an object is moving or initially stable.
+
+Impact of the confounder estimate. Our model does not rely on any supervision of the confounders; we do, however, explore what effect supervision could have on performance, as shown in Table 6 (Middle). Adding the supervision increases the performance of the model for  $K = 3$  but the difference seems marginal (0.004 for  $K = 3$  and 0.020 for  $K = 4$ ). Directly feeding the confounder quantities as input leads to better performance, which is expected (but not comparable).
+
+Model architecture. All design choices of CoPhyNet are ablated in Table 6 (Left) to fully illustrate the impact of each submodule. Estimating the stability once for the whole sequence  $\mathbf{D}$  decreases the performance by 0.020 for  $K = 3$  and 0.018 for  $K = 4$  compared to predicting the stability per object at each time step. Replacing the GCN by a MLP (i.e. concatenating the object representation) hurts the performance of the overall system by increasing the MSE by 0.286 when tested in the  $K = 4$  setting. Finally we compare our approach against a single-step counterfactual prediction. Non-surprisingly, predicting the future autoregressively in a step-by-step fashion turns out to be more effective than predicting the whole sequence at once.
+
+Confounder estimation. After training for predicting the target CF sequences, we evaluate the quality of the learned latent representation. In this experiment, we predict the confounder quantities of each object (mass, friction coefficient) from their latent representation by training a simple linear classifier, freezing the weights of the whole network. The obtained results are shown in Table 5. A
+
+<table><tr><td>Train→Test</td><td>Copy C</td><td>Copy B</td><td>IN</td><td>NPE</td><td>CoPhyNet</td><td>IN sup.</td></tr><tr><td>4 → 2</td><td>7.271</td><td>3.267</td><td>5.060</td><td>4.989</td><td>2.307</td><td>2.109</td></tr><tr><td>4 → 3</td><td>6.820</td><td>2.865</td><td>4.895</td><td>4.901</td><td>1.990</td><td>1.886</td></tr><tr><td>4 → 4</td><td>6.538</td><td>2.688</td><td>4.785</td><td>4.821</td><td>1.978</td><td>2.069</td></tr><tr><td>4 → 5</td><td>6.221</td><td>2.568</td><td>4.732</td><td>4.817</td><td>1.958</td><td>2.346</td></tr><tr><td>4 → 6</td><td>6.045</td><td>2.488</td><td>4.661</td><td>4.668</td><td>1.899</td><td>2.564</td></tr></table>
+
+Table 3: BallsSCF: MSE on 2D pose average over time. IN sup. methods in the last column exploit the ground truth confounder quantities as input and thus is not directly comparable.  
+
+<table><tr><td>Train→Test</td><td>Copy C</td><td>Copy B</td><td>IN</td><td>NPE</td><td>CoPhyNet</td><td>IN sup.</td></tr><tr><td>all→all</td><td>4.370</td><td>0.665</td><td>0.701</td><td>0.697</td><td>0.173</td><td>0.332</td></tr><tr><td>sphere→cylinder</td><td>4.245</td><td>0.481</td><td>0.715</td><td>0.710</td><td>0.220</td><td>0.435</td></tr><tr><td>cylinder→sphere</td><td>4.571</td><td>0.932</td><td>0.720</td><td>0.699</td><td>0.152</td><td>0.586</td></tr></table>
+
+Table 4: CollisionCF: MSE on 3D pose average over time. IN sup. methods in the last column exploit the ground truth confounder quantities as input and thus is not directly comparable (still showing inferior performance).  
+Table 5: Ablations on BlockTowerCF: confounder prediction (masses, friction coefficients) from the joint latent representation  $U$ . Metric: 4-way classification accuracy: Random=random classification.  
+
+<table><tr><td>Method</td><td>3 → 3</td><td>3 → 4</td></tr><tr><td>Static gating</td><td>0.305</td><td>0.496</td></tr><tr><td>GCN replaced by MLP</td><td>0.289</td><td>0.764</td></tr><tr><td>Single-step prediction</td><td>0.295</td><td>0.492</td></tr><tr><td>CoPhyNet</td><td>0.285</td><td>0.478</td></tr></table>
+
+<table><tr><td rowspan="2">Subset</td><td colspan="2">Feedforward</td><td colspan="2">Counterfactual</td></tr><tr><td>confounders: 
+input</td><td>-</td><td>confounders: 
+supervision</td><td>-</td></tr><tr><td>K=4</td><td>0.248</td><td>0.349</td><td>0.281</td><td>0.285</td></tr><tr><td>K=3</td><td>0.410</td><td>0.552</td><td>0.458</td><td>0.478</td></tr></table>
+
+<table><tr><td>Method</td><td>3 → 3</td><td>4 → 4</td></tr><tr><td>Random</td><td>25.0</td><td>25.0</td></tr><tr><td>CoPhyNet</td><td>65.7</td><td>68.9</td></tr></table>
+
+<table><tr><td>Method</td><td>3 → 3</td><td>3 → 4</td></tr><tr><td>Copy C</td><td>71.0</td><td>69.8</td></tr><tr><td>Copy B</td><td>69.9</td><td>68.5</td></tr><tr><td>GCN(C)</td><td>71.8</td><td>70.1</td></tr><tr><td>CoPhyNet</td><td>76.8</td><td>73.8</td></tr></table>
+
+Table 6: Ablation study on BlockTowerCF: (Left) Impact of each component of our model (MSE on 3D pose average over time). (Middle) Impact of the confounder estimate (MSE on 3D pose average over time, validation set). Feedforward methods do not estimate the confounder, counterfactual methods do. We compare against soft upper bounds, which use the ground truth confounder as input or supervise its estimation. (Right) Stability prediction (accuracy per block). With the ground truth confounder values as input, graph convolutional networks (GCN(C)) reach a performance of 85.4 and 77.3 in the  $3\rightarrow 3$  and  $3\rightarrow 4$  settings respectively (a soft upper bound, not comparable).
+
+prediction is correct if both the mass and the friction coefficients are correctly predicted. Our model outperforms the random baseline by a large margin suggesting that the confounder quantities are correctly encoded into the latent representation of each object during training.
+
+Stability prediction. We studied the performance of the stability estimation module in the BlockTowerCF scenario and compared it to several baselines, as shown in Table 6 (Right). Our method predicts stability of each block from the confounder estimate  $U$  and the frame  $\mathbf{C}$ . It outperforms the baselines estimating stability from a single input  $\mathbf{C}$  or from the sequence  $(\mathbf{A}, \mathbf{B})$  by a large margin, further indicating the efficiency of the confounder estimation and the complimentarity of this non-visual information w.r.t. the visual observation  $\mathbf{C}$ .
+
+# 6 CONCLUSION
+
+We formulated a new task of counterfactual reasoning for learning intuitive physics from images, developed a large-scale benchmark suite and proposed a practical approach for this problem. The task requires to predict alternative outcomes of a physical problem given the original past and outcome and an alternative past after do-intervention. Our challenging benchmarks cannot be solved by classical methods predicting by extrapolation, as the alternative future depends on confounder variables, which are unobservable from a single image of the alternative past. We train a neural model by supervising the do-operator, but not the confounders. Our experiments show that the CF setting outperforms conventional forecasting, and that the latent representation is related to the GT confounder quantities. We report human performance on this task, show its challenging nature and importance of CF reasoning.
+
+We believe that counterfactual reasoning in high-dimensional spaces is a key component of AI and hope that our task will spawn new research in this area and thus contribute to bridging the gap between causal reasoning and deep learning. We also expect the benchmark to become a testbed in model based RL, which employs predictive models of an environment for learning agent behavior. Forward models are classically used in this context, but we conjecture that CF reasoning will contribute to disentangling representations and inferring causal relationships between different factors of variation.
+
+Acknowledgements. This work was funded by grant Deepvision (ANR-15-CE23-0029, STPGP-479356-15), a joint French/Canadian call by ANR & NSERC.
+
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+
+# A APPENDIX
+
+# A.1 NEURAL DE-RENDERING
+
+We train a convolutional neural network (CNN) to detect the  $K$  blocks and estimate their positions in 3D, denoted as  $O = \{o^k\}$ ,  $k = 0\dots K - 1$  where  $o^k$  corresponds to the position of object  $k$ . The network is inspired by classical region-based methods for object detection (He et al., 2017) and takes as an input an RGB image of resolution  $224\times 224$ .
+
+We define a double convolution module as a stack of a convolutional layer, batch normalization and ReLu activation, repeated two times. The resulting CNN includes three such modules with 64, 128 and 256 channels respectively, separated by  $2 \times 2$  max pooling, which produces output feature maps of size  $256 \times 56 \times 56$ .
+
+We design  $K$  different heads by splitting the final feature maps channel-wise into  $K$  feature maps and perform a double convolution module with 1 output channel. Each head outputs a feature map of size  $1 \times 56 \times 56$  which is transformed into a vector to regress the object positions.
+
+We first pre-train the de-rendering module alone without the model parts responsible for reasoning in the graph space. In particular, we de-render images  $\hat{X}$  randomly sampled from  $\mathbf{A}$ ,  $\mathbf{B}$ ,  $\mathbf{C}$  and  $\mathbf{D}$  into its object representations  $O = \{o^k\}$  and train with the following supervised loss:
+
+$$
+\mathcal {L} _ {\text {d e r e n d e r}} = \sum_ {k = 1} ^ {K} \mathcal {L} _ {\text {m s e}} \left(o ^ {k}, o ^ {k *}\right) \tag {6}
+$$
+
+where  $\hat{\sigma}^{k*}$  are the ground truth 3D positions and  $\mathcal{L}_{mse}$  corresponds to the mean square error. The rest of the model is trained end-to-end as described in section 4.2, paragraph "Training".
+
+# A.2 OTHER ARCHITECTURES
+
+Below are the descriptions of other networks used in our pipeline:
+
+$\mathbf{f}, \mathbf{g}$  are implemented as MLPs with 4 and 2 layers respectively, with hidden layers of size 32 and ReLu activations.
+
+$\phi, \psi$  are implemented as GRU modules with 2 layers and a hidden state of dimension 32.
+
+**Confounders (mass and friction coefficients)** are predicted with a single fully connected layer on top of the "confounder" representation of each object denoted  $u^k$ .
+
+# A.3 FEEDFORWARD BASELINES
+
+We compare our approach against two recent feedforward approaches, namely IN (Battaglia et al., 2016) and NPE (Chang et al., 2017). Both methods assume that GT object positions are available as input at training and test time, so they directly work on GT positions. They both predict the next position of each object using a Graph Convolution Network. IN is modeling object pairwise interaction between all objects in the scene, while NPE is taking into account only neighbouring objects for estimating the object interactions.
+
+# A.4 TRAINING FROM ESTIMATED POSITIONS
+
+In Table 7 we report the impact of the de-rendering module on performance. In particular, we compare performance of our model (CoPhyNet w/o GT, as described in the main part of the paper) with a version where we use ground-truth positions (CoPhyNet GT) for training. During training time, GT positions are fed to the main model. For testing, we do, however, use positions estimated by the de-rendering module in both versions.
+
+We can see that training using the GT positions gives slightly better performance than training from estimated positions, which is expected.
+
+<table><tr><td>Train→Test</td><td>Copy C</td><td>Copy B</td><td>CoPhyNet GT</td><td>CoPhyNet w/o GT (=ours)</td></tr><tr><td>3 → 3</td><td>0.470</td><td>0.601</td><td>0.294</td><td>0.309</td></tr><tr><td>3 → 3 †</td><td>0.365</td><td>0.592</td><td>0.289</td><td>0.298</td></tr><tr><td>3 → 4</td><td>0.754</td><td>0.846</td><td>0.482</td><td>0.504</td></tr></table>
+
+Table 7: BlocktowerCF: MSE on 3D pose averaged over time. We evaluate different types of training: from the ground-truth positions (GT) or from the estimated positions (w/o GT).  ${}^{ \dagger  }$  Test confounder configurations not seen during training (50/50 split).
\ No newline at end of file
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+# CROSS-DOMAIN FEW-SHOT CLASSIFICATION VIA LEARNED FEATURE-WISE TRANSFORMATION
+
+Hung-Yu Tseng
+
+University of California, Merced
+
+htseng6@ucmerced.edu
+
+Hsin-Ying Lee
+
+University of California, Merced
+
+hlee246@ucmerced.edu
+
+Jia-Bin Huang
+
+Virginia Tech
+
+jbhuang@vt.edu
+
+Ming-Hsuan Yang
+
+University of California, Merced
+
+Google Research
+
+Yonsei University
+
+mhyang@ucmerced.edu
+
+# ABSTRACT
+
+Few-shot classification aims to recognize novel categories with only few labeled images in each class. Existing metric-based few-shot classification algorithms predict categories by comparing the feature embeddings of query images with those from a few labeled images (support examples) using a learned metric function. While promising performance has been demonstrated, these methods often fail to generalize to unseen domains due to large discrepancy of the feature distribution across domains. In this work, we address the problem of few-shot classification under domain shifts for metric-based methods. Our core idea is to use feature-wise transformation layers for augmenting the image features using affine transforms to simulate various feature distributions under different domains in the training stage. To capture variations of the feature distributions under different domains, we further apply a learning-to-learn approach to search for the hyper-parameters of the feature-wise transformation layers. We conduct extensive experiments and ablation studies under the domain generalization setting using five few-shot classification datasets: mini-[ImageNet, CUB, Cars, Places, and Plantae. Experimental results demonstrate that the proposed feature-wise transformation layer is applicable to various metric-based models, and provides consistent improvements on the few-shot classification performance under domain shift.
+
+# 1 INTRODUCTION
+
+Few-shot classification (Lake et al., 2015) aims to recognize instances from novel categories (query instances) with only few labeled examples in each class (support examples). Among various recent approaches for addressing the few-shot classification problem, metric-based meta-learning methods (Garcia & Bruna, 2018; Sung et al., 2018; Vinyals et al., 2016; Snell et al., 2017; Oreshkin et al., 2018) have received considerable attention due to their simplicity and effectiveness. In general, metric-based few-shot classification methods make the prediction based on the similarity between the query image and support examples. As illustrated in Figure 1, metric-based approaches consist of 1) a feature encoder and 2) a metric function. Given an input task consisting of few labeled images (the support set) and unlabeled images (the query set) from novel classes, the encoder first extracts the image features. The metric function then takes the features of both the labeled and unlabeled images as input and predicts the category of the query images. Despite the success of recognizing novel classes sampled from the same domain as in the training stage (e.g., both training and testing are on mini-ImageNet classes), Chen et al. (Chen et al., 2019a) recently raise the issue that existing metric-based approaches often do not generalize well to categories from different domains. The generalization ability to unseen domains, however, is of critical importance due to
+
+![](images/b7b912c0194e9602e0297d6760c6c6a827e484c41882731d11b776c585d261ce.jpg)
+
+![](images/66da930b1aaa4d4a0b5dad2a2ea30184ffb18d0c81e9b2d8cb6b6785064501a0.jpg)
+
+![](images/e3bbe8ea14a270c30a723868e7e9d7f723446fa330ec58311fc0b79e1767da6b.jpg)  
+Figure 1: Problem formulation and motivation. Metric-based meta-learning models usually consist of a feature encoder  $E$  and metric function  $M$ . We aim to improve the generalization ability of the models training from seen domains to arbitrary unseen domains. The key observation is that the distributions of the image features extracted from tasks in the unseen domains are significantly different from those in the seen domains.
+
+![](images/cadf6f1a3acc6fdb67f4b5c82dbe2bf88bdbb3106aed7e4b63bb9dd07250434e.jpg)
+
+the difficulty to construct large training datasets for rare classes (e.g., recognizing rare bird species in a fine-grained classification setting). As a result, understanding and addressing the domain shift problem for few-shot classification is of great interest.
+
+To alleviate the domain shift issue, numerous unsupervised domain adaptation techniques have been proposed (Pan & Yang, 2010; Chen et al., 2018; Tzeng et al., 2017). These methods focus on adapting the classifier of the same category from the source to the target domain. Building upon the domain adaptation formulation, Dong and Xing (Dong & Xing, 2018) relax the constraint and transfer knowledge across domains for recognizing novel category in the one-shot setting. However, unsupervised domain adaptation approaches assume that numerous unlabeled images are available in the target domain during training. In many cases, this assumption may not be realistic. For example, the cost and efforts of collecting numerous images of rare bird species can be prohibitively high. On the other hand, domain generalization methods have been developed (Blanchard et al., 2011; Li et al., 2019) to learn classifiers that generalize well to multiple unseen domains without requiring the access to data from those domains. Yet, existing domain generalization approaches aim at recognizing instance from the same category in the training stage.
+
+In this paper, we tackle the domain generalization problem for recognizing novel category in the few-shot classification setting. As shown in Figure 1(c), our key observation is that the distributions of the image features extracted from the tasks in different domains are significantly different. As a result, during the training stage, the metric function may overfit to the feature distributions encoded only from the seen domains and thus fail to generalize to unseen domains. To address the issue, we propose to integrate feature-wise transformation layer to modulate the feature activations with affine transformations into the feature encoder. The use of these feature-wise transformation layers allows us to simulate various distributions of image features during the training stage, and thus improve the generalization ability of the metric function in the testing phase. Nevertheless, the hyper-parameters of the feature-wise transformation layers may require meticulous hand-tuning due to the difficulty to model the complex variation of the image feature distributions across various domains. In light of this, we develop a learning-to-learn algorithm to optimize the proposed feature-wise transformation layers. The core idea is to optimize the feature-wise transformation layers so that the model can work well on the unseen domains after training the model using the seen domains. We make the source code and datasets public available to simulate future research in this field.
+
+We make the following three contributions in this work:
+
+- We propose to use feature-wise transformation layers to simulate various image feature distributions extracted from the tasks in different domains. Our feature-wise transformation
+
+layers are method-agnostic and can be applied to various metric-based few-shot classification approaches for improving their generalization to unseen domains.
+
+- We develop a learning-to-learn method to optimize the hyper-parameters of the feature-wise transformation layers. In contrast to the exhaustive parameter hand-tuning process, the proposed learning-to-learn algorithm is capable of finding the hyper-parameters for the feature-wise transformation layers to capture the variation of image feature distribution across various domains.  
+- We evaluate the performance of three metric-based few-shot classification models (including MatchingNet (Vinyals et al., 2016), RelationNet (Sung et al., 2018), and Graph Neural Networks (Garcia & Bruna, 2018)) with extensive experiments under the domain generalization setting. We show that the proposed feature-wise transformation layers can effectively improve the generalization ability of metric-based models to unseen domains. We also demonstrate further performance improvement with our learning-to-learn scheme for learning the feature-wise transformation layers.
+
+# 2 RELATED WORK
+
+Few-shot classification. Few-shot classification aims to learn to recognize novel categories with a limited number of labeled examples in each class. Significant progress has been made using the meta-learning based formulation. There are three main classes of meta-learning approaches for addressing the few-shot classification problem. First, recurrent-based frameworks (Rezende et al., 2016; Santoro et al., 2016) sequentially process and encode the few labeled images of novel categories. Second, optimization-based schemes (Finn et al., 2017; Rusu et al., 2019; Tseng et al., 2019; Vuorio et al., 2019) learn to fine-tune the model with the few example images by integrating the fine-tuning process in the meta-training stage. Third, metric-based methods (Koch et al., 2015; Vinyals et al., 2016; Snell et al., 2017; Oreshkin et al., 2018; Sung et al., 2018; Lifchitz et al., 2019) classify the query images by computing the similarity between the query image and few labeled images of novel categories.
+
+Among these three classes, metric-based methods have attracted considerable attention due to their simplicity and effectiveness. Metric-based few-shot classification approaches consist of 1) a feature encoder to extract features from both the labeled and unlabeled images and 2) a metric function that takes image features as input and predict the category of unlabeled images. For example, MatchingNet (Vinyals et al., 2016) applies cosine similarity along with a recurrent network, ProtoNet (Snell et al., 2017) utilizes euclidean distance, RelationNet (Sung et al., 2018) uses CNN modules, GNN (Garcia & Bruna, 2018) employs graph convolution modules as the metric functions. However, these metric functions may fail to generalize to unseen domains since the distributions of the image features extracted from the task in various domains can be drastically different. Chen et al. (Chen et al., 2019a) recently show that the performance of existing few-shot classification methods degrades significantly under domain shifts. Our work focuses on improving the generalization ability of metric-based few-shot classification models to unseen domains. Very recently, Triantafillou et al. (Triantafillou et al., 2020) also target on the cross-domain few-shot classification problem. We encourage the readers to review for a more complete picture.
+
+Domain adaptation. Domain adaptation methods (Pan & Yang, 2010) aim to reduce the domain shift between the source and target domains. Since the emergence of domain adversarial neural networks (DANN) (Ganin et al., 2016), numerous frameworks have been proposed to apply adversarial training to align the source and target distributions on the feature-level (Tzeng et al., 2017; Chen et al., 2018; Hsu et al., 2020) or on the pixel-level (Tsai et al., 2018; Hoffman et al., 2018; Bousmalis et al., 2017; Chen et al., 2019b; Lee et al., 2018). Most domain frameworks, however, target at adapting knowledge of the same category learned from the source to target domain and thus are less effective to handle novel category as in the few-shot classification scenarios. One exception is the work by Dong and Xing (Dong & Xing, 2018) that address the domain shift issue in the one-shot learning setting. Nevertheless, these domain adaptation methods require access to the unlabeled images in the target domain during the training. Such an assumption may not be feasible in many applications due to the difficulty of collecting abundant examples of rare categories (e.g., rare bird species).
+
+Domain generalization. In contrast to the domain adaptation frameworks, domain generalization (Blanchard et al., 2011) methods aim at generalizing from a set of seen domains to the unseen domain without accessing instances from the unseen domain during the training stage. Before the emerging of learning-to-learn (i.e., meta-learning) (Ravi & Larochelle, 2017; Finn et al., 2017) approaches, several methods have been proposed for tackling the domain generalization problem. Examples include extracting domain-invariant features from various seen domains (Blanchard et al., 2011; Li et al., 2018b; Muandet et al., 2013), improving the classifiers by fusing classifiers learned from seen domains (Niu et al., 2015a;b), and decomposing the classifiers into domain-specific and domain-invariant components (Khosla et al., 2012; Li et al., 2017a). Another stream of work learns to augment the input data with adversarial learning (Shankar et al., 2018; Volpi et al., 2018). Most recently, a number of methods apply the learning-to-learn strategy to simulate the generalization process in the training stage (Balaji et al., 2018; Li et al., 2018a; 2019). Our method adopts a similar approach to train the proposed feature-wise transformation layers. The application context, however, differs from prior work as we focus on recognizing novel category from unseen domains in few-shot classification. The goal of this work is to make few-shot classification algorithms robust to domain shifts.
+
+Learning-based data augmentation. Data augmentation methods are designed to increase the diversity of data for the training process. Unlike the hand-crafted approaches such as horizontal flipping and random cropping, several recent approaches have been proposed to learn the data augmentation (Cubuk et al., 2019; DeVries & Taylor, 2017a; Lemley et al., 2017; Perez & Wang, 2017; Sixt et al., 2018; Tran et al., 2017). For instance, the SmartAugmentation (Lemley et al., 2017) scheme trains a network that combines multiple images from the same category. The Bayesian DA (Tran et al., 2017) method augments the data according to the distribution learned from the training set, and the RenderGAN (Sixt et al., 2018) model simulates realistic images using generative adversarial networks. In addition, the AutoAugment (Cubuk et al., 2019) algorithm learns the augmentation via reinforcement learning. Two recent frameworks (Shankar et al., 2018; Volpi et al., 2018) target at augmenting the data by modeling to the variation across different domains with adversarial learning. Similar to these approaches for capturing the variations across multiple domains, we develop a learning-to-learn process to optimize the proposed feature-wise transformation layers for simulating various distributions of image features encoded from different domains.
+
+Conditional normalization. Conditional normalization aims to modulate the activation via a learned affine transformation conditioned on external data (e.g., an image of an artwork for capturing a specific style). Conditional normalization methods, including Conditional Batch Normalization (Dumoulin et al., 2017), Adaptive Instance Normalization (Huang & Belongie, 2017), and SPADE (Park et al., 2019), are widely used in the style transfer and image synthesis tasks (Karras et al., 2019; Lee et al., 2020; AlBahar & Huang, 2019). In addition to image stylization and generation, conditional normalization has also been applied to align different data distributions for domain adaptation (Cariucci et al., 2017; Li et al., 2017b). In particular, the TADAM method (Oreshkin et al., 2018) applies conditional batch normalization to metric-based models for the few-shot classification task. The TADAM method aims to model the training task distribution under the same domain. In contrast, we focus on simulating various features distributions from different domains.
+
+Regularization for neural networks. Adding some form of randomness in the training stage is an effective way to improve generalization (Srivastava et al., 2014; Wan et al., 2013; Larsson et al., 2017; DeVries & Taylor, 2017b; Zhang et al., 2018; Ghiasi et al., 2018). The proposed feature-wise transformation layer for modulating the feature activations of intermediate layers (by applying random affine transformations) can also be viewed as a way to regularize network training.
+
+# 3 METHODOLOGY
+
+# 3.1 PRELIMINARIES
+
+Few-shot classification and metric-based method. The few-shot classification problem is typically characterized as  $N_{w}$  way (number of categories) and  $N_{s}$  shot (number of labeled examples for each category). Figure 1 shows an example of how the metric-based frameworks operate in the 3-way 3-shot few-shot classification task. A metric-based algorithm generally contains a feature
+
+encoder  $E$  and a metric function  $M$ . For each iteration during the training stage, the algorithm randomly samples  $N_{w}$  categories and constructs a task  $T$ . We denote the collection of input images as  $\mathcal{X} = \{\mathbf{x}_1,\mathbf{x}_2,\dots ,\mathbf{x}_n\}$  and the corresponding categorical labels as and  $\mathcal{Y} = \{y_1,y_2,\dots ,y_n\}$ . A task  $T$  consists of a support set  $\mathbf{S} = \{(\mathcal{X}_s,\mathcal{Y}_s)\}$  and a query set  $\mathbf{Q} = \{(\mathcal{X}_q,\mathcal{Y}_q)\}$ . The support set  $\mathbf{S}$  and query set  $\mathbf{Q}$  are respectively formed by randomly selecting  $N_{s}$  and  $N_{q}$  samples for each of the  $N_{w}$  categories.
+
+The feature encoder  $E$  first extracts the features from both the support and query images. The metric function  $M$  then predicts the categories of the query images  $\mathcal{X}_q$  according to the label of support images  $\mathcal{Y}_s$ , the encoded query image  $E(\mathbf{x}^q)$ , and the encoded support images  $E(\mathcal{X}_s)$ . The process can be formulated as
+
+$$
+\hat {\mathcal {Y}} _ {q} = M \left(\mathcal {Y} _ {s}, E \left(\mathcal {X} _ {s}\right), E \left(\mathcal {X} _ {q}\right)\right). \tag {1}
+$$
+
+Finally, the training objective of a metric-based framework is the classification loss of the images in the query set,
+
+$$
+L = L _ {\mathrm {c l s}} \left(\mathcal {Y} _ {q}, \hat {\mathcal {Y}} _ {q}\right). \tag {2}
+$$
+
+The main difference between various metric-based algorithms lies in the design choice for the metric function  $M$ . For instance, the MatchingNet (Vinyals et al., 2016) method utilizes long-short-term memories (LSTM), the RelationNet (Sung et al., 2018) model applies convolutional neural networks (CNN), and the GNN (Garcia & Bruna, 2018) scheme uses graph convolutional networks.
+
+Problem setting. In this work, we address the few-shot classification problem under the domain generalization setting. We denote a domain consisting of a collection of few-shot classification tasks as  $\mathcal{T} = \{T_1,T_2,\dots ,T_n\}$ . We assume  $N$  seen domains  $\{\mathcal{T}_1^{\mathrm{seen}},\mathcal{T}_2^{\mathrm{seen}},\dots ,\mathcal{T}_N^{\mathrm{seen}}\}$  available in the training phase. The goal is to learn a metric-based few-show classification model using the seen domains, such that the model can generalize well to an unseen domain  $\mathcal{T}^{\mathrm{unseen}}$ . For example, one can train the model with the mini-ImageNet (Ravi & Larochelle, 2017) dataset as well as some public available fine-grained few-shot classification domains, e.g., CUB (Welinder et al., 2010), and then evaluate the generalization ability of the model on an unseen plants domain. Note that our problem formulation does not access images in the unseen domain at the training stage.
+
+# 3.2 FEATURE-WISE TRANSFORMATION LAYER
+
+Our focus in this work is to improve the generalization ability of metric-based few-shot classification models to arbitrary unseen domains. As shown in Figure 1, due to the discrepancy between the feature distributions extracted from the task in the seen and unseen domains, the metric function  $M$  may overfit to the seen domains and fail to generalize to the unseen domains. To address the problem, we propose to integrate a feature-wise transformation to augment the intermediate feature activations with affine transformations into the feature encoder  $E$ . Intuitively, the feature encoder  $E$  integrated with the feature-wise transformation layers can produce more diverse feature distributions which improve the generalization ability of the metric function  $M$ . As shown in Figure 2(b), we insert the feature-wise transformation layer after the batch normalization layer in the feature encoder  $E$ . The hyper-parameters  $\theta_{\gamma} \in R^{C \times 1 \times 1}$  and  $\theta_{\beta} \in R^{C \times 1 \times 1}$  indicate the standard deviations of the Gaussian distributions for sampling the affine transformation parameters. Given an intermediate feature activation map  $\mathbf{z}$  in the feature encoder with the dimension of  $C \times H \times W$ , we first sample the scaling term  $\gamma$  and bias term  $\beta$  from Gaussian distributions,
+
+$$
+\gamma \sim N (\mathbf {1}, \text {s o f t p l u s} (\theta_ {\gamma})) \quad \beta \sim N (\mathbf {0}, \text {s o f t p l u s} (\theta_ {\beta})). \tag {3}
+$$
+
+We then compute the modulated activation  $\hat{\mathbf{z}}$  as
+
+$$
+\hat {z} _ {c, h, w} = \gamma_ {c} \times z _ {c, h, w} + \beta_ {c}, \tag {4}
+$$
+
+where  $\hat{z}_{c,h,w} \in \hat{\mathbf{z}}$  and  $z_{c,h,w} \in \mathbf{z}$ . In practice, we insert the feature-wise transformation layers to the feature encoder  $E$  at multiple levels.
+
+# 3.3 LEARNING THE FEATURE-WISE TRANSFORMATION LAYERS
+
+While we can empirically determine hyper-parameters  $\theta_f = \{\theta_\gamma, \theta_\beta\}$  of the feature-wise transformation layer, it remains challenging to hand-tune a generic set of parameters which are effective
+
+![](images/09d987287acd406e2f0d67e9cfba64b3275341dd85f353c37114c091f7ecb587.jpg)  
+Figure 2: Method overview. (a) We propose a feature-wise transformation layer to modulate intermediate feature activation  $\mathbf{z}$  in the feature encoder  $E$  with the scaling and bias terms sampled from the Gaussian distributions parameterized by the hyper-parameters  $\theta_{\gamma}$  and  $\theta_{\beta}$ . During the training phase, we insert a collection of feature-wise transformation layers into the feature encoder to simulate feature distributions extracted from the tasks in various domains. (b) We design a learning-to-learn algorithm to optimize the hyper-parameters  $\theta_{\gamma}$  and  $\theta_{\beta}$  of feature-wise transformation layers by maximizing the performance of the applied metric-based model on the pseudo-unseen domain (bottom) after it is optimized on the pseudo-seen domain (top).
+
+on different settings (i.e., different metric-based frameworks and different seen domains). To address this problem, we design a learning-to-learn algorithm to optimize the hyper-parameters of the feature-wise transformation layer. The core idea is that training the metric-based model integrated with the proposed layers on the seen domains should improve the performance of the model on the unseen domains.
+
+We illustrate the process in Figure 2(b) and Algorithm 1. In each training iteration  $t$ , we sample a pseudo-seen domain  $\mathcal{T}^{\mathrm{ps}}$  and a pseudo-unseen domain  $\mathcal{T}^{\mathrm{pu}}$  from a set of seen domains  $\{\mathcal{T}_1^{\mathrm{seen}},\mathcal{T}_2^{\mathrm{seen}},\dots ,\mathcal{T}_N^{\mathrm{seen}}\}$ . Given a metric-based model with feature encoder  $E_{\theta_e^t}$  and metric function  $M_{\theta_m^t}$ , we first integrate the proposed layers with hyper-parameters  $\theta_f^t = \{\theta_\gamma^t,\theta_\beta^t\}$  into the feature encoder (i.e.,  $E_{\theta_e^t,\theta_f^t}$ ). We then use the loss in equation 2 to update the parameters in the metric-based model with the pseudo-seen task  $T^{\mathrm{ps}} = \{(\mathcal{X}_s^{\mathrm{ps}},\mathcal{Y}_s^{\mathrm{ps}}),(\mathcal{X}_q^{\mathrm{ps}},\mathcal{Y}_q^{\mathrm{ps}})\} \in \mathcal{T}^{\mathrm{ps}}$ , namely
+
+$$
+\left(\theta_ {e} ^ {t + 1}, \theta_ {m} ^ {t + 1}\right) = \left(\theta_ {e} ^ {t}, \theta_ {m} ^ {t}\right) - \alpha \bigtriangledown_ {\theta_ {e} ^ {t}, \theta_ {m} ^ {t}} L _ {\mathrm {c l s}} \left(\mathcal {Y} _ {q} ^ {\mathrm {p s}}, M _ {\theta_ {m} ^ {t}} \left(\mathcal {Y} _ {s} ^ {\mathrm {p s}}, E _ {\theta_ {e} ^ {t}, \theta_ {f} ^ {t}} \left(\mathcal {X} _ {s} ^ {\mathrm {p s}}\right), E _ {\theta_ {e} ^ {t}, \theta_ {f} ^ {t}} \left(\mathcal {X} _ {q} ^ {\mathrm {p s}}\right)\right)\right), \quad (5)
+$$
+
+where  $\alpha$  is the learning rate. We then measure the generalization ability of the updated metric-based model by 1) removing the feature-wise transformation layers from the model and 2) computing the classification loss of the updated model on the pseudo-unseen task  $T^{\mathrm{pu}} = \{(\mathcal{X}_s^{\mathrm{pu}},\mathcal{Y}_s^{\mathrm{pu}}),(\mathcal{X}_q^{\mathrm{pu}},\mathcal{Y}_q^{\mathrm{pu}})\} \in T^{\mathrm{pu}}$ , namely
+
+$$
+L ^ {\mathrm {p u}} = L _ {\operatorname {c l s}} \left(\mathcal {Y} _ {q} ^ {\mathrm {p u}}, M _ {\theta_ {m} ^ {t + 1}} \left(\mathcal {Y} _ {s} ^ {\mathrm {p u}}, E _ {\theta_ {e} ^ {t + 1}} \left(\mathcal {X} _ {s} ^ {\mathrm {p u}}\right), E _ {\theta_ {e} ^ {t + 1}} \left(\mathcal {X} _ {q} ^ {\mathrm {p u}}\right)\right)\right). \tag {6}
+$$
+
+Finally, as the loss  $L^{\mathrm{pu}}$  reflects the effectiveness of the feature-wise transformation layers, we optimize the hyper-parameters  $\theta_f$  by
+
+$$
+\theta_ {f} ^ {t + 1} = \theta_ {f} ^ {t} - \alpha \bigtriangledown_ {\theta_ {f} ^ {t}} L ^ {\mathrm {p u}}. \tag {7}
+$$
+
+Note that the metric-based model and feature-wise transformation layers are jointly optimized in the training stage.
+
+Algorithm 1: Learning-to-Learn Feature-Wise Transformation.  
+Require: Seen domains  $\{\mathcal{T}_1^{\mathrm{seen}},\mathcal{T}_2^{\mathrm{seen}},\dots ,\mathcal{T}_n^{\mathrm{seen}}\}$  , learning rate  $\alpha$    
+Randomly initialize  $\theta_{e},\theta_{m}$  and  $\theta_{f}$    
+while training do   
+Randomly sample non-overlapping pseudo-seen  $\mathcal{T}^{\mathrm{ps}}$  and psuedo-unseen  $\mathcal{T}^{\mathrm{pu}}$  domains from the seen domains   
+Sample a peso-seen task  $T^{\mathrm{ps}}\in T^{\mathrm{ps}}$  and a pseudo-unseen task  $T^{\mathrm{pu}}\in T^{\mathrm{pu}}$    
+// Update metric-based model with pseudo-seen task:   
+Obtain  $\theta_e^{t + 1},\theta_m^{t + 1}$  using equation 5   
+// Update feature-wise transformation layers with pseudo-unseen task:   
+Obtain  $\theta_f^{t + 1}$  using equation 6 and equation 7   
+end
+
+# 4 EXPERIMENTAL RESULTS
+
+# 4.1 EXPERIMENTAL SETUPS
+
+We validate the efficacy of the proposed feature-wise transformation layer with three existing metric-based algorithms (Vinyals et al., 2016; Sung et al., 2018; Garcia & Bruna, 2018) under two experimental settings. First, we empirically determine the hyper-parameters  $\theta_{f} = \{\theta_{\gamma},\theta_{\beta}\}$  of the featurewise transformation layers and analyze the impact of the feature-wise transformation layers. We train the few-shot classification model on the mini-[ImageNet (Bousmalis et al., 2017) domain and evaluate the trained model on four different domains: CUB (Welinder et al., 2010), Cars (Krause et al., 2013), Places (Zhou et al., 2017), and Plantae (Van Horn et al., 2018). Second, we demonstrate the importance of the proposed learning-to-learn scheme for optimizing the hyper-parameters of feature-wise transformation layers. We adopt the leave-one-out setting by selecting an unseen domain from CUB, Cars, Places, and Plantae domains. The mini-[ImageNet (Bousmalis et al., 2017) and the remaining domains then serve as the seen domains for training both the metric-based model and feature-wise transformation layers using Algorithm 1. After the training, we evaluate the trained model on the selected unseen domain.
+
+Datasets. We conduct experiments using five datasets: mini-ImageNet (Ravi & Larochelle, 2017), CUB (Welinder et al., 2010), Cars (Krause et al., 2013), Places (Zhou et al., 2017), and Plantae (Van Horn et al., 2018). Since the mini-ImageNet dataset serves as the seen domain for all experiments, we select the training iterations with the best accuracy on the validation set of the mini-ImageNet dataset for evaluation. More details of dataset processing are presented in Appendix A.1.
+
+Implementation details. We apply the feature-wise transformation layers to three metric-based frameworks: MatchingNet (Vinyals et al., 2016), RelationNet (Sung et al., 2018), and GNN (Garcia & Bruna, 2018). We use the public implementation from Chen et al. (Chen et al., 2019a) to train both the MatchingNet and RelationNet model. For the GNN approach, we integrate the official implementation for graph convolutional network into Chen's implementation. In all experiments, we adopt the ResNet-10 (He et al., 2016) model as the backbone network for our feature encoder  $E$ .
+
+We present the average results over 1,000 trials for all the experiments. In each trial, we randomly sample  $N_{w}$  categories (e.g., 5 classes for 5-way classification). For each category, we randomly select  $N_{s}$  images (e.g., 1-shot or 5-shot) for the support set  $\mathcal{X}_s$  and 16 images for the query set  $\mathcal{X}_q$ . We discuss the implementation details in Appendix A.2.
+
+Pre-trained feature encoder. Prior to the few-shot classification training stage, we first pre-train the feature encoder  $E$  by minimizing the standard cross-entropy classification loss on the 64 training categories in the mini-ImageNet dataset. This strategy can significantly improve the performance of
+
+Table 1: Few-shot classification results trained with the mini-ImageNet dataset. We train the model on the mini-ImageNet domain and evaluate the trained model on another domain. FT indicates that we apply the feature-wise transformation layers with empirically determined hyperparameters to train the model.  
+
+<table><tr><td>5-way 1-Shot</td><td>FT</td><td>mini-ImageNet</td><td>CUB</td><td>Cars</td><td>Places</td><td>Plantae</td></tr><tr><td rowspan="2">MatchingNet</td><td>-</td><td>59.10 ± 0.64%</td><td>35.89 ± 0.51%</td><td>30.77 ± 0.47%</td><td>49.86 ± 0.79%</td><td>32.70 ± 0.60%</td></tr><tr><td>✓</td><td>58.76 ± 0.61%</td><td>36.61 ± 0.53%</td><td>29.82 ± 0.44%</td><td>51.07 ± 0.68%</td><td>34.48 ± 0.50%</td></tr><tr><td rowspan="2">RelationNet</td><td>-</td><td>57.80 ± 0.88%</td><td>42.44 ± 0.77%</td><td>29.11 ± 0.60%</td><td>48.64 ± 0.85%</td><td>33.17 ± 0.64%</td></tr><tr><td>✓</td><td>58.64 ± 0.85%</td><td>44.07 ± 0.77%</td><td>28.63 ± 0.59%</td><td>50.68 ± 0.87%</td><td>33.14 ± 0.62%</td></tr><tr><td rowspan="2">GNN</td><td>-</td><td>60.77 ± 0.75%</td><td>45.69 ± 0.68%</td><td>31.79 ± 0.51%</td><td>53.10 ± 0.80%</td><td>35.60 ± 0.56%</td></tr><tr><td>✓</td><td>66.32 ± 0.80%</td><td>47.47 ± 0.75%</td><td>31.61 ± 0.53%</td><td>55.77 ± 0.79%</td><td>35.95 ± 0.58%</td></tr><tr><td>5-way 5-Shot</td><td>FT</td><td>mini-ImageNet</td><td>CUB</td><td>Cars</td><td>Places</td><td>Plantae</td></tr><tr><td rowspan="2">MatchingNet</td><td>-</td><td>70.96 ± 0.65%</td><td>51.37 ± 0.77%</td><td>38.99 ± 0.64%</td><td>63.16 ± 0.77%</td><td>46.53 ± 0.68%</td></tr><tr><td>✓</td><td>72.53 ± 0.69%</td><td>55.23 ± 0.83%</td><td>41.24 ± 0.65%</td><td>64.55 ± 0.75%</td><td>41.69 ± 0.63%</td></tr><tr><td rowspan="2">RelationNet</td><td>-</td><td>71.00 ± 0.69%</td><td>57.77 ± 0.69%</td><td>37.33 ± 0.68%</td><td>63.32 ± 0.76%</td><td>44.00 ± 0.60%</td></tr><tr><td>✓</td><td>73.78 ± 0.64%</td><td>59.46 ± 0.71%</td><td>39.91 ± 0.69%</td><td>66.28 ± 0.72%</td><td>45.08 ± 0.59%</td></tr><tr><td rowspan="2">GNN</td><td>-</td><td>80.87 ± 0.56%</td><td>62.25 ± 0.65%</td><td>44.28 ± 0.63%</td><td>70.84 ± 0.65%</td><td>52.53 ± 0.59%</td></tr><tr><td>✓</td><td>81.98 ± 0.55%</td><td>66.98 ± 0.68%</td><td>44.90 ± 0.64%</td><td>73.94 ± 0.67%</td><td>53.85 ± 0.62%</td></tr></table>
+
+metric-based models and is widely adopted in several recent frameworks (Rusu et al., 2019; Gidaris & Komodakis, 2018; Lifchitz et al., 2019).
+
+# 4.2 FEATURE-WISE TRANSFORMATION WITH MANUAL PARAMETER TUNING
+
+We train the model using the mini-[ImageNet dataset and evaluate the trained model with four other unseen domains: CUB, Cars, Places, and Plantae. We add the proposed feature-wise transformation layers after the last batch normalization layer of all the residual blocks in the feature encoder  $E$  during the training stage. We empirically set  $\theta_{\gamma}$  and  $\theta_{\beta}$  in all feature-wise transformation layers to be 0.3 and 0.5, respectively. Table 1 shows the metric-based model trained with the feature-wise transformation layers performs favorably against the individual baselines. We attribute the improvement of generalization to the use of the proposed layers for making the feature encoder  $E$  produce more diverse feature distributions in the training stage. As a by-product, we also observe the improvement on the seen domain (i.e., mini-[ImageNet) since there is still a slight discrepancy between the feature distributions extracted from the training and testing sets of the same domain. It is noteworthy that we also compare the proposed method with several recent approaches (e.g., Lee et al. (2019)) in Table 8 and Table 9. With the proposed feature-wise transformation layers, the GNN (Garcia & Bruna, 2018) model performs favorably against the state-of-the-art frameworks on both the seen domain (i.e., mini-[ImageNet) and unseen domains.
+
+# 4.3 GENERALIZATION FROM MULTIPLE DOMAINS
+
+Here we validate the effectiveness of the proposed learning-to-learn algorithm for optimizing the hyper-parameters of the feature-wise transformation layers. We compare the metric-model trained with the proposed learning procedure to the model trained with the pre-determined feature-wise transformation layers. The leave-one-out setting is used to select one domain from the CUB, Cars, Places, and Plantae as the unseen domain for the evaluation. The mini-[ImageNet] and the remaining domains serve as the seen domains for training the model. Since we select the training iteration according to the validation performance on the mini-[ImageNet] domain for evaluation, we do not consider the mini-[ImageNet] as the unseen domain. We present the results in Table 2. We denote FT and LFT as applying pre-determined feature-wise transformation layers and those layers optimized with the proposed learning-to-learn algorithm, respectively. The models optimized with proposed learning scheme outperforms those trained with the pre-determined feature-wise transformation layers since the optimized feature-wise transformation layers can better capture the variation of feature distributions across different domains. Table 1 and Table 2 show that the proposed feature-wise transformation layers together with the learning-to-learn algorithm effectively mitigate the domain shift problem for metric-based frameworks.
+
+Table 2: Few-shot classification results trained with multiple datasets. We use the leave-one-out setting to select the unseen domain and train the model as well as the feature-wise transformation layers using Algorithm 1. FT and LFT indicate applying the pre-determined and learning-to-learned feature-wise transformation, respectively.  
+
+<table><tr><td>5-way 1-Shot</td><td></td><td>CUB</td><td>Cars</td><td>Places</td><td>Plantae</td></tr><tr><td rowspan="3">MatchingNet</td><td>-</td><td>37.90 ± 0.55%</td><td>28.96 ± 0.45%</td><td>49.01 ± 0.65%</td><td>33.21 ± 0.51%</td></tr><tr><td>FT</td><td>41.74 ± 0.59%</td><td>28.30 ± 0.44%</td><td>48.77 ± 0.65%</td><td>32.15 ± 0.50%</td></tr><tr><td>LFT</td><td>43.29 ± 0.59%</td><td>30.62 ± 0.48%</td><td>52.51 ± 0.67%</td><td>35.12 ± 0.54%</td></tr><tr><td rowspan="3">RelationNet</td><td>-</td><td>44.33 ± 0.59%</td><td>29.53 ± 0.45%</td><td>47.76 ± 0.63%</td><td>33.76 ± 0.52%</td></tr><tr><td>FT</td><td>44.67 ± 0.58%</td><td>30.38 ± 0.47%</td><td>48.40 ± 0.64%</td><td>35.40 ± 0.53%</td></tr><tr><td>LFT</td><td>48.38 ± 0.63%</td><td>32.21 ± 0.51%</td><td>50.74 ± 0.66%</td><td>35.00 ± 0.52%</td></tr><tr><td rowspan="3">GNN</td><td>-</td><td>49.46 ± 0.73%</td><td>32.95 ± 0.56%</td><td>51.39 ± 0.80%</td><td>37.15 ± 0.60%</td></tr><tr><td>FT</td><td>48.24 ± 0.75%</td><td>33.26 ± 0.56%</td><td>54.81 ± 0.81%</td><td>37.54 ± 0.62%</td></tr><tr><td>LFT</td><td>51.51 ± 0.80%</td><td>34.12 ± 0.63%</td><td>56.31 ± 0.80%</td><td>42.09 ± 0.68%</td></tr><tr><td>5-way 5-Shot</td><td></td><td>CUB</td><td>Cars</td><td>Places</td><td>Plantae</td></tr><tr><td rowspan="3">MatchingNet</td><td>-</td><td>51.92 ± 0.80%</td><td>39.87 ± 0.51%</td><td>61.82 ± 0.57%</td><td>47.29 ± 0.51%</td></tr><tr><td>FT</td><td>56.29 ± 0.80%</td><td>39.58 ± 0.54%</td><td>62.32 ± 0.58%</td><td>46.48 ± 0.52%</td></tr><tr><td>LFT</td><td>61.41 ± 0.57%</td><td>43.08 ± 0.55%</td><td>64.99 ± 0.59%</td><td>48.32 ± 0.57%</td></tr><tr><td rowspan="3">RelationNet</td><td>-</td><td>62.13 ± 0.74%</td><td>40.64 ± 0.54%</td><td>64.34 ± 0.57%</td><td>46.29 ± 0.56%</td></tr><tr><td>FT</td><td>63.64 ± 0.77%</td><td>42.24 ± 0.57%</td><td>65.42 ± 0.58%</td><td>47.81 ± 0.51%</td></tr><tr><td>LFT</td><td>64.99 ± 0.54%</td><td>43.44 ± 0.59%</td><td>67.35 ± 0.54%</td><td>50.39 ± 0.52%</td></tr><tr><td rowspan="3">GNN</td><td>-</td><td>69.26 ± 0.68%</td><td>48.91 ± 0.67%</td><td>72.59 ± 0.67%</td><td>58.36 ± 0.68%</td></tr><tr><td>FT</td><td>70.37 ± 0.68%</td><td>47.68 ± 0.63%</td><td>74.48 ± 0.70%</td><td>57.85 ± 0.68%</td></tr><tr><td>LFT</td><td>73.11 ± 0.68%</td><td>49.88 ± 0.67%</td><td>77.05 ± 0.65%</td><td>58.84 ± 0.66%</td></tr></table>
+
+![](images/f5a646bb7cc007aa62876ee892aed10fb77cc4f82358dc4ce8f0f9b5dff809da.jpg)  
+(a) RelationNet
+
+![](images/9d12d9881d0e6f1b31e4884773e40ff79b91e77baa190a78ed2715174cb51179.jpg)  
+Figure 3: T-SNE visualization of the image features extracted from tasks in different domains. We show the t-SNE visualization of the features extracted by the (a) original feature encoder  $E$ , (b) feature encoder with pre-determined feature-wise transformation layers, and (c) feature encoder with learning-to-learned feature-wise transformation.
+
+![](images/3e779dd4d978800bc87167935748ea670b9b363124cee5ea83e5535a6b64bfbc.jpg)  
+(b) RelationNet + FT  
+(c) RelationNet + LFT
+
+# Seen domains
+
+mini-ImageNet  
+Cars  
+Places  
+Plantae
+
+# Unseen domain
+
+CUB
+
+Note since the proposed learning-to-learn approach optimizes the hyper-parameters via stochastic gradient descent, it may not find the global minimum that achieves the best performance. It is certainly possible to manually find another set of hyper-parameter settings that achieves better performance. However, this requires meticulous and computationally expensive hyper-parameter tuning. Specifically, the dimension of the hyper-parameters  $\theta_{\gamma}$  and  $\theta_{\beta}$  is  $c_{i} \times 1 \times 1$  for the  $i$ -th feature-wise transformation layer in the feature encoder, where  $c_{i}$  is the number of feature channels. As there are  $n$  feature-wise transformation layers in the feature encoder  $E$ , we need to perform the hyperparameter search in a  $(c_{1} + c_{2} \cdots + c_{n}) \times 2$ -dimensional space. In practice, the dimension of the search space is 1920.
+
+Visualizing feature-wise transformed features. To demonstrate that the proposed feature-wise transformation layers can simulate various feature distributions extracted from the task in different domains, we show the t-SNE visualizations of the image features extracted by the feature encoder in the RelationNet (Sung et al., 2018) model in Figure 3. The model is trained with 5-way 5-shot classification setting on the mini-ImageNet, Cars, Places, and Plantae domains (i.e., corresponding
+
+![](images/ca3b6e05c6b7dbe627ea6c0c0a0f0f6e44cf4dcef5872ec0654157f29b838b08.jpg)  
+Figure 4: Visualization of the feature-wise transformation layers. We show the quartile visualization of the activations  $\text{softplus}(\theta_{\gamma})$  and  $\text{softplus}(\theta_{\beta})$  from each feature-wise transformation layer that are optimized by the proposed learning-to-learn algorithm.
+
+![](images/d4098240c7dadfac42e116da2be5902499fa8b518091de207311748241b56d65.jpg)
+
+to the fifth block of the second column in Table 2). We observe that the distance between features extracted from different domains becomes smaller with the help of feature-wise transformation layers. Furthermore, the proposed learning-to-learn scheme can further help the feature-wise transformation layers capture the variation of feature distributions from various domains, thus close the domain gap and improve the generalization ability of metric-based models.
+
+Visualizing feature-wise transformation layers. To better understand how the learned feature-wise transformation layers operate, we show the values of the softplus  $(\theta_{\gamma})$  and softplus  $(\theta_{\gamma})$  in the feature-wise transformation layer. Figure 4 presents the visualization. The values of scaling terms softplus  $(\theta_{\gamma})$  tend to become smaller in the deeper layers, particularly for those in the last residual block. On the other hand, the depth of the layer does not seem to have an apparent impact on the distributions of the bias terms softplus  $(\theta_{\beta})$ . The distributions are also different across different metric-based classification methods. These results suggest the importance of the proposed learning-to-learn algorithm because there does not exist a set of optimal hyper-parameters of the feature-wise transformation layers which work well with all metric-based approaches.
+
+# 5 CONCLUSIONS
+
+We propose a method to effectively enhance metric-based few-shot classification frameworks under domain shifts. The core idea of our method lies in using the feature-wise transformation layer to simulate various feature distributions extracted from the tasks in different domains. We develop a learning-to-learn approach for optimizing the hyper-parameters of the feature-wise transformation layers by simulating the generalization process using multiple seen domains. From extensive experiments, we demonstrate that our technique is applicable to different metric-based few-shot classification algorithms and show consistent improvement over the baselines.
+
+# 6 ACKNOWLEDGEMENTS
+
+This work is supported in part by the NSF CAREER Grant #1149783, the NSF Grant #1755785, and gifts from Google.
+
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+
+# A APPENDIX
+
+# A.1 DATASET COLLECTION
+
+We use five few-shot classification datasets in all of our experiments: mini-ImageNet, CUB, Cars, Places, and Plantae. We follow the setting in Ravi & Larochelle (2017) and Hilliard et al. (2018) to process mini-ImageNet and CUB datasets. As for the other datasets, we manually process the dataset by random splitting the classes. The number of training, validation, testing categories for each dataset are summarized in Table 3.
+
+Table 3: Summarization of the datasets (domains). We additionally collect and split the Cars, Places, and Plantae datasets.  
+
+<table><tr><td>Datasets</td><td>mini-ImageNet</td><td>CUB</td><td>Cars</td><td>Places</td><td>Plantae</td></tr><tr><td>Source</td><td>Deng et al. (2009)</td><td>Welinder et al. (2010)</td><td>Krause et al. (2013)</td><td>Zhou et al. (2017)</td><td>Van Horn et al. (2018)</td></tr><tr><td># Training categories</td><td>64</td><td>100</td><td>98</td><td>183</td><td>100</td></tr><tr><td># Validation categories</td><td>16</td><td>50</td><td>49</td><td>91</td><td>50</td></tr><tr><td># Testing categories</td><td>20</td><td>50</td><td>49</td><td>91</td><td>50</td></tr><tr><td>Split setting</td><td>Ravi &amp; Larochelle (2017)</td><td>Hilliard et al. (2018)</td><td>randomly split</td><td>randomly split</td><td>randomly split</td></tr></table>
+
+# A.2 ADDITIONAL IMPLEMENTATION DETAILS
+
+We use the implementation and adopt the setting of hyper-parameters from Chen et al. (Chen et al., 2019a). We train the metric-based model and feature-wise transformation layers with a learning rate of 0.001 and 40,000 iterations. For feature-wise transformation layers, we apply L2 regularization with a weight of  $10^{-8}$ . The number of inner iterations adopted in the learning-to-learn scheme is set to be 1.
+
+Matching Network. We cannot utilize the MatchingNet implementation from Chen et al. (Chen et al., 2019a) since they applied the Pytorch built-in LSTM module, which does not support second-order backpropagation. Without the second-order backpropagation, we are unable to optimize the feature augmentation layers using the proposed learning-to-learn algorithm. As a result, we implement the LSTM module for the MatchingNet model. To verify the correctness of our implementation, we evaluate the 5-way 5-shot performance with the ResNet-10 (He et al., 2016) backbone network on the mini-ImageNet dataset (Ravi & Larochelle, 2017). Our implementation reports  $68.88 \pm 0.69\%$  accuracy, which is similar to the result (i.e.,  $68.82 \pm 0.65\%$ ) reported by Chen et al. (Chen et al., 2019a). We make the source code and datasets public available to foster future progress in this field. $^{5}$
+
+# A.3 ADDITIONAL EXPERIMENTAL RESULTS
+
+Ablation study on pre-trained metric encoder. As described in Section 4.1, we pre-trained the metric encoder  $E$  by minimizing the cross-entropy classification loss using the 64 training categories from the mini-[ImageNet] dataset. To understand the impact of the pre-training, we conduct an ablation study using the leave-one-out experiment illustrated in Section 4.3. As shown in Table 4, pre-training the metric encoder  $E$  substantially improve the few-shot classification performance of metric-based frameworks. Note that such a pre-training process is also adopted by several recent frameworks (Rusu et al., 2019; Gidaris & Komodakis, 2018; Lifchitz et al., 2019) to boost the few-shot classification performance.
+
+Number of ways in testing stage. In this experiment, we consider a practical scenario that the number of ways  $N_{w}$  in the testing phase is different from that in the training stage. Since the GNN (Garcia & Bruna, 2018) framework requires the numbers of ways to be consistent in the training and testing, we evaluate the MatchingNet (Vinyals et al., 2016) and RelationNet (Sung et al., 2018) model with this setting. Table 5 reports the performances of the models trained on the miniImageNet, Cars, Places, and Plantae domains under the 5-way 5-shot setting (i.e., corresponding
+
+Table 4: Ablation study on pre-trained metric encoder. We conduct leave-one-out setting to select the unseen domain to study the effectiveness of pre-training the feature encoder  $E$  on the mini-ImageNet dataset.  
+
+<table><tr><td>1-Shot</td><td>Pre-trained</td><td>CUB</td><td>Cars</td><td>Places</td><td>Plantae</td></tr><tr><td rowspan="2">MatchingNet</td><td>-</td><td>37.37 ± 0.55%</td><td>30.60 ± 0.51%</td><td>41.42 ± 0.59%</td><td>31.93 ± 0.51%</td></tr><tr><td>✓</td><td>37.90 ± 0.55%</td><td>28.96 ± 0.45%</td><td>49.01 ± 0.65%</td><td>33.21 ± 0.51%</td></tr><tr><td rowspan="2">RelationNet</td><td>-</td><td>38.46 ± 0.56%</td><td>30.77 ± 0.51%</td><td>37.49 ± 0.58%</td><td>32.86 ± 0.53%</td></tr><tr><td>✓</td><td>44.33 ± 0.59%</td><td>29.53 ± 0.45%</td><td>47.76 ± 0.63%</td><td>33.76 ± 0.52%</td></tr><tr><td rowspan="2">GNN</td><td>-</td><td>37.21 ± 0.63%</td><td>29.01 ± 0.56%</td><td>36.06 ± 0.62%</td><td>34.99 ± 0.63%</td></tr><tr><td>✓</td><td>49.46 ± 0.73%</td><td>32.95 ± 0.56%</td><td>51.39 ± 0.80%</td><td>37.15 ± 0.60%</td></tr><tr><td>5-Shot</td><td>Pre-trained</td><td>CUB</td><td>Cars</td><td>Places</td><td>Plantae</td></tr><tr><td rowspan="2">MatchingNet</td><td>-</td><td>49.83 ± 0.55%</td><td>39.41 ± 0.53%</td><td>59.18 ± 0.60%</td><td>43.53 ± 0.53%</td></tr><tr><td>✓</td><td>51.92 ± 0.80%</td><td>39.87 ± 0.51%</td><td>61.82 ± 0.57%</td><td>47.29 ± 0.51%</td></tr><tr><td rowspan="2">RelationNet</td><td>-</td><td>55.85 ± 0.55%</td><td>42.55 ± 0.58%</td><td>59.85 ± 0.54%</td><td>45.24 ± 0.55%</td></tr><tr><td>✓</td><td>62.13 ± 0.74%</td><td>40.64 ± 0.54%</td><td>64.34 ± 0.57%</td><td>46.29 ± 0.56%</td></tr><tr><td rowspan="2">GNN</td><td>-</td><td>60.13 ± 0.64%</td><td>43.60 ± 0.67%</td><td>56.67 ± 0.64%</td><td>49.17 ± 0.62%</td></tr><tr><td>✓</td><td>69.26 ± 0.68%</td><td>48.91 ± 0.67%</td><td>72.59 ± 0.67%</td><td>58.36 ± 0.68%</td></tr></table>
+
+Table 5: Few-shot classification results under various numbers of ways in testing stage. We compare the 5-shot performance under various number of ways in the testing phase. The CUB dataset is select as the testing (unseen) domain. All the models are trained with 5-way 5-shot setting.  
+
+<table><tr><td>5-Shot</td><td></td><td>CUB 2-way</td><td>CUB 5-way</td><td>CUB 10-way</td><td>CUB 20-way</td></tr><tr><td rowspan="3">MatchingNet</td><td>-</td><td>78.46 ± 0.78%</td><td>51.92 ± 0.80%</td><td>38.22 ± 0.38%</td><td>26.17 ± 0.24%</td></tr><tr><td>FT</td><td>80.74 ± 0.77%</td><td>56.29 ± 0.80%</td><td>41.09 ± 0.39%</td><td>29.19 ± 0.24%</td></tr><tr><td>LFT</td><td>83.88 ± 0.72%</td><td>61.41 ± 0.57%</td><td>45.69 ± 0.39%</td><td>32.81 ± 0.23%</td></tr><tr><td rowspan="3">RelationNet</td><td>-</td><td>84.25 ± 0.72%</td><td>62.13 ± 0.74%</td><td>47.15 ± 0.40%</td><td>34.52 ± 0.24%</td></tr><tr><td>FT</td><td>85.48 ± 0.69%</td><td>63.64 ± 0.77%</td><td>48.35 ± 0.38%</td><td>35.30 ± 0.24%</td></tr><tr><td>LFT</td><td>85.44 ± 0.72%</td><td>64.99 ± 0.54%</td><td>49.90 ± 0.40%</td><td>37.20 ± 0.25%</td></tr></table>
+
+to the fourth and fifth block of the second column in Table 2). Our proposed learning-to-learned feature-wise transformation layers are capable of improving the generalization of metric-based models to the unseen domain under various numbers of ways in the testing stage.
+
+Pre-determined hyper-parameters of feature-wise transformation layers. We demonstrate the difficulty to hand-tune the hyper-parameters of the proposed feature-wise transformation layers in this experiment. Different from the setting described in Section 4.2, we set the hyper-parameters  $\theta_{\gamma}$  and  $\theta_{\beta}$  in all feature-wise transformation layers to be 1. The model is trained under 5-way setting using the mini-[ImageNet domain, and evaluate it on the other domains. We report the 1-shot and 5-shot performance in Table 6. We denote applying feature-wise transformation layers with  $\{\theta_{\gamma},\theta_{\beta}\} = \{0.3,0.5\}$  as FT and those with  $\{\theta_{\gamma},\theta_{\beta}\} = \{1,1\}$  as FT*. We observe that the metric-based models applied with FT perform favorably against those applied with FT*. In several cases, applying FT* even yields inferior results compared to the original training without the feature-wise transformation layers. This suggests the difficulty of hand-tuning the hyper-parameters and the importance of the proposed learning-to-learn scheme for optimizing the hyper-parameters of the feature-wise transformation layers.
+
+Hyper-parameter initialization for learning-to-learn. For all the experiments, we initialize the parameters  $\theta_{\gamma}$  and  $\theta_{\beta}$  to 0.3 and 0.5, which we empirically determine in Section 4.2, to train the feature-wise transformation layers. In practice, we find that the cross-domain performance is not sensitive as long as the initialized values are within the same order (e.g., 0.1 and 0.3). Here we report the results of training the RelationNet with the initialization  $\{\theta_{\gamma}, \theta_{\beta}\} = \{0.1, 0.3\}$ . In this experiment, we use the CUB dataset as the unseen domain for evaluation and conduct the training
+
+Table 6: Few-shot classification results by applying different pre-determined hyper-parameters of feature-wise transformation layers. We train the model on the mini-ImageNet with a different set of pre-determined hyper-parameters of feature-wise transformation layers. FT and FT* indicate that we apply the feature-wise transformation layers with hyper-parameters  $\{\theta_{\gamma},\theta_{\beta}\}$  to be  $\{0.3,0.5\}$  and  $\{1,1\}$ , respectively.  
+
+<table><tr><td>1-Shot</td><td></td><td>mini-ImageNet</td><td>CUB</td><td>Cars</td><td>Places</td><td>Plantae</td></tr><tr><td rowspan="2">MatchingNet</td><td>FT</td><td>58.76 ± 0.61%</td><td>36.61 ± 0.53%</td><td>29.82 ± 0.44%</td><td>51.07 ± 0.68%</td><td>33.48 ± 0.50%</td></tr><tr><td>FT*</td><td>51.66 ± 0.64%</td><td>31.74 ± 0.51%</td><td>27.08 ± 0.41%</td><td>45.04 ± 0.64%</td><td>28.73 ± 0.42%</td></tr><tr><td rowspan="2">RelationNet</td><td>FT</td><td>58.64 ± 0.85%</td><td>44.07 ± 0.77%</td><td>28.63 ± 0.59%</td><td>50.68 ± 0.87%</td><td>33.14 ± 0.62%</td></tr><tr><td>FT*</td><td>57.45 ± 0.66%</td><td>40.20 ± 0.53%</td><td>29.15 ± 0.45%</td><td>49.40 ± 0.64%</td><td>33.21 ± 0.47%</td></tr><tr><td rowspan="2">GNN</td><td>FT</td><td>66.32 ± 0.80%</td><td>47.47 ± 0.75%</td><td>31.61 ± 0.53%</td><td>55.77 ± 0.79%</td><td>35.95 ± 0.58%</td></tr><tr><td>FT*</td><td>62.63 ± 0.76%</td><td>44.61 ± 0.66%</td><td>31.56 ± 0.52%</td><td>53.39 ± 0.74%</td><td>36.73 ± 0.57%</td></tr><tr><td>5-Shot</td><td></td><td>mini-ImageNet</td><td>CUB</td><td>Cars</td><td>Places</td><td>Plantae</td></tr><tr><td rowspan="2">MatchingNet</td><td>FT</td><td>72.53 ± 0.69%</td><td>55.23 ± 0.83%</td><td>41.24 ± 0.65%</td><td>64.55 ± 0.75%</td><td>41.69 ± 0.63%</td></tr><tr><td>FT*</td><td>64.93 ± 0.60%</td><td>42.83 ± 0.61%</td><td>32.19 ± 0.48%</td><td>59.47 ± 0.63%</td><td>39.61 ± 0.49%</td></tr><tr><td rowspan="2">RelationNet</td><td>FT</td><td>73.78 ± 0.64%</td><td>59.46 ± 0.71%</td><td>39.91 ± 0.69%</td><td>66.28 ± 0.72%</td><td>45.08 ± 0.59%</td></tr><tr><td>FT*</td><td>72.79 ± 0.64%</td><td>59.18 ± 0.57%</td><td>40.54 ± 0.54%</td><td>65.73 ± 0.52%</td><td>43.64 ± 0.49%</td></tr><tr><td rowspan="2">GNN</td><td>FT</td><td>81.98 ± 0.55%</td><td>66.98 ± 0.68%</td><td>44.90 ± 0.64%</td><td>73.94 ± 0.67%</td><td>53.85 ± 0.62%</td></tr><tr><td>FT*</td><td>82.40 ± 0.58%</td><td>66.33 ± 0.73%</td><td>47.63 ± 0.64%</td><td>75.48 ± 0.65%</td><td>51.92 ± 0.59%</td></tr></table>
+
+Table 7: Few-shot classification results by applying the learning-to-learn approach trained with a single seen domain. We attempt to conduct the proposed learning-to-learn trainin with as single seen domain, denoted as LFT*. We train the model using the mini-ImageNet dataset and report the 5-way 5-shot classification accuracy.  
+
+<table><tr><td colspan="2">5-way 5-Shot</td><td>mini-ImageNet</td><td>CUB</td><td>Cars</td><td>Places</td><td>Plantae</td></tr><tr><td>RelationNet</td><td>FT</td><td>73.78 ± 0.64%</td><td>59.46 ± 0.71%</td><td>39.91 ± 0.69%</td><td>66.28 ± 0.72%</td><td>45.08 ± 0.59%</td></tr><tr><td>RelationNet</td><td>LFT*</td><td>73.50 ± 0.50%</td><td>58.19 ± 0.52%</td><td>39.35 ± 0.54%</td><td>66.17 ± 0.57%</td><td>46.75 ± 0.51%</td></tr></table>
+
+described in Algorithm 1. The 5-way 5-shot classification accuracy on the CUB dataset is  $64.79 \pm 0.55\%$ , which is similar to the one we report in Table 2 (i.e.,  $64.99 \pm 0.54\%$ ).
+
+Learning-to-learn using a single domain. The proposed learning-to-learn method requires multiple domains for training. Here we apply the learning-to-learn method based on one single domain. More specifically, we randomly sample two different tasks from the mini-[ImageNet dataset in each iteration of the training process described in Algorithm 1. One task serves as the pseudo-seen task, while the other one serves as the pseudo-unseen task. We train the RelationNet model with the above-mentioned setting on 5-way 5-shot classification using the mini-[ImageNet dataset only. As shown in Table 7, the performance improvement of the RelationNet model is not significant compared to the model trained with pre-determined hyper-parameters  $\{\theta_{\gamma},\theta_{\beta}\} = \{0.3,0.5\}$ . This suggests that utilizing one single domain for learning the feature-wise transformation is not as effective as that using multiple domains (demonstrated in Table 2). The reason is that during the training phase, the discrepancy of the feature distributions extracted from the pseudo-seen and pseudo-unseen tasks is not as significant since these two tasks are sampled from the same domain. As a result, the hyper-parameters  $\{\theta_{\gamma},\theta_{\beta}\}$  cannot be effectively optimized to capture the variation of feature distributions sampled from various domains.
+
+Comparison with the state-of-the-art few-shot classification on the mini-[ImageNet. We compare the metric-based frameworks applied with the proposed feature-wise transformation layers to the state-of-the-art few-shot classification methods in Table 8. In this experiment, we train the model with the pre-determined hyper-parameters of feature-wise transformation layers on the training set of the mini-[ImageNet (Ravi & Larochelle, 2017) dataset. Note that we do not use the learned version of the feature-wise transformation layers in the training to ensure fair comparison. Combining Table 4 and Table 8, we observe that the metric-based frameworks train with 1) pre-trained feature encoder, and 2) feature-wise transformation layers with carefully hand-tuned hyper-parameters can demonstrate competitive performance.
+
+Table 8: Comparison to the state-of-the-art few-shot classification algorithms. We compare the metric-based frameworks applied with the proposed feature-wise transformation layers using predetermined hyper-parameter  $\{\theta_{\gamma},\theta_{\beta}\} = \{0.3,0.5\}$  (denoted as FT) to other state-of-the-art few-shot classification methods. Note that all the methods are trained only on the mini-ImageNet dataset. To ensure fair comparisons with other methods, we are unable to use the learned version of the feature-wise transformation layers described in Section 3.3. By augmenting existing metric-based few-shot classification models with the proposed feature-wise transformation layer, we obtain competitive performance when compared with many recent and more complicated methods. The best results in each block are highlighted in bold.
+
+<table><tr><td>backbone</td><td colspan="2">method</td><td>5-way 1-shot</td><td>5-way 5-shot</td></tr><tr><td rowspan="4">ResNet-12</td><td colspan="2">TADAM (Oreshkin et al., 2018)</td><td>58.50 ± 0.30%</td><td>76.70 ± 0.30%</td></tr><tr><td colspan="2">DC (Lifchitz et al., 2019)</td><td>62.53 ± 0.19%</td><td>78.95 ± 0.13%</td></tr><tr><td colspan="2">DC + IMP (Lifchitz et al., 2019)</td><td>-</td><td>79.77 ± 0.19%</td></tr><tr><td colspan="2">MetaOptNet-SVM-trainval (Lee et al., 2019)</td><td>64.09 ± 0.62%</td><td>80.00 ± 0.45%</td></tr><tr><td rowspan="2">WRN-28</td><td colspan="2">Qiao et al. (2018)</td><td>59.60 ± 0.41%</td><td>77.74 ± 0.19%</td></tr><tr><td colspan="2">LEO (Rusu et al., 2019)</td><td>61.76 ± 0.08%</td><td>77.59 ± 0.12%</td></tr><tr><td rowspan="6">ResNet-10</td><td rowspan="2">MatchingNet</td><td>-</td><td>59.10 ± 0.64%</td><td>70.96 ± 0.65%</td></tr><tr><td>FT</td><td>58.76 ± 0.61%</td><td>72.53 ± 0.69%</td></tr><tr><td rowspan="2">RelationNet</td><td>-</td><td>57.80 ± 0.88%</td><td>71.00 ± 0.69%</td></tr><tr><td>FT</td><td>58.64 ± 0.85%</td><td>73.78 ± 0.64%</td></tr><tr><td rowspan="2">GNN</td><td>-</td><td>60.77 ± 0.75%</td><td>80.87 ± 0.56%</td></tr><tr><td>FT</td><td>66.32 ± 0.80%</td><td>81.98 ± 0.55%</td></tr></table>
+
+Table 9: Evaluation with the state-of-the-art approach under the cross-domain setting. We evaluate the metric-based frameworks with the proposed feature-wise transformation layers using pre-determined hyper-parameter  $\{\theta_{\gamma},\theta_{\beta}\} = \{0.3,0.5\}$  (denoted as FT) against the state-of-the-art MetaOptNet-SVM-trainval Lee et al. (2019) method. Note that all the methods are trained only on the mini-[ImageNet] dataset. To ensure fair comparisons with other methods, we do not use the learned version of the feature-wise transformation layers described in Section 3.3. By augmenting the existing metric-based few-shot classification models with the proposed feature-wise transformation layer, we obtain competitive performance when compared with recent and more complicated methods. The best results are highlighted in bold.
+
+<table><tr><td>method</td><td></td><td>CUB</td><td>Cars</td><td>Places</td><td>Plantae</td></tr><tr><td colspan="2">MetaOptNet-SVM-trainval</td><td>54.67 ± 0.56%</td><td>45.90 ± 0.49%</td><td>65.83 ± 0.57%</td><td>46.48 ± 0.52%</td></tr><tr><td rowspan="2">MatchingNet</td><td>-</td><td>51.37 ± 0.77%</td><td>38.99 ± 0.64%</td><td>63.16 ± 0.77%</td><td>46.53 ± 0.68%</td></tr><tr><td>FT</td><td>55.23 ± 0.83%</td><td>41.24 ± 0.65%</td><td>64.55 ± 0.75%</td><td>41.69 ± 0.63%</td></tr><tr><td rowspan="2">RelationNet</td><td>-</td><td>57.77 ± 0.69%</td><td>37.33 ± 0.68%</td><td>63.32 ± 0.76%</td><td>44.00 ± 0.60%</td></tr><tr><td>FT</td><td>59.46 ± 0.71%</td><td>39.91 ± 0.69%</td><td>66.28 ± 0.72%</td><td>45.08 ± 0.59%</td></tr><tr><td rowspan="2">GNN</td><td>-</td><td>62.25 ± 0.65%</td><td>44.28 ± 0.63%</td><td>70.84 ± 0.65%</td><td>52.53 ± 0.59%</td></tr><tr><td>FT</td><td>66.98 ± 0.68%</td><td>44.90 ± 0.64%</td><td>73.94 ± 0.67%</td><td>53.85 ± 0.62%</td></tr></table>
+
+Comparison to the state-of-the-art few-shot classification under domain shift. We evaluate the metric-based frameworks with the proposed feature-wise transformation layers and the state-of-the-art MetaOptNet approach Lee et al. (2019). We use the model trained on the mini-ImageNet dataset provided by the authors for the evaluation on the other datasets. As shown in Table 9, while the MetaOptNet method achieves state-of-the-art performance on the mini-ImageNet dataset, this approach also suffers from the domain shifts in the cross-domain setting. Training the GNN framework with the pre-trained feature encoder and the proposed feature-wise transformation layers performs favorably against the MetaOptNet method under the cross-domain setting.
\ No newline at end of file
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+# DDSP: DIFFERENTIABLE DIGITAL SIGNAL PROCESSING
+
+Jesse Engel, Lamtharn Hantrakul, Chenjie Gu, & Adam Roberts
+
+Google Research, Brain Team
+
+Mountain View, CA 94043, USA
+
+{jesseengel,hanoih,gcj,adarob}@google.com
+
+# ABSTRACT
+
+Most generative models of audio directly generate samples in one of two domains: time or frequency. While sufficient to express any signal, these representations are inefficient, as they do not utilize existing knowledge of how sound is generated and perceived. A third approach (vocoders/synthesizers) successfully incorporates strong domain knowledge of signal processing and perception, but has been less actively researched due to limited expressivity and difficulty integrating with modern auto-differentiation-based machine learning methods. In this paper, we introduce the Differentiable Digital Signal Processing (DDSP) library, which enables direct integration of classic signal processing elements with deep learning methods. Focusing on audio synthesis, we achieve high-fidelity generation without the need for large autoregressive models or adversarial losses, demonstrating that DDSP enables utilizing strong inductive biases without losing the expressive power of neural networks. Further, we show that combining interpretable modules permits manipulation of each separate model component, with applications such as independent control of pitch and loudness, realistic extrapolation to pitches not seen during training, blind dereverberation of room acoustics, transfer of extracted room acoustics to new environments, and transformation of timbre between disparate sources. In short, DDSP enables an interpretable and modular approach to generative modeling, without sacrificing the benefits of deep learning. The library is publicly available<sup>1</sup> and we welcome further contributions from the community and domain experts.
+
+# 1 INTRODUCTION
+
+Neural networks are universal function approximators in the asymptotic limit (Hornik et al., 1989), but their practical success is largely due to the use of strong structural priors such as convolution (LeCun et al., 1989), recurrence (Sutskever et al., 2014; Williams & Zipser, 1990; Werbos, 1990), and self-attention (Vaswani et al., 2017). These architectural constraints promote generalization and data efficiency to the extent that they align with the data domain. From this perspective, end-to-end learning relies on structural priors to scale, but the practitioner's toolbox is limited to functions that can be expressed differentiably. Here, we increase the size of that toolbox by introducing the Differentiable Digital Signal Processing (DDSP) library, which integrates interpretable signal processing elements into modern automatic differentiation software (TensorFlow). While this approach has broad applicability, we highlight its potential in this paper through exploring the example of audio synthesis.
+
+Objects have a natural tendency to periodically vibrate. Small shape displacements are usually restored with elastic forces that conserve energy (similar to a canonical mass on a spring), leading to harmonic oscillation between kinetic and potential energy (Smith, 2010). Accordingly, human hearing has evolved to be highly sensitive to phase-coherent oscillation, decomposing audio into spectrotemporal responses through the resonant properties of the basilar membrane and tonotopic
+
+mappings into the auditory cortex (Moerel et al., 2012; Chi et al., 2005; Theunissen & Elie, 2014). However, neural synthesis models often do not exploit this periodic structure for generation and perception.
+
+# 1.1 CHALLENGES OF NEURAL AUDIO SYNTHESIS
+
+As shown in Figure 1, most neural synthesis models generate waveforms directly in the time domain, or from their corresponding Fourier coefficients in the frequency domain. While these representations are general and can represent any waveform, they are not free from bias. This is because they often apply a prior over generating audio with aligned wave packets rather than oscillations. For example, stripped convolution models—such as SING (Defossez et al., 2018), MCNN (Arik et al., 2019), and WaveGAN (Donahue et al., 2019)—generate waveforms directly with overlapping frames. Since audio oscillates at many frequencies, all with different periods from the fixed frame hop size, the model must precisely align waveforms between different frames and learn filters to cover all possible phase variations. This challenge is visualized on the left of Figure 1.
+
+Fourier-based models—such as Tacotron (Wang et al., 2017) and GANSynth (Engel et al., 2019)—also suffer from the phase-alignment problem, as the Short-time Fourier Transform (STFT) is a representation over windowed wave packets. Additionally, they must contend with spectral leakage, where sinusoids at multiple neighboring frequencies and phases must be combined to represent a single sinusoid when Fourier basis frequencies do not perfectly match the audio. This effect can be seen in the middle diagram of Figure 1.
+
+Autoregressive waveform models—such as WaveNet (Oord et al., 2016), SampleRNN (Mehri et al., 2016), and WaveRNN (Kalchbrenner et al., 2018)—avoid these issues by generating the waveform a single sample at a time. They are not constrained by the bias over generating wave packets and can express arbitrary waveforms. However, they require larger and more data-hungry networks, as they do not take advantage of a bias over oscillation (size comparisons can be found in Table B.6). Furthermore, the use of teacher-forcing during training leads to exposure bias during generation, where errors with feedback can compound. It also makes them incompatible with perceptual losses such as spectral features (Defossez et al., 2018), pretrained models (Dosovitskiy & Brox, 2016), and discriminators (Engel et al., 2019). This adds further inefficiency to these models, as a waveform's shape does not perfectly correspond to perception. For example, the three waveforms on the right of Figure 1 sound identical (a relative phase offset of the harmonics) but would present different losses to an autoregressive model.
+
+![](images/6f006243c3dd8c50635f86c66b4ff42dbde3e406a05016ce82db3c71cb8b7fd9.jpg)  
+Strided Convolution  
+(WaveGAN, SING)  
+Figure 1: Challenges of neural audio synthesis. Full description provided in Section 1.1.
+
+![](images/5e2ed5c3fabdcdea155524e4728edd56e589a6b1770cead2703eaf88846eeeb3.jpg)  
+Phase Alignment  
+Fourier Representation  
+Spectral Leakage  
+(Tacotron, GANSynth)
+
+![](images/5b7dd7dc33c0f6ff8cbf2776791a85a0d7ea997f6c3e931e5dbedfc56a64e199.jpg)  
+Autoregressive  
+Waveform != Perception  
+(WaveNet, SampleRNN)
+
+# 1.2 OSCILLATOR MODELS
+
+Rather than predicting waveforms or Fourier coefficients, a third model class directly generates audio with oscillators. Known as vocoders or synthesizers, these models are physically and perceptually motivated and have a long history of research and applications (Beauchamp, 2007; Morise et al., 2016). These "analysis/synthesis" models use expert knowledge and hand-tuned heuristics to extract synthesis parameters (analysis) that are interpretable (loudness and frequencies) and can be used by the generative algorithm (synthesis).
+
+Neural networks have been used previously to some success in modeling pre-extracted synthesis parameters (Blaauw & Bonada, 2017; Chandna et al., 2019), but these models fall short of end-to-end learning. The analysis parameters must still be tuned by hand and gradients cannot flow through the synthesis procedure. As a result, small errors in parameters can lead to large errors in
+
+the audio that cannot propagate back to the network. Crucially, the realism of vocoders is limited by the expressivity of a given analysis/synthesis pair.
+
+# 1.3 CONTRIBUTIONS
+
+In this paper, we overcome the limitations outlined above by using the DDSP library to implement fully differentiable synthesizers and audio effects. DDSP models combine the strengths of the above approaches, benefiting from the inductive bias of using oscillators, while retaining the expressive power of neural networks and end-to-end training.
+
+We demonstrate that models employing DDSP components are capable of generating high-fidelity audio without autoregressive or adversarial losses. Further, we show the interpretability and modularity of these models enable:
+
+- Independent control over pitch and loudness during synthesis.  
+Realistic extrapolation to pitches not seen during training.  
+- Blind dereverberation of audio through seperate modelling of room acoustics.  
+- Transfer of extracted room acoustics to new environments.  
+- Timbre transfer between disparate sources, converting a singing voice into a violin.  
+- Smaller network sizes than comparable neural synthesizers.
+
+Audio samples for all examples and figures are provided in the online supplement<sup>2</sup>. We highly encourage readers to listen to the samples as part of reading the paper.
+
+# 2 RELATED WORK
+
+Vocoders. Vocoders come in several varieties. Source-filter/subtractive models are inspired by the human vocal tract and dynamically filter a harmonically rich source signal (Flanagan, 2013), while sinusoidal/additive models generate sound as the combination of a set of time-varying sine waves (McAulay & Quatieri, 1986; Serra & Smith, 1990). Additive models are strictly more expressive than subtractive models but have more parameters as each sinusoid has its own time-varying loudness and frequency. This work builds a differentiable synthesizer off the Harmonic plus Noise model (Serra & Smith, 1990; Beauchamp, 2007): an additive synthesizer combines sinusoids in harmonic (integer) ratios of a fundamental frequency alongside a time-varying filtered noise signal.
+
+Synthesizers. A separate thread of research has tried to estimate parameters for commercial synthesizers using gradient-free methods (Huang et al., 2014; Hoffman & Cook, 2006). Synthesizer outputs modeled with a variational autoencoder were recently used as a "world model" (Ha & Schmidhuber, 2018) to pass approximate gradients to a controller during learning (Esling et al., 2019). DDSP differs from black-box approaches to modeling existing synthesizers; it is a toolkit of differentiable DSP components for end-to-end learning.
+
+Neural Source Filter (NSF). Perhaps closest to this work, promising speech synthesis results were recently achieved using a differentiable waveshaping synthesizer (Wang et al., 2019). The NSF can be seen as a specific DDSP model, that uses convolutional waveshaping of a sinusoidal oscillator to create harmonic content, rather than additive synthesis explored in this work. Both works also generate audio in the time domain and impose multi-scale spectrograms losses in the frequency domain. A key contribution of this work is to highlight how these models are part of a common family of techniques and to release a modular library that makes them accessible by leveraging automatic differentiation to easily mix and match components at a high level.
+
+# 3 DDSP COMPONENTS
+
+Many DSP operations can be expressed as functions in modern automatic differentiation software. We express core components as feedforward functions, allowing efficient implementation on parallel
+
+hardware such as GPUs and TPUs, and generation of samples during training. These components include oscillators, envelopes, and filters (linear-time-varying finite-impulse-response, LTV-FIR).
+
+# 3.1 SPECTRAL MODELING SYNTHESIS
+
+Here, as an example DDSP model, we implement a differentiable version of Spectral Modeling Synthesis (SMS) Serra & Smith (1990). This model generates sound by combining an additive synthesizer (adding together many sinusoids) with a subtractive synthesizer (filtering white noise). We choose SMS because, despite being parametric, it is a highly expressive model of sound, and has found widespread adoption in tasks as diverse as spectral morphing, time stretching, pitch shifting, source separation, transcription, and even as a general purpose audio codec in MPEG-4 (Tellman et al., 1995; Klapuri et al., 2000; Purnhagen & Meine, 2000).
+
+As we only consider monophonic sources in these experiments, we use the Harmonic plus Noise model, that further constrains sinusoids to be integer multiples of a fundamental frequency (Beauchamp, 2007). One of the reasons that SMS is more expressive than many other parametric models because it has so many more parameters. For example, in the 4 seconds of  $16\mathrm{kHz}$  audio in the datasets considered here, the synthesizer coefficients actually have  $\sim 2.5$  times more dimensions than the audio waveform itself ((1 amplitude + 100 harmonics + 65 noise band magnitudes) * 1000 timesteps = 165,000 dimensions, vs. 64,000 audio samples). This makes them amenable to control by a neural network, as it would be difficult to realistically specify all these parameters by hand.
+
+# 3.2 HARMONIC OSCILLATOR / ADDITIVE SYNTHESIZER
+
+At the heart of the synthesis techniques explored in this paper is the sinusoidal oscillator. A bank of oscillators that outputs a signal  $x(n)$  over discrete time steps,  $n$ , can be expressed as:
+
+$$
+x (n) = \sum_ {k = 1} ^ {K} A _ {k} (n) \sin \left(\phi_ {k} (n)\right), \tag {1}
+$$
+
+where  $A_{k}(n)$  is the time-varying amplitude of the  $k$ -th sinusoidal component and  $\phi_{k}(n)$  is its instantaneous phase. The phase  $\phi_{k}(n)$  is obtained by integrating the instantaneous frequency  $f_{k}(n)$ :
+
+$$
+\phi_ {k} (n) = 2 \pi \sum_ {m = 0} ^ {n} f _ {k} (m) + \phi_ {0, k}, \tag {2}
+$$
+
+where  $\phi_{0,k}$  is the initial phase that can be randomized, fixed, or learned.
+
+For a harmonic oscillator, all the sinusoidal frequencies are harmonic (integer) multiples of a fundamental frequency,  $f_{0}(n)$ , i.e.,  $f_{k}(n) = kf_{0}(n)$ . Thus the output of the harmonic oscillator is entirely parameterized by the time-varying fundamental frequency  $f_{0}(n)$  and harmonic amplitudes  $A_{k}(n)$ . To aid interpretability we further factorize the harmonic amplitudes:
+
+$$
+A _ {k} (n) = A (n) c _ {k} (n). \tag {3}
+$$
+
+into a global amplitude  $A(n)$  that controls the loudness and a normalized distribution over harmonics  $c(n)$  that determines spectral variations, where  $\sum_{k=0}^{K} c_k(n) = 1$  and  $c_k(n) \geq 0$ . We also constrain both amplitudes and harmonic distribution components to be positive through the use of a modified sigmoid nonlinearity as described in the appendix. Figure 6 provides a graphical example of the additive synthesizer. Audio is provided in our online supplement<sup>2</sup>.
+
+# 3.3 ENVELOPES
+
+The oscillator formulation above requires time-varying amplitudes and frequencies at the audio sample rate, but our neural networks operate at a slower frame rate. For instantaneous frequency upsampling, we found bilinear interpolation to be adequate. However, the amplitudes and harmonic distributions of the additive synthesizer required smoothing to prevent artifacts. We are able to achieve
+
+this with a smoothed amplitude envelope by adding overlapping Hamming windows at the center of each frame and scaled by the amplitude. For these experiments we found a 4ms (64 timesteps) hop size and 8 ms frame size (50% overlap) to be responsive to changes while removing artifacts.
+
+# 3.4 FILTER DESIGN: FREQUENCY SAMPLING METHOD
+
+Linear filter design is a cornerstone of many DSP techniques. Standard convolutional layers are equivalent to linear time invariant finite impulse response (LTI-FIR) filters. However, to ensure interpretability and prevent phase distortion, we employ the frequency sampling method to convert network outputs into impulse responses of linear-phase filters.
+
+Here, we design a neural network to predict the frequency-domain transfer functions of a FIR filter for every output frame. In particular, the neural network outputs a vector  $\mathbf{H}_l$  (and accordingly  $\mathbf{h}_l = \mathrm{IDFT}(\mathbf{H}_l)$ ) for the  $l$ -th frame of the output. We interpret  $\mathbf{H}_l$  as the frequency-domain transfer function of the corresponding FIR filter. We therefore implement a time-varying FIR filter.
+
+To apply the time-varying FIR filter to the input, we divide the audio into non-overlapping frames  $\boldsymbol{x}_l$  to match the impulse responses  $h_l$ . We then perform frame-wise convolution via multiplication of frames in the Fourier domain:  $\boldsymbol{Y}_l = \boldsymbol{H}_l\boldsymbol{X}_l$  where  $\boldsymbol{X}_l = \mathrm{DFT}(\boldsymbol{x}_l)$  and  $\boldsymbol{Y}_l = \mathrm{DFT}(\boldsymbol{y}_l)$  is the output. We recover the frame-wise filtered audio,  $\boldsymbol{y}_l = \mathrm{IDFT}(\boldsymbol{Y}_l)$ , and then overlap-add the resulting frames with the same hop size and rectangular window used to originally divide the input audio. The hop size is given by dividing the audio into equally spaced frames for each frame of conditioning. For 64000 samples and 250 frames, this corresponds to a hop size of 256.
+
+In practice, we do not use the neural network output directly as  $H_{l}$ . Instead, we apply a window function  $W$  on the network output to compute  $H_{l}$ . The shape and size of the window can be decided independently to control the time-frequency resolution trade-off of the filter. In our experiments, we default to a Hann window of size 257. Without a window, the resolution implicitly defaults to a rectangular window which is not ideal for many cases. We take care to shift the IR to zero-phase (symmetric) form before applying the window and revert to causal form before applying the filter.
+
+# 3.5 FILTERED NOISE / SUBTRACTIVE SYNTHESIZER
+
+Natural sounds contain both harmonic and stochastic components. The Harmonic plus Noise model captures this by combining the output of an additive synthesizer with a stream of filtered noise (Serra & Smith, 1990; Beauchamp, 2007). We are able to realize a differentiable filtered noise synthesizer by simply applying the LTV-FIR filter from above to a stream of uniform noise  $\mathbf{Y}_l = \mathbf{H}_l\mathbf{N}_l$  where  $\mathbf{N}_l$  is the DFT of uniform noise in domain [-1, 1].
+
+# 3.6 REVERB: LONG IMPULSE RESPONSES
+
+Room reverberation (reverb) is an essential characteristic of realistic audio, which is usually implicitly modeled by neural synthesis algorithms. In contrast, we gain interpretability by explicitly factorizing the room acoustics into a post-synthesis convolution step. A realistic room impulse response (IR) can be as long as several seconds, corresponding to extremely long convolutional kernel sizes ( $\sim 10$ -100k timesteps). Convolution via matrix multiplication scales as  $\mathcal{O}(n^3)$ , which is intractable for such large kernel sizes. Instead, we implement reverb by explicitly performing convolution as multiplication in the frequency domain, which scales as  $\mathcal{O}(n\log n)$  and does not bottleneck training.
+
+# 4 EXPERIMENTS
+
+For empirical verification of this approach, we test two DDSP autoencoder variants—supervised and unsupervised—on two different musical datasets: NSynth (Engel et al., 2017) and a collection of solo violin performances. The supervised DDSP autoencoder is conditioned on fundamental frequency (F0) and loudness features extracted from audio, while the unsupervised DDSP autoencoder learns F0 jointly with the rest of the network.
+
+![](images/e0a324f001f7c27e8944bab6e4804bb54fa7d8eea6babeed290c672e95d2c5b7.jpg)  
+Figure 2: Autoencoder architecture. Red components are part of the neural network architecture, green components are the latent representation, and yellow components are deterministic synthesizers and effects. Components with dashed borders are not used in all of our experiments. Namely,  $z$  is not used in the model trained on solo violin, and reverb is not used in the models trained on NSynth. See the appendix for more detailed diagrams of the neural network components.
+
+# 4.1 DDSP AUTOENCODER
+
+DDSP components do not put constraints on the choice of generative model (GAN, VAE, Flow, etc.), but we focus here on a deterministic autoencoder to investigate the strength of DDSP components independent of any particular approach to adversarial training, variational inference, or Jacobian design. Just as autoencoders utilizing convolutional layers outperform fully-connected autoencoders on images, we find DDSP components are able to dramatically improve autoencoder performance in the audio domain. Introducing stochastic latents (such as in GAN, VAE, and Flow models) will likely further improve performance, but we leave that to future work as it is orthogonal to the core question of DDSP component performance that we investigate in this paper.
+
+In a standard autoencoder, an encoder network  $f_{\mathrm{enc}}(\cdot)$  maps the input  $x$  to a latent representation  $z = f_{\mathrm{enc}}(x)$  and a decoder network  $f_{\mathrm{dec}}(\cdot)$  attempts to directly reconstruct the input  $\hat{x} = f_{\mathrm{dec}}(z)$ . Our architecture (Figure 2) contrasts with this approach through the use of DDSP components and a decomposed latent representation.
+
+Encoders: Detailed descriptions of the encoders are given in Section B.1. For the supervised autoencoder, the loudness  $l(t)$  is extracted directly from the audio, a pretrained CREPE model with fixed weights (Kim et al., 2018) is used as an  $f(t)$  encoder to extract the fundamental frequency, and optional encoder extracts a time-varying latent encoding  $z(t)$  of the residual information. For the  $z(t)$  encoder, MFCC coefficients (30 per a frame) are first extracted from the audio, which correspond to the smoothed spectral envelope of harmonics (Beauchamp, 2007), and transformed by a single GRU layer into 16 latent variables per a frame.
+
+For the unsupervised autoencoder, the pretrained CREPE model is replaced with a Resnet architecture (He et al., 2016) that extracts  $f(t)$  from a mel-scaled log spectrogram of the audio, and is jointly trained with the rest of the network.
+
+Decoder: A detailed description of the decoder network is given in Section B.2. The decoder network maps the tuple  $(f(t), l(t), z(t))$  to control parameters for the additive and filtered noise synthesizers described in Section 3. The synthesizers generate audio based on these parameters, and a reconstruction loss between the synthesized and original audio is minimized. The network architecture is chosen to be fairly generic (fully connected, with a single recurrent layer) to demonstrate that it is the DDSP components, and not other modeling decisions, that enables the quality of the work.
+
+Also unique to our approach, the latent  $f(t)$  is fed directly to the additive synthesizer as it has structural meaning for the synthesizer outside the context of any given dataset. As shown later in Section 5.2, this disentangled representation enables the model to both interpolate within and ex
+
+<table><tr><td></td><td>Loudness (L1)</td><td>F0 (L1)</td><td>F0 Outliers</td></tr><tr><td>Supervised</td><td></td><td></td><td></td></tr><tr><td>WaveRNN (Hantrakul et al., 2019)</td><td>0.10</td><td>1.00</td><td>0.07</td></tr><tr><td>DDSP Autoencoder</td><td>0.07</td><td>0.02</td><td>0.003</td></tr><tr><td>Unsupervised</td><td></td><td></td><td></td></tr><tr><td>DDSP Autoencoder</td><td>0.09</td><td>0.80</td><td>0.04</td></tr></table>
+
+Table 1: Resynthesis accuracies. Comparison of DDSP models to SOTA WaveRNN model provided the same conditioning information. The supervised DDSP Autoencoder and WaveRNN models use the fundamental frequency from a pretrained CREPE model, while the unsupervised DDSP autoencoder learns to infer the frequency from the audio during training.
+
+trapolate outside the data distribution. Indeed, recent work support incorporation of strong inductive biases as a prerequisite for learning disentangled representations (Locatello et al., 2018).
+
+Model Size: Table B.6, compares parameter counts for the DDSP models and comparable models including GANSynth (Engel et al., 2019), WaveRNN (Hantrakul et al., 2019), and a WaveNet Autoencoder (Engel et al., 2017). The DDSP models have the fewest parameters (up to 10 times less), despite no effort to minimize the model size in these experiments. Initial experiments with very small models (240k parameters, 300x smaller than a WaveNet Autoencoder) have less realistic outputs than the full models, but still have fairly high quality and are promising for low-latency applications, even on CPU or embedded devices. Audio samples are available in the online supplement<sup>2</sup>.
+
+# 4.2 DATASETS
+
+NSynth: We focus on a smaller subset of the NSynth dataset (Engel et al., 2017) consistent with other work (Engel et al., 2019; Hantrakul et al., 2019). It totals 70,379 examples comprised mostly of strings, brass, woodwinds and mallets with pitch labels within MIDI pitch range 24-84. We employ a 80/20 train/test split shuffling across instrument families. For the NSynth experiments, we use the autoencoder as described above (with the  $z(t)$  encoder). We experiment with both the supervised and unsupervised variants.
+
+Solo Violin: The NSynth dataset does not capture aspects of a real musical performance. Using the MusOpen royalty free music library, we collected 13 minutes of expressive, solo violin performances<sup>4</sup>. We purposefully selected pieces from a single performer (John Garner), that were monophonic and shared a consistent room environment to encourage the model to focus on performance. Like NSynth, audio is converted to mono  $16\mathrm{kHz}$  and divided into 4 second training examples (64000 samples total). Code to process the audio files into a dataset is available online.<sup>5</sup>
+
+For the solo violin experiments, we use the supervised variant of the autoencoder (without the  $z(t)$  encoder), and add a reverb module to the signal processor chain to account for room reverberation. While the room impulse response could be produced as an output of the decoder, given that the solo violin dataset has a single acoustic environment, we use a single fixed variable (4 second reverb corresponding to 64000 dimensions) for the impulse response.
+
+# 4.2.1 MULTI-SCALE SPECTRAL LOSS
+
+The primary objective of the autoencoder is to minimize reconstruction loss. However, for audio waveforms, point-wise loss on the raw waveform is not ideal, as two perceptually identical audio samples may have distinct waveforms, and point-wise similar waveforms may sound very different.
+
+Instead, we use a multi-scale spectral loss-similar to the multi-resolution spectral amplitude distance in Wang et al. (2019)-defined as follows. Given the original and synthesized audio, we compute their (magnitude) spectrogram  $S_{i}$  and  $\hat{S}_i$ , respectively, with a given FFT size  $i$ , and define the loss as the
+
+![](images/708de64634f0e2afe98953cf6892919211853b731ffd9d78649777301837d9bb.jpg)  
+Figure 3: Separate interpolations over loudness, pitch, and timbre. The conditioning features (solid lines) are extracted from two notes and linearly mixed (dark to light coloring). The features of the resynthesized audio (dashed lines) closely follow the conditioning. On the right, the latent vectors,  $z(t)$ , are interpolated, and the spectral centroid of resulting audio (thin solid lines) smoothly varies between the original samples (dark solid lines).
+
+![](images/c6ef8b9df672541b3b785283c6e37dc792231e2d6fbecba7adb6cc85ef343cdf.jpg)
+
+![](images/16863463473a9f04b6f80683ced940ac83fdc167c95a8332acd1d02121b599a8.jpg)
+
+sum of the L1 difference between  $S_{i}$  and  $\hat{S}_i$  as well as the L1 difference between  $\log S_{i}$  and  $\log \hat{S}_i$ .
+
+$$
+L _ {i} = \left\| S _ {i} - \hat {S} _ {i} \right\| _ {1} + \alpha \| \log S _ {i} - \log \hat {S} _ {i} \| _ {1}. \tag {4}
+$$
+
+where  $\alpha$  is a weighting term set to 1.0 in our experiments. The total reconstruction loss is then the sum of all the spectral losses,  $L_{\mathrm{reconstruction}} = \sum_{i} L_{i}$ . In our experiments, we used FFT sizes (2048, 1024, 512, 256, 128, 64), and the neighboring frames in the Short-Time Fourier Transform (STFT) overlap by  $75\%$ . Therefore, the  $L_{i}$ 's cover differences between the original and synthesized audios at different spatial-temporal resolutions.
+
+# 5 RESULTS
+
+# 5.1 HIGH-FIDELITY SYNTHESIS
+
+As shown in Figure 5, the DDSP autoencoder learns to very accurately resynthesize the solo violin dataset. Again, we highly encourage readers to listen to the samples provided in the online supplement<sup>2</sup>. A full decomposition of the components is provided Figure 5. High-quality neural audio synthesis has previously required very large autoregressive models (Oord et al., 2016; Kalchbrenner et al., 2018) or adversarial loss functions (Engel et al., 2019). While amenable to an adversarial loss, the DDSP autoencoder achieves these results with a straightforward L1 spectrogram loss, a small amount of data, and a relatively simple model. This demonstrates that the model is able to efficiently exploit the bias of the DSP components, while not losing the expressive power of neural networks.
+
+For the NSynth dataset, we quantitatively compare the quality of DDSP resynthesis with that of a state-of-the-art baseline using WaveRNN (Hantrakul et al., 2019). The models are comparable as they are trained on the same data, provided the same conditioning, and both targeted towards realtime synthesis applications. In Table 1, we compute loudness and fundamental frequency (F0)  $L_{1}$  metrics described in Section C of the appendix. Despite the strong performance of the baseline, the supervised DDSP autoencoder still outperforms it, especially in F0  $L_{1}$ . This is not unexpected, as the additive synthesizer directly uses the conditioning frequency to synthesize audio.
+
+The unsupervised DDSP autoencoder must learn to infer its own F0 conditioning signal directly from the audio. As described in Section B.4, we improve optimization by also adding a perceptual loss in the form of a pretrained CREPE network (Kim et al., 2018). While not as accurate as the supervised DDSP version, the model does a fair job at learning to generate sounds with the correct frequencies without supervision, outperforming the supervised WaveRNN model.
+
+# 5.2 INDEPENDENT CONTROL OF LOUDNESS AND PITCH
+
+Interpolation: Interpretable structure allows for independent control over generative factors. Each component of the factorized latent variables  $(f(t), l(t), z(t))$  independently alters samples along a matching perceptual axis. For example, Figure 3 shows an interpolation between two sound in the loudness conditioning  $l(t)$ . With other variables held constant, loudness of the synthesized audio closely matches the interpolated input. Similarly, the model reliably matches intermediate pitches between a high pitched  $f(t)$  and low pitched  $f(t)$ . In Table C.2 of the appendix, we quantitatively demonstrate how across interpolations, conditioning independently controls the corresponding characteristics of the audio.
+
+![](images/f7b3e5efeeda43a14e6099dbfa61fc9ebfc1c9cbb90413e30409031f2a58055e.jpg)  
+Figure 4: Timbre transfer from singing voice to violin. F0 and loudness features are extracted from the voice and resynthesized with a DDSP autoencoder trained on solo violin.
+
+With loudness and pitch explicitly controlled by  $(f(t), l(t))$ , the model should use the residual  $z(t)$  to encode timbre. Although architecture and training do not strictly enforce this encoding, we qualitatively demonstrate how varying  $z$  leads to a smooth change in timbre. In Figure 3, we use the smooth shift in spectral centroid, or "center of mass" of a spectrum, to illustrate this behavior.
+
+Extrapolation: As described in Section 4.1,  $f(t)$  directly controls the additive synthesizer and has structural meaning outside the context of any given dataset. Beyond interpolating between datapoints, the model can extrapolate to new conditions not seen during training. The rightmost plot of Figure 7 demonstrates this by resynthesizing a clip of solo violin after shifting  $f(t)$  down an octave and outside the range of the training data. The audio remains coherent and resembles a related instrument such as a cello.  $f(t)$  is only modified for the synthesizer, as the decoder is still bounded by the nearby distribution of the training data and produces unrealistic harmonic content if conditioned far outside that distribution.
+
+# 5.3 DEREVERBERATION AND ACOUSTIC TRANSFER
+
+Removing reverb in a "blind" setting, where only reverberated audio is available, is a standing problem in acoustics (Naylor & Gaubitch, 2010). However, a benefit of our modular approach to generative modeling is that it becomes possible to completely separate the source audio from the effect of the room. For the solo violin dataset, the DDSP autoencoder is trained with an additional reverb module as shown in Figure 2 and described in Section 3.6. Figure 7 (left) demonstrates that bypassing the reverb module during resynthesis results in completely dereverberated audio, similar to recording in an anechoic chamber. The quality of the approach is limited by the underlying generative model, which is quite high for our autoencoder. Similarly, Figure 7 (center) demonstrates that we can also apply the learned reverb model to new audio, in this case singing, and effectively transfer the acoustic environment of the solo violin recordings.
+
+# 5.4 TIMBRE TRANSFER
+
+Figure 4 demonstrates timbre transfer, converting the singing voice of an author into a violin. F0 and loudness features are extracted from the singing voice and the DDSP autoencoder trained on solo violin used for resynthesis. To better match the conditioning features, we first shift the fundamental frequency of the singing up by two octaves to fit a violin's typical register. Next, we transfer the room acoustics of the violin recording (as described in Section 5.3) to the voice before extracting loudness, to better match the loudness contours of the violin recordings. The resulting audio captures many subtleties of the singing with the timbre and room acoustics of the violin dataset. Note the interesting "breathing" artifacts in the silence corresponding to unvoiced syllables from the singing.
+
+# 6 CONCLUSION
+
+The DDSP library fuses classical DSP with deep learning, providing the ability to take advantage of strong inductive biases without losing the expressive power of neural networks and end-to-end
+
+learning. We encourage contributions from domain experts and look forward to expanding the scope of the DDSP library to a wide range of future applications.
+
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+
+# A APPENDIX
+
+![](images/1718d86b9f0f4481a7df2af6d9e2ceae9b88d375fe268ed41f6e0cff026537e6.jpg)  
+Full Resynthesis
+
+![](images/187611bdc1e01f8e1b28b29dcd6018c902a71d210528bc824dfe647517801eb9.jpg)  
+Combined Audio
+
+![](images/de5b93306eb24ef5e3b1ea109f61da1addb75173940ff8dced90c40a848e37f7.jpg)  
+Additive Audio
+
+![](images/9a77fbf3791d92b14bab026f66668e0d7b062371d3ac4a2aa38fe829440b1f71.jpg)  
+Impulse Response
+
+![](images/070cc31c997ef7566ccafe1cf1a995eb40e13e887681db1a9193d7328ce268a3.jpg)  
+Filtered Noise Audio
+
+![](images/9a2588b9ca2ebdcab601284fdf7288b5197c1b7fa7032e32357cbfb047d23c90.jpg)  
+Amplitude
+
+![](images/5f70449ac911d6ea6407cfa785b07211f3460023d023116c2f1ad2191bd502cc.jpg)  
+Harmonic Distribution
+
+![](images/22cf13f0634615e59375db1bf6ee2dbb748a1f2a7afdb61caeeef64f6329bc71.jpg)  
+Noise Magnitudes
+
+![](images/8621627b5f3de8ae44fa8666060e2ad3ee61f1b90ab61fb706b351f561b1619c.jpg)  
+Loudness
+
+![](images/65f0bb3fede1f5b35ce69fba6ea573c21e518150d02df02cd442922366c3c085.jpg)  
+Fundamental Frequency
+
+![](images/23666947b12dd3ecde97aa5cf158fc16fd167eb54ef145367fc1a7a8a9712b69.jpg)  
+Original Audio  
+Figure 5: Decomposition of a clip of solo violin. Audio is visualized with log magnitude spectrograms. Loudness and fundamental frequency signals are extracted from the original audio. The loudness curve does not exhibit clear note segmentations because of the effects of the room acoustics. The DDSP autoencoder takes those conditioning signals and predicts amplitudes, harmonic distributions, and noise magnitudes. Note that the amplitudes are clearly segmented along note boundaries without supervision and that the harmonic and noise distributions are complex and dynamic despite the simple conditioning signals. Finally, the extracted impulse response is applied to the combined audio from the synthesizers to give the full resynthesis audio.
+
+![](images/1b78ae8c1fa4250e20f8c0014237c4f5e6b226ab8e3f8922922ac1fd92629398.jpg)  
+Figure 6: Diagram of the Additive Synthesizer component. The synthesizer generates audio as a sum of sinusoids at harmonic (integer) multiples of the fundamental frequency. The neural network is then tasked with emitting time-varying synthesizer parameters (fundamental frequency, amplitude, harmonic distribution). In this example linear-frequency log-magnitude spectrograms show how the harmonics initially follow the frequency contours of the fundamental. We then factorize the harmonic amplitudes into an overall amplitude envelope that controls the loudness, and a normalized distribution among the different harmonics that determines spectral variations.
+
+![](images/b06d177f242c14d94fa88871cd8bcc1019a446ed4053fb6edfc470a98fa5d7eb.jpg)  
+Figure 7: Interpretable generative models enables disentanglement and extrapolation. Spectrograms of audio examples include dereverberation of a clip of solo violin playing (left), transfer of the extracted room response to new audio (center), and transposition below the range of training data (right).
+
+# B MODEL DETAILS
+
+# B.1 ENCODERS
+
+The model has three encoders:  $f$ -encoder that outputs fundamental frequency  $f(t)$ ,  $l$ -encoder that outputs loudness  $l(t)$ , and a  $z$ -encoder that outputs residual vector  $z(t)$ .
+
+$f$ -encoder: We use a pretrained CREPE pitch detector (Kim et al., 2018) as the  $f$ -encoder to extract ground truth fundamental frequencies (F0) from the audio. We used the "large" variant of CREPE, which has SOTA accuracy for monophonic audio samples of musical instruments. For our supervised autoencoder experiments, we fixed the weights of the  $f$ -encoder like (Hantrakul et al., 2019), and for our unsupervised autoencoder experiments we jointly learn the weights of a resnet model fed log mel spectrograms of the audio. Full details of the resnet architecture are show in Table 2.
+
+$l$ -encoder: We use identical computational steps to extract loudness as (Hantrakul et al., 2019). Namely, an A-weighting of the power spectrum, which puts greater emphasis on higher frequencies, followed by log scaling. The vector is then centered according to the mean and standard deviation of the dataset.
+
+$\mathbf{z}$ -encoder: As shown in Figure 8, the encoder first calculates MFCC's (Mel Frequency Cepstrum Coefficients) from the audio. MFCC is computed from the log-mel-spectrogram of the audio with a FFT size of 1024, 128 bins of frequency range between  $20\mathrm{Hz}$  to  $8000\mathrm{Hz}$ , overlap of  $75\%$ . We use only the first 30 MFCCs that correspond to a smoothed spectral envelope. The MFCCs are then passed through a normalization layer (which has learnable shift and scale parameters) and a 512-unit GRU. The GRU outputs (over time) fed to a 512-unit linear layer to obtain  $\mathbf{z}(t)$ . The  $\mathbf{z}$  embedding reported in this model has 16 dimensions across 250 time-steps.
+
+<table><tr><td>Residual Block</td><td>Output Size</td><td>kTime</td><td>kFreq</td><td>sFreq</td><td>kFilters</td></tr><tr><td>layer norm + relu</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>conv</td><td>-</td><td>1</td><td>1</td><td>1</td><td>kFilters / 4</td></tr><tr><td>layer norm + relu</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>conv</td><td>-</td><td>3</td><td>3</td><td>sFreq</td><td>kFilters / 4</td></tr><tr><td>layer norm + relu</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>conv</td><td>-</td><td>1</td><td>1</td><td>1</td><td>kFilters</td></tr><tr><td>add residual</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Resnet</td><td>Output Size</td><td>kTime</td><td>kFreq</td><td>sFreq</td><td>kFilters</td></tr><tr><td>LogMelSpectrogram</td><td>(125, 229, 1)</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>conv2d</td><td>(125, 115, 64)</td><td>7</td><td>7</td><td>2</td><td>64</td></tr><tr><td>max pool</td><td>(125, 58, 64)</td><td>1</td><td>3</td><td>2</td><td>-</td></tr><tr><td>residual block</td><td>(125, 58, 128)</td><td>3</td><td>3</td><td>1</td><td>128</td></tr><tr><td>residual block</td><td>(125, 57, 128)</td><td>3</td><td>3</td><td>1</td><td>128</td></tr><tr><td>residual block</td><td>(125, 29, 256)</td><td>3</td><td>3</td><td>2</td><td>256</td></tr><tr><td>residual block</td><td>(125, 29, 256)</td><td>3</td><td>3</td><td>1</td><td>256</td></tr><tr><td>residual block</td><td>(125, 29, 256)</td><td>3</td><td>3</td><td>1</td><td>256</td></tr><tr><td>residual block</td><td>(125, 15, 512)</td><td>3</td><td>3</td><td>2</td><td>512</td></tr><tr><td>residual block</td><td>(125, 15, 512)</td><td>3</td><td>3</td><td>1</td><td>512</td></tr><tr><td>residual block</td><td>(125, 15, 512)</td><td>3</td><td>3</td><td>1</td><td>512</td></tr><tr><td>residual block</td><td>(125, 15, 512)</td><td>3</td><td>3</td><td>1</td><td>512</td></tr><tr><td>residual block</td><td>(125, 8, 1024)</td><td>3</td><td>3</td><td>2</td><td>1024</td></tr><tr><td>residual block</td><td>(125, 8, 1024)</td><td>3</td><td>3</td><td>1</td><td>1024</td></tr><tr><td>residual block</td><td>(125, 8, 1024)</td><td>3</td><td>3</td><td>1</td><td>1024</td></tr><tr><td>dense</td><td>(125, 1, 128)</td><td>-</td><td>-</td><td>128</td><td>1</td></tr><tr><td>upsample time</td><td>(1000, 1, 128)</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>softplus and normalize</td><td>(1000, 1, 128)</td><td>-</td><td>-</td><td>-</td><td>-</td></tr></table>
+
+Table 2: Model architecture for the f(t) encoder using a Resnet on log mel spectrograms. Spectrograms have a frame size of 2048 and a hop size of 512, and are upsampled at the end to have the same time resolution as other latents (4ms per a frame). All convolutions use "same" padding and a temporal stride of 1. Each residual block uses a bottleneck structure (He et al., 2016). The final output is a normalized probability distribution over 128 frequency values (logarithmically scaled between  $8.2\mathrm{Hz}$  and  $13.3\mathrm{kHz}$  (https://wwwInspiredacoustics.com/en/MIDI-note_number_and_center_frequencies)). The finally frequency value is the weighted sum of each frequency by its probability.
+
+# B.2 DECODER
+
+The decoder's input is the latent tuple  $(f(t), l(t), z(t))$  (250 timesteps). Its outputs are the parameters required by the synthesizers. For example, in the case of the harmonic synthesizer and filtered noise synthesizer setup, the decoder outputs  $a(t)$  (amplitudes of the harmonics) for the harmonic synthesizer (note that  $f(t)$  is fed directly from the latent), and  $H$  (transfer function of the FIR filter) for the filtered noise synthesizer.
+
+![](images/81759fc93b2e5a36e94efd1a7f933a3f46f1afce047adc61ed875a4387f5f40f.jpg)  
+Figure 8: Diagram of the  $z$ -encoder.
+
+As shown in Figure 9, we use a "shared-bottom" architecture, which computes a shared embedding from the latent tuple, and then have one head for each of the  $(\mathbf{a}(t),\mathbf{H})$  outputs.
+
+In particular, we apply separate MLPs to each of the  $(f(t), l(t), z(t))$  input. The outputs of the MLPs are concatenated and passed to a 512-unit GRU. We concatenate the GRU outputs with the outputs of the  $f(t)$  and  $l(t)$  MLPs (in the channel dimension) and pass it through a final MLP and Linear layer to get the decoder outputs.
+
+![](images/e48c2e0f825308a2ac26cd3a65e4a40a8475faa4e40c1727e15182011060870d.jpg)  
+Figure 9: Diagram of the decoder for the harmonic synthesizer and the filtered noise synthesizer.
+
+The MLP architecture, shown in Figure 10, is a standard MLP with a layer normalization (tf.contrib.layers(layer_norm) before the RELU nonlinearity. In Figure 9, all the MLPs have 3 layers and each layer has 512 units.
+
+![](images/1d8c3576cdbf90f393f182f06259f8d74e8258d02e60714a19a86b9db6555bc3.jpg)  
+Figure 10: MLP in the decoder.
+
+# B.3 TRAINING
+
+Because all DDSP components are differentiable, the model is differentiable end-to-end. Therefore, we can apply any SGD optimizer to train the model. We used ADAM optimizer with learning rate 0.001 and exponential learning rate decay 0.98 every 10,000 steps.
+
+# B.4 PERCEPTUAL LOSS
+
+To help guide the DDSP autoencoder that must predict  $f(t)$  on the NSynth dataset, we also added an additional perceptual loss using pretrained models, such as the CREPE (Kim et al., 2018) pitch estimator and the encoder of the WaveNet autoencoder (Engel et al., 2017). Compared to the L1 loss on the spectrogram, the activations of different layers in these models correlate better with the perceptual quality of the audio. After a large-scale hyperparameter search, we obtained our best results by using the L1 distance between the activations of the small CREPE model's fifth max pool layer with a weighting of  $5 \times 10^{-5}$  relative to the spectral loss.
+
+# B.5 SYNTHESIZERS
+
+Harmonic synthesizer / Additive Synthesis: We use 101 harmonics in the harmonic synthesizer (i.e.,  $a(t)$ 's dimension is 101). Amplitude and harmonic distribution parameters are upsampled with overlapping Hamming window envelopes whose frame size is 128 and hop size is 64. Initial phases are all fixed to zero, as neither the spectrogram loss functions or human perception are sensitive to absolute offsets in harmonic phase. We also do not include synthesis elements to model DC components to signals as they are inaudible and not reflected in the spectrogram losses.
+
+We force the amplitudes, harmonic distributions, and filtered noise magnitudes to be non-negative by applying a sigmoid nonlinearity to network outputs. We find a slight improvement in training stability by modifying the sigmoid to have a scaled output, larger slope by exponentiating, and threshold at a minimum value:
+
+$$
+y = 2. 0 \cdot \operatorname {s i g m o i d} (x) ^ {\log 1 0} + 1 0 ^ {- 7} \tag {5}
+$$
+
+Filtered noise synthesizer: We use 65 network output channels as magnitude inputs to the FIR filter of the filtered noise synthesizer.
+
+# B.6 PARAMETER COUNTS
+
+<table><tr><td>Model</td><td>Parameters</td></tr><tr><td>WaveNet Autoencoder (Engel et al., 2017)</td><td>75M</td></tr><tr><td>WaveRNN (Hantrakul et al., 2019)</td><td>23M</td></tr><tr><td>GANsynth (Engel et al., 2019)</td><td>15M</td></tr><tr><td>DDSP Autoencoder (Unsupervised)</td><td>12M</td></tr><tr><td>DDSP Autoencoder (Supervised, NSynth)</td><td>7M</td></tr><tr><td>DDSP Autoencoder (Supervised, Solo Violin)</td><td>6M</td></tr><tr><td>DDSP Autoencoder Tiny (Supervised, Solo Violin)</td><td>0.24M</td></tr></table>
+
+Table 3: Parameter counts for different models. All models trained on NSynth dataset except for those marked (Solo Violin). Autoregressive models have the most parameters with GANs requiring less. The DDSP models examined in this paper (which have not been optimized at all for size) require 2 to 3 times less parameters than GANSynth. The unsupervised model has more parameters because of the CREPE (small)  $f(t)$  encoder, and the autoencoder has additional parameters for the  $z(t)$  encoder. Initial experiments with extremely small models (single GRU, 256 units), have slightly less realistic outputs, but still relatively high quality (as can be heard in the supplemental audio).
+
+# C EVALUATION DETAILS
+
+# C.1 METRICS
+
+Loudness  $L_{1}$  distance: The loudness vector is extracted from the synthesized audio and  $L_{1}$  distance computed against the input's conditioning loudness vector (ground truth). A better model will produce lower  $L_{1}$  distances, indicating input and generated loudness vectors closely match. Note this distance is not back-propagated through the network as a training objective.
+
+F0  $L_{1}$  distance: The F0  $L_{1}$  distance is reported in MIDI space for easier interpretation; an average F0  $L_{1}$  of 1.0 corresponds to a semitone difference. We use the same confidence threshold of 0.85 in (Hantrakul et al., 2019) to select portions where there was detectable pitch content, and compute the metric only in these areas.
+
+F0 Outliers: Pitch tracking using CREPE, like any pitch tracker, is not completely reliable. Instabilities in pitch tracking, such as sudden octave jumps at low volumes, can result errors not due to model performance and need to be accounted for. F0 outliers accounts for pitch tracking imperfections in CREPE (Kim et al., 2018) vs. genuinely bad samples generated by the trained model. CREPE outputs both an F0 value as well as a F0 confidence. Samples with confidences below a threshold of 0.85 in (Hantrakul et al., 2019) are labeled as outliers and usually indicate the sample was mostly noise with no pitch or harmonic component. As the model outputs better quality audio, the number of outliers decrease, thus lower scores indicate better performance.
+
+# C.2 INTERPOLATION METRICS
+
+Loudness  $L_{1}$  and F0  $L_{1}$  are shown for different interpolation tasks in table C.2. In reconstruction, the model is supplied with the standard  $(f(t)_A,l(t)_A,z(t)_A)$ . Loudness  $(L_{1})$  and F0  $(L_{1})$  are computed against the ground truth inputs. In loudness interpolation, the model is supplied with  $(f(t)_A,l(t)_B,z(t)_A)$ , and Loudness  $(L_{1})$  is calculated using  $l(t)_B$  as ground truth instead of  $(l(t)_A)$ . For F0 interpolation, the model is supplied with  $(f(t)_B,l(t)_A,z(t)_A)$  and F0  $(L_{1})$  is calculated using  $f(t)_B$  as ground truth instead of  $f(t)_A$ . For Z interpolation, the model is supplied with  $(f(t)_A,l(t)_A,z(t)_B)$ . The low and constant Loudness  $(L_{1})$  and F0  $(L_{1})$  metrics across these interpolations indicate the model is able independently vary these variables without affecting other components.
+
+<table><tr><td>Task</td><td>Loudness (L1)</td><td>F0 (L1)</td></tr><tr><td>Reconstruction</td><td>0.042</td><td>0.060</td></tr><tr><td>Loudness l(t) interp.</td><td>0.061</td><td>0.060</td></tr><tr><td>F0 f(t) interp.</td><td>0.048</td><td>0.070</td></tr><tr><td>Z z(t) interp.</td><td>0.063</td><td>0.065</td></tr></table>
+
+Table 4: Loudness and F0 metrics for different interpolation tasks.
+
+# OTHER NOTES
+
+# D.1 SINUSOIDAL MODELS
+
+While unconstrained sinusoidal oscillator banks are strictly more expressive than harmonic oscillators, we restricted ourselves to harmonic synthesizers for the time being to focus the problem domain. However, this is not a fundamental limitation of the technique and perturbations such as inharmonicity can also be incorporated to handle phenomena such as stiff strings (Smith, 2010).
+
+# D.2 REGULARIZATION
+
+It is worth mentioning that the modular structure of the synthesizers also makes it possible to define additional losses in terms of different synthesizer outputs and parameters. For example, we may impose an SNR loss to penalize outputs with too much noise if we know the training data consists of mostly clean data. We have not experimented too much with such engineered losses, but we believe they can make training more efficient, even though such engineering methods deviates from the end-to-end training paradigm,
\ No newline at end of file
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+# DEEP LEARNING FOR SYMBOLIC MATHEMATICS
+
+Guillaume Lample*
+
+Facebook AI Research
+
+glample@fb.com
+
+François Charton*
+
+Facebook AI Research
+
+fcharton@fb.com
+
+# ABSTRACT
+
+Neural networks have a reputation for being better at solving statistical or approximate problems than at performing calculations or working with symbolic data. In this paper, we show that they can be surprisingly good at more elaborated tasks in mathematics, such as symbolic integration and solving differential equations. We propose a syntax for representing mathematical problems, and methods for generating large datasets that can be used to train sequence-to-sequence models. We achieve results that outperform commercial Computer Algebra Systems such as Matlab or Mathematica.
+
+# 1 INTRODUCTION
+
+A longstanding tradition in machine learning opposes rule-based inference to statistical learning (Rumelhart et al., 1986), and neural networks clearly stand on the statistical side. They have proven to be extremely effective in statistical pattern recognition and now achieve state-of-the-art performance on a wide range of problems in computer vision, speech recognition, natural language processing (NLP), etc. However, the success of neural networks in symbolic computation is still extremely limited: combining symbolic reasoning with continuous representations is now one of the challenges of machine learning.
+
+Only a few studies investigated the capacity of neural network to deal with mathematical objects, and apart from a small number of exceptions (Zaremba et al., 2014; Loos et al., 2017; Allamanis et al., 2017; Arabshahi et al., 2018b), the majority of these works focus on arithmetic tasks like integer addition and multiplication (Zaremba & Sutskever, 2014; Kaiser & Sutskever, 2015; Trask et al., 2018). On these tasks, neural approaches tend to perform poorly, and require the introduction of components biased towards the task at hand (Kaiser & Sutskever, 2015; Trask et al., 2018).
+
+In this paper, we consider mathematics, and particularly symbolic calculations, as a target for NLP models. More precisely, we use sequence-to-sequence models (seq2seq) on two problems of symbolic mathematics: function integration and ordinary differential equations (ODEs). Both are difficult, for trained humans and computer software. For integration, humans are taught a set of rules (integration by parts, change of variable, etc.), that are not guaranteed to succeed, and Computer Algebra Systems use complex algorithms (Geddes et al., 1992) that explore a large number of specific cases. For instance, the complete description of the Risch algorithm (Risch, 1970) for function integration is more than 100 pages long.
+
+Yet, function integration is actually an example where pattern recognition should be useful: detecting that an expression is of the form  $yy'(y^2 + 1)^{-1/2}$  suggests that its primitive will contain  $\sqrt{y^2 + 1}$ . Detecting this pattern may be easy for small expressions  $y$ , but becomes more difficult as the number of operators in  $y$  increases. However, to the best of our knowledge, no study has investigated the ability of neural networks to detect patterns in mathematical expressions.
+
+We first propose a representation of mathematical expressions and problems that can be used by seq2seq models, and discuss the size and structure of the resulting problem space. Then, we show how to generate datasets for supervised learning of integration and first and second order differential equations. Finally, we apply seq2seq models to these datasets, and show that they achieve a better performance than state-of-the-art computer algebra programs, namely Matlab and Mathematica.
+
+# 2 MATHEMATICS AS A NATURAL LANGUAGE
+
+# 2.1 EXPRESSIONS AS TREES
+
+Mathematical expressions can be represented as trees, with operators and functions as internal nodes, operands as children, and numbers, constants and variables as leaves. The following trees represent expressions  $2 + 3 \times (5 + 2)$ ,  $3x^{2} + \cos(2x) - 1$ , and  $\frac{\partial^2 \psi}{\partial x^2} - \frac{1}{\nu^2} \frac{\partial^2 \psi}{\partial t^2}$ :
+
+![](images/de6db57abc83650d9f76e322bd0f9b974e57f5748bed22c89dc5b622db09c1ae.jpg)
+
+![](images/cc9b52ea96a72806de932f16bb6d38c69329ae412b78a7c5182ed08054620782.jpg)
+
+![](images/d50356fe3a0c9222881ede81dd15d2c9b5d21e3b261165fb7faba01216b5da50.jpg)
+
+Trees disambiguate the order of operations, take care of precedence and associativity and eliminate the need for parentheses. Up to the addition of meaningless symbols like spaces, punctuation or redundant parentheses, different expressions result in different trees. With a few assumptions, discussed in Section A of the appendix, there is a one-to-one mapping between expressions and trees.
+
+We consider expressions as sequences of mathematical symbols.  $2 + 3$  and  $3 + 2$  are different expressions, as are  $\sqrt{4} x$  and  $2x$ , and they will be represented by different trees. Most expressions represent meaningful mathematical objects.  $x / 0$ ,  $\sqrt{-2}$  or  $\log (0)$  are also legitimate expressions, even though they do not necessarily make mathematical sense.
+
+Since there is a one-to-one correspondence between trees and expressions, equality between expressions will be reflected over their associated trees, as an equivalence: since  $2 + 3 = 5 = 12 - 7 = 1 \times 5$ , the four trees corresponding to these expressions are equivalent.
+
+Many problems of formal mathematics can be reframed as operations over expressions, or trees. For instance, expression simplification amounts to finding a shorter equivalent representation of a tree. In this paper, we consider two problems: symbolic integration and differential equations. Both boil down to transforming an expression into another, e.g. mapping the tree of an equation to the tree of its solution. We regard this as a particular instance of machine translation.
+
+# 2.2 TREES AS SEQUENCES
+
+Machine translation systems typically operate on sequences (Sutskever et al., 2014; Bahdanau et al., 2015). Alternative approaches have been proposed to generate trees, such as Tree-LSTM (Tai et al., 2015) or Recurrent Neural Network Grammars (RNNG) (Dyer et al., 2016; Eriguchi et al., 2017). However, tree-to-tree models are more involved and much slower than their seq2seq counterparts, both at training and at inference. For the sake of simplicity, we use seq2seq models, which were shown to be effective at generating trees, e.g. in the context of constituency parsing (Vinyals et al., 2015), where the task is to predict a syntactic parse tree of input sentences.
+
+Using seq2seq models to generate trees requires to map trees to sequences. To this effect, we use prefix notation (also known as normal Polish notation), writing each node before its children, listed from left to right. For instance, the arithmetic expression  $2 + 3 * (5 + 2)$  is represented as the sequence  $[+2 * 3 + 52]$ . In contrast to the more common infix notation  $2 + 3 * (5 + 2)$ , prefix sequences need no parentheses and are therefore shorter. Inside sequences, operators, functions or variables are represented by specific tokens, and integers by sequences of digits preceded by a sign. As in the case between expressions and trees, there exists a one-to-one mapping between trees and prefix sequences.
+
+# 2.3 GENERATING RANDOM EXPRESSIONS
+
+To create training data, we need to generate sets of random mathematical expressions. However, sampling uniformly expressions with  $n$  internal nodes is not a simple task. Naive algorithms (such as
+
+recursive methods or techniques using fixed probabilities for nodes to be leaves, unary, or binary) tend to favour deep trees over broad trees, or left-leaning over right leaning trees. Here are examples of different trees that we want to generate with the same probability.
+
+![](images/e6c919b7bca26dce71b27b6111519aea39583a5d1f55b543cbccca2e082a68da.jpg)
+
+![](images/8142be368c296ef45b78a311568f5d3e5e9eb2e2f7ef56c3db8aeabf80252e1d.jpg)
+
+![](images/5a26bc7a33064aaaaecf161c7445b846333a44c3fe38c1dacedbfb0e06d068a3.jpg)
+
+![](images/d1a82bea6ec04d768926236950b0ff58121d13c05a38a020a55b4eaca3502036.jpg)
+
+In Section C of the appendix, we present an algorithm to generate random trees and expressions, where the four expression trees above are all generated with the same probability.
+
+# 2.4 COUNTING EXPRESSIONS
+
+We now investigate the number of possible expressions. Expressions are created from a finite set of variables (i.e. literals), constants, integers, and a list of operators that can be simple functions (e.g. cos or exp) or more involved operators (e.g. differentiation or integration). More precisely, we define our problem space as:
+
+- trees with up to  $n$  internal nodes  
+- a set of  $p_1$  unary operators (e.g. cos, sin, exp, log)  
+- a set of  $p_2$  binary operators (e.g.  $+, -, \times, \mathrm{pow}$ )  
+- a set of  $L$  leaf values containing variables (e.g.  $x, y, z$ ), constants (e.g.  $e, \pi$ ), integers (e.g.  $\{-10, \dots, 10\}$ )
+
+If  $p_1 = 0$ , expressions are represented by binary trees. The number of binary trees with  $n$  internal nodes is given by the  $n$ -th Catalan numbers  $C_n$  (Sloane, 1996). A binary tree with  $n$  internal nodes has exactly  $n + 1$  leaves. Each node and leaf can take respectively  $p_2$  and  $L$  different values. As a result, the number of expressions with  $n$  binary operators can be expressed by:
+
+$$
+E _ {n} = C _ {n} p _ {2} ^ {n} L ^ {n + 1} \approx \frac {4 ^ {n}}{n \sqrt {\pi n}} p _ {2} ^ {n} L ^ {n + 1} \quad \text {w i t h} \quad C _ {n} = \frac {1}{n + 1} \binom {2 n} {n}
+$$
+
+If  $p_1 > 0$ , expressions are unary-binary trees, and the number of trees with  $n$  internal nodes is the  $n$ -th large Schroeder number  $S_{n}$  (Sloane, 1996). It can be computed by recurrence using the following equation:
+
+$$
+(n + 1) S _ {n} = 3 (2 n - 1) S _ {n - 1} - (n - 2) S _ {n - 2} \tag {1}
+$$
+
+Finally, the number  $E_{n}$  of expressions with  $n$  internal nodes,  $p_1$  unary operator,  $p_2$  binary operators and  $L$  possible leaves is recursively computed as
+
+$$
+(n + 1) E _ {n} = \left(p _ {1} + 2 L p _ {2}\right) (2 n - 1) E _ {n - 1} - p _ {1} (n - 2) E _ {n - 2} \tag {2}
+$$
+
+If  $p_1 = p_2 = L = 1$ , Equation 2 boils down to Equation 1. If  $p_2 = L = 1, p_1 = 0$ , we have  $(n + 1)E_n = 2(2n - 1)E_{n - 1}$  which is the recurrence relation satisfied by Catalan numbers. The derivations and properties of all these formulas are provided in Section B of the appendix.
+
+In Figure 1, we represent the number of binary trees  $(C_n)$  and unary-binary trees  $(S_{n})$  for different numbers of internal nodes. We also represent the number of possible expressions  $(E_{n})$  for different sets of operators and leaves.
+
+![](images/87fa7d115b4aab506d0d2f08bb738f786141a98209d14e0ed9b985fc2160c823.jpg)  
+Figure 1: Number of trees and expressions for different numbers of operators and leaves.  $p_1$  and  $p_2$  correspond to the number of unary and binary operators respectively, and  $L$  to the number of possible leaves. The bottom two curves correspond to the number of binary and unary-binary trees (enumerated by Catalan and Schroeder numbers respectively). The top two curves represent the associated number of expressions. We observe that adding leaves and binary operators significantly increases the size of the problem space.
+
+# 3 GENERATING DATASETS
+
+Having defined a syntax for mathematical problems and techniques to randomly generate expressions, we are now in a position to build the datasets our models will use. In the rest of the paper, we focus on two problems of symbolic mathematics: function integration and solving ordinary differential equations (ODE) of the first and second order.
+
+To train our networks, we need datasets of problems and solutions. Ideally, we want to generate representative samples of the problem space, i.e. randomly generate functions to be integrated and differential equations to be solved. Unfortunately, solutions of random problems sometimes do not exist (e.g. the integrals of  $f(x) = \exp (x^{2})$  or  $f(x) = \log (\log (x))$  cannot be expressed with usual functions), or cannot be easily derived. In this section, we propose techniques to generate large training sets for integration and first and second order differential equations.
+
+# 3.1 INTEGRATION
+
+We propose three approaches to generate functions with their associated integrals.
+
+Forward generation (FWD). A straightforward approach is to generate random functions with up to  $n$  operators (using methods from Section 2) and calculate their integrals with a computer algebra system. Functions that the system cannot integrate are discarded. This generates a representative sample of the subset of the problem space that can be successfully solved by an external symbolic mathematical framework.
+
+Backward generation (BWD). An issue with the forward approach is that the dataset only contains functions that symbolic frameworks can solve (they sometimes fail to compute the integral of integrable functions). Also, integrating large expressions is time expensive, which makes the overall method particularly slow. Instead, the backward approach generates a random function  $f$ , computes its derivative  $f'$ , and adds the pair  $(f', f)$  to the training set. Unlike integration, differentiation is always possible and extremely fast even for very large expressions. As opposed to the forward approach, this method does not depend on an external symbolic integration system.
+
+Backward generation with integration by parts (IBP). An issue with the backward approach is that it is very unlikely to generate the integral of simple functions like  $f(x) = x^3 \sin(x)$ . Its integral,  $F(x) = -x^3 \cos(x) + 3x^2 \sin(x) + 6x \cos(x) - 6 \sin(x)$ , a function with 15 operators, has a very low probability of being generated randomly. Besides, the backward approach tends to generate examples where the integral (the solution) is shorter than the derivative (the problem), while forward generation favors the opposite (see Figure 2 in Section E in the Appendix). To address this issue, we
+
+leverage integration by parts: given two randomly generated functions  $F$  and  $G$ , we compute their respective derivatives  $f$  and  $g$ . If  $fG$  already belongs to the training set, we know its integral, and we can compute the integral of  $Fg$  as:
+
+$$
+\int F g = F G - \int f G
+$$
+
+Similarly, if  $Fg$  is in the training set, we can infer the integral of  $fG$ . Whenever we discover the integral of a new function, we add it to the training set. If none of  $fG$  or  $Fg$  are in the training set, we simply generate new functions  $F$  and  $G$ . With this approach, we can generate the integrals of functions like  $x^{10} \sin(x)$  without resorting to an external symbolic integration system.
+
+Comparing different generation methods. Table 1 in Section 4.1 summarizes the differences between the three generation methods. The FWD method tends to generate short problems with long solutions (that computer algebras can solve). The BWD approach, on the other hand, generates long problems with short solutions. IBP generates datasets comparable to FWD (short problems and long solutions), without an external computer algebra system. A mixture of BWD and IBP generated data should therefore provide a better representation of problem space, without resorting to external tools. Examples of functions / integrals for the three approaches are given in Table 9 of the Appendix.
+
+# 3.2 FIRST ORDER DIFFERENTIAL EQUATION (ODE 1)
+
+We now present a method to generate first order differential equations with their solutions. We start from a bivariate function  $F(x,y)$  such that the equation  $F(x,y) = c$  (where  $c$  is a constant) can be analytically solved in  $y$ . In other words, there exists a bivariate function  $f$  that satisfies  $\forall (x,c), F(x,f(x,c)) = c$ . By differentiation with respect to  $x$ , we have that  $\forall x, c$ :
+
+$$
+\frac {\partial F (x , f _ {c} (x))}{\partial x} + f _ {c} ^ {\prime} (x) \frac {\partial F (x , f _ {c} (x))}{\partial y} = 0
+$$
+
+where  $f_{c} = x \mapsto f(x, c)$ . As a result, for any constant  $c$ ,  $f_{c}$  is solution of the first order differential equation:
+
+$$
+\frac {\partial F (x , y)}{\partial x} + y ^ {\prime} \frac {\partial F (x , y)}{\partial y} = 0 \tag {3}
+$$
+
+With this approach, we can use the method described in Section C of the appendix to generate arbitrary functions  $F(x,y)$  analytically solvable in  $y$ , and create a dataset of differential equations with their solutions.
+
+Instead of generating a random function  $F$ , we can generate a solution  $f(x, c)$ , and determine a differential equation that it satisfies. If  $f(x, c)$  is solvable in  $c$ , we compute  $F$  such that  $F\big(x, f(x, c)\big) = c$ . Using the above approach, we show that for any constant  $c$ ,  $x \mapsto f(x, c)$  is a solution of differential Equation 3. Finally, the resulting differential equation is factorized, and we remove all positive factors from the equation.
+
+A necessary condition for this approach to work is that the generated functions  $f(x, c)$  can be solved in  $c$ . For instance, the function  $f(x, c) = c \times \log (x + c)$  cannot be analytically solved in  $c$ , i.e. the function  $F$  that satisfies  $F\big(x, f(x, c)\big) = c$  cannot be written with usual functions. Since all the operators and functions we use are invertible, a simple condition to ensure the solvability in  $c$  is to guarantee that  $c$  only appears once in the leaves of the tree representation of  $f(x, c)$ . A straightforward way to generate a suitable  $f(x, c)$  is to sample a random function  $f(x)$  by the methods described in Section C of the appendix, and to replace one of the leaves in its tree representation by  $c$ . Below is an example of the whole process:
+
+Generate a random function  $f(x) = x \log (c / x)$
+
+Solve in  $c$
+
+Differentiate in  $x$
+
+Simplify  $xy' - y + x = 0$
+
+# 3.3 SECOND ORDER DIFFERENTIAL EQUATION (ODE 2)
+
+Our method for generating first order equations can be extended to the second order, by considering functions of three variables  $f(x, c_1, c_2)$  that can be solved in  $c_2$ . As before, we derive a function of three variables  $F$  such that  $F\big(x, f(x, c_1, c_2), c_1\big) = c_2$ . Differentiation with respect to  $x$  yields a first order differential equation:
+
+$$
+\left. \frac {\partial F (x , y , c _ {1})}{\partial x} + f _ {c _ {1}, c _ {2}} ^ {\prime} (x) \frac {\partial F (x , y , c _ {1})}{\partial y} \right| _ {y = f _ {c _ {1}, c _ {2}} (x)} = 0
+$$
+
+where  $f_{c_1,c_2} = x \mapsto f(x,c_1,c_2)$ . If this equation can be solved in  $c_1$ , we can infer another three-variable function  $G$  satisfying  $\forall x, G\big(x,f_{c_1,c_2}(x),f_{c_1,c_2}'(x)\big) = c_1$ . Differentiating with respect to  $x$  a second time yields the following equation:
+
+$$
+\frac{\partial G(x,y,z)}{\partial x} +f^{\prime}_{c_{1},c_{2}}(x)\frac{\partial G(x,y,z)}{\partial y} +f^{\prime \prime}_{c_{1},c_{2}}(x)\frac{\partial G(x,y,z)}{\partial z}\bigg|_{\substack{y = f_{c_{1},c_{2}}(x)\\ z = f^{\prime}_{c_{1},c_{2}}(x)}} = 0
+$$
+
+Therefore, for any constants  $c_{1}$  and  $c_{2}$ ,  $f_{c_1,c_2}$  is solution of the second order differential equation:
+
+$$
+\frac {\partial G (x , y , y ^ {\prime})}{\partial x} + y ^ {\prime} \frac {\partial G (x , y , y ^ {\prime})}{\partial y} + y ^ {\prime \prime} \frac {\partial G (x , y , y ^ {\prime})}{\partial z} = 0
+$$
+
+Using this approach, we can create pairs of second order differential equations and solutions, provided we can generate  $f(x, c_1, c_2)$  is solvable in  $c_2$ , and that the corresponding first order differential equation is solvable in  $c_1$ . To ensure the solvability in  $c_2$ , we can use the same approach as for first order differential equation, e.g. we create  $f_{c_1, c_2}$  so that  $c_2$  has exactly one leaf in its tree representation. For  $c_1$ , we employ a simple approach where we simply skip the current equation if we cannot solve it in  $c_1$ . Although naive, we found that the differentiation equation can be solved in  $c_1$  about 50% the time. As an example:
+
+Generate a random function  $f(x) = c_{1}e^{x} + c_{2}e^{-x}$
+
+Solve in  $c_{2}$
+
+Differentiate in  $x$  （20  $e^{x}\big(f^{\prime}(x) + f(x)\big) - 2c_{1}e^{2x} = 0$
+
+Solve in  $c_{1}$
+
+Differentiate in  $x$
+
+Simplify  $y^{\prime \prime} - y = 0$
+
+# 3.4 DATASET CLEANING
+
+Equation simplification In practice, we simplify generated expressions to reduce the number of unique possible equations in the training set, and to reduce the length of sequences. Also, we do not want to train our model to predict  $x + 1 + 1 + 1 + 1 + 1$  when it can simply predict  $x + 5$ . As a result, sequences  $[+2 + x3]$  and  $[+3 + 2x]$  will both be simplified to  $[+x5]$  as they both represent the expression  $x + 5$ . Similarly, the expression  $\log (e^{x + 3})$  will be simplified to  $x + 3$ , the expression  $\cos^2 (x) + \sin^2 (x)$  will be simplified to 1, etc. On the other hand,  $\sqrt{(x - 1)^2}$  will not be simplified to  $x - 1$  as we do not make any assumption on the sign of  $x - 1$ .
+
+Coefficients simplification In the case of first order differential equations, we modify generated expressions by equivalent expressions up to a change of variable. For instance,  $x + x \tan(3) + cx + 1$  will be simplified to  $cx + 1$ , as a particular choice of the constant  $c$  makes these two expressions identical. Similarly,  $\log(x^2) + c \log(x)$  becomes  $c \log(x)$ .
+
+We apply a similar technique for second order differential equations, although simplification is sometimes a bit more involved because there are two constants  $c_{1}$  and  $c_{2}$ . For instance,  $c_{1} - c_{2}x / 5 + c_{2} + 1$  is simplified to  $c_{1}x + c_{2}$ , while  $c_{2}e^{c_{1}}e^{c_{1}xe - 1}$  can be expressed with  $c_{2}e^{c_{1}x}$ , etc.
+
+We also perform transformations that are not strictly equivalent, as long as they hold under specific assumptions. For instance, we simplify  $\tan (\sqrt{c_2} x) + \cosh (c_1 + 1) + 4$  to  $c_{1} + \tan (c_{2}x)$ , although the constant term can be negative in the second expression, but not the first one. Similarly  $e^3 e^{c_1x}e^{c_1\log (c_2)}$  is transformed to  $c_{2}e^{c_{1}x}$ .
+
+Invalid expressions Finally, we also remove invalid expressions from our dataset. For instance, expressions like  $\log (0)$  or  $\sqrt{-2}$ . To detect them, we compute in the expression tree the values of subtrees that do not depend on  $x$ . If a subtree does not evaluate to a finite real number (e.g.  $-\infty$ ,  $+\infty$  or a complex number), we discard the expression.
+
+# 4 EXPERIMENTS
+
+# 4.1 DATASET
+
+For all considered tasks, we generate datasets using the method presented in Section 3, with:
+
+- expressions with up to  $n = 15$  internal nodes  
+-  $L = 11$  leaf values in  $\{x\} \cup \{-5,\dots ,5\} \setminus \{0\}$  
+-  $p_2 = 4$  binary operators:  $+, -, \times, /$  
+-  $p_1 = 15$  unary operators: exp, log, sqrt, sin, cos, tan,  $\sin^{-1}$ ,  $\cos^{-1}$ ,  $\tan^{-1}$ , sinh, cosh, tanh,  $\sinh^{-1}$ ,  $\cosh^{-1}$ ,  $\tanh^{-1}$
+
+Statistics about our datasets are presented in Table 1. As discussed in Section 3.1, we observe that the backward approach generates derivatives (i.e. inputs) significantly longer than the forward generator. We discuss this in more detail in Section E of the appendix.
+
+<table><tr><td></td><td>Forward</td><td>Backward</td><td>Integration by parts</td><td>ODE 1</td><td>ODE 2</td></tr><tr><td>Training set size</td><td>20M</td><td>40M</td><td>20M</td><td>40M</td><td>40M</td></tr><tr><td>Input length</td><td>18.9±6.9</td><td>70.2±47.8</td><td>17.5±9.1</td><td>123.6±115.7</td><td>149.1±130.2</td></tr><tr><td>Output length</td><td>49.6±48.3</td><td>21.3±8.3</td><td>26.4±11.3</td><td>23.0±15.2</td><td>24.3±14.9</td></tr><tr><td>Length ratio</td><td>2.7</td><td>0.4</td><td>2.0</td><td>0.4</td><td>0.1</td></tr><tr><td>Input max length</td><td>69</td><td>450</td><td>226</td><td>508</td><td>508</td></tr><tr><td>Output max length</td><td>508</td><td>75</td><td>206</td><td>474</td><td>335</td></tr></table>
+
+Table 1: Training set sizes and length of expressions (in tokens) for different datasets. FWD and IBP tend to generate examples with outputs much longer than the inputs, while the BWD approach generates shorter outputs. Like in the BWD case, ODE generators tend to produce solutions much shorter than their equations.
+
+# 4.2 MODEL
+
+For all our experiments, we train a seq2seq model to predict the solutions of given problems, i.e. to predict a primitive given a function, or predict a solution given a differential equation. We use a transformer model (Vaswani et al., 2017) with 8 attention heads, 6 layers, and a dimensionality of 512. In our experiences, using larger models did not improve the performance. We train our models with the Adam optimizer (Kingma & Ba, 2014), with a learning rate of  $10^{-4}$ . We remove expressions with more than 512 tokens, and train our model with 256 equations per batch.
+
+At inference, expressions are generated by a beam search (Koehn, 2004; Sutskever et al., 2014), with early stopping. We normalize the log-likelihood scores of hypotheses in the beam by their sequence length. We report results with beam widths of 1 (i.e. greedy decoding), 10 and 50.
+
+During decoding, nothing prevents the model from generating an invalid prefix expression, e.g.  $[+2 * 3]$ . To address this issue, Dyer et al. (2016) use constraints during decoding, to ensure
+
+that generated sequences can always be converted to valid expression trees. In our case, we found that model generations are almost always valid and we do not use any constraint. When an invalid expression is generated, we simply consider it as an incorrect solution and ignore it.
+
+# 4.3 EVALUATION
+
+At the end of each epoch, we evaluate the ability of the model to predict the solutions of given equations. In machine translation, hypotheses given by the model are compared to references written by human translators, typically with metrics like the BLEU score (Papineni et al., 2002) that measure the overlap between hypotheses and references. Evaluating the quality of translations is a very difficult problem, and many studies showed that a better BLEU score does not necessarily correlate with a better performance according to human evaluation. Here, however, we can easily verify the correctness of our model by simply comparing generated expressions to their reference solutions.
+
+For instance, for the given differential equation  $xy' - y + x = 0$  with a reference solution  $x\log (c / x)$  (where  $c$  is a constant), our model may generate  $x\log (c) - x\log (x)$ . We can check that these two solutions are equal, although they are written differently, using a symbolic framework like SymPy (Meurer et al., 2017).
+
+However, our model may also generate  $xc - x\log (x)$  which is also a valid solution, that is actually equivalent to the previous one for a different choice of constant  $c$ . In that case, we replace  $y$  in the differential equation by the model hypothesis. If  $xy^{\prime} - y + x = 0$ , we conclude that the hypothesis is a valid solution. In the case of integral computation, we can simply differentiate the model hypothesis, and compare it with the function to integrate. For the three problems, we measure the accuracy of our model on equations from the test set.
+
+Since we can easily verify the correctness of generated expressions, we consider all hypotheses in the beam, and not only the one with the highest score. We verify the correctness of each hypothesis, and consider that the model successfully solved the input equation if one of them is correct. As a result, results with "Beam size 10" indicate that at least one of the 10 hypotheses in the beam was correct.
+
+# 4.4 RESULTS
+
+Table 2 reports the accuracy of our model for function integration and differential equations. For integration, the model achieves close to  $100\%$  performance on a held-out test set, even with greedy decoding (beam size 1). This performance is consistent over the three integration datasets (FWD, BWD, and IBP). Greedy decoding (beam size 1) does not work as well for differential equations. In particular, we observe an improvement in accuracy of almost  $40\%$  when using a large beam size of 50 for second order differential equations. Unlike in machine translation, where increasing the beam size does not necessarily increase the performance (Ott et al., 2018), we always observe significant improvements with wider beams. Typically, using a beam size of 50 provides an improvement of  $8\%$  accuracy compared to a beam size of 10. This makes sense, as increasing the beam size will provide more hypotheses, although a wider beam may displace a valid hypothesis to consider invalid ones with better log-probabilities.
+
+<table><tr><td></td><td>Integration (FWD)</td><td>Integration (BWD)</td><td>Integration (IBP)</td><td>ODE (order 1)</td><td>ODE (order 2)</td></tr><tr><td>Beam size 1</td><td>93.6</td><td>98.4</td><td>96.8</td><td>77.6</td><td>43.0</td></tr><tr><td>Beam size 10</td><td>95.6</td><td>99.4</td><td>99.2</td><td>90.5</td><td>73.0</td></tr><tr><td>Beam size 50</td><td>96.2</td><td>99.7</td><td>99.5</td><td>94.0</td><td>81.2</td></tr></table>
+
+Table 2: Accuracy of our models on integration and differential equation solving. Results are reported on a held out test set of 5000 equations. For differential equations, using beam search decoding significantly improves the accuracy of the model.
+
+# 4.5 COMPARISON WITH MATHEMATICAL FRAMEWORKS
+
+We compare our model with three popular mathematical frameworks: Mathematica (Wolfram-Research, 2019), Maple and Matlab (MathWorks, 2019)<sup>1</sup>. Prefix sequences in our test set are
+
+converted back to their infix representations, and given as input to the computer algebra. For a specific input, the computer algebra either returns a solution, provides no solution (or a solution including integrals or special functions), or, in the case of Mathematica, times out after a preset delay. When Mathematica times out, we conclude that it is not able to compute a solution (although it might have found a solution given more time). For integration, we evaluate on the BWD test set. By construction, the FWD data only consists of integrals generated by computer algebra systems, which makes comparison uninteresting.
+
+In Table 3, we present accuracy for our model with different beam sizes, and for Mathematica with a timeout delay of 30 seconds. Table 8 in the appendix provides detailed results for different values of timeout, and explains our choice of 30 seconds. In particular, we find that with 30 seconds, only  $20\%$  of failures are due to timeouts, and only  $10\%$  when the timeout is set to 3 minutes. Even with timeout limits, evaluation would take too long on our 5000 test equations, so we only evaluate on a smaller test subset of 500 equations, on which we also re-evaluate our model.
+
+<table><tr><td></td><td>Integration (BWD)</td><td>ODE (order 1)</td><td>ODE (order 2)</td></tr><tr><td>Mathematica (30s)</td><td>84.0</td><td>77.2</td><td>61.6</td></tr><tr><td>Matlab</td><td>65.2</td><td>-</td><td>-</td></tr><tr><td>Maple</td><td>67.4</td><td>-</td><td>-</td></tr><tr><td>Beam size 1</td><td>98.4</td><td>81.2</td><td>40.8</td></tr><tr><td>Beam size 10</td><td>99.6</td><td>94.0</td><td>73.2</td></tr><tr><td>Beam size 50</td><td>99.6</td><td>97.0</td><td>81.0</td></tr></table>
+
+On all tasks, we observe that our model significantly outperforms Mathematica. On function integration, our model obtains close to  $100\%$  accuracy, while Mathematica barely reaches  $85\%$ . On first order differential equations, Mathematica is on par with our model when it uses a beam size of 1, i.e. with greedy decoding. However, using a beam search of size 50 our model accuracy goes from  $81.2\%$  to  $97.0\%$ , largely surpassing Mathematica. Similar observations can be made for second order differential equations, where beam search is even more critical since the number of equivalent solutions is larger. On average, Matlab and Maple have slightly lower performance than Mathematica on the problems we tested.
+
+Table 4 shows examples of functions that our model was able to solve, on which Mathematica and Matlab did not find a solution. The denominator of the function to integrate,  $-16x^{8} + 112x^{7} - 204x^{6} + 28x^{5} - x^{4} + 1$ , can be rewritten as  $1 - (4x^{4} - 14x^{3} + x^{2})^{2}$ . With the simplified input:
+
+$$
+\frac {1 6 x ^ {3} - 4 2 x ^ {2} + 2 x}{\left(1 - (4 x ^ {4} - 1 4 x ^ {3} + x ^ {2}) ^ {2}\right) ^ {1 / 2}}
+$$
+
+integration becomes easier and Mathematica is able to find the solution.
+
+Table 3: Comparison of our model with Mathematica, Maple and Matlab on a test set of 500 equations. For Mathematica we report results by setting a timeout of 30 seconds per equation. On a given equation, our model typically finds the solution in less than a second.  
+
+<table><tr><td>Equation</td><td>Solution</td></tr><tr><td>y&#x27; = 16x3 - 42x2 + 2x / (-16x8 + 112x7 - 204x6 + 28x5 - x4 + 1)1/2</td><td>y = sin-1(4x4 - 14x3 + x2)</td></tr><tr><td>3xy cos(x) - √9x2 sin(x)2 + 1y&#x27; + 3y sin(x) = 0</td><td>y = cexp(sinh-1(3x sin(x)))</td></tr><tr><td>4x4yy&#x27;&#x27; - 8x4y&#x27;2 - 8x3yy&#x27; - 3x3y&#x27;&#x27; - 8x2y2 - 6x2y&#x27; - 3x2y&#x27;&#x27; - 9xy&#x27; - 3y = 0</td><td>y = (c1 + 3x + 3log(x)) / x (c2 + 4x)</td></tr></table>
+
+Table 4: Examples of problems that our model is able to solve, on which Mathematica and Matlab were not able to find a solution. For each equation, our model finds a valid solution with greedy decoding.
+
+# 4.6 EQUIVALENT SOLUTIONS
+
+An interesting property of our model is that it is able to generate solutions that are exactly equivalent, but written in different ways. For instance, we consider the following first order differential equation, along with one of its solutions:
+
+$$
+1 6 2 x \log (x) y ^ {\prime} + 2 y ^ {3} \log (x) ^ {2} - 8 1 y \log (x) + 8 1 y = 0 y = \frac {9 \sqrt {x} \sqrt {\frac {1}{\log (x)}}}{\sqrt {c + 2 x}}
+$$
+
+In Table 5, we report the top 10 hypotheses returned by our model for this equation. We observe that all generations are actually valid solutions, although they are expressed very differently. They are however not all equal: merging the square roots within the first and third equations would give the same expression except that the third one would contain a factor 2 in front of the constant  $c$ , but up to a change of variable, these two solutions are actually equivalent. The ability of the model to recover equivalent expressions, without having been trained to do so, is very intriguing.
+
+<table><tr><td>Hypothesis</td><td>Score</td><td>Hypothesis</td><td>Score</td></tr><tr><td>9√x√1/ log(x)/√c+2x</td><td>-0.047</td><td>9/√c log(x)/x + 2 log(x)</td><td>-0.124</td></tr><tr><td>9√x/√c+2x√log(x)</td><td>-0.056</td><td>9√x/√c log(x) + 2x log(x)</td><td>-0.139</td></tr><tr><td>9√2√x√1/ log(x)/2√c+x</td><td>-0.115</td><td>9/√c/x + 2√log(x)</td><td>-0.144</td></tr><tr><td>9√x√1/c log(x) + 2x log(x)</td><td>-0.117</td><td>9√1/c log(x)/x + 2 log(x)</td><td>-0.205</td></tr><tr><td>9√2√x/2√c+x√log(x)</td><td>-0.124</td><td>9√x√1/c log(x) + 2x log(x) + log(x)</td><td>-0.232</td></tr></table>
+
+Table 5: Top 10 generations of our model for the first order differential equation  ${162x}\log \left( x\right) {y}^{\prime } + 2{y}^{3}\log {\left( x\right) }^{2} -$ ${81y}\log \left( x\right)  + {81y} = 0$  ,generated with a beam search. All hypotheses are valid solutions, and are equivalent up to a change of the variable  $c$  . Scores are log-probabilities normalized by sequence lengths.
+
+# 4.7 GENERALIZATION ACROSS GENERATORS
+
+Models for integration achieve close to  $100\%$  performance on held-out test samples generated with the same method as their training data. In Table 6, we compare the accuracy on the FWD, BWD and IBP test sets for 4 models trained using different combinations of training data. When the test set is generated with the same generator as the training set, the model performs extremely well. For instance, the three models trained either on BWD, BWD + IBP or BWD + IBP + FWD achieve  $99.7\%$  accuracy on the BWD test set with a beam size of 50.
+
+On the other hand, even with a beam size of 50, a FWD-trained model only achieves  $17.2\%$  accuracy on the BWD test set, and a BWD-trained model achieves  $27.5\%$  on the FWD test set. This results from the very different structure of the FWD and BWD data sets (cf. Table 1 and the discussion in Section E of the appendix). Overall, a model trained on BWD samples learns that integration tends to shorten expressions, a property that does not hold for FWD samples. Adding diversity to the training set improves the results. For instance, adding IBP-generated examples to the BWD-trained model raises the FWD test accuracy from  $27.5\%$  to  $56.1\%$ , and with additional FWD training data the model reaches  $94.3\%$  accuracy. Generalization is further discussed in Section E of the appendix.
+
+<table><tr><td rowspan="2">Training data</td><td colspan="3">Forward (FWD)</td><td colspan="3">Backward (BWD)</td><td colspan="3">Integration by parts (IBP)</td></tr><tr><td>Beam 1</td><td>Beam 10</td><td>Beam 50</td><td>Beam 1</td><td>Beam 10</td><td>Beam 50</td><td>Beam 1</td><td>Beam 10</td><td>Beam 50</td></tr><tr><td>FWD</td><td>93.6</td><td>95.6</td><td>96.2</td><td>10.9</td><td>13.9</td><td>17.2</td><td>85.6</td><td>86.8</td><td>88.9</td></tr><tr><td>BWD</td><td>18.9</td><td>24.6</td><td>27.5</td><td>98.4</td><td>99.4</td><td>99.7</td><td>42.9</td><td>54.6</td><td>59.2</td></tr><tr><td>BWD + IBP</td><td>41.6</td><td>54.9</td><td>56.1</td><td>98.2</td><td>99.4</td><td>99.7</td><td>96.8</td><td>99.2</td><td>99.5</td></tr><tr><td>BWD + IBP + FWD</td><td>89.1</td><td>93.4</td><td>94.3</td><td>98.1</td><td>99.3</td><td>99.7</td><td>97.2</td><td>99.4</td><td>99.7</td></tr></table>
+
+# 4.8 GENERALIZATION BEYOND THE GENERATOR - SYMPY
+
+Our forward generator, FWD, generates a set of pairs  $(f,F)$  of functions with their integrals. It relies on an external symbolic framework, SymPy (Meurer et al., 2017), to compute the integral of randomly generated functions. SymPy is not perfect, and fails to compute the integral of many integrable functions. In particular, we found that the accuracy of SymPy on the BWD test set is only  $30\%$ . Our FWD-trained model only obtains an accuracy of  $17.2\%$  on BWD. However, we observed that the FWD-trained model is sometimes able to compute the integral of functions that SymPy cannot compute. This means that by only training on functions that SymPy can integrate, the model was able to generalize to functions that SymPy cannot integrate. Table 7 presents examples of such functions with their integrals.
+
+Table 6: Accuracy of our models on function integration. We report the accuracy of our model on the three integration datasets: forward (FWD), backward (BWD), and integration by parts (IBP), for four models trained with different combinations of training data. We observe that a FWD-trained model performs poorly when it tries to integrate functions from the BWD dataset. Similarly, a BWD-trained model only obtains  $27.5\%$  accuracy on the FWD dataset, as it fails to integrate simple functions like  $x^5 \sin(x)$ . On the other hand, training on both the BWD + IBP datasets allows the model to reach up to  $56.1\%$  accuracy on FWD. Training on all datasets allows the model to perform well on the three distributions.  
+
+<table><tr><td>x2(tan2(x)+1)+2x tan(x)+1</td><td>x2tan(x)+x</td></tr><tr><td>1+2cos(2x)/√sin2(2x)+1</td><td>x+asinh(sin(2x))</td></tr><tr><td>x tan(x)+log(x cos(x))-1/√log(x cos(x))2</td><td>x/√log(x cos(x))</td></tr><tr><td>-2x cos(asin2(x)) asin(x)/√1-x2sin2(asin2(x))+1/sin(asin2(x))</td><td>x/sin(asin2(x))</td></tr><tr><td>√x+x(2x/√x4+1)+1/2√x)+x+asin(x2)</td><td>x(√x+x+asin(x2))</td></tr><tr><td>-3-3(-3x2sin(x3)+1/2√x)/√x+cos(x3)/(x+log(√x+cos(x3))))2</td><td>3/x+log(√x+cos(x3))</td></tr><tr><td>-2tan2(log(log(x))-2/√log(x)tan2(log(log(x))))+2/tan(log(log(x)))</td><td>2x/tan(log(log(x)))</td></tr></table>
+
+Table 7: Examples of functions / integrals that the FWD-trained model can integrate, but not SymPy. Although the FWD model was only trained on a subset of functions that SymPy can integrate, it learned to generalize to functions that SymPy cannot integrate.
+
+# 5 RELATED WORK
+
+Computers were used for symbolic mathematics since the late 1960s (Moses, 1974). Computer algebra systems (CAS), such as Matlab, Mathematica, Maple, PARI and SAGE, are used for a variety of mathematical tasks (Gathen & Gerhard, 2013). Modern methods for symbolic integration are based on Risch algorithm (Risch, 1970). Implementations can be found in Bronstein (2005) and Geddes et al. (1992). However, the complete description of the Risch algorithm takes more than 100 pages, and is not fully implemented in current mathematical framework.
+
+Deep learning networks have been used to simplify treelike expressions. Zaremba et al. (2014) use recursive neural networks to simplify complex symbolic expressions. They use tree representations for expressions, but provide the model with problem related information: possible rules for simplification. The neural network is trained to select the best rule. Allamanis et al. (2017) propose a framework called neural equivalence networks to learn semantic representations of algebraic expressions. Typically, a model is trained to map different but equivalent expressions (like the 10 expressions proposed in Table 5) to the same representation. However, they only consider Boolean and polynomial expressions. More recently, Arabshahi et al. (2018a,b) used tree-structured neural networks to verify the correctness of given symbolic entities, and to predict missing entries in incomplete mathematical equations. They also showed that these networks could be used to predict whether an expression is a valid solution of a given differential equation.
+
+Most attempts to use deep networks for mathematics have focused on arithmetic over integers (sometimes over polynomials with integer coefficients). For instance, Kaiser & Sutskever (2015) proposed the Neural-GPU architecture, and train networks to perform additions and multiplications of numbers given in their binary representations. They show that a model trained on numbers with up-to 20 bits can be applied to much larger numbers at test time, while preserving a perfect accuracy. Freivalds & Liepins (2017) proposed an improved version of the Neural-GPU by using hard non-linear activation functions, and a diagonal gating mechanism.
+
+Saxton et al. (2019) use LSTMs (Hochreiter & Schmidhuber, 1997) and transformers on a wide range of problems, from arithmetic to simplification of formal expressions. However, they only consider polynomial functions, and the task of differentiation, which is significantly easier than integration. Trask et al. (2018) propose the Neural arithmetic logic units, a new module designed to learn systematic numerical computation, and that can be used within any neural network. Like Kaiser & Sutskever (2015), they show that at inference their model can extrapolate on numbers orders of magnitude larger than the ones seen during training.
+
+# 6 CONCLUSION
+
+In this paper, we show that standard seq2seq models can be applied to difficult tasks like function integration, or solving differential equations. We propose an approach to generate arbitrarily large datasets of equations, with their associated solutions. We show that a simple transformer model trained on these datasets can perform extremely well both at computing function integrals, and solving differential equations, outperforming state-of-the-art mathematical frameworks like Matlab or Mathematica that rely on a large number of algorithms and heuristics, and a complex implementation (Risch, 1970). Results also show that the model is able to write identical expressions in very different ways.
+
+These results are surprising given the difficulty of neural models to perform simpler tasks like integer addition or multiplication. However, proposed hypotheses are sometimes incorrect, and considering multiple beam hypotheses is often necessary to obtain a valid solution. The validity of a solution itself is not provided by the model, but by an external symbolic framework (Meurer et al., 2017). These results suggest that in the future, standard mathematical frameworks may benefit from integrating neural components in their solvers.
+
+# REFERENCES
+
+Miltiadis Allamanis, Pankajan Chanthirasegaran, Pushmeet Kohli, and Charles Sutton. Learning continuous semantic representations of symbolic expressions. In Proceedings of the 34th International Conference on Machine Learning - Volume 70, ICML'17, pp. 80-88. JMLR.org, 2017.  
+Forough Arabshahi, Sameer Singh, and Animashree Anandkumar. Combining symbolic expressions and black-box function evaluations for training neural programs. In International Conference on Learning Representations, 2018a.  
+Forough Arabshahi, Sameer Singh, and Animashree Anandkumar. Towards solving differential equations through neural programming. 2018b.  
+D. Bahdanau, K. Cho, and Y. Bengio. Neural machine translation by jointly learning to align and translate. In International Conference on Learning Representations (ICLR), 2015.  
+M. Bronstein. *Symbolic Integration I: Transcendental Functions*. Algorithms and combinatorics. Springer, 2005. ISBN 978-3-540-21493-9.  
+Chris Dyer, Adhiguna Kuncoro, Miguel Ballesteros, and Noah A Smith. Recurrent neural network grammars. In Proceedings of the 2016 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, pp. 199-209, 2016.  
+Akiko Eriguchi, Yoshimasa Tsuruoka, and Kyunghyun Cho. Learning to parse and translate improves neural machine translation. In Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics (Volume 2: Short Papers), pp. 72-78, 2017.  
+Philippe Flajolet and Andrew M. Odlyzko. Singularity analysis of generating functions. SIAM J. Discrete Math., 3(2):216-240, 1990.  
+Philippe Flajolet and Robert Sedgewick. Analytic Combinatorics. Cambridge University Press, New York, NY, USA, 1 edition, 2009. ISBN 0521898064, 9780521898065.  
+Karlis Freivalds and Renars Liepins. Improving the neuralgpu architecture for algorithm learning. ArXiv, abs/1702.08727, 2017.  
+Joachim von zur Gathen and Jurgen Gerhard. Modern Computer Algebra. Cambridge University Press, New York, NY, USA, 3rd edition, 2013. ISBN 1107039037, 9781107039032.  
+Keith O. Geddes, Stephen R. Czapor, and George Labahn. Algorithms for Computer Algebra. Kluwer Academic Publishers, Norwell, MA, USA, 1992. ISBN 0-7923-9259-0.  
+Sepp Hochreiter and Jürgen Schmidhuber. Long short-term memory. Neural computation, 9(8): 1735-1780, 1997.  
+Lukasz Kaiser and Ilya Sutskever. Neural gpus learn algorithms. CoRR, abs/1511.08228, 2015.  
+Diederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.  
+Donald E. Knuth. The Art of Computer Programming, Volume 1 (3rd Ed.): Fundamental Algorithms. Addison Wesley Longman Publishing Co., Inc., Redwood City, CA, USA, 1997. ISBN 0-201-89683-4.  
+Philipp Koehn. Pharaoh: a beam search decoder for phrase-based statistical machine translation models. In Conference of the Association for Machine Translation in the Americas, pp. 115-124. Springer, 2004.  
+Sarah Loos, Geoffrey Irving, Christian Szegedy, and Cezary Kaliszyk. Deep network guided proof search. arXiv preprint arXiv:1701.06972, 2017.  
+MathWorks. Matlab optimization toolbox (r2019a), 2019. The MathWorks, Natick, MA, USA.
+
+Aaron Meurer, Christopher P. Smith, Mateusz Paprocki, Ondrej Čertík, Sergey B. Kirpichev, Matthew Rocklin, AMiT Kumar, Sergiu Ivanov, Jason K. Moore, Sartaj Singh, Thilina Rathnayake, Sean Vig, Brian E. Granger, Richard P. Muller, Francesco Bonazzi, Harsh Gupta, Shivam Vats, Fredrik Johansson, Fabian Pedregosa, Matthew J. Curry, Andy R. Terrel, Štepan Roučka, Ashutosh Saboo, Isuru Fernando, Sumith Kulal, Robert Cimrman, and Anthony Scopatz. Sympy: symbolic computing in python. PeerJ Computer Science, 3:e103, January 2017. ISSN 2376-5992. doi: 10.7717/peerj-cs.103. URL https://doi.org/10.7717/peerj-cs.103.  
+Joel Moses. Macsyma - the fifth year. SIGSAM Bull., 8(3):105-110, August 1974. ISSN 0163-5824.  
+Myle Ott, Michael Auli, David Grangier, et al. Analyzing uncertainty in neural machine translation. In International Conference on Machine Learning, pp. 3953-3962, 2018.  
+Kishore Papineni, Salim Roukos, Todd Ward, and Wei-Jing Zhu. Bleu: a method for automatic evaluation of machine translation. In Proceedings of the 40th annual meeting on association for computational linguistics, pp. 311-318. Association for Computational Linguistics, 2002.  
+Robert H. Risch. The solution of the problem of integration in finite terms. Bull. Amer. Math. Soc., 76(3):605-608, 05 1970.  
+David E. Rumelhart, James L. McClelland, and CORPORATE PDP Research Group (eds.). Parallel Distributed Processing: Explorations in the Microstructure of Cognition, Vol. 1: Foundations. MIT Press, Cambridge, MA, USA, 1986. ISBN 0-262-68053-X.  
+David Saxton, Edward Grefenstette, Felix Hill, and Pushmeet Kohli. Analysing mathematical reasoning abilities of neural models. In International Conference on Learning Representations, 2019.  
+N. J. A. Sloane. The encyclopedia of integer sequences, 1996.  
+Richard P. Stanley. Enumerative Combinatorics: Volume 1. Cambridge University Press, New York, NY, USA, 2nd edition, 2011. ISBN 1107602629, 9781107602625.  
+Ilya Sutskever, Oriol Vinyals, and Quoc V Le. Sequence to sequence learning with neural networks. In Advances in Neural Information Processing Systems, pp. 3104-3112, 2014.  
+Kai Sheng Tai, Richard Socher, and Christopher D Manning. Improved semantic representations from tree-structured long short-term memory networks. In Proceedings of the 53rd Annual Meeting of the Association for Computational Linguistics and the 7th International Joint Conference on Natural Language Processing (Volume 1: Long Papers), pp. 1556–1566, 2015.  
+Andrew Trask, Felix Hill, Scott E Reed, Jack Rae, Chris Dyer, and Phil Blunsom. Neural arithmetic logic units. In Advances in Neural Information Processing Systems, pp. 8035-8044, 2018.  
+Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in Neural Information Processing Systems, pp. 6000-6010, 2017.  
+Oriol Vinyals, Łukasz Kaiser, Terry Koo, Slav Petrov, Ilya Sutskever, and Geoffrey Hinton. Grammar as a foreign language. In Advances in neural information processing systems, pp. 2773-2781, 2015.  
+H.S. Wilf. generatingfunctionology: Third Edition. CRC Press, 2005. ISBN 978-1-4398-6439-5. URL https://www.math.upenn.edu/ wilf/gfologyLinked2.pdf.  
+Wolfram-Research.Mathematica,version12.0,2019.Champaign,IL,2019.  
+Wojciech Zaremba and Ilya Sutskever. Learning to execute. arXiv preprint arXiv:1410.4615, 2014.  
+Wojciech Zaremba, Karol Kurach, and Rob Fergus. Learning to discover efficient mathematical identities. In Proceedings of the 27th International Conference on Neural Information Processing Systems - Volume 1, NIPS'14, pp. 1278-1286, Cambridge, MA, USA, 2014. MIT Press.
+
+# A A SYNTAX FOR MATHEMATICAL EXPRESSIONS
+
+We represent mathematical expressions as trees with operators as internal nodes, and numbers, constants or variables, as leaves. By enumerating nodes in prefix order, we transform trees into sequences suitable for seq2seq architectures.
+
+For this representation to be efficient, we want expressions, trees and sequences to be in a one-to-one correspondence. Different expressions will always result in different trees and sequences, but for the reverse to hold, we need to take care of a few special cases.
+
+First, expressions like sums and products may correspond to several trees. For instance, the expression  $2 + 3 + 5$  can be represented as any one of those trees:
+
+![](images/63a629478a73ccb35f97453c2645dd6daf9b4805a30dddcaa2746323605f9807.jpg)  
+-2
+
+![](images/c92291ea7fdd3afcad70b024f1c64c321b2884f19a80f6acc1de61848481ada4.jpg)
+
+![](images/b0ffe416060149137e5aae6c2c4ba40ca402f50033c9d367afb4a51d5edc1c59.jpg)
+
+We will assume that all operators have at most two operands, and that, in case of doubt, they are associative to the right.  $2 + 3 + 5$  would then correspond to the rightmost tree.
+
+Second, the distinction between internal nodes (operators) and leaves (mathematical primitive objects) is somewhat arbitrary. For instance, the number  $-2$  could be represented as a basic object, or as a unary minus operator applied to the number 2. Similarly, there are several ways to represent  $\sqrt{5}$ ,  $42x^5$ , or the function  $\log_{10}$ . For simplicity, we only consider numbers, constants and variables as possible leaves, and avoid using a unary minus. In particular, expressions like  $-x$  are represented as  $-1 \times x$ . Here are the trees for  $-2$ ,  $\sqrt{5}$ ,  $42x^5$  and  $-x$ :
+
+![](images/962d80daebbb6b92062add585f21b808d49f334e568dd033c1e285034c21f0b0.jpg)
+
+![](images/8af5b477748f51c6f8b0c4e833310083c0b4d403c79e711090996acd24c66c75.jpg)
+
+![](images/c4b1491ced2f1204f59382587ec393d707ddc4d94388d6cb8d7e7b99328f64e5.jpg)
+
+Integers are represented in positional notation, as a sign followed by a sequence of digits (from 0 to 9 in base 10). For instance, 2354 and  $-34$  are represented as  $+2$  3 5 4 and  $-3$  4. For zero, a unique representation is chosen  $(+0$  or  $-0)$ .
+
+# B MATHEMATICAL DERIVATIONS OF THE PROBLEM SPACE SIZE
+
+In this section, we investigate the size of the problem space by computing the number of expressions with  $n$  internal nodes. We first deal with the simpler case where we only have binary operators  $(p_1 = 0)$ , then consider trees and expressions composed of unary and binary operators. In each case, we calculate a generating function (Flajolet & Sedgewick, 2009; Wilf, 2005) from which we derive a closed formula or recurrence on the number of expressions, and an asymptotic expansion.
+
+# B.1 BINARY TREES AND EXPRESSIONS
+
+The main part of this derivation follows (Knuth, 1997) (pages 388-389).
+
+Generating function Let  $b_{n}$  be the number of binary trees with  $n$  internal nodes. We have  $b_{0} = 1$  and  $b_{1} = 1$ . Any binary tree with  $n$  internal nodes can be generated by concatenating a left and a right subtree with  $k$  and  $n - 1 - k$  internal nodes respectively. By summing over all possible values of  $k$ , we have that:
+
+$$
+b _ {n} = b _ {0} b _ {n - 1} + b _ {1} b _ {n - 2} + \dots + b _ {n - 2} b _ {1} + b _ {n - 1} b _ {0}
+$$
+
+Let  $B(z)$  be the generating function of  $b_{n}$ ,  $B(z) = b_{0} + b_{1}z + b_{2}z^{2} + b_{3}z^{3} + \ldots$ .
+
+$$
+\begin{array}{l} B (z) ^ {2} = b _ {0} ^ {2} + \left(b _ {0} b _ {1} + b _ {1} b _ {0}\right) z + \left(b _ {0} b _ {2} + b _ {1} b _ {1} + b 2 b _ {0}\right) z ^ {2} + \dots \\ = b _ {1} + b _ {2} z + b _ {3} z ^ {2} + \dots \\ = \frac {B (z) - b _ {0}}{z} \\ \end{array}
+$$
+
+So,  $zB(z)^2 - B(z) + 1 = 0$ . Solving for  $B(z)$  gives:
+
+$$
+B (z) = \frac {1 \pm \sqrt {1 - 4 z}}{2 z}
+$$
+
+and since  $B(0) = b_{0} = 1$ , we derive the generating function for sequence  $b_{n}$
+
+$$
+B (z) = \frac {1 - \sqrt {1 - 4 z}}{2 z}
+$$
+
+We now derive a closed formula for  $b_{n}$ . By the binomial theorem,
+
+$$
+\begin{array}{l} B (z) = \frac {1}{2 z} \left(1 - \sum_ {k = 0} ^ {\infty} \binom {1 / 2} {k} (- 4 z) ^ {k}\right) \\ = \frac {1}{2 z} \left(1 + \sum_ {k = 0} ^ {\infty} \frac {1}{2 k - 1} \binom {2 k} {k} z ^ {k}\right) \\ = \frac {1}{2 z} \sum_ {k = 1} ^ {\infty} \frac {1}{2 k - 1} \binom {2 k} {k} z ^ {k} \\ = \sum_ {k = 1} ^ {\infty} \frac {1}{2 (2 k - 1)} \binom {2 k} {k} z ^ {k - 1} \\ = \sum_ {k = 0} ^ {\infty} \frac {1}{2 (2 k + 1)} \binom {2 k + 2} {k + 1} z ^ {k} \\ = \sum_ {k = 0} ^ {\infty} \frac {1}{k + 1} \binom {2 k} {k} z ^ {k} \\ \end{array}
+$$
+
+Therefore
+
+$$
+b _ {n} = \frac {1}{n + 1} \binom {2 n} {n} = \frac {(2 n) !}{(n + 1) ! n !}
+$$
+
+These are the Catalan numbers, a closed formula for the number of binary trees with  $n$  internal nodes. We now observe that a binary tree with  $n$  internal nodes has exactly  $n + 1$  leaves. Since each node in a binary tree can represent  $p_2$  operators, and each leaf can take  $L$  values, we have that a tree with  $n$  nodes can take  $p_2^n L^{n+1}$  possible combinations of operators and leaves. As a result, the number of binary expressions with  $n$  operators is given by:
+
+$$
+E _ {n} = \frac {(2 n) !}{(n + 1) ! n !} p _ {2} ^ {n} L ^ {n + 1}
+$$
+
+Asymptotic estimate To derive an asymptotic approximation of  $b_{n}$ , we apply the Stirling formula:
+
+$$
+n! \approx \sqrt {2 \pi n} \left(\frac {n}{e}\right) ^ {n} \quad \mathrm {s o} \quad \binom {2 n} {n} \approx \frac {4 ^ {n}}{\sqrt {\pi n}} \quad \mathrm {a n d} \quad b _ {n} \approx \frac {4 ^ {n}}{n \sqrt {\pi n}}
+$$
+
+Finally, we have the following formulas for the number of expressions with  $n$  internal nodes:
+
+$$
+E _ {n} \approx \frac {1}{n \sqrt {\pi n}} (4 p _ {2}) ^ {n} L ^ {n + 1}
+$$
+
+# B.2 UNARY-BINARY TREES
+
+Generating function Let  $s_n$  be the number of unary-binary trees (i.e. trees where internal nodes can have one or two children) with  $n$  internal nodes. We have  $s_0 = 1$  and  $s_1 = 2$  (the only internal node is either unary or binary).
+
+Any tree with  $n$  internal nodes is obtained either by adding a unary internal node at the root of a tree with  $n - 1$  internal nodes, or by concatenating with a binary operator a left and a right subtree with  $k$  and  $n - 1 - k$  internal nodes respectively. Summing up as before, we have:
+
+$$
+s _ {n} = s _ {n - 1} + s _ {0} s _ {n - 1} + s _ {1} s _ {n - 2} + \dots + s _ {n - 1} s _ {0}
+$$
+
+Let  $S(z)$  be the generating function of the  $s_n$ . The above formula translates into
+
+$$
+S (z) ^ {2} = \frac {S (z) - s _ {0}}{z} - S (z)
+$$
+
+$$
+z S (z) ^ {2} + (z - 1) S (z) + 1 = 0
+$$
+
+solving and taking into account the fact that  $S(0) = 1$ , we obtain the generating function of the  $s_n$
+
+$$
+S (z) = \frac {1 - z - \sqrt {1 - 6 z + z ^ {2}}}{2 z}
+$$
+
+The numbers  $s_n$  generated by  $S(z)$  are known as the Schroeder numbers (OEIS A006318) (Sloane, 1996). They appear in different combinatorial problems (Stanley, 2011). Notably, they correspond to the number of paths from  $(0,0)$  to  $(n,n)$  of a  $n \times n$  grid, moving north, east, or northeast, and never rising above the diagonal.
+
+Calculation Schroeder numbers do not have a simple closed formula, but a recurrence allowing for their calculation can be derived from their generating function. Rewriting  $S(z)$  as
+
+$$
+2 z S (z) + z - 1 = - \sqrt {1 - 6 z + z ^ {2}}
+$$
+
+and differentiating, we have
+
+$$
+2 z S ^ {\prime} (z) + 2 S (z) + 1 = \frac {3 - z}{\sqrt {1 - 6 z + z ^ {2}}} = \frac {3 - z}{1 - 6 z + z ^ {2}} (1 - z - 2 z S (z))
+$$
+
+$$
+2 z S ^ {\prime} (z) + 2 S (z) \left(1 + \frac {3 z - z ^ {2}}{1 - 6 z + z ^ {2}}\right) = \frac {(3 - z) (1 - z)}{1 - 6 z + z ^ {2}} - 1
+$$
+
+$$
+2 z S ^ {\prime} (z) + 2 S (z) \frac {1 - 3 z}{1 - 6 z + z ^ {2}} = \frac {2 + 2 z}{1 - 6 z + z ^ {2}}
+$$
+
+$$
+z (1 - 6 z + z ^ {2}) S ^ {\prime} (z) + (1 - 3 z) S (z) = 1 + z
+$$
+
+Replacing  $S(z)$  and  $S'(z)$  with their n-th coefficient yields, for  $n > 1$
+
+$$
+n s _ {n} - 6 (n - 1) s _ {n - 1} + (n - 2) s _ {n - 2} + s _ {n} - 3 s _ {n - 1} = 0
+$$
+
+$$
+(n + 1) s _ {n} = 3 (2 n - 1) s _ {n - 1} - (n - 2) s _ {n - 2}
+$$
+
+Together with  $s_0 = 1$  and  $s_1 = 2$ , this allows for fast  $(O(n))$  calculation of Schroeder numbers.
+
+Asymptotic estimate To derive an asymptotic formula of  $s_n$ , we develop the generating function around its smallest singularity (Flajolet & Odlyzko, 1990), i.e. the radius of convergence of the power series. Since
+
+$$
+1 - 6 z + z ^ {2} = \left(1 - (3 - \sqrt {8}) z\right) \left(1 - (3 + \sqrt {8}) z\right)
+$$
+
+The smallest singular value is
+
+$$
+r _ {1} = \frac {1}{(3 + \sqrt {8})}
+$$
+
+and the asymptotic formula will have the exponential term
+
+$$
+r _ {1} ^ {- n} = (3 + \sqrt {8}) ^ {n} = (1 + \sqrt {2}) ^ {2 n}
+$$
+
+In a neighborhood of  $r_1$ , the generating function can be rewritten as
+
+$$
+S (z) \approx (1 + \sqrt {2}) \left(1 - 2 ^ {1 / 4} \sqrt {1 - (3 + \sqrt {8}) z}\right) + O (1 - (3 + \sqrt {8}) z) ^ {3 / 2}
+$$
+
+Since
+
+$$
+[ z _ {n} ] \sqrt {1 - a z} \approx - \frac {a ^ {n}}{\sqrt {4 \pi n ^ {3}}}
+$$
+
+where  $[z_n]F(z)$  denotes the n-th coefficient in the formal series of F, we have
+
+$$
+s _ {n} \approx \frac {(1 + \sqrt {2}) (3 + \sqrt {8}) ^ {n}}{2 ^ {3 / 4} \sqrt {\pi n ^ {3}}} = \frac {(1 + \sqrt {2}) ^ {2 n + 1}}{2 ^ {3 / 4} \sqrt {\pi n ^ {3}}}
+$$
+
+Comparing with the number of binary trees, we have
+
+$$
+s _ {n} \approx 1. 4 4 (1. 4 6) ^ {n} b _ {n}
+$$
+
+# B.3 UNARY-BINARY EXPRESSIONS
+
+In the binary case, the number of expressions can be derived from the number of trees. This cannot be done in the unary-binary case, as the number of leaves in a tree with  $n$  internal nodes depends on the number of binary operators  $(n_{2} + 1)$ .
+
+Generating function The number of trees with  $n$  internal nodes and  $n_2$  binary operators can be derived from the following observation: any unary-binary tree with  $n_2$  binary internal nodes can be generated from a binary tree by adding unary internal nodes. Each node in the binary tree can receive one or several unary parents.
+
+Since the binary tree has  $2n_{2} + 1$  nodes and the number of unary internal nodes to be added is  $n - n_{2}$ , the number of unary-binary trees that can be created from a specific binary tree is the number of multisets with  $2n_{2} + 1$  elements on  $n - n_{2}$  symbols, that is
+
+$$
+\left( \begin{array}{c} n + n _ {2} \\ n - n _ {2} \end{array} \right) = \left( \begin{array}{c} n + n _ {2} \\ 2 n _ {2} \end{array} \right)
+$$
+
+If  $b_{q}$  denotes the q-th Catalan number, the number of trees with  $n_2$  binary operators among  $n$  is
+
+$$
+\left( \begin{array}{c} n + n _ {2} \\ 2 n _ {2} \end{array} \right) b _ {n _ {2}}
+$$
+
+Since such trees have  $n_2 + 1$  leaves, with  $L$  leaves,  $p_2$  binary and  $p_1$  unary operators to choose from, the number of expressions is
+
+$$
+E (n, n _ {2}) = \left( \begin{array}{c} n + n _ {2} \\ 2 n _ {2} \end{array} \right) b _ {n _ {2}} p _ {2} ^ {n _ {2}} p _ {1} ^ {n - n _ {2}} L ^ {n _ {2} + 1}
+$$
+
+Summing over all values of  $n_2$  (from 0 to  $n$ ) yields the number of different expressions
+
+$$
+E _ {n} = \sum_ {n _ {2} = 0} ^ {n} \binom {n + n _ {2}} {2 n _ {2}} b _ {n _ {2}} p _ {2} ^ {n _ {2}} p _ {1} ^ {n - n _ {2}} L ^ {n _ {2} + 1} z ^ {n}
+$$
+
+Let  $E(z)$  be the corresponding generating function.
+
+$$
+\begin{array}{l} E (z) = \sum_ {n = 0} ^ {\infty} E _ {n} z ^ {n} \\ = \sum_ {n = 0} ^ {\infty} \sum_ {n _ {2} = 0} ^ {n} \binom {n + n _ {2}} {2 n _ {2}} b _ {n _ {2}} p _ {2} ^ {n _ {2}} p _ {1} ^ {n - n _ {2}} L ^ {n _ {2} + 1} z ^ {n} \\ = L \sum_ {n = 0} ^ {\infty} \sum_ {n _ {2} = 0} ^ {n} \binom {n + n _ {2}} {2 n _ {2}} b _ {n _ {2}} \left(\frac {L p _ {2}}{p _ {1}}\right) ^ {n _ {2}} p _ {1} ^ {n} z ^ {n} \\ = L \sum_ {n = 0} ^ {\infty} \sum_ {n _ {2} = 0} ^ {\infty} \binom {n + n _ {2}} {2 n _ {2}} b _ {n _ {2}} \left(\frac {L p _ {2}}{p _ {1}}\right) ^ {n _ {2}} (p _ {1} z) ^ {n} \\ \end{array}
+$$
+
+since  $\binom{n+n_2}{2n_2} = 0$  when  $n > n_2$
+
+$$
+\begin{array}{l} E (z) = L \sum_ {n _ {2} = 0} ^ {\infty} b _ {n _ {2}} \left(\frac {L p _ {2}}{p _ {1}}\right) ^ {n _ {2}} \sum_ {n = 0} ^ {\infty} \binom {n + n _ {2}} {2 n _ {2}} (p _ {1} z) ^ {n} \\ = L \sum_ {n _ {2} = 0} ^ {\infty} b _ {n _ {2}} \left(\frac {L p _ {2}}{p _ {1}}\right) ^ {n _ {2}} \sum_ {n = 0} ^ {\infty} \binom {n + 2 n _ {2}} {2 n _ {2}} (p _ {1} z) ^ {n + n _ {2}} \\ = L \sum_ {n _ {2} = 0} ^ {\infty} b _ {n _ {2}} (L p _ {2} z) ^ {n _ {2}} \sum_ {n = 0} ^ {\infty} \binom {n + 2 n _ {2}} {2 n _ {2}} (p _ {1} z) ^ {n} \\ \end{array}
+$$
+
+applying the binomial formula
+
+$$
+\begin{array}{l} E (z) = L \sum_ {n _ {2} = 0} ^ {\infty} b _ {n _ {2}} (L p _ {2} z) ^ {n _ {2}} \frac {1}{(1 - p _ {1} z) ^ {2 n _ {2} + 1}} \\ = \frac {L}{1 - p _ {1} z} \sum_ {n _ {2} = 0} ^ {\infty} b _ {n _ {2}} \left(\frac {L p _ {2} z}{(1 - p _ {1} z) ^ {2}}\right) ^ {n _ {2}} \\ \end{array}
+$$
+
+applying the generating function for binary trees
+
+$$
+\begin{array}{l} E (z) = \frac {L}{1 - p _ {1} z} \left(\frac {1 - \sqrt {1 - 4 \frac {L p _ {2} z}{(1 - p _ {1} z) ^ {2}}}}{2 \frac {L p _ {2} z}{(1 - p _ {1} z) ^ {2}}}\right) \\ = \frac {1 - p _ {1} z}{2 p _ {2} z} \left(1 - \sqrt {1 - 4 \frac {L p _ {2} z}{\left(1 - p _ {1} z\right) ^ {2}}}\right) \\ = \frac {1 - p _ {1} z - \sqrt {(1 - p _ {1} z) ^ {2} - 4 L p _ {2} z}}{2 p _ {2} z} \\ \end{array}
+$$
+
+Reducing, we have
+
+$$
+E (z) = \frac {1 - p _ {1} z - \sqrt {1 - 2 (p _ {1} + 2 L p _ {2} k) z + p _ {1} z ^ {2}}}{2 p _ {2} z}
+$$
+
+Calculation As before, there is no closed simple formula for  $E_{n}$ , but we can derive a recurrence formula by differentiating the generating function, rewritten as
+
+$$
+2 p _ {2} z E (z) + p _ {1} z - 1 = - \sqrt {1 - 2 \left(p _ {1} + 2 p _ {2} L\right) z + p _ {1} z ^ {2}}
+$$
+
+$$
+2 p _ {2} z E ^ {\prime} (z) + 2 p _ {2} E (z) + p _ {1} = \frac {p _ {1} + 2 p _ {2} L - p _ {1} z}{\sqrt {1 - 2 \left(p _ {1} + 2 p _ {2} L\right) z + p _ {1} z ^ {2}}}
+$$
+
+$$
+2 p _ {2} z E ^ {\prime} (z) + 2 p _ {2} E (z) + p _ {1} = \frac {(p _ {1} + 2 p _ {2} L - p _ {1} z) (1 - p _ {1} z - 2 p _ {2} z E (z))}{1 - 2 (p _ {1} + 2 p _ {2} L) z + p _ {1} z ^ {2}}
+$$
+
+$$
+2 p _ {2} z E ^ {\prime} (z) + 2 p _ {2} E (z) \left(1 + \frac {z \left(p _ {1} + 2 p _ {2} L - p _ {1} z\right)}{1 - 2 \left(p _ {1} + 2 p _ {2} L\right) z + p _ {1} z ^ {2}}\right) = \frac {\left(p _ {1} + 2 p _ {2} L - p _ {1} z\right) \left(1 - p _ {1} z\right)}{1 - 2 \left(p _ {1} + 2 p _ {2} L\right) z + p _ {1} z ^ {2}} - p _ {1}
+$$
+
+$$
+2 p _ {2} z E ^ {\prime} (z) + 2 p _ {2} E (z) \left(\frac {1 - (p _ {1} + 2 p _ {2} L) z}{1 - 2 (p _ {1} + 2 p _ {2} L) z + p _ {1} z ^ {2}}\right) = \frac {2 p _ {2} L (1 + p _ {1} z) + p _ {1} (p _ {1} - 1) z}{1 - 2 (p _ {1} + 2 p _ {2} L) z + p _ {1} z ^ {2}}
+$$
+
+$$
+2 p _ {2} z E ^ {\prime} (z) \left(1 - 2 \left(p _ {1} + 2 p _ {2} L\right) z + p _ {1} z ^ {2}\right) + 2 p _ {2} E (z) \left(1 - \left(p _ {1} + 2 p _ {2} L\right) z\right) = \left(2 p _ {2} L \left(1 + p _ {1} z\right) + p _ {1} \left(p _ {1} - 1\right) z\right)
+$$
+
+replacing  $E(z)$  and  $E'(z)$  with their coefficients
+
+$$
+2 p _ {2} \left(n E _ {n} - 2 \left(p _ {1} + 2 p _ {2} L\right) (n - 1) E _ {n - 1} + p _ {1} (n - 2) E (n - 2)\right) + 2 p _ {2} \left(E _ {n} - \left(p _ {1} + 2 p _ {2} L\right) E _ {n - 1}\right) = 0
+$$
+
+$$
+(n + 1) E _ {n} - \left(p _ {1} + 2 p _ {2} L\right) (2 n - 1) E _ {n - 1} + p _ {1} (n - 2) E _ {n - 2} = 0
+$$
+
+$$
+(n + 1) E _ {n} = \left(p _ {1} + 2 p _ {2} L\right) (2 n - 1) E _ {n - 1} - p _ {1} (n - 2) E _ {n - 2}
+$$
+
+which together with
+
+$$
+E _ {0} = L
+$$
+
+$$
+E _ {1} = \left(p _ {1} + p _ {2} L\right) L
+$$
+
+provides a formula for calculating  $E_{n}$ .
+
+Asymptotic estimate As before, approximations of  $E_{n}$  for large  $n$  can be found by developing  $E(z)$  in the neighbourhood of the root with the smallest module of
+
+$$
+1 - 2 \left(p _ {1} + 2 p _ {2} L\right) z + p _ {1} z ^ {2}
+$$
+
+The roots are
+
+$$
+r _ {1} = \frac {p _ {1}}{p _ {1} + 2 p _ {2} L - \sqrt {p _ {1} ^ {2} + 4 p _ {2} ^ {2} L ^ {2} + 4 p _ {2} p _ {1} L - p _ {1}}}
+$$
+
+$$
+r _ {2} = \frac {p _ {1}}{p _ {1} + 2 p _ {2} L + \sqrt {p _ {1} ^ {2} + 4 p _ {2} ^ {2} L ^ {2} + 4 p _ {2} p _ {1} L - p _ {1}}}
+$$
+
+both are positive and the smallest one is  $r_2$
+
+To alleviate notation, let
+
+$$
+\delta = \sqrt {p _ {1} ^ {2} + 4 p _ {2} ^ {2} L ^ {2} + 4 p _ {2} p _ {1} L - p _ {1}}
+$$
+
+$$
+r _ {2} = \frac {p _ {1}}{p _ {1} + 2 p _ {2} L + \delta}
+$$
+
+developing  $E(z)$  near  $r_2$
+
+$$
+E (z) \approx \frac {1 - p _ {1} r _ {2} - \sqrt {1 - r _ {2} \left(\frac {p _ {1} + 2 p _ {2} L - \delta}{p _ {1}}\right)} \sqrt {1 - \frac {z}{r _ {2}}}}{2 p _ {2} r _ {2}} + O \left(1 - \frac {z}{r _ {2}}\right) ^ {3 / 2}
+$$
+
+$$
+E (z) \approx \frac {p _ {1} + 2 p _ {2} L + \delta - p _ {1} ^ {2} - \sqrt {p _ {1} + 2 p _ {2} L + \delta} \sqrt {2 \delta} \sqrt {1 - \frac {z}{r _ {2}}}}{2 p _ {2} p _ {1}} + O (1 - \frac {z}{r _ {2}}) ^ {3 / 2}
+$$
+
+and therefore
+
+$$
+E _ {n} \approx \frac {\sqrt {\delta} r _ {2} ^ {- n - \frac {1}{2}}}{2 p _ {2} \sqrt {2 \pi p _ {1} n ^ {3}}} = \frac {\sqrt {\delta}}{2 p _ {2} \sqrt {2 \pi n ^ {3}}} \frac {(p _ {1} + 2 p _ {2} L + \delta) ^ {n + \frac {1}{2}}}{p _ {1} ^ {n + 1}}
+$$
+
+# C GENERATING RANDOM EXPRESSIONS
+
+In this section we present algorithms to generate random expressions with  $n$  internal nodes. We achieve this by generating random trees, and selecting randomly their nodes and leaves. We begin with the simpler binary case ( $p_1 = 0$ ).
+
+# C.1 BINARY TREES
+
+To generate a random binary tree with  $n$  internal nodes, we use the following one-pass procedure. Starting with an empty root node, we determine at each step the position of the next internal nodes among the empty nodes, and repeat until all internal nodes are allocated.
+
+Start with an empty node, set  $e = 1$
+
+while  $n > 0$  do
+
+Sample a position  $k$  from  $K(e,n)$ ;
+
+Sample the  $k$  next empty nodes as leaves;
+
+Sample an operator, create two empty children;
+
+Set  $e = e - k + 1$  and  $n = n - 1$ ;
+
+end
+
+Algorithm 1: Generate a random binary tree
+
+We denote by  $e$  the number of empty nodes, by  $n > 0$  the number of operators yet to be generated, and by  $K(e,n)$  the probability distribution of the position (0-indexed) of the next internal node to allocate.
+
+To calculate  $K(e, n)$ , let us define  $D(e, n)$ , the number of different binary subtrees that can be generated from  $e$  empty elements, with  $n$  internal nodes to generate. We have
+
+$$
+D (0, n) = 0
+$$
+
+$$
+D (e, 0) = 1
+$$
+
+$$
+D (e, n) = D (e - 1, n) + D (e + 1, n - 1)
+$$
+
+The first equation states that no tree can be generated with zero empty node and  $n > 0$  operators. The second equation says that if no operator is to be allocated, empty nodes must all be leaves and there is only one possible tree. The last equation states that if we have  $e > 0$  empty nodes, the first one is either a leaf (and there are  $D(e - 1, n)$  such trees) or an internal node ( $D(e + 1, n - 1)$  trees). This allows us to compute  $D(e, n)$  for all  $e$  and  $n$ .
+
+To calculate distribution  $K(e, n)$ , observe that among the  $D(e, n)$  trees with  $e$  empty nodes and  $n$  operators,  $D(e + 1, n - 1)$  have a binary node in their first position. Therefore
+
+$$
+P (K (e, n) = 0) = \frac {D (e + 1 , n - 1)}{D (e , n)}
+$$
+
+Of the remaining  $D(e - 1, n)$  trees,  $D(e, n - 1)$  have a binary node in their first position (same argument for  $e - 1$ ), that is
+
+$$
+P (K (e, n) = 1) = \frac {D (e , n - 1)}{D (e , n)}
+$$
+
+By induction over  $k$ , we have the general formula
+
+$$
+P \big (K (e, n) = k \big) = \frac {D (e - k + 1 , n - 1)}{D (e , n)}
+$$
+
+# C.2 UNARY-BINARY TREES
+
+In the general case, internal nodes can be of two types: unary or binary. We adapt the previous algorithm by considering the two-dimensional probability distribution  $L(e,n)$  of position (0-indexed) and arity of the next internal node (i.e.  $P(L(e,n) = (k,a)$  is the probability that the next internal node is in position  $k$  and has arity  $a$ ).
+
+Start with an empty node, set  $e = 1$
+
+while  $n > 0$  do
+
+Sample a position  $k$  and arity  $a$  from  $L(e,n)$  (if  $a = 1$  the next internal node is unary); Sample the  $k$  next empty nodes as leaves; if  $a = 1$  then Sample a unary operator; Create one empty child; Set  $e = e - k$  end else Sample a binary operator; Create two empty children; Set  $e = e - k + 1$  end Set  $n = n - 1$  end
+
+To compute  $L(e, n)$ , we derive  $D(e, n)$ , the number of subtrees with  $n$  internal nodes that can be generated from  $e$  empty nodes. We have, for all  $n > 0$  and  $e$ :
+
+$$
+\begin{array}{l} D (0, n) = 0 \\ D (e, 0) = 1 \\ D (e, n) = D (e - 1, n) + D (e, n - 1) + D (e + 1, n - 1) \\ \end{array}
+$$
+
+The first equation states that no tree can be generated with zero empty node and  $n > 0$  operators. The second says that if no operator is to be allocated, empty nodes must all be leaves and there is only one possible tree. The third equation states that with  $e > 0$  empty nodes, the first one will either be a leaf  $(D(e - 1, n)$  possible trees), a unary operator  $(D(e, n - 1)$  trees), or a binary operator  $(D(e + 1, n - 1)$  trees).
+
+To derive  $L(e, n)$ , we observe that among the  $D(e, n)$  subtrees with  $e$  empty nodes and  $n$  internal nodes to be generated,  $D(e, n - 1)$  have a unary operator in position zero, and  $D(e + 1, n - 1)$  have a binary operator in position zero. As a result, we have
+
+$$
+P \big (L (e, n) = (0, 1) \big) = \frac {D (e , n - 1)}{D (e , n)} \quad \text {a n d} \quad P \big (L (e, n) = (0, 2) \big) = \frac {D (e + 1 , n - 1)}{D (e , n)}
+$$
+
+As in the binary case, we can generalize these probabilities to all positions  $k$  in  $\{0\ldots e - 1\}$
+
+$$
+P \big (L (e, n) = (k, 1) \big) = \frac {D (e - k , n - 1)}{D (e , n)} \quad \text {a n d} \quad P \big (L (e, n) = (k, 2) \big) = \frac {D (e - k + 1 , n - 1)}{D (e , n)}
+$$
+
+# C.3 SAMPLING EXPRESSIONS
+
+To generate expressions, we sample random trees (binary, or unary binary), that we "decorate" by randomly selecting their internal nodes and leaves from a list of possible operators or mathematical entities (integers, variables, constants).
+
+Nodes and leaves can be selected uniformly, or according to a prior probability. For instance, integers between  $-a$  and  $a$  could be sampled so that small absolute values are more frequent than large ones. For operators, addition and multiplication could be more common than subtraction and division.
+
+If all  $L$  leaves,  $p_1$  and  $p_2$  operators are equiprobable, an alternative approach to generation can be defined by computing  $D(e,n)$  as
+
+$$
+\begin{array}{l} D (0, n) = 0 \\ D (e, 0) = L ^ {e} \\ D (e, n) = L D (e - 1, n) + p _ {1} D (e, n - 1) + p _ {2} D (e + 1, n - 1) \\ \end{array}
+$$
+
+and normalizing the probabilities  $P(L(e, n))$  as
+
+$$
+P \big (L (e, n) = (k, 1) \big) = \frac {L ^ {e} D (e - k , n - 1)}{D (e , n)} \quad \text {a n d} \quad P \big (L (e, n) = (k, 2) \big) = \frac {L ^ {e} D (e - k + 1 , n - 1)}{D (e , n)}
+$$
+
+Samples then become dependent on the number of possible leaves and operators.
+
+# D IMPACT OF TIMEOUT ON MATHEMATICA
+
+In the case of Mathematica, we use function DSolve to solve differential equations, and function Integrate to integrate functions. Since computations can take a long time, we set a finite timeout to limit the time spent on each equation. Table 8 shows the impact of the timeout value on the accuracy with Mathematica. Increasing the timeout delay naturally improves the accuracy. With a timeout of 30 seconds, Mathematica times out on  $20\%$  of unsolved equations. With a limit of 3 minutes, timeouts represent about  $10\%$  of failed equations. This indicates that even in the ideal scenario where Mathematica would succeed on all equations where it times out, the accuracy would not exceed  $86.2\%$ .
+
+<table><tr><td>Timeout (s)</td><td>Success</td><td>Failure</td><td>Timeout</td></tr><tr><td>5</td><td>77.8</td><td>9.8</td><td>12.4</td></tr><tr><td>10</td><td>82.2</td><td>11.6</td><td>6.2</td></tr><tr><td>30</td><td>84.0</td><td>12.8</td><td>3.2</td></tr><tr><td>60</td><td>84.4</td><td>13.4</td><td>2.2</td></tr><tr><td>180</td><td>84.6</td><td>13.8</td><td>1.6</td></tr></table>
+
+Table 8: Accuracy of Mathematica on 500 functions to integrate, for different timeout values. As the timeout delay increases, the percentage of failures due to timeouts decreases. With a limit of 3 minutes, timeouts only represent  $10\%$  of failures. As a result, the accuracy without timeout would not exceed  $86.2\%$ .
+
+# E GENERALIZATION ACROSS GENERATORS
+
+On the integration problem, we achieve (c.f. Table 6) near perfect performance when the training and test data are generated by the same method (either FWD, BWD, or IBP). Given the relatively small size of the training set (4.10<sup>7</sup> examples), the model cannot overfit to the entire problem space (10<sup>34</sup> possible expressions). This shows that:
+
+- Our model generalizes well to functions created by the training generator.  
+- This property holds for the three considered generators, FWD, BWD, and IBP.
+
+Table 6 also measures the ability of our model to generalize across generators. A FWD-trained model achieves a low performance (17.2% with beam 50) on a BWD-generated test set. A BWD-trained model does a little better on the FWD test set (27.5%), but accuracy remains low. On the other hand, FWD-trained models achieve very good accuracy over an IBP-generated test set (88.9%), and BWD-trained models stand in the middle (59.2%).
+
+Figure 2 provides an explanation for these results. The input/output pairs produced by FWD and BWD have very different distributions: integration tends to shorten BWD generated expressions, and to expand FWD generated expressions. As a result, a model trained on BWD generated data will learn this shortening feature of integration, which will prove wrong on a FWD test set. Similar problems will happen on a FWD trained model with a BWD test set. Since IBP keeps average expression lengths unchanged, BWD and FWD-trained models will generalize better to IBP test sets (and be more accurate on FWD-trained models, since their input length distributions are closer).
+
+![](images/129a497567bcdda9cc424b2b0ed9d04f685284ee92987f5d9a465875c51b04d2.jpg)  
+Figure 2: Distribution of input and output lengths for different integration datasets. The FWD generator produces short problems with long solutions. Conversely, the BWD generator creates long problems, with short solutions. The IBP approach stands in the middle, and generates short problems with short solutions.
+
+![](images/05e2664c86d91b130285a7d574cfcd7ccbbded5e85d5d8d99740d2bc0061494f.jpg)
+
+This suggests that what looks at first glance like a generalization problem (bad accuracy of BWD-trained models on FWD generated sets, and the converse) is in fact a consequence of data generation. BWD and FWD methods generate training sets with specific properties, that our model will learn. But this can be addressed by adding IBP or FWD data to the BWD dataset, as shown in the two last lines of Table 6. In practice, a better approach could be implemented with self-supervised learning, where new training examples are generated by the model itself.
+
+<table><tr><td colspan="2">Functions and their primitives generated with the forward approach (FWD)</td></tr><tr><td>cos-1(x)</td><td>x cos-1(x) - √1-x2</td></tr><tr><td>x (2x + cos (2x))</td><td>2x3/3 + x sin (2x)/2 + cos (2x)/4</td></tr><tr><td>x (x + 4)/x + 2</td><td>x2/2 + 2x - 4 log (x + 2)</td></tr><tr><td>cos (2x)/sin (x)</td><td>log (cos (x) - 1)/2 - log (cos (x) + 1)/2 + 2 cos (x)</td></tr><tr><td>3x2sinh-1(2x)</td><td>x3sinh-1(2x) - x2√4x2+1/6 + √4x2+1/12</td></tr><tr><td>x3log (x2)4</td><td>x4log (x2)4/4 - x4log (x2)3/2 + 3x4log (x2)2/4 - 3x4log (x2)/4 + 3x4/8</td></tr></table>
+
+<table><tr><td colspan="2">Functions and their primitives generated with the backward approach (BWD)</td></tr><tr><td>cos(x) + tan2(x) + 2</td><td>x + sin(x) + tan(x)</td></tr><tr><td>1/x2√x-1√x+1</td><td>√x-1√x+1/x</td></tr><tr><td>(2x/cos2(x) + tan(x)) tan(x)</td><td>x tan2(x)</td></tr><tr><td>x tan(e^x/x) + (x-1)e^x/cos^2(e^x/x)/x</td><td>x tan(e^x/x)</td></tr><tr><td>1 + 1/log(log(x)) - 1/log(x) log(log(x))^2</td><td>x + x/√log(log(x))</td></tr><tr><td>-2x^2sin(x^2) tan(x) + x (tan^2(x) + 1) cos(x^2) + cos(x^2) tan(x)</td><td>x cos(x^2) tan(x)</td></tr></table>
+
+<table><tr><td colspan="2">Functions and their primitives generated with the integration by parts approach (IBP)</td></tr><tr><td>x(x + log(x))</td><td>x2(4x + 6 log(x) - 3)/12</td></tr><tr><td>x/(x + 3)2</td><td>-x + (x + 3) log(x + 3)/x + 3</td></tr><tr><td>x + √2/cos2(x)</td><td>(x + √2) tan(x) + log(cos(x))</td></tr><tr><td>x(2x + 5)(3x + 2 log(x) + 1)</td><td>x2(27x2 + 24x log(x) + 94x + 90 log(x))/18</td></tr><tr><td>(x - 2x/sin2(x) + 1/tan(x)) log(x)/sin(x)</td><td>x log(x) + tan(x)/sin(x) tan(x)</td></tr><tr><td>x3sinh(x)</td><td>x3cosh(x) - 3x2sinh(x) + 6x cosh(x) - 6 sinh(x)</td></tr></table>
+
+Table 9: Examples of functions with their integrals, generated by our FWD, BWD and IBP approaches. We observe that the FWD and IBP approaches tend to generate short functions, with long integrals, while the BWD approach generates short functions with long derivatives.
\ No newline at end of file
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+# DEEP NEUROETHOLOGY OF A VIRTUAL RODENT
+
+Josh Merel\*, Diego Aldarondo\*, Jesse Marshall\*, Yuval Tassa\*, Greg Wayne\*, Bence Olveczky\*
+
+$^{1}$ DeepMind, London, UK.  
+$^{2}$ Program in Neuroscience,  $^{3}$ Center for Brain Science,  $^{4}$ Department of Organismic and Evolutionary Biology, Harvard University, Cambridge, MA 02138, USA.
+
+jsmerel@google.com, diegoaldarondo@g.harvard.edu,
+
+jesse_d-marshall@fas.harvard.edu
+
+# ABSTRACT
+
+Parallel developments in neuroscience and deep learning have led to mutually productive exchanges, pushing our understanding of real and artificial neural networks in sensory and cognitive systems. However, this interaction between fields is less developed in the study of motor control. In this work, we develop a virtual rodent as a platform for the grounded study of motor activity in artificial models of embodied control. We then use this platform to study motor activity across contexts by training a model to solve four complex tasks. Using methods familiar to neuroscientists, we describe the behavioral representations and algorithms employed by different layers of the network using a neuroethological approach to characterize motor activity relative to the rodent's behavior and goals. We find that the model uses two classes of representations which respectively encode the task-specific behavioral strategies and task-invariant behavioral kinematics. These representations are reflected in the sequential activity and population dynamics of neural subpopulations. Overall, the virtual rodent facilitates grounded collaborations between deep reinforcement learning and motor neuroscience.
+
+# 1 INTRODUCTION
+
+Animals have nervous systems that allow them to coordinate their movement and perform a diverse set of complex behaviors. Mammals, in particular, are generalists in that they use the same general neural network to solve a wide variety of tasks. This flexibility in adapting behaviors towards many different goals far surpasses that of robots or artificial motor control systems. Hence, studies of the neural underpinnings of flexible behavior in mammals could yield important insights into the classes of algorithms capable of complex control across contexts and inspire algorithms for flexible control in artificial systems (Merel et al., 2019b).
+
+Recent efforts at the interface of neuroscience and machine learning have sparked renewed interest in constructive approaches in which artificial models that solve tasks similar to those solved by animals serve as normative models of biological intelligence. Researchers have attempted to leverage these models to gain insights into the functional transformations implemented by neurobiological circuits, prominently in vision (Khaligh-Razavi & Kriegeskorte, 2014; Yamins et al., 2014; Kar et al., 2019), but also increasingly in other areas, including audition (Kell et al., 2018) and navigation (Banino et al., 2018; Cueva & Wei, 2018). Efforts to construct models of biological locomotion systems have informed our understanding of the mechanisms and evolutionary history of bodies and behavior (Grillner et al., 2007; Ijspeert et al., 2007; Ramdya et al., 2017; Nyakatura et al., 2019). Neural control approaches have also been applied to the study of reaching movements, though often in constrained behavioral paradigms (Lillicrap & Scott, 2013), where supervised training is possible (Sussillo et al., 2015; Michaels et al., 2019).
+
+While these approaches model parts of the interactions between animals and their environments (Chiel & Beer, 1997), none attempt to capture the full complexity of embodied control, involving how an animal uses its senses, body and behaviors to solve challenges in a physical environment.
+
+The development of models of embodied control is valuable to the field of motor neuroscience, which typically focuses on restricted behaviors in controlled experimental settings. It is also valuable for AI research, where flexible models of embodied control could be applicable to robotics.
+
+Here, we introduce a virtual model of a rodent to facilitate grounded investigation of embodied motor systems. The virtual rodent affords a new opportunity to directly compare principles of artificial control to biological data from real-world rodents, which are more experimentally accessible than humans. We draw inspiration from emerging deep reinforcement learning algorithms which now allow artificial agents to perform complex and adaptive movement in physical environments with sensory information that is increasingly similar to that available to animals (Peng et al., 2016; 2017; Heess et al., 2017; Merel et al., 2019a;c). Similarly, our virtual rodent exists in a physical world, equipped with a set of actuators that must be coordinated for it to behave effectively. It also possesses a sensory system that allows it to use visual input from an egocentric camera located on its head and proprioceptive input to sense the configuration of its body in space.
+
+There are several questions one could answer using the virtual rodent platform. Here we focus on the problem of embodied control across multiple tasks. While some efforts have been made to analyze neural activity in reduced systems trained to solve multiple tasks (Song et al., 2017; Yang et al., 2019), those studies lacked the important element of motor control in a physical environment. Our rodent platform presents the opportunity to study how representations of movements as well as sequences of movements change as a function of goals and task contexts.
+
+To address these questions, we trained our virtual rodent to solve four complex tasks within a physical environment, all requiring the coordinated control of its body. We then ask "Can a neuroscientist understand a virtual rodent?" – a more grounded take on the originally satirical "Can a biologist fix a radio?" (Lazebnik, 2002) or the more recent "Could a neuroscientist understand a microprocessor?" (Jonas & Kording, 2017). We take a more sanguine view of the tremendous advances that have been made in computational neuroscience in the past decade, and posit that the supposed 'failure' of these approaches in synthetic systems is partly a misdirection. Analysis approaches in neuroscience were developed with the explicit purpose of understanding sensation and action in real brains, and often implicitly rooted in the types of architectures and processing that are thought relevant in biological control systems. With this philosophy, we use analysis approaches common in neuroscience to explore the types of representations and dynamics that the virtual rodent's neural network employs to coordinate multiple complex movements in the service of solving motor and cognitive tasks.
+
+# 2 APPROACH
+
+# 2.1 VIRTUAL RODENT BODY
+
+![](images/9b86dd6696f80d6603e62781234ea787fdf8ba2fb0054a9aba9a4d1fe1b2ca79.jpg)  
+Figure 1: (A) Anatomical skeleton of a rodent (as reference; not part of physical simulation). (B) A body designed around the skeleton to match the anatomy and model collisions with the environment. (C) Purely cosmetic skin to cover the body. (D) Semi-transparent visualization of (A)-(C) overlain.
+
+![](images/79e89649d0912b4f39e9d611acee3b5330ffee899dcfb91a4509b506c43db0d9.jpg)
+
+![](images/a0760267b297aa53807b42d94f935f0a19c6c34b2ece02f651b1fb759a96ce08.jpg)
+
+![](images/28b2e98208ae27273be8e889d42e03014d37d49521e862646bb2be1f06e4d519.jpg)
+
+We implemented a virtual rodent body (Figure 1) in MuJoCo (Todorov et al., 2012), based on measurements of laboratory rats (see Appendix A.1). The rodent body has 38 controllable degrees of freedom. The tail, spine, and neck consist of multiple segments with joints, but are controlled by tendons that co-activate multiple joints (spatial tendons in MuJoCo). The rodent will be released as part of dm_control/locomotion.
+
+The virtual rodent has access to proprioceptive information as well as "raw" egocentric RGB-camera  $(64\times 64$  pixels) input from a head-mounted camera. The proprioceptive inputs include internal
+
+joint angles and angular velocities, the positions and velocities of the tendons that provide actuation, egocentric vectors from the root (pelvis) of the body to the positions of the head and paws, a vestibular-like upright orientation vector, touch or contact sensors in the paws, as well as egocentric acceleration, velocity, and 3D angular velocity of the root.
+
+# 2.2 VIRTUAL RODENT TASKS
+
+![](images/19dde6e94dd2093570af178fd7f419c512e1ae6370a18519148b41b554cc733d.jpg)  
+Figure 2: Visualizations of four tasks the virtual rodent was trained to solve: (A) jumping over gaps ("gaps run"), (B) foraging in a maze ("maze forage"), (C) escaping from a hilly region ("bowl escape"), and (D) touching a ball twice with a forepaw with a precise timing interval between touches ("two-tap").
+
+![](images/bb1250ffb6f14eaf66bbf39a2112e94c351369e576cb2e5363e728bd6cab3205.jpg)
+
+![](images/f20dca6abdf883fc90f0b4628d1ef97da5b782a9d7b3fcadb65ede37897870f5.jpg)
+
+![](images/1d19540fb2ad0a415f21129583639bc8f1350bf12f68af08312016f9fc3338bf.jpg)
+
+We implemented four tasks adapted from previous work in deep reinforcement learning and motor neuroscience (Merel et al., 2019a; Tassa et al., 2018; Kawai et al., 2015) to encourage diverse motor behaviors in the rodent. The tasks are as follows: (1) Run along a corridor, over "gaps", with a reward for traveling along the corridor at a target velocity (Figure 2A). (2) Collect all the blue orbs in a maze, with a sparse reward for each orb collected (Figure 2B). (3) Escape a bowl-shaped region by traversing hilly terrain, with a reward proportional to distance from the center of the bowl (Figure 2C). (4) Approach orbs in an open field, activate them by touching them with a forepaw, and touch them a second time after a precise interval of  $800\mathrm{ms}$  with a tolerance of  $\pm 100\mathrm{ms}$ ; there is a time-out period if the touch is not within the tolerated window and rewards are provided sparsely on the first and second touch (Figure 2D). We did not provide the agent with a cue or context indicating its task. Rather, the agent had to infer the task from the visual input and behave appropriately.
+
+# 2.3 TRAINING A MULTI-TASK POLICY
+
+![](images/f51ee016ebbe47d0ba529f65fd107fd55c1cf152772020f5f1ba4962af3e798c.jpg)  
+Figure 3: The virtual rodent agent architecture. Egocentric visual image inputs are encoded into features via a small residual network (He et al., 2016) and proprioceptive state observations are encoded via a small multi-layer perceptron. The features are passed into a recurrent LSTM module (Hochreiter & Schmidhuber, 1997). The core module is trained by backpropagation during training of the value function. The outputs of the core are also passed as features to the policy module (with the dashed arrow indicating no backpropagation along this path during training) along with shortcut paths from the proprioceptive observations as well as encoded features. The policy module consists of one or more stacked LSTMs (with or without skip connections) which then produce the actions via a stochastic policy.
+
+Emboldened by recent results in which end-to-end RL produces a single terrain-adaptive policy (Peng et al., 2016; 2017; Heess et al., 2017), we trained a single architecture on the multiple motor-control-reliant tasks (see Figure 3). To train a single policy to perform all four tasks, we used an IMPALA-style setup for actor-critic DeepRL (Espeholt et al., 2018); parallel workers collected
+
+rollouts, logged them to a replay, from which a central learner sampled data to perform updates. The value-function critic was trained using off-policy correction via V-trace. To update the actor, we used a variant of MPO (Abdolmaleki et al., 2018) where the E-step is performed using advantages determined from the empirical returns and the value-function, instead of the Q-function (Song et al., 2019). Empirically, we found that the "escape" task was more challenging to learn during interleaved training relative to the other tasks. Consequently, we present results arising from training a single-task expert on the escape task and training the multi-task policies using kickstarting for that task (Schmitt et al., 2018), with a weak coefficient (.001 or .005). Kickstarting on this task made the seeds more reliably solve all four tasks, facilitating comparison of the multi-task policies with different architectures (i.e. the policy having 1, 2, or 3 layers, with or without skip connections across those layers). The procedure yields a single neural network that uses visual inputs to determine how to behave and coordinates its body to move in ways required to solve the tasks. See video examples of a single policy solving episodes of each task: gaps, forage, escape, and two-tap.
+
+# 3 ANALYSIS
+
+![](images/8eda16527921e59287bc3e1f461259963fb0bfa9641772cb106696b805865bf0.jpg)  
+A  
+B  
+Figure 4: Ethology of the virtual rodent. (A) Example jumping sequence in gaps run task with a representative subset of recorded behavioral features. Dashed lines denote the time of the corresponding frames (top). (B) tSNE embedding of 60 behavioral features describing the pose and kinematics of the virtual rodent allows identification of rodent behaviors. Points are colored by hand-labeling of behavioral clusters identified by watershed clustering. (C) The first two principal components of different behavioral features reveals that behaviors are more shared across tasks at short,  $5 - 25\mathrm{Hz}$  timescales (fast kinematics), but no longer  $0.3 - 5\mathrm{Hz}$  timescales (slow kinematics).
+
+![](images/59da37f58a25e1762e22dbe63de385115aaf9a126927c72561fcde04dc5f3029.jpg)
+
+![](images/4aa5082bdcec566c466a917abe3c2d28951e1f581d648bd77f71ccbc65496ddd.jpg)  
+C
+
+We analyzed the virtual rodent's neural network activity in conjunction with its behavior to characterize how it solves multiple tasks (Figure 4A). We used analyses and perturbation techniques adapted from neuroscience, where a range of techniques have been developed to highlight the properties of real neural networks. Biological neural networks have been hypothesized to control, select, and modulate movement through a variety of debated mechanisms, ranging from explicit neural representations of muscle forces and behavioral primitives, to more abstract production of neural dynamics that could underly movement (Graziano, 2006; Kalaska, 2009; Churchland et al., 2012). A challenge with nearly all of these models however is that they have largely been inspired by findings from individual behavioral tasks, making it unclear how to generalize them to a broader range of naturalistic behaviors. To provide insight into mechanisms underlying movement in the virtual rodent, and to potentially give insight by proxy into the mechanisms underlying behavior in real rats, we thus systematically tested how the different network layers encoded and generated different aspects of movement.
+
+For all analyses we logged the virtual rodent's kinematics, joint angles, computed forces, sensory inputs, and the cell unit activity of the LSTMs in core and policy layers during 25 trials per task from each network architecture.
+
+# 3.1 VIRTUAL RODENTS EXHIBIT BEHAVIORAL FLEXIBILITY.
+
+We began our analysis by quantitatively describing the behavioral repertoire of the virtual rodent. A challenge in understanding the neural mechanisms underlying behavior is that it can be described at many timescales. On short timescales, one could describe rodent locomotion using a set of actuators that produce joint-specific patterns of forces and kinematics. However on longer timescales, these force patterns are organized into coordinated, re-used movements, such as running, jumping, and turning. These movements can be further combined to form behavioral strategies or goal-directed behaviors. Relating neural representations to motor behaviors therefore requires analysis methods that span multiple timescales of behavioral description. To systematically examine the classes of behaviors these networks learn to generate and how they are differentially deployed across tasks, we developed sets of behavioral features that describe the kinematics of the animal on fast (5-25 Hz), intermediate (1-25 Hz) or slow (0.3-5 Hz) timescales (Appendix A.2, A.3). As validation that these features reflected meaningful differences across behaviors, embedding these features using tSNE (Maaten & Hinton, 2008) produced a behavioral map in which virtual rodent behaviors, were segregated to different regions of the map (Figure 4B)(see video). This behavioral repertoire of the virtual rodent consisted of many behaviors observed in rodents, such as rearing, jumping, running, climbing and spinning. While the exact kinematics of the virtual rodent's behaviors did not exactly match those observed in real rats, they did reproduce unexpected features. For instance the stride frequency of the virtual rodent during galloping matches that observed in rats (Appendix A.3).
+
+We next investigated how these behaviors were used by the virtual rodent across tasks. On short timescales, low-level motor features like joint speed and actuator forces occupied similar regions in principal component space (Figure 4C). In contrast, behavioral kinematics, especially on long,  $0.3 - 5\mathrm{Hz}$  timescales, were more differentiated across tasks. Similar results held when examining overlap in other dimensions using multidimensional scaling. Overall this suggests that the network learned to adapt similar movements in a selective manner for different tasks, suggesting that the agent exhibited a form of behavioral flexibility.
+
+# 3.2 NETWORKS PRIMARILY REFLECT BEHAVIORS, NOT FORCES
+
+![](images/1acdc04d295d8f0990d78fe431bf659480fa35db1cfc23c1f4cb63b37e954e71.jpg)  
+Figure 5: Representational structure of the rodent's neural network. (A) Example similarity matrices of neural networks and behavioral descriptors. We grouped behavioral descriptors into 50 clusters that and we computed the average neural population vector during each cluster (AppendixA.4). Similarity was assessed by computing the dot product of either the neural population vector or the behavioral feature vector within each cluster. (B) Centered Kernel Alignment (CKA) index of neural and behavioral feature similarity matrices for 3 and 1 policy layer architectures. (C) CKA index of feature similarity matrices across all pairs of network layers. (D) Average CKA index between core and policy layers and behavioral features, compared across architectures. Points show values from individual network seeds. Policy values are averaged across layers.
+
+![](images/63a666cd6871625726b73b83b9929668305199a7d4b54e21de72d3f2cc45152c.jpg)
+
+![](images/3b9e703e28b0ab8cd8459744d56d85550d71a2a4c9cf06d6b04eb429aff57a51.jpg)
+
+We next examined the neural activity patterns underlying the virtual rodent's behavior to test if networks produced behaviors through explicit representations of forces, kinematics or behaviors. As
+
+expected, core and policy units operate on distinct timescales (See Appendix A.3, Figure 9). Units in the core typically fluctuated over timescales of 1-10 seconds, likely representing variables associated with context and reward. In contrast, units in policy layers were more active over subsecond timescales, potentially encoding motor and behavioral features.
+
+To quantify which aspects of behavior were encoded in the core and policy layers, and how these patterns varied across layers, we used representational similarity analysis (RSA) (Kriegeskorte et al., 2008; Kriegeskorte & Diedrichsen, 2019). RSA provides a global measure of how well different features are encoded in layers of a neural network by analyzing the geometries of network activity upon exposure to several stimuli, such as objects. To apply RSA, first a representational similarity (or equivalently, dissimilarity) matrix is computed that quantifies the similarity of neural population responses to a set of stimuli. To test if different neural populations show similar stimulus encodings, these similarity matrices can then be directly compared across different network layers. Multiple metrics, such as the matrix correlation or dot product can be used to compare these neural representational similarity matrices. Here we used the linear centered kernel alignment (CKA) index, which shows invariance to orthonormal rotations of population activity (Kornblith et al., 2019).
+
+RSA can also be used to directly test how well a particular stimulus feature is encoded in a population. If each stimuli can be quantitively described by one or more feature vectors, a similarity matrix can also be computed across the set of stimuli themselves. The strength of encoding of a particular set of features can be measured by comparing the correlation of the stimulus feature similarity matrix and the neuronal similarity matrix. The correlation strength directly reflects the ability of a linear decoder trained on the neuronal population vector to distinguish different stimuli (Kriegeskorte & Diedrichsen, 2019). Unlike previous applications of RSA in the analysis of discrete stimuli such as objects, (Khaligh-Razavi & Kriegeskorte, 2014; Yamins et al., 2014) behavior evolves continuously. To adapt RSA to behavioral analysis, we partitioned time by discretizing each behavioral feature into 50 clusters (Appendix A.4).
+
+As expected, RSA revealed that core and policy layers encoded somewhat distinct behavioral features. Policy layers contained greater information about fast timescale kinematics in a manner that was largely conserved across layers, while core layers showed more moderate encoding of kinematics that was stronger for slow behavioral features (Figure 5B,C). This difference in encoding was largely consistent across all architectures tested (Figure 5D).
+
+The feature encoding of policy networks was somewhat consistent with the emergence of a hierarchy of behavioral abstraction. In networks trained with three policy layers, representations were distributed in timescales across layers, with the last layer (policy 2) showing stronger encoding of fast behavioral features, and the first layer (policy 0) instead showing stronger encoding of slow behavioral features. However, policy layer activity, even close to the motor periphery, did not show strong explicit encoding of behavioral kinematics or forces.
+
+# 3.3 BEHAVIORAL REPRESENTATIONS ARE SHARED ACROSS TASKS
+
+We then investigated the degree to which the rodent's neural networks used the same neural representations to produce behaviors, such as running or spinning, that were shared across tasks. Embedding population activity into two-dimensions using multidimensional scaling revealed that core neuron representations were highly distinct across all tasks, while policy layers contained more overlap (Figure 6A), suggesting that some behavioral representations were re-used. Comparison of representational similarity matrices for behaviors that were shared across tasks revealed that policy layers tended to possess a relatively similar encoding of behavioral features, especially fast behavioral features, over tasks (Figure 6C; Appendix A.4). This was validated by inspection of neural activity during individual behaviors shared across tasks (Appendix A.5, Figure 10). Core layer representations across almost all behavioral categories were more variable across tasks, consistent with encoding behavioral sequences or task variables.
+
+Interestingly, when comparing this cross-task encoding similarity across architectures, we found that one layer networks showed a marked increase in the similarity of behavioral encoding across tasks (Figure 6D). This suggests that in networks with lower computational capacity, animals must rely on a smaller, shared behavioral representation across tasks.
+
+![](images/ad03717d3e8dc6040a6ef871c425bc4f5ca85813c6ce4974fdbf468d5683ff72.jpg)  
+A  
+Figure 6: Policy representations are shared across tasks. (A) Two-dimensional multidimensional scaling embeddings of core and policy activity shows that while policy representations overlap across some tasks, core representations are largely distinct. (B) CKA index of the policy 2 and core network representations of behavioral features during behaviors shared across different tasks (Appendix A.4). Policy 2, but not core networks show similar encoding patterns across the across the maze forage and two-tap tasks, as well as the gaps run and maze forage tasks, consistent with the shared behaviors used across these tasks. (C) The similarity of behavioral feature encoding (CKA index) across different architectures demonstrates that networks with fewer layers show greater similarity across tasks. Points show values from individual seeds.
+
+![](images/39d1c2f571af6f8f481d5505786b31b35f420d0fa16b1e7f8a3c17d31f6819ed.jpg)  
+B  
+C
+
+![](images/8551b4185304f6eb3011b2ebf0b6a939e291a34cb3aebc5163e40616633acaed.jpg)
+
+![](images/6545302d0f6424bd97c4744d7034811c64d37afd097f00b08940993fe0e65aea.jpg)
+
+# 3.4 NEURAL POPULATION DYNAMICS ARE SYNCHRONIZED WITH BEHAVIOR
+
+![](images/ba50e1c648681cb2e430ebaba466fa0e4775196bbe3cb72c92e2d0848a3acf5b.jpg)  
+Figure 7: Neurons in core and policy networks show sequential activity during stereotyped behavior. (A) Example video stills showing the virtual rodent engaged in the two-tap task (B) Average absolute z-scored activity traces of all 128 neurons in each layer during performance of the two-tap sequence. Traces are sorted by the time of peak average firing rate. Dashed lines indicate the times of first and second taps. Sequential neural activity is present during the two-tap sequence.
+
+While RSA described which behavioral features were represented in core and policy activity, we were also interested in describing how neural activity changes over time to produce different behaviors. We began by analyzing neural activity during the production of stereotyped behaviors. Activity
+
+patterns in the two-tap task showed peak activity in core and policy units that was sequentially organized (Figure 7), uniformly tiling time between both taps of the two-tap sequence. This sequential activation was observed across tasks and behaviors in the policy network, including during running (see video) where, consistent with policy networks encoding short-timescale kinematic features in a task-invariant manner, neural activity sequences were largely conserved across tasks (See Appendix A.5, Figure 10). These sequences were reliably repeated across instances of the respective behaviors, and in the case of the two-tap sequence, showed reduced neural variability relative to surrounding timepoints (See Appendix A.6, Figure 11).
+
+![](images/d9fb849d66f34ddb686f4b534e627125d3d228493bcea5b40a5f4af764517ba6.jpg)  
+Figure 8: Latent network dynamics within tasks reflect rodent behavior on different timescales. (A) Vector field representation of the first two principal components of neural activity in the core and final policy layers during the two-tap task. PC spaces show signatures of rotational dynamics. (B) Vector field representation of first two jPC planes for the core and final policy layers during the two-tap task. Apparent rotations within the different planes are associated with behaviors and behavioral features of different timescales, labeled above. Columns denote layer (as in (A)), while rows denote jPC plane. (C) Characteristic frequency of rotations within each jPC plane. Groups of three points respectively indicate the first, second, and third jPC planes for a given layer. Rotations in the core are slower than those in the policy. (D) Variance explained by each jPC plane.
+
+![](images/9e988d40de29103dbfc0a9b844f2d2873f5e1bcb9ee741e69e00aa9bcea5b8b5.jpg)
+
+![](images/0e1b34c0158eebeca317927066c37fa033635824c7e8d747bc82547ed8094eef.jpg)
+
+The finding of sequential activity hints at a putative mechanism for the rodent's behavioral production. We next hoped to systematically quantify the types of sequential and dynamical activity present in core and policy networks without presupposing the behaviors of interest. To describe population dynamics in relation to behavior, we first applied principal components analysis (PCA) to the activity during the performance of single tasks, and visualized the gradient of the population vector as a vector field. Figure 8A shows such a vector field representation of the first two principal components of the core and final policy layer during the two-tap task. We generated vector fields by discretizing the PC space into a two-dimensional grid and calculating the average neural activity gradient with respect to time for each bin.
+
+The vector fields showed strong signatures of rotational dynamics across all layers, likely a signature of previously described sequential activity. To extract rotational patterns, we used jPCA, a dimensionality reduction method that extracts latent rotational dynamics in neural activity (Churchland et al., 2012). The resulting jPCs form an orthonormal basis that spans the same space as the first six traditional PCs, while maximally emphasizing rotational dynamics. Figure 8B shows the vector
+
+fields of the first two jPC planes for the core and final policy layers along with their characteristic frequency. Consistent with our previous findings, jPC planes in the core have lower characteristic frequencies than those in policy layers across tasks (Figure 8C). The jPC planes also individually explained a large percentage of total neural variability (Figure 8D).
+
+These rotational dynamics in the policy and core jPC planes were respectively associated with the production of behaviors and the reward structure of the task. For example, in the two-tap task, rotations in the fastest jPC plane in the core were concurrent with the approach to reward, while rotations in the second fastest jPC were concurrent with long timescale transitions between running to the orb and performing the two-tap sequence. Similarly, the fastest jPC in policy layers was correlated with the phase of running, while the second fastest was correlated with the phase of the two-tap sequence (video). This trend of core and policy neural dynamics respectively reflecting behavioral and task-related features was also present in other tasks. For example, in the maze forage task, the first two jPC planes in the core respectively correlated with reaching the target orb and discovering the location of new orbs, while those in the policy were correlated with low-level locomotor features such as running phase (video). Along with RSA, these findings support a model in which the core layer transforms sensory information into a contextual signal in a task-specific manner. This signal then modulates activity in the policy toward different trajectories that generate appropriate behaviors in a more task-independent fashion. For a more complete set of behaviors with neural dynamics visualizations overlaid, see Appendix A.7.
+
+# 3.5 NEURAL PERTURBATIONS CORROBORATE DISTINCT ROLES ACROSS LAYERS
+
+To causally demonstrate the differing roles of core and policy units in respectively encoding task-relevant features and movement, we performed silencing and activation of different neuronal subsets in the two-tap task. We identified two stereotyped behaviors (rears and spinning jumps) that were reliably used in two different seeds of the agent to reach the orb in the task. We ranked neurons according to the degree of modulation of their z-scored activity during the performance of these behaviors. We then inactivated subsets of neurons by clamping activity to the mean values between the first and second taps and observed the effects of inactivation on trial success and behavior.
+
+In both seeds analyzed, inactivation of policy units had a stronger effect on motor behavior than the inactivation of core units. For instance, in the two-tap task, ablation of 64 neurons in the final policy layer disrupts the performance of the spinning jump (Appendix A.8 Figure 12B video). In contrast, ablation of behavior-modulated core units did not prevent the production of the behavior, but mildly affected the way in which the behavior is directed toward objects in the environment. For example, ablation of a subset of core units during the performance of a spinning jump had a limited effect, but sometimes resulted in jumps that missed the target orbs (video; See Appendix A.8, Figure 12C).
+
+We also performed a complementary perturbation aimed to elicit behaviors by overwriting the cell state of neurons in each layer with the average time-varying trajectory of neural activity measured during natural performance of a target behavior. The efficacy of stimulation was found to depend on the gross body posture and behavioral state of an animal, but was nevertheless successful in some cases. For example, during the two-tap sequence, we were able to elicit spinning movements common to searching behaviors in the forage task (video; See Appendix A.8, Figure 12D, E). The efficacy of this activation was more reliable in layers closer to the motor output (Figure 12D). In fact, activation of core units rarely elicited spins, but rather elicited sporadic dashes reminiscent of the searching strategy of many models during the forage task (video).
+
+# 4 DISCUSSION
+
+For many computational neuroscientists and artificial intelligence researchers, an aim is to reverse-engineer the nervous system at an appropriate level of abstraction. In the motor system, such an effort requires that we build embodied models of animals equipped with artificial nervous systems capable of controlling their synthetic bodies across a range of behavior. Here we introduced a virtual rodent capable of performing a variety of complex locomotor behaviors to solve multiple tasks using a single policy. We then used this virtual nervous system to study principles of the neural control of movement across contexts and described several commonalities between the neural activity of artificial control and previous descriptions of biological control.
+
+A key advantage of this approach relative to experimental approaches in neuroscience is that we can fully observe sensory inputs, neural activity, and behavior, facilitating more comprehensive testing of theories related to how behavior can be generated. Furthermore, we have complete knowledge of the connectivity, sources of variance, and training objectives of each component of the model, providing a rare ground truth to test the validity of our neural analyses. With these advantages in mind, we evaluated our analyses based on their capacity to both describe the algorithms and representations employed by the virtual rodent and recapitulate the known functional objectives underlying its creation without prior knowledge.
+
+To this end, our description of core and policy as respectively representing value and motor production is consistent with the model's actor-critic training objectives. But beyond validation, our analyses provide several insights into how these objectives are reached. RSA revealed that the cell activity of core and policy layers had greater similarity with behavioral and postural features than with short-timescale actuators. This suggests that the representation of behavior is useful in the moment-to-moment production of motor actions in artificial control, a model that has been previously proposed in biological action selection and motor control (Mink, 1996; Graziano, 2006). These behavioral representations were more consistent across tasks in the policy than in the core, suggesting that task context and value activity in the core engaged task-specific behavioral strategies through the reuse of shared motor activity in the policy.
+
+Our analysis of neural dynamics suggests that reused motor activity patterns are often organized as sequences. Specifically, the activity of policy units uniformly tiles time in the production of several stereotyped behaviors like running, jumping, spinning, and the two-tap sequence. This finding is consistent with reports linking sequential neural activity to the production of stereotyped motor and task-oriented behavior in rodents (Berke et al., 2009; Rueda-Orozco & Robbe, 2015; Dhawale et al., 2019), including during task delay periods (Akhlaghpour et al., 2016), as well as in singing birds (Albert & Margoliash, 1996; Hahnloser et al., 2002). Similarly, by relating rotational dynamics to the virtual rodent's behavior, we found that different behaviors were seemingly associated with distinct rotations in neural activity space that evolved at different timescales. These findings are consistent with a hierarchical control scheme in which policy layer dynamics that generate reused behaviors are activated and modulated by sensorimotor signals from the core.
+
+This work represents an early step toward the constructive modeling of embodied control for the purpose of understanding the neural mechanisms behind the generation of behavior. Incrementally and judiciously increasing the realism of the model's embodiment, behavioral repertoire, and neural architecture is a natural path for future research. Our virtual rodent possesses far fewer actuators and touch sensors than a real rodent, uses a vastly different sense of vision, and lacks integration with olfactory, auditory, and whisker-based sensation (see Zhuang et al., 2017). While the virtual rodent is capable of locomotor behaviors, an increased diversity of tasks involving decision making, memory-based navigation, and working memory could give insight into "cognitive" behaviors of which rodents are capable. Furthermore, biologically-inspired design of neural architectures and training procedures should facilitate comparisons to real neural recordings and manipulations. We expect that this comparison will help isolate residual elements of animal behavior generation that are poorly captured by current models of motor control, and encourage the development of artificial neural architectures that can produce increasingly realistic behavior.
+
+# AUTHOR CONTRIBUTIONS
+
+Josh and Yuval built the rodent MuJoCo model, with measurements collected by Diego and Jesse. Josh trained the virtual rodent model. Jesse performed behavioral and neural representation analyses. Diego performed neural dynamics analyses. Josh, Jesse, and Diego drafted the manuscript. All authors contributed to the conception of the project.
+
+# ACKNOWLEDGMENTS
+
+The rodent skeleton reference model was purchased fromleo3Dmodelsn on TurboSquid. Thanks to Max Cant for the rodent skin, and Marcus Wainwright for the skybox and ground textures. D.A. was supported by NSF GRFP DGE1745303. J.D.M was supported by a fellowship from the Helen Hay Whitney foundation sponsored by Vertex and a K99/R00 award from the NINDS.
+
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+
+# A APPENDIX
+
+# A.1 RAT MEASUREMENTS
+
+To construct the virtual rodent model, we obtained the mass and lengths of the largest body segments that influence the physical properties of the virtual rodent. First, we dissected cadavers of two female Long-Evans rats, and measured the mass of relevant limb segments and organs. Next, we measured the lengths of body segments over the skin of animals anesthetized with  $2\%$  v/v isoflurane anesthesia in oxygen. We confirmed that these skin based measurements approximated bone lengths by measuring bone lengths in a third cadaver. The care and experimental manipulation of all animals were reviewed and approved by the appropriate Institutional Animal Care and Use Committee.
+
+Animal (#)  
+63 64  
+
+<table><tr><td>Body part</td><td colspan="2">Mass (g)</td><td>Average mass (g)</td></tr><tr><td>Hindlimb L</td><td>21</td><td>26</td><td>23.5</td></tr><tr><td>Hindlimb R</td><td>21</td><td>26</td><td>23.5</td></tr><tr><td>Tail</td><td>8</td><td>10</td><td>9</td></tr><tr><td>Forelimb R</td><td>11</td><td>14</td><td>12.5</td></tr><tr><td>Forelimb L</td><td>12</td><td>13</td><td>12.5</td></tr><tr><td>Full torso</td><td>176</td><td>187</td><td>181.5</td></tr><tr><td>Head</td><td>26</td><td>26</td><td>26</td></tr><tr><td>Upper torso</td><td>78</td><td>71</td><td>74.5</td></tr><tr><td>Lower torso</td><td>98</td><td>114</td><td>106</td></tr><tr><td>Torso without organs</td><td>54</td><td>58</td><td>56</td></tr><tr><td>Intestines and stomach</td><td>22</td><td>32</td><td>27</td></tr><tr><td>Liver</td><td>26</td><td>17</td><td>21.5</td></tr><tr><td>Pelvis and kidneys</td><td>74</td><td>80</td><td>77</td></tr><tr><td>Jaw</td><td>2.43</td><td>4.70</td><td>3.57</td></tr><tr><td>Skull</td><td>23</td><td>21</td><td>22</td></tr><tr><td>Tail (base to mid)</td><td>5.92</td><td>7.20</td><td>6.56</td></tr><tr><td>Tail (mid to tip)</td><td>1.78</td><td>2.30</td><td>2.04</td></tr><tr><td>Scapula L</td><td>3.19</td><td>4.70</td><td>3.94</td></tr><tr><td>Humerus L</td><td>6.25</td><td>4.70</td><td>5.48</td></tr><tr><td>Radius/ulna L</td><td>2.61</td><td>2.8</td><td>2.70</td></tr><tr><td>Forepaw L</td><td>0.53</td><td>0.5</td><td>0.52</td></tr><tr><td>Scapula R</td><td>2.23</td><td>3.9</td><td>3.07</td></tr><tr><td>Humerus R</td><td>6.08</td><td>6.7</td><td>6.39</td></tr><tr><td>Radius/ulna R</td><td>2.17</td><td>3.3</td><td>2.74</td></tr><tr><td>Forepaw R</td><td>0.53</td><td>0.5</td><td>0.52</td></tr><tr><td>Hindpaw L</td><td>1.66</td><td>1.7</td><td>1.68</td></tr><tr><td>Tibia L</td><td>9</td><td>9</td><td>9</td></tr><tr><td>Femur L</td><td>13</td><td>16</td><td>14.5</td></tr><tr><td>Hindpaw R</td><td>1.81</td><td>1.6</td><td>1.71</td></tr><tr><td>Tibia R</td><td>5</td><td>6</td><td>5.5</td></tr><tr><td>Femur R</td><td>13</td><td>18</td><td>15.5</td></tr><tr><td>Total</td><td>281</td><td>301</td><td>291</td></tr></table>
+
+Table 1: Before weighing, limb segments were divided at their respective joints. Mass of all segments includes all bones, skin, muscle, fascia and adipose layers. L and R refer to the left and right sides of the animal. Precision of measurements listed without decimal places is  $\pm 0.5\mathrm{g}$
+
+<table><tr><td rowspan="2"></td><td colspan="8">Animal (#)</td></tr><tr><td>48</td><td>62</td><td>55</td><td>56</td><td>64</td><td>63</td><td>62*</td><td>Average ± std</td></tr><tr><td>Age (days)</td><td>382</td><td>82</td><td>330</td><td>330</td><td>83</td><td>83</td><td>83</td><td></td></tr><tr><td>Mass (g)</td><td>325</td><td>273</td><td>389</td><td>348</td><td>283</td><td>269</td><td>273</td><td>309 ± 47</td></tr><tr><td>Body part</td><td colspan="8">Length (mm)</td></tr><tr><td>Ankle to claw L</td><td>40.2</td><td>39.5</td><td>39.7</td><td>37.8</td><td>39.9</td><td>41.5</td><td>39.8</td><td>39.8 ± 1.1</td></tr><tr><td>Ankle to toe L</td><td>38.4</td><td>38.12</td><td>37.7</td><td>35.6</td><td>36.6</td><td>39.3</td><td>38</td><td>37.7 ± 1.2</td></tr><tr><td>Ankle to pad L</td><td>23.4</td><td>22.2</td><td>23</td><td>22.12</td><td>22.5</td><td>23.3</td><td>6.4</td><td>20.4 ± 6.2</td></tr><tr><td>Ankle to claw R</td><td></td><td>38.2</td><td>40.4</td><td>38.3</td><td>39.3</td><td>39.6</td><td>38.3</td><td>39.0 ± 0.9</td></tr><tr><td>Ankle to toe R</td><td></td><td>37</td><td>38.7</td><td>36.3</td><td>37.7</td><td>38.6</td><td>36.2</td><td>37.4 ± 1.1</td></tr><tr><td>Ankle to pad R</td><td></td><td>22.4</td><td>23.3</td><td>21.9</td><td>21.8</td><td>23.1</td><td>24.1</td><td>22.8 ± 0.9</td></tr><tr><td>Tibia L</td><td>50</td><td>36.3</td><td>38.5</td><td>49.2</td><td>35.8</td><td>38.7</td><td>34.1</td><td>40.4 ± 6.5</td></tr><tr><td>Femur L</td><td>44.5</td><td>31.6</td><td>32.1</td><td>37.9</td><td>33.4</td><td>35.35</td><td>32.4</td><td>35.3 ± 4.6</td></tr><tr><td>Tibia R</td><td></td><td>36.7</td><td>39.1</td><td>37.9</td><td>35.1</td><td>38.4</td><td>36.18</td><td>37.2 ± 1.5</td></tr><tr><td>Femur R</td><td></td><td>32.9</td><td>32.1</td><td>38.7</td><td>31.9</td><td>32.1</td><td>32.6</td><td>33.4 ± 2.6</td></tr><tr><td>Pelvis</td><td>25.8</td><td>32</td><td>31.7</td><td>30.2</td><td>26.7</td><td>27.2</td><td></td><td>28.9 ± 2.7</td></tr><tr><td>Wrist to claw L</td><td>15</td><td>18.8</td><td>17.6</td><td>18.6</td><td>16</td><td>19.02</td><td>19.2</td><td>17.7 ± 1.6</td></tr><tr><td>Wrist to finger L</td><td></td><td>16</td><td>15.8</td><td>17.4</td><td>15.5</td><td>17.07</td><td>17.6</td><td>16.6 ± 0.9</td></tr><tr><td>Wrist to pad L</td><td></td><td>6</td><td>6.4</td><td>8.34</td><td>4.9</td><td>6.1</td><td>6.4</td><td>6.4 ± 1.1</td></tr><tr><td>Wrist to olecranon L</td><td>29.1</td><td>34</td><td>32.5</td><td>31.7</td><td>33.9</td><td>32.1</td><td>29.9</td><td>31.9 ± 1.9</td></tr><tr><td>Humerus L</td><td>31.9</td><td>29.52</td><td>31</td><td>28.2</td><td>27</td><td>31.2</td><td>25.4</td><td>29.2 ± 2.4</td></tr><tr><td>Scapula L</td><td>22.7</td><td>24</td><td>26.4</td><td>29.3</td><td>25.9</td><td>29.1</td><td>26.2</td><td>26.2 ± 2.4</td></tr><tr><td>Wrist to claw R</td><td></td><td>16.8</td><td>17</td><td>17.8</td><td>15.9</td><td>16.3</td><td>18.1</td><td>17.0 ± 0.8</td></tr><tr><td>Wrist to finger R</td><td></td><td>14.1</td><td>13</td><td>15.6</td><td>15.6</td><td>15.3</td><td>16.9</td><td>15.1 ± 1.4</td></tr><tr><td>Wrist to pad R</td><td></td><td>5.6</td><td>5.8</td><td>6.55</td><td>5.2</td><td>5</td><td>5.8</td><td>5.7 ± 0.5</td></tr><tr><td>Wrist to olecranon R</td><td></td><td>30.6</td><td>33.5</td><td>31.2</td><td>30.4</td><td>31.8</td><td>29.9</td><td>31.2 ± 1.3</td></tr><tr><td>Humerus R</td><td></td><td>28.2</td><td>33.5</td><td>28.8</td><td>25</td><td>28.2</td><td>25.2</td><td>28.1 ± 3.1</td></tr><tr><td>Scapula R</td><td></td><td>23.8</td><td>29.5</td><td>25.9</td><td>26.2</td><td>28.8</td><td>24.4</td><td>26.4 ± 2.3</td></tr><tr><td>Headcap width</td><td>39</td><td></td><td></td><td></td><td></td><td></td><td></td><td>39</td></tr><tr><td>Headcap length</td><td>30</td><td></td><td></td><td></td><td></td><td></td><td></td><td>30</td></tr><tr><td>Skull width</td><td>38.8</td><td>23.35</td><td>23</td><td>21.8</td><td>22.8</td><td>23.9</td><td>22.2</td><td>25.1 ± 6.1</td></tr><tr><td>Skull length</td><td>57</td><td>51.1</td><td>61</td><td>56.48</td><td>53.16</td><td>58.13</td><td>48</td><td>55.0 ± 4.5</td></tr><tr><td>Skull height</td><td></td><td></td><td></td><td></td><td>21.59</td><td>21.5</td><td>21</td><td>21.4 ± 0.3</td></tr><tr><td>Head to thoracic</td><td></td><td>48.6</td><td>71.4</td><td>68.68</td><td>65</td><td>60.4</td><td>71.2</td><td>64.2 ± 8.7</td></tr><tr><td>Thoracic to sacral</td><td></td><td>73.1</td><td>73.6</td><td>62.9</td><td>65.04</td><td>64.7</td><td>68.8</td><td>68.0 ± 4.6</td></tr><tr><td>Head to sacral</td><td>145</td><td>126</td><td>145.5</td><td>127.05</td><td>127.2</td><td>123.7</td><td>140.9</td><td>133.6 ± 9.7</td></tr><tr><td>Head width</td><td>53.4</td><td></td><td></td><td></td><td></td><td></td><td></td><td>53.4</td></tr><tr><td>Ear</td><td>18</td><td>17.55</td><td>19.3</td><td>17.9</td><td>19.2</td><td>18.8</td><td></td><td>18.5 ± 0.7</td></tr><tr><td>Eye</td><td>7.2</td><td>8.25</td><td>8.6</td><td>8.8</td><td>8.2</td><td>8.3</td><td></td><td>8.2 ± 0.6</td></tr></table>
+
+Table 2: Length measurements of limb segments used to construct the virtual rodent model from 7 female Long-Evans rats. Measurements were performed using calipers either over the skin or over dissected bones (*). Thoracic and sacral refer to vertebral segments. L and R refer to the left and right sides of the animal's body.
+
+# A.2 BEHAVIORAL ANALYSIS
+
+We generated features describing the whole-body pose and kinematics of the virtual rodent on fast, intermediate, and slow temporal scales. To describe the whole-body pose, we took the top 15 principal components of the virtual rodent's joint angles and joint positions to yield two 15 dimensional sets of eigenpostures (Stephens et al., 2008). We combined these into a 30 dimensional set of postural features. To describe the animal's whole-body kinematics, we computed the continuous wavelet transform of each eigenposture using a Morlet wavelet spanning 25 scales. For each set of eigenpostures this yielded a 375 dimensional time-frequency representation of the underlying kinematics. We then computed the top 15 principal components of each 375 dimensional time-frequency representation and combined them to yield a 30 dimensional representational description of the animal's behavioral kinematics. To facilitate comparison of kinematics to neural representations on different timescales, we used three sets of wavelet frequencies on 1 to  $25\mathrm{Hz}$  (intermediate), 0.3 to  $5\mathrm{Hz}$  (slow) or  $5 - 25\mathrm{Hz}$  (fast) timescales. In separate work, we have found that combining postural and kinematic information improves separation of animal behaviors in behavioral embeddings. Therefore, we combined postural and dynamical features, the later on intermediate timescales, to yield a 60 dimensional set of 'behavioral features' that we used to map the animal's behavior using tSNE (Figure 4C) (Berman et al., 2014). tSNEs were made using the Barnes-Hut approximation with a perplexity of 30.
+
+# A.3 POWER SPECTRAL DENSITY OF BEHAVIOR AND NETWORK ACTIVITY
+
+![](images/4a85c7aefeeca18528b7f489683abb6b822c7e561357cc7085b6a06d5c7d17d8.jpg)  
+Figure 9: (A) Power spectral density estimates of four different features describing animal behavior, computed by averaging the spectral density of the top ten principal components of each feature, weighted by the variance they explain. (B) Power spectral density estimates of four different network layers, computed by averaging the spectral density of the top ten principal components of each matrix of activations, weighted by the variance they explain. Notice that policy layers have more power in high frequency bands than core layers. Arrows mark peaks in the power spectra corresponding to locomotion. Notably, the  $4 - 5\mathrm{Hz}$  frequency of galloping in the virtual rat matches that measured in laboratory rats (Heglund & Taylor, 1988). Power spectral density was computed using Welch's method using a 10 s window size and 5 s overlap.
+
+![](images/7550f941ab86dd25d0395268966d2ef93c7984ff386bb376d0b85e24a52ff8e1.jpg)
+
+# A.4 REPRESENTATIONAL SIMILARITY ANALYSIS
+
+We used representational similarity analysis to compare population representations across different network layers and to compute the encoding strength of different features describing animal behavior in the population. Representational similarity analysis has in the past been used to compare neural population responses in tasks where behavioral stimuli are discrete, for instance corpuses of objects or faces (Kriegeskorte et al., 2008; Kriegeskorte & Diedrichsen, 2019). A challenge in scaling such approaches to neural analysis in the context of behavior is that behavior unfolds continuously in time. It is thus a priori unclear how to discretize behavior into discrete chunks in which to compare representations.
+
+Formally, we defined eight sets of features  $B_{i = 1\dots 8}$  describing the behavior of the animal on different timescales. These included features such as joint angles, the angular speed of the joint angles, eigenposture coefficients, and actuator forces that vary on short timescales, as well as behavioral kinematics, which vary on longer timescales and 'behavioral features', which consisted of both kinematics and eigenpostures. Each feature set is a matrix  $B_{i}\in \mathbb{R}^{Mxq_{i}}$  where  $M$  is the number of timepoints in the experiment and  $q_{i}$  is the number of features in the set. We discretized each set  $B_{i}$  using k-means clustering with  $k = 50$  to yield a partition of the timepoints in the experiment  $P_{i}$ .
+
+Using the discretization defined in  $P_{i}$ , we can perform representational similarity analysis to compare the structure of population responses across neural network layers  $L_{m}$  and  $L_{n}$  or between a given network layer and features of the behavior  $B_{i}$ . Following notation in (Kornblith et al., 2019) we let  $X \in \mathbb{R}^{kxp}$  be a matrix of population responses across  $p$  neurons and the  $k$  behavioral categories in  $P_{i}$ . We let  $Y \in \mathbb{R}^{kq}$  be either the matrix of population responses from  $q$  neurons in a distinct network layer, or a set of  $q$  features describing the behavior of the animal in the feature set  $B_{i}$ .
+
+After computing the response matrices in a given behavioral partition, we compared the representational structure of the matrices  $XX^T$  and  $YY^T$ . To do so, we compute the similarity between these matrices using the linear Centered Kernel Alignment index, which is invariant under orthonormal rotations of the population activity. Following (Kornblith et al., 2019), the CKA coefficient is:
+
+$$
+C K A \left(X X ^ {T}, Y Y ^ {T}\right) = \frac {\| X Y ^ {T} \| _ {F}}{\| X X ^ {T} \| _ {F} \| Y Y ^ {T} \| _ {F}} \tag {1}
+$$
+
+Where  $\| \cdot \|_F$  is the Frobenius norm. For centered  $X$  and  $Y$ , the numerator is equivalent to the dot-product between the vectorized responses  $\| XY^T \|_F = \langle \mathrm{vec}(XX^T), \mathrm{vec}(YY^T) \rangle$ .
+
+For a given network layer  $L_{m}$ , and a behavioral partition  $P_{i}$ , we can denote  $XX^{T} = D_{P_{i}}^{L_{m}} = D_{i}^{m}$ . Similarly, for a given feature set  $B_{i}$ , let  $D_{P_{i}}^{B_{i}} = D_{i}^{i}$ . Thus we are interested in characterizing both
+
+$$
+C K A \left(D _ {i} ^ {m}, D _ {i} ^ {n}\right) \tag {2}
+$$
+
+and
+
+$$
+C K A \left(D _ {i} ^ {m}, D _ {i} ^ {i}\right). \tag {3}
+$$
+
+The former equation describes the similarity across two layers of the network, and the later describes the similarity of the network activity to a set of behavioral descriptors.
+
+An additional challenge comes when restricting this analysis to comparing the neural representations of behavioral across different tasks  $T_{a}, T_{b}$ , where not all behaviors are necessarily used in each task. To make such a comparison, we denote  $B_{i}(T_{a})$  to be the set of behavioral clusters observed in task  $T_{a}$ , and  $B_{i}^{T_{a}T_{b}} = B_{i}(T_{a}) \cap B_{i}(T_{a})$  to be the set of behaviors used in each of the two tasks. We can then define a restricted partition of timepoints for each task  $P_{i}^{T_{a},T_{b}}$  or  $P_{i}^{T_{b},T_{a}}$  that includes only these behaviors, and compute the representational similarity between the same layer across tasks:
+
+$$
+C K A \left(D _ {i, T _ {a}} ^ {m}, D _ {i, T _ {b}} ^ {m}\right). \tag {4}
+$$
+
+We have presented a means of performing representational similarity analysis across continuous time domains, where the natural units of discretization are unclear and likely manifold. While we focused on analyzing responses on the population level, it is likely that different subspaces of the population may encode information about distinct behavioral features at different timescales, which is still an emerging domain in representational similarity analysis techniques.
+
+# A.5 NEURAL POPULATION ACTIVITY ACROSS TASKS DURING RUNNING
+
+![](images/2845df0e10c8ae9b5bbfb6fda180568cb52c143b10eae15916e2b32e1eaf364c.jpg)  
+Figure 10: Average activity in the final policy layer (policy 2) during running cycles across different tasks. In each heatmap, rows correspond to the absolute averaged z-scored activity for individual neurons, while columns denote time relative to the mid stance of the running phase. Across heatmaps, neurons are sorted by the time of peak activity in the tasks denoted on the left, such that each column of heatmaps contains the same average activity information with rearranged rows. Aligned running bouts were acquired by manually segmenting the the principal component space of policy 2 activity to find instances of mid-stance running and analyzing the surrounding  $200\mathrm{ms}$ .
+
+# A.6 STEREOTYPED BEHAVIOR INITIATION AND NEURAL VARIABILITY
+
+During the execution of stereotyped behaviors, neural variability was reduced (Figure 11). Recall that in our setting, neurons have no intrinsic noise, but inherit motor noise through observations of the state (i.e. via sensory reafference). This effect loosely resembles, and perhaps informs one line of interpretation of the widely reported phenomenon of neural variability reducing with stimulus or task onset (Churchland et al., 2010). Our reproduction of this effect, which simply emerges from training, suggests that variance modulation may partly arise from moments in a task that benefit from increased behavioral precision (Renart & Machens, 2014).
+
+![](images/9e772115f48ce54ac958869aef72564bc5b305e528c32001d8ba00f2937e63da.jpg)  
+A
+
+![](images/8cf551d73a8483fd7c41a775ab355c309660be32851f280d0ac4986b37fe9cca.jpg)  
+B  
+Figure 11: Quantification of neural variability in inter-tap interval of two-tap task relative to the second tap. (A) Example normalized activity traces of ten randomly selected neurons in the final policy layer. Lines indicate mean normalized activity while shaded regions range from the 20th percentile to the 80th percentile. Dashed lines indicate the times of first and second taps. (B) Standard deviation of normalized activity across all neurons in the final policy layer as a function of time relative to the second tap. Lines indicate the mean standard deviation while shaded regions range from the 20th percentile to the 80th percentile. Observe that variability is reduced during the two-tap interval.
+
+# A.7 NEURAL DYNAMICS VISUALIZED DURING TASK BEHAVIOR
+
+For completeness, we provide links to videos of a few variants of neural dynamics for each task.
+
+<table><tr><td>Network</td><td>Visualization</td><td>Task (link)</td></tr><tr><td rowspan="4">1-layer policy</td><td>PCA</td><td>gaps</td></tr><tr><td>PCA</td><td>forage</td></tr><tr><td>PCA</td><td>escape</td></tr><tr><td>PCA</td><td>two-tap</td></tr><tr><td rowspan="4">3-layer policy</td><td>PCA</td><td>gaps</td></tr><tr><td>PCA</td><td>forage</td></tr><tr><td>PCA</td><td>escape</td></tr><tr><td>PCA</td><td>two-tap</td></tr><tr><td rowspan="4">3-layer policy</td><td>jPCA</td><td>gaps</td></tr><tr><td>jPCA</td><td>forage</td></tr><tr><td>jPCA</td><td>escape</td></tr><tr><td>jPCA</td><td>two-tap</td></tr></table>
+
+Table 3: Links to representative visualizations of neural dynamics and behavior
+
+# A.8 PERTURBATION RESULTS
+
+![](images/853eff4f2a4626879c2f9f96f7e3b7148c1b4265277dbc7d640f57f52b63e627.jpg)
+
+![](images/70145b77f16b1e963956a0d07dd1ca4634f69cdd4a5f12143fac10992183f87b.jpg)
+
+![](images/74c337994b5c270e9114fb4e23bc0f6840cafd30e13ad46e0bba70d785624dff.jpg)
+
+![](images/f427d1ebd9f1da83b6b943943abc7488ce2b972bb393112fe120e6151e27906d.jpg)
+
+![](images/43618f5f229878a655eaabb85cb0c3c66119915e04aaa5608053b7b5a4cb2ece.jpg)
+
+![](images/67671d93adfed1d9b52d30a368c14649dd7764c466fca849f0e20a97907c3457.jpg)
+
+![](images/e85773f29d1af29e1bed83f879d804225f034df7449e9708ed2bcc24be081beb.jpg)
+
+![](images/61df72d140c396575fde87305d02f93785d2eafadbe09aedd51a933c79aa954b.jpg)
+
+![](images/2868a71f5f05654d5b50e3a9c21f2437ede993bbfd98cb90731b1a793cd47a08.jpg)  
+Figure 12: Causal manipulations reveal distinct roles for core and policy layers in the production of behavior. (A) Two-tap accuracy during the inactivation of units modulated by idiosyncratic behaviors within the two-tap sequence. Core inactivation has a weaker negative effect on trial success than policy inactivation for several levels of inactivation. (B) Representative example of a failed trial during inactivation of the final policy layer in a model that performs a spinning jump during the two-tap sequence. The model is incapable of producing the spinning jump behavior while inactivated. (C) Representative example of a failed trial during core inactivation in a model that performs a spinning jump during the two-tap sequence. The model is still able to perform the spinning jump behavior, but misses the orb. (D) Proportion of attempts at stimulation that successfully elicited spin behavior during the two-tap task. The efficacy of this activation was more reliable in layers closer to the motor output. (E) Representative example of a single trial in which an extra spin occurs after policy 2 activation.
+
+![](images/d7ba5822b66622a0e5fd35a662f4d8d629d74be757010a14b4bbf5cecf408d5b.jpg)
+
+![](images/55f980ce7c35ed8bddc3ffe6a3a4ec060039762d99c3054745095f2b87eee094.jpg)
+
+![](images/9e9d8919a5ea9fee2acb5c790a4f78085de8b15f6008873b3296c942f76abab2.jpg)
\ No newline at end of file
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+# DEEPSPHERE: A GRAPH-BASED SPHERICAL CNN
+
+# Michaël Defferrard, Martino Milani & Frédérique Gusset
+
+École Polytechnique Fédérale de Lausanne (EPFL), Switzerland
+
+{michael.defferrard,martino.milani,frederick.gusset}@epfl.ch
+
+# Nathanael Perraudin
+
+Swiss Data Science Center (SDSC), Switzerland
+
+nathanael.perraudin@sdsc.ethz.ch
+
+# ABSTRACT
+
+Designing a convolution for a spherical neural network requires a delicate tradeoff between efficiency and rotation equivariance. DeepSphere, a method based on a graph representation of the sampled sphere, strikes a controllable balance between these two desiderata. This contribution is twofold. First, we study both theoretically and empirically how equivariance is affected by the underlying graph with respect to the number of vertices and neighbors. Second, we evaluate DeepSphere on relevant problems. Experiments show state-of-the-art performance and demonstrate the efficiency and flexibility of this formulation. Perhaps surprisingly, comparison with previous work suggests that anisotropic filters might be an unnecessary price to pay. Our code is available at https://github.com/deepsphere.
+
+# 1 INTRODUCTION
+
+Spherical data is found in many applications (figure 1). Planetary data (such as meteorological or geological measurements) and brain activity are examples of intrinsically spherical data. The observation of the universe, LIDAR scans, and the digitalization of 3D objects are examples of projections due to observation. Labels or variables are often to be inferred from them. Examples are the inference of cosmological parameters from the distribution of mass in the universe (Perraudin et al., 2019), the segmentation of omnidirectional images (Khasanova & Frossard, 2017), and the segmentation of cyclones from Earth observation (Mudigonda et al., 2017).
+
+![](images/1b1e349cfd902086c0e9c526a3e92895ea99e4e07bd0ccf066dc557cd094c755.jpg)  
+(a)
+
+![](images/0f550218e2b9b11b56ec724f42c9814d848c48826b8e13bab5cf78648094598c.jpg)  
+(b)
+
+![](images/b8a820f8c110f671128dfa71fdf6c45ba2a2d688d1d246953ed06ee78cc753c8.jpg)  
+(c)  
+Figure 1: Examples of spherical data: (a) brain activity recorded through magnetoencephalography (MEG), $^{1}$ (b) the cosmic microwave background (CMB) temperature from Planck Collaboration (2016), (c) hourly precipitation from a climate simulation (Jiang et al., 2019), (d) daily maximum temperature from the Global Historical Climatology Network (GHCN). $^{2}$  A rigid full-sphere sampling is not ideal: brain activity is only measured on the scalp, the Milky Way's galactic plane masks observations, climate scientists desire a variable resolution, and the position of weather stations is arbitrary and changes over time. (e) Graphs can faithfully and efficiently represent sampled spherical data by placing vertices where it matters.
+
+![](images/64f56a35925594e05866d2d70693acd6c4e91117056f0c0f199742446ce7e047.jpg)  
+(d)
+
+![](images/e212a37444f820f83091abdbabea8f8cfad0b9077bbd9c40234f3ce1a532c47d.jpg)  
+(e)
+
+As neural networks (NNs) have proved to be great tools for inference, variants have been developed to handle spherical data. Exploiting the locally Euclidean property of the sphere, early attempts used standard 2D convolutions on a grid sampling of the sphere (Boomsma & Frellsen, 2017; Su & Grauman, 2017; Coors et al., 2018). While simple and efficient, those convolutions are not equivariant to rotations. On the other side of this tradeoff, Cohen et al. (2018) and Esteves et al. (2018) proposed to perform proper spherical convolutions through the spherical harmonic transform. While equivariant to rotations, those convolutions are expensive (section 2).
+
+As a lack of equivariance can penalize performance (section 4.2) and expensive convolutions prohibit their application to some real-world problems, methods standing between these two extremes are desired. Cohen et al. (2019) proposed to reduce costs by limiting the size of the representation of the symmetry group by projecting the data from the sphere to the icosahedron. The distortions introduced by this projection might however hinder performance (section 4.3).
+
+Another approach is to represent the sampled sphere as a graph connecting pixels according to the distance between them (Bruna et al., 2013; Khasanova & Frossard, 2017; Perraudin et al., 2019). While Laplacian-based graph convolutions are more efficient than spherical convolutions, they are not exactly equivariant (Defferrard et al., 2019). In this work, we argue that graph-based spherical CNNs strike an interesting balance, with a controllable tradeoff between cost and equivariance (which is linked to performance). Experiments on multiple problems of practical interest show the competitiveness and flexibility of this approach.
+
+# 2 METHOD
+
+DeepSphere leverages graph convolutions to achieve the following properties: (i) computational efficiency, (ii) sampling flexibility, and (iii) rotation equivariance (section 3). The main idea is to model the sampled sphere as a graph of connected pixels: the length of the shortest path between two pixels is an approximation of the geodesic distance between them. We use the graph CNN formulation introduced in (Defferrard et al., 2016) and a pooling strategy that exploits hierarchical samplings of the sphere.
+
+Sampling. A sampling scheme  $\mathcal{V} = \{x_i\in \mathbb{S}^2\}_{i = 1}^n$  is defined to be the discrete subset of the sphere containing the  $n$  points where the values of the signals that we want to analyse are known. For a given continuous signal  $f$ , we represent such values in a vector  $\pmb {f}\in \mathbb{R}^n$ . As there is no analogue of uniform sampling on the sphere, many samplings have been proposed with different tradeoffs. In this work, depending on the considered application, we will use the equiangular (Driscoll & Healy, 1994), HEALPix (Gorski et al., 2005), and icosahedral (Baumgardner & Frederickson, 1985) samplings.
+
+Graph. From  $\mathcal{V}$ , we construct a weighted undirected graph  $\mathcal{G} = (\mathcal{V}, w)$ , where the elements of  $\mathcal{V}$  are the vertices and the weight  $w_{ij} = w_{ji}$  is a similarity measure between vertices  $x_i$  and  $x_j$ . The combinatorial graph Laplacian  $\pmb{L} \in \mathbb{R}^{n \times n}$  is defined as  $\pmb{L} = \pmb{D} - \pmb{A}$ , where  $\pmb{A} = (w_{ij})$  is the weighted adjacency matrix,  $\pmb{D} = (d_{ii})$  is the diagonal degree matrix, and  $d_{ii} = \sum_{j} w_{ij}$  is the weighted degree of vertex  $x_i$ . Given a sampling  $\mathcal{V}$ , usually fixed by the application or the available measurements, the freedom in constructing  $\mathcal{G}$  is in setting  $w$ . Section 3 shows how to set  $w$  to minimize the equivariance error.
+
+**Convolution.** On Euclidean domains, convolutions are efficiently implemented by sliding a window in the signal domain. On the sphere however, there is no straightforward way to implement a convolution in the signal domain due to non-uniform samplings. Convolutions are most often performed in the spectral domain through a spherical harmonic transform (SHT). That is the approach taken by Cohen et al. (2018) and Esteves et al. (2018), which has a computational cost of  $\mathcal{O}(n^{3/2})$  on isolatitude samplings (such as the HEALPix and equiangular samplings) and  $\mathcal{O}(n^2)$  in general.
+
+On the other hand, following Defferrard et al. (2016), graph convolutions can be defined as
+
+$$
+h (\boldsymbol {L}) \boldsymbol {f} = \left(\sum_ {i = 0} ^ {P} \alpha_ {i} \boldsymbol {L} ^ {i}\right) \boldsymbol {f}, \tag {1}
+$$
+
+where  $P$  is the polynomial order (which corresponds to the filter's size) and  $\alpha_{i}$  are the coefficients to be optimized during training.3 Those convolutions are used by Khasanova & Frossard (2017) and Perraudin et al. (2019) and cost  $\mathcal{O}(n)$  operations through a recursive application of  $\pmb{L}$ .4
+
+Pooling. Down- and up-sampling is natural for hierarchical samplings, where each subdivision divides a pixel in (an equal number of) child sub-pixels. To pool (down-sample), the data supported on the sub-pixels is summarized by a permutation invariant function such as the maximum or the average. To unpool (up-sample), the data supported on a pixel is copied to all its sub-pixels.
+
+Architecture. All our NNs are fully convolutional, and employ a global average pooling (GAP) for rotation invariant tasks. Graph convolutional layers are always followed by batch normalization and ReLU activation, except in the last layer. Note that batch normalization and activation act on the elements of  $f$  independently, and hence don't depend on the domain of  $f$ .
+
+# 3 GRAPH CONVOLUTION AND EQUIVARIANCE
+
+While the graph framework offers great flexibility, its ability to faithfully represent the underlying sphere — for graph convolutions to be rotation equivariant — highly depends on the sampling locations and the graph construction.
+
+# 3.1 PROBLEM FORMULATION
+
+A continuous function  $f: \mathcal{C}(\mathbb{S}^2) \supset F_{\mathcal{V}} \to \mathbb{R}$  is sampled as  $T_{\mathcal{V}}(f) = \pmb{f}$  by the sampling operator  $T_{\mathcal{V}}: C(\mathbb{S}^2) \supset F_{\mathcal{V}} \to \mathbb{R}^n$  defined as  $\pmb{f}: f_i = f(x_i)$ . We require  $F_{\mathcal{V}}$  to be a suitable subspace of continuous functions such that  $T_{\mathcal{V}}$  is invertible, i.e., the function  $f \in F_{\mathcal{V}}$  can be unambiguously reconstructed from its sampled values  $\pmb{f}$ . The existence of such a subspace depends on the sampling  $\mathcal{V}$  and its characterization is a common problem in signal processing (Driscoll & Healy, 1994). For most samplings, it is not known if  $F_{\mathcal{V}}$  exists and hence if  $T_{\mathcal{V}}$  is invertible. A special case is the equiangular sampling where a sampling theorem holds, and thus a closed-form of  $T_{\mathcal{V}}^{-1}$  is known. For samplings where no such sampling formula is available, we leverage the discrete SHT to reconstruct  $f$  from  $\pmb{f} = T_{\mathcal{V}}f$ , thus approximating  $T_{\mathcal{V}}^{-1}$ . For all theoretical considerations, we assume that  $F_{\mathcal{V}}$  exists and  $f \in F_{\mathcal{V}}$ .
+
+By definition, the (spherical) graph convolution is rotation equivariant if and only if it commutes with the rotation operator defined as  $R(g), g \in SO(3)$ :  $R(g)f(x) = f(g^{-1}x)$ . In the context of this work, graph convolution is performed by recursive applications of the graph Laplacian (1). Hence, if  $R(g)$  commutes with  $L$ , then, by recursion, it will also commute with the convolution  $h(L)$ . As a result,  $h(L)$  is rotation equivariant if and only if
+
+$$
+\boldsymbol {R} _ {\mathcal {V}} (g) \boldsymbol {L} \boldsymbol {f} = \boldsymbol {L} \boldsymbol {R} _ {\mathcal {V}} (g) \boldsymbol {f}, \quad \forall f \in F _ {\mathcal {V}} \text {a n d} \forall g \in S O (3),
+$$
+
+where  $R_{\mathcal{V}}(g) = T_{\mathcal{V}}R(g)T_{\mathcal{V}}^{-1}$ . For an empirical evaluation of equivariance, we define the normalized equivariance error for a signal  $\pmb{f}$  and a rotation  $g$  as
+
+$$
+E _ {\boldsymbol {L}} (\boldsymbol {f}, g) = \left(\frac {\| \boldsymbol {R} _ {\mathcal {V}} (g) \boldsymbol {L} \boldsymbol {f} - \boldsymbol {L R} _ {\mathcal {V}} (g) \boldsymbol {f} \|}{\| \boldsymbol {L} \boldsymbol {f} \|}\right) ^ {2}. \tag {2}
+$$
+
+More generally for a class of signals  $f \in C \subset F_{\mathcal{V}}$ , the mean equivariance error defined as
+
+$$
+\bar {E} _ {\boldsymbol {L}, C} = \mathbb {E} _ {\boldsymbol {f} \in C, g \in S O (3)} E _ {\boldsymbol {L}} (\boldsymbol {f}, g) \tag {3}
+$$
+
+represents the overall equivariance error. The expected value is obtained by averaging over a finite number of random functions and random rotations.
+
+![](images/aab213c0cd13bf3270e20aa91768a762df9d068567b47ac8189b71c52dca4d47.jpg)  
+Figure 2: Mean equivariance error (3). There is a clear tradeoff between equivariance and computational cost, governed by the number of vertices  $n$  and edges  $kn$ .
+
+![](images/8e20075f38abe3b897813caba11e763deb3444eb12b7558678eb1c0b3f09dec5.jpg)  
+Figure 3: Kernel widths.
+
+![](images/96f715570bd1fd35baf0a998c2ff1be5c5ac1319917b0249e47cea9934dd7180.jpg)  
+Figure 4: 3D object represented as a spherical depth map.
+
+![](images/bb100e6cbadfdcef6c02f7740592b0ee27d07242d88118e8953d7af173973c3f.jpg)  
+Figure 5: Power spectral densities.
+
+# 3.2 FINDING THE OPTIMAL WEIGHTING SCHEME
+
+Considering the equiangular sampling and graphs where each vertex is connected to 4 neighbors (north, south, east, west), Khasanova & Frossard (2017) designed a weighting scheme to minimize (3) for longitudinal and latitudinal rotations. Their solution gives weights inversely proportional to Euclidean distances:
+
+$$
+w _ {i j} = \frac {1}{\left\| x _ {i} - x _ {j} \right\|}. \tag {4}
+$$
+
+While the resulting convolution is not equivariant to the whole of  $SO(3)$  (figure 2), it is enough for omnidirectional imaging because, as gravity consistently orients the sphere, objects only rotate longitudinally or latitudinally.
+
+To achieve equivariance to all rotations, we take inspiration from Belkin & Niyogi (2008). They prove that for a random uniform sampling, the graph Laplacian  $\pmb{L}$  built from weights
+
+$$
+w _ {i j} = e ^ {- \frac {1}{4 t} \| x _ {i} - x _ {j} \| ^ {2}} \tag {5}
+$$
+
+converges to the Laplace-Beltrami operator  $\Delta_{\mathbb{S}^2}$  as the number of samples grows to infinity. This result is a good starting point as  $\Delta_{\mathbb{S}^2}$  commutes with rotation, i.e.,  $\Delta_{\mathbb{S}^2}R(g) = R(g)\Delta_{\mathbb{S}^2}$ . While the weighting scheme is full (i.e., every vertex is connected to every other vertex), most weights are small due to the exponential. We hence make an approximation to limit the cost of the convolution (1) by only considering the  $k$  nearest neighbors ( $k$ -NN) of each vertex. Given  $k$ , the optimal kernel width  $t$  is found by searching for the minimizer of (3). Figure 3 shows the optimal kernel widths found for various resolutions of the HEALPix sampling. As predicted by the theory,  $t_n \propto n^\beta, \beta \in \mathbb{R}$ . Importantly however, the optimal  $t$  also depends on the number of neighbors  $k$ .
+
+Considering the HEALPix sampling, Perraudin et al. (2019) connected each vertex to their 8 adjacent vertices in the tiling of the sphere, computed the weights with (5), and heuristically set  $t$  to half the average squared Euclidean distance between connected vertices. This heuristic however overestimates  $t$  (figure 3) and leads to an increased equivariance error (figure 2).
+
+# 3.3 ANALYSIS OF THE PROPOSED WEIGHTING SCHEME
+
+We analyze the proposed weighting scheme both theoretically and empirically.
+
+Theoretical convergence. We extend the work of (Belkin & Niyogi, 2008) to a sufficiently regular, deterministic sampling. Following their setting, we work with the extended graph Laplacian operator as the linear operator  $L_{n}^{t}:L^{2}(\mathbb{S}^{2})\to L^{2}(\mathbb{S}^{2})$  such that
+
+$$
+L _ {n} ^ {t} f (y) := \frac {1}{n} \sum_ {i = 1} ^ {n} e ^ {- \frac {\left\| x _ {i} - y \right\| ^ {2}}{4 t}} \left(f (y) - f \left(x _ {i}\right)\right). \tag {6}
+$$
+
+This operator extends the graph Laplacian with the weighting scheme (5) to each point of the sphere (i.e.,  $L_{n}^{t}\pmb {f} = T_{\mathcal{V}}L_{n}^{t}f$ ). As the radius of the kernel  $t$  will be adapted to the number of samples, we scale the operator as
+
+![](images/331d43ea79e24db3715e598f214372642bfc8e1adca751d7992a7bb160567fb4.jpg)  
+Figure 6: Patch.
+
+$\hat{L}_n^t \coloneqq |\mathbb{S}^2| (4\pi t^2)^{-1} L_n^t$ . Given a sampling  $\mathcal{V}$ , we define  $\sigma_i$  to be the patch of the surface of the sphere corresponding to  $x_i$ ,  $A_i$  its corresponding area, and  $d_i$  the largest distance between the center  $x_i$  and any point on the surface  $\sigma_i$ . Define  $d^{(n)} \coloneqq \max_{i=1,\dots,n} d_i$  and  $A^{(n)} \coloneqq \max_{i=1,\dots,n} A_i$ .
+
+Theorem 3.1. For a sampling  $\mathcal{V}$  of the sphere that is equi-area and such that  $d^{(n)}\leq \frac{C}{n^{\alpha}}$ $\alpha \in (0,1 / 2]$ , for all  $f:\mathbb{S}^2\to \mathbb{R}$  Lipschitz with respect to the Euclidean distance in  $\mathbb{R}^3$ , for all  $y\in \mathbb{S}^2$ , there exists a sequence  $t_n = n^\beta$ ,  $\beta \in \mathbb{R}$  such that
+
+$$
+\lim _ {n \to \infty} \hat {L} _ {n} ^ {t _ {n}} f (y) = \Delta_ {\mathbb {S} ^ {2}} f (y).
+$$
+
+This is a major step towards equivariance, as the Laplace-Beltrami operator commutes with rotation. Based on this property, we show the equivariance of the scaled extended graph Laplacian.
+
+Theorem 3.2. Under the hypothesis of theorem 3.1, the scaled graph Laplacian commutes with any rotation, in the limit of infinite sampling, i.e.,
+
+$$
+\forall y \in \mathbb {S} ^ {2} \quad \left| R (g) \hat {L} _ {n} ^ {t _ {n}} f (y) - \hat {L} _ {n} ^ {t _ {n}} R (g) f (y) \right| \xrightarrow {n \to \infty} 0.
+$$
+
+From this theorem, it follows that the discrete graph Laplacian will be equivariant in the limit of  $n \to \infty$  as by construction  $\pmb{L}_n^t\pmb{f} = T_{\mathcal{V}}L_n^t\pmb{f}$  and as the scaling does not affect the equivariance property of  $\pmb{L}_n^t$ .
+
+Importantly, the proof of Theorem 3.1 (in Appendix A) inspires our construction of the graph Laplacian. In particular, it tells us that  $t$  should scale as  $n^{\beta}$ , which has been empirically verified (figure 3). Nevertheless, it is important to keep in mind the limits of Theorem 3.1 and 3.2. Both theorems present asymptotic results, but in practice we will always work with finite samplings. Furthermore, since this method is based on the capability of the eigenvectors of the graph Laplacian to approximate the spherical harmonics, a stronger type of convergence of the graph Laplacian would be preferable, i.e., spectral convergence (that is proved for a full graph in the case of random sampling for a class of Lipschitz functions in (Belkin & Niyogi, 2007)). Finally, while we do not have a formal proof for it, we strongly believe that the HEALPix sampling does satisfy the hypothesis  $d^{(n)} \leq \frac{C}{n^\alpha}$ ,  $\alpha \in (0,1/2]$ , with  $\alpha$  very close or equal to  $1/2$ . The empirical results discussed in the next paragraph also points in this direction. This is further discussed in Appendix A.
+
+Empirical convergence. Figure 2 shows the equivariance error (3) for different parameter sets of DeepSphere for the HEALPix sampling as well as for the graph construction of Khasanova & Frossard (2017) for the equiangular sampling. The error is estimated as a function of the sampling resolution and signal frequency. The resolution is controlled by the number of pixels  $n = 12N_{side}^2$  for HEALPix and  $n = 4b^2$  for the equiangular sampling. The frequency is controlled by setting the set  $C$  to functions  $f$  made of spherical harmonics of a single degree  $\ell$ . To allow for an almost perfect implementation (up to numerical errors) of the operator  $R_{\mathcal{V}}$ , the degree  $\ell$  was chosen in the range  $(0, 3N_{side} - 1)$  for HEALPix and  $(0, b)$  for the equiangular sampling (Gorski et al., 1999). Using these parameters, the measured error is mostly due to imperfections in the empirical approximation of the Laplace-Beltrami operator and not to the sampling.
+
+<table><tr><td></td><td colspan="2">performance</td><td>size</td><td colspan="2">speed</td></tr><tr><td></td><td>F1</td><td>mAP</td><td>params</td><td>inference</td><td>training</td></tr><tr><td>Cohen et al. (2018) (b = 128)</td><td>-</td><td>67.6</td><td>1400 k</td><td>38.0 ms</td><td>50 h</td></tr><tr><td>Cohen et al. (2018) (simplified,9b = 64)</td><td>78.9</td><td>66.5</td><td>400 k</td><td>12.0 ms</td><td>32 h</td></tr><tr><td>Esteves et al. (2018) (b = 64)</td><td>79.4</td><td>68.5</td><td>500 k</td><td>9.8 ms</td><td>3 h</td></tr><tr><td>DeepSphere (equiangular, b = 64)</td><td>79.4</td><td>66.5</td><td>190 k</td><td>0.9 ms</td><td>50 m</td></tr><tr><td>DeepSphere (HEALPix, Nside = 32)</td><td>80.7</td><td>68.6</td><td>190 k</td><td>0.9 ms</td><td>50 m</td></tr></table>
+
+Table 1: Results on SHREC'17 (3D shapes). DeepSphere achieves similar performance at a much lower cost, suggesting that anisotropic filters are an unnecessary price to pay.
+
+Figure 2 shows that the weighting scheme (4) from (Khasanova & Frossard, 2017) does indeed not lead to a convolution that is equivariant to all rotations  $g \in SO(3)$ . For  $k = 8$  neighbors, selecting the optimal kernel width  $t$  improves on (Perraudin et al., 2019) at no cost, highlighting the importance of this parameter. Increasing the resolution decreases the equivariance error in the high frequencies, an effect most probably due to the sampling. Most importantly, the equivariance error decreases when connecting more neighbors. Hence, the number of neighbors  $k$  gives us a precise control of the tradeoff between cost and equivariance.
+
+# 4 EXPERIMENTS
+
+# 4.1 3D OBJECTS RECOGNITION
+
+The recognition of 3D shapes is a rotation invariant task: rotating an object doesn't change its nature. While 3D shapes are usually represented as meshes or point clouds, representing them as spherical maps (figure 4) naturally allows a rotation invariant treatment.
+
+The SHREC'17 shape retrieval contest (Savva et al., 2017) contains 51,300 randomly oriented 3D models from ShapeNet (Chang et al., 2015), to be classified in 55 categories (tables, lamps, airplanes, etc.). As in (Cohen et al., 2018), objects are represented by 6 spherical maps. At each pixel, a ray is traced towards the center of the sphere. The distance from the sphere to the object forms a depth map. The cos and sin of the surface angle forms two normal maps. The same is done for the object's convex hull. The maps are sampled by an equiangular sampling with bandwidth  $b = 64$  ( $n = 4b^2 = 16$ , 384 pixels) or an HEALPix sampling with  $N_{side} = 32$  ( $n = 12N_{side}^2 = 12$ , 288 pixels).
+
+The equiangular graph is built with (4) and  $k = 4$  neighbors (following Khasanova & Frossard, 2017). The HEALPix graph is built with (5),  $k = 8$ , and a kernel width  $t$  set to the average of the distances (following Perraudin et al., 2019). The NN is made of 5 graph convolutional layers, each followed by a max pooling layer which down-samples by 4. A GAP and a fully connected layer with softmax follow. The polynomials are all of order  $P = 3$  and the number of channels per layer is 16, 32, 64, 128, 256, respectively. Following Esteves et al. (2018), the cross-entropy plus a triplet loss is optimized with Adam for 30 epochs on the dataset augmented by 3 random translations. The learning rate is  $5 \cdot 10^{-2}$  and the batch size is 32.
+
+Results are shown in table 1. As the network is trained for shape classification rather than retrieval, we report the classification F1 alongside the mAP used in the retrieval contest. $^{10}$  DeepSphere achieves the same performance as Cohen et al. (2018) and Esteves et al. (2018) at a much lower cost, suggesting that anisotropic filters are an unnecessary price to pay. As the information in those spherical maps resides in the low frequencies (figure 5), reducing the equivariance error didn't translate into improved performance. For the same reason, using the more uniform HEALPix sampling or lowering the resolution down to  $N_{side} = 8$  ( $n = 768$  pixels) didn't impact performance either.
+
+<sup>7</sup>We however verified that the convolution is equivariant to longitudinal and latitudinal rotations, as intended.  
+Albeit we didn't observe much improvement by using the convex hull.  
+<sup>7</sup>As implemented in https://github.com/jonas-koehler/s2cnn.  
+<sup>10</sup>We omit the F1 for Cohen et al. (2018) as we didn't get the mAP reported in the paper when running it.
+
+<table><tr><td></td><td>accuracy</td><td>time</td></tr><tr><td>Perraudin et al. (2019), 2D CNN baseline</td><td>54.2</td><td>104 ms</td></tr><tr><td>Perraudin et al. (2019), CNN variant, k = 8</td><td>62.1</td><td>185 ms</td></tr><tr><td>Perraudin et al. (2019), FCN variant, k = 8</td><td>83.8</td><td>185 ms</td></tr><tr><td>k = 8 neighbors, t from section 3.2</td><td>87.1</td><td>185 ms</td></tr><tr><td>k = 20 neighbors, t from section 3.2</td><td>91.3</td><td>250 ms</td></tr><tr><td>k = 40 neighbors, t from section 3.2</td><td>92.5</td><td>363 ms</td></tr></table>
+
+Table 2: Results on the classification of partial convergence maps. Lower equivariance error translates to higher performance.
+
+![](images/89e712641408acc8709e6a5a00c32fe4ead915bb5b85b3c18dab6ae9830a7695.jpg)  
+Figure 7: Tradeoff between cost and accuracy.
+
+# 4.2 COSMOLOGICAL MODEL CLASSIFICATION
+
+Given observations, cosmologists estimate the posterior probability of cosmological parameters, such as the matter density  $\Omega_{m}$  and the normalization of the matter power spectrum  $\sigma_{8}$ . Those parameters are typically estimated by likelihood-free inference, which requires a function to predict the parameters from simulations. As that is complicated to setup, prediction methods are typically benchmarked on the classification of spherical maps instead (Schmelzle et al., 2017). We used the same task, data, and setup as Perraudin et al. (2019): the classification of 720 partial convergence maps made of  $n\approx 10^{6}$  pixels ( $1/12\approx 8\%$ ) of a sphere at  $N_{side} = 1024$  from two  $\Lambda$ CDM cosmological models,  $(\Omega_{m} = 0.31, \sigma_{8} = 0.82)$  and  $(\Omega_{m} = 0.26, \sigma_{8} = 0.91)$ , at a relative noise level of 3.5 (i.e., the signal is hidden in noise of 3.5 times higher standard deviation). Convergence maps represent the distribution of over- and under-densities of mass in the universe (see Bartelmann, 2010, for a review of gravitational lensing).
+
+Graphs are built with (5),  $k = 8, 20, 40$  neighbors, and the corresponding optimal kernel widths  $t$  given in section 3.2. Following Perraudin et al. (2019), the NN is made of 5 graph convolutional layers, each followed by a max pooling layer which down-samples by 4. A GAP and a fully connected layer with softmax follow. The polynomials are all of order  $P = 4$  and the number of channels per layer is 16, 32, 64, 64, 64, respectively. The cross-entropy loss is optimized with Adam for 80 epochs. The learning rate is  $2 \cdot 10^{-4} \cdot 0.999^{\mathrm{step}}$  and the batch size is 8.
+
+Unlike on SHREC'17, results (table 2) show that a lower equivariance error on the convolutions translates to higher performance. That is probably due to the high frequency content of those maps (figure 5). There is a clear cost-accuracy tradeoff, controlled by the number of neighbors  $k$  (figure 7). This experiment moreover demonstrates DeepSphere's flexibility (using partial spherical maps) and scalability (competing spherical CNNs were tested on maps of at most 10,000 pixels).
+
+# 4.3 CLIMATE EVENT SEGMENTATION
+
+We evaluate our method on a task proposed by (Mudigonda et al., 2017): the segmentation of extreme climate events, Tropical Cyclones (TC) and Atmospheric Rivers (AR), in global climate simulations (figure 1c). The data was produced by a 20-year run of the Community Atmospheric Model v5 (CAM5) and consists of 16 channels such as temperature, wind, humidity, and pressure at multiple altitudes. We used the pre-processed dataset from (Jiang et al., 2019). There is 1,072,805 spherical maps, down-sampled to a level-5 icosahedral sampling ( $n = 10 \cdot 4^l + 2 = 10,242$  pixels). The labels are heavily unbalanced with  $0.1\%$  TC,  $2.2\%$  AR, and  $97.7\%$  background (BG) pixels.
+
+The graph is built with (5),  $k = 6$  neighbors, and a kernel width  $t$  set to the average of the distances. Following Jiang et al. (2019), the NN is an encoder-decoder with skip connections. Details in section C.3. The polynomials are all of order  $P = 3$ . The cross-entropy loss (weighted or non-weighted) is optimized with Adam for 30 epochs. The learning rate is  $1 \cdot 10^{-3}$  and the batch size is 64.
+
+Results are shown in table 3 (details in tables 6, 7 and 8). The mean and standard deviation are computed over 5 runs. Note that while Jiang et al. (2019) and Cohen et al. (2019) use a weighted cross-entropy loss, that is a suboptimal proxy for the mAP metric. DeepSphere achieves state-of
+
+<table><tr><td></td><td>accuracy</td><td>mAP</td></tr><tr><td>Jiang et al. (2019) (rerun)</td><td>94.95</td><td>38.41</td></tr><tr><td>Cohen et al. (2019) (S2R)</td><td>97.5</td><td>68.6</td></tr><tr><td>Cohen et al. (2019) (R2R)</td><td>97.7</td><td>75.9</td></tr><tr><td>DeepSphere (weighted loss)</td><td>97.8 ± 0.3</td><td>77.15 ± 1.94</td></tr><tr><td>DeepSphere (non-weighted loss)</td><td>87.8 ± 0.5</td><td>89.16 ± 1.37</td></tr></table>
+
+Table 3: Results on climate event segmentation: mean accuracy (over TC, AR, BG) and mean average precision (over TC and AR). DeepSphere achieves state-of-the-art performance.  
+
+<table><tr><td rowspan="2">order P</td><td colspan="3">temp. (from past temp.)</td><td colspan="3">day (from temperature)</td><td colspan="3">day (from precipitations)</td></tr><tr><td>MSE</td><td>MAE</td><td>R2</td><td>MSE</td><td>MAE</td><td>R2</td><td>MSE</td><td>MAE</td><td>R2</td></tr><tr><td>0</td><td>10.88</td><td>2.42</td><td>0.896</td><td>0.10</td><td>0.10</td><td>0.882</td><td>0.58</td><td>0.42</td><td>-0.980</td></tr><tr><td>4</td><td>8.20</td><td>2.11</td><td>0.919</td><td>0.05</td><td>0.05</td><td>0.969</td><td>0.50</td><td>0.18</td><td>0.597</td></tr></table>
+
+Table 4: Prediction results on data from weather stations. Structure always improves performance.
+
+the-art performance, suggesting again that anisotropic filters are unnecessary. Note that results from Mudigonda et al. (2017) cannot be directly compared as they don't use the same input channels.
+
+Compared to Cohen et al. (2019)'s conclusion, it is surprising that S2R does worse than DeepSphere (which is limited to S2S). Potential explanations are (i) that their icosahedral projection introduces harmful distortions, or (ii) that a larger architecture can compensate for the lack of generality. We indeed observed that more feature maps and depth led to higher performance (section C.3).
+
+# 4.4 UNEVEN SAMPLING
+
+To demonstrate the flexibility of modeling the sampled sphere by a graph, we collected historical measurements from  $n \approx 10,000$  weather stations scattered across the Earth. The spherical data is heavily non-uniformly sampled, with a much higher density of weather stations over North America than the Pacific (figure 1d). For illustration, we devised two artificial tasks. A dense regression: predict the temperature on a given day knowing the temperature on the previous 5 days. A global regression: predict the day (represented as one period of a sine over the year) from temperature or precipitations. Predicting from temperature is much easier as it has a clear yearly pattern.
+
+The graph is built with (5),  $k = 5$  neighbors, and a kernel width  $t$  set to the average of the distances. The equivariance property of the resulting graph has not been tested, and we don't expect it to be good due to the heavily non-uniform sampling. The NN is made of 3 graph convolutional layers. The polynomials are all of order  $P = 0$  or 4 and the number of channels per layer is 50, 100, 100, respectively. For the global regression, a GAP and a fully connected layer follow. For the dense regression, a graph convolutional layer follows instead. The MSE loss is optimized with RMSprop for 250 epochs. The learning rate is  $1 \cdot 10^{-3}$  and the batch size is 64.
+
+Results are shown in table 4. While using a polynomial order  $P = 0$  is like modeling each time series independently with an MLP, orders  $P > 0$  integrate neighborhood information. Results show that using the structure induced by the spherical geometry always yields better performance.
+
+# 5 CONCLUSION
+
+This work showed that DeepSphere strikes an interesting, and we think currently optimal, balance between desiderata for a spherical CNN. A single parameter, the number of neighbors  $k$  a pixel is connected to in the graph, controls the tradeoff between cost and equivariance (which is linked to performance). As computational cost and memory consumption scales linearly with the number of pixels, DeepSphere scales to spherical maps made of millions of pixels, a required resolution
+
+to faithfully represent cosmological and climate data. Also relevant in scientific applications is the flexibility offered by a graph representation (for partial coverage, missing data, and non-uniform samplings). Finally, the implementation of the graph convolution is straightforward, and the ubiquity of graph neural networks — pushing for their first-class support in DL frameworks — will make implementations even easier and more efficient.
+
+A potential drawback of graph Laplacian-based approaches is the isotropy of graph filters, reducing in principle the expressive power of the NN. Experiments from Cohen et al. (2019) and Boscaini et al. (2016) indeed suggest that more general convolutions achieve better performance. Our experiments on 3D shapes (section 4.1) and climate (section 4.3) however show that DeepSphere's isotropic filters do not hinder performance. Possible explanations for this discrepancy are that NNs somehow compensate for the lack of anisotropic filters, or that some tasks can be solved with isotropic filters. The distortions induced by the icosahedral projection in (Cohen et al., 2019) or the leakage of curvature information in (Boscaini et al., 2016) might also alter performance.
+
+Developing graph convolutions on irregular samplings that respect the geometry of the sphere is another research direction of importance. Practitioners currently interpolate their measurements (coming from arbitrarily positioned weather stations, satellites or telescopes) to regular samplings. This practice either results in a waste of resolution or computational and storage resources. Our ultimate goal is for practitioners to be able to work directly on their measurements, however distributed.
+
+# ACKNOWLEDGMENTS
+
+We thank Pierre Vandergheynst for advices, and Taco Cohen for his inputs on the intriguing results of our comparison with Cohen et al. (2019). We thank the anonymous reviewers for their constructive feedback. The following software packages were used for computation and plotting: PyGSP (Defferrard et al.), healpy (Zonca et al., 2019), matplotlib (Hunter, 2007), SciPy (Virtanen et al., 2020), NumPy (Walt et al., 2011), TensorFlow (Abadi et al., 2015).
+
+# REFERENCES
+
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+Angel X Chang, Thomas Funkhouser, Leonidas Guibas, Pat Hanrahan, Qixing Huang, Zimo Li, Silvio Savarese, Manolis Savva, Shuran Song, Hao Su, et al. Shapenet: An information-rich 3d model repository. arXiv:1512.03012, 2015.  
+Taco S Cohen, Mario Geiger, Jonas Koehler, and Max Welling. Spherical cnns. In International Conference on Learning Representations (ICLR), 2018. URL https://arxiv.org/abs/1801.10130.  
+Taco S. Cohen, Maurice Weiler, Berkay Kicanaoglu, and Max Welling. Gauge equivariant convolutional networks and the icosahedral cnn. In International Conference on Machine Learning (ICML), 2019. URL http://arxiv.org/abs/1902.04615.  
+Benjamin Coors, Alexandru Paul Condurache, and Andreas Geiger. Spherenet: Learning spherical representations for detection and classification in omnidirectional images. In European Conference on Computer Vision, 2018.  
+Michaël Defferrard, Lionel Martin, Rodrigo Pena, and Nathanaël Perraudin. Pygsp: Graph signal processing in python. URL https://github.com/epfl-lts2/pygsp/.  
+Michaël Defferrard, Xavier Bresson, and Pierre Vandergheynst. Convolutional neural networks on graphs with fast localized spectral filtering. In Advances in Neural Information Processing Systems, 2016. URL https://arxiv.org/abs/1606.09375.  
+Michaël Defferrard, Nathanaël Perraudin, Tomasz Kacprzak, and Raphael Sgier. Deepsphere: towards an equivariant graph-based spherical cnn. In ICLR Workshop on Representation Learning on Graphs and Manifolds, 2019. URL https://arxiv.org/abs/1904.05146.  
+J. R. Driscoll and D. M. Healy. Computing fourier transforms and convolutions on the 2-sphere. Adv. Appl. Math., 1994. URL http://dx.doi.org/10.1006/aama.1994.1008.  
+Carlos Esteves, Christine Allen-Blanchette, Ameesh Makadia, and Kostas Daniilidis. Learning so(3) equivariant representations with spherical cnns. In Proceedings of the European Conference on Computer Vision (ECCV), 2018. URL https://arxiv.org/abs/1711.06721.  
+Krzysztof M Gorski, Benjamin D Wandelt, Frode K Hansen, Eric Hivon, and Anthony J Banday. The healpix primer. arXiv preprint astro-ph/9905275, 1999.  
+Krzysztof M Gorski, Eric Hivon, AJ Banday, Benjamin D Wandelt, Frode K Hansen, Mstvos Reinecke, and Matthia Bartelmann. Healpix: a framework for high-resolution discretization and fast analysis of data distributed on the sphere. The Astrophysical Journal, 2005.  
+J. D. Hunter. Matplotlib: A 2d graphics environment. Computing in Science & Engineering, 9(3): 90-95, 2007. doi: 10.1109/MCSE.2007.55.  
+Chiyu "Max" Jiang, Jingwei Huang, Karthik Kashinath, Prabhat, Philip Marcus, and Matthias Niessner. Spherical cnns on unstructured grids. In International Conference on Learning Representations (ICLR), 2019. URL https://arxiv.org/abs/1901.02039.  
+Renata Khasanova and Pascal Frossard. Graph-based classification of omnidirectional images. In Proceedings of the IEEE International Conference on Computer Vision, 2017. URL https://arxiv.org/abs/1707.08301.  
+Mayur Mudigonda, Sookyung Kim, Ankur Mahesh, Samira Kahou, Karthik Kashinath, Dean Williams, Vincen Michalski, Travis O'Brien, and Mr Prabhat. Segmenting and tracking extreme climate events using neural networks. In Deep Learning for Physical Sciences (DLPS) Workshop, held with NIPS Conference, 2017. URL https://dl4physicalsciences.github.io/files/nips_dlps_2017_20.pdf.  
+Nathanael Perraudin, Michael Defferrard, Tomasz Kacprzak, and Raphael Sgier. Deepsphere: Efficient spherical convolutional neural network with healpix sampling for cosmological applications. *Astronomy and Computing*, 2019. URL https://arxiv.org/abs/1810.12186.  
+Planck Collaboration. Planck 2015 results. i. overview of products and scientific results. *Astronomy & Astrophysics*, 2016.
+
+Manolis Savva, Fisher Yu, Hao Su, Asako Kanezaki, Takahiko Furuya, Ryutarou Ohbuchi, Zhichao Zhou, Rui Yu, Song Bai, Xiang Bai, et al. Large-scale 3d shape retrieval from shapenet core55: Shrec'17 track. In Eurographics Workshop on 3D Object Retrieval, 2017.  
+J. Schmelzle, A. Lucchi, T. Kacprzak, A. Amara, R. Sgier, A. Réfrégier, and T. Hofmann. Cosmological model discrimination with deep learning. arxiv:1707.05167, 2017.  
+Yu-Chuan Su and Kristen Grauman. Learning spherical convolution for fast features from 360 imagery. In Advances in Neural Information Processing Systems, 2017.  
+Pauli Virtanen, Ralf Gommers, Travis E. Oliphant, Matt Haberland, Tyler Reddy, David Cournapeau, Evgeni Burovski, PEARU Peterson, Warren Weckesser, Jonathan Bright, Stéfan J. van der Walt, Matthew Brett, Joshua Wilson, K. Jarrod Millman, Nikolay Mayorov, Andrew R. J. Nelson, Eric Jones, Robert Kern, Eric Larson, CJ Carey, Ilhan Polat, Yu Feng, Eric W. Moore, Jake Vand erPlas, Denis Laxalde, Josef Perktold, Robert Cirmrnan, Ian Henriksen, E. A. Quintero, Charles R Harris, Anne M. Archibald, Antonio H. Ribeiro, Fabian Pedregosa, Paul van Mulbregt, and SciPy 1.0 Contributors. SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python. Nature Methods, 2020. doi: https://doi.org/10.1038/s41592-019-0686-2.  
+Stéfan van der Walt, S Chris Colbert, and Gael Varoquaux. The numpy array: a structure for efficient numerical computation. Computing in Science & Engineering, 13(2):22-30, 2011.  
+Andrea Zonca, Leo Singer, Daniel Lenz, Martin Reinecke, Cyrille Rosset, Eric Hivon, and Krzysztof Gorski. healpy: equal area pixelization and spherical harmonics transforms for data on the sphere in python. Journal of Open Source Software, 4(35):1298, March 2019. doi: 10.21105/joss.01298. URL https://doi.org/10.21105/joss.01298.
+
+# SUPPLEMENTARY MATERIAL
+
+# A PROOF OF THEOREM 3.1
+
+Preliminaries. The proof of theorem 3.1 is inspired from the work of Belkin & Niyogi (2008). As a result, we start by restating some of their results. Given a sampling  $\mathcal{V} = \{x_{i}\in \mathcal{M}\}_{i = 1}^{n}$  of a closed, compact and infinitely differentiable manifold  $\mathcal{M}$ , a smooth  $(\in \mathcal{C}_\infty (\mathcal{M}))$  function  $f:\mathcal{M}\to \mathbb{R}$ , and defined the vector  $\pmb{f}$  of samples of  $f$  as follows:  $T_{\mathcal{V}}f = \pmb {f}\in \mathbb{R}^n$ ,  $\pmb {f}_i = f(x_i)$ . The proof is constructed by leveraging 3 different operators:
+
+- The extended graph Laplacian operator, already presented in (6), is a linear operator  $L_{n}^{t}: L^{2}(\mathcal{M}) \to L^{2}(\mathcal{M})$  defined as
+
+$$
+L _ {n} ^ {t} f (y) := \frac {1}{n} \sum_ {i = 1} ^ {n} e ^ {- \frac {\left\| x _ {i} - y \right\| ^ {2}}{4 t}} \left(f (y) - f \left(x _ {i}\right)\right). \tag {7}
+$$
+
+Note that we have the following relation  $L_{n}^{t}\pmb {f} = T_{\mathcal{V}}L_{n}^{t}f$
+
+- The functional approximation to the Laplace-Beltrami operator is a linear operator  $L^t: L^2(\mathcal{M}) \to L^2(\mathcal{M})$  defined as
+
+$$
+L ^ {t} f (y) = \int_ {\mathcal {M}} e ^ {- \frac {\| x - y \| ^ {2}}{4 t}} \left(f (y) - f (x)\right) d \mu (x), \tag {8}
+$$
+
+where  $\mu$  is the uniform probability measure on the manifold  $\mathcal{M}$ , and  $\mathrm{vol}(\mathcal{M})$  is the volume of  $\mathcal{M}$ .
+
+- The Laplace-Beltrami operator  $\Delta_{\mathcal{M}}$  is defined as the divergence of the gradient
+
+$$
+\Delta_ {\mathcal {M}} f (y) := - \operatorname {d i v} (\nabla_ {\mathcal {M}} f) \tag {9}
+$$
+
+of a differentiable function  $f: \mathcal{M} \to \mathbb{R}$ . The gradient  $\nabla f: \mathcal{M} \to T_p\mathcal{M}$  is a vector field defined on the manifold pointing towards the direction of steepest ascent of  $f$ , where  $T_p\mathcal{M}$  is the affine space of all vectors tangent to  $\mathcal{M}$  at  $p$ .
+
+Leveraging these three operators, Belkin & Niyogi (2008; 2007) have built proofs of both pointwise and spectral convergence of the extended graph Laplacian towards the Laplace-Beltrami operator in the general setting of any compact, closed and infinitely differentiable manifold  $\mathcal{M}$ , where the sampling  $\mathcal{V}$  is drawn randomly on the manifold. For this reason, their results are all to be interpreted in a probabilistic sense. Their proofs consist in establishing that (6) converges in probability towards (8) as  $n\to \infty$  and (8) converges towards (9) as  $t\to 0$ . In particular, this second step is given by the following:
+
+Proposition 1 (Belkin & Niyogi (2008), Proposition 4.4). Let  $\mathcal{M}$  be a  $k$ -dimensional compact smooth manifold embedded in some Euclidean space  $\mathbb{R}^N$ , and fix  $y \in \mathcal{M}$ . Let  $f \in C_{\infty}(\mathcal{M})$ . Then
+
+$$
+\frac {1}{t} \frac {1}{(4 \pi t) ^ {k / 2}} L ^ {t} f (y) \xrightarrow {t \rightarrow 0} \frac {1}{v o l (\mathcal {M})} \Delta_ {\mathcal {M}} f (y). \tag {10}
+$$
+
+Building the proof. As the sphere is a compact smooth manifold embedded in  $\mathbb{R}^3$ , we can reuse proposition 1. Thus, our strategy to prove Theorem 3.1 is to (i) show that
+
+$$
+\lim  _ {n \rightarrow \infty} L _ {n} ^ {t} f (y) = L ^ {t} (y) \tag {11}
+$$
+
+for a particular class of deterministic samplings, and (ii) apply Proposition 1.
+
+We start by proving that for smooth functions, for any fixed  $t$ , the extended graph Laplacian  $L_{n}^{t}$  converges towards its continuous counterpart  $L^{t}$  as the sampling increases in size.
+
+Proposition 2. For an equal area sampling  $\{x_{i}\in \mathbb{S}^{2}\}_{i = 1}^{n}:A_{i} = A_{j}\forall i,j$  of the sphere it is true that for all  $f:\mathbb{S}^2\to \mathbb{R}$  Lipschitz with respect to the Euclidean distance  $\| \cdot \|$  with Lipschitz constant  $C_f$
+
+$$
+\left| \int_ {\mathbb {S} ^ {2}} f (x) d \mu (x) - \frac {1}{n} \sum_ {i} f (x _ {i}) \right| \leq C _ {f} d ^ {(n)}.
+$$
+
+Furthermore, for all  $y \in \mathbb{S}^2$  the Heat Kernel Graph Laplacian operator  $L_{n}^{t}$  converges pointwise to the functional approximation of the Laplace Beltrami operator  $L^{t}$
+
+$$
+L _ {n} ^ {t} f (y) \xrightarrow {n \to \infty} L ^ {t} f (y).
+$$
+
+Proof. Assuming  $f: \mathbb{S}^2 \to \mathbb{R}$  is Lipschitz with Lipschitz constant  $C_f$ , we have
+
+$$
+\left| \int_ {\sigma_ {i}} f (x) \mathrm {d} \mu (x) - \frac {1}{n} f (x _ {i}) \right| \leq C _ {f} d ^ {(n)} \frac {1}{n},
+$$
+
+where  $\sigma_{i} \subset \mathbb{S}^{2}$  is the subset of the sphere corresponding to the patch around  $x_{i}$ . Remember that the sampling is equal area. Hence, using the triangular inequality and summing all the contributions of the  $n$  patches, we obtain
+
+$$
+\left| \int_ {\mathbb {S} ^ {2}} f (x) \mathrm {d} \mu (x) - \frac {1}{n} \sum_ {i} f (x _ {i}) \right| \leq \sum_ {i} \left| \frac {1}{4 \pi^ {2}} \int_ {\sigma_ {i}} f (x) \mathrm {d} \mu (x) - \frac {1}{n} f (x _ {i}) \right| \leq n C _ {f} d ^ {(n)} \frac {1}{n} = C _ {f} d ^ {(n)}
+$$
+
+A direct application of this result leads to the following pointwise convergences
+
+$$
+\forall f \mathrm {L i p s c h i t z}, \forall y \in \mathbb {S} ^ {2}, \frac {1}{n} \sum_ {i} e ^ {- \frac {\| x _ {i} - y \| ^ {2}}{4 t}} \to \int e ^ {- \frac {\| x - y \| ^ {2}}{4 t}} d \mu (x)
+$$
+
+$$
+\forall f \mathrm {L i p s c h i t z}, \forall y \in \mathbb {S} ^ {2}, \frac {1}{n} \sum_ {i} e ^ {- \frac {| | x _ {i} - y | | ^ {2}}{4 t}} f (x _ {i}) \to \int e ^ {- \frac {\| x - y \| ^ {2}}{4 t}} f (x) d \mu (x)
+$$
+
+Definitions 6 and 8 end the proof.
+
+![](images/d962f7961756b403eaf9e8fcee1cfe5ea25d49e837a8c03e43e017a02b4f2cbb.jpg)
+
+The last proposition shows that for a fixed  $t$ ,  $L_{n}^{t}f(x) \to 1 / 4\pi^{2}L^{t}f(x)$ . To utilize Proposition 1 and complete the proof, we need to find a sequence of  $t_n$  for which this holds as  $t_n \to 0$ . Furthermore, this should hold with a faster decay than  $\frac{1}{4\pi t_n^2}$ .
+
+Proposition 3. Given a sampling regular enough, i.e., for which we assume  $A_{i} = A_{j}\forall i,j$  and  $d^{(n)}\leq \frac{C}{n^{\alpha}}$ ,  $\alpha \in (0,1 / 2]$ , a Lipschitz function  $f$  and a point  $y\in \mathbb{S}^2$  there exists a sequence  $t_n = n^\beta ,\beta < 0$  such that
+
+$$
+\forall f L i p s c h i t z, \forall x \in \mathbb {S} ^ {2} \quad \left| \frac {1}{4 \pi t _ {n} ^ {2}} \left(L _ {n} ^ {t _ {n}} f (x) - L ^ {t _ {n}} f (x)\right) \right| \xrightarrow {n \to \infty} 0.
+$$
+
+Proof. To ease the notation, we define
+
+$$
+K ^ {t} (x, y) := e ^ {- \frac {\| x - y \| ^ {2}}{4 t}} \tag {12}
+$$
+
+$$
+\phi^ {t} (x; y) := e ^ {- \frac {\| x - y \| ^ {2}}{4 t}} (f (y) - f (x)). \tag {13}
+$$
+
+We start with the following inequality
+
+$$
+\begin{array}{l} \left\| L _ {n} ^ {t} f - L ^ {t} f \right\| _ {\infty} = \max  _ {y \in \mathbb {S} ^ {2}} \left| L _ {n} ^ {t} f (y) - L ^ {t} f (y) \right| \\ = \max  _ {y \in \mathbb {S} ^ {2}} \left| \frac {1}{n} \sum_ {i = 1} ^ {n} \phi^ {t} (x _ {i}; y) - \int_ {\mathbb {S} ^ {2}} \phi^ {t} (x; y) d \mu (x) \right| \\ \leq \max _ {y \in \mathbb {S} ^ {2}} \sum_ {i = 1} ^ {n} \left| \frac {1}{n} \phi^ {t} (x _ {i}; y) - \int_ {\sigma_ {i}} \phi^ {t} (x; y) d \mu (x) \right| \\ \leq d ^ {(n)} \max  _ {y \in \mathbb {S} ^ {2}} C _ {\phi_ {y} ^ {t}}, \tag {14} \\ \end{array}
+$$
+
+where  $C_{\phi_y^t}$  is the Lipschitz constant of  $x \to \phi^t(x, y)$  and the last inequality follows from Proposition 2. Using the assumption  $d^{(n)} \leq \frac{C}{\sqrt{n}}$  we find
+
+$$
+\| L _ {n} ^ {t} f - L ^ {t} f \| _ {\infty} \leq \frac {C}{\sqrt {n}} \max  _ {y \in \mathbb {S} ^ {2}} C _ {\phi_ {y} ^ {t}}
+$$
+
+We now find the explicit dependence between  $t$  and  $C_{\phi_y^t}$
+
+$$
+\begin{array}{l} C _ {\phi_ {y} ^ {t}} = \left\| \partial_ {x} \phi^ {t} (\cdot ; y) \right\| _ {\infty} \\ = \left\| \partial_ {x} \left(K ^ {t} (\cdot ; y) f\right) \right\| _ {\infty} \\ = \| \partial_ {x} K ^ {t} (\cdot ; y) f + K ^ {t} (\cdot ; y) \partial_ {x} f \| _ {\infty} \\ \leq \left\| \partial_ {x} K ^ {t} (\cdot ; y) f \right\| _ {\infty} + \left\| K ^ {t} (\cdot ; y) \partial_ {x} f \right\| _ {\infty} \\ \leq \left\| \partial_ {x} K ^ {t} (\cdot ; y) \right\| _ {\infty} \| f \| _ {\infty} + \left\| K ^ {t} (\cdot ; y) \right\| _ {\infty} \| \partial_ {x} f \| _ {\infty} \\ = \| \partial_ {x} K ^ {t} (\cdot ; y) \| _ {\infty} \| f \| _ {\infty} + \| \partial_ {x} f \| _ {\infty} \\ = C _ {K _ {y} ^ {t}} \| f \| _ {\infty} + \| \partial_ {x} f \| _ {\infty} \\ = C _ {K _ {y} ^ {t}} \| f \| _ {\infty} + C _ {f} \\ \end{array}
+$$
+
+where  $C_{K_y^t}$  is the Lipschitz constant of the function  $x \to K^{t}(x;y)$ . We note that this constant does not depend on  $y$ :
+
+$$
+C _ {K _ {y} ^ {t}} = \left\| \partial_ {x} e ^ {- \frac {x ^ {2}}{4 t}} \right\| _ {\infty} = \left\| \frac {x}{2 t} e ^ {- \frac {x ^ {2}}{4 t}} \right\| _ {\infty} = \left. \frac {x}{2 t} e ^ {- \frac {x ^ {2}}{4 t}} \right| _ {x = \sqrt {2 t}} = (2 e t) ^ {- \frac {1}{2}} \propto t ^ {- \frac {1}{2}}.
+$$
+
+Hence we have
+
+$$
+\begin{array}{l} \frac {C}{\sqrt {n}} \max  _ {y \in \mathbb {S} ^ {2}} C _ {\phi_ {y} ^ {t}} \leq \frac {C}{\sqrt {n}} \left((2 e t) ^ {- \frac {1}{2}} \| f \| _ {\infty} + C _ {f}\right) \\ \leq \frac {C \| f \| _ {\infty}}{n ^ {\alpha} (2 e t) ^ {1 / 2}} + \frac {C}{n ^ {\alpha}} C _ {f}. \\ \end{array}
+$$
+
+Inculding this result in (14) and rescaling by  $1 / 4\pi t^2$ , we obtain
+
+$$
+\begin{array}{l} \left\| \frac {1}{4 \pi t ^ {2}} \left(L _ {n} ^ {t} f - L ^ {t} f\right) \right\| _ {\infty} \leq \frac {1}{4 \pi t ^ {2}} \left\| \left(L _ {n} ^ {t} f - L ^ {t} f\right) \right\| _ {\infty} \\ \leq \frac {C}{4 \pi} \left[ \frac {\| f \| _ {\infty}}{\sqrt {2 e}} \frac {1}{n ^ {\alpha} t ^ {5 / 2}} + \frac {C _ {f}}{n ^ {\alpha} t ^ {2}} \right]. \\ \end{array}
+$$
+
+In order for  $\frac{C}{4\pi} \left[ \frac{\|f\|_{\infty}}{\sqrt{2e}} \frac{1}{n^{\alpha} t^{5/2}} + \frac{C_f}{n^{\alpha} t^2} \right] \xrightarrow[n \to 0]{n \to \infty} 0$ , we need  $\left\{ \begin{array}{l} n^{\alpha} t^{5/2} \to \infty \\ n^{\alpha} t^2 \to \infty \end{array} \right.$
+
+It happens if  $\left\{ \begin{array}{ll}t(n) = n^{\beta}, & \beta \in (-\frac{2\alpha}{5},0)\\ t(n) = n^{\beta}, & \beta \in (-\frac{\alpha}{2},0) \end{array} \right.\Rightarrow t(n) = n^{\beta},\quad \beta \in (-\frac{2\alpha}{5},0).$
+
+Indeed, we have
+
+$$
+n ^ {a} l p h a t ^ {5 / 2} = n ^ {5 / 2 \beta + \alpha} \xrightarrow {n \to \infty} \infty \text {s i n c e} \frac {5}{2} \beta + \alpha > 0 \Longleftrightarrow \beta > - \frac {2 \alpha}{5}
+$$
+
+$$
+\text {a n d} n ^ {\alpha} t ^ {2} = n ^ {2 \beta + \alpha} \xrightarrow {n \rightarrow \infty} \infty \text {s i n c e} 2 \beta + \alpha > 0 \Longleftrightarrow \beta > - \frac {\alpha}{2}.
+$$
+
+As a result, for  $t = n^{\beta}$  with  $\beta \in (-\frac{1}{5},0)$  we have  $\left\{ \begin{array}{l}(t_n)\xrightarrow{n\to\infty}0\\ \left\| \frac{1}{4\pi t_n^2} L_n^{t_n}f - \frac{1}{4\pi t_n^2} L^{t_n}f\right\|_{\infty}\xrightarrow{n\to\infty}0, \end{array} \right.$
+
+which concludes the proof.
+
+Theorem 3.1, is then an immediate consequence of Proposition 3 and 1.
+
+Proof of Theorem 3.1. Thanks to Proposition 3 and Proposition 1 we conclude that  $\forall y\in \mathbb{S}^2$
+
+$$
+\lim _ {n \to \infty} \frac {1}{4 \pi t _ {n} ^ {2}} L _ {n} ^ {t _ {n}} f (y) = \lim _ {n \to \infty} \frac {1}{4 \pi t _ {n} ^ {2}} L ^ {t _ {n}} f (y) = \frac {1}{| \mathbb {S} ^ {2} |} \Delta_ {\mathbb {S} ^ {2}} f (y)
+$$
+
+![](images/58d8806421fdd3f28dcf82f717ea4ec507a47c412955b71ab39354de958f0861.jpg)
+
+In (Belkin & Niyogi, 2008), the sampling is drawn from a uniform random distribution on the sphere, and their proof heavily relies on the uniformity properties of the distribution from which the sampling is drawn. In our case the sampling is deterministic, and this is indeed a problem that we need to overcome by imposing the regularity conditions above.
+
+<table><tr><td rowspan="2"></td><td colspan="4">micro (label average)</td><td colspan="4">macro (instance average)</td></tr><tr><td>P@N</td><td>R@N</td><td>F1@N</td><td>mAP</td><td>P@N</td><td>R@N</td><td>F1@N</td><td>mAP</td></tr><tr><td>Cohen et al. (2018) (b = 128)</td><td>0.701</td><td>0.711</td><td>0.699</td><td>0.676</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Cohen et al. (2018) (simplified, b = 64)</td><td>0.704</td><td>0.701</td><td>0.696</td><td>0.665</td><td>0.430</td><td>0.480</td><td>0.429</td><td>0.385</td></tr><tr><td>Esteves et al. (2018) (b = 64)</td><td>0.717</td><td>0.737</td><td>-</td><td>0.685</td><td>0.450</td><td>0.550</td><td>-</td><td>0.444</td></tr><tr><td>DeepSphere (equiangular b = 64)</td><td>0.709</td><td>0.700</td><td>0.698</td><td>0.665</td><td>0.439</td><td>0.489</td><td>0.439</td><td>0.403</td></tr><tr><td>DeepSphere (HEALPix Nside = 32)</td><td>0.725</td><td>0.717</td><td>0.715</td><td>0.686</td><td>0.475</td><td>0.508</td><td>0.468</td><td>0.428</td></tr></table>
+
+Table 5: Official metrics from the SHREC'17 object retrieval competition.
+
+To conclude, we see that the result obtained is of similar form than the result obtained in (Belkin & Niyogi, 2008). Given the kernel density  $t(n) = n^{\beta}$ , Belkin & Niyogi (2008) proved convergence in the random case for  $\beta \in (-\frac{1}{4}, 0)$  and we proved convergence in the deterministic case for  $\beta \in (-\frac{2\alpha}{5}, 0)$ , where  $\alpha \in (0, 1/2]$  (for the spherical manifold).
+
+# B PROOF OF THEOREM 3.2
+
+Proof. Fix  $x \in \mathbb{S}^2$ . Since any rotation  $R(g)$  is an isometry, and the Laplacian  $\Delta$  commutes with all isometries of a Riemannian manifold, and defining  $R(g)f \eqqcolon f'$  for ease of notation, we can write that
+
+$$
+\begin{array}{l} \left| R (g) \hat {L} _ {n} ^ {t _ {n}} f (x) - \hat {L} _ {n} ^ {t _ {n}} R (g) f (x) \right| \leq \left| R (g) \hat {L} _ {n} ^ {t _ {n}} f (x) - R (g) \Delta_ {\mathbb {S} ^ {2}} f (x) \right| + \left| R (g) \Delta_ {\mathbb {S} ^ {2}} f (x) - \hat {L} _ {n} ^ {t _ {n}} R (g) f (x) \right| = \\ = \left| R (g) \left(\hat {L} _ {n} ^ {t _ {n}} f - \Delta_ {\mathbb {S} ^ {2}} f\right) (x) \right| + \left| \Delta_ {\mathbb {S} ^ {2}} f ^ {\prime} (x) - \hat {L} _ {n} ^ {t _ {n}} f ^ {\prime} (x) \right| \leq \\ \leq \left| (\hat {L} _ {n} ^ {t _ {n}} f - \Delta_ {\mathbb {S} ^ {2}} f) (g ^ {- 1} (x)) \right| + \left| \Delta_ {\mathbb {S} ^ {2}} f ^ {\prime} (x) - \hat {L} _ {n} ^ {t _ {n}} f ^ {\prime} (x) \right| \\ \end{array}
+$$
+
+Since  $g^{-1}(x) \in \mathbb{S}^2$  and  $f'$  still satisfies hypothesis, we can apply theorem 3.1 to say that
+
+$$
+\left| \left(\hat {L} _ {n} ^ {t _ {n}} f - \Delta_ {\mathbb {S} ^ {2}} f\right) \left(g ^ {- 1} (x)\right) \right| \xrightarrow {n \to \infty} 0
+$$
+
+$$
+\left| \Delta_ {\mathbb {S} ^ {2}} f ^ {\prime} (x) - \hat {L} _ {n} ^ {t _ {n}} f ^ {\prime} (x) \right| \xrightarrow {n \to \infty} 0
+$$
+
+to conclude that
+
+$$
+\forall x \in \mathbb {S} ^ {2} \quad \left| R (g) \hat {L} _ {n} ^ {t _ {n}} f (x) - \hat {L} _ {n} ^ {t _ {n}} R (g) f (x) \right| \xrightarrow {n \to \infty} 0
+$$
+
+![](images/15d5d9b96e706e58e1ef26302d483f75cb2a8792b69eb4f212ddead5538de890.jpg)
+
+# C EXPERIMENTAL DETAILS
+
+# C.1 3D OBJECTS RECOGNITION
+
+Table 5 shows the results obtained from the SHREC'17 competition's official evaluation script.
+
+$$
+\begin{array}{l} \left[ G C _ {1 6} + B N + R e L U \right] _ {n s i d e 3 2} + \text {P o o l} + \left[ G C _ {3 2} + B N + R e L U \right] _ {n s i d e 1 6} + \text {P o o l} \\ + \left[ G C _ {6 4} + B N + R e L U \right] _ {n s i d e 8} + \text {P o o l} + \left[ G C _ {1 2 8} + B N + R e L U \right] _ {n s i d e 4} \\ + \text {P o o l} + \left[ G C _ {2 5 6} + B N + R e L U \right] _ {n s i d e 2} + \text {P o o l} + G A P + F C N + \text {s o f t m a x} \\ \end{array}
+$$
+
+# C.2 COSMOLOGICAL MODEL CLASSIFICATION
+
+$$
+\begin{array}{l} \left[ G C _ {1 6} + B N + R e L U \right] _ {n s i d e 1 0 2 4} + \text {P o o l} + \left[ G C _ {3 2} + B N + R e L U \right] _ {n s i d e 5 1 2} \\ + \text {P o o l} + \left[ G C _ {6 4} + B N + R e L U \right] _ {n s i d e 2 5 6} + \text {P o o l} \tag {16} \\ + \left[ G C _ {6 4} + B N + R e L U \right] _ {n s i d e 1 2 8} + \text {P o o l} + \left[ G C _ {6 4} + B N + R e L U \right] _ {n s i d e 6 4} \\ + \text {P o o l} + \left[ G C _ {2} \right] _ {n s i d e 3 2} + G A P + \text {s o f t m a x} \\ \end{array}
+$$
+
+<table><tr><td></td><td>TC</td><td>AR</td><td>BG</td><td>mean</td></tr><tr><td>Mudigonda et al. (2017)</td><td>74</td><td>65</td><td>97</td><td>78.67</td></tr><tr><td>Jiang et al. (2019) (paper)</td><td>94</td><td>93</td><td>97</td><td>94.67</td></tr><tr><td>Jiang et al. (2019) (rerun)</td><td>93.9</td><td>95.7</td><td>95.2</td><td>94.95</td></tr><tr><td>Cohen et al. (2019) (S2R)</td><td>97.8</td><td>97.3</td><td>97.3</td><td>97.5</td></tr><tr><td>Cohen et al. (2019) (R2R)</td><td>97.9</td><td>97.8</td><td>97.4</td><td>97.7</td></tr><tr><td>DS (Jiang architecture, weighted loss)</td><td>97.1</td><td>97.6</td><td>96.5</td><td>97.1</td></tr><tr><td>DS (weighted loss)</td><td>97.4 ± 1.1</td><td>97.7 ± 0.7</td><td>98.2 ± 0.5</td><td>97.8 ± 0.3</td></tr><tr><td>DS (wider architecture, weighted loss)</td><td>91.5</td><td>93.4</td><td>99.0</td><td>94.6</td></tr><tr><td>DS (Jiang architecture, non-weighted loss)</td><td>33.6</td><td>93.6</td><td>99.3</td><td>75.5</td></tr><tr><td>DS (non-weighted loss)</td><td>69.2 ± 3.7</td><td>94.5 ± 2.9</td><td>99.7 ± 0.1</td><td>87.8 ± 0.5</td></tr><tr><td>DS (wider architecture, non-weighted loss)</td><td>73.4</td><td>92.7</td><td>99.8</td><td>88.7</td></tr></table>
+
+Table 6: Results on climate event segmentation: accuracy. Tropical cyclones (TC) and atmospheric rivers (AR) are the two positive classes, against the background (BG). Mudigonda et al. (2017) is not directly comparable as they don't use the same input feature maps. Note that a non-weighted cross-entropy loss is not optimal for the accuracy metric.  
+
+<table><tr><td></td><td>TC</td><td>AR</td><td>mean</td></tr><tr><td>Jiang et al. (2019) (rerun)</td><td>11.08</td><td>65.21</td><td>38.41</td></tr><tr><td>Cohen et al. (2019) (S2R)</td><td>-</td><td>-</td><td>68.6</td></tr><tr><td>Cohen et al. (2019) (R2R)</td><td>-</td><td>-</td><td>75.9</td></tr><tr><td>DS (Jiang architecture, non-weighted loss)</td><td>46.2</td><td>93.9</td><td>70.0</td></tr><tr><td>DS (non-weighted loss)</td><td>80.86 ± 2.42</td><td>97.45 ± 0.38</td><td>89.16 ± 1.37</td></tr><tr><td>DS (wider architecture, non-weighted loss)</td><td>84.71</td><td>98.05</td><td>91.38</td></tr><tr><td>DS (Jiang architecture, weighted loss)</td><td>49.7</td><td>89.2</td><td>69.5</td></tr><tr><td>DS (weighted loss)</td><td>58.88 ± 3.17</td><td>95.41 ± 1.51</td><td>77.15 ± 1.94</td></tr><tr><td>DS (wider architecture, weighted loss)</td><td>52.80</td><td>94.78</td><td>73.79</td></tr></table>
+
+Table 7: Results on climate event segmentation: average precision. Tropical cyclones (TC) and atmospheric rivers (AR) are the two positive classes. Note that a weighted cross-entropy loss is not optimal for the average precision metric.
+
+# C.3 CLIMATE EVENT SEGMENTATION
+
+Table 6, 7, and 8 show the accuracy, mAP, and efficiency of all the NNs we ran.
+
+The experiment with the model from Jiang et al. (2019) was rerun in order to obtain the AP metrics, but with a batch size of 64 instead of 256 due to GPU memory limit.
+
+Several experiments were run with different architectures for DeepSphere (DS). Jiang architecture use a similar one as Jiang et al. (2019), with only the convolutional operators replaced. DeepSphere only is the original architecture giving the best results, deeper and with four times more feature maps than Jiang architecture. And the wider architecture is the same as the previous one with two times the number of feature maps.
+
+Regarding the weighted loss, the weights are chosen with scikit-learn function compute_class_weight on the training set.
+
+<table><tr><td></td><td>size</td><td colspan="2">speed</td></tr><tr><td></td><td>params</td><td>inference</td><td>training</td></tr><tr><td>Jiang et al. (2019)</td><td>330 k</td><td>10 ms</td><td>10 h</td></tr><tr><td>DeepSphere (Jiang architecture)</td><td>590 k</td><td>5 ms</td><td>3 h</td></tr><tr><td>DeepSphere</td><td>13 M</td><td>33 ms</td><td>13 h</td></tr><tr><td>DeepSphere (wider architecture)</td><td>52 M</td><td>50 ms</td><td>20 h</td></tr></table>
+
+Table 8: Results on climate event segmentation: size and speed.
+
+DeepSphere with Jiang architecture encoder:
+
+$$
+\begin{array}{l} \left[ G C _ {8} + B N + R e L U \right] _ {L 5} + \text {P o o l} + \left[ G C _ {1 6} + B N + R e L U \right] _ {L 4} + \text {P o o l} \tag {17} \\ + \left[ G C _ {3 2} + B N + R e L U \right] _ {L 3} + \text {P o o l} + \left[ G C _ {6 4} + B N + R e L U \right] _ {L 2} + \text {P o o l} \\ + \left[ G C _ {1 2 8} + B N + R e L U \right] _ {L 1} + \text {P o o l} + \left[ G C _ {1 2 8} + B N + R e L U \right] _ {L 0} \\ \end{array}
+$$
+
+decoder:
+
+$$
+\begin{array}{l} \operatorname {U n p o o l} + \left[ G C _ {1 2 8} + B N + R e L U \right] _ {L 1} + \operatorname {c o n c a t} + \left[ G C _ {1 2 8} + B N + R e L U \right] _ {L 1} \\ + \text {U n p o o l} + \left[ G C _ {6 4} + B N + R e L U \right] _ {L 2} + \text {c o n c a t} \\ + \left[ G C _ {6 4} + B N + R e L U \right] _ {L 2} + \text {U n p o o l} + \left[ G C _ {3 2} + B N + R e L U \right] _ {L 3} \tag {18} \\ + \text {c o n c a t} + \left[ G C _ {3 2} + B N + R e L U \right] _ {L 3} + \text {U n p o o l} \\ + \left[ G C _ {1 6} + B N + R e L U \right] _ {L 4} + \text {c o n c a t} + \left[ G C _ {1 6} + B N + R e L U \right] _ {L 4} + \text {U n p o o l} \\ + \left[ G C _ {8} + B N + R e L U \right] _ {L 5} + \text {c o n c a t} + \left[ G C _ {8} + B N + R e L U \right] _ {L 5} + \left[ G C _ {3} \right] _ {L 5} \\ \end{array}
+$$
+
+Concat is the operation that concatenate the results of the corresponding encoder layer.
+
+Original DeepSphere architecture with encoder decoder encoder:
+
+$$
+\begin{array}{l} \left[ G C _ {3 2} + B N + R e L U \right] _ {L 5} + \left[ G C _ {6 4} + B N + R e L U \right] _ {L 5} \\ + \text {P o o l} + \left[ G C _ {1 2 8} + B N + R e L U \right] _ {L 4} + \text {P o o l} \tag {19} \\ + \left[ G C _ {2 5 6} + B N + R e L U \right] _ {L 3} + \text {P o o l} + \left[ G C _ {5 1 2} + B N + R e L U \right] _ {L 2} \\ + \operatorname {P o o l} + \left[ G C _ {5 1 2} + B N + R e L U \right] _ {L 1} + \operatorname {P o o l} + \left[ G C _ {5 1 2} \right] _ {L 0} \\ \end{array}
+$$
+
+decoder:
+
+$$
+\begin{array}{l} \operatorname {U n p o o l} + \left[ G C _ {5 1 2} + B N + R e L U \right] _ {L 1} + \operatorname {c o n c a t} + \left[ G C _ {5 1 2} + B N + R e L U \right] _ {L 1} \\ + \text {U n p o o l} + \left[ G C _ {2 5 6} + B N + R e L U \right] _ {L 2} + \text {c o n c a t} \\ + \left[ G C _ {2 5 6} + B N + R e L U \right] _ {L 2} + \text {U n p o o l} + \left[ G C _ {1 2 8} + B N + R e L U \right] _ {L 3} \tag {20} \\ + \text {c o n c a t} + \left[ G C _ {1 2 8} + B N + R e L U \right] _ {L 3} + \text {U n p o o l} \\ + \left[ G C _ {6 4} + B N + R e L U \right] _ {L 4} + \text {c o n c a t} + \left[ G C _ {6 4} + B N + R e L U \right] _ {L 4} \\ + \text {U n p o o l} + \left[ G C _ {3 2} + B N + R e L U \right] _ {L 5} + \left[ G C _ {3} \right] _ {L 5} \\ \end{array}
+$$
+
+# C.4 UNEVEN SAMPLING
+
+Architecture for dense regression:
+
+$$
+\left[ G C _ {5 0} + B N + R e L U \right] + \left[ G C _ {1 0 0} + B N + R e L U \right] + \left[ G C _ {1 0 0} + B N + R e L U \right] + \left[ G C _ {1} \right] \tag {21}
+$$
+
+Architecture for global regression:
+
+$$
+\begin{array}{l} \left[ G C _ {5 0} + B N + R e L U \right] + \left[ G C _ {1 0 0} + B N + R e L U \right] \tag {22} \\ + \left[ G C _ {1 0 0} + B N + R e L U \right] + G A P + F C N \\ \end{array}
+$$
\ No newline at end of file
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+# DEFENDING AGAINST PHYSICALLY REALIZABLE ATTACKS ON IMAGE CLASSIFICATION
+
+Tong Wu, Liang Tong & Yevgeniy Vorobeychik
+
+Department of Computer Science and Engineering
+
+Washington University in St. Louis
+
+{tongwu, liangtong, yvorobeychik}@wustl.edu
+
+# ABSTRACT
+
+We study the problem of defending deep neural network approaches for image classification from physically realizable attacks. First, we demonstrate that the two most scalable and effective methods for learning robust models, adversarial training with PGD attacks and randomized smoothing, exhibit limited effectiveness against three of the highest profile physical attacks. Next, we propose a new abstract adversarial model, rectangular occlusion attacks, in which an adversary places a small adversarily crafted rectangle in an image, and develop two approaches for efficiently computing the resulting adversarial examples. Finally, we demonstrate that adversarial training using our new attack yields image classification models that exhibit high robustness against the physically realizable attacks we study, offering the first effective generic defense against such attacks.
+
+# 1 INTRODUCTION
+
+State-of-the-art effectiveness of deep neural networks has made it the technique of choice in a variety of fields, including computer vision (He et al., 2016), natural language processing (Sutskever et al., 2014), and speech recognition (Hinton et al., 2012). However, there have been a myriad of demonstrations showing that deep neural networks can be easily fooled by carefully perturbing pixels in an image through what have become known as adversarial example attacks (Szegedy et al., 2014; Goodfellow et al., 2015; Carlini & Wagner, 2017b; Vorobeychik & Kantarcioglu, 2018). In response, a large literature has emerged on defending deep neural networks against adversarial examples, typically either proposing techniques for learning more robust neural network models (Wong & Kolter, 2018; Wong et al., 2018; Raghunathan et al., 2018b; Cohen et al., 2019; Madry et al., 2018), or by detecting adversarial inputs (Metzen et al., 2017; Xu et al., 2018).
+
+Particularly concerning, however, have been a number of demonstrations that implement adversarial perturbations directly in physical objects that are subsequently captured by a camera, and then fed through the deep neural network classifier (Boloor et al., 2019; Eykholt et al., 2018; Athalye et al., 2018b; Brown et al., 2018; Bhattad et al., 2020). Among the most significant of such physical attacks on deep neural networks are three that we specifically consider here: 1) the attack which fools face recognition by using adversarially designed eyeglass frames (Sharif et al., 2016), 2) the attack which fools stop sign classification by adding adversarially crafted stickers (Eykholt et al., 2018), and 3) the universal adversarial patch attack, which causes targeted misclassification of any object with the adversarially designed sticker (patch) (Brown et al., 2018). Oddly, while considerable attention has been devoted to defending against adversarial perturbation attacks in the digital space, there are no effective methods specifically to defend against such physical attacks.
+
+Our first contribution is an empirical evaluation of the effectiveness of conventional approaches to robust ML against two physically realizable attacks: the eyeglass frame attack on face recognition (Sharif et al., 2016) and the sticker attack on stop signs (Eykholt et al., 2018). Specifically, we study the performance on adversarial training and randomized smoothing against these attacks, and show that both have limited effectiveness in this context (quite ineffective in some settings, and somewhat more effective, but still not highly robust, in others), despite showing moderate effectiveness against  $l_{\infty}$  and  $l_{2}$  attacks, respectively.
+
+Our second contribution is a novel abstract attack model which more directly captures the nature of common physically realizable attacks than the conventional  $l_{p}$ -based models. Specifically, we consider a simple class of rectangular occlusion attacks in which the attacker places a rectangular sticker onto an image, with both the location and the content of the sticker adversarially chosen. We develop several algorithms for computing such adversarial occlusions, and use adversarial training to obtain neural network models that are robust to these. We then experimentally demonstrate that our proposed approach is significantly more robust against physical attacks on deep neural networks than adversarial training and randomized smoothing methods that leverage  $l_{p}$ -based attack models.
+
+Related Work While many approaches for defending deep learning in vision applications have been proposed, robust learning methods have been particularly promising, since alternatives are often defeated soon after being proposed (Madry et al., 2018; Raghunathan et al., 2018a; Wong & Kolter, 2018; Vorobeychik & Kantarcioglu, 2018). The standard solution approach for this problem is an adaptation of Stochastic Gradient Descent (SGD) where gradients are either with respect to the loss at the optimal adversarial perturbation for each  $i$  (or approximation thereof, such as using heuristic local search (Goodfellow et al., 2015; Madry et al., 2018) or a convex over-approximation (Raghunathan et al., 2018b; Wang et al., 2018)), or with respect to the dual of the convex relaxation of the attacker maximization problem (Raghunathan et al., 2018a; Wong & Kolter, 2018; Wong et al., 2018). Despite these advances, adversarial training a la Madry et al. (2018) remains the most practically effective method for hardening neural networks against adversarial examples with  $l_{\infty}$ -norm perturbation constraints. Recently, randomized smoothing emerged as another class of techniques for obtaining robustness (Lecuyer et al., 2019; Cohen et al., 2019), with the strongest results in the context of  $l_{2}$ -norm attacks. In addition to training neural networks that are robust by construction, a number of methods study the problem of detecting adversarial examples (Metzen et al., 2017; Xu et al., 2018), with mixed results (Carlini & Wagner, 2017a). Of particular interest is recent work on detecting physical adversarial examples (Chou et al., 2018). However, detection is inherently weaker than robustness, which is our goal, as even perfect detection does not resolve the question of how to make decisions on adversarial examples. Finally, our work is in the spirit of other recent efforts that characterize robustness of neural networks to physically realistic perturbations, such as translations, rotations, blurring, and contrast (Engstrom et al., 2019; Hendrycks & Dietterich, 2019).
+
+# 2 BACKGROUND
+
+# 2.1 ADVERSARIAL EXAMPLES IN THE DIGITAL AND PHYSICAL WORLD
+
+Adversarial examples involve modifications of input images that are either invisible to humans, or unsuspected, and that cause systematic misclassification by state-of-the-art neural networks (Szegedy et al., 2014; Goodfellow et al., 2015; Vorobeychik & Kantarcioglu, 2018). Commonly, approaches for generating adversarial examples aim to solve an optimization problem of the following form:
+
+$$
+\underset {\delta} {\arg \max } L (f (\boldsymbol {x} + \boldsymbol {\delta}; \boldsymbol {\theta}), y) \quad s. t. \| \boldsymbol {\delta} \| _ {p} \leq \epsilon , \tag {1}
+$$
+
+where  $\pmb{x}$  is the original input image,  $\delta$  is the adversarial perturbation,  $L(\cdot)$  is the adversary's utility function (for example, the adversary may wish to maximize the cross-entropy loss), and  $\| \cdot \|_p$  is some  $l_p$  norm. While a host of such digital attacks have been proposed, two have come to be viewed as state of the art: the attack developed by Carlini & Wagner (2017b), and the projected gradient descent attack (PGD) by Madry et al. (2018).
+
+While most of the work to date has been on attacks which modify the digital image directly, we focus on a class of physical attacks which entail modifying the actual object being photographed in order to fool the neural network that subsequently takes its digital representation as input. The attacks we will focus on will have three characteristics:
+
+1. The attack can be implemented in the physical space (e.g., modifying the stop sign);  
+2. the attack has low suspiciousness; this is operationalized by modifying only a small part of the object, with the modification similar to common "noise" that obtains in the real world; for example, stickers on a stop sign would appear to most people as vandalism, but covering the stop sign with a printed poster would look highly suspicious; and  
+3. the attack causes misclassification by state-of-the-art deep neural network.
+
+Since our ultimate purpose is defense, we will not concern ourselves with the issue of actually implementing the physical attacks. Instead, we will consider the digital representation of these attacks, ignoring other important issues, such as robustness to many viewpoints and printability. For example, in the case where the attack involves posting stickers on a stop sign, we will only be concerned with simulating such stickers on digital images of stop signs. For this reason, we refer to such attacks physically realizable attacks, to allude to the fact that it is possible to realize them in practice. It is evident that physically realizable attacks represent a somewhat stronger adversarial model than their actual implementation in the physical space. Henceforth, for simplicity, we will use the terms physical attacks and physically realizable attacks interchangeably.
+
+![](images/9bd182a8efa77d3e939211b86decb413ada9ca71b9022c93245bc5d86628cdcc.jpg)
+
+![](images/f9b75b4b7514711f56f7c8d476dfef7d3aede6f1c4c525275b208d90b97de584.jpg)  
+(a)  
+Figure 1: (a) An example of the eyeglass frame attack. Left: original face input image. Middle: modified input image (adversarial eyeglasses superimposed on the face). Right: an image of the predicted individual with the adversarial input in the middle image. (b) An example of the stop sign attack. Left: original stop sign input image. Middle: adversarial mask. Right: stop sign image with adversarial stickers, classified as a speed limit sign.
+
+![](images/29f7ed4bc5ca5bea12f1bd65f5e27467222daa0bc26d2a67c83f178fe9e2fcdb.jpg)
+
+![](images/cdd787d101d9db2f1d8c39080c91c409a6efba97a4326df285b0654a6aad8029.jpg)
+
+![](images/6498653bc22ef5db3757cfcdb00820e032f1572a2f67391543beedd3463521b2.jpg)  
+(b)
+
+![](images/3895b2878ba11c5578f1b7c4e83db9ebbd24a93c8bf1985364e39f9e5fb1d6c3.jpg)
+
+We consider three physically realizable attacks. The first is the attack on face recognition by Sharif et al. (2016), in which the attacker adds adversarial noise inside printed eyeglass frames that can subsequently be put on to fool the deep neural network (Figure 1a). The second attack posts adversarially crafted stickers on a stop sign to cause it to be misclassified as another road sign, such as the speed limit sign (Figure 1b) (Eykholt et al., 2018). The third, adversarial patch, attack designs a patch (a sticker) with adversarial noise that can be placed onto an arbitrary object, causing that object to be misclassified by a deep neural network (Brown et al., 2018).
+
+# 2.2 ADVERSARIALLY ROBUST DEEP LEARNING
+
+While numerous approaches have been proposed for making deep learning robust, many are heuristic and have soon after been defeated by more sophisticated attacks (Carlini & Wagner, 2017b; He et al., 2017; Carlini & Wagner, 2017a; Athalye et al., 2018a). Consequently, we focus on principled approaches for defense that have not been broken. These fall broadly into two categories: robust learning and randomized smoothing. We focus on a state-of-the-art representative from each class.
+
+Robust Learning The goal of robust learning is to minimize a robust loss, defined as follows:
+
+$$
+\boldsymbol {\theta} ^ {*} = \arg \min  _ {\boldsymbol {\theta}} \mathbb {E} _ {(\boldsymbol {x}, y) \sim \mathcal {D}} \left[ \max  _ {\| \boldsymbol {\delta} \| _ {p} \leq \epsilon} L (f (\boldsymbol {x} + \boldsymbol {\delta}; \boldsymbol {\theta}), y) \right], \tag {2}
+$$
+
+where  $\mathcal{D}$  denotes the training data set. In itself this is a highly intractable problem. Several techniques have been developed to obtain approximate solutions. Among the most effective in practice is the adversarial training approach by Madry et al. (2018), who use the PGD attack as an approximation to the inner optimization problem, and then take gradient descent steps with respect to the associated adversarial inputs. In addition, we consider a modified version of this approach termed curriculum adversarial training (Cai et al., 2018). Our implementation of this approach proceeds as follows: first, apply adversarial training for a small  $\epsilon$ , then increase  $\epsilon$  and repeat adversarial training, and so on, increasing  $\epsilon$  until we reach the desired level of adversarial noise we wish to be robust to.
+
+Randomized Smoothing The second class of techniques we consider works by adding noise to inputs at both training and prediction time. The key idea is to construct a smoothed classifier  $g(\cdot)$  from a base classifier  $f(\cdot)$  by perturbing the input  $x$  with isotropic Gaussian noise with variance  $\sigma$ . The prediction is then made by choosing a class with the highest probability measure with respect to the induced distribution of  $f(\cdot)$  decisions:
+
+$$
+g (\boldsymbol {x}) = \arg \max  _ {c} P (f (\boldsymbol {x} + \boldsymbol {\sigma}) = c), \quad \boldsymbol {\sigma} \sim \mathcal {N} (0, \sigma^ {2} I). \tag {3}
+$$
+
+To achieve provably robust classification in this manner one typically trains the classifier  $f(\cdot)$  by adding Gaussian noise to inputs at training time (Lecuyer et al., 2019; Cohen et al., 2019).
+
+# 3 ROBUSTNESS OF CONVENTIONAL ROBUST ML METHODS AGAINST PHYSICAL ATTACKS
+
+Most of the approaches for endowing deep learning with adversarial robustness focus on adversarial models in which the attacker introduces  $l_{p}$ -bounded adversarial perturbations over the entire input. Earlier we described two representative approaches in this vein: adversarial training, commonly focused on robustness against  $l_{\infty}$  attacks, and randomized smoothing, which is most effective against  $l_{2}$  attacks (although certification bounds can be extended to other  $l_{p}$  norms as well). We call these methods conventional robust ML.
+
+In this section, we ask the following question:
+
+Are conventional robust ML methods robust against physically realizable attacks?
+
+This is similar to the question was asked in the context of malware classifier evasion by Tong et al. (2019), who found that  $l_{p}$ -based robust ML methods can indeed be successful in achieving robustness against realizable evasion attacks. Ours is the first investigation of this issue in computer vision applications and for deep neural networks, where attacks involve adversarial masking of objects.
+
+We study this issue experimentally by considering two state-of-the-art approaches for robust ML: adversarial training a-la-Madry et al. (2018), along with its curriculum learning variation (Cai et al., 2018), and randomized smoothing, using the implementation by Cohen et al. (2019). These approaches are applied to defend against two physically realizable attacks described in Section 2.1: an attack on face recognition which adds adversarial eyeglass frames to faces (Sharif et al., 2016), and an attack on stop sign classification which adds adversarial stickers to a stop sign to cause misclassification (Eykholt et al., 2018).
+
+We consider several variations of adversarial training, as a function of the  $l_{\infty}$  bound,  $\epsilon$ , imposed on the adversary. Just as Madry et al. (2018), adversarial instances in adversarial training were generated using PGD. We consider attacks with  $\epsilon \in \{4, 8\}$  (adversarial training failed to make progress when we used  $\epsilon = 16$ ). For curriculum adversarial training, we first performed adversarial training with  $\epsilon = 4$ , then doubled  $\epsilon$  to 8 and repeated adversarial training with the model robust to  $\epsilon = 4$ , then doubled  $\epsilon$  again, and so on. In the end, we learned models for  $\epsilon \in \{4, 8, 16, 32\}$ . For all versions of adversarial training, we consider 7 and 50 iterations of the PGD attack. We used the learning rate of  $\epsilon / 4$  for the former and 1 for the latter. In all cases, pixels are in  $0 \sim 255$  range and retraining was performed for 30 epochs using the ADAM optimizer.
+
+For randomized smoothing, we consider noise levels  $\sigma \in \{0.25, 0.5, 1\}$  as in Cohen et al. (2019), and take 1,000 Monte Carlo samples at test time.
+
+# 3.1 ADVERSARIAL EYEGLASSES IN FACE RECOGNITION
+
+We applied white-box dodging (untargeted) attacks on the face recognition systems (FRS) from Sharif et al. (2016). We used both the VGGFace data and transferred VGGFace CNN model for the face recognition task, subselecting 10 individuals, with 300-500 face images for each. Further details about the dataset, CNN architecture, and training procedure are in Appendix A. For the attack, we used identical frames as in Sharif et al. (2016) occupying  $6.5\%$  of the pixels. Just as Sharif et al. (2016), we compute attacks (that is, adversarial perturbations inside the eyeglass frame area) by using the learning rate 20 as well as momentum value 0.4, and vary the number of attack iterations between 0 (no attack) and 300.
+
+Figure 2 presents the results of classifiers obtained from adversarial training (left) as well as curriculum adversarial training (middle), in terms of accuracy (after the attack) as a function of the number of iterations of the Sharif et al. (2016) eyeglass frame attack. First, it is clear that none of the variations of adversarial training are particularly effective once the number of physical attack iterations is
+
+![](images/63cd64ed7b37ea4118cf947547c826c9866e5bd58e7708a494bd63cb3acdd8f9.jpg)  
+Figure 2: Performance of adversarial training (left), curriculum adversarial training (with 7 PGD iterations) (middle), and randomized smoothing (right) against the eyeglass frame attack.
+
+![](images/f87a17085609045eccb37486babdb475688a62ffc6716d0349f0eb7aeda8b59a.jpg)
+
+![](images/75b58e708670cb9951274e61ae7827af0ed403c1ac7ffa734cf5cc820b46662e.jpg)
+
+above 20. The best performance in terms of adversarial robustness is achieved by adversarial training with  $\epsilon = 8$ , for approaches using either 7 or 50 PGD iterations (the difference between these appears negligible). However, non-adversarial accuracy for these models is below  $70\%$ , a  $\sim 20\%$  drop in accuracy compared to the original model. Moreover, adversarial accuracy is under  $40\%$  for sufficiently strong physical attacks. Curriculum adversarial training generally achieves significantly higher non-adversarial accuracy, but is far less robust, even when trained with PGD attacks that use  $\epsilon = 32$ .
+
+Figure 2 (right) shows the performance of randomized smoothing when faced with the eyeglass frames attack. It is readily apparent that randomized smoothing is ineffective at deflecting this physical attack: even as we vary the amount of noise we add, accuracy after attacks is below  $20\%$  even for relatively weak attacks, and often drops to nearly 0 for sufficiently strong attacks.
+
+# 3.2 ADVERSARIAL STICKERS ON STOP SIGNS
+
+Following Eykholt et al. (2018), we use the LISA traffic sign dataset for our experiments, and 40 stop signs from this dataset as our test data and perform untargeted attacks (this is in contrast to the original work, which is focused on targeted attacks). For the detailed description of the data and the CNN used for traffic sign prediction, see Appendix A. We apply the same settings as in the original attacks and use ADAM optimizer with the same parameters. Since we observed few differences in performance between running PGD for 7 vs. 50 iterations, adversarial training methods in this section all use 7 iterations of PGD.
+
+![](images/4eb14803c29093848a63b79b1a112bdc5cd6cae3aaa6617272df7378e08d12d8.jpg)  
+Figure 3: Performance of adversarial training (left), curriculum adversarial training (with 7 PGD iterations) (middle), and randomized smoothing (right) against the stop sign attack.
+
+![](images/9f0efb7a747c1297369d5de88cb54009ffd9a80ea2395df3c191a25ce213a29f.jpg)
+
+![](images/de328b8d4a71680a5eb06b8e38a7a1e8ca4a8a1b41c0334f12fab6e417238f40.jpg)
+
+Again, we begin by considering adversarial training (Figure 3, left and middle). In this case, both the original and curriculum versions of adversarial training with PGD are ineffective when  $\epsilon = 32$  (error rates on clean data are above  $90\%$ ); these are consequently omitted from the plots. Curriculum adversarial training with  $\epsilon = 16$  has the best performance on adversarial data, and works well on clean data. Surprisingly, most variants of adversarial training perform at best marginally better than the original model against the stop sign attack. Even the best variant has relatively poor performance, with robust accuracy under  $50\%$  for stronger attacks.
+
+Figure 3 (right) presents the results for randomized smoothing. In this set of experiments, we found that randomized smoothing performs inconsistently. To address this, we used 5 random seeds to repeat the experiments, and use the resulting mean values in the final results. Here, the best variant
+
+uses  $\sigma = 0.25$ , and, unlike experiments with the eyeglass frame attack, significantly outperforms adversarial training, reaching accuracy slightly above  $60\%$  even for the stronger attacks. Nevertheless, even randomized smoothing results in significant degradation of effectiveness on adversarial instances (nearly  $40\%$ , compared to clean data).
+
+# 3.3 DISCUSSION
+
+There are two possible reasons why conventional robust ML perform poorly against physical attacks: 1) adversarial models involving  $l_{p}$ -bounded perturbations are too hard to enable effective robust learning, and 2) the conventional attack model is too much of a mismatch for realistic physical attacks. In Appendix B, we present evidence supporting the latter. Specifically, we find that conventional robust ML models exhibit much higher robustness when faced with the  $l_{p}$ -bounded attacks they are trained to be robust to.
+
+# 4 PROPOSED APPROACH:DEFENSE AGAINST OCCLUSION ATTACKS (DOA)
+
+As we observed in Section 3, conventional models for making deep learning robust to attack can perform quite poorly when confronted with physically realizable attacks. In other words, the evidence strongly suggests that the conventional models of attacks in which attackers can make  $l_{p}$ -bounded perturbations to input images are not particularly useful if one is concerned with the main physical threats that are likely to be faced in practice. However, given the diversity of possible physical attacks one may perpetrate, is it even possible to have a meaningful approach for ensuring robustness against a broad range of physical attacks? For example, the two attacks we considered so far couldn't be more dissimilar: in one, we engineer eyeglass frames; in another, stickers on a stop sign. We observe that the key common element in these attacks, and many other physical attacks we may expect to encounter, is that they involve the introduction of adversarial occlusions to a part of the input. The common constraint faced in such attacks is to avoid being suspicious, which effectively limits the size of the adversarial occlusion, but not necessarily its shape or location. Next, we introduce a simple abstract model of occlusion attacks, and then discuss how such attacks can be computed and how we can make classifiers robust to them.
+
+# 4.1 ABSTRACT ATTACK MODEL: RECTANGULAR OCCLUSION ATTACKS (ROA)
+
+We propose the following simple abstract model of adversarial occlusions of input images. The attacker introduces a fixed-dimension rectangle. This rectangle can be placed by the adversary anywhere in the image, and the attacker can furthermore introduce  $l_{\infty}$  noise inside the rectangle with an exogenously specified high bound  $\epsilon$  (for example,  $\epsilon = 255$ , which effectively allows addition of arbitrary adversarial noise). This model bears some similarity to  $l_{0}$  attacks, but the rectangle imposes a contiguity constraint, which reflects common physical limitations. The model is clearly abstract: in practice, for example, adversarial occlusions need not be rectangular or have fixed dimensions (for example, the eyeglass frame attack is clearly not rectangular), but at the same time cannot usually be arbitrarily superimposed on an image, as they are implemented in the physical environment. Nevertheless, the model reflects some of the most important aspects common to many physical attacks, such as stickers placed on an adversially chosen portion of the object we wish to identify. We call our attack model a rectangular occlusion attack (ROA). An important feature of this attack is that it is untargeted: since our ultimate goal is to defend against physical attacks whatever their target, considering untargeted attacks obviates the need to have precise knowledge about the attacker's goals. For illustrations of the ROA attack, see Appendix C.
+
+# 4.2 COMPUTING ATTACKS
+
+The computation of ROA attacks involves 1) identifying a region to place the rectangle in the image, and 2) generating fine-grained adversarial perturbations restricted to this region. The former task can be done by an exhaustive search: consider all possible locations for the upper left-hand corner of the rectangle, compute adversarial noise inside the rectangle using PGD for each of these, and choose the worst-case attack (i.e., the attack which maximizes loss computed on the resulting image). However, this approach would be quite slow, since we need to perform PGD inside the rectangle for every possible position. Our approach, consequently, decouples these two tasks. Specifically, we first
+
+perform an exhaustive search using a grey rectangle to find a position for it that maximizes loss, and then fix the position and apply PGD inside the rectangle.
+
+An important limitation of the exhaustive search approach for ROA location is that it necessitates computations of the loss function for every possible location, which itself requires full forward propagation each time. Thus, the search itself is still relatively slow. To speed the process up further, we use the gradient of the input image to identify candidate locations. Specifically, we select a subset of  $C$  locations for the sticker with the highest magnitude of the gradient, and only exhaustively search among these  $C$  locations.  $C$  is exogenously specified to be small relative to the number of pixels in the image, which significantly limits the number of loss function evaluations. Full details of our algorithms for computing ROA are provided in Appendix D.
+
+# 4.3 DEFENDING AGAINST ROA
+
+Once we are able to compute the ROA attack, we apply the standard adversarial training approach for defense. We term the resulting classifiers robust to our abstract adversarial occlusion attacks Defense against Occlusion Attacks (DOA), and propose these as an alternative to conventional robust ML for defending against physical attacks. As we will see presently, this defense against ROA is quite adequate for our purposes.
+
+# 5 EFFECTIVENESS OF DOA AGAINST PHYSICALLY REALIZABLE ATTACKS
+
+We now evaluate the effectiveness of DOA—that is, adversarial training using the ROA threat model we introduced—against physically realizable attacks (see Appendix G for some examples that defeat conventional methods but not DOA). Recall that we consider only digital representations of the corresponding physical attacks. Consequently, we can view our results in this section as a lower bound on robustness to actual physical attacks, which have to deal with additional practical constraints, such as being robust to multiple viewpoints. In addition to the two physical attacks we previously considered, we also evaluate DOA against the adversarial patch attack, implemented on both face recognition and traffic sign data.
+
+# 5.1 DOA AGAINST ADVERSARIAL EYEGLASSES
+
+We consider two rectangle dimensions resulting in comparable area:  $100 \times 50$  and  $70 \times 70$ , both in pixels. Thus, the rectangles occupy approximately  $10\%$  of the  $224 \times 224$  face images. We used  $\{30, 50\}$  iterations of PGD with  $\epsilon = 255/2$  to generate adversarial noise inside the rectangle, and with learning rate  $\alpha = \{8, 4\}$  correspondingly. For the gradient version of ROA, we choose  $C = 30$ . DOA adversarial training is performed for 5 epochs with a learning rate of 0.0001.
+
+![](images/9a5756cac3c055148303201393eb56f9e050836048bc4c463c7db0e99af78084.jpg)  
+Figure 4: Performance of DOA (using the  $100 \times 50$  rectangle) against the eyeglass frame attack in comparison with conventional methods. Left: comparison between DOA, adversarial training, and randomized smoothing (using the most robust variants of these). Middle/Right: Comparing DOA performance for different rectangle dimensions and numbers of PGD iterations inside the rectangle. Middle: using exhaustive search for ROA; right: using the gradient-based heuristic for ROA.
+
+![](images/080bb4c472990559cdd528d19c3050f4d2205dea59b6a8a4b1c493011277c02a.jpg)
+
+![](images/344a7b76e272e0197fbbda8019f5dade91afec5f392a87263c293c91c472b560.jpg)
+
+Figure 4 (left) presents the results comparing the effectiveness of DOA against the eyeglass frame attack on face recognition to adversarial training and randomized smoothing (we took the most robust variants of both of these). We can see that DOA yields significantly more robust classifiers for this domain. The gradient-based heuristic does come at some cost, with performance slightly worse
+
+than when we use exhaustive search, but this performance drop is relatively small, and the result is still far better than conventional robust ML approaches. Figure 4 (middle and right) compares the performance of DOA between two rectangle variants with different dimensions. The key observation is that as long as we use enough iterations of PGD inside the rectangle, changing its dimensions (keeping the area roughly constant) appears to have minimal impact.
+
+# 5.2 DOA AGAINST THE STOP SIGN ATTACK
+
+We now repeat the evaluation with the traffic sign data and the stop sign attack. In this case, we used  $10 \times 5$  and  $7 \times 7$  rectangles covering  $\sim 5\%$  of the  $32 \times 32$  images. We set  $C = 10$  for the gradient-based ROA. Implementation of DOA is otherwise identical as in the face recognition experiments above.
+
+We present our results using the square rectangle, which in this case was significantly more effective; the results for the  $10 \times 5$  rectangle DOA attacks are in Appendix F. Figure 5 (left) compares the effectiveness of DOA against the stop sign attack on traffic sign data with the best variants of adversarial training and randomized smoothing. Our results here are for 30 iterations of PGD; in Appendix F, we study the impact of varying the number of PGD iterations. We can observe that DOA is again significantly more robust, with robust accuracy over  $90\%$  for the exhaustive search variant, and  $\sim 85\%$  for the gradient-based variant, even for stronger attacks. Moreover, DOA remains  $100\%$  effective at classifying stop signs on clean data, and exhibits  $\sim 95\%$  accuracy on the full traffic sign classification task.
+
+![](images/23f635916ad60b4b8428bd2ed14a31a9a4107680cfbe6a9755f8650252ca5293.jpg)  
+Figure 5: Performance of the best case of adversarial training, randomized smoothing, and DOA against the stop sign attack (left) and adversarial patch attack (right).
+
+![](images/a1968030172250f3cc60769ccdab6ff566f9cd66e17df9a1547e809707e3aa70.jpg)
+
+# 5.3 DOA AGAINST ADVERSARIAL PATCH ATTACKS
+
+Finally, we evaluate DOA against the adversarial patch attacks. In these attacks, an adversarial patch (e.g., sticker) is designed to be placed on an object with the goal of inducing a target prediction. We study this in both face recognition and traffic sign classification tasks. Here, we present the results for face recognition; further detailed results on both datasets are provided in Appendix F.
+
+As we can see from Figure 5 (right), adversarial patch attacks are quite effective once the attack region (fraction of the image) is  $10\%$  or higher, with adversarial training and randomized smoothing both performing rather poorly. In contrast, DOA remains highly robust even when the adversarial patch covers  $20\%$  of the image.
+
+# 6 CONCLUSION
+
+As we have shown, conventional methods for making deep learning approaches for image classification robust to physically realizable attacks tend to be relatively ineffective. In contrast, a new threat model we proposed, rectangular occlusion attacks (ROA), coupled with adversarial training, achieves high robustness against several prominent examples of physical attacks. While we explored a number of variations of ROA attacks as a means to achieve robustness against physical attacks, numerous questions remain. For example, can we develop effective methods to certify robustness against ROA, and are the resulting approaches as effective in practice as our method based on a combination of heuristically computed attacks and adversarial training? Are there other types of occlusions that are more effective? Answers to these and related questions may prove a promising
+
+path towards practical robustness of deep learning when deployed for downstream applications of computer vision such as autonomous driving and face recognition.
+
+# ACKNOWLEDGMENTS
+
+This work was partially supported by the NSF (IIS-1905558, IIS-1903207), ARO (W911NF-19-1-0241), and NVIDIA.
+
+# REFERENCES
+
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+Anish Athalye, Logan Engstrom, Andrew Ilyas, and Kevin Kwok. Synthesizing robust adversarial examples. In ICML, 2018b.  
+Anand Bhattachad, Min Jin Chong, Kaizhao Liang, Bo Li, and David Forsyth. Unrestricted adversarial examples via semantic manipulation. In International Conference on Learning Representations, 2020. URL https://openreview.net/forum?id=Sye_OgHFwH.  
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+Tom B. Brown, Dandelion Mané, Aurko Roy, Martín Abadi, and Justin Gilmer. Adversarial patch. CoRR, abs/1712.09665, 2018.  
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+Jeremy M. Cohen, Elan Rosenfeld, and J. Zico Kolter. Certified adversarial robustness via randomized smoothing. In International Conference on Machine Learning, 2019.  
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+Zico Kolter and Aleksander Madry. Adversarial robustness - theory and practice, 2019. https://adversarial-ml-tutorial.org/.  
+M. Lecuyer, V. Atlidakis, R. Geambasu, D. Hsu, and S. Jana. Certified robustness to adversarial examples with differential privacy. In IEEE Symposium on Security and Privacy, 2019.  
+Aleksander Madry, Aleksandar Makelov, Ludwig Schmidt, Dimitris Tsipras, and Adrian Vladu. Towards deep learning models resistant to adversarial attacks. In International Conference on Learning Representations, 2018.  
+Jan Hendrik Metzen, Tim Genewein, Volker Fischer, and Bastian Bischoff. On detecting adversarial perturbations. In International Conference on Learning Representations, 2017.  
+Andreas Møgelmose, Mohan Manubhai Trivedi, and Thomas B. Moeslund. Vision-based traffic sign detection and analysis for intelligent driver assistance systems: Perspectives and survey. IEEE Transactions on Intelligent Transportation Systems, 13:1484-1497, 2012.  
+Nicolas Papernot, Patrick D. McDaniel, Somesh Jha, Matt Fredrikson, Z. Berkay Celik, and Ananthram Swami. The limitations of deep learning in adversarial settings. 2016 IEEE European Symposium on Security and Privacy (EuroS&P), pp. 372-387, 2015.  
+Omkar M. Parkhi, Andrea Vedaldi, and Andrew Zisserman. Deep face recognition. In BMVC, 2015.  
+Aditi Raghunathan, Jacob Steinhardt, and Percy Liang. Certified defenses against adversarial examples. In International Conference on Learning Representations, 2018a.  
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+Mahmood Sharif, Sruti Bhagavatula, Lujo Bauer, and Michael K. Reiter. Accessorize to a crime: Real and stealthy attacks on state-of-the-art face recognition. In ACM SIGSAC Conference on Computer and Communications Security, pp. 1528-1540, 2016.  
+Ilya Sutskever, Oriol Vinyals, and Quoc V. Le. Sequence to sequence learning with neural networks. In NIPS, 2014.  
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+Liang Tong, Bo Li, Chen Hajaj, Chaowei Xiao, Ning Zhang, and Yevgeniy Vorobeychik. Improving robustness of ml classifiers against realizable evasion attacks using conserved features. In USENIX Security Symposium, pp. 285-302, 2019.  
+Yevgeniy Vorobeychik and Murat Kantarcioglu. Adversarial Machine Learning. Morgan Claypool, 2018.  
+S. Wang, K. Pei, J. Whitehouse, J. Yang, and S. Jana. Efficient formal safety analysis of neural networks. In Neural Information Processing Systems, 2018.  
+Eric Wong and J. Zico Kolter. Provable defenses against adversarial examples via the convex outer adversarial polytope. In International Conference on Machine Learning, 2018.  
+Eric Wong, Frank Schmidt, Jan Hendrik Metzen, and J. Zico Kolter. Scaling provable adversarial defenses. In Neural Information Processing Systems, 2018.  
+Weilin Xu, David Evans, and Yanjun Qi. Feature squeezing: Detecting adversarial examples in deep neural networks. In Network and Distributed Systems Security Symposium, 2018.
+
+# A DESCRIPTION OF DATASETS AND DEEP LEARNING CLASSIFIERS
+
+# A.1 FACE RECOGNITION
+
+The VGGFace dataset<sup>3</sup> (Parkhi et al., 2015) is a benchmark for face recognition, containing 2622 subjets with 2.6 million images in total. We chose ten subjects: A. J. Buckley, A. R. Rahman, Aamir Khan, Aaron Staton, Aaron Tveit, Aaron Yoo, Abbie Cornish, Abel Ferrara, Abigail Breslin, and Abigail Spencer, and subselected face images pertaining only to these individuals. Since approximately half of the images cannot be downloaded, our final dataset contains 300-500 images for each subject.
+
+We used the standard corp-and-resize method to process the data to be  $224 \times 224$  pixels, and split the dataset into training, validation, and test according to a 7:2:1 ratio for each subject. In total, the data set has 3178 images in the training set, 922 images in the validation set, and 470 images in the test set.
+
+We use the VGGFace convolutional neural network (Parkhi et al., 2015) model, a variant of the VGG16 model containing 5 convolutional layer blocks and 3 fully connected layers. We make use of standard transfer learning as we only classify 10 subjects, keeping the convolutional layers as same as VGGFace structure, but changing the fully connected layer to be  $1024 \rightarrow 1024 \rightarrow 10$  instead of  $4096 \rightarrow 4096 \rightarrow 2622$ . Specifically, in our Pytorch implementation, we convert the images from RGB to BGR channel orders and subtract the mean value [129.1863, 104.7624, 93.5940] in order to use the pretrained weights from VGG-Face on convolutional layers. We set the batch size to be 64 and use Pytorch built-in Adam Optimizer with an initial learning rate of  $10^{-4}$  and default parameters in Pytorch. We drop the learning rate by 0.1 every 10 epochs. Additionally, we used validation set accuracy to keep track of model performance and choose a model in case of overfitting. After 30 epochs of training, the model successfully obtains  $98.94\%$  on test data.
+
+# A.2 TRAFFIC SIGN CLASSIFICATION
+
+To be consistent with (Eykholt et al., 2018), we select the subset of LISA which contains 47 different U.S. traffic signs (Møgelmose et al., 2012). To alleviate the problem of imbalance and extremely blurry data, we picked 16 best quality signs with 3509 training and 1148 validation data points. From the validation data, we obtain the test data that includes only 40 stop signs to evaluate performance with respect to the stop sign attack, as done by Eykholt et al. (2018). In the main body of the paper, we present results only on this test data to evaluate robustness to stop sign attacks. In the appendix below, we also include performance on the full validation set without adversarial manipulation.
+
+All the data was processed by standard crop-and-resize to  $32 \times 32$  pixels. We use the LISA-CNN architecture defined in (Eykholt et al., 2018), and construct a convolutional neural network containing three convolutional layers and one fully connected layer. We use the Adam Optimizer with initial learning rate of  $10^{-1}$  and default parameters  $^5$ , dropping the learning rate by 0.1 every 10 epochs. We set the batch size to be 128. After 30 epochs, we achieve the  $98.69\%$  accuracy on the validation set, and  $100\%$  accuracy in identifying the stop signs in our test data.
+
+# B EFFECTIVENESS OF CONVENTIONAL ROBUST ML METHODS AGAINST  $l_{\infty}$  AND  $l_{2}$  ATTACKS
+
+In this appendix, we show that adversarial training and randomized smoothing degrade more gracefully when faced with attacks that they are designed for. In particular, we consider here variants of projected gradient descent (PGD) for both the  $l_{\infty}$  and  $l_{2}$  attacks Madry et al. (2018). In particular, the form of PGD for the  $l_{\infty}$  attack is
+
+$$
+x _ {t + 1} = \operatorname {P r o j} (x _ {t} + \alpha \operatorname {s g n} (\nabla L (x _ {t}; \theta))),
+$$
+
+where  $\mathrm{Proj}$  is a projection operator which clips the result to be feasible,  $x_{t}$  the adversarial example in iteration  $t$ ,  $\alpha$  the learning rate, and  $L(\cdot)$  the loss function. In the case of an  $l_{2}$  attack, PGD becomes
+
+$$
+x _ {t + 1} = \operatorname {P r o j} \left(x _ {t} + \alpha \frac {\nabla L (x _ {t} ; \theta)}{\| \nabla L (x _ {t} ; \theta) \| _ {2}}\right),
+$$
+
+where the projection operator normalizes the perturbation  $\delta = x_{t + 1} - x_t$  to have  $\| \delta \| _2\leq \epsilon$  if it doesn't already Kolter & Madry (2019).
+
+The experiments were done on the face recognition and traffic sign datasets, but unlike physical attacks on stop signs, we now consider adversarial perturbations to all sign images.
+
+# B.1 FACE RECOGNITION
+
+Table 1: Curriculum Adversarial Training against 7 Iterations  ${L}_{\infty }$  Attacks on Face Recognition  
+
+<table><tr><td rowspan="2"></td><td colspan="5">Attack Strength</td></tr><tr><td>ε = 0</td><td>ε = 2</td><td>ε = 4</td><td>ε = 8</td><td>ε = 16</td></tr><tr><td>Clean Model</td><td>98.94%</td><td>57.87%</td><td>13.62%</td><td>0%</td><td>0%</td></tr><tr><td>CAdv. Training: ε = 4</td><td>97.45%</td><td>94.68%</td><td>87.02%</td><td>65.11%</td><td>17.23%</td></tr><tr><td>CAdv. Training: ε = 8</td><td>96.17%</td><td>93.40%</td><td>89.36%</td><td>75.53%</td><td>31.49%</td></tr><tr><td>CAdv. Training: ε = 16</td><td>90.64%</td><td>88.09%</td><td>84.04%</td><td>75.96%</td><td>45.74%</td></tr><tr><td>CAdv. Training: ε = 32</td><td>80.85%</td><td>76.60%</td><td>74.04%</td><td>65.10%</td><td>47.87%</td></tr></table>
+
+Table 2: Curriculum Adversarial Training against 20 Iterations  ${L}_{\infty }$  Attacks on Face Recognition  
+
+<table><tr><td rowspan="2"></td><td colspan="5">Attack Strength</td></tr><tr><td>ε = 0</td><td>ε = 2</td><td>ε = 4</td><td>ε = 8</td><td>ε = 16</td></tr><tr><td>Clean Model</td><td>98.94%</td><td>44.04%</td><td>1.70%</td><td>0%</td><td>0%</td></tr><tr><td>CAdv. Training: ε = 4</td><td>97.45%</td><td>94.68%</td><td>85.74%</td><td>46.60%</td><td>5.11%</td></tr><tr><td>CAdv. Training: ε = 8</td><td>96.17%</td><td>93.19%</td><td>88.94%</td><td>69.36%</td><td>9.57%</td></tr><tr><td>CAdv. Training: ε = 16</td><td>90.64%</td><td>88.08%</td><td>83.83%</td><td>73.62%</td><td>35.96%</td></tr><tr><td>CAdv. Training: ε = 32</td><td>80.85%</td><td>76.17%</td><td>74.04%</td><td>63.82%</td><td>43.62%</td></tr></table>
+
+We begin with our results on the face recognition dataset. Tables 1 and 2 present results for (curriculum) adversarial training for varying  $\epsilon$  of the  $l_{\infty}$  attacks, separately for training and evaluation. As we can see, curriculum adversarial training with  $\epsilon = 16$  is generally the most robust, and remains reasonably effective for relatively large perturbations. However, we do observe a clear tradeoff between accuracy on non-adversarial data and robustness, as one would expect.
+
+Table 3: Randomized Smoothing against 20 Iterations  $L_{2}$  Attacks on Face Recognition  
+
+<table><tr><td rowspan="2"></td><td colspan="7">Attack Strength</td></tr><tr><td>ε = 0</td><td>ε = 0.5</td><td>ε = 1</td><td>ε = 1.5</td><td>ε = 2</td><td>ε = 2.5</td><td>ε = 3</td></tr><tr><td>Clean Model</td><td>98.94%</td><td>93.19%</td><td>70.85%</td><td>44.68%</td><td>22.13%</td><td>8.29%</td><td>3.19%</td></tr><tr><td>RS: σ = 0.25</td><td>98.51%</td><td>97.23%</td><td>95.53%</td><td>91.70%</td><td>81.06%</td><td>67.87%</td><td>52.97%</td></tr><tr><td>RS: σ = 0.5</td><td>97.65%</td><td>94.25%</td><td>93.61%</td><td>91.70%</td><td>87.87%</td><td>82.55%</td><td>71.70%</td></tr><tr><td>RS: σ = 1</td><td>92.97%</td><td>93.19%</td><td>91.70%</td><td>91.06%</td><td>88.51%</td><td>85.53%</td><td>82.98%</td></tr></table>
+
+Table 3 presents the results of using randomized smoothing on face recognition data, when facing the  $l_{2}$  attacks. Again, we observe a high level of robustness and, in most cases, relatively limited drop in performance, with  $\sigma = 0.5$  perhaps striking the best balance.
+
+# B.2 TRAFFIC SIGN CLASSIFICATION
+
+Table 4: Curriculum Adversarial Training against 7 Iterations  ${L}_{\infty }$  Attacks on Traffic Signs  
+
+<table><tr><td rowspan="2"></td><td colspan="5">Attack Strength</td></tr><tr><td>ε = 0</td><td>ε = 2</td><td>ε = 4</td><td>ε = 8</td><td>ε = 16</td></tr><tr><td>Clean Model</td><td>98.69%</td><td>90.24%</td><td>65.24%</td><td>33.80%</td><td>10.10%</td></tr><tr><td>CAdv. Training: ε = 4</td><td>99.13%</td><td>97.13%</td><td>93.47%</td><td>63.85%</td><td>20.73%</td></tr><tr><td>CAdv. Training: ε = 8</td><td>98.72%</td><td>96.86%</td><td>93.90%</td><td>81.70%</td><td>38.24%</td></tr><tr><td>CAdv. Training: ε = 16</td><td>96.95%</td><td>95.03%</td><td>92.77%</td><td>87.63%</td><td>64.02%</td></tr><tr><td>CAdv. Training: ε = 32</td><td>65.63%</td><td>53.83%</td><td>50.87%</td><td>46.69%</td><td>38.07%</td></tr></table>
+
+Table 5: Curriculum Adversarial Training against 20 Iterations  ${L}_{\infty }$  Attacks on Traffic Signs  
+
+<table><tr><td></td><td colspan="5">Attack Strength</td></tr><tr><td></td><td>ε = 0</td><td>ε = 2</td><td>ε = 4</td><td>ε = 8</td><td>ε = 16</td></tr><tr><td>Clean Model</td><td>98.69%</td><td>89.54%</td><td>61.58%</td><td>24.65%</td><td>5.14%</td></tr><tr><td>CAdv. Training: ε = 4</td><td>99.13%</td><td>96.95%</td><td>91.90%</td><td>56.53%</td><td>12.02%</td></tr><tr><td>CAdv. Training: ε = 8</td><td>98.72%</td><td>96.68%</td><td>93.64%</td><td>76.22%</td><td>28.13%</td></tr><tr><td>CAdv. Training: ε = 16</td><td>96.95%</td><td>95.03%</td><td>92.51%</td><td>86.76%</td><td>54.01%</td></tr><tr><td>CAdv. Training: ε = 32</td><td>65.63%</td><td>53.83%</td><td>50.78%</td><td>46.08%</td><td>36.49%</td></tr></table>
+
+Tables 4 and 5 present evaluation on traffic sign data for curriculum adversarial training against the  $l_{\infty}$  attack for varying  $\epsilon$ . As with face recognition data, we can observe that the approaches tend to be relatively robust, and effective on non-adversarial data for adversarial training methods using  $\epsilon < 32$ .
+
+Table 6: Randomized Smoothing against 20 Iterations  $L_{2}$  Attacks on Traffic Signs  
+
+<table><tr><td rowspan="2"></td><td colspan="7">Attack Strength</td></tr><tr><td>ε = 0</td><td>ε = 0.5</td><td>ε = 1</td><td>ε = 1.5</td><td>ε = 2</td><td>ε = 2.5</td><td>ε = 3</td></tr><tr><td>Clean Model</td><td>98.69%</td><td>61.67%</td><td>25.78%</td><td>11.50%</td><td>7.84%</td><td>4.97%</td><td>3.57%</td></tr><tr><td>RS: σ = 0.25</td><td>98.22%</td><td>89.08%</td><td>55.69%</td><td>34.06%</td><td>23.46%</td><td>18.61%</td><td>14.75%</td></tr><tr><td>RS: σ = 0.5</td><td>96.28%</td><td>90.80%</td><td>76.13%</td><td>52.64%</td><td>35.31%</td><td>23.43%</td><td>16.52%</td></tr><tr><td>RS: σ = 1</td><td>88.21%</td><td>83.68%</td><td>75.49%</td><td>64.90%</td><td>50.03%</td><td>36.53%</td><td>26.22%</td></tr></table>
+
+The results of randomized smoothing on traffic sign data are given in Table 6. Since images are smaller here than in VGGFace, lower values of  $\epsilon$  for the  $l_{2}$  attacks are meaningful, and for  $\epsilon \leq 1$  we
+
+generally see robust performance on randomized smoothing, with  $\sigma = 0.5$  providing a good balance between non-adversarial accuracy and robustness, just as before.
+
+# C EXAMPLES OF ROA
+
+![](images/504081c79e50fd2982ef88818cdc16a22afb8e030d20176e3fd8d71ba1a71604.jpg)
+
+![](images/9c8b3b70b9826cfc1fe96f3d9e86bcb144e7d0be389de09ff16992b8dceccc11.jpg)  
+(a)  
+Figure 6: Examples of the ROA attack on face recognition, using a rectangle of size  $100 \times 50$ . (a) Left: the original A. J. Buckley's image. Middle: modified input image (ROA superimposed on the face). Right: an image of the predicted individual who is Aaron Tveit with the adversarial input in the middle image. (b) Left: the original Abigail Spencer's image. Middle: modified input image (ROA superimposed on the face). Right: an image of the predicted individual who is Aaron Yoo with the adversarial input in the middle image.
+
+![](images/79474d744f65d5556252550c58c481b49de649113616373e6b27e5d2274d8fb0.jpg)
+
+![](images/400fb651c620c82cb53755330a0bfcb9223f961d2e4250761bff82c6a2d6e937.jpg)
+
+![](images/8e33cc02bfbc02ac2331a5f09a3252e498bfde0854e98c6aa7c55d174f5f721c.jpg)  
+(b)
+
+![](images/1ca268105608afcf3f710222f99eb3ee505e35c66ddce6c8fb697ed3423e59b1.jpg)
+
+Figure 6 provides several examples of the ROA attack in the context of face recognition. Note that in these examples, the adversaries choose to occlude the noise on upper lip and eye areas of the image, and, indeed, this makes the face more challenging to recognize even to a human observer.
+
+![](images/e045e27e55c45c1e932417893f7947168c96bb5fbf7393d640732b34d6f67209.jpg)  
+Figure 7: Examples of the ROA attack on traffic sign, using a rectangle of size  $7 \times 7$ . (a) Left: the original Speedlimit45 sign. Middle: modified input image (ROA superimposed on the sign). Right: an image of the predicted which is Speedlimit30 with the adversarial input in the middle image. (b) Left: the original Stop sign. Middle: modified input image (ROA superimposed on the sign). Right: an image of the predicted Yield sign with the adversarial input in the middle image.
+
+![](images/ad99506a32a2ce17ad76c4344c336b3b113ab508954ccb29a3dddc076ee60eb1.jpg)  
+(a)
+
+![](images/186a388106933fb341d2ff02400d8e6d60b95fadc6f970d771f15ed6f754d27f.jpg)
+
+![](images/969adc03cf117accf3f29aa4058c9d7ad09b36d5f662df81d613ee80be8b00b0.jpg)
+
+![](images/fb83de6349e9085e3cef535d6ff57b31895c99a71a43b6a36f75aa424bf37f19.jpg)  
+(b)
+
+![](images/92e816150c8f7d4f8addfebd27a51106f080e6ba8cd80187b6be411b43e0762d.jpg)
+
+# D DETAILED DESCRIPTION OF THE ALGORITHMS FOR COMPUTING THE RECTANGULAR OCCLUSION ATTACKS
+
+Our basic algorithm for computing rectangular occlusion attacks (ROA) proceeds through the following two steps:
+
+1. Iterate through possible positions for the rectangle's upper left-hand corner point in the image. Find the position for a grey rectangle (RGB value  $= [127.5, 127.5, 127.5]$ ) in the image that maximizes loss.  
+2. Generate high- $\epsilon$ $l_{\infty}$  noise inside the rectangle at the position computed in step 1.
+
+Algorithm 1 presents the full algorithm for identifying the ROA position, which amounts to exhaustive search through the image pixel region. This algorithm has several parameters. First, we assume that images are squares with dimensions  $N^2$ . Second, we introduce a stride parameter  $S$ . The purpose of this parameter is to make location computation faster by only considering every other  $S$ th pixel during the search (in other words, we skip  $S$  pixels each time). For our implementation of ROA attacks, we choose the stride parameter  $S = 5$  for face recognition and  $S = 2$  for traffic sign classification.
+
+Algorithm 1 Computation of ROA position using exhaustive search.  
+Input:  
+Data:  $X_i, y_i$ ; Test data shape:  $N \times N$ ; Target model parameters:  $\theta$ ; Stride:  $S$ ; Output:  
+ROA Position:  $(j', k')$   
+1. function ExhaustiveSearching(Model,  $X_i, y_i, N, S$ )  
+2. for j in range(N/S) do:  
+3. for k in range(N/S) do:  
+4. Generate the adversarial  $X_i^{adv}$  image by:  
+5. place a grey rectangle onto the image with top-left corner at  $(j \times S, k \times S)$ ;  
+6. if L( $X_i^{adv}, y_i, \theta$ ) is higher than previous loss:  
+7. Update  $(j', k') = (j, k)$   
+8. end for  
+9. end for  
+10. return  $(j', k')$
+
+Algorithm 2 Computation of ROA position using gradient-based search.  
+Input:  
+Data:  $X_i, y_i$ ; Test data shape:  $N \times N$ ; Target Model:  $\theta$ ; Stride:  $S$ ;  
+Number of Potential Candidates:  $C$ ;  
+Output:  
+Best Sticker Position:  $(j', k')$   
+1.function GradientBasedSearch( $X_i, y_i, N, S, C, \theta$ )  
+2. Calculate the gradient  $\nabla L$  of  $Loss(X_i, y_i, \theta)$  w.r.t.  $X_i$   
+3.  $\mathbb{J}, \mathbb{K} = \text{HelperSearching}(\nabla L, N, S, C)$   
+4. for  $j, k$  in  $\mathbb{J}, \mathbb{K}$  do:  
+5. Generate the adversarial  $X_i^{adv}$  image by:  
+6. put the sticker on the image where top-left corner at  $(j \times S, k \times S)$ ;  
+7. if  $Loss(X_i^{adv}, y_i, \theta)$  is higher than previous loss:  
+8. Update  $(j', k') = (j, k)$   
+9. end for  
+10. return  $(j', k')$   
+1.function HelperSearching $(\nabla L, N, S, C)$   
+2. for  $j$  in range(N/S) do:  
+3. for  $k$  in range(N/S) do:  
+4. Calculate the Sensitivity value  $L = \sum_{i \in \text{rectangle}} (\nabla L_i)^2$  where top-left corner at  $(j \times S, k \times S)$ ;  
+5. if the Sensitivity value  $L$  is in top  $C$  of previous values:  
+6. Put  $(j, k)$  in  $\mathbb{J}, \mathbb{K}$  and discard  $(j_s, k_s)$  with lowest  $L$   
+7. end for  
+8. end for  
+9. return  $\mathbb{J}, \mathbb{K}$
+
+Despite introducing the tunable stride parameter, the search for the best location for ROA still entails a large number of loss function evaluations, which are somewhat costly (since each such evaluation means a full forward pass through the deep neural network), and these costs add up quickly. To speed things up, we consider using the magnitude of the gradient of the loss as a measure of sensitivity of particular regions to manipulation. Specifically, suppose that we compute a gradient  $\nabla L$ , and let  $\nabla L_{i}$  be the gradient value for a particular pixel  $i$  in the image. Now, we can iterate over the possible ROA locations, but for each location compute the gradient of the loss at that location corresponding to the rectangular region. We do this by adding squared gradient values  $(\nabla L_{i})^{2}$  over pixels  $i$  in the rectangle. We use this approach to find the top  $C$  candidate locations for the rectangle. Finally, we consider each of these, computing the actual loss for each location, to find the position of ROA. The full algorithm is provided as Algorithm 2.
+
+Once we've found the place for the rectangle, our next step is to introduce adversarial noise inside it. For this, we use the  $l_{\infty}$  version of the PGD attack, restricting perturbations to the rectangle. We used  $\{7, 20, 30, 50\}$  iterations of PGD to generate adversarial noise inside the rectangle, and with learning rate  $\alpha = \{32, 16, 8, 4\}$  correspondingly.
+
+![](images/71904717a3a552ab9f9f7d1c006bc06d8d68689835175aec5da57c330452a021.jpg)  
+Original image
+
+![](images/a5eb58853e07b3d83f9b082f301aaca14b90e4fed2499a54bf8d129f0b9a48df.jpg)  
+Grad. plot
+
+![](images/2858b58e4e3f33207c9736f965f251ac836ae15105560896e8d8743477ac8552.jpg)  
+Grad. searching
+
+![](images/3dab413883dcd03c15f1572bf1ae8c9f5524d2ac23371e800b21921f091f4860.jpg)  
+Exh. searching
+
+![](images/f3388768baba33bd7c6592418b1e5b05defd7016b289980c0956dba6afce8dc7.jpg)  
+Figure 8: Examples of different search techniques. From left to right: 1) the original input image, 2) the plot of input gradient, 3) face with ROA location identified using gradient-based search, 4) face with ROA location identified using exhaustive search. Each row is a different example.
+
+![](images/8c4f98585e861fce3205217c1f7d2cab5b75bf16a204b6dc1eeb140ea7d3f9b0.jpg)  
+Figure 8 offers a visual illustration of how gradient-based search compares to exhaustive search for computing ROA.
+
+![](images/7c53b20b8e389d28fd83eb31d11821934b11f0895067a58004ebb3e15435cf43.jpg)
+
+![](images/6d0f4b0f712f391b990be032136b6bfdf645929dcf64581c81604943e5d9bf8f.jpg)
+
+# E DETAILS OF PHYSICALLY REALIZABLE ATTACKS
+
+Physically realizable attacks that we study have a common feature: first, they specify a mask, which is typically precomputed, and subsequently introduce adversarial noise inside the mask area. Let  $M$  denote the mask matrix constraining the area of the perturbation  $\delta$ ;  $M$  has the same dimensions as the input image and contains 0s where no perturbation is allowed, and 1s in the area which can be perturbed. The physically realizable attacks we consider then solve an optimization problem of the following form:
+
+$$
+\underset {\delta} {\arg \max } L (f (\boldsymbol {x} + M \delta ; \boldsymbol {\theta}), y). \tag {4}
+$$
+
+Next, we describe the details of the three physical attacks we consider in the main paper.
+
+# E.1 EYEGLASS FRAME ATTACKS ON FACE RECOGNITION
+
+Following Sharif et al. (2016), we first initialized the eyeglass frame with 5 different colors, and chose the best starting color by calculating the cross-entropy loss. For each update step, we divided the gradient value by its maximum value before multiplying by the learning rate which is 20. Then we only kept the gradient value of eyeglass frame area. Finally, we clipped and rounded the pixel value to keep it in the valid range.
+
+# E.2 STICKER ATTACKS ON STOP SIGNS
+
+Following Eykholt et al. (2018), we initialized the stickers on the stop signs with random noise. For each update step, we used the Adam optimizer with 0.1 learning rate and with default parameters. Just as for other attacks, adversarial perturbations were restricted to the mask area exogenously specified; in our case, we used the same mask as Eykholt et al. (2018)—a collection of small rectangles.
+
+# E.3 ADVERSARIAL PATCH ATTACK
+
+We used gradient ascent to maximize the log probability of the targeted class  $P[y_{\text{target}}|x]$ , as in the original paper (Brown et al., 2018). When implementing the adversarial patch, we used a square patch rather than the circular patch in the original paper; we don't anticipate this choice to be practically consequential. We randomly chose the position and direction of the patch, used the learning rate of 5, and fixed the number of attack iterations to 100 for each image. We varied the attack region (mask)  $R \in \{0\%, 5\%, 10\%, 15\%, 20\%, 25\% \}$ .
+
+For the face recognition dataset, we used 27 images (9 classes (without targeted class)  $\times$  3 images in each class) to design the patch, and then ran the attack over 20 epochs. For the smaller traffic sign dataset, we used 15 images (15 classes (without targeted class)  $\times$  1 image in each class) to design the patch, and then ran the attack over 5 epochs. Note that when evaluating the adversarial patch, we used the validation set without the targeted class images.
+
+# F ADDITIONAL EXPERIMENTS WITH DOA
+
+# F.1 FACE RECOGNITION AND EYEGLASS FRAME ATTACK
+
+![](images/68ae6b07ab76492aa8714767ce0d44c5d23b3cef8bd472aadb10a27451501048.jpg)  
+Figure 9: Effectiveness of DOA using the gradient-based method and the  $100 \times 50$  region against the eyeglass frame attack, varying the number of PGD iterations for adversarial perturbations inside the rectangle. Left: using exhaustive search. Right: using gradient-based search.
+
+![](images/be8a1fdddce42e6bf81d29790293ddf8d7f62b3a1bfb3eca2c8d8a20f9ea84ab.jpg)
+
+![](images/3467e6999ccbbcd63367dfcf272aea35aa2c04847a444a00313657256d9718d9.jpg)  
+Figure 10: Effectiveness of DOA using the gradient-based method and the  $70 \times 70$  region against the eyeglass frame attack, varying the number of PGD iterations for adversarial perturbations inside the rectangle. Left: using exhaustive search. Right: using gradient-based search.
+
+![](images/5dbc2feef6504f7eeefee46a96d364a31a78cf3a060086906105a01dd956390c.jpg)
+
+# F.2 TRAFFIC SIGN CLASSIFICATION AND THE STOP SIGN ATTACK
+
+Table 7: Comparison of effectiveness of different approaches against the stop sign attack. Best parameter choices were made for each.  
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Validation Set</td><td colspan="4">Attack Iterations</td></tr><tr><td>0</td><td>101</td><td>102</td><td>103</td></tr><tr><td>Clean Model</td><td>98.69%</td><td>100%</td><td>42.5%</td><td>32.5%</td><td>25%</td></tr><tr><td>CAdv. Training: ε = 16</td><td>96.95%</td><td>97.5%</td><td>57.5%</td><td>45%</td><td>42.5%</td></tr><tr><td>Randomized Smoothing: σ = 0.25</td><td>98.22%</td><td>100%</td><td>70%</td><td>64.5%</td><td>62.5%</td></tr><tr><td>DOA (exhaustive; 7 × 7; 30 PGD)</td><td>92.51%</td><td>100%</td><td>92%</td><td>92%</td><td>90.5%</td></tr><tr><td>DOA (gradient-based; 7 × 7; 30 PGD)</td><td>95.82%</td><td>100%</td><td>85%</td><td>85.5%</td><td>84%</td></tr></table>
+
+Table 8: Effectiveness of  ${10} \times  5$  DOA using exhaustive search with different numbers of PGD iterations against the stop sign attack.  
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Validation Set</td><td colspan="4">Attack Iterations</td></tr><tr><td>0</td><td>101</td><td>102</td><td>103</td></tr><tr><td>Clean Model</td><td>98.69%</td><td>100%</td><td>42.5%</td><td>32.5%</td><td>25%</td></tr><tr><td>Exhaustive Search with 7 PGD</td><td>96.50%</td><td>100%</td><td>73%</td><td>70.5%</td><td>71%</td></tr><tr><td>Exhaustive Search with 20 PGD</td><td>95.86%</td><td>100%</td><td>86%</td><td>85%</td><td>85%</td></tr><tr><td>Exhaustive Search with 30 PGD</td><td>95.59%</td><td>100%</td><td>86.5%</td><td>84%</td><td>82.5%</td></tr><tr><td>Exhaustive Search with 50 PGD</td><td>95.87%</td><td>100%</td><td>77%</td><td>76.5%</td><td>73.5%</td></tr></table>
+
+Table 9: Effectiveness of  ${10} \times  5$  gradient-based DOA with different numbers of PGD iterations against the stop sign attack.  
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Validation Set</td><td colspan="4">Attack Iterations</td></tr><tr><td>0</td><td>101</td><td>102</td><td>103</td></tr><tr><td>Clean Model</td><td>98.69%</td><td>100%</td><td>42.5%</td><td>32.5%</td><td>25%</td></tr><tr><td>Gradient Based Search with 7 PGD</td><td>97.53%</td><td>100%</td><td>83.5%</td><td>78%</td><td>78%</td></tr><tr><td>Gradient Based Search with 20 PGD</td><td>97.11%</td><td>100%</td><td>82.5%</td><td>81.5%</td><td>80.5%</td></tr><tr><td>Gradient Based Search with 30 PGD</td><td>96.83%</td><td>100%</td><td>83.5%</td><td>79.5%</td><td>81%</td></tr><tr><td>Gradient Based Search with 50 PGD</td><td>96.46%</td><td>100%</td><td>82.5%</td><td>82.5%</td><td>81.5%</td></tr></table>
+
+Table 10: Effectiveness of  $7 \times 7$  DOA using exhaustive search with different numbers of PGD iterations against the stop sign attack.  
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Validation Set</td><td colspan="4">Attack Iterations</td></tr><tr><td>0</td><td>101</td><td>102</td><td>103</td></tr><tr><td>Clean Model</td><td>98.69%</td><td>100%</td><td>42.5%</td><td>32.5%</td><td>25%</td></tr><tr><td>Exhaustive Search with 7 PGD</td><td>94.16%</td><td>100%</td><td>86.5%</td><td>85%</td><td>85%</td></tr><tr><td>Exhaustive Search with 20 PGD</td><td>92.74%</td><td>100%</td><td>83%</td><td>80%</td><td>79%</td></tr><tr><td>Exhaustive Search with 30 PGD</td><td>92.51%</td><td>100%</td><td>92%</td><td>92%</td><td>90.5%</td></tr><tr><td>Exhaustive Search with 50 PGD</td><td>92.94%</td><td>100%</td><td>89.5%</td><td>89%</td><td>88.5%</td></tr></table>
+
+Table 11: Effectiveness of  $7 \times 7$  gradient-based DOA with different numbers of PGD iterations against the stop sign attack.  
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Validation Set</td><td colspan="4">Attack Iterations</td></tr><tr><td>0</td><td>101</td><td>102</td><td>103</td></tr><tr><td>Clean Model</td><td>98.69%</td><td>100%</td><td>42.5%</td><td>32.5%</td><td>25%</td></tr><tr><td>Gradient Based Search with 7 PGD</td><td>96.52%</td><td>100%</td><td>81.5%</td><td>80%</td><td>79.5%</td></tr><tr><td>Gradient Based Search with 20 PGD</td><td>95.51%</td><td>100%</td><td>83.5%</td><td>82.5%</td><td>83%</td></tr><tr><td>Gradient Based Search with 30 PGD</td><td>95.82%</td><td>100%</td><td>85%</td><td>85.5%</td><td>84%</td></tr><tr><td>Gradient Based Search with 50 PGD</td><td>95.59%</td><td>100%</td><td>81%</td><td>81%</td><td>81.5%</td></tr></table>
+
+# F.3 EVALUATION WITH THE ADVERSARIAL PATCH ATTACK
+
+# F.3.1 FACE RECOGNITION
+
+Table 12: Comparison of effectiveness of different approaches against the adversarial patch attack on the face recognition data. Best parameter choices were made for each.  
+
+<table><tr><td rowspan="2"></td><td colspan="6">Size of Attacking Region</td></tr><tr><td>0%</td><td>5%</td><td>10%</td><td>15%</td><td>20%</td><td>25%</td></tr><tr><td>Clean Model</td><td>99.30%</td><td>80.69%</td><td>41.21%</td><td>19.41%</td><td>9.76%</td><td>5.75%</td></tr><tr><td>CAdv. Training: ε = 16</td><td>94.15%</td><td>48.26%</td><td>34.38%</td><td>28.85%</td><td>23.54%</td><td>19.02%</td></tr><tr><td>Randomized Smoothing: σ = 1</td><td>98.83%</td><td>61.99%</td><td>36.26%</td><td>33.80%</td><td>24.68%</td><td>16.02%</td></tr><tr><td>DOA (exh. ; 100 × 50; 50 PGD)</td><td>99.30%</td><td>98.72%</td><td>97.83%</td><td>97.18%</td><td>81.24%</td><td>24.19%</td></tr><tr><td>DOA (exh. ; 70 × 70; 50 PGD)</td><td>99.41%</td><td>98.37%</td><td>97.29%</td><td>96.53%</td><td>63.23%</td><td>44.14%</td></tr></table>
+
+Table 13: Effectiveness of adversarial training against the adversarial patch attack on face recognition data.  
+
+<table><tr><td rowspan="2"></td><td colspan="6">Size of Attacking Region</td></tr><tr><td>0%</td><td>5%</td><td>10%</td><td>15%</td><td>20%</td><td>25%</td></tr><tr><td>Clean Model</td><td>99.30%</td><td>80.69%</td><td>41.21%</td><td>19.41%</td><td>9.76%</td><td>5.75%</td></tr><tr><td>7 Iterations Adv. Training: ε = 4</td><td>92.28%</td><td>57.05%</td><td>39.15%</td><td>34.71%</td><td>10.52%</td><td>7.81%</td></tr><tr><td>7 Iterations Adv. Training: ε = 8</td><td>66.54%</td><td>27.98%</td><td>23.54%</td><td>21.26%</td><td>18.55%</td><td>17.03%</td></tr><tr><td>50 Iterations Adv. Training: ε = 4</td><td>90.29%</td><td>57.38%</td><td>53.58%</td><td>36.01%</td><td>28.85%</td><td>16.05%</td></tr><tr><td>50 Iterations Adv. Training: ε = 8</td><td>53.68%</td><td>21.80%</td><td>21.15%</td><td>17.79%</td><td>17.79%</td><td>16.70%</td></tr></table>
+
+Table 14: Effectiveness of curriculum adversarial training against the adversarial patch attack on face recognition data.  
+
+<table><tr><td rowspan="2"></td><td colspan="6">Size of Attacking Region</td></tr><tr><td>0%</td><td>5%</td><td>10%</td><td>15%</td><td>20%</td><td>25%</td></tr><tr><td>Clean Model</td><td>99.30%</td><td>80.69%</td><td>41.21%</td><td>19.41%</td><td>9.76%</td><td>5.75%</td></tr><tr><td>CAdv. Training: ε = 4</td><td>97.89%</td><td>52.49%</td><td>45.44%</td><td>30.91%</td><td>16.05%</td><td>10.85%</td></tr><tr><td>CAdv. Training: ε = 8</td><td>96.84%</td><td>44.79%</td><td>29.18%</td><td>23.86%</td><td>12.47%</td><td>10.20%</td></tr><tr><td>CAdv. Training: ε = 16</td><td>94.15%</td><td>48.26%</td><td>34.38%</td><td>28.85%</td><td>23.54%</td><td>19.20%</td></tr><tr><td>CAdv. Training: ε = 32</td><td>81.05%</td><td>42.95%</td><td>40.46%</td><td>32.00%</td><td>20.93%</td><td>17.14%</td></tr></table>
+
+Table 15: Effectiveness of randomized smoothing against the adversarial patch attack on face recognition data.  
+
+<table><tr><td rowspan="2"></td><td colspan="6">Size of Attacking Region</td></tr><tr><td>0%</td><td>5%</td><td>10%</td><td>15%</td><td>20%</td><td>25%</td></tr><tr><td>Clean Model</td><td>99.30%</td><td>80.69%</td><td>41.21%</td><td>19.41%</td><td>9.76%</td><td>5.75%</td></tr><tr><td>Randomized Smoothing: σ = 0.25</td><td>98.95%</td><td>60.35%</td><td>26.67%</td><td>13.80%</td><td>6.67%</td><td>5.15%</td></tr><tr><td>Randomized Smoothing: σ = 0.5</td><td>97.66%</td><td>81.05%</td><td>58.83%</td><td>33.92%</td><td>26.55%</td><td>14.74%</td></tr><tr><td>Randomized Smoothing: σ = 1</td><td>98.83%</td><td>61.99%</td><td>36.26%</td><td>33.80%</td><td>24.68%</td><td>16.02%</td></tr></table>
+
+Table 16: Effectiveness of DOA (50 PGD iterations) against the adversarial patch attack on face recognition data.  
+
+<table><tr><td rowspan="2"></td><td colspan="6">Size of Attacking Region</td></tr><tr><td>0%</td><td>5%</td><td>10%</td><td>15%</td><td>20%</td><td>25%</td></tr><tr><td>Clean Model</td><td>99.30%</td><td>80.69%</td><td>41.21%</td><td>19.41%</td><td>9.76%</td><td>5.75%</td></tr><tr><td>(100 × 50)Exhaustive Search</td><td>99.30%</td><td>98.70%</td><td>97.83%</td><td>97.18%</td><td>81.24%</td><td>24.19%</td></tr><tr><td>(100 × 50)Gradient Based Search</td><td>99.53%</td><td>98.41%</td><td>96.75%</td><td>87.42%</td><td>57.48%</td><td>24.62%</td></tr><tr><td>(70 × 70)Exhaustive Search</td><td>99.41%</td><td>98.37%</td><td>97.29%</td><td>96.53%</td><td>63.23%</td><td>44.14%</td></tr><tr><td>(70 × 70)Gradient Based Search</td><td>98.83%</td><td>97.51%</td><td>96.31%</td><td>93.82%</td><td>92.19%</td><td>31.34%</td></tr></table>
+
+# F.3.2 TRAFFIC SIGN CLASSIFICATION
+
+Table 17: Comparison of effectiveness of different approaches against the adversarial patch attack on the traffic sign data. Best parameter choices were made for each.  
+
+<table><tr><td rowspan="2"></td><td colspan="6">Size of Attacking Region</td></tr><tr><td>0%</td><td>5%</td><td>10%</td><td>15%</td><td>20%</td><td>25%</td></tr><tr><td>Clean Model</td><td>98.38%</td><td>75.87%</td><td>58.62%</td><td>40.68%</td><td>33.10%</td><td>22.47%</td></tr><tr><td>CAdv. Training: ε = 16</td><td>96.77%</td><td>86.93%</td><td>70.21%</td><td>66.11%</td><td>51.13%</td><td>40.77%</td></tr><tr><td>Randomized Smoothing: σ = 0.5</td><td>95.39%</td><td>83.28%</td><td>68.17%</td><td>53.86%</td><td>48.79%</td><td>33.10%</td></tr><tr><td>DOA (grad. ; 7 × 7; 30 PGD)</td><td>94.69%</td><td>90.68%</td><td>88.41%</td><td>81.79%</td><td>72.30%</td><td>58.71%</td></tr><tr><td>DOA (grad. ; 7 × 7; 50 PGD)</td><td>94.69%</td><td>90.33%</td><td>89.29%</td><td>78.92%</td><td>70.21%</td><td>58.54%</td></tr></table>
+
+Table 18: Effectiveness of adversarial training against the adversarial patch attack on traffic sign data.  
+
+<table><tr><td rowspan="2"></td><td colspan="6">Size of Attacking Region</td></tr><tr><td>0%</td><td>5%</td><td>10%</td><td>15%</td><td>20%</td><td>25%</td></tr><tr><td>Clean Model</td><td>98.38%</td><td>75.87%</td><td>58.62%</td><td>40.68%</td><td>33.10%</td><td>22.47%</td></tr><tr><td>Adv. Training: ε = 4</td><td>98.96%</td><td>82.14%</td><td>64.46%</td><td>45.73%</td><td>31.36%</td><td>23.78%</td></tr><tr><td>Adv. Training: ε = 8</td><td>95.62%</td><td>76.74%</td><td>62.89%</td><td>46.60%</td><td>37.37%</td><td>25.09%</td></tr></table>
+
+Table 19: Effectiveness of curriculum adversarial training against the adversarial patch attack on traffic sign data.  
+
+<table><tr><td rowspan="2"></td><td colspan="6">Size of Attacking Region</td></tr><tr><td>0%</td><td>5%</td><td>10%</td><td>15%</td><td>20%</td><td>25%</td></tr><tr><td>Clean Model</td><td>98.38%</td><td>75.87%</td><td>58.62%</td><td>40.68%</td><td>33.10%</td><td>22.47%</td></tr><tr><td>CAdv. Training: ε = 4</td><td>98.96%</td><td>77.18%</td><td>57.32%</td><td>39.02%</td><td>38.94%</td><td>31.10%</td></tr><tr><td>CAdv. Training: ε = 8</td><td>98.96%</td><td>86.24%</td><td>68.64%</td><td>60.19%</td><td>42.42%</td><td>35.10%</td></tr><tr><td>CAdv. Training: ε = 16</td><td>96.77%</td><td>86.93%</td><td>70.21%</td><td>66.11%</td><td>51.13%</td><td>40.77%</td></tr><tr><td>CAdv. Training: ε = 32</td><td>63.78%</td><td>59.93%</td><td>51.57%</td><td>45.38%</td><td>37.37%</td><td>27.79%</td></tr></table>
+
+Table 20: Effectiveness of randomized smoothing against the adversarial patch attack on traffic sign data.  
+
+<table><tr><td rowspan="2"></td><td colspan="6">Size of Attacking Region</td></tr><tr><td>0%</td><td>5%</td><td>10%</td><td>15%</td><td>20%</td><td>25%</td></tr><tr><td>Clean Model</td><td>98.38%</td><td>75.87%</td><td>58.62%</td><td>40.68%</td><td>33.10%</td><td>22.47%</td></tr><tr><td>Randomized Smoothing: σ = 0.25</td><td>98.27%</td><td>82.24%</td><td>66.78%</td><td>54.67%</td><td>40.37%</td><td>33.79%</td></tr><tr><td>Randomized Smoothing: σ = 0.5</td><td>95.39%</td><td>83.28%</td><td>68.17%</td><td>53.86%</td><td>48.79%</td><td>33.10%</td></tr><tr><td>Randomized Smoothing: σ = 1</td><td>85.47%</td><td>74.39%</td><td>54.44%</td><td>45.44%</td><td>40.48%</td><td>29.06%</td></tr></table>
+
+Table 21: Effectiveness of DOA (30 PGD iterations) against the adversarial patch attack on traffic sign data.  
+
+<table><tr><td rowspan="2"></td><td colspan="6">Size of Attacking Region</td></tr><tr><td>0%</td><td>5%</td><td>10%</td><td>15%</td><td>20%</td><td>25%</td></tr><tr><td>Clean Model</td><td>98.38%</td><td>75.87%</td><td>58.62%</td><td>40.68%</td><td>33.10%</td><td>22.47%</td></tr><tr><td>(10 × 5)Exhaustive Search</td><td>94.00%</td><td>91.46%</td><td>86.67%</td><td>76.39%</td><td>68.12%</td><td>55.57%</td></tr><tr><td>(10 × 5)Gradient Based Search</td><td>93.77%</td><td>86.67%</td><td>80.57%</td><td>75.00%</td><td>64.90%</td><td>54.44%</td></tr><tr><td>(7 × 7)Exhaustive Search</td><td>92.27%</td><td>89.02%</td><td>82.93%</td><td>78.40%</td><td>66.90%</td><td>55.92%</td></tr><tr><td>(7 × 7)Gradient Based Search</td><td>94.69%</td><td>90.68%</td><td>88.41%</td><td>81.79%</td><td>72.30%</td><td>58.71%</td></tr></table>
+
+Table 22: Effectiveness of DOA (50 PGD iterations) against the adversarial patch attack on traffic sign data.  
+
+<table><tr><td rowspan="2"></td><td colspan="6">Size of Attacking Region</td></tr><tr><td>0%</td><td>5%</td><td>10%</td><td>15%</td><td>20%</td><td>25%</td></tr><tr><td>Clean Model</td><td>98.38%</td><td>75.87%</td><td>58.62%</td><td>40.68%</td><td>33.10%</td><td>22.47%</td></tr><tr><td>(10 × 5)Exhaustive Search</td><td>96.42%</td><td>93.90%</td><td>90.42%</td><td>80.66%</td><td>67.77%</td><td>53.92%</td></tr><tr><td>(10 × 5)Gradient Based Search</td><td>93.54%</td><td>90.33%</td><td>85.80%</td><td>79.79%</td><td>69.86%</td><td>49.74%</td></tr><tr><td>(7 × 7)Exhaustive Search</td><td>87.08%</td><td>82.49%</td><td>75.96%</td><td>70.99%</td><td>60.10%</td><td>52.53%</td></tr><tr><td>(7 × 7)Gradient Based Search</td><td>94.69%</td><td>90.33%</td><td>89.29%</td><td>78.92%</td><td>70.21%</td><td>58.54%</td></tr></table>
+
+# G EXAMPLES OF PHYSICALLY REALIZABLE ATTACK AGAINST ALL DEFENSE MODELS
+
+# G.1 FACE RECOGNITION
+
+![](images/ee95538974a05217c231bd6ba6f4e4366c43375a862de04a28801c490c6a4418.jpg)  
+Original image
+
+![](images/515392bb00546eac4bb729f681a63e07419595cfb4e54f288e58f590f831a39c.jpg)  
+Adv. image
+
+![](images/56be58ec299ae9caa65e26089364e7610fe7500736d48ad3d5cd556bc883c07b.jpg)  
+Adv. training  
+A.R. Rahman
+
+![](images/b09c1a0ba94ccff7da843ffd21cb35991b53a48a258523c358430385276e6f20.jpg)  
+Rand. smoothing  
+Aaron Yoo
+
+![](images/7165d0b50918e5ddc555634766e3870887ee0dc0e980fda9f9009bdf0403334d.jpg)  
+DOA  
+Aaron Tveit
+
+![](images/0d0e765a201b00bd758851d5375a76c071972de9aceed17eb1131c15a1a9d813.jpg)  
+Aaron Tveit  
+Abbie Cornish  
+Figure 11: Examples of the eyeglass attack on face recognition. From left to right: 1) the original input image, 2) image with adversarial eyeglass frames, 3) face predicted by a model generated through adversarial training, 4) face predicted by a model generated through randomized smoothing, 5) face predicted (correctly) by a model generated through DOA. Each row is a separate example.
+
+![](images/fd4f09f554f2b9bb89f75ee808a8c0aeccfe3b70dbb19429a856426d1f76993e.jpg)  
+Aaron Tveit  
+Abbie Cornish
+
+![](images/804d2be12780bf2562f240cf9ecd6d9c99206defea1c42b4739176d66439b91f.jpg)  
+Abigail Breslin
+
+![](images/3fc0e7a6e871378bf6a95652e0cfc45023e37e43afa5328dc6a69297b14b1987.jpg)  
+Aaron Tveit
+
+![](images/aa7957639e27d3a480b464992ef29cec131bde6cc67d4c36427b98c4bf50b936.jpg)  
+Abigail Cornish
+
+# G.2 TRAFFIC SIGN CLASSIFICATION
+
+![](images/9fce9479c74e134ee8ffbf92f565a6cf18cf6e1952a617bd938e7e52273967aa.jpg)  
+Original image  
+Stop sign
+
+![](images/4f54eaeb7d9a903ddbeb64d6e4af5688282f315c0cd30a5385459160f8045560.jpg)  
+Adv. image  
+Stop sign
+
+![](images/017bf966d1d79b85a8e534a8497a87d7c60b0be9552685a85c3f21787039d3a5.jpg)  
+Adv. training  
+Added lane
+
+![](images/310cdac32a7b5f4df9ce1b1283021cf67c1ef187716af46b53430d9d614e7790.jpg)  
+Rand. smoothing  
+Speed limit 35
+
+![](images/302753eeee2e75c0bdd1fe4711be4350a21be7f83ca40833750fb0c40206579c.jpg)  
+DOA  
+Stop sign
+
+![](images/1ebb0fda846e1910f0a649a4254f0f719254a2b858f110f997a5e2a49e2840b4.jpg)  
+Stop sign  
+Figure 12: Examples of the stop sign attack. From left to right: 1) the original input image, 2) image with adversarial eyeglass frames, 3) face predicted by a model generated through adversarial training, 4) face predicted by a model generated through randomized smoothing, 5) face predicted (correctly) by a model generated through DOA. Each row is a separate example.
+
+![](images/a0b5880db65937a05a9e8e5e0b37c88e2211aef2a27738ed52267859f45c250c.jpg)  
+Stop sign
+
+![](images/35a69361d3f68860699b8042f5487e226d7f0bb8a22bdb56d7f4dff4e0b383ae.jpg)  
+Speed limit 35
+
+![](images/bcb843cc3d275c07945140f7d90eb7b76118e6a95e2d4a07eac588e236bd9386.jpg)  
+Speed limit 35
+
+![](images/a055b793bb83cf0e224e4fa317595b00464ebf5d841a4be99a2c7bcba2d8a444.jpg)  
+Stop sign
+
+# H EFFECTIVENESS OF DOA METHODS AGAINST  $l_{\infty}$  ATTACKS
+
+For completeness, this section includes evaluation of DOA in the context of  $l_{\infty}$ -bounded attacks implemented using PGD, though these are outside the scope of our threat model.
+
+Table 23: Effectiveness of DOA (50 PGD iterations) against 20 Iterations  $L_{\infty}$  Attacks on Face Recognition  
+
+<table><tr><td rowspan="2"></td><td colspan="5">Attack Strength</td></tr><tr><td>ε = 0</td><td>ε = 2</td><td>ε = 4</td><td>ε = 8</td><td>ε = 16</td></tr><tr><td>Clean Model</td><td>98.94%</td><td>44.04%</td><td>1.70%</td><td>0%</td><td>0%</td></tr><tr><td>(100 × 50)Exhaustive Search</td><td>98.72%</td><td>39.79%</td><td>1.06%</td><td>0%</td><td>0%</td></tr><tr><td>(100 × 50)Gradient Based Search</td><td>98.51%</td><td>40.64%</td><td>3.40%</td><td>0%</td><td>0%</td></tr><tr><td>(70 × 70)Exhaustive Search</td><td>98.51%</td><td>35.74%</td><td>0.43%</td><td>0%</td><td>0%</td></tr><tr><td>(70 × 70)Gradient Based Search</td><td>97.45%</td><td>33.40%</td><td>0.43%</td><td>0%</td><td>0%</td></tr></table>
+
+Table 24: Effectiveness of DOA (50 PGD iterations) against 20 Iterations  $L_{\infty}$  Attacks on Traffic Sign Classification  
+
+<table><tr><td rowspan="2"></td><td colspan="5">Attack Strength</td></tr><tr><td>ε = 0</td><td>ε = 2</td><td>ε = 4</td><td>ε = 8</td><td>ε = 16</td></tr><tr><td>Clean Model</td><td>98.69%</td><td>89.54%</td><td>61.58%</td><td>24.65%</td><td>5.14%</td></tr><tr><td>(10 × 5)Exhaustive Search</td><td>95.87%</td><td>91.55%</td><td>76.57%</td><td>39.02%</td><td>7.75%</td></tr><tr><td>(10 × 5)Gradient Based Search</td><td>96.46%</td><td>91.81%</td><td>78.83%</td><td>46.86%</td><td>7.84%</td></tr><tr><td>(7 × 7)Exhaustive Search</td><td>92.94%</td><td>89.54%</td><td>77.09%</td><td>42.77%</td><td>5.83%</td></tr><tr><td>(7 × 7)Gradient Based Search</td><td>95.59%</td><td>91.20%</td><td>79.52%</td><td>46.86%</td><td>6.70%</td></tr></table>
+
+Table 23 presents results of several variants of DOA in the context of PGD attacks in the context of face recognition, while Table 24 considers these in traffic sign classification. The results are quite consistent with intuition: DOA is largely unhelpful against these attacks. The reason is that DOA fundamentally assumes that the attacker only modifies a relatively small proportion ( $\sim 5\%$ ) of the scene (and the resulting image), as otherwise the physical attack would be highly suspicious.  $l_{\infty}$  bounded attacks, on the other hand, modify all pixels.
+
+# I EFFECTIVENESS OF DOA METHODS AGAINST  $l_{0}$  ATTACKS
+
+In addition to considering physical attacks, we evaluate the effectiveness of DOA against Jacobian-based saliency map attacks (JSMA) (Papernot et al., 2015) for implementing  $l_{0}$ -constraint adversarial examples. As Figure 13 shows, in both the face recognition and traffic sign classification, DOA is able to improve classification robustness compared to the original model.
+
+![](images/e0387e8f1f5aaa53421cf0ee80ef09cd8a61d2fff53a750c256592c9b11a783d.jpg)  
+Face Recognition
+
+![](images/157938808bec67553b81230e442b82a0acd3d3e0c9c8194173577a60204d4551.jpg)  
+Traffic Sign Classification  
+Figure 13:  $L_{0}$  attacks on face recognition and traffic sign classification.
+
+# J EFFECTIVENESS OF DOA AGAINST OTHER MASK-BASED ATTACKS
+
+To further illustrate the ability of DOA to generalize, we evaluate its effectiveness in the context of three additional occlusion patterns: a union of triangles and circle, a single larger triangle, and a heart pattern.
+
+As the results in Figures 14 and 15 suggest, DOA is able to generalize successfully to a variety of physical attack patterns. It is particularly noteworthy that the larger patterns (large triangle—middle of the figure, and large heart—right of the figure) are actually quite suspicious (particularly the heart pattern), as they occupy a significant fraction of the image (the heart mask, for example, accounts for  $8\%$  of the face).
+
+![](images/ec20a7c7c20911153a9fe0023f2fbeb68e9c81d361799b9a759348741d74a402.jpg)  
+Mask1
+
+![](images/5914996fa6822a64d59f75bb642bd8fdbd339d0d3dc2c6ae14ce4194a4053848.jpg)  
+Abbie Cornish
+
+![](images/ed4fe69f2f317cd7bab66338fd6c2c7eb8241d79e93439f7767ca5310c788c8a.jpg)  
+Mask2
+
+![](images/66ef70b5e1377064565bf8f666a9a56823772f732dc5831dc4d8d47ebac24f3a.jpg)  
+Abbie Cornish
+
+![](images/2dd9d2f0838778c08bd7f68f7472d2f99d40be14a994977ffdfc72986a4c4b2d.jpg)  
+Mask3
+
+![](images/1327840d35d8f7e55fa4e7c9d902ede50792e0e0c7ade32d2f0cdf578d427356.jpg)  
+Abbie Cornish
+
+![](images/8bd5cc81d28ca55d6a0ff47f8648fc8d27998ef2fc5cca35c405579bdd4a1f0f.jpg)  
+Figure 14: Additional mask-based attacks on face recognition.
+
+![](images/b2fd42013b9e69e9b104e2ed1ca1a39b066fddf9cc79852ee1b1cc0bc0f2852b.jpg)
+
+![](images/42295f1c2017e22221e2b74b6fb5cb0340e4ee05463b9798961db5332baabd12.jpg)
+
+![](images/41dbfd42353dcd7aad89cd3c47d8aa3ee1003eae33030ff41e4f80b266a0eec3.jpg)  
+Mask1
+
+![](images/1dd23b05d13048c4fa0449440ba12d5b54ae7e17b0a8586124eb20f846f246f7.jpg)  
+Keep Right
+
+![](images/20c9902ff71d50252c224313e09f24d1ac324a5c6c18cb2d3e8fd7758f3b1019.jpg)  
+Mask2
+
+![](images/f0a7144705217f61f471ce4f5f75c42e2b099f703fcb2d634d5bd569ed6cdb1d.jpg)  
+Keep Right
+
+![](images/02e7f1eefedf1a92b36ef13a2197222eb3f2a91d907ad8396e1c742f475f6787.jpg)  
+Mask3
+
+![](images/84efea936c39c6e642e5d1a7e8db463f1c794e9267f9b1208b70147c2770fd47.jpg)  
+Keep Right
+
+![](images/0f4c51f8ebd394f35720199c75ce183c1805eda295fa067945b4f73f42c351df.jpg)  
+Figure 15: Additional mask-based attacks on traffic sign classification.
+
+![](images/9428cf8d039c7ee5d3c3ff56457080b881b1292542db8d1ea930dfedf470d5af.jpg)
+
+![](images/e975ed95b81109a8929aeca65e0e0f52f13da5f15ac3c323c584f84f27224144.jpg)
\ No newline at end of file
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+# DEPTH-WIDTH TRADE-OFFS FOR RELU NETWORKS VIA SHARKOVSKY'S THEOREM
+
+Vaggos Chatziafratis
+
+Department of Computer Science
+
+Stanford University
+
+Sai Ganesh Nagarajan & Ioannis Panageas & Xiao Wang
+
+Singapore University of Technology and Design
+
+# ABSTRACT
+
+Understanding the representational power of Deep Neural Networks (DNNs) and how their structural properties (e.g., depth, width, type of activation unit) affect the functions they can compute, has been an important yet challenging question in deep learning and approximation theory. In a seminal paper, Telgarsky highlighted the benefits of depth by presenting a family of functions (based on simple triangular waves) for which DNNs achieve zero classification error, whereas shallow networks with fewer than exponentially many nodes incur constant error. Even though Telgarsky's work reveals the limitations of shallow neural networks, it doesn't inform us on why these functions are difficult to represent and in fact he states it as a tantalizing open question to characterize those functions that cannot be well-approximated by smaller depths.
+
+In this work, we point to a new connection between DNNs expressivity and Sharkovsky's Theorem from dynamical systems, that enables us to characterize the depth-width trade-offs of ReLU networks for representing functions based on the presence of a generalized notion of fixed points, called periodic points (a fixed point is a point of period 1). Motivated by our observation that the triangle waves used in Telgarsky's work contain points of period 3 - a period that is special in that it implies chaotic behaviour based on the celebrated result by Li-Yorke - we proceed to give general lower bounds for the width needed to represent periodic functions as a function of the depth. Technically, the crux of our approach is based on an eigenvalue analysis of the dynamical systems associated with such functions.
+
+# 1 INTRODUCTION
+
+In approximation theory, one typically tries to understand how to best approximate a complicated family of functions using simpler functions as building blocks. For instance, Weierstrass (1885) proved a general result stating that every continuous function can be uniformly approximated as closely as desired by a polynomial. It wasn't until later that Vitushkin (1959) gave quantitative bounds between the approximation error and the polynomial's degree. Drifting away from polynomials and given the recent breakthroughs of deep learning in a variety of difficult tasks like image classification, natural language processing, game playing and self-driving cars, researchers have tried to understand the approximation theory that governs neural networks. This question of neural network expressivity, i.e. how architectural properties like the depth, width or the activation units affect the functions it can compute, has been a fundamental ongoing challenge with a rich history. A classical result by Cybenko (1989), Hornik et al. (1989), Fukushima (1980) demonstrates the expressive power of neural networks: it states that even two layered neural networks (using well known activation functions) can approximate any continuous function on a bounded domain. The caveat is that the size of such networks may be exponential in the dimension of the input, which makes them highly susceptible to overfitting as well as impractical, since one can always add extra layers in their model aiming at increasing the representational power of the neural network.
+
+More recently, in a seminal paper by Telgarsky (2016), it was shown that there exist functions that can be represented by DNNs, i.e., by some particular choice of weights on their edges (and for a wide variety of standard activation units in their layers), yet cannot be approximated by shallow networks unless they are exponentially large. More concretely, he showed that for any positive integer  $k$ , there exist neural networks with  $\Theta(k^3)$  layers,  $\Theta(1)$  nodes per layer, and  $\Theta(1)$  distinct parameters which cannot be approximated by networks with  $\mathcal{O}(k)$  layers, unless they have  $\Omega(2^k)$  nodes. At a high level, he uses the number of oscillations present in certain functions as a notion of "complexity" that distinguishes between deep and shallow networks' representation capabilities via the following three facts: a) functions with few oscillations poorly approximate functions with many oscillations, b) functions computed by networks with few layers must have few oscillations and c) functions computed by networks with many layers can have many oscillations.
+
+Our main contribution is a novel connection between the theory of dynamical systems and the representational power of DNNs via the well-studied notion of periodic points, a notion that captures the important notion of fixed points of a continuous function.
+
+Definition 1 (Period). We say that a (continuous) Lipschitz function  $f:[0,1]\to [0,1]$  contains a point of period  $n\geq 1$  if there exists a point  $x_0\in [0,1]$  such that:
+
+$$
+f ^ {n} \left(x _ {0}\right) = x _ {0} \quad a n d \quad \text {(p o i n t o f p e r i o d} n)
+$$
+
+$$
+f ^ {k} \left(x _ {0}\right) \neq x _ {0}, \forall 1 \leq k \leq n - 1.
+$$
+
+In particular, all numbers in  $C = \{x_0, f(x_0), f(f(x_0)), \ldots, f^{n-1}(x_0)\}$  are distinct, each of which is a point of period  $n$  and the set  $C$  is called a cycle (or orbit) of period  $n$ . Observe that since  $f: [0,1] \to [0,1]$  is continuous, it certainly has at least one point of period 1, which is called a fixed point.
+
+For the rest of this paper, we focus on (continuous) Lipschitz functions  $f:[0,1]\to [0,1]$ , unless otherwise stated. Note that the choice of interval [0, 1] is for simplicity of our presentation and that our results will hold for any closed interval  $[a,b]$ .
+
+As we observe, points of period 3 are contained in both Telgarsky (2016) and Schmitt (2000) constructions and this could as well have been a coincidence, however we show that the existence of periodic points of certain periods are actually one of the reasons explaining why depth is needed to represent functions that contain them (otherwise exponential width is required). Towards this direction, we will make use of a deep result in the literature of iterated dynamical systems, that gave rise to the field of Combinatorial Dynamics (Alseda et al. (2000)), called Sharkovsky's Theorem (Sharkovsky (1964; 1965)).
+
+# 1.1 SHARKOVSKY'S THEOREM
+
+Consider the set of positive natural numbers  $\mathbb{N}^* = \{1, 2, \ldots\}$  and define the following (decreasing) ordering  $\triangleright$  called Sharkovsky's ordering as follows:
+
+$$
+\begin{array}{c} 3 \triangleright 5 \triangleright 7 \triangleright \dots \triangleright (\text {o d d n u m b e r s b i g g e r t h a n o n e}) \\ \triangleright 2 \cdot 3 \triangleright 2 \cdot 5 \triangleright 2 \cdot 7 \triangleright \dots \triangleright (\text {o d d m u l t i p l e s o f t w o b u t n o t t w o}) \\ \triangleright 2 ^ {2} \cdot 3 \triangleright 2 ^ {2} \cdot 5 \triangleright 2 ^ {2} \cdot 7 \triangleright \dots \triangleright (\text {o d d m u l t i p l e s o f f o u r b u t n o t f o u r}) \end{array}
+$$
+
+$$
+\begin{array}{c} \vdots \\ \triangleright \dots \triangleright 2 ^ {4} \triangleright 2 ^ {3} \triangleright 2 ^ {2} \triangleright 2 \triangleright 1 (\text {p o w e r s o f t w o i n d e c r e a s i n g o r d e r}). \end{array}
+$$
+
+This is a total ordering; we write  $l \triangleright r$  or  $r \triangleleft l$  whenever  $l$  is to the left of  $r$ . Sharkovsky showed that this ordering describes which numbers can be periods for a continuous map on an interval; allowed periods need to be a suffix of the Sharkovsky ordering:
+
+Theorem 1.1 (Sharkovsky's "Forcing" Theorem, Sharkovsky (1964; 1965)). Let  $I$  be a closed interval and  $f: I \to I$  be a continuous map. If  $n$  is a period for  $f$  and  $n \triangleright n'$ , then  $n'$  is also a period for  $f$ .
+
+Remark 1.1. Note that the number 3 is the maximum period according to Sharkovsky's ordering, so an important corollary is that a function having a point of period 3, must also have points of any
+
+period. This special corollary is a weaker version of Sharkovsky's theorem and was proved some years later<sup>2</sup> in a celebrated result by Li & Yorke (1975), who coined the term “chaos” as used in Mathematics.
+
+We conclude the subsection with the definition of a prime period of a function  $f$ .
+
+Definition 2 (Prime period). A function  $f$  has prime period  $n$  as long as it has a cycle of period  $n$ , but has no cycles with period greater than  $n$  according to the Sharkovsky ordering.
+
+For example, in the interval  $[0,1]$ , the function  $f(x) = 1 - x$  has prime period 2, since  $f(f(x)) = 1 - (1 - x) = x$  so all points are periodic with period 2, except the fixed point at  $1/2$ .
+
+Before formally stating our main theorems, we present an illustrative example inspired from Telgarsky's triangle wave construction and we connect it to DNNs' sensitivity to weight perturbations and their representational power.
+
+# 1.2 SENSITIVITY ANALYSIS - A MOTIVATING EXAMPLE
+
+An important ingredient in Telgarsky's proof, was the "triangular wave" function (sometimes referred to as the tent map or sawtooth) depicted in Figure 1b and given by:
+
+$$
+t (x; 2) = \left\{ \begin{array}{l l} 2 x, & \text {i f} 0 \leq x \leq \frac {1}{2} \\ 2 (1 - x), & \text {i f} \frac {1}{2} <   x \leq 1 \\ 0, & \text {o t h e r w i s e} \end{array} \right.
+$$
+
+He shows that the composition of  $t(x;2)$  with itself  $k$  times (denoted by  $t^k (x;2)$ ), will create exponentially (in  $k$ ) many oscillations and as a result he is able to show a separation for the classification error when using a shallow vs a deep neural network as a predictor.
+
+Our starting point is the observation that the triangular wave function  $t(x;2)$  contains points of period 3, e.g.  $\left(\frac{2}{9} \rightarrow \frac{4}{9} \rightarrow \frac{8}{9} \rightarrow \frac{2}{9}\right)$ . It follows in particular, that  $t(x;2)$  exhibits Li-Yorke Chaos (Li & Yorke (1975)) in the sense that it contains all periods. The compositions of such functions will look highly complex (see Figure 2) and in fact Telgarsky heavily relied on the highly oscillatory behavior of  $t(x;2)$  to prove his depth separation result.
+
+However, his result doesn't inform us on what would happen if one used a slightly modified version of the triangle wave  $t(x;2)$ . Observe that since a simple neural network with one hidden layer can represent the function  $t(x;2)$ , the question is basically equivalent to asking how modifying the weights on the edges of the neural network can affect its representational power (see Figure 1), hence the title of the current subsection. The main question is can we have a general theory that informs us on when will the function composition be hard to represent and when not? Our paper's main point is to provide an answer by checking if the function at hand has a simple property, relating to the presence of chaotic behavior.
+
+To illustrate our point, consider the generalized triangle wave function  $t(x;\mu)$  parameterized by  $\mu$ :
+
+$$
+t (x; \mu) = \left\{ \begin{array}{l l} \mu x, & \text {i f} 0 \leq x \leq \frac {1}{2} \\ \mu (1 - x), & \text {i f} \frac {1}{2} <   x \leq 1 \\ 0, & \text {o t h e r w i s e} \end{array} \right.
+$$
+
+This function parameterized by  $\mu$  ranges from  $[0, \mu / 2]$  and is closely related to the logistic map  $f(x) := rx(1 - x)$  used in Schmitt (2000) and exhibits a variety of limiting behaviors: for instance, it converges to a stable fixed point when  $\mu \leq 1$ , it exhibits chaos when  $\mu = 2$  etc. Instead of  $\mu = 2$ , if we set  $\mu = 1$ , we get the network depicted in Figure 1c, 1d.
+
+![](images/7bffa8fc3353f027d6ee5d64d21f375c7ef9ede70f28646b868c8b9aefb6ef9b.jpg)  
+(a) Neural net with the appropriate weights and biases (the value indicated inside the hidden and output neurons).
+
+![](images/07601ef7bc5eef6a74d43c1988da45367c323e11ff78665c8910f30d83ac0b73.jpg)  
+(b) Tent map with  $\mu = 2$
+
+![](images/dd54a25bcbeeccf59441e153f18553e7f41a95660020f4953ec4a09d48d492fe.jpg)  
+(c) Neural net with the appropriate weights and biases (the value indicated inside the hidden and output neurons).
+
+![](images/8fef3ebb98e6135c389880a74ce9c0fdffdf51d6cdb3fc40dfa4429962be7347.jpg)  
+(d) Tent map with  $\mu = 1$  
+Figure 1: The neural network instantiations that are used to create two different tent maps which vary only in the maximum value. This is effected by a small change of weights in the output layer. All activation functions are ReLU's.
+
+Note that compositions of  $t(x;1)$  (created by the same neural network architecture but with slightly different weights), behave completely differently since in the  $\mu = 1$  case, we will not get a highly oscillatory behavior. This can be seen in Figure 3. One difference between the two cases is the relative position of the map with the line  $y = x$  and this seems to be pointing that fixed points and their generalizations i.e. periodic orbits play an important role when dealing with function compositions. Indeed, despite the wide range of possibilities one can expect by composing such functions, as we show, their behavior can be characterized using tools from dynamical systems; the exponential growth in complexity (or lack thereof) of these compositions can be explained by invoking a fundamental property of these continuous functions on bounded intervals which is the existence (or not) of periodic points of certain periods.
+
+Similarly, we can argue about changing the parameters of the logistic map which is given by  $f(x; r) \coloneqq rx(1 - x)$  used in Schmitt (2000) for sigmoidal networks (where  $f(x; 4)$  was used). The properties of the logistic map are well known and was first studied by Robert May and Mitchell Feigenbaum (May (1976) and Feigenbaum (1976)). It is known that as one varies the parameter  $r$ , the logistic map gives rise to a plethora of different behaviors, hence the same is true for when one slightly perturbs the weights of a neural net used to represent the map. Please refer to Appendix C for some figures that illuminate these differences in the logistic map.
+
+# 1.3 INFORMAL STATEMENTS OF MAIN THEOREMS
+
+We demonstrate that a simple property of  $f$  governs the depth-width trade-offs in order to represent it and we give quantitative bounds for them. This simple property has to do with the periods that the function  $f$  contains. Informally, our first main theorem states that if a function  $f$  contains periodic points with certain periods, then composing  $f$  with itself many times, will result in exponentially many oscillations, giving rise to complicated behaviors and chaos:
+
+Theorem 1.2. Let  $f:[0,1] \to [0,1]$  be a continuous function. Assume that there exists a cycle of period  $n$ , where  $n = m \cdot p$  with  $p$  being an odd number greater than one and with  $m$  being a
+
+![](images/f5c4fa8bdcc260937942869aabb47416f19491fb39eaafb277c3e1c4809c4137.jpg)  
+(a)  $f(x) \coloneqq 3.9 * x(1 - x)$  on the interval [0, 1]
+
+![](images/628e76e9abb624ef5e93aef9fa65750ba89cb9cbc3bf665ef1ccd94fb0bf9e28.jpg)  
+(b)  $f^{10}(x)$  of the map on the left (zoomed in)  
+Figure 2: Compositions of the logistic map  $f(x) = 3.9x(1 - x)$  defined on the interval [0, 1]. This map is well known to exhibit chaos and in the above figure has non-vanishing oscillations that grow with the number of compositions, albeit irregularly.
+
+![](images/ea5db6de88a6bdbaa7692970d9512e888b67dcff7d873af4d00251c7edb51c7e.jpg)  
+(a)  $t(x;\mu)$  for the tent map with  $\mu = 2$  (blue) and  $\mu = 1$  (red)  
+Figure 3: Compositions of  $t(x; \mu)$  with different parameters  $\mu = 2$  and  $\mu = 1$  are shown. The compositions create (exponential) non-vanishing oscillations when  $\mu = 2$ , however the compositions remain unchanged when  $\mu = 1$ .
+
+![](images/1d87a23c7f98946f484c8cb330c7cc6785663808d72495a7e13aff1ba22c8d33.jpg)  
+(b)  $t^6 (x;\mu)$  for the tent map with  $\mu = 2$  (blue) and  $\mu = 1$  (red).
+
+power of two (it might be  $m = 1$ ). Then, there exist  $x, y \in [0,1]$  such that the function  $f^{mt}$  (taking mt compositions of  $f$  with itself) "oscillates" (also look Definition 4) at least  $\rho^t$  times between  $x$  and  $y$  for all  $t \in \mathbb{N}^*$ , where  $\rho$  is the positive root greater than one of the polynomial equation  $\lambda^{p - 1} - \lambda^{p - 2} - 1 = 0$ .
+
+Our second main theorem then draws the connection between the number of oscillations a function has and the depth-width trade-offs needed:
+
+Theorem 1.3. Let  $k$  be a positive integer and  $f$  be a function as above. We set  $\rho$  to be the positive root greater than one of the polynomial equation  $\lambda^{p - 1} - \lambda^{p - 2} - 1 = 0$ . We can construct a sequence of points  $(x_{i},y_{i})_{i = 1}^{2n}$  with  $n\coloneqq \frac{|\rho^k|}{2}$  such that the classification error of the function  $f^{mk}$  is zero, whereas the classification error of any neural network with  $l$  layers and  $u$  nodes per layer, where  $u\leq \frac{\rho^{\frac{k}{l}}}{8}$ , necessarily has classification error  $\geq \frac{1}{4}$ .
+
+Formal statements for the two theorems can be found in Section 3 and Section 4.
+
+Using these theorems, we draw connections with previous results Telgarsky (2016), Schmitt (2000) in a unified way, thus identifying chaotic behavior as the main underlying thread for depth-width trade-offs. Technically, our approach is based on an eigenvalue analysis of certain matrices associated with such periodic functions.
+
+# 1.4 OTHER RELATED WORK
+
+Understanding the benefits of depths on the expressive power a specific computational model can have, is an important area of research spanning different computational models and results come in the flavor of depth separation arguments. Roughly speaking, many of the results in this area rely on a suitably defined notion of "complexity" of a function we would like to represent, and then proceed by proving that under this notion, deep models have significantly more power than shallower models. For example, if the computational model of interest is the family of boolean or threshold circuits, depth lower bounds are given in Hastad (1986); Rossman et al. (2015); Håstad (1987); Parberry et al. (1994); Kane & Williams (2016). Furthermore, people have analyzed sum-product networks (summation and product nodes) and studied trade-offs for depth (Delalleau & Bengio (2011); Martens & Medabalimi (2014)).
+
+Coming closer to neural networks computation where the activation units can be general real-valued functions, important previous results include Eldan & Shamir (2016); Telgarsky (2015; 2016); Schmitt (2000); Montufar et al. (2014); Malach & Shalev-Shwartz (2019); Poole et al. (2016); Raghu et al. (2017); Arora et al. (2016); Liang & Srikant (2016); Kileel et al. (2019). Regarding the aforementioned notions of "complexity" used in depth separation arguments, examples include the notion of global curvature (Poole et al. (2016)), trajectory length (Raghu et al. (2017)), number of oscillations (Telgarsky (2015; 2016) and Schmitt (2000)), number of linear regions (Montufar et al. (2014)), fractals (Malach & Shalev-Shwartz (2019)), dimensions for algebraic varieties (Kileel et al. (2019)) and more. Our work is more closely related to Telgarsky (2015; 2016), and Schmitt (2000) since it is easy to see that their maps are chaotic, but we conjecture that many of the notions of complexity introduced in this line of research to showcase benefits of depth actually arise due to chaotic behavior. In this sense, we conjecture that chaotic behavior is the main culprit for the failure of neural networks to represent certain functions, unless they are sufficiently deep (or have exponential width). Moreover, other works that have exploited the powerful result by Li-Yorke (in online learning frameworks) are Palaiopanos et al. (2017); Chotibut et al. (2019).
+
+# 2 FURTHER BACKGROUND: THE COVERING LEMMA
+
+The crux of the proof of Sharkovsky's theorem provided by Burns & Hasselblatt (2011) contains a covering lemma that will be our starting point to prove our main results. Before we proceed with the statement of the Covering Lemma, we provide one more important definition.
+
+Definition 3 (Covering relation). Let  $f$  be a function and  $I_1, I_2$  be two closed intervals. We say that  $I_1$  covers  $I_2$  under  $f$ , denoted by  $I_1 \xrightarrow{f} I_2$  as long as  $I_2 \subseteq f(I_1)$ .
+
+![](images/442c366ba2e59cf67b619392001e56e979fd09e8e4d71310689a179ac2440a52.jpg)  
+(a) Covering Lemma 1 relations for cycle of period three  $(r = 1)$ .
+
+![](images/79f10f72fd1792a228c60484421f956b664408c94f5e2365d2e0b877e7da740b.jpg)  
+(b) Covering Lemma 1 relations for cycle of odd period at least three.  
+Figure 4: The covering relations of intervals  $J_0, \dots, J_r$  from Lemma 1. Observe that the graph is a directed cycle with a self loop at interval  $J_0$ . Note that there might be more relations ("edges").
+
+For example, the triangle wave  $t(x;2)$  that has the period 3 point  $\frac{2}{9}$  (recall  $\frac{2}{9} \to \frac{4}{9} \to \frac{8}{9} \to \frac{2}{9}$ ) naturally defines two intervals  $I_1 = [\frac{2}{9},\frac{4}{9}]$  and  $I_2 = [\frac{4}{9},\frac{8}{9}]$  with the covering relations:  $I_1 \xrightarrow{f} I_2$ ,  $I_2 \xrightarrow{f} I_2$  and  $I_2 \xrightarrow{f} I_1$ .
+
+Lemma 1 (Covering Lemma for odd periods). Let  $f:[0,1]\to [0,1]$  be a continuous function and assume  $f$  has a cycle  $C$  of period  $n$ , where  $n > 1$  is an odd number. Denote  $\beta_0,\dots,\beta_{n - 1}\in C$  the elements of the cycle in increasing order and define the sequence of closed intervals  $I_0,\ldots ,I_{n - 2}$  where  $I_{i} = [\beta_{i},\beta_{i + 1}]$  (they have pairwise disjoint interiors). Then, there exists a sub-collection of the aforementioned intervals (not necessarily in the same ordering)  $J_{0},\ldots J_{r}$  with  $1\leq r\leq n - 2$  such that the following covering relation holds:
+
+1.  $J_{i}\xrightarrow{f}J_{i + 1},$  for  $1\leq i\leq r - 1,$  
+2.  $J_{r}\stackrel {f}{\to}J_{0}$  and  $J_0\stackrel {f}{\to}J_0\cup J_1$
+
+For a pictorial illustration of the Covering Lemma, see Figure 4. In particular, observe that for  $n = 3$  we get  $r = 1$  so the covering relation is as in Figure 4. We conclude this section with the formal definition of crossings (or oscillations) and we refer the reader to Figure 5 for some examples.
+
+Definition 4 (Crossings). We say that a continuous function  $f:[0,1] \to [0,1]$  crosses the interval  $[x,y]$  with  $x,y \in [0,1]$  if there exist  $a,b$  such that  $f(a) = x$  and  $f(b) = y$ . Moreover we denote  $C_{x,y}(f)$  the number of times  $f$  crosses  $[x,y]$ . That is  $C_{x,y}(f) = t$  if there exist numbers  $a_1,b_1 < a_2,b_2 < \dots < a_t,b_t$  in  $[0,1]$  so that  $f(a_i) = x$  and  $f(b_i) = y$  for all  $1 \leq i \leq t$ . Observe that if  $\mathcal{I}_{f,x,y}$  is used to denote the number of intervals the function  $\tilde{f}_{x,y}(z) := 1[f(z) \geq \frac{x + y}{2}]$  is piecewise constant and partitions  $[0,1]$ , then  $C_{x,y}(f) \leq \mathcal{I}_{f,x,y}$ .
+
+# 3 PERIODS DETERMINE THE NUMBER OF CROSSINGS
+
+# 3.1 PERIOD THAT IS NOT A POWER OF TWO IMPLIES EXPONENTIAL CROSSINGS
+
+In this section, we prove our main theorem, the statement of which is given below. Technically, we make use of the Lemma 1 (Covering Lemma) to show the exponential growth of the number of crossings.
+
+Theorem 3.1. Let  $f:[0,1] \to [0,1]$  be a continuous function. Assume that there exists a cycle of period  $n$  where  $n = m \cdot p$ ,  $p$  is an odd number greater than one and  $m$  being a power of two (it might be  $m = 1$ ). It holds that there exist  $x, y \in [0,1]$  so that  $C_{x,y}(f^{mt})$  is  $c^t$  for all  $t \in \mathbb{N}^*$ , where  $c$  is the positive root greater than one of the polynomial equation  $\lambda^{p - 1} - \lambda^{p - 2} - 1 = 0$ .
+
+Counting the number of oscillations. For a given continuous function  $f:[0,1]\to [0,1]$ , let  $J_0,\ldots ,J_r$ , where  $1\leq r\leq n - 2$ , be the intervals as promised from Lemma 1. We define a sequence of vectors  $\delta^t\in \mathbb{N}^{r + 1}$  such that  $\delta_i^t$  is defined as the number of times the function  $f^t$  crosses the interval  $J_{i}$  for all  $0\leq i\leq r$ . In particular we define  $f^0$  to be the identity function and hence  $\delta^0 = (1,\dots ,1)$  (all ones vector). For what follows, we will try to express recursively  $\delta^t$  in terms of  $\delta^{t - 1}$  and in the end we will show that  $\delta_0^k$  is  $\Omega (c^{k})$  where  $c$  is some constant that depends on  $r$ . To build some intuition, we first analyze the case of period three and then we prove the general case.
+
+# 3.1.1 WARM UP: THE CASE OF PERIOD 3 AND THE FIBONACCI SEQUENCE
+
+Assume that  $f$  has a cycle of period 3, that is the numbers  $\{x_0, f(x_0), f^2(x_0)\}$  are distinct and  $f^3(x_0) = x_0$  for some  $x_0 \in [0,1]$ . Let  $\beta_0 < \beta_1 < \beta_2$  be the numbers  $x_0, f(x_0), f^2(x_0)$  in increasing order. We define  $I_0 = [\beta_0, \beta_1]$  and  $I_1 = [\beta_1, \beta_2]$ . From Lemma 1, when  $n = 3$ , we can see that  $r = 1$  and thus we have the following possibilities for the covering relations:
+
+Either  $I_0 \xrightarrow{f} I_0 \cup I_1$ ,  
+or  $I_1 \xrightarrow{f} I_0 \cup I_1$ .
+
+We define  $J_0$  to be the interval among  $I_0, I_1$  that involves the self-loop covering and  $J_1$  to be the remaining interval. Define  $\delta^t \in \mathbb{N}^2$  as above, and so we get that:
+
+$$
+\left( \begin{array}{l} \delta_ {0} ^ {t + 1} \\ \delta_ {1} ^ {t + 1} \end{array} \right) \geq \left( \begin{array}{l l} 1 & 1 \\ 1 & 0 \end{array} \right) \left( \begin{array}{l} \delta_ {0} ^ {t} \\ \delta_ {1} ^ {t} \end{array} \right), \tag {3.1}
+$$
+
+where  $\delta_0^0 = 1$  and  $\delta_1^0 = 1$ . The matrix  $A := \left( \begin{array}{ll}1 & 1\\ 1 & 0 \end{array} \right)$  can be interpreted as the adjacency matrix that corresponds to the covering relations between  $J_{0}, J_{1}$  (which consists of a directed cycle with a self-loop at vertex  $J_{0}$ ). The reason we have an inequality instead of an equality is because the Covering Lemma only guarantees that the number of times  $J_{0}$  "covers"  $J_{0}$  and  $J_{1}$  is at least one and not necessarily exactly one.
+
+We set  $\alpha^0 = \delta^0$  and we define  $\alpha^{t + 1} = A\alpha^t$ . It is clear that  $\delta^t\geq \alpha^t$  (entry-wise) for all  $t\in \mathbb{N}$ . Moreover,  $\alpha_0^t$  is the well-known Fibonacci sequence  $F_{t + 1}$  (with  $F_{0} = F_{1} = 1$ ), therefore  $\alpha_0^t = \frac{\left(\frac{1 + \sqrt{5}}{2}\right)^{t + 2} - \left(\frac{1 - \sqrt{5}}{2}\right)^{t + 2}}{\sqrt{5}}$ . We conclude that  $\delta_0^t\geq \left(\frac{1 + \sqrt{5}}{2}\right)^t$ . See also Figure 5 for a pictorial illustration about the proof for  $t = 1,2,3,4$ .
+
+# 3.1.2 EVERY PERIOD GREATER THAN 3 BUT NOT POWER OF TWO
+
+Assume that  $f$  has a cycle of period  $n > 3$  with  $n$  odd, that is the numbers  $\{x_0, f(x_0), f^2(x_0), \ldots, f^{n-1}(x_0)\}$  are distinct and  $f^n(x_0) = x_0$  for some  $x_0 \in [0,1]$ . Let  $\beta_0 < \beta_1 < \beta_2 < \ldots < \beta_{n-1}$  be the numbers  $x_0, f(x_0), f^2(x_0), \ldots, f^{n-1}(x_0)$  in increasing order. We define  $I_i = [\beta_i, \beta_{i+1}]$  for  $0 \leq i \leq n-2$ . From Lemma 1 it follows that there is a subcollection of the intervals  $I_0, \ldots, I_{n-2}$  (with not necessarily the same ordering)  $J_0, \ldots, J_r$  ( $1 \leq r \leq n-2$ ) such that
+
+1.  $J_{i}\xrightarrow{f}J_{i + 1}$  , for  $1\leq i\leq r - 1$  
+2.  $J_{r}\stackrel {f}{\to}J_{0}$  and  $J_0\stackrel {f}{\to}J_0\cup J_1$
+
+The interval  $J_0$  is the one that involves the self-loop covering. As in the case for  $n = 3$ , we define  $\delta^t$  which is in  $\mathbb{N}^{r + 1}$ , with  $\delta_i^t$  capturing the number of times  $f^t$  crosses the interval  $J_{i}$ . We get that:
+
+$$
+\left( \begin{array}{c} \delta_ {0} ^ {t + 1} \\ \delta_ {1} ^ {t + 1} \\ \vdots \\ \delta_ {r} ^ {t + 1} \end{array} \right) \geq A \left( \begin{array}{c} \delta_ {0} ^ {t} \\ \delta_ {1} ^ {t} \\ \vdots \\ \delta_ {r} ^ {t} \end{array} \right), \tag {3.2}
+$$
+
+where  $\delta^0 = (1,\ldots ,1)$  (all ones vector) and  $A\in \mathbb{R}^{(r + 1)\times (r + 1)}$  is defined to be:
+
+$$
+\left\{ \begin{array}{l} A _ {j i} = 1, \text {i f} i = 0, j = 0 \\ A _ {j i} = 1, \text {i f} j = i + 1 \text {a n d} 0 \leq i \leq r - 1 \\ A _ {j i} = 1, \text {i f} i = r, j = 0 \\ A _ {j i} = 0, \text {o t h e r w i s e} \end{array} \right. \tag {3.3}
+$$
+
+In words,  $A$  is the adjacency matrix of a graph with  $r + 1$  nodes that is a directed cycle that involves a self-loop at vertex  $J_0$ . We define  $\alpha^t$  in a similar way as in the case for period three, i.e.,  $\alpha^{t + 1} = A\alpha^t$  and  $\alpha^0 = \delta^0$  so that  $\delta^t\geq \alpha^t$  (entry-wise) for all  $t\in \mathbb{N}$ . We can easily observe that the following holds:  $\alpha^{t + 1} = A^{t + 1}\alpha^0$ .
+
+Our next plan is to compute a lower bound on the spectral radius of the matrix  $A^{\top}$  (denoted by  $\mathrm{sp}(A^{\top})$ ) with the following claim (proof in Appendix A).
+
+Claim 3.1. The characteristic polynomial of  $A^\top$  is:
+
+$$
+\pi (\lambda) = \lambda^ {r + 1} - \lambda^ {r} - 1. \tag {3.4}
+$$
+
+Let us call  $\rho_r$  the largest root in absolute value of the polynomial  $\pi(\lambda)$  in A.1. Since  $A$  is a nonnegative matrix, the largest root in absolute value is actually a positive real number (by the Perron-Frobenius theorem). It is easy to see that the polynomial in A.1 has always a root greater than one and less than two (by Bolzano's theorem, see  $\pi(1) = -1 < 0$  and  $\pi(2) = 2^{r+1} - 2^r - 1 = 2^r - 1 > 0$ ).
+
+Hence we have  $\mathfrak{sp}(A) = \rho_r > 1$ . Furthermore, it is easy to see that since  $A$  is a non-negative matrix (and powers of  $A$  are also non-negative), it holds that
+
+$$
+\left\| A ^ {t} \right\| _ {\infty} = \sum_ {j = 0} ^ {r} A _ {0 j} ^ {t}
+$$
+
+for all  $t \geq 1$ , that is the row with the largest sum of its entries is the first row (row for  $i = 0$ ). Using the fact that
+
+$$
+\left\| A ^ {t} \right\| _ {\infty} \geq \mathrm {s p} (A ^ {t}) = \rho_ {r} ^ {t},
+$$
+
+that is the spectral radius of a matrix is always at most any matrix norm, we conclude that  $\sum_{j=0}^{r} A_{0j}^{t} \geq \rho_{r}^{t}$ .
+
+The case of odd period greater than three follows by noting that  $\sum_{j=0}^{r} A_{0j}^{t} = \alpha_{0}^{t}$ , thus  $\delta_{0}^{t} \geq \alpha_{0}^{t} \geq \rho_{r}^{t}$ . Observe that for period three, we have that  $r = 1$  and also  $\rho_{1} = \frac{1 + \sqrt{5}}{2}$  (the largest root of  $\lambda^2 - \lambda - 1 = 0$ ).
+
+We would like to make the following two remarks:
+
+Remark 3.1. The spectral radius  $\rho_r$  is strictly decreasing in  $r$ : this is easy to see since  $\rho_r > 1$  and is satisfying the equation  $x^{r+1} - x^r = 1$  (note that  $x^{r+1} - x^r$  is increasing in  $r$  for  $x > 1$ ). This implies that smaller odd periods can potentially have a number of crossings that grows at faster rates than larger odd periods, hence giving rise to more complex behaviors. See also Remark 4.1.
+
+Remark 3.2 (The case of even period but not power of two). Our result above is applied for cycles of period  $n = m \cdot n'$  where  $m$  is a power of two and  $n'$  is an odd number greater than one. The trick is to observe that if a function has cycle of period  $n$ , then  $f^m$  has a cycle of period  $n'$  (which is an odd number greater than one). Therefore, the number of oscillations  $C_{x,y}(f^{mt})$  with  $x, y$  being the endpoints of  $J_0$ , is at least  $\rho_{n'-2}^t$  for  $t \in \mathbb{N}$ .
+
+Proof of Theorem 3.1. The proof now follows from the case analysis carried out in Sections 3.1.1, 3.1.2 and Remark 3.2.
+
+# 3.2 PERIOD THAT IS A POWER OF TWO MAY HAVE POLYNOMIAL CROSSINGS
+
+Lemma 2 (Period power of two - proof in Appendix A). There exist continuous functions  $f$  with prime period  $n$  that is a power of two so that the number of crossings  $C_{x,y}(f^t)$  scales at most polynomially with  $t$  for any  $x, y \in [0,1]$ .
+
+# 4 PERIOD-DEPENDENT LOWER BOUNDS FOR DNNS
+
+Building on Telgarsky (2015; 2016), the representation power of different networks will be measured via the classification error. For a given collection of  $n$  points  $(x_{i},y_{i})_{i = 1}^{n}$  with  $y_{i}\in \{0,1\}$ , one can define the classification error of a function  $g$  to be  $(\tilde{g}$  is just the thresholded value of  $g)$ :
+
+$$
+\mathcal {R} (g) = \frac {1}{n} \sum_ {i = 1} ^ {n} \mathbf {1} [ \tilde {g} (x _ {i}) \neq y _ {i} ]
+$$
+
+In this section, we argue that functions with cycles of period not a power of two, will have compositions for which any shallow neural network will have classification error a positive constant.
+
+Assume we are given a continuous function  $f:[0,1]\to [0,1]$  so that  $f$  has a cycle of period  $m\times p$  where  $p$  is an odd number greater than one and  $m$  is a power of two. From Theorem 3.1, there exist  $x,y\in [0,1]$  so that  $C_{x,y}(f^{tm})$  is at least  $\frac{\rho_{p - 2}^t}{2}$ , where  $\rho_r$  is defined to be the root that is greater than one of the polynomial equation  $\lambda^{r + 1} - \lambda^r -1 = 0$ . We set  $\rho \coloneqq \rho_{p - 2}$ ,  $h\coloneqq f^{k\cdot m}$  and assume that  $g:[0,1]\to [0,1]$  is a neural network with  $l$  layers and  $u$  nodes (ReLU activations) per layer. In Lemma 2.1 of Telgarsky (2015), it is proved that a neural network with  $u$  ReLU units per layer and with  $l$  layers is piecewise affine with at most  $(2m)^l$  pieces.
+
+We define as  $\tilde{h}(z) = \mathbf{1}[h(z) \geq \frac{x + y}{2}]$  and  $\tilde{g}(z) = \mathbf{1}[g(z) \geq \frac{x + y}{2}]$  (note that we changed the threshold to be  $\frac{x + y}{2}$  instead of  $\frac{1}{2}$  that was used in Telgarsky (2015)).
+
+Since  $C_{x,y}(h)$  is at least  $\rho^k$ , it holds that there exist points  $(x_i, y_i)_{i=1}^{2n}$  with  $n := \frac{|\rho^k|}{2}$  such that  $h(x_j) = x$ ,  $y_j = 0$  for  $j$  odd and  $h(x_j) = y$ ,  $y_j = 1$  for  $j$  even. It is clear that for this collection of points the classification error of the function  $h$  is zero, whereas the classification error for function  $g$  is bounded from below by
+
+$$
+\mathcal {R} (g) \geq \frac {n - 4 (2 u) ^ {l}}{2 n} = \frac {1}{2} - \frac {(2 u) ^ {l}}{n}.
+$$
+
+The above inequality is an application of Lemma 2.2 of Telgarsky (2015) (with careful counting it has been slightly improved). By choosing  $u$  to be at most  $\frac{\rho\frac{k}{T}}{8}$  it holds that the classification error  $\mathcal{R}(g)\geq \frac{1}{4}$  for any neural network  $g$  with  $u$  ReLUs and  $l$  layers.
+
+The above discussion implies the following theorem:
+
+Theorem 4.1 (Classification Error Theorem). Let  $k$  be a positive integer and  $f$  be a function of period  $m \times p$  with  $p$  an odd number greater than one and  $m$  being a power of two (it might hold  $m = 1$ ). We set  $\rho$  to be the positive root greater than one of the polynomial equation  $\lambda^{p - 1} - \lambda^{p - 2} - 1 = 0$ . We can construct a sequence of points  $(x_{i},y_{i})_{i = 1}^{2n}$  with  $n := \frac{|\rho^k|}{2}$  so that the classification error of function  $f^{mk}$  is zero, whereas the classification error of any neural network of  $l$  layers and  $u$  nodes per layer with  $u \leq \frac{\rho^{\frac{k}{l}}}{8}$  satisfies  $\mathcal{R}(g) \geq \frac{1}{4}$ .
+
+Remark 4.1. Observe that if the number of units  $u$  per layer is constant and the number of layers  $l$  is  $o(k)$ , then the classification error is always a positive constant for any neural network (whereas for  $f^{mk}$  is zero). Moreover, observe that since  $\rho$  is decreasing in  $p$  (recall  $p$  is the odd factor of the period), it holds that the classification error decreases as  $p$  increases (with fixed number of layers and nodes per layer). This indicates that the composition of functions with large odd period is simpler than of functions with small odd period (period greater than one) following the intuition we have from the Sharkovsky's ordering.
+
+# ACKNOWLEDGMENTS
+
+Vaggos Chatziafratis is partially supported by an Onassis Foundation Scholarship. Sai Ganesh Nagarajan would like to acknowledge SUTD President's Graduate Fellowship (SUTD-PGF). Ioannis Panageas and Xiao Wang would like to acknowledge NRF-NRFFAI1-2019-0003, SRG ISTD 2018 136 and ANR NRF 0095 ALIAS. Part of this project happened while the authors were visiting the Simons Foundation program "Foundations of Deep Learning" at Berkeley and would like to thank the organizers for their hospitality.
+
+# REFERENCES
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+Guido F Montufar, Razvan Pascanu, Kyunghyun Cho, and Yoshua Bengio. On the number of linear regions of deep neural networks. In Advances in neural information processing systems, pp. 2924-2932, 2014.  
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+Maithra Raghu, Ben Poole, Jon Kleinberg, Surya Ganguli, and Jascha Sohl Dickstein. On the expressive power of deep neural networks. In Proceedings of the 34th International Conference on Machine Learning-Volume 70, pp. 2847-2854. JMLR.org, 2017.  
+Benjamin Rossman, Rocco A Servedio, and Li-Yang Tan. An average-case depth hierarchy theorem for boolean circuits. In 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, pp. 1030-1048. IEEE, 2015.  
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+
+# A APPENDIX
+
+Claim A.1. The characteristic polynomial of  $A^\top$  is:
+
+$$
+\pi (\lambda) = \lambda^ {r + 1} - \lambda^ {r} - 1. \tag {A.1}
+$$
+
+Proof. Let  $I$  denote the identity matrix of size  $(r + 1) \times (r + 1)$ . We consider the matrix:
+
+$$
+A ^ {\top} - \lambda I = \left( \begin{array}{c c c c c c c} 1 - \lambda & 1 & 0 & 0 & 0 & \ldots & 0 \\ 0 & - \lambda & 1 & 0 & 0 & \ldots & 0 \\ 0 & 0 & - \lambda & 1 & 0 & \ldots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & \ldots & 0 & - \lambda & 1 \\ 1 & 0 & 0 & 0 & \ldots & 0 & - \lambda \end{array} \right).
+$$
+
+Observe that  $\lambda = 0,1$  are not eigenvalues of the matrix  $A^{\top}$ , hence we can multiply the first row by  $\frac{1}{\lambda - 1}$ , the second row by  $\frac{1}{\lambda(\lambda - 1)}$ , the third row by  $\frac{1}{\lambda^2(\lambda - 1)},\ldots$ , the  $i$ -th row by  $\frac{1}{\lambda^{i - 1}(\lambda - 1)}$  (and so on) and add them to the last row. Let  $B$  be the resulting matrix:
+
+$$
+B = \left( \begin{array}{c c c c c c c} 1 - \lambda & 1 & 0 & 0 & 0 & \dots & 0 \\ 0 & - \lambda & 1 & 0 & 0 & \dots & 0 \\ 0 & 0 & - \lambda & 1 & 0 & \dots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & \dots & 0 & - \lambda & 1 \\ 0 & 0 & 0 & 0 & \dots & 0 & - \lambda + \frac {1}{\lambda^ {r - 1} (\lambda - 1)} \end{array} \right).
+$$
+
+It is clear that  $\operatorname{det}(B) = 0$  as an equation has the same roots as  $\operatorname{det}(A^\top - \lambda I) = 0$ . Since  $B$  is an upper triangular matrix, it follows that
+
+$$
+\det  (B) = (- \lambda) ^ {r - 1} (1 - \lambda) \left(- \lambda + \frac {1}{\lambda^ {r - 1} (\lambda - 1)}\right).
+$$
+
+We conclude that the eigenvalues of  $A^\top$  (and hence of  $A$ ) must be roots of  $(\lambda^r - \lambda^{r-1})\lambda - 1$  and the claim follows.
+
+Lemma 3 (Period power of two). There exist continuous functions  $f$  with prime period  $n$  that is a power of two so that the number of crossings  $C_{x,y}(f^t)$  scales at most polynomially with  $t$  for any  $x, y \in [0,1]$ .
+
+Proof. The easiest example one can construct is the function  $f:[0,1] \to [0,1]$  that is defined  $f(x) = 1 - x$ . Observe that for any  $a \in [0,1]$  one has  $f(f(a)) = a$  and moreover if  $a \neq \frac{1}{2}$  then  $f(a) \neq a$ . Hence  $f$  is a function of prime period two. It is also clear that  $f^t(x) = x$  if  $t$  is even and  $f^t(x) = 1 - x$  if  $t$  is odd, so the number of crossings is always one for all  $t \in \mathbb{N}^*$ .
+
+One other less trivial example is the following function (see also Figure 6):
+
+$$
+f (x) = \left\{ \begin{array}{l l} - x + 5, & \quad 1 \leq x \leq 2 \\ - 2 x + 7, & \quad 2 \leq x \leq 3 \\ x - 2, & \quad 3 \leq x \leq 4. \end{array} \right.
+$$
+
+It is not hard to see that this function has prime period four  $(f(1) = 4, f(4) = 2, f(2) = 3, f(3) = 1)$ . Let  $J_0 = [1, 2]$ ,  $J_1 = [2, 3]$ ,  $J_2 = [3, 4]$ . It is clear that
+
+$$
+\bullet f (J _ {0}) = J _ {2}, f (J _ {1}) = J _ {0} \cup J _ {1} \text {a n d} f (J _ {2}) = J _ {0}.
+$$
+
+By letting  $\delta_i^t$  be the number of crossings of the function  $f$  for the interval  $J_{i}$ $(i\in \{0,1,2\})$ , one has recursively
+
+$$
+\left( \begin{array}{l} \delta_ {0} ^ {t + 1} \\ \delta_ {1} ^ {t + 1} \\ \delta_ {2} ^ {t + 1} \end{array} \right) = \left( \begin{array}{l l l} 0 & 1 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \left( \begin{array}{l} \delta_ {0} ^ {t} \\ \delta_ {1} ^ {t} \\ \delta_ {2} ^ {t} \end{array} \right) \tag {A.2}
+$$
+
+where  $\delta^0 = (1,1,1)$  (all ones vector). It is easy to observe that the matrix  $A = \left( \begin{array}{ccc}0 & 1 & 1\\ 0 & 1 & 0\\ 1 & 0 & 0 \end{array} \right)$  has spectral radius one (as opposed to the case of odd period greater than one) and moreover it holds
+
+![](images/0eb8a8a59641063815eec709896cd1beb46d9cd8c33055900798a24bee351baa.jpg)  
+(a) This figure captures one composition of function  $f$ . Observe that  $f$  crosses the interval [2, 3] two times (once for  $x \in [1, 2]$  and once for  $x \in [2, 3]$ ) and it crosses the interval [1, 2] once. In particular,  $\delta^1 = (2, 1)$ .
+
+![](images/83ac6314ccae200fe30f49d7ad05911224ae6ac1d7785976accaa24b7e19217f.jpg)  
+(b) This figure captures two compositions of function  $f$ . Observe that  $f$  crosses the interval [2, 3] three times (two times for  $x \in [2, 3]$  and once for  $x \in [1, 2]$ ) and it crosses the interval [1, 2] two times (once for  $x \in [1, 2]$  and once for  $x \in [2, 3]$ ). In particular,  $\delta^2 = (3, 2)$ .
+
+![](images/11ff94f0639c220edcabe33fee277be778b01af25eaf5aa76e3a50ba08ee5b44.jpg)  
+(c) This figure captures three compositions of function  $f$ . Observe that  $f$  crosses the interval [2, 3] five times (three times for  $x \in [2, 3]$  and twice for  $x \in [1, 2]$ ) and it crosses the interval [1, 2] three times (once for  $x \in [1, 2]$  and twice for  $x \in [2, 3]$ ). In particular,  $\delta^3 = (5, 3)$ .
+
+![](images/dfbb8efca9e26e285caf0e12b87fc5635df664a5b5a3d92e24a682a4b0eadff3.jpg)  
+(d) This figure captures four compositions of function  $f$ . Observe that  $f$  crosses the interval [2, 3] eight times (five times for  $x \in [2, 3]$  and three times for  $x \in [1, 2]$ ) and it crosses the interval [1, 2] five times (twice for  $x \in [1, 2]$  and three times for  $x \in [2, 3]$ ). In particular,  $\delta^4 = (8, 5)$ .
+
+![](images/a1b53e0176040862e42a5b64d6f35e2d81d0e77f2cbf94bc3c85baa5b6e4665e.jpg)  
+Figure 5: Compositions of a piecewise linear function that has a point of period 3.  
+Figure 6: A piecewise linear function  $f:[1,4]\to [1,4]$  that has prime period four.
+
+![](images/e88472273f22607fabb48d6f04efccedb41101359db80585ad28421f383a802a.jpg)  
+(a)  $\delta^1 = (1,1,2,2)$
+
+![](images/468da54519f1bf7cd5d28064ac6c6bcf29eb0655ae1e63062ae35ab64e62db8a.jpg)  
+(b)  $\delta^2 = (2,2,3,2)$
+
+![](images/5d23e47a1b73beb33ed23bc6ebcd420b580f1e0c12428a7d39d4bee4dbe8b5d5.jpg)  
+(c)  $\delta^3 = (2,3,5,4)$
+
+![](images/19227da4ef6879f48baef71843749d95dbaf2992b5cc9f72f9776ba8c9dd34d0.jpg)  
+(d)  $\delta^4 = (4,5,7,5)$  
+Figure 7: Compositions of a piecewise linear function that has a point of period 5. We start with the all ones vector for  $\delta^0$  and each composition arises from the covering relation between the sets.
+
+that  $\sum_{i=0}^{2}\sum_{j=0}^{2}A_{ij}^{t}=t+3$  for all  $t\in\mathbb{N}^*$ . We conclude that  $\alpha_0^t+\alpha_1^t+\alpha_2^t=t+3$ , therefore the number of crossings for  $J_0,J_1,J_2$  of the function  $f^t$  grows linearly with  $t$  (and not exponentially). Since the function we defined is of prime period four and is piecewise monotone (and so is any composition with itself) in each interval  $J_0,J_1,J_2$ , we conclude that the number of crossings of  $f^t$  for any possible pairs of values is at most linear in  $t$ .
+
+# B AN EXAMPLE WHICH IS PERIOD 5 BUT NOT PERIOD 3 (LI & YORKE (1975))
+
+Here, we show an example function, that has a point of period 5, but not period 3, thereby respecting the Sharkovsky ordering. Our proof approach for general odd periods is similar to the case of period 3, by using the induced covering graph and counting the crossings over each interval. This is illustrated in Figure 7.
+
+# C THE HETEROGENEITY OF THE LOGISTIC MAP
+
+In this section, we illustrate how the compositions of the logistic map  $f(x; r) \coloneqq rx(1 - x)$  behaves as  $r$  varies slightly. We give certain examples in the form of Figure 8. It is known that the map when  $r = 3.9$ , has a point of period 3. In contrast when  $r$  is reduced to 3.5 the map has a point of period 4 and further bringing  $r$  down to 3.2 will ensure that the map has a point of period 2. The figures below illustrate how the oscillations grow under these scenarios.
+
+![](images/17ba17b8d400dac11fab017dc27dbe7dd5cafb31c87e01d9c6fa990d66d27f37.jpg)  
+(a) Here  $f(x; 3.9) \coloneqq 3.9x(1 - x)$  is shown.
+
+![](images/8d4a10b325faba4f7e406ef3e65ba9fbe86cefaa3f2c89c0c262e930759aa728.jpg)  
+(b) Here  $f^6 (x;3.9)$  is shown.
+
+![](images/3edc00430898b412d41e43569ac5f11fb2463389536273ff5d2bc0f4fed94949.jpg)  
+(c) Here  $f(x; 3.5) \coloneqq 3.5x(1 - x)$  is shown.
+
+![](images/5a93dd678a38004371e7ec7442f42f6fd5cec44397b39315cd627650e7b2d6e4.jpg)  
+(d) Here  $f^6 (x;3.5)$  is shown.
+
+![](images/2377aa79709de6525035194d4bfc3c4d02bf6ebee7737be12483dba3f09e83ce.jpg)  
+(e) Here  $f(x; 3.2) \coloneqq 3.2x(1 - x)$  is shown.
+
+![](images/882e111bf938ca762c620604459e56b70a21605f702838f1d860dc079a5bb9d3.jpg)  
+(f) Here  $f^6 (x;3.2)$  is shown.  
+Figure 8: The compositions of the logistic map  $f(x; r) \coloneqq rx(1 - x)$  with different parameter values are shown here. The left column has the functions themselves while the right column shows the corresponding compositions. We can see that oscillations in these family of functions vary vastly with changes in  $r$  and these changes are made in the weights of an appropriate neural network (see Telgarsky (2016), Schmitt (2000)).
+
+![](images/3d1040900a8b5ebb5dfd9ea36634ec29adae3d685db6f700a617ddd7c5ef61c9.jpg)  
+(a) The numerical solutions to  $f^3 (x) = x$  are shown here.
+
+![](images/c9d1478a7bce50a23bdb57061e6d214c7c0860922f28181d19a79d9b0712c7e5.jpg)  
+(b) The numerical solutions to  $g^3 (x) = x$  are shown here.  
+Figure 9: We see that  $f^3(x) = x$  has solutions other than just the fixed point, since it has intersections in 3 other places, other than the fixed point. However,  $g^3(x) = x$  does not have any solutions other than the fixed point, as there are no other intersections.
+
+# D FURTHER DISCUSSIONS
+
+In this section, we provide some additional theoretical and experimental remarks on our characterization.
+
+# D.1 INCORPORATING BIAS TERMS
+
+If we add a bias term in the ReLU activation unit, e.g., use  $\max(v, \epsilon)$  instead of  $\max(v, 0)$  for the activation gates, where  $\epsilon$  is a small number (positive or negative), then our results do not change; in particular our trade-off in Theorem 4.1 still holds (since the Lemma 2.2 from Telgarsky (2015) is for general sawtooth functions). But, if one adds the bias term to the function  $f$  itself, then things get more interesting indeed: Suppose  $f$  has some period  $p$  where  $p$  is not a power of two; due to bifurcation phenomena (i.e., phenomena arising because we are at critical regimes of parameters such as the parameter  $\mu$  in our generalized triangle wave function), then the compositions of the function  $(f + \text{bias term})$  with itself may give rise to qualitatively different behaviors compared to  $f$ . In particular, the function  $(f + \text{bias term})$  might not have period  $p$  anymore. Intuitively, one can think that the small bias term is amplified after many compositions and is not negligible anymore.
+
+One such example is the triangle function  $f(x) = \phi x$  for  $0 \leq x \leq 0.5$  and  $\phi(1 - x)$  for  $1/2 \leq x \leq 1$ , where  $\phi = (1 + \sqrt{5})/2$  is the golden ratio. This function has period 3, see Figure 9a. However, if we consider the function  $g(x) = (\phi - \epsilon)x$  for  $0 \leq x \leq 0.5$  and  $(\phi - \epsilon)(1 - x)$  for  $0.5 \leq x \leq 1$  with  $\epsilon > 0$  (arbitrarily small positive) then  $g$  does not have period 3, see Figure 9b. In this sense, period as a property can be brittle to numerical changes if we are at the critical point.
+
+# D.2 SOME EXPERIMENTAL EVIDENCE
+
+In this section, we provide experimental evidence for our depth separation results by training a neural network of constant width, but with increasing depth on a classification task that closely resembles the  $n$ -alternating points problem that appeared in Telgarsky (2015) and is the foundation of our separation results as well. As mentioned before, this is a specific instance of a function that has a point of period 3. For simplicity, we do not consider this original problem exactly but rather a "smoothed" variant of it, in order to make it more amenable to the training procedure. Our goal is to create a diagram showing how the classification error drops as a function of the depth of the network for a fixed value of the width.
+
+We create 8000 equally spaced points from [0,1] (in increasing order), where the first 1000 points are of label 0, the second 1000 are label 1 and this label alternates every 1000 points. This is what we call a "smoothed" alternating point problem. Although, the theory would have used the classical 8-alternating points to argue about the lower bounds, in practice, performing training of deep (4 and above layers) and narrow networks (hidden layers with less than 4 neurons) with very few data points is a major challenge, see for instance Lu et al. (2018). Apart from the separation results that we show in theory, we show empirically that deep networks generally do improve the accuracy in
+
+![](images/deb5570f5698bfad619ad4dd9f4b7e15eac61c579d6664c994b0b79e100a09eb.jpg)  
+Figure 10: We see that depth does reduce the classification error for this particular task and when depth is 5, the classification error is close to 0. The saturation in between may be attributed to the general uncertainties in the training/optimization.
+
+this task compared to the shallow network and in fact a deep network with 5 layers can reach an accuracy of  $99.04\%$ . Any additional uncertainties in the error is generally attributed to the training procedure.
+
+To perform the experiments, we vary the depth of the neural network (excluding the input and the output layer) as  $d = 1,2,3,4,5$ . In addition, we fix the neurons for each layer to be 6. All activations are ReLU's, while the last layer is the classifier that uses a sigmoid to output probabilities. Each model adds one extra hidden layer and we make use of the same hyper-parameters to train all networks. Moreover, we require the training error or the classification error to tend to 0 during the training procedure, i.e., we will try and overfit the data (as we try to demonstrate a representation result, rather than a statistical/generalization result). Thus, for the actual training we use the same parameters to train all the different models using the "ADAM" optimizer Kingma & Ba (2014) and make the epochs to be 200 in order to enable overfitting. To record the training error, we verify that the training saturates by seeing the performance over the epochs and report by default the error in the last epoch. The results are shown in Figure 10.
+
+# D.3 PERIOD AS A NATURAL CHARACTERIZATION
+
+In a nutshell, our paper provides a "natural" property of a function (periodic points of certain periods) and then derive depth-width trade-offs based on it. This addresses some questions raised not only in Telgarsky (2016; 2015)'s works, but also in the paper Poole et al. (2016) that seeks to provide a natural, general measure of functional complexity helping us understand the benefits of depth. On the contrary, many of the previous depth separation results take a worst case approach for the representation question (showing that there exist functions implemented by deep networks that are hard to approximate with a shallow net). However, it is not clear whether such analysis applies to the typical instances arising in practice of neural-networks. We believe that our work together with Telgarsky (2016; 2015) and the paper Eldan & Shamir (2016) show a depth separation argument for very natural functions, such as the triangle waves or the indicator function of the unit ball.
+
+Given a specific prediction task in practice, how could one assess the period? We believe that this would be extremely useful yet a very difficult question that seems to be outside the reach of current techniques in the literature. Previous works and our work so far are able to present depth separation for representing certain functions.
+
+We point out that, intuitively, our characterization result consists of a certificate informing us qualitatively and quantitatively about which functions have complicated compositions and which not. Similar to computational problems in class NP, if one is given the certificate (the points  $(x_{1},\ldots ,x_{p})$ , then one can easily verify (if we have oracle access to evaluate the function  $f$ ), if the given function has a  $p$ -periodic cycle with points  $(x_{1},\ldots ,x_{p})$ . Nevertheless, we believe that
+
+finding the certificate for arbitrary continuous functions is not a straightforward problem, except maybe for particular restricted classes of functions. Having said that, we want to emphasize that in many prediction problems that are inspired by physics, one may a priori expect to have complicated dynamics behavior and hence require deeper networks for better performance. Such examples include efforts to solve the notorious 3-body problem or turbulent flows showing empirical evidence that complex physical processes require deep networks (see for instance, Ling et al. (2016) and Breen et al. (2019) that uses a 10 layered neural network).
\ No newline at end of file
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+# DIFFERENTIATION OF BLACKBOX COMBINATORIAL SOLVERS
+
+Marin Vlastelica $^{1*}$ , Anselm Paulus $^{1*}$ , Vít Musil $^{2}$ , Georg Martius $^{1}$ , Michal Rolínek $^{1}$
+
+<sup>1</sup> Max-Planck-Institute for Intelligent Systems, Tübingen, Germany  
+<sup>2</sup> Università degli Studi di Firenze, Italy
+
+{marin.vastelica, anselm.paulus, georg.martius, michal.rolinek} @tuebingen.mpg.de vit.musil@unifi.it
+
+# ABSTRACT
+
+Achieving fusion of deep learning with combinatorial algorithms promises transformative changes to artificial intelligence. One possible approach is to introduce combinatorial building blocks into neural networks. Such end-to-end architectures have the potential to tackle combinatorial problems on raw input data such as ensuring global consistency in multi-object tracking or route planning on maps in robotics. In this work, we present a method that implements an efficient backward pass through blackbox implementations of combinatorial solvers with linear objective functions. We provide both theoretical and experimental backing. In particular, we incorporate the Gurobi MIP solver, Blossom V algorithm, and Dijkstra's algorithm into architectures that extract suitable features from raw inputs for the traveling salesman problem, the min-cost perfect matching problem and the shortest path problem. The code is available at
+
+https://github.com/martius-lab/blackbox-backprop.
+
+# 1 INTRODUCTION
+
+The toolbox of popular methods in computer science currently sees a split into two major components. On the one hand, there are classical algorithmic techniques from discrete optimization – graph algorithms, SAT-solvers, integer programming solvers – often with heavily optimized implementations and theoretical guarantees on runtime and performance. On the other hand, there is the realm of deep learning allowing data-driven feature extraction as well as the flexible design of end-to-end architectures. The fusion of deep learning with combinatorial optimization is desirable both for foundational reasons – extending the reach of deep learning to data with large combinatorial complexity – and in practical applications. These often occur for example in computer vision problems that require solving a combinatorial sub-task on top of features extracted from raw input such as establishing global consistency in multi-object tracking from a sequence of frames.
+
+The fundamental problem with constructing hybrid architectures is differentiability of the combinatorial components. State-of-the-art approaches pursue the following paradigm: introduce suitable approximations or modifications of the objective function or of a baseline algorithm that eventually yield a differentiable computation. The resulting algorithms are often sub-optimal in terms of runtime, performance and optimality guarantees when compared to their unmodified counterparts. While the sources of sub-optimality vary from example to example, there is a common theme: any differentiable algorithm in particular outputs continuous values and as such it solves a relaxation of the original problem. It is well-known in combinatorial optimization theory that even strong and practical convex relaxations induce lower bounds on the approximation ratio for large classes of problems (Raghavendra, 2008; Thapper & Zivny, 2017) which makes them inherently sub-optimal. This inability to incorporate the best implementations of the best algorithms is unsatisfactory.
+
+In this paper, we propose a method that, at the cost of one hyperparameter, implements a backward pass for a blackbox implementation of a combinatorial algorithm or a solver that optimizes a linear
+
+![](images/fda88b11de5cf10647769dbac0a636dc454807d00f9349918be0753d75a93d74.jpg)  
+Figure 1: Architecture design enabled by Theorem 1. Blackbox combinatorial solver embedded into a neural network.
+
+objective function. This effectively turns the algorithm or solver into a composable building block of neural network architectures, as illustrated in Fig. 1. Suitable problems with linear objective include classical problems such as SHORTEST-PATH, TRAVELING-SALESMAN (TSP), MIN-COST-PERFECT-MATCHING, various cut problems as well as entire frameworks such as integer programs (IP), Markov random fields (MRF) and conditional random fields (CRF).
+
+The main technical challenge boils down to providing an informative gradient of a piecewise constant function. To that end, we are able to heavily leverage the minimization structure of the underlying combinatorial problem and efficiently compute a gradient of a continuous interpolation. While the roots of the method lie in loss-augmented inference, the employed mathematical technique for continuous interpolation is novel. The computational cost of the introduced backward pass matches the cost of the forward pass. In particular, it also amounts to one call to the solver.
+
+In experiments, we train architectures that contain unmodified implementations of the following efficient combinatorial algorithms: general-purpose mixed-integer programming solver Gurobi (Gurobi Optimization, 2019), state-of-the-art C implementation of MIN-COST-PERFECT-MATCHING algorithm - Blossom V (Kolmogorov, 2009) and Dijkstra's algorithm (Dijkstra, 1959) for SHORTEST-PATH. We demonstrate that the resulting architectures train without sophisticated tweaks and are able to solve tasks that are beyond the capabilities of conventional neural networks.
+
+# 2 RELATED WORK
+
+Multiple lines of work lie at the intersection of combinatorial algorithms and deep learning. We primarily distinguish them by their motivation.
+
+Motivated by applied problems. Even though computer vision has seen a substantial shift from combinatorial methods to deep learning, some problems still have a strong combinatorial aspect and require hybrid approaches. Examples include multi-object tracking (Schulter et al., 2017), semantic segmentation (Chen et al., 2018), multi-person pose estimation (Pishchulin et al., 2016; Song et al., 2018), stereo matching (Knobelreiter et al., 2017) and person re-identification (Ye et al., 2017). The combinatorial algorithms in question are typically Markov random fields (MRF) (Chen et al., 2015), conditional random fields (CRF) (Marin et al., 2019), graph matching (Ye et al., 2017) or integer programming (Schulter et al., 2017). In recent years, a plethora of hybrid end-to-end architectures have been proposed. The techniques used for constructing the backward pass range from employing various relaxations and approximations of the combinatorial problem (Chen et al., 2015; Zheng et al., 2015) over differentiating a fixed number of iterations of an iterative solver (Paschalidou et al., 2018; Thompson et al., 2014; Liu et al., 2015) all the way to relying on the structured SVM framework (Tsochantaridis et al., 2005; Chen et al., 2015).
+
+Motivated by "bridging the gap". Building links between combinatorics and deep learning can also be viewed as a foundational problem; for example, (Battaglia et al., 2018) advocate that "combinatorial generalization must be a top priority for AI". One such line of work focuses on designing architectures with algorithmic structural prior – for example by mimicking the layout of a Turing machine (Sukhbaatar et al., 2015; Vinyals et al., 2015; Graves et al., 2014; 2016) or by promoting behaviour that resembles message-passing algorithms as it is the case in Graph Neural Networks and
+
+related architectures (Scarselli et al., 2009; Li et al., 2016; Battaglia et al., 2018). Another approach is to provide neural network building blocks that are specialized to solve some types of combinatorial problems such as satisfiability (SAT) instances (Wang et al., 2019), mixed integer programs (Ferber et al., 2019), sparse inference (Niculae et al., 2018), or submodular maximization (Tschiatschek et al., 2018). A related mindset of learning inputs to an optimization problem gave rise to the "predict-and-optimize" framework and its variants (Elmachtoub & Grigas, 2017; Demirovic et al., 2019; Mandi et al., 2019). Some works have directly addressed the question of learning combinatorial optimization algorithms such as the TRAVELING-SALESMAN-PROBLEM in (Bello et al., 2017) or its vehicle routing variants (Nazari et al., 2018). A recent approach also learns combinatorial algorithms via a clustering proxy (Wilder et al., 2019).
+
+There are also efforts to bridge the gap in the opposite direction; to use deep learning methods to improve state-of-the-art combinatorial solvers, typically by learning (otherwise hand-crafted) heuristics. Some works have again targeted the TRAVELING-SALESMAN-PROBLEM (Kool et al., 2019; Deudon et al., 2018; Bello et al., 2017) as well as other NP-Hard problems (Li et al., 2018). Also, more general solvers received some attention; this includes SAT-solvers (Selsam & Bjørner, 2019; Selsam et al., 2019), integer programming solvers (often with learning branch-and-bound rules) (Khalil et al., 2016; Balcan et al., 2018; Gasse et al., 2019) and SMT-solvers (satisfiability modulo theories)(Balunovic et al., 2018).
+
+# 3 METHOD
+
+Let us first formalize the notion of a combinatorial solver. We expect the solver to receive continuous input  $w \in W \subseteq \mathbb{R}^N$  (e.g. edge weights of a fixed graph) and return discrete output  $y$  from some finite set  $Y$  (e.g. all traveling salesman tours on a fixed graph) that minimizes some cost  $\mathbf{c}(w, y)$  (e.g. length of the tour). More precisely, the solver maps
+
+$$
+w \mapsto y (w) \quad \text {s u c h t h a t} \quad y (w) = \underset {y \in Y} {\arg \min } \mathbf {c} (w, y). \tag {1}
+$$
+
+We will restrict ourselves to objective functions  $\mathbf{c}(w,y)$  that are linear, namely  $\mathbf{c}(w,y)$  may be represented as
+
+$$
+\mathbf {c} (w, y) = w \cdot \phi (y) \quad \text {f o r} w \in W \text {a n d} y \in Y \tag {2}
+$$
+
+in which  $\phi \colon Y \to \mathbb{R}^N$  is an injective representation of  $y \in Y$  in  $\mathbb{R}^N$ . For brevity, we omit the mapping  $\phi$  and instead treat elements of  $Y$  as discrete points in  $\mathbb{R}^N$ .
+
+Note that such definition of a solver is still very general as there are no assumptions on the set of constraints or on the structure of the output space  $Y$ .
+
+Example 1 (Encoding shortest-path problem). If  $G = (V, E)$  is a given graph with vertices  $s, t \in V$ , the combinatorial solver for the  $(s, t)$ -SHORTEST-PATH would take edge weights  $w \in W = \mathbb{R}^{|E|}$  as input and produce the shortest path  $y(w)$  represented as  $\phi(y) \subseteq \{0, 1\}^{|E|}$  an indicator vector of the selected edges. The cost function is then indeed the inner product  $\mathbf{c}(w, y) = w \cdot \phi(y)$ .
+
+The task to solve during back-propagation is the following. We receive the gradient  $\mathrm{d}L / \mathrm{d}y$  of the global loss  $L$  with respect to solver output  $y$  at a given point  $\hat{y} = y(\hat{w})$ . We are expected to return  $\mathrm{d}L / \mathrm{d}w$ , the gradient of the loss with respect to solver input  $w$  at a point  $\hat{w}$ .
+
+Since  $Y$  is finite, there are only finitely many values of  $y(w)$ . In other words, this function of  $w$  is piecewise constant and the gradient is identically zero or does not exist (at points of jumps). This should not come as a surprise; if one does a small perturbation to edge weights of a graph, one usually does not change the optimal TSP tour and on rare occasions alters it drastically. This has an important consequence:
+
+The fundamental problem with differentiating through combinatorial solvers is not the lack of differentiability; the gradient exists almost everywhere. However, this gradient is a constant zero and as such is unhelpful for optimization.
+
+Accordingly, we will not rely on standard techniques for gradient estimation (see (Mohamed et al., 2019) for a comprehensive survey).
+
+![](images/c02aebd4d7e11487299c1dfea101a216440022442357d1733856eba4f6db7b7c.jpg)  
+(a)
+
+![](images/d2161db815a4b7d2060ad726039779cfedfa9e7da8a650f0383e2abf2ab8ce9d.jpg)  
+(b)  
+Figure 2: Continuous interpolation of a piecewise constant function. (a)  $f_{\lambda}$  for a small value of  $\lambda$ ; the set  $W_{\mathrm{eq}}^{\lambda}$  is still substantial and only two interpolators  $g_{1}$  and  $g_{2}$  are incomplete. Also, all interpolators are 0-interpolators. (b)  $f_{\lambda}$  for a high value of  $\lambda$ ; most interpolators are incomplete and we also encounter a  $\delta$ -interpolator  $g_{3}$  (between  $y_{1}$  and  $y_{2}$ ) which attains the value  $f(y_{1})$ $\delta$ -away from the set  $P_{1}$ . Despite losing some local structure for high  $\lambda$ , the gradient of  $f_{\lambda}$  is still informative.
+
+First, we simplify the situation by considering the linearization  $f$  of  $L$  at the point  $\hat{y}$ . Then for
+
+$$
+f (y) = L (\hat {y}) + \frac {\mathrm {d} L}{\mathrm {d} y} (\hat {y}) \cdot (y - \hat {y}) \quad \text {w e h a v e} \quad \frac {\mathrm {d} f (y (w))}{\mathrm {d} w} = \frac {\mathrm {d} L}{\mathrm {d} w}
+$$
+
+and therefore it suffices to focus on differentiating the piecewise constant function  $f(y(w))$ .
+
+If the piecewise constant function at hand was arbitrary, we would be forced to use zero-order gradient estimation techniques such as computing finite differences. These require prohibitively many function evaluations particularly for high-dimensional problems.
+
+However, the function  $f(y(w))$  is a result of a minimization process and it is known that for smooth spaces  $Y$  there are techniques for such "differentiation through argmin" (Schmidt & Roth, 2014; Samuel & Tappen, 2009; Foo et al., 2008; Domke, 2012; Amos et al., 2017; Amos & Kolter, 2017). It turns out to be possible to build – with different mathematical tools – a viable discrete analogy. In particular, we can efficiently construct a function  $f_{\lambda}(w)$ , a continuous interpolation of  $f(y(w))$ , whose gradient we return (see Fig. 2). The hyper-parameter  $\lambda > 0$  controls the trade-off between "informativeness of the gradient" and "faithfulness to the original function".
+
+Before diving into the formalization, we present the final algorithm as listed in Algo. 1. It is simple to implement and the backward pass indeed only runs the solver once on modified input. Providing the justification, however, is not straightforward, and it is the subject of the rest of the section.
+
+<table><tr><td colspan="2">Algorithm 1 Forward and Backward Pass</td></tr><tr><td>function FORWARDPASS(ˆw)</td><td>function BACKWARDPASS(dL/dy(ˆy), λ)</td></tr><tr><td>ˆy := Solver(ˆw) //ˆy = y(ˆw)</td><td>loadˆw andˆy from forward pass</td></tr><tr><td>saveˆw andˆy for backward pass</td><td>w&#x27; :=ˆw + λ·dL/dy(ˆy)</td></tr><tr><td>returnˆy</td><td>// Calculate perturbed weights</td></tr><tr><td></td><td>yλ := Solver(w&#x27;)</td></tr><tr><td></td><td>return ∇wfλ(ˆw) := -1/λ[ˆy - yλ]</td></tr><tr><td></td><td>// Gradient of continuous interpolation</td></tr></table>
+
+# 3.1 CONSTRUCTION AND PROPERTIES OF  $f_{\lambda}$
+
+Before we give the exact definition of the function  $f_{\lambda}$ , we formulate several requirements on it. This will help us understand why  $f_{\lambda}(w)$  is a reasonable replacement for  $f(y(w))$  and, most importantly, why its gradient captures changes in the values of  $f$ .
+
+Property A1. For each  $\lambda > 0$ ,  $f_{\lambda}$  is continuous and piecewise affine.
+
+The second property describes the trade-off induced by changing the value of  $\lambda$ . For  $\lambda > 0$ , we define sets  $W_{\mathrm{eq}}^{\lambda}$  and  $W_{\mathrm{dif}}^{\lambda}$  as the sets where  $f(y(w))$  and  $f_{\lambda}(w)$  coincide and where they differ, i.e.
+
+$$
+W _ {\mathrm {e q}} ^ {\lambda} = \left\{w \in W: f _ {\lambda} (w) = f \big (y (w) \big) \right\} \quad \text {a n d} \quad W _ {\mathrm {d i f}} ^ {\lambda} = W \setminus W _ {\mathrm {e q}} ^ {\lambda}.
+$$
+
+Property A2. The sets  $W_{\mathrm{dif}}^{\lambda}$  are monotone in  $\lambda$  and they vanish as  $\lambda \to 0^{+}$ , i.e.
+
+$$
+W _ {\text {d i f}} ^ {\lambda_ {1}} \subseteq W _ {\text {d i f}} ^ {\lambda_ {2}} \quad \text {f o r} 0 <   \lambda_ {1} \leq \lambda_ {2} \quad \text {a n d} \quad W _ {\text {d i f}} ^ {\lambda} \to \emptyset \quad \text {a s} \lambda \to 0 ^ {+}.
+$$
+
+In other words, Property A2 tells us that  $\lambda$  controls the size of the set where  $f_{\lambda}$  deviates from  $f$  and where  $f_{\lambda}$  has meaningful gradient. This behaviour of  $f_{\lambda}$  can be seen in Fig. 2.
+
+In the third and final property, we want to capture the interpolation behavior of  $f_{\lambda}$ . For that purpose, we define a  $\delta$ -interpolator of  $f$ . We say that  $g$ , defined on a set  $G \subset W$ , is a  $\delta$ -interpolator of  $f$  between  $y_{1}$  and  $y_{2} \in Y$ , if
+
+-  $g$  is non-constant affine function;  
+- the image  $g(G)$  is an interval with endpoints  $f(y_{1})$  and  $f(y_{2})$ ;  
+-  $g$  attains the boundary values  $f(y_{1})$  and  $f(y_{2})$  at most  $\delta$ -far away from where  $f(y(w))$  does. In particular, there is a point  $w_{k} \in G$  for which  $g(w_{k}) = f(y_{k})$  and  $\mathrm{dist}(w_k,P_k) \leq \delta$ , where  $P_{k} = \{w \in W : y(w) = y_{k}\}$ , for  $k = 1,2$ .
+
+In the special case of a 0-interpolator  $g$ , the graph of  $g$  connects (in a topological sense) two components of the graph of  $f\big(y(w)\big)$ . In the general case,  $\delta$  measures displacement of the interpolator (see also Fig. 2 for some examples). This displacement on the one hand loosens the connection to  $f\big(y(w)\big)$  but on the other hand allows for less local interpolation which might be desirable.
+
+Property A3. The function  $f_{\lambda}$  consists of finitely many (possibly incomplete)  $\delta$ -interpolators of  $f$  on  $W_{\mathrm{dif}}^{\lambda}$  where  $\delta \leq C\lambda$  for some fixed  $C$ . Equivalently, the displacement is linearly controlled by  $\lambda$ .
+
+Intuitively, the consequence of Property A3 is that  $f_{\lambda}$  has reasonable gradients everywhere since it consists of elementary affine interpolators.
+
+For defining the function  $f_{\lambda}$ , we need a solution of a perturbed optimization problem
+
+$$
+y _ {\lambda} (w) = \underset {y \in Y} {\arg \min } \left\{\mathbf {c} (w, y) + \lambda f (y) \right\}. \tag {3}
+$$
+
+Theorem 1. Let  $\lambda >0$ . The function  $f_{\lambda}$  defined by
+
+$$
+f _ {\lambda} (w) = f \left(y _ {\lambda} (w)\right) - \frac {1}{\lambda} \left[ \mathbf {c} (w, y (w)) - \mathbf {c} (w, y _ {\lambda} (w)) \right] \tag {4}
+$$
+
+satisfies Properties A1, A2, A3.
+
+Let us remark that already the continuity of  $f_{\lambda}$  is not apparent from its definition as the first term  $f(y_{\lambda}(w))$  is still a piecewise constant function. Proof of this result, along with geometrical description of  $f_{\lambda}$ , can be found in section A.2. Fig. 3 visualizes  $f_{\lambda}$  for different values if  $\lambda$ .
+
+Now, since  $f_{\lambda}$  is ensured to be differentiable, we have
+
+$$
+\nabla f _ {\lambda} (w) = - \frac {1}{\lambda} \left[ \frac {\mathrm {d} \mathbf {c}}{\mathrm {d} w} (w, y (w)) - \frac {\mathrm {d} \mathbf {c}}{\mathrm {d} w} (w, y _ {\lambda} (w)) \right] = - \frac {1}{\lambda} [ y (w) - y _ {\lambda} (w) ]. \tag {5}
+$$
+
+The second equality then holds due to (2). We then return  $\nabla f_{\lambda}$  as a loss gradient.
+
+Remark 1. The roots of the method we propose lie in loss-augmented inference. In fact, the update rule from (5) (but not the function  $f_{\lambda}$  or any of its properties) was already proposed in a different context in (Hazan et al., 2010; Song et al., 2016) and was later used in (Lorberbom et al., 2018; Mohapatra et al., 2018). The main difference to our work is that only the case of  $\lambda \to 0^{+}$  is recommended and studied, which in our situation computes the correct but uninformative zero gradient. Our analysis implies that larger values of  $\lambda$  are not only sound but even preferable. This will be seen in experiments where we use values  $\lambda \approx 10 - 20$ .
+
+![](images/0f92661ee1052fad94c912665fa4253a4e0e47eca29cdfc604dcd5a9ee9721cf.jpg)  
+Figure 3: Example  $f_{\lambda}$  for  $w \in \mathbb{R}^2$  and  $\lambda = 3, 10, 20$  (left to right). As  $\lambda$  changes, the interpolation  $f_{\lambda}$  is less faithful to the piecewise constant  $f(y(w))$  but provides reasonable gradient on a larger set.
+
+![](images/93fd19617cb0a82601a714667cd97af0f0e39ddf289128ebeddc8c1657d52d3c.jpg)
+
+![](images/279e1fd203ce1f00b304dbf9aad3c7e3d1d31d2c9289af7cfb11bce091a19228.jpg)
+
+# 3.2 EFFICIENT COMPUTATION OF  $f_{\lambda}$
+
+Computing  $y_{\lambda}$  in (3) is the only potentially expensive part of evaluating (5). However, the linear interplay of the cost function and the gradient trivially gives a resolution.
+
+Proposition 1. Let  $\hat{w} \in W$  be fixed. If we set  $w' = \hat{w} + \lambda \frac{\mathrm{d}L}{\mathrm{d}y} (\hat{y})$ , we can compute  $y_{\lambda}$  as
+
+$$
+y_{\lambda}(\hat{w}) = \operatorname *{arg  min}_{y\in Y}\mathbf{c}(w^{\prime},y).
+$$
+
+In other words,  $y_{\lambda}$  is the output of calling the solver on input  $w'$ .
+
+# 4 EXPERIMENTS
+
+In this section, we experimentally validate a proof of concept: that architectures containing exact blackbox solvers (with backward pass provided by Algo. 1) can be trained by standard methods.
+
+Table 1: Experiments Overview.  
+
+<table><tr><td>Graph Problem</td><td>Solver</td><td>Solver instance size</td><td>Input format</td></tr><tr><td>Shortest path</td><td>Dijkstra</td><td>up to 900 vertices</td><td>(image) up to 240 × 240</td></tr><tr><td>Min Cost PM</td><td>Blossom V</td><td>up to 1104 edges</td><td>(image) up to 528 × 528</td></tr><tr><td>Traveling Salesman</td><td>Gurobi</td><td>up to 780 edges</td><td>up to 40 images (20 × 40)</td></tr></table>
+
+To that end, we solve three synthetic tasks as listed in Tab. 1. These tasks are designed to mimic practical examples from Section 2 and solving them anticipates a two-stage process: 1) extract suitable features from raw input, 2) solve a combinatorial problem over the features. The dimensionalities of input and of intermediate representations also aim to mirror practical problems and are chosen to be prohibitively large for zero-order gradient estimation methods. Guidelines of setting the hyperparameter  $\lambda$  are given in section A.1.
+
+We include the performance of ResNet18 (He et al., 2016) as a sanity check to demonstrate that the constructed datasets are too complex for standard architectures.
+
+Remark 2. The included solvers have very efficient implementations and do not severely impact runtime. All models train in under two hours on a single machine with 1 GPU and no more than 24 utilized CPU cores. Only for the large TSP problems the solver's runtime dominates.
+
+# 4.1 WARCRAFT SHORTEST PATH
+
+Problem input and output. The training dataset for problem  $\mathrm{SP}(k)$  consists of 10000 examples of randomly generated images of terrain maps from the Wcraft II tileset (Guyomarch, 2017). The maps have an underlying grid of dimension  $k\times k$  where each vertex represents a terrain with a fixed cost that is unknown to the network. The shortest (minimum cost) path between top left and bottom right vertices is encoded as an indicator matrix and serves as a label (see also Fig. 4). We consider datasets  $\mathrm{SP}(k)$  for  $k\in \{12,18,24,30\}$ . More experimental details are provided in section A.3.
+
+![](images/07a6c20690290131ffb256ce6db66ae2a1456139e2e41670116f74e43ba007a9.jpg)  
+Figure 4: The  $\mathrm{SP}(k)$  dataset. (a) Each input is a  $k \times k$  grid of tiles corresponding to a Wcraft II terrain map, the respective label is a matrix indicating the shortest path from top left to bottom right. (b) is a different map with correctly predicted shortest path.
+
+![](images/b7328d72f6f1e87aee298dad44e0799dd60b4c8821d0e469b966b151977c8c58.jpg)
+
+Architecture. An image of the terrain map is presented to a convolutional neural network which outputs a  $k \times k$  grid of vertex costs. These costs are then the input to the Dijkstra algorithm to compute the predicted shortest path for the respective map. The loss used for computing the gradient update is the Hamming distance between the true shortest path and the predicted shortest path.
+
+Results. Our method learns to predict the shortest paths with high accuracy and generalization capability, whereas the ResNet18 baseline unsurprisingly fails to generalize already for small grid sizes of  $k = 12$ . Since the shortest paths in the maps are often nonunique (i.e. there are multiple shortest paths with the same cost), we report the percentage of shortest path predictions that have optimal cost. The results are summarized in Tab. 2.
+
+Table 2: Results for Wcraft shortest path. Reported is the accuracy, i.e. percentage of paths with the optimal costs. Standard deviations are over five restarts.  
+
+<table><tr><td rowspan="2">k</td><td colspan="2">Embedding Dijkstra</td><td colspan="2">ResNet18</td></tr><tr><td>Train %</td><td>Test %</td><td>Train %</td><td>Test %</td></tr><tr><td>12</td><td>99.7 ± 0.0</td><td>96.0 ± 0.3</td><td>100.0 ± 0.0</td><td>23.0 ± 0.3</td></tr><tr><td>18</td><td>98.9 ± 0.2</td><td>94.4 ± 0.2</td><td>99.9 ± 0.0</td><td>0.7 ± 0.3</td></tr><tr><td>24</td><td>97.8 ± 0.2</td><td>94.4 ± 0.6</td><td>100.0 ± 0.0</td><td>0.0 ± 0.0</td></tr><tr><td>30</td><td>97.4 ± 0.1</td><td>94.0 ± 0.3</td><td>95.6 ± 0.5</td><td>0.0 ± 0.0</td></tr></table>
+
+# 4.2 GLOBE TRAVELING SALESMAN PROBLEM
+
+Problem input and output. The training dataset for problem  $\mathrm{TSP}(k)$  consists of 10000 examples where the input for each example is a  $k$ -element subset of fixed 100 country flags and the label is the shortest traveling salesman tour through the capitals of the corresponding countries. The optimal tour is represented by its adjacency matrix (see also Fig. 5). We consider datasets  $\mathrm{TSP}(k)$  for  $k \in \{5, 10, 20, 40\}$ .
+
+![](images/4896326e9b9454b2ceefa8e5b7f1a92f72961bfe4e074d4db36af6a996fc8b4d.jpg)  
+Figure 5: The TSP  $(k)$  problem. (a) illustrates the dataset. Each input is a sequence of  $k$  flags and the corresponding label is the adjacency matrix of the optimal TSP tour around the corresponding capitals. (b) displays the learned locations of 10 country capitals in southeast Asia and Australia, accurately recovering their true position.
+
+![](images/2d3335c7e13329f80f1254c2e4e01438ce40d9e5c44ebf274320da783d6e9a31.jpg)
+
+Architecture. Each of the  $k$  flags is presented to a convolutional network that produces  $k$  three-dimensional vectors. These vectors are projected onto the unit sphere in  $\mathbb{R}^3$ ; a representation of the globe. The TSP solver receives a matrix of pairwise distances of the  $k$  computed locations. The loss of the network is the Hamming distance between the true and the predicted TSP adjacency matrix. The architecture is expected to learn the correct representations of the flags (i.e. locations of the respective countries' capitals on Earth, up to rotations of the sphere). The employed Gurobi solver optimizes a mixed-integer programming formulation of TSP using the cutting plane method (Marchand et al., 2002) for lazy sub-tour elimination.
+
+Results. This architecture not only learns to extract the correct TSP tours but also learns the correct representations. Quantitative evidence is presented in Tab. 3, where we see that the learned locations generalize well and lead to correct TSP tours also on the test set and also on somewhat large instances (note that there are  $39! \approx 10^{46}$  admissible TSP tours for  $k = 40$ ). The baseline architecture
+
+Table 3: Results for Globe TSP. Reported is the full tour accuracy. Standard deviations are over five restarts.  
+
+<table><tr><td rowspan="2">k</td><td colspan="2">Embedding TSP Solver</td><td colspan="2">ResNet18</td></tr><tr><td>Train %</td><td>Test %</td><td>Train %</td><td>Test %</td></tr><tr><td>5</td><td>99.8 ± 0.0</td><td>99.2 ± 0.1</td><td>100.0 ± 0.0</td><td>1.9 ± 0.6</td></tr><tr><td>10</td><td>99.8 ± 0.1</td><td>98.7 ± 0.2</td><td>99.0 ± 0.1</td><td>0.0 ± 0.0</td></tr><tr><td>20</td><td>99.1 ± 0.1</td><td>98.4 ± 0.4</td><td>98.8 ± 0.3</td><td>0.0 ± 0.0</td></tr><tr><td>40</td><td>97.4 ± 0.2</td><td>96.7 ± 0.4</td><td>96.9 ± 0.3</td><td>0.0 ± 0.0</td></tr></table>
+
+only memorizes the training set. Additionally, we can extract the suggested locations of world capitals and compare them with reality. To that end, we present Fig. 5b, where the learned locations of 10 capitals in Southeast Asia are displayed.
+
+# 4.3 MNIST MIN-COST PERFECT MATCHING
+
+Problem input and output. The training dataset for problem  $\mathrm{PM}(k)$  consists of 10000 examples where the input to each example is a set of  $k^2$  digits drawn from the MNIST dataset arranged in a  $k\times k$  grid. For computing the label, we consider the underlying  $k\times k$  grid graph (without diagonal edges) and solve a MIN-COST-PERFECT-MATCHING problem, where edge weights are given simply by reading the two vertex digits as a two-digit number (we read downwards for vertical edges and from left to right for horizontal edges). The optimal perfect matching (i.e. the label) is encoded by an indicator vector for the subset of the selected edges, see example in Fig. 6.
+
+Architecture. The grid image is the input of a convolutional neural network which outputs a grid of vertex weights. These weights are transformed into edge weights as described above and given to the solver. The loss function is Hamming distance between solver output and the true label.
+
+Results. The architecture containing the solver is capable of good generalizations suggesting that the correct representation is learned. The performance is good even on larger instances and despite the presence of noise in supervision - often there are many optimal matchings. In contrast, the ResNet18 baseline only achieves reasonable performance for the simplest case  $\mathrm{PM}(4)$ . The results are summarized in Tab. 4.
+
+Table 4: Results for MNIST Min-cost perfect matching. Reported is the accuracy of predicting an optimal matching. Standard deviations are over five restarts.  
+
+<table><tr><td rowspan="2">k</td><td colspan="2">Embedding Blossom V</td><td colspan="2">ResNet18</td></tr><tr><td>Train %</td><td>Test %</td><td>Train %</td><td>Test %</td></tr><tr><td>4</td><td>99.97 ± 0.01</td><td>98.32 ± 0.24</td><td>100.0 ± 0.0</td><td>92.5 ± 0.3</td></tr><tr><td>8</td><td>99.95 ± 0.04</td><td>99.92 ± 0.01</td><td>100.0 ± 0.0</td><td>8.3 ± 0.8</td></tr><tr><td>16</td><td>99.02 ± 0.84</td><td>99.06 ± 0.57</td><td>100.0 ± 0.0</td><td>0.0 ± 0.0</td></tr><tr><td>24</td><td>95.63 ± 5.49</td><td>92.06 ± 7.97</td><td>96.1 ± 0.5</td><td>0.0 ± 0.0</td></tr></table>
+
+# 5 DISCUSSION
+
+We provide a unified mathematically sound algorithm to embed combinatorial algorithms into neural networks. Its practical implementation is straightforward and training succeeds with standard deep learning techniques. The two main branches of future work are: 1) exploring the potential of newly enabled architectures, 2) addressing standing real-world problems. The latter case requires
+
+![](images/28623a1a089f70d63e98f3ce0a8811758321f026e32be7f2db63352084d9776f.jpg)  
+(a)
+
+![](images/cde2ba76bf152015f60122f8c2aa64e03d15db37bbf8c5b89f98aae886b38573.jpg)  
+(b)  
+Figure 6: Visualization of the PM dataset. (a) shows the case of  $\mathrm{PM}(4)$ . Each input is a  $4\times 4$  grid of MNIST digits and the corresponding label is the indicator vector for the edges in the min-cost perfect matching. (b) shows the correct min-cost perfect matching output from the network. The cost of the matching is 348 ( $46 + 12$  horizontally and  $27 + 45 + 40 + 67 + 78 + 33$  vertically).
+
+embedding approximate solvers (that are common in practice). This breaks some of our theoretical guarantees but given their strong empirical performance, the fusion might still work well in practice.
+
+# ACKNOWLEDGEMENT
+
+We thank the International Max Planck Research School for Intelligent Systems (IMPRS-IS) for supporting Marin Vlastelica. We acknowledge the support from the German Federal Ministry of Education and Research (BMBF) through the Tbingen AI Center (FKZ: 01IS18039B). Additionally, we would like to thank Paul Swoboda and Alexander Kolesnikov for valuable feedback on an early version of the manuscript.
+
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+
+# A APPENDIX
+
+# A.1 GUIDELINES FOR SETTING THE VALUES OF  $\lambda$ .
+
+In practice,  $\lambda$  has to be chosen appropriately, but we found its exact choice uncritical (no precise tuning was required). Nevertheless, note that  $\lambda$  should cause a noticeable disruption in the optimization problem from equation (3), otherwise it is too likely that  $y(w) = y_{\lambda}(w)$  resulting in a zero gradient. In other words,  $\lambda$  should roughly be of the magnitude that brings the two terms in the definition of  $w^{\prime}$  in Prop. 1 to the same order:
+
+$$
+\lambda \approx \frac {\langle w \rangle}{\left\langle \frac {\mathrm {d} L}{\mathrm {d} y} \right\rangle}
+$$
+
+where  $\langle \cdot \rangle$  stands for the average. This again justifies that  $\lambda$  is a true hyperparameter and that there is no reason to expect values around  $\lambda \rightarrow 0^{+}$ .
+
+# A.2 PROOFS
+
+Proof of Proposition 1. Let us write  $L = L(\hat{y})$  and  $\nabla L = \frac{\mathrm{d}L}{\mathrm{d}y} (\hat{y})$ , for brevity. Thanks to the linearity of  $\mathbf{c}$  and the definition of  $f$ , we have
+
+$$
+\mathbf {c} (\hat {w}, y) + \lambda f (y) = \hat {w} y + \lambda \left(L + \nabla L (y - \hat {y})\right) = (\hat {w} + \lambda \nabla L) y + \lambda L - \lambda \nabla L \hat {y} = \mathbf {c} (w ^ {\prime}, y) + \mathbf {c} _ {0},
+$$
+
+where  $\mathbf{c}_0 = \lambda L - \lambda \nabla L\hat{y}$  and  $w^{\prime} = \hat{w} +\lambda \nabla L$  as desired. The conclusion about the points of minima then follows.
+
+Before we prove Theorem 1, we make some preliminary observations. To start with, due to the definition of the solver, we have the fundamental inequality
+
+$$
+\mathbf {c} (w, y) \geq \mathbf {c} (w, y (w)) \quad \text {f o r e v e r y} w \in W \text {a n d} y \in Y. \tag {6}
+$$
+
+Observation 1. The function  $w \mapsto \mathbf{c}(w, y(w))$  is continuous and piecewise linear.
+
+Proof. Since  $\mathbf{c}$ 's are linear and distinct,  $\mathbf{c}\big(w,y(w)\big)$ , as their pointwise minimum, has the desired properties.
+
+Analogous fundamental inequality
+
+$$
+\mathbf {c} (w, y) + \lambda f (y) \geq \mathbf {c} \left(w, y _ {\lambda} (w)\right) + \lambda f \left(y _ {\lambda} (w)\right) \quad \text {f o r e v e r y} w \in W \text {a n d} y \in Y \tag {7}
+$$
+
+follows from the definition of the solution to the optimization problem (3).
+
+A counterpart of Observation 1 reads as follows.
+
+Observation 2. The function  $w \mapsto \mathbf{c}\big(w, y_{\lambda}(w)\big) + \lambda f\big(y_{\lambda}(w)\big)$  is continuous and piecewise affine.
+
+Proof. The function under inspection is a pointwise minimum of distinct affine functions  $w \mapsto \mathbf{c}(w, y) + \lambda f(y)$  as  $y$  ranges  $Y$ .
+
+As a consequence of above-mentioned fundamental inequalities, we obtain the following two-sided estimates on  $f_{\lambda}$ .
+
+Observation 3. The following inequalities hold for  $w \in W$
+
+$$
+f \big (y _ {\lambda} (w) \big) \leq f _ {\lambda} (w) \leq f \big (y (w) \big).
+$$
+
+Proof. Inequality (6) implies that  $\mathbf{c}\big(w,y(w)\big) - \mathbf{c}\big(w,y_{\lambda}(w)\big)\leq 0$  and the first inequality then follows simply from the definition of  $f_{\lambda}$ . As for the second one, it suffices to apply (7) to  $y = y(w)$ .
+
+Now, let us introduce few notions that will be useful later in the proofs. For a fixed  $\lambda$ ,  $W$  partitions into maximal connected sets  $P$  on which  $y_{\lambda}(w)$  is constant (see Fig. 7). We denote this collection of sets by  $\mathcal{W}_{\lambda}$  and set  $\mathcal{W} = \mathcal{W}_0$ .
+
+For  $\lambda \in \mathbb{R}$  and  $y_{1} \neq y_{2} \in Y$ , we denote
+
+$$
+F _ {\lambda} \left(y _ {1}, y _ {2}\right) = \left\{w \in W: c \left(w, y _ {1}\right) + \lambda f \left(y _ {1}\right) = c \left(w, y _ {2}\right) + \lambda f \left(y _ {2}\right) \right\}.
+$$
+
+We write  $F(y_{1},y_{2}) = F_{0}(y_{1},y_{2})$ , for brevity. For technical reasons, we also allow negative values of  $\lambda$  here.
+
+![](images/787d047c7a2f9fe404bcc40f6fa04cec93b40768332512369efea2018ca2c020.jpg)  
+(a) The situation for  $\lambda = 0$ . We can see the polytope  $P$  on which  $y(w)$  attains  $y_{1} \in Y$ . The boundary of  $P$  is composed of segments of lines  $F(y_{1}, y_{k})$  for  $k = 2, \ldots, 5$ .  
+Figure 7: The family  $\mathcal{W}_{\lambda}$  of all maximal connected sets  $P$  on which  $y_{\lambda}$  is constant.
+
+![](images/6cbc5d1007a7e1ed4667ad946df44b1f420a7cb0eb4f727636bfc6b1502d7b5c.jpg)  
+(b) The same situation is captured for some relatively small  $\lambda > 0$ . Each line  $F_{\lambda}(y_1, y_k)$  is parallel to its corresponding  $F(y_1, y_k)$  and encompasses a convex polytope in  $\mathcal{W}_{\lambda}$ .
+
+Note, that if  $W = \mathbb{R}^N$ , then  $F_{\lambda}$  is a hyperplane since  $\mathbf{c}$ 's are linear. In general,  $W$  may just be a proper subset of  $\mathbb{R}^N$  and, in that case,  $F_{\lambda}$  is just the restriction of a hyperplane onto  $W$ . Consequently, it may happen that  $F_{\lambda}(y_1, y_2)$  will be empty for some pair of  $y_1, y_2$  and some  $\lambda \in \mathbb{R}$ . To emphasize this fact, we say "hyperplane in  $W$ ". Analogous considerations should be taken into account for all other linear objects. The note "in  $W$ " stands for the intersection of these linear objects with the set  $W$ .
+
+Observation 4. Let  $P \in \mathcal{W}_{\lambda}$  and let  $y_{\lambda}(w) = y$  for  $w \in P$ . Then  $P$  is a convex polytope in  $W$ , where the facets consist of parts of finitely many hyperplanes  $F_{\lambda}(y, y_k)$  in  $W$  for some  $\{y_k\} \subset Y$ .
+
+Proof. Assume that  $W = \mathbb{R}^N$ . The values of  $y_{\lambda}$  may only change on hyperplanes of the form  $F_{\lambda}(y, y')$  for some  $y' \in Y$ . Then  $P$  is an intersection of corresponding half-spaces and therefore  $P$  is a convex polytope. If  $W$  is a proper subset of  $\mathbb{R}^N$  the claim follows by intersecting all the objects with  $W$ .
+
+Observation 5. Let  $y_{1},y_{2}\in Y$  be distinct. If nonempty, the hyperplanes  $F(y_{1},y_{2})$  and  $F_{\lambda}(y_1,y_2)$  are parallel and their distance is equal to  $|\lambda |K(y_1,y_2)$ , where
+
+$$
+K (y _ {1}, y _ {2}) = \frac {| f (y _ {1}) - f (y _ {2}) |}{\| y _ {1} - y _ {2} \|}.
+$$
+
+Proof. If we define a function  $c(w) = \mathbf{c}(w, y_1) - \mathbf{c}(w, y_2) = w(y_1 - y_2)$  and a constant  $C = f(y_2) - f(y_1)$ , then our objects rewrite to
+
+$$
+F (y _ {1}, y _ {2}) = \{w \in W: c (w) = 0 \} \quad \text {a n d} \quad F _ {\lambda} (y _ {1}, y _ {2}) = \{w \in W: c (w) = \lambda C \}.
+$$
+
+Since  $c$  is linear, these sets are parallel and  $F(y_{1},y_{2})$  intersects the origin. Thus, the required distance is the distance of the hyperplane  $F_{\lambda}(y_1,y_2)$  from the origin, which equals to  $|\lambda C| / \| y_1 - y_2\|$ .
+
+As the set  $Y$  is finite, there is a uniform upper bound  $K$  on all values of  $K(y_{1},y_{2})$ . Namely
+
+$$
+K = \max  _ {\substack {y _ {1}, y _ {2} \in Y \\ y _ {1} \neq y _ {2}}} K (y _ {1}, y _ {2}). \tag{8}
+$$
+
+# A.2.1 PROOF OF THEOREM 1
+
+Proof of Property A1. Now, Property A1 follows, since
+
+$$
+f _ {\lambda} (w) = \frac {1}{\lambda} \left[ \mathbf {c} \left(w, y _ {\lambda} (w)\right) + \lambda f \left(y _ {\lambda} (w)\right) \right] - \frac {1}{\lambda} \mathbf {c} \left(w, y (w)\right)
+$$
+
+and  $f_{\lambda}$  is a difference of continuous and piecewise affine functions.
+
+Proof of Property A2. Let  $0 < \lambda_1 \leq \lambda_2$  be given. We show that  $W_{\mathrm{eq}}^{\lambda_2} \subseteq W_{\mathrm{eq}}^{\lambda_1}$  which is the same as showing  $W_{\mathrm{dif}}^{\lambda_1} \subseteq W_{\mathrm{dif}}^{\lambda_2}$ . Assume that  $w \in W_{\mathrm{eq}}^{\lambda_2}$ , that is, by the definition of  $W_{\mathrm{eq}}^{\lambda_2}$  and  $f_{\lambda}$ ,
+
+$$
+\mathbf {c} (w, y (w)) + \lambda_ {2} f (y (w)) = \mathbf {c} (w, y _ {2}) + \lambda_ {2} f (y _ {2}), \tag {9}
+$$
+
+in which we denoted  $y_{2} = y_{\lambda_{2}}(w)$ . Our goal is to show that
+
+$$
+\mathbf {c} (w, y (w)) + \lambda_ {1} f (y (w)) = \mathbf {c} (w, y _ {1}) + \lambda_ {1} f (y _ {1}), \tag {10}
+$$
+
+where  $y_{1} = y_{\lambda_{1}}(w)$  as this equality then guarantees that  $w \in W_{\mathrm{eq}}^{\lambda_1}$ . Observe that (7) applied to  $\lambda = \lambda_{1}$  and  $y = y(w)$ , yields the inequality “ $\geq$ ” in (10).
+
+Let us show the reversed inequality. By Observation 3 applied to  $\lambda = \lambda_1$ , we have
+
+$$
+f (y (w)) \geq f (y _ {1}). \tag {11}
+$$
+
+We now use (7) with  $\lambda = \lambda_{2}$  and  $y = y_{1}$ , followed by equality (9) to obtain
+
+$$
+\begin{array}{l} \mathbf {c} (w, y _ {1}) + \lambda_ {1} f (y _ {1}) = \mathbf {c} (w, y _ {1}) + \lambda_ {2} f (y _ {1}) + (\lambda_ {1} - \lambda_ {2}) f (y _ {1}) \\ \geq \mathbf {c} (w, y _ {2}) + \lambda_ {2} f (y _ {2}) + (\lambda_ {1} - \lambda_ {2}) f (y _ {1}) \\ = \mathbf {c} (w, y (w)) + \lambda_ {2} f (y (w)) + (\lambda_ {1} - \lambda_ {2}) f (y _ {1}) \\ = \mathbf {c} (w, y (w)) + \lambda_ {1} f (y (w)) + (\lambda_ {2} - \lambda_ {1}) [ f (y (w)) - f (y _ {1}) ] \\ \geq \mathbf {c} (w, y (w)) + \lambda_ {1} f (y (w)) \\ \end{array}
+$$
+
+where the last inequality holds due to (11).
+
+Next, we have to show that  $W_{\mathrm{dif}}^{\lambda} \to \emptyset$  as  $\lambda \to 0^{+}$ , i.e. that for almost every  $w \in W$ , there is a  $\lambda > 0$  such that  $w \notin W_{\mathrm{dif}}^{\lambda}$ . To this end, let  $w \in W$  be given. We can assume that  $y(w)$  is a unique solution of solver (1), since two solutions, say  $y_{1}$  and  $y_{2}$ , coincide only on the hyperplane  $F(y_{1},y_{2})$  in  $W$ , which is of measure zero. Thus, since  $Y$  is finite, the constant
+
+$$
+c = \min_{\substack{y\in Y\\ y\neq y(w)}}\left\{\mathbf{c}(w,y) - \mathbf{c}\bigl(w,y(w)\bigr)\right\}
+$$
+
+is positive. Denote
+
+$$
+d = \max  _ {y \in Y} \left\{f (y (w)) - f (y) \right\}. \tag {12}
+$$
+
+If  $d > 0$ , set  $\lambda < c / d$ . Then, for every  $y \in Y$  such that  $f(y(w)) > f(y)$ , we have
+
+$$
+\lambda <   \frac {\mathbf {c} (w , y) - \mathbf {c} (w , y (w))}{f (y (w)) - f (y)}
+$$
+
+which rewrites
+
+$$
+\mathbf {c} (w, y (w)) + \lambda f (y (w)) <   \mathbf {c} (w, y) + \lambda f (y). \tag {13}
+$$
+
+For the remaining  $y$ 's, (13) holds trivially for every  $\lambda > 0$ . Therefore,  $y(w)$  is a solution of the minimization problem (3), whence  $y_{\lambda}(w) = y(w)$ . This shows that  $w \in W_{\mathrm{eq}}^{\lambda}$  as we wished. If  $d = 0$ , then  $f(y(w)) \leq f(y)$  for every  $y \in Y$  and (13) follows again.
+
+Proof of Property A3. Let  $y_{1} \neq y_{2} \in Y$  be given. We show that on the component of the set
+
+$$
+\{w \in W: y (w) = y _ {1} \text {a n d} y _ {\lambda} (w) = y _ {2} \} \tag {14}
+$$
+
+the function  $f_{\lambda}$  agrees with a  $\delta$ -interpolator, where  $\delta \leq C\lambda$  and  $C > 0$  is an absolute constant. The claim follows as there are only finitely many sets and their components of the form (14) in  $W_{\mathrm{dif}}^{\lambda}$ .
+
+Let us set
+
+$$
+h (w) = \mathbf {c} (w, y _ {1}) - \mathbf {c} (w, y _ {2}) \quad \text {f o r} w \in W
+$$
+
+and
+
+$$
+g (w) = f \left(y _ {2}\right) - \frac {1}{\lambda} h (w).
+$$
+
+The condition on  $\mathbf{c}$  tells us that  $h$  is a non-constant affine function. It follows by the definition of  $F(y_{1},y_{2})$  and  $F_{\lambda}(y_1,y_2)$  that
+
+$$
+h (w) = 0 \quad \text {i f a n d o n l y i f} \quad w \in F \left(y _ {1}, y _ {2}\right) \tag {15}
+$$
+
+and
+
+$$
+h (w) = \lambda \left(f \left(y _ {2}\right) - f \left(y _ {1}\right)\right) \quad \text {i f a n d o n l y i f} \quad w \in F _ {\lambda} \left(y _ {1}, y _ {2}\right). \tag {16}
+$$
+
+By Observation 5, the sets  $F$  and  $F_{\lambda}$  are parallel hyperplanes. Denote by  $G$  the nonempty intersection of their corresponding half-spaces in  $W$ . We show that  $g$  is a  $\delta$ -interpolator of  $f$  on  $G$  between  $y_{1}$  and  $y_{2}$ , with  $\delta$  being linearly controlled by  $\lambda$ .
+
+We have already observed that  $g$  is the affine function ranging from  $f(y_{1})$  - on the set  $F_{\lambda}(y_1,y_2)$  to  $f(y_{2})$  - on the set  $F(y_{1},y_{2})$ . It remains to show that  $g$  attains both the values  $f(y_{1})$  and  $f(y_{2})$  at most  $\delta$ -far from the sets  $P_{1}$  and  $P_{2}$ , respectively, where  $P_{k}\in \mathcal{W}$  denotes a component of the set  $\{w\in W:y(w) = y_k\} ,k = 1,2.$
+
+Consider  $y_{1}$  first. By Observation 4, there are  $z_{1},\ldots ,z_{\ell}\in Y$ , such that facets of  $P_{1}$  are parts of hyperplanes  $F(y_{1},z_{1}),\ldots ,F(y_{1},z_{\ell})$  in  $W$ . Each of them separates  $W$  into two half-spaces, say  $W_{k}^{+}$  and  $W_{k}^{-}$ , where  $W_{k}^{-}$  is the half-space which contains  $P_{1}$  and  $W_{k}^{+}$  is the other one. Let us denote
+
+$$
+c _ {k} (w) = \mathbf {c} \left(w, y _ {1}\right) - \mathbf {c} \left(w, z _ {k}\right) \quad \text {f o r} w \in W \text {a n d} k = 1, \dots , \ell .
+$$
+
+Every  $c_k$  is a non-zero linear function which is negative on  $W_k^-$  and positive on  $W_k^+$ . By the definition of  $y_1$ , we have
+
+$$
+\mathbf {c} (w, y _ {1}) + \lambda f (y _ {1}) \leq \mathbf {c} (w, z _ {k}) + \lambda f (z _ {k}) \quad \text {f o r} w \in P _ {1} \text {a n d f o r} k = 1, \dots , \ell ,
+$$
+
+that is
+
+$$
+c _ {k} (w) \leq \lambda \left(f \left(z _ {k}\right) - f \left(y _ {1}\right)\right) \quad \text {f o r} w \in P _ {1} \text {a n d f o r} k = 1, \dots , \ell .
+$$
+
+Figure 8: The polytopes  $P_{1}$  and  $P_{1}^{\lambda}$  and the interpolator  $g$ .  
+![](images/ea1262fe63523bc53e80be8f1c75ac016aa7c1f353e4106fc5b51f00737a759f.jpg)  
+(a) The facets of  $P_{1}$  consist of parts of hyperplanes  $F(y_{1},z_{k})$  in  $W$ . Each facet  $F(y_{1},z_{k})$  has its corresponding shifts  $F_{\lambda}$  and  $F_{-\lambda}$ , from which only one intersects  $P$ . The polytope  $P_{1}^{\lambda}$  is then bounded by those outer shifts.
+
+![](images/e87dcf38af5c648c178a14a0717cf15e1bc0746176f253a979877a5c4354cb4f.jpg)  
+(b) The interpolator  $g$  attains the value  $f(y_{1})$  on a part of  $F_{\lambda}(y_1,y_2)$  -a border of the domain  $G$  The value  $f(y_{2})$  is attained on a part of  $F(y_{1},y_{2})$  - the second border of the strip  $G$
+
+Now, denote
+
+$$
+W _ {k} ^ {\lambda} = \left\{w \in W: c _ {k} (w) \leq \lambda \left| f \left(z _ {k}\right) - f \left(y _ {1}\right) \right| \right\} \quad \text {f o r} k = 1, \dots , \ell .
+$$
+
+Each  $W_{k}^{\lambda}$  is a half-space in  $W$  containing  $W_{k}^{-}$  and hence  $P_{1}$ . Let us set  $P_{1}^{\lambda} = \bigcap_{k=1}^{\ell} W_{k}^{\lambda}$ . Clearly,  $P_{1} \subseteq P_{1}^{\lambda}$  (see Fig. 8). By Observation 5, the distance of the hyperplane  $\{w \in W : c_{k}(w) = \lambda | f(z_{k}) - f(y_{1})|\}$  from  $P_{1}$  is at most  $\lambda K$ , where  $K$  is given by (8). Therefore, since all the facets of  $P_{1}^{\lambda}$  are at most  $\lambda K$  far from  $P_{1}$ , there is a constant  $C$  such that each point of  $P_{1}^{\lambda}$  is at most  $C\lambda$  far from  $P_{1}$ .
+
+Finally, choose any  $w_{1} \in P_{1}^{\lambda} \cap F_{\lambda}(y_{1}, y_{2})$ . By (16), we have  $g(w_{1}) = f(y_{1})$ , and by the definition of  $P_{1}^{\lambda}$ ,  $w_{1}$  is no farther than  $C\lambda$  away from  $P_{1}$ .
+
+Now, let us treat  $y_{2}$  and define the set  $P_{2}^{\lambda}$  analogous to  $P_{1}^{\lambda}$ , where each occurrence of  $y_{1}$  is replaced by  $y_{2}$ . Any  $w_{2} \in P_{2}^{\lambda} \cap F(y_{1},y_{2})$  has desired properties. Indeed, (15) ensures that  $g(w_{2}) = f(y_{2})$  and  $w_{2}$  is at most  $C\lambda$  far away from  $P_{2}$ .
+
+# A.3 DETAILS OF EXPERIMENTS
+
+# A.3.1 WARCRAFT SHORTEST PATH
+
+The maps for the dataset have been generated with a custom random generation process by using 142 tiles from the Wcraft II tileset (Guyomarch, 2017). The costs for the different terrain types range from 0.8-9.2. Some example maps of size  $18 \times 18$  are presented in Fig. 9a together with a histogram of the shortest path lengths. We used the first five layers of ResNet18 followed by a max-pooling operation to extract the latent costs for the vertices.
+
+Optimization was carried out via Adam optimizer (Kingma & Ba, 2014) with scheduled learning rate drops dividing the learning rate by 10 at epochs 30 and 40. Hyperparameters and model details are listed in Tab. 5
+
+Table 5: Experimental setup for Wrcraft Shortest Path.  
+
+<table><tr><td>k</td><td>Optimizer(LR)</td><td>Architecture</td><td>Epochs</td><td>Batch Size</td><td>λ</td></tr><tr><td>12, 18, 24, 30</td><td>Adam(5 × 10-4)</td><td>subset of ResNet18</td><td>50</td><td>70</td><td>20</td></tr></table>
+
+![](images/16832c51301b6cc4b14707753c1cd77dca738aa88ebaf9cfcc9b6f7e28f94630.jpg)  
+(a) Three random example maps.
+
+![](images/ec1ec963e8ee1b0e0169e32797817e932e84628b4b3b52cea173053c35a65e41.jpg)  
+Figure 9:Warcraft SP(18) dataset.
+
+![](images/45c46e0ba6b1faa280c5fb8906fce373ba39062197ab98c47847681535fd5169.jpg)  
+(b) the shortest path distribution in the training set. All possible path lengths (18-35) occur.
+
+![](images/90b16b5b107a4ac01a6cf99274a0a6ba854fd17e5bc5bad8f38feb1179234ecf.jpg)
+
+# A.3.2 MNIST MIN-COST PERFECT MATCHING
+
+The dataset consists of randomly generated grids of MNIST digits that are sampled from a subset of 1000 digits of the full MNIST dataset. We trained a fully convolutional neural network with two convolutional layers followed by a max-pooling operation that outputs a  $k \times k$  grid of vertex costs for each example. The vertex costs are transformed into the edge costs via the known cost function and the edge costs are then the inputs to the Blossom V solver (Edmonds, 1965) as implemented in (Kolmogorov, 2009).
+
+Regarding the optimization procedure, we employed the Adam optimizer along with scheduled learning rate drops dividing the learning rate by 10 at epochs 10 and 20, respectively. Other training details are in Tab. 6. Lower batch sizes were used to reduce GPU memory requirements.
+
+Table 6: Experimental setup for MNIST Min-cost Perfect Matching.  
+
+<table><tr><td>k</td><td>Optimizer(LR)</td><td>Architecture
+[channels, kernel size, stride]</td><td>Epochs</td><td>Batch Size</td><td>λ</td></tr><tr><td>4,8</td><td>Adam(10-3)</td><td>[[20,5,1], [20,5,1]]</td><td>30</td><td>70</td><td>10</td></tr><tr><td>16</td><td>Adam(10-3)</td><td>[[50,5,1], [50,5,1]]</td><td>30</td><td>40</td><td>10</td></tr><tr><td>24</td><td>Adam(10-3)</td><td>[[50,5,1], [50,5,1]]</td><td>30</td><td>30</td><td>10</td></tr></table>
+
+# A.3.3 GLOBE TRAVELING SALESMAN PROBLEM
+
+For the Globe Traveling Salesman Problem we used a convolutional neural network architecture of three convolutional layers and two fully connected layers. The last layer outputs a vector of dimension  $3k$  containing the  $k$  3-dimensional representations of the respective countries' capital cities. These representations are projected onto the unit sphere and the matrix of pairwise distances is fed to the TSP solver.
+
+The high combinatorial complexity of TSP has negative effects on the loss landscape and results in many local minima and high sensitivity to random restarts. For reducing sensitivity to restarts, we set Adam parameters to  $\beta_{1} = 0.5$  (as it is done for example in GAN training (Radford et al., 2015)) and  $\epsilon = 10^{-3}$ .
+
+The local minima correspond to solving planar TSP as opposed to spherical TSP. For example, if all cities are positioned to almost identical locations, the network can still make progress but it will never have the incentive to spread the cities apart in order to reach the global minimum. To mitigate that, we introduce a repellent force between epochs 15 and 30. In particular, we set
+
+$$
+L_{\text{rep}} = \underset {i\neq j}{\mathbb{E}}e^{-\| x_{i} - x_{j}\|}
+$$
+
+where  $x_{i} \in \mathbb{R}^{3}$  for  $i = 1, \ldots, k$  are the positions of the  $k$  cities on the unit sphere. The regularization constants  $C_k$  were chosen as 2.0, 3.0, 6.0, and 20.0 for  $k \in \{5, 10, 20, 40\}$ .
+
+For fine-tuning we also introduce scheduled learning rate drops where we divide the learning rate by 10 at epochs 80 and 90.
+
+Table 7: Experimental setup for the Globe Traveling Salesman Problem.  
+
+<table><tr><td>k</td><td>Optimizer(LR)</td><td>Architecture
+[channels, kernel size, stride], linear layer size</td><td>Epochs</td><td>Batch Size</td><td>λ</td></tr><tr><td>5, 10, 20</td><td>Adam(10-4)</td><td>[20, 4, 2], [50, 4, 2], 500]</td><td>100</td><td>50</td><td>20</td></tr><tr><td>40</td><td>Adam(5 × 10-5)</td><td>[20, 4, 2], [50, 4, 2], 500]</td><td>100</td><td>50</td><td>20</td></tr></table>
+
+In Fig. 5b, we compare the true city locations with the ones learned by the hybrid architecture. Due to symmetries of the sphere, the architecture can embed the cities in any rotated or flipped fashion. We resolve this by computing "the most favorable" isometric transformation of the suggested locations. In particular, we solve the orthogonal Procrustes problem (Gower & Dijksterhuis, 2004)
+
+$$
+R^{*} = \operatorname *{arg  min}_{\substack{R:R^{T}\\ R = I}}\| RX - Y\|^{2}
+$$
+
+where  $X$  are the suggested locations,  $Y$  the true locations, and  $R^*$  the optimal transformation to apply. We report the resulting offsets in kilometers in Tab. 8.
+
+Table 8: Average errors of city placement on the Earth.  
+
+<table><tr><td>k</td><td>5</td><td>10</td><td>20</td><td>40</td></tr><tr><td>Location offset (km)</td><td>69 ± 11</td><td>19 ± 5</td><td>11 ± 5</td><td>58 ± 7</td></tr></table>
+
+# A.4 TRAVELING SALESMAN WITH AN APPROXIMATE SOLVER
+
+Since approximate solvers often appear in practice where the combinatorial instances are too large to be solved exactly in reasonable time, we test our method also in this setup. In particular, we use the approximate solver (OR-Tools (ort, 2019)) for the Globe TSP. We draw two conclusions from the numbers presented below in Tab. 9.
+
+(i) The choice of the solver matters. Even if OR-Tools is fed with the ground truth representations (i.e. true locations) it does not achieve perfect results on the test set (see the right column). We expect, that also in practical applications, running a suboptimal solver (e.g. a differentiable relaxation) substantially reduces the maximum attainable performance.  
+(ii) The suboptimality of the solver didn't harm the feature extraction – the point of our method. Indeed, the learned locations yield performance that is close to the upper limit of what the solver allows (compare the middle and the right column).
+
+Table 9: Perfect path accuracy for Globe TSP using the approximate solver OR-Tools (ort, 2019). The maximal achievable performance is in the right column, where the solver uses the ground truth city locations.  
+
+<table><tr><td rowspan="2">k</td><td colspan="2">Embedding</td><td>OR-tools on GT locations</td></tr><tr><td>Train %</td><td>Test %</td><td>Test %</td></tr><tr><td>5</td><td>99.8 ± 0.0</td><td>99.3 ± 0.1</td><td>100.0</td></tr><tr><td>10</td><td>84.3 ± 0.2</td><td>84.4 ± 0.2</td><td>88.6</td></tr><tr><td>20</td><td>49.2 ± 0.2</td><td>48.6 ± 0.8</td><td>54.4</td></tr><tr><td>40</td><td>14.6 ± 0.1</td><td>15.1 ± 0.3</td><td>15.2</td></tr></table>
\ No newline at end of file
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+# DIRECTIONALMESSAGEPASSINGFORMOLECULAR GRAPHS
+
+Johannes Gasteiger, Janek Groß & Stephan Gunnemann
+
+Technical University of Munich, Germany
+
+{j.gasteiger,grossja,guennemann}@in.tum.de
+
+# ABSTRACT
+
+Graph neural networks have recently achieved great successes in predicting quantum mechanical properties of molecules. These models represent a molecule as a graph using only the distance between atoms (nodes). They do not, however, consider the spatial direction from one atom to another, despite directional information playing a central role in empirical potentials for molecules, e.g. in angular potentials. To alleviate this limitation we propose directional message passing, in which we embed the messages passed between atoms instead of the atoms themselves. Each message is associated with a direction in coordinate space. These directional message embeddings are rotationally equivariant since the associated directions rotate with the molecule. We propose a message passing scheme analogous to belief propagation, which uses the directional information by transforming messages based on the angle between them. Additionally, we use spherical Bessel functions and spherical harmonics to construct theoretically well-founded, orthogonal representations that achieve better performance than the currently prevalent Gaussian radial basis representations while using fewer than  $1/4$  of the parameters. We leverage these innovations to construct the directional message passing neural network (DimeNet). DimeNet outperforms previous GNNs on average by  $76\%$  on MD17 and by  $31\%$  on QM9. Our implementation is available online.
+
+# 1 INTRODUCTION
+
+In recent years scientists have started leveraging machine learning to reduce the computation time required for predicting molecular properties from a matter of hours and days to mere milliseconds. With the advent of graph neural networks (GNNs) this approach has recently experienced a small revolution, since they do not require any form of manual feature engineering and significantly outperform previous models (Gilmer et al., 2017; Schutt et al., 2017). GNNs model the complex interactions between atoms by embedding each atom in a high-dimensional space and updating these embeddings by passing messages between atoms. By predicting the potential energy these models effectively learn an empirical potential function. Classically, these functions have been modeled as the sum of four parts: (Leach, 2001)
+
+$$
+E = E _ {\text {b o n d s}} + E _ {\text {a n g l e}} + E _ {\text {t o r s i o n}} + E _ {\text {n o n - b o n d e d}}, \tag {1}
+$$
+
+where  $E_{\mathrm{bonds}}$  models the dependency on bond lengths,  $E_{\mathrm{angle}}$  on the angles between bonds,  $E_{\mathrm{torsion}}$  on bond rotations, i.e. the dihedral angle between two planes defined by pairs of bonds, and  $E_{\mathrm{non-bonded}}$  models interactions between unconnected atoms, e.g. via electrostatic or van der Waals interactions. The update messages in GNNs, however, only depend on the previous atom embeddings and the pairwise distances between atoms – not on directional information such as bond angles and rotations. Thus, GNNs lack the second and third terms of this equation and can only model them via complex higher-order interactions of messages. Extending GNNs to model them directly is not straightforward since GNNs solely rely on pairwise distances, which ensures their invariance to translation, rotation, and inversion of the molecule, which are important physical requirements.
+
+In this paper, we propose to resolve this restriction by using embeddings associated with the directions to neighboring atoms, i.e. by embedding atoms as a set of messages. These directional message
+
+embeddings are equivariant with respect to the above transformations since the directions move with the molecule. Hence, they preserve the relative directional information between neighboring atoms. We propose to let message embeddings interact based on the distance between atoms and the angle between directions. Both distances and angles are invariant to translation, rotation, and inversion of the molecule, as required. Additionally, we show that the distance and angle can be jointly represented in a principled and effective manner by using spherical Bessel functions and spherical harmonics. We leverage these innovations to construct the directional message passing neural network (DimeNet). DimeNet can learn both molecular properties and atomic forces. It is twice continuously differentiable and solely based on the atom types and coordinates, which are essential properties for performing molecular dynamics simulations. DimeNet outperforms previous GNNs on average by  $76\%$  on MD17 and by  $31\%$  on QM9. Our paper's main contributions are:
+
+1. Directional message passing, which allows GNNs to incorporate directional information by connecting recent advances in the fields of equivariance and graph neural networks as well as ideas from belief propagation and empirical potential functions such as Eq. 1.  
+2. Theoretically principled orthogonal basis representations based on spherical Bessel functions and spherical harmonics. Bessel functions achieve better performance than Gaussian radial basis functions while reducing the radial basis dimensionality by 4x or more.  
+3. The Directional Message Passing Neural Network (DimeNet): A novel GNN that leverages these innovations to set the new state of the art for molecular predictions and is suitable both for predicting molecular properties and for molecular dynamics simulations.
+
+# 2 RELATED WORK
+
+ML for molecules. The classical way of using machine learning for predicting molecular properties is combining an expressive, hand-crafted representation of the atomic neighborhood (Bartók et al., 2013) with Gaussian processes (Bartók et al., 2010; 2017; Chmiela et al., 2017) or neural networks (Behler & Parrinello, 2007). Recently, these methods have largely been superseded by graph neural networks, which do not require any hand-crafted features but learn representations solely based on the atom types and coordinates molecules (Duvenaud et al., 2015; Gilmer et al., 2017; Schütt et al., 2017; Hy et al., 2018; Unke & Meuwly, 2019). Our proposed message embeddings can also be interpreted as directed edge embeddings or embeddings on the line graph (Chen et al., 2019b). (Undirected) edge embeddings have already been used in previous GNNs for molecules (Jørgensen et al., 2018; Chen et al., 2019a). However, these GNNs use both node and edge embeddings and do not leverage any directional information.
+
+Graph neural networks. GNNs were first proposed in the 90s (Baskin et al., 1997; Sperduti & Starita, 1997) and 00s (Gori et al., 2005; Scarselli et al., 2009). General GNNs have been largely inspired by their application to molecular graphs and have started to achieve breakthrough performance in various tasks at around the same time the molecular variants did (Kipf & Welling, 2017; Gasteiger et al., 2019; Zambaldi et al., 2019). Some recent progress has been focused on GNNs that are more powerful than the 1-Weisfeiler-Lehman test of isomorphism (Morris et al., 2019; Maron et al., 2019). However, for molecular predictions these models are significantly outperformed by GNNs focused on molecules (see Sec. 7). Some recent GNNs have incorporated directional information by considering the change in local coordinate systems per atom (Ingraham et al., 2019). However, this approach breaks permutation invariance and is therefore only applicable to chain-like molecules (e.g. proteins).
+
+Equivariant neural networks. Group equivariance as a principle of modern machine learning was first proposed by Cohen & Welling (2016). Following work has generalized this principle to spheres (Cohen et al., 2018), molecules (Thomas et al., 2018), volumetric data (Weiler et al., 2018), and general manifolds (Cohen et al., 2019). Equivariance with respect to continuous rotations has been achieved so far by switching back and forth between Fourier and coordinate space in each layer (Cohen et al., 2018) or by using a fully Fourier space model (Kondor et al., 2018; Anderson et al., 2019). The former introduces major computational overhead and the latter imposes significant constraints on model construction, such as the inability of using non-linearities. Our proposed solution does not suffer from either of those limitations.
+
+# 3 REQUIREMENTS FOR MOLECULAR PREDICTIONS
+
+In recent years machine learning has been used to predict a wide variety of molecular properties, both low-level quantum mechanical properties such as potential energy, energy of the highest occupied molecular orbital (HOMO), and the dipole moment and high-level properties such as toxicity, permeability, and adverse drug reactions (Wu et al., 2018). In this work we will focus on scalar regression targets, i.e. targets  $t \in \mathbb{R}$ . A molecule is uniquely defined by the atomic numbers  $z = \{z_{1},\ldots ,z_{N}\}$  and positions  $\pmb {X} = \{\pmb {x}_1,\dots ,\pmb {x}_N\}$ . Some models additionally use auxiliary information  $\Theta$  such as bond types or electronegativity of the atoms. We do not include auxiliary features in this work since they are hand-engineered and non-essential. In summary, we define an ML model for molecular prediction with parameters  $\theta$  via  $f_{\theta}:\{X,z\} \to \mathbb{R}$ .
+
+Symmetries and invariances. All molecular predictions must obey some basic laws of physics, either explicitly or implicitly. One important example of such are the fundamental symmetries of physics and their associated invariances. In principle, these invariances can be learned by any neural network via corresponding weight matrix symmetries (Ravanbakhsh et al., 2017). However, not explicitly incorporating them into the model introduces duplicate weights and increases training time and complexity. The most essential symmetries are translational and rotational invariance (follows from homogeneity and isotropy), permutation invariance (follows from the indistinguishability of particles), and symmetry under parity, i.e. under sign flips of single spatial coordinates.
+
+Molecular dynamics. Additional requirements arise when the model should be suitable for molecular dynamics (MD) simulations and predict the forces  $\pmb{F}_i$  acting on each atom. The force field is a conservative vector field since it must satisfy conservation of energy (the necessity of which follows from homogeneity of time (Noether, 1918)). The easiest way of defining a conservative vector field is via the gradient of a potential function. We can leverage this fact by predicting a potential instead of the forces and then obtaining the forces via backpropagation to the atom coordinates, i.e.  $\pmb{F}_i(\pmb{X},\pmb{z}) = -\frac{\partial}{\partial\pmb{x}_i} f_\theta (\pmb{X},\pmb{z})$ . We can even directly incorporate the forces in the training loss and directly train a model for MD simulations (Pukrittayakamee et al., 2009):
+
+$$
+\mathcal {L} _ {\mathrm {M D}} (\boldsymbol {X}, \boldsymbol {z}) = \left| f _ {\theta} (\boldsymbol {X}, \boldsymbol {z}) - \hat {t} (\boldsymbol {X}, \boldsymbol {z}) \right| + \frac {\rho}{3 N} \sum_ {i = 1} ^ {N} \sum_ {\alpha = 1} ^ {3} \left| - \frac {\partial f _ {\theta} (\boldsymbol {X} , \boldsymbol {z})}{\partial \boldsymbol {x} _ {i \alpha}} - \hat {F} _ {i \alpha} (\boldsymbol {X}, \boldsymbol {z}) \right|, \tag {2}
+$$
+
+where the target  $\hat{t} = \hat{E}$  is the ground-truth energy (usually available as well),  $\hat{\pmb{F}}$  are the ground-truth forces, and the hyperparameter  $\rho$  sets the forces' loss weight. For stable simulations  $\pmb{F}_i$  must be continuously differentiable and the model  $f_{\theta}$  itself therefore twice continuously differentiable. We hence cannot use discontinuous transformations such as ReLU non-linearities. Furthermore, since the atom positions  $\pmb{X}$  can change arbitrarily we cannot use pre-computed auxiliary information  $\Theta$  such as bond types.
+
+# 4 DIRECTIONALMESSAGEPASSING
+
+Graph neural networks. Graph neural networks treat the molecule as a graph, in which the nodes are atoms and edges are defined either via a predefined molecular graph or simply by connecting atoms that lie within a cutoff distance  $c$ . Each edge is associated with a pairwise distance between atoms  $d_{ij} = \| \pmb{x}_i - \pmb{x}_j\|_2$ . GNNs implement all of the above physical invariances by construction since they only use pairwise distances and not the full atom coordinates. However, note that a predefined molecular graph or a step function-like cutoff cannot be used for MD simulations since this would introduce discontinuities in the energy landscape. GNNs represent each atom  $i$  via an atom embedding  $h_i \in \mathbb{R}^H$ . The atom embeddings are updated in each layer by passing messages along the molecular edges. Messages are usually transformed based on an edge embedding  $e_{(ij)} \in \mathbb{R}^{H_e}$  and summed over the atom's neighbors  $\mathcal{N}_i$ , i.e. the embeddings are updated in layer  $l$  via
+
+$$
+\boldsymbol {h} _ {i} ^ {(l + 1)} = f _ {\text {u p d a t e}} \left(\boldsymbol {h} _ {i} ^ {(l)}, \sum_ {j \in \mathcal {N} _ {i}} f _ {\text {i n t}} \left(\boldsymbol {h} _ {j} ^ {(l)}, \boldsymbol {e} _ {(i j)} ^ {(l)}\right)\right), \tag {3}
+$$
+
+with the update function  $f_{\mathrm{update}}$  and the interaction function  $f_{\mathrm{int}}$ , which are both commonly implemented using neural networks. The edge embeddings  $e_{(ij)}^{(l)}$  usually only depend on the interatomic distances, but can also incorporate additional bond information (Gilmer et al., 2017) or be recursively updated in each layer using the neighboring atom embeddings (Jorgensen et al., 2018).
+
+Directionality. In principle, the pairwise distance matrix contains the full geometrical information of the molecule. However, GNNs do not use the full distance matrix since this would mean passing messages globally between all pairs of atoms, which increases computational complexity and can lead to overfitting. Instead, they usually use a cutoff distance  $c$ , which means they cannot distinguish between certain molecules (Xu et al., 2019). E.g. at a cutoff of roughly  $2\AA$  a regular GNN would not be able to distinguish between a hexagonal (e.g. Cyclohexane) and two triangular molecules (e.g. Cyclopropane) with the same bond lengths since the neighborhoods of each atom are exactly the same for both (see Appendix, Fig. 6). This problem can be solved by modeling the directions to neighboring atoms instead of just their distances. A principled way of doing so while staying invariant to a transformation group  $G$  (such as described in Sec. 3) is via group-equivariance (Cohen & Welling, 2016). A function  $f: X \to Y$  is defined as being equivariant if  $f(\varphi_g^X(x)) = \varphi_g^Y(f(x))$ , with the group action in the input and output space  $\varphi_g^X$  and  $\varphi_g^Y$ . However, equivariant CNNs only achieve equivariance with respect to a discrete set of rotations (Cohen & Welling, 2016). For a precise prediction of molecular properties we need continuous equivariance with respect to rotations, i.e. to the SO(3) group.
+
+Directional embeddings. We solve this problem by noting that an atom by itself is rotationally invariant. This invariance is only broken by neighboring atoms that interact with it, i.e. those inside the cutoff  $c$ . Since each neighbor breaks up to one rotational invariance they also introduce additional degrees of freedom, which we need to represent in our model. We can do so by generating a separate embedding  $m_{ji}$  for each atom  $i$  and neighbor  $j$  by applying the same learned filter in the direction of each neighboring atom (in contrast to equivariant CNNs, which apply filters in fixed, global directions). These directional embeddings are equivariant with respect to global rotations since the associated directions rotate with the molecule and hence conserve the relative directional information between neighbors.
+
+Representation via joint 2D basis. We use the directional information associated with each embedding by leveraging the angle  $\alpha_{(kj,ji)} = \angle x_kx_jx_i$  when aggregating the neighboring embeddings  $m_{kj}$  of  $m_{ji}$ . We combine the angle with the interatomic distance  $d_{kj}$  associated with the incoming message  $m_{kj}$  and jointly represent both in  $a_{\mathrm{SBF}}^{(kj,ji)} \in \mathbb{R}^{N_{\mathrm{SHBF}} \cdot N_{\mathrm{SRBF}}}$  using a 2D representation based on spherical Bessel functions and spherical harmonics, as explained in Sec. 5. We empirically found that this basis representation provides a better inductive bias than the raw angle alone. Note that by only using interatomic distances and angles our model becomes invariant to rotations.
+
+Message embeddings. The directional embedding  $m_{ji}$  associated with the atom pair  $ji$  can be thought of as a message being sent from atom  $j$  to atom  $i$ . Hence, in analogy to belief propagation, we embed each atom  $i$  using a set of incoming messages  $m_{ji}$ , i.e.  $h_i = \sum_{j \in \mathcal{N}_i} m_{ji}$ , and update the message  $m_{ji}$  based on the incoming messages  $m_{kj}$  (Yedidia et al., 2003). Hence, as illustrated in Fig. 1, we define the update function and aggregation scheme for message embeddings as
+
+![](images/692566f7170bc9a8f89875f6ed8833d64ac4ef39419ab11654bea95c85cfe2f6.jpg)  
+Figure 1: Aggregation scheme for message embeddings.
+
+$$
+\boldsymbol {m} _ {j i} ^ {(l + 1)} = f _ {\text {u p d a t e}} \left(\boldsymbol {m} _ {j i} ^ {(l)}, \sum_ {k \in \mathcal {N} _ {j} \backslash \{i \}} f _ {\text {i n t}} \left(\boldsymbol {m} _ {k j} ^ {(l)}, \boldsymbol {e} _ {\mathrm {R B F}} ^ {(j i)}, \boldsymbol {a} _ {\mathrm {S B F}} ^ {(k j, j i)}\right)\right), \tag {4}
+$$
+
+where  $e_{\mathrm{RBF}}^{(ji)}$  denotes the radial basis function representation of the inter-
+
+atomic distance  $d_{ji}$ , which will be discussed in Sec. 5. We found this aggregation scheme to not only have a nice analogy to belief propagation, but also to empirically perform better than alternatives. Note that since  $f_{\mathrm{int}}$  now incorporates the angle between atom pairs, or bonds, we have enabled our model to directly learn the angular potential  $E_{\mathrm{angle}}$ , the second term in Eq. 1. Moreover, the message embeddings are essentially embeddings of atom pairs, as used by the provably more powerful GNNs based on higher-order Weisfeiler-Lehman tests of isomorphism. Our model can therefore provably distinguish molecules that a regular GNN cannot (e.g. the previous example of a hexagonal and two triangular molecules) (Morris et al., 2019).
+
+# 5 PHYSICALLY BASED REPRESENTATIONS
+
+Representing distances and angles. For the interaction function  $f_{\mathrm{int}}$  in Eq. 4 we use a joint representation  $a_{\mathrm{SBF}}^{(kj,ji)}$  of the angles  $\alpha_{(kj,ji)}$  between message embeddings and the interatomic
+
+distances  $d_{kj} = \| \pmb{x}_k - \pmb{x}_j\| _2$ , as well as a representation  $e_{\mathrm{RBF}}^{(ji)}$  of the distances  $d_{ji}$ . Earlier works have used a set of Gaussian radial basis functions to represent interatomic distances, with tightly spaced means that are distributed e.g. uniformly (Schutt et al., 2017) or exponentially (Unke & Meuwly, 2019). Similar in spirit to the functional bases used by steerable CNNs (Cohen & Welling, 2017; Cheng et al., 2019) we propose to use an orthogonal basis instead, which reduces redundancy and thus improves parameter efficiency. Furthermore, a basis chosen according to the properties of the modeled system can even provide a helpful inductive bias. We therefore derive a proper basis representation for quantum systems next.
+
+From Schrödinger to Fourier-Bessel. To construct a basis representation in a principled manner we first consider the space of possible solutions. Our model aims at approximating results of density functional theory (DFT) calculations, i.e. results given by an electron density  $\langle \Psi (d)|\Psi (d)\rangle$ , with the electron wave function  $\Psi (d)$  and  $d = x_{k} - x_{j}$ . The solution space of  $\Psi (d)$  is defined by the time-independent Schrödinger equation  $\left(-\frac{\hbar^2}{2m}\nabla^2 +V(d)\right)\Psi (d) = E\Psi (d)$ , with constant mass  $m$  and energy  $E$ . We do not know the potential  $V(d)$  and so choose it in an uninformative way by simply setting it to 0 inside the cutoff distance  $c$  (up to which we pass messages between atoms) and to  $\infty$  outside. Hence, we arrive at the Helmholtz equation  $(\nabla^{2} + k^{2})\Psi (d) = 0$ , with the wave number  $k = \frac{\sqrt{2mE}}{\hbar}$  and the boundary condition  $\Psi (c) = 0$  at the cutoff  $c$ . Separation of variables in polar coordinates  $(d,\alpha ,\varphi)$  yields the solution (Griffiths & Schroeter, 2018)
+
+$$
+\Psi (d, \alpha , \varphi) = \sum_ {l = 0} ^ {\infty} \sum_ {m = - l} ^ {l} \left(a _ {l m} j _ {l} (k d) + b _ {l m} y _ {l} (k d)\right) Y _ {l} ^ {m} (\alpha , \varphi), \tag {5}
+$$
+
+with the spherical Bessel functions of the first and second kind  $j_{l}$  and  $y_{l}$  and the spherical harmonics  $Y_{l}^{m}$ . As common in physics we only use the regular solutions, i.e. those that do not approach  $-\infty$  at the origin, and hence set  $b_{lm} = 0$ . Recall that our first goal is to construct a joint 2D basis for  $d_{kj}$  and  $\alpha_{(kj,ji)}$ , i.e. a function that depends on  $d$  and a single angle  $\alpha$ . To achieve this we set  $m = 0$  and obtain  $\Psi_{\mathrm{SBF}}(d,\alpha) = \sum_{l}a_{l}j_{l}(kd)Y_{l}^{0}(\alpha)$ . The boundary conditions are satisfied by setting  $k = \frac{z_{ln}}{c}$ , where  $z_{ln}$  is the  $n$ -th root of the  $l$ -order Bessel function, which are precomputed numerically. Normalizing  $\Psi_{\mathrm{SBF}}$  inside the cutoff distance  $c$  yields the 2D spherical Fourier-Bessel basis  $\tilde{\pmb{a}}_{\mathrm{SBF}}^{(kj,ji)} \in \mathbb{R}^{N_{\mathrm{SBF}} \cdot N_{\mathrm{SRBF}}}$ , which is illustrated in Fig. 2 and defined by
+
+$$
+\tilde {a} _ {\mathrm {S B F}, l n} (d, \alpha) = \sqrt {\frac {2}{c ^ {3} j _ {l + 1} ^ {2} \left(z _ {l n}\right)}} j _ {l} \left(\frac {z _ {l n}}{c} d\right) Y _ {l} ^ {0} (\alpha), \tag {6}
+$$
+
+with  $l \in [0 \dots N_{\mathrm{SHBF}} - 1]$  and  $n \in [1 \dots N_{\mathrm{SRBF}}]$ . Our second goal is constructing a radial basis for  $d_{ji}$ , i.e. a function that solely depends on  $d$  and not on the angles  $\alpha$  and  $\varphi$ . We achieve this by setting  $l = m = 0$  and obtain  $\Psi_{\mathrm{RBF}}(d) = aj_0(\frac{z_{0,n}}{c} d)$ , with roots at  $z_{0,n} = n\pi$ . Normalizing this function on  $[0, c]$  and using  $j_0(d) = \sin(d)/d$  gives the radial basis  $\tilde{e}_{\mathrm{RBF}} \in \mathbb{R}^{N_{\mathrm{RBF}}}$ , as shown in Fig. 3 and defined by
+
+$$
+\tilde {e} _ {\mathrm {R B F}, n} (d) = \sqrt {\frac {2}{c}} \frac {\sin \left(\frac {n \pi}{c} d\right)}{d}, \tag {7}
+$$
+
+with  $n \in [1 \dots N_{\mathrm{RBF}}]$ . Both of these bases are purely real-valued and orthogonal in the domain of interest. They furthermore enable us to
+
+bound the highest-frequency components by  $\omega_{\alpha} \leq \frac{N_{\mathrm{SHBF}}}{2\pi}$ ,  $\omega_{d_{kj}} \leq \frac{N_{\mathrm{SRBF}}}{c}$ , and  $\omega_{d_{ji}} \leq \frac{N_{\mathrm{RBF}}}{c}$ . This restriction is an effective way of regularizing the model and ensures that predictions are stable to small perturbations. We found  $N_{\mathrm{SRBF}} = 6$  and  $N_{\mathrm{RBF}} = 16$  radial basis functions to be more than sufficient. Note that  $N_{\mathrm{RBF}}$  is 4x lower than PhysNet's 64 (Unke & Meuwly, 2019) and 20x lower than SchNet's 300 radial basis functions (Schütt et al., 2017).
+
+Continuous cutoff.  $\tilde{a}_{\mathrm{SBF}}^{(kj,ji)}$  and  $\tilde{e}_{\mathrm{RBF}}(d)$  are not twice continuously differentiable due to the step function cutoff at  $c$ . To alleviate this problem we introduce an envelope function  $u(d)$  that has a root of multiplicity 3 at  $d = c$ , causing the final functions  $a_{\mathrm{RBF}}(d) = u(d)\tilde{a}_{\mathrm{RBF}}(d)$  and  $e_{\mathrm{RBF}}(d) =$
+
+![](images/2c42fcf7683c03699807dbab641abb5b2681496391fbc2aed5285a5a6fe1ac52.jpg)  
+Figure 2: 2D spherical Fourier-Bessel basis  $\tilde{a}_{\mathrm{SBF},ln}(d,\alpha)$
+
+![](images/1d2a9d60cdd6c82c3dfddb4d6c45d156fefc704f71ef4844ce0c47df73601c73.jpg)  
+Figure 3: Radial Bessel basis for  $N_{\mathrm{RBF}} = 5$
+
+![](images/0775d5e0c5080c4811815be6964ad9e38899cebf1dd9abd4e215bfee725e7c5a.jpg)  
+Figure 4: The DimeNet architecture.  $\square$  denotes the layer's input and  $\parallel$  denotes concatenation. The distances  $d_{ji}$  are represented using spherical Bessel functions and the distances  $d_{kj}$  and angles  $\alpha_{(kj,ji)}$  are jointly represented using a 2D spherical Fourier-Bessel basis. An embedding block generates the initial message embeddings  $m_{ji}$ . These embeddings are updated in multiple interaction blocks via directional message passing, which uses the neighboring messages  $m_{kj}, k \in \mathcal{N}_j \setminus \{i\}$ , the 2D representations  $a_{\mathrm{SBF}}^{(kj,ji)}$ , and the distance representations  $e_{\mathrm{RBF}}^{(ji)}$ . Each block passes the resulting embeddings to an output block, which transforms them using the radial basis  $e_{\mathrm{RBF}}^{(ji)}$  and sums them up per atom. Finally, the outputs of all layers are summed up to generate the prediction.
+
+$u(d)\tilde{e}_{\mathrm{RBF}}(d)$  and their first and second derivatives to go to 0 at the cutoff. We achieve this with the polynomial
+
+$$
+u (d) = 1 - \frac {(p + 1) (p + 2)}{2} d ^ {p} + p (p + 2) d ^ {p + 1} - \frac {p (p + 1)}{2} d ^ {p + 2}, \tag {8}
+$$
+
+where  $p \in \mathbb{N}_0$ . We did not find the model to be sensitive to different choices of envelope functions and choose  $p = 6$ . Note that using an envelope function causes the bases to lose their orthonormality, which we did not find to be a problem in practice. We furthermore fine-tune the Bessel wave numbers  $k_n = \frac{n\pi}{c}$  used in  $\tilde{\boldsymbol{e}}_{\mathrm{RBF}} \in \mathbb{R}^{N_{\mathrm{RBF}}}$  via backpropagation after initializing them to these values, which we found to give a small boost in prediction accuracy.
+
+# 6 DIRECTIONALMESSAGEPASSINGNEURALNETWORK(DIMENET)
+
+The Directional Message Passing Neural Network's (DimeNet) design is based on a streamlined version of the PhysNet architecture (Unke & Meuwly, 2019), in which we have integrated directional message passing and spherical Fourier-Bessel representations. DimeNet generates predictions that are invariant to atom permutations and translation, rotation and inversion of the molecule. DimeNet is suitable both for the prediction of various molecular properties and for molecular dynamics (MD) simulations. It is twice continuously differentiable and able to learn and predict atomic forces via backpropagation, as described in Sec. 3. The predicted forces fulfill energy conservation by construction and are equivariant with respect to permutation and rotation. Model differentiability in combination with basis representations that have bounded maximum frequencies furthermore guarantees smooth predictions that are stable to small deformations. Fig. 4 gives an overview of the architecture.
+
+**Embedding block.** Atomic numbers are represented by learnable, randomly initialized atom type embeddings  $\pmb{h}_i^{(0)} \in \mathbb{R}^F$  that are shared across molecules. The first layer generates message embeddings from these and the distance between atoms via
+
+$$
+\boldsymbol {m} _ {j i} ^ {(1)} = \sigma \left(\left[ \boldsymbol {h} _ {j} ^ {(0)} \| \boldsymbol {h} _ {i} ^ {(0)} \| \boldsymbol {e} _ {\mathrm {R B F}} ^ {(j i)} \right] \boldsymbol {W} + \boldsymbol {b}\right), \tag {9}
+$$
+
+where  $\parallel$  denotes concatenation and the weight matrix  $W$  and bias  $\pmb{b}$  are learnable.
+
+Table 1: MAE on QM9. DimeNet sets the state of the art on 11 targets, outperforming the second-best model on average by  $31\%$  (mean std. MAE).  
+
+<table><tr><td>Target</td><td>Unit</td><td>PPGN</td><td>SchNet</td><td>PhysNet</td><td>MEGNet-s</td><td>Cormorant</td><td>DimeNet</td></tr><tr><td>μ</td><td>D</td><td>0.047</td><td>0.033</td><td>0.0529</td><td>0.05</td><td>0.13</td><td>0.0286</td></tr><tr><td>α</td><td>a03</td><td>0.131</td><td>0.235</td><td>0.0615</td><td>0.081</td><td>0.092</td><td>0.0469</td></tr><tr><td>εHOMO</td><td>meV</td><td>40.3</td><td>41</td><td>32.9</td><td>43</td><td>36</td><td>27.8</td></tr><tr><td>εLUMO</td><td>meV</td><td>32.7</td><td>34</td><td>24.7</td><td>44</td><td>36</td><td>19.7</td></tr><tr><td>Δε</td><td>meV</td><td>60.0</td><td>63</td><td>42.5</td><td>66</td><td>60</td><td>34.8</td></tr><tr><td>〈R2〉</td><td>a02</td><td>0.592</td><td>0.073</td><td>0.765</td><td>0.302</td><td>0.673</td><td>0.331</td></tr><tr><td>ZPVE</td><td>meV</td><td>3.12</td><td>1.7</td><td>1.39</td><td>1.43</td><td>1.98</td><td>1.29</td></tr><tr><td>U0</td><td>meV</td><td>36.8</td><td>14</td><td>8.15</td><td>12</td><td>28</td><td>8.02</td></tr><tr><td>U</td><td>meV</td><td>36.8</td><td>19</td><td>8.34</td><td>13</td><td>-</td><td>7.89</td></tr><tr><td>H</td><td>meV</td><td>36.3</td><td>14</td><td>8.42</td><td>12</td><td>-</td><td>8.11</td></tr><tr><td>G</td><td>meV</td><td>36.4</td><td>14</td><td>9.40</td><td>12</td><td>-</td><td>8.98</td></tr><tr><td>cv</td><td>cal/mol K</td><td>0.055</td><td>0.033</td><td>0.0280</td><td>0.029</td><td>0.031</td><td>0.0249</td></tr><tr><td>std. MAE</td><td>%</td><td>1.84</td><td>1.76</td><td>1.37</td><td>1.80</td><td>2.14</td><td>1.05</td></tr><tr><td>logMAE</td><td>-</td><td>-4.64</td><td>-5.17</td><td>-5.35</td><td>-5.17</td><td>-4.75</td><td>-5.57</td></tr></table>
+
+Interaction block. The embedding block is followed by multiple stacked interaction blocks. This block implements  $f_{\mathrm{int}}$  and  $f_{\mathrm{update}}$  of Eq. 4 as shown in Fig. 4. Note that the 2D representation  $a_{\mathrm{SBF}}^{(kj,ji)}$  is first transformed into an  $N_{\mathrm{bilinear}}$ -dimensional representation via a linear layer. The main purpose of this is to make the dimensionality of  $a_{\mathrm{SBF}}^{(kj,ji)}$  independent of the subsequent bilinear layer, which uses a comparatively large  $N_{\mathrm{bilinear}} \times F \times F$ -dimensional weight tensor. We have also experimented with using a bilinear layer for the radial basis representation, but found that the element-wise multiplication  $e_{\mathrm{RBF}}^{(ji)} W \odot m_{kj}$  performs better, which suggests that the 2D representations require more complex transformations than radial information alone. The interaction block transforms each message embedding  $m_{ji}$  using multiple residual blocks, which are inspired by ResNet (He et al., 2016) and consist of two stacked dense layers and a skip connection.
+
+Output block. The message embeddings after each block (including the embedding block) are passed to an output block. The output block transforms each message embedding  $m_{ji}$  using the radial basis  $e_{\mathrm{RBF}}^{(ji)}$ , which ensures continuous differentiability and slightly improves performance. Afterwards the incoming messages are summed up per atom  $i$  to obtain  $h_i = \sum_j m_{ji}$ , which is then transformed using multiple dense layers to generate the atom-wise output  $t_i^{(l)}$ . These outputs are then summed up to obtain the final prediction  $t = \sum_i \sum_l t_i^{(l)}$ .
+
+Continuous differentiability. Multiple model choices were necessary to achieve twice continuous model differentiability. First, DimeNet uses the self-gated Swish activation function  $\sigma(x) = x \cdot \operatorname{sigmoid}(x)$  (Ramachandran et al., 2018) instead of a regular ReLU activation function. Second, we multiply the radial basis functions  $\tilde{e}_{\mathrm{RBF}}(d)$  with an envelope function  $u(d)$  that has a root of multiplicity 3 at the cutoff  $c$ . Finally, DimeNet does not use any auxiliary data but relies on atom types and positions alone.
+
+# 7 EXPERIMENTS
+
+Models. For hyperparameter choices and training setup see Appendix B. We use 6 state-of-the-art models for comparison: SchNet (Schütt et al., 2017), PhysNet (results based on the reference implementation) (Unke & Meuwly, 2019), provably powerful graph networks (PPGN, results provided by the original authors) (Maron et al., 2019), MEGNet-simple (without auxiliary information) (Chen et al., 2019a), Cormorant (Anderson et al., 2019), and symmetrized gradient-domain machine learning (sGDML) (Chmiela et al., 2018). Note that sGDML cannot be used for QM9 since it can only be trained on a single molecule.
+
+QM9. We test DimeNet's performance for predicting molecular properties using the common QM9 benchmark (Ramakrishnan et al., 2014). It consists of roughly 130 000 molecules in equilibrium
+
+Table 2: MAE on MD17 using 1000 training samples (energies in  $\frac{\mathrm{kcal}}{\mathrm{mol}}$ , forces in  $\frac{\mathrm{kcal}}{\mathrm{mol}\mathring{\mathrm{A}}}$ ). DimeNet outperforms SchNet by a large margin and performs roughly on par with sGDML.  
+
+<table><tr><td></td><td></td><td>sGDML</td><td>SchNet</td><td>DimeNet</td></tr><tr><td rowspan="2">Aspirin</td><td>Energy</td><td>0.19</td><td>0.37</td><td>0.204</td></tr><tr><td>Forces</td><td>0.68</td><td>1.35</td><td>0.499</td></tr><tr><td rowspan="2">Benzene</td><td>Energy</td><td>0.10</td><td>0.08</td><td>0.078</td></tr><tr><td>Forces</td><td>0.06</td><td>0.31</td><td>0.187</td></tr><tr><td rowspan="2">Ethanol</td><td>Energy</td><td>0.07</td><td>0.08</td><td>0.064</td></tr><tr><td>Forces</td><td>0.33</td><td>0.39</td><td>0.230</td></tr><tr><td rowspan="2">Malonaldehyde</td><td>Energy</td><td>0.10</td><td>0.13</td><td>0.104</td></tr><tr><td>Forces</td><td>0.41</td><td>0.66</td><td>0.383</td></tr><tr><td rowspan="2">Naphthalene</td><td>Energy</td><td>0.12</td><td>0.16</td><td>0.122</td></tr><tr><td>Forces</td><td>0.11</td><td>0.58</td><td>0.215</td></tr><tr><td rowspan="2">Salicylic acid</td><td>Energy</td><td>0.12</td><td>0.20</td><td>0.134</td></tr><tr><td>Forces</td><td>0.28</td><td>0.85</td><td>0.374</td></tr><tr><td rowspan="2">Toluene</td><td>Energy</td><td>0.10</td><td>0.12</td><td>0.102</td></tr><tr><td>Forces</td><td>0.14</td><td>0.57</td><td>0.216</td></tr><tr><td rowspan="2">Uracil</td><td>Energy</td><td>0.11</td><td>0.14</td><td>0.115</td></tr><tr><td>Forces</td><td>0.24</td><td>0.56</td><td>0.301</td></tr><tr><td rowspan="2">std. MAE (%)</td><td>Energy</td><td>2.53</td><td>3.32</td><td>2.49</td></tr><tr><td>Forces</td><td>1.01</td><td>2.38</td><td>1.10</td></tr></table>
+
+![](images/e52d57c86062a80035d6bacf04a505ceb62446143cf7d6ef3b390a4677887dcf.jpg)  
+Figure 5: Examples of DimeNet filters. They exhibit a clear 2D structure. For details see Appendix D.
+
+Table 3: Ablation studies using multi-task learning on QM9. All of our contributions have a significant impact on performance.  
+
+<table><tr><td>Variation</td><td>MAE MAE DimeNet</td><td>ΔlogMAE</td></tr><tr><td>Gaussian RBF</td><td>110 %</td><td>0.10</td></tr><tr><td>\(N_{\text{SHBF}} = 1\)</td><td>126 %</td><td>0.11</td></tr><tr><td>Node embeddings</td><td>168 %</td><td>0.45</td></tr></table>
+
+with up to 9 heavy C, O, N, and F atoms. We use 110 000 molecules in the training, 10 000 in the validation and 10 831 in the test set. We only use the atomization energy for  $U_{0}$ ,  $U$ ,  $H$ , and  $G$ , i.e. subtract the atomic reference energies, which are constant per atom type, and perform the training using eV. In Table 1 we report the mean absolute error (MAE) of each target and the overall mean standardized MAE (std. MAE) and mean standardized logMAE (for details see Appendix C). We predict  $\Delta \epsilon$  simply by taking  $\epsilon_{\mathrm{LUMO}} - \epsilon_{\mathrm{HOMO}}$ , since it is calculated in exactly this way by DFT calculations. We train a separate model for each target, which significantly improves results compared to training a single shared model for all targets (see App. E). DimeNet sets the new state of the art on 11 out of 12 targets and decreases mean std. MAE by  $31\%$  and mean logMAE by 0.22 compared to the second-best model.
+
+MD17. We use MD17 (Chmiela et al., 2017) to test model performance in molecular dynamics simulations. The goal of this benchmark is predicting both the energy and atomic forces of eight small organic molecules, given the atom coordinates of the thermalized (i.e. non-equilibrium, slightly moving) system. The ground truth data is computed via molecular dynamics simulations using DFT. A separate model is trained for each molecule, with the goal of providing highly accurate individual predictions. This dataset is commonly used with 50 000 training and 10 000 validation and test samples. We found that DimeNet can match state-of-the-art performance in this setup. E.g. for Benzene, depending on the force weight  $\rho$ , DimeNet achieves  $0.035\mathrm{kcal mol}^{-1}$  MAE for the energy or  $0.07\mathrm{kcal mol}^{-1}$  and  $0.17\mathrm{kcal mol}^{-1}\mathring{\mathrm{A}}^{-1}$  for energy and forces, matching the results reported by Anderson et al. (2019) and Unke & Meuwly (2019). However, this accuracy is two orders of magnitude below the DFT calculation's accuracy (approx.  $2.3\mathrm{kcal mol}^{-1}$  for energy (Faber et al., 2017)), so any remaining difference to real-world data is almost exclusively due to errors in the DFT simulation. Truly reaching better accuracy can therefore only be achieved with more precise ground-truth data, which requires far more expensive methods (e.g. CCSD(T)) and thus ML models that are more sample-efficient (Chmiela et al., 2018). We therefore instead test our model on the harder task of using only 1000 training samples. As shown in Table 2 DimeNet outperforms SchNet by a large margin and performs roughly on par with sGDML. However, sGDML uses hand-engineered descriptors that provide a strong advantage for small datasets, can only be trained on a single molecule (a fixed set of atoms), and does not scale well with the number of atoms or training samples.
+
+Ablation studies. To test whether directional message passing and the Fourier-Bessel basis are the actual reason for DimeNet's improved performance, we ablate them individually and compare the mean standardized MAE and logMAE for multi-task learning on QM9. Table 3 shows that both of our contributions have a significant impact on the model's performance. Using 64 Gaussian RBFs
+
+instead of 16 and 6 Bessel basis functions to represent  $d_{ji}$  and  $d_{kj}$  increases the error by  $10\%$ , which shows that this basis does not only reduce the number of parameters but additionally provides a helpful inductive bias. DimeNet's error increases by around  $26\%$  when we ignore the angles between messages by setting  $N_{\mathrm{SHBF}} = 1$ , showing that directly incorporating directional information does indeed improve performance. Using node embeddings instead of message embeddings (and hence also ignoring directional information) has the largest impact and increases MAE by  $68\%$ , at which point DimeNet performs worse than SchNet. Furthermore, Fig. 5 shows that the filters exhibit a structurally meaningful dependence on both the distance and angle. For example, some of these filters are clearly being activated by benzene rings ( $120^{\circ}$  angle,  $1.39\mathring{\mathrm{A}}$  distance). This further demonstrates that the model learns to leverage directional information.
+
+# 8 CONCLUSION
+
+In this work we have introduced directional message passing, a more powerful and expressive interaction scheme for molecular predictions. Directional message passing enables graph neural networks to leverage directional information in addition to the interatomic distances that are used by normal GNNs. We have shown that interatomic distances can be represented in a principled and effective manner using spherical Bessel functions. We have furthermore shown that this representation can be extended to directional information by leveraging 2D spherical Fourier-Bessel basis functions. We have leveraged these innovations to construct DimeNet, a GNN suitable both for predicting molecular properties and for use in molecular dynamics simulations. We have demonstrated DimeNet's performance on QM9 and MD17 and shown that our contributions are the essential ingredients that enable DimeNet's state-of-the-art performance. DimeNet directly models the first two terms in Eq. 1, which are known as the important "hard" degrees of freedom in molecules (Leach, 2001). Future work should aim at also incorporating the third and fourth terms of this equation. This could improve predictions even further and enable the application to molecules much larger than those used in common benchmarks like QM9.
+
+# ACKNOWLEDGMENTS
+
+This research was supported by the German Federal Ministry of Education and Research (BMBF), grant no. 01IS18036B, and by the Deutsche Forschungsgemeinschaft (DFG) through the Emmy Noether grant GU 1409/2-1 and the TUM International Graduate School of Science and Engineering (IGSSE), GSC 81. The authors of this work take full responsibilities for its content.
+
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+Taco S. Cohen, Mario Geiger, Jonas Kohler, and Max Welling. Spherical CNNs. In ICLR, 2018.  
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+Johannes Gasteiger, Aleksandar Bojchevski, and Stephan Gunnemann. Predict then Propagate: Graph Neural Networks Meet Personalized PageRank. In ICLR, 2019.  
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+M. Gori, G. Monfardini, and F. Scarselli. A new model for learning in graph domains. In IEEE International Joint Conference on Neural Networks, July 2005.  
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+Truong Son Hy, Shubhendu Trivedi, Horace Pan, Brandon M. Anderson, and Risi Kondor. Predicting molecular properties with covariant compositional networks. The Journal of Chemical Physics, 148(24):241745, June 2018.  
+John Ingraham, Vikas K. Garg, Regina Barzilay, and Tommi S. Jaakkola. Generative Models for Graph-Based Protein Design. In ICLR workshop, 2019.  
+Peter Bjørn Jørgensen, Karsten Wedel Jacobsen, and Mikkel N. Schmidt. Neural Message Passing with Edge Updates for Predicting Properties of Molecules and Materials. CoRR, 1806.03146, 2018.  
+Thomas N. Kipf and Max Welling. Semi-Supervised Classification with Graph Convolutional Networks. In ICLR, 2017.  
+Risi Kondor, Zhen Lin, and Shubhendu Trivedi. Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network. In NeurIPS, 2018.  
+Andrew R. Leach. Molecular Modelling: Principles and Applications. 2001.
+
+Haggai Maron, Heli Ben-Hamu, Hadar Serviansky, and Yaron Lipman. Provably Powerful Graph Networks. In NeurIPS, 2019.  
+Christopher Morris, Martin Ritzert, Matthias Fey, William L. Hamilton, Jan Eric Lenssen, Gaurav Rattan, and Martin Grohe. Weisfeiler and Leman Go Neural: Higher-Order Graph Neural Networks. In AAAI, 2019.  
+Emmy Noether. Invariante Variationsprobleme. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 1918:235-257, 1918.  
+A. Pukrittayakamee, M. Malshe, M. Hagan, L. M. Raff, R. Narulkar, S. Bukkapatnum, and R. Komanduri. Simultaneous fitting of a potential-energy surface and its corresponding force fields using feedforward neural networks. The Journal of Chemical Physics, 130(13):134101, April 2009.  
+Prajit Ramachandran, Barret Zoph, and Quoc V. Le. Searching for Activation Functions. In ICLR workshop, 2018.  
+Raghunathan Ramakrishnan, Pavlo O. Dral, Matthias Rupp, and O. Anatole von Lilienfeld. Quantum chemistry structures and properties of 134 kilo molecules. Scientific Data, 1(1):1-7, August 2014.  
+Siamak Ravanbakhsh, Jeff G. Schneider, and Barnabás Póczos. Equivalence Through Parameter-Sharing. In ICML, 2017.  
+Sashank J. Reddi, Satyen Kale, and Sanjiv Kumar. On the Convergence of Adam and Beyond. In ICLR, 2018.  
+F. Scarselli, M. Gori, Ah Chung Tsoi, M. Hagenbuchner, and G. Monfardini. The Graph Neural Network Model. IEEE Transactions on Neural Networks, 20(1):61-80, January 2009.  
+Kristof Schütt, Pieter-Jan Kindermans, Huziel Enoc Sauceda Felix, Stefan Chmiela, Alexandre Tkatchenko, and Klaus-Robert Müller. SchNet: A continuous-filter convolutional neural network for modeling quantum interactions. In NIPS, 2017.  
+A. Sperduti and A. Starita. Supervised neural networks for the classification of structures. IEEE Transactions on Neural Networks, 8(3):714-735, May 1997.  
+Nathaniel Thomas, Tess Smidt, Steven M. Kearnes, Lusann Yang, Li Li, Kai Kohlhoff, and Patrick Riley. Tensor Field Networks: Rotation- and Translation-Equivariant Neural Networks for 3D Point Clouds. CoRR, 1802.08219, 2018.  
+Oliver T. Unke and Markus Meuwly. PhysNet: A Neural Network for Predicting Energies, Forces, Dipole Moments, and Partial Charges. Journal of Chemical Theory and Computation, 15(6): 3678-3693, June 2019.  
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+
+![](images/4270560a829d1d83f538ac7cf114b6a8efdb84cfff132ff86862859431f53071.jpg)  
+A INDISTINGUISHABLE MOLECULES
+
+![](images/c074a3d5a5bb9995c4015d2f2bdbba347ebc8d8880092c8dbcdef9bf32ff78f8.jpg)  
+Figure 6: A standard non-directional GNN cannot distinguish between a hexagonal (left) and two triangular molecules (right) with the same bond lengths, since the neighborhood of each atom is exactly the same. An example of this would be Cyclohexane and two Cyclopropane molecules with slightly stretched bonds, when the GNN either uses the molecular graph or a cutoff distance of  $c \leq 2.5\mathring{\mathrm{A}}$ . Directional message passing solves this problem by considering the direction of each bond.
+
+# B EXPERIMENTAL SETUP
+
+The model architecture and hyperparameters were optimized using the QM9 validation set. We use 6 stacked interaction blocks and embeddings of size  $F = 128$  throughout the model. For the basis functions we choose  $N_{\mathrm{SHBF}} = 7$  and  $N_{\mathrm{SRBF}} = N_{\mathrm{RBF}} = 6$ . For the weight tensor in the interaction block we use  $N_{\mathrm{bilinear}} = 8$ . We did not find the model to be very sensitive to these values as long as they were large enough (i.e. at least 4).
+
+We found the cutoff  $c = 5\AA$  and the learning rate  $1\times 10^{-3}$  to be rather important hyperparameters. We optimized the model using AMSGrad (Reddi et al., 2018) with 32 molecules per mini-batch. We use a linear learning rate warm-up over 3000 steps and an exponential decay with ratio 0.1 every 2000000 steps. The model weights for validation and test were obtained using an exponential moving average (EMA) with decay rate 0.999. For MD17 we use the loss function from Eq. 2 with force weight  $\rho = 100$ , like previous models Schutt et al. (2017). Note that  $\rho$  presents a trade-off between energy and force accuracy. It should be chosen rather high since the forces determine the dynamics of the chemical system (Unke & Meuwly, 2019). We use early stopping on the validation loss. On QM9 we train for at most 3000000 and on MD17 for at most 100000 steps.
+
+# C SUMMARY STATISTICS
+
+We summarize the results across different targets using the mean standardized MAE
+
+$$
+\text {s t d .} \mathrm {M A E} = \frac {1}{M} \sum_ {m = 1} ^ {M} \left(\frac {1}{N} \sum_ {i = 1} ^ {N} \frac {\left| f _ {\theta} ^ {(m)} \left(\boldsymbol {X} _ {i} , \boldsymbol {z} _ {i}\right) - \hat {t} _ {i} ^ {(m)} \right|}{\sigma_ {m}}\right), \tag {10}
+$$
+
+and the mean standardized logMAE
+
+$$
+\log \mathrm {M A E} = \frac {1}{M} \sum_ {m = 1} ^ {M} \log \left(\frac {1}{N} \sum_ {i = 1} ^ {N} \frac {\left| f _ {\theta} ^ {(m)} \left(\boldsymbol {X} _ {i} , \boldsymbol {z} _ {i}\right) - \hat {t} _ {i} ^ {(m)} \right|}{\sigma_ {m}}\right), \tag {11}
+$$
+
+with target index  $m$ , number of targets  $M = 12$ , dataset size  $N$ , ground truth values  $\hat{\pmb{t}}^{(m)}$ , model  $f_{\theta}^{(m)}$ , inputs  $\mathbf{X}_i$  and  $\mathbf{z}_i$ , and standard deviation  $\sigma_m$  of  $\hat{\pmb{t}}^{(m)}$ . Std. MAE reflects the average error compared to the standard deviation of each target. Since this error is dominated by a few difficult targets (e.g.  $\epsilon_{\mathrm{HOMO}}$ ) we also report logMAE, which reflects every relative improvement equally but is sensitive to outliers, such as SchNet's result on  $\langle R^2\rangle$ .
+
+# D DIMENET FILTERS
+
+To illustrate the filters learned by DimeNet we separate the spatial dependency in the interaction function  $f_{\mathrm{int}}$  via
+
+$$
+f _ {\text {i n t}} (\boldsymbol {m}, d _ {j i}, d _ {k j}, \alpha_ {(k j, j i)}) = \sum_ {n} [ \sigma (\boldsymbol {W} \boldsymbol {m} + \boldsymbol {b}) ] _ {n} f _ {\text {f i l t e r 1 , n}} (d _ {j i}) f _ {\text {f i l t e r 2 , n}} (d _ {k j}, \alpha_ {(k j, j i)}). \tag {12}
+$$
+
+The filters  $f_{\mathrm{filter}1,n}:\mathbb{R}^+\to \mathbb{R}$  and  $f_{\mathrm{filter}2,n}:\mathbb{R}^{+}\times [0,2\pi ]\to \mathbb{R}^{F}$  are given by
+
+$$
+f _ {\text {f i l t e r} 1, n} (d) = \left(\boldsymbol {W} _ {\mathrm {R B F}} \boldsymbol {e} _ {\mathrm {R B F}} (d)\right) _ {n}, \tag {13}
+$$
+
+$$
+f _ {\text {f i l t e r} 2, n} (d, \alpha) = \left(\boldsymbol {W} _ {\mathrm {S B F}} \boldsymbol {a} _ {\mathrm {S B F}} (d, \alpha)\right) ^ {T} \boldsymbol {W} _ {n}, \tag {14}
+$$
+
+where  $W_{\mathrm{RBF}}$ ,  $W_{\mathrm{SBF}}$ , and  $W$  are learned weight matrices/tensors,  $e_{\mathrm{RBF}}(d)$  is the radial basis representation, and  $a_{\mathrm{SBF}}(d,\alpha)$  is the 2D spherical Fourier-Bessel representation. Fig. 5 shows how the first 15 elements of  $f_{\mathrm{filter2},n}(d,\alpha)$  vary with  $d$  and  $\alpha$  when choosing the tensor slice  $n = 1$  (with  $\alpha = 0$  at the top of the figure).
+
+# E MULTI-TARGET RESULTS
+
+Table 4: MAE on QM9 with multi-target learning. Single-target learning significantly improves performance on all targets. Using a separate output block per target slightly reduces this difference with little impact on training time.  
+
+<table><tr><td>Target</td><td>Unit</td><td>Multi-target</td><td>Sep. output blocks</td><td>Single-target</td></tr><tr><td>μ</td><td>D</td><td>0.0775</td><td>0.0815</td><td>0.0286</td></tr><tr><td>α</td><td>a03</td><td>0.0649</td><td>0.0616</td><td>0.0469</td></tr><tr><td>εHOMO</td><td>meV</td><td>45.1</td><td>45.5</td><td>27.8</td></tr><tr><td>εLUMO</td><td>meV</td><td>41.1</td><td>33.9</td><td>19.7</td></tr><tr><td>Δε</td><td>meV</td><td>59.2</td><td>63.6</td><td>34.8</td></tr><tr><td>〈R2〉</td><td>a02</td><td>0.345</td><td>0.348</td><td>0.331</td></tr><tr><td>ZPVE</td><td>meV</td><td>2.87</td><td>1.44</td><td>1.29</td></tr><tr><td>U0</td><td>meV</td><td>12.9</td><td>10.6</td><td>8.02</td></tr><tr><td>U</td><td>meV</td><td>13.0</td><td>10.5</td><td>7.89</td></tr><tr><td>H</td><td>meV</td><td>13.0</td><td>10.4</td><td>8.11</td></tr><tr><td>G</td><td>meV</td><td>13.8</td><td>10.8</td><td>8.98</td></tr><tr><td>cv</td><td>cal/molK</td><td>0.0309</td><td>0.0283</td><td>0.0249</td></tr><tr><td>std. MAE</td><td>%</td><td>1.92</td><td>1.90</td><td>1.05</td></tr><tr><td>logMAE</td><td>-</td><td>-5.07</td><td>-5.21</td><td>-5.57</td></tr></table>
\ No newline at end of file
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+# DISAGREEMENT-REGULARIZED IMITATION LEARNING
+
+Kianté Brantley *
+
+University of Maryland
+
+kdrbrant@cs.umd.edu
+
+Wen Sun
+
+Microsoft Research
+
+sun.wen@microsoft.com
+
+Mikael Henaff
+
+Microsoft Research
+
+mihenaff@microsoft.com
+
+# ABSTRACT
+
+We present a simple and effective algorithm designed to address the covariate shift problem in imitation learning. It operates by training an ensemble of policies on the expert demonstration data, and using the variance of their predictions as a cost which is minimized with RL together with a supervised behavioral cloning cost. Unlike adversarial imitation methods, it uses a fixed reward function which is easy to optimize. We prove a regret bound for the algorithm which is linear in the time horizon multiplied by a coefficient which we show to be low for certain problems on which behavioral cloning fails. We evaluate our algorithm empirically across multiple pixel-based Atari environments and continuous control tasks, and show that it matches or significantly outperforms behavioral cloning and generative adversarial imitation learning.
+
+# 1 INTRODUCTION
+
+Training artificial agents to perform complex tasks is essential for many applications in robotics, video games and dialogue. If success on the task can be accurately described using a reward or cost function, reinforcement learning (RL) methods offer an approach to learning policies which has proven to be successful in a wide variety of applications (Mnih et al., 2015; 2016; Lillicrap et al., 2016; Hessel et al., 2018). However, in other cases the desired behavior may only be roughly specified and it is unclear how to design a reward function to characterize it. For example, training a video game agent to adopt more human-like behavior using RL would require designing a reward function which characterizes behaviors as more or less human-like, which is difficult.
+
+Imitation learning (IL) offers an elegant approach whereby agents are trained to mimic the demonstrations of an expert rather than optimizing a reward function. Its simplest form consists of training a policy to predict the expert's actions from states in the demonstration data using supervised learning. While appealingly simple, this approach suffers from the fact that the distribution over states observed at execution time can differ from the distribution observed during training. Minor errors which initially produce small deviations become magnified as the policy encounters states further and further from its training distribution. This phenomenon, initially noted in the early work of (Pomerleau, 1989), was formalized in the work of (Ross & Bagnell, 2010) who proved a quadratic  $\mathcal{O}(\epsilon T^2)$  bound on the regret and showed that this bound is tight. The subsequent work of (Ross et al., 2011) showed that if the policy is allowed to further interact with the environment and make queries to the expert policy, it is possible to obtain a linear bound on the regret. However, the ability to query an expert can often be a strong assumption.
+
+In this work, we propose a new and simple algorithm called DRIL (Disagreement-Regularized Imitation Learning) to address the covariate shift problem in imitation learning, in the setting where the agent is allowed to interact with its environment. Importantly, the algorithm does not require any additional interaction with the expert. It operates by training an ensemble of policies on the demonstration data, and using the disagreement in their predictions as a cost which is optimized through RL together with a supervised behavioral cloning cost. The motivation is that the policies in the ensemble will tend to agree on the set of states covered by the expert, leading to low cost, but are more likely to disagree on states not covered by the expert, leading to high cost. The RL cost
+
+thus guides the agent back towards the distribution of the expert, while the supervised cost ensures that it mimics the expert within the expert's distribution.
+
+Our theoretical results show that, subject to realizability and optimization oracle assumptions<sup>1</sup>, our algorithm obtains a  $\mathcal{O}(\epsilon \kappa T)$  regret bound, where  $\kappa$  is a measure which quantifies a tradeoff between the concentration of the demonstration data and the diversity of the ensemble outside the demonstration data. We evaluate DRIL empirically across multiple pixel-based Atari environments and continuous control tasks, and show that it matches or significantly outperforms behavioral cloning and generative adversarial imitation learning, often recovering expert performance with only a few trajectories.
+
+# 2 PRELIMINARIES
+
+We consider episodic finite horizon MDP in this work. Denote by  $S$  the state space,  $A$  the action space, and  $\Pi$  the class of policies the learner is considering. Let  $T$  denote the task horizon and  $\pi^{\star}$  the expert policy whose behavior the learner is trying to mimic. For any policy  $\pi$ , let  $d_{\pi}$  denote the distribution over states induced by following  $\pi$ . Denote  $C(s,a)$  the expected immediate cost of performing action  $a$  in state  $s$ , which we assume is bounded in  $[0,1]$ . In the imitation learning setting, we do not necessarily know the true costs  $C(s,a)$ , and instead we observe expert demonstrations. Our goal is to find a policy  $\pi$  which minimizes an observed surrogate loss  $\ell$  between its actions and the actions of the expert under its induced distribution of states, i.e.
+
+$$
+\hat {\pi} = \arg \min  _ {s \sim d _ {\pi}} \mathbb {E} _ {s \sim d _ {\pi}} [ \ell (\pi (s), \pi^ {\star} (s)) ] \tag {1}
+$$
+
+For the following, we will assume  $\ell$  is the total variation distance (denoted by  $\| \cdot \|$ ), which is an upper bound on the  $0 - 1$  loss. Our goal is thus to minimize the following quantity, which represents the distance between the actions taken by our policy  $\pi$  and the expert policy  $\pi^{\star}$ :
+
+$$
+J _ {\exp} (\pi) = \mathbb {E} _ {s \sim d _ {\pi}} \left[ \| \pi (\cdot | s) - \pi^ {\star} (\cdot | s) \| \right] \tag {2}
+$$
+
+Denote  $Q_{t}^{\pi}(s,a)$  as the standard Q-function of the policy  $\pi$ , which is defined as  $Q_{t}^{\pi}(s,a) = \mathbb{E}\left[\sum_{\tau = t}^{T}C(s_{\tau},a_{\tau})|(s_{t},a_{t}) = (s,a),a_{\tau}\sim \pi \right]$ . The following result shows that if  $\ell$  is an upper bound on the  $0 - 1$  loss and  $C$  satisfies certain smoothness conditions, then minimizing this loss within  $\epsilon$  translates into an  $\mathcal{O}(\epsilon T)$  regret bound on the true task cost  $J_{\mathrm{C}}(\pi) = \mathbb{E}_{s,a\sim d_{\pi}}[C(s,a)]$ :
+
+Theorem 1. (Ross et al., 2011) If  $\pi$  satisfies  $J_{\mathrm{exp}}(\pi) = \epsilon$ , and  $Q_{T - t + 1}^{\pi^{\star}}(s,a) - Q_{T - t + 1}^{\pi^{\star}}(s,\pi^{\star}) \leq u$  for all time steps  $t$ , actions  $a$  and states  $s$  reachable by  $\pi$ , then  $J_{\mathrm{C}}(\pi) \leq J_{\mathrm{C}}(\pi^{\star}) + uT\epsilon$ .
+
+Unfortunately, it is often not possible to optimize  $J_{\mathrm{exp}}$  directly, since it requires evaluating the expert policy on the states induced by following the current policy. The supervised behavioral cloning cost  $J_{\mathrm{BC}}$ , which is computed on states induced by the expert, is often used instead:
+
+$$
+J _ {\mathrm {B C}} (\pi) = \mathbb {E} _ {s \sim d _ {\pi^ {*}}} \left[ \| \pi^ {*} (\cdot | s) - \pi (\cdot | s) \| \right] \tag {3}
+$$
+
+Minimizing this loss within  $\epsilon$  yields a quadratic regret bound on regret:
+
+Theorem 2. (Ross & Bagnell, 2010) Let  $J_{\mathrm{BC}}(\pi) = \epsilon$ , then  $J_{\mathrm{C}}(\pi) \leq J_{\mathrm{C}}(\pi^{\star}) + T^{2}\epsilon$ .
+
+Furthermore, this bound is tight: as we will discuss later, there exist simple problems which match the worst-case lower bound.
+
+# 3 ALGORITHM
+
+Our algorithm is motivated by two criteria: i) the policy should act similarly to the expert within the expert's data distribution, and ii) the policy should move towards the expert's data distribution
+
+Algorithm 1 Disagreement-Regularized Imitation Learning (DRIL)  
+1: Input: Expert demonstration data  $\mathcal{D} = \{(s_i,a_i)\}_{i = 1}^N$    
+2: Initialize policy  $\pi$  and policy ensemble  $\Pi_{\mathrm{E}} = \{\pi_1,\dots,\pi_E\}$    
+3: for  $e = 1,E$  do   
+4: Sample  $\mathcal{D}_e\sim \mathcal{D}$  with replacement, with  $|\mathcal{D}_e| = |\mathcal{D}|$    
+5: Train  $\pi_e$  to minimize  $J_{\mathrm{BC}}(\pi_e)$  on  $\mathcal{D}_e$  to convergence.   
+6: end for   
+7: for  $i = 1,\ldots$  do   
+8: Perform one gradient update to minimize  $J_{\mathrm{BC}}(\pi)$  using a minibatch from  $\mathcal{D}$    
+9: Perform one step of policy gradient to minimize  $\mathbb{E}_{s\sim d_{\pi},a\sim \pi (\cdot |s)}[C_{\mathrm{U}}^{\mathrm{clip}}(s,a)]$    
+10: end for
+
+if it is outside of it. These two criteria are addressed by combining two losses: a standard behavior cloning loss, and an additional loss which represents the variance over the outputs of an ensemble  $\Pi_{\mathrm{E}} = \{\pi_1,\dots,\pi_E\}$  of policies trained on the demonstration data  $\mathcal{D}$ . We call this the uncertainty cost, which is defined as:
+
+$$
+C _ {\mathrm {U}} (s, a) = \operatorname {V a r} _ {\pi \sim \Pi_ {\mathrm {E}}} (\pi (a | s)) = \frac {1}{E} \sum_ {i = 1} ^ {E} \left(\pi_ {i} (a | s) - \frac {1}{E} \sum_ {i = 1} ^ {E} \pi_ {i} (a | s)\right) ^ {2}
+$$
+
+The motivation is that the variance over plausible policies is high outside the expert's distribution, since the data is sparse, but it is low inside the expert's distribution, since the data there is dense. Minimizing this cost encourages the policy to return to regions of dense coverage by the expert. Intuitively, this is what we would expect the expert policy  $\pi^{\star}$  to do as well. The total cost which the algorithm optimizes is given by:
+
+$$
+J _ {\mathrm {a l g}} (\pi) = \underbrace {\mathbb {E} _ {s \sim d _ {\pi^ {\star}}} \left[ | | \pi^ {\star} (\cdot | s) - \pi (\cdot | s) | | \right]} _ {J _ {\mathrm {B C}} (\pi)} + \underbrace {\mathbb {E} _ {s \sim d _ {\pi} , a \sim \pi (\cdot | s)} [ C _ {\mathrm {U}} (s , a) ]} _ {J _ {\mathrm {U}} (\pi)}
+$$
+
+The first term is a behavior cloning loss and is computed over states generated by the expert policy, of which the demonstration data  $\mathcal{D}$  is a representative sample. The second term is computed over the distribution of states generated by the current policy and can be optimized using policy gradient.
+
+Note that the demonstration data is fixed, and this ensemble can be trained once offline. We then interleave the supervised behavioral cloning updates and the policy gradient updates which minimize the variance of the ensemble. The full algorithm is shown in Algorithm 1. We also found that dropout (Srivastava et al., 2014), which has been proposed as an approximate form of ensembling, worked well (see Appendix D).
+
+In practice, for the supervised loss we optimize the KL divergence between the actions predicted by the policy and the expert actions, which is an upper bound on the total variation distance due to Pinsker's inequality. We also found it helpful to use a clipped uncertainty cost:
+
+$$
+C _ {\mathrm {U}} ^ {\mathrm {c l i p}} (s, a) = \left\{ \begin{array}{l l} - 1 & \text {i f} C _ {\mathrm {U}} (s, a) \leq q \\ + 1 & \text {e l s e} \end{array} \right.
+$$
+
+where the threshold  $q$  is a top quantile of the raw uncertainty costs computed over the demonstration data. The threshold  $q$  defines a normal range of uncertainty based on the demonstration data, and values above this range incur a positive cost (or negative reward).
+
+The RL cost can be optimized using any policy gradient method. In our experiments we used advantage actor-critic (A2C) (Mnih et al., 2016) or PPO (Schulman et al., 2017), which estimate the expected cost using rollouts from multiple parallel actors all sharing the same policy (see Appendix C for details). We note that model-based RL methods could in principle be used as well if sample efficiency is a constraint.
+
+# 4 ANALYSIS
+
+# 4.1 COVERAGE COEFFICIENT
+
+We now analyze DRIL for MDPs with discrete action spaces and potentially large or infinite state spaces. We will show that, subject to assumptions that the policy class contains an optimal policy and that we are able to optimize costs within  $\epsilon$  of their global minimum, our algorithm obtains a regret bound which is linear in  $\kappa T$ , where  $\kappa$  is a quantity which depends on the environment dynamics, the expert distribution  $d_{\pi}^{\star}$ , and our learned ensemble. Intuitively,  $\kappa$  represents a tradeoff between how concentrated the demonstration data is and how high the variance of the ensemble is outside the expert distribution.
+
+Assumption 1. (Realizability)  $\pi^{\star} \in \Pi$ .
+
+Assumption 2. (Optimization Oracle) For any given cost function  $J$ , our minimization procedure returns a policy  $\hat{\pi} \in \Pi$  such that  $J(\hat{\pi}) \leq \arg \min_{\pi \in \Pi} J(\pi) + \epsilon$ .
+
+The motivation behind our algorithm is that the policies in the ensemble agree inside the expert's distribution and disagree outside of it. This defines a reward function which pushes the learner back towards the expert's distribution if it strays away. However, what constitutes inside and outside the distribution, or sufficient agreement or disagreement, is ambiguous. Below we introduce quantities which makes these ideas precise.
+
+Definition 1. For any set  $\mathcal{U} \subseteq \mathcal{S}$ , define the concentrability inside of  $\mathcal{U}$  as  $\alpha(\mathcal{U}) = \max_{\pi \in \Pi} \sup_{s \in \mathcal{U}} \frac{d_{\pi}(s)}{d_{\pi^*(s)}}$ .
+
+The notion of concentrability has been previously used to give bounds on the performance of value iteration (Munos & Szepesvári, 2008). For a set  $\mathcal{U}$ ,  $\alpha(\mathcal{U})$  will be low if the expert distribution has high mass at the states in  $\mathcal{U}$  that are reachable by policies in the policy class.
+
+Definition 2. Define the minimum variance of the ensemble outside of  $\mathcal{U}$  as  $\beta (\mathcal{U}) = \min_{s\notin \mathcal{U},a\in \mathcal{A}}\mathrm{Var}_{\pi \sim \Pi_{\mathrm{E}}}[\pi (a|s)]$
+
+We now define the  $\kappa$  coefficient as the minimum ratio of these two quantities over all possible subsets of  $S$ .
+
+Definition 3. We define  $\kappa = \min_{\mathcal{U}\subseteq \mathcal{S}}\frac{\alpha(\mathcal{U})}{\beta(\mathcal{U})}.$
+
+We can view  $\kappa$  as the quantity which minimizes the tradeoff over different subsets  $\mathcal{U}$  between coverage by the expert policy inside of  $\mathcal{U}$ , and variance of the ensemble outside of  $\mathcal{U}$ .
+
+# 4.2 REGRETBOUND
+
+We now establish a relationship between the  $\kappa$  coefficient just defined, the cost our algorithm optimizes, and  $J_{\mathrm{exp}}$  defined in Equation (2) which we would ideally like to minimize and which translates into a regret bound. All proofs can be found in Appendix A.
+
+Lemma 1. For any  $\pi \in \Pi$ , we have  $J_{\mathrm{exp}}(\pi) \leq \kappa J_{\mathrm{alg}}(\pi)$ .
+
+This result shows that if  $\kappa$  is not too large, and we are able to make our cost function  $J_{\mathrm{alg}}(\pi)$  small, then we can ensure  $J_{\mathrm{exp}}(\pi)$  is also small. This result is only useful if our cost function can indeed achieve a small minimum. The next lemma shows that this is the case.
+
+Lemma 2.  $\min_{\pi \in \Pi}J_{\mathrm{alg}}(\pi)\leq 2\epsilon$
+
+Here  $\epsilon$  is the threshold specified in Assumption 2. Combining these two lemmas with the previous result of Ross et al. (2011), we get a regret bound which is linear in  $\kappa T$ .
+
+Theorem 3. Let  $\hat{\pi}$  be the result of minimizing  $J_{\mathrm{alg}}$  using our optimization oracle, and assume that  $Q_{T - t + 1}^{\pi^{\star}}(s,a) - Q_{T - t + 1}^{\pi^{\star}}(s,\pi^{\star})\leq u$  for all actions  $a$ , time steps  $t$  and states  $s$  reachable by  $\pi$ . Then  $\hat{\pi}$  satisfies  $J_{\mathrm{C}}(\hat{\pi})\leq J_{\mathrm{C}}(\pi^{\star}) + 3u\kappa \epsilon T$ .
+
+Our bound is an improvement over that of behavior cloning if  $\kappa$  is less than  $\mathcal{O}(T)$ . Note that DRIL does not require knowledge of  $\kappa$ . The quantity  $\kappa$  is problem-dependent and depends on the
+
+![](images/24e46fca034f86513d6c63fe18f13e8dc7ba044771e8ed0b8ea361efb9602b4d.jpg)  
+Figure 1: Example of a problem where behavioral cloning incurs quadratic regret.
+
+environment dynamics, the expert policy and the policies in the learned ensemble. We next compute  $\kappa$  exactly for a problem for which behavior cloning is known to perform poorly, and show that it is independent of  $T$ .
+
+Example 1. Consider the tabular MDP given in (Ross & Bagnell, 2010) as an example of a problem where behavioral cloning incurs quadratic regret, shown in Figure 1. There are 3 states  $S = (s_0, s_1, s_2)$  and two actions  $(a_1, a_2)$ . Each policy  $\pi$  can be represented as a set of probabilities  $\pi(a_1|s)$  for each state  $s \in S^2$ . Assume the models in our ensemble are drawn from a posterior  $p(\pi(a_1|s)|\mathcal{D})$  given by a Beta distribution with parameters  $\text{Beta}(n_1 + 1, n_2 + 1)$  where  $n_1, n_2$  are the number of times the pairs  $(s, a_1)$  and  $(s, a_2)$  occur, respectively, in the demonstration data  $\mathcal{D}$ . The agent always starts in  $s_0$  and the expert's policy is given by  $\pi^\star(a_1|s_0) = 1$ ,  $\pi^\star(a_1|s_1) = 0$ ,  $\pi^\star(a_1|s_2) = 1$ . For any  $(s, a)$  pair, the task cost is  $C(s, a) = 0$  if  $a = \pi^\star(s)$  and 1 otherwise. Here  $d_\pi^\star = \left( \frac{1}{T}, \frac{T-1}{T}, 0 \right)$ . For any  $\pi$ ,  $d_\pi(s_0) = \frac{1}{T}$  and  $d_\pi(s_1) \leq \frac{T-1}{T}$  due to the dynamics of the MDP, so  $\frac{d_\pi(s)}{d_\pi^\star(s)} \leq 1$  for  $s \in \{s_0, s_1\}$ . Writing out  $\alpha(\{s_0, s_1\})$ , we get:  $\alpha(\{s_0, s_1\}) = \max_{\pi \in \Pi} \sup_{s \in \{s_0, s_1\}} \frac{d_\pi(s)}{d_\pi^\star(s)} \leq 1$ .
+
+Furthermore, since  $s_2$  is never visited in the demonstration data, for each policy  $\pi_i$  in the ensemble we have  $\pi_i(a_1|s_2), \pi_i(a_2|s_2) \sim Beta(1, 1) = \text{Uniform}(0, 1)$ . It follows that  $\text{Var}_{\pi \sim \Pi_E}(\pi(a|s_2))$  is approximately equal to the variance of a uniform distribution over  $[0, 1]$ , i.e.  $\frac{1}{12}$ . Therefore:
+
+$$
+\kappa = \min _ {\mathcal {U} \subseteq \mathcal {S}} \frac {\alpha (\mathcal {U})}{\beta (\mathcal {U})} \leq \frac {\alpha (\{s _ {0} , s _ {1} \})}{\beta (\{s _ {0} , s _ {1} \})} \lesssim \frac {1}{\frac {1}{1 2}} = 1 2
+$$
+
+Applying our result from Theorem 3, we see that our algorithm obtains an  $\mathcal{O}(\epsilon T)$  regret bound on this problem, in contrast to the  $\mathcal{O}(\epsilon T^2)$  regret of behavioral cloning<sup>4</sup>.
+
+# 5 RELATED WORK
+
+The idea of learning through imitation dates back at least to the work of (Pomerleau, 1989), who trained a neural network to imitate the steering actions of a human driver using images as input. The problem of covariate shift was already observed, as the author notes: "the network must not solely be shown examples of accurate driving, but also how to recover once a mistake has been made".
+
+This issue was formalized in the work of (Ross & Bagnell, 2010), who on one hand proved an  $\mathcal{O}(\epsilon T^2)$  regret bound, and on the other hand provided an example showing this bound is tight. The subsequent work (Ross et al., 2011) proposed the DAGGER algorithm which obtains linear regret, provided the agent can both interact with the environment, and query the expert policy. Our approach also requires environment interaction, but importantly does not need to query the expert. Also of
+
+Note is the work of (Venkatraman et al., 2015), which extended DAGGER to time series prediction problems by using the true targets as expert corrections.
+
+Imitation learning has been used within the context of modern RL to help improve sample efficiency (Chang et al., 2015; Ross & Bagnell, 2014; Sun et al., 2017; Hester et al., 2018; Le et al., 2018; Cheng & Boots, 2018) or overcome exploration (Nair et al., 2017). These settings assume the reward is known and that the policies can then be fine-tuned with reinforcement learning. In this case, covariate shift is less of an issue since it can be corrected using the reinforcement signal.
+
+The work of (Luo et al., 2019) also proposed a method to address the covariate shift problem when learning from demonstrations when the reward is known, by conservatively extrapolating the value function outside the training distribution using negative sampling. This addresses a different setting from ours, and requires generating plausible states which are off the manifold of training data, which may be challenging when the states are high dimensional such as images. The work of (Reddy et al., 2019) proposed to treat imitation learning within the Q-learning framework, setting a positive reward for all transitions inside the demonstration data and zero reward for all other transitions in the replay buffer. This rewards the agent for repeating (or returning to) the expert's transitions. The work of (Sasaki et al., 2019) also incorporates a mechanism for reducing covariate shift by fitting a Q-function that classifies whether the demonstration states are reachable from the current state. Random Expert Distillation (Wang et al., 2019) uses Random Network Distillation (RND) (Burda et al., 2019) to estimate the support of the expert's distribution in state-action space, and minimizes an RL cost designed to guide the agent towards the expert's support. This is related to our method, but differs in that it minimizes the RND prediction error rather than the ensemble variance and does not include a behavior cloning cost. The behavior cloning cost is essential to our theoretical results and avoids certain failure modes, see Appendix B for more discussion.
+
+Generative Adversarial Imitation Learning (GAIL) (Ho & Ermon, 2016) is a state-of-the-art algorithm which addresses the same setting as ours. It operates by training a discriminator network to distinguish expert states from states generated by the current policy, and the negative output of the discriminator is used as a reward signal to train the policy. The motivation is that states which are outside the training distribution will be assigned a low reward while states which are close to it will be assigned a high reward. This encourages the policy to return to the expert distribution if it strays away from it. However, the adversarial training procedure means that the reward function is changing over time, which can make the algorithm unstable or difficult to tune. In contrast, our approach uses a simple fixed reward function. We include comparisons to GAIL in our experiments.
+
+Using disagreement between models in an ensemble to represent uncertainty has recently been explored in several contexts. The works of (Shyam et al., 2018; Pathak et al., 2019; Henaff, 2019) used disagreement between different dynamics models to drive exploration in the context of model-based RL. Conversely, (Henaff et al., 2019) used variance across different dropout masks to prevent policies from exploiting error in dynamics models. Ensembles have also been used to represent uncertainty over Q-values in model-free RL in order to encourage exploration (Osband et al., 2016). Within the context of imitation learning, the work of (Menda et al., 2018) used the variance of the ensemble together with the DAGGER algorithm to decide when to query the expert demonstrator to minimize unsafe situations. Here, we use disagreement between different policies trained on demonstration data to address covariate shift in the context of imitation learning.
+
+# 6 EXPERIMENTS
+
+# 6.1 TABULAR MDPS
+
+As a first experiment, we applied DRIL to the tabular MDP of (Ross & Bagnell, 2010) shown in Figure 1. We computed the posterior over the policy parameters given the demonstration data using a separate Beta distribution for each state  $s$  with parameters determined by the number of times each action was performed in  $s$ . For behavior cloning, we sampled a single policy from this posterior. For DRIL, we sampled an ensemble of 5 policies and used their negative variance to define an additional reward function. We combined this with a reward which was the probability density function of a given state-action pair under the posterior distribution, which corresponds to the supervised learning loss, and used tabular Q-learning to optimize the sum of these two reward functions. This experiment
+
+![](images/c48780c2ef9c4c2429117ca5ed609b90fa300b929aeec36134ca510072708130.jpg)  
+Figure 2: Results on tabular MDP from (Ross & Bagnell, 2010). Shaded region represents range between  $5^{\text{th}}$  and  $95^{\text{th}}$  quantiles, computed across 500 trials. Behavior cloning exhibits poor worst-case regret, whereas DRIL has low regret across all trials.
+
+![](images/cbd56ed4c23f021a9e0e5d26666e6d3c0636577339524d6cced14806251adf64.jpg)
+
+![](images/436aa71c4fa9adf2763892c6de3b1d1e3519560101cf36f1c67a64a73121d8c0.jpg)
+
+was repeated 500 times for time horizon lengths up to 500 and  $N = 1, 5, 10$  expert demonstration trajectories.
+
+Figure 2 shows plots of the regret over the 500 different trials across different time horizons. Although BC achieves good average performance, it exhibits poor worst-case performance with some trials incurring very high regret, especially when using fewer demonstrations. Our method has low regret across all trials, which stays close to constant independently of the time horizon, even with a single demonstration. This performance is better than that suggested by our analysis, which showed a worst-case linear bound with respect to time horizon.
+
+# 6.2 ATARI ENVIRONMENTS
+
+We next evaluated our approach on six different Atari environments. We used pretrained PPO (Schulman et al., 2017) agents from the stable baselines repository (Hill et al., 2018) to generate  $N = \{1,3,5,10,15,20\}$  expert trajectories. We compared against two other methods: standard behavioral cloning (BC) and Generative Adversarial Imitation Learning (GAIL). Results are shown in Figure 3a. DRIL outperforms behavioral cloning across most environments and numbers of demonstrations, often by a substantial margin. In many cases, our method is able to match the expert's performance using a small number of trajectories. Figure 3b shows the evolution of the uncertainty cost and the policy reward throughout training. In all cases, the reward improves while the uncertainty cost decreases.
+
+We were not able to obtain meaningful performance for GAIL on these domains, despite performing a hyperparameter search across learning rates for the policy and discriminator, and across different numbers of discriminator updates. We additionally experimented with clipping rewards in an effort to stabilize performance. These results are consistent with those of (Reddy et al., 2019), who also reported negative results when running GAIL on images. While improved performance might be possible with more sophisticated adversarial training techniques, we note that this contrasts with our method which uses a fixed reward function obtained through simple supervised learning.
+
+In Appendix D we provide ablation experiments examining the effects of the cost function clipping and the role of the BC loss. We also compare the ensemble approach to a dropout-based approximation and show that DRIL works well in both cases.
+
+# 6.3 CONTINUOUS CONTROL
+
+We next report results of running our method on 6 different continuous control tasks from the PyBullet $^5$  and OpenAI Gym (Brockman et al., 2016) environments. We again used pretrained agents to generate expert demonstrations, and compared to Behavior Cloning and GAIL.
+
+Results for all methods are shown in Figure 4. In these environments we found Behavior Cloning to be a much stronger baseline than for the Atari environments: in several tasks it was able to match expert performance using as little as 3 trajectories, suggesting that covariate shift may be less of an issue. Our method performs similarly to Behavior Cloning on most tasks, except on Walker2D, where it yields improved performance for  $N = 1, 3, 5$  trajectories. GAIL performs
+
+![](images/5d6da37f13108fb7f8fb7ce71fcb7954cad1f82213116b4b932d29e1440774dd.jpg)
+
+![](images/44b9efaffc44be3a553936a38397e7a76c2e2277bdd43d56e279a353e58c87a0.jpg)
+
+![](images/8be4a424e69f08550cbdb3f908466ba4bdb95cb27e326c94ec96309571554506.jpg)
+
+![](images/1269200864f17600c63f0f588c19a54418696b4c3efe23cd5f9b5d2f7a378e73.jpg)
+
+![](images/8a18efba73747b8a269815706d7e2d8b0ee8628f571a21c090efbe6ef6aaaffa.jpg)
+
+![](images/e0ed93840e1c3494bf472482dd79b5d9569162dd8a370dd991b7d9fc5394dfb2.jpg)
+
+![](images/68b585dac789e79ddc6c319f5b3aaa0566853f3c59cec6fd4827f5ba4eb6fd0a.jpg)
+
+![](images/ce3dd3da58765455806093966e397afe466f2a46c9588ff36b1dde81f67e6c31.jpg)  
+a)
+
+![](images/9a534ac82239be10342183946a3cbe8f36d6f4bc7f3cf8bc24d97d8893da5880.jpg)  
+b)
+
+![](images/c462653071522689a3c97c472b3bcdf2f8a33bdd4c33d523a16ed123fbe16b7f.jpg)  
+Figure 3: Results on Atari environments. a) Median final policy performance for different numbers of expert trajectories, taken over 4 seeds (shaded regions are min/max performance) b) Evolution of policy reward and uncertainty cost during training with  $N = 3$  trajectories.
+
+somewhat better than DRIL on HalfCheetah and Walker2D, but performs worse than both DRIL and BC on LunarLander and BipedalWalkerHardcore. The fact that DRIL is competitive across all tasks provides evidence of its robustness.
+
+# 7 CONCLUSION
+
+Addressing covariate shift has been a long-standing challenge in imitation learning. In this work, we have proposed a new method to address this problem by penalizing the disagreement between an ensemble of different policies trained on the demonstration data. Importantly, our method requires no additional labeling by an expert. Our experimental results demonstrate that DRIL can often match expert performance while using only a small number of trajectories across a wide array of tasks, ranging from tabular MDPs to pixel-based Atari games and continuous control tasks. On the theoretical side, we have shown that our algorithm can provably obtain a low regret bound for problems in which the  $\kappa$  parameter is low.
+
+![](images/e62876bf3b42038060c17f0866d09b1968ef3cb23dee9df60597cc4191b6e282.jpg)
+
+![](images/f699bc75432a28d26973ceef77640a7379df38dc36af452d3b24ddcaf0fab52f.jpg)
+
+![](images/a8b62d71d3abb6347702c7e90af08fb663a99b34c01409147ccf02b092fb5291.jpg)
+
+![](images/7b526a4f7f282d06149442a31488bca6f686ae0e340eb68c2d37f4f461d4f61f.jpg)  
+Figure 4: Results on continuous control tasks.
+
+![](images/45b28bfc3a463aaf29aac2873a4931830213b0c1f9d2fbab107279a036475456.jpg)
+
+![](images/c4240eb985d3434c39ad054cc35c741ac512913b6a024b1f42130f3b6ed11af0.jpg)
+
+There are multiple directions for future work. On the theoretical side, characterizing the  $\kappa$  parameter on a larger array of problems would help to better understand the settings where our method can expect to do well. Empirically, there are many other settings in structured prediction (Daumé et al., 2009) where covariate shift is an issue and where our method could be applied. For example, in dialogue and language modeling it is common for generated text to become progressively less coherent as errors push the model off the manifold it was trained on. Our method could potentially be used to fine-tune language or translation models (Cho et al., 2014; Welleck et al., 2019) after training by applying our uncertainty-based cost function to the generated text.
+
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+Ashvin Nair, Bob McGrew, Marcin Andrychowicz, Wojciech Zaremba, and Pieter Abbeel. Overcoming exploration in reinforcement learning with demonstrations. 2018 IEEE International Conference on Robotics and Automation (ICRA), pp. 6292-6299, 2017.  
+Ian Osband, Charles Blundell, Alexander Pritzel, and Benjamin Van Roy. Deep exploration via bootstrapped DQN. CoRR, abs/1602.04621, 2016. URL http://arxiv.org/abs/1602.04621.  
+Deepak Pathak, Dhiraj Gandhi, and Abhinav Gupta. Self-supervised exploration via disagreement. In ICML, 2019.  
+Dean A. Pomerleau. Alvinn: An autonomous land vehicle in a neural network. In D. S. Touretzky (ed.), Advances in Neural Information Processing Systems 1, pp. 305-313. Morgan-Kaufmann, 1989. URL http://papers.nips.cc/paper/95-alvinn-an-autonomous-land-vehicle-in-a-neural-network.pdf.  
+Siddharth Reddy, Anca D. Dragan, and Sergey Levine. SQL: imitation learning via regularized behavioral cloning. CoRR, abs/1905.11108, 2019. URL http://arxiv.org/abs/1905.11108.  
+Stephane Ross and Drew Bagnell. Efficient reductions for imitation learning. In Yee Whye Teh and Mike Titterington (eds.), Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, volume 9 of Proceedings of Machine Learning Research, pp. 661-668, Chia Laguna Resort, Sardinia, Italy, 13-15 May 2010. PMLR. URL http://proceedings.mlr.press/v9/ross10a.html.  
+Stephane Ross and J Andrew Bagnell. Reinforcement and imitation learning via interactive no-regret learning. arXiv preprint arXiv:1406.5979, 2014.
+
+Stephane Ross, Geoffrey Gordon, and Drew Bagnell. A reduction of imitation learning and structured prediction to no-regret online learning. In Geoffrey Gordon, David Dunson, and Miroslav Dudk (eds.), Proceedings of the Fourteenth International Conference on Artificial Intelligence and Statistics, volume 15 of Proceedings of Machine Learning Research, pp. 627-635, Fort Lauderdale, FL, USA, 11-13 Apr 2011. PMLR. URL http://proceedings.mlr.press/v15/ross11a.html.  
+Fumihiro Sasaki, Tetsuya Yohira, and Atsuo Kawaguchi. Sample efficient imitation learning for continuous control. In International Conference on Learning Representations, 2019. URL https://openreview.net/forum?id=BkN5UoAqF7.  
+John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal policy optimization algorithms. CoRR, abs/1707.06347, 2017. URL http://arxiv.org/abs/1707.06347.  
+Pranav Shyam, Wojciech Jaskowski, and Faustino Gomez. Model-based active exploration. CoRR, abs/1810.12162, 2018.  
+Nitish Srivastava, Geoffrey E. Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. Dropout: a simple way to prevent neural networks from overfitting. Journal of Machine Learning Research, 15(1):1929-1958, 2014. URL http://www.cs.toronto.edu/~rsalakhu/papers/srivastava14a.pdf.  
+Wen Sun, Arun Venkatraman, Geoffrey J Gordon, Byron Boots, and J Andrew Bagnell. Deeply aggravated: Differentiable imitation learning for sequential prediction. In Proceedings of the 34th International Conference on Machine Learning-Volume 70, pp. 3309-3318. JMLR.org, 2017.  
+Arun Venkatraman, Martial Hebert, and J. Andrew Bagnell. Improving multi-step prediction of learned time series models. In Proceedings of the Twenty-Ninth AAAI Conference on Artificial Intelligence, AAAI'15, pp. 3024-3030. AAAI Press, 2015. ISBN 0-262-51129-0. URL http://dl.acm.org/citation.cfm?id=2888116.2888137.  
+Ruohan Wang, Carlo Ciliberto, Pierlugi Amadori, and Yiannis Demiris. Random expert distillation: Imitation learning via expert policy support estimation. In Proceedings of International Conference on Machine Learning. ACM, 2019.  
+Sean Welleck, Kianté Brantley, Hal Daumé III, and Kyunghyun Cho. Non-monotonic sequential text generation. CoRR, abs/1902.02192, 2019. URL http://arxiv.org/abs/1902.02192.
+
+# A PROOFS
+
+Lemma 1. For any  $\pi \in \Pi$  we have  $J_{\mathrm{exp}}(\pi) \leq \kappa J_{\mathrm{alg}}(\pi)$
+
+Proof. We will first show that for any  $\pi \in \Pi$  and  $\mathcal{U} \subseteq S$ , we have  $J_{\exp}(\pi) \leq \frac{\alpha(\mathcal{U})}{\beta(\mathcal{U})} J_{\mathrm{alg}}(\pi)$ . We can rewrite this as:
+
+$$
+\begin{array}{l} J _ {\exp} (\pi) = \mathbb {E} _ {s \sim d _ {\pi}} \left[ \| \pi (\cdot | s) - \pi^ {\star} (\cdot | s) \| \right] \\ = \mathbb {E} _ {s \sim d _ {\pi}} \left[ \mathbb {I} (s \in \mathcal {U}) \| \pi (\cdot | s) - \pi^ {\star} (\cdot | s) \| \right] + \mathbb {E} _ {s \sim d _ {\pi}} \left[ \mathbb {I} (s \notin \mathcal {U}) \| \pi (\cdot | s) - \pi^ {\star} (\cdot | s) \| \right] \\ \end{array}
+$$
+
+We begin by bounding the first term:
+
+$$
+\begin{array}{l} \mathbb {E} _ {s \sim d _ {\pi}} \left[ \mathbb {I} (s \in \mathcal {U}) \| \pi (\cdot | s) - \pi^ {\star} (\cdot | s) \| \right] = \sum_ {s \in \mathcal {U}} d _ {\pi} (s) \| \pi (\cdot | s) - \pi^ {\star} (\cdot | s) \| \\ = \sum_ {s \in \mathcal {U}} \frac {d _ {\pi} (s)}{d _ {\pi^ {\star}} (s)} d _ {\pi^ {\star}} (s) \| \pi (\cdot | s) - \pi^ {\star} (\cdot | s) \| \\ \leq \sum_{s\in \mathcal{U}}\underbrace{\left(\max\sup_{\pi^{\prime}\in\Pi}\sup_{s\in\mathcal{U}}\frac{d_{\pi^{\prime}(s)}}{d_{\pi^{\star}}(s)}\right)}_{\alpha (\mathcal{U})}d_{\pi^{\star}}(s)\| \pi (\cdot |s) - \pi^{\star}(\cdot |s)\| \\ = \alpha (\mathcal {U}) \sum_ {s \in \mathcal {U}} d _ {\pi^ {\star}} (s) \| \pi (\cdot | s) - \pi^ {\star} (\cdot | s) \| \\ \leq \alpha (\mathcal {U}) \sum_ {s \in \mathcal {S}} d _ {\pi^ {\star}} (s) \| \pi (\cdot | s) - \pi^ {\star} (\cdot | s) \| \\ = \alpha (\mathcal {U}) \mathbb {E} _ {s \sim d _ {\pi^ {\star}}} \left[ \| \pi (\cdot | s) - \pi^ {\star} (\cdot | s) \| \right] \\ = \alpha (\mathcal {U}) J _ {\mathrm {B C}} (\pi) \\ \end{array}
+$$
+
+We next bound the second term:
+
+$$
+\begin{array}{l} \mathbb {E} _ {s \sim d _ {\pi}} \left[ \mathbb {I} (s \notin \mathcal {U}) \| \pi (\cdot | s) - \pi^ {\star} (\cdot | s) \| \right] \leq \mathbb {E} _ {s \sim d _ {\pi}} \left[ \mathbb {I} (s \notin \mathcal {U}) \right] \\ \leq \mathbb {E} _ {s \sim d _ {\pi}} \left[ \mathbb {I} (s \notin \mathcal {U}) \frac {\operatorname* {m i n} _ {a \in \mathcal {A}} \operatorname {V a r} _ {\pi_ {i} \sim \Pi_ {\mathrm {E}}} [ \pi_ {i} (a | s) ]}{\beta (\mathcal {U})} \right] \\ = \frac {1}{\beta (\mathcal {U})} \mathbb {E} _ {s \sim d _ {\pi}} \left[ \mathbb {I} (s \notin \mathcal {U}) \sum_ {a \in \mathcal {A}} \pi (a | s) \operatorname {V a r} _ {\pi_ {i} \sim \Pi_ {\mathrm {E}}} [ \pi_ {i} (a | s) ] \right] \\ = \frac {1}{\beta (\mathcal {U})} \underbrace {\sum_ {s \notin \mathcal {U}} d _ {\pi} (s) \sum_ {a \in \mathcal {A}} \pi (a | s) \operatorname {V a r} _ {\pi_ {i} \sim \Pi_ {\mathrm {E}}} [ \pi_ {i} (a | s) ]} _ {A (\pi)} \\ \end{array}
+$$
+
+Now observe we can decompose the RL cost as follows:
+
+$$
+\begin{array}{l} J _ {\mathrm {U}} (\pi) = \mathbb {E} _ {s \sim d _ {\pi}, a \sim \pi (\cdot | s)} \left[ \operatorname {V a r} _ {\pi_ {i} \sim \Pi_ {\mathrm {E}}} \pi_ {i} (a | s) \right] \\ = \sum_ {s} d _ {\pi} (s) \sum_ {a} \pi (a | s) \left[ \mathrm {V a r} _ {\pi_ {i} \sim \Pi_ {\mathrm {E}}} \pi_ {i} (a | s) \right] \\ = \underbrace {\sum_ {s \in U} d _ {\pi} (s) \sum_ {a} \pi (a | s) \left[ \operatorname {V a r} _ {\pi_ {i} \sim \Pi_ {\mathrm {E}}} \pi_ {i} (a | s) \right]} _ {B (\pi)} + \underbrace {\sum_ {s \notin \mathcal {U}} d _ {\pi} (s) \sum_ {a} \pi (a | s) \left[ \operatorname {V a r} _ {\pi_ {i} \sim \Pi_ {\mathrm {E}}} \pi_ {i} (a | s) \right]} _ {A (\pi)} \\ \end{array}
+$$
+
+Putting these together, we get the following:
+
+$$
+\begin{array}{l} J _ {\exp} (\pi) \leq \alpha (\mathcal {U}) J _ {\mathrm {B C}} (\pi) + \frac {1}{\beta (\mathcal {U})} A (\pi) \\ = \frac {\alpha (\mathcal {U}) \beta (\mathcal {U})}{\beta (\mathcal {U})} J _ {\mathrm {B C}} (\pi) + \frac {\alpha (\mathcal {U})}{\alpha (\mathcal {U}) \beta (\mathcal {U})} A (\pi) \\ \leq \frac {\alpha (\mathcal {U})}{\beta (\mathcal {U})} J _ {\mathrm {B C}} (\pi) + \frac {\alpha (\mathcal {U})}{\beta (\mathcal {U})} A (\pi) \\ \leq \frac {\alpha (\mathcal {U})}{\beta (\mathcal {U})} \left(J _ {\mathrm {B C}} (\pi) + A (\pi)\right) \\ \leq \frac {\alpha (\mathcal {U})}{\beta (\mathcal {U})} \left(J _ {\mathrm {B C}} (\pi) + J _ {\mathrm {U}} (\pi)\right) \\ = \frac {\alpha (\mathcal {U})}{\beta (\mathcal {U})} J _ {\mathrm {a l g}} (\pi) \\ \end{array}
+$$
+
+Here we have used the fact that  $\beta (\mathcal{U})\leq 1$  since  $0\leq \pi (a|s)\leq 1$  and  $\alpha (\mathcal{U})\geq \sup_{s\in \mathcal{U}}\frac{d^*_{\pi}(s)}{d^*_{\pi}(s)} = 1$  hence  $\frac{1}{\alpha(\mathcal{U})}\leq 1$ . Taking the minimum over subsets  $\mathcal{U}\subseteq S$ , we get  $J_{\mathrm{exp}}(\pi)\leq \kappa J_{\mathrm{alg}}(\pi)$ .
+
+![](images/5c1a4478502c9ca3435bd4cc93e4274954e4499f18fff9da23d7abf57f6fccd0.jpg)
+
+Lemma 2.  $\min_{\pi \in \Pi}J_{\mathrm{alg}}(\pi)\leq 2\epsilon$
+
+Proof. Plugging the optimal policy into  $J_{\mathrm{alg}}$ , we get:
+
+$$
+\begin{array}{l} J _ {\mathrm {a l g}} (\pi^ {\star}) = J _ {\mathrm {B C}} (\pi^ {\star}) + J _ {\mathrm {U}} (\pi^ {\star}) \\ = 0 + \mathbb {E} _ {s \sim d _ {\pi^ {\star}}, a \sim \pi^ {\star} (\cdot | s)} \left[ \operatorname {V a r} _ {\pi_ {i} \sim \Pi_ {\mathrm {E}}} \left[ \pi_ {i} (a | s) \right] \right] \\ = \mathbb {E} _ {s \sim d _ {\pi^ {\star}}, a \sim \pi^ {\star} (\cdot | s)} \left[ \frac {1}{E} \sum_ {i = 1} ^ {E} \left(\pi_ {i} (a | s) - \bar {\pi} (a | s)\right) ^ {2} \right] \\ \leq \mathbb {E} _ {s \sim d _ {\pi^ {\star}}, a \sim \pi^ {\star} (\cdot | s)} \left[ \frac {1}{E} \sum_ {i = 1} ^ {E} \left(\pi_ {i} (a | s) - \pi^ {\star} (a | s)\right) ^ {2} + \left(\bar {\pi} (a | s) - \pi^ {\star} (a | s)\right) ^ {2} \right] \\ = \underbrace{\mathbb{E}_{s\sim d_{\pi^{\star}},a\sim\pi^{\star}(\cdot|s)}\left[\frac{1}{E}\sum_{i = 1}^{E}\left(\pi_{i}(a|s) - \pi^{\star}(a|s)\right)^{2}\right]}_{\text{Term1}} + \underbrace{\mathbb{E}_{s\sim d_{\pi^{\star}},a\sim\pi^{\star}(\cdot|s)}\left[\left(\bar{\pi} (a|s) - \pi^{\star}(a|s)\right)^{2}\right]}_{\text{Term2}} \\ \end{array}
+$$
+
+We will first bound Term 1:
+
+$$
+\begin{array}{l} \mathbb {E} _ {s \sim d _ {\pi^ {\star}}, a \sim \pi^ {\star} (\cdot | s)} \left[ \frac {1}{E} \sum_ {i = 1} ^ {E} \left(\pi_ {i} (a | s) - \pi^ {\star} (a | s)\right) ^ {2} \right] = \frac {1}{E} \mathbb {E} _ {s \sim d _ {\pi^ {\star}}} \left[ \sum_ {a \in \mathcal {A}} \pi^ {\star} (a | s) \sum_ {i = 1} ^ {E} \left(\pi_ {i} (a | s) - \pi^ {\star} (a | s)\right) ^ {2} \right] \\ \leq \frac {1}{E} \mathbb {E} _ {s \sim d _ {\pi^ {\star}}} \left[ \sum_ {a \in \mathcal {A}} \pi^ {\star} (a | s) \sum_ {i = 1} ^ {E} \left| \pi_ {i} (a | s) - \pi^ {\star} (a | s) \right| \right] \\ \leq \frac {1}{E} \mathbb {E} _ {s \sim d _ {\pi^ {\star}}} \left[ \sum_ {i = 1} ^ {E} \sum_ {a \in \mathcal {A}} \left| \pi_ {i} (a | s) - \pi^ {\star} (a | s) \right| \right] \\ \leq \frac {1}{E} \sum_ {i = 1} ^ {E} \mathbb {E} _ {s \sim d _ {\pi^ {\star}}} \left[ \left\| \pi_ {i} (\cdot | s) - \pi^ {\star} (\cdot | s) \right\| \right] \\ \leq \frac {1}{E} \sum_ {i = 1} ^ {E} \epsilon \\ = \epsilon \\ \end{array}
+$$
+
+We will next bound Term 2:
+
+$$
+\begin{array}{l} \mathbb {E} _ {s \sim d _ {\pi^ {\star}}, a \sim \pi^ {\star} (\cdot | s)} \left[ \left(\bar {\pi} (a | s) - \pi^ {\star} (a | s)\right) ^ {2} \right] = \mathbb {E} _ {s \sim d _ {\pi^ {\star}}, a \sim \pi^ {\star} (\cdot | s)} \left[ \left(\pi^ {\star} (a | s) - \frac {1}{E} \sum_ {i = 1} ^ {E} \pi_ {i} (a | s)\right) ^ {2} \right] \\ = \mathbb {E} _ {s \sim d _ {\pi^ {\star}}, a \sim \pi^ {\star} (\cdot | s)} \left[ \left(\frac {1}{E} \sum_ {i = 1} ^ {E} \pi^ {\star} (a | s) - \frac {1}{E} \sum_ {i = 1} ^ {E} \pi_ {i} (a | s)\right) ^ {2} \right] \\ = \mathbb {E} _ {s \sim d _ {\pi^ {\star}}, a \sim \pi^ {\star} (\cdot | s)} \left[ \left(\frac {1}{E} \sum_ {i = 1} ^ {E} \left(\pi^ {\star} (a | s) - \pi_ {i} (a | s)\right)\right) ^ {2} \right] \\ \leq \mathbb {E} _ {s \sim d _ {\pi^ {\star}}, a \sim \pi^ {\star} (\cdot | s)} \left[ \frac {1}{E ^ {2}} E \sum_ {i = 1} ^ {E} \left(\pi^ {\star} (a | s) - \pi_ {i} (a | s)\right) ^ {2} \right] (\text {C a u c h y - S c h w a r z}) \\ = \frac {1}{E} \sum_ {i = 1} ^ {E} \mathbb {E} _ {s \sim d _ {\pi^ {\star}}, a \sim \pi^ {\star} (\cdot | s)} \left[ \left(\pi^ {\star} (a | s) - \pi_ {i} (a | s)\right) ^ {2} \right] \\ \leq \frac {1}{E} \sum_ {i = 1} ^ {E} \mathbb {E} _ {s \sim d _ {\pi^ {\star}}, a \sim \pi^ {\star} (\cdot | s)} \left[ \left| \pi^ {\star} (a | s) - \pi_ {i} (a | s) \right| \right] \\ \leq \frac {1}{E} \sum_ {i = 1} ^ {E} \mathbb {E} _ {s \sim d _ {\pi^ {\star}}} \left[ \| \pi^ {\star} (\cdot | s) - \pi_ {i} (\cdot | s) \| \right] \\ = \frac {1}{E} \sum_ {i = 1} ^ {E} J _ {\mathrm {B C}} \left(\pi_ {i}\right) \\ \leq \epsilon \\ \end{array}
+$$
+
+The last step follows from our optimization oracle assumption:  $0 \leq \min_{\pi \in \Pi} J_{\mathrm{BC}}(\pi) \leq J_{\mathrm{BC}}(\pi^{\star}) = 0$ , hence  $J_{\mathrm{BC}}(\pi_i) \leq 0 + \epsilon = \epsilon$ . Combining the bounds on the two terms, we get  $J_{\mathrm{alg}}(\pi^{\star}) \leq 2\epsilon$ . Since  $\pi^{\star} \in \Pi$ , the result follows.
+
+![](images/bcf6ef44707c64fdc5a40d85732231785906ae47eea0c259ac798e063fd3f358.jpg)
+
+Theorem 1. Let  $\hat{\pi}$  be the result of minimizing  $J_{\mathrm{alg}}$  using our optimization oracle, and assume that  $Q_{T - t + 1}^{\pi^*}(s,a) - Q_{T - t + 1}^{\pi^*}(s,\pi^*) \leq u$  for all  $a \in \mathcal{A}, t \in \{1,2,\dots,T\}, d_{\pi}^{t}(s) > 0$ . Then  $\hat{\pi}$  satisfies  $J(\hat{\pi}) \leq J(\pi^*) + 3u\kappa \epsilon T$ .
+
+Proof. By our optimization oracle and Lemma 2, we have
+
+$$
+\begin{array}{l} J _ {\mathrm {a l g}} (\hat {\pi}) \leq \min  _ {\pi \in \Pi} J _ {\mathrm {a l g}} (\pi) + \epsilon \\ \leq 2 \epsilon + \epsilon \\ = 3 \epsilon \\ \end{array}
+$$
+
+Combining with Lemma 1, we get:
+
+$$
+\begin{array}{l} J _ {\mathrm {e x p}} (\hat {\pi}) \leq \kappa J _ {\mathrm {a l g}} (\hat {\pi}) \\ \leq 3 \kappa \epsilon \\ \end{array}
+$$
+
+Applying Theorem 1 from (Ross et al., 2011), we get  $J(\hat{\pi}) \leq J(\pi^{\star}) + 3u\kappa \epsilon T$ .
+
+![](images/620ac166fc07c0672ebfd75b8fb3c59768847bbe272925e748992903ed3887f1.jpg)
+
+# B IMPORTANCE OF BEHAVIOR CLONING COST
+
+The following example shows how minimizing the uncertainty cost alone without the BC cost can lead to highly sub-optimal policies if the demonstration data is generated by a stochastic policy which is only slightly suboptimal. Consider the following deterministic chain MDP:
+
+![](images/ff09c2fcab93eea9c30c794a02c73bbc7b9256d3f2e0ebe11417d0188a0ed523.jpg)
+
+The agent always starts in  $s_1$ , and gets a reward of 1 in  $s_3$  and 0 elsewhere. The optimal policy is given by:
+
+$$
+\begin{array}{l} \pi^ {\star} (\cdot | s _ {0}) = (0, 1) \\ \pi^ {\star} (\cdot | s _ {1}) = (0, 1) \\ \pi^ {\star} (\cdot | s _ {2}) = (0, 1) \\ \pi^ {\star} (\cdot \mid s _ {3}) = (0, 1) \\ \end{array}
+$$
+
+Assume the demonstration data is generated by the following policy, which is only slightly suboptimal:
+
+$$
+\begin{array}{l} \pi_ {\mathrm {d e m o}} (\cdot | s _ {0}) = (0, 1) \\ \pi_ {\mathrm {d e m o}} (\cdot | s _ {1}) = (0, 1) \\ \pi_ {\mathrm {d e m o}} (\cdot | s _ {2}) = (0. 1, 0. 9) \\ \pi_ {\mathrm {d e m o}} (\cdot | s _ {3}) = (0, 1) \\ \end{array}
+$$
+
+Let us assume realizability and perfect optimization for simplicity. If both transitions  $(s_2, a_0)$  and  $(s_2, a_1)$  appear in the demonstration data, then Random Expert Distillation (RED) will assign zero
+
+cost to both transitions. If we do not use bootstrapped samples to train the ensemble, then DRIL without the BC cost (we will call this UO-DRIL for Uncertainty-Only DRIL) will also assign zero cost to both transitions since all models in the ensemble would recover the Bayes optimal solution given the demonstration data. If we are using bootstrapped samples, then the Bayes optimal solution for each bootstrapped sample may differ and thus the different policies in the ensemble might disagree in their predictions, although given enough demonstration data we would expect these differences (and thus the uncertainty cost) to be small.
+
+Note also that since no samples at the state  $s_0$  occur in the demonstration data, both RED and UO-DRIL will likely assign high uncertainty costs to state-action pairs at  $(s_0, a_0), (s_0, a_1)$  and thus avoid highly suboptimal policies which get stuck at  $s_0$ .
+
+Now consider policies  $\hat{\pi}_1, \hat{\pi}_2$  given by:
+
+$$
+\hat {\pi} _ {1} (\cdot | s _ {0}) = (0, 1)
+$$
+
+$$
+\hat {\pi} _ {1} (\cdot | s _ {1}) = (0, 1)
+$$
+
+$$
+\hat {\pi} _ {1} (\cdot | s _ {2}) = (1, 0)
+$$
+
+$$
+\hat {\pi} _ {1} (\cdot | s _ {3}) = (0, 1)
+$$
+
+and
+
+$$
+\hat {\pi} _ {2} (\cdot | s _ {0}) = (0, 1)
+$$
+
+$$
+\hat {\pi} _ {2} (\cdot | s _ {1}) = (0, 1)
+$$
+
+$$
+\hat {\pi} _ {2} (\cdot | s _ {2}) = (0. 2, 0. 8)
+$$
+
+$$
+\hat {\pi} _ {2} (\cdot | s _ {3}) = (0, 1)
+$$
+
+Both of these policies only visit state-action pairs which are visited by the demonstration policy. In the case described above, both RED and UO-DRIL will assign  $\hat{\pi}_1$  and  $\hat{\pi}_2$  similarly low costs. However,  $\hat{\pi}_1$  will cycle forever between  $s_1$  and  $s_2$ , never collecting reward, while  $\hat{\pi}_2$  will with high probability reach  $s_3$  and stay there, thus achieving high reward. This shows that minimizing the uncertainty cost alone does not necessarily distinguish between good and bad policies. However,  $\hat{\pi}_1$  will incur a higher BC cost than  $\hat{\pi}_2$ , since  $\hat{\pi}_2$  more closely matches the demonstration data at  $s_2$ . This shows that including the BC cost can be important for further disambiguating between policies which all stay within the distribution of the demonstration data, but have different behavior within that distribution.
+
+# C EXPERIMENTAL DETAILS
+
+# C.1 ATARI ENVIRONMENTS
+
+All behavior cloning models were trained to minimize the negative log-likelihood classification loss on the demonstration data for 500 epochs using Adam (Kingma & Ba, 2014) and a learning rate of  $2.5 \cdot 10^{-4}$ . We stopped training once the validation error did not improve for 20 epochs. For our method, we initially performed a hyperparameter search on Space Invaders over the values shown in Table 1
+
+Table 1: Hyperparameters for DRIL  
+
+<table><tr><td>Hyperparameter</td><td>Values Considered</td><td>Final Value</td></tr><tr><td>Policy Learning rate</td><td>2.5 · 10-2, 2.5 · 10-3, 2.5 · 10-4</td><td>2.5 · 10-3</td></tr><tr><td>Quantile cutoff</td><td>0.8, 0.9, 0.95, 0.98</td><td>0.98</td></tr><tr><td>Number of supervised updates</td><td>1, 5</td><td>1</td></tr><tr><td>Number of policies in ensemble</td><td>5</td><td>5</td></tr><tr><td>Gradient clipping</td><td>0.1</td><td>0.1</td></tr><tr><td>Entropy coefficient</td><td>0.01</td><td>0.01</td></tr><tr><td>Value loss coefficient</td><td>0.5</td><td>0.5</td></tr><tr><td>Number of steps</td><td>128</td><td>128</td></tr><tr><td>Parallel Environments</td><td>16</td><td>16</td></tr></table>
+
+We then chose the best values and kept those hyperparameters fixed for all other environments. All other A2C hyperparameters follow the default values in the repo (Kostrikov, 2018): policy networks consisted of 3-layer convolutional networks with  $8 - 32 - 64$  feature maps followed by a single-layer MLP with 512 hidden units.
+
+For GAIL, we used the implementation in (Kostrikov, 2018) and replaced the MLP discriminator by a CNN discriminator with the same architecture as the policy network. We initially performed a hyperparameter search on Breakout with 10 demonstrations over the values shown in Table 2. However, we did not find any hyperparameter configuration which performed better than behavioral cloning.
+
+Table 2: Hyperparameters for GAIL  
+
+<table><tr><td>Hyperparameter</td><td>Values Considered</td><td>Final Value</td></tr><tr><td>Policy Learning rate</td><td>2.5 · 10-2, 2.5 · 10-3, 2.5 · 10-4</td><td>2.5 · 10-3</td></tr><tr><td>Discriminator Learning rate</td><td>2.5 · 10-2, 2.5 · 10-3, 2.5 · 10-4</td><td>2.5 · 10-3</td></tr><tr><td>Number of discriminator updates</td><td>1, 5, 10</td><td>5</td></tr><tr><td>Gradient clipping</td><td>0.1</td><td>0.1</td></tr><tr><td>Entropy coefficient</td><td>0.01</td><td>0.01</td></tr><tr><td>Value loss coefficient</td><td>0.5</td><td>0.5</td></tr><tr><td>Number of steps</td><td>128</td><td>128</td></tr><tr><td>Parallel Environments</td><td>16</td><td>16</td></tr></table>
+
+# C.2 CONTINUOUS CONTROL
+
+All behavior cloning and ensemble models were trained to minimize the mean-squared error regression loss on the demonstration data for 500 epochs using Adam (Kingma & Ba, 2014) and a learning rate of  $2.5 \cdot 10^{-4}$ . Policy networks were 2-layer fully-connected MLPs with tanh activations and 64 hidden units.
+
+Table 3: Hyperparameters (our method)  
+
+<table><tr><td>Hyperparameter</td><td>Values Considered</td><td>Final Value</td></tr><tr><td>Policy Learning rate</td><td>2.5 · 10-3, 2.5 · 1 0-4, 1 · 10-4, 5 · 10-5</td><td>2.5 · 10-5</td></tr><tr><td>Quantile cutoff</td><td>0.98</td><td>0.98</td></tr><tr><td>Number of supervised updates</td><td>1</td><td>1</td></tr><tr><td>Number of policies in ensemble</td><td>5</td><td>5</td></tr><tr><td>Gradient clipping</td><td>0.1</td><td>0.1</td></tr><tr><td>Entropy coefficient</td><td>0.01</td><td>0.01</td></tr><tr><td>Value loss coefficient</td><td>0.5</td><td>0.5</td></tr><tr><td>Number of steps</td><td>128</td><td>128</td></tr><tr><td>Parallel Environments</td><td>16</td><td>16</td></tr></table>
+
+# D ABLATION EXPERIMENTS
+
+In this section we provide ablation experiments examining the effects of the cost function clipping and the role of the BC loss. We also compare the ensemble approach to a dropout-based approximation and show that DRIL works well in both cases.
+
+Table 4: Ablation Experiments with 3 expert trajectories  
+
+<table><tr><td>Environment</td><td>SpaceInvaders</td><td>Breakout</td><td>BeamRider</td></tr><tr><td>DRIL (ensemble)</td><td>555.7</td><td>286.7</td><td>2033.4</td></tr><tr><td>DRIL (dropout)</td><td>581.4</td><td>205.4</td><td>2124.5</td></tr><tr><td>DRIL (raw cost)</td><td>421.8</td><td>70.9</td><td>1265.5</td></tr><tr><td>DRIL (no BC cost)</td><td>102.1</td><td>78.3</td><td>538.4</td></tr><tr><td>BC</td><td>257.0</td><td>2.7</td><td>689.7</td></tr></table>
+
+Results are shown in Figure 4. First, switching from the clipped cost in  $\{-1, +1\}$  to the raw cost causes a drop in performance. One explanation may be that since the raw costs are always positive (which corresponds to a reward which is always negative), the agent may learn to terminate the episode early in order to minimize the total cost incurred. Using a cost/reward which has both positive and negative values avoids this behavior.
+
+Second, optimizing the pure BC cost performs better than the pure uncertainty cost for some environments (SpaceInvaders, BeamRider) while optimizing the pure uncertainty cost performs better than BC in Breakout. DRIL, which optimizes both, has robust performance and performs the best over all environments.
+
+For the dropout approximation we trained a single policy network with a dropout rate of 0.1 applied to all layers except the last, and estimated the variance for each state-action pair using 5 different dropout masks. Similarly to the ensemble approach, we computed the  $98^{\mathrm{th}}$  quantile of the variance on the demonstration data and used this value in our clipped cost. MC-dropout performs similarly to the ensembling approach, which shows that our method can be paired with different approaches to posterior estimation.
\ No newline at end of file
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+# DISENTANGLEMENT BY NONLINEAR ICA WITH GENERAL INCOMPRESSIBLE-FLOW NETWORKS (GIN)
+
+Peter Sorrenson, Carsten Rother, Ullrich Köthe
+
+Visual Learning Lab
+
+Heidelberg University
+
+# ABSTRACT
+
+A central question of representation learning asks under which conditions it is possible to reconstruct the true latent variables of an arbitrarily complex generative process. Recent breakthrough work by Khemakhem et al. (2019) on nonlinear ICA has answered this question for a broad class of conditional generative processes. We extend this important result in a direction relevant for application to real-world data. First, we generalize the theory to the case of unknown intrinsic problem dimension and prove that in some special (but not very restrictive) cases, informative latent variables will be automatically separated from noise by an estimating model. Furthermore, the recovered informative latent variables will be in one-to-one correspondence with the true latent variables of the generating process, up to a trivial component-wise transformation. Second, we introduce a modification of the RealNVP invertible neural network architecture (Dinh et al., 2016) which is particularly suitable for this type of problem: the General Incompressible-flow Network (GIN). Experiments on artificial data and EMNIST demonstrate that theoretical predictions are indeed verified in practice. In particular, we provide a detailed set of exactly 22 informative latent variables extracted from EMNIST.
+
+# 1 INTRODUCTION
+
+Deep latent-variable models promise to unlock the key factors of variation within a dataset, opening a window to interpretation and granting the power to manipulate data in an intuitive fashion. The theory of identifiability in linear independent component analysis (ICA) (Comon, 1994) tells us when this is possible, if we restrict the model to a linear transformation, but until recently there was no corresponding theory for the highly nonlinear models needed to manipulate complex data. This changed with the recent breakthrough work by Khemakhem et al. (2019), which showed that under relatively mild conditions, it is possible to recover the joint data and latent space distribution, up to a simple transformation in the latent space. The key requirement is that the generating process is conditioned on a variable which is observed along with the data. This condition could be a class label, time index of a time series, or any other piece of information additional to the data. They interpret their theory as a nonlinear version of ICA.
+
+This work extends this theory in a direction relevant for application to real-world data. The existing theory assumes knowledge of the intrinsic problem dimension, but this is unrealistic for anything but artificially generated datasets. Here, we show that in the special case of Gaussian latent space distributions, the intrinsic problem dimension can be discovered. The important latent variables are organically separated from noise variables by the estimating model. Furthermore, the variables discovered correspond to the true generating latent variables, up to a trivial component-wise translation and scaling. Very similar results exist for other members of the exponential family with two parameters, such as the beta and gamma distributions.
+
+We introduce a variant of the RealNVP (Dinh et al., 2016) invertible neural network: the General Incompressible-flow Network (GIN). The flow is called incompressible in reference to fluid dynamics, since it preserves volumes: the Jacobian determinant is simply unity. We emphasise its generality and increased expressive power in comparison to previous volume-preserving flows, such as NICE (Dinh et al., 2014). As already noted in Khemakhem et al. (2019), flow-based generative models are a natural fit for the theory of nonlinear ICA, as are the variational autoencoders (VAEs) (Kingma &
+
+Welling, 2013) used in that work. For us, major advantages of invertible architectures over VAEs are the ability to specify volume preservation and directly optimize the likelihood, and freedom from the requirement to specify the dimension of the model's latent space. An INN always has a latent space of the same dimension as the data. In addition, the forward and backward models share parameters, saving the effort of learning separate models for each direction.
+
+In summary, our work makes the following contributions:
+
+- We extend the theory of nonlinear ICA to allow for unknown intrinsic problem dimension. Doing so, we find that this dimension can be discovered and a one-to-one correspondence between generating and estimated latent variables established.  
+- We propose as an implementation an invertible neural network obtained by modifying the RealNVP architecture. We call our new architecture GIN: the General Incompressible-flow Network.  
+- We demonstrate the viability of the model on artificial data and the EMNIST dataset. We extract 22 meaningful variables from EMNIST, encoding both global and local features.
+
+# 2 RELATED WORK
+
+The basic goals of nonlinear ICA stem from the original work on linear ICA. An influential formulation, as well as the first identifiability results, were given in Comon (1994). These stated the conditions which allow the generating latent variables to be discovered, when the mixing function is a linear transformation. However, it was shown in Hyvarinen & Pajunen (1999) that this approach to identifiability does not extend to general nonlinear functions.
+
+The first identifiability results in nonlinear ICA came in Hyvarinen & Morioka (2016) and Hyvarinen & Morioka (2017), applied to time series, and implemented via a discriminative model and semi-supervised learning. A more general formulation, valid for other forms of data, was given in Hyvarinen et al. (2018) and the theory was extended to generative models in Khemakhem et al. (2019), where experiments were implemented by a VAE.
+
+Many authors have addressed the general problem of disentanglement, and proposed models to learn disentangled features. Prominent among these is  $\beta$ -VAE (Higgins et al., 2017) and its variations (e.g. Chen et al., 2018) which augment the standard ELBO loss with tunable hyperparameters to encourage disentanglement. There are also attempts to modify the GAN framework (Goodfellow et al., 2014) such as InfoGAN (Chen et al., 2016), which tries to maximize the mutual information between some dimensions of the latent space and the observed data. Many of these approaches are unsupervised. However, as pointed out and empirically demonstrated in Locatello et al. (2018), unsupervised models without conditioning in the latent space are in general unidentifiable.
+
+Several unsupervised VAE models implement conditioning in the latent space by means of Gaussian mixtures (Johnson et al., 2016; Dilokthanakul et al., 2016; Zhao et al., 2019). Our work differs mainly by (i) only considering supervised tasks, therefore being safely covered by the theory of Khemakhem et al. (2019), and (ii) enforcing volume-preservation, not possible in a VAE.
+
+The invertible neural networks in this work build upon the NICE framework (Dinh et al., 2014) and its extension in RealNVP (Dinh et al., 2016). A similar network design to ours is Ardizzzone et al. (2019). This is a conditioned INN based on RealNVP, however the conditioning information is applied as a parameter of the network. The authors find in experiments with MNIST that meaningful variables are present in their latent space, but are rotated such that they are not aligned with the axes of the space. In this work, the conditioning is only present as a parameter of the latent space distributions. As a result, it is covered by the theory of Khemakhem et al. (2019) and its extension here, which results in non-rotated, meaningful latent variables.
+
+# 3 THEORY
+
+# 3.1 EXISTING THEORY OF NONLINEAR ICA
+
+This section adapts theoretical results from Khemakhem et al. (2019) to the context of invertible neural networks. Suppose the existence of the following three random variables: a latent generating
+
+![](images/dd62fa1314b628bc2198bca306cadf3cee78b634012b37ae6788c77aa3fa47d0.jpg)  
+Figure 1: Relationship between the variables and functions defined in Sec. 3.1, as well as indication of the training scheme.
+
+variable  $\mathbf{z} \in \mathbb{R}^n$ , a condition  $\mathbf{u} \in \mathbb{R}^m$  and a data point  $\mathbf{x} \in \mathbb{R}^d$ , where  $n \leq d$ . If  $n < d$ , also suppose the existence of a noise variable  $\varepsilon \in \mathbb{R}^{d - n}$ . The variables  $\mathbf{z}$  and  $\varepsilon$  make up the generating latent space, and are not necessarily known, whereas  $\mathbf{u}$  and  $\mathbf{x}$  are observable and therefore always known.
+
+The distribution of  $\mathbf{z}$  is a factorial member of the exponential family with  $k$  sufficient statistics, conditioned on  $\mathbf{u}$ . In its most general form the distribution can be written as
+
+$$
+p _ {\mathrm {z}} (\mathrm {z} | \mathrm {u}) = \prod_ {i = 1} ^ {n} \frac {Q _ {i} \left(z _ {i}\right)}{Z _ {i} (\mathrm {u})} \exp \left[ \sum_ {j = 1} ^ {k} T _ {i, j} \left(z _ {i}\right) \lambda_ {i, j} (\mathrm {u}) \right] \tag {1}
+$$
+
+where the  $T_{i,j}$  are the sufficient statistics, the  $\lambda_{i,j}$  their coefficients and  $Z_{i}$  the normalizing constant.  $Q_{i}$  is called the base measure, which in many cases is simply 1.
+
+The distribution of  $\varepsilon$  must not depend on u and its probability density function must be always finite.
+
+The variable  $\mathbf{x}$  is the result of an arbitrarily complex, invertible, deterministic transformation from the generating latent space to the data space:  $\mathrm{x} = f(\mathrm{z},\varepsilon)$ . This can alternatively be formulated as an injective, non-deterministic transformation from the lower-dimensional  $\mathbf{Z}$ -space to the higher-dimensional  $\mathbf{X}$ -space:  $\mathrm{x} = f_{\varepsilon}(\mathrm{z})$ .
+
+In general, an observed dataset  $\mathcal{D}$  will consist only of instances of  $\mathbf{x}$  and  $\mathbf{u}$ . The task of nonlinear ICA is to disentangle the data to recover the generating latent variables  $\mathbf{z}$ , as well as the form of the function  $f$  and its inverse. We can try to achieve this with a sufficiently general, invertible function approximator  $g$ , which maps from a latent variable  $\mathbf{w} \in \mathbb{R}^d$  to the data space:  $\mathbf{x} = g(\mathbf{w};\theta)$ . Here  $\theta$  denotes the parameters of  $g$ . Note that the dimensions of the latent space and data space are the same due to the invertibility of  $g$ . We assume that  $\mathbf{w}$  follows a conditionally independent exponential probability distribution, conditioned on the same condition  $\mathbf{u}$  as  $\mathbf{z}$ :
+
+$$
+p _ {\mathrm {w}} (\mathrm {w} | \mathrm {u}) = \prod_ {i = 1} ^ {d} \frac {Q _ {i} ^ {\prime} \left(w _ {i}\right)}{Z _ {i} ^ {\prime} (\mathrm {u})} \exp \left[ \sum_ {j = 1} ^ {k} T _ {i, j} ^ {\prime} \left(w _ {i}\right) \lambda_ {i, j} ^ {\prime} (\mathrm {u}) \right]. \tag {2}
+$$
+
+The coefficients of the sufficient statistics  $\lambda_{i,j}^{\prime}(\mathfrak{u})$  are not restricted, which means that a given variable  $i$  of the estimated latent space is allowed to lose its dependence on  $\mathbf{u}$ . If this occurs, we will consider this variable to encode noise, playing a role equivalent to  $\varepsilon$  in the generating latent space.
+
+In addition to this specification of the generative and estimating models, some conditions are necessary to ensure the latent variables can be recovered. The most important of these concerns the variability of  $\lambda_{i,j}(\mathbf{u})$  under  $\mathbf{u}$ . However, as long as the  $\lambda_{i,j}$  are randomly and independently generated, and there are at least  $nk + 1$  distinct conditions  $\mathbf{u}$ , this condition is almost surely fulfilled. See Appendix A.1 for further details.
+
+In the limit of infinite data and perfect convergence, the estimating model will give the same conditional likelihood to all data points as the true generating model:
+
+$$
+p _ {\mathbf {T}, \boldsymbol {\lambda}, f, \varepsilon} (\mathrm {x} | \mathrm {u}) = p _ {\mathbf {T} ^ {\prime}, \boldsymbol {\lambda} ^ {\prime}, g} (\mathrm {x} | \mathrm {u}). \tag {3}
+$$
+
+If this is the case, the vector of sufficient statistics  $\mathbf{T}$  from the generating latent space will be related to that of the estimated latent space by an affine transformation:
+
+$$
+\mathbf {T} (\mathbf {z}) = \boldsymbol {A} \mathbf {T} ^ {\prime} (\mathbf {w}) + \boldsymbol {c}, \tag {4}
+$$
+
+where  $\mathbf{A}$  is some constant, full-rank,  $nk \times dk$  matrix and  $\mathbf{c} \in \mathbb{R}^{nk}$  some constant vector. The relationship holds for all values of  $\mathbf{z}$  and  $\mathbf{w}$ . The proof of this result can be found in Appendix A.
+
+This relationship is a generalization of that derived in Khemakhem et al. (2019). In that version, they assume knowledge of  $n$ , the dimension of the generating latent space, and give their estimating model a latent space of the same dimension. In this context the matrix  $A$  defines a relationship between latent spaces of the same dimension, making it square and invertible.
+
+# 3.2 NONLINEAR ICA WITH A GAUSSIAN LATENT SPACE
+
+Given the relationship in equation (4), we can ask whether any stronger results hold for particular special cases. We hope in particular to induce sparsity in the matrix  $\mathbf{A}$ , so that the estimated latent space is related to the true one by as simple a transformation as possible. We show (proof in Appendix B) that when both the generating and estimated latent spaces follow a Gaussian distribution, the generating latent space variables are recovered up to a trivial translation and scaling. Furthermore, the dimension of the generating latent space is recovered.
+
+More precisely, each generating latent variable  $z_{i}$  is related to exactly one estimated latent variable  $w_{j}$ , for some  $j$ , as  $z_{i} = a_{i}w_{j} + b_{i}$ , for some constant  $a_{i}$  and  $b_{i}$ . Furthermore, each estimated latent variable  $w_{j}$  is related to at most one  $z_{i}$ . If the estimating latent space has higher dimension than the dimension of the generating latent space, some estimating latent variables are not related to any generating latent variables and so must encode only noise. This is the dimension discovery mechanism, since the estimated latent space organically splits into informative and non-informative parts, with the dimension of the informative part equal to the unknown intrinsic dimension of the generating latent space. Very similar results can be derived for all common continuous two-parameter members of the exponential family, including the gamma and beta distributions. See Appendix C.
+
+# 3.3 VOLUME PRESERVATION VIA INCOMPRESSIBLE FLOW
+
+In a multivariate Gaussian distribution with diagonal covariance, the standard deviations are directly proportional to the principal axes of the ellipsoid defining a surface of constant probability. In classical PCA, this property is used to assign importance to each principal component based on its standard deviation. In PCA we can think of the rotation of the data into the basis defined by the principal components as the latent space. In a deep latent variable model the transformation between the latent space and the data is significantly more complex, but if this transformation preserves volumes like the rotation in PCA, it will retain the desirable correspondence between the standard deviation of a latent space variable and its importance in explaining the data. Because invertible neural networks (normalizing flows) have a tractable Jacobian determinant, they open the possibility to constrain it to unity. This is equivalent to volume preservation and is the rationale behind the General Incompressible-flow Network.
+
+# 4 EXPERIMENTS
+
+Experiments on artificial datasets confirm the theory for a normally distributed latent space, as well as identifying potential causes of failure. Experiments on the EMNIST dataset (Cohen et al., 2017) demonstrate the ability of GIN to estimate independent and interpretable latent variables from real-world data.
+
+# 4.1 MODEL DESCRIPTION
+
+The GIN model is similar in form to RealNVP (Dinh et al., 2016) and shares its flexibility, but retains the volume-preserving properties of the NICE framework (Dinh et al., 2014).
+
+RealNVP coupling layers split  $D$ -dimensional input  $x$  into two parts,  $x_{1:d}$  and  $x_{d+1:D}$  where  $d < D$ . The output of the layer is the concatenation of  $y_{1:d}$  and  $y_{d+1:D}$  with
+
+$$
+y _ {1: d} = x _ {1: d} \tag {5}
+$$
+
+$$
+y _ {d + 1: D} = x _ {d + 1: D} \odot \exp (s \left(x _ {1: d}\right)) + t \left(x _ {1: d}\right) \tag {6}
+$$
+
+![](images/00b5b1877310755b4d3cc3d80a114e0eb41a4c4eea4e4a6f9bb07033b428efed.jpg)  
+Figure 2: Successful reconstruction by GIN of the two informative latent variables out of ten in total. The other eight are correctly identified as noise. The observed data is ten-dimensional (projection into two dimensions shown here). The spectrum shows the standard deviation of each variable of the reconstruction (in black, log scale) which quantifies its importance. Ground truth is in gray. There is a clear distinction between the two informative dimensions and the noise dimensions, showing that GIN has correctly detected a two-dimensional manifold in the ten-dimensional data presented to it.
+
+where addition, multiplication and exponentiation are applied component-wise. The logarithm of the Jacobian determinant of a coupling layer is simply the sum of the scaling function  $s(x_{1:d})$ . Volume preservation is achieved by setting the final component  $s(x_d)$  to the negative sum of the previous components, so the total sum of  $s(x_{1:d})$  is zero (in contrast, NICE sets  $s(x_{1:d}) \equiv 0$ ). Hence the Jacobian determinant is unity and volume is preserved. The network is free to allow the volume contribution of some dimensions of the output of any coupling layer to grow, but only by shrinking the other dimensions of the output in direct proportion. As well as enforcing a strong correspondence between the importance of a latent variable and its standard deviation, we believe volume-preservation has a regularizing effect, since it is a very strong constraint, comparable to orthonormality in linear transformations.
+
+# 4.2 OPTIMIZATION
+
+The experiments in this section deal with labeled data, where each data point belongs to one of  $M$  different classes. This class label is used as the condition u associated with each data point x. In the estimated latent space, all data instances with the same label should belong to the same Gaussian distribution. Hence we are learning a Gaussian mixture in the estimated latent space, with  $M$  mixture components. Since the distribution for each class in the estimated latent space is required to be factorial, the variance of each mixture component is diagonal, and we can write  $\sigma_i^2 (\mathbf{u})$  for the variance in the  $i$ -th dimension.
+
+Given a set of data, condition pairs  $\mathcal{D} = \{(\mathrm{x}^{(1)},\mathrm{u}^{(1)}),\dots ,(\mathrm{x}^{(N)},\mathrm{u}^{(N)})\}$  and model  $g$ , parameterized by  $\pmb{\theta}$  (where  $g$  maps from the latent space to the data space) we can construct a loss from the log-likelihood. Using the change of variables formula, we have  $\log p(\mathrm{x}|\mathrm{u}) = \log p(\mathrm{w}|\mathrm{u})$ , with no Jacobian term since the transformation  $\mathrm{w} = g^{-1}(\mathrm{x};\pmb{\theta})$  is volume-preserving. To maximize the likelihood of  $\mathcal{D}$ , we minimize the negative log-likelihood of  $\mathrm{w}$  in the estimated latent space:
+
+$$
+\mathcal {L} (\boldsymbol {\theta}) = \mathbb {E} _ {(\mathrm {x}, \mathrm {u}) \in \mathcal {D}} \left[ \frac {1}{n} \sum_ {i = 1} ^ {n} \left(\frac {\left(g _ {i} ^ {- 1} (\mathrm {x} ; \boldsymbol {\theta}) - \mu_ {i} (\mathrm {u} ; \boldsymbol {\theta})\right) ^ {2}}{2 \sigma_ {i} ^ {2} (\mathrm {u} ; \boldsymbol {\theta})} + \log \left(\sigma_ {i} (\mathrm {u}; \boldsymbol {\theta})\right)\right) \right]. \tag {7}
+$$
+
+# 4.3 ARTIFICIAL DATA
+
+# 4.3.1 EXPERIMENT 1: CONDITIONS OF THEORY FULFILLED
+
+Samples are generated in two dimensions, conditioned on five different cluster labels, see Fig. 2. The means of the clusters are chosen independently from a uniform distribution on  $[-5,5]$  and variances from a uniform distribution on  $[.5,3]$ . This data is then concatenated with independent Gaussian noise in eight dimensions to make a ten-dimensional generating latent space, where only the first two variables are informative. The noise is scaled by 0.01 to be small in comparison to the informative dimensions. The latent space samples are then passed through a RealNVP network
+
+with 8 fully connected coupling blocks with randomly initialized weights to produce the observed data. This acts as a highly nonlinear mixing which can only be successfully treated with nonlinear methods.
+
+GIN is used as the estimating model, with 8 fully connected coupling blocks (full details in Appendix D). Training converges quickly and stably using the Adam optimizer (Kingma & Ba, 2014) with initial learning rate  $10^{-2}$  and other values set to the usual recommendations. Batch size is 1,000 and the data is augmented with Gaussian noise  $(\sigma = 0.01)$  at each iteration. After convergence of the loss, the learning rate is reduced by a factor of 10 and trained again until convergence.
+
+Over a number of experiments we made the following observations:
+
+- The model converges stably and gives importance (quantified by standard deviation) to only two variables in the estimated latent space, provided there is sufficient overlap between the mixture components in the generating latent space.  
+- Where there is not enough overlap in the generating latent space, the model cannot recognize common variables across all the different classes, and tends to split one genuine dimension of variation into two or more in its estimated latent space. This appears to be a problem of finite data. We have observed that when this behaviour occurs, and if the gap between mixture components is not too large, it can be prevented by increasing the number of samples so that the space between the mixture components is better filled (see Fig. 6 and 7 in Appendix E). This is consistent with the theory, where equation (4) is true asymptotically. Since the latent space distributions are members of the exponential family, they have support across the entire domain of the latent space, hence gaps can never remain in the limit of infinite samples.  
+- Choice of learning rate is important. If the initial learning rate is too low, training gets stuck in bad local optima, where single true variables are split into several latent dimensions.
+
+# 4.3.2 EXPERIMENT 2: CONDITIONS OF THEORY NOT FULFILLED
+
+Samples are generated as in Experiment 1, but with only three mixture components. Since there are two sufficient statistics per dimension, and two dimensions of variation, according to the theory we need at least  $nk + 1 = 5$  distinct conditions u for equation (4) to hold (see section 3.1). Therefore, we might not expect successful experiments. Nonetheless, we observe essentially the same results as for the previous experiments (see Fig. 8 in Appendix E), with the same caveats regarding gaps between the mixture components in the generating latent space. This suggests that the conditions derived in Khemakhem et al. (2019), although sufficient for disentanglement, are not necessary.
+
+# 4.4 EMNIST
+
+# 4.4.1 EXPERIMENT
+
+![](images/87a08e43cd267d582b9df8010469e50405633e0d1c1ba3e996657a6c1c79bf66.jpg)  
+Figure 3: Spectrum of sorted standard deviations derived from training GIN on EMNIST. On the right is the equivalent spectrum from PCA, a linear method, on MNIST. The nonlinear spectrum exhibits a sharp knee not obtained by linear methods. In the nonlinear spectrum, the first 22 latent variables encode information about the shape of a digit, while the rest of the latent variables encode noise. This distinction is marked with a dotted line in the left and center figures (see Sec. 4.4.2 for explanation of the choice of this cut-off). Within the first 22 variables, the first eight encode global information, such as slant and width, whereas the following 14 encode more local information. This distinction is marked in the center figure only.
+
+The data comes from the EMNIST Digits training set of 240,000 images of handwritten digits with labels (Cohen et al., 2017). EMNIST is a larger version of the well-known MNIST dataset, and also includes handwritten letters. Here we use only the digits. The digit label is used as the condition u, hence we construct a Gaussian mixture in the estimated latent space with 10 mixture components. According to the theory (see 3.1), these are only enough conditions to guarantee identifiability if there are only four informative latent variables in the generating latent space. We expect that the true number of informative variables is somewhat higher, so as in Experiment 2 above, we are operating outside of the guarantees of the theory. In addition, the true generative process of human handwriting may not exactly fulfill our method's assumptions, and we might still lack data on rare or subtle variations, despite the large size of the dataset in comparison to MNIST.
+
+The estimating model is a GIN which uses convolutional coupling blocks and fully connected coupling blocks to transform the data to the latent space (full details in Appendix D). Optimization is with the Adam optimizer, with initial learning rate 3e-4. Batch size is 240 and the data is augmented with Gaussian noise ( $\sigma = 0.01$ ) at each iteration. The model is trained for 45 epochs, then for a further 50 epochs with the learning rate reduced by a factor of 10.
+
+# 4.4.2 RESULTS
+
+![](images/cb8c30c35299c10d8fbd85a818da96c64fcd2a7b35f3b4b92d7aad7b594212eb.jpg)  
+(a) Variable 1: upper width
+
+![](images/c7622551d9407056d42f101eda22913a0ce3d5c894f694c9c2df5c5716a5296c.jpg)  
+(b) Variable 8: lower width
+
+![](images/3a25ff8d5ffe24d8a5970b91436eb2d0439b3db6d8214953a73353c4131a19b3.jpg)  
+(c) Variable 3: height
+
+![](images/62d3927b578f577a83d1f6005d3add519ecd27602dfc2070065eed0d79019f76.jpg)  
+(d) Variable 4: bend
+
+![](images/149bdc0fe67ae6862a21886c7ff06de34f684667a6ab2a142b3a5dd3a27ed437.jpg)  
+Figure 4: Selection of global latent variables found by GIN. Each row is conditioned on a different digit label. The variable runs from -2 to +2 standard deviations across the columns, with all other variables held constant at their mean value. The rightmost column shows a heatmap of the image areas most affected by the variable, computed as the absolute pixel difference between -1 and +1 standard deviations. Variable 1 controls the width of the top half of a digit, whereas variable 8 controls the width of the bottom half. Width in both cases is somewhat entangled with slant. Variable 3 controls the height and variable 4 controls how bent the digit is. Full set of variables for all digits in Appendix F.  
+(a) Variable 11  
+Figure 5: Selection of local latent variables found by GIN. Refer to Figure 4 above for explanation of the layout. Variable 11 controls the shape of curved lines in the middle right. Digits without such lines are not affected. Variable 12 controls extension towards the upper right. Variable 13 modifies the top left of 2, 3 and 7 only (7 not shown here) and variable 16 modifies only the lower right stroke of a 2. Full set of variables for all digits in Appendix F.
+
+![](images/102c247df82f7118725e1bb4d3e0752692efe3afa73f2f5021f99b98d8d02fd6.jpg)  
+(b) Variable 12
+
+![](images/8a68d271bc9e7fa5f57edc2dcb97184496edb54dff0b4c694bb88a35da216e23.jpg)  
+(c) Variable 13
+
+![](images/cd7ac316850f2270821a8502c7bf3faba6b860b5a59ffc9b17f1ffeea36f2eae.jpg)  
+(d) Variable 16
+
+The model encodes information into 22 broadly interpretable latent space variables. The eight variables with the highest standard deviation encode global information, such as slant, line thickness and width which affect all digits more or less equally. The remaining 14 meaningful variables encode more local information, which does not affect all digits equally. We interpret all other variables as encoding noise. This is motivated mainly by our observation of an abrupt end to any observable change in the reconstructed digits when any variable after the first 22 is changed. This can be seen clearly in Fig. 15 in Appendix F. Some variables, particularly the global ones, are not entirely disentangled from another. Nevertheless, the results are compelling and suggest that the assumptions required by the theory are approximately met.
+
+We observed some global features which are not usually seen in disentanglement experiments on MNIST (e.g. Chen et al., 2016; Dupont, 2018). These experiments usually obtain digit slant, width and line thickness as the major global independent variables. We too observe slant and line thickness as independent variables (variables 2 and 5), but find width to be split into two variables, one governing the width of the upper half of a digit (variable 1) and the other the lower half (variable 8). This makes sense, since these can in fact vary independently. We also observe the height of a digit (variable 3) (see Fig. 4). This does not usually appear in disentanglement experiments, possibly because it is too subtle a variation for those experiments, but possibly because it is not present or not discoverable in the smaller MNIST dataset but is in EMNIST.
+
+Local features are also usually not observed in such experiments, so the variables which control these are particularly interesting. These variables modify only a region of the digit, leaving the rest of the digit untouched. In addition, digits which do not have the feature which is being modified in that region are left alone. Examples include variable 13, which changes the orientation of the top-left stroke in 2, 3 and 7, and variable 16, which modifies only the lower-right stroke of a 2 (see Fig. 5). The full set of local and global variables can be seen in Fig. 10 to Fig. 15 in Appendix F.
+
+# 4.4.3 NOTE ON THE CHOICE OF CONDITIONING VARIABLE
+
+The concept of a "true generative process" rests on the assumption that conditioning variables u act as causes for the latent variables z, such that the elements of the latter become independent given the former. In the case of handwritten digits, u represents a person's decision to write a particular digit, and z determines the hand motions to produce this digit. Modelling u in terms of digit labels is a plausible proxy for the unknown causal variables in the human brain, and the success of the resulting model suggests that the assumptions are approximately fulfilled. Identifying promising conditioning variables u in less obvious situations is an important open problem for future work.
+
+# 4.4.4 POTENTIAL IMPROVEMENTS
+
+Increasing the number of conditioning variables would bring this work closer in line with the existing theory. At the current value of ten, the theory only applies if there are at most four informative generating latent variables, which is almost certainly too low a number. One option to increase the number of conditions is to relax the labels from hard to soft, i.e. compute posterior probabilities of class membership given a data example. This could be achieved by information distillation as in Hinton et al. (2015). Another possibility is to split existing labels into sub-labels, for example making a distinction between sevens with and without crossbars. This may also aid the generation of more realistic samples, since we observed no sevens with crossbars in our generated samples, even though they make up approximately  $5\%$  of the sevens in the dataset.
+
+# 5 CONCLUSION AND OUTLOOK
+
+We have expanded the theory of nonlinear ICA to cover problems with unknown intrinsic dimension and demonstrated an implementation with GIN, a new volume-preserving modification of the RealNVP invertible network architecture. The variables discovered by GIN in EMNIST are interpretable and detailed enough to suggest that the assumptions made about the generating process are approximately true. Furthermore, our experiments with EMNIST demonstrate the viability of applying models inspired by the new theory of nonlinear ICA to real-world data, even when not all of the conditions of the theory are met.
+
+It is not clear if the methods from this work will scale to larger problems. However, given the recent advances of similar flow-based generative models in density estimation on larger datasets such as CelebA and ImageNet (e.g. Kingma & Dhariwal, 2018; Ho et al., 2019), it is a plausible prospect. In addition, it is not clear whether the method can successfully be extended to the context of semi-supervised learning, or ultimately, unsupervised learning.
+
+# REFERENCES
+
+Lynton Ardizzone, Carsten Luth, Jakob Kruse, Carsten Rother, and Ullrich Köthe. Guided image generation with conditional invertible neural networks. arXiv:1907.02392, 2019.  
+Tian Qi Chen, Xuechen Li, Roger B Grosse, and David K Duvenaud. Isolating sources of disentanglement in variational autoencoders. In Advances in Neural Information Processing Systems, pp. 2610-2620, 2018.  
+Xi Chen, Yan Duan, Rein Houthooft, John Schulman, Ilya Sutskever, and Pieter Abbeel. Infogan: Interpretable representation learning by information maximizing generative adversarial nets. In Advances in neural information processing systems, pp. 2172-2180, 2016.  
+Gregory Cohen, Saeed Afshar, Jonathan Tapson, and André van Schaik. EMNIST: an extension of MNIST to handwritten letters. arXiv:1702.05373, 2017.  
+Pierre Comon. Independent component analysis, a new concept? Signal processing, 36(3):287-314, 1994.  
+Nat Dilokthanakul, Pedro AM Mediano, Marta Garnelo, Matthew CH Lee, Hugh Salimbeni, Kai Arulkumaran, and Murray Shanahan. Deep unsupervised clustering with gaussian mixture variational autoencoders. arXiv preprint arXiv:1611.02648, 2016.  
+Laurent Dinh, David Krueger, and Yoshua Bengio. Nice: Non-linear independent components estimation. arXiv:1410.8516, 2014.  
+Laurent Dinh, Jascha Sohl-Dickstein, and Samy Bengio. Density estimation using Real NVP. arXiv:1605.08803, 2016.  
+Emilien Dupont. Learning disentangled joint continuous and discrete representations. In Advances in Neural Information Processing Systems, pp. 710-720, 2018.  
+Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Advances in neural information processing systems, pp. 2672-2680, 2014.  
+Irina Higgins, Loic Matthew, Arka Pal, Christopher Burgess, Xavier Glorot, Matthew Botvinick, Shakir Mohamed, and Alexander Lerchner.  $\beta$ -VAE: Learning basic visual concepts with a constrained variational framework. *ICLR*, 2(5):6, 2017.  
+Geoffrey Hinton, Oriol Vinyals, and Jeff Dean. Distilling the knowledge in a neural network. arXiv:1503.02531, 2015.  
+Jonathan Ho, Xi Chen, Aravind Srinivas, Yan Duan, and Pieter Abbeel. Flow++: Improving flow-based generative models with variational dequantization and architecture design. arXiv:1902.00275, 2019.  
+Aapo Hyvarinen and Hiroshi Morioka. Unsupervised feature extraction by time-contrastive learning and nonlinear ICA. In Advances in Neural Information Processing Systems, pp. 3765-3773, 2016.  
+Aapo Hyvarinen and Hiroshi Morioka. Nonlinear ICA of temporally dependent stationary sources. In Proceedings of Machine Learning Research, 2017.  
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+Aapo Hyvärinen, Hiroaki Sasaki, and Richard E Turner. Nonlinear ICA using auxiliary variables and generalized contrastive learning. arXiv:1805.08651, 2018.  
+Jörn-Henrik Jacobsen, Arnold Smeulders, and Edouard Oyallon. i-RevNet: Deep invertible networks. arXiv:1802.07088, 2018.  
+Matthew J Johnson, David K Duvenaud, Alex Wiltschko, Ryan P Adams, and Sandeep R Datta. Composing graphical models with neural networks for structured representations and fast inference. In Advances in neural information processing systems, pp. 2946-2954, 2016.
+
+Ilyes Khemakhem, Diederik P Kingma, and Aapo Hyvarinen. Variational autoencoders and nonlinear ICA: A unifying framework. arXiv:1907.04809, 2019.  
+Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv:1412.6980, 2014.  
+Diederik P Kingma and Max Welling. Auto-encoding variational bayes. arXiv:1312.6114, 2013.  
+Durk P Kingma and Prafulla Dhariwal. Glow: Generative flow with invertible 1x1 convolutions. In Advances in Neural Information Processing Systems, pp. 10215-10224, 2018.  
+Francesco Locatello, Stefan Bauer, Mario Lucic, Sylvain Gelly, Bernhard Schölkopf, and Olivier Bachem. Challenging common assumptions in the unsupervised learning of disentangled representations. arXiv:1811.12359, 2018.  
+Qingyu Zhao, Nicolas Honnorat, Ehsan Adeli, Adolf Pfefferbaum, Edith V Sullivan, and Kilian M Pohl. Variational autoencoder with truncated mixture of gaussians for functional connectivity analysis. In International Conference on Information Processing in Medical Imaging, pp. 867-879. Springer, 2019.
+
+# SUPPLEMENTARY MATERIAL
+
+# A PROOF OF IDENTIFIABILITY
+
+This section reproduces a proof from Khemakhem et al. (2019) with some modifications to adapt it to the context of invertible neural networks (normalizing flows).
+
+Define the domain of  $f_{\varepsilon}$  as  $\mathcal{Z} = \mathcal{Z}_1 \times \dots \times \mathcal{Z}_n$ . Define the vector of sufficient statistics  $\mathbf{T}(\mathbf{z}) = (T_{1,1}(z_1), \ldots, T_{n,k}(z_n))$  and the vector of their coefficients  $\lambda(\mathfrak{u}) = (\lambda_{1,1}(\mathfrak{u}), \ldots, \lambda_{n,k}(\mathfrak{u}))$ .
+
+Theorem 1 Assume we observe data distributed according to the generative model defined in Sec. 3.1. Further suppose the following:
+
+(i) The sufficient statistics  $T_{i,j}(z)$  are differentiable almost everywhere and their derivatives  $\mathrm{d}T_{i,j}(z) / \mathrm{d}z$  are nonzero almost surely for all  $z \in \mathcal{Z}_i$  and all  $1 \leq i \leq n$  and  $1 \leq j \leq k$ .  
+(ii) There exist  $nk + 1$  distinct conditions  $\mathfrak{u}_0,\ldots ,\mathfrak{u}_{nk}$  such that the matrix
+
+$$
+\boldsymbol {L} = \left(\boldsymbol {\lambda} \left(\mathrm {u} _ {1}\right) - \boldsymbol {\lambda} \left(\mathrm {u} _ {0}\right), \dots , \boldsymbol {\lambda} \left(\mathrm {u} _ {n k}\right) - \boldsymbol {\lambda} \left(\mathrm {u} _ {0}\right)\right) \tag {8}
+$$
+
+of size  $nk \times nk$  is invertible.
+
+Then the sufficient statistics of the generating latent space are related to those of the estimated latent space by the following relationship:
+
+$$
+\mathbf {T} (\mathrm {z}) = \mathbf {A T} ^ {\prime} (\mathrm {w}) + \mathbf {c} \tag {9}
+$$
+
+where  $\mathbf{A}$  is a constant, full-rank  $nk \times dk$  matrix and  $c \in \mathbb{R}^{nk}$  a constant vector.
+
+Proof: The conditional probabilities  $p_{\mathbf{T}, \boldsymbol{\lambda}, f, \varepsilon}(\mathbf{x}|\mathbf{u})$  and  $p_{\mathbf{T}', \boldsymbol{\lambda}', g}(\mathbf{x}|\mathbf{u})$  are assumed to be the same in the limit of infinite data. By expanding these expressions via the change of variables formula and taking the logarithm we find
+
+$$
+\left. \log p _ {\mathbf {T}, \boldsymbol {\lambda}} (\mathrm {z} | \mathrm {u}) + \log p (\varepsilon) + \log \left| \det J _ {f ^ {- 1}} (\mathrm {x}) \right| = \log p _ {\mathbf {T} ^ {\prime}, \boldsymbol {\lambda} ^ {\prime}} (\mathrm {w} | \mathrm {u}) + \log \left| \det J _ {g ^ {- 1}} (\mathrm {x}) \right| \right. \tag {10}
+$$
+
+where  $J$  is the Jacobian matrix. Let  $\mathfrak{u}_0, \ldots, \mathfrak{u}_{nk}$  be the conditions from (ii) above. We can subtract this expression for  $\mathfrak{u}_0$  from the expression for some condition  $\mathfrak{u}_l$ . The Jacobian terms and the term involving  $\varepsilon$  will vanish, since they do not depend on  $\mathbf{u}$ :
+
+$$
+\log p _ {\mathrm {z}} (\mathrm {z} | \mathrm {u} _ {l}) - \log p _ {\mathrm {z}} (\mathrm {z} | \mathrm {u} _ {0}) = \log p _ {\mathrm {w}} (\mathrm {w} | \mathrm {u} _ {l}) - \log p _ {\mathrm {w}} (\mathrm {w} | \mathrm {u} _ {0}). \tag {11}
+$$
+
+Since the conditional probabilities of  $z$  and  $w$  are exponential family members, we can write this expression as
+
+$$
+\begin{array}{l} \sum_ {i = 1} ^ {n} \left[ \log \frac {Z _ {i} (\mathbf {u} _ {0})}{Z _ {i} (\mathbf {u} _ {l})} + \sum_ {j = 1} ^ {k} T _ {i, j} (\mathbf {z}) \left(\lambda_ {i, j} (\mathbf {u} _ {l}) - \lambda_ {i, j} (\mathbf {u} _ {0})\right) \right] \\ = \sum_ {i = 1} ^ {d} \left[ \log \frac {Z _ {i} ^ {\prime} \left(\mathrm {u} _ {0}\right)}{Z _ {i} ^ {\prime} \left(\mathrm {u} _ {l}\right)} + \sum_ {j = 1} ^ {k} T _ {i, j} ^ {\prime} (\mathrm {w}) \left(\lambda_ {i, j} ^ {\prime} \left(\mathrm {u} _ {l}\right) - \lambda_ {i, j} ^ {\prime} \left(\mathrm {u} _ {0}\right)\right) \right] \tag {12} \\ \end{array}
+$$
+
+where the base measures  $Q_{i}$  have cancelled out, since they do not depend on u. Defining  $\bar{\lambda}(\mathfrak{u}) = \lambda(\mathfrak{u}) - \lambda(\mathfrak{u}_0)$  and writing the above in terms of inner products, we find
+
+$$
+\left\langle \mathbf {T} (\mathbf {z}), \bar {\boldsymbol {\lambda}} \right\rangle + \sum_ {i} \log \frac {Z _ {i} \left(\mathbf {u} _ {0}\right)}{Z _ {i} \left(\mathbf {u} _ {l}\right)} = \left\langle \mathbf {T} ^ {\prime} (\mathbf {w}), \bar {\boldsymbol {\lambda}} ^ {\prime} \right\rangle + \sum_ {i} \log \frac {Z _ {i} ^ {\prime} \left(\mathbf {u} _ {0}\right)}{Z _ {i} ^ {\prime} \left(\mathbf {u} _ {l}\right)} \tag {13}
+$$
+
+for  $1 \leq l \leq nk$ . Combining the  $nk$  expressions into a single matrix equation we can write this in terms of  $\pmb{L}$  from condition (ii) above and an analogously defined  $\pmb{L}^{\prime} \in \mathbb{R}^{dk \times nk}$ :
+
+$$
+\boldsymbol {L} ^ {T} \mathbf {T} (\mathrm {z}) = \boldsymbol {L} ^ {\prime T} \mathbf {T} ^ {\prime} (\mathrm {w}) + \boldsymbol {b} \tag {14}
+$$
+
+where  $b_{l} = \sum_{i}\log \frac{Z_{i}^{\prime}(\mathrm{u}_{0})Z_{i}(\mathrm{u}_{l})}{Z_{i}(\mathrm{u}_{0})Z_{i}^{\prime}(\mathrm{u}_{l})}$ . Since  $L^{T}$  is invertible, we can multiply this expression by its inverse from the left to get
+
+$$
+\mathbf {T} (\mathrm {z}) = \mathbf {A T} ^ {\prime} (\mathrm {w}) + \mathbf {c} \tag {15}
+$$
+
+where  $\mathbf{A} = \mathbf{L}^{-T}\mathbf{L}^{\prime T}$  and  $\pmb {c} = \pmb{L}^{-T}\pmb{b}$ . It now remains to show that  $\mathbf{A}$  has full rank.
+
+According to a lemma from Khemakhem et al. (2019), there exist  $k$  distinct values  $z_{i}^{1}$  to  $z_{i}^{k}$  such that  $\left(\frac{\mathrm{d}T_i}{\mathrm{d}z_i}(z_i^1),\dots ,\frac{\mathrm{d}T_i}{\mathrm{d}z_i}(z_i^k)\right)$  are linearly independent in  $\mathbb{R}^k$ , for all  $1\leq i\leq n$ . Define  $k$  vectors  $\mathbf{Z}^l = (z_1^l,\ldots ,z_n^l)$  from the points given by this lemma. Take the derivative of equation (15) evaluated at each of these  $k$  vectors and concatenate the resulting Jacobians as  $\pmb {Q} = [J_{\mathbf{T}}(\mathbf{z}^{1}),\dots ,J_{\mathbf{T}}(\mathbf{z}^{k})]$ . Each Jacobian has size  $nk\times n$ , hence  $\pmb{Q}$  has size  $nk\times nk$  and is invertible by the lemma and the fact that each component of  $\mathbf{T}$  is univariate. We can construct a corresponding matrix  $\pmb{Q}'$  made up of the Jacobians of  $\mathbf{T}'(g^{-1}\circ f_{\varepsilon}(\mathbf{z}))$  evaluated at the same points and write
+
+$$
+Q = A Q ^ {\prime}. \tag {16}
+$$
+
+Since  $Q$  is full rank, both  $\mathbf{A}$  and  $Q^{\prime}$  must also be full rank.
+
+# A.1 REMARK REGARDING CONDITION (II)
+
+If the coefficients of the sufficient statistics  $\lambda$  are generated randomly and independently, then condition (ii) is almost surely fulfilled. In this case, we can ignore the dependence on  $u$  to consider the  $\lambda(u_{l})$  as independent random variables. Then condition (ii) states that instances of these random variables are in general position in  $\mathbb{R}^{nk}$ , which is true almost surely.
+
+# A.2 REMARK REGARDING NOISE VARIABLES IN THE ESTIMATED LATENT SPACE
+
+Any variables in the estimated latent space whose distribution does not depend on the conditioning variable  $u$  is considered as encoding only noise. Such variables will be cancelled out in equation (12) and the corresponding column of  $L'^T$  will contain only zeros. This means the corresponding column of  $A$  will also contain only zeros, so any variation in such a noise variable has no effect on  $z$ . The reverse is also true: if a column of  $A$  contains only zeros, the corresponding variable in the estimated latent space does not depend on  $u$  and must encode only noise.
+
+# B SPARSITY IN UNMIXING MATRIX: GAUSSIAN DISTRIBUTION
+
+Suppose samples  $\mathbf{z}$  from the generating latent space follow a conditional Gaussian distribution. Suppose the estimating model  $g$  faithfully reproduces the observed conditional density  $p(\mathrm{x}|\mathrm{u})$  and samples  $\mathbf{w}$  from its latent space also follow a conditional Gaussian distribution. Then we can apply equation (4) to relate the generating and estimating latent spaces.
+
+The sufficient statistics of a normal distribution with free mean and variance are  $z$  and  $z^2$ . Hence the relationship between the latent spaces becomes
+
+$$
+\binom {z} {z ^ {2}} = A \binom {w} {w ^ {2}} + c \tag {17}
+$$
+
+where the squaring is applied element-wise. We can write  $A$  in block matrix form as
+
+$$
+\boldsymbol {A} = \left( \begin{array}{l l} \boldsymbol {A} ^ {(1)} & \boldsymbol {A} ^ {(2)} \\ \boldsymbol {A} ^ {(3)} & \boldsymbol {A} ^ {(4)} \end{array} \right) \tag {18}
+$$
+
+and  $c$  as
+
+$$
+\boldsymbol {c} = \left( \begin{array}{l} \boldsymbol {c} ^ {(1)} \\ \boldsymbol {c} ^ {(2)} \end{array} \right) \tag {19}
+$$
+
+Then:
+
+$$
+\mathbf {z} = \boldsymbol {A} ^ {(1)} \mathbf {w} + \boldsymbol {A} ^ {(2)} \mathbf {w} ^ {2} + \boldsymbol {c} ^ {(1)} \tag {20}
+$$
+
+$$
+\mathrm {z} ^ {2} = \boldsymbol {A} ^ {(3)} \mathrm {w} + \boldsymbol {A} ^ {(4)} \mathrm {w} ^ {2} + \boldsymbol {c} ^ {(2)} \tag {21}
+$$
+
+so we can write for each dimension  $i$  of  $\mathbf{z}$
+
+$$
+z _ {i} = \sum_ {j} A _ {i j} ^ {(1)} w _ {j} + \sum_ {j} A _ {i j} ^ {(2)} w _ {j} ^ {2} + c _ {i} ^ {(1)} \tag {22}
+$$
+
+$$
+z _ {i} ^ {2} = \sum_ {j} A _ {i j} ^ {(3)} w _ {j} + \sum_ {j} A _ {i j} ^ {(4)} w _ {j} ^ {2} + c _ {i} ^ {(2)} \tag {23}
+$$
+
+In order to compare the equations, we need to square (22). To do so, we will have to square the second term on the right hand side, involving  $w_{j}^{2}$ . There is no matching term in (23), so we have to set all entries of  $A^{(2)}$  to zero. In more detail:
+
+$$
+\begin{array}{l} \left(\sum_ {j} A _ {i j} ^ {(2)} w _ {j} ^ {2}\right) ^ {2} = \sum_ {j} \sum_ {j ^ {\prime}} A _ {i j} ^ {(2)} A _ {i j ^ {\prime}} ^ {(2)} w _ {j} ^ {2} w _ {j ^ {\prime}} ^ {2} (24) \\ = \sum_ {j} \left(A _ {i j} ^ {(2)}\right) ^ {2} w _ {j} ^ {4} + \sum_ {j \neq j ^ {\prime}} A _ {i j} ^ {(2)} A _ {i j ^ {\prime}} ^ {(2)} w _ {j} ^ {2} w _ {j ^ {\prime}} ^ {2} (25) \\ \end{array}
+$$
+
+The first term with  $w_{j}^{4}$  matches no term in (23), so we have to set  $A_{ij}^{(2)} = 0$  for all  $i$  and  $j$ . This simplifies the earlier equation:
+
+$$
+z _ {i} = \sum_ {j} A _ {i j} ^ {(1)} w _ {j} + c _ {i} ^ {(1)} \tag {26}
+$$
+
+The square of the first term on the right hand side involves terms with  $w_{j}w_{j^{\prime}}$  cross terms:
+
+$$
+\left(\sum_ {j} A _ {i j} ^ {(1)} w _ {j}\right) ^ {2} = \sum_ {j} A _ {i j} ^ {(1)} w _ {j} ^ {2} + \sum_ {j \neq j ^ {\prime}} A _ {i j} ^ {(1)} A _ {i j ^ {\prime}} ^ {(1)} w _ {j} w _ {j ^ {\prime}} \tag {27}
+$$
+
+so we have to set  $A_{ij}^{(1)}A_{ij'}^{(1)} = 0$  for all  $j \neq j'$ . This means that the  $i$ -th row of  $A^{(1)}$  can have at most one nonzero entry. It must also have at least one nonzero entry, since if the row were all zero, a row of  $A$  would be all zero (since  $A^{(2)} = 0$ ), but  $A$  has full rank. Since there are as many or fewer rows than columns ( $n \leq d$ ), each row of  $A$  is linearly independent, so it is not possible for one to be zero. Hence each row of  $A^{(1)}$  has exactly one nonzero entry. Moreover, no two rows of  $A^{(1)}$  have their nonzero entries in the same column. If they did, the two rows would not be linearly independent, but they must be since  $A$  has full rank. Therefore we can write
+
+$$
+z _ {i} = a _ {i} w _ {j} + b _ {i} \tag {28}
+$$
+
+where  $a_{i} = A_{ij}^{(1)}$  and  $b_{i} = c_{i}^{(1)}$ . That is, the generating latent variable  $z_{i}$  is linearly related to some latent variable of the estimating model  $w_{j}$ . This estimated latent variable is uniquely associated with  $z_{i}$  and any estimated latent variables not associated with a generating latent variable  $z_{i}$  (in the case  $d > n$ ) encode no information about the generating latent space. So the model has decoded the original latent variables  $z$  up to an affine transformation and permutation as a subset of variables in its estimated latent space and has encoded no information (only noise) into the remaining latent variables.
+
+# C SPARSITY IN UNMIXING MATRIX: TWO-PARAMETER EXPONENTIAL FAMILY MEMBERS
+
+The results of Appendix B can be extended to other members of the exponential family with 2 parameters. In the general case, writing  $T_{i,1} = T_1$  and  $T_{i,2} = T_2$  for all  $i$ , equations (22) and (23) become
+
+$$
+T _ {1} \left(z _ {i}\right) = \sum_ {j} A _ {i j} ^ {(1)} T _ {1} \left(w _ {j}\right) + \sum_ {j} A _ {i j} ^ {(2)} T _ {2} \left(w _ {j}\right) + c _ {i} ^ {(1)} \tag {29}
+$$
+
+$$
+T _ {2} \left(z _ {i}\right) = \sum_ {j} A _ {i j} ^ {(3)} T _ {1} \left(w _ {j}\right) + \sum_ {j} A _ {i j} ^ {(4)} T _ {2} \left(w _ {j}\right) + c _ {i} ^ {(2)} \tag {30}
+$$
+
+which, with the definition  $t = T_2 \circ T_1^{-1}$  becomes
+
+$$
+T _ {1} \left(z _ {i}\right) = \sum_ {j} A _ {i j} ^ {(1)} T _ {1} \left(w _ {j}\right) + \sum_ {j} A _ {i j} ^ {(2)} t \left(T _ {1} \left(w _ {j}\right)\right) + c _ {i} ^ {(1)} \tag {31}
+$$
+
+$$
+t \left(T _ {1} \left(w _ {j}\right)\right) = \sum_ {j} A _ {i j} ^ {(3)} T _ {1} \left(w _ {j}\right) + \sum_ {j} A _ {i j} ^ {(4)} t \left(T _ {1} \left(w _ {j}\right)\right) + c _ {i} ^ {(2)}. \tag {32}
+$$
+
+We can combine these equations to get
+
+$$
+t \left(\sum_ {j} A _ {i j} ^ {(1)} T _ {1} \left(w _ {j}\right) + \sum_ {j} A _ {i j} ^ {(2)} t \left(T _ {1} \left(w _ {j}\right)\right) + c _ {i} ^ {(1)}\right) = \sum_ {j} A _ {i j} ^ {(3)} T _ {1} \left(w _ {j}\right) + \sum_ {j} A _ {i j} ^ {(4)} t \left(T _ {1} \left(w _ {j}\right)\right) + c _ {i} ^ {(2)}. \tag {33}
+$$
+
+This equation has many summed terms on the right. Suppose that  $t$  has a convergent Taylor expansion in some region of its domain. We can take this expansion of the term on the left and compare coefficients. Since  $t$  cannot be linear, there will be terms in the expansion of order two or higher. As in the Gaussian case, these polynomial terms create cross terms which are impossible to reconcile with those on the right hand side of the equation. The only consistent solutions can be found by setting all coefficients except those of functions of  $w_{j}$  (for some  $j$ ) to zero:
+
+$$
+t \left(A _ {i j} ^ {(1)} T _ {1} \left(w _ {j}\right) + A _ {i j} ^ {(2)} t \left(T _ {1} \left(w _ {j}\right)\right) + c _ {i} ^ {(1)}\right) = A _ {i j} ^ {(3)} T _ {1} \left(w _ {j}\right) + A _ {i j} ^ {(4)} t \left(T _ {1} \left(w _ {j}\right)\right) + c _ {i} ^ {(2)} \tag {34}
+$$
+
+We can further simplify this equation by examining higher-order terms, but need to know the form of  $t$  to do so. In any case, we can now write equation (29) as
+
+$$
+T _ {1} \left(z _ {i}\right) = A _ {i j} ^ {(1)} T _ {1} \left(w _ {j}\right) + A _ {i j} ^ {(2)} t \left(T _ {1} \left(w _ {j}\right)\right) + c _ {i} ^ {(1)} \tag {35}
+$$
+
+showing that each generating latent variable  $z_{i}$  is related to exactly one estimated latent variable  $w_{j}$ . As in the Gaussian case, we can use the full rank property of  $A$  to see that each estimated latent variable is associated with at most one generating latent variable, and any estimated latent variables not associated with any generating latent variables must encode only noise. The task now is to check the form of  $t$  for each two-parameter member of the exponential family, to see what further constraints we can derive from equation (34). The results are stated in Table 1.
+
+Table 1: Two-parameter exponential family members and selected properties.  
+
+<table><tr><td>Distribution</td><td>Sufficient Statistics</td><td>t(x)</td><td>Latent Space Relationship</td></tr><tr><td>Normal</td><td>(z, z2)</td><td>x2</td><td>zi = awj + c</td></tr><tr><td>Lognormal</td><td>(logz, (logz)2)</td><td>x2</td><td>log zi = a logwj + c</td></tr><tr><td>Inverse Gaussian</td><td>(z, 1/z)</td><td>1/x</td><td>zi = awj or zi = awj</td></tr><tr><td>Gamma</td><td>(logz, z)</td><td>exp(x)</td><td>zi = awj</td></tr><tr><td>Inverse Gamma</td><td>(logz, 1/z)</td><td>exp(-x)</td><td>zi = awj</td></tr><tr><td>Beta</td><td>(logz, log(1 - z))</td><td>log(1 - exp(x))</td><td>log zi = logwj or logzi = log(1 -wj)</td></tr></table>
+
+# D NETWORK ARCHITECTURE
+
+The estimating model  $g$  is built in the reverse direction for practical purposes, so the models described here are  $g^{-1}$  which maps from the data space to the latent space. The type and number of coupling blocks for the different experiments are shown below. The affine coupling function is the concatenation of the scale function  $s$  and the translation function  $t$ , computed together for efficiency, as in Kingma & Dhariwal (2018). It is implemented as either a fully connected network (MLP) or convolutional network, with the specified layer widths and a ReLU activation after all but the final layers. For the convolutional coupling blocks, the splits are along the channel dimension. The scale function  $s$  is passed through a clamping function  $2\tanh(s)$ , which limits the output to the range (-2, 2), as in Ardizzone et al. (2019). Two affine coupling functions are applied per block, as described in Dinh et al. (2016). Downsampling increases the number of channels by a factor of 4 and decreases the image width and height by a factor of 2, done in a checkerboard-like manner, as described in Jacobsen et al. (2018). The dimensions are permuted in a random but fixed way before application of each fully connected coupling block and likewise the channels for the convolutional coupling blocks. The network for the artificial data experiments has 4,480 learnable parameters and the network for the EMNIST experiments has 2,620,192 learnable parameters.
+
+Table 2: Network architecture for artificial data experiments  
+
+<table><tr><td>Type of block</td><td>Number</td><td>Input shape</td><td>Affine coupling function layer widths</td></tr><tr><td>Fully Connected Coupling</td><td>8</td><td>10</td><td>5 → 10 → 10 → 10</td></tr></table>
+
+Table 3: Network architecture for EMNIST experiments  
+
+<table><tr><td>Type of block</td><td>Number</td><td>Input shape</td><td>Affine coupling function layer widths</td></tr><tr><td>Downsampling</td><td>1</td><td>(1, 28, 28)</td><td></td></tr><tr><td>Convolutional Coupling</td><td>4</td><td>(4, 14, 14)</td><td>2 → 16 → 16 → 4</td></tr><tr><td>Downsampling</td><td>1</td><td>(4, 14, 14)</td><td></td></tr><tr><td>Convolutional Coupling</td><td>4</td><td>(16, 7, 7)</td><td>8 → 32 → 32 → 16</td></tr><tr><td>Flattening</td><td>1</td><td>(16, 7, 7)</td><td></td></tr><tr><td>Fully Connected Coupling</td><td>2</td><td>784</td><td>392 → 392 → 392 → 784</td></tr></table>
+
+# D.1 NOTE ON OPTIMIZATION METHOD
+
+In the experiments described in this paper, the mean and variance of a mixture component was updated at each iteration as the mean and variance of the transformations to latent space of all data points belonging to that mixture component in a minibatch of data  $\mathcal{D}$ :
+
+$$
+\mu_ {i} \left(\mathbf {u} ^ {\prime}; \boldsymbol {\theta}\right) \leftarrow \mathbb {E} _ {\mathcal {D}: \mathrm {u} = \mathrm {u} ^ {\prime}} \left(g _ {i} ^ {- 1} (\mathrm {x}; \boldsymbol {\theta})\right) \tag {36}
+$$
+
+$$
+\sigma_ {i} ^ {2} \left(\mathbf {u} ^ {\prime}; \boldsymbol {\theta}\right) \leftarrow \operatorname {V a r} _ {\mathcal {D}: \mathrm {u} = \mathrm {u} ^ {\prime}} \left(g _ {i} ^ {- 1} (\mathrm {x}; \boldsymbol {\theta})\right). \tag {37}
+$$
+
+Hence the parameters of the mixture components would change in each batch, according to the data present. The notation  $\mu_{i}(\mathbf{u};\pmb{\theta})$  does not indicate that  $\mu$  is directly parameterized by  $\pmb{\theta}$  and learned, instead it indicates that  $\mu$  is a function of  $g^{-1}$  which is parameterized by  $\pmb{\theta}$ . A change in  $\pmb{\theta}$  will also change the output of  $\mu$ , given the same mini-batch of data  $\mathcal{D}$ . The same holds for  $\sigma_{i}(\mathbf{u};\pmb{\theta})$ .
+
+We expect that specifying the means and variances as learnable parameters updated in tandem with the model weights would have worked equally well.
+
+# E FIGURES FROM THE ARTIFICIAL DATA EXPERIMENTS
+
+![](images/c96a514831171ad3ddf5b5e195281ba220e82fae3d3a206f6950931384e127c5.jpg)  
+Figure 6: Experiment 1: Five mixture components and 100,000 data points. GIN successfully reconstructs the generating latent space and gives importance to only two of its ten latent variables, reflecting the two-dimensional nature of the generating latent space.
+
+![](images/6b283bc4ad17ab22d82c69f16d101613692339c4b28472673f24938fe764919f.jpg)  
+Figure 7: Same experiment as in Figure 6 with the number of data points reduced to 10,000. GIN fails to successfully estimate the ground truth latent variables, due to limited data: The first variable  $(x$ -axis) is well approximated in the reconstruction, but the second variable  $(y$ -axis) is split into two in the reconstruction, one capturing mainly information about the lower two clusters (shown here) and another information about the other mixture components (not shown). We also observe a less clear spectrum, where three variables are given more importance than the rest, not faithfully reflecting the two-dimensional nature of the generating latent space.
+
+![](images/73da98dba2758c052e5fc4df9293c053f68b5dabb0479b9c95bfec3056867255.jpg)  
+Figure 8: Experiment 2: Only three mixture components (not sufficient for identifiability according to the theory). Nevertheless, GIN successfully reconstructs the ground truth latent variables. This suggests that the current theory of nonlinear ICA relies on sufficient, but not necessary, conditions for identifiability.
+
+# F EMNIST FIGURES
+
+![](images/1d57d945a2d1c036521d1c6b0ac3ab3a4561532a143ffd7dc3116dbc7eca0584.jpg)  
+(a) Full Temperature  
+Figure 9: Full and reduced temperature samples from the model trained on EMNIST. Reduced temperature samples are made by sampling from a Gaussian distribution where the standard deviation is reduced by the temperature factor. The 22 most significant variables are sampled, with the others kept to their mean value. This eliminates noise from the images but preserves the full variability of digit shapes. Each row has the same latent code (whitened value) but is conditioned on a different class in each column, hence the style of the digits is consistent across rows.
+
+![](images/32d798eb4ec70cc027361f9795a437a1f78e0ebabd58c380ce97ef083ffbcb8b.jpg)  
+(b) Reduced Temperature  $(\mathrm{T} = 0.8)$
+
+![](images/3317d875a433cbf6a0df320c1dcfee84ffe345cf3401ee1a091ba306b1015190.jpg)  
+(a) Variable 1: width of top half
+
+![](images/0a5648282686fe15137b9c2602bd7a1bc9aef90ca56108b2aaba69a0ee8b6734.jpg)  
+(b) Variable 2: slant/angle
+
+![](images/ab4790eed2b39c594f81ada3a51a9c16f5e685dbe99dab9c2a3792fa296be526.jpg)  
+(c) Variable 3: height
+
+![](images/e2e902b098ce41fa2d52813c7564b6ed0c52d48be777b276f6968cc1807f8393.jpg)  
+(d) Variable 4: bend through center  
+Figure 10: Most significant latent variables 1 to 4. Each row is conditioned on a different digit label. The variable runs from -2 to +2 standard deviations across the columns, with all other variables held constant at their mean value. The rightmost column shows a heatmap of the image areas most affected by the variable, computed as the absolute pixel difference between -1 and +1 standard deviations.
+
+![](images/78756ada58c53ede3579e2fd589f5d28148ea852d4ad37187822d36998e2842c.jpg)  
+(a) Variable 5: line thickness
+
+![](images/3b45608a90033e669b063b16ab941e1656dfe0c7677cc3d419b450951ea08c21.jpg)  
+(b) Variable 6: slant of vertical bar in 4,7 and 9
+
+![](images/651c10704cdbe5f4a07c1c6783591146343d02af01dea39cd0078e148ade2167.jpg)  
+(c) Variable 7: height of horizontal bar
+
+![](images/02a08f2d77e3d3f0756811c5967cf87623600b577dd6ccf4e16d388b88f075f5.jpg)  
+(d) Variable 8: width of bottom half  
+Figure 11: Most significant latent variables 5 to 8. Each row is conditioned on a different digit label. The variable runs from -2 to +2 standard deviations across the columns, with all other variables held constant at their mean value. The rightmost column shows a heatmap of the image areas most affected by the variable, computed as the absolute pixel difference between -1 and +1 standard deviations.
+
+![](images/44d6d39782721324ec6ed22b81d11fe8d3070ef1c6acdafb56d9e86ed4efc8fe.jpg)  
+(a) Variable 9: extension of center left feature towards the left
+
+![](images/da9dd2ac9eefc300bc9c36c09fc269ff6346edb93e6967108c77a806a385f102.jpg)  
+(b) Variable 10: openness of lower loop
+
+![](images/3d580bf4a6b87efa2946dfe9ad1f6f60e4e27631351282b938ed22cbc57e52f4.jpg)  
+(c) Variable 11: shape of center right feature  
+Figure 12: Most significant latent variables 9 to 12. Each row is conditioned on a different digit label. The variable runs from -2 to +2 standard deviations across the columns, with all other variables held constant at their mean value. The rightmost column shows a heatmap of the image areas most affected by the variable, computed as the absolute pixel difference between -1 and +1 standard deviations.
+
+![](images/56921a077d0a6a63724c10af51d795c7b0896b9433fe330de024d5ded56df3dc.jpg)  
+(d) Variable 12: top right corner
+
+![](images/9d62b6cae9eda8102782601bb4ee470d3406737e672fff38b364fd8639fda89a.jpg)  
+(a) Variable 13: orientation of stroke in top left corner for 2, 3 and 7
+
+![](images/dc306131d38c47934abdfb46e63d5bf80fba56308b5d2af943fea6aaed3b3cab.jpg)  
+(b) Variable 14: shape of top part of bottom loop
+
+![](images/c8a56590628dec5e4a1e712de9cc3c02dbf727770d9232a5339b13f43b7fd1a4.jpg)  
+(c) Variable 15: angle of centrally located horizontal features  
+Figure 13: Most significant latent variables 13 to 16. Each row is conditioned on a different digit label. The variable runs from -2 to +2 standard deviations across the columns, with all other variables held constant at their mean value. The rightmost column shows a heatmap of the image areas most affected by the variable, computed as the absolute pixel difference between -1 and +1 standard deviations.
+
+![](images/f2693668164282993391a83c9e63e308f0c5e24e0379728995f823c4feb8d111.jpg)  
+(d) Variable 16: bottom right stroke of 2
+
+![](images/8983a72b6b6807881dce58197b8f2e85991da2fed0ddafbb12520727d53cef99.jpg)  
+(a) Variable 17: top left stroke of 4
+
+![](images/3b11bdc210eb7777822abddb156935f44ebbaf4e0739305e2df3a2b302383c03.jpg)  
+(b) Variable 18: top stroke of 5 and 7
+
+![](images/d045773ec16b7e36cc9b4858dedeefa237208fdb420572fc5fc697bad7e2fc26.jpg)  
+(c) Variable 19: extension towards top right
+
+![](images/d126ab51f29c4f7ae4f4029d8157c206e7e5b71365306fcc3773dea65e24ecbc.jpg)  
+(d) Variable 20: curvature of vertical stroke of 4  
+Figure 14: Most significant latent variables 17 to 20. Each row is conditioned on a different digit label. The variable runs from -2 to +2 standard deviations across the columns, with all other variables held constant at their mean value. The rightmost column shows a heatmap of the image areas most affected by the variable, computed as the absolute pixel difference between -1 and +1 standard deviations.
+
+![](images/d74241262fa3ee1acd97fe5d6b26d23c3b79c31c884b854686ad569dc9294d68.jpg)  
+(a) Variable 21: thickness of upper loop
+
+![](images/c550dd24dc1943e511660041987c191411f7eaf8493d582ead8cc496a559fc5a.jpg)  
+(b) Variable 22: extension towards top left
+
+![](images/38245bb08f5943a074fd1db8e3ff817d1da198a410561dd12d58f2d498d9dc70.jpg)  
+(c) Variable 23: no effect  
+Figure 15: Most significant latent variables 21 to 24. Each row is conditioned on a different digit label. The variable runs from -2 to +2 standard deviations across the columns, with all other variables held constant at their mean value. The rightmost column shows a heatmap of the image areas most affected by the variable, computed as the absolute pixel difference between -1 and +1 standard deviations.
+
+![](images/add14b0949d2fb2b1715cd38c1469c7b057f4b76261203ef277f227e6288d77f.jpg)  
+(d) Variable 24: no effect
\ No newline at end of file
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+# DISENTANGLING NEURAL MECHANISMS FOR PERCEPTUAL GROUPING
+
+Junkyung Kim†‡, Drew Linsley†, Kalpit Thakkar & Thomas Serre
+
+Department of Cognitive, Linguistic and Psychological Sciences
+
+Brown University
+
+Providence, RI 02912, USA
+
+junkyung@google.com
+
+{drew_linsley, kalpit_thakkar, thomas_serre}@brown.edu
+
+# ABSTRACT
+
+Forming perceptual groups and individuating objects in visual scenes is an essential step towards visual intelligence. This ability is thought to arise in the brain from computations implemented by bottom-up, horizontal, and top-down connections between neurons. However, the relative contributions of these connections to perceptual grouping are poorly understood. We address this question by systematically evaluating neural network architectures featuring combinations bottom-up, horizontal, and top-down connections on two synthetic visual tasks, which stress low-level "Gestalt" vs. high-level object cues for perceptual grouping. We show that increasing the difficulty of either task strains learning for networks that rely solely on bottom-up connections. Horizontal connections resolve straining on tasks with Gestalt cues by supporting incremental grouping, whereas top-down connections rescue learning on tasks with high-level object cues by modifying coarse predictions about the position of the target object. Our findings dissociate the computational roles of bottom-up, horizontal and top-down connectivity, and demonstrate how a model featuring all of these interactions can more flexibly learn to form perceptual groups.
+
+# 1 INTRODUCTION
+
+The ability to form perceptual groups and segment scenes into a discrete set of objects constitutes a fundamental component of visual intelligence. Decades of research in biological vision have suggested that biological visual systems extract these objects through visual routines for perceptual grouping, which rely on feedforward (or bottom-up) and potentially feedback (or recurrent) processes (Roelfsema, 2006; Roelfsema & Houtkamp, 2011; Wyatte et al., 2014). Feedforward processes group scenes by encoding increasingly more complex feature conjunctions through a cascade of filtering, rectification and normalization operations. As shown in Fig. 1a, these computations can be sufficient to detect and localize objects in scenes with little or no background clutter, or when an object "pops out" because of color, contrast, etc. (Nothdurft, 1991). However, as illustrated in Fig. 1b, visual scenes are usually complex and contain objects interposed in background clutter. When this is the case, feedforward processes alone are often insufficient for perceptual grouping (Herzog & Clarke, 2014; Pelli et al., 2004; Freeman et al., 2012; Freeman & Pelli, 2010), and it has been suggested that our visual system leverages feedback mechanisms to refine an initially coarse scene segmentation (Ullman, 1984; Roelfsema & Houtkamp, 2011; Treisman & Gelade, 1980; Lamme & Roelfsema, 2000).
+
+To rough approximation, there are two distinct types of perceptual grouping routines that are associated with feedback computations. One routine involves grouping low-level visual features with their neighbors according to Gestalt laws like similarity, good continuation, etc. (Fig. 1b, top; Jolicoeur et al., 1986; 1991; Pringle & Egeth, 1988; Roelfsema et al., 1999; Houtkamp & Roelfsema, 2010; Houtkamp et al., 2003; Roelfsema et al., 2002). Another routine is object-based and mediated by
+
+![](images/1790dbda87f8446c77fc3d9b21821c736b6246db090808ce20547c6dba90d7ca.jpg)  
+Easy
+
+![](images/d66c96f2b81fc6ef7080717186b13673452458355fa40bea8d204b6178a658e5.jpg)  
+(a)
+
+![](images/dfcfed2ab5162649aaf1e206eb84dd6a4d56cd49413f5ab7cff75a38787d5cf2.jpg)  
+Hard  
+Low-level / "Gestalt"
+
+![](images/d2ecfa87a028705a2555e71b76ceea6b7c97fe1bba48a80cfb442b6a5db75e49.jpg)  
+Object-level / Semantics  
+(b)  
+Figure 1: "Are both dots on the same object?" (a) Bottom-up grouping. Our visual system can segregate an object from its background with rapid, bottom-up mechanisms, when the object is dissimilar to its background. In this case, the object is said to "pop-out" from its background, making it trivial to judge whether the two dots are on the same object surface or not. (b) Non-bottom-up processes. Perceptual grouping mechanisms help segment a visual scene. (b, Top) The elements of a behaviorally relevant perceptual object, such as a path formed by tracks in the snow, are transitively grouped together according to low-level "Gestalt" principles. This allows the observer to trace a path from one end to the other. (b, Bottom) Alternatively, observers may rely on prior knowledge or semantic cues to segment an object from a cluttered background. (c) Synthetic grouping tasks. (c, Top) The Pathfinder challenge (reproduced with permission from Linsley et al. (2018b) and inspired by Houtkamp & Roelfsema 2010) involves answering a simple question: "Are the two dots connected by a path?" This task can be easily solved with Gestalt grouping strategies. (c, Bottom) Here, we introduce a novel "cluttered ABC" (cABC) challenge, inspired by Vecera & Farah (1997). The cABC challenge asks the same question on visual stimuli for which Gestalt strategies are ineffective. Instead, cABC taps into object-based strategies for perceptual grouping.
+
+![](images/8f674defa2965e00f844d6af1ef985026ed034afa3d45d1ba1e02bd6e220a10e.jpg)  
+Biological vs. artificial vision
+
+![](images/9d9b15b9bd09ccc3d97a78b3f849c2d897fa752f85a166c77125c9439e308422.jpg)
+
+![](images/8d610107f4765deba5121b8e8d58edf07f89084416257297706d51c2643f74a1.jpg)  
+Vecera & Farah (1997)  
+(c)
+
+![](images/7250ec3b27e51413503a39257bea7767de46251be49817345949b3c929ee31aa.jpg)  
+Kim*, Linsley*, & Serre (2019)
+
+high-level expectations about the shape and structure of perceptual objects. In this routine, feedback refines a coarse initial feedforward analysis of a scene with high-level hypotheses about the objects it contains (Fig. 1b, bottom; Vecera & Farah, 1997; Vecera & O'Reilly, 1998; Vecera, 1993; Zemel et al., 2002). Both of these feedback routines are iterative and rely on recurrent computations.
+
+What are the neural circuits that implement Gestalt vs. object-based routines for perceptual grouping? Visual neuroscience studies have suggested that these routines emerge from specific types of neural interactions: (i) horizontal connections between neurons within an area, spanning spatial locations and potentially feature selectivity (Stettler et al., 2002; Gilbert & Wiesel, 1989; McManus et al., 2011; Bosking et al., 1997; Schmidt et al., 1997; Wannig et al., 2011), and (ii) descending top-down connections from neurons in higher-to-lower areas (Ko & von der Heydt, 2018; Gilbert & Li, 2013; Tang et al., 2018; Lamme et al., 1998; Murray et al., 2004; 2002). The anatomical and functional properties of these feedback connections have been well-documented (see Gilbert & Li 2013 for a review), but the relative contributions of horizontal vs. top-down connections for perceptual grouping remains an open question.
+
+Contributions Here we investigate the function of horizontal vs. top-down connections during perceptual grouping. We evaluate each network on two perceptual grouping "challenges" that are designed to be solved by either Gestalt or object-based visual routines. Our contributions are:
+
+- We introduce a biologically-inspired deep recurrent neural network (RNN) architecture featuring feedforward and feedback connections (the TD+H-CNN). We develop a range of lesioned variants of this network for measuring the relative contributions of feedforward
+
+and feedback processing by selectively disabling either horizontal (TD-CNN), top-down (H-CNN), or both types of connections (BU-CNN).
+
+- We introduce the cluttered ABC challenge (cABC), a synthetic visual reasoning challenge that requires object-based grouping strategies (Fig. 1c, bottom-right). We pair this dataset with the Pathfinder challenge (Linsley et al. 2018b, Fig. 1c, top-right) which features low-level Gestalt cues. Both challenges allow for a parametric control over image variability. With these challenges, we investigate whether each of our models can learn to leverage either object-level of Gestalt cues for grouping.  
+- We find that top-down connections are key for grouping objects when semantic cues are present, whereas horizontal connections support grouping when they are not. Of the network models tested (which included ResNets and U-Nets), only our recurrent model with the full array of connections (the TD+H-CNN) solves both tasks across levels of difficulty.  
+- We compare model predictions to human psychophysics data on the same visual tasks and show that human judgements are significantly more consistent with image predictions from our TD+H-CNN than ResNet and U-Net models. This indicates that the strategies used by the human visual system on difficult segmentation tasks are best matched by a highly-recurrent model with bottom-up, horizontal, and top-down connections.
+
+# 2 RELATED WORK
+
+Recurrent neural networks Recurrent neural networks (RNNs) are a class of models that can be trained with gradient descent to approximate discrete-time dynamical systems. RNNs are classically featured in sequence learning, but have also begun to show promise in computer vision tasks. One example of this includes autoregressive RNNs that learn the statistical relationship between neighboring pixels for image generation (Van Den Oord et al., 2016). Another approach, successfully applied to object recognition and super-resolution tasks, is to incorporate convolutional kernels into RNNs (O'Reilly et al., 2013; Liang & Hu, 2015; Liao & Poggio, 2016; Kim et al., 2016; Zamir et al., 2017). These convolutional RNNs share kernels across processing timesteps, allowing them to achieve processing depth with a fraction of the parameters needed for a CNN of equivalent depth. Such convolutional RNN models have also been used by multiple groups as a foundation for vision models with biologically-inspired feedback mechanisms (Lotter et al., 2016; Spoerer et al., 2017; Kietzmann et al., 2019; Wen et al., 2018; Linsley et al., 2018b; Nayebi et al., 2018a; Kar et al., 2019).
+
+Here, we construct convolutional-RNN architectures using the feedback gated recurrent unit (fGRU) proposed by Linsley et al. (2018a). The fGRU extends the horizontal gated recurrent unit (hGRU, Linsley et al. 2018b), to implement (i) horizontal connections between units in a processing layer separated by spatial location and/or feature channel, and (ii) top-down connections from units in higher-to-lower network layers. The fGRU was applied to contour detection in natural and electron microscopy images, where it was found to be more sample efficient than state-of-the-art models (Linsley et al., 2020). We leverage a modified version of this architecture to study the relative contributions of horizontal and top-down interactions for perceptual grouping.
+
+Synthetic visual tasks There is a long history of using synthetic visual recognition challenges for evaluating computer vision algorithms (Ullman, 1996; Fleuret et al., 2011). There is a popular dataset of simple geometric shapes used to evaluate instance segmentation algorithms (featured in Reichert & Serre 2014; Greff et al. 2016), and variants of the cluttered MNIST dataset and CAPTCHA datasets have been helpful for testing model tolerance to visual clutter (e.g., Gregor et al. 2015; Sabbah et al. 2017; Spoerer et al. 2017; George et al. 2017a; Jaderberg et al. 2015; Sabour et al. 2017). Recent synthetic challenges have sought to significantly ramp up task difficulty (Michaelis et al., 2018), and/or to parameterically control intra-class image variability (Kim et al., 2018; Linsley et al., 2018b). The current work follows this trend of developing novel synthetic stimuli that are tightly controlled for low-level biases (unlike e.g., MNIST), and to design parametric controls (e.g., intra-class variability) for task difficulty to try to strain network models.
+
+In the current study, we begin with the "Pathfinder" challenge (Linsley et al., 2018b), which consists of images with two white markers that may – or may not – be connected by a long path (Fig. 2). The challenge illustrated how a shallow network with horizontal connections could recognize a connected path by learning an efficient incremental grouping routine, while feedforward architectures (CNNs)
+
+![](images/944315a7fe030d2557194f7792bd7d9d936f6f97e9eb194cd0477df0716e7dc5.jpg)  
+Figure 2: An overview of the two synthetic visual reasoning challenges used: the " Pathfinder" (top two rows) and the "cluttered ABC" (cABC, bottom two rows). On both tasks, a model must judge whether the two white markers fall on the same or different paths/letters. For both, we parametrically controlled task difficulty by adjusting intra-class image variability in the image dataset. In the Pathfinder challenge, we increased the length of the target curve; in cABC, we decreased the correlation between geometric transformations applied to the two letters while also increasing the relative positional variability between letters.
+
+with orders of magnitude more free parameters struggled to learn the same task. Our novel "cluttered ABC" (cABC) challenge complements the Pathfinder challenge by posing the same task on images that feature semantic objects rather than curves. Thus, cABC may be difficult to solve using the same incremental grouping routine as Pathfinder, which could capture both objects in a single group (Fig. 2).
+
+# 3 SYNTHETIC PERCEPTUAL GROUPING CHALLENGES
+
+Pathfinder and cABC challenges share a key similarity in their stimulus design: white shapes are placed on a black background along with two circular and white "markers" (Fig. 2). These markers are either placed on two different objects or the same object, and the task posed in both datasets is to discriminate between these two alternatives. The two challenges depict different types of objects. Images in the Pathfinder challenge contain two flexible curves, whereas the cABC challenge uses overlapping English-alphabet characters that are transformed in appearance and position.
+
+Local vs. object-level cues for perceptual grouping Differences in the types of shapes used in the two challenges make them ideally suited for different feedback grouping routines. The Pathfinder challenge features smooth and flexible curves, making it well suited for incremental Gestalt-based grouping (Linsley et al., 2018b). Conversely, the English-alphabet characters in the cABC challenge make it a good match for a grouping routine that relies on semantic-level object cues.
+
+The two synthetic challenges are designed to cause a high computational burden in models that rely on sub-optimal grouping routines. For instance, cluttered, flexible curves in the Pathfinder challenge are well-characterized by local continuity but have no set "global shape" or any semantic-level cues. This makes the task difficult for feedforward models like ResNets because the number of fixed templates that they must learn to solve it increases with the number of possible shapes the flexible paths may take. In contrast, letters in cABC images are globally stable but locally degenerate (due to warping and pixelation; see Appendix A for details of its design.), and letters may overlap with each other. Because the strokes of overlapping letters form arbitrary conjunctions, a feedforward network needs to learn a large number of templates to discriminate real from spurious stroke conjunctions.
+
+Overall, we consider these challenges two extremes on a continuum of perceptual grouping cues found in nature. On one hand, Pathfinder curves provide reliable Gestalt cues without any semantic information. On the other hand, cABC letters are semantic but do not have local grouping cues. As we describe in the following sections, human observers easily solve both challenges with minimal training, indicating that biological vision is capable of relying on a single routine for perceptual grouping.
+
+Parameterization of image variability It is difficult to disentangle visual strategies of neural network architecture choices using standard computer vision datasets. For instance, the relative advantages of visual routines implemented by horizontal vs. top-down connections can be washed out by neural networks latching onto trivial dataset biases. More importantly, the typical training regime for deep learning involves big datasets, and in this regime, the ability of neural networks to act as universal function approximators can make inductive biases less important.
+
+We overcome these limitations with our synthetic challenges. Each challenge consists of three limited-size datasets of different task difficulties (easy, intermediate and hard; characterized by increasing levels of intra-class variability). We train and test network architectures on each of the three datasets and measure how accuracy changes when the need for stronger generalization capabilities arises as difficulty increases. This method, called "straining" (Kim et al., 2018; Linsley et al., 2018b), dissociates an architecture's expressiveness from other incidental factors tied to a particular image domain. Architectures with appropriate inductive biases are more likely to generalize as difficulty/intra-class variability increases compared to architectures that rely on low-bias routines that can resemble "rote memorization" (Zhang et al., 2016).
+
+Difficulty of Pathfinder challenge is parameterized by the length of target curves in the images (6-, 9-, or 14-length; Linsley et al. 2018b). cABC challenge difficulty is controlled in two ways: first, by increasing the number of possible relative positional arrangements between the two letters. Second, by decreasing the correlation between random transformation parameters (rotation, scale and shear) applied to each letter in an image. For instance, on the "Easy" difficulty dataset, letter transformations are correlated: the centers of two letters are displaced by a distance uniformly sampled in an interval between 25 and 30 pixels, and the same affine transformation is applied to both letters. In contrast, letter transformations are independent: letters are randomly separated by 15 to 40 pixels, and random affine transformations are independently sampled and applied to each. We balance the variation of these random transformation parameters across the three difficulty levels so that the total variability of individual letter shape remains constant (i.e., the number of transformations applied to each letter in a dataset is the same across difficulty levels). Harder cABC datasets lead to increasingly varied conjunctions between the strokes of overlapping letters, making them well suited for models that can iteratively process semantic groups instead of learning fixed feature templates. See Appendix A for more details about cABC generation.
+
+# 4 NETWORK ARCHITECTURES
+
+We test the role of horizontal vs. top-down connections for perceptual grouping using different configurations of the same recurrent CNN architecture (Fig. 3b). This model is equipped with recurrent modules called fGRUs (Linsley et al., 2018a), which can implement top-down and/or horizontal feedback (Fig. 3b). By selectively disabling top-down connections, horizontal connections, or both types of feedback, we directly compared the relative contributions of these connections for solving Pathfinder and cABC tasks over a basic deep convolutional network "backbone" capable of only feedforward computations. As an additional reference, we test representative feedforward
+
+architectures for computer vision: residual networks (He et al., 2016) and a U-Net (Ronneberger et al., 2015).
+
+The fGRU module The fGRU takes input from two unit populations: an external drive,  $\mathbf{X}$ , and an internal state,  $\mathbf{H}$ . The fGRU integrates these two inputs to get an updated state for a given processing timestep  $\mathbf{H}[t]$ , which is passed to the next layer. Each layer's recurrent state is sequentially updated during a single timestep, letting the model perform bottom-up to top-down sweeps of processing. When applied to static images, timesteps of processing allow feedback connections to spread and evolve, allowing units in different spatial positions and layers to interact with each other. This iterative spread of activity is critical for implementing dynamic grouping strategies. On the final timestep of processing, the fGRU hidden state  $\mathbf{H}$  closest to image resolution is sent to a readout module to compute a binary category prediction.
+
+In our recurrent architecture (the TD+H-CNN), an fGRU module implements either horizontal or top-down feedback. When implementing horizontal feedback, an fGRU takes as external drive  $(\mathbf{X})$  the activities from a preceding convolutional layer and iteratively updates its evolving hidden state  $\mathbf{H}$  using spatial (horizontal) kernels (Fig. 3b, green fGRU).
+
+In an alternative configuration, an fGRU can implement top-down feedback by taking as input hidden state activities from two other fGRUs, one from a lower layer and the other from a high layer (Fig. 3b, red fGRU). This top-down fGRU uses a local  $(1\times 1)$  kernel to first suppress the low-level  $\mathbf{H}^{(\ell)}[t]$  with the activity in  $\mathbf{H}^{(\ell +1)}[t]$ , then a separate kernel to "facilitate" the residual activity, supporting operations like sharpening or "filling-in". Additional details regarding the fGRU module are in Appendix C.
+
+![](images/4abece1adfaad52ca8a1647a44302a0cd466fe6e30bbd9e8a89c9eee912cab53.jpg)  
+Figure 3: We define a convolutional RNN model that can learn feedforward (gray), horizontal (green), and/or top-down (red) connections to disentangle the contributions of each type of connections for perceptual grouping. (a) A conceptual diagram of the TD+H-CNN architecture, depicting three layers of activity and feedforward, horizontal, and top-down connections. (b) A deep learning diagram of the TD+H-CNN, unraveled over processing timesteps. Bottom-up to top-down passes of the input image allow recurrent horizontal (green fGRUs) and top-down (red fGRU) interactions between units to evolve. This allows the model to implement complex grouping strategies, like transitively linking target features together or selecting whole objects from clutter.
+
+Recurrent networks with feedback Our full model architecture, which we call TD+H-CNN (Fig. 3b), introduces three fGRU modules into a CNN to implement top-down (TD) and horizontal (H) feedback. The architecture consists of a downsampling (Conv-pool) pathway and an upsampling (Transpose Conv) pathway as in the U-Net (Ronneberger et al., 2015), and processes images in a bottom-up to top-down loop through its entire architecture at every timestep. The downsampling pathway lets the model implement feedforward and horizontal connections, and the upsampling pathway lets it implement top-down interactions (conceptualized in Fig. 3b, red fGRU). The first two fGRUs in the model learn horizontal connections at the lowest- and highest-level feature processing layers, whereas the third fGRU learns top-down interactions between these two fGRUs.
+
+We define three variants of the TD+H-CNN by lesioning fGRU modules for horizontal and/or top-down connections: a top-down-only architecture (TD-CNN); a horizontal-only architecture (H-CNN); and a feedforward-only architecture ("bottom-up" BU-CNN; see Appendix C for details).
+
+![](images/c08fdc53af7c04d29f44c229c61fb360690491b78a7f6d9da53cfdaaef94568f.jpg)  
+Figure 4: Pathfinder and cABC challenges disentangle the relative contributions of horizontal vs. top-down connections for perceptual grouping. (a) Accuracies of human and neural network models on three levels of the Pathfinder and cABC challenges. While human participants and the TD+H-CNN model are able to solve all difficulties of each challenge, Pathfinder strains the TD-CNN and cABC strains the H-CNN. (b) We visualized the computations learned by our networks using images containing single markers (See Appendix B for details.) We found that H-CNN and TD-CNN learn distinct strategies to solve Pathfinder and cABC challenges, respectively. The H-CNN solves Pathfinder by iteratively grouping segments of a target path while suppressing background clutter. The TD-CNN solves cABC by globally facilitating activity of a target letter while suppressing the other letter. Model strategies are visualized by plotting the per-timestep L2 difference between hidden states of the fGRU responsible for model predictions.
+
+Reference networks We also measure the performance of popular neural network architectures on Pathfinder and cABC. We tested three "Residual Network" (ResNets, He et al. 2016) with 18, 50, and 152 layers and a "U-Net" (Ronneberger et al., 2015) which uses a VGG16 encoder. Both model types utilize skip connections to mix information between processing layers. However, the U-Net uses skip connections to pass activities between layers of a downsampling encoder and an upsampling decoder. This pattern of residual connections effectively makes the U-Net equivalent to a network with top-down feedback that is simulated for one timestep.
+
+# 5 EXPERIMENTS
+
+Training In initial pilot experiments, we calibrated the difficulty of Pathfinder and cABC challenges by identifying the smallest dataset size of the "easy" version of the challenges for which all models exceeded  $90\%$  validation accuracy. For Pathfinder this was 60K images, and for cABC this was 45K, and we used these image budgets for all difficulty levels of the respective challenges (with  $10\%$  held out for evaluation).
+
+Our objective was to understand whether models could learn these challenges on their image budgets. We were interested in evaluating model sample efficiency, not absolute accuracy on a single dataset nor speed of training ("wall-time"). Thus, we adopted early stopping criteria during training: training continued until 10 straight epochs had passed without a reduction in validation loss or if 48 epochs of training had elapsed. Models were trained with batches of 32 images and the Adam optimizer. We trained each model with five random weight initializations and searched over learning rates  $[1e^{-3}, 1e^{-4}, 1e^{-5}, 1e^{-6}]$ . We report the best performance achieved by each model (according to validation accuracy) out of the 20 training runs.
+
+Screening feedback computations Our study revealed three main findings. First, we found that human participants performed well on both Pathfinder and cABC challenges with no significant
+
+![](images/2aef443dff4b7f1ab5c74be362ee7ee84fbf69e6d457e1a54b05b7cb94839872.jpg)
+
+![](images/893f6246afc4ba276a731bb918d158f7f158aa407c7b9ef7c377ea5630a41f84.jpg)
+
+![](images/2baa560e5eec75ef6f477f1f88733a4add9356c5b2b85fecb3e5b4649b200110.jpg)
+
+![](images/4f739e2af967ad827e7f9f51a9c6d0d09709da19ff6136421fd8fdce028d2585.jpg)  
+(a)
+
+![](images/7a3ed42ad023f1caf2bf48547ad90d43156d447289db9c25b50601b7cef9e34f.jpg)  
+Figure 5: Pathfinder and cABC challenges are efficiently solved by visual strategies that depend on feedback – not feedforward – processing. (a) Performance of feedforward models on each difficulty of the two challenges. Deeper ResNet models solved Pathfinder, whereas the ResNet-18 and U-Net solved cABC, indicating a trade-off between capacity vs. parameter efficiency for feedforward strategies to Pathfinder vs. cABC, respectively (failures to generalize reflect overfitting). (b) The number of parameters of each model (plotted as a multiple of the number of parameters in the H-CNN) plotted against model performance on the hard dataset of each challenge. The  $\mathrm{H + TD - CNN}$  solves both challenges with an order-of-magnitude fewer parameters than reference ResNet and U-Net models. (c) Correlations between human and leading model decisions on the hard dataset of each challenge. The  $\mathrm{TD + H - CNN}$  and its successful lesioned variants were significantly more correlated with humans than successful feedforward reference models. Significance comparisons with the worst feedforward reference model omitted due to space constraints. P-values are derived from bootstrapped comparisons of model performance (Edgington, 1964). Error bars depict SEM;  $^{*}p < 0.05$ ;  $^{**}p < 0.01$ .
+
+![](images/ade6e9206ed0f7fb993cd2a3cf23480ec038c32cc97eef84195e7ff83a2f503b.jpg)  
+(b)  
+Free parameter multiplier (relative to H-CNN)
+
+![](images/62b0af4bc671440d74b1f06ca1ff7641ede572ab5229bfa37e030f57b8e09d63.jpg)  
+(c)
+
+drop in accuracy as difficulty increased (Fig. 4a). The fact that each challenge has been designed to feature a single cue for particular grouping (e.g., local Gestalt for Pathfinder and global semantics for cABC) suggests that human participants were able to utilize a different grouping routine for each challenge. Second, the pattern of accuracies from our recurrent architectures reveal complementary contributions of horizontal vs. top-down connections (Fig. 4b). The H-CNN solved the Pathfinder challenge with minimal straining as difficulty increased, but it struggled on the easy cABC dataset. On the other hand, the TD-CNN performed well on the entire cABC challenge but struggled on the Pathfinder challenge as difficulty increased. Together, these results suggest that top-down interactions (from higher-to-lower layers) help process object-level grouping cues, whereas horizontal feedback (between units in different feature columns/spatial locations in a layer) help process local Gestalt grouping cues. Our recurrent architecture which possessed both types of feedback, the TD+H-CNN, was able to solve both challenges at all levels of difficulties, roughly matching human performance.
+
+All three of these recurrent models relied on recurrence to achieve their performance on these datasets. In particular, an H-CNN with one processing timestep could not solve Pathfinder, and a TD-CNN with one processing timestep could not solve cABC (Fig. S7). Separately, performance of the TD+H-CNN was relatively robust to different model configurations, and deeper versions of the model still performed well on both challenges (Fig. S8).
+
+We also tested models on two controlled variants of the cABC challenge: one where the two letters never touch or overlap ("position control") and another where the two letters are rendered in different pixel intensities ("luminance control"). No significant straining was found on these tasks for any of our networks, confirming our initial hypothesis that local ambiguities and occlusion between the letters makes top-down feedback important for grouping on cABC images (Fig. S5).
+
+In contrast to our recurrent architectures, its strictly-feedforward variant, the BU-CNN, struggled on both challenges, demonstrating the importance of feedback for perceptual grouping (Fig. 5a). Of the deep feedforward networks we tested, only the ResNet-50 and ResNet-152 were able to solve the entire Pathfinder challenge, but these same architectures failed to solve cABC. In contrast, the ResNet-18 and U-Net solved the cABC challenge, but strained on difficult Pathfinder datasets. Overall, no feedforward architecture was general enough to learn both types of grouping strategies efficiently, suggesting that these networks face a trade-off between Pathfinder vs. cABC performance.
+
+We explored this tradeoff faced by ResNets in two ways. First, we tested whether the big ResNets which failed to learn the hard cABC challenge could be rescued with larger training datasets. We found that the ResNet-50 but not ResNet-152 was able to solve this task when trained on a dataset that was  $4 \times$  as large, demonstrating that ResNets are very sensitive to overfitting on cABC (and that the ResNet-152 needs more than  $4 \times$  as much data as a ResNet-18 to learn cABC). Second, we tested whether the ResNet-18, which failed to learn the hard Pathfinder challenge, could be rescued by increasing its capacity. Indeed, we found that a wide ResNet-18, with a similar number of parameters as a ResNet-50, was able to perform better (although still far worse than the ResNet-50) on the hard Pathfinder challenge (Fig. S8).
+
+Modeling biological feedback Only the  $\mathrm{H + TD - CNN}$  and human observers solved all levels of the Pathfinder and cABC challenges. To what extent do the visual strategies learned by the  $\mathrm{H + TD - CNN}$  and other reference models in our experiments match those of humans for solving these tasks? We investigated this with a large-scale psychophysics study, in which human participants categorized exemplars from Pathfinder or cABC that were also viewed by the models (see Appendix D for experiment details).
+
+We recruited 648 participants on Amazon Mechanical Turk to complete a web-based psychophysics experiments. Participants were given between 800ms-1300ms to categorize each image in Pathfinder, and 800ms-1600ms to categorize each image in cABC (324 per challenge). Each participant viewed a subset of the images in a challenge, which spanned all three levels of difficulty. The experiment took approximately 2 minutes, and no participant viewed images from more than one challenge.
+
+We gathered 20 human decisions for every image in the Pathfinder and cABC challenges, and measured human performance on each image as a logit of the average accuracy across participants (using average accuracy as a stand-in for "human confidence"). Focusing on the hard-difficulty dataset, we used a two-step procedure to measure how much human decision variance each model explained. First, we estimated a ceiling for human inter-rater reliability. We randomly (without replacement) split participants for every image into half, recorded per-image accuracy for each group, then recorded the correlation between groups. This correlation was then corrected for the split-half procedure using a spearman-brown correction (Spearman, 1910). By repeating this procedure 1,000 times we built a distribution of inter-rater correlations. We took the  $95^{th}$  percentile score of this distribution to represent the ceiling for inter-rater reliability. Second, we recorded the correlation between each model's logits and human logits, and calculated explained variance of human decisions as  $\frac{model}{human\_ceiling}$ .
+
+Both the TD+H-CNN and the H-CNN explained a significantly greater fraction of human decision variance on Pathfinder images than the 50- and 152-layer ResNets, which were also able to solve this challenge (Fig. 5c; H-CNN > ResNet 152:  $p < 0.001$ ; H-CNN > ResNet 50:  $p < 0.001$ ; TD+H-CNN > ResNet 152:  $p < 0.01$ ; TD+H-CNN > ResNet 50:  $p < 0.001$ ; p-values derived from a bootstrap test, Edginton 1964). We also computed partial correlations between human and model scores on every image, which controlled for model accuracy in the measured association. These partial correlations also indicated that human decisions were significantly more correlated with the TD+H-CNN and H-CNN than they were with either ResNet model (Explained variance/Partial correlation; H-CNN: 0.721/0.487; TD+H-CNN: 0.618/0.425; ResNet-152: 0.426/0.290; ResNet-50: 0.234/0.159).
+
+We repeated this analysis for the cABC challenge. Once again, the recurrent models explained a significantly greater fraction of human decisions than either the feedforward U-Net or ResNet 18 that also solved the challenge (Fig. 5c; TD-CNN > U-Net:  $p < 0.01$ ; TD-CNN > ResNet-18:  $p < 0.001$ ; TD+H-CNN > U-Net:  $p < 0.001$ ; TD+H-CNN > ResNet-18:  $p < 0.001$ ). Partial correlations also reflected a similar pattern of associations (Explained variance/Partial correlation; TD+H-CNN: 0.633/0.437; TD-CNN: 0.538/0.376; U-Net: 0.324/0.190; ResNet-18: 0.249/0.176).
+
+# 6 DISCUSSION
+
+Perceptual grouping is essential for reasoning about the visual world. Although it is known that bottom-up, horizontal and top-down connections in the visual system are important for perceptual grouping, their relative contributions are not well understood. We directly tested a long-held theory related to the role of horizontal vs. top-down connections for perceptual grouping by screening neural network architectures on controlled synthetic visual tasks. Without specifying any role for feedback connections a priori, we found a dissociation between horizontal vs. top-down feedback connections which emerged from training network architectures for classification. Our study provides direct computational evidence for the distinct roles played by these cortical mechanisms.
+
+Our study also demonstrates a clear limitation of network models that rely solely on feedforward processing, including ResNets of arbitrary depths, which are strained by perceptual grouping tasks that involve cluttered visual stimuli. Deep ResNets performed better on the Pathfinder challenge, whereas the shallower ResNet-18 performed better on the cABC challenge. With more data, deep ResNets performed better on cABC; with more width, the ResNet-18 performed better on Pathfinder. We take this pattern of results as evidence that the architectures of feedforward models like ResNets must be carefully tuned on complex visual tasks. Because Pathfinder does not have semantic cues, it presents a greater computational burden for feedforward models than cABC, making it a better fit for very deep feedforward architectures; because cABC has relatively small datasets, deeper ResNets were quicker to overfit. The highly-recurrent TD+H-CNN was far more flexible and efficient than the reference feedforward models, and made decisions on Pathfinder and cABC images that were significantly more similar to those of human observers. Our study thus adds to a growing body of literature (George et al., 2017b; Nayebi et al., 2018b; Linsley et al., 2018b;a; Kar et al., 2019) which suggests that recurrent circuits are necessary to explain complex visual recognition processes.
+
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+
+# A CLUTTERED ABC
+
+The goal of the cluttered ABC (cABC) challenge is to test the ability of models and humans to make perceptual grouping judgements based solely on object-level semantic cues. Images in the challenge depict a pair of letters positioned sufficiently close to each other to ensure frequent overlap. To further minimize local image cues which might permit trivial solutions to the problem, we use fonts as letter images that contain uniform and regular strokes with no distinct decorations. Two publicly available fonts are chosen from the website https://www.dafont.com: "Futurist fixed-width" and "Instruction".
+
+Random transformations including affine transformations, geometric distortions, and pixelation are applied before placing each letter on the image to further eliminate local category cues. Positioning and transformations of letters are applied randomly, and their variances are used to control image variability in each difficulty level of the challenge.
+
+Letter transformations Although each letter in the cABC challenge is sampled from just two different fonts, we ensure that each letter appears in sufficiently varied forms by applying random linear transformations to each letter. We use three different linear transformations in our challenge: rotation, scaling, and shearing. Each type of transformation applied to an image uses three different sets of parameters: two sets that are sampled and applied separately to each letter, and an additional one that is sampled once commonly applied to both letters. We call the first type of transformation parameters "letter-wise" and the second type "common" transformation parameters. For example, rotations applied to the first and the second letter are each described by the sum of letter-wise and common rotational parameters,  $\phi_1 + \phi_c$  and  $\phi_2 + \phi_c$ , respectively. Scaling is described by the product of letter-wise and common scale parameters,  $\mathrm{S}_1\mathrm{S}_c$  and  $\mathrm{S}_2\mathrm{S}_c$ . Shearing is described by the sum of letter-wise and common shear parameters,  $\mathrm{E}_1 + \mathrm{E}_c$  and  $\mathrm{E}_2 + \mathrm{E}_c$ . Each shear parameter specifies the value of the off-diagonal shear matrix applied to each letter image. Either vertical or horizontal shear is applied to both letters in each image with shear axis randomly chosen with equal probability. The random transformation procedure is visually detailed in Fig. S1.
+
+We decompose each transformation to letter-wise and common transformation to independently increase variability at the level of an image while keeping letter-wise variability constant. This is achieved by gradually increasing variance of the letter-wise transformation parameters while decreasing variance of the common parameters. See S1 for a summary of the distributions of random transformation parameters used in the three levels of difficulty.
+
+We apply additional, nonlinear transformations to letter images. First, geometric warping is applied independently to each affine-transformed letter image by computing a "warp template" for each image. A warp template is an image in same dimensions as each letter image, consisting of 10 randomly placed Gaussians with sigma (width) of 20 pixels. Each pixel in a letter image is translated by a displacement proportional to the gradient of the warp template in the corresponding location. This process is depicted in Fig. S2. Second, the resulting letter image is "pixelated" by first partitioning the letter image into a grid of  $5 \times 5$  pixels. Each grid is filled with 255s (white) if and only if more than  $30\%$  of the pixels are white, resulting in a binary letter image in  $\frac{1}{5}$  of the original resolution. The squares then undergo random translation with displacement independently sampled for each axis. We use normal distribution with standard deviation of 2 pixels truncated at two standard deviations.
+
+Letter positioning We sample the positions of letters by first defining an invisible circle of radius  $r$  placed at the center of the image. Two additional random variables,  $\theta$  and  $\Delta \theta$ , are sampled to specify two angular positions of the letters on the circle,  $\theta$  and  $\theta + \Delta \theta$ .  $r$ ,  $\theta$  and  $\Delta \theta$  are sampled from uniform distributions whose ranges increase with difficulty level (Table S1). By sampling these parameters from increasingly larger intervals, we increase relative positional variability of letters in harder datasets. In easy difficulty, we choose  $r \sim U(50, 60)$ . In intermediate and hard difficulties, we choose  $R \sim U(40, 70)$  and  $R \sim U(30, 80)$ , respectively. For  $\theta$  and  $\Delta \theta$ , we chose  $\theta_1 \sim U(-30, 30)$  and  $\Delta \theta \sim U(170, 190)$  for the easy difficulty,  $\theta_1 \sim U(-60, 60)$  and  $\Delta \theta \sim U(110, 250)$  for the intermediate difficulty, and  $\theta_1 \sim U(-90, 90)$  and  $\Delta \theta \sim U(70, 290)$  for the hard difficulty.
+
+Letter images are placed on the sampled positions by aligning the positions of each letter's "center of mass", which is defined by the average  $(\mathrm{x},\mathrm{y})$  coordinate of all the foreground pixels in each letter image, with each of the sampled coordinates in an image.
+
+![](images/3713c53b3cc1a722c346bb74c5472929c95190c617eba7d6399561a1445bd66b.jpg)  
+Figure S1: Visual depiction of the cABC image generation algorithm. (Top row) Generation procedure of individual letter images. Two independently sampled letter categories are rendered in a randomly sampled (but same) font. Each letter image undergoes three linear transformations – scale, rotation and shear – as well as two steps of nonlinear transformations – warp and pixelation. Transformation magnitudes are determined by random variables whose distributions differ between levels of difficulties. (Bottom row) Positional sampling procedure. Two letters are placed on two randomly chosen points on an invisible circle of radius  $r$ . Two angular positions on the circle are sampled by first sampling the position of the first letter,  $\theta$ , and then sampling the angular displacement of the second letter relative to the first letter,  $\Delta \theta$ . Each letter is positioned by aligning its “center of mass”, which is the average coordinate of white pixels in a letter image, with the coordinate of the sampled position on the circle.
+
+![](images/33ecd28bbb91bfcc8683a4b641f2833aaf8b5879dfd6094a0de6787e73798c78.jpg)  
+Figure S2: Two example letter images undergoing geometric warping. A "warp template" for each image is sampled by randomly overlaying 10 Gaussian images. Each pixel in a letter image is translated by a displacement proportional to the gradient of its warp template in the corresponding location.
+
+<table><tr><td>Parameters</td><td>Easy</td><td>Intermediate</td><td>Hard</td></tr><tr><td>θ (degrees)</td><td>U(-30,30)</td><td>U(-60,60)</td><td>U(-90,90)</td></tr><tr><td>Δθ (degrees)</td><td>U(170,180)</td><td>U(110,180)</td><td>U(70,180)</td></tr><tr><td>r (pixels)</td><td>U(25,30)</td><td>U(20,35)</td><td>U(15,40)</td></tr><tr><td>φ1, φ2 (degrees)</td><td>0</td><td>N(0, 30/√2)</td><td>N(0,30)</td></tr><tr><td>φc (degrees)</td><td>N(0,30)</td><td>N(0,30/√2)</td><td>0</td></tr><tr><td>S1, S2</td><td>1</td><td>logN(1.5,0,1/2√2)</td><td>logN(1.5,0,1/2)</td></tr><tr><td>Sc</td><td>logN(1.5,0,1/2)</td><td>logN(1.5,0,1/2√2)</td><td>1</td></tr><tr><td>E1, E2</td><td>0</td><td>N(0,2/10√2)</td><td>N(0,2/10)</td></tr><tr><td>Ec</td><td>N(0,2/10)</td><td>N(0,2/10√2)</td><td>0</td></tr></table>
+
+Table S1: Distributions of transformation parameters used in cluttered ABC challenge.  $U(a,b)$  denotes uniform distribution with range  $[a,b]$ ;  $\mathcal{N}(\mu, \sigma)$  denotes normal distribution with mean  $\mu$  and standard deviation  $\sigma$ ;  $\log \mathcal{N}(z,\mu,\sigma)$  denotes log-normal distribution with base  $z$ , mean  $\mu$  and standard deviation  $\sigma$ .
+
+![](images/ac405b4cba686fd4914a2619394cf2f8d1cb1d3eac84c7faa5077429eb3bb8fa.jpg)  
+Luminance control
+
+![](images/8fcfcecda84caad7858b4095c0a392d6a292f3a6d3f4ac84a0fd766e735ed7a4.jpg)  
+Figure S3: Two control cABC challenges. In luminance control, two letters are rendered in different, randomly sampled pixel intensity values. In positional control, two letters are always rendered without touching or overlapping each other.
+
+![](images/eedf94332d0979bfc31916582e34716d3330c5deb033ef882b7ff02e8dc96ef1.jpg)
+
+![](images/edb569039bb91388533522d0821bbc2a71524f67b27cb4dadafdfb0843b8c671.jpg)  
+Position control
+
+![](images/21e0e3ef5a33ffb588651e75185d72a8e743b48916734099632a5b433a829732.jpg)
+
+![](images/cd361bd429d5122d470e06ee89493dd35956cc89ff44246587b58776e14bfcaa.jpg)
+
+![](images/ef3abd4bbf24fcf3c56d5c64ee620a91ed25e0e44f18d8249b916eef5e165c32.jpg)  
+Figure S4: Segmentation version of the two challenges. Here, a model is tasked with producing as output an image which contains only the object which is marked in the input image.
+
+![](images/f5523adf30c6a861d6e7f39a30a8cdadc44265b6a6f65c3b3acbf2da3ca3efa0.jpg)
+
+# B ADDITIONAL EXPERIMENTS
+
+Control cABC challenges To further validate our implementation of the cABC, we added two control cABC challenges in our study (Fig. S3). In the "luminance control" challenge, two letters are rendered with a pair of uniformly sampled random pixel intensity values that are greater than 128 but always differ by at least 40 between the two letters. In the "positional control" challenge, the letters are disallowed from touching or overlapping. Relative to the original challenge, the two controls provide additional local grouping cues with which to determine the extent of each letter. The luminance control challenge permits additional pixel-level cue with which a model can infer image category by comparing the values of letter pixels surrounding two markers. Positional control challenge provides unambiguous local Gestalt cue to solve the task. Ensuring that the two letters are spatially separated has an additional effect of minimizing the amount of clutter to interfere the feature extraction process in each letter.
+
+In both control challenges, we found no straining effect in any of the models we tested. This strengthens our initial assumption behind the design of cABC that the only way to efficiently solve the default cABC challenge is to employ object-based grouping strategy. Because all networks successfully solved the control challenges, the relative difficulty suffered by the H-CNN or reference
+
+![](images/2c16d2295922aa5ddd392d9873da081a1b28a3a97558bd96824e00bf91bc9e36.jpg)  
+Positive
+
+![](images/daabe0b1a2fcbca39ac89001bd1c988bb634efd233072001424ec0fd8251f404.jpg)  
+Negative
+
+![](images/d38ccff49f870e1579882d0d1218c52fb52252196afb56bef933001711fbe25b.jpg)  
+(b)
+
+![](images/7d86f0553d64ef2ba26b6d01b9494fd73c0a657e628d3efa3d51a3fd00c4b63d.jpg)  
+(d)
+
+![](images/2f3d435a7e3dda67b6763e52761c528d297526afc98334639c20d536745587d5.jpg)  
+(a)  
+Positive  
+Luminance Control
+
+![](images/9e25cd61a26586a32ede705629381a9e1da77e2d559ff897b0ae916e18d01f89.jpg)  
+Position Control
+
+![](images/a200393bcdc1c4f1cf7ea97f706971237686c5c7b63aad1aadd89a16e9cca7fc.jpg)  
+ABC-control challenges. Top-down feedback is critical when it comes to  $\mathbf{b}$ . We verify this by constructing two controlled versions of the fault grouping cues and providing in the form of spatial segregation of form of luminance segregation (b). In both cases, the architectures of mechanism can solve the challenge at all levels of difficulties.  
+Figure S6: fGRUs outperform GRUs and LSTMs on Pathfinder and cABC. Our recurrent hierarchical architectures performed better on Pathfinder and cABC challenges when they used fGRUs instead of either LSTMs (Hochreiter & Schmidhuber, 1997) or GRUs (Cho et al., 2014a).
+
+feedforward networks on the original cABC couldn't have been caused by idiosyncracies of image features we used such as the average size of letters. These networks lack the mechanisms to separate the mutually occluding letter strokes according to high-level hypothesis about the categories of letters present in each image.
+
+Segmentation challenges To further understand and visualize the nature of computations employed by our recurrent architectures when solving the Pathfinder and the cABC challenge, we construct a variant of each dataset where we pose a segmentation problem. Here, we modify each dataset by placing only one marker per image instead of two. The task is to output a per-pixel prediction of the object tagged by the marker (Figure S4). We train each model using 10 thousand images of the cABC challenge and 40 thousand images of the Pathfinder challenge and validate using 400 images in both challenges. We only use hard difficulty in each challenge. We use only the TD- and H-CNN in this experiment. We replace the readout module in each of our recurrent architecture by two  $1 \times 1$  convolution layers with bias and rectification after each layer. We also include the "reference" recurrent architectures in which the fGRU modules have been replaced by either LSTM Hochreiter & Schmidhuber (1997) or GRU Cho et al. (2014a). Because the U-Net is a dense prediction architecture by design, we simply remove the readout block that we used in the main categorization challenge.
+
+![](images/ef272eee6b56501638944d7aa1455c20a50c14507fe415ec75470f3a0f5ddf43.jpg)  
+Figure S7: Recurrence improves fGRU model performance on Pathfinder and cABC. Each of our recurrent models performed better when given 8-timesteps of recurrent processing instead of 1 timestep.
+
+![](images/e826e0f4b326e3413249d63f26a886474bc54d87cec1edd459a25e3e0c8019cd.jpg)  
+Figure S8: A comparison of different parameterizations of H+TD-CNN and ResNet-18 models on Pathfinder and cABC. (a) We compared the original H+TD-CNN to a deeper version (5 instead of 3 convolutional blocks between its low- and high-level fGRU modules). This deeper H+TD-CNN performed similarly to the original version. We also tested whether widening the ResNet-18 could rescue its performance on Pathfinder. Indeed, the wider ResNet-18 (matched in number of parameters to ResNet-50) performed far better on Pathfinder than the original version, but was still far worse on the task than a normal ResNet-50. (b) Performance of each of these models is plotted as a function of their total free parameters. Variants of the TD+H-CNN are far more parameter efficient at solving Pathfinder and cABC than variants of ResNet-18. Ellipses group variants of each model class.
+
+We found that the target curves in the Pathfinder challenge is best segmented by H-CNN (0.79 f1 score), followed by H-CNN with GRU (0.72) and H-CNN with LSTM (0.69). No top-down-only architecture was able to successfully solve this challenge (we set a cutoff of greater than 0.6 f1). The cABC challenge is best segmented by the TD-CNN (0.83), followed by TD-CNN with LSTM (0.64) and TD-CNN with GRU (0.62). Consistent with the main challenge, no purely horizontal architecture was able to successfully solve this challenge. In short, we were able to reproduce the dissociation in performance we found between horizontal and top-down connections in other recurrent architectures (LSTM and GRU) as well, although these recurrent modules were less successful than the fGRU.
+
+We visualize the internal update performed by our recurrent networks by plotting the difference of norm of each per-pixel feature vector at every timestep in the low-level recurrent state,  $\mathbf{H}^{(1)}$  (closest to image resolution), vs. its immediately preceding timestep. Of course, this method does not reveal the full extent of the computations taking place in the fGRU, but it at least serves as a proxy for inferring the spatial layout of how activities evolve over timesteps (4b). More examples of both successful and mistaken segmentations and the temporal evolution of internal activities of H-CNN (on the Pathfinder challenge) TD-CNN (on the cABC challenge) are shown in Fig.S10 and Fig.S11
+
+Using conventional recurrent layers We tested the performance of modifications of TD+H-CNN by replacing the fGRU modules with conventional recurrent modules - LSTMs (Hochreiter & Schmidhuber, 1997) and GRUs (Cho et al., 2014a). We found that neither of these architectures matched the performance of our fGRU-based architecture on either grouping challenges S6. The degradation of performance was most pronounced in Pathfinder. Similar to the findings in Linsley et al. (2018b), we believe that the use of both multiplicative and additive interactions as well as the separation of the suppressive and facilitatory stages in fGRU are critical for learning stable transitive grouping in Pathfinder.
+
+# C NETWORK ARCHITECTURES
+
+# C.1 THE FGRU MODULE
+
+At timestep  $t$ , the fGRU module takes two inputs, an instantaneous external drive  $\mathbf{X} \in \mathbb{R}^{H \times W \times K}$  and a persistent recurrent state  $\mathbf{H}[t - 1] \in \mathbb{R}^{H \times W \times K}$  and produces as output an updated recurrent state  $\mathbf{H}[t]$ . Detailed description of its internal operation over a single iteration from  $[t - 1]$  to  $[t]$  is defined by the following formulae. For clarity, tensors are bolded but kernels and parameters are not:
+
+$$
+\mathbf {G} ^ {I} = \operatorname {s i g m o i d} (\operatorname {B N} (U ^ {I} * \mathbf {H} [ t - 1 ]))
+$$
+
+$$
+\mathbf {C} ^ {I} = \operatorname {B N} \left(W ^ {I} * (\mathbf {H} [ t - 1 ] \odot \mathbf {G} ^ {I})\right)
+$$
+
+$$
+\mathbf {Z} = \left[ \mathbf {X} - \left[ (\alpha \mathbf {H} [ t - 1 ] + \mu) \mathbf {C} ^ {I} \right] _ {+} \right] _ {+}
+$$
+
+$$
+\mathbf {G} ^ {E} = \operatorname {s i g m o i d} (\mathrm {B N} (U ^ {E} * \mathbf {Z}))
+$$
+
+$$
+\mathbf {C} ^ {E} = \operatorname {B N} \left(W ^ {E} * \mathbf {Z}\right) \tag {1}
+$$
+
+$$
+\tilde {\mathbf {H}} = \left[ \kappa (\mathbf {C} ^ {E} + \mathbf {Z}) + \omega (\mathbf {C} ^ {E} * \mathbf {Z}) \right] _ {+}
+$$
+
+$$
+\mathbf {H} [ t ] = (1 - \mathbf {G} ^ {E}) \odot \mathbf {H} [ t - 1 ] + \mathbf {G} ^ {E} \odot \tilde {\mathbf {H}}
+$$
+
+$$
+\text {w h e r e} \operatorname {B N} (\mathbf {r}; \delta , \nu) = \nu + \delta \odot \frac {\mathbf {r} - \widehat {\mathbb {E}} [ \mathbf {r} ]}{\sqrt {\operatorname {V a r} [ \mathbf {r} ] + \eta}}.
+$$
+
+The evolution of the recurrent state  $\mathbf{H}$  can be broadly described by two discrete stages. During the first stage,  $\mathbf{H}$  is combined with an extrinsic drive  $\mathbf{X}$ . Interactions between units in the hidden state are computed in  $\mathbf{C}^I$  by convolving  $\mathbf{H}[t - 1]$  with a kernel  $W^{I}$ . These interactions are next linearly and multiplicatively combined with  $\mathbf{H}[t - 1]$  via the  $k$ -dimensional learnable coefficients  $\mu$  and  $\alpha$  to compute the intermediate "suppressed" activity,  $\mathbf{Z}$ .
+
+The second stage involves updating  $\mathbf{H}$  with a transformation of the intermediate activity  $\mathbf{Z}$ . Here,  $\mathbf{Z}$  is convolved with the kernel  $W^{E}$  to compute  $\mathbf{C}^{E}$ . A candidate output  $\tilde{\mathbf{H}}[t]$  is calculated via the  $k$ -dimensional learnable coefficients  $\kappa$  and  $\omega$ , which control the linear and multiplicative terms of self-interactions between  $\mathbf{C}^{E}$  and  $\mathbf{Z}$ .
+
+Two gates modulate the dynamics of both of these stages: the "gain"  $\mathbf{G}^I[t]$ , which modulates channels in  $\mathbf{H}[t-1]$  during the first stage, and the "mix"  $\mathbf{G}^E[t]$ , which mixes a candidate  $\tilde{\mathbf{H}}[t]$  with the persistent  $\mathbf{H}[t-1]$  during the second stage. Both the gain and mix are transformed into the range  $[0,1]$  by a sigmoid nonlinearity. This two-stage computational structure in the fGRU captures complex nonlinear interactions between units in  $\mathbf{X}$  and  $\mathbf{H}$ . Brackets [.] denote linear rectification. Separate
+
+applications of batch-norm are used on every timestep, where  $\mathbf{r} \in \mathbb{R}^d$  is the vector of activations that will be normalized. The parameters  $\delta$ ,  $\nu \in \mathbb{R}^d$  control the scale and bias of normalized activities,  $\eta$  is a regularization hyperparameter, and  $\odot$  is elementwise multiplication.
+
+Learnable gates, like those in the fGRU, support RNN training. But there are multiple other heuristics that also help optimize performance. We use several heuristics to train our models, including Chronos initialization of fGRU gate biases (Tallec & Ollivier, 2018). We also initialized the learnable scale parameter  $\delta$  of fGRU normalizations to 0.1, since values near 0 optimize the dynamic range of gradients passing through its sigmoidal gates (Cooijmans et al., 2017). Similarly, fGRU parameters for learning additive suppression/facilitation  $(\mu, \kappa)$  were initialized to 0, and parameters for learning multiplicative suppression/facilitation  $(\alpha, \omega)$  were initialized to 0.1. Finally, when implementing top-down connections, we incorporated an extra learnable gate, which we found improved the stability of training. Consider the horizontal activities in two layers,  $\mathbf{H}^{(l)}, \mathbf{H}^{(2)}$ , and the function fGRU, which implements top down connections. The introduction of this extra gate means that top-down connections are learned as:  $\mathbf{H}^{(l)} = (1 - \text{sigmoid}(\beta)) \odot \text{fGRU}(\mathbf{H}^{(l)}, \mathbf{H}^{(l+1)}) + \text{sigmoid}(\beta) \odot \mathbf{H}^{(l)}$ , where  $\beta \in \mathbb{R}^K$  is initialized to 0 and learns how to incorporate/ignore top-down connections.
+
+The broad structure of the fGRU is related to the Gated Recurrent Unit (GRU, (Cho et al., 2014b)). Nevertheless, one of the main aspects in which the fGRU diverges from the GRU is that its state update is carried out via two discrete steps of processing instead of one. This is inspired by a computational neuroscience model of contextual effects in visual cortex (see Mély et al. 2018 for details). Unlike that model, the fGRU allows for both additive and multiplicative combinations following each step of convolution, which potentially encourages a more diverse range of nonlinear computations to be learned (Linsley et al., 2018b;a).
+
+# C.2 ARCHITECTURE DETAILS
+
+Using the fGRU module, we construct three recurrent convolutional architectures as well as one feedforward variant. First, the  $\mathrm{H + TD - CNN}$  (Fig. S9a) is equipped with both top-down and horizontal connections. The TD-CNN (Fig. S9b) is constructed by selectively lesioning the horizontal connections in the  $\mathrm{H + TD - CNN}$ . The H-CNN (Fig. S9c) is constructed by selectively lesioning the top-down connections in the  $\mathrm{H + TD - CNN}$ , essentially making recurrence only take place within the low-level fGRU. Lastly the BU-CNN (Fig. S9d) is constructed by disabling both types of recurrence, turning our recurrent architecture into a feedforward encoder-decoder network. All experiments were run with NVIDIA Titan X GPUs.
+
+$\mathbf{TD} + \mathbf{H}$  CNN Our main recurrent network architecture,  $\mathrm{TD} + \mathrm{H}$  CNN, is built on a convolutional and transpose-convolutional "backbone". An input image  $X$  is first processed by a "Conv" block which consists of two convolutional layers of  $7\times 7$  kernels with 20 output channels and a ReLU activation applied between. The resulting feature activity is sent to the first fGRU module (fGRU(1)), which is equipped with  $15\times 15$  kernels with 20 output channels to carry out horizontal interactions. fGRU(1) iteratively updates its recurrent state,  $\mathbf{H}^{(1)}$ . The output undergoes a batch normalization and a pooling layer with a  $2\times 2$  pool kernel with  $2\times 2$  strides which then passes through the downsampling block consisting of two convolutional "stacks" (Fig. S9a, The gray box named "DS"). Each stack consists of three convolutional layers with  $3\times 3$  kernels, each followed by a ReLU activations and a batch normalization layer. The convolutional stacks increase the number of output channels from 20 to 32 to 128, respectively, while they progressively downsample the activity via a  $2\times 2$  pool-stride layer after each stack. The resulting activity is fed to another fGRU module in the top layer, fGRU(2), where it is integrated with the higher-level recurrent state,  $\mathbf{H}^{(2)}$ . The kernels in fGRU(2), unlike fGRU(1), is strictly local with  $1\times 1$  filter size. This essentially makes fGRU(2) a persistent memory module without performing any spatial integration over timesteps. The output of this fGRU module then passes through the "upsampling" block which consists of two transpose convolutional stacks (Fig. S9a, The gray box named "US"). Each stack consists of one transpose convolutional layer of  $4\times 4$  kernels. Each stack also reduces the number of output channels from 128 to 32 to 12, respectively, while it progressively upsamples the activity via a  $2\times 2$  stride. Each transpose convolutional layer is followed by a ReLU activations and a batch normalization layer. The resulting activity, which now has the same shape as the output of the initial fGRU, is sent to the third fGRU module (fGRU(3)) where it is integrated with the recurrent state of fGRU(1),  $\mathbf{H}^{(1)}$ . This module plays the role of integrating
+
+top-down activity (originating form "US") and bottom-up activity (originating from  $\mathrm{fGRU}^{(1)}$ ) via  $1 \times 1$  kernels. To summarize,  $\mathrm{fGRU}^{(1)}$  in our architecture implements horizontal feedback in the first layer while the  $\mathrm{fGRU}^{(3)}$  implements top-down feedback.
+
+The above steps repeat over 8 timesteps, and in the  $8^{th}$  timestep, the final recurrent activity from  $\mathrm{fGRU}^{(1)},\mathbf{H}^{(1)}$  , is extracted from the network and passed through batch normalization. The resulting activity is processed by the "Readout" block which consists of two convolutional layers of  $1\times 1$  kernels with a global max-pooling layer between. Each convolutional layer has one output channel and uses ReLU activations.
+
+Lesioned architectures Our top-down-only architecture, the TD-CNN, is identical to the TD+H-CNN, but its horizontal connections have been disabled by replacing the spatial kernel of  $\mathrm{fGRU}^{(1)}$  with a  $1\times 1$  kernel, thereby preventing spatial propagation of activities within the fGRU altogether. Thus, its recurrent updates in the low-level layer are only driven by top-down feedback (in  $\mathrm{fGRU}^{(3)}$ ). The H-CNN is implemented by disabling  $\mathrm{fGRU}^{(3)}$ . This effectively lesions both the downsampling and upsampling pathways entirely from contributing to the final output. As a result, its final category decision is determined entirely by low-level horizontal propagation of activities in the netwrok.
+
+Lastly, we construct the purely feedforward architecture, called the BU-CNN, by lesioning both horizontal and top-down feedback. It replaces the spatial kernel in  $\mathrm{fGRU}^{(1)}$  by a  $1\times 1$  kernel and disables  $\mathrm{fGRU}^{(3)}$ . By running for only one timestep, the BU-CNN is equivalent to a deep convolutional encoder-decoder network.
+
+Feedforward reference networks Because U-Nets (Ronneberger et al., 2015) are designed for dense image predictions, we attach the same readout block that we use to decode fGRU activities  $(\mathbf{H}^{(1)})$  to the output layer of a U-Net, which pairs a VGG16 as its downsampling architecture, with upsampling + convolutional layers to transform its encodings into the same resolution as the input image.
+
+# D HUMAN EXPERIMENTS
+
+We devised psychophysics categorization experiments to understand the visual processes underlying the ability to solve tasks that encourage perceptual grouping.
+
+These visual categorization experiments adopted a paradigm similar to Eberhardt et al. (2016), where stimuli were flashed quickly and participants had to respond within a fixed period of time. We conducted separate experiments for the "Pathfinder" and the "cluttered ABC" challenge datasets. Participants had several possible response time (RT) widows for responding to stimuli in each dataset. For "Pathfinder", we considered  $\{800, 1050, 1300\}$  ms RT windows, while for the "cluttered ABC", we consider  $\{800, 1200, 1600\}$  ms. We recruited 324 participants for each experiment (648 overall) from Amazon Mechanical Turk (www.mturk.com).
+
+The "Pathfinder" and the "cluttered ABC" are binary categorization challenges with "Same" and "Different" as the possible responses. For both experiments, we present 90 stimuli to each participant and assign a single RT to that participant. These 90 stimuli consist of three sets of 30, corresponding to the three level of difficulties present in the challenge datasets. We randomize the order of the three difficulties (sets of 30 stimuli) across participants as well as the assignment of keys to class "Same" and "Different". Participant accuracy is shown in Fig. 4.
+
+Stimulus generation We generated a short video for each stimuli image from the challenge dataset based on the response time allowed from the onset of stimuli. We generated three stimulus videos for each challenge image, corresponding to the three different response times chosen. In each stimulus video, we included an image of a cross (black plus sign overlaid on a white background) for  $1000\mathrm{ms}$  before the onset of stimulus that is shown for RT ms. The stimuli were stored as .webm videos and presented for  $1000 + \mathrm{RT}$  ms during the experiments. We created a video so that the cross image and the stimulus image are seamlessly combined and shown without delays, which would otherwise be caused by loading their separate images in the browser, which introduces noise in stimulus timing.
+
+![](images/01a3d9ff7e44d53e932733c9f67f2ab449e20ab21e41a3151ca9c34ec0f59423.jpg)  
+(a) TD+H-CNN
+
+![](images/982f05ff38102be096126783b14edc894daa2c72bf36a31727638a4323cbe88a.jpg)  
+(b) TD-CNN  
+(d) BU-CNN
+
+![](images/636c2eb4c06027f7edbcf77a3d401e4ad4f0fb923fe02756c16f07e4d8a3357e.jpg)  
+(c) H-CNN  
+Figure S9: Our recurrent convolutional architectures. Feedforward or bottom-up connections (gray) pass information from lower-order to higher-order layers; horizontal connections (green) pass information between units in the same layer separated by spatial location and/or channels; top-down connections (red) pass information from higher-order to lower-order layers. We use "feedback" as an umbrella term for both top-down and horizontal connections as these can only be implemented in a model with recurrent activity that supports incremental updates to model units.
+
+We selected a set of 200 images per difficulty in both challenges (600/challenge). Each participant responded to 30 images from each difficulty. We collected responses to the 600 images in each challenge from 18 different participants.
+
+Psychophysics experiment We conducted separate experiments for: (1) Pathfinder and (2) cABC images. The image dataset used for each experiment consisted of three difficulties - easy, intermediate and hard. Participants were assigned three blocks of trials, each having stimuli from a single difficulty level. The order of these three blocks is randomized and balanced across participants. The keys for responses to the stimuli are set to be + and -. Their assignment to "Same" and "Different" class label is randomized across participants as well.
+
+Each experiment began with a training phase, where participants responded to 6 trials without a time limit. This phase ensured that participants understood the task and the type of image stimuli. Next,
+
+participants went through 15 trials, 5 from each difficulty level, where they responded by pressing + or - within a limited response period. For this set of images, participants were given feedback on whether their response was correct or incorrect. After this block of "training" trials, each participant completed 90 "test" trials and responded with either "Same" or "Different" in the stipulated amount of response time, without feedback on their performance. Participants were given time to rest and the difficulty level of the viewed images changed after every 30 trials. Before starting a new block of trials, a representative image from that difficulty level was shown and instructions were given to respond as quickly as possible.
+
+Experiments were implemented using jsPsych and custom javascript functions. We used the .webm format to ensure fast loading times. Videos were rendered at  $256 \times 256$  pixels.
+
+Processing Responses To generate the final results from human responses on the two categorization tasks, we filtered responses rendered in less than  $450~\mathrm{ms}$ , which we deem unreliable responses following the criteria of Eberhardt et al. (2016).
+
+![](images/1de2e308dbe252577af8e49ec48b22049cbcdf427ebf8094af12a9180fedcc36.jpg)  
+Figure S10: Activity time-courses for the Pathfinder challenge (Hard).
+
+![](images/337aee46cc455342eef6e231c6031e80bdbe964882435604ef91182bd9a9d34a.jpg)  
+Figure S11: Activity time-courses for the cABC challenge (Hard).
\ No newline at end of file
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+# DOUBLY ROBUST BIAS REDUCTION IN INFINITE HORIZON OFF-POLICY ESTIMATION
+
+Ziyang Tang
+
+The University of Texas at Austin
+
+ztang@cs.utexas.edu
+
+Yihao Feng *
+
+The University of Texas at Austin  
+yihao@cs.utexas.edu
+
+Lihong Li
+
+Google Research
+
+lihong@google.com
+
+Dengyong Zhou
+
+Google Research
+
+dennyzhou@google.com
+
+Qiang Liu
+
+The University of Texas at Austin
+
+lqiang@cs.utexas.edu
+
+# ABSTRACT
+
+Infinite horizon off-policy policy evaluation is a highly challenging task due to the excessively large variance of typical importance sampling (IS) estimators. Recently, Liu et al. (2018a) proposed an approach that substantially reduces the variance of infinite horizon off-policy evaluation by estimating the stationary density ratio, but at the cost of introducing biases due to errors in density ratio estimation. In this paper, we develop a bias-reduced augmentation of their method, which can take advantage of a learned value function to improve accuracy. Our method is doubly robust in that the bias vanishes when either the density ratio or value function estimation is perfect. In general, when either of them is accurate, the bias can also be reduced. Both theoretical and empirical results show that our method yields significant advantages over previous methods.
+
+# 1 INTRODUCTION
+
+A key problem in reinforcement learning (RL) (Sutton & Barto, 1998) is off-policy policy evaluation: given a fixed target policy of interest, estimating the average reward garnered by an agent that follows the policy, by only using data collected from different behavior policies. This problem is widely encountered in many real-life applications (e.g., Murphy et al., 2001; Li et al., 2011; Bottou et al., 2013; Thomas et al., 2017), where online experiments are expensive and high-quality simulators are difficult to build. It also serves as a key algorithmic component of off-policy policy optimization (e.g., Dudík et al., 2011; Jiang & Li, 2016; Thomas & Brunskill, 2016; Liu et al., 2019b).
+
+There are two major families of approaches to policy evaluation. The first approach is to build a simulator that mimics the reward and next-state transitions of the real environment (e.g., Fonteneau et al., 2013). While straightforward, this approach strongly relies on the model assumptions in building the simulator, which may invalidate evaluation results. The second approach is to use importance sampling to correct the sampling bias in off-policy data, so that an (almost) unbiased estimator can be obtained (Liu, 2001; Strehl et al., 2010; Bottou et al., 2013). A major limitation, however, is that importance sampling can become inaccurate due to high variance. In particular, most existing IS-based estimators compute the weight as the product of the importance ratios of many steps in the trajectory, causing excessively high variance for problems with long or infinite horizon, yielding a curse of horizon (Liu et al., 2018a).
+
+Recently, Liu et al. (2018a) proposes a new estimator for infinite-horizon off-policy evaluation, which presents significant advantages to standard importance sampling methods. Their method directly estimates the density ratio between the state stationary distributions of the target and behavior policies, instead of the trajectories, thus avoiding exponential blowup of variance in the horizon. While Liu et al.'s method shows much promise by significantly reducing the variance, in practice, it may suffer from high bias due to the error or model misspecification when estimating the density ratio function.
+
+In this paper, we develop a doubly robust estimator for infinite horizon off-policy estimation, by integrating Liu et al.'s method with information from an additional value function estimation. This significantly reduces the bias of Liu et al.'s method once either the density ratio, or the value function estimation is accurate (hence doubly robust). Since Liu et al.'s method already promises low variance, our additional bias reduction allows us to achieve significantly better accuracy for practical problems.
+
+Technically, our new bias reduction method provides a new angle of double robustness for off-policy evaluation, orthogonal to the existing literature of doubly robust policy evaluation that solely devotes to variance reduction (Jiang & Li, 2016; Thomas & Brunskill, 2016; Farajtabar et al., 2018), mostly based on the idea of control variates (e.g. Asmussen & Glynn, 2007). Our double robustness for bias reduction is significantly different, and instead yields an intriguing connection with the fundamental primal-dual relations between the density (ratio) functions and value functions (e.g., Bertsekas, 2000; Puterman, 2014). This new perspective can inspire more efficient algorithms for policy evaluation, and lead to unified frameworks for these types of double robustness in future work.
+
+# 2 BACKGROUND
+
+Infinite Horizon Off-Policy Estimation Let  $M = \langle S, \mathcal{A}, r, T, \mu_0 \rangle$  be a Markov decision process (MDP) with state space  $S$ , action space  $\mathcal{A}$ , reward function  $r$ , transition probability function  $T$ , and initial-state distribution  $\mu_0$ . A policy  $\pi$  maps states to distributions over  $\mathcal{A}$ , with  $\pi(a|s)$  being the probability of selecting  $a$  given  $s$ . The average discounted reward for  $\pi$ , with a given discount  $\gamma \in (0,1)^1$ , is defined as
+
+$$
+R ^ {\pi} := \lim _ {T \to \infty} \mathbb {E} _ {\tau \sim \pi} \left[ \frac {\sum_ {t = 0} ^ {T} \gamma^ {t} r _ {t}}{\sum_ {t = 0} ^ {T} \gamma^ {t}} \right],
+$$
+
+where  $\tau = \{s_t, a_t, r_t\}_{0 \leq t \leq T}$  is a trajectory with states, actions, and rewards collected by following policy  $\pi$  in the MDP, starting from  $s_0 \sim \mu_0$ . Given a set of  $n$  trajectories,  $\mathcal{D} = \{s_t^{(i)}, a_t^{(i)}, r_t^{(i)}\}_{1 \leq i \leq n, 0 \leq t \leq T}$ , collected under a behavior policy  $\pi_0(a|s)$ , the off-policy evaluation problem aims to estimate the average discounted reward  $R^{\pi}$  for another target policy  $\pi(a|s)$ .
+
+Estimation via Value Function The value function for policy  $\pi$  is defined as the expected accumulated discounted future rewards started from a certain state:  $V^{\pi}(s) = \mathbb{E}_{\tau \sim \pi}[\sum_{t=0}^{\infty} \gamma^{t} r_{t}|s_{0} = s]$ . We use  $r^{\pi}(s) = \mathbb{E}_{a \sim \pi(\cdot|s)}[r(s, a)]$  to denote the average reward for state  $s$  given policy  $\pi$ . Under the definition, the value function can be seen as a fixed point of the Bellman equation:
+
+$$
+V ^ {\pi} (s) = r ^ {\pi} (s) + \gamma \mathcal {P} ^ {\pi} V ^ {\pi} (s), \quad \mathcal {P} ^ {\pi} V ^ {\pi} (s) := \mathbb {E} _ {a \sim \pi (\cdot | s), s ^ {\prime} \sim \boldsymbol {T} (\cdot | s, a)} \left[ V ^ {\pi} \left(s ^ {\prime}\right) \right], \quad \forall s \in \mathcal {S}, \tag {1}
+$$
+
+where  $\mathcal{P}^{\pi}V(s)$  is the average of the next value function given the current state  $s$  and policy  $\pi$ ; see Appendix A.1 for details.
+
+The value function and the expected reward  $R^{\pi}$  is related in the following straightforward way
+
+$$
+R ^ {\pi} = (1 - \gamma) \mathbb {E} _ {s \sim \mu_ {0}} \left[ V ^ {\pi} (s) \right], \tag {2}
+$$
+
+where the expectation is with respect to the distribution  $\mu_0(s)$  of the initial states  $s_0$  at time  $t$ . Therefore, given an approximation  $\widehat{V}$  of  $V^{\pi}$ , and samples  $\mathcal{D}_0 \coloneqq \{s_0^{(i)}\}_{1 \leq i \leq n_0}$  drawn from  $\mu_0(s)$ , we can estimate  $R^{\pi}$  by
+
+$$
+\widehat {R} _ {\mathrm {V A L}} ^ {\pi} [ \widehat {V} ] = \frac {(1 - \gamma)}{n _ {0}} \sum_ {i = 1} ^ {n _ {0}} \widehat {V} (s _ {0} ^ {(i)}).
+$$
+
+Note that this estimator is off-policy in nature, since it requires no samples from the target policy  $\pi$ .
+
+Estimation via State Density Function Denote  $d_{\pi ,t}(\cdot)$  as average visitation of  $s_t$  in time step  $t$ . The state density function, or the discounted average visitation, is defined as:
+
+$$
+d _ {\pi} (s) := \lim  _ {T \to \infty} \frac {\sum_ {t = 0} ^ {T} \gamma^ {t} d _ {\pi , t} (s)}{\sum_ {t = 0} ^ {T} \gamma^ {t}} = (1 - \gamma) \sum_ {t = 0} ^ {\infty} \gamma^ {t} d _ {\pi , t} (s),
+$$
+
+where  $(1 - \gamma)$  can be viewed as the normalization factor introduced by  $\sum_{t=0}^{\infty} \gamma^{t}$ .
+
+Similar to Bellman equation for value function, the state density function can also be viewed as a fixed point to the following recursive equation (Liu et al., 2018a, Lemma 3):
+
+$$
+d _ {\pi} \left(s ^ {\prime}\right) = (1 - \gamma) \mu_ {0} \left(s ^ {\prime}\right) + \gamma \mathcal {T} ^ {\pi} d _ {\pi} \left(s ^ {\prime}\right), \quad \text {w h e r e} \quad \mathcal {T} ^ {\pi} d _ {\pi} \left(s ^ {\prime}\right) := \sum_ {s, a} \boldsymbol {T} \left(s ^ {\prime} \mid s, a\right) \pi (a \mid s) d _ {\pi} (s). \tag {3}
+$$
+
+The operator  $\mathcal{T}^{\pi}$  is an adjoint operator of  $\mathcal{P}^{\pi}$  used in (1); see Appendix A.1 for a discussion.
+
+If the density function  $d_{\pi}$  is known, it provides an alternative way for estimating the expected reward  $R^{\pi}$ , by noting that
+
+$$
+R ^ {\pi} = \mathbb {E} _ {s \sim d _ {\pi}, a \sim \pi (\cdot | s)} [ r (s, a) ]. \tag {4}
+$$
+
+We can see that both density function  $d_{\pi}$  and value function  $V^{\pi}$  can be used to estimate the expected reward  $R^{\pi}$ . We clarify the connection in detail in Appendix A.1.
+
+Off-Policy State Visitation Importance Sampling Equation (4) can not be directly used for off-policy estimation, since it involves expectation under the behavior policy  $\pi$ . Liu et al. (2018a) addressed this problem by introducing a change of measures via importance sampling:
+
+$$
+R ^ {\pi} = \mathbb {E} _ {s \sim d _ {\pi_ {0}}, a \sim \pi_ {0} (\cdot | s)} \left[ w _ {\pi / \pi_ {0}} (s) \frac {\pi (a | s)}{\pi_ {0} (a | s)} r (s, a) \right], \quad \text {w i t h} \quad w _ {\pi / \pi_ {0}} (s) = \frac {d _ {\pi} (s)}{d _ {\pi_ {0}} (s)}, \tag {5}
+$$
+
+where  $w_{\pi/\pi_0}(s)$  is the density ratio function of  $d_{\pi}$  and  $d_{\pi_0}$ .
+
+Given an approximation  $\widehat{w}$  of  $w_{\pi/\pi_0}$ , and samples  $\mathcal{D} = \{s_t^{(i)}, a_t^{(i)}, r_t^{(i)}\}_{1 \leq i \leq n, 0 \leq t \leq T}$  collected from policy  $\pi_0$ , we can estimate  $R^{\pi}$  as:
+
+$$
+\widehat {R} _ {\mathrm {S I S}} ^ {\pi} [ \widehat {w} ] = \frac {1}{Z} \sum_ {i = 1} ^ {n} \sum_ {t = 0} ^ {T} \gamma^ {t} \widehat {w} \left(s _ {t} ^ {(i)}\right) \frac {\pi \left(a _ {i} \mid s _ {i}\right)}{\pi_ {0} \left(a _ {i} \mid s _ {i}\right)} r _ {i}, \quad Z = \sum_ {i = 1} ^ {n} \sum_ {t = 0} ^ {T} \gamma^ {t} \widehat {w} \left(s _ {t} ^ {(i)}\right) \frac {\pi \left(a _ {t} ^ {(i)} \mid s _ {t} ^ {(i)}\right)}{\pi_ {0} \left(a _ {t} ^ {(i)} \mid s _ {t} ^ {(i)}\right)}, \tag {6}
+$$
+
+where  $Z$  is the normalized constant of the importance weights.
+
+# 3 DOUBLY ROBUST ESTIMATOR
+
+Doubly robust estimator is first proposed into reinforcement learning community to solve contextual bandit problem by Dudík et al. (2011) as an estimator combining inverse propensity score (IPS) estimator and direct method (DM) estimator.
+
+Jiang & Li (2016) introduce the idea of doubly robust estimator into off-policy evaluation in reinforcement learning. It incorporates an approximate value function as a control variate to reduce the variance of importance sampling estimator. Inspired by previous works, we propose a new doubly robust estimator based on our infinite horizon off-policy estimator  $\widehat{R}_{\mathrm{SIS}}^{\pi}$ .
+
+# 3.1 DOUBLY ROBUST ESTIMATOR FOR INFINITE HORIZON MDP
+
+The value-based estimator  $\widehat{R}_{\mathrm{VAL}}^{\pi}[\widehat{V}]$  and density-ratio-based estimator  $\widehat{R}_{\mathrm{SIS}}^{\pi}[\widehat{w}]$  are expected to be accurate when  $\widehat{V}$  and  $\widehat{w}$  are accurate, respectively. Our goal is to combine their advantages, obtaining a doubly robust estimator that is accurate once either  $\widehat{V}$  or  $\widehat{w}$  or is accurate.
+
+To simplify the problem, it is useful to examine the limit of infinite samples, with which  $\widehat{R}_{\mathrm{VAL}}^{\pi}[\widehat{V}]$  and  $\widehat{R}_{\mathrm{SIS}}^{\pi}[\widehat{w}]$  converge to the following limit of expectations:
+
+$$
+R _ {\mathrm {S I S}} ^ {\pi} [ \widehat {w} ] := \lim  _ {n, T \rightarrow \infty} \widehat {R} _ {\mathrm {S I S}} ^ {\pi} [ \widehat {w} ] = \sum_ {s} r ^ {\pi} (s) d _ {\pi_ {0}} (s) \widehat {w} (s), \tag {7}
+$$
+
+$$
+R _ {\mathrm {V A L}} ^ {\pi} [ \widehat {V} ] := \lim  _ {n _ {0} \rightarrow \infty} \widehat {R} _ {\mathrm {V A L}} ^ {\pi} [ \widehat {V} ] = (1 - \gamma) \sum_ {s} \widehat {V} (s) \mu_ {0} (s). \tag {8}
+$$
+
+Here and throughout this work, we assume  $\widehat{V}$  and  $\widehat{w}$  are fixed pre-defined approximations, and only consider the randomness from the data  $\mathcal{D}$ .
+
+A first observation is that we expect to have  $r^{\pi} \approx \widehat{V} - \gamma \mathcal{P}^{\pi} \widehat{V}$  by Bellman equation (1), when  $\hat{V}$  approximates the true value  $V^{\pi}$ . Plugging this into  $R_{\mathrm{SIS}}^{\pi}[\widehat{w}]$  in Equation (7), we obtain the following "bridge estimator", which incorporates information from both  $\widehat{w}$  and  $\widehat{V}$ :
+
+$$
+R _ {\mathrm {b r i d g e}} ^ {\pi} [ \widehat {V}, \widehat {w} ] = \sum_ {s} \left(\widehat {V} (s) - \gamma \mathcal {P} ^ {\pi} \widehat {V} (s)\right) d _ {\pi_ {0}} (s) \widehat {w} (s), \tag {9}
+$$
+
+where operator  $\mathcal{P}^{\pi}$  is defined in Bellman equation (1). The corresponding empirical estimator is defined by
+
+$$
+\widehat {R} _ {\mathrm {b r i d g e}} ^ {\pi} [ \widehat {V}, \widehat {w} ] = \sum_ {i = 1} ^ {n} \sum_ {t = 0} ^ {T - 1} \left(\frac {1}{Z _ {1}} \gamma^ {t} \widehat {w} \left(s _ {t} ^ {(i)}\right) \widehat {V} \left(s _ {t} ^ {(i)}\right) - \frac {1}{Z _ {2}} \gamma^ {t + 1} \widehat {w} \left(s _ {t} ^ {(i)}\right) \frac {\pi \left(a _ {i} \mid s _ {i}\right)}{\pi_ {0} \left(a _ {i} \mid s _ {i}\right)} \widehat {V} \left(s _ {t + 1} ^ {(i)}\right)\right), \tag {10}
+$$
+
+where  $Z_{1} = \sum_{i=1}^{n} \sum_{t=0}^{T-1} \gamma^{t} \widehat{w}(s_{t}^{(i)})$  and  $Z_{2} = \sum_{i=1}^{n} \sum_{t=0}^{T-1} \gamma^{t+1} \widehat{w}(s_{t}^{(i)}) \beta_{\pi/\pi_{0}}(a_{t}^{(i)}|s_{t}^{(i)})$  are self-normalized constant of important weights each empirical estimation.
+
+However, directly estimating  $R^{\pi}$  using the bridge estimator  $\widehat{R}_{\mathrm{bridge}}^{\pi}[\widehat{V}, \widehat{w}]$  yields a poor estimation, because it includes the errors from both  $\widehat{w}$  and  $\widehat{V}$  and can be "doubly worse". However, we can construct our "doubly robust" estimator by canceling  $R_{\mathrm{bridge}}^{\pi}[\widehat{V}, \widehat{w}]$  out from  $R_{\mathrm{SIS}}^{\pi}[\widehat{w}]$  and  $R_{\mathrm{VAL}}^{\pi}[\widehat{V}]$ :
+
+$$
+R _ {\mathrm {D R}} ^ {\pi} [ \widehat {V}, \widehat {w} ] = \underbrace {\sum_ {s} r ^ {\pi} (s) d _ {\pi_ {0}} (s) \widehat {w} (s)} _ {R _ {\mathrm {S I S}} ^ {\pi} [ \widehat {w} ]} + \underbrace {(1 - \gamma) \sum_ {s} \widehat {V} (s) \mu_ {0} (s)} _ {R _ {\mathrm {V A L}} ^ {\pi} [ \widehat {V} ]} - \underbrace {\sum_ {s} \left(\widehat {V} (s) - \gamma \mathcal {P} ^ {\pi} \widehat {V} (s)\right) d _ {\pi_ {0}} (s) \widehat {w} (s)} _ {R _ {\mathrm {b r i d g e}} ^ {\pi} [ \widehat {V}, \widehat {w} ]}. \tag {11}
+$$
+
+And its corresponding empirical estimator can be written as:
+
+$$
+\widehat {R} _ {\mathrm {D R}} ^ {\pi} [ \widehat {V}, \widehat {w} ] = \widehat {R} _ {\mathrm {S I S}} ^ {\pi} [ \widehat {w} ] + \widehat {R} _ {\mathrm {V A L}} ^ {\pi} [ \widehat {V} ] - \widehat {R} _ {\mathrm {b r i d g e}} ^ {\pi} [ \widehat {V}, \widehat {w} ].
+$$
+
+Doubly Robust Bias Reduction The double robustness of  $\widehat{R}_{\mathrm{DR}}^{\pi}[\widehat{V},\widehat{w} ]$  is reflected in the following key theorem, which shows that it is accurate once either  $\widehat{V}$  or  $\widehat{w}$  is accurate.
+
+Theorem 3.1 (Doubly Robustness). Let  $R_{DR}^{\pi}[\widehat{V}, \widehat{w}] \coloneqq \lim_{n_0, n, T \to \infty} \widehat{R}_{DR}^{\pi}[\widehat{V}, \widehat{w}]$  be the limit of  $\widehat{R}_{DR}^{\pi}$  when it has infinite samples. Following the definition above, we have
+
+$$
+R _ {D R} ^ {\pi} [ \widehat {V}, \widehat {w} ] - R ^ {\pi} = \mathbb {E} _ {s \sim d _ {\pi_ {0}}} \left[ \varepsilon_ {\widehat {w}} (s) \varepsilon_ {\widehat {V}} (s) \right], \tag {12}
+$$
+
+where  $\varepsilon_{\widehat{V}}$  and  $\varepsilon_{\widehat{w}}$  are errors of  $\widehat{V}$  and  $\widehat{w}$ , respective, defined as follows
+
+$$
+\varepsilon_ {\widehat {w}} = \frac {d _ {\pi} (s)}{d _ {\pi_ {0}} (s)} - \widehat {w} (s), \qquad \qquad \varepsilon_ {\widehat {V}} (s) = \widehat {V} (s) - r ^ {\pi} (s) - \gamma \mathcal {P} ^ {\pi} \widehat {V} (s).
+$$
+
+The error  $\varepsilon_{\widehat{w}}$  of  $\widehat{w}$  is measured by the difference with the true density ratio  $d_{\pi}(s) / d_{\pi_0}(s)$ , and the error  $\varepsilon_{\widehat{V}}$  of  $\widehat{V}$  is measured using the Bellman residual.
+
+From the theorem we can see that, if  $\widehat{V}$  is exact  $(\widehat{V} \equiv V^{\pi})$ , we have  $\varepsilon_{\widehat{V}} \equiv 0$ ; if  $\widehat{w}$  is exact  $(\widehat{w} \equiv d_{\pi} / d_{\pi_0})$ , we have  $\varepsilon_{\widehat{w}} \equiv 0$ . Therefore, our estimator is consistent (i.e.,  $\lim_{n,n_0 \to \infty} \widehat{R}_{\mathrm{DR}}^{\pi}[\widehat{V}, \widehat{w}] = R^{\pi}$ ) if either  $\widehat{V}$  or  $\widehat{w}$  are exact. The estimator is thus doubly robust in this sense. In contrast,  $\widehat{R}_{\mathrm{SIS}}^{\pi}[\widehat{w}]$  and  $\widehat{R}_{\mathrm{VAL}}^{\pi}[\widehat{V}]$  can be more sensitive to the error of  $\widehat{w}$  and  $\widehat{V}$ , respectively:
+
+$$
+R _ {\mathrm {S I S}} ^ {\pi} [ \widehat {w} ] - R ^ {\pi} = \mathbb {E} _ {s \sim d _ {\pi_ {0}}} \left[ \varepsilon_ {\widehat {w}} (s) r ^ {\pi} (s) \right], \qquad R _ {\mathrm {V A L}} ^ {\pi} [ \widehat {V} ] - R ^ {\pi} = \mathbb {E} _ {s \sim d _ {\pi_ {0}}} \left[ w _ {\pi / \pi_ {0}} (s) \varepsilon_ {\widehat {V}} (s) \right].
+$$
+
+Variance Analysis Different from the bias reduction, the doubly robust estimator does not guarantee to reduce the variance over  $\widehat{R}_{\mathrm{SIS}}^{\pi}[\widehat{w}]$  or  $\widehat{R}_{\mathrm{VAL}}^{\pi}[\widehat{V}]$  in general. However, as we show in the following result, we can break the variance of  $\widehat{R}_{\mathrm{DR}}^{\pi}[\widehat{V},\widehat{w}]$  into two parts. The first is the variance of  $\widehat{R}_{\mathrm{VAL}}^{\pi}[\widehat{V}]$  which is often relatively small. The second is generally no greater than the variance of  $\widehat{R}_{\mathrm{SIS}}^{\pi}[\widehat{w}]$ , and can be much smaller when  $\widehat{V}\approx V^{\pi}$ . Moreover,  $\widehat{R}_{\mathrm{VAL}}^{\pi}[\widehat{V}]$  and  $\widehat{R}_{\mathrm{SIS}}^{\pi}[\widehat{w}]$  avoid the curse of horizon by design, so their variances tend to be much smaller than the corresponding estimators that apply IPS correction on the trajectories.
+
+# Algorithm 1 Infinite Horizon Doubly Robust Estimator
+
+Input: Transition data  $\mathcal{D}_{\pi_0} = \{s_t^{(i)},a_t^{(i)},r_t^{(i)}\}_{1\leq i\leq n,0\leq t\leq T}$  from policy  $\pi_0$  .  $\mathcal{D}_0 = \{s_0^{(j)}\}_{1\leq j\leq n_0}$  be samples from initial distribution  $\mu_0$  target policy  $\pi$  , value function estimate  $\widehat{V}$  ; density ratio estimate  $\widehat{w}$
+
+Estimation: Use  $\widehat{R}_{\mathrm{DR}}^{\pi}$  in (11) to estimate  $R^{\pi}$  using sample from  $\mathcal{D}$  and  $\mathcal{D}_0$
+
+Proposition 3.1 (Variance Analysis). Assume  $\widehat{R}_{DR}^{\pi}[\widehat{V},\widehat{w}]$  is estimated based on sample  $\mathcal{D}_0\sim \mu_0$  and  $\mathcal{D}_{\pi_0}\sim d_{\pi_0}$ , which we assume to be independent of each other. We have
+
+$$
+\operatorname {V a r} _ {\mathcal {D} _ {0}, \mathcal {D} _ {\pi_ {0}}} \left[ \widehat {R} _ {D R} ^ {\pi} [ \widehat {V}, \widehat {w} ] \right] = \operatorname {V a r} _ {\mathcal {D} _ {0}} \left[ \widehat {R} _ {V A L} ^ {\pi} [ \widehat {V} ] \right] + \operatorname {V a r} _ {\mathcal {D} _ {\pi_ {0}}} \left[ \widehat {R} _ {r e s} ^ {\pi} [ \widehat {V}, \widehat {w} ] \right], \tag {13}
+$$
+
+with  $\widehat{R}_{res}^{\pi}[\widehat{V},\widehat{w} ]\coloneqq \widehat{R}_{SIS}^{\pi}[\widehat{w} ] - \widehat{R}_{bridge}^{\pi}[\widehat{V},\widehat{w} ] = \widehat{\mathbb{E}}_{\mathcal{D}_{\pi_0}}\left[\widehat{\varepsilon}_{\hat{V}}(s)\widehat{w} (s)\right]$  and  $\widehat{\varepsilon}_{\hat{V}}(s)$  the TD error of  $\widehat{V}$ ,
+
+$$
+\widehat {\varepsilon} _ {\widehat {V}} (s) = \widehat {r ^ {\pi}} (s) - \widehat {V} (s) + \gamma \widehat {\mathcal {P} ^ {\pi}} \widehat {V} (s),
+$$
+
+with  $\widehat{r^{\pi}}(s) = r(s, a)\pi(a|s)/\pi_0(a|s), \widehat{\mathcal{P}^{\pi}}\widehat{V}(s) = \pi(a|s)/\pi_0(a|s)\widehat{V}(s')$ . Therefore,  $Var_{\mathcal{D}_{\pi_0}}(\widehat{R}_{res}^{\pi}[\widehat{V}, \widehat{w}])$  can be small when  $\widehat{V}$  is close to the true value  $V^{\pi}$ , or  $\widehat{\varepsilon}_{\widehat{V}}(s) \approx 0$ .
+
+From the proposition we can see that when  $\widehat{V}$  is close to the true value  $V^{\pi}$ , the variance of the residual  $\widehat{\varepsilon}_{\widehat{V}}$  may be negligibly small compared to the variance of  $\widehat{R}_{\mathrm{SIS}}^{\pi}[\widehat{w}]$ . A further comparison of the variances between  $\widehat{R}_{\mathrm{res}}^{\pi}[\widehat{V},\widehat{w}]$  and  $\widehat{R}_{\mathrm{SIS}}^{\pi}[\widehat{w}]$  is provided in Appendix A.3. In the case when the TD error  $\widehat{\varepsilon}_{\widehat{V}}$  is negligible, we have  $\operatorname{Var}_{\mathcal{D}_0,\mathcal{D}_{\pi_0}}\left[\widehat{R}_{\mathrm{DR}}^{\pi}[\widehat{V},\widehat{w}]\right] \approx \operatorname{Var}_{\mathcal{D}_0}\left[\widehat{R}_{\mathrm{VAL}}^{\pi}[\widehat{V}]\right]$ . Typically, the variance of  $\widehat{R}_{\mathrm{VAL}}^{\pi}[\widehat{V}]$  is smaller than  $\widehat{R}_{\mathrm{SIS}}^{\pi}[\widehat{w}]$  since the variance of importance sampling methods heavily depends on the effective sample size, which is less controllable compared to  $\widehat{R}_{\mathrm{VAL}}^{\pi}[\widehat{V}]$ . Therefore, the variance of our doubly robust estimator may even be smaller than that of  $\widehat{R}_{\mathrm{SIS}}^{\pi}[\widehat{w}]$  in practice.
+
+The variance in (13) is a sum of two terms because of the assumption that samples from  $\mu_0$  and  $d_{\pi_0}$  are independent. In practice, they have dependency but it is possible to couple the samples from  $\mu_0$  and  $d_{\pi_0}$  in a certain way to further decrease the variance, which we leave as future work.
+
+Proposed Algorithm for Off-Policy Evaluation Suppose we have already obtained  $\widehat{V}$  and  $\widehat{w}$ , as estimations of  $V^{\pi}$  and  $w_{\pi/\pi_0}$ , respectively, we can directly use equation (11) to estimate  $R^{\pi}$ . A detail procedure is described in Algorithm 1.
+
+# 4 DOUBLE ROBUSTNESS AND LAGRANGIAN DUALITY
+
+In this section, we reveal a surprising connection between our double robustness and Lagrangian duality. We show that our doubly robust estimator is equivalent to the Lagrangian function of a primal-dual formulation of policy evaluation. This connection is of interest in itself, and may provide a foundation for deriving more effective algorithms.
+
+We start with the following classic, optimization formulation of policy evaluation (Puterman, 2014):
+
+$$
+R ^ {\pi} = \min  _ {V} \left\{(1 - \gamma) \sum_ {s} \mu_ {0} (s) V (s) \quad \text {s . t .} \quad V (s) \geq r ^ {\pi} (s) + \gamma \mathcal {P} ^ {\pi} V (s), \quad \forall s \right\}, \tag {14}
+$$
+
+where we find  $V$  to maximize its average value, subject to an inequality constraint on the Bellman equation. It can be shown that the solution of (14) is achieved by the true value function  $V^{\pi}$ , hence yielding a true expected reward  $R^{\pi}$ .
+
+Introducing a Lagrangian multiplier  $\rho \geq 0$ , we can derive the Lagrangian function  $L(V,\rho)$  of (14),
+
+$$
+L (V, \rho) = (1 - \gamma) \sum_ {s} \mu_ {0} (s) V (s) - \sum_ {s} \rho (s) \left(V (s) - r ^ {\pi} (s) + \gamma \mathcal {P} ^ {\pi} V (s)\right). \tag {15}
+$$
+
+Comparing  $L(V, \rho)$  with our estimator  $\widehat{R}_{\mathrm{DR}}^{\pi}[\widehat{V}, \widehat{w}]$  in (11), we can see that they are in fact equivalent in expectation.
+
+Theorem 4.1 (Primal Dual). I) Define  $w_{\rho/\pi_0}(s) = \frac{\rho(s)}{d_{\pi_0}(s)}$ . We have
+
+$$
+L (V, \rho) = R _ {D R} ^ {\pi} [ V, w _ {\rho / \pi_ {0}} ], \qquad a n d h e n c e \qquad L (V, d _ {\pi}) = L (V ^ {\pi}, \rho) = R ^ {\pi}, \forall V, \rho ,
+$$
+
+which suggests that  $L(V, \rho)$  is "doubly robust" in that it equals  $R^{\pi}$  if either  $V = V^{\pi}$  or  $\rho = d_{\pi}$ .
+
+II) The primal problem (14) forms a strong duality with the following dual problem,
+
+$$
+R ^ {\pi} = \max  _ {\rho \geq 0} \left\{\sum_ {s} \rho (s) r ^ {\pi} (s) \quad s. t. \quad \rho \left(s ^ {\prime}\right) = (1 - \gamma) \mu_ {0} \left(s ^ {\prime}\right) + \gamma \mathcal {T} ^ {\pi} \rho \left(s ^ {\prime}\right), \quad \forall s ^ {\prime} \right\}, \tag {16}
+$$
+
+where  $\mathcal{T}^{\pi}$  is defined in (3).
+
+This shows that the dual problem is equivalent to constraint  $\rho$  using the fixed point equation (3) and maximize the average reward given distribution  $\rho$ . Since the unique fixed point of (3) is  $d_{\pi}(s)$ , the solution of (16) naturally yields the true reward  $R^{\pi}$ , hence forming a zero duality gap with (14).
+
+It is natural to intuit the double robustness of the Lagrangian function. From (15),  $L(V, \rho)$  can be viewed as estimating the reward  $R^{\pi}$  using value function with a correction of Bellman residual  $(V - r^{\pi} - \gamma \mathcal{P}^{\pi} V)$ . If  $V = V^{\pi}$ , the estimation equals the true reward and the correction equals zero. From the dual problem (16),  $L(V, \rho)$  can be viewed as estimating  $R^{\pi}$  using density function  $\rho$ , corrected by the residual  $(\rho - (1 - \gamma) \mu_0 - \gamma \mathcal{T}^{\pi} \rho)$ . We again get the true reward if  $\rho = d_{\pi}$ .
+
+It turns out that we can use the primal-dual formula when  $\gamma = 1$  to obtain the double robust estimator for the average reward case. We clarify it in appendix B.
+
+Remark The fact that the density function  $d_{\pi}$  forms a dual variable of the value function  $V^{\pi}$  is widely known in the optimal control and reinforcement learning literature (e.g., Bertsekas, 2000; Puterman, 2014; de Farias & Van Roy, 2003), and has been leveraged in various works for policy optimization. However, it does not seem to be well exploited in the literature of off-policy policy evaluation.
+
+# 5 RELATED WORK
+
+Off-Policy Value Evaluation The problem of off-policy value evaluation has been studied extensively in contextual bandits and MDPs (Fonteneau et al., 2013; Li et al., 2015; Jiang & Li, 2016; Thomas & Brunskill, 2016; Liu et al., 2018b; Farajtabar et al., 2018; Hanna et al., 2019; Xie et al., 2019; Rowland et al., 2019). However, most of the existing works are based on importance sampling (IS) to correct the mismatch between the distribution of the whole trajectories induced by the behavior and target policies, which faces the "curse of horizon" (Liu et al., 2018a) when extended to long-horizon (or infinite-horizon) problems.
+
+Several other works (Guo et al., 2017; Hallak & Mannor, 2017; Liu et al., 2018a; Gelada & Bellemare, 2019; Nachum et al., 2019; Liu et al., 2019a) have been proposed to address the high variance issue in the long-horizon problems. Liu et al. (2018a) apply importance sampling on the average visitation distribution of state-action pairs, instead of the distribution of the whole trajectories, providing an effective approach to break "the curse of horizon". However, they require to learn a density ratio function over the whole state-action pairs, which may induce large bias. Our work incorporates the density ratio and value function estimation, which significantly reduces the induced bias of two estimators, resulting a doubly robust estimator.
+
+Our work is also closely related to DR techniques used in finite horizon problems (Murphy et al., 2001; Dudík et al., 2011; Jiang & Li, 2016; Thomas & Brunskill, 2016; Farajtabar et al., 2018), which incorporate an approximate value function as control variates to IS estimators. Different from existing DR approaches, our work is related to the well known duality between the density and the value function, which reveals the relationship between density (ratio) learning (Liu et al., 2018a) and value function learning. Based on this interesting observation, we further obtain the doubly robust estimator for estimating average reward in infinite-horizon problems.
+
+A recent work (Xie et al., 2019) has also explored the idea of tracking marginal state distribution shifts at every single time step in an episode, instead of the stationary state distribution across all time steps (Liu et al., 2018a). It provides additional benefits for understanding and analyzing problems
+
+![](images/a5c1584ad702bec39aa39bad94a65f3953817ed57f4d6f881b8ac8a52f66f7bc.jpg)
+
+![](images/9fc38024b1ee6e1abb18bbc4dd64c15961552c3d1630038bb56df024cfdb82a8.jpg)
+
+![](images/8b8cc1318bbbc79b4ffbc560766e0185c5edecc8748c5cb03bbcd3dfcbd7f7bb.jpg)
+
+![](images/9dd31d20363ed239accb6cc13f2050a531d2cc502e2d8f566f8b0d06b36d793c.jpg)  
+(a) MSE  
+(d) MSE with  $H$  changes  
+Figure 1: Off Policy Evaluation Results on Taxi. Default parameter, discounted factor  $\gamma = 0.99$ , mixed ratio  $\alpha = \beta = 1$ , horizon length  $H = 600$ . For (a)-(c) the x-axis is the number of trajectories and y-axis corresponds to MSE, Bias Square and Variance, respectively. For (d) we fix the total number of samples (number of trajectories times horizon length) and change the horizon length as x-axis and observe the MSE. (e) and (f) show the change the mixed ratio of  $\alpha, \beta$  with the change of bias. We repeat each experiment for 1000 runs.
+
+![](images/5617b522f73e6cdac5f6a02f36d8a600ca7393b26f3b6ea7f7f0172c17c7384f.jpg)  
+(b) Bias Square  
+(e) Bias square with  $\alpha$  changes
+
+![](images/2e6e56d930be33e8bf7e8a50a7c42a8133d000f2559540d40e14590858cbba20.jpg)  
+(c) Variance  
+(f) Bias square with  $\beta$  changes
+
+such as short horizon and time-variant MDPs. They call this approach marginalized importance sampling (MIS). Later on, Kallus & Uehara (2019a) incorporate the DR technique to improve the MIS estimator. Independent of this work, Kallus & Uehara (2019b); Uehara et al. (2019) extend previous works to propose estimators can be viewed as a variant of ours.
+
+Primal-Dual Value Learning Primal-dual optimization techniques have been widely used for off-policy value function learning and policy optimization (Liu et al., 2015; Chen & Wang, 2016; Dai et al., 2017; 2018; Feng et al., 2019). Nevertheless, the duality between density and value function has not been well explored in the literature of off policy value estimation. Our work proposes a new doubly robustness technique for off-policy value estimation, which can be naturally viewed as the Lagrangian function of the primal-dual formulation of policy evaluation, providing an alternative unified view for off policy value evaluation.
+
+# 6 EXPERIMENT
+
+In this section, we conduct simulation experiments on different environmental settings to compare our new doubly robust estimator with existing methods. We mainly compare with infinite horizon based estimator including state importance sampling estimator (Liu et al. (2018a)) and value function estimator. We do not report results on the vanilla trajectory-based importance sampling estimators because of their significant higher variance, but we do compare with the doubly robust version induced by Thomas & Brunskill (2016) (self-normalized variant of Jiang & Li (2016)). In all experiments we compare with Monte Carlo and naive average as Liu et al. (2018a) suggested. The ground truth for each environment is calculated by averaging Monte Carlo estimation with a very large sample size.
+
+![](images/988eb344cccba0146003aa4388181a52b5c45d99c8655f8429e5b4f27ff46b90.jpg)  
+Figure 2: Off Policy Evaluation Results on Puck-Mountain. We set discounted factor  $\gamma = 0.995$  as default. For (a)-(c) we set the horizon  $H = 1000$  and the x-axis is the number of trajectories for used for evaluation. For (d) we fix the total number of samples and change the horizon length.
+
+![](images/e1f6845a260739aed7f31797a0710613a99463777bed2f73d1b10b0d3e2078c4.jpg)  
+(a) MSE
+
+![](images/68eaf437c939c216be3bdc6ef425174ca002dcef43476a869c255f13a9bb931c.jpg)  
+(b) Bias Square
+
+![](images/5000d6581248065d7575478a93f753216ef0385c5a021566bb0e13edeca83b68.jpg)  
+(c) Variance  
+Figure 3: Off Policy Evaluation Results on InvertedPendulum-v2. We set discounted factor  $\gamma = 0.995$  as default. For (a)-(c) we set the horizon  $H = 1000$  and the x-axis is the number of trajectories for used for evaluation. For (d) we fix the total number of samples and change the horizon length.
+
+![](images/7a38a14fb7ea7728023e017b4af3f0ce6a75ea943362ef29f56f023d5f2b1f2e.jpg)  
+(d) MSE with  $H$  changes
+
+Taxi Environment We follow Liu et al. (2018a)'s tabular environment Taxi, which has 2000 states and 6 actions in total. For more experimental details, please check appendix C.1.
+
+We pre-train two different  $\widehat{V}$  and  $\tilde{V}$  trained with a small and fairly large size of samples, respectively, where  $\tilde{V}$  is very close to true value function  $V^{\pi}$  but  $\widehat{V}$  is relatively further from it. Similarly we pre-train  $\widehat{\rho}$  and  $\tilde{\rho} \approx d_{\pi}$ . For estimation we use a mixed ratio  $\alpha, \beta$  to control the bias of the input  $V, \rho$ , where  $V = \alpha \widehat{V} + (1 - \alpha) \tilde{V}$  and  $\rho = \beta \widehat{\rho} + (1 - \beta) \tilde{\rho}$ .
+
+Figure 1(a)-(c) show results of comparison for different methods as we changing the number of trajectories. We can see that the MSE performance of value function  $(\widehat{R}_{\mathrm{VAL}}^{\pi})$  and state visitation importance sampling  $(\widehat{R}_{\mathrm{SIS}}^{\pi})$  estimators are mainly impeded by their large biases, while our method has much less bias thus it can keep decreasing as sample size increase and achieves same performance as on policy estimator. Figure 1(d) shows results if we change the horizon length. Notice that here we keep the number of samples to be the same, so if we increase our horizon length we will decrease the number of trajectories in the same time. We can see that our method alongside with all infinite horizon methods will get better result as horizon length increase. Figure 1(e)-(f) indicate the "double robustness" of our method, where our method benefits from either a better  $V$  or a better  $\rho$ .
+
+Puck-Mountain Puck-Mountain is an environment similar to Mountain-Car, except that the goal of Puck-Mountain is to push the puck as high as possible in a local valley, which has a continuous state space of  $\mathbb{R}^2$  and a discrete action space similar to Mountain-Car. We use the softmax functions of an optimal Q-function as both target policy and behavior policy, where the temperature of the behavior policy is higher (encouraging exploration). For more details of constructing policies and training algorithms for density ratio and value functions, please check appendix C.2.
+
+Figure 2(a)-(c) show results of comparison for different methods as we changing the number of trajectories. Similar to taxi, we find our method has much lower bias than density ratio and value function estimation, which yields a better MSE. In Figure 2(d) the performance for all infinite horizon estimator will not degenerate as horizon increases, while finite horizon method such as finite weighted horizon doubly robust will suffer from larger variance as horizon increases.
+
+InvertedPendulum InvertedPendulum is a pendulum that has its center of mass above its pivot point. We use the implementation of InvertedPendulum from OpenAI gym (Brockman et al., 2016), which is a continuous control task with state space in  $\mathcal{R}^4$  and we discrete the action space to be  $\{-1, -0.3, -0.2, 0, 0.2, 0.3, 1\}$ . More experiment details can be found in appendix C.2.
+
+In Figure 3(a)-(c) our method again significantly reduces the bias, which yields a better MSE comparing with value and density estimation. Figure 3(d) also shows that our method consistently outperforms all other methods as the horizon increases with a fixed total timesteps.
+
+# 7 CONCLUSION
+
+In this paper, we develop a new doubly robust estimator based on the infinite horizon density ratio and off policy value estimation. Our new proposed doubly robust estimator can be accurate as long as one of the estimators are accurate, which yields a significant advantage comparing to previous estimators. Future directions include deriving more novel optimization algorithms to learn value function and density(ratio) function by using the primal dual framework.
+
+# ACKNOWLEDGEMENT
+
+This work is supported in part by NSF CRII 1830161 and NSF CAREER 1846421. We would like to acknowledge Google Cloud for their support.
+
+# REFERENCES
+
+Søren Asmussen and Peter W Glynn. Stochastic simulation: algorithms and analysis, volume 57. Springer Science & Business Media, 2007.  
+Dimitri P. Bertsekas. Dynamic Programming and Optimal Control. Athena Scientific, 2nd edition, 2000. ISBN 1886529094.  
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+Bo Dai, Albert Shaw, Lihong Li, Lin Xiao, Niao He, Zhen Liu, Jianshu Chen, and Le Song. SBEED: Convergent reinforcement learning with nonlinear function approximation. In Proceedings of the Thirty-Fifth International Conference on Machine Learning (ICML), pp. 1133-1142, 2018.  
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+
+Mehrdad Farajtabar, Yinlam Chow, and Mohammad Ghavamzadeh. More robust doubly robust off-policy evaluation. In Proceedings of the 35th International Conference on Machine Learning (ICML), pp. 1446-1455, 2018.  
+Yihao Feng, Lihong Li, and Qiang Liu. A kernel loss for solving the bellman equation. Neural Information Processing Systems (NeurIPS), 2019.  
+Raphael Fonteneau, Susan A. Murphy, Louis Wehenkel, and Damien Ernst. Batch mode reinforcement learning based on the synthesis of artificial trajectories. Annals of Operations Research, 208(1): 383-416, 2013.  
+Carles Gelada and Marc G Bellemare. Off-policy deep reinforcement learning by bootstrapping the covariate shift. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 33, pp. 3647-3655, 2019.  
+Zhaohan Guo, Philip S. Thomas, and Emma Brunskill. Using options and covariance testing for long horizon off-policy policy evaluation. In Advances in Neural Information Processing Systems 30 (NIPS), pp. 2489-2498, 2017.  
+Assaf Hallak and Shie Mannor. Consistent on-line off-policy evaluation. In Proceedings of the 34th International Conference on Machine Learning (ICML), pp. 1372-1383, 2017.  
+Josiah Hanna, Scott Niekum, and Peter Stone. Importance sampling policy evaluation with an estimated behavior policy. In Proceedings of the 36th International Conference on Machine Learning, volume 97, pp. 2605-2613, 2019.  
+Nan Jiang and Lihong Li. Doubly robust off-policy evaluation for reinforcement learning. In Proceedings of the 23rd International Conference on Machine Learning (ICML), pp. 652-661, 2016.  
+Nathan Kallus and Masatoshi Uehara. Double reinforcement learning for efficient off-policy evaluation in markov decision processes. arXiv preprint arXiv:1908.08526, 2019a.  
+Nathan Kallus and Masatoshi Uehara. Efficiently breaking the curse of horizon: Double reinforcement learning in infinite-horizon processes. arXiv preprint arXiv:1909.05850, 2019b.  
+Lihong Li, Wei Chu, John Langford, and Xuanhui Wang. Unbiased offline evaluation of contextual-bandit-based news article recommendation algorithms. In Proceedings of the 4th International Conference on Web Search and Data Mining (WSDM), pp. 297-306, 2011.  
+Lihong Li, Rémi Munos, and Csaba Szepesvári. Toward minimax off-policy value estimation. In Proceedings of the 18th International Conference on Artificial Intelligence and Statistics (AISTATS), pp. 608-616, 2015.  
+Bo Liu, Ji Liu, Mohammad Ghavamzadeh, Sridhar Mahadevan, and Marek Petrik. Finite-sample analysis of proximal gradient td algorithms. In UAI, pp. 504-513. Citeseer, 2015.  
+Jun S. Liu. Monte Carlo Strategies in Scientific Computing. Springer Series in Statistics. Springer-Verlag, 2001. ISBN 0387763694.  
+Qiang Liu, Lihong Li, Ziyang Tang, and Dengyong Zhou. Breaking the curse of horizon: Infinite-horizon off-policy estimation. In Advances in Neural Information Processing Systems, pp. 5361-5371, 2018a.  
+Yao Liu, Omer Gottesman, Aniruddh Raghu, Matthieu Komorowski, Aldo A Faisal, Finale Doshi-Velez, and Emma Brunskill. Representation balancing mdps for off-policy policy evaluation. In Advances in Neural Information Processing Systems, pp. 2644-2653, 2018b.  
+Yao Liu, Pierre-Luc Bacon, and Emma Brunskill. Understanding the curse of horizon in off-policy evaluation via conditional importance sampling. arXiv preprint arXiv:1910.06508, 2019a.  
+Yao Liu, Adith Swaminathan, Alekh Agarwal, and Emma Brunskill. Off-policy policy gradient with state distribution correction. arXiv preprint arXiv:1904.08473, 2019b.
+
+Susan A. Murphy, Mark van der Laan, and James M. Robins. Marginal mean models for dynamic regimes. Journal of the American Statistical Association, 96(456):1410-1423, 2001.  
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+Martin L Puterman. Markov Decision Processes.: Discrete Stochastic Dynamic Programming. John Wiley & Sons, 2014.  
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+
+# A PROOF
+
+# A.1 TRANSITION OPERATOR FOR BELLMAN EQUATION
+
+For simplicity, we define the following two operators thorough our proofs to simplify our notations.
+
+Definition A.1. Given a policy  $\pi$  and the unknown environment transition  $T$ , we define  $\mathcal{T}^{\pi}$  and  $\mathcal{P}^{\pi}$  over any function  $f: S \to \mathcal{R}$  as
+
+$$
+\left(\mathcal {T} ^ {\pi} f\right) \left(s ^ {\prime}\right) = \sum_ {s, a} \boldsymbol {T} \left(s ^ {\prime} | s, a\right) \pi (a | s) f (s)
+$$
+
+$$
+\left(\mathcal {P} ^ {\pi} f\right) (s) = \sum_ {s ^ {\prime}, a} \boldsymbol {T} \left(s ^ {\prime} | s, a\right) \pi (a | s) f \left(s ^ {\prime}\right)
+$$
+
+Using these operator notations, we can rewrite the above two recursive equations as:
+
+$$
+V ^ {\pi} = r ^ {\pi} + \gamma \mathcal {P} ^ {\pi} V ^ {\pi},
+$$
+
+$$
+d _ {\pi} = (1 - \gamma) \mu_ {0} + \gamma T ^ {\pi} d _ {\pi},
+$$
+
+where  $r^{\pi}(s) = \mathbb{E}_{a\sim \pi (\cdot |s)}[r(s,a)]$
+
+These transition operators have the following nice adjoint property.
+
+Lemma A.2. For two function  $f$  and  $g$ , if the following summation is finite, we will have
+
+$$
+\sum_ {s} \left(\mathcal {P} ^ {\pi} f\right) (s) g (s) = \sum_ {s} f (s) \left(\mathcal {T} ^ {\pi} g\right) (s). \tag {17}
+$$
+
+Proof.
+
+$$
+\begin{array}{l} \sum_ {s} \left(\mathcal {P} ^ {\pi} f\right) (s) g (s) = \sum_ {s} \left(\sum_ {s ^ {\prime}, a} \boldsymbol {T} \left(s ^ {\prime} \mid s, a\right) \pi (a \mid s) f \left(s ^ {\prime}\right)\right) g (s) \\ = \sum_ {s ^ {\prime}} f \left(s ^ {\prime}\right) \left(\sum_ {s, a} \boldsymbol {T} \left(s ^ {\prime} \mid s, a\right) \pi (a \mid s) g (s)\right) \\ = \sum_ {s ^ {\prime}} f (s ^ {\prime}) \left(\mathcal {T} ^ {\pi} g\right) (s ^ {\prime}) \\ \end{array}
+$$
+
+![](images/daa7a2c57c506447ea4dacd4b2db68750fc3f95052d227bdaad3ec34d08a0d3e.jpg)
+
+Using this property we can actually using Bellman Equations to re-derive the two different way to get  $R^{\pi}$ .
+
+$$
+\begin{array}{l} R ^ {\pi} = \lim _ {T \to \infty} \mathbb {E} _ {\tau^ {(i)} \sim \pi} [ \frac {\sum_ {t = 0} ^ {T} \gamma^ {t} r _ {t}}{\sum_ {t = 0} ^ {T} \gamma^ {t}} ] \\ = \sum_ {s} V ^ {\pi} (s) (1 - \gamma) \mu_ {0} (s) \\ = \sum_ {s} V ^ {\pi} (s) (\mathcal {I} - \gamma \mathcal {T} ^ {\pi}) d _ {\pi} (s) \\ = \sum_ {s} \left(\mathcal {I} - \gamma \mathcal {P} ^ {\pi}\right) V ^ {\pi} (s) d _ {\pi} (s) \\ = \sum_ {s} r ^ {\pi} (s) d _ {\pi} (s). \\ \end{array}
+$$
+
+# A.2 PROOF OF THEOREM 3.1
+
+Theorem 3.1 (Doubly Robustness). Let  $R_{DR}^{\pi}[\widehat{V}, \widehat{w}] \coloneqq \lim_{n_0, n, T \to \infty} \widehat{R}_{DR}^{\pi}[\widehat{V}, \widehat{w}]$  be the limit of  $\widehat{R}_{DR}^{\pi}$  when it has infinite samples. Following the definition above, we have
+
+$$
+R _ {D R} ^ {\pi} [ \widehat {V}, \widehat {w} ] - R ^ {\pi} = \mathbb {E} _ {s \sim d _ {\pi_ {0}}} \left[ \varepsilon_ {\widehat {w}} (s) \varepsilon_ {\widehat {V}} (s) \right], \tag {12}
+$$
+
+where  $\varepsilon_{\widehat{V}}$  and  $\varepsilon_{\widehat{w}}$  are errors of  $\widehat{V}$  and  $\widehat{w}$ , respective, defined as follows
+
+$$
+\varepsilon_ {\widehat {w}} = \frac {d _ {\pi} (s)}{d _ {\pi_ {0}} (s)} - \widehat {w} (s), \quad \varepsilon_ {\widehat {V}} (s) = \widehat {V} (s) - r ^ {\pi} (s) - \gamma \mathcal {P} ^ {\pi} \widehat {V} (s).
+$$
+
+The error  $\varepsilon_{\widehat{w}}$  of  $\widehat{w}$  is measured by the difference with the true density ratio  $d_{\pi}(s) / d_{\pi_0}(s)$ , and the error  $\varepsilon_{\widehat{V}}$  of  $\widehat{V}$  is measured using the Bellman residual.
+
+Proof. Using the property of the operator, we can rewrite  $(1 - \gamma)\mu_0(s)$  using Bellman equation as  $d_{\pi} - \gamma T^{\pi}d_{\pi}$ , thus we have
+
+$$
+\begin{array}{l} R _ {\mathrm {V A L}} ^ {\pi} [ \widehat {V} ] = (1 - \gamma) \sum_ {s} \widehat {V} (s) \mu_ {0} (s) \\ = \sum_ {s} \widehat {V} (s) \left(d _ {\pi} - \gamma \mathcal {T} ^ {\pi} d _ {\pi}\right) (s) \\ = \sum_ {s} (I - \gamma \mathcal {P} ^ {\pi}) \widehat {V} (s) d _ {\pi} (s). \\ \end{array}
+$$
+
+and similarly if we break  $r^\pi$  as  $(I - \gamma \mathcal{P}^{\pi})V^{\pi}$ , for  $R_{\mathrm{SIS}}^{\pi}[\widehat{w} ]$  we have:
+
+$$
+R _ {\mathrm {S I S}} ^ {\pi} [ \widehat {w} ] = \sum_ {s} (I - \gamma \mathcal {P} ^ {\pi}) V ^ {\pi} (s) d _ {\widehat {w}} (s),
+$$
+
+where  $d_{\hat{w}} = d_{\pi_0}\widehat{w}$  for short.
+
+Compare with  $R^{\pi} = \sum_{s}\left(I - \gamma \mathcal{P}^{\pi}\right)V^{\pi}(s)d_{\pi}(s)$ , we can see the main difference between  $R_{\mathrm{SIS}}^{\pi}$  and  $R_{\mathrm{VAL}}^{\pi}$  with  $R^{\pi}$  are they replace  $d_{\pi}$  and  $d_{\widehat{w}}$  and  $V^{\pi}$  with  $\widehat{V}$  respectively. If we add them together and minus the connection estimator, we have we will have:
+
+$$
+\begin{array}{l} R _ {\mathrm {D R}} ^ {\pi} [ \widehat {V}, \widehat {w} ] - R ^ {\pi} = R _ {\mathrm {S I S}} ^ {\pi} [ \widehat {w} ] + R _ {\mathrm {V A L}} ^ {\pi} [ \widehat {V} ] - R _ {\mathrm {b r i d g e}} ^ {\pi} [ \widehat {V}, \widehat {w} ] - R ^ {\pi} \\ = \sum_ {s} \left(\left(I - \gamma \mathcal {P} ^ {\pi}\right) V ^ {\pi} (s) d _ {\widehat {w}} (s) + \left(I - \gamma \mathcal {P} ^ {\pi}\right) \widehat {V} (s) d _ {\pi} (s) \right. \\ \left. - \left(I - \gamma \mathcal {P} ^ {\pi}\right) \widehat {V} (s) d _ {\widehat {w}} (s) - \left(I - \gamma \mathcal {P} ^ {\pi}\right) V ^ {\pi} (s) d _ {\pi} (s)\right) \\ = \sum_ {s} (I - \gamma \mathcal {P} ^ {\pi}) (V ^ {\pi} - \widehat {V}) (s) (d _ {\widehat {w}} - d _ {\pi}) (s) \\ = \sum_ {s} d _ {\pi_ {0}} (s) \varepsilon_ {\widehat {V}} (s) \varepsilon_ {\widehat {w}} (s) \\ = \mathbb {E} _ {s \sim d _ {\pi_ {0}}} \left[ \varepsilon_ {\widehat {V}} (s) \varepsilon_ {\widehat {w}} (s) \right], \\ \end{array}
+$$
+
+where the third equation is because  $(I - \gamma \mathcal{P}^{\pi})(V^{\pi} - \widehat{V}) = (I - \gamma \mathcal{P}^{\pi})V^{\pi} - (I - \gamma \mathcal{P}^{\pi})\widehat{V} = r^{\pi} - (I - \gamma \mathcal{P}^{\pi})\widehat{V}$ . Similarly, the bias for  $R_{\mathrm{DR}}^{\pi}$  and  $R_{\mathrm{SIS}}^{\pi}$  can be calculated as:
+
+$$
+\begin{array}{l} R _ {\mathrm {S I S}} ^ {\pi} [ \hat {w} ] - R ^ {\pi} = \sum_ {s} (I - \gamma \mathcal {P} ^ {\pi}) V ^ {\pi} (s) (d _ {\hat {w}} (s) - d _ {\pi} (s)) \\ = \sum_ {s} r ^ {\pi} (s) d _ {\pi_ {0}} (s) \varepsilon_ {\hat {w}} (s) \\ = \mathbb {E} _ {s \sim d _ {\pi_ {0}}} \left[ \varepsilon_ {\widehat {w}} (s) r ^ {\pi} (s) \right], \\ \end{array}
+$$
+
+and
+
+$$
+\begin{array}{l} R _ {\text {V A L}} ^ {\pi} [ \widehat {V} ] - R ^ {\pi} = \sum_ {s} (I - \gamma \mathcal {P} ^ {\pi}) (\widehat {V} (s) - V ^ {\pi} (s)) d _ {\pi} (s) \\ = \sum_ {s} \varepsilon_ {\widehat {V}} (s) w _ {\pi / \pi_ {0}} (s) d _ {\pi_ {0}} (s) \\ = \mathbb {E} _ {s \sim d _ {\pi_ {0}}} \left[ w _ {\pi / \pi_ {0}} (s) \varepsilon_ {\hat {V}} (s) \right]. \\ \end{array}
+$$
+
+![](images/ab5f31ca11b044463bacc0f6d604f3b489f3f30e1ef7b26f3989cc6aacee798a.jpg)
+
+# A.3 MORE DISCUSSIONS ON THE VARIANCE IN PROPOSITION 3.1
+
+Recall the proposition 3.1.
+
+Proposition 3.1 (Variance Analysis). Assume  $\widehat{R}_{DR}^{\pi}[\widehat{V},\widehat{w}]$  is estimated based on sample  $\mathcal{D}_0\sim \mu_0$  and  $\mathcal{D}_{\pi_0}\sim d_{\pi_0}$ , which we assume to be independent of each other. We have
+
+$$
+\operatorname {V a r} _ {\mathcal {D} _ {0}, \mathcal {D} _ {\pi_ {0}}} \left[ \widehat {R} _ {D R} ^ {\pi} [ \widehat {V}, \widehat {w} ] \right] = \operatorname {V a r} _ {\mathcal {D} _ {0}} \left[ \widehat {R} _ {V A L} ^ {\pi} [ \widehat {V} ] \right] + \operatorname {V a r} _ {\mathcal {D} _ {\pi_ {0}}} \left[ \widehat {R} _ {r e s} ^ {\pi} [ \widehat {V}, \widehat {w} ] \right], \tag {13}
+$$
+
+with  $\widehat{R}_{res}^{\pi}[\widehat{V},\widehat{w} ]:= \widehat{R}_{SIS}^{\pi}[\widehat{w}] - \widehat{R}_{bridge}^{\pi}[\widehat{V},\widehat{w}] = \widehat{\mathbb{E}}_{\mathcal{D}_{\pi_0}}\left[\widehat{\varepsilon}_{\hat{V}}(s)\widehat{w} (s)\right]$  and  $\widehat{\varepsilon}_{\hat{V}}(s)$  the TD error of  $\widehat{V}$ ,
+
+$$
+\widehat {\varepsilon} _ {\widehat {V}} (s) = \widehat {r ^ {\pi}} (s) - \widehat {V} (s) + \gamma \widehat {\mathcal {P} ^ {\pi}} \widehat {V} (s),
+$$
+
+with  $\widehat{r^{\pi}}(s) = r(s, a)\pi(a|s)/\pi_0(a|s), \widehat{\mathcal{P}^{\pi}}\widehat{V}(s) = \pi(a|s)/\pi_0(a|s)\widehat{V}(s')$ . Therefore,  $Var_{\mathcal{D}_{\pi_0}}(\widehat{R}_{res}^{\pi}[\widehat{V}, \widehat{w}])$  can be small when  $\widehat{V}$  is close to the true value  $V^{\pi}$ , or  $\widehat{\varepsilon}_{\widehat{V}}(s) \approx 0$ .
+
+Proof. Since  $\widehat{R}_{\mathrm{DR}}^{\pi}[\widehat{V},\widehat{w}] = \widehat{R}_{\mathrm{VAL}}^{\pi}[\widehat{V}] + \widehat{R}_{\mathrm{res}}^{\pi}[\widehat{V},\widehat{w}]$  and we have the assumption that the samples from  $\mathcal{D}_0$  and  $\mathcal{D}_{\pi_0}$  are independent, the proposition comes immediately.
+
+Intuitively, the variance of  $\widehat{R}_{\mathrm{res}}^{\pi}[\widehat{V},\widehat{w} ]$  is less than  $\widehat{R}_{\mathrm{SIS}}^{\pi}[\widehat{w} ]$ . We want to quantitatively study every pieces of orthogonal randomness of  $\widehat{R}_{\mathrm{res}}^{\pi}[\widehat{V},\widehat{w} ]$  and  $\widehat{R}_{\mathrm{SIS}}^{\pi}[\widehat{w} ]$  in details, and we state the analysis as the following theorem.
+
+Theorem A.3. Let  $\text{Var}_{\mathcal{D}_{\pi_0}}\left[\widehat{R}_{res}^{\pi}[\widehat{V},\widehat{w}]\right]$  be defined in Proposition 3.1, and suppose the normalization is constant (and hence an ordinary importance sampling which is unbiased). Then we can further break it into two terms
+
+$$
+\left. \right. \operatorname {V a r} _ {\mathcal {D} _ {\pi_ {0}}} \left[ \widehat {R} _ {r e s} ^ {\pi} [ \widehat {V}, \widehat {w} ] \right] = \frac {1}{n} \left(\operatorname {V a r} \left[ \widehat {w} (s) \varepsilon_ {\widehat {V}} (s) \right] + \mathbb {E} \left[ \widehat {w} (s) ^ {2} \left(\delta_ {1} (s, a) + \gamma \delta_ {2} (s, a, s ^ {\prime})\right) ^ {2} \right]\right), \tag {18}
+$$
+
+where  $\varepsilon_{\widehat{V}}(s) = \widehat{V}(s) - r^{\pi}(s) - \gamma \mathcal{P}^{\pi} \widehat{V}(s')$  is the Bellman residual (not the empirical one),  $\delta_1(s, a) = \frac{\pi(a|s)}{\pi_0(a|s)} r(s, a) - r^{\pi}(s)$  is the randomness for action and  $\delta_2(s, a, s') = \frac{\pi(a|s)}{\pi_0(a|s)} \widehat{V}(s') - \mathcal{P}^{\pi} \widehat{V}(s)$  is the randomness for transition operator over function  $\widehat{V}$ . Both  $\delta_1$  and  $\delta_2$  is zero mean if we condition over  $s$ .
+
+Compared with Var  $\left[\widehat{R}_{SIS}^{\pi}[\widehat{w}]\right]$  we have:
+
+$$
+\operatorname {V a r} \left[ \widehat {R} _ {S I S} ^ {\pi} [ \widehat {w} ] \right] = \frac {1}{n} \left(\operatorname {V a r} \left[ \widehat {w} (s) r ^ {\pi} (s) \right] + \mathbb {E} \left[ \widehat {w} (s) ^ {2} \delta_ {1} (s, a) ^ {2} \right]\right) \tag {19}
+$$
+
+Proof.  $\widehat{R}_{\mathrm{res}}^{\pi}[\widehat{V},\widehat{w} ]$  can be written as
+
+$$
+\hat {R} _ {\mathrm {r e s}} ^ {\pi} [ \hat {V}, \hat {w} ] = \frac {1}{n} \sum \hat {w} (s) \left(\beta_ {\pi / \pi_ {0}} (a | s) (r + \gamma \hat {V} (s ^ {\prime})) - \hat {V} (s)\right),
+$$
+
+where  $\beta_{\pi/\pi_0}(a|s)$  is short for  $\frac{\pi(a|s)}{\pi_0(a|s)}$ . We can break  $\beta_{\pi/\pi_0}(a|s)(r + \gamma \widehat{V}(s')) - \widehat{V}(s)$  into
+
+$$
+\begin{array}{l} \beta_ {\pi / \pi_ {0}} (a | s) (r (s, a) + \gamma \widehat {V} (s ^ {\prime})) - \widehat {V} (s) \\ = (- \widehat {V} (s) + r ^ {\pi} (s) + \gamma \mathcal {P} ^ {\pi} \widehat {V} (s)) + \underbrace {(\beta_ {\pi / \pi_ {0}} (a | s) r (s , a) - r ^ {\pi} (s))} _ {\delta_ {1}} + \gamma \underbrace {(\beta_ {\pi / \pi_ {0}} (a | s) \widehat {V} (s ^ {\prime}) - \mathcal {P} ^ {\pi} \widehat {V} (s))} _ {\delta_ {2}} \\ = - \varepsilon_ {\widehat {V}} (s) + \delta_ {1} (s, a) + \gamma \delta_ {2} (s, a, s ^ {\prime}). \\ \end{array}
+$$
+
+where  $\varepsilon_{\widehat{V}} = \widehat{V} - r^{\pi} - \mathcal{P}^{\pi}\widehat{V}$  is the Bellman residual and the if we condition over  $s$  we have the expectations for  $\delta_1$  and  $\delta_2$  are 0. Also notice that if we condition over  $s$  then  $\varepsilon_{\widehat{V}}(s)$  becomes a constant thus it is independent to  $\delta_1$  and  $\delta_2$ . Thus we have:
+
+$$
+\begin{array}{l} \operatorname {V a r} \left[ \widehat {w} (s) \left(\beta_ {\pi / \pi_ {0}} (a | s) \left(r + \gamma \widehat {V} \left(s ^ {\prime}\right)\right) - \widehat {V} (s)\right) \right] \\ = \operatorname {V a r} \left[ \widehat {w} (s) \left(- \varepsilon_ {\widehat {V}} (s) + \delta_ {1} (s, a) + \gamma \delta_ {2} (s, a, s ^ {\prime})\right) \right] \\ = \operatorname {V a r} \left[ \widehat {w} (s) \varepsilon_ {\widehat {V}} (s) \right] + \mathbb {E} \left[ \widehat {w} (s) ^ {2} \left(\delta_ {1} (s, a) + \gamma \delta_ {2} (s, a, s ^ {\prime})\right) ^ {2} \right] \\ \end{array}
+$$
+
+Therefore we have:
+
+$$
+\operatorname {V a r} \left[ \widehat {R} _ {\mathrm {D R}} ^ {\pi} [ \widehat {V}, \widehat {w} ] \right] = \frac {(1 - \gamma) ^ {2}}{n _ {0}} \operatorname {V a r} [ \widehat {V} (s _ {0}) ] + \frac {1}{n} (\operatorname {V a r} \left[ \widehat {w} (s) \varepsilon_ {\widehat {V}} (s) \right] + \mathbb {E} \left[ \widehat {w} (s) ^ {2} \left(\delta_ {1} (s, a) + \gamma \delta_ {2} (s, a, s ^ {\prime})\right) ^ {2} \right]).
+$$
+
+For  $\operatorname{Var}\left[\widehat{R}_{\mathrm{SIS}}^{\pi}[\widehat{w}]\right]$  we have:
+
+$$
+\begin{array}{l} \operatorname {V a r} \left[ \widehat {R} _ {\mathrm {S I S}} ^ {\pi} [ \widehat {w} ] \right] = \frac {1}{n} \operatorname {V a r} \left[ \widehat {w} (s) \beta_ {\pi / \pi_ {0}} (a | s) r (s, a) \right] \\ = \frac {1}{n} \operatorname {V a r} [ \widehat {w} (s) r ^ {\pi} (s) + \widehat {w} (s) \delta_ {1} (s, a) ] \\ = \frac {1}{n} (\operatorname {V a r} [ \widehat {w} (s) r ^ {\pi} (s) ] + \mathbb {E} [ \widehat {w} ^ {2} (s) \delta_ {1} (s, a) ^ {2} ]). \\ \end{array}
+$$
+
+![](images/0fd01ad14a1398224c096575648a56755eaeb11f527b757a5372cab08d905e03.jpg)
+
+From the theorem we can see that the variance of residual comes from two parts, the majority part relies on the variance of  $|\varepsilon_{\widehat{V}}(s)|$  is usually much smaller than  $r^{\pi}$  as the majority variance of state visitation importance sampling. When  $\widehat{V} \approx V^{\pi}$ ,  $\varepsilon_{\widehat{V}}(s) \approx 0$ , the variance of residual only comes from  $\delta_1(s,a)$  and  $\delta_2(s,a,s')$ .
+
+In practice when we get trajectories data from  $\pi_0$ , to get samples uniformly from  $d_{\pi_0}$ , we can either draw sample  $s_t$  depends on its discounted factor  $\gamma^t$  or add an importance weight  $\gamma^t$  as we did in our empirical estimator in equation (6) and equation (10).
+
+# A.4 PROOF OF THEOREM 4.1
+
+Theorem 4.1 (Primal Dual). I) Define  $w_{\rho/\pi_0}(s) = \frac{\rho(s)}{d_{\pi_0}(s)}$ . We have
+
+$$
+L (V, \rho) = R _ {D R} ^ {\pi} [ V, w _ {\rho / \pi_ {0}} ], \quad \text {a n d h e n c e} \quad L (V, d _ {\pi}) = L (V ^ {\pi}, \rho) = R ^ {\pi}, \forall V, \rho ,
+$$
+
+which suggests that  $L(V, \rho)$  is "doubly robust" in that it equals  $R^{\pi}$  if either  $V = V^{\pi}$  or  $\rho = d_{\pi}$ .
+
+II) The primal problem (14) forms a strong duality with the following dual problem,
+
+$$
+R ^ {\pi} = \max  _ {\rho \geq 0} \left\{\sum_ {s} \rho (s) r ^ {\pi} (s) \quad s. t. \quad \rho \left(s ^ {\prime}\right) = (1 - \gamma) \mu_ {0} \left(s ^ {\prime}\right) + \gamma \mathcal {T} ^ {\pi} \rho \left(s ^ {\prime}\right), \quad \forall s ^ {\prime} \right\}, \tag {16}
+$$
+
+where  $\mathcal{T}^{\pi}$  is defined in (3).
+
+Proof. The Lagrangian can be written as:
+
+$$
+\begin{array}{l} L (V, \rho) = (1 - \gamma) \sum_ {s} \mu_ {0} (s) V (s) - \sum_ {s} \rho (s) \left(V (s) - r ^ {\pi} (s) - \gamma \mathcal {P} ^ {\pi} V (s)\right) \\ = \underbrace {\sum_ {s} (1 - \gamma) \mu_ {0} (s) V (s)} _ {= R _ {\mathrm {V A L}} ^ {\pi} [ V ]} - \underbrace {\sum_ {s} \rho (s) \left(I - \gamma \mathcal {P} ^ {\pi}\right) V (s)} _ {= R _ {\mathrm {b r i d g e}} ^ {\pi} [ V, w _ {\rho / \pi_ {0}} ]} + \underbrace {\sum_ {s} \rho (s) r ^ {\pi} (s)} _ {= R _ {\mathrm {S I S}} ^ {\pi} [ w _ {\rho / \pi_ {0}} ]} \\ = \sum_ {s} (1 - \gamma) \mu_ {0} (s) V (s) - \sum_ {s} \left(I - \gamma \mathcal {T} ^ {\pi}\right) \rho (s) V (s) + \sum_ {s} \rho (s) r ^ {\pi} (s) \\ = \sum_ {s} \left((1 - \gamma) \mu_ {0} (s) - (I - \gamma \mathcal {T} ^ {\pi}) \rho (s)\right) V (s) + \sum_ {s} \rho (s) r ^ {\pi} (s). \\ \end{array}
+$$
+
+We can see that the Lagrangian  $L(V,\rho)$  is actually our doubly robust estimator  $R_{\mathrm{DR}}^{\pi}[V,w_{\rho /\pi_0}]$ .
+
+From the last equation we can derive our dual as:
+
+$$
+\begin{array}{l} \max _ {\rho \geq 0} \sum_ {s} \rho (s) r ^ {\pi} (s) \\ \begin{array}{l} \text {s . t .} \quad \rho (s) = (1 - \gamma) \mu_ {0} (s) + \gamma \mathcal {T} ^ {\pi} \rho (s), \forall s. \end{array} \\ \end{array}
+$$
+
+![](images/ccd7b5453826451635fcea87dd2fa30e811a35efdb0dc15965af5bd5f0968d6c.jpg)
+
+# B DOUBLY ROBUST ESTIMATOR FOR AVERAGE CASE
+
+# B.1 PRIMAL DUAL FRAMEWORK
+
+We start from primal dual framework to get our doubly robust estimator similar to section 4. To estimate the average reward of a given policy  $\pi$ , we can consider solve the following linear programming:
+
+$$
+\begin{array}{l} \max  _ {\rho \geq 0} \sum_ {s} \rho (s) r ^ {\pi} (s) \\ \text {s . t .} \sum_ {s} \rho (s) = 1, \quad \rho (s) = \mathcal {T} ^ {\pi} \rho (s), \forall s, \tag {20} \\ \end{array}
+$$
+
+where  $\rho(s)$  is the stationary distribution of states under  $\mathcal{P}^{\pi}$ , and the objective is the average reward given  $\pi$ .
+
+Consider the Lagrangian of above linear programming:
+
+$$
+\begin{array}{l} L (V, \rho , \bar {v}) = \sum_ {s} \rho (s) r ^ {\pi} (s) - \sum_ {s} V (s) (\rho (s) - \mathcal {T} ^ {\pi} \rho (s)) - \bar {v} (\sum_ {s} \rho (s) - 1) \\ = \bar {v} - \sum_ {s} \rho (s) (V (s) - r ^ {\pi} (s) - \mathcal {P} ^ {\pi} V (s) + \bar {v}). \tag {21} \\ \end{array}
+$$
+
+From Equation (21) we can get the dual formula as:
+
+$$
+\begin{array}{l} \min  _ {V, \bar {v}} \bar {v} \\ \text {s . t .} \bar {v} + V (s) - \mathcal {P} ^ {\pi} V (s) - r ^ {\pi} (s) \geq 0, \forall s, \tag {22} \\ \end{array}
+$$
+
+where  $V(s)$  is the value function and  $\bar{v}$  is the average reward we want to optimize.
+
+Notice that in average case,  $V^{\pi}(s)$  can be viewed as the fixed-point solution to the following Bellman equation:
+
+$$
+V ^ {\pi} (s) - \mathbb {E} _ {s ^ {\prime}, a | s \sim d _ {\pi}} \left[ V ^ {\pi} \left(s ^ {\prime}\right) \right] = \mathbb {E} _ {a | s \sim \pi} \left[ r (s, a) - \bar {v} \right].
+$$
+
+Note that this explains the constraint and only if we pick  $\bar{v} = R^{\pi}$ , we can find a  $V$  to guarantee the constraint  $\bar{v} + V(s) - \mathcal{P}^{\pi}V(s) - r^{\pi}(s) \geq 0$  holds true.
+
+# B.2 DOUBLY ROBUST ESTIMATOR
+
+We want to build the doubly robust estimator via the lagrangian. However, the Lagrangian consists of three term  $\rho, V$  and  $\bar{v}$ . It is counter-intuitive if we already given an estimator of  $\bar{v} \approx R^{\pi}$  into our estimator.
+
+A better way to solve this problem is to remove the constraint  $\sum \rho(s) = 1$ , but we divide it as an self-normalization. Then our Lagrangian becomes
+
+$$
+L (V, \rho) = \frac {\sum_ {s} \rho (s) r ^ {\pi} (s) - \sum_ {s} V (s) (\rho (s) - \mathcal {T} ^ {\pi} \rho (s))}{\sum \rho (s)}.
+$$
+
+which can be utilized to define the doubly robust estimator for average reward.
+
+Definition B.1. Given a learned value function  $\hat{V}(s)$  for policy  $\pi$  and an estimated density ratio  $\hat{w}(s)$  for  $w_{\pi/\pi_0}(s)$ , we define
+
+$$
+\widehat {R} _ {D R} ^ {\pi} [ \widehat {V}, \widehat {w} ] := \frac {\sum_ {s , a , r , s ^ {\prime} \in \mathcal {D}} \widehat {w} (s) \left(\beta_ {\pi / \pi_ {0}} (a | s) (r + \widehat {V} (s ^ {\prime})) - \widehat {V} (s)\right)}{\sum_ {s \in \mathcal {D}} \widehat {w} (s)},
+$$
+
+where  $\beta_{\pi /\pi_0}(a|s) = \frac{\pi(a|s)}{\pi_0(a|s)}$
+
+Similarly to Theorem 3.1 we will have our double robustness for our average doubly robust estimator:
+
+Theorem B.2. Suppose we have infinite samples and we can get
+
+$$
+R _ {D R} ^ {\pi} [ \widehat {V}, \widehat {w} ] = \frac {\mathbb {E} _ {s \sim d _ {\pi_ {0}}} \left[ \widehat {w} (s) \left(r _ {\pi} (s) - \widehat {V} (s) + \mathcal {P} ^ {\pi} \widehat {V} (s)\right) \right]}{\mathbb {E} _ {s \sim d _ {\pi_ {0}}} [ \widehat {w} (s) ]}.
+$$
+
+Then we have
+
+$$
+R _ {D R} ^ {\pi} [ \widehat {V}, \widehat {w} ] - R ^ {\pi} = \mathbb {E} _ {s \sim d _ {\pi_ {0}}} \left[ \varepsilon_ {\widehat {w}} (s) \varepsilon_ {\widehat {V}} (s) \right], \tag {23}
+$$
+
+where  $\varepsilon_{\widehat{V}}$  and  $\varepsilon_{\widehat{w}}$  are errors of  $\widehat{V}$  and  $\widehat{w}$ , respective, defined as follows
+
+$$
+\varepsilon_ {\widehat {w}} = \frac {\widehat {w} (s)}{\mathbb {E} _ {s \sim d _ {\pi_ {0}}} [ \widehat {w} (s) ]} - \frac {d _ {\pi} (s)}{d _ {\pi_ {0}} (s)}, \qquad \varepsilon_ {\widehat {V}} (s) = r _ {\pi} - \widehat {V} + \mathcal {P} ^ {\pi} \widehat {V} - R ^ {\pi}.
+$$
+
+Proof. A key observation is that
+
+$$
+\begin{array}{l} \mathbb {E} _ {s \sim d _ {\pi_ {0}}} [ w _ {\pi / \pi_ {0}} (s) \varepsilon_ {\widehat {V}} (s) ] = \mathbb {E} _ {s \sim d _ {\pi}} [ r ^ {\pi} (s) - \widehat {V} (s) + \mathcal {P} ^ {\pi} \widehat {w} (s) - R ^ {\pi} ] \\ = \left(\mathbb {E} _ {s \sim d _ {\pi}} \left[ r ^ {\pi} (s) \right] - R ^ {\pi}\right) + \mathbb {E} _ {s \sim d _ {\pi}} \left[ - \widehat {V} (s) + \mathcal {P} ^ {\pi} \widehat {w} (s) \right] \\ = 0 \\ \end{array}
+$$
+
+Thus we have
+
+$$
+\begin{array}{l} R _ {\mathrm {D R}} ^ {\pi} [ \widehat {V}, \widehat {w} ] - R ^ {\pi} = \mathbb {E} _ {s \sim d _ {\pi_ {0}}} \left[ \frac {\widehat {w} (s)}{\mathbb {E} _ {s \sim d _ {\pi_ {0}}} [ \widehat {w} (s) ]} \left(R ^ {\pi} + \varepsilon_ {\widehat {V}} (s)\right) \right] - R ^ {\pi} \\ = \mathbb {E} _ {s \sim d _ {\pi_ {0}}} \left[ \frac {\widehat {w} (s)}{\mathbb {E} _ {s \sim d _ {\pi_ {0}}} [ \widehat {w} (s) ]} \varepsilon_ {\widehat {V}} (s) \right] \\ = \mathbb {E} _ {s \sim d _ {\pi_ {0}}} \left[ \frac {\widehat {w} (s)}{\mathbb {E} _ {s \sim d _ {\pi_ {0}}} [ \widehat {w} (s) ]} \varepsilon_ {\widehat {V}} (s) \right] - \mathbb {E} _ {s \sim d _ {\pi_ {0}}} [ w _ {\pi / \pi_ {0}} (s) \varepsilon_ {\widehat {V}} (s) ] \\ = \mathbb {E} _ {s \sim d _ {\pi_ {0}}} \left[ \varepsilon_ {\widehat {w}} (s) \varepsilon_ {\widehat {V}} (s) \right]. \\ \end{array}
+$$
+
+Similar to discounted case we have  $R_{\mathrm{DR}}^{\pi}[\widehat{V}, \widehat{w}] = R^{\pi}$  if either  $\widehat{w}$  or  $\widehat{V}$  is accurate.
+
+# C EXPERIMENTAL DETAILS
+
+# C.1 TABULAR CASE: TAXI
+
+Behavior and Target Policies Choosing We use an on-policy Q-learning to get a sequence of policy  $\pi_0, \pi_1, \dots, \pi_{19}$  as data size increases. We pick the last policy  $\pi_{19}$  (almost optimum) as our target policy and  $\pi_{18}$  as our behavior policy to guarantee that those policies are not far away. We set our discounted factor  $\gamma = 0.99$ .
+
+Train  $\widehat{V}$  and  $\widehat{\rho}$  Separate from testing, we use a set of independent sample to first train a value function  $\widehat{V}$  and a density function  $\widehat{\rho}$ . Both  $\widehat{V}$  and  $\widehat{\rho}$  have bias due to finite sample approximation.
+
+For training  $\widehat{V}$  and  $\widehat{\rho}$ , we choose to use Monte Carlo method to estimate  $\widehat{V}$  and  $\widehat{\rho}$ . We first use the finite samples to get an estimated model  $\widehat{T}(s'|s,a)$  and rewards function  $\widehat{r}(s,a)$  and  $\widehat{d}_0$ . Then we solve the following linear equation (by iteration like power method, which is actually Monte Carlo):
+
+$$
+\begin{array}{l} \widehat {V} (s) = \sum_ {a} \pi (a | s) \widehat {Q} ^ {\pi} (s, a), \\ \widehat {Q} ^ {\pi} (s, a) = \widehat {r} (s, a) + \gamma \sum_ {s ^ {\prime}} \widehat {T} (s ^ {\prime} | s, a) \widehat {V} (s ^ {\prime}), \\ \widehat {\nu} (s, a) = \widehat {\rho} (s) \pi (a | s), \\ \widehat {\rho} (s) = (1 - \gamma) \widehat {d _ {0}} (s) + \gamma \sum_ {s, a} \widehat {T} (s ^ {\prime} | s, a) \widehat {\nu} (s, a). \\ \end{array}
+$$
+
+Estimate  $R^{\pi}$  Using  $\widehat{V}$  and  $\widehat{\rho}$  We put  $\widehat{V}$  and  $\widehat{\rho}$  into the Lagrangian as equation (15) as our doubly robust estimator. For those states we haven't visited, we set  $\widehat{V}(s)$  and  $\widehat{\rho}(s)$  as 0 and we self-normalized the  $\widehat{\rho}$  to get a fair estimation.
+
+# C.2 CONTINUOUS STATES OFF-POLICY EVALUATION
+
+Evaluation Environments We evaluate our method on two infinite horizon environments: Puck-Mountain and InvertedPendulum.
+
+Puck-Mountain is an environment similar to Mountain-car, except that the goal of the task is to push the puck as high as possible in a local valley whose initial position is at the bottom of the valley. If the ball reaches the top sides of the valley, it will hit a roof and changes the speed to its opposite direction with half of its original speeds. The rewards was determined by the current velocity and height of the puck.
+
+InvertedPendulum is a pendulum that has its center of mass above its pivot point. It is unstable and without additional help will fall over. We train a near optimal policy that can make the pendulum balance for a long horizon. For both behavior and target policies, we assume they are good enough to keep the pendulum balance and will never fall down until they reach the maximum timesteps. We use the implementation from OpenAI Gym (Brockman et al., 2016) and changing the dynamic by adding some additional zero mean Gaussian noise to the transition dynamic.
+
+Behavior and Target Policies Learning We use the open source implementation $^2$  of deep Q-learning to train a  $32 \times 32$  MLP parameterized Q-function to converge. We then use the softmax policy of learned the Q-function as the target policy  $\pi$ , which has a default temperature  $\tau = 1$ . For the behavior policy  $\pi_0$ , we set a relative large temperature which encourages exploration. We set the temperature of the behavior policy  $\tau_0 = 1.88$  for Puck-Mountain and  $\tau_0 = 1.50$  for InvertedPendulum respectively.
+
+Training of density ratio  $\hat{w}(s)$  and value function  $\hat{V}(s)$  We use a separate training dataset with 200 trajectories whose horizon length is 1000 to learn the density ratio  $\hat{w}(s)$  and the value function
+
+Algorithm 2 Optimization of density ratio  $\widehat{w}$
+
+Input: Transition data  $\mathcal{D} = \{s_i, a_i, s_i', r_i\}_{i=1}^n$  from the behavior policy  $\pi_0$ ; a target policy  $\pi$  for which we want to estimate the expected reward. Discount factor  $\gamma \in (0,1)$ , starting state  $\mathcal{D}_0 = \{s_j^{(0)}\}_{j=1}^m$  from initial distribution.
+
+Initial the density ratio  $w(s) = w_{\theta}(s)$  to be a neural network parameterized by  $\theta$ ,  $f(s) = f_{\beta}(s)$  to be a neural network parameterized by  $\beta$ . We need to ensure that the final layer of  $\theta$  is a softmax layer.
+
+for iteration  $= 1,2,\dots,T$  do
+
+Randomly choose a batch  $\mathcal{M} \subseteq \{1, \dots, n\}$  uniformly from the transition data  $\mathcal{D}$  and a batch  $\mathcal{M}_0 \subseteq \{1, \dots, m\}$  uniformly from start states  $\mathcal{D}_0$ .
+
+for iteration  $= 1,2,\dots$  ,K do
+
+Update the parameter  $\beta$  by  $\beta \gets \beta +\epsilon_{\beta}\nabla_{\beta}\hat{L} (w_{\theta},\phi_{\beta})$  , where
+
+$$
+\hat {L} (w _ {\theta}, f _ {\beta}) = \frac {1}{| \mathcal {M} |} \sum_ {i \in \mathcal {M}} \left((w (s _ {i} ^ {\prime}) - \gamma w (s _ {i}) \frac {\pi (a _ {i} | s _ {i})}{\pi_ {0} (a _ {i} | s _ {i})} - \frac {1}{2} f (s _ {i} ^ {\prime})) f (s _ {i} ^ {\prime})\right) - (1 - \gamma) \frac {1}{| \mathcal {M} _ {0} |} \sum_ {j \in \mathcal {M} _ {0}} f (s _ {j})
+$$
+
+end for
+
+Update the parameter  $\theta$  by  $\theta \gets \theta -\epsilon_{\theta}\nabla_{\theta}\hat{L} (w_{\theta},f_{\beta})$
+
+end for
+
+Output: the density ratio  $\widehat{w} = w_{\theta}$
+
+Algorithm 3 Optimization of value function  $\widehat{V}$
+
+Input: Transition data  $\mathcal{D} = \{s_i, a_i, s_i', r_i\}_{i=1}^n$  from the behavior policy  $\pi_0$ ; a target policy  $\pi$  for which we want to estimate the expected reward. Discount factor  $\gamma \in (0,1)$ .
+
+Initial the value function  $V(s) = V_{\phi}(s)$  to be a neural network parameterized by  $\phi$ ,  $f(s) = f_{\beta}(s)$  to be a neural network parameterized by  $\beta$ .
+
+for iteration  $= 1,2,\dots,T$  do
+
+Randomly choose a batch  $\mathcal{M} \subseteq \{1, \dots, n\}$  uniformly from the transition data  $\mathcal{D}$ .
+
+for iteration  $= 1,2,\dots$  ,K do
+
+Update the parameter  $\beta$  by  $\beta \gets \beta +\epsilon_{\beta}\nabla_{\beta}\hat{L} (V_{\phi},\phi_{\beta})$  , where
+
+$$
+\hat {L} (V _ {\phi}, f _ {\beta}) = \frac {1}{| \mathcal {M} |} \sum_ {i \in \mathcal {M}} \left(\left(V _ {\phi} (s _ {i}) - \frac {\pi (a _ {i} | s _ {i})}{\pi_ {0} (a _ {i} | s _ {i})} (r _ {i} + \gamma V _ {\phi} (s _ {i} ^ {\prime}))\right) f _ {\beta} (s _ {i}) - \frac {1}{2} f _ {\beta} (s _ {i}) ^ {2}\right)
+$$
+
+end for
+
+Update the parameter  $\phi$  by  $\phi \gets \phi -\epsilon_{\phi}\nabla_{\phi}\hat{L} (V_{\phi},f_{\beta})$
+
+end for
+
+Output: the density ratio  $\widehat{V} = V_{\phi}$
+
+$\hat{V}(s)$ . For the training of density ratio, we adapt the algorithm 2 in Liu et al. (2018a) to train a neural network parameterized  $w_{\theta}(s)$ . Instead of taking the test function  $f(s)$  into an RKHS  $\mathcal{H}_{\mathcal{K}}$ , we parameterize the test function  $f(s) = f_{\beta}(s)$  to be a neural network with parameter  $\beta$ , and perform minimax optimization over  $\mathrm{param}$ $\theta$  and  $\beta$ . A detail description can be found in Algorithm 2.
+
+Similarly, for the training of value function, we use primal-dual based optimization methods (Dai et al., 2018; Feng et al., 2019) to minimize the bellman residual:
+
+$$
+\min _ {\phi} \max _ {f _ {\beta} \in \mathcal {F}} \frac {1}{| \mathcal {M} |} \sum_ {i \in \mathcal {M}} \left(\left(V _ {\phi} (s _ {i}) - \frac {\pi (a _ {i} | s _ {i})}{\pi_ {0} (a _ {i} | s _ {i})} (r _ {i} + \gamma V _ {\phi} (s _ {i} ^ {\prime}))\right) f _ {\beta} (s _ {i}) - \frac {1}{2} f _ {\beta} (s _ {i}) ^ {2}\right),
+$$
+
+where  $V_{\phi}(s)$  is the parameterized value function and  $f_{\beta}(s)$  is the test function. We also perform minimax over parameter  $\phi$  and  $\beta$ . A detail description can be found in Algorithm 3.
+
+For the network structures, we use  $32 \times 32$  feed forward neural networks to parameterize value function  $V_{\phi}$  and density ratio  $w_{\theta}(s)$ , and we use one hidden neural network with 10 units to parameterize the test function  $f_{\beta}(s)$ . We use Adam Optimizer for all our experiments.
+
+![](images/11910a8b57fcea7aa05f62cc809e1b180a250c381ebaa94120f7ccf91e5ee612.jpg)  
+(a) MSE
+
+![](images/5fca61320732dbdc127386b6bcdfe55b0aec00661e22b1225585b8530f71b13a.jpg)  
+(b) Bias Square
+
+![](images/5d633c350ee47dec30cdf477511b4a3db05805d9fb1cca31378889aeb684379a.jpg)  
+(c) Variance  
+Figure 4: Additional results on Taxi, with an extra curve for model based method.
+
+Estimate  $R^{\pi}$  using  $\widehat{V}$  and  $\widehat{w}$  Given data samples from the policy  $\pi_0$ , We can directly use  $\widehat{R}_{\mathrm{DR}}^{\pi}$  in equation (11) to estimate  $R^{\pi}$ .
+
+# D ADDITIONAL EXPERIMENTAL RESULTS
+
+We add a standard model-based baseline in Figure 4 following the same procedure as Liu et al. (2018a).<sup>3</sup> Figure 4 shows that the performance of the model-based approach largely depends on the size of the data, which is consistent with that in Liu et al. (2018a).
\ No newline at end of file
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+# DRAWING EARLY-BIRD TICKETS: TOWARDS MORE EFFICIENT TRAINING OF DEEP NETWORKS
+
+Haoran You, Chaojian Li, Pengfei Xu, Yonggan Fu, Yue Wang, Richard G. Baraniuk & Yingyan Lin*
+
+Department of Electrical and Computer Engineering
+
+Rice University
+
+Houston, TX 77005, USA
+
+{hy34, cl114, px5, yf22, yw68, yingyan.lin, richb}@rice.edu
+
+# Xiaohan Chen & Zhangyang Wang*
+
+Department of Computer Science and Engineering
+
+Texas A&M University
+
+College Station, TX 77843, USA
+
+{chernxh, atlaswang}@tamu.edu
+
+# ABSTRACT
+
+(Frankle & Carbin, 2019) shows that there exist winning tickets (small but critical subnetworks) for dense, randomly initialized networks, that can be trained alone to achieve a comparable accuracy to the latter in a similar number of iterations. However, the identification of these winning tickets still requires the costly train-prune-retrain process, limiting their practical benefits. In this paper, we discover for the first time that the winning tickets can be identified at a very early training stage, which we term as Early-Bird (EB) tickets, via low-cost training schemes (e.g., early stopping and low-precision training) at large learning rates. Our finding on the existence of EB tickets is consistent with recently reported observations that the key connectivity patterns of neural networks emerge early. Furthermore, we propose a mask distance metric that can be used to identify EB tickets with a low computational overhead, without needing to know the true winning tickets that emerge after the full training. Finally, we leverage the existence of EB tickets and the proposed mask distance to develop efficient training methods, which are achieved by first identifying EB tickets via low-cost schemes, and then continuing to train merely the EB tickets towards the target accuracy. Experiments based on various deep networks and datasets validate: 1) the existence of EB tickets and the effectiveness of mask distance in efficiently identifying them; and 2) that the proposed efficient training via EB tickets can achieve up to  $5.8 \times \sim 10.7 \times$  energy savings while maintaining comparable or even better accuracy as compared to the most competitive state-of-the-art training methods, demonstrating a promising and easily adopted method for tackling the often cost-prohibitive deep network training. Codes available at https://github.com/RICE-EIC/Early-Bird-Tickets
+
+# 1 INTRODUCTION
+
+The recent record-breaking predictive performance achieved by deep neural networks (DNNs) motivates a tremendously growing demand to bring DNN-powered intelligence into numerous applications (Xu et al., 2020). However, the excellent performance of modern DNNs comes at an often prohibitive training cost due to the required vast volume of training data and model parameters. As an illustrative example of the computational complexity of DNN training, one forward pass of the ResNet50 (He et al., 2016a) model requires 4 GFLOPs (FLOPs: floating point operations) of computations and training requires  $10^{18}$  FLOPs, which takes 14 days on one state-of-the-art NVIDIA M40 GPU (You et al., 2018). As a result, training a state-of-the-art DNN model often demands considerable energy, along with the associated financial and environmental costs. For example, a recent report shows that training a single DNN can cost over $10K US dollars and emit as much carbon as five cars in their lifetimes (Strubell et al., 2019), limiting the rapid development of DNN innovations and raising various environmental concerns.
+
+The recent trends of improving DNN efficiency mostly focus on compressing models and accelerating inference. An empirically adopted practice is the so-called progressive pruning and training
+
+routine, i.e., training a large model fully, pruning it, and then retraining the pruned model to restore the performance (the process can be iterated several rounds). While this has been a standard practice for model compression (Han et al., 2015), some recent efforts start empirically linking it to the potential of more efficient training. Notably, the latest series of works (Frankle & Carbin, 2019; Liu et al., 2018b) reveals that dense, randomly-initialized networks contain small subnetworks which can match the test accuracy of original networks when trained alone themselves. These subnetworks are called winning tickets. Despite their insightful findings, there remains to be a major gap between the winning ticket observation and the goal of more efficient training, since winning tickets were only identified by pruning unimportant connections after fully training a dense network.
+
+This paper closes this gap by demonstrating the Early-Bird (EB) tickets phenomenon: the winning tickets can be drawn very early in training and with aggressively low-cost training algorithms. Through a range of experiments on different DNNs and datasets, we observe the consistent existence of EB tickets and the cheap costs needed to reliably draw them, and develop a novel mask distance metric to detect their emergence. After being identified, re-training those EB tickets (using standard training) leads to comparable or even better final accuracies, compared to either standard training, or re-training the "ground-truth" winning tickets drawn after full training as in (Frankle & Carbin, 2019). Our observations seem to coincide with the recent findings by (Achille et al., 2019; Li et al., 2019) about the two-stage optimization trajectory in training. Taking advantage of EB tickets, we propose an efficient DNN training scheme termed EB Train. To our best knowledge, this is the first step taken towards exploiting winning tickets for a realistic efficient training goal.
+
+Our contribution can be summarized as follow:
+
+1. We discover the Early-Bird (EB) tickets, and show that they 1) consistently exist across DNN models and datasets; 2) can emerge very early in training; and 3) stay robust under (and sometimes even favor) various aggressive and low-cost training schemes (in addition to early stopping), including large learning rates and low-precision training.  
+2. We propose a practical, easy-to-compute mask distance as an indicator to draw EB tickets without accessing the "ground-truth" winning tickets (drawn after full training), fixing a major paradox for connecting winning tickets with the efficient training goal.  
+3. We design a novel efficient training framework based on EB tickets (EB Train). Experiments in state-of-the-art benchmarks and models show that EB Train can achieve up to  $5.8 \times 10.7 \times$  energy savings, while maintaining the same or even better accuracy, compared to training with the original winning tickets.
+
+# 2 RELATED WORKS
+
+Winning Ticket Hypothesis. The lottery ticket hypothesis (Frankle & Carbin, 2019) first points out that a small subnetwork, called the winning ticket, can be identified by pruning a fully trained dense network; when training it in isolation with the same weight initialization once assigned to the corresponding weights in the dense network, one can restore the comparable test accuracy to the dense network. However, finding winning tickets hinged on costly (iterative) pruning and retraining. (Morcos et al., 2019) studies the reuse of winning tickets, transferable across different datasets. (Zhou et al., 2019) discovers the existence of supermasks that can be applied to an untrained, randomly-initialized network. (Liu et al., 2018b) argues that the weight initialization might make less difference when trained with a large learning rate, while the searched connectivity is more of the winning ticket's core value. It also explores the usage of both unstructured and (more hardware-friendly) structured pruning and shows that both lead to the emergence of winning tickets.
+
+Another related work (Lee et al., 2019) prunes a network at single-shot with one mini-batch, in which the irrelevant connections are identified by a connection sensitivity criterion. Comparing to (Frankle & Carbin, 2019), the authors show their method to be more efficient in finding the good subnetwork (not the winning ticket), although its re-training accuracy/efficiency is found to be inferior, compared to training the "ground truth" winning ticket.
+
+Other Relevant Observations in Training. (Rahaman et al., 2019; Xu et al., 2019) argue that deep networks will first learn low-complexity (lower-frequency) functional components, before absorbing high-frequency features: the former being more robust to perturbations. An important hint can be found in (Achille et al., 2019): the early stage of training seems to first discover the important connections and the connectivity patterns between layers, which becomes relatively fixed in the
+
+later training stage. That seems to imply that the critical sub-network (connectivity) can be identified independent of, and seemingly also ahead of, the (final best) weights. Finally, Li et al. (2019) demonstrates that training a deep network with a large initial learning rate helps the model focus on memorizing easier-to-fit, more generalizable pattern faster and better – a direct inspiration for us to try drawing EB tickets using large learning rates.
+
+Efficient Inference and Training. Model compression has been extensively studied for lighter-weight inference. Popular means include pruning (Li et al., 2017; Liu et al., 2017; He et al., 2018; Wen et al., 2016; Luo et al., 2017; Liu et al., 2018a), weight factorization (Denton et al., 2014), weight sharing (Wu et al., 2018a), quantization (Hubara et al., 2017), dynamic inference (Wang et al., 2018b; Shen et al., 2020), network architecture search (Zoph & Le, 2017), among many others (Wang et al., 2018c;d). On the other hand, the literature on efficient training appears to be much sparser. A handful of works (Goyal et al., 2017; Cho et al., 2017; You et al., 2018; Akiba et al., 2017; Jia et al., 2018; Gupta et al., 2015) focus on reducing the total training time in paralleled, communication-efficient distributed settings. In contrast, our goal is to shrink the total resource cost for in-situ, resource-constrained training, as (Wang et al., 2019a) advocated. (Banner et al., 2018; Wang et al., 2018a) presented low-precision training, which is aligned with our goal and can be incorporated into EB Train (see later).
+
+# 3 DRAWING EARLY-BIRD TICKETS: HYPOTHESIS AND EXPERIMENTS
+
+We hypothesize that the winning tickets can emerge at a very early training stage, which we term as an Early-Bird (EB) ticket. Consider a dense, randomly-initialized network  $f(x; \theta)$ ,  $f$  reaches a minimum validation loss  $f_{\mathrm{loss}}$  at the  $i$ -th iteration with a test accuracy  $f_{\mathrm{acc}}$ , when optimized with stochastic gradient descent (SGD) on a training set. In addition, consider subnetworks  $f(x; m \odot \theta)$  with a mask  $m \in \{0, 1\}$  indicates the pruned and unpruned connections in  $f(x; \theta)$ . When being optimized with SGD on the same training set,  $f(x; m \odot \theta_t)$  reach a minimum validation loss  $f_{\mathrm{loss}}'$  with a test accuracy  $f_{\mathrm{acc}}'$ , where  $\theta_t$  denotes the weights at the  $t$ -th iteration of training. The EB tickets hypothesis articulates that there exists  $m$  such that  $f_{\mathrm{acc}}' \approx f_{\mathrm{acc}}$  (even  $\geq$ ), i.e., same or better generalization, with  $t \ll i$  (e.g., early stopping) and a sparse  $m$  (i.e., much reduced parameters).
+
+Section 3 addresses three key questions pertaining to the EB ticket hypothesis. We first show via an extensive set of experiments, that EB tickets can be observed across popular models and datasets (Section 3.1). We then try to be more aggressive to see if high-quality EB tickets still emerge under "cheaper" training (Section 3.2). We finally reveal that EB tickets can be identified using a novel mask distance between consecutive epochs, thus no full training needed (Section 3.3).
+
+# 3.1 DO EARLY-BIRD TICKETS ALWAYS EXIST?
+
+We perform ablation simulations using two representative deep models: VGG16 (Simonyan & Zisserman, 2014) and pre-activation residual networks-101 (PreResNet101) (He et al., 2016b), on two popular datasets: CIFAR-10 and CIFAR-100. For drawing the tickets, we adopt a standard training protocol (Liu et al., 2018b) for both CIFAR-10 and CIFAR-100: the training takes 160 epochs in total and the batch size of 256; the initial learning rate is set to 0.1, and is divided by 10 at the 80th and 120th epochs, respectively; the SGD solver is adopted with a momentum of 0.9 and a weight decay of  $10^{-4}$ . For retraining the tickets, we keep the same setting by default.
+
+We follow the main idea of (Frankle & Carbin, 2019), but instead prune networks trained at much earlier points (before the accuracies reach their final top values), to see if reliable tickets can still be drawn. We adopt the same channel pruning in (Liu et al., 2017) for all experiments since it is hardware friendly and aligns with our end goal of efficient training (Wen et al., 2016). Figure 1 reports the accuracies achieved by re-training the tickets drawn from different early epochs. All results consistently endorse that there exist high-quality tickets, at as early as the 20th epoch (w.r.t. a total of 160 epochs), that can achieve strong re-training accuracies. Comparing among different pruning ratios  $p$ , it is not too surprising to see over-pruning (e.g.,  $p = 70\%$ ) makes drawing good tickets harder, indicating a balance that we need to calibrate between accuracy and training efficiency.
+
+Two more striking observations from Figure 1: 1) there consistently exist EB tickets drawn at certain early epoch ranges, that outperform those drawn in a later stages, including the "ground-truth" winning tickets drawn at the 160th epoch. That intriguing phenomenon implies the possible "over-cooking" when networks try to identify connectivity patterns at later stage (Achille et al., 2019),
+
+![](images/c439969107dcd51f76d8c67bd6419818a50d80071e99827e59d0a5623e49441c.jpg)  
+Figure 1: Retraining accuracy vs. epoch numbers at which the subnetworks are drawn, for both PreResNet101 and VGG16 on the CIFAR-10/100 datasets, where  $p$  indicates the channel pruning ratio and the dashed line shows the accuracy of the corresponding dense model on the same dataset,  $\star$  denotes the retraining accuracies of subnetworks drawn from the epochs with the best search accuracies, and error bars show the minimum and maximum of three runs.
+
+![](images/7eb8265cfe23c3ae8fad3964114c9768f02937237935fe126d0152182b54d2b4.jpg)
+
+that might hamper generalization; 2) some EB tickets are able to outperform even their unpruned, fully-trained models (e.g., the dashlines), potentially thanks to the sparse regularization learned by EB tickets.
+
+Table 1: The retraining accuracy of subnetworks drawn at different training epochs using different learning rate schedules, with a pruning ratio of 0.5. Here [0, 100] represents  $[0_{LR\to 0.01},100_{LR\to 0.001}]$  while [80, 120] denotes  $[80_{LR\to 0.01},120_{LR\to 0.001}]$ , for compactness.  
+
+<table><tr><td rowspan="2"></td><td rowspan="2">LR Schedule</td><td colspan="4">Retrain acc. (%) (CIFAR-100)</td><td colspan="4">Retrain acc. (%) (CIFAR-10)</td></tr><tr><td>10</td><td>20</td><td>40</td><td>final</td><td>10</td><td>20</td><td>40</td><td>final</td></tr><tr><td rowspan="2">VGG16</td><td>[0, 100]</td><td>66.70</td><td>67.15</td><td>66.96</td><td>69.72</td><td>92.88</td><td>93.03</td><td>92.80</td><td>92.64</td></tr><tr><td>[80, 120]</td><td>71.11</td><td>71.07</td><td>69.14</td><td>69.74</td><td>93.26</td><td>93.34</td><td>93.20</td><td>92.96</td></tr><tr><td rowspan="2">PreResNet101</td><td>[0, 100]</td><td>69.68</td><td>69.69</td><td>69.79</td><td>70.96</td><td>92.41</td><td>92.72</td><td>92.42</td><td>93.05</td></tr><tr><td>[80, 120]</td><td>71.58</td><td>72.67</td><td>72.67</td><td>71.52</td><td>93.60</td><td>93.46</td><td>93.56</td><td>93.42</td></tr></table>
+
+# 3.2 DO EARLY-BIRD TICKETS STILL EMERGE UNDER LOWER-COST TRAINING?
+
+For EB tickets, only the important connections (connectivity) found by them matter, while the weights are to be re-trained anyway. One might hence come up with an idea, that more aggressive and "cheaper" training methods might be applicable to further shrink the cost of finding EB tickets (on top of the aforementioned early stopping of training thanks to the identification of EB tickets), as long as the same significant connections still emerge. We experimentally investigate the impacts of two schemes (which are using appropriate large learning rates and training in lower precision): EB tickets are observed to survive well under both of them.
+
+Appropriate large learning rates are important to the emergence of EB tickets. We first vary the learning rate schedule in the above experiments. The original schedule is denoted as  $[80_{LR\rightarrow 0.01},120_{LR\rightarrow 0.001}]$ , i.e., starting from 0.1, decayed to 0.01 at the 80th epoch, and further decayed to 0.001 at the 120th epoch. In comparison, we test a new learning rate schedule  $[0_{LR\rightarrow 0.01},100_{LR\rightarrow 0.001}]$ : starting from 0.01 (the 0th epoch), and decayed to 0.001 at the 100th epoch. After drawing tickets, we re-train them all using the same learning rate schedule  $[80_{LR\rightarrow 0.01},120_{LR\rightarrow 0.001}]$  for sufficient training and a fair comparison. We can see from Table 1 that high-quality EB tickets always emerge earlier when searching with the schedule of a larger learning rates, i.e.,  $[80_{LR\rightarrow 0.01},120_{LR\rightarrow 0.001}]$ , whose final accuracies are also better. Note that the earlier emergence of good EB tickets contributes to lowering the training costs. It was observed that larger learning rates are beneficial for training the dense model fully to draw the winning tickets in (Frankle & Carbin, 2019; Liu et al., 2018b): here we show this benefit extends to EB tickets too. More experiments with larger learning rates can be found in Appendix A.2.
+
+Low-precision training does not destroy EB tickets. We next examine the emergence of EB tickets within a state-of-the-art low-precision training algorithm (Wu et al., 2018b), where all model weights, activations, gradients and errors are quantized to 8 bits throughout training. Note that here we only apply low-precision training to the stage of identifying EB tickets, and afterwards the tickets are trained in the same full-precision as in Section 3.1. Figure 2 shows the retraining accuracy and total number of FLOPs (EB ticket search (8 bits) + retraining (32 bits floating points)) for VGG16 and CIFAR-10/100. We can see that EB tickets still emerge when using aggressively low-precision for identifying EB tickets: the general trends resemble the full-precision training cases in Figure 1, except the emergence of good EB tickets seem to become even earlier up to initial epochs. In this
+
+![](images/9309905769817e9534d0c5dc609275ed3116c668d100d268e74daf631b0b497c.jpg)  
+Figure 2: Retraining accuracy and total training FLOPs comparison vs. epoch number at which the subnetwork is drawn, when using 8 bits precision during the stage of identifying EB tickets based on the VGG16 model and CIFAR-10/100 datasets, where  $p$  indicates the channel-wise pruning ratio and the dashed line shows the accuracy of the corresponding dense model on the same dataset.
+
+![](images/3cd589367938871726c71adb7e2024685ceeba09f8875556e490376e6616baa5.jpg)
+
+![](images/9501982f61c579a03ba87ab528ef24a06d6d1b1cbd9d6281153124f047373aaf.jpg)
+
+![](images/5491aa17b7065096748ad3b8649caee9813b459b29c6b2280bd6fa14811275b8.jpg)
+
+way, it will lead to cost savings in finding EB tickets, since low-precision updates can aggressively save energy compared to their full-precision baseline, as shown in the Table 2.
+
+# 3.3 HOW TO IDENTIFY EARLY-BIRD TICKETS PRACTICALLY?
+
+Distance between Ticket Masks. For each time of pruning, we define a binary mask of the drawn ticket (pruned subnetwork) w.r.t. the full dense network. We follow (Liu et al., 2017) to consider the scaling factor  $r$  in batch normalization (BN) layers as indicators of the corresponding channels' significance. Given a target pruning ratio  $p$ , we then prune the channels with top  $p$ -percentage smallest  $r$  values. Denote the pruned channels as 0 while the kept ones as 1, the original network can be mapped into a binary "ticket mask". For any two sub-networks pruned from the same dense model, we calculate their mask distance via the Hamming distance between their two ticket masks.
+
+Detecting EB Tickets via Mask Distances. We first visualize and observe the global behaviors of mask distances between consecutive epochs. Figure 3 plots the pairwise mask distance matrices  $(160 \times 160)$  of the VGG16 and PreResNet101 experiments on CIFAR-100 (from Section 3.1), at different pruning ratios  $p$ , where  $(i,j)$ -th element in a matrix denotes the mask distance between subnetworks drawn from the  $i$ -th and  $j$ -th epochs in that corresponding experiment (160 epochs in total for all). For the ease of visualization, all elements in a matrix are linearly normalized between 0 and 1; Therefore, in the resulting matrix, a lower value (close to 0) indicates a smaller mask distance and is highlighted with a warmer color (same hereinafter).
+
+Figure 3 demonstrates fairly consistent behaviors. Taking VGG16 on CIFAR-100  $(p = 0.2)$  for an example: 1) at the very beginning, the mask distances change rapidly between epochs, manifested by the quickly "cooling" colors (yellow  $\rightarrow$  green), from the diagonal line (lowest since that is comparing an epoch's mask with itself), to off-diagonal (comparing different epochs; the more an element deviates from the diagonal, the more distant the two epochs are away from each other); 2) after  $\sim 10$  epochs, the off-diagonal elements become "yellow" too, and the color transition becomes smoother from diagonal to off-diagonal, indicating the masks change mildly only after passing this point; 3)
+
+![](images/f1dabacbff1cffb00d659e9f3055a1d19e655d15c0c7af84dc12aa2e91ab82db.jpg)  
+Figure 3: Visualization of the pairwise mask distance matrix for VGG16 and PreResNet101 on CIFAR-100.
+
+after  $\sim 80$  epochs, the mask distances almost remain unchanged across epochs. Similar trends are observed in other plots too. It seems to concur the hypothesis in (Achille et al., 2019) that a network first learns important connectivity patterns and then fixes them and further tune their weights.
+
+Our observation that the ticket masks quickly become stable and hardly changed in early training stages supports drawing EB tickets. We therefore measure the mask distance between the consecutive epochs, and draw EB tickets when such distance is smaller than a threshold  $\epsilon$ . Practically, to improve the reliability of EB tickets, we will stop to draw EB tickets when the last five recorded mask distances are all smaller than  $\epsilon$ , to avoid some irregular fluctuation in early training stages. In Figure 3, the red lines indicate the identification of EB tickets when  $\epsilon$  is set to 0.1.
+
+# 4 EFFICIENT TRAINING VIA EARLY BIRD TICKETS
+
+In this section, we present an efficient DNN training scheme, termed as EB Train, which leverages 1) the existence of EB tickets and 2) the proposed mask distance that can detect the emergence of EB tickets for time- and energy-efficient training. We will first provide a conceptual overview, and then describe the routine of EB Train, and finally show the evaluation performance of EB Train by benchmarking it with state-of-the-art methods of obtaining compressed DNN models based on representative datasets and DNNs.
+
+![](images/08b241d71eb38cfbb76b7ad3d2ba4261e03df13c61612e8bc8d6912a0c008dd0.jpg)  
+Figure 4: A high-level overview of the commonly adopted progressive pruning and training scheme and our EB Train.
+
+# 4.1 WHY IS EB TRAIN MORE EFFICIENT?
+
+EB Train vs. Progressive Pruning and Training. Figure 4 illustrates an overview of our proposed EB Train and the progressive pruning and training scheme, e.g., as in (Frankle & Carbin, 2019). In particular, the progressive pruning and training scheme adopts a three-step routine of 1) training a large and dense model, 2) pruning it, and 3) then retraining the pruned model to restore performance, and these three steps can be iterated (Han et al., 2015). The first step often dominates (e.g., occupy  $75\%$  training FLOPs when using the PreResNet101 model and CIFAR-10 dataset) in terms of training energy and time costs as it involves training a large and dense model. The key feature of our EB Train scheme is that it replaces the aforementioned steps 1 and 2 with a lower-cost step of detecting the EB tickets, i.e., enabling early stopping during the time- and energy-dominant step of training the large and dense model, thus promising large savings in both training time and energy. For example, assuming the first step of the progressive pruning and training scheme requires  $N$  epochs to sufficiently train the model for achieving the target accuracy, the proposed EB Train needs only  $N_{EB}$  epochs to identify the EB tickets by making use of the mask distance that can detect the emergence of EB tickets, where  $N_{EB} \ll N$  (e.g.,  $N_{EB} / N = 12.5\%$  in the experiments summarized in Figure 1 and  $N_{EB} / N = 6.25\%$  in the experiment summarized in Table 1).
+
+Algorithm 1: The Algorithm for Searching EB Tickets  
+1: Initialize the weights  $W$ , scaling factor  $r$ , pruning ratio  $p$ , and the FIFO queue  $Q$  with length  $l$ ;  
+2: while  $t$  (epoch)  $< t_{max}$  do  
+3: Update  $W$  and  $r$  using SGD training;  
+4: Perform structured pruning based on  $r_t$  towards the target ratio  $p$ ;  
+5: Calculate the mask distance between the current and last subnetworks and add to  $Q$ .  
+6:  $t = t + 1$   
+7: if  $\mathbf{Max}(Q) < \epsilon$  then  
+8:  $t^* = t$   
+9: Return  $f(x; m_t^* \odot W)$  (EB ticket);  
+10: end if  
+11: end while
+
+![](images/1761a8c38079b5f51316e45d7508d932b884ec3d457eaed775a40e9d4b992595.jpg)  
+Figure 5: The total training FLOPs vs. the epochs at which the subnetworks are drawn from, for both the PreResNet101 and VGG16 models on the CIFAR-10 and CIFAR-100 datasets, where  $p$  indicates the channel-wise pruning ratio for extracting the subnetworks. Note that the EB tickets at all cases achieve comparable or higher accuracies and consume less FLOPs than those of the "ground-truth" winning tickets (drawn after the full training of 160 epochs).
+
+![](images/aad4df26b074760cdfb28e00e57e108eb30fdb00e83cc775924aa103ed68fd5e.jpg)
+
+# 4.2 HOW TO IMPLEMENT EB TRAIN?
+
+From Figure 1, we can see that the EB Train scheme consists of a training (searching) step to identify EB tickets and a retraining step to retrain the EB tickets for achieving the target accuracy. Algorithm 1 describes the step of searching for EB tickets. Specifically, the EB searching process 1) first initializes the weights  $\mathbf{W}$  and scaling factors  $\mathbf{r}$ ; 2) iterates the structured pruning process as in (Liu et al. (2017)) to calculate the mask distances between the consecutive resulting subnetworks, storing them into a first-in-first-out (FIFO) queue with a length of  $l = 5$ ; and 3) exits when the maximum mask distance in the FIFO is smaller than a specified threshold  $\epsilon$  (default 0.1 with the normalized distances of [0,1]). The output is the resulting EB ticket denoted as  $f(x; m_t^* \odot W)$ , which will be retrained further to reach the target accuracy. Note that unlike the lottery ticket training in (Frankle & Carbin, 2019), EB Train inherits the unpruned weights from the drawn EB ticket instead of rewinding to the original initialization, as it has been shown that deeper networks are not robust to reinitialization with untrained weights (Frankle et al., 2020).
+
+# 4.3 HOW DOES EB TRAIN PERFORM COMPARED TO STATE-OF-THE-ART TECHNIQUES?
+
+Experiment Setting. We consider training the VGG16 and PreResNet101 models on both CIFAR-10/100 and ImageNet datasets following the basic setting of (Liu et al., 2018b) (i.e., training 160 epochs for CIFAR and 90 epochs for ImageNet). We measure the training energy costs from real-device operations as the energy cost consists of both computational and data movement costs, the latter of which is often dominant but can not be captured by the commonly used metrics, such as the number of FLOPs (Chen et al., 2017), we evaluate the proposed techniques against the baselines in terms of accuracy and real measured energy consumption. Specifically, all the energy consumption of full-precision models are obtained by training the corresponding models in an embedded GPU (NVIDIA JETSON TX2). The GPU measurement setup can be found in the Appendix. Note that the energy measurement results include the overhead of using the mask distance to detect the emergence of EB tickets, which is found negligible as compared to the total training cost. For example, for the VGG16 model, the overhead caused by computing mask distances is  $\leq 0.029\%$  in memory storage size,  $\leq 0.026\%$  in FLOPs, and  $\leq 0.07\%$  in real-measured energy costs. This is because 1) the mask distance evaluation only involves simple hamming distance calculations and 2) each epoch only calculates the distance once.
+
+Results and Analysis. We first compare the computational savings with the baselines using a progressive pruning and training scheme (Liu et al., 2017) in terms of the total training FLOPs, when drawing subnetworks at different epochs. Figure 5 summarizes the results of the PreRes-Net101/VGG16 models and CIFAR-10/100 datasets, corresponding to the same set of experiments as in Figure 1. We see that EB Train can achieve  $2.2 \sim 2.4 \times$  FLOPs reduction over the baselines, while leading to comparable or even better accuracy ( $-0.81\% \sim +2.38\%$  over the baseline).
+
+Table 2 compares the retraining accuracy and consumed FLOPs/energy of EB Train with four state-of-the-art progressive pruning and training techniques: the original lottery ticket (LT) training (Frankle & Carbin, 2019), network slimming (NS) (Liu et al., 2017), ThiNet (Luo et al., 2017) and SNIP (Lee et al., 2019). While all of them involve the process of training a dense network, pruning it, and retraining, they apply different pruning criteria, e.g., NS imposes  $L_{1}$ -sparsity on channel-wise
+
+Table 2: Comparing the accuracy and energy/FLOPs of EB Train (including its variants), NS (Liu et al. (2017)), LT (Frankle & Carbin, 2019), SNIP (Lee et al., 2019), and ThiNet (Luo et al. (2017)).  
+
+<table><tr><td rowspan="2">Setting</td><td rowspan="2">Methods</td><td colspan="3">Retrain acc.</td><td colspan="3">Energy cost (KJ)/FLOPs (P)</td></tr><tr><td>p=30%</td><td>p=50%</td><td>p=70%</td><td>p=30%</td><td>p=50%</td><td>p=70%</td></tr><tr><td rowspan="9">PreResNet -101 CIFAR-10</td><td>LT (one-shot)</td><td>93.70</td><td>93.21</td><td>92.78</td><td>6322/14.9</td><td>6322/14.9</td><td>6322/14.9</td></tr><tr><td>SNIP</td><td>93.76</td><td>93.31</td><td>92.76</td><td>3161/7.40</td><td>3161/7.40</td><td>3161/7.40</td></tr><tr><td>NS</td><td>93.83</td><td>93.42</td><td>92.49</td><td>5270/13.9</td><td>4641/12.7</td><td>4211/11.0</td></tr><tr><td>ThiNet</td><td>93.39</td><td>93.07</td><td>91.42</td><td>3579/13.2</td><td>2656/10.6</td><td>1901/8.65</td></tr><tr><td>EB Train (re-init)</td><td>93.88</td><td>93.29</td><td>92.39</td><td>2817/7.75</td><td>2382/7.05</td><td>1565/3.77</td></tr><tr><td>EB Train (FF)</td><td>93.91</td><td>93.90</td><td>92.49</td><td>2370/6.50</td><td>1970/5.70</td><td>1452/3.50</td></tr><tr><td>EB Train (LF)</td><td>93.48</td><td>93.31</td><td>92.24</td><td>2265/6.45</td><td>1667/5.39</td><td>1338/3.44</td></tr><tr><td>EB Train (LL)</td><td>93.24</td><td>92.85</td><td>92.12</td><td>489.4/6.45</td><td>410.9/5.39</td><td>281.8/3.44</td></tr><tr><td>EB Train Improv.</td><td>0.08</td><td>0.48</td><td>-0.29</td><td>6.5×/1.1×</td><td>6.5×/1.4×</td><td>6.7×/2.2×</td></tr><tr><td rowspan="9">VGG16 CIFAR-10</td><td>LT (one-shot)</td><td>93.18</td><td>93.25</td><td>93.28</td><td>746.2/30.3</td><td>746.2/30.3</td><td>746.2/30.3</td></tr><tr><td>SNIP</td><td>93.20</td><td>92.71</td><td>92.30</td><td>373.1/15.1</td><td>373.1/15.1</td><td>373.1/15.1</td></tr><tr><td>NS</td><td>93.05</td><td>92.96</td><td>92.70</td><td>617.1/27.4</td><td>590.7/25.7</td><td>553.8/23.8</td></tr><tr><td>ThiNet</td><td>92.82</td><td>91.92</td><td>90.40</td><td>298.0/22.6</td><td>383.9/19.0</td><td>380.1/16.6</td></tr><tr><td>EB Train (re-init)</td><td>93.11</td><td>93.23</td><td>92.71</td><td>290.4/14.4</td><td>237.3/12.0</td><td>200.5/9.45</td></tr><tr><td>EB Train (FF)</td><td>93.39</td><td>93.26</td><td>92.71</td><td>256.4/12.7</td><td>213.4/10.8</td><td>184.2/9.85</td></tr><tr><td>EB Train (LF)</td><td>93.20</td><td>93.19</td><td>92.91</td><td>250.1/12.8</td><td>199.4/10.7</td><td>170.3/8.51</td></tr><tr><td>EB Train (LL)</td><td>93.25</td><td>93.13</td><td>92.60</td><td>56.1/12.8</td><td>43.1/10.7</td><td>36.5/8.51</td></tr><tr><td>EB Train Improv.</td><td>0.19</td><td>0.01</td><td>-0.57</td><td>6.6×/1.2×</td><td>8.6×/1.4×</td><td>10.2×/1.8×</td></tr><tr><td rowspan="9">PreResNet -101 CIFAR-100</td><td>LT (one-shot)</td><td>71.90</td><td>71.60</td><td>69.95</td><td>6095/14.9</td><td>6095/14.9</td><td>6095/14.9</td></tr><tr><td>SNIP</td><td>72.34</td><td>71.63</td><td>70.01</td><td>3047/7.40</td><td>3047/7.40</td><td>3047/7.40</td></tr><tr><td>NS</td><td>72.80</td><td>71.52</td><td>68.46</td><td>4851/13.7</td><td>4310/12.5</td><td>3993/10.3</td></tr><tr><td>ThiNet</td><td>73.10</td><td>70.92</td><td>67.29</td><td>3603/13.2</td><td>2642/10.6</td><td>1893/8.65</td></tr><tr><td>EB Train (re-init)</td><td>73.23</td><td>73.36</td><td>71.05</td><td>2413/7.35</td><td>2016/6.25</td><td>1392/3.53</td></tr><tr><td>EB Train (FF)</td><td>73.52</td><td>73.15</td><td>72.29</td><td>2020/6.40</td><td>1769/5.45</td><td>1294/3.28</td></tr><tr><td>EB Train (LF)</td><td>73.41</td><td>73.02</td><td>70.72</td><td>2038/6.42</td><td>1614/5.45</td><td>1171/2.99</td></tr><tr><td>EB Train (LL)</td><td>73.04</td><td>71.82</td><td>69.45</td><td>434.4/6.42</td><td>366.5/5.45</td><td>247.3/2.99</td></tr><tr><td>EB Train Improv.</td><td>0.42</td><td>1.73</td><td>2.28</td><td>7.0×/1.2×</td><td>7.2×/1.4×</td><td>7.6×/2.5×</td></tr><tr><td></td><td></td><td>p=10%</td><td>p=30%</td><td>p=50%</td><td>p=10%</td><td>p=30%</td><td>p=50%</td></tr><tr><td rowspan="9">VGG16 CIFAR-100</td><td>LT (one-shot)</td><td>72.62</td><td>71.31</td><td>70.96</td><td>741.2/30.3</td><td>741.2/30.3</td><td>741.2/30.3</td></tr><tr><td>SNIP</td><td>71.55</td><td>70.83</td><td>70.35</td><td>370.6/15.1</td><td>370.6/15.1</td><td>370.6/15.1</td></tr><tr><td>NS</td><td>71.24</td><td>71.28</td><td>69.74</td><td>636.5/29.3</td><td>592.3/27.1</td><td>567.8/24.0</td></tr><tr><td>ThiNet</td><td>70.83</td><td>69.57</td><td>67.22</td><td>632.2/27.4</td><td>568.5/22.6</td><td>381.4/19.0</td></tr><tr><td>EB Train (re-init)</td><td>71.65</td><td>71.48</td><td>69.66</td><td>345.3/16.3</td><td>300.0/13.7</td><td>246.8/10.6</td></tr><tr><td>EB Train (FF)</td><td>71.81</td><td>72.17</td><td>71.28</td><td>287.7/14.1</td><td>262.2/12.2</td><td>221.7/9.85</td></tr><tr><td>EB Train (LF)</td><td>71.60</td><td>71.50</td><td>70.27</td><td>270.5/14.0</td><td>262.7/12.8</td><td>208.7/10.1</td></tr><tr><td>EB Train (LL)</td><td>71.34</td><td>70.53</td><td>69.91</td><td>54.6/14.0</td><td>64.4/12.8</td><td>44.4/10.1</td></tr><tr><td>EB Train Improv.</td><td>-0.81</td><td>0.86</td><td>0.32</td><td>6.8×/1.1×</td><td>5.8×/1.2×</td><td>10.7×/1.5×</td></tr></table>
+
+scaling factors from BN layers, and ThiNet greedily prunes the channel that has the smallest effect on the next layer's activation values. For EB Train, we by default follow (Liu et al., 2017) to inherit the same weights when re-training the searched ticket, adopting floating points for both the search and retrain stages (i.e., EB Train FF). We also notice existing debates (Liu et al., 2018b) on the initialization re-use, and thus also compare with a variant by re-training the ticket from a new random initialization, termed as EB Train (re-init). The comparisons in Table 2 further show that inheriting weights from the EB tickets favor the generalization of retraining as compared to both the random initialization and "over-cooked" weights, aligning well with the recent discussion between rewinding and fine-tuning (Renda et al., 2020). Furthermore, we apply the proposed EB Train on top of the low-precision training method (Yang et al., 2019a) and obtain experiment results of another two variants of EB Train: 1) EB Train with low-precision search and full-precision retrain (i.e., EB Train LF), and 2) EB Train with both low-precision search and retrain (i.e., EB Train LL).
+
+Table 2 demonstrates that EB Train consistently outperforms all competitors in terms of saving training energy and computational costs, meanwhile improving the final accuracy in most cases. We use EB Train Improv. to record the performance margin (either accuracy or energy/computation) between EB Train and the strongest competitor among the four state-of-the-art baselines. Specifically, EB Train with full-precision floating point (FP32) search and retrain outperforms those pruning
+
+Table 3: Comparing the accuracy and total training FLOPs of EB Train, Network Slimming (Liu et al., 2017), ThiNet (Luo et al., 2017), and SFP (He et al., 2018). The "Acc. Improv." is the accuracy of the pruned model minus that of the unpruned model, so a positive number means the pruned model has a higher accuracy.  
+
+<table><tr><td>Models</td><td>Methods</td><td>Pruning ratio</td><td>Top-1 Acc. (%)</td><td>Top-1 Acc. Improv. (%)</td><td>Top-5 Acc. (%)</td><td>Top-5 Acc. Improv. (%)</td><td>Total Training FLOPs (P)</td><td>Total Training Energy (MJ)</td></tr><tr><td rowspan="6">ResNet18ImageNet</td><td>Unpruned</td><td>-</td><td>69.57</td><td>-</td><td>89.24</td><td>-</td><td>1259.13</td><td>98.14</td></tr><tr><td rowspan="2">NS</td><td>10%</td><td>69.65</td><td>+0.08</td><td>89.20</td><td>-0.04</td><td>2424.86</td><td>193.51</td></tr><tr><td>30%</td><td>67.85</td><td>-1.72</td><td>88.07</td><td>-1.17</td><td>2168.89</td><td>180.92</td></tr><tr><td>SFP</td><td>30%</td><td>67.10</td><td>-2.47</td><td>87.78</td><td>-1.46</td><td>1991.94</td><td>158.14</td></tr><tr><td rowspan="2">EB Train</td><td>10%</td><td>69.84</td><td>+0.27</td><td>89.39</td><td>+0.15</td><td>1177.15</td><td>95.71</td></tr><tr><td>30%</td><td>68.28</td><td>-1.29</td><td>88.28</td><td>-0.96</td><td>952.46</td><td>84.65</td></tr><tr><td rowspan="7">ResNet50ImageNet</td><td>Unpruned</td><td>-</td><td>75.99</td><td>-</td><td>92.98</td><td>-</td><td>2839.96</td><td>280.72</td></tr><tr><td rowspan="3">Thinet</td><td>30%</td><td>72.04</td><td>-3.55</td><td>90.67</td><td>-2.31</td><td>4358.53</td><td>456.13</td></tr><tr><td>50%</td><td>71.01</td><td>-4.58</td><td>90.02</td><td>-2.96</td><td>3850.03</td><td>431.73</td></tr><tr><td>70%</td><td>68.42</td><td>-7.17</td><td>88.30</td><td>-4.68</td><td>3431.48</td><td>416.44</td></tr><tr><td rowspan="3">EB Train</td><td>30%</td><td>73.86</td><td>-1.73</td><td>91.52</td><td>-1.46</td><td>2242.30</td><td>232.18</td></tr><tr><td>50%</td><td>73.35</td><td>-2.24</td><td>91.36</td><td>-1.62</td><td>1718.78</td><td>188.18</td></tr><tr><td>70%</td><td>70.16</td><td>-5.43</td><td>89.55</td><td>-3.43</td><td>890.65</td><td>121.15</td></tr></table>
+
+methods by up to  $1.2 \sim 4.7 \times$  and  $1.1 \sim 4.5 \times$  in terms of the energy consumption and computational FLOPs, while always achieving comparable or even better accuracies, across three pruning ratios, two DNN models and two datasets. Moreover, EB Train with both low-precision (8 bits block floating point (FP8)) search and retrain outperforms the baselines by up to  $5.8 \sim 24.6 \times$  and  $1.1 \sim 5.0 \times$  in terms of energy consumption estimated using real-measured unit energy from (Yang et al., 2019b) and computational FLOPs. Note that FP8 and FP32 are counted as the same FLOPs count units (Sohoni et al., 2019). In addition, comparing with the re-init variant endorses the effectiveness of initialization inheritance in EB Train. As an additional highlight, EB Train naturally leads to more efficient inference of the pruned DNN models, unifying the boost of efficiency throughout the full learning lifecycle.
+
+ResNet18/50 on ImageNet. To study the performance of EB Train in a harder dataset, we conduct experiments using ResNet18/50 and ImageNet, and compare its resulting accuracy, total training FLOPs and energy cost with those of the method in (He et al., 2016a), NS (Liu et al., 2017), ThiNet (Luo et al., 2017) and SFP (He et al., 2018) as summarized in Table 3. We have three observations: 1) When training ResNet18 on ImageNet, EB Train achieves a better accuracy  $(+0.27\%)$  over the unpruned one while introducing  $10\%$  channel sparsity; 2) EB Train outperforms other baselines for both ResNet18 and ResNet50 by reducing the total training costs while achieving a comparable or better accuracy. Specifically, EB Train achieves a reduced training FLOPs of  $51.5\% \sim 56.1\%$  and  $48.6\% \sim 74.0\%$  and a reduced training energy of  $46.5\% \sim 53.2\%$  and  $49.1\% \sim 70.9\%$ , while leading to a better top-1 accuracy of  $+0.19\% \sim +1.18\%$  and  $+1.74\% \sim +2.34\%$  for ResNet18 and ResNet50, respectively. For example, when training ResNet50 on ImageNet, EB Train with a pruning ratio of  $70\%$  achieves better  $(+1.74\%)$  accuracy over ThiNet while saving  $74\%$  total training FLOPs and  $71\%$  training energy; 3) Different from the results on CIFAR-10/100 shown in Table 2, all methods performed on ImageNet (a harder dataset) start to yield accuracy reductions  $(-1.72\% \sim -3.55\%)$  for ResNet18/50 with a pruning ratio of only  $30\%$ , compared with the unpruned one. Note that in Table 3, the unpruned results are based on the official implementation in (He et al., 2016a); The SFP results are obtained from their original paper (He et al., 2018), which does not provide results at a pruning ratio of  $10\%$ ; all the ThiNet results under various pruning ratios are obtained from the original paper (Luo et al., 2017); and the NS results are obtained by conducting the experiments ourselves.
+
+# 5 DISCUSSION AND CONCLUSION
+
+We have demonstrated that winning tickets can be drawn at the very early training stage, i.e., EB tickets exist, in both the standard training and several lower-cost training schemes. That motivates a practical success of applying EB tickets to efficient training, whose results compare favorably against state-of-the-arts. Moreover, experiments show that EB Train with low-precision search and retraining achieve more efficient training. We believe many promising problems still remain open to be addressed. An immediate future work is to test low-precision EB Train algorithms on larger models/datasets. We are also curious whether more lower-cost training techniques could be associated with EB Train. Finally, sometimes high pruning ratios (e.g,  $p \geq 0.7$ ) can hurt the quality of EB tickets and the retrained networks. We look forward to automating the choice of  $p$  for different models/datasets, unleashing higher efficiency.
+
+# 6 ACKNOWLEDGEMENT
+
+The work is supported by the National Science Foundation (NSF) through the Real-Time Machine Learning (RTML) program (Award number: 1937592, 1937588).
+
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+
+# A APPENDIX
+
+# A.1 MEASUREMENT OF ENERGY COST
+
+Figure 6 shows our GPU measurement setup, in which the GPU board is connected to a laptop and a power meter. In particular, the training settings are downloaded from the laptop to the GPU board, and the real-measured energy consumption is obtained via the power meter and runtime measurement for the whole training process.
+
+![](images/b7193fd732dbee7450ce766b94fa8719394f378e12b1ed0e500ca96ef0e646e4.jpg)  
+Figure 6: The energy measurement setup (from left to right) with a MAC Air latpstop, an embedded GPU (JETSON TX2 (NVIDIA Inc.)), and a power meter.
+
+# A.2 APPROPRIATE LARGE LEARNING RATE FAVORS THE EMERGENCE OF EB TICKETS
+
+Here we show the effect of larger learning rates setting. The original schedule is denoted as  $[80_{LR\to 0.01},120_{LR\to 0.001}]$ , i.e., starting from 0.1, decayed to 0.01 at the 80th epoch, and further decayed to 0.001 at the 120th epoch. In comparison, we test a larger initial learning rate, staring from 0.5 and 0.2 for VGG16 and PreResNet101 datasets, respectively. After drawing tickets, we retrain them all using the same learning rate schedule with their searching phase for sufficient training and a fair comparison. We see from Table 4 that high-quality EB tickets also emerge early when searching with the larger initial learning rate, whose final accuracies are even better.
+
+Table 4: The retraining accuracy of subnetworks drawn at different training epochs using different initial learning rate, where the pruning ratio is 0.5.  
+
+<table><tr><td rowspan="2"></td><td rowspan="2">LR Initial</td><td colspan="4">Retrain acc.(%) (CIFAR-100)</td><td colspan="4">Retrain acc.(%) (CIFAR-10)</td></tr><tr><td>10</td><td>20</td><td>40</td><td>final</td><td>10</td><td>20</td><td>40</td><td>final</td></tr><tr><td rowspan="2">VGG16</td><td>0.1</td><td>71.11</td><td>71.07</td><td>69.14</td><td>69.74</td><td>93.26</td><td>93.34</td><td>93.20</td><td>92.96</td></tr><tr><td>0.5</td><td>70.69</td><td>71.65</td><td>71.94</td><td>71.58</td><td>93.49</td><td>93.45</td><td>93.44</td><td>93.29</td></tr><tr><td rowspan="2">PreResNet101</td><td>0.1</td><td>71.58</td><td>72.67</td><td>72.67</td><td>71.52</td><td>93.60</td><td>93.46</td><td>93.56</td><td>93.42</td></tr><tr><td>0.2</td><td>72.58</td><td>72.90</td><td>72.86</td><td>72.71</td><td>93.40</td><td>93.46</td><td>93.87</td><td>93.69</td></tr></table>
\ No newline at end of file
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+# DREAM TO CONTROL: LEARNING BEHAVIORS BY LATENT IMAGINATION
+
+Danijar Hafner
+
+University of Toronto
+
+Google Brain
+
+Timothy Lillicrap
+
+DeepMind
+
+Jimmy Ba
+
+University of Toronto
+
+Mohammad Norouzi
+
+Google Brain
+
+# Abstract
+
+Learned world models summarize an agent's experience to facilitate learning complex behaviors. While learning world models from high-dimensional sensory inputs is becoming feasible through deep learning, there are many potential ways for deriving behaviors from them. We present Dreamer, a reinforcement learning agent that solves long-horizon tasks from images purely by latent imagination. We efficiently learn behaviors by propagating analytic gradients of learned state values back through trajectories imagined in the compact state space of a learned world model. On 20 challenging visual control tasks, Dreamer exceeds existing approaches in data-efficiency, computation time, and final performance.
+
+# 1 INTRODUCTION
+
+Intelligent agents can achieve goals in complex environments even though they never encounter the exact same situation twice. This ability requires building representations of the world from past experience that enable generalization to novel situations. World models offer an explicit way to represent an agent's knowledge about the world in a parametric model that can make predictions about the future.
+
+When the sensory inputs are high-dimensional images, latent dynamics models can abstract observations to predict forward in compact state spaces (Watter et al., 2015; Oh et al., 2017; Gregor et al., 2019). Compared to predictions in image space, latent states have a small memory footprint that enables imagining thousands of trajectories in parallel. Learning effective latent dynamics models is becoming feasible through advances in deep learning and latent variable models (Krishnan et al., 2015; Karl et al., 2016; Doerr et al., 2018; Buesing et al., 2018).
+
+Behaviors can be derived from dynamics models in many ways. Often, imagined rewards are maximized with a parametric policy (Sutton, 1991; Ha and Schmidhuber, 2018; Zhang et al., 2019) or by online planning (Chua et al., 2018; Hafner et al., 2018). However, considering only rewards within a fixed imagination horizon results in shortsighted behaviors (Wang et al., 2019). Moreover, prior work commonly resorts to derivative-free optimization for robustness to model errors (Ebert et al., 2017; Chua et al., 2018; Parmas et al., 2019), rather than leveraging analytic gradients offered by neural network dynamics (Henaff et al., 2019; Srinivas et al., 2018).
+
+We present Dreamer, an agent that learns long-horizon behaviors from images purely by latent imagination. A novel actor critic algorithm accounts for rewards beyond the imagination horizon while making efficient use of the neural network dynamics. For this, we predict state values and actions in the learned latent space as summarized in Figure 1. The values optimize Bellman consistency for imagined rewards and the policy maximizes the values by propagating their analytic gradients back through the dynamics.
+
+In comparison to actor critic algorithms that learn online or by experience replay (Lillicrap et al., 2015; Mnih et al., 2016; Schulman et al., 2017; Haarnoja et al., 2018; Lee et al., 2019), world models can interpolate past experience and offer analytic gradients of multi-step returns for efficient policy optimization.
+
+![](images/67c55628e3285d1c69d03dc3f391afd42be5b02d4c5f9f1b6d9ddcd7a53f41f4.jpg)  
+Dataset of Experience
+
+![](images/58209e72ab5b0760a6a10ef8c45572a5d5cfb897d61b0e186c97a01995159e95.jpg)  
+Learned Latent Dynamics
+
+![](images/19e9b3acba9d601c1d7ca8a3d28e45f9119cc5fbff3b799eba700cd7ff64acf9.jpg)  
+Value and Action Learned by Latent Imagination  
+Figure 1: Dreamer learns a world model from past experience and efficiently learns farsighted behaviors in its latent space by backpropagating value estimates back through imagined trajectories.
+
+![](images/96ca066b4a962fb843311a36c80d1783887c20648e721280c98bec10641d50f3.jpg)  
+(a) Cup
+
+![](images/d76aae8b25470d76813e1f78aaee58294910b8010d7c8a6537adf202c41a0777.jpg)  
+(b) Acrobot
+
+![](images/b2d098109ef7014397c13825f649a43bc9a075fe66b9e38b77531182c90e6153.jpg)  
+(c) Hopper  
+Figure 2: Image observations for 5 of the 20 visual control tasks used in our experiments. The tasks pose a variety of challenges including contact dynamics, sparse rewards, many degrees of freedom, and 3D environments. Several of these tasks could previously not be solved through world models.
+
+![](images/6042e6d32ceb3840048f7d36f694e46fa632ce652f4402de3a26c32e07e3b89b.jpg)  
+(d) Walker
+
+![](images/fe3c40fd9aeb9e0416e30c22888374e47099a8e7bc1e6a41f135423e920231c7.jpg)  
+(e) Quadruped
+
+The key contributions of this paper are summarized as follows:
+
+- Learning long-horizon behaviors by latent imagination Model-based agents can be short-sighted if they use a finite imagination horizon. We approach this limitation by predicting both actions and state values. Training purely by imagination in a latent space lets us efficiently learn the policy by propagating analytic value gradients back through the latent dynamics.  
+- Empirical performance for visual control We pair Dreamer with existing representation learning methods and evaluate it on the DeepMind Control Suite with image inputs, illustrated in Figure 2. Using the same hyper parameters for all tasks, Dreamer exceeds previous model-based and model-free agents in terms of data-efficiency, computation time, and final performance.
+
+# 2 CONTROL WITH WORLD MODELS
+
+Reinforcement learning We formulate visual control as a partially observable Markov decision process (POMDP) with discrete time step  $t \in [1; T]$ , continuous vector-valued actions  $a_{t} \sim p(a_{t} \mid o_{\leq t}, a_{<t})$  generated by the agent, and high-dimensional observations and scalar rewards  $o_{t}, r_{t} \sim p(o_{t}, r_{t} \mid o_{<t}, a_{<t})$  generated by the unknown environment. The goal is to develop an agent that maximizes the expected sum of rewards  $\mathrm{E}_{p}\left(\sum_{t=1}^{T} r_{t}\right)$ . Figure 2 shows a selection of our tasks.
+
+Agent components The classical components of agents that learn in imagination are dynamics learning, behavior learning, and environment interaction (Sutton, 1991). In the case of Dreamer, the behavior is learned by predicting hypothetical trajectories in the compact latent space of the world model. As outlined in Figure 3 and detailed in Algorithm 1, Dreamer performs the following operations throughout the agent's life time, either interleaved or in parallel:
+
+- Learning the latent dynamics model from the dataset of past experience to predict future rewards from actions and past observations. Any learning objective for the world model can be incorporated with Dreamer. We review existing methods for learning latent dynamics in Section 4.  
+- Learning action and value models from predicted latent trajectories, as described in Section 3. The value model optimizes Bellman consistency for imagined rewards and the action model is updated by propagating gradients of value estimates back through the neural network dynamics.  
+- Executing the learned action model in the world to collect new experience for growing the dataset.
+
+Latent dynamics Dreamer uses a latent dynamics model that consists of three components. The representation model encodes observations and actions to create continuous vector-valued model states  $s_t$  with Markovian transitions (Watter et al., 2015; Zhang et al., 2019; Hafner et al., 2018). The transition model predicts future model states without seeing the corresponding observations that will later cause them. The reward model predicts the rewards given the model states,
+
+Representation model:  $p(s_{t} \mid s_{t-1}, a_{t-1}, o_{t})$
+
+Transition model:  $q(s_{t}\mid s_{t - 1},a_{t - 1})$  (1)
+
+Reward model:  $q(r_{t} \mid s_{t})$ .
+
+We use  $p$  for distributions that generate samples in the real environment and  $q$  for their approximations that enable latent imagination. Specifically, the transition model lets us predict ahead in the compact latent space without having to observe or imagine the corresponding images. This results in a low memory footprint and fast predictions of thousands of imagined trajectories in parallel.
+
+The model mimics a non-linear Kalman filter (Kalman, 1960), latent state space model, or HMM with real-valued states. However, it is conditioned on actions and predicts rewards, allowing the agent to imagine the outcomes of potential action sequences without executing them in the environment.
+
+![](images/deb695380a4a7215adc9183c3e1216ed65c542c07196cf1fe3013bc95480a78f.jpg)  
+(a) Learn dynamics from experience
+
+![](images/5596106e1e6a646bb9b5ee46debfaa685ddaf39081ab78f554b55a43e83cfc07.jpg)  
+(b) Learn behavior in imagination  
+Figure 3: Components of Dreamer. (a) From the dataset of past experience, the agent learns to encode observations and actions into compact latent states  $(\bullet)$ , for example via reconstruction, and predicts environment rewards  $(\diamond)$ . (b) In the compact latent space, Dreamer predicts state values  $(\text{空})$  and actions  $(\text{空})$  that maximize future value predictions by propagating gradients back through imagined trajectories. (c) The agent encodes the history of the episode to compute the current model state and predict the next action to execute in the environment. See Algorithm 1 for pseudo code of the agent.
+
+![](images/febc8aa63c69ebf3b664e9633cf58002ac092d2beb6221346ad827147e4accb9.jpg)  
+(c) Act in the environment
+
+# 3 LEARNING BEHAVIORS BY LATENT IMAGINATION
+
+Dreamer learns long-horizon behaviors in the compact latent space of a learned world model by efficiently leveraging the neural network latent dynamics. For this, we propagate stochastic gradients of multi-step returns through neural network predictions of actions, states, rewards, and values using reparameterization. This section describes the main contribution of our paper.
+
+Imagination environment The latent dynamics define a Markov decision process (MDP; Sutton, 1991) that is fully observed because the compact model states  $s_t$  are Markovian. We denote imagined quantities with  $\tau$  as the time index. Imagined trajectories start at the true model states  $s_t$  of observation sequences drawn from the agent's past experience. They follow predictions of the transition model  $s_\tau \sim q(s_\tau \mid s_{\tau-1}, a_{\tau-1})$ , reward model  $r_\tau \sim q(r_\tau \mid s_\tau)$ , and a policy  $a_\tau \sim q(a_\tau \mid s_\tau)$ . The objective is to maximize expected imagined rewards  $\mathrm{E}_q\left(\sum_{\tau=t}^{\infty} \gamma^{\tau-t} r_\tau\right)$  with respect to the policy.
+
+# Algorithm 1: Dreamer
+
+Initialize dataset  $\mathcal{D}$  with  $S$  random seed episodes. Initialize neural network parameters  $\theta, \phi, \psi$  rando while not converged do
+
+for update step  $c = 1..C$  do
+
+// Dynamics learning
+
+Draw  $B$  data sequences  $\{(a_t, o_t, r_t)\}_{t = k}^{k + L} \sim \mathcal{D}$ .
+
+Compute model states  $s_t \sim p_\theta(s_t \mid s_{t-1}, a_{t-1}, o_t)$ .
+
+Update  $\theta$  using representation learning.
+
+// Behavior learning
+
+Imagine trajectories  $\{(s_{\tau},a_{\tau})\}_{\tau = t}^{t + H}$  from each  $s_t$
+
+Predict rewards  $\operatorname{E}\left(q_{\theta}(r_{\tau} \mid s_{\tau})\right)$  and values  $v_{\psi}(s_{\tau})$
+
+Compute value estimates  $\mathrm{V}_{\lambda}(s_{\tau})$  via Equation 6.
+
+Update  $\phi \gets \phi +\alpha \nabla_{\phi}\sum_{\tau = t}^{t + H}\mathrm{V}_{\lambda}(s_{\tau})$
+
+Update  $\psi \gets \psi -\alpha \nabla_{\psi}\sum_{\tau = t}^{t + H}\frac{1}{2}\big||v_{\psi}(s_{\tau}) - V_{\lambda}(s_{\tau})\big||^2.$
+
+# Model components
+
+Representation  $p_{\theta}(s_t \mid s_{t-1}, a_{t-1}, o_t)$
+
+Transition  $q_{\theta}(s_t \mid s_{t-1}, a_{t-1})$
+
+Reward  $q_{\theta}(r_t \mid s_t)$
+
+Action
+
+Value  $v_{\psi}(s_t)$
+
+# Hyper parameters
+
+Seed episodes
+
+Collect interval C
+
+Batch size B
+
+Sequence length L
+
+Imagination horizon H
+
+Learning rate  $\alpha$
+
+// Environment interaction  $o_1\gets$  env.reset()
+
+for time step  $t = 1..T$  do
+
+Compute  $s_t \sim p_\theta(s_t \mid s_{t-1}, a_{t-1}, o_t)$  from history.
+
+Compute  $a_{t} \sim q_{\phi}(a_{t} \mid s_{t})$  with the action model.
+
+Add exploration noise to action.
+
+$r_t, o_{t+1} \gets \text{env}. \text{step}(a_t)$ .
+
+Add experience to dataset  $\mathcal{D}\gets \mathcal{D}\cup \{(o_t,a_t,r_t)_{t = 1}^T\}$ .
+
+![](images/4371a017222e0d5baf20a18fc5a69b6bc6efc3f445b1513544105538e5c645e6.jpg)  
+Figure 4: Imagination horizons. We compare the final performance of Dreamer, learning an action model without value prediction, and online planning using PlaNet. Learning a state value model to estimate rewards beyond the imagination horizon makes Dreamer more robust to the horizon length. The agents use pixel reconstruction for representation learning and an action repeat of  $R = 2$ .
+
+![](images/fbeaa6d242ebfcfa9d26e8234942c8f29f6da9f24f41a4a2c1a1f2b82b5e5033.jpg)
+
+![](images/5170cf994178a928dbae78351698e8101e9ee17daf50928023d9b8698273480c.jpg)
+
+![](images/6c387eb35ffca9437b44b233e8c408e5e1bc18c241f1fcff804c49d1326df0ee.jpg)
+
+Action and value models Consider imagined trajectories with a finite horizon  $H$ . Dreamer uses an actor critic approach to learn behaviors that consider rewards beyond the horizon. We learn an action model and a value model in the latent space of the world model for this. The action model implements the policy and aims to predict actions that solve the imagination environment. The value model estimates the expected imagined rewards that the action model achieves from each state  $s_{\tau}$ ,
+
+Action model:
+
+$$
+a _ {\tau} \sim q _ {\phi} (a _ {\tau} \mid s _ {\tau})
+$$
+
+Value model:
+
+$$
+v _ {\psi} \left(s _ {\tau}\right) \approx \mathrm {E} _ {q \left(\cdot \mid s _ {\tau}\right)} \left(\sum_ {\tau = t} ^ {t + H} \gamma^ {\tau - t} r _ {\tau}\right). \tag {2}
+$$
+
+The action and value models are trained cooperatively as typical in policy iteration: the action model aims to maximize an estimate of the value, while the value model aims to match an estimate of the value that changes as the action model changes.
+
+We use dense neural networks for the action and value models with parameters  $\phi$  and  $\psi$ , respectively. The action model outputs a tanh-transformed Gaussian (Haarnoja et al., 2018) with sufficient statistics predicted by the neural network. This allows for reparameterized sampling (Kingma and Welling, 2013; Rezende et al., 2014) that views sampled actions as deterministically dependent on the neural network output, allowing us to backpropagate analytic gradients through the sampling operation,
+
+$$
+a _ {\tau} = \tanh  \left(\mu_ {\phi} \left(s _ {\tau}\right) + \sigma_ {\phi} \left(s _ {\tau}\right) \epsilon\right), \quad \epsilon \sim \operatorname {N o r m a l} (0, \mathbb {I}). \tag {3}
+$$
+
+Value estimation To learn the action and value models, we need to estimate the state values of imagined trajectories  $\{s_{\tau},a_{\tau},r_{\tau}\}_{\tau = t}^{t + H}$ . These trajectories branch off of the model states  $s_t$  of sequence batches drawn from the agent's dataset of experience and predict forward for the imagination horizon  $H$  using actions sampled from the action model. State values can be estimated in multiple ways that trade off bias and variance (Sutton and Barto, 2018),
+
+$$
+\mathrm {V} _ {\mathrm {R}} \left(s _ {\tau}\right) \doteq \mathrm {E} _ {q _ {\theta}, q _ {\phi}} \left(\sum_ {n = \tau} ^ {t + H} r _ {n}\right), \tag {4}
+$$
+
+$$
+\mathrm {V} _ {\mathrm {N}} ^ {k} \left(s _ {\tau}\right) \doteq \mathrm {E} _ {q _ {\theta}, q _ {\phi}} \left(\sum_ {n = \tau} ^ {h - 1} \gamma^ {n - \tau} r _ {n} + \gamma^ {h - \tau} v _ {\psi} \left(s _ {h}\right)\right) \quad \text {w i t h} \quad h = \min  (\tau + k, t + H), \tag {5}
+$$
+
+$$
+\mathrm {V} _ {\lambda} \left(s _ {\tau}\right) \doteq (1 - \lambda) \sum_ {n = 1} ^ {H - 1} \lambda^ {n - 1} \mathrm {V} _ {\mathrm {N}} ^ {n} \left(s _ {\tau}\right) + \lambda^ {H - 1} \mathrm {V} _ {\mathrm {N}} ^ {H} \left(s _ {\tau}\right), \tag {6}
+$$
+
+where the expectations are estimated under the imagined trajectories.  $\mathrm{V}_{\mathrm{R}}$  simply sums the rewards from  $\tau$  until the horizon and ignores rewards beyond it. This allows learning the action model without a value model, an ablation we compare to in our experiments.  $\mathrm{V}_{\mathrm{N}}^{k}$  estimates rewards beyond  $k$  steps with the learned value model. Dreamer uses  $\mathrm{V}_{\lambda}$ , an exponentially-weighted average of the estimates for different  $k$  to balance bias and variance. Figure 4 shows that learning a value model in imagination enables Dreamer to solve long-horizon tasks while being robust to the imagination horizon. The experimental details and results on all tasks are described in Section 6.
+
+![](images/6926cb930c74c6fceb59544689a1adeca3eb270182864ca00be9ef6f544d62c6.jpg)  
+Figure 5: Reconstructions of long-term predictions. We apply the representation model to the first 5 images of two hold-out trajectories and predict forward for 45 steps using the latent dynamics, given only the actions. The recurrent state space model (RSSM; Hafner et al., 2018) performs accurate long-term predictions, enabling Dreamer to learn successful behaviors in a compact latent space.
+
+Learning objective To update the action and value models, we first compute the value estimates  $\mathrm{V}_{\lambda}(s_{\tau})$  for all states  $s_{\tau}$  along the imagined trajectories. The objective for the action model  $q_{\phi}(a_{\tau} \mid s_{\tau})$  is to predict actions that result in state trajectories with high value estimates. The objective for the value model  $v_{\psi}(s_{\tau})$ , in turn, is to regress the value estimates,
+
+$$
+\max  _ {\phi} \mathrm {E} _ {q _ {\theta}, q _ {\phi}} \left(\sum_ {\tau = t} ^ {t + H} \mathrm {V} _ {\lambda} (s _ {\tau})\right), \quad (7) \quad \min  _ {\psi} \mathrm {E} _ {q _ {\theta}, q _ {\phi}} \left( \right.\sum_ {\tau = t} ^ {t + H} \frac {1}{2} \left\| \right. v _ {\psi} (s _ {\tau}) - \mathrm {V} _ {\lambda} (s _ {\tau})\left. \right)\left. \right\| ^ {2}\left. \right). \tag {8}
+$$
+
+The value model is updated to regress the targets, around which we stop the gradient as typical (Sutton and Barto, 2018). The action model uses analytic gradients through the learned dynamics to maximize the value estimates. To understand this, we note that the value estimates depend on the reward and value predictions, which depend on the imagined states, which in turn depend on the imagined actions. Since all steps are implemented as neural networks, we analytically compute  $\nabla_{\phi}\mathrm{E}_{q_{\theta},q_{\phi}}\left(\sum_{\tau = t}^{t + H}\mathrm{V}_{\lambda}(s_{\tau})\right)$  by stochastic backpropagation (Kingma and Welling, 2013; Rezende et al., 2014). We use reparameterization for continuous actions and latent states and straight-through gradients (Bengio et al., 2013) for discrete actions. The world model is fixed while learning behaviors. In tasks with early termination, the world model also predicts the discount factor from each latent state to weigh the time steps in Equations 7 and 8 by the cumulative product of the predicted discount factors, so terms are weighted down based on how likely the imagined trajectory would have ended.
+
+Comparison to actor critic methods Agents using Reinforce gradients (Williams, 1992), such as A3C and PPO (Mnih et al., 2016; Schulman et al., 2017), employ value baselines to reduce gradient variance, while Dreamer backpropagates through the value model. This is similar to deterministic or reparameterized actor critics (Silver et al., 2014), such as DDPG and SAC (Lillicrap et al., 2015; Haarnoja et al., 2018). However, these do not leverage gradients through transitions and only maximize immediate Q-values. MVE and STEVE (Feinberg et al., 2018; Buckman et al., 2018) extend them to multi-step Q-learning with learned dynamics to provide more accurate Q-value targets. We predict state values, which is sufficient for policy optimization since we backpropagate through the dynamics. Refer to Section 5 for a more detailed comparison to related work.
+
+# 4 LEARNING LATENT DYNAMICS
+
+Learning behaviors in imagination requires a world model that generalizes well. We focus on latent dynamics models that predict forward in a compact latent space, facilitating long-term predictions and allowing the agent to imagine thousands of trajectories in parallel. Several objectives for learning representations for control have been proposed (Watter et al., 2015; Jaderberg et al., 2016; Oord et al., 2018; Eslami et al., 2018). We review three approaches for learning representations to use with Dreamer: reward prediction, image reconstruction, and contrastive estimation.
+
+Reward prediction Latent imagination requires a representation model  $p(s_{t} \mid s_{t-1}, a_{t-1}, o_{t})$ , transition model  $q(s_{t} \mid s_{t-1}, a_{t-1})$ , and reward model  $q(r_{t} \mid s_{t})$ , as described in Section 2. In principle, this could be achieved by simply learning to predict future rewards given actions and past observations (Oh et al., 2017; Gelada et al., 2019; Schrittwieser et al., 2019). With a large and diverse dataset, such representations should be sufficient for solving a control task. However, with a finite dataset and especially when rewards are sparse, learning about observations that correlate with rewards is likely to improve the world model (Jaderberg et al., 2016; Gregor et al., 2019).
+
+![](images/1d0e9127dc840181b25559ef7ce747df68408ea44472ce22dcb31285c5cd20b9.jpg)  
+Figure 6: Performance comparison to existing methods. Dreamer inherits the data-efficiency of PlaNet while exceeding the asymptotic performance of the best model-free agents. After  $5 \times 10^{6}$  environment steps, Dreamer reaches an average performance of 823 across tasks, compared to PlaNet at 332 and the top model-free D4PG agent at 786 after  $10^{9}$  steps. Results are averages over 5 seeds.
+
+Reconstruction We first describe the world model used by PlaNet (Hafner et al., 2018) that learns latent dynamics by reconstructing images as shown in Figure 3a. The world model consists of the following components, where the observation model is only used to provide a learning signal,
+
+Representation model:  $p_{\theta}(s_t \mid s_{t-1}, a_{t-1}, o_t)$
+
+Observation model:  $q_{\theta}(o_t\mid s_t)$  Reward model:  $q_{\theta}(r_t\mid s_t)$  (9)
+
+Transition model:  $q_{\theta}(s_t \mid s_{t-1}, a_{t-1})$ .
+
+The components are optimized jointly to increase the variational lower bound (ELBO; Jordan et al., 1999) or more generally the variational information bottleneck (VIB; Tishby et al., 2000; Alemi et al., 2016). As derived in Appendix B, the bound includes reconstruction terms for observations and rewards and a KL regularizer. The expectation is taken under the dataset and representation model,
+
+$$
+\mathcal {J} _ {\mathrm {R E C}} \doteq \mathrm {E} _ {p} \left(\sum_ {t} \left(\mathcal {J} _ {\mathrm {O}} ^ {t} + \mathcal {J} _ {\mathrm {R}} ^ {t} + \mathcal {J} _ {\mathrm {D}} ^ {t}\right)\right) + \text {c o n s t} \quad \mathcal {J} _ {\mathrm {O}} ^ {t} \doteq \ln q \left(o _ {t} \mid s _ {t}\right) \tag {10}
+$$
+
+$$
+\mathcal {J} _ {\mathrm {R}} ^ {t} \doteq \ln q (r _ {t} \mid s _ {t}) \qquad \mathcal {J} _ {\mathrm {D}} ^ {t} \doteq - \beta \operatorname {K L} \left(p (s _ {t} \mid s _ {t - 1}, a _ {t - 1}, o _ {t}) \parallel q (s _ {t} \mid s _ {t - 1}, a _ {t - 1})\right).
+$$
+
+We implement the transition model as a recurrent state space model (RSSM; Hafner et al., 2018), the representation model by combining the RSSM with a convolutional neural network (CNN; LeCun et al., 1989) applied to the image observation, the observation model as a transposed CNN, and the reward model as a dense network. The combined parameter vector  $\theta$  is updated by stochastic backpropagation (Kingma and Welling, 2013; Rezende et al., 2014). Figure 5 shows video predictions of this model. We refer to Appendix A and Hafner et al. (2018) model details.
+
+Contrastive estimation Predicting pixels can require high model capacity. We can also encourage mutual information between model states and observations by instead predicting the states from the images (Guo et al., 2018). This replaces the observation model with a state model,
+
+$$
+\text {S t a t e m o d e l :} \quad q _ {\theta} \left(s _ {t} \mid o _ {t}\right). \tag {11}
+$$
+
+While the reconstruction objective used the fact that the observation marginal is a constant, we now face the state marginal. As shown in Appendix B, this can be estimated via noise contrastive estimation (NCE; Gutmann and Hyvarinen, 2010; Oord et al., 2018) by averaging the state model over observations  $o^{\prime}$  of the current sequence batch. Intuitively,  $q(s_{t} \mid o_{t})$  makes the state predictable from the current image while  $\ln \sum_{o^{\prime}} q(s_{t} \mid o^{\prime})$  keeps it diverse to prevent collapse,
+
+$$
+\mathcal {J} _ {\mathrm {N C E}} \doteq \mathrm {E} \left(\sum_ {t} \left(\mathcal {J} _ {\mathrm {S}} ^ {t} + \mathcal {J} _ {\mathrm {R}} ^ {t} + \mathcal {J} _ {\mathrm {D}} ^ {t}\right)\right) \quad \mathcal {J} _ {\mathrm {S}} ^ {t} \doteq \ln q \left(s _ {t} \mid o _ {t}\right) - \ln \left(\sum_ {o ^ {\prime}} q \left(s _ {t} \mid o ^ {\prime}\right)\right). \tag {12}
+$$
+
+We implement the state model as a CNN and again optimize the bound with respect to the combined parameter vector  $\theta$  using stochastic backpropagation. While avoiding pixel prediction, the amount of information this bound can extract efficiently is limited (McAllester and States, 2018). We empirically compare reward, reconstruction, and contrastive objectives in our experiments in Figure 8.
+
+![](images/bda44ef2ee5b14fbc9ad34fe25252c2667a36f807642ddf7ed9fec67b7ad1e7d.jpg)
+
+![](images/d74539370f066f223d80135fbc4cde98106d7d515e4f19087f3beaf8924e542b.jpg)
+
+![](images/eced283b7bf62da3f3b60a07fa15d1671ee37d751cc788a958fac326b7ddb285.jpg)
+
+![](images/63d63c1b63096b570be3099d87bf59eb82f48f29f488fc144ff16b3a44c7600f.jpg)
+
+Figure 7: Dreamer succeeds at visual control tasks that require long-horizon credit assignment, such as the acrobot and hopper tasks. Optimizing only imagined rewards within the horizon via an action model or by online planning yields shortsighted behaviors that only succeed in reactive tasks, such as in the walker domain. The performance on all 20 tasks is summarized in Figure 6 and training curves are shown in Appendix D. See Tassa et al. (2018) for performance curves of D4PG and A3C.  
+![](images/c3a75904794f55a92eb248bc23b5be6ff403e48b4387f1255041875484854796.jpg)  
+Dreamer No value PlaNet D4PG (1e9 steps) A3C (1e9 steps, proprio)
+
+![](images/7654c35f5e3709252b2114a09b60dad550dc001b30a46fba1183c599472dbd84.jpg)
+
+![](images/7db8ea5683bd64d74b55ea1bf546f00e3d8b1f45596d91e9e9bec08430b398a1.jpg)
+
+![](images/f2a705cbe854fedb770e4917c2dec9e617ae8093e8743f7187271077335bda6f.jpg)
+
+# 5 RELATED WORK
+
+Prior works learn latent dynamics for visual control by derivative-free policy learning or online planning, augment model-free agents with multi-step predictions, or use analytic gradients of Q-values or multi-step rewards, often for low-dimensional tasks. In comparison, Dreamer uses analytic gradients to efficiently learn long-horizon behaviors for visual control purely by latent imagination.
+
+Control with latent dynamics E2C (Watter et al., 2015) and RCE (Banijamali et al., 2017) embed images to predict forward in a compact space to solve simple tasks. World Models (Ha and Schmidhuber, 2018) learn latent dynamics in a two-stage process to evolve linear controllers in imagination. PlaNet (Hafner et al., 2018) learns them jointly and solves visual locomotion tasks by latent online planning. SOLAR (Zhang et al., 2019) solves robotic tasks via guided policy search in latent space. I2A (Weber et al., 2017) hands imagined trajectories to a model-free policy, while Lee et al. (2019) and Gregor et al. (2019) learn belief representations to accelerate model-free agents.
+
+Imagined multi-step returns VPN (Oh et al., 2017), MVE (Feinberg et al., 2018), and STEVE (Buckman et al., 2018) learn dynamics for multi-step Q-learning from a replay buffer. AlphaGo (Silver et al., 2017) combines predictions of actions and state values with planning, assuming access to the true dynamics. Also assuming access to the dynamics, POLO (Lowrey et al., 2018) plans to explore by learning a value ensemble. MuZero (Schrittwieser et al., 2019) learns task-specific reward and value models to solve challenging tasks but requires large amounts of experience. PETS (Chua et al., 2018), VisualMPC (Ebert et al., 2017), and PlaNet (Hafner et al., 2018) plan online using derivative-free optimization. POPLIN (Wang and Ba, 2019) improves over online planning by self-imitation. Piergiovanni et al. (2018) learn robot policies by imagination with a latent dynamics model. Planning with neural network gradients was shown on small problems (Schmidhuber, 1990; Henaff et al., 2018) but has been challenging to scale (Parmas et al., 2019).
+
+Analytic value gradients DPG (Silver et al., 2014), DDPG (Lillicrap et al., 2015), and SAC (Haarnoja et al., 2018) leverage gradients of learned immediate action values to learn a policy by experience replay. SVG (Heess et al., 2015) reduces the variance of model-free on-policy algorithms by analytic value gradients of one-step model predictions. Concurrent work by Byravan et al. (2019) uses latent imagination with deterministic models for navigation and manipulation tasks. ME-TRPO (Kurutach et al., 2018) accelerates an otherwise model-free agent via gradients of predicted rewards for proprioceptive inputs. DistGBP (Henaff et al., 2017; 2019) uses model gradients for online planning in simple tasks.
+
+![](images/7cc1cc84cd073de23edb62451cf0b8f4830c2e9a7abf5617ff0ca8e6a60022ad.jpg)
+
+![](images/c4e8cd119dcd80614ccbc901ff9431d5edec2a8064d61cf44221960cdc573a28.jpg)
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+![](images/301c058c91502a34e3acc6b2100e14a5ec8d416b71ca918f9af2ab81f641e8ed.jpg)
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+![](images/08dbb32f6e07fed4421f4bd5c8c3c8c5034302d63d15752e8c67942a75072f14.jpg)
+
+Figure 8: Comparison of representation learning objectives to be used with Dreamer. Pixel reconstruction performs best for the majority of tasks. The contrastive objective solves about half of the tasks, while predicting rewards alone was not sufficient in our experiments. The results suggest that future developments in learning representations are likely to translate into improved task performance for Dreamer. The performance curves for all tasks are included in Appendix E.  
+![](images/99277299319f6325585798b1cc7d1f449c25fce3e660244f8bf55c702fdeeb5a.jpg)  
+Dreamer + Reconstruction - Dreamer + Contrastive - Dreamer + Reward only D4PG (1e9 steps) A3C (1e9 steps, proprio)
+
+![](images/c2ea2adbbe4e2c939b4f1f87a58e539e82ddeece6c066f8945cf14a917280b53.jpg)
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+![](images/4036e6052a20a23fd301c8dacaafe429b0f6b520bdc02eec7a24850b3c3d888c.jpg)
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+![](images/0afb0cdfb15bfbdc6c420e32c12645ba1aa6d560a5147154a002b2f6e218284d.jpg)
+
+# 6 EXPERIMENTS
+
+We experimentally evaluate Dreamer on a variety of control tasks. We designed the experiments to compare Dreamer to current best methods in the literature, and to evaluate its ability to solve tasks with long horizons, continuous actions, discrete actions, and early termination. We further compare the orthogonal choice of learning objective for the world model. The source code for all our experiments and videos of Dreamer are available at https://danijar.com/dreamer.
+
+Control tasks We evaluate Dreamer on 20 visual control tasks of the DeepMind Control Suite (Tassa et al., 2018), illustrated in Figure 2. These tasks pose a variety of challenges, including sparse rewards, contact dynamics, and 3D scenes. We selected the tasks on which Tassa et al. (2018) report non-zero performance from image inputs. Agent observations are images of shape  $64 \times 64 \times 3$ , actions range from 1 to 12 dimensions, rewards range from 0 to 1, episodes last for 1000 steps and have randomized initial states. We use a fixed action repeat of  $R = 2$  across tasks. We further evaluate the applicability of Dreamer to discrete actions and early termination on a subset of Atari games (Bellemare et al., 2013) and DeepMind Lab levels (Beattie et al., 2016) as detailed in Appendix C.
+
+Implementation Our implementation uses TensorFlow Probability (Dillon et al., 2017). We use a single Nvidia V100 GPU and 10 CPU cores for each training run. The training time for our Dreamer implementation is below 5 hours per  $10^{6}$  environment steps on the control suite, compared to 11 hours for online planning using PlaNet, and the 24 hours used by D4PG to reach similar performance. We use the same hyper parameters across all continuous tasks, and similarly across all discrete tasks, detailed in Appendix A. The world models are learned via reconstruction unless specified.
+
+Baseline methods The highest reported performance on the continuous tasks is achieved by D4PG (Barth-Maron et al., 2018), an improved variant of DDPG (Lillicrap et al., 2015) that uses distributed collection, distributional Q-learning, multi-step returns, and prioritized replay. We include the scores for D4PG with pixel inputs and A3C (Mnih et al., 2016) with state inputs from Tassa et al. (2018). PlaNet (Hafner et al., 2018) learns the same world model as Dreamer and selects actions via online planning without an action model and drastically improves over D4PG and A3C in data efficiency. We run PlaNet with  $R = 2$  for a unified experimental setup. For Atari, we show the final performance of SimPLe (Kaiser et al., 2019), DQN (Mnih et al., 2015) and Rainbow (Hessel et al., 2018) reported by Castro et al. (2018), and for DeepMind Lab that of IMPALA (Espeholt et al., 2018) as a guideline.
+
+Performance To evaluate the performance of Dreamer, we compare it to state-of-the-art reinforcement learning agents. The results are summarized in Figure 6. With an average score of 823 across tasks after  $5 \times 10^{6}$  environment steps, Dreamer exceeds the performance of the strong model-free D4PG agent that achieves an average of 786 within  $10^{9}$  environment steps. At the same time, Dreamer inherits the data-efficiency of PlaNet, confirming that the learned world model can help to generalize from small amounts of experience. The empirical success of Dreamer shows that learning behaviors by latent imagination with world models can outperform top methods based on experience replay.
+
+Long horizons To investigate its ability to learn long-horizon behaviors, we compare Dreamer to alternatives for deriving behaviors from the world model at various horizon lengths. For this, we learn an action model to maximize imagined rewards without a value model and compare to online planning using PlaNet. Figure 4 shows the final performance for different imagination horizons, confirming that the value model makes Dreamer more robust to the horizon and performs well even for short horizons. Performance curves for all 19 tasks with horizon of 20 are shown in Appendix D, where Dreamer outperforms the alternatives on 16 of 20 tasks, with 4 ties.
+
+Representation learning Dreamer can be used with any differentiable dynamics model that predicts future rewards given actions and past observations. Since the representation learning objective is orthogonal to our algorithm, we compare three natural choices described in Section 4: pixel reconstruction, contrastive estimation, and pure reward prediction. Figure 8 shows clear differences in task performance for different representation learning approaches, with pixel reconstruction outperforming contrastive estimation on most tasks. This suggests that future improvements in representation learning are likely to translate to higher task performance with Dreamer. Reward prediction alone was not sufficient in our experiments. Further ablations are included in the appendix of the paper.
+
+# 7 CONCLUSION
+
+We present Dreamer, an agent that learns long-horizon behaviors purely by latent imagination. For this, we propose an actor critic method that optimizes a parametric policy by propagating analytic gradients of multi-step values back through learned latent dynamics. Dreamer outperforms previous methods in data-efficiency, computation time, and final performance on a variety of challenging continuous control tasks with image inputs. We further show that Dreamer is applicable to tasks with discrete actions and early episode termination. Future research on representation learning can likely scale latent imagination to environments of higher visual complexity.
+
+Acknowledgements We thank Simon Kornblith, Benjamin Eysenbach, Ian Fischer, Amy Zhang, Geoffrey Hinton, Shane Gu, Adam Kosiorek, Jacob Buckman, Calvin Luo, and Rishabh Agarwal, and our anonymous reviewers for feedback and discussions. We thank Yuval Tassa for adding the quadruped environment to the control suite.
+
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+
+# A HYPER PARAMETERS
+
+Model components We use the convolutional encoder and decoder networks from Ha and Schmidhuber (2018), the RSSM of Hafner et al. (2018), and implement all other functions as three dense layers of size 300 with ELU activations (Clevert et al., 2015). Distributions in latent space are 30-dimensional diagonal Gaussians. The action model outputs a tanh mean scaled by a factor of 5 and a softplus standard deviation for the Normal distribution that is then transformed using tanh (Haarnoja et al., 2018). The scaling factor allows the agent to saturate the action distribution.
+
+Learning updates We draw batches of 50 sequences of length 50 to train the world model, value model, and action model models using Adam (Kingma and Ba, 2014) with learning rates  $6 \times 10^{-4}$ ,  $8 \times 10^{-5}$ ,  $8 \times 10^{-5}$ , respectively and scale down gradient norms that exceed 100. We do not scale the KL regularizers ( $\beta = 1$ ) but clip them below 3 free nats as in PlaNet. The imagination horizon is  $H = 15$  and the same trajectories are used to update both action and value models. We compute the  $\mathrm{V}_{\lambda}$  targets with  $\gamma = 0.99$  and  $\lambda = 0.95$ . We did not find latent overshooting for learning the model, an entropy bonus for the action model, or target networks for the value model necessary.
+
+Environment interaction The dataset is initialized with  $S = 5$  episodes collected using random actions. We iterate between 100 training steps and collecting 1 episode by executing the predicted mode action with Normal(0, 0.3) exploration noise. Instead of manually selecting the action repeat for each environment as in Hafner et al. (2018) and Lee et al. (2019), we fix it to 2 for all environments. See Figure 12 for an assessment of the robustness to different action repeat values.
+
+Discrete control For experiments on Atari games and DeepMind Lab levels, the action model predicts the logits of a categorical distribution. We use straight-through gradients for the sampling step during latent imagination. The action noise is epsilon greedy where  $\epsilon$  is linearly scheduled from  $0.4\rightarrow 0.1$  over the first 200,000 gradient steps. To account for the higher complexity of these tasks, we use an imagination horizon of  $H = 10$ , scale the KL regularizers by  $\beta = 0.1$ , and bound rewards using tanh. We predict the discount factor from the latent state with a binary classifier that is trained towards the soft labels of 0 and  $\gamma$ .
+
+# B DERIVATIONS
+
+We define the information bottleneck objective (Tishby et al., 2000) for latent dynamics models,
+
+$$
+\max  \mathrm {I} \left(s _ {1: T}; \left(o _ {1: T}, r _ {1: T}\right) \mid a _ {1: T}\right) - \beta \mathrm {I} \left(s _ {1: T}, i _ {1: T} \mid a _ {1: T}\right), \tag {13}
+$$
+
+where  $\beta$  is scalar and  $i_t$  are dataset indices that determine the observations  $p(o_t \mid i_t) \doteq \delta(o_t - \bar{o}_t)$  as in Alemi et al. (2016).
+
+Maximizing the objective leads to model states that can predict the sequence of observations and rewards while limiting the amount of information extracted at each time step. This encourages the model to reconstruct each image by relying on information extracted at preceding time steps to the extent possible, and only accessing additional information from the current image when necessary. As a result, the information regularizer encourages the model to learn long-term dependencies.
+
+For the generative objective, we lower bound the first term using the non-negativity of the KL divergence and drop the marginal data probability as it does not depend on the representation model,
+
+$$
+\begin{array}{l} \mathrm {I} \left(s _ {1: T}; \left(o _ {1: T}, r _ {1: T}\right) \mid a _ {1: T}\right) \\ = \operatorname {E} _ {p \left(o _ {1: T}, r _ {1: T}, s _ {1: T}, a _ {1: T}\right)} \left(\sum_ {t} \ln p \left(o _ {1: T}, r _ {1: T} \mid s _ {1: T}, a _ {1: T}\right) - \underbrace {\ln p \left(o _ {1 : T} , r _ {1 : T} \mid a _ {1 : T}\right)} _ {\text {c o n s t}}\right) \\ \stackrel {\pm} {=} \mathrm {E} \left(\sum_ {t} \ln p \left(o _ {1: T}, r _ {1: T} \mid s _ {1: T}, a _ {1: T}\right)\right) \\ \geq \operatorname {E} \left(\sum_ {t} \ln p \left(o _ {1: T}, r _ {1: T} \mid s _ {1: T}, a _ {1: T}\right)\right) - \operatorname {K L} \left(p \left(o _ {1: T}, r _ {1: T} \mid s _ {1: T}, a _ {1: T}\right) \| \prod_ {t} q \left(o _ {t} \mid s _ {t}\right) q \left(r _ {t} \mid s _ {t}\right)\right) \\ = \mathrm {E} \left(\sum_ {t} \ln q \left(o _ {t} \mid s _ {t}\right) + \ln q \left(r _ {t} \mid s _ {t}\right)\right). \tag {14} \\ \end{array}
+$$
+
+For the contrastive objective, we subtract the constant marginal probability of the data under the variational encoder, apply Bayes rule, and use the InfoNCE mini-batch bound (Poole et al., 2019),
+
+$$
+\begin{array}{l} \mathrm {E} \left(\ln q \left(o _ {t} \mid s _ {t}\right) + \ln q \left(r _ {t} \mid s _ {t}\right)\right) \\ \stackrel {\pm} {=} \mathrm {E} \left(\ln q \left(o _ {t} \mid s _ {t}\right) - \ln q \left(o _ {t}\right) + \ln q \left(r _ {t} \mid s _ {t}\right)\right) \\ = \mathrm {E} \left(\ln q \left(s _ {t} \mid o _ {t}\right) - \ln q \left(s _ {t}\right) + \ln q \left(r _ {t} \mid s _ {t}\right)\right) \tag {15} \\ \geq \operatorname {E} \bigg (\ln q (s _ {t} \mid o _ {t}) - \ln \sum_ {o ^ {\prime}} q (s _ {t} \mid o ^ {\prime}) + \ln q (r _ {t} \mid s _ {t}) \bigg). \\ \end{array}
+$$
+
+For the second term, we use the non-negativity of the KL divergence to obtain an upper bound,
+
+$$
+\begin{array}{l} \mathrm {I} \left(s _ {1: T}; i _ {1: T} \mid a _ {1: T}\right) \\ = \operatorname {E} _ {p \left(o _ {1: T}, r _ {1: T}, s _ {1: T}, a _ {1: T}, i _ {1: T}\right)} \left(\sum_ {t} \ln p \left(s _ {t} \mid s _ {t - 1}, a _ {t - 1}, i _ {t}\right) - \ln p \left(s _ {t} \mid s _ {t - 1}, a _ {t - 1}\right)\right) \\ = \mathrm {E} \left(\sum_ {t} \ln p \left(s _ {t} \mid s _ {t - 1}, a _ {t - 1}, o _ {t}\right) - \ln p \left(s _ {t} \mid s _ {t - 1}, a _ {t - 1}\right)\right) \tag {16} \\ \leq \mathrm {E} \left(\sum_ {t} \ln p \left(s _ {t} \mid s _ {t - 1}, a _ {t - 1}, o _ {t}\right) - \ln q \left(s _ {t} \mid s _ {t - 1}, a _ {t - 1}\right)\right) \\ = \operatorname {E} \left(\sum_ {t} \operatorname {K L} \left(p \left(s _ {t} \mid s _ {t - 1}, a _ {t - 1}, o _ {t}\right) \| q \left(s _ {t} \mid s _ {t - 1}, a _ {t - 1}\right)\right)\right). \\ \end{array}
+$$
+
+This lower bounds the objective.
+
+# C DISCRETE CONTROL
+
+We evaluate Dreamer on a subset of tasks with discrete actions from the Atari suite (Bellemare et al., 2013) and DeepMind Lab (Beattie et al., 2016). While agents that purely learn through world models are not yet competitive in these domains (Kaiser et al., 2019), the tasks offer a diverse test bed with visual complexity, sparse rewards, and early termination. Agents observe  $64 \times 64 \times 3$  images and select one of between 3 and 18 actions. For Atari, we follow the evaluation protocol of Machado et al. (2018) with sticky actions. Refer to Figure 9 for these experiments.
+
+![](images/4fb17bef2dcd37d8e29188d1822b41853ea30d902ee542d4157d7ed25677f326.jpg)
+
+![](images/eab468e5ab18a0c5dc1b505aae31d36832623d568ae9c82683205e28cc1a7052.jpg)
+
+![](images/43f2fef170d38d920cdae8ce303814d32b29b4c7c9c7f0022109f24aaadee0a5.jpg)
+
+![](images/c45ae4e89247e44e1486fa2e6ceef65b246e36956453b2cb5b6c5d1016e30e17.jpg)
+
+![](images/472a5dd2eba0546b23efeb1087a99d66db997051f7d73883e825e1822125e508.jpg)
+
+![](images/fbf2018e2edd6408c9c50b6b850ee67b723ea610aeb4dba3ff85694e824175f9.jpg)
+
+![](images/22078cfcb23bc451ee5d1e96bfade9815e433937327aa2cc37e9b8f9cf684cf2.jpg)
+
+![](images/d58dbfa4aab3f89de46b29a076b6e54094f1870e58d9c4a14083946a7ff0e734.jpg)
+
+![](images/e8d6d4a2c33230d81af2f85e74c1d59eff3673dc54d1576558dda2a404b436cc.jpg)
+
+![](images/33a4d0f4717d6caf52a31a17cd442d2e4a0d8b3d4ad3797e47de82784fe10377.jpg)
+
+![](images/6ca32274cd0b9ae6eba79c3100c96b8869aa8b21794ec4d2f1080555bed5368e.jpg)
+
+![](images/fae1dbe18b278ad68ca2e9e7d4ba67e6b385fc583c9476b076d3d91f1e13e65e.jpg)
+
+![](images/1f4ed7db446684f75ba0a33c9a7e432e41c82a06cd8d8a4608f70f5e231c5429.jpg)  
+Dreamer - SimPLe (1e5 steps) - DQN (2e8 steps) - Rainbow (2e8 steps) - IMPALA (1e10 steps) - Random
+
+![](images/b19b6175f9094d4fdae74bfb644d33fb114cadba0da120e4983ff02da955789d.jpg)  
+Figure 9: Performance of Dreamer in environments with discrete actions and early termination. Dreamer learns successful behaviors on this subset of Atari games and the object collection level of DMLab. We highlight representation learning for these environments as a direction of future work that could enable competitive performance across all Atari games and DMLab levels using Dreamer.
+
+![](images/8b838485c91458858a40ebe80f66be5fda60fe3ebf65e73f5501787cbf04d854.jpg)
+
+![](images/42c1c1cdcf06ed0a5a37142277bc9ccc1228d4e6d30c80f2c28ca71473be0b74.jpg)
+
+# D BEHAVIOR LEARNING
+
+![](images/06ad6ee07dbb8b9ce592f06f7041db19e47f752ae56922e8acf60896fe4a433c.jpg)
+
+![](images/cc75c828be5495d361d60f66ae3cf1393ceed673945de328b295b7bdc95829d9.jpg)
+
+![](images/0343130c26298f78c227f32fe46562b04c5c237d08fce93b1324ec8b4797d639.jpg)
+
+![](images/b39d9ad8ec14c7deb93869e37a2720c573b95f786abddbdc9f6b16f1e5e42d98.jpg)
+
+![](images/bac200d33e6758953bf298f798ed73cb1914f96c12c1534e1c7d9770ab3af14c.jpg)
+
+![](images/e00c61b48c1e23f969ce8a940ffc2ddf3f7a59056d8538b0c2d2276e1dd60086.jpg)
+
+![](images/4368b48dd2eb20338132f5dd17463f8672c8a2fb6a46b15cab889580eb6c5dfe.jpg)
+
+![](images/7490531ca030d0adc9a695cc885c73a804f59d44e3621ca164896a10f3f3b7de.jpg)
+
+![](images/97f87a1a1ce81b23797d498aa383862fff7e98dfda688961da2cb30e593f3a01.jpg)
+
+![](images/b2c2461bd6da1c783f92f157cc4fc2a563f86caac9f4ba01ba1eb9a5cf0817ec.jpg)
+
+![](images/e9d2cf8a6baad9d92a6521192ee11a31d8232d879b22312ddce59097c7800f11.jpg)
+
+![](images/565a6a092e5e15b6e55c0043ad53f29ab92b8f801a066df9fbdcb8a984da32a6.jpg)
+
+![](images/3a9f9262855d6ec722030bcab32652a786aef880fca36ac784f875a52593e21e.jpg)
+
+![](images/9bd6aee37578475c822a0583456f0e99e07da6a57652044224577c68f4951a1f.jpg)
+
+![](images/3afd50b383a85327a4890c32cb4d81868bdcbaffc5c4c70d5d1e835d2d4a4139.jpg)
+
+![](images/79ad8f65c791c8ed4079f3d6a04dce6db39afefdb024f4ffb00c23affb4705fc.jpg)
+
+Figure 10: Comparison of action selection schemes on the continuous control tasks of the DeepMind Control Suite from pixel inputs. The lines show mean scores over environment steps and the shaded areas show the standard deviation across 5 seeds. We compare Dreamer that learns both actions and values in imagination, to only learning actions in imagination, and Planet that selects actions by online planning instead of learning a policy. The baselines include the top model-free algorithm D4PG, the well-known A3C agent, and the hybrid SLAC agent.  
+![](images/aa8360719200083135f1a638fa35a7b56bc062779a3d2f1580ef2f4374f2a8d3.jpg)  
+Dreamer No value PlaNet D4PG (1e9 steps) A3C (1e9 steps, proprio) SLAC (3e6 steps)
+
+![](images/e97ab358157a0be0965def7749114dbb5b608f195c0bc5bbe0e56cfbf65f58aa.jpg)
+
+![](images/5e66295dd5885632cc09b4f985bc5bf08ccf951eca055211215ad513627aa2fd.jpg)
+
+![](images/c25a0a329ca1f050e759ca5f3f587c34415a113ab6058d3337a69efecf334dc7.jpg)
+
+# E REPRESENTATION LEARNING
+
+![](images/7fae1e2bd6e517752b72b628a15103b2a823c3eba2bebe0010475dc29a562b2b.jpg)
+
+![](images/6ef77a9767d4e91ae8afbe872f1d965a9398f4d8a512c15808f70f82623ecc15.jpg)
+
+![](images/61fbd8b9746b2c321cdcc45003cc3f4ed54313ca120596e4ba332ac4fb5759c8.jpg)
+
+![](images/e35efdacc48058b33a0f15bd81e81b40999b214046e001d5e5c7ca5085e77e7e.jpg)
+
+![](images/620a2626ebf01e96dc7424db0b7d4821f4ce984b93d8e00811491eadc8a14861.jpg)
+
+![](images/0afb4d0fc68109fe851feb916451950589344961e7012b92b7edce2b72c2c8b9.jpg)
+
+![](images/46e25c6fbe63223162e94684b5a02e18754b4a139df0de619ed8984f41643fa6.jpg)
+
+![](images/2f827bc892054994944056340b1a70e756fe98440e5ee0c81f2bd1b6c8011b62.jpg)
+
+![](images/99b24ec35cb05db668982a33e67be327f39d65c129746cf07739e39816d1b963.jpg)
+
+![](images/729475dd5c56fa4e586e5108d69c14d6842a9cf4d736b8bf67a34b98116f2d93.jpg)
+
+![](images/b1936e0f1ba9c3c399bc37811fbc5b692c012088b51e71855202e0cbb2616f4d.jpg)
+
+![](images/8eeda3d538df1723166a3acd883293d92264dabbee435a369e070a925bc9e9d8.jpg)
+
+![](images/17de4515e9821a3d8bf1b6df93c745b3c6e2e937157d16ef7a99cddcc1dc6fa5.jpg)
+
+![](images/1b92d4fd22d36df916576ca7ea243aa86a0311fa45777688202d2c453544cb1f.jpg)
+
+![](images/edf8c8f43f5d139e3d587515c618f1bfe7a8c98b7a85b6a962167e2d4ab7fbba.jpg)
+
+![](images/1a7b4b4e22979417d1d7f8ea5715adc348feb8e319c912e3f37c8ac99133eb04.jpg)
+
+Figure 11: Comparison of representation learning methods for Dreamer. The lines show mean scores and the shaded areas show the standard deviation across 5 seeds. We compare generating both images and rewards, generating rewards and using a contrastive loss to learn about the images, and only predicting rewards. Image reconstruction provides the best learning signal across most of the tasks, followed by the contrastive objective. Learning purely from rewards was not sufficient in our experiments and might require larger amounts of experience.  
+![](images/ca3a64ffc67a4f1ae9b9c2afb866709c04d1e36861a9609692083e4117f1002c.jpg)  
+Dreamer + Reconstruction - Dreamer + Contrastive - Dreamer + Reward only D4PG (1e9 steps) A3C (1e9 steps, proprioio)
+
+![](images/69650522082f17d02c5b653439fdbc8a206429e88aba023de662bb955e2ccaf9.jpg)
+
+![](images/e035d1eff99518a4e893d4c3298ca9381f60d23c3456406790630e3718f0e17c.jpg)
+
+![](images/51c11cc852b2f1aa62c49adbcfc90d235415d5b497681d244a703857478046a1.jpg)
+
+# F ACTION REPEAT
+
+![](images/4b7187e88ca74c723d9d216cabae9783d7c73949f29d9084083a99c4766d20a7.jpg)
+
+![](images/5e00e5ad2c2613ac2628623f874ab9ea8403423837d91bed3abe7372e8ad2b87.jpg)
+
+![](images/8cc4faa333143f0a36eec553c6431a75098e64dafa3079363fbfc61add477ac5.jpg)
+
+![](images/5d94d1d678838ac71c368b270ac3499a09e07ace7504da8e79d3e10b5e61d675.jpg)
+
+![](images/981a2b6100191093000c41624fd44d940840133b9b7ed6465cd2969089f7ae92.jpg)
+
+![](images/3472301a15a6d2c9de11ac296d13b5e895db71c17a3b6dde6ab33fdce2fa52e9.jpg)
+
+![](images/b0a8532cd9106297a3215754ff6d55044a166673b138713ab201931420a8df2d.jpg)
+
+![](images/2b2f16f216a801606fc78626bd9bfcd0f98e846f528f7f46aa4b0c05759d11d3.jpg)
+
+![](images/4cf7157ca5c8c6aed20897a6936bf9a0221b110bdaab23d8b95ea1e35df1a0a0.jpg)
+
+![](images/fd24fce1118ab24a137e441bf5dc4fb9b8c2a364e868d9d678735ba1501bd3ef.jpg)
+
+![](images/21e42a9e6b33317c86f7b891f7e7293a4676a2ebfc9573dfbce08aa887c6e83f.jpg)
+
+![](images/85d09ac0fdcc2ba3ff5b532870ba93bdcc93cd8ad614afcc4a3fc35b04b7f4e8.jpg)
+
+![](images/31959870aadc808452288f0a5c23bc88961cde4b7ab9ab8faed3c0f580b82068.jpg)
+
+![](images/064bd2ef5f00ec1f0704fe5e7a3bcb527af655ba7f8087c24272a39724adf232.jpg)
+
+![](images/4c781f173e8573ef046659aa63854249c110a12f7eae2a884be2df2a6b804ae6.jpg)
+
+![](images/8dfcb1312834dfe273f3db6bd3919ecd00487aa1342926815c1948a4fb3e72ed.jpg)
+
+Figure 12: Robustness of Dreamer to different control frequencies. Reinforcement learning methods can be sensitive to this hyper parameter, which could be amplified when learning dynamics models at the control frequency of the environment. For this experiment, we train Dreamer with different amounts of action repeat. The areas show one standard deviation across 2 seeds. We used a previous hyper parameter setting for this experiment. We find that a value of  $R = 2$  works best across tasks.  
+![](images/f2e482330228b415957268482e3c27a38992a855212255f1e478cd7eb3a46c20.jpg)  
+Repeat 1 Repeat 2 Repeat 4 A3C (1e9 steps, proprio)
+
+![](images/bd1e21de6bac3d6cd624b626cd8a665fd4e8605fb98d05e4fe5123309dc12ef3.jpg)  
+D4PG (1e9 steps) -- PlaNet (1e6 steps) -- SLAC (3e6 steps)
+
+![](images/71b81dd04f427489f70a2ceae8a54b549131441e38badd2a2132719de71d1c26.jpg)
+
+![](images/cc47c606cbb0cd847095ce4e2062f481f1ad700465ec42f9c8550dfa0968500d.jpg)
+
+G CONTINUOUS CONTROL SCORES  
+
+<table><tr><td></td><td>A3C</td><td>D4PG</td><td>PlaNet1</td><td>Dreamer</td></tr><tr><td>Modality</td><td>proprio</td><td>pixels</td><td>pixels</td><td>pixels</td></tr><tr><td>Steps</td><td>\(10^9\)</td><td>\(10^9\)</td><td>\(5 \times 10^6\)</td><td>\(5 \times 10^6\)</td></tr><tr><td>Acrobot Swingup</td><td>41.90</td><td>91.70</td><td>3.21</td><td>365.26</td></tr><tr><td>Cartpole Balance</td><td>951.60</td><td>992.80</td><td>452.56</td><td>979.56</td></tr><tr><td>Cartpole Balance Sparse</td><td>857.40</td><td>1000.00</td><td>164.74</td><td>941.84</td></tr><tr><td>Cartpole Swingup</td><td>558.40</td><td>862.00</td><td>312.56</td><td>833.66</td></tr><tr><td>Cartpole Swingup Sparse</td><td>179.80</td><td>482.00</td><td>0.64</td><td>812.22</td></tr><tr><td>Cheetah Run</td><td>213.90</td><td>523.80</td><td>496.12</td><td>894.56</td></tr><tr><td>Cup Catch</td><td>104.70</td><td>980.50</td><td>455.98</td><td>962.48</td></tr><tr><td>Finger Spin</td><td>129.40</td><td>985.70</td><td>495.25</td><td>498.88</td></tr><tr><td>Finger Turn Easy</td><td>167.30</td><td>971.40</td><td>451.22</td><td>825.86</td></tr><tr><td>Finger Turn Hard</td><td>88.70</td><td>966.00</td><td>312.55</td><td>891.38</td></tr><tr><td>Hopper Hop</td><td>0.50</td><td>242.00</td><td>0.37</td><td>368.97</td></tr><tr><td>Hopper Stand</td><td>27.90</td><td>929.90</td><td>5.96</td><td>923.72</td></tr><tr><td>Pendulum Swingup</td><td>48.60</td><td>680.90</td><td>3.27</td><td>833.00</td></tr><tr><td>Quadruped Run</td><td>-</td><td>-</td><td>280.45</td><td>888.39</td></tr><tr><td>Quadruped Walk</td><td>-</td><td>-</td><td>238.90</td><td>931.61</td></tr><tr><td>Reacher Easy</td><td>95.60</td><td>967.40</td><td>468.50</td><td>935.08</td></tr><tr><td>Reacher Hard</td><td>39.70</td><td>957.10</td><td>187.02</td><td>817.05</td></tr><tr><td>Walker Run</td><td>191.80</td><td>567.20</td><td>626.25</td><td>824.67</td></tr><tr><td>Walker Stand</td><td>378.40</td><td>985.20</td><td>759.19</td><td>977.99</td></tr><tr><td>Walker Walk</td><td>311.00</td><td>968.30</td><td>944.70</td><td>961.67</td></tr><tr><td>Average</td><td>243.70</td><td>786.32</td><td>332.97</td><td>823.39</td></tr></table>
\ No newline at end of file
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+# DURATION-OF-STAY STORAGE ASSIGNMENT UNDER UNCERTAINTY
+
+Michael Lingzhi Li
+
+Operation Research Center
+
+Massachusetts Institute of Technology
+
+Cambridge, MA 02139
+
+mlli@mit.edu
+
+Elliott Wolf
+
+Lineage Logistics
+
+San Francisco, California
+
+ewolf@lineagelogistics.com
+
+Daniel Wintz
+
+Lineage Logistics
+
+San Francisco, California
+
+dwintz@lineageLogistics.com
+
+# ABSTRACT
+
+Storage assignment, the act of choosing what goods are placed in what locations in a warehouse, is a central problem of supply chain logistics. Past literature has shown that the optimal method to assign pallets is to arrange them in increasing duration of stay (DoS) in the warehouse (the DoS method), but the methodology requires perfect prior knowledge of DoS for each pallet, which is unknown and uncertain under realistic conditions. Attempts to predict DoS have largely been unfruitful due to the multi-valuedness nature (every shipment contains identical pallets with different DoS) and data sparsity induced by lack of matching historical conditions. In this paper, we introduce a new framework for storage assignment that provides a solution to the DoS prediction problem through a distributional reformulation and a novel neural network, ParallelNet. Through collaboration with a world-leading cold storage company, we show that the system is able to predict DoS with a MAPE of  $29\%$ , a decrease of  $\sim 30\%$  compared to a CNN-LSTM model, and suffers less performance decay into the future. The framework is then integrated into a first-of-its-kind Storage Assignment system, which is being deployed in warehouses across United States, with initial results showing up to  $21\%$  in labor savings. We also release the first publicly available set of warehousing records to facilitate research into this central problem.
+
+# 1 INTRODUCTION
+
+The rise of the modern era has been accompanied by ever-shortening product life cycles, straining the entire supply chain and demanding efficiency at every node. One integral part of any supply chain is warehousing (storage); warehouse operations often have major impacts downstream on the capability to deliver product on time.
+
+One of the largest cold storage companies in the world is looking to improve the efficiency of their warehouses by optimizing the scheduling of storage systems. According to Hausman et al. (1976), the scheduling of labor in warehouses can be divided into three main components:
+
+- Pallet Assignment: The assignment of multiple items to the same pallet.  
+- Storage Assignment: The assignment of pallets to a storage location.  
+- Interleaving: The overarching rules for dealing with concurrent inbound and outbound requests.
+
+For this particular paper, we focus on the problem of storage assignment. Various papers such as Goetschalckx & Ratliff (1990) show labor efficiency to be a bottleneck. In a modern warehouse, the process of storage assignment usually involves forklift drivers moving inbound pallets from the
+
+staging area of the warehouse to the storage location, so a sub-optimal assignment system causes unnecessary long travel times to store the pallet. Unfortunately, the inefficiency is quadrupled when the return of the forklift and the retrieval of the pallet are considered.
+
+To increase the efficiency of the warehouse, we would thus like to minimize the total travel time needed to store a set of shipments from the staging area. Many different theoretical frameworks exist, and the details of such frameworks are contained in Appendix 9.1. The ones of chief interest are turnover-based, class-based, and Duration-of-Stay (DoS) based strategies.
+
+Turnover-based strategies (e.g. Hausman et al. (1976), Yu & De Koster (2009)) assign locations so that the travel distance is inversely proportional to the turnover of the product. Class-based strategies (e.g. Hausman et al. (1976), Schwarz et al. (1978)) separate products into  $k$  classes, with each class assigned a dedicated area of storage. DoS-based strategies (e.g. Goetschalckx & Ratliff (1990), Chen et al. (2016)) assign pallets to locations with travel distance proportional to the duration-of-stay.
+
+Simulation experiments in Goetschalckx & Ratliff (1990) and Kulturel et al. (1999) demonstrated that under complex stochastic environments, DoS-based strategies outperform other methodologies significantly. However, among all three categories, the most commonly used strategy is class-based, as pointed out by Yu et al. (2015) and Yu & De Koster (2013). The authors and industry evidence suggest that this is due to the fact that class-based systems are relatively easy to implement, but also because DoS is not known in advance. To utilize a DoS system realistically would therefore require an accurate prediction model using the features available at shipment entry to the warehouse.
+
+However, even with the availability of modern high-powered predictive methods including Gradient Boosted Trees and Neural Networks, there has been no documented progress in employing DoS-based methods. This reflects the following significant challenges in a dynamic, real warehouse:
+
+- Multi-valuedness: It is common for  $10+$  identical pallets of the same product to arrive in a single shipment, and then leave the warehouse at different times depending on the consumption of the end consumer. This causes the ground truth for the DoS of a product entering at a given time to be ill-defined.  
+- Data Sparsity: A large warehouse would have significant available historical DoS data, but such data is scattered across thousands of products/SKUs, and different operating conditions (e.g. time of the year, day of the week, shipment size). Given strong variation of DoS in a warehouse, it is very unlikely that all environment-product combinations would exist in data for the historical average to be valid for future DoS. Furthermore, new SKUs are created relatively frequently, and the predictive algorithm needs to be robust against that as well.
+
+To solve such difficulties, we reformulate the DoS as a distribution and develop a new framework based on nonparametric estimation. Then we combine it with a parallel architecture of Residual Deep Convolutional Networks (He et al. (2015)) and Gated Recurrent Unit (GRU) networks (Cho et al. (2014)) to provide strong estimation results. As far as the authors know, this is the first documented attempt to predict DoS in warehousing systems. We further release the first public dataset of warehousing records to enable future research into this problem.
+
+This neural network is then integrated into the larger framework, which is being implemented in live warehouses. We illustrate how initial results from the ground show appreciable labor savings.
+
+Specifically, our contributions in this paper include:
+
+- We develop an novel end-to-end framework for optimizing warehouse storage assignment using the distribution of DoS.  
+- We release a curated version of a large historical dataset of warehousing records that can be used to build and test models that predicts DoS.  
+- We introduce a type of neural network architecture, ParallelNet, that achieves state-of-the-art performance in estimating the DoS distribution.  
+- Most importantly, we present real-life results of implementing the framework with ParallelNet in live warehouses, and show labor savings by up to  $21\%$ .
+
+The structure of the paper is as followed. In Section 2, we introduce the storage assignment problem formally, and how this leads to estimating DoS in a distributional way. In Section 3, we develop the
+
+storage assignment framework. We would introduce the dataset in Section 4 and Section 5 contains the implementation with ParallelNet, and its results compared to strong baselines. Section 6 shows the computational results, while real-life evidence is provided in Section 7.
+
+# 2 SOLVING THE STORAGE ASSIGNMENT PROBLEM
+
+The general storage assignment problem asks for an assignment function that outputs a storage location given a pallet and warehouse state so the total travel time for storage is minimized. We formalize such statement below and show how it naturally leads to the problem of estimating DoS.
+
+Let warehouse  $\mathcal{W}$  have locations labeled  $\{1, \dots, N\}$  where the expected travel time from the loading dock to location  $i$  is  $t_i$ . We assume pallets arrive in discrete time periods  $\{1, 2, \dots, T\}$  where  $T$  is the lifetime of the warehouse and each of the pallets has an integer DoS in  $\{1, \dots, P\}$ . The warehouse uses an assignment function  $\mathcal{A}$  that assigns each arriving pallet to an available location in  $\{1, \dots, N\}$ . Define  $n_i^{\mathcal{A}}(t)$  as the number of pallets (0 or 1) stored in location  $i$  in time period  $t$  under assignment function  $\mathcal{A}$ . Then our optimization problem can be stated as:
+
+$$
+\min  _ {\mathcal {A}} \sum_ {t = 1} ^ {T} \sum_ {i = 1} ^ {N} 4 t _ {i} n _ {i} ^ {\mathcal {A}} (t) \tag {1}
+$$
+
+The constant 4 is to allow for the 4 distances traveled during storage/retrieval. Given that future pallet arrivals could be arbitrary, we need additional assumptions to make the theoretical analysis tractable. In particular, we assume the warehouse is in steady state or perfect balance:
+
+Definition 1. Let  $n_p(t)$  be the number of pallets arriving at time period  $t$  that has a DoS of  $p$ . A warehouse is in perfect balance iff for all  $p$  and  $t > p$  we have  $n_p(t - p) = n_p(t)$ .
+
+This means that in each time period  $t > p$ , the  $n_p(t - p)$  outgoing pallets with DoS of  $p$  are replaced exactly by incoming pallets with DoS of  $p$ , so the number of pallets with DoS of  $p$  in the warehouse remains constant at  $z_{p} = \sum_{i=1}^{p} n_{p}(i)$  for every period  $t > p$ . For a real-life large warehouse, this steady state assumption is usually a good approximation except when shock events happen. Under such assumption, ideally we only need  $z_{p}$  positions to store all the pallets with DoS of  $p$ , or in total  $\sum_{p=1}^{P} z_{p}$  locations. Goetschalckx & Ratliff (1990) showed that the DoS strategy below is optimal:
+
+Theorem 1 (Goetschalckx & Ratliff (1990)). Assume the warehouse is in perfect balance, and  $P$  is large enough so that  $z_{p} \leq 1$  for every  $p$ . Define  $W(t) = \frac{\sum_{p < t} z_{p}}{\sum_{p=1}^{P} z_{p}}$  as the cumulative distribution of pallet DoS. Let  $r = \frac{\sum_{p=1}^{P} z_{p}}{N}$  be the average occupancy rate of the warehouse. Then the optimal solution to (1),  $\mathcal{A}^*$ , would assign a pallet with DoS of  $p$  to the  $NrW(p)$ th location.
+
+In other words, the locations are arranged in increasing order of DoS. We can always make  $P$  large enough so that  $z_{p} \leq 1$  by separating time periods to finer intervals. With Theorem 1, we can implement  $\mathcal{A}^*$  in a real warehouse using three approximations:
+
+1. We approximate  $W(t)$  with the historical pallet cumulative DoS distribution  $\hat{W}(t)$ .  
+2. We approximate  $r$  with the historical average occupancy rate of the warehouse  $\hat{r}$ .  
+3. We approximate the DoS  $p$  with a prediction  $\hat{p}$ .
+
+Both  $\hat{W}(t)$  and  $\hat{r}$  are easily retrieved from the historical data, and  $N$  is known. We stress however that  $\hat{W}(t)$  needs to be size-debiased. This is because pallets of DoS  $p$  would appear with a frequency  $\propto p z_{p}$  in a period of time, and thus the historical data needs to be corrected to match the definition above, which is  $\propto z_{p}$ . This is done through standard theory of size-biased distributions (Gove (2003)).
+
+Therefore, in the next subsection, we would focus on the task of generating a prediction  $\hat{p}$  of the DoS.
+
+# 2.1 PREDICTING DURATION OF STAY
+
+To utilize Theorem 1 above, we need to predict the DoS of a pallet at its arrival. However, in most real-life warehouses, a product  $P_{i}$  enters the warehouse with multiple  $z_{i} > 1$  identical pallets,
+
+which leave at different times  $t_{i1}, \dots, t_{iz_i}$ . Thus, assuming the product came in at time 0, the DoS of incoming pallets of  $P_i$  could be any of the numbers  $t_{i1}, \dots, t_{izi}$ , which makes the quantity ill-defined. The uncertainty can further be very large - our collaborating company had an average variance of DoS within a shipment of 10 days when the median DoS was just over 12 days.
+
+Therefore, to account for such uncertainty, we would assume that for every shipment  $S$  which contains multiple pallets, the DoS of a random pallet follows a cumulative distribution  $F_{S}(t)$ . Furthermore, we assume that such distribution can be identified using the characteristics  $X_{S}$  of the shipment known at arrival. As in, there exists a function  $g$  such that:
+
+$$
+F _ {S} (t) = g (X _ {S}) (t)
+$$
+
+for all possible shipments  $S$ . This assumption is not as strong as it may seem - the most important variables that affect DoS usually are the time and the good that is being transported, both of which are known at arrival. As a simple example, if the product is ice cream and it is the summer, then we expect the DoS distribution to be right skewed, as ice cream is in high demand during the summer. Moreover, the experimental results in Section 6 are indicative that  $g$  exists and is highly estimable.
+
+If the above assumption holds true, then we can estimate  $g$  using machine learning. Assume we have an optimal estimate  $\tilde{g}$  of  $g$  relative to a loss function  $l(F_S, \tilde{g}(X_S))$  denoting the loss between the actual and predicted distribution. Since by our assumption  $F_S$  is identified by  $X_S$ , we cannot obtain any further information about DoS relative to this loss function. Thus, for each shipment with features  $X_S$ , we take  $\hat{p}$  to be a random sample from the distribution  $\tilde{F}_S = \tilde{g}(X_S)$ . In the next section, we would outline our storage assignment framework based on using such prediction  $\hat{p}$ .
+
+# 3 OVERVIEW OF STORAGE ASSIGNMENT FRAMEWORK
+
+With  $\hat{W}(t)$ ,  $\hat{r}$ , and  $\hat{p}$  defined, we can now utilize the DoS strategy  $\mathcal{A}^*$ . For a pallet with predicted DoS  $\hat{p}$ , our storage assignment function  $A: \mathbb{R} \to \mathcal{W}$  is:
+
+$$
+A(\hat{p}) = \arg \min_{w\in \tilde{\mathcal{W}}}d(N\hat{r}\hat{W} (\hat{p}),w) + c(w)
+$$
+
+Where  $d(v, w)$  is the distance between location  $v$  and  $w$  in the warehouse, and  $c(w)$  are other costs associated with storing at this position, including anti-FIFO orders, item mixing, height mismatch, and others.  $\tilde{\mathcal{W}}$  is the set of positions that are available when the pallet enters the warehouse.
+
+The approximate optimal position of DoS optimal location is  $N\hat{r}\hat{W}(\hat{p})$ . However, it is probable that such location is not ideal, either because it is not available or other realistic concerns. For the collaborating company, one important factor is item mixing due to potential cross-contamination of food allergens in close pallets. These terms are highly dependent on the specific storage company, so we include them as a general cost  $c(w)$  to add to the cost  $d(N\hat{r}\hat{W}(\hat{p}), w)$  of not storing the pallet at the DoS optimal position. The resulting location is chosen based on the combination of the two costs.
+
+In summary, our framework consists of three steps:
+
+1. Utilize machine learning to provide an estimate  $\tilde{g}$  of  $g$  with respect to some loss function  $l$ .  
+2. For a shipment  $S$ , calculate  $\tilde{F}_S = \tilde{g}(X_S)$  and generate a random sample  $\hat{p}$  from  $\tilde{F}_S$ .  
+3. Apply the assignment function  $A$  defined above to determine a storage location  $A(\hat{p})$ .
+
+# 4 WAREHOUSING DATASET OVERVIEW AND CONSTRUCTION
+
+In this section, we introduce the historical dataset from the cold storage company to test out the framework and model introduced in Section 3.
+
+# 4.1 OVERVIEW OF THE DATA
+
+The data consists of all warehouse storage records from 2016.1 to 2018.1, with a total of 8,443,930 records from 37 different facilities. Each record represents a single pallet and one shipment of goods usually contain multiple identical pallets (which have different DoS). On average there are 10.6 pallets per shipment in the dataset. The following covariates are present:
+
+- Non-sequential Information: Date of Arrival, Warehouse Location, Customer Type, Product Group, Pallet Weight, Inbound Location, Outbound Location  
+- Sequential Information: Textual description of product in pallets.
+
+Inbound and Outbound location refers to where the shipment was coming from and will go to. The records are mainly food products, with the most common categories being (in decreasing order): chicken, beef, pork, potato, and dairy. However, non-food items such as cigarettes are also present.
+
+The item descriptions describe the contents of the pallet, but most of them are not written in a human readable form, such as "NY TX TST CLUB PACK". Acronyms are used liberally due to the length restriction of item descriptions in the computer system. Furthermore, the acronyms do not necessarily represent the same words: "CKN WG" means "chicken wing" while "WG CKN" means "WG brand chicken". Therefore, even though the descriptions are short, the order of the words is important.
+
+To enable efficient use of the descriptions, we decoded common acronyms by hand (e.g.  $\mathrm{tst} \rightarrow$  toast). However, the resulting dataset is not perfectly clean (intentionally so to mimic realistic conditions) and contains many broken phrases, misspelled words, unidentified acronyms, and other symbols.
+
+# 4.2 PUBLIC RELEASE VERSION
+
+We release the above dataset, which as far as the authors know, is the first publicly available dataset of warehousing records.
+
+The collaborating company transports an appreciable amount ( $>30\%$ ) of the entire US refrigerated food supply, so US law prohibits the release of the full detail of transported shipments. Furthermore, NDA agreements ban any mentioning of the brand names. Thus, for the public version, we removed all brands and detailed identifying information in the item descriptions. The testing in the section below is done on the private version to reflect the full reality in the warehouse, but the results on the public version are similar (in Appendix 9.4) and the conclusions carry over to the public version.
+
+# 5 IMPLEMENTING THE FRAMEWORK
+
+The Framework described in Section 3 requires the knowledge of four parameters:  $X_{S}$  and  $F_{S}$  for pallets in the training data, the loss function  $l(F_{S},\tilde{F}_{S})$ , and the machine learning model estimate  $\tilde{g}$ .
+
+$X_{S}$  is immediately available in the form of the non-sequential and sequential information. For the textual description, we encode the words using GloVe embeddings. (Pennington et al. (2014)) We limit the description to the first five words with zero padding.
+
+For  $F_{S}$ , we exploit that each shipment arriving usually contains  $p \gg 1$  units of the good. We treat these  $p$  units as  $p$  copies of a one-unit shipment, denoted  $S_{1}, \dots, S_{p}$ . Then by using the DoS for each of these  $p$  units  $(T_{1}, \dots, T_{p})$ , we can create an empirical distribution function  $\hat{F}_{S}(t) = \frac{1}{p} \sum_{i=1}^{n} \mathbf{1}_{T_{i} \leq t}$  for the DoS of the shipment. This is treated as the ground truth for the training data.
+
+To obtain a loss function for the distribution, we selected the  $5\%$ ,  $\dots$ $95\%$  percentile points of the CDF, which forms a 19-dimensional output. This definition provides a more expressive range of CDFs than estimating the coefficients for a parametric distribution, as many products' CDF do not follow any obvious parametric distribution. Then we chose the mean squared logarithmic error (MSLE) as our loss function to compare each percentile point with those predicted. This error function is chosen as the error in estimating DoS affects the positioning of the unit roughly logarithmically in the warehouse under the DoS policy. For example, estimating a shipment to stay 10 days rather than the true value of 1 day makes about the same difference in storage position compared to estimating a shipment to stay 100 days rather than a truth of 10. This is due to a pseudo-lognormal distribution of the DoS in the entire warehouse as seen in the historical data.
+
+![](images/9bfdaecc3319434030bd27cb145fd79e50a6c99454394b55aea93d60deea6eb9.jpg)  
+Figure 1: ParallelNet Simplified Architecture. Green boxes are inputs and the red box is the output. We separate inputs into sequential data and non-sequential data to exploit different types of data.
+
+Thus, our empirical loss function is defined as:
+
+$$
+L (\hat {F} _ {S}, \tilde {F} _ {S}) = \frac {1}{1 9} \sum_ {i = 1} ^ {1 9} \left(\log (\hat {F} _ {S} ^ {- 1} (0. 0 5 i) + 1) - \log (\tilde {F} _ {S} ^ {- 1} (0. 0 5 i) + 1)\right) ^ {2}
+$$
+
+Now, we would introduce the machine learning algorithm  $\tilde{g}$  to approximate  $F_{S}$ .
+
+# 5.1 INTRODUCTION OF PARALLELNET
+
+For the dataset introduced in Section 4, the textual description carries important information relating to the DoS distribution. The product identity usually determines its seasonality and largely its average DoS, and therefore it would be desirable to extract as much information as possible through text.
+
+We would utilize both convolutional neural networks (CNN) and recurrent neural networks (RNN) to model the textual description. As illustrated in Section 3, word order is important in this context, and RNNs are well equipped to capture such ordering. As the textual information is critical to the DoS prediction, we would supplement the RNN prediction with a CNN architecture in a parallel manner, as presented in Figure 1. We then utilized a residual output layer to produce 19 percentile points  $(\tilde{F}_S^{-1}(0.05),\tilde{F}_S^{-1}(0.1),\dots ,\tilde{F}_S^{-1}(0.95))$  that respect the increasing order of these points.
+
+This architecture is similar to ensembling, which is well known for reducing bias in predictions (see Opitz & Maclin (1999)). However, it has the additional advantage of a final co-training output layer that allows a complex combination of the output of two models, compared to the averaging done for ensembling. This is particularly useful for our purpose of predicting 19 quantile points of a distribution, as it is likely the CNN and RNN would be better at predicting different points, and thus a simple average would not fully exploit the information contained in both of the networks. We would see in Section 6 that this allows the model to further improve its ability to predict the DoS distribution. We also further note that this is similar to the LSTM-CNN framework proposed by Donahue et al. (2014), except that the LSTM-CNN architecture stagcs the CNN and RNN in a sequential manner. We would compare with such framework in our evaluation in Section 6.
+
+In interest of brevity, we omit the detailed architecture choice in RNN and CNN along with the output layer structure and include it in Appendix 9.2. Hyperparameters are contained in Appendix 9.3.
+
+# 6 COMPUTATION RESULTS
+
+In this section, we test the capability of ParallelNet to estimate the DoS distributions  $F_{S}$  on the dataset introduced in Section 4. We separate the dataset introduced in Section 4 into the following:
+
+- Training Set: All shipments that exited the warehouse before 2017/06/30, consisting about  $60\%$  of the entire dataset.  
+- Testing Set: All shipments that arrived at the warehouse after 2017/06/30 and left the warehouse before 2017/07/30, consisting about  $7\%$  of the entire dataset.  
+- Extended Testing Set: All shipments that arrived at the warehouse after 2017/09/30 and left the warehouse before 2017/12/31, consisting about  $14\%$  of the entire dataset.
+
+We then trained five separate neural networks and two baselines to evaluate the effectiveness of ParallelNet. Specifically, for neural networks, we evaluated the parallel combination of CNN and RNNs against a vertical combination (introduced in Donahue et al. (2014)), a pure ensembling model, and the individual network components. We also compare against two classical baselines, gradient-boosted trees (GBM) (Friedman (2001)) and linear regression (LR), as below:
+
+- CNN-LSTM This implements the model in Donahue et al. (2014). To ensure the best comparability, we use a ResNet CNN and a 2-layer GRU, same as ParallelNet.  
+- CNN+LSTM This implements an ensembling model of the two architectures used in ParallelNet, where the two network's final output is averaged.  
+- ResNet (CNN) We implement the ResNet arm of ParallelNet.  
+- GRU We implement the GRU arm of ParallelNet.  
+- Fully-connected Network (FNN) We implement a 3-layer fully-connected network.  
+- Gradient Boosted Trees with Text Features (GBM) We utilize Gradient Boosted Trees (Friedman, 2001) as a benchmark outside of neural networks. As gradient boosted trees cannot self-generate features from the text data, we utilize 1-gram and 2-gram features.  
+- Linear Regression with Text Features (LR) We implement multiple linear regression. Similar to GBM, we generate 1-gram and 2-gram features. The  $y$  variable is log-transformed as we assume the underlying duration distribution is approximately log-normal.
+
+All neural networks are trained on Tensorflow 1.9.0 with Adam optimizer (Kingma & Ba, 2014). The learning rate, decay, and number of training epochs are 10-fold cross-validated. The GBM is trained on R 3.4.4 with the lightgbm package, and number of trees 10-fold cross-validated over the training set. The Linear Regression is trained with the lm function on R 3.4.4. We used a 6-core i7-5820K, GTX1080 GPU, and 16GB RAM. The results on the Testing Set are as followed:
+
+<table><tr><td rowspan="2">Architecture</td><td colspan="2">Testing Set</td><td colspan="2">Extended Testing Set</td></tr><tr><td>MSLE</td><td>MAPE</td><td>MSLE</td><td>MAPE</td></tr><tr><td>ParallelNet</td><td>0.4419</td><td>29%</td><td>0.7945</td><td>51%</td></tr><tr><td>CNN-LSTM</td><td>0.4812</td><td>41%</td><td>0.9021</td><td>80%</td></tr><tr><td>CNN+LSTM</td><td>0.5024</td><td>47%</td><td>0.9581</td><td>91%</td></tr><tr><td>CNN</td><td>0.6123</td><td>70%</td><td>1.0213</td><td>124%</td></tr><tr><td>GRU</td><td>0.5305</td><td>47%</td><td>1.1104</td><td>122%</td></tr><tr><td>FNN</td><td>0.8531</td><td>120%</td><td>1.0786</td><td>130%</td></tr><tr><td>GBM</td><td>1.1325</td><td>169%</td><td>1.2490</td><td>187%</td></tr><tr><td>LR</td><td>0.9520</td><td>132%</td><td>1.1357</td><td>140%</td></tr></table>
+
+<table><tr><td rowspan="2">Architecture</td><td colspan="2">Testing Set</td></tr><tr><td>MSLE</td><td>MAPE</td></tr><tr><td>ParallelNet</td><td>0.4419</td><td>29%</td></tr><tr><td>Without Product Group</td><td>0.6149</td><td>65%</td></tr><tr><td>Without Date of Arrival</td><td>0.5723</td><td>61%</td></tr><tr><td>Without Customer Type</td><td>0.5687</td><td>58%</td></tr><tr><td>Without All Nontextual Features</td><td>0.8312</td><td>110%</td></tr><tr><td>Without Textual Features</td><td>0.9213</td><td>125%</td></tr></table>
+
+Table 1: Table of Prediction Results for Different Machine Learning Architectures
+
+Table 2: Ablation Studies
+
+We can see that ParallelNet comfortably outperforms other architectures. Its loss is lower than the vertical stacking CNN-LSTM, by  $8\%$ . The result of 0.4419 shows that on average, our prediction in the 19 percentiles is  $44\%$  away from the true value. We also note that its loss is about  $15\%$  less than the pure ensembling architecture, indicating that there is a large gain from the final co-training layer.
+
+We also note that the baselines (GBM, LR) significantly underperformed neural network architectures, indicating that advanced modeling of text is important. We also conducted ablation studies to find that product group, date of arrival, and customer type are important for the task, which is intuitive. Furthermore, both nontextual and textual features are required for good performance.
+
+We then look at a different statistic: the Median Absolute Percentage Error (MAPE). For every percentile in every sample, the Absolute Percentage Error (APE) of the predicted number of days  $\hat{T}$  and the actual number of days  $T$  is defined as:
+
+$$
+\mathbf {A P E} (\hat {T}, T) = \frac {| \hat {T} - T |}{T}
+$$
+
+Then MAPE is defined as the median value of the APE across all 19 percentiles and all samples in the testing set. This statistic is more robust to outliers in the data.
+
+As seen in Table 2, ParallelNet has a MAPE of  $29\%$ . This is highly respectable given the massive innate fluctuations of a dynamic warehouse, as this means the model could predict  $50\%$  of all
+
+![](images/698d3c6b1c89acb9f6d54f2c75f7a5bba96f4214ac049d481bf08775466001ed.jpg)
+
+![](images/f3afd57156b972f0c0b51afbb423b3c4b67d4c942ddfacde16de9be8e5b07995.jpg)
+
+![](images/3c5e6e8afff9bb7a19433766831f27d7ac4093fa704df40edbad769272fa1ce3.jpg)  
+Figure 2: Placement percentile of putaway pallets before and after using a DoS system for 5 months in two selected facilities. Red line denotes the mean and blue line denotes the median.
+
+![](images/837e14bc94e38320b62da22835dc7652d8e7b96c27add1b149af612723ca536c.jpg)
+
+percentiles with an error less than  $29\%$ . The result also compares well with the other methods, as ParallelNet reduces the MAPE by  $29.3\%$  when compared to the best baseline of CNN-LSTM.
+
+If we look further out into the future in the Extended Testing Set, the performance of all algorithms suffer. This is expected as our information in the training set is outdated. Under this scenario, we see that ParallelNet still outperforms the other comparison algorithms by a significant amount. In fact, the difference between the pairs (CNN-LSTM, ParallelNet) and (CNN+LSTM, ParallelNet) both increase under the MSLE and MAPE metrics. This provides evidence that a parallel co-training framework like that of ParallelNet is able to generalize better. We hypothesize that this is due to the reduction in bias due to ensembling-like qualities leading to more robust answers.
+
+# 7 REAL-LIFE IMPLEMENTATION RESULTS
+
+With the favorable computational results, the collaborating company is implementing the framework with ParallelNet across their warehouses, and in this section we would analyze the initial results.
+
+The graphs in Figure 2 records the empirical distribution of the placement percentiles before and after the DoS framework was online. The placement percentile is the percentile of the distance from the staging area to the placement location. Thus,  $40\%$  means a pallet is put at the 40th percentile of the distance from staging. The distance distribution of locations is relatively flat, so this is a close proxy of driving distance between staging and storage locations, and thus time spent storing the pallets.
+
+Additionally, we plotted two curves obtained by simulating placement percentiles under a perfect DoS strategy (blue) and a greedy strategy that allocates the closest available position to any pallet ignoring DoS (black). These simulations assumed all locations in the warehouse are open to storage and ignored all other constraints in the placement; thus they do not reflect the full complex workings of a warehouse. However, this simple simulation show clearly the advantages of the DoS strategy as it is able to store more pallets in the front of the warehouse. We observe that in both facilities, the real placement percentiles behaved relatively closely to the blue curve. This is strong evidence that the implemented DoS framework is effective in real-life and provides efficiency gains. We note that there is a drop in the lower percentiles compared to the DoS curve - this is due to some locations close to the staging area being reserved for packing purposes and thus not actually available for storage.
+
+Specifically, Facility A had an average placement percentile of  $51\%$  before, and  $41\%$  after, while Facility B had an average placement percentile of  $50\%$  before, and  $39\%$  after. On average, we record a  $10.5\%$  drop in absolute terms or  $21\%$  in relative terms. This means that the labor time spent on storing pallets has roughly declined by  $21\%$ . An unpaired  $t$ -test on the average putaway percentile shows the change is statistically significant on the  $1\%$  level for both facilities. This provides real-life evidence that the system is able to generate real labor savings in the warehouses.
+
+# 8 CONCLUSION AND REMARKS
+
+In conclusion, we have introduced a comprehensive framework for storage assignment under an uncertain DoS. We produced an implementation of this framework using a parallel formulation of two effective neural network architectures. We showed how the parallel formulation has favorable generalization behavior and out-of-sample testing results compared with sequential stacking and assembling. This has allowed this framework to be now implemented in live warehouses all around the country, and results show appreciable labor savings on the ground. We also release the first dataset of warehousing records to stimulate research in this central problem for storage assignment.
+
+# REFERENCES
+
+Lu Chen, Andr Langevin, and Diane Riopel. The storage location assignment and interleaving problem in an automated storage/retrieval system with shared storage. International Journal of Production Research, 48(4):991-1011, 2010. doi: 10.1080/00207540802506218. URL https://doi.org/10.1080/00207540802506218.  
+Zhuxi Chen, Xiaoping Li, and Jatinder N.D. Gupta. Sequencing the storages and retrievals for flow-rack automated storage and retrieval systems with duration-of-stay storage policy. International Journal of Production Research, 54(4):984-998, 2016. doi: 10.1080/00207543.2015.1035816. URL https://doi.org/10.1080/00207543.2015.1035816.  
+Kyunghyun Cho, Bart van Merrienboer, Caglar Gülçehre, Fethi Bougares, Holger Schwenk, and Yoshua Bengio. Learning phrase representations using RNN encoder-decoder for statistical machine translation. CoRR, abs/1406.1078, 2014. URL http://arxiv.org/abs/1406.1078.  
+Junyoung Chung, Caglar Gulcehre, KyungHyun Cho, and Yoshua Bengio. Empirical evaluation of gated recurrent neural networks on sequence modeling. arXiv preprint arXiv:1412.3555, 2014.  
+Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. Bert: Pre-training of deep bidirectional transformers for language understanding. arXiv preprint arXiv:1810.04805, 2018.  
+Jeff Donahue, Lisa Anne Hendricks, Sergio Guadarrama, Marcus Rohrbach, Subhashini Venugopalan, Kate Saenko, and Trevor Darrell. Long-term recurrent convolutional networks for visual recognition and description. CoRR, abs/1411.4389, 2014. URL http://arxiv.org/abs/1411.4389.  
+Jerome H Friedman. Greedy function approximation: a gradient boosting machine. Annals of statistics, pp. 1189-1232, 2001.  
+Marc Goetschalckx and H. Donald Ratliff. Shared storage policies based on the duration stay of unit loads. Management Science, 36(9):1120-1132, 1990. doi: 10.1287/mnsc.36.9.1120. URL https://doi.org/10.1287/mnsc.36.9.1120.  
+Jeffery H. Gove. Estimation and applications of size-biased distributions in forestry. Modeling Forest Systems, 2003.  
+Warren H. Hausman, Leroy B. Schwarz, and Stephen C. Graves. Optimal storage assignment in automatic warehousing systems. Management Science, 22(6):629-638, 1976. doi: 10.1287/mnsc.22.6.629. URL https://doi.org/10.1287/mnsc.22.6.629.  
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. CoRR, abs/1512.03385, 2015. URL http://arxiv.org/abs/1512.03385.  
+Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.  
+Sadan Kulturel, Nur E Ozdemirel, Canan Sepil, and Zafer Bozkurt. Experimental investigation of shared storage assignment policies in automated storage/retrieval systems. IIE transactions, 31(8): 739-749, 1999.  
+Charles J Malmborg. Interleaving models for the analysis of twin shuttle automated storage and retrieval systems. International Journal of Production Research, 38(18):4599-4610, 2000.
+
+David Opitz and Richard Maclin. Popular ensemble methods: An empirical study. Journal of artificial intelligence research, 11:169-198, 1999.  
+Jeffrey Pennington, Richard Socher, and Christopher D. Manning. Glove: Global vectors for word representation. In Empirical Methods in Natural Language Processing (EMNLP), pp. 1532-1543, 2014. URL http://www.aclweb.org/anthology/D14-1162.  
+Charles G Petersen, Gerald R Aase, and Daniel R Heiser. Improving order-picking performance through the implementation of class-based storage. International Journal of Physical Distribution & Logistics Management, 34(7):534-544, 2004.  
+Alec Radford, Jeffrey Wu, Rewon Child, David Luan, Dario Amodei, and Ilya Sutskever. Language models are unsupervised multitask learners.  
+Meir J Rosenblatt and Amit Eynan. Notederiving the optimal boundaries for class-based automatic storage/retrieval systems. Management Science, 35(12):1519-1524, 1989.  
+Mike Schuster and Kuldip K Paliwal. Bidirectional recurrent neural networks. IEEE Transactions on Signal Processing, 45(11):2673-2681, 1997.  
+Leroy B. Schwarz, Stephen C. Graves, and Warren H. Hausman. Scheduling policies for automatic warehousing systems: Simulation results. A I E Transactions, 10(3):260-270, 1978. doi: 10. 1080/05695557808975213. URL https://doi.org/10.1080/05695557808975213.  
+Nitish Srivastava, Geoffrey Hinton, Alex Krizhevsky, Ilya Sutskever, and Ruslan Salakhutdinov. Dropout: a simple way to prevent neural networks from overfitting. The Journal of Machine Learning Research, 15(1):1929-1958, 2014.  
+Yugang Yu and MBM De Koster. Designing an optimal turnover-based storage rack for a 3d compact automated storage and retrieval system. International journal of production research, 47(6): 1551-1571, 2009.  
+Yugang Yu and René BM De Koster. On the suboptimality of full turnover-based storage. International Journal of Production Research, 51(6):1635-1647, 2013.  
+Yugang Yu, René BM de Koster, and Xiaolong Guo. Class-based storage with a finite number of items: Using more classes is not always better. Production and operations management, 24(8): 1235-1247, 2015.
+
+# 9 APPENDIX
+
+# 9.1 LITERATURE REVIEW ON STORAGE ASSIGNMENT
+
+Since Hausman et al. (1976), many different theoretical frameworks have been introduced, which can roughly be separated into two classes: dedicated storage systems and shared storage systems.
+
+# 9.1.1 DEDICATED STORAGE SYSTEMS
+
+For this class of storage systems, each product gets assigned fixed locations in the warehouse. When the product comes in, it is always assigned to one of the pre-determined locations. Under this constraint, it is optimal to dedicate positions with travel distance inversely proportional to the turnover of the product, as shown in Goetschalckx & Ratliff (1990). Turnover of a product is defined as the inverse of Cube per Order Index (COI), which is the ratio of the size of the location it needs to the frequency of retrieval needed. Heuristically, those products with the smallest surface footprint and the highest frequency should be put closest to the warehouse entry, so that those locations are maximally utilized.
+
+# 9.1.2 SHARED STORAGE SYSTEMS
+
+This class of storage systems allows multiple pallets to occupy the same position in the warehouse (at different times). It is widely considered to be superior than dedicated storage systems due to its savings on travel time and smaller need for storage space, as shown by Yu et al. (2015), Malmborg (2000). Within this category, there are mainly three strategies:
+
+- Turnover (Cube-per-Order, COI) Based: Products coming into the warehouse are assigned locations so that the resultant travel distance is inversely proportional to the turnover of the product. Examples of such work include Hausman et al. (1976), Yu & De Koster (2009), and Yu & De Koster (2013).  
+- Class Based: Products are first separated into  $k$  classes, with each class assigned a dedicated area of storage. The most popular type of class assignment is called ABC assignment, which divides products into three classes based on their turnover within the warehouse. Then within each class, a separate system is used to sort the pallets (usually random or turnover-based). It was introduced by Hausman et al. (1976), and he showed that a simple framework saves on average  $20 - 25\%$  of time compared to the dedicated storage policy in simulation. Implementation and further work in this area include Rosenblatt & Eynan (1989), Schwarz et al. (1978) and Petersen et al. (2004).  
+- Duration-of-Stay (DoS) Based: Individual products are assigned locations with travel distance proportional to the duration of stay. Goetschalckx & Ratliff (1990) proved that if DoS is known in advance and the warehouse is completely balanced in input/output, then the DoS policy is theoretically optimal. The main work in this area was pioneered by Goetschalckx & Ratliff (1990). Recently, Chen et al. (2016) and Chen et al. (2010) reformulated the DoS-based assignment problem as a mixed-integer optimization problem in an automated warehouse under different configurations. Both papers assume that the DoS is known exactly ahead of time.
+
+# 9.2 ARCHITECTURE CHOICE
+
+In modeling text, there are three popular classes of architectures: convolutional neural networks (CNN), recurrent neural networks (RNN), and transformers. In particular recently transformers have gained popularity due to their performance in machine translation and generation tasks (e.g. Devlin et al. (2018), Radford et al.). However, we argue that transformers are not the appropriate model for the textual descriptions here. The words often do not form a coherent phrase so there is no need for attention, and there is a lack of long-range dependency due to the short length of the descriptions.
+
+# 9.2.1 BIDIRECTIONAL GRU LAYERS
+
+Gated Recurrent Units, introduced by Cho et al. (2014), are a particular implementation of RNN intended to capture long-range pattern information. In the proposed system, we further integrate bi-directionality, as detailed in Schuster & Paliwal (1997), to improve feature extraction by training the sequence both from the start to end and in reverse.
+
+The use of GRU rather than LSTM is intentional. Empirically GRU showed better convergence properties, which has also been observed by Chung et al. (2014), and better stability when combined with the convolutional neural network.
+
+# 9.2.2 RESNET CONVOLUTIONAL LAYERS
+
+In a convolutional layer, many independent filters are used to find favorable combinations of features that leads to higher predictive power. Further to that, they are passed to random dropout layers introduced in Srivastava et al. (2014) to reduce over-fitting and improve generalization. Dropout layers randomly change some outputs in  $c_{0}, \dots, c_{i}$  to zero to ignore the effect of the network at some nodes, reducing the effect of over-fitting.
+
+Repeated blocks of convolution layers and random dropout layers are used to formulate a deep convolution network to increase generalization capabilities of the network.
+
+However, a common problem with deep convolutional neural networks is the degradation of training accuracy with increasing number of layers, even though theoretically a deeper network should perform at least as well as a shallow one. To prevent such issues, we introduce skip connections in the convolution layers, introduced by Residual Networks in He et al. (2015). The residual network introduces identity connections between far-away layers. This effectively allows the neural network to map a residual mapping into the layers in between, which is empirically shown to be easier to train.
+
+# 9.2.3 OUTPUT LAYER
+
+We designed the output layer as below in Figure 3 to directly predict the 19 percentile points  $(\tilde{F}_S^{-1}(0.05),\tilde{F}_S^{-1}(0.1),\dots ,\tilde{F}_S^{-1}(0.95))$  ..
+
+![](images/c18dc76eb14a8884767b6cca0dac2fecc69eb7568f671766f419359c6fb7900c.jpg)  
+Figure 3: Output Layer. Here  $t_1, \dots, t_{19}$  corresponds to the  $5\%, \dots, 95\%$  percentile points.
+
+Note that the 19 percentile points are always increasing. Thus the output is subjected to 19 separate 1-neuron dense layers with ReLu, and the output of the previous dense layer is added to the next one, creating a residual output layer in which each (non-negative) output from the 1-neuron dense layer is only predicting the residual increase  $t_{i+1} - t_i$ .
+
+# 9.3 DETAILED SETTINGS OF IMPLEMENTED ARCHITECTURE
+
+![](images/53e7ced249a0bb61c52c5f63b8b507b498666c437f1d11f79f2cf70c901e7eb1.jpg)  
+Figure 4: ParallelNet Architecture
+
+# 9.4 TESTING RESULTS ON PUBLIC DATASET
+
+<table><tr><td rowspan="2">Architecture</td><td colspan="2">Testing Set</td><td colspan="2">Extended Testing Set</td></tr><tr><td>MSLE</td><td>MAPE</td><td>MSLE</td><td>MAPE</td></tr><tr><td>ParallelNet</td><td>0.4602</td><td>34%</td><td>0.7980</td><td>50%</td></tr><tr><td>CNN</td><td>0.6203</td><td>73%</td><td>1.0087</td><td>122%</td></tr><tr><td>GBM</td><td>1.1749</td><td>175%</td><td>1.2605</td><td>190%</td></tr></table>
+
+Table 3: Results on Public Dataset for Selected Architectures
\ No newline at end of file
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+# EMERGENCE OF FUNCTIONAL AND STRUCTURAL PROPERTIES OF THE HEAD DIRECTION SYSTEM BY OPTIMIZATION OF RECURRENT NEURAL NETWORKS
+
+Christopher J. Cueva\*, Peter Y. Wang, Matthew Chin, Xue-Xin Wei
+
+Columbia University
+
+New York, NY 10027, USA
+
+# ABSTRACT
+
+Recent work suggests goal-driven training of neural networks can be used to model neural activity in the brain. While response properties of neurons in artificial neural networks bear similarities to those in the brain, the network architectures are often constrained to be different. Here we ask if a neural network can recover both neural representations and, if the architecture is unconstrained and optimized, the anatomical properties of neural circuits. We demonstrate this in a system where the connectivity and the functional organization have been characterized, namely, the head direction circuits of the rodent and fruit fly. We trained recurrent neural networks (RNNs) to estimate head direction through integration of angular velocity. We found that the two distinct classes of neurons observed in the head direction system, the Compass neurons and the Shifter neurons, emerged naturally in artificial neural networks as a result of training. Furthermore, connectivity analysis and in-silico neurophysiology revealed structural and mechanistic similarities between artificial networks and the head direction system. Overall, our results show that optimization of RNNs in a goal-driven task can recapitulate the structure and function of biological circuits, suggesting that artificial neural networks can be used to study the brain at the level of both neural activity and anatomical organization.
+
+# 1 INTRODUCTION
+
+Artificial neural networks have been increasingly used to study biological neural circuits. In particular, recent work in vision demonstrated that convolutional neural networks (CNNs) trained to perform visual object classification provide state-of-the-art models that match neural responses along various stages of visual processing (Yamins et al., 2014; Khaligh-Razavi & Kriegeskorte, 2014; Yamins & DiCarlo, 2016; Cadieu et al., 2014; Güçlü & van Gerven, 2015; Kriegeskorte, 2015). Recurrent neural networks (RNNs) trained on cognitive tasks have also been used to account for neural response characteristics in various domains (Mante et al., 2013; Sussillo et al., 2015; Song et al., 2016; Cueva & Wei, 2018; Banino et al., 2018; Remington et al., 2018; Wang et al., 2018; Orhan & Ma, 2019; Yang et al., 2019). While these results provide important insights on how information is processed in neural circuits, it is unclear whether artificial neural networks have converged upon similar architectures as the brain to perform either visual or cognitive tasks. Answering this question requires understanding the functional, structural, and mechanistic properties of artificial neural networks and of relevant neural circuits.
+
+We address these challenges using the brain's internal compass - the head direction system, a system that has accumulated substantial amounts of functional and structural data over the past few decades in rodents and fruit flies (Taube et al., 1990a,b; Turner-Evans et al., 2017; Green et al., 2017; Seelig & Jayaraman, 2015; Stone et al., 2017; Lin et al., 2013; Finkelstein et al., 2015; Wolff et al., 2015; Green & Maimon, 2018). We trained RNNs to perform a simple angular velocity (AV) integration task (Etienne & Jeffery, 2004) and asked whether the anatomical and functional features that have emerged as a result of stochastic gradient descent bear similarities to biological networks sculpted
+
+by long evolutionary time. By leveraging existing knowledge of the biological head direction (HD) systems, we demonstrate that RNNs exhibit striking similarities in both structure and function. Our results suggest that goal-driven training of artificial neural networks provide a framework to study neural systems at the level of both neural activity and anatomical organization.
+
+![](images/bbdb11ae253315af10330ab3223ca428d4367b58e515a1597ead8240de2c58ff.jpg)
+
+![](images/6fa19940a069b038bc25cb952721e1ddfab490454867845f5e5a5ea2ac2db4ba.jpg)
+
+![](images/3ab99f37bdcda47ee5a0866aa12d195139f7b3afbfbf6d6bd2f9746e3521c215.jpg)
+
+![](images/e154f15639d380ebefb6014e2d4055517a237c4c88a91e7746c7c90be37ea69b.jpg)
+
+![](images/25831ec83d5a6c082256dbe551e05ae4e1bd239f2d95bcb39f9f82aa9798cc6c.jpg)
+
+![](images/2b6208c73681c8c4f3f57bbf678fc78bf6284d5d7abc47b6631a2ec959ab8b07.jpg)  
+Figure 1: Overview of head direction system in rodents, fruit flies, and the RNN model. a, c) tuning curve of an example head direction (HD) cell in rodents (a, adapted from Taube (1995)) and fruit flies (c, adapted from Turner-Evans et al. (2017)). b, d) Tuning curve of an example angular velocity (AV) selective cell in rodents (b, adapted from Sharp et al. (2001)) and fruit flies (d, adapted from Turner-Evans et al. (2017)). e) The brain structures in the fly central complex that are crucial for maintaining and updating heading direction, including the protocerebral bridge (PB) and the ellipsoid body (EB). f) The RNN model. All connections within the RNN are randomly initialized. g) After training, the output of the RNN accurately tracks the current head direction.
+
+![](images/83f681fbc71269683c5880944c773dea9091b63a4f980b2d01eaf7ef1a5e3cbd.jpg)
+
+# 2 MODEL
+
+# 2.1 MODEL STRUCTURE
+
+We trained our networks to estimate the agent's current head direction by integrating angular velocity over time (Fig. 1f). Our network model consists of a set of recurrently connected units ( $N = 100$ ), which are initialized to be randomly connected, with no self-connections allowed during training. The dynamics of each unit in the network  $r_i(t)$  is governed by the standard continuous-time RNN equation:
+
+$$
+\tau \frac {d x _ {i} (t)}{d t} = - x _ {i} (t) + \sum_ {j} W _ {i j} ^ {\text {r e c}} r _ {j} (t) + \sum_ {k} W _ {i k} ^ {\text {i n}} I _ {k} (t) + b _ {i} + \xi_ {i} (t) \tag {1}
+$$
+
+for  $i = 1, \dots, N$ . The firing rate of each unit,  $r_i(t)$ , is related to its total input  $x_i(t)$  through a rectified tanh nonlinearity,  $r_i(t) = \max(0, \tanh(x_i(t)))$ . Every unit in the RNN receives input from all other units through the recurrent weight matrix  $W^{\mathrm{rec}}$  and also receives external input,  $I(t)$ , through the weight matrix  $W^{\mathrm{in}}$ . These weight matrices are randomly initialized so no structure is a priori introduced into the network. Each unit has an associated bias,  $b_i$  which is learned and an associated noise term,  $\xi_i(t)$ , sampled at every timestep from a Gaussian with zero mean and constant variance. The network was simulated using the Euler method for  $T = 500$  timesteps of duration  $\tau / 10$  ( $\tau$  is set to be 250ms throughout the paper).
+
+Let  $\theta$  be the current head direction. Input to the RNN is composed of three terms: two inputs encode the initial head direction in the form of  $\sin (\theta_0)$  and  $\cos (\theta_0)$ , and a scalar input encodes both clockwise (CW, negative) and counterclockwise, (CCW, positive) angular velocity at every timestep. The RNN
+
+is connected to two linear readout neurons,  $y_{1}(t)$  and  $y_{2}(t)$ , which are trained to track current head direction in the form of  $\sin(\theta)$  and  $\cos(\theta)$ . The activities of  $y_{1}(t)$  and  $y_{2}(t)$  are given by:
+
+$$
+y _ {j} (t) = \sum_ {i} W _ {j i} ^ {\text {o u t}} r _ {i} (t) \tag {2}
+$$
+
+# 2.2 INPUT STATISTICS
+
+Velocity at every timestep (assumed to be  $25\mathrm{ms}$ ) is sampled from a zero-inflated Gaussian distribution (see Fig. 5). Momentum is incorporated for smooth movement trajectories, consistent with the observed animal behavior in flies and rodents. More specifically, we updated the angular velocity as  $\mathrm{AV}(t) = \sigma X + \mathrm{momentum*AV}(t - 1)$ , where  $X$  is a zero mean Gaussian random variable with standard deviation of one. In the Main condition, we set  $\sigma = 0.03$  radians/timestep and the momentum to be 0.8, corresponding to a mean absolute AV of  $\sim 100$  deg/s. These parameters are set to roughly match the angular velocity distribution of the rat and fly (Stackman & Taube, 1998; Sharp et al., 2001; Bender & Dickinson, 2006; Raudies & Hasselmo, 2012). In section 4, we manipulate the magnitude of AV by changing  $\sigma$  to see how the trained RNN may solve the integration task differently.
+
+# 2.3 TRAINING
+
+We optimized the network parameters  $W^{\mathrm{rec}}$ ,  $W^{\mathrm{in}}$ ,  $b$  and  $W^{\mathrm{out}}$  to minimize the mean-squared error in equation (3) between the target head direction and the network outputs generated according to equation (2), plus a metabolic cost for large firing rates ( $L_{2}$  regularization on  $r$ ).
+
+$$
+E = \sum_ {t, j} \left(y _ {j} (t) - y _ {j} ^ {\text {t a r g e t}} (t)\right) ^ {2} \tag {3}
+$$
+
+Parameters were updated with the Hessian-free algorithm (Martens & Sutskever, 2011). Similar results were also obtained using Adam (Kingma & Ba, 2015).
+
+# 3 FUNCTIONAL AND STRUCTURAL PROPERTIES EMERGED IN THE NETWORK
+
+We found that the trained network could accurately track the angular velocity (Fig. 1g). We first examined the functional and structural properties of model units in the trained RNN and compared them to the experimental data from the head direction system in rodents and flies.
+
+# 3.1 EMERGENCE OF HD CELLS (COMPASS UNITS) AND  $\mathrm{HD}\times \mathrm{AV}$  CELLS (SHIFTERS)
+
+# Emergence of different classes of neurons with distinct tuning properties
+
+We first plotted the neural activity of each unit as a function of HD and AV (Fig. 2a). This revealed two distinct classes of units based on the strength of their HD and AV tuning (see Appendix Fig. 6a,b,c). Units with essentially zero activity are excluded from further analyses. The first class of neurons exhibited HD tuning with minimal AV tuning (Fig. 2f). The second class of neurons were tuned to both HD and AV and can be further subdivided into two populations - one with high firing rate when animal performs CCW rotation (positive AV), the other favoring CW rotation (negative AV) (CW tuned cell shown in Fig. 2g). Moreover, the preferred head direction of each sub-population of neurons tile the complete angular space (Fig. 2b). Embedding the model units into 3D space using t-SNE reveals a clear compass-like structure, with the three classes of units being separated (Fig. 2c).
+
+# Mapping the functional architecture of RNN to neurophysiology
+
+Neurons with HD tuning but not AV tuning have been widely reported in rodents (Taube et al., 1990a; Blair & Sharp, 1995; Stackman & Taube, 1998), although the  $\mathrm{HD^{*}AV}$  tuning profiles of neurons are rarely shown (but see Lozano et al. (2017)). By re-analyzing the data from Peyrache et al. (2015), we find that neurons in the anterodorsal thalamic nucleus (ADN) of the rat brain are selectively tuned to HD but not AV (Fig. 2d, also see Lozano et al. (2017)), with  $\mathrm{HD^{*}AV}$  tuning profile similar to what our model predicts. Preliminary evidence suggests that this might also be true for ellipsoid body (EB) compass neurons in the fruit fly HD system (Green et al., 2017; Turner-Evans et al., 2017).
+
+![](images/e5776ea66b28eea0685652d951cfebe85fd11a431a131e750c1a1d2f5fa48c4b.jpg)  
+a  
+b
+
+![](images/f0f14646a81749747b44303e4f36162ea967ccb0ddb3c9256068a645922bda6f.jpg)
+
+![](images/37f20a62d63c2bb60552a1fe2be4a14ae96038d20c40ee638a377a63cd3fbce0.jpg)  
+C
+
+![](images/a93854683558c4989966501a368ab7a4d6fef66f28a99b69880b188e48a3fe8b.jpg)  
+d
+
+![](images/2b27c5a5470b8b2b378391e3bf660d6778457f919e73bc04b2fbb00dab68af32.jpg)  
+f
+
+![](images/b1635839a0928368d1d4ee7f3760ac6a3c2da5c1df1ebefac7c6a061780c2127.jpg)  
+h
+
+![](images/983b4ba7aef508bcfc1ed31c585f2ae9ffecc93e268fe068b5818a533078c27a.jpg)  
+e  
+Figure 2: Emergence of different functional cell types in the trained RNN. a) Joint HD*AV tuning plots for individual units in the RNN. Units are arranged by functional type (Compass, CCW Shifters, CW Shifters). The activity of each unit is shown as a function of head direction (x-axis) and angular velocity (y-axis). For each unit, AV ranges from -550 to 550 deg/s, and HD ranges from 0 to 360 deg. b) Preferred HD of each unit within each functional type. Approximately uniform tiling of preferred HD for each functional type of model neurons is observed. c) 3D embedding of model neurons using t-SNE, with the distance between two units defined as one minus their firing rate correlation, exhibits a compass-like structure from one view angle (left) and are segregated according to AV in another view angle (right). Each dot represents one unit. d) Joint HD*AV tuning plots for two example HD neurons in the rat anterodorsal thalamic nucleus, or ADN (plotted based on data from Peyrache et al. (2015), downloaded from CRCNS website). White indicates high firing rate. e) Joint HD*AV tuning plots for two example neurons from the PB of the fly central complex, adapted from Turner-Evans et al. (2017). Red indicates high firing rate. (f,g,h,i) Detailed tuning properties of model neurons match neural data. f) HD tuning curves for model Compass units exhibit shifted peaks at high CW (green) and CCW rotations (blue). g) HD tuning curves for model Shifters exhibit peak shift and gain changes when comparing CW (green) and CCW (blue) rotations. h) HD tuning curves for CW (solid) and CCW (dashed) conditions for two example neurons in the postsubiculum of rats, adapted from Stackman & Taube (1998). i) HD tuning curves for CW (solid) and CCW (dashed) conditions for two example neurons in the lateral mammillary nuclei of rats, adapted from Stackman & Taube (1998).
+
+![](images/cd0c223c2df3bbec79178b1c3b745d3d9f6f2bce00d1b1bf3a830626f5ccfc4a.jpg)  
+g
+
+![](images/078690b21044fb73e136efa5ee8e84c56b036337938202392649f94eecd97fc6.jpg)  
+i
+
+Neurons tuned to both HD and AV tuning have also been reported previously in rodents and fruit flies (Sharp et al., 2001; Stackman & Taube, 1998; Bassett & Taube, 2001), although the joint HD*AV tuning profiles of neurons have only been documented anecdotally with a few cells (Turner-Evans et al. (2017)). In rodents, certain cells are also observed to display HD and AV tuning (Fig. 2e). In addition, in the fruit fly heading system, neurons on the two sides of the protocerebral bridge (PB) (Pfeiffer & Homberg, 2014) are also tuned to CW and CCW rotation, respectively, and tile the complete angular space, much like what has been observed in our trained network (Green et al., 2017;
+
+Turner-Evans et al., 2017). These observations collectively suggest that neurons that are HD but not AV selective in our model can be tentatively mapped to "Compass" units in the EB, and the two sub-populations of neurons tuned to both HD and AV map to "Shifter" neurons on the left PB and right PB, respectively. We will correspondingly refer to our model neurons as either 'Compass' units or 'CW/CCW Shifters' (Further justification of the terminology will be given in sections 3.2 & 3.3)
+
+# Tuning properties of model neurons match experimental data
+
+We next sought to examine the tuning properties of both Compass units and Shifters of our network in greater detail. First, we observe that for both Compass units and Shifters, the HD tuning curve varies as a function of AV (see example Compass unit in Fig. 2f and example Shifter in Fig. 2g). Population summary statistics concerning the amount of tuning shift are shown in Appendix Fig. 7a. The preferred HD tuning is biased towards a more CW angle at CW angular velocities, and vice versa for CCW angular velocities. Consistent with this observation, the HD tuning curves in rodents were also dependent upon AV (see example neurons in Fig. 2h,i) (Blair & Sharp, 1995; Stackman & Taube, 1998; Taube & Muller, 1998; Blair et al., 1997; 1998). Second, the AV tuning curves for the Shifters exhibit graded response profiles, consistent with the measured AV tuning curves in flies and rodents (see Fig. 1b,d). Across neurons, the angular velocity tuning curves show substantial diversity (see Appendix Fig. 6b), also consistent with experimental reports (Turner-Evans et al., 2017).
+
+In summary, the majority of units in the trained RNN could be mapped onto the biological head direction system in both general functional architecture and also in detailed tuning properties. Our model unifies a diverse set of experimental observations, suggesting that these neural response properties are the consequence of a network solving an angular integration task optimally.
+
+# 3.2 CONNECTIVITY STRUCTURE OF THE NETWORK
+
+![](images/03a9f8fba26334d1ef018a08fb43cc757701e47e49e49e245a2801991cc8f757.jpg)  
+a
+
+![](images/ee817c7e2aa26fb577d2eb03da4a133bee183dedeb59ab6a5fecdda2b35a0eaf.jpg)  
+b  
+Figure 3: Connectivity of the trained network is structured and exhibits similarities with the connectivity in the fly central complex. a) Pixels represent connections from the units in each column to the units in each row. Excitatory connections are in red, and inhibitory connections are in blue. Units are first sorted by functional classes, and then are further sorted by their preferred HD within each class. The black box highlights recurrent connections to the Compass units from Compass units, from CCW Shifters, and from CW Shifters. b) Ensemble connectivity from each functional cell type to the Compass units as highlighted in a), in relation to the architecture of the PB & EB in the fly central complex. Plots show the average connectivity (shaded area indicates one s.d.) as a function of the difference between the preferred HD of the cell and the Compass unit it is connecting to. Compass units connect strongly to units with similar HD tuning and inhibit units with dissimilar HD tuning. CCW Shifters connect strongly to Compass units with preferred head directions that are slightly CCW-shifted to its own, and CW Shifters connect strongly to Compass units with preferred head directions that are slightly CW-shifted to its own. Refer to Appendix Fig. 8b for the full set of ensemble connectivity between different classes.
+
+Previous experiments have detailed a subset of connections between EB and PB neurons in the fruit fly. We next analyzed the connectivity of Compass units and Shifters in the trained RNN to ask whether it recapitulates these connectivity patterns - a test which has never been done to our knowledge in any system between artificial and biological neural networks (see Fig. 3).
+
+# Compass units exhibit local excitation and long-range inhibition
+
+We ordered Compass units, CCW Shifters, and CW Shifters by their preferred head direction tuning and plotted their connection strengths (Fig. 3a). This revealed highly structured connectivity patterns within and between each class of units. We first focused on the connections between individual Compass units and observed a pattern of local excitation and global inhibition. Neurons that have similar preferred head directions are connected through positive weights and neurons whose preferred head directions are anti-phase are connected through negative weights (Fig. 3b). This pattern is consistent with the connectivity patterns inferred in recent work based on detailed calcium imaging and optogenetic perturbation experiments (Kim et al., 2017), with one caveat that the connectivity pattern inferred in this study is based on the effective connectivity rather than anatomical connectivity. We conjecture that Compass units in the trained RNN serve to maintain a stable activity bump in the absence of inputs (see section 3.3), as proposed in previous theoretical models (Turing, 1952; Amari, 1977; Zhang, 1996).
+
+# Asymmetric connectivity from Shifters to Compass units
+
+We then analyzed the connectivity between Compass units and Shifters. We found that CW Shifters excite Compass units with preferred head directions that are clockwise to its own, and inhibit Compass units with preferred head directions counterclockwise to its own (Fig. 3b). The opposite pattern is observed for CCW Shifters. Such asymmetric connections from Shifters to the Compass units are consistent with the connectivity pattern observed between the PB and the EB in the fruit fly central complex (Lin et al., 2013; Green et al., 2017; Turner-Evans et al., 2017), and also in agreement with previously proposed mechanisms of angular integration (Skaggs et al., 1995; Green et al., 2017; Turner-Evans et al., 2017; Zhang, 1996) (Fig. 3b). We note that while the connectivity between PB Shifters and EB Compass units are one-to-one (Lin et al., 2013; Wolff et al., 2015; Green et al., 2017), the connectivity profile in our model is broad, with a single CW Shifter exciting multiple Compass units with preferred HDs that are clockwise to its own, and vice versa for CCW Shifters.
+
+In summary, the RNN developed several anatomical features that are consistent with structures reported or hypothesized in previous experimental results. A few novel predictions are worth mentioning. First, in our model the connectivity between CW and CCW Shifters exhibit specific recurrent connectivity (Fig. 8). Second, the connections from Shifters to Compass units exhibit not only excitation in the direction of heading motion, but also inhibition that is lagging in the opposite direction. This inhibitory connection has not been observed in experiments yet but may facilitate the rotation of the neural bump in the Compass units during turning (Wolff et al., 2015; Franconville et al., 2018; Green et al., 2017; Green & Maimon, 2018). In the future, EM reconstructions together with functional imaging and optogenetics should allow direct tests of these predictions.
+
+# 3.3 PROBING THE COMPUTATION IN THE NETWORK
+
+We have segregated neurons into Compass and Shifter populations according to their HD and AV tuning, and have shown that they exhibit different connectivity patterns that are suggestive of different functions. Compass units putatively maintain the current heading direction and Shifter units putatively rotate activity on the compass according to the direction of angular velocity. To substantiate these functional properties, we performed a series of perturbation experiments by lesioning specific subsets of connections.
+
+# Perturbation while holding a constant head direction
+
+We first lesioned connections when there is zero angular velocity input. Normally, the network maintains a stable bump of activity within each class of neurons, i.e., Compass units, CW Shifters, and CCW Shifters (see Fig. 4a,b). We first lesioned connections from Compass units to all units and found that the activity bumps in all three classes disappeared and were replaced by diffuse activity in a large proportion of units. As a consequence, the network could not report an accurate estimate of its current heading direction. Furthermore, when the connections were restored, a bump formed
+
+![](images/23be359c630a2058b52ea1dbb008cfce43c66e0593c8d29e1708a0f311e097d5.jpg)  
+Figure 4: Probing the functional role of different classes of model neurons. a-j) Perturbation analysis in the case of maintaining a constant HD. a) Under normal conditions, the RNN output matches the target output. b) Population activity for the trial shown in a), sorted by the preferred HD and the class of each unit, i.e., Compass units, CCW Shifters, CW Shifters. c-j) RNN output and population activity when a specific set of connections are set to zero during the period indicated by blue arrows in i) and j). k-v) Perturbation analysis in the case of a shifting HD. For each manipulation, CW rotation and CCW rotation are tested, resulting in two trials. k-n) Normal case without any perturbation. o-v) RNN output and population activity when connections from CW Shifters (o-r) and CCW Shifters (s-v) are set to zero. Refer to the main text for the interpretation of the results.
+
+again without any external input (Fig. 4d), suggesting the network can spontaneously generate an activity bump through recurrent connections mediated by Compass units.
+
+We then lesioned connections from CW Shifters to all units and found that all three bumps exhibit a CCW rotation, and the read-out units correspondingly reported a CCW rotation of heading direction (Fig. 4e,f). Analogous results were obtained with lesions of CCW Shifters, which resulted in a CW drifting bump of activity (Fig. 4g,h). These results are consistent with the hypothesis that CW and CCW Shifters simultaneously activate the compass, with mutually cancelling signals, even when the heading direction is stationary. When connections are lesioned from both CW and CCW Shifters to all units, we observe that Compass units are still capable of holding a stable HD activity bump (Fig. 4i,j), consistent with the predictions that while CW/CCW Shifters are necessary for updating heading during motion, Compass units are responsible for maintaining heading.
+
+# Perturbation while integrating constant angular velocity
+
+We next lesioned connections during either constant CW or CCW angular velocity. Normally, the network can integrate AV accurately (Fig. 4k-n). As expected, during CCW rotation, we observe a corresponding rotation of the activity bump in Compass units and in CCW Shifters, but CW Shifters display low levels of activity. The converse is true during CW rotation. We first lesioned connections from CW Shifters to all units, and found that it significantly impaired rotation in the CW direction,
+
+and also increased the rotation speed in the CCW direction. Lesioning of CCW Shifters to all units had the opposite effect, significantly impairing rotation in the CCW direction. These results are consistent with the hypothesis that CW/CCW Shifters are responsible for shifting the bump in a CW and CCW direction, respectively, and are consistent with the data in Green et al. (2017), which shows that inhibition of Shifter units in the PB of the fruit fly heading system impairs the integration of HD. Our lesion experiments further support the segregation of units into modular components that function to separately maintain and update heading during angular motion.
+
+![](images/fe2212b0e85b6630bd1045e05f0b811a787a6c961bbbd7eeeb1dc4d7a3fff372.jpg)
+
+![](images/a746f69c673467bbd293768882dc9a83dd02d537f1d03ad9e11f7caf84fc9409.jpg)
+
+![](images/515c16b85c3cc6fb4b4419a80fb4c664152bfd0f11f785d56aa841242d6b9833.jpg)
+
+![](images/3f80cae80d12ce9421c42c972bbf1a76d172860361961a485c08da179c278a82.jpg)  
+Figure 5: Representations in the trained RNN vary as the input statistics change. a) The AV distribution used to train the RNN in the low angular velocity condition. b) A histogram of the slopes of the AV tuning curves for individual units in the low AV condition. c) Heatmaps of the joint AV and HD tuning for each unit in the low AV condition. Units are arranged by functional type (Compass, CCW Shifters, CW Shifters). The activity of each unit is shown as a function of head direction (x-axis) and angular velocity (y-axis). d,e,f) Same convention as a-c, but for the main condition (f is the same as Fig. 2a). g,h,i) Same convention as a-c, but for the high angular velocity condition.
+
+![](images/e4a886a15a2ca89f8b469429b5efdd7570f1d8fa0a46931a8bb7952a9adb84a8.jpg)
+
+![](images/815c160167b9e7cb0ea13ea98ad035107cb4a2475e6c7332d196c737a0a597c7.jpg)
+
+# 4 ADAPTATION OF NETWORK PROPERTIES TO INPUT STATISTICS
+
+Optimal computation requires the system to adapt to the statistical structure of the inputs (Barlow, 1961; Attneave, 1954). In order to understand how the statistical properties of the input trajectories affect how a network solves the task, we trained RNNs to integrate inputs generated from low and high AV distributions.
+
+When networks are trained with small angular velocities, we observe the presence of more units with strong head direction tuning but minimal angular velocity tuning. Conversely, when networks are trained with large AV inputs, fewer Compass units emerge and more units become Shifter-like and exhibit both HD and AV tuning (Fig. 5c,f,i). We sought to quantify the overall AV tuning under each velocity regime by computing the slope of each neuron's AV tuning curve at its preferred HD angle. We found that by increasing the magnitude of AV inputs, more neurons developed strong AV tuning (Fig. 5b,e,h). In summary, with a slowly changing head direction trajectory, it is advantageous to allocate more resources to hold a stable activity bump, and this requires more Compass units. In contrast, with quickly changing inputs, the system must rapidly update the activity bump to integrate head direction, requiring more Shifter units. This prediction may be relevant for understanding the diversity of the HD systems across different animal species, as different species exhibit different overall head turning behavior depending on the ecological demand (Stone et al., 2017; Seelig & Jayaraman, 2015; Heinze, 2017; Finkelstein et al., 2018).
+
+# 5 DISCUSSION
+
+Previous work in the sensory systems have mainly focused on obtaining an optimal representation (Barlow, 1961; Laughlin, 1981; Linsker, 1988; Olshausen & Field, 1996; Simoncelli & Olshausen, 2001; Yamins et al., 2014; Khaligh-Razavi & Kriegeskorte, 2014) with feedforward models. Several recent studies have probed the importance of recurrent connections in understanding neural computation by training RNNs to perform tasks (e.g., Mante et al. (2013); Sussillo et al. (2015); Cueva & Wei (2018)), but the relation of these trained networks to the anatomy and function of brain circuits are not mapped. Using the head direction system, we demonstrate that goal-driven optimization of recurrent neural networks can be used to understand the functional, structural and mechanistic properties of neural circuits. While we have mainly used perturbation analysis to reveal the dynamics of the trained RNN, other methods could also be applied to analyze the network. For example, in Appendix Fig. 10, using fixed point analysis (Sussillo & Barak, 2013; Maheswaranathan et al., 2019), we found evidence consistent with attractor dynamics. Due to the limited amount of experimental data available, comparisons regarding tuning properties and connectivity are largely qualitative. In the future, studies of the relevant brain areas using Neuropixel probes (Jun et al., 2017) and calcium imaging (Denk et al., 1990) will provide a more in-depth characterization of the properties of HD circuits, and will facilitate a more quantitative comparison between model and experiment.
+
+In the current work, we did not impose any additional structural constraint on the RNNs during training, asides from prohibiting self-connections. We have chosen to do so in order to see what structural properties would emerge as a consequence of optimizing the network to solve the task. It is interesting to consider how additional structural constraints affect the representation and computation in the trained RNNs. One possibility would be to have the input or output units only connect to a subset of the RNN units. Another possibility would be to freeze a subset of connections during training. Future work should systematically explore these issues.
+
+Recent work suggests it is possible to obtain tuning properties in RNNs with random connections (Sederberg & Nemenman, 2019). We found that training was necessary for the joint HD*AV tuning (see Appendix Fig. 9) to emerge. While Sederberg & Nemenman (2019) consider a simple binary classification task, our integration task is computationally more complicated. Stable HD tuning requires the system to keep track of HD by accurate integration of AV, and to stably store these values over time. This computation might be difficult for a random network to perform and, more generally, completely random networks may lead to different memory representations than the attractor geometry we observe in Fig. 10 (Cueva et al., 2019).
+
+Our approach contrasts with previous network models for the HD system, which are based on hand-crafted connectivity (Zhang, 1996; Skaggs et al., 1995; Xie et al., 2002; Green et al., 2017; Kim et al., 2017; Knierim & Zhang, 2012; Song & Wang, 2005; Kakaria & de Bivort, 2017; Stone et al., 2017). Our modeling approach optimizes for task performance through stochastic gradient descent. We found that different input statistics lead to different heading representations in an RNN, suggesting that the optimal architecture of a neural network varies depending on the task demand - an insight that would be difficult to obtain using the traditional approach of hand-crafting network solutions. Although we have focused on a simple integration task, this framework should be of general relevance to other neural systems as well, providing a new approach to understand neural computation at multiple levels.
+
+Our model may be used as a building block for AI systems to perform general navigation (Pei et al., 2019). In order to effectively navigate in complex environments, the agent would need to construct a cognitive map of the surrounding environment and update its own position during motion. A circuit that performs heading integration will likely be combined with another circuit to integrate the magnitude of motion (speed) to perform dead reckoning. Training RNNs to perform more challenging navigation tasks such as these, along with multiple sources of inputs, i.e., vestibular, visual, auditory, will be useful for building robust navigational systems and for improving our understanding of the computational mechanisms of navigation in the brain (Cueva & Wei, 2018; Banino et al., 2018).
+
+# ACKNOWLEDGMENTS
+
+Research supported by NSF NeuroNex Award DBI-1707398 and the Gatsby Charitable Foundation. We would like to thank Kenneth Kay for careful reading of an earlier version of the paper, and Rong Zhu for help preparing panel d in Figure 2.
+
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+Adrien Peyrache, Marie M Lacroix, Peter C Petersen, and György Buzsáki. Internally organized mechanisms of the head direction sense. Nature neuroscience, 18(4):569, 2015.  
+Keram Pfeiffer and Uwe Homberg. Organization and functional roles of the central complex in the insect brain. Annual review of entomology, 59:165-184, 2014.  
+Florian Raudies and Michael E Hasselmo. Modeling boundary vector cell firing given optic flow as a cue. PLoS computational biology, 8(6):e1002553, 2012.  
+Evan D Remington, Devika Narain, Eghbal A Hosseini, and Mehrdad Jazayeri. Flexible sensorimotor computations through rapid reconfiguration of cortical dynamics. Neuron, 2018.  
+Audrey J Sederberg and Ilya Nemenman. Randomly connected networks generate emergent selectivity and predict decoding properties of large populations of neurons. arXiv preprint arXiv:1909.10116, 2019.  
+Johannes D Seelig and Vivek Jayaraman. Neural dynamics for landmark orientation and angular path integration. Nature, 521(7551):186, 2015.  
+Patricia E Sharp, Hugh T Blair, and Jeiwon Cho. The anatomical and computational basis of the rat head-direction cell signal. Trends in neurosciences, 24(5):289-294, 2001.  
+Eero P Simoncelli and Bruno A Olshausen. Natural image statistics and neural representation. Annual review of neuroscience, 24(1):1193-1216, 2001.  
+William E Skaggs, James J Knierim, Hemant S Kudrimoti, and Bruce L McNaughton. A model of the neural basis of the rat's sense of direction. In Advances in neural information processing systems, pp. 173-180, 1995.  
+H Francis Song, Guangyu R Yang, and Xiao-Jing Wang. Training excitatory-inhibitory recurrent neural networks for cognitive tasks: a simple and flexible framework. PLoS computational biology, 12(2):e1004792, 2016.  
+Pengcheng Song and Xiao-Jing Wang. Angular path integration by moving "hill of activity": a spiking neuron model without recurrent excitation of the head-direction system. Journal of Neuroscience, 25(4):1002-1014, 2005.  
+Robert W Stackman and Jeffrey S Taube. Firing properties of rat lateral mammillary single units: head direction, head pitch, and angular head velocity. Journal of Neuroscience, 18(21):9020-9037, 1998.  
+Thomas Stone, Barbara Webb, Andrea Adden, Nicolai Ben Weddig, Anna Honkanen, Rachel Templin, William Wcislo, Luca Scimeca, Eric Warrant, and Stanley Heinze. An anatomically constrained model for path integration in the bee brain. Current Biology, 27(20):3069-3085, 2017.  
+David Sussillo and Omri Barak. Opening the black box: low-dimensional dynamics in high-dimensional recurrent neural networks. Neural computation, 25(3):626-649, 2013.  
+David Sussillo, Mark M Churchland, Matthew T Kaufman, and Krishna V Shenoy. A neural network that finds a naturalistic solution for the production of muscle activity. Nature neuroscience, 18(7): 1025-1033, 2015.  
+Jeffrey S Taube. Head direction cells recorded in the anterior thalamic nuclei of freely moving rats. Journal of Neuroscience, 15(1):70-86, 1995.  
+Jeffrey S Taube and Robert U Muller. Comparisons of head direction cell activity in the postsubiculum and anterior thalamus of freely moving rats. Hippocampus, 8(2):87-108, 1998.  
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+
+Jeffrey S Taube, Robert U Muller, and James B Ranck. Head-direction cells recorded from the postsubiculum in freely moving rats. ii. effects of environmental manipulations. Journal of Neuroscience, 10(2):436-447, 1990b.  
+Alan Turing. The chemical basis of morphogenesis. Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, 237(641):37-72, 1952.  
+Daniel Turner-Evans, Stephanie Wegener, Herve Rouault, Romain Franconville, Tanya Wolff, Johannes D Seelig, Shaul Druckmann, and Vivek Jayaraman. Angular velocity integration in a fly heading circuit. *Elife*, 6:e23496, 2017.  
+Jing Wang, Devika Narain, Eghbal A Hosseini, and Mehrdad Jazayeri. Flexible timing by temporal scaling of cortical responses. Nature Neuroscience, 2018.  
+Tanya Wolff, Nirmala A Iyer, and Gerald M Rubin. Neuroarchitecture and neuroanatomy of the drosophila central complex: A gal4-based dissection of protocerebral bridge neurons and circuits. Journal of Comparative Neurology, 523(7):997-1037, 2015.  
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+Daniel LK Yamins, Ha Hong, Charles F Cadieu, Ethan A Solomon, Darren Seibert, and James J DiCarlo. Performance-optimized hierarchical models predict neural responses in higher visual cortex. Proceedings of the National Academy of Sciences, 111(23):8619-8624, 2014.  
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+Kochen Zhang. Representation of spatial orientation by the intrinsic dynamics of the head-direction cell ensemble: a theory. Journal of Neuroscience, 16(6):2112-2126, 1996.
+
+# A APPENDIX
+
+![](images/b8252108f50737a3e65f66ca68f4efbefaed1f39e76775a572667ddba3d40e20.jpg)  
+a
+
+![](images/98e9977c8814bc110e1bbedfbca820cc178551902c27b2c089057ef0a1fdeb19.jpg)  
+b
+
+![](images/14c429aec54cf01fc0723402ff04bf31ef0633fb937315312f7612cddba9dc6a.jpg)  
+Figure 6: Tuning properties and unit classification. a) HD tuning curves for all 100 units in the RNN based on the Main condition. Units are arranged by functional type (Compass, CCW Shifters, CW Shifters). The activity of each unit is shown as a function of head direction. The 12 units at the bottom were not active, did not contribute to the network, and were not included in the analyses. All 88 units with some activity were included in the analyses. b) Similar to a), but for AV tuning curves. c) Classification into different populations using HD and AV tuning strength.
+
+![](images/191357c350881a82cd162ee588697a132901dfe7f9f0898f8d9b95c26546f4d7.jpg)  
+Figure 7: Population summary of tuning shift and gain change when comparing the CW and CCW rotations. a) Population summary of tuning shift. b) Population summary of change of the peak firing rate of HD tuning curves.
+
+![](images/c2d64628b9575acfbbf613769137dd39e53d68302442ac0c65be10e7428f7e09.jpg)
+
+![](images/52959f8786f3dcf4813fefbd14c370bcca1df8ec545d411a30f37e1e84ad6a26.jpg)  
+Figure 8: Connectivity of the trained network. a) Pixels represent connections from the units in each column to the units in each row. Units are first sorted by functional classes, and then are further sorted by their preferred HD within each class. Excitatory connections are in red, and inhibitory connections are in blue. b) Ensemble connectivity from each functional cell type. Plots show the average connectivity (shaded area indicates one standard deviation) as a function of the difference in preferred HD.
+
+![](images/5f246e4f9cf1527b2c4b0783bb0b474ac65284fc5edefe0c1bfa3131f139fb30.jpg)
+
+![](images/b7b16d7a32dc7ac1a3f28a90d76ef7aaf166fcb1f21a7de4c172812c36fb9d6b.jpg)  
+a
+
+![](images/9bed32785d916d84a68ff8af612188f2e1e47424a956bff54892f1d061e8df36.jpg)  
+b  
+Figure 9: Joint  $\mathrm{HD} \times \mathrm{AV}$  tuning of the initial, randomly connected network and the final trained network. a) Before training, the 100 units in the network do not have pronounced joint  $\mathrm{HD} \times \mathrm{AV}$  tuning. The color scale is different for each unit (blue = minimum activity, yellow = maximum activity) to maximally highlight any potential variation in the untrained network. b) After training, the units are tuned to  $\mathrm{HD} \times \mathrm{AV}$ , with the exception of 12 units (shown at the bottom) which are not active and do not influence the network.
+
+![](images/e02c5743691f525c0f2e1e6fbe883a6647e6c163dee8978eccc96299b7d83612.jpg)  
+a
+
+![](images/be1c07ef0bfe55d98710ae468c03191b326eb4cdbcf6fa4bc840b026ef16f915.jpg)  
+b
+
+![](images/10f016736ddd68e550ada6e46d9b2ce700bdc36059fdd36984a809999c5b89c4.jpg)  
+C
+
+![](images/ab0edd3c7db3bf6754f1ead666ad07f51bbaacca874dc5576c23ac20caeaaae8.jpg)  
+d  
+Figure 10: Attractor structure of the trained network. To store angular information the neural activity settles into a compass-like geometry, with the position along the compass determined by the angle. a) The RNN is able to store angular information in the presence of noise. In this example sequence an initial input to the RNN specifies the initial heading direction as 270 degrees. The angular velocity input to the RNN is zero and so the network must continue to store this value of 270 degrees throughout the sequence. b) The activity of all units in the RNN are shown for 180 sequences, similar to a), after projecting onto the two principal components capturing most of the variance (91%). Each line shows the neural trajectory over time for a single sequence. For example, the neural activity that produces the output shown in a) is highlighted with black dots (1 dot for each timestep). The activity of the RNN is initialized to be the same for all sequences, i.e.  $x(0)$  in equation 1 is the same, so the activity for all sequences starts at the same location, namely, the origin of the figure. The activity quickly settles into a compass-like geometry, with the location around the compass determined by the heading direction. (c,d) When the RNN is integrating a nonzero angular velocity it appears to transition between the attractors shown in b), i.e. the attractors found when the network was only storing a constant angle. Panel c) shows an example sequence with a nonzero angular velocity. In d) the neural trajectory from c) is superimposed over the attractor geometry from b).
\ No newline at end of file
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+# EMERGENT TOOL USE FROM MULTI-AGENT AUTOCURRICULA
+
+Bowen Baker*
+
+OpenAI
+
+bowen@openai.com
+
+Ingmar Kanitscheider*
+
+OpenAI
+
+ingmar@openai.com
+
+Todor Markov*
+
+OpenAI
+
+todor@openai.com
+
+Yi Wu*
+
+OpenAI
+
+jxwuyi@openai.com
+
+Glenn Powell*
+
+OpenAI
+
+glenn@openai.com
+
+Bob McGrew*
+
+OpenAI
+
+bmcgrew@openai.com
+
+Igor Mordatch*†
+
+Google Brain
+
+imordatch@google.com
+
+# ABSTRACT
+
+Through multi-agent competition, the simple objective of hide-and-seek, and standard reinforcement learning algorithms at scale, we find that agents create a self-supervised autocurriculum inducing multiple distinct rounds of emergent strategy, many of which require sophisticated tool use and coordination. We find clear evidence of six emergent phases in agent strategy in our environment, each of which creates a new pressure for the opposing team to adapt; for instance, agents learn to build multi-object shelters using moveable boxes which in turn leads to agents discovering that they can overcome obstacles using ramps. We further provide evidence that multi-agent competition may scale better with increasing environment complexity and leads to behavior that centers around far more human-relevant skills than other self-supervised reinforcement learning methods such as intrinsic motivation. Finally, we propose transfer and fine-tuning as a way to quantitatively evaluate targeted capabilities, and we compare hide-and-seek agents to both intrinsic motivation and random initialization baselines in a suite of domain-specific intelligence tests.
+
+# 1 INTRODUCTION
+
+Creating intelligent artificial agents that can solve a wide variety of complex human-relevant tasks has been a long-standing challenge in the artificial intelligence community. Of particular relevance to humans will be agents that can sense and interact with objects in a physical world. One approach to creating these agents is to explicitly specify desired tasks and train a reinforcement learning (RL) agent to solve them. On this front, there has been much recent progress in solving physically grounded tasks, e.g. dexterous in-hand manipulation (Rajeswaran et al., 2017; Andrychowicz et al., 2018) or locomotion of complex bodies (Schulman et al., 2015; Heess et al., 2017). However, specifying reward functions or collecting demonstrations in order to supervise these tasks can be
+
+![](images/7588ec2b0be5e04bf89224981486cdc66251c419b03f2958321bc8e964c582a6.jpg)  
+Figure 1: Emergent Skill Progression From Multi-Agent Autocurricula. Through the reward signal of hide-and-seek (shown on the y-axis), agents go through 6 distinct stages of emergence. (a) Seekers (red) learn to chase hiders, and hiders learn to crudely run away. (b) Hiders (blue) learn basic tool use, using boxes and sometimes existing walls to construct forts. (c) Seekers learn to use ramps to jump into the hiders' shelter. (d) Hiders quickly learn to move ramps to the edge of the play area, far from where they will build their fort, and lock them in place. (e) Seekers learn that they can jump from locked ramps to unlocked boxes and then surf the box to the hiders' shelter, which is possible because the environment allows agents to move together with the box regardless of whether they are on the ground or not. (f) Hiders learn to lock all the unused boxes before constructing their fort. We plot the mean over 3 independent training runs with each individual seed shown with a dotted line. Please see openai.com/blog/emergent-tool-use for example videos.
+
+time consuming and costly. Furthermore, the learned skills in these single-agent RL settings are inherently bounded by the task description; once the agent has learned to solve the task, there is little room to improve.
+
+Due to the high likelihood that direct supervision will not scale to unboundedly complex tasks, many have worked on unsupervised exploration and skill acquisition methods such as intrinsic motivation. However, current undirected exploration methods scale poorly with environment complexity and are drastically different from the way organisms evolve on Earth. The vast amount of complexity and diversity on Earth evolved due to co-evolution and competition between organisms, directed by natural selection (Dawkins & Krebs, 1979). When a new successful strategy or mutation emerges, it changes the implicit task distribution neighboring agents need to solve and creates a new pressure
+
+for adaptation. These evolutionary arms races create implicit autocurricula (Leibo et al., 2019a) whereby competing agents continually create new tasks for each other. There has been much success in leveraging multi-agent autocurricula to solve multi-player games, both in classic discrete games such as Backgammon (Tesauro, 1995) and Go (Silver et al., 2017), as well as in continuous real-time domains such as Dota (OpenAI, 2018) and Starcraft (Vinyals et al., 2019). Despite the impressive emergent complexity in these environments, the learned behavior is quite abstract and disembodied from the physical world. Our work sees itself in the tradition of previous studies that showcase emergent complexity in simple physically grounded environments (Sims, 1994a; Bansal et al., 2018; Jaderberg et al., 2019; Liu et al., 2019); the success in these settings inspires confidence that inducing autocurricula in physically grounded and open-ended environments could eventually enable agents to acquire an unbounded number of human-relevant skills.
+
+We introduce a new mixed competitive and cooperative physics-based environment in which agents compete in a simple game of hide-and-seek. Through only a visibility-based reward function and competition, agents learn many emergent skills and strategies including collaborative tool use, where agents intentionally change their environment to suit their needs. For example, hiders learn to create shelter from the seekers by barricading doors or constructing multi-object forts, and as a counter strategy seekers learn to use ramps to jump into hiders' shelter. Moreover, we observe signs of dynamic and growing complexity resulting from multi-agent competition and standard reinforcement learning algorithms; we find that agents go through as many as six distinct adaptations of strategy and counter-strategy, which are depicted in Figure 1. We further present evidence that multi-agent co-adaptation may scale better with environment complexity and qualitatively centers around more human-interpretable behavior than intrinsically motivated agents.
+
+However, as environments increase in scale and multi-agent autocurriculum become more open-ended, evaluating progress by qualitative observation will become intractable. We therefore propose a suite of targeted intelligence tests to measure capabilities in our environment that we believe our agents may eventually learn, e.g. object permanence (Baillargeon & Carey, 2012), navigation, and construction. We find that for a number of the tests, agents pretrained in hide-and-seek learn faster or achieve higher final performance than agents trained from scratch or pretrained with intrinsic motivation; however, we find that the performance differences are not drastic, indicating that much of the skill and feature representations learned in hide-and-seek are entangled and hard to fine-tune.
+
+The main contributions of this work are: 1) clear evidence that multi-agent self-play can lead to emergent autocurricula with many distinct and compounding phase shifts in agent strategy, 2) evidence that when induced in a physically grounded environment, multi-agent autocurricula can lead to human-relevant skills such as tool use, 3) a proposal to use transfer as a framework for evaluating agents in open-ended environments as well as a suite of targeted intelligence tests for our domain, and 4) open-sourced environments and code<sup>1</sup> for environment construction to encourage further research in physically grounded multi-agent autocurricula.
+
+# 2 RELATED WORK
+
+There is a long history of using self-play in multi-agent settings. Early work explored self-play using genetic algorithms (Paredis, 1995; Pollack et al., 1997; Rosin & Belew, 1995; Stanley & Miikkulainen, 2004). Sims (1994a) and Sims (1994b) studied the emergent complexity in morphology and behavior of creatures that coevolved in a simulated 3D world. Open-ended evolution was further explored in the environments Polyworld (Yaeger, 1994) and Geb (Channon et al., 1998), where agents compete and mate in a 2D world, and in Tierra (Ray, 1992) and Avida (Ofria & Wilke, 2004), where computer programs compete for computational resources. More recent work attempted to formulate necessary preconditions for open-ended evolution (Taylor, 2015; Soros & Stanley, 2014). Co-adaptation between agents and environments can also give rise to emergent complexity (Florensa et al., 2017; Sukhbaatar et al., 2018; Wang et al., 2019). In the context of multi-agent RL, Tesauro (1995), Silver et al. (2016), OpenAI (2018), Jaderberg et al. (2019) and Vinyals et al. (2019) used self-play with deep RL techniques to achieve super-human performance in Backgammon, Go, Dota, Capture-the-Flag and Starcraft, respectively. Bansal et al. (2018) trained agents in a simulated 3D physics environment to compete in various games such as sumo wrestling and soccer goal shooting. In Liu et al. (2019), agents learn to manipulate a soccer ball in a 3D soccer environment and discover
+
+emergent behaviors such as ball passing and interception. In addition, communication has also been shown to emerge from multi-agent RL (Sukhbaatar et al., 2016; Foerster et al., 2016; Lowe et al., 2017; Mordatch & Abbeel, 2018).
+
+Intrinsic motivation methods have been widely studied in the literature (Chentanez et al., 2005; Singh et al., 2010). One example is count-based exploration, where agents are incentivized to reach infrequently visited states by maintaining state visitation counts (Strehl & Littman, 2008; Bellemare et al., 2016; Tang et al., 2017) or density estimators (Ostrovski et al., 2017; Burda et al., 2019b). Another paradigm is transition-based methods, in which agents are rewarded for high prediction error in a learned forward or inverse dynamics model (Schmidhuber, 1991; Stadie et al., 2015; Mohamed & Rezende, 2015; Houthooft et al., 2016; Achiam & Sastry, 2017; Pathak et al., 2017; Burda et al., 2019a; Haber et al., 2018). Jaques et al. (2019) consider multi-agent scenarios and adopt causal influence as a motivation for coordination. In our work, we utilize intrinsic motivation methods as an alternative exploration baseline to multi-agent autocurricula. Similar comparisons have also been made in Haber et al. (2018) and Leibo et al. (2019b).
+
+Tool use is a hallmark of human and animal intelligence (Hunt, 1996; Shumaker et al., 2011); however, learning tool use in RL settings can be a hard exploration problem when rewards are unaligned. For example, in Forestier et al. (2017); Xie et al. (2019) a real-world robot learns to solve various tasks requiring tools. In Bapat et al. (2019), an agent solves construction tasks in a 2-D environment using both model-based and model-free methods. Allen et al. (2019) uses a combination of human-designed priors and model-based policy optimization to solve a collection of physics-based puzzles requiring tool use. However, in each of these works, agents were explicitly incentivized to interact with and use tools, whereas in our environment agents implicitly create this incentive through multi-agent competition.
+
+# 3 HIDE AND SEEK
+
+Agents are tasked with competing in a two-team hide-and-seek game in a physics-based environment. The hiders are tasked with avoiding line of sight from the seekers, and the seekers are tasked with keeping vision of the hiders. There are objects scattered throughout the environment that the agents can grab and also lock in place. There are also randomly generated immovable rooms and walls that the agents must learn to navigate. Before the game of hide-and-seek begins, the hiders are given a preparation phase where the seekers are immobilized, giving the hiders a chance to run away or change their environment.
+
+There are no explicit incentives for agents to interact with objects in the environment; the only supervision given is through the hide-and-seek objective. Agents are given a team based reward; hiders are given a reward of 1 if all hiders are hidden and -1 if any hider is seen by a seeker. Seekers are given the opposite reward, -1 if all hiders are hidden and +1 otherwise. To confine agent behavior to a reasonable space, agents are penalized with a reward of -10 if they go too far outside of the play area (outside an 18 meter square). An episode lasts 240 timesteps, the first  $40\%$  of which are the preparation phase where all agents are given zero reward.
+
+We simulate the environment in the MUJOCO physics engine (Todorov et al., 2012). The world is populated with 1 to 3 hiders, 1 to 3 seekers, 3 to 9 movable boxes of which at least 3 are elongated, 2 movable ramps, and randomly generated static walls and rooms. We also experiment with a simpler, less randomized environment described in Appendix A.3. Agents observe the position, velocity, and size (in the case of the randomly shaped boxes) of objects and other agents. If entities are not in line-of-sight of the agent or not in a 135 degree cone in front of the agent, then they are masked out in the policy. Agents also have 30 range sensors arrayed evenly around them, similar to a lidar. In addition, each agent observes its own team and other agents' teams as well as how much time is left in the preparation phase.
+
+Agents are simulated as spherical objects and have 3 action types that can be chosen simultaneously at each time step. They may move by setting a discretized force along their  $x$  and  $y$  axis and torque around their  $z$ -axis. They have a single binary action to grab objects, which binds the agent to the closest object while the action is enabled. Agents may also lock objects in place with a single binary action. Objects may be unlocked only by agents on the team of the agent who originally locked the object. Agents may only grab or lock objects that are in front of them and within a small radius.
+
+# 4 POLICY OPTIMIZATION
+
+Agents are trained using self-play, which acts as a natural curriculum as agents always play opponents of an appropriate level.
+
+Agent policies are composed of two separate networks with different parameters - a policy network which produces an action distribution and a critic network which predicts the discounted future returns. Policies are optimized using Proximal Policy Optimization (PPO) (Schulman et al., 2017) and Generalized Advantage Estimation (GAE) (Schulman et al., 2015), and training is performed using rapid (OpenAI, 2018), a large-scale distributed RL framework. We utilize decentralized execution and centralized training. At execution time, each agent acts given only its own observations and memory state. At optimization time, we use a centralized omniscient value function for each agent, which has access to the full environment state without any information masked due to visibility, similar to Pinto et al. (2017); Lowe et al. (2017); Foerster et al. (2018).
+
+In all reported experiments, agents share the same policy parameters but act and observe independently; however, we found using separate policy parameters per agent also achieved all six stages of emergence but at reduced sample efficiency.
+
+![](images/d730b1f6c57e14fe8cd13896cb8b1e4dd11a76f0bb18e88542f17f77eedf2eea.jpg)  
+Figure 2: Agent Policy Architecture. All entities are embedded with fully connected layers with shared weights across entity types, e.g. all box entities are encoded with the same function. The policy is ego-centric so there is only one embedding of "self" and (#agents - 1) embeddings of other agents. Embeddings are then concatenated and processed with masked residual self-attention and pooled into a fixed sized vector (all of which admits a variable number of entities).  $x$  and  $v$  stand for state (position and orientation) and velocity.
+
+We utilize entity-centric observations (Džeroski et al., 2001; Diuk et al., 2008) and use attention mechanisms to capture object-level information (Duan et al., 2017; Zambaldi et al., 2018). As shown in Figure 2 we use a self-attention (Vaswani et al., 2017) based policy architecture over entities, which is permutation invariant and generalizes to varying number of entities. More details can be found in Appendix B.
+
+![](images/85b247a99f3835a49717aaa337f44ce24ba43395020dbe3bc3164183d92683a4.jpg)  
+Figure 3: Environment specific statistics used to track stages of emergence in hide-and-seek. We plot the mean across 3 seeds with each individual seed shown in a dotted line, and we overlay the 6 emergent phases of strategy: (1) Running and Chasing, (2) Fort Building, (3) Ramp Use, (4) Ramp Defense, (5) Box Surfing, (6) Surf Defense. We track the maximum movement of any box or ramp during the game as well as during the preparation phase (denoted with "Prep"). We similarly track how many objects of each type were locked at the end of the episode and preparation phase. As agents train, their interaction with the tools in their environment changes. For instance, as the agents learn to build forts they move boxes and lock boxes much more during the preparation phase.
+
+![](images/f572d9e039de67cde24e10fcf0766d5123347828b7f3b55048dcb4fdbe999d21.jpg)
+
+# 5 AUTO-CURRICULA AND EMERGENT BEHAVIOR
+
+As agents train against each other in hide-and-seek, as many as six distinct strategies emerge, each of which creates a previously non-existing pressure for agents to progress to the next stage. Note that there are no direct incentives for agents to interact with objects or to explore, but rather the emergent strategies are solely a result of the autocurriculum induced by multi-agent competition. Figure 1 shows the progression of emergent strategies agents learn in our environment (see Appendix A.1 for trajectory traces of each strategy $^2$ ).
+
+Initially, hiders and seekers learn to crudely run away and chase. After approximately 25 million episodes of hide-and-seek, the hiders learn to use the tools at their disposal and intentionally modify their environment. They begin to construct secure shelters in which to hide by moving many boxes together or against walls and locking them in place. After another 75 million episodes, the seekers also learn rudimentary tool use; they learn to move and use ramps to jump over obstacles, allowing them to enter the hiders' shelter. 10 million episodes later, the hiders learn to defend against this strategy; the hiders learn to bring the ramps to the edge of the play area and lock them in place, seemingly removing the only tool the seekers have at their disposal.
+
+We originally believed defending against ramp use would be the last stage of emergence in this environment; however, we were surprised to find that yet two more qualitatively new strategies emerged. After 380 million total episodes of training, the seekers learn to bring a box to the edge of the play area where the hiders have locked the ramps. The seekers then use the ramp to move on top of the box and surf it to the hiders' shelter. This is possible because the agents' movement action allows them to apply a force on themselves regardless of whether they are on the ground or not; if they do this while grabbing the box under them, the box will move with while they are on top of it. In response, the hiders learn to lock all of the boxes in place before building their shelter.<sup>3</sup>
+
+In all stages of strategy agents must learn to coordinate within their team. Similar to Liu et al. (2019), we use team-based rewards such that agents are required to collaborate in order to succeed; however, in our work we require neither population-based training (Jaderberg et al., 2017) or evolved dense rewards (Jaderberg et al., 2019). Notably, hiders learn efficient division of labor; for instance, when constructing shelter they often separately bring their own box to the construction area. Furthermore,
+
+as hiders attempt to defend against box surfing by locking boxes in place during the preparation phase, we find that 2 and 3 hinder teams are able to lock  $25\%$  and  $36\%$  more boxes, respectively, than a single hinder team, indicating that larger teams are able to divide labor and accomplish an objective a smaller team could not.
+
+Many stages of emergent strategy can be mapped to behavioral shifts in the way agents interact with the tools in their environment, similar to Leibo et al. (2017); Perolat et al. (2017). We therefore track basic statistics about the agents' interaction with objects during training, shown in Figure 3. For instance, as the hiders learn to build forts, they move and lock boxes much more during the preparation phase. Similarly, as the seekers learn to move and use ramps, the ramp movement in the main phase of the game increases, and as they learn to "box surf" there is a slight increase in the box movement during the main phase of the game. Finally, as the hiders learn to defend against this strategy by locking all boxes in place, the number of locked boxes in the preparation phase increases.
+
+We found that scale plays a critical role in enabling progression through the emergent autocurricula in hide-and-seek. The default model, which uses a batch size of 64,000 and 1.6 million parameters, requires 132.3 million episodes (31.7 billion frames) over 34 hours of training to reach stage 4 of the skill progression, i.e. ramp defense. In Figure 4 we show the effect of varying the batch size in our agents ability to reach stage 4. We find that larger batch sizes lead to much quicker training time by virtue of reducing the number of required optimization steps, while only marginally affecting sample efficiency down to a batch size of 32,000; however, we found that experiments with batch sizes of 16,000 and 8,000 never converged.
+
+We find the emergent autocurriculum to be fairly robust as long as we randomize the environment during training. If randomization is
+
+![](images/4f17e3fa9f9d894d7c8b9edd3fe7c3c88e99dabecc107a331b0301d033300ba9.jpg)  
+Figure 4: Effect of Scale on Emergent Autocurriculum. Number of episodes (blue) and wall clock time (orange) required to achieve stage 4 (ramp defense) of the emergent skill progression presented in Figure 1. Batch size denotes number of chunks, each of which consists of 10 contiguous transitions (the truncation length for backpropagation through time).
+
+reduced, we find that fewer stages of the skill progression emerges, and at times less sophisticated strategies emerge instead (e.g. hiders can learn to run away and use boxes as moveable shields.); see Appendix A.2 for more details. In addition, we find that design choices such as the minimum number of elongated boxes or giving each agent their own locking mechanism instead of a team based locking mechanism can drastically increase the sample complexity. We also experimented with adding additional objects and objectives to our hide-and-seek environment as well as with several game variants instead of hide-and-seek (see Appendix A.6). We find that these alternative environments also lead to emergent tool use, providing further evidence that multi-agent interaction is a promising path towards self-supervised skill acquisition.
+
+# 6 EVALUATION
+
+In the previous section we presented evidence that hide-and-seek induces a multi-agent autocurriculum such that agents continuously learn new skills and strategies. As is the case with many unsupervised reinforcement learning methods, the objective being optimized does not directly incentivize the learned behavior, making evaluation of those behaviors nontrivial. Tracking reward is an insufficient evaluation metric in multi-agent settings, as it can be ambiguous in indicating whether agents are improving evenly or have stagnated. Metrics like ELO (Elo, 1978) or Trueskill (Herbrich et al., 2007) can more reliably measure whether performance is improving relative to previous policy versions or other policies in a population; however, these metrics still do not give insight into whether improved performance stems from new adaptations or improving previously learned skills. Finally, using environment specific statistics such as object movement (see Figure 3) can also be ambiguous, e.g. the choice to track absolute movement does not illuminate which direction agents moved, and designing sufficient metrics will become difficult and costly as environments scale.
+
+![](images/f826ec4624767270fd770ef6af75e9373c3e5e8da5908dee8fe9a288552dcf92.jpg)  
+Figure 5: Behavioral Statistics from Count-Based Exploration Variants and Random Network Distillation (RND) Across 3 Seeds. We compare net box movement and maximum agent movement between state representations for count-based exploration: Single agent, 2-D box location (blue); Single agent, box location, rotation and velocity (green); 1-3 agents, full observation space (red). Also shown is RND for 1-3 agents with full observation space (purple). We train all agents to convergence as measured by their behavioral statistics.
+
+In Section 6.1, we first qualitatively compare the behaviors learned in hide-and-seek to those learned from intrinsic motivation, a common paradigm for unsupervised exploration and skill acquisition. In Section 6.2, we then propose a suite of domain-specific intelligence tests to quantitatively measure and compare agent capabilities.
+
+# 6.1 COMPARISON TO INTRINSIC MOTIVATION
+
+Intrinsic motivation has become a popular paradigm for incentivizing unsupervised exploration and skill discovery, and there has been recent success in using intrinsic motivation to make progress in sparsely rewarded settings (Bellemare et al., 2016; Burda et al., 2019b). Because intrinsically motivated agents are incentivized to explore uniformly, it is conceivable that they may not have meaningful interactions with the environment (as with the "noisy-TV" problem (Burda et al., 2019a)). As a proxy for comparing meaningful interaction in the environment, we measure agent and object movement over the course of an episode.
+
+We first compare behaviors learned in hide-and-seek to a count-based exploration baseline (Strehl & Littman, 2008) with an object invariant state representation, which is computed in a similar way as in the policy architecture in Figure 2. Count-based objectives are the simplest form of state density based incentives, where one explicitly keeps track of state visitation counts and rewards agents for reaching infrequently visited states (details can be found in Appendix D). In contrast to the original hide-and-seek environment where the initial locations of agents and objects are randomized, we restrict the initial locations to a quarter of the game area to ensure that the intrinsically motivated agents receive additional rewards for exploring.
+
+We find that count-based exploration leads to the largest agent and box movement if the state representation only contains the 2-D location of boxes: the agent consistently interacts with objects and learns to navigate. Yet, when using progressively higher-dimensional state representations, such as box location, rotation and velocity or 1-3 agents with full observation space, agent movement and, in particular, box movement decrease substantially. This is a severe limitation because it indicates that, when faced with highly complex environments, count-based exploration techniques require identifying by hand the "interesting" dimensions in state space that are relevant for the behaviors one would like the agents to discover. Conversely, multi-agent self-play does not need this degree of supervision. We also train agents with random network distillation (RND) (Burda et al., 2019b), an intrinsic motivation method designed for high dimensional observation spaces, and find it to perform slightly better than count-based exploration in the full state setting.
+
+# 6.2 TRANSFER AND FINE-TUNING AS EVALUATION
+
+We propose to use transfer to a suite of domain-specific tasks in order to assess agent capabilities. To this end, we have created 5 benchmark intelligence tests that include both supervised and reinforce-
+
+![](images/e8e4baca1fc2f10ae7cde2b05eed72afd3d2309001e52470f48f72329e5065b0.jpg)  
+Figure 6: Fine-tuning Results. We plot the mean normalized performance and  $90\%$  confidence interval across 3 seeds smoothed with an exponential moving average, except for Blueprint Construction where we plot over 6 seeds due to higher training variance.
+
+ment learning tasks. The tests use the same action space, observation space, and types of objects as in the hide-and-seek environment. We examine whether pretraining agents in our multi-agent environment and then fine-tuning them on the evaluation suite leads to faster convergence or improved overall performance compared to training from scratch or pretraining with count-based intrinsic motivation. We find that on 3 out of 5 tasks, agents pretrained in the hide-and-seek environment learn faster and achieve a higher final reward than both baselines.
+
+We categorize the 5 intelligence tests into 2 domains: cognition and memory tasks, and manipulation tasks. We briefly describe the tasks here; for the full task descriptions, see Appendix C. For all tasks, we reinitialize the parameters of the final dense layer and layernorm for both the policy and value networks.
+
+# Cognition and memory tasks:
+
+In the Object Counting supervised task, we aim to measure whether the agents have a sense of object permanence; the agent is pinned to a location and watches as 6 boxes each randomly move to the right or left where they eventually become obscured by a wall. It is then asked to predict how many boxes have gone to each side for many timesteps after all boxes have disappeared. The agent's policy parameters are frozen and we initialize a classification head off of the LSTM hidden state. In the baseline, the policy network has frozen random parameters and only the classification head off of the LSTM hidden state is trained.
+
+In Lock and Return we aim to measure whether the agent can remember its original position while performing a new task. The agent must navigate an environment with 6 random rooms and 1 box, lock the box, and return to its starting position.
+
+In Sequential Lock there are 4 boxes randomly placed in 3 random rooms without doors but with a ramp in each room. The agent needs to lock all the boxes in a particular order — a box is only lockable when it is locked in the correct order — which is unobserved by the agent. The agent must discover the order, remember the position and status of visited boxes, and use ramps to navigate between rooms in order to finish the task efficiently.
+
+Manipulation tasks: With these tasks we aim to measure whether the agents have any latent skill or representation useful for manipulating objects.
+
+In the Construction From Blueprint task, there are 8 cubic boxes in an open room and between 1 and 4 target sites. The agent is tasked with placing a box on each target site.
+
+In the Shelter Construction task there are 3 elongated boxes, 5 cubic boxes, and one static cylinder. The agent is tasked with building a shelter around the cylinder.
+
+Results: In Figure 6 we show the performance on the suite of tasks for the hide-and-seek, count-based, and trained from scratch policies across 3 seeds. The hide-and-seek pretrained policy performs slightly better than both the count-based and the randomly initialized baselines in Lock and Return, Sequential Lock and Construction from Blueprint; however, it performs slightly worse than the count-based baseline on Object Counting, and it achieves the same final reward but learns slightly slower than the randomly initialized baseline on Shelter Construction.
+
+We believe the cause for the mixed transfer results is rooted in agents learning skill representations that are entangled and difficult to fine-tune. We conjecture that tasks where hide-and-seek pretraining outperforms the baseline are due to reuse of learned feature representations, whereas better-than-baseline transfer on the remaining tasks would require reuse of learned skills, which is much more difficult. This evaluation metric highlights the need for developing techniques to reuse skills effectively from a policy trained in one environment to another. In addition, as future environments become more diverse and agents must use skills in more contexts, we may see more generalizable skill representations and more significant signal in this evaluation approach.
+
+In Appendix A.5 we further evaluate policies sampled during each phase of emergent strategy on the suite of targeted intelligence tasks, by which we can gain intuition as to whether the capabilities we measure improve with training, are transient and accentuated during specific phases, or generally uncorrelated to progressing through the autocurriculum. Notably, we find the agent's memory improves through training as indicated by performance in the navigation tasks; however, performance in the manipulation tasks is uncorrelated, and performance in object counting changes seems transient with respect to source hide-and-seek performance.
+
+# 7 DISCUSSION AND FUTURE WORK
+
+We have demonstrated that simple game rules, multi-agent competition, and standard reinforcement learning algorithms at scale can induce agents to learn complex strategies and skills. We observed emergence of as many as six distinct rounds of strategy and counter-strategy, suggesting that multiagent self-play with simple game rules in sufficiently complex environments could lead to open-ended growth in complexity. We then proposed to use transfer as a method to evaluate learning progress in open-ended environments and introduced a suite of targeted intelligence tests with which to compare agents in our domain.
+
+Our results with hide-and-seek should be viewed as a proof of concept showing that multi-agent autocurricula can lead to physically grounded and human-relevant behavior. We acknowledge that the strategy space in this environment is inherently bounded and likely will not surpass the six modes presented as is; however, because it is built in a high-fidelity physics simulator it is physically grounded and very extensible. In order to support further research in multi-agent autocurricula, we are open-sourcing our environment code.
+
+Hide-and-seek agents require an enormous amount of experience to progress through the six stages of emergence, likely because the reward functions are not directly aligned with the resulting behavior. While we have found that standard reinforcement learning algorithms are sufficient, reducing sample complexity in these systems will be an important line of future research. Better policy learning algorithms or policy architectures are orthogonal to our work and could be used to improve sample efficiency and performance on transfer evaluation metrics.
+
+We also found that agents were very skilled at exploiting small inaccuracies in the design of the environment, such as seekersurfing on boxes without touching the ground, hiders running away from the environment while shielding themselves with boxes, or agents exploiting inaccuracies of the physics simulations to their advantage. Investigating methods to generate environments without these unwanted behaviors is another import direction of future research (Amodei et al., 2016; Lehman et al., 2018).
+
+# ACKNOWLEDGMENTS
+
+We thank Pieter Abbeel, Rewon Child, Jeff Clune, Harri Edwards, Jessica Hamrick, Joel Liebo, John Schulman and Peter Welinder for their insightful comments on this manuscript. We also thank Alex Ray for writing parts of our open sourced code.
+
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+Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in neural information processing systems, pp. 5998-6008, 2017.  
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+
+# Appendix
+
+# Table of Contents
+
+# A Further Emergence Results 17
+
+A.1 Trajectory Traces From Each Stage of Emergent Strategy 17  
+A.2 Dependence of Skill Emergence on Randomness in the Training Distribution of Environments 17  
+A.3 Quadrant Environment 18  
+A.4 Further Ablations 19  
+A.5 Evaluating Agents at Different Phases of Emergence 19  
+A.6 Alternative Games to Hide-and-Seek with Secondary Objectives 20  
+A.7 Zero-shot generalization 22
+
+# B Optimization Details 23
+
+B.1 Notation 23  
+B.2 Proximal Policy Optimization (PPO) 23  
+B.3 Generalized advantage estimation 23  
+B.4 Normalization of observations, advantage targets and value function targets 24  
+B.5 Optimization setup 24  
+B.6 Optimization hyperparameters 24  
+B.7 Policy architecture details 24
+
+# C Intelligence Test Suite Details 25
+
+C.1 Cognition and memory task 25  
+C.2 Manipulation task 25
+
+# D Intrinsic Motivation Methods 26
+
+D.1 Counted-Based Exploration 26  
+D.2 Random Network Distillation 27  
+D.3 Fine-tuning From Intrinsic Motivation Variants 27
+
+# A FURTHER EMERGENCE RESULTS
+
+# A.1 TRAJECTORY TRACES FROM EACH STAGE OF EMERGENT STRATEGY
+
+![](images/54f401175cdfab4e371ba5a6abe01bcd212a78fc853e75789f9ca833fe190e80.jpg)  
+Figure A.1: Trajectory Traces From Each Stage of Emergent Strategy. Rows correspond to different strategies, columns to characteristic snapshots in chronological order within an episode that showcase the strategy. Running and chasing: Hiders (blue) try to hide behind walls and moveable objects, and seekers (red) chase and search for hiders. Fort building: During the preparation phase (first 3 snapshots), hiders build a fort out of 3 boxes and hide inside. Seekers are not able to overcome this barrier during the remainder of the trial. Ramp use: During the preparation phase, hiders build a fort out of 2 boxes and a wall (1st and 2nd snapshot). After the preparation phase, a seeker grabs a ramp (3rd snapshot), drags it towards the fort (4th snapshot) and uses it to enter the fort (5th snapshot). Ramp defense: Hiders prevent ramp use by dragging ramps away and locking them in place during the preparation phase (1st, 2nd and 3rd snapshot). They then build a fort in safe distance from the ramps (4th snapshot). After the preparation phase, seekers are not able to enter the fort (5th snapshot). Box surfing: A single hider locks both ramps in place and hides inside a fort built from 2 boxes and an interior wall during the preparation phase (1st and 2nd snapshot). The seeker grabs a square box (3rd snapshot) and drags it next to one of the locked ramps to jump on top (4th snapshot). It then "surfs" on the box to the edge of the fort and jumps inside (5th snapshot). Surf defense: During the preparation phase, hiders lock all ramps and boxes in place (1st, 2nd and 3rd snapshot) and construct a fort out of 3 boxes (4th snapshot). After the preparation phase, seekers are not able to enter the fort (5th snapshot).
+
+# A.2 DEPENDENCE OF SKILL EMERGENCE ON RANDOMNESS IN THE TRAINING DISTRIBUTION OF ENVIRONMENTS
+
+Discovering tool use is a difficult exploration problem, as only a very specific set of actions will lead to an increase in reward. During training, we find that randomization over many components of our environment, such as the number of agents in each team, the number of boxes, the shape of boxes (square or elongated), the initial location of agents and objects and the presence and location of interior and exterior walls helps emergence, despite leading to a broader training distribution. As we successively reduce the amount of randomization, we find that fewer stages of skill progression emerges, and with at times less sophisticated strategies (e.g. hiders learn to run away and use boxes as moveable shields), Table A.1.
+
+<table><tr><td>team size</td><td># boxes</td><td>box shape</td><td>initial location</td><td>walls</td><td>emergence</td></tr><tr><td>1-3</td><td>3-9</td><td>random</td><td>random</td><td>random</td><td>6 stages</td></tr><tr><td>1</td><td>3-9</td><td>random</td><td>random</td><td>random</td><td>6 stages</td></tr><tr><td>1</td><td>7</td><td>fixed</td><td>random</td><td>random</td><td>6 stages</td></tr><tr><td>1-3</td><td>3-9</td><td>random</td><td>random</td><td>no walls</td><td>4 stages</td></tr><tr><td>1</td><td>3-9</td><td>random</td><td>random</td><td>no walls</td><td>2 stages*</td></tr><tr><td>1</td><td>7</td><td>fixed</td><td>random</td><td>no walls</td><td>2 stages*</td></tr><tr><td>1</td><td>7</td><td>fixed</td><td>fixed</td><td>no walls</td><td>2 stages</td></tr></table>
+
+* Hiders run away and use boxes as movable shields.
+
+Table A.1: Number of stages of emergence for different levels of environmental randomization (batch size is held constant).  
+
+<table><tr><td>Running and chasing</td><td></td><td></td><td></td><td></td></tr><tr><td>Fort building</td><td></td><td></td><td></td><td></td></tr><tr><td>Ramp use</td><td></td><td></td><td></td><td></td></tr><tr><td>Ramp defense</td><td></td><td></td><td></td><td></td></tr></table>
+
+Figure A.2: Sample Trajectory Traces From Each Stage of Emergent Strategy in the Quadrant Environment. Rows correspond to different strategies, columns to characteristic snapshots in chronological order within an episode that showcase the strategy. Running and chasing: Hiders (blue) try to hide behind walls and moveable objects, seekers (red) chase and search for hiders. Fort building: During the preparation phase (first 3 snapshots), hiders push the two boxes into the two doors of the lower right room and lock them in place. Seekers are not able to overcome this barrier during the remainder of the trial. Ramp use: During the preparation phase, hiders secure the room using the two boxes (1st and 2nd snapshot). After the preparation phase, a seeker grabs the ramp (3rd snapshot), drags it towards the walls of the room (4th snapshot) and both seekers use the ramp to enter the room (5th snapshot). Ramp defense: Hiders prevent ramp use by grabbing the ramp during the preparation phase (2nd snapshot), pulling it inside the room (3rd snapshot), and then secure the doors using the 2 blocks (4th snapshot). After the preparation phase, the seekers are not able to enter the room (5th snapshot). Also, note that while one hider brings the ramp into the room, the other hider drags a box out of corner and then passes it to its team mate (3rd and 4th snapshot), evidence of sophisticated coordination between hiders.
+
+# A.3 QUADRANT ENVIRONMENT
+
+As well as the default random rooms environment described above, we also train and evaluate agents in the much simpler quadrant environment (Figure A.2) for illustration purposes. This environment consists of a room in the lower right corner that is connected by 1-2 doors to the remainder of the environment. Unlike in the main environment, the interior walls are fixed and only the location of the doors vary from episode to episode. Hiders are spawned randomly in the environment, but seekers are only spawned outside the room. Two cubic boxes are spawned inside the room and one ramp is spawned inside or outside the room. Both hiders and seekers receive a negative reward if they leave the environment or if they bring an object outside the environment. Also, in contrast to the main environment, only boxes, but not ramps can be locked in place.
+
+The quadrant environment gives rise to a similar skill progression as in the main environment (Figure A.2): basic running and chasing, then fort building by hiders, ramp use by seekers, and ramp defense by hiders. Hiders create shelter by pushing boxes into the doors of the room in the lower right corner, preventing seekers from entering. Seekers drag ramps to the exterior walls of the room and then use them to jump inside. As ramps are not lockable in this environment, hiders instead defend against ramp use by pulling the ramp inside the room before securing the wall opening with boxes.
+
+# A.4 FURTHER ABLATIONS
+
+![](images/b63f248686ea0a3064207b7f99c6260f6830251740000bcf1be42680f8d04d3b.jpg)  
+Figure A.3: Effect of Omniscent Value Function and Pooling Architectures on Emergent Autocurriculum. In blue we show the number of episodes required and in orange the wall clock time required to achieve stage 4 (ramp defense) of the emergent skill progression presented in Figure 1 for varying batch and model sizes. Note that in our ablation comparing omniscient and masked value functions, the experiment with a masked value function never reached stage 4 in the allotted time, so here we compare timing to stage 3.
+
+![](images/ee4777af02411b0e866648894b8f029157f525554c1320c00501261490198671.jpg)
+
+In Figure A.3 we compare the performance between a masked and omniscient value function as well a purely pooling architecture versus self-attention. We find that using an omniscient value function, meaning that the value function has access to the full state of the unobscured environment, is critical to progressing through the emergent autocurricula at the given scale. We found that with the same compute budget, training with a masked value function never progressed past stage 3 (ramp usage). We further found that our self-attention architecture increases sample efficiency as compared to an architecture that only embeds and then pools entities together with a similar number of parameters. However, because self-attention requires more compute despite having the same number of parameters, the wall-clock time to convergence is slightly slower.
+
+# A.5 EVALUATING AGENTS AT DIFFERENT PHASES OF EMERGENCE
+
+![](images/b8a8c465cdc047cfeb7ade98c692732a0437de5d3cfdd5508a2086a0d083e1e1.jpg)  
+Figure A.4: Fine-tuning From Different Phases in the Emergent Autocurriculum. We plot the mean performance on the transfer suite and  $90\%$  confidence interval across 3 seeds and smooth performance and confidence intervals with an exponential moving average. We show the fine-tuning performance for a policy sampled at each of the six phases of emergent strategy (see Figure 1).
+
+In Figure A.4 we evaluate policies sampled during each phase of emergent strategy on the suite of targeted intelligence tasks, by which we can gain intuition as to whether the capabilities we measure improve with training, are transient and accentuated during specific phases, or generally uncorrelated to progressing through the autocurriculum. We find that the hide-and-seek agent improves on the navigation and memory tasks as it progresses; notably on Lock and Return, the performance monotonically increases with emergence phase, and the policy from the phase 6 performs  $20\%$  better than the policy from phase 1. However, performance on Object Counting is transient; during
+
+![](images/5be1294917502e45600445b1648e1fee6f9711eaf2c5a1b36e5de64067cae1ba.jpg)  
+Figure A.5: Example trajectory in hide-and-seek environment with additional food reward. During the preparation phase, hiders lock the ramps at the boundary of the play area (2nd snapshot) and construct a fort around the food reward (3rd and 4th snapshot). After the preparation phase, hiders can eat food and receive the additional food reward, because they are hidden from seekers inside the fort.
+
+phase 1 the hide-and-seek agent performs extremely well, much better than all baselines and other phases but loses this ability in later stages. Finally, we find that performance on the manipulation tasks is relatively uncorrelated to the phases of emergence, and surprisingly the policy transferred from phase 1, the phase before any tool use emerges, performs comparably well to other phases.
+
+# A.6 ALTERNATIVE GAMES TO隐-AND-SEEK WITH SECONDARY OBJECTIVES
+
+# A.6.1 HIDE-AND-SEEK WITH FOOD REWARD
+
+In the main hide-and-seek environment, hiders build forts at locations that would give them the best chance of staying hidden from seekers. To test whether hiders could be incentivized by a secondary objective to adapt the location of forts, we add additional food rewards and test whether hiders would bias the location of their forts towards the location of the food. In this environment, hiders can eat food and receive food rewards only under the following conditions: after the preparation phase, when all hiders are hidden from seekers and when the food is sufficiently close and visible to them. Therefore, just chasing after food would not be an effective strategy for hiders, because hiders receive neither hide-and-seek nor food rewards if one or more hiders are seen by a seeker. Instead, hiders are incentivized to build forts around the location of food and then eat the food while being unobserved by seekers (figure A.5).
+
+Food is distributed in the form of 5 food pellets that are spawned close to each other in a rectangular area in the center of the environment whose side length is  $1/4$  of the room size. Each food pellet can provide a positive food reward of  $+1$  for each time step. Food rewards are shared between hiders, irrespective of which hider eats a food item. Hiders (as well as seekers) can observe the locations of food pellets as separate entities; if food pellets are obstructed or outside the field of view they are masked out like other type of objects.
+
+As shown in figure A.6, this environment gives rise to four levels of skill progression, similar to the one of the main environment: basic running and chasing, then fort building by hiders, ramp use by seekers, and ramp defense by hiders. Moreover, hiders consume food, and food consumption is highly correlated with their ability to construct stable forts; food consumption decreases during the initial phase as seekers get better at chasing and therefore prevent hiders from eating food. Food consumption then increases again as hiders learn to construct forts and shield themselves from the view of seekers, but plateaus once seekers learn to use ramps. Finally, food consumption rises again as hiders get better at defending against ramp use.
+
+# A.6.2 ※HIDE-AND-SEEK WITH DYNAMIC FOOD
+
+In the food variant introduce in Sec. A.6.1, the location of food pellets is fixed throughout the episode. Here we consider a dynamic food variant such that a food pellet will be eaten up, i.e., disappear, when a hider is close to it, and then a new food pellet will show up in a different location but still within the center region. More precisely, the game area is simply an empty room without outside walls containing 2 seekers, 3 hiders, 1 dynamic food and 8 elongated boxes. The food will be always located within a square in the center of the environment with side length  $1/5$  of the game area size. We inherit the same reward and policy structures from the previous game.
+
+In this game, merely building a center fort is not sufficient for obtaining the highest reward since the food might disappear and respawn outside the fort. The agent must ensure that the fort is large
+
+![](images/de118a5229f28762e44d1a9beb5f283353307d4f3cdc29acf4902c762fb1f6f5.jpg)
+
+![](images/7032b3ed2cc02860745720d174c56045ef70b121b3fdb0fadba580336da42204.jpg)
+
+![](images/980e0f484555193133f3c2e31c9cecac454bbc57ed2684c63461a5b5d953b66a.jpg)  
+Figure A.6: Reward and environment specific statistics during emergence in hide-and-seek with secondary food rewards for hiders.
+
+![](images/b29fa22499518ea867086426da5ba20e3b0a77f4b2cabec85c6ed01cdf0bf6aa.jpg)
+
+enough such that all the possible spawning positions of the food will be inside the fort. Such a behavior does emerge after training for around  $4.5 \times 10^{10}$  samples.
+
+We also experimented on variants where the food spawning region has different side length. When the side length is reduced to 1/6 of the game area, the same behavior emerges faster taking  $1.5 \times 10^{10}$  samples. However, when the side length of the dynamic food region is increased to 1/4 of the game area, hiders converge to a policy that ignores the food and only builds a small fort to protect themselves.
+
+# A.6.3 FOOD PROTECTION GAME
+
+In the previous two variants, we introduce extra food reward to the hider in addition to the original hide-and-seek reward for promoting more goal-oriented behavior. Now we consider a different game rule such that the competition between hiders and seekers only depends on the food collecting reward and show that this rule can also lead to tool use and complex behavior.
+
+We consider an empty game area surrounded by walls that contains 50 food pellets randomly distributed in a center square of size  $2/3$  of the game environment. There are 3 hiders, 2 seekers and 7 elongated boxes in the game. The only goal for seekers is to collect food. Once a food pellet is collected by any of the seeker, a  $+3$  reward will be given to all the seekers and then the food will disappear permanently from the game. The goal for the hider is to protect the food from seekers and their reward is simply the negative value of seekers. Each episode consists of 200 time steps. The preparation period, in which only hiders can move, extends to the first 100 time steps. The last 60 steps of the game corresponds to a food collecting period, during which the hiders cannot move. Additionally, after the preparation phase we also add a -1 "boundary penalty" for hiders when they are too close to the wall to ensure they stay within the food region.
+
+Our initial motivation for this task was to promote hiders to learn to construct complex and large fort structures with a more direct competition pressure. For example, we expected that the agents would learn to build a large fort that would cover as much food as possible. Unfortunately, such behavior did not emerge. However, after around  $7 \times 10^{10}$  samples of training, the hiders surprisingly discover
+
+![](images/834c99a0f1468b605abbefb00230b902c80d4cf2fec4fa0d0f81ddb694676768.jpg)  
+Figure A.7: Example trajectory in the food protection game. During the preparation phase, the hiders first explore the room to find seekers, then physically push the seekers to the corner and eventually collaboratively grab elongated boxes to build a small fort surrounding the seekers to trap them there. To secure the trap, they even learn to build the fort using two elongated boxes to prevent seekers from escaping.
+
+![](images/c095eea7a526617d765ebe5630d0284f95f548e7d24087142e50b7e850daa67e.jpg)
+
+![](images/6bc21b507e9370bce56e278ac3168a21f858da13ef6680f9c79c647a31381fc9.jpg)
+
+![](images/5e33904e7189b49bfbfef6f2c50ff5bb15ac3bc3a2b23f25817006f4c1ebedef.jpg)
+
+![](images/634a1f2b7defe401d9b8147801eed6d469efb49c40b5043e80c80a44523a3458.jpg)
+
+![](images/47c41b247ea42080693f6504058101d89e0e2e62fd1f8dc7c4ad0043be1d05ce.jpg)  
+Figure A.8: Zero-shot generalization to a larger number of hiders (left), ramps (center) and boxes (right). The dotted line denotes the boundary of the training distribution (1-3 hiders, 2 ramps, 1-9 boxes). Error bars denote standard error of the mean.
+
+![](images/4c0a8f70362281f754fb4b4a1c2f5b98dadc47017585477eae51635da0d8a481.jpg)
+
+![](images/d1be9be1e4edf384deb027b2a5c1eef4d57e027bd88d6f3028298760741f2c8d.jpg)
+
+an extremely efficient and effective strategy: they learn to physically 'push' the immobile seekers to the corner of the environment during the preparation phase and then collaboratively use elongated boxes to trap the seekers in the corner. Figure A.7 illustrates a trajectory corresponding to this strategy. Interestingly, the hiders demonstrate strong collaborations. Since there are 2 seekers and 3 hiders, when the 2 hiders are pushing the seekers to the corner, the 3rd hider will follow the 'pushing' hiders distantly, and once the pushing hiders finish the job, this 3rd hider will quickly grab a box to build the trap without blocking the escaping route of those 2 pushing hiders. We emphasize that this behavior was completely surprising to us and further indicates that complex behavior can emerge from multi-agent self-play with a wide range of competition rules.
+
+Lastly, we also experimented with different versions of the close-to-wall "boundary penalties". In all cases, we observe similar fort-building behaviors, namely blocking the seekers to the corner, with tiny variations. When we remove the penalty, the behavior emerges twice faster ( $3.5 \times 10^{10}$  samples) and 1 hider will stay in the fort with the seekers instead of getting outside. When we add the penalty throughout the episode, the hiders even learned to stay in the center region while grabbing an elongated box as a tool to push the seekers towards the corner. However, the behavior emerges twice slower in this setting, i.e.,  $1.4 \times 10^{11}$  samples required.
+
+# A.7 ZERO-SHOT GENERALIZATION
+
+We find that the trained policies can zero-shot generalize to environments with slightly larger number of hiders, ramps or boxes in Figure A.8. The hider reward provides a reasonably good measure for generalization performance, since the modifications in question make constructing safe shelter more difficult. First, we looked at increasing the number of hiders from 1-3 hiders, as in the training distribution, to 4-6 hiders; increasing the number of hiders decreases the hider reward because all hiders need to stay hidden from seekers. However, the decline in hider reward is very gradual, indicating that the policy generalizes well to more than 4 hiders. A similar effect occurs when increasing the number of ramps because hiders need to secure more ramps from seekers. If we increase the number of ramps from 2 to 3 or 4 the hider reward drops only gradually. Finally, we find hider performance is remarkably stable, though still slowly declines, when increasing the number of boxes.
+
+# B OPTIMIZATION DETAILS
+
+# B.1 NOTATION
+
+We consider the standard multi-agent reinforcement learning formalism of  $N$  agents interacting with each other in an environment. This interaction is defined by a set of states  $\mathcal{S}$  describing the state of the world and configurations of all agents, a set of observations  $\mathcal{O}^1,\ldots ,\mathcal{O}^N$  of all agents, a set of actions  $\mathcal{A}^1,\dots ,\mathcal{A}^N$  of all agents, a transition function  $\mathcal{T}:S\times \mathcal{A}^1\dots \mathcal{A}^N\to \mathcal{S}$  determining the distribution over next states, and a reward for each agent  $i$  which is a function of the state and the agent's action. Agents choose their actions according to a stochastic policy  $\pi_{\theta_i}:\mathcal{O}^i\times \mathcal{A}^i\rightarrow [0,1]$ , where  $\theta_{i}$  are the parameters of the policy. In our formulation, policies are shared between agents,  $\pi_{\theta_i} = \pi_\theta$  and the set of observations  $\mathcal{O}$  contains information for which role (e.g. hider or seeker) the agent will be rewarded. Each agent  $i$  aims to maximize its total expected discounted return  $R^{i} = \sum_{t = 0}^{H}\gamma^{t}r_{t}^{i}$ , where  $H$  is the horizon length and  $\gamma$  is a time discounting factor that biases agents towards preferring short term rewards to long term rewards. The action-value function is defined as  $Q^{\pi_i}(s_t,a_t^i) = \mathbb{E}[R_t^i |s_t,a_t^i]$ , while the state-value function is defined as  $V^{\pi_i}(s_t) = \mathbb{E}[R_t|s_t]$ . The advantage function  $A^{\pi_i}(s_t,a_t^i)\coloneqq Q^{\pi_i}(s_t,a_t^i) - V^{\pi_i}(s_t)$  describes whether taking action  $a_{t}^{i}$  is better or worse for agent  $i$  when in state  $s_t$  than the average action of policy  $\pi_{i}$ .
+
+# B.2 PROXIMAL POLICY OPTIMIZATION (PPO)
+
+Policy gradient methods aim to estimate the gradient of the policy parameters with respect to the discounted sum of rewards, which is often non-differentiable. A typical estimator of the policy gradient is  $g \coloneqq \mathbb{E}[\hat{A}_t\nabla_\theta \log \pi_\theta]$ , where  $\hat{A}$  is an estimate of the advantage function. PPO (Schulman et al., 2017), a policy gradient variant, penalizes large changes to the policy to prevent training instabilities. PPO optimizes the objective  $L = \mathbb{E}\left[\min(l_t(\theta)\hat{A}_t,\mathrm{clip}(l_t(\theta),1 - \epsilon,1 + \epsilon)\hat{A}_t\right]$ , where  $l_t(\theta) = \frac{\pi_\theta(a_t|s_t)}{\pi_{old}(a_t|s_t)}$  denotes the likelihood ratio between new and old policies and  $\mathrm{clip}(l_t(\theta),1 - \epsilon,1 + \epsilon)$  clips  $l_t(\theta)$  in the interval  $[1 - \epsilon,1 + \epsilon]$ .
+
+# B.3 GENERALIZED ADVANTAGE ESTIMATION
+
+We use Generalized Advantage Estimation (Schulman et al., 2015) with horizon length  $H$  to estimate the advantage function. This estimator is given by:
+
+$$
+\hat {A} _ {t} ^ {H} = \sum_ {l = 0} ^ {H} (\gamma \lambda) ^ {l} \delta_ {t + l}, \qquad \delta_ {t + l} := r _ {t + l} + \gamma V (s _ {t + l + 1}) - V (s _ {t + l})
+$$
+
+where  $\delta_{t + l}$  is the TD residual,  $\gamma ,\lambda \in [0,1]$  are discount factors that control the bias-variance tradeoff of the estimator,  $V(s_{t})$  is the value function predicted by the value function network and we set  $V(s_{t}) = 0$  if  $s_t$  is the last step of an episode. This estimator obeys the reverse recurrence relation  $\hat{A}_t^H = \delta_t + \gamma \lambda \hat{A}_{t + 1}^{H - 1}$ .
+
+We calculate advantage targets by concatenating episodes from policy rollouts and truncating them to windows of  $T = 160$  time steps (episodes contain 240 time steps). If a window  $(s_0, \ldots, s_{T-1})$  was generated in a single episode we use the advantage targets  $(\hat{A}_0^{H=T}, \hat{A}_1^{H=T-1}, \ldots, \hat{A}_{T-1}^{H=1})$ . If a new episode starts at time step  $j$  we use the advantage targets  $(\hat{A}_0^{H=j}, \hat{A}_1^{H=j-1}, \ldots, \hat{A}_{j-1}^{H=1}, \hat{A}_j^{H=T-j}, \ldots, \hat{A}_{T-1}^{H=1})$ .
+
+Similarly we use as targets for the value function  $(\hat{G}_0^{H = T},\hat{G}_1^{H = T - 1},\dots ,\hat{G}_{T - 1}^{H = 1})$  for a window generated by a single episode and  $(\hat{G}_0^{H = j},\hat{G}_1^{H = j - 1},\dots ,\hat{G}_{j - 1}^{H = 1},\hat{G}_j^{H = T - j},\dots ,\hat{G}_{T - 1}^{H = 1})$  if a new episode starts at time step  $j$ , where the return estimator is given by  $\hat{G}_t^H \coloneqq \hat{A}_t^H + V(s_t)$ . This value function estimator corresponds to the TD( $\lambda$ ) estimator (Sutton & Barto, 2018).
+
+# B.4 NORMALIZATION OF OBSERVATIONS, ADVANTAGE TARGETS AND VALUE FUNCTION TARGETS
+
+We normalize observations, advantage targets and value function targets. Advantage targets are z-scored over each buffer before each optimization step. Observations and value function targets are z-scored using a mean and variance estimator that is obtained from a running estimator with decay parameter  $1 - 10^{-5}$  per optimization substep.
+
+# B.5 OPTIMIZATION SETUP
+
+Training is performed using the distributed rapid framework (OpenAI, 2018). Using current policy and value function parameters, CPU machines roll out the policy in the environment, collect rewards, and compute advantage and value function targets. Rollouts are cut into windows of 160 timesteps and reformatted into 16 chunks of 10 timesteps (the BPTT truncation length). The rollouts are then collected in a training buffer of 320,000 chunks. Each optimization step consists of 60 SGD substeps using Adam with mini-batch size 64,000. One rollout chunk is used for at most 4 optimization steps. This ensures that the training buffer stays sufficiently on-policy.
+
+# B.6 OPTIMIZATION HYPERPARAMETERS
+
+Our optimization hyperparameter settings are as follows:
+
+<table><tr><td>Buffer size</td><td>320,000</td></tr><tr><td>Mini-batch size</td><td>64,000 chunks of 10 timesteps</td></tr><tr><td>Learning rate</td><td>3·10-4</td></tr><tr><td>PPO clipping parameter ε</td><td>0.2</td></tr><tr><td>Gradient clipping</td><td>5</td></tr><tr><td>Entropy coefficient</td><td>0.01</td></tr><tr><td>γ</td><td>0.998</td></tr><tr><td>λ</td><td>0.95</td></tr><tr><td>Max GAE horizon length T</td><td>160</td></tr><tr><td>BPTT truncation length</td><td>10</td></tr></table>
+
+# B.7 POLICY ARCHITECTURE DETAILS
+
+Lidar observations are first passed through a circular 1D-convolution and concatenated onto the agents representation of self,  $x_{\mathrm{self}}$ . Each object is concatenated with  $x_{\mathrm{self}}$  and then embedded with a dense layer where parameters are shared between objects of the same type, e.g. all boxes share the same embedding weights. All the embedded entities are then passed through a residual self-attention block, similar to Vaswani et al. (2017) but without position embeddings, in the form of  $y = \mathrm{dense}(\mathrm{self\_attention}(x)) + x$ . We then average-pool entity embeddings and concatenate this pooled representation to  $x_{\mathrm{self}}$ . Note that in the policy network the entities not observed by each agent are masked away through self-attention and pooling. Finally, this pooled representation is passed through another dense layer and an LSTM (Hochreiter & Schmidhuber, 1997) before pulling off separate action heads for each of the 3 action types described in Section 3. We also add layer normalization (Ba et al., 2016) to every hidden layer of the policy network except the 1D-convolution layer. We empirically observe that layer normalization leads to faster training and better transfer performance.
+
+<table><tr><td>Size of embedding layer</td><td>128</td></tr><tr><td>Size of MLP layer</td><td>256</td></tr><tr><td>Size of LSTM layer</td><td>256</td></tr><tr><td>Residual attention layer</td><td>4 attention heads of size 32</td></tr><tr><td>Weight decay coefficient</td><td>10-6</td></tr></table>
+
+# C INTELLIGENCE TEST SUITE DETAILS
+
+All evaluation tasks in the intelligence test suite are single-agent tasks. In order to align the observations between the hide-and-seek environment and the evaluation tasks, we add fake hinder observations, preparation phase observations and set them all to 0.
+
+# C.1 COGNITION AND MEMORY TASK
+
+All the variations of cognition and memory tasks have a horizon of 120 timesteps, and all boxes are cuboids.
+
+# Object Counting:
+
+An agent is pinned in place and watches as 6 boxes move either to left or right at random. The boxes disappear behind walls such that agent can no longer see them, and the agent is asked to predict how many boxes went left or right far after all the boxes have disappeared. In this test we evaluate the quality of the existing representation by holding the agent's policy fixed and only train a new classification head from the agent's LSTM state. The classification head first passes the LSTM state through a layernorm and a single dense layer with 64 units. We then do a 7-class classification predicting whether 0 through 6 boxes have gone to the left.
+
+# Lock and Return:
+
+In this game, the agent needs to navigate towards a hidden box, lock it, and then return to its starting position.
+
+The game area has 6 randomly generated connected rooms with static walls and 1 box. When the box is locked, the agent will be given a reward of  $+5$ . If the agent unlocks the box during the episode, a -5 penalty will be given. Additionally, if the box remains unlocked at the end of the episode, the agent will be given another -5 penalty. A success is determined when the agent returns to its starting location within 0.1 radius and with the box locked. For promoting fast task accomplishment, we give the agent a  $+1$  reward for each timestep of success. We also introduce shaped reward with coefficient of 0.5 for easier learning: at each time step, the shaped reward is the decrement in the distance between the agent location towards the target (either the unlocked box or the starting location).
+
+# Sequential Lock:
+
+There are 4 boxes and the agent needs to lock all the boxes in an unobserved order sequentially. A box can be locked only if it is locked in the right order.
+
+The game area is randomly partition into three rooms with 2 walls. The 4 boxes are randomly placed in the game area. Each room has a ramp. The agent has to utilize the ramps to navigate between rooms. When a box is successfully locked (according to the order), a  $+5$  bonus is given. If a box is unlocked, -5 penalty will be added. When all the boxes get locked, the agent will receive a  $+1$  per-timestep success bonus. We also use the same shaped distance reward as the lock and return task here.
+
+# C.2 MANIPULATION TASK
+
+All variations of the manipulation task have 8 boxes, but no ramps.
+
+# Construction from Blueprint:
+
+The horizon is at most 240 timesteps, but an episode can end early if the agent successfully finishes the construction. The game area is an empty room. The locations of the construction sites are sampled uniformly at random (we use rejection sampling to ensure that construction sites do not overlap).
+
+For each construction site, agents observe its position and that of its 4 corners. Since there are no construction sites in the hide-and-seek game and the count-based baseline environments, we need to change our policy architecture to integrate the new observations. Each construction site observation is concatenated with  $x_{self}$  and then embedded through a new dense layer shared across all sites. This
+
+dense layer is randomly initialized and added to the multi-agent and count-based policies before the start of training. The embedded construction site representations are then concatenated with all other embedded object representations before the residual self-attention block, and the rest of the architecture is the same as the one used in the hide-and-seek game.
+
+The reward at each timestep is equal to a reward scale constant times the mean of the smooth minimum of the distances between each construction site corner and every box corner. Let there be  $k$  construction sites and  $n$  boxes, and let  $d_{ij}$  be the distance between construction site corner  $i$  and box corner  $j$ , and let  $d_i$  be the smooth minimum of the distances from construction site corner  $i$  to all box corners. The reward at each timestep follows the following formula:
+
+$$
+d _ {i} = \left(\sum_ {j = 1} ^ {4 n} d _ {i j} e ^ {\alpha d _ {i j}}\right) / \sum_ {j = 1} ^ {4 n} e ^ {\alpha d _ {i j}} \forall i = 1, 2, \dots , 4 k
+$$
+
+$$
+r e w = s _ {d} \left(\sum_ {i = 1} ^ {4 k} d _ {i}\right) / 4 k
+$$
+
+Here,  $s_d$  is the reward scale parameter and  $\alpha$  is the smoothness hyperparameter ( $\alpha$  must be nonpositive;  $\alpha = 0$  gives us the mean, and  $\alpha \rightarrow -\infty$  gives us the regular min function). In addition, when all construction sites have a box placed within a certain distance  $d_{min}$  of them, and all construction site corners have a box corner located within  $d_{min}$  of them, the episode ends and all agents receive reward equal to  $s_c * k$ , where  $s_c$  is a separate reward scale parameter. For our experiment,  $n = 8$  and  $k$  is randomly sampled between 1 and 4 (inclusive) every episode. The hyperparameter values we use for the reward are the following:
+
+$$
+\alpha = - 1. 5
+$$
+
+$$
+s _ {d} = 0. 0 5
+$$
+
+$$
+d _ {m i n} = 0. 1
+$$
+
+$$
+s _ {c} = 3
+$$
+
+Shelter construction: The goal of the task is to build a shelter around a cylinder that is randomly placed in the play area. The horizon is 150 timesteps, and the game area is an empty room. The location of the cylinder is uniformly sampled at random a minimum distance away from the edges of the room (this is because if the cylinder is too close to the external walls of the room, the agents are physically unable to complete the whole shelter). The diameter of the cylinder is uniformly randomly sampled between  $d_{min}$  and  $d_{max}$ . There are 3 movable elongated boxes and 5 movable square boxes. There are 100 rays that originate from evenly spaced locations on the bounding walls of the room and target the cylinder placed within the room. The reward at each timestep is  $(-n*s)$ , where  $n$  is the number of raycasts that collide with the cylinder that timestep and  $s$  is the reward scale hyperparameter.
+
+We use the following hyperparameters:
+
+$$
+s = 0. 0 0 1
+$$
+
+$$
+d _ {m i n} = 1. 5
+$$
+
+$$
+d _ {m a x} = 2
+$$
+
+# D INTRINSIC MOTIVATION METHODS
+
+We inherit the same policy architecture as well as optimization hyperparameters as used in the hide-and-seek game.
+
+Note that only the Sequential Lock task in the transfer suite contains ramps, so for the other 4 tasks we remove ramps in the environment for training intrinsic motivation agents.
+
+# D.1 COUNTED-BASED EXPLORATION
+
+For each real value from the continuous state of interest, we discretize it into 30 bins. Then we randomly project each of these discretized integers into a discrete embedding of dimension 16 with
+
+integer value ranging from 0 to 9. Here we use discrete embeddings for the purpose of accurate hashing. For each input entity, we concatenate all its obtained discrete embeddings as this entity's feature embedding. An max-pooling is performed over the feature embeddings of all the entities belonging to each object type (i.e., agent, lidar, box and ramp) to obtain a entity-invariant object representation. Finally, concatenating all the derived object representations results in the final state representation to count.
+
+We run a decentralized version of the count-based exploration where each parallel rollout worker shares the same random projection for computing embeddings but maintains its own counts. Let  $N(S)$  denote the counts for state  $S$  in a particular rollout worker. Then the intrinsic reward is calculated by  $\frac{0.1}{\sqrt{N(S)}}$ .
+
+# D.2 RANDOM NETWORK DISTILLATION
+
+Random Network Distillation (RND) (Burda et al., 2019b) uses a fixed random network, i.e., a target network, to produce a random projection for each state while learns anther network, i.e., a predictor network, to fit the output from the target network on visited states. The prediction error between two networks is used as the intrinsic motivation.
+
+For the random target network, we use the same architecture as the value network except that we remove the LSTM layer and project the final layer to a 64 dimensional vector instead of a single value. The architecture is the same for predictor network. We use the squared difference between predictor and target network output with an coefficient of 1.0 as the intrinsic reward.
+
+# D.3 FINE-TUNING FROM INTRINSIC MOTIVATION VARIANTS
+
+We compare the performances of different policies pretrained by intrinsic motivation variants on our intelligence test suites in Figure D.1. The pretraining methods of consideration include RND and 3 different count-based exploration variants with different state representations. For policies trained by count-based variants, as discussed in the Section 6.1, we know that more concise state representation for counts leads to better emergent object manipulation skills, i.e., only using object position is better than using position and velocity information while using all the input as state representation performs the worst. In our transfer task suites, we observe that polices with better pretrained skills perform better in 3 of the 5 tasks except the Object Counting task and the Construction from Blueprint task. In Construction from Blueprint, policies with better skills adapt better in the early phase but may have a higher chance of failure later in this challenging task. In Object Counting, the polices with better skills perform worse. Interestingly, this observation is consistent with the transfer result in the main paper (Figure 6), where the policy pretrained by count-based exploration outperforms the multi-agent pretrained policy. We conjecture that the Object Counting task examines some factors of the agents that may not strongly relate to the quality of emergent skills, such as navigation and tool use.
+
+![](images/1da8f3dbe59dbeb7c863f0e42938fafbbf1c29875b3abe23f1855dcfd4c3d5a0.jpg)  
+Figure D.1: Fine-tuning From Intrinsic Motivation Variants. We plot the mean performance on the suite of transfer tasks and  $90\%$  confidence interval across 3 seeds and smooth performance and confidence intervals with an exponential moving average. We vary the state representation used for collecting counts: in blue we show the performance for a single agent where the state is defined on box 2-D positions, in green we show the performance of a single agent where the state is box position but also box rotation and velocity, in red we show the performance of 1-3 agents with the full observation space given to a hide-and-seek policy, and finally in purple we show the performance also with 1-3 agents and a full observation space but with RND which should scale better than count-based exploration.
\ No newline at end of file
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+# ENCODING WORD ORDER IN COMPLEX EMBEDDINGS
+
+Benyou Wang *
+
+University of Padua
+
+wang@dei.unipd.it
+
+Donghao Zhao *
+
+Tianjin University
+
+zhaodh@tju.edu.cn
+
+Christina Lioma
+
+University of Copenhagen
+
+chrh@di.ku.dk
+
+Qiuchi Li
+
+University of Padua
+
+qiuchili@dei.unipd.it
+
+Peng Zhang
+
+Tianjin University
+
+pzhang@tju.edu.cn
+
+Jakob Grue Simonsen
+
+University of Copenhagen
+
+simonsen@di.ku.dk
+
+# ABSTRACT
+
+Sequential word order is important when processing text. Currently, neural networks (NNs) address this by modeling word position using position embeddings. The problem is that position embeddings capture the position of individual words, but not the ordered relationship (e.g., adjacency or precedence) between individual word positions. We present a novel and principled solution for modeling both the global absolute positions of words and their order relationships. Our solution generalizes word embeddings, previously defined as independent vectors, to continuous word functions over a variable (position). The benefit of continuous functions over variable positions is that word representations shift smoothly with increasing positions. Hence, word representations in different positions can correlate with each other in a continuous function. The general solution of these functions is extended to complex-valued domain due to richer representations. We extend CNN, RNN and Transformer NNs to complex-valued versions to incorporate our complex embedding (we make all code available). Experiments 1 on text classification, machine translation and language modeling show gains over both classical word embeddings and position-enriched word embeddings. To our knowledge, this is the first work in NLP to link imaginary numbers in complex-valued representations to concrete meanings (i.e., word order).
+
+# 1 INTRODUCTION
+
+When processing text, the sequential structure of language is important, but can be computationally costly to model with neural networks (NNs) (Socher et al., 2011) due to the difficulty in parallelization. This has been alleviated by modeling word sequence not on the NN architecture level, but by adding position embeddings on the feature level. This has been done by the convolutional sequence model (ConvSeq) (Gehring et al., 2017) and the Transformer model (Vaswani et al., 2017) that replaces recurrent and convolution operations with purely attention mechanisms. More generally, vanilla position embeddings (Gehring et al., 2017) assume that individual word positions are independent and do not consider relations between neighboring word positions. We posit that both the global absolute positions of words and their inner sequential and adjacent relationships are crucial in language. This is supported by recent empirical findings by Shaw et al. (2018) and Dai et al. (2019) who show the importance of modeling distance between sequential elements, and explicitly use extra relative position encodings to capture the relative-distance relationship of words.
+
+We present a novel and principled approach to model both the global absolute positions of words and their inner sequential and adjacent relationships as follows: we extend each word embedding, previously defined as an independent vector, as a continuous function over an independent variable i.e., position. The benefit of continuous functions over variable positions is that word representations shift smoothly with increasing positions. Hence, word representations in different positions
+
+![](images/de38ebb28cdeeba61b23f68d57c1297d7df699380765b5dac454d8c95ca5b562.jpg)  
+Figure 1: 3-dimensional complex embedding for a single word in different positions. The three wave functions (setting the initial phases as zero) show the real part of the embedding; the imaginary part has a  $\frac{\pi}{2}$  phase difference and shows the same curves with its real-valued counterpart. The x-axis denotes the absolute position of a word and the y-axis denotes the value of each element in its word vector. Colours mark different dimensions of the embedding. The three cross points between the functions and each vertical line (corresponding to a specific position pos) represent the embedding for this word in the pos-th position.
+
+can correlate with each other in a continuous function. Fig. 1 illustrates this type of word representation with a three-dimensional complex-valued embedding, where the amplitudes  $\{r_1, r_2, r_3\}$  denote semantic aspects corresponding to classical word vectors, and periods  $\{p_1, p_2, p_3\}$  denote how sensitive the word is to positional information. We further discuss the necessary properties of these functions to model sequential information and obtain a general solution in the form of a complex-valued embedding. Interestingly, there is a direct connection between a specific case of our general embedding and the well-known positional encoding in Vaswani et al. (2017) (see App. A).
+
+We contribute (i) a novel paradigm that extends word vectors as continuous functions over changing variables like word position, and (ii) a general word embedding that models word order in a mathematically-sound manner. We integrate our complex word embeddings in state-of-the-art (SOTA) NN architectures (CNN, RNN, Transformer and experimentally find that it yields gains over both classical word embeddings and position-enriched word embeddings in text classification, machine translation and language modeling. Note that this is the first work in NLP to link imaginary numbers in complex-valued representation to concrete meanings (i.e., word order).
+
+# 2 MODELLING WORD ORDER IN EMBEDDING SPACE
+
+A Word Embedding (WE) generally defines a map  $f_{we} : \mathbb{N} \to \mathbb{R}^D$  from a discrete word index to a  $D$ -dimensional real-valued vector and  $\mathbb{N} = \{0, 1, 2, \ldots\}$ . Similarly, a Position Embedding (PE) (Gehring et al., 2017; Vaswani et al., 2017) defines another map  $f_{pe} : \mathbb{N} \to \mathbb{R}^D$  from a discrete position index to a vector. The final embedding for word  $w_j$  ( $w_j \in \mathbb{W}$  with index  $j$  in a given vocabulary  $\mathbb{W}$ ) in the pos-th position in a sentence is usually constructed by the sum
+
+$$
+f (j, p o s) = f _ {w e} (j) + f _ {p e} (p o s), \tag {1}
+$$
+
+and  $f(j, pos) \in \mathbb{R}^D$ . Since both the word embedding map  $f_w$  and the position embedding map  $f_p$  only take integer values as word indexes or position indexes, embedding vectors for individual words or positions are trained independently. The independent training for each word vector is reasonable, since a word index is based on the order of a given arbitrary vocabulary and does not capture any specific sequential relationship with its neighboring words. However, the position index captures an ordered relationship, for instance adjacency or precedence, leading to the problem that position embeddings in individual positions (Gehring et al., 2017) are independent of each other; the ordered relationship between positions is not modelled. We refer to this as the position independence problem. This problem becomes more crucial when position embeddings are used in position-insensitive NNs, e.g., FastText (Mikolov et al., 2013b), ConvSeq (Gehring et al., 2017) and Transformer (Vaswani et al., 2017), because it is hard for such position-insensitive NNs with vanilla position embeddings (Gehring et al., 2017) to infer that  $w_{j_1}$  in the pos-th position is close to  $w_{j_2}$  in the pos + 1-th position, or that  $w_{j_1}$  precedes  $w_{j_2}$ ; instead, it is only inferred that  $w_{j_1}$  and  $w_{j_2}$  are in different positions, while the relative distance between them is almost unknown. Thus vanilla position embeddings (Gehring et al., 2017) cannot fully capture the sequential aspect of language.
+
+Next, we first introduce the necessary properties to model word order in embeddings, and then give a unique solution to meet such properties.
+
+# 2.1 EXTENDING VECTORS TO FUNCTIONS
+
+In the general definition in Eq. 1, each dimension of the position embedding is obtained based on the discrete position indexes  $\{0,1,2,\dots,\mathrm{pos},\dots\}$ . This makes it difficult to model the ordered relationship between the positions. One solution to this problem is to build continuous functions over a variable (i.e., position index) to represent a specific word in an individual dimension. Formally, we define a general embedding as
+
+$$
+f (j, \operatorname {p o s}) = \boldsymbol {g} _ {j} (\operatorname {p o s}) \in \mathbb {R} ^ {D}, \tag {2}
+$$
+
+where  $\pmb{g}_j$  is short for  $g_{we}(j) \in (\mathcal{F})^D$ , indicating  $D$  functions over position index pos, and  $g_{we}(\cdot): \mathbb{N} \to (\mathcal{F})^D$  is a mapping from a word index to  $D$  functions. By expanding the  $D$  dimension of  $\pmb{g}_j$ , a word  $w_j$  in the pos-th position can be represented as a  $D$ -dimensional vector as shown in
+
+$$
+\left[ g _ {j, 1} (\operatorname {p o s}), g _ {j, 2} (\operatorname {p o s}), \dots , g _ {j, D} (\operatorname {p o s}) \right] \in \mathbb {R} ^ {D}, \tag {3}
+$$
+
+in which  $\forall g_{j,d}(\cdot)\in \mathcal{F}:\mathbb{N}\to \mathbb{R},d\in \{1,2,\dots,D\}$  is a function over the position index pos. To move the word  $w_{j}$  from the current position pos to another one pos', it needs only replace the variable pos to pos' without changing  $\pmb{g}_j$
+
+Functions for words, especially continuous functions, are expected to capture smooth transformation from a position to its adjacent position therefore modeling word order. The position-independent position embedding (Gehring et al., 2017) can be considered as a special case of our definition when it only takes independent values for individual positions in the embedding function.
+
+# 2.2 PROPERTIES FOR THE FUNCTIONS TO CAPTURE WORD ORDER
+
+Relative distance is hard to compute because position indices are not visible in NNs after vector embedding (discrete position indices are necessarily embedded as vectors like words to be backpropagated with the gradient). Hence, we claim that the modeling of relative distance in NNs should be position-free: absolute position indices cannot be directly accessed in intermediate layers. Instead of processing position-free operations in NNs to capture relative distance between words, prior work (Shaw et al., 2018; Dai et al., 2019) first calculates the relative distance between words, and then feeds the relative distance as an additional feature or as embeddings/weights to NNs, instead of directly feeding with the raw position indices.
+
+Assume that words are embedded into  $\mathbb{R}^D$ , and let, for  $1 \leq d \leq D$ , the function  $g_{j,d}: \mathbb{N} \to \mathbb{R}$  be the embedding function giving the  $d$ -th coordinate of the representation of word  $w_j$  (i.e.,  $g_{j,d}(\mathrm{pos})$  is the  $d$ -th coordinate of the embedding of  $w_j$  if it occurs at position pos. In the following, we simply write  $g$  instead of  $g_{j,d}$  when there is no risk of confusion. Ideally, one would like there to exist a function  $\mathrm{Transform}_n: \mathbb{R} \to \mathbb{R}$  that transforms the embedding of any word at some position pos to the embedding of a word at position pos + n such that  $\mathrm{Transform}_n$  is only dependent on the embedded value itself, but independent of the position pos, that is  $\forall \mathrm{pos}: g(\mathrm{pos} + n) = \mathrm{Transform}_n(g(\mathrm{pos}))$ .
+
+Prior work in NLP (Li et al., 2019), Information Retrieval (Van Rijsbergen, 2004) and Machine Learning (Trabelsi et al., 2017) has shown the usefulness of complex numbers as richer representations. Complex word embeddings (Wang et al., 2019; Li et al., 2019; Li et al., 2018) have been used to model language. To investigate the potential of complex-valued representation, we extend the target domains of  $g(\cdot)$  from  $\mathbb{R}^D$  to  $\mathbb{C}^D$  without losing generality, since real-valued numbers are specific complex numbers with their imaginary part being zero. This property regarding "position-free offset transformation" in complex-valued domains is formally defined in Property 1 below.
+
+Property 1. Position-free offset transformation: An embedding function  $g: \mathbb{N} \to \mathbb{C}$  is said to be a position-free offset transformation if there exists a function  $\operatorname{Transform}: \mathbb{N} \times \mathbb{C} \to \mathbb{C}$  (called the witness) such that for all  $n \geq 1$ , the function  $\operatorname{Transform}_n(\cdot) = \operatorname{Transform}(n, \cdot)$  satisfies  $\forall \mathrm{pos} \in \mathbb{N}: g(\mathrm{pos} + n) = \operatorname{Transform}_n(g(\mathrm{pos}))$ . A position-free offset transformation  $g$  is said to be linearly witnessed if there is a function  $w: \mathbb{N} \to \mathbb{C}$  such that  $g$  has a witness Transform satisfying, for all  $n$ ,  $\operatorname{Transform}(n, \mathrm{pos}) = \operatorname{Transform}_n(\mathrm{pos}) = w(n)$  (i.e., each  $\operatorname{Transform}_n$  is a linear function).
+
+Additionally, a boundedness property is necessary to ensure that the position embedding can deal with text of any length (pos could be large in a long document).
+
+Property 2. Boundedness: The function over the variable position should be bounded, i.e.  $\exists \delta \in \mathbb{R}^+, \forall \mathrm{pos} \in \mathbb{N}, |g(\mathrm{pos})| \leq \delta$ .
+
+Formally, we prove the following claim that there is a unique solution that meets Properties 1 and 2 under the condition that the embedding function is linearly witnessed. We use linear functions because they are well-understood and simple with a single floating-point operation in NNs.
+
+Claim 1. A function  $g: \mathbb{N} \to \mathbb{C}$  is a bounded and linearly witnessed position-free offset transformation iff it is on the form  $g(pos) = z_2z_1^{pos}$  for  $z_1, z_2 \in \mathbb{C}$  with  $|z_1| \leq 1$ .
+
+Proof. Assume that  $g$  is a bounded and linearly witnessed position-free offset transformation. Then, by linear witnessing, we have for all pos,  $n_1, n_2 \in \mathbb{N}$ :
+
+$$
+\begin{array}{l} w \left(n _ {1}\right) w \left(n _ {2}\right) g (\mathrm {p o s}) = w \left(n _ {2}\right) g (\mathrm {p o s} + n _ {1}) = g (\mathrm {p o s} + n _ {1} + n _ {2}) \\ = \operatorname {T r a n s f o r m} _ {n _ {1} + n _ {2}} (g (\mathrm {p o s})) = w (n _ {1} + n _ {2}) g (\mathrm {p o s}) \\ \end{array}
+$$
+
+whence  $w(n_{1} + n_{2}) = w(n_{1})w(n_{2})$ . Write  $w(1) = z_{1}$  and  $g(0) = z_{2}$ . As  $n_{1}, n_{2} \in \mathbb{N}$  were arbitrary, we have  $w(n) = (w(1))^{n} = z_{1}^{n}$  for all  $n \in \mathbb{N}$ . But then  $g(\mathrm{pos} + n) = w(n)g(\mathrm{pos}) = z_{1}^{n}g(\mathrm{pos})$ . Furthermore, observe that for  $\mathrm{pos} \geq 1$ , we have  $g(\mathrm{pos}) = g(1 + \mathrm{pos} - 1) = w(\mathrm{pos})g(0) = z_{1}^{\mathrm{pos}}z_{2} = z_{2}z_{1}^{\mathrm{pos}}$ . For  $\mathrm{pos} = 0$ ,  $g(0) = z_{2} = z_{2}z_{1}^{0}$ , whence  $g(\mathrm{pos}) = z_{2}z_{1}^{\mathrm{pos}}$ , as desired. Observe that if  $|z_{1}| > 1$ , then  $g(\mathrm{pos})$  is unbounded, whence we have  $|z_{1}| \leq 1$ . Conversely, assume that  $g$  is on the form  $g(\mathrm{pos}) = z_{2}z_{1}^{\mathrm{pos}}$  with  $|z_{1}| \leq 1$ . Then,  $|g(\mathrm{pos})| \leq |z_{2}z_{1}^{\mathrm{pos}}| \leq |z_{2}||z_{1}^{\mathrm{pos}}| \leq |z_{2}|$ , whence  $g$  is bounded. Define, for each  $n \in \mathbb{N}$ ,  $w(n) = z_{1}^{n}$  and  $\mathrm{Transform}_{n}(\mathrm{pos}) = w(n)\mathrm{pos}$ . Then, for all  $\mathrm{pos}, n \in \mathbb{N}$ ,
+
+$$
+g (\mathrm {p o s} + n) = z _ {2} z _ {1} ^ {\mathrm {p o s} + n} = z _ {2} z _ {1} ^ {\mathrm {p o s}} z _ {1} ^ {n} = g (\mathrm {p o s}) z _ {1} ^ {n} = \operatorname {T r a n s f o r m} _ {n} (g (\mathrm {p o s}))
+$$
+
+showing that  $g$  is a linearly witnessed position-free offset transformation.
+
+![](images/02758025b4ea580933a41c682624e366b4d6694e9d6f75fafba1076424935ef2.jpg)
+
+For any  $z \in \mathbb{C}$ , we may write  $z = r e^{i\theta} = r (\cos \theta + i \sin \theta)$ . Thus, for the general form of the embedding  $g$  from Theorem 1, we have:
+
+$$
+g (\mathrm {p o s}) = z _ {2} z _ {1} ^ {\mathrm {p o s}} = r _ {2} e ^ {i \theta_ {2}} \left(r _ {1} e ^ {i \theta_ {1}}\right) ^ {\mathrm {p o s}} = r _ {2} r _ {1} ^ {\mathrm {p o s}} e ^ {i \left(\theta_ {2} + \theta_ {1} \mathrm {p o s}\right)} \text {s u b j e c t t o} | r _ {1} | \leq 1 \tag {4}
+$$
+
+In implementations, the above definition of  $g$  will lead to an optimization problem due to the constraint  $|r_1| \leq 1$ . A natural and simple way to avoid this is to fix  $r_1 = 1$ ; note that  $|e^{ix}| \equiv 1$ , thus automatically satisfying the constraint, in contrast to a real-valued embedding where one would need to explicitly devise functions satisfying the constraint. Finally, Eq. 4 can be written in the simplified form:  $g(\mathrm{pos}) = re^{i(\omega \mathrm{pos} + \theta)}$ . Thus, one can think of  $g$  as embedding positions counterclockwise on a complex circle of radius  $r$  with a fixed period ( $r$  is the amplitude term,  $\theta$  is the initial phase term,  $\frac{\omega}{2\pi}$  is the frequency, and  $\frac{2\pi}{\omega}$  is the period term).
+
+# 2.3 COMPLEX-VALUED WORD EMBEDDING
+
+We now define our complex-valued word embedding  $g$  as a map taking a word index  $j$  and position word index pos to  $\mathbb{C}^D$ . For a word  $w_j$  in position pos, our general complex-valued embedding is defined as  $f(j, \mathrm{pos}) = g_j(\mathrm{pos}) = r_j e^{i(\omega_j \mathrm{pos} + \theta_j)}$ . Therefore,  $f(j, \mathrm{pos})$  is defined as:
+
+$$
+\left[ r _ {j, 1} e ^ {i \left(\omega_ {j, 1} \mathrm {p o s} + \theta_ {j, 1}\right)}, \dots , r _ {j, 2} e ^ {i \left(\omega_ {j, 2} \mathrm {p o s} + \theta_ {j, 2}\right)}, \dots , r _ {j, D} e ^ {i \left(\omega_ {j, D} \mathrm {p o s} + \theta_ {j, D}\right)} \right] \tag {5}
+$$
+
+Note that each coordinate  $d$  ( $1 \leq d \leq D$ ) has a separate amplitude  $r_{j,d}$ , period  $p_{j,d} = \frac{2\pi}{\omega_{j,d}}$ , and initial phase  $\theta_{j,d}$ . In Fig. 1 each dimension is represented as a wave which is parameterized by an amplitude, a period/frequency, and an initial phase. The trainable parameters of the embedding are the amplitudes vector  $\boldsymbol{r}_j = [r_{j,1}, \dots, r_{j,D}]$ , the period/frequency related weights  $\omega_j = [\omega_{j,1}, \dots, \omega_{j,D}]$  and the initial phase vector  $\boldsymbol{\theta}_j = [\theta_{j,1}, \dots, \theta_{j,D}]$ . Note that the mean values of  $f(j, \cdot)$  over all positions are linearly dependent on the amplitude. Observe that the period/frequency determines to what degree the word is sensitive to the position. With an extremely long period (i.e.,  $\omega_j$  very small), the complex-valued embedding is approximately constant for all possible values of pos, and hence approximates a standard word embedding. Conversely, if the period is short, the embedding will be highly sensitive to the position argument.
+
+In our embedding, the mean vectors of  $f(j,\cdot)$  taken over all positions are linearly correlated to the amplitude embedding  $\boldsymbol{r}_j = [r_{j,1},\dots,r_{j,K}]$  with a coefficient  $\frac{2}{\pi}$ . The amplitude  $r_{j,d}$  of our embedding depends only on the word  $w_j$  (and coordinate  $d$ ), not on the position of the word, whence one can
+
+<table><tr><td>Dataset</td><td>train</td><td>test</td><td>vocab.</td><td>task</td><td>Classes</td></tr><tr><td>CR (Hu &amp; Liu, 2014)</td><td>4K</td><td>CV</td><td>6K</td><td>product reviews</td><td>2</td></tr><tr><td>MPQA (Wiebe et al., 2005)</td><td>11k</td><td>CV</td><td>6K</td><td>opinion polarity</td><td>2</td></tr><tr><td>SUBJ (Pang &amp; Lee, 2005)</td><td>10k</td><td>CV</td><td>21k</td><td>subjectivity</td><td>2</td></tr><tr><td>MR (Pang &amp; Lee, 2005)</td><td>11.9k</td><td>CV</td><td>20k</td><td>movie reviews</td><td>2</td></tr><tr><td>SST (Socher et al., 2013)</td><td>67k</td><td>2.2k</td><td>18k</td><td>movie reviews</td><td>2</td></tr><tr><td>TREC (Li &amp; Roth, 2002)</td><td>5.4k</td><td>0.5k</td><td>10k</td><td>Question</td><td>6</td></tr></table>
+
+Table 1: Dataset Statistics. CV means 10-fold cross validation. The last 2 datasets come with train/dev/test splits.
+
+think of the vector  $g_{pe}(j, \mathrm{pos}) = [e^{i(\omega_{j,1} \mathrm{pos} + \theta_{j,1})}, \dots, e^{i(\omega_{j,D} \mathrm{pos} + \theta_{j,D})}]$  as a "purely" positional embedding. Consequently, our complex embedding can be considered an element-wise multiplication between the word embedding  $g_{we}(j) = [r_{j,1}, \dots, r_{j,K}]$  and position embedding  $g_{pe}$ .
+
+$$
+f (j, \operatorname {p o s}) = g _ {w e} (j) \odot g _ {p e} (j, \operatorname {p o s}) \tag {6}
+$$
+
+Prior work (Gehring et al., 2017; Vaswani et al., 2017) uses mean-weight addition between word embeddings  $f_{we}$  and position embeddings  $f_{pe}$  (all words share the weights). In our work, word embeddings and position embeddings are decoupled to some extent by element-wise multiplication and therefore the frequency/period terms (related to  $\omega_{j,d}$ ) can adaptively adjust the importance between semantic and position information for each word and each dimension. In particular, with higher frequency (i.e., large  $\omega_{j,d}$ ), the final embedding will change dramatically with the changing positions, while it can be fixed for any positions with an extremely-small frequency (i.e., small  $\omega_{j,d}$ ). Interestingly, the well-known position embedding in Transformer (Vaswani et al., 2017) can be seen as a degraded version of one of our specific complex word embeddings (see the Appendix A).
+
+# 3 EXPERIMENTAL EVALUATION
+
+We evaluate our embeddings in text classification, machine translation and language modeling.
+
+# 3.1 TEXT CLASSIFICATION
+
+Experimental Setup. We use six popular text classification datasets: CR, MPQA, SUBJ, MR, SST, and TREC (see Tab. 1). We use accuracy as evaluation measure based on fixed train/dev/test splits or cross validation, as per prior work. We use Fasttext (Joulin et al., 2016), CNN (Kim, 2014), LSTM and Transformer (Vaswani et al., 2017) as NN baselines $^{2}$ . We use each of them: (1) without positional information; (2) with Vanilla Position Embeddings (PE) (randomly initialized and updated during training using the sum between word and position vectors (Gehring et al., 2017); (3) with Trigonometric Position Embeddings (TPE) (defining position embeddings as trigonometric functions as per Eq. 7); (4) with Complex-vanilla word embeddings (where the amplitude embedding is initialized by the pre-trained word vectors, and the phrase embedding is randomly initialized in a range from  $-\pi$  to  $\pi$  without considering word order (Wang et al., 2019)); and (5) with our order-aware complex-valued word embeddings, Complex-order (which encode position in the phase parts, train the periods, and where the amplitude embedding is also initialized by pretrained word vectors). For more details on the complex-valued extensions of NNs, see App. B and App. C.
+
+Our embedding generally has  $3 \times D \times |\mathbb{W}|$  parameters with D-dimensional word vectors and  $|\mathbb{W}|$  words, while previous work (Mikolov et al., 2013b; Pennington et al., 2014) usually employs only  $D \times |\mathbb{W}|$  parameters for embedding lookup tables. To increase efficiency and facilitate fair comparison with previous work we set initial phases  $\theta_{j} = [\theta_{j,1},\dots,\theta_{j,D}]$  to a shared constant value (such as zero). Furthermore, the period vectors  $\omega_{j,d}$  depend on word index  $j$  with length  $|\mathbb{W}|$  and the coordinate index  $d$  with length  $D$ . To decrease the number of parameters, one can either use a word-sharing scheme (i.e.,  $\omega_{j,d} = \omega_{.,d}$ ), or a dimension-sharing scheme ( $\omega_{j,d} = \omega_{j,}$ ), leading to  $|\mathbb{W}|*D + |\mathbb{W}|$  and  $|\mathbb{W}|*D + D$  parameters in total for the embedding layer.
+
+Table 2: Text classification accuracy without position embeddings, with random position embeddings (PE), with trigonometric position embeddings (TPE), with complex-valued NNs without position embeddings (complex-vanilla), and with our complex-order embeddings. Superscripts  $\S$ ,  $\dagger$ ,  $\ddagger$  and * mean a significant improvement over a baseline without position embeddings  $\S$ ,  $\mathrm{PE}^{\dagger}$ ,  $\mathrm{TPE}^{\ddagger}$  and Complex-vanilla * using Wilcoxon's signed-rank test  $p < 0.05$ .  
+
+<table><tr><td>Method</td><td>MR</td><td>SUBJ</td><td>CR</td><td>MPQA</td><td>SST</td><td>TREC</td></tr><tr><td>Fasttext</td><td>0.765</td><td>0.916</td><td>0.789</td><td>0.874</td><td>0.788</td><td>0.874</td></tr><tr><td>Fasttext-PE</td><td>0.774</td><td>0.922</td><td>0.789</td><td>0.882</td><td>0.791</td><td>0.874</td></tr><tr><td>Fasttext-TPE</td><td>0.776</td><td>0.921</td><td>0.796</td><td>0.884</td><td>0.792</td><td>0.88</td></tr><tr><td>Fasttext-Complex-vanilla</td><td>0.773</td><td>0.918</td><td>0.79</td><td>0.867</td><td>0.803</td><td>0.872</td></tr><tr><td>Fasttext-Complex-order</td><td>0.787$†‡*</td><td>0.929$†‡*</td><td>0.800$†‡*</td><td>0.889$†‡*</td><td>0.809$†‡*</td><td>0.892$†‡*</td></tr><tr><td>LSTM</td><td>0.775</td><td>0.896</td><td>0.813</td><td>0.887</td><td>0.807</td><td>0.858</td></tr><tr><td>LSTM-PE</td><td>0.778</td><td>0.915</td><td>0.822</td><td>0.889</td><td>0.811</td><td>0.858</td></tr><tr><td>LSTM-TPE</td><td>0.776</td><td>0.912</td><td>0.814</td><td>0.888</td><td>0.813</td><td>0.865</td></tr><tr><td>LSTM-Complex-vanilla</td><td>0.765</td><td>0.907</td><td>0.810</td><td>0.823</td><td>0.784</td><td>0.784</td></tr><tr><td>LSTM-Complex-order</td><td>0.790$†‡*</td><td>0.926$†‡*</td><td>0.828$†‡*</td><td>0.897$†‡*</td><td>0.819$†‡*</td><td>0.869$†‡*</td></tr><tr><td>CNN</td><td>0.809</td><td>0.928</td><td>0.830</td><td>0.894</td><td>0.856</td><td>0.898</td></tr><tr><td>CNN-PE</td><td>0.816</td><td>0.938</td><td>0.831</td><td>0.897</td><td>0.856</td><td>0.890</td></tr><tr><td>CNN-TPE</td><td>0.815</td><td>0.938</td><td>0.836</td><td>0.896</td><td>0.838</td><td>0.918</td></tr><tr><td>CNN-Complex-vanilla</td><td>0.811</td><td>0.937</td><td>0.825</td><td>0.878</td><td>0.823</td><td>0.900</td></tr><tr><td>CNN-Complex-order</td><td>0.825$†‡*</td><td>0.951$†‡*</td><td>0.852$†‡*</td><td>0.906$†‡*</td><td>0.864$†‡*</td><td>0.939$†‡*</td></tr><tr><td>Transformer w/o position embedding</td><td>0.669</td><td>0.847</td><td>0.735</td><td>0.716</td><td>0.736</td><td>0.802</td></tr><tr><td>Transformer-PE</td><td>0.737</td><td>0.859</td><td>0.751</td><td>0.722</td><td>0.753</td><td>0.820</td></tr><tr><td>Transformer-TPE (Vaswani et al., 2017)</td><td>0.731</td><td>0.863</td><td>0.762</td><td>0.723</td><td>0.761</td><td>0.834</td></tr><tr><td>Transformer-Complex-vanilla</td><td>0.715</td><td>0.848</td><td>0.753</td><td>0.786</td><td>0.742</td><td>0.856</td></tr><tr><td>Transformer-Complex-order</td><td>0.746$†‡*</td><td>0.895$†‡*</td><td>0.806$†‡*</td><td>0.863$†‡*</td><td>0.813$†‡*</td><td>0.896$†‡*</td></tr></table>
+
+Table 3: Text classification accuracy.  $\star$  means that scores are reported from other papers.  
+
+<table><tr><td>Method</td><td>MR</td><td>SUBJ</td><td>CR</td><td>MPQA</td><td>SST</td><td>TREC</td></tr><tr><td>Word2vec Bow (Conneau et al., 2017)</td><td>0.777</td><td>0.909</td><td>0.798</td><td>0.883</td><td>0.797</td><td>0.836</td></tr><tr><td>Sent2Vec (Pagliardini et al., 2017)</td><td>0.763</td><td>0.912</td><td>0.791</td><td>0.872</td><td>0.802</td><td>0.858</td></tr><tr><td>QuickThoughts (Logeswaran &amp; Lee, 2018)</td><td>0.824</td><td>0.948</td><td>0.860</td><td>0.902</td><td>-</td><td>0.928</td></tr><tr><td>InferSent (Conneau et al., 2017)</td><td>0.811</td><td>0.924</td><td>0.863</td><td>0.902</td><td>0.846</td><td>0.882</td></tr><tr><td>QPDN (Wang et al., 2019)</td><td>0.801</td><td>0.927</td><td>0.810</td><td>0.870</td><td>0.839</td><td>0.882</td></tr></table>
+
+We search the hyper parameters from a parameter pool, with batch size in  $\{32,64,128\}$ , learning rate in  $\{0.001,0.0001,0.00001\}$ , L2-regularization rate in  $\{0,0.001,0.0001\}$ , and number of hidden layer units in  $\{120,128\}$ . We use pre-trained 300-dimensional vectors from word2vec (Mikolov et al., 2013a) in all models except for Transformers. The models with trainable trigonometric position embedding produce nearly identical results compared to the non-trainable version, therefore we report the result of fixed position embeddings as per Vaswani et al. (2017). We adopt narrow convolution and max pooling in CNN, with number of filters in  $\{64,128\}$ , and size of filters in  $\{3,4,5\}$ . In all Transformer models, we only use the encoder layer to extract feature information, where the layer is 1, dimension of word and inner hidden are 256 and 512 respectively, and head number is 8.
+
+Results. The results are shown in Tab. 2. Our complex-order embeddings outperform all other variations at all times. This gain in effectiveness comes at a negligible (or non-existent) cost in efficiency (it varies per NN architecture – see Fig. 2). CNNs are the best performing NN as expected following Bai et al. (2018). Tranformer NNs benefit the most from our complex-order embeddings, most likely because they are our weakest baseline. To contextualise these results, Tab. 3 shows classification accuracy of five typical approaches on the same datasets (as reported in the original papers). Our complex-order embeddings outperform all methods, except for the CR dataset, where InferSent is marginally better. Overall, our approach is on a par with the SOTA in embeddings.
+
+We perform an ablation test (Tab. 4) on Transformer because it is the most common NN to be used with position embeddings. The two period-sharing schemas (dimension-sharing and word-sharing)
+
+![](images/b4555df2bd9de7610a6e7aa8de3404f0b727ddf966d85b3deea52904d7cb9ce1.jpg)  
+Figure 2: Computation time (seconds) per epoch in Tensorflow on TITAN X GPU.
+
+Table 4: Ablation test for Transformer, showing the effect of (i) the definition of embedding layer  $(f_{d}(j,\mathrm{pos}))$ , and (ii) whether the real-part and imaginary transition share the weights, i.e.,  $\Re(W^{Q/K/V}) = \Im(W^{Q/K/V})$ .  
+
+<table><tr><td>Method</td><td>\(f_d(j, pos)\)</td><td>Setting share in \(W^{Q/K/V}\)</td><td>Params</td><td>Accuracy</td><td>\(\Delta\)</td></tr><tr><td>Transformer-complex-order</td><td>\(r_{j,d}e^{i(\omega_j,dpos)}\)</td><td>×</td><td>8.33M</td><td>0.813</td><td>-</td></tr><tr><td>adding initial phases</td><td>\(r_{j,d}e^{i(\omega_j,dpos+\theta_j,d)}\)</td><td>×</td><td>11.89M</td><td>0.785</td><td>-0.028</td></tr><tr><td>dimension-sharing period schema</td><td>\(r_{j,d}e^{i\omega_j, pos}\)</td><td>×</td><td>5.82M</td><td>0.797</td><td>-0.016</td></tr><tr><td>word-sharing period schema</td><td>\(r_{j,d}e^{i\omega_{\cdot,dpos}}\)</td><td>×</td><td>5.81M</td><td>0.805</td><td>-0.008</td></tr><tr><td>dimension-sharing amplitude schema</td><td>\(r_{j,\cdot}e^{i\omega_j, pos}\)</td><td>×</td><td>5.82M</td><td>0.798</td><td>-0.015</td></tr><tr><td>word-sharing amplitude schema</td><td>\(r_{\cdot,d}e^{i\omega_{\cdot,dpos}}\)</td><td>×</td><td>5.81M</td><td>0.804</td><td>-0.009</td></tr><tr><td>w/t encoding positions (complex-vanilla)</td><td>\(r_{j,d}e^{i\omega_j,d}\)</td><td>×</td><td>9.38M</td><td>0.764</td><td>-0.049</td></tr><tr><td>dimension-sharing period schema</td><td>\(r_{j,d}e^{i\omega_j, pos}\)</td><td>✓</td><td>4.77M</td><td>0.794</td><td>-0.019</td></tr><tr><td>word-sharing period schema</td><td>\(r_{j,d}e^{i\omega_{\cdot,dpos}}\)</td><td>✓</td><td>4.76M</td><td>0.797</td><td>-0.016</td></tr><tr><td>dimension-sharing amplitude schema</td><td>\(r_{j,\cdot}e^{i\omega_j, pos}\)</td><td>✓</td><td>4.77M</td><td>0.792</td><td>-0.021</td></tr><tr><td>word-sharing amplitude schema</td><td>\(r_{\cdot,d}e^{i\omega_{\cdot,dpos}}\)</td><td>✓</td><td>4.76M</td><td>0.801</td><td>-0.012</td></tr><tr><td>w/t encoding positions (complex-vanilla)</td><td>\(r_{j,d}e^{i\omega_j,d}\)</td><td>✓</td><td>8.33M</td><td>0.743</td><td>-0.07</td></tr><tr><td>vanilla Transformer (Vaswani et al., 2017)</td><td>\(WE_{j,d}+PE_{d}\)</td><td>-</td><td>4.1M</td><td>0.761</td><td>-0.052</td></tr></table>
+
+slightly drop performance, because fewer parameters limit the representative power. Adding initial phases also hurts performance, although we observed that the loss could decrease faster in early epochs compared to the setting without offset. The negative effect of initial phases may be due to periodicity, and  $\omega$  cannot be directly regularized with L2-norm penalties. The sharing schemes slightly decrease the performance with less parameters. More details of the learned periods/frequencies (e.g. the distributions of periods/frequencies and case studies) are shown in App. D.
+
+Note that the word-sharing schema outperform the Vanilla Transformer, (both have a comparable number of parameters). If we choose  $\Re(W^{Q/K/V}) = \Im(W^{Q/K/V})$ , the additional parameters in the embedding layers will affect much less the whole parameter scale in the multiple-layer Transformer, since a embedding layer is only used in the first layer instead of the following Transformer layers.
+
+# 3.2 MACHINE TRANSLATION
+
+Experimental Setup. We use the standard WMT 2016 English-German dataset (Sennrich et al., 2016), whose training set consists of 29,000 sentence pairs. We use four baselines: basic Attentional encoder-decoder (AED) (Bahdanau et al., 2014); AED with Byte-pair encoding (BPE) subword segmentation for open-vocabulary translation (Sennrich et al., 2016); AED with extra linguistic features (morphological, part-of-speech, and syntactic dependency labels) (Sennrich & Haddow, 2016); and a 6-layer Transformer. Our approach (Transformer Complex-order) uses a batch size of 64, a head of 8, 6 layers, the rate of dropout is 0.1, and the dimension of the word embedding is 512. The embedding layer does not use initial phases, i.e., following  $f(j, pos) = r_j e^{i(\omega_j pos)}$ . We evaluate MT performance with the Bilingual Evaluation Understudy (BLEU) measure.
+
+Table 6: Language modeling results.  $\star$  marks scores reported from other papers.  
+
+<table><tr><td>Method</td><td>BLEU</td></tr><tr><td>AED (Bahdanau et al., 2014) ★</td><td>26.8</td></tr><tr><td>AED+Linguistic (Sennrich &amp; Haddow, 2016) ★</td><td>28.4</td></tr><tr><td>AED+BPE (Sennrich et al., 2016) ★</td><td>34.2</td></tr><tr><td>Transformer (Ma et al., 2019) ★</td><td>34.5</td></tr><tr><td>Transformer complex vanilla</td><td>34.7</td></tr><tr><td>Transformer Complex-order</td><td>35.8</td></tr></table>
+
+Table 5: Machine translation results.  $\star$  marks scores reported from other papers.  
+
+<table><tr><td>Method</td><td>BPC</td></tr><tr><td>BN-LSTM (Cooijmans et al., 2016)</td><td>1.36</td></tr><tr><td>LN HM-LSTM (Chung et al., 2016)</td><td>1.29</td></tr><tr><td>RHN (Zilly et al., 2017)</td><td>1.27</td></tr><tr><td>Large mLSTM (Krause et al., 2016)</td><td>1.27</td></tr><tr><td>Transformer XL 6L (Dai et al., 2019)</td><td>1.29</td></tr><tr><td>Transformer complex vanilla</td><td>1.30</td></tr><tr><td>Transformer XL Complex-order 6L</td><td>1.26</td></tr></table>
+
+Results Tab. 5 shows the MT results. Our approach outperforms all baselines. Two things are worth noting: (1) Both the vanilla Transformer and our Transformer Complex-order outperform the three Attentional encoder-decoder baselines which are based on an LSTM encoder and decoder, even when AED uses additional features. (2) Our Transformer Complex-Order outperforms the Vanilla Transformer and complex-vanilla Transformer by 1.3 and 1.1 in absolute BLEU score respectively.
+
+# 3.3 LANGUAGE MODELING
+
+Experimental Setup. We use the text8 (Mahoney, 2011) dataset, consisting of English Wikipedia articles. The text is lowercased from a to z, and space. The dataset contains 100M characters (90M for training, 5M for dev, and 5M for testing, as per Mikolov et al. (2012)). We use as baselines BN-LSTM, LN HM-LSTM RHN and Large mLSTM, which are typical recurrent NNs for language modeling in this dataset. We evaluate performance with the Bits Per Character (BPC) measure, (the lower, the better). We run the coder in Dai et al. (2019) with 6 layers for Transformer XL 6L; our model, named Transformer XL complex-order, directly replaces the word embedding with our proposed embedding under the same setting. We choose 6 layers due to limitations in computing resources. For Transformer XL Complex-order, all other parameter settings are as for Transformer XL (Dai et al., 2019). Our complex-order model does not use initial phases.
+
+Results. We see in Tab. 6 that our method outperforms all baselines. The first four baselines (BN-LSTM, LN HM-LSTM, RHN and Large mLSTM) are based on recurrent NNs and rely on different regulation methods to become applicable in multiple-layer recurrent architectures. Transformer-based architectures can easily be stacked with multiple layers due to their advantages in parallelization, however the vanilla Transformer does not outperform the multiple-layer recurrent baselines, most likely due to its limitation of 6 layers. Our Transformer XL Complex-order outperforms its vanilla counterpart under the 6-layer setting (and also strong recurrent network baselines), demonstrating that our embedding also generalizes well in tasks with long-term dependency. With limited resources, slightly increasing the parameters in the feature layer like our proposed embedding could be more beneficial than stacking more layers with linearly increasing parameters.
+
+# 4 RELATED WORK
+
+Complex-valued NNs are not new (Georgiou & Koutsougeras, 1992; Kim & Adali, 2003; Hirose, 2003). Complex-valued weights have been used in NNs, motivated by biology (Reichert & Serre, 2013), and also as signal processing in speech recognition (Shi et al., 2006). More recently, Arjovsky et al. (2016) shifted RNNs into the complex domain and Wolter & Yao (2018) proposed a novel complex gated recurrent cell. Trabelsi et al. (2017) also developed a complex-valued NN for computer vision and audio processing.
+
+Complex numbers have also been applied to text processing like (Van Rijsbergen, 2004; Melucci, 2015; Blacoe et al., 2013). Trouillon et al. (2016) adopt complex embedding for entities in Knowledge Graph Completion to represent antisymmetric relations with Hermitian dot product. Li et al. (2019); Wang et al. (2019) extend word embeddings to complex-valued fashion in quantum probability driven NNs, seeing the overview in Wang et al. (2019). However, the physical meaning of both the complex-valued entity and word embeddings is unknown, since a complex number was considered as two real numbers in a black-box learning paradigm. Our work first links the phase in complex numbers to word position to define concrete physical meaning in document representations.
+
+# 5 CONCLUSIONS
+
+We extended word vectors to word functions with a variable i.e. position, to model the smooth shift among sequential word positions and therefore implicitly capture relative distances between words. These functions are well-defined to model the relative distances, therefore we derive a general solution in complex-valued fashion. Interestingly, the position embedding in Vaswani et al. (2017) can be considered a simplified version of our approach. We extend CNN, RNN and Transformer NNs to complex-valued versions to incorporate our complex embedding. Experiments on text classification, machine translation and language modeling show that our embeddings are more effective than vanilla position embeddings (Vaswani et al., 2017).
+
+# ACKNOWLEDGMENTS
+
+We thank Massimo Melucci and Emanuele Di Buccio for their helpful comments, Xindian Ma for his detailed experimental suggestions.
+
+This work is supported by the Quantum Access and Retrieval Theory (QUARTZ) project, which has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 721321. Peng Zhang and Donghao Zhang are supported in part by Natural Science Foundation of China (grant No. 61772363, U1636203)
+
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+
+# A LINKING TO THE POSITION EMBEDDINGS IN (VASWANI ET AL., 2017)
+
+Vaswani et al. (2017) proposed a new initialization for position embedding, resulting in comparable performance with previous one (Gehring et al., 2017) even without fine-tuning. The position embedding is empirically selected as
+
+$$
+P E _ {2 k} (\cdot , p o s) = \sin (p o s / 1 0 0 0 0 ^ {2 k / d _ {\text {m o d e l}}})
+$$
+
+$$
+P E _ {2 k + 1} (\cdot , p o s) = \cos (p o s / 1 0 0 0 0 ^ {2 k / d _ {\text {m o d e l}}}) \tag {7}
+$$
+
+Where  $pos$  is the position index,  $2k$  and  $2k + 1$  is the dimension index and  $d_{model}$  is the dimension size of embedding. The reason for choosing this position embedding was not well-explained and its general extension is unknown, leading to some difficulties to improve it.
+
+We claim that the proposed position embedding in (Vaswani et al., 2017) is a degraded version of one of our specific complex word embedding in word-sharing schema (i.e.,  $\omega_{j,d} = \omega_{\cdot ,d}$ ), in which  $p_{j,k} = 2\pi \times 10000^{2k / d_{model}}$  and the initial phases are set as zero. In our complex-valued position embedding, let  $f_{pe,k}(\cdot ,pos) = e^{i\times 10000^{2k / d_{model}}} = \cos (10000^{2k / d_{model}}pos) + i\sin (10000^{2k / d_{model}}pos)$ . Note that there exists a bi-jection between  $PE(\cdot ,pos)$  and  $f_{pe,k}(\cdot ,pos)$ :
+
+$$
+P E _ {2 k} (\cdot , p o s) = \Im (f _ {p e, k} (\cdot , p o s)),
+$$
+
+$$
+P E _ {2 k + 1} (\cdot , p o s) = \Re (f _ {p e, k} (\cdot , p o s)) \tag {8}
+$$
+
+where  $\Re$  and  $\Im$  are the operations to take the real and imaginary part of a complex-valued number. Its inverse transformation is
+
+$$
+f _ {p e, k} (\cdot , p o s) = P E _ {2 k + 1} (\cdot , p o s) + i P E _ {2 k} (\cdot , p o s) \tag {9}
+$$
+
+In our overall embedding, each dimension  $f_{k}(j, pos) = f_{we,k}(j) \odot f_{pe,k}(\cdot, pos)$  in our approaches, while it is  $E_{k}(j, pos) = WE_{k}(j) + PE_{k}(\cdot, pos)$  in (Gehring et al., 2017; Vaswani et al., 2017). Hence the position embedding in (Vaswani et al., 2017) is equivalent, albeit not identical, to our complex-valued position embedding with  $p_{j,k} = 2\pi \times 10000^{2k / d_{model}}$ . It is therefore a particular case of our complex-valued position embedding, the word-sharing schema in which all words share the same period at a certain dimension, i.e.,  $p_{j,k} = p_{\cdot,k}$  is irrelevant to the choice of  $j$ .
+
+# B INTEGRATING COMPLEX-VALUED EMBEDDING TO GENERAL NEURAL NETWORKS
+
+Neural networks are typically given real numbers as inputs and return real numbers as outputs. To accommodate complex numbers as in- and output, we devise a complex-valued version of various neural network layers i.e. complex-valued FastText with dense layer, CNN, and RNN. Unlike existing complex-valued neural networks (Trabelsi et al., 2017; Wolter & Yao, 2018), our feature layers are also converted into complex-valued layers.
+
+Complex-valued FastText FastTest (Joulin et al., 2016) is a simple and efficient neural network architecture using a dense layer over the sum of all word embeddings for general text classification. For a linear dense layer, i.e.,  $z = \text{dense}(x + iy)$ , where  $x + iy$  and  $z$  denote the complex-valued in- and output, respectively. Let  $W = A + iB$  and  $b = c + id$  be complex-valued linear weights and bias, respectively. Then, the complex-valued dense layer is given by:
+
+$$
+\boldsymbol {z} = \sigma (\boldsymbol {A} \boldsymbol {x} - \boldsymbol {B} \boldsymbol {y} + \boldsymbol {c}) + i \sigma (\boldsymbol {B} \boldsymbol {x} + \boldsymbol {A} \boldsymbol {y} + \boldsymbol {d}) \tag {10}
+$$
+
+where  $\sigma$  is a real-valued activation function such as the sigmoid function. By rewriting (10) in matrix form (Trabelsi et al., 2017), we obtain:
+
+$$
+\left[ \begin{array}{l} \Re (z) \\ \Im (z) \end{array} \right] = \left[ \begin{array}{l} \sigma (\boldsymbol {A x} - \boldsymbol {B y} + \boldsymbol {c}) \\ \sigma (\boldsymbol {B x} + \boldsymbol {A y} + \boldsymbol {d}) \end{array} \right] \tag {11}
+$$
+
+where, for  $z = x + iy$ ,  $\Re(z) = x$  and  $\Im(z) = y$ . To save parameters and fairly compare with our real-valued baselines, the weights for real-part and imaginary-part input can be shared, i.e.,  $A = B$ ,  $c = d$ .
+
+Complex-valued CNN For the complex-valued version of the convolution operation Trabelsi et al. (2017), we similarly define a complex-valued convolution with separate real and imaginary kernels  $A$  and  $B$ , to compute convolutions on the real and imaginary parts of the input in Eq. 10. A complex-valued CNN network is constructed by stacking the operations based on a complex-valued convolution kernel, adding a complex dense layer in (10), and taking their norm as the final prediction in the last layer.
+
+Complex-valued RNN The basic complex RNN formulation is:
+
+$$
+h _ {t} ^ {C} = f \left(\boldsymbol {W} ^ {h} \boldsymbol {h} _ {t - 1} + \boldsymbol {W} ^ {z} \boldsymbol {z} _ {t} + \boldsymbol {b}\right) \tag {12}
+$$
+
+where  $\mathbf{z}_t$  and  $h_t$  represent the complex-valued input and complex-value hidden state vectors at time  $t$ ,  $\mathbf{b}$  is a complex-valued bias,  $W^h$  and  $W^z$  are complex-valued weight transitions for hidden state and input state, and  $f(\mathbf{z}) = \sigma(\Re(\mathbf{z})) + i\sigma(\Im(\mathbf{z}))$  is the activation function. The multiplication  $W^h h_{t-1}$  and  $W^z z_t$  is computed as defined in (10) above. Similarly, the complex-valued gates are used in LSTM via operations as in (12). In the final layer, a l2-norm operation is adopted to obtain a real-valued loss for backpropagation.
+
+Complex-valued Transformer The main components in the Transformer are self-attention sublayers and position-wise feed-forward (FFN) sublayers. A self-attention sublayer employs  $h$  attention heads and the concatenation of all heads is used as the output followed by a parameterized linear transformation. For a sequence embedded as complex-valued vector  $input = \{w_1, w_2, \dots, w_n\}$ , the output of each head is computed as a weighted sum of a linear transformation of the input sequence itself, namely
+
+$$
+o u t p u t _ {i} = \sum_ {j} a _ {i, j} \boldsymbol {w} _ {j} \boldsymbol {W} ^ {V}, \tag {13}
+$$
+
+where  $\boldsymbol{w}_j$  is a complex-valued vector and  $\boldsymbol{W}^V$  is a complex linear transformation; therefore output $_i$  is also complex. Hence, output is a sequence of complex-valued vectors with the same shape as input. The weight coefficient,  $a_{i,j}$ , which is defined as in real-valued domain, is calculated as the softmax of the product between complex-valued query vectors and key vectors:  $a_{i,j} = \mathrm{softmax}\frac{e_{i,j}}{\sum_{i=1}^{n} e_{i,j}}$ , and
+
+$$
+e _ {i, j} = \sqrt {\frac {\Re (z) ^ {2} + \Im (z) ^ {2}}{n}}, z = \left(\boldsymbol {w} _ {i} \boldsymbol {W} ^ {Q}\right) \left(\boldsymbol {w} _ {j} \boldsymbol {W} ^ {K}\right) ^ {\dagger}, \tag {14}
+$$
+
+$z$  is a complex number since  $\boldsymbol{w}_i$  and  $\boldsymbol{w}_j$  are complex-valued vectors and  $W^Q, W^K$  are complex-valued transformation. To extend another variant of Transformer called Transformer XL, we keep its original relative position embedding and additionally replace its word embedding with our proposed embedding.
+
+Correspondingly, the FFN sublayer can easily be extended to a complex-valued version by replacing the real-valued layers with complex-valued ones. We use batch-normalization separately for the real and imaginary parts.
+
+# C IMPLEMENTATION OF THE PROPOSED EMBEDDING
+
+Words functions are implemented in neural networks by storing the function parameters  $\{r, \omega, \theta\}$  and then construct the values based on the arguments. Based on the definition, the implementation of the proposed embedding can easily be implemented with only modifying the embedding layer. We list the basic code to construct our general embedding as below:
+
+import torch   
+import math   
+class ComplexNN(torch nnModule): def__init__(self,opt): super(ComplexNN,self).__init_() self word_emb  $=$  torch nn. Embedding(opt.n_token，opt.d_model) self frequency_emb  $=$  torch nn. Embedding(opt.n_token，opt.d_model) self.initial_phase_emb  $=$  torch nn. Embedding(opt.n_token，opt.d_model)
+
+![](images/ef4ed817909f941feb6d3c689fc10f4e8b6a4cd18b9a950d8952b3f9a3ac8b45.jpg)  
+Figure 3: The distribution of the  $\delta_{j}$ . Higher values mean that the word representations are more sensitive to the word positions.
+
+```python
+def get_embedding(self, x):  
+    amplitude = self.textemb(x)  
+    frequency = self.textfrequencyemb(x)  
+    self.initial_phase_embweight = torch.nn.Parameters(self.initial_phase_emb.weight % (2 * math.pi))  
+    sent_len=x.size(-1)  
+    pos_seq = torch.arange(1, sent_len + 1, 1.0, device=amplitude_device)  
+    pos_seq = pos_seq unsqueeze(0).unsqueeze(-1)  
+    pos_seq = pos_seqrepeat([x.size(0),1,amplitude.size(-1)])  
+    dimension_bais = self.initial_phase_emb(x)  
+    enc_output_phase = torch.mul(pos_seq, frequency)+ dimension_bais  
+    enc_output_real = amplitude * torch.cos(enc_output_phase)  
+    enc_output_image = amplitude * torch.sin(enc_output_phase)  
+    # return torch.cat([enc_output_real, enc_output_image], -1)  
+    return enc_output_real, enc_output_image  
+def forward(self, x):  
+    return self.get_embedding(x)
+```
+
+Note that both the frequency vectors  $\omega$  and initial-phase vectors  $\theta$  can be shared between words or dimensions, to save parameters. The proposed embedding can be also used in real-valued neural networks if one directly concatenates the real-part numbers and imaginary-part numbers as a double-size real-valued vector; therefore it could easily be extended in any existing networks without any complex-valued components. For instance, it could be a good extension for Transformer based pretrained models like (Devlin et al., 2018) by enriching the feature layer.
+
+# D VISUALIZATION OF FREQUENCIES/PERIODS
+
+After training, we obtain the frequency vector  $\omega$  for each word. For each word, the mean value of the absolute frequency values, i.e.,  $\delta_{j} = \frac{1}{|D|}\sum_{d = 1}^{D}|\omega_{j,d}|$  is considered as a metric to test the positional sensitivities of the word, since a period value could be negative during training. The density of the  $\delta_{j}$  is shown in Fig. 3.
+
+Words with the 50 greatest, and the 50 smallest, frequencies in the SST dataset are shown in Tab. 7. For the words with greatest frequencies, most of them are strong sentiment words like "worst"
+
+“stupid” and “powerful”; a reason for this may be that such words appear in many positions in many documents during training, and thus they are more sensitive to the positions. Conversely, there are fewer words expressing strong sentiment among words with smaller frequencies, as shown in the second row.
+
+<table><tr><td></td><td>words</td></tr><tr><td>greatest frequencies in descending order</td><td>worst solid stupid powerful mess wonderful remarkable suffers intoxicating thoughtful rare captures portrait gem frontal terrific unique wannabe witty lousy pointless contrived none worse refreshingly charming inventive amazing junk incoherent refreshing mediocre unfunny thinks enjoyed heartbreaking delightfully crisp brilliant heart spirit perfectly nowhere mistake engrossing fashioned excellent unexpected wonderfully means</td></tr><tr><td>smallest frequencies in ascending order</td><td>slowly proposal schemes roiling juliette titles fabric superstar ah wow choreographed tastelessness beg fabulous muccino jacobi legendary jae rate example code sensation counter deaths hall eun drug mctiernan storylines cellophane wild motion ups trick comedy entertained mission frightening witnesses snoots liners african groan satisfaction calm saturday estranged holm refuses inquisitive</td></tr></table>
+
+Table 7: Words with greatest frequencies and frequencies periods (based on  $\delta_j$ ) in SST (a sentiment classification task), all words are converted to lower-case. The strong sentiment words are bold based on manual labeling.
\ No newline at end of file
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+# ENERGY-BASED MODELS FOR ATOMIC-RESOLUTION PROTEIN CONFORMATIONS
+
+Yilun Du*
+
+Massachusetts Institute of Technology
+
+Cambridge, MA
+
+yilundu@mit.edu
+
+Joshua Meier
+
+Facebook AI Research
+
+New York, NY
+
+jmeier@fb.com
+
+Jerry Ma
+
+Facebook AI Research
+
+Menlo Park, CA
+
+maj@fb.com
+
+Rob Fergus
+
+Facebook AI Research & New York University
+
+New York, NY
+
+robfergus@fb.com
+
+Alexander Rives
+
+New York University
+
+New York, NY
+
+arives@cs.nyu.edu
+
+# ABSTRACT
+
+We propose an energy-based model (EBM) of protein conformations that operates at atomic scale. The model is trained solely on crystallized protein data. By contrast, existing approaches for scoring conformations use energy functions that incorporate knowledge of physical principles and features that are the complex product of several decades of research and tuning. To evaluate the model, we benchmark on the rotamer recovery task, the problem of predicting the conformation of a side chain from its context within a protein structure, which has been used to evaluate energy functions for protein design. The model achieves performance close to that of the Rosetta energy function, a state-of-the-art method widely used in protein structure prediction and design. An investigation of the model's outputs and hidden representations finds that it captures physicochemical properties relevant to protein energy.
+
+# 1 INTRODUCTION
+
+Methods for the rational design of proteins make use of complex energy functions that approximate the physical forces that determine protein conformations (Cornell et al., 1995; Jorgensen et al., 1996; MacKerell Jr et al., 1998), incorporating knowledge about statistical patterns in databases of protein crystal structures (Boas & Harbury, 2007). The physical approximations and knowledge-derived features that are included in protein design energy functions have been developed over decades, building on results from a large community of researchers (Alford et al., 2017).
+
+In this work<sup>1</sup>, we investigate learning an energy function for protein conformations directly from protein crystal structure data. To this end, we propose an energy-based model using the Transformer architecture (Vaswani et al., 2017), that accepts as inputs sets of atoms and computes an energy for their configuration. Our work is a logical extension of statistical potential methods (Tanaka & Scheraga, 1976; Sippl, 1990; Lazaridis & Karplus, 2000) that fit energetic terms from data, which, in combination with physically motivated force
+
+fields, have contributed to the feasibility of de novo design of protein structures and functions (Kuhlman et al., 2003; Ambroggio & Kuhlman, 2006; Jiang et al., 2008; King et al., 2014).
+
+To date, energy functions for protein design have incorporated extensive feature engineering, encoding knowledge of physical and biochemical principles (Boas & Harbury, 2007; Alford et al., 2017). Learning from data can circumvent the process of developing knowledge-based potential functions by automatically discovering features that contribute to the protein's energy, including terms that are unknown or are difficult to express with rules or simple functions. Since energy functions are additive, terms learned by neural energy-based models can be naturally composed with those derived from physical knowledge.
+
+In principle, neural networks have the ability to identify and represent non-additive higher order dependencies that might uncover features such as hydrogen bonding networks. Such features have been shown to have important roles in protein structure and function (Guo & Salahub, 1998; Redzic & Bowler, 2005; Livesay et al., 2008), and are important in protein design (Boyken et al., 2016). Incorporation of higher order terms has been an active research area for energy function design (Maguire et al., 2018).
+
+Evaluations of molecular energy functions have used as a measure of fidelity, the ability to identify native side chain configurations (rotamers) from crystal structures where the ground-truth configuration has been masked out (Jacobson et al., 2002; Bower et al., 1997). Leaver-Fay et al. (2013) introduced a set of benchmarks for the Rosetta energy function that includes the task of rotamer recovery. In the benchmark, the ground-truth configuration of the side chain is masked and rotamers (possible configurations of the side chain) are sampled and evaluated within the surrounding molecular context (the rest of the atoms in the protein structure not belonging to the side chain). The energy function is scored by comparing the lowest-energy rotamer (as determined by the energy function) against the rotamer that was observed in the empirically-determined crystal structure.
+
+This work takes an initial step toward fully learning an atomic-resolution energy function from data. Prediction of native rotamers from their context within a protein is a restricted problem setting for exploring how neural networks might be used to learn an atomic-resolution energy function for protein design. We compare the model to the Rosetta energy function, as detailed in Leaver-Fay et al. (2013), and find that on the rotamer recovery task, deep learning-based models obtain results approaching the performance of Rosetta. We investigate the outputs and representations of the model toward understanding its representation of molecular energies and exploring relationships to physical properties of proteins.
+
+Our results open for future work the more general problem settings of combinatorial side chain optimization for a fixed backbone (Tuffery et al., 1991; Holm & Sander, 1992) and the inverse folding problem (Pabo, 1983) – the recovery of native sequences for a fixed backbone – which has also been used in benchmarking and development of molecular energy functions for protein design (Leaver-Fay et al., 2013).
+
+# 2 BACKGROUND
+
+Protein conformation Proteins are linear polymers composed of an alphabet of twenty canonical amino acids (residues), each of which shares a common backbone moiety responsible for formation of the linear polymeric backbone chain, and a differing side chain moiety with biochemical properties that vary from amino acid to amino acid. The energetic interplay of tight packing of side chains within the core of the protein and exposure of polar residues at the surface drives folding of proteins into stable molecular conformations (Richardson & Richardson, 1989; Dill, 1990).
+
+The conformation of a protein can be described through two interchangeable coordinate systems. Each atom has a set of spatial coordinates, which up to an arbitrary rotation and translation of all coordinates describes a unique conformation. In the internal coordinate system, the conformation is described by a sequence of rigid-body motions from each atom to the next, structured as a kinematic tree. The major degrees of freedom in protein conformation are the dihedral rotations (Richardson & Richardson, 1989), about the backbone
+
+bonds termed phi  $(\phi)$  and psi  $(\psi)$  angles, and the dihedral rotations about the side chain bonds termed chi  $(\chi)$  angles.
+
+Within folded proteins, the side chains of amino acids preferentially adopt configurations that are determined by their molecular structure. A relatively small number of configurations separated by high energetic barriers are accessible to each side chain (Janin et al., 1978). These configurations are called rotamers. In Rosetta and other protein design methods, rotamers are commonly represented by libraries that estimate a probability distribution over side chain configurations, conditioned on the backbone  $\phi$  and  $\psi$  torsion angles. We use the Dunbrack library (Shapovalov & Dunbrack Jr, 2011) for rotamer configurations.
+
+Energy-based models A variety of methods have been proposed for learning distributions of high-dimensional data, e.g. generative adversarial networks (Goodfellow et al., 2014) and variational autoencoders (Kingma & Welling, 2013). In this work, we adopt energy-based models (EBMs) (Dayan et al., 1995; Hinton & Salakhutdinov, 2006; LeCun et al., 2006). This is motivated by their simplicity and scalability, as well as their compelling results in other domains, such as image generation (Du & Mordatch, 2019).
+
+In EBMs, a scalar parametric energy function  $E_{\theta}(x)$  is fit to the data, with  $\theta$  set through a learning procedure such that the energy is low in regions around  $x$  and high elsewhere. The energy function maps to a probability density using the Boltzmann distribution:  $p_{\theta}(x) = \exp (-E_{\theta}(x)) / Z(\theta)$ , where  $Z = \int \exp (-E_{\theta}(x))dx$  denotes the partition function.
+
+EBMs are typically trained using the maximum-likelihood method (ML), in which  $\theta$  is adjusted to minimize  $\mathrm{KL}(p_D(x)||p_\theta (x))$ , the Kullback-Leibler divergence between the data and the model distribution. This corresponds to maximizing the log-likelihood of the data under the model:
+
+$$
+L _ {\mathrm {M L}} (\theta) = \mathbb {E} _ {x \sim p _ {D}} [ \log p _ {\theta} (x) ] = \mathbb {E} _ {x \sim p _ {D}} [ E _ {\theta} (x) - \log Z (\theta) ]
+$$
+
+Following Carreira-Perpinan & Hinton (2005), the gradient of this objective can be written as:
+
+$$
+\nabla_ {\theta} L _ {\mathrm {M L}} \approx \mathbb {E} _ {x ^ {+} \sim p _ {D}} [ \nabla_ {\theta} E _ {\theta} (x ^ {+}) ] - \mathbb {E} _ {x ^ {-} \sim p _ {\theta}} [ \nabla_ {\theta} E _ {\theta} (x ^ {-}) ]
+$$
+
+Intuitively, this gradient decreases the energy of samples from the data distribution  $x^{+}$  and increases the energy of samples drawn from the model  $x^{-}$ . Sampling from  $p_{\theta}$  can be done in a variety of ways, such as Markov chain Monte Carlo or Gibbs sampling (Hinton & Salakhutdinov, 2006), possibly accelerated using Langevin dynamics (Du & Mordatch, 2019). Our method uses a simpler scheme to approximate  $\nabla_{\theta}L_{\mathrm{ML}}$ , detailed in Section 3.4.
+
+# 3 METHOD
+
+Our goal is to score molecular configurations of the protein side chains given a fixed target backbone structure. We define an architecture for an energy-based model and describe its training procedure.
+
+# 3.1 MODEL
+
+The model calculates scalar functions,  $f_{\theta}(A)$ , of size-  $k$  subsets,  $A$ , of atoms within a protein.
+
+Selection of atom subsets In our experiments, we choose  $A$  to be nearest-neighbor sets around the residues of the protein and set  $k = 64$ . For a given residue, we construct  $A$  to be the  $k$  atoms that are nearest to the residue's beta carbon.
+
+Atom input representations Each atom in  $A$  is described by its 3D Cartesian coordinates and categorical features: (i) the identity of the atom (N, C, O, S); (ii) an ordinal label of the atom in the side chain (i.e. which specific carbon, nitrogen, etc. atom it is in the side chain) and (iii) the amino acid type (which of the 20 types of amino acids the atom belongs to). The coordinates are normalized to have zero mean across the  $k$
+
+![](images/43285e9072d57377353e135facc8af452997efd3d160b9f83dc25e7aa85f8709.jpg)  
+Figure 1: Overview of the model. The model takes as input a set of atoms,  $A$ , consisting of the rotamer to be predicted (shown in green) and surrounding atoms (shown in dark grey). The Cartesian coordinates and attributes of each atom are embedded. The set of embeddings is processed by Transformer blocks, and the final hidden representations are pooled over the atoms to produce a vector. The vector is passed through a two-layer MLP to output a scalar energy value,  $f_{\theta}(A)$ .
+
+atoms. Each categorical feature is embedded into 28 dimensions, and the spatial coordinates are projected into 172 dimensions², which are then concatenated into a 256-dimensional atom representation. The parameters for the input embeddings and projections of spatial information are learned via training. During training, a random rotation is applied to the coordinates in order to encourage rotational invariance of the model. For visualizations, a fixed number of random rotations (100) is applied and the results are averaged.
+
+Architecture In our proposed approach,  $f_{\theta}(A)$  takes the form of a Transformer model (Vaswani et al., 2017) that processes a set of atom representations. The self-attention layers allow each atom to attend to the representations of other atoms in the set, modeling the energy of the molecular configuration as a non-linear interaction of single, pairwise, and higher-order interactions between the atoms. The final hidden representations of the Transformer are pooled across the atoms to produce a single vector, which is finally passed to a two-layer multilayer perceptron (MLP) that produces the scalar output of the model. Figure 1 illustrates the model.
+
+For all experiments, we use a 6-layer Transformer with embedding dimension of 256 (split over 8 attention heads) and feed-forward dimension of 1024. The final MLP contains 256 hidden units. The models are trained without dropout. Layer normalization (Ba et al., 2016) is applied before the attention blocks.
+
+# 3.2 PARAMETERIZATION OF PROTEIN CONFORMATIONS
+
+The structure of a protein can be represented by two parameterizations: (1) absolute Cartesian coordinates of the set of atoms, and (2) internal coordinates of the atoms encoded as a set of in-plane/out-of-plane rotations and displacements relative to each atom's reference frame. Out-of-plane rotations are parameterized by  $\chi$  angles which are the primary degrees of freedom in the rotamer configurations. The coordinate systems are interchangeable.
+
+# 3.3 USAGE AS AN ENERGY FUNCTION
+
+We specify our energy function  $E_{\theta}(x,c)$  to take an input set composed of two parts: (1) the atoms belonging to a rotamer to be predicted,  $x$ , and (2) the atoms of the surrounding molecular context,  $c$ . The energy function is defined as follows:
+
+$$
+E _ {\theta} (x, c) = f _ {\theta} (A (x, c))
+$$
+
+where  $A(x,c)$  is the set of embeddings from  $k$  atoms nearest to the rotamer's beta carbon.
+
+# 3.4 TRAINING AND LOSS FUNCTIONS
+
+In all experiments, the energy function is trained to learn the conditional distribution of the rotamer given its context by approximately maximizing the log likelihood of the data.
+
+$$
+\mathcal {L} (\theta) = - E _ {\theta} (x, c) - \log Z _ {\theta} (c)
+$$
+
+To estimate the partition function, we note that:
+
+$$
+\log Z _ {\theta} (c) = \log \int e ^ {- E _ {\theta} (x, c)} d x = \log (\mathbb {E} _ {q (x | c)} [ \frac {e ^ {- E _ {\theta} (x , c)}}{q (x | c)} ])
+$$
+
+for some importance sampler  $q(x|c)$ . Furthermore, if we assume  $q(x|c)$  is uniformly distributed on supported configurations, we obtain a simplified maximum likelihood objective given by
+
+$$
+\mathcal {L} (\theta) = - E _ {\theta} (x, c) - \log \left(\mathbb {E} _ {q (x ^ {i} | c)} [ e ^ {- E _ {\theta} (x ^ {i}, c)} ]\right)
+$$
+
+for some context dependent importance sampler  $q(x|c)$ . We choose our sampler  $q(x|c)$  to be an empirically collected rotamer library (Shapovalov & Dunbrack Jr, 2011) conditioned on the amino acid identity and the backbone  $\phi$  and  $\psi$  angles. We write the importance sampler as a function of atomic coordinates which are interchangeable with the angular coordinates in the rotamer library. The library consists of lists of means and standard deviations of possible  $\chi$  angles for each 10 degree interval for both  $\phi$  and  $\psi$ . We sample rotamers uniformly from this library, given by a continuous  $\phi$  and  $\psi$ , by sampling from a weighted mixture of Gaussians of  $\chi$  angles at each of the four surrounding bins, with weights given by distance to the bins via bilinear interpolation. Every candidate rotamer at each bin is assigned uniform probability. To ensure our context dependent importance sampler effectively samples high likelihood areas in the model, we further add the real context as a sample from  $q(x|c)$ .
+
+Training setup Models were trained for 180 thousand parameter updates using 32 NVIDIA V100 GPUs, a batch size of 16,384, and the Adam optimizer  $(\alpha = 2 \cdot 10^{-4}, \beta_{1} = 0.99, \beta_{2} = 0.999)$ . We evaluated training progress using a held-out  $5\%$  subset of the training data as a validation set.
+
+# 4 EXPERIMENTS
+
+# 4.1 DATASETS
+
+We constructed a curated dataset of high-resolution PDB structures using the CullPDB database, with the following criteria: resolution finer than  $1.8\AA$ ; sequence identity less than  $90\%$ ; and R value less than 0.25 as defined in Wang & R. L. Dunbrack (2003). To test the model on rotamer recovery, we use the test set of structures from Leaver-Fay et al. (2013). To prevent training on structures that are similar to those in the test set, we ran BLAST on sequences derived from the PDB structures and removed all train structures with more than  $25\%$  sequence identity to sequences in the test dataset. Ultimately, our train dataset consisted of 12,473 structures and our test dataset consisted of 129 structures.
+
+<table><tr><td>Model</td><td>Avg</td><td>Buried</td><td>Surface</td></tr><tr><td>Rosetta score12 (rotamer-trials)</td><td>72.2 (72.6)</td><td>-</td><td>-</td></tr><tr><td>Rosetta ref2015 (rotamer-trials)</td><td>73.6</td><td>-</td><td>-</td></tr><tr><td>Atom Transformer</td><td>70.4</td><td>87.0</td><td>58.3</td></tr><tr><td>Atom Transformer (ensemble)</td><td>71.5</td><td>89.2</td><td>59.9</td></tr></table>
+
+Table 1: Rotamer recovery of energy functions under the discrete rotamer sampling method detailed in Section 4.2.1. Parentheses denote value reported by Leaver-Fay et al. (2013).
+
+# 4.2 BASELINES
+
+We compare to three baseline neural network architectures: a fully-connected network, the architecture for embedding sets in the set2set paper (Vinyals et al., 2015); and a graph neural network (Velicković et al., 2017). All models have around 10 million parameters. Details of the baseline architectures are given in Appendix A.1.2.
+
+Results are also compared to Rosetta. We ran Rosetta using score12 and and ref15 energy functions using the rotamer trials and rtmin protocols with default settings.
+
+# 4.2.1 EVALUATION
+
+For the comparison of the model to Rosetta in Table 1, we reimplement the sampling scheme that Rosetta uses for rotamer trials evaluation. We take discrete samples from the rotamer library, with bilinear interpolation of the mean and standard deviations using the four grid points surrounding the backbone  $\phi$  and  $\psi$  angles for the residue. We take discrete samples of the rotamers at  $\mu$ , except that for buried residues we sample  $\chi_{1}$  and  $\chi_{2}$  at  $\mu$  and  $\mu \pm \sigma$  as was done in Leaver-Fay et al. (2013). We define buried residues to have  $\geq 24C_{\beta}$  neighbors within  $10\AA$  of the residue's  $C_{\beta}$  ( $C_{\alpha}$  for glycine residues). For buried positions we accumulate rotamers up to  $98\%$  of the distribution, and for other positions the accumulation is to  $95\%$ . We score a rotamer as recovered correctly if all  $\chi$  angles are within  $20^{\circ}$  of the ground-truth residue.
+
+We also use a continuous sampling scheme which approximates the empirical conditional distribution of the rotamers using a mixture of Gaussians with means and standard deviations computed by bilinear interpolation as above. Instead of sampling discretely, the component rotamers are sampled with the probabilities given by the library, and a sample is generated with the corresponding mean and standard deviation. This is the same sampling scheme used to train models, but with component rotamers now weighted by probability as opposed to uniform sampling.
+
+# 4.3 ROTAMER RECOVERY RESULTS
+
+Table 1 directly compares our EBM model (which we refer to as the Atom Transformer) with two versions of the Rosetta energy function. We run Rosetta on the set of 152 proteins from the benchmark of Leaver-Fay et al. (2013). We also include published performance on the same test set from Leaver-Fay et al. (2013). As discussed above, comparable sampling strategies are used to evaluate the models, enabling a fair comparison of the energy functions. We find that a single model evaluated on the benchmark performs slightly worse than both versions of the Rosetta energy function. An ensemble of 10 models improves the results.
+
+Table 2 evaluates the performance of the energy function under alternative sampling strategies with the goal of optimizing recovery rates. We indicate performance of the Rosetta energy function on recovery rates using the rtmin protocol for continuous minimization. We evaluate the learned energy function with the continuous sampling from a mixture of Gaussians conditioned on the  $\phi/\psi$  settings of the backbone angles as detailed
+
+<table><tr><td>Model</td><td>Avg</td><td>Buried</td><td>Surface</td></tr><tr><td>Fully-connected</td><td>39.1</td><td>54.4</td><td>30.0</td></tr><tr><td>Set2set</td><td>43.2</td><td>60.3</td><td>31.7</td></tr><tr><td>GraphNet</td><td>69.0</td><td>94.3</td><td>54.2</td></tr><tr><td>Atom Transformer</td><td>73.1</td><td>91.1</td><td>58.3</td></tr><tr><td>Atom Transformer (ensemble)</td><td>74.1</td><td>91.2</td><td>59.5</td></tr><tr><td>Rosetta score12 (rt-min)</td><td>75.4 (74.2)</td><td>-</td><td>-</td></tr><tr><td>Rosetta ref2015 (rt-min)</td><td>76.4</td><td>-</td><td>-</td></tr></table>
+
+Table 2: Rotamer recovery of energy functions under continuous optimization schemes. Rosetta continuous optimization is performed with the rtmin protocol. Parentheses denote value reported by Leaver-Fay et al. (2013).  
+
+<table><tr><td>Amino Acid</td><td>R</td><td>K</td><td>M</td><td>I</td><td>L</td><td>S</td><td>T</td><td>V</td></tr><tr><td>Atom Transformer</td><td>37.2</td><td>31.7</td><td>53.0</td><td>93.3</td><td>82.6</td><td>79.0</td><td>96.5</td><td>94.0</td></tr><tr><td>Rosetta score12</td><td>26.7</td><td>31.7</td><td>49.6</td><td>85.4</td><td>87.5</td><td>72.5</td><td>92.6</td><td>94.3</td></tr><tr><td>Amino Acid</td><td>N</td><td>D</td><td>Q</td><td>E</td><td>H</td><td>W</td><td>F</td><td>Y</td></tr><tr><td>Atom Transformer</td><td>67.4</td><td>76.0</td><td>40.8</td><td>49.8</td><td>65.5</td><td>83.5</td><td>80.3</td><td>77.6</td></tr><tr><td>Rosetta score12</td><td>56.8</td><td>60.4</td><td>30.7</td><td>33.6</td><td>55.0</td><td>85.0</td><td>85.4</td><td>82.9</td></tr></table>
+
+Table 3: Comparison of rotamer recovery rates by amino acid between Rosetta and the ensembled energy-based model under discrete rotamer sampling. The model appears to perform well on polar amino acids glutamine, serine, asparagine, and threonine, while Rosetta performs better on larger amino acids phenylalanine, tyrosine, and tryptophan and the common amino acid, leucine. The numbers reported for Rosetta are from Leaver-Fay et al. (2013).
+
+above. We find that with ensembling the model performance is close to that of the Rosetta energy functions. We also compare to three baselines for embedding sets with similar numbers of parameters to the Atom Transformer model and find that they have weaker performance.
+
+Buried residues are more constrained in their configurations by tight packing of the side chains within the core of the protein. In comparison, surface residues are more free to vary. Therefore we also report performance separately on both categories. We find that the ensembled Atom Transformer has a  $91.2\%$  rotamer recovery rate for buried residues, compared to  $59.5\%$  for surface residues.
+
+Table 3 reports recovery rates by residue comparing the Rosetta score12 results reported in Leaver-Fay et al. (2013) to the Atom Transformer model using the Rosetta discrete sampling method. The Atom Transformer model appears to perform well on smaller rotameric amino acids as well as polar amino acids such as glutamate/aspartate while Rosetta performs better on larger amino acids like phenylalanine and tryptophan and more common ones like leucine.
+
+# 4.4 VISUALIZING ENERGIES
+
+In this section, we visualize and understand how the Atom Transformer models the energy of rotamers in their native contexts. We explore the response of the model to perturbations in the configuration of side chains away from their native state. We retrieve all protein structures in the test set and individually perturb rotameric  $\chi$  angles across the unit circle, plotting results in Figures 2, 3, and 4.
+
+![](images/e75bdb8d1bb1932b9cfbd27b8a80ffe820d64ca2a6ce75900b67a241c4f5f72f.jpg)  
+Figure 2: The energy function models distinct behavior between core and surface residues. Core residues are more sensitive to perturbations away from the native state in the  $\chi_{1}$  torsion angle. On average, residues closer to the core have a steeper energy well.
+
+![](images/d2279e3a247d54496e28da300df5caf7185afca3e7fe9667bcdb9d88af0bec5f.jpg)  
+Figure 3: There is a relation between the residue size and the depth of the energy well, with larger amino acids (e.g. Trp, Phe, Thr, Lys) having steeper wells.
+
+Core/Surface Energies Figure 2 shows that steeper response to variations away from the native state is observed for residues in the core of the protein (having  $\geq 24$  contacting side chains) than for residues on the surface ( $\leq 16$ ), consistent with the observation that buried side chains are tightly packed (Richardson & Richardson, 1989).
+
+Rotameric Energies Figure 3 shows a relation between the residue size and the depth of the energy well, with larger amino acids having steeper wells (more sensitive to perturbations). Furthermore Figure 4 shows that the model learns the symmetries of amino acids. We find that responses to perturbations of the  $\chi_{2}$  angle for the residues Tyr, Asp, and Phe are symmetric about  $\chi_{2}$ . A  $180^{\circ}$  periodicity is observed, in contrast to the non-symmetric residues.
+
+Embeddings of Atom Sets Building on the observation of a relation between the depth of the residue and its response to perturbation from the native state, we ask whether core and surface residues are clustered within the representations of the model. To visualize the final hidden representation of the molecular contexts within a protein, we compute the final vector embedding for the 64 atom context around the carbon-  $\beta$  atom (or for glycine, the carbon-  $\alpha$  atom) for each residue. We find that a projection of these representations by t-SNE (Maaten & Hinton, 2008) into 2 dimensions shows a clear clustering between representations of core residues and surface residues. A representative example is shown in Figure 5.
+
+![](images/032570dbcbe6a3aaaf167f6d5d8e05191a9b4d4cbf14dc3b1ea7f7fdea594a3a.jpg)  
+Figure 4: Note the periodicity for the amino acids Tyr, Asp, and Phe with terminal symmetry about  $\chi_{2}$ .
+
+Saliency Map The 10-residue protease-binding loop in a chymotrypsin inhibitor from barley seeds is highly structured due to the presence of backbone-backbone and backbone-sidechain hydrogen bonds in the same residue (Das, 2011). To visualize the dependence of the energy function on individual atoms, we compute the energy of the 64 atom context centered around the backbone carbonyl oxygen of residue 39 (isoleucine) in PDB: 2CI2 (McPhalen & James, 1987) and derive the gradients with respect to the input atoms. Figure 6 overlays the gradient magnitudes on the structure, indicating the model attends to both sidechain and backbone atoms, which participate in hydrogen bonds.
+
+![](images/3d5b0ea5a875dfc11ea99040d43af6ae1e2d2c87676b1e1558c63c9121cc53cc.jpg)
+
+![](images/60b132afa827ac39ee69285dc77f6cec160e63237c1edc1dad0810e16e9e1691.jpg)
+
+![](images/c0b4fcd0c419bb292a9f0dc4cbeaa9c1ccac61c107c7fc2013b8904ee56d7d52.jpg)  
+Figure 5: Left: 3-dimensional representation of CcmG reducing oxidoreductase (PDB ID 1KNG; Edeling et al., 2002), a protein from the test set. Atoms are colored dark blue (buried), orange (exposed), or neither (not colored). Right: t-SNE (Maaten & Hinton, 2008) projection of EBM hidden representation when focused on the alpha carbon atom for each residue in the hidden representation. In the embedding space, buried and surface residues are distinguished.  
+Figure 6: The model's saliency map applied to the test protein, serine proteinase inhibitor (PDB ID: 2CI2; McPhalen & James (1987)). The 64 atom context is centered on the carbonyl oxygen of residue 39 (isoleucine). Atoms in the context are labeled red with color saturation proportional to gradient magnitude (interaction strength). Hydrogen bonds with the carbonyl oxygen are shown by dotted lines.
+
+# 5 RELATED WORK
+
+Energy functions have been widely used in the modeling of protein conformations and the design of protein sequences and structures (Boas & Harbury, 2007). Rosetta, for example, uses a combination of physically motivated terms and knowledge-based potentials (Alford et al., 2017) to model proteins and other macromolecules.
+
+Leaver-Fay et al. (2013) proposed optimizing the feature weights and parameters of the terms of an energy function for protein design; however their method used physical features designed with expert knowledge and data analysis. Our work draws on their development of rigorous benchmarks for energy functions, but in contrast automatically learns complex features from data.
+
+Neural networks have also been explored for protein folding. Xu (2018) developed a deep residual network that predicts the pairwise distances between residues in the protein structure from evolutionary covariation information. Senior et al. (2018) used evolutionary covariation to predict pairwise distance distributions,
+
+using maximization of the probability of the backbone structure with respect to the predicted distance distribution to fold the protein. Ingraham et al. (2018) proposed learning an energy function for protein folding by backpropagating through a differentiable simulator. AlQuraishi (2019) investigated predicting protein structure from sequence without using co-evolution.
+
+Deep learning has shown practical utility in the related field of small molecule chemistry. Gilmer et al. (2017) achieved state-of-the-art performance on a suite of molecular property benchmarks. Similarly, Feinberg et al. (2018) achieved state-of-the-art performance on predicting the binding affinity between proteins and small molecules using graph convolutional networks. Mansimov et al. (2019) used a graph neural network to learn an energy function for small molecules. In contrast to our work, these methods operate over small molecular graphs and were not applied to large macromolecules, like proteins.
+
+In parallel, recent work proposes that generative models pre-trained on protein sequences can transfer knowledge to downstream supervised tasks (Bepler & Berger, 2019; Alley et al., 2019; Yang et al., 2019; Rives et al., 2019). These methods have also been explored for protein design (Wang et al., 2018).
+
+Generative models of protein structures have also been proposed for generating protein backbones (Anand & Huang, 2018) and for the inverse protein folding problem (Ingraham et al., 2019).
+
+# 6 DISCUSSION
+
+In this work we explore the possibility of learning an energy function of protein conformations at atomic resolution. We develop and evaluate the method in the benchmark problem setting of recovering protein side chain conformations from their native context, finding that a learned energy function nears the performance in this restricted domain to energy functions that have been developed through years of research into approximation of the physical forces guiding protein conformation and engineering of statistical terms.
+
+The method developed here models sets of atoms and can discover and represent the energetic contribution of high order dependencies within its inputs. We find that learning an energy function from the data of protein crystal structures automatically discovers features relevant to computing molecular energies; and we observe that the model responds to its inputs in ways that are consistent with an intuitive understanding of protein conformation and energy.
+
+Generative biology proposes that the design principles used by nature can be automatically learned from data and can be harnessed to generate and design new biological molecules and systems (Rives et al., 2019). High-fidelity generative modeling for proteins, operating at the level of structures and sequences, can enable generative protein design. To create new proteins outside the space of those discovered by nature, it is necessary to use design principles that generalize to all proteins. Huang et al. (2016) have argued that since the physical principles that govern protein conformation apply to all proteins, encoding knowledge of these physical and biochemical principles into an energy function will make it possible to design de novo new protein structures and functions that have not appeared before in nature.
+
+Learning features from data with generative methods is a possible direction for realizing this goal to enable design in the space of sequences not visited by evolution. The generalization of neural energy functions to harder problem settings used in the protein design community, e.g. combinatorial side chain optimization (Tuffery et al., 1991; Holm & Sander, 1992), and inverse-folding (Pabo, 1983), is a direction for future work. The methods explored here have the potential for extension into these settings.
+
+# ACKNOWLEDGMENTS
+
+We thank Kyunghyun Cho, Siddharth Goyal, Andrew Leaver-Fay, and Yann LeCun for helpful discussions. Alexander Rives was supported by NSF Grant #1339362.
+
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+
+# A.1 APPENDIX
+
+# A.1.1 PSEUDOCODE FOR THE TRAINING ALGORITHM
+
+Pseudocode for the training algorithm is given below in Algorithm 1.
+
+Algorithm 1 Training Procedure for the EBM  
+Input: Rotamer library  $q(x|c)$ , Training set of proteins  $D$   
+for Protein  $d_i$  of  $D$  do  
+ $\triangleright$  Sample random amino acid from  $d_i$ $R \sim d_i$ $\triangleright$  Set positive sample to 64 nearest neighbor atoms of carbon beta of  $R$ $c^+ \gets \mathrm{NN}_{64}(R)$ $\triangleright$  Generate  $N$  negative samples from the rotamer library  
+ $c^- \gets q(x|c^+)$ $\triangleright$  Compute loss of model (logsumexp across all negative samples)  
+ $L_{ml} = E(c^+;\theta) + \log \mathrm{sumexp}(-E(c^+;\theta), -E(c_0^-;\theta), -E(c_1^-;\theta), \ldots, -E(c_N^-;\theta))$ $\triangleright$  Minimization step of  $L_{ml}$  using Adam optimizer  
+ $\theta \gets \theta - \nabla_\theta L_{ml}$   
+end for
+
+# A.1.2 MODEL ARCHITECTURE
+
+Architectural descriptions are provided below for each the three neural network baselines as well as for the Atom Transformer. For Set2set models, we use 6 processing steps of computation. For graph networks, we add residual connections between each layer.
+
+<table><tr><td>Embed Each Atom to 256 Dim</td></tr><tr><td>Flatten</td></tr><tr><td>Dense → 1024</td></tr><tr><td>1024 → 1024</td></tr><tr><td>1024 → 1024</td></tr><tr><td>ResBlock down 256</td></tr><tr><td>Global Mean Pooling</td></tr><tr><td>Dense → 1</td></tr></table>
+
+(a) Fully Connected Model  
+
+<table><tr><td>Embed Each Atom to 256 Dim</td></tr><tr><td>Dense → 1024</td></tr><tr><td>Repeat (6x):</td></tr><tr><td>LSTM 2048
+Attention 2048 → 128 → 1</td></tr><tr><td>End Repeat</td></tr><tr><td>Dense → 1024</td></tr><tr><td>1024 → 1</td></tr></table>
+
+(b) Set2Set Model (6 Permutation Invariant Blocks)
+
+Figure A1: Architectures for Fully Connected and Set2Set Baselines
+
+![](images/ac9aa768e1204c7fc5c86080ef38356338ac51e98136ab54cfe5ddc66f4f598e.jpg)  
+Figure A2: Graph Network Model Architecture (9 Graph Attention Blocks)
+
+<table><tr><td>Embed Each Atom to 256 Dim</td></tr><tr><td>Transformer Encoder Block (8 heads, feedforward dim 1024, 256 encoder dim)</td></tr><tr><td>Transformer Encoder Block (8 heads, feedforward dim 1024, 256 encoder dim)</td></tr><tr><td>Transformer Encoder Block (8 heads, feedforward dim 1024, 256 encoder dim)</td></tr><tr><td>Transformer Encoder Block (8 heads, feedforward dim 1024, 256 encoder dim)</td></tr><tr><td>Transformer Encoder Block (8 heads, feedforward dim 1024, 2.56 encoder dim)</td></tr><tr><td>Transformer Encoder Block (8 heads, feedforward dim 1024, 256 encoder dim)</td></tr><tr><td>Global Max Pooling</td></tr><tr><td>dense → 1</td></tr></table>
+
+Figure A3: Atom Transformer Model (6 Transformer Encoder Blocks)
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+# ENHANCING ADVERSARIAL DEFENSE BY  $k$ -WINNERS-TAKE-ALL
+
+Chang Xiao Peilin Zhong Changxi Zheng
+
+Columbia University
+
+{chang, peilin, cxz}@cs.columbia.edu
+
+# ABSTRACT
+
+We propose a simple change to existing neural network structures for better defending against gradient-based adversarial attacks. Instead of using popular activation functions (such as ReLU), we advocate the use of  $k$ -Winners-Take-All ( $k$ -WTA) activation, a  $C^0$  discontinuous function that purposely invalidates the neural network model's gradient at densely distributed input data points. The proposed  $k$ -WTA activation can be readily used in nearly all existing networks and training methods with no significant overhead. Our proposal is theoretically rationalized. We analyze why the discontinuities in  $k$ -WTA networks can largely prevent gradient-based search of adversarial examples and why they at the same time remain innocuous to the network training. This understanding is also empirically backed. We test  $k$ -WTA activation on various network structures optimized by a training method, be it adversarial training or not. In all cases, the robustness of  $k$ -WTA networks outperforms that of traditional networks under white-box attacks.
+
+# 1 INTRODUCTION
+
+In the tremendous success of deep learning techniques, there is a grain of salt. It has become well-known that deep neural networks can be easily fooled by adversarial examples (Szegedy et al., 2014). Those deliberately crafted input samples can mislead the networks to produce an output drastically different from what we expect. In many important applications, from face recognition authorization to autonomous cars, this vulnerability gives rise to serious security concerns (Barreno et al., 2010; 2006; Sharif et al., 2016; Thys et al., 2019).
+
+Attacking the network is straightforward. Provided a labeled data item  $(x, y)$ , the attacker finds a perturbation  $x'$  perceptually similar to  $x$  but misleading enough to cause the network to output a label different from  $y$ . By far, the most effective way of finding such a perturbation (or adversarial example) is by exploiting the gradient information of the network with respect to its input: the gradient indicates how to perturb  $x$  to trigger the maximal change of  $y$ .
+
+The defense, however, is challenging. Recent studies showed that adversarial examples always exist if one tends to pursue a high classification accuracy—adversarial robustness seems at odds with the accuracy (Tsipras et al., 2018; Shafahi et al., 2019a; Su et al., 2018a; Weng et al., 2018; Zhang et al., 2019). This intrinsic difficulty of eliminating adversarial examples suggests an alternative path: can we design a network whose adversarial examples are evasive rather than eliminated? Indeed, along with this thought is a series of works using obfuscated gradients as a defense mechanism (Athalye et al., 2018). Those methods hide the network's gradient information by artificially discretizing the input (Buckman et al., 2018; Lin et al., 2019) or introducing certain randomness to the input (Xie et al., 2018a; Guo et al., 2018) or the network structure (Dhillon et al., 2018; Cohen et al., 2019) (see more discussion in Sec. 1.1). Yet, the hidden gradient in those methods can still be approximated, and as recently pointed out by Athalye et al. (2018), those methods remain vulnerable.
+
+Technical contribution I) Rather than obfuscating the network's gradient, we make the gradient undefined. This is achieved by a simple change to the standard neural network structure: we advocate the use of a  $C^0$  discontinuous activation function, namely the  $k$ -Winners-Take-All ( $k$ -WTA) activation, to replace the popular activation functions such as rectified linear units (ReLU). This is the only change we propose to a deep neural network. All other components (such as BatchNorm, convolution, and pooling) as well as the training methods remain unaltered. With no significant overhead,  $k$ -WTA activation can be readily used in nearly all existing networks and training methods.
+
+$k$ -WTA activation takes as input the entire output of a layer, retains its  $k$  largest values and deactivates all others to zero. As we will show in this paper, even an infinitesimal perturbation to the input may cause a complete change to the network neurons' activation pattern, thereby resulting in a large jump in the network's output. This means that, mathematically, if we use  $f(\pmb{x}; \pmb{w})$  to denote a  $k$ -WTA network taking an input  $\pmb{x}$  and parameterized by weights  $\pmb{w}$ , then the gradient  $\nabla_{\pmb{x}} f(\pmb{x}; \pmb{w})$  at certain  $\pmb{x}$  is undefined— $f(\pmb{x}; \pmb{w})$  is  $C^0$  discontinuous.
+
+Technical contribution II) More intriguing than the mere replacement of the activation function is why  $k$ -WTA helps improve the adversarial robustness. We offer our theoretical reasoning of its behavior from two perspectives. On the one hand, we show that the discontinuities of  $f(\pmb{x}; \pmb{w})$  is densely distributed in the space of  $\pmb{x}$ . Dense enough such that a tiny perturbation from any  $\pmb{x}$  almost always comes across some discontinuities, where the gradients are undefined and thus the attacker's search of adversarial examples becomes blinded (see Figure 1).
+
+On the other hand, a paradox seemingly exists. The discontinuities in the activation function also renders  $f(\pmb{x}; \pmb{w})$  discontinuous with respect to the network weights  $\pmb{w}$  (at
+
+certain  $\boldsymbol{w}$  values). But training the network relies on the presumption that the gradient with respect to the weights is almost always available. Interestingly, we show that, under  $k$ -WTA activation, the discontinuities of  $f(\boldsymbol{x}; \boldsymbol{w})$  is rather sparse in the space of  $\boldsymbol{w}$ , intuitively because the dimension of  $\boldsymbol{w}$  (in parameter space) is much larger than the dimension of  $\boldsymbol{x}$  (in data space). Thus, the network can be trained successfully.
+
+![](images/d5daaa30c4f51f5212f0ce6bc0f09fe16aa9d9cf367deaf38493459b3f325994.jpg)  
+Figure 1: 1D illustration. Fit a 1D function (green dotted curve) using a  $k$ -WTA model provided with a set of points (red). The resulting model is piecewise continuous (blue curve), and the discontinuities can be dense.
+
+Summary of results. We conducted extensive experiments on multiple datasets under different network architectures, including ResNet (He et al., 2016), DenseNet (Huang et al., 2017), and Wide ResNet (Zagoruyko & Komodakis, 2016), that are optimized by regular training as well as various adversarial training methods (Madry et al., 2017; Zhang et al., 2019; Shafahi et al., 2019b).
+
+In all these setups, we compare the robustness performance of using the proposed  $k$ -WTA activation with commonly used ReLU activation under several white-box attacks, including PGD (Kurakin et al., 2016), Deepfool (Moosavi-Dezfooli et al., 2016), C&W (Carlini & Wagner, 2017), MIM (Dong et al., 2018), and others. In all tests,  $k$ -WTA networks outperform ReLU networks.
+
+The use of  $k$ -WTA activation is motivated for defending against gradient-based adversarial attacks. Our experiments suggest that the robustness improvement gained by simply switching to  $k$ -WTA activation is universal, not tied to specific network architectures or training methods. To promote reproducible research, we will release our implementation of  $k$ -WTA networks, along with our experiment code, configuration files and pre-trained models<sup>1</sup>.
+
+# 1.1 RELATED WORK: OBFUSCATED GRADIENTS AS A DEFENSE MECHANISM
+
+Before delving into  $k$ -WTA details, we review prior adversarial defense methods that share the same philosophy with our method and highlight our advantages. For a review of other attack and defense methods, we refer to Appendix A.
+
+Methods aiming for concealing the gradient information from the attacker has been termed as obfuscated gradients (Athalye et al., 2018) or gradient masking (Papernot et al., 2017; Tramère et al., 2017) techniques. One type of such methods is by exploiting randomness, either randomly transforming the input before feeding it to the network (Xie et al., 2018a; Guo et al., 2018) or introducing stochastic layers in the network (Dhillon et al., 2018). However, the gradient information in these methods can be estimated by taking the average over multiple trials (Athalye et al., 2018; 2017). As a result, these methods are vulnerable.
+
+Another type of obfuscated gradient methods relies on the so-called shattered gradient (Athalye et al., 2018), which aims to make the network gradients nonexistent or incorrect to the attacker, by purposely discretizing the input (Buckman et al., 2018; Ma et al., 2018) or artificially raising numerical instability for gradient evaluation (Song et al., 2018; Samangouei et al., 2018). Unfortunately, these methods are also vulnerable. As shown by Athalye et al. (2018), they can be compromised by backward pass
+
+![](images/affb89ded4dc05c771c54fde0f6f04b784e91b2ac03c606aa8a44375988c110e.jpg)  
+Figure 2: Different activation functions. ReLU: all neurons with negative activation values will be set to zero. Max-pooling: only the largest activation in each group is transmitted to the next layer, and this effectively downsample the output. LWTA: the largest activation in each group retains its value when entering the next layer, others are set to zero.  $k$ -WTA: the  $k$  largest activations in the entire layer retain their values when entering the next layer, others are set to zero ( $k = 3$  in this example). Note that the output is not downsampled through ReLU, LWTA and  $k$ -WTA.
+
+![](images/0198bcc573332d2b577b153be6e1a2c4194fa9c6285300ee9e613e01b2e30855.jpg)
+
+![](images/301cb23db311fbb9a92a60fa495907370ddcbcdc32895516d80b961764f1e21e.jpg)
+
+![](images/a86a2bf9023a73684cc808c305e7ec31968a17459088e5db421378da87c5d735.jpg)
+
+differentiable approximation (BPDA). Suppose  $f_{i}(\pmb{x})$  is a non-differentiable component of a network expressed by  $f = f_{1} \circ f_{2} \circ \dots \circ f_{n}$ . The gradient  $\nabla_{\pmb{x}}f$  can be estimated as long as one can find a smooth delegate  $g$  that approximates well  $f_{i}$  (i.e.,  $g(x) \approx f_{i}(x)$ ).
+
+In stark contrast to all those methods, a slight change of the  $k$ -WTA activation pattern in an earlier layer of a network can cause a radical reorganization of activation patterns in later layers (as shown in Sec. 3). Thereby,  $k$ -WTA activation not just obfuscates the network's gradients but destroys them at certain input samples, introducing discontinuities densely distributed in the input data space. We are not aware of any possible smooth approximation of a  $k$ -WTA network to launch BPDA attacks.
+
+# 2  $k$ -WINNERS-TAKE-ALL ACTIVATION
+
+The debut of the Winner-Takes-All (WTA) activation on the stage of neural networks dates back to 1980s, when Grossberg (1982) introduced shunting short-term memory equations in on-center off-surface networks and showed the ability to identify the largest of  $N$  real numbers. Later, Majani et al. (1989) generalized the WTA network to identify the  $K$  largest of  $N$  real numbers, and they termed the network as the K-Winners-Take-All (KWTA) network. These early WTA-type activation functions output only boolean values, mainly motivated by the properties of biological neural circuits. In particular, Maass (2000a;b) has proved that any boolean function can be computed by a single KWTA unit. Yet, the boolean nature of these activation functions differs starkly from the modern activation functions, including the one we use.
+
+# 2.1 DEEP NEURAL NETWORKS ACTIVATED BY  $k$ -WINNERS-TAKE-ALL
+
+We propose to use  $k$ -Winners-Take-All ( $k$ -WTA) activation, a natural generalization of the boolean KWTA $^2$  (Majani et al., 1989).  $k$ -WTA retains the  $k$  largest values of an  $N \times 1$  input vector and sets all others to be zero before feeding the vector to the next network layer, namely,
+
+$$
+\phi_ {k} (\boldsymbol {y}) _ {j} = \left\{ \begin{array}{l l} y _ {j}, & y _ {j} \in \{k \text {l a r g e s t e l e m e n t s o f} \boldsymbol {y} \}, \\ 0, & \text {O t h e r w i s e .} \end{array} \right. \tag {1}
+$$
+
+Here  $\phi_k: \mathbb{R}^N \to \mathbb{R}^N$  is the  $k$ -WTA function (parameterized by an integer  $k$ ),  $\pmb{y} \in \mathbb{R}^N$  is the input to the activation, and  $\phi_k(\pmb{y})_j$  denote the  $j$ -th element of the output  $\phi_k(\pmb{y})$  (see the rightmost subfigure of Figure 2). Note that if  $\pmb{y}$  has multiple elements that are equally  $k$ -th largest, we break the tie by retaining the element with smaller indices until the  $k$  slots are taken.
+
+When using  $k$ -WTA activation, we need to choose  $k$ . Yet it makes no sense to fix  $k$  throughout all layers of the neural network, because these layers often have different output dimensions; a small  $k$  to one layer's dimension can be relatively large to the other. Instead of specifying  $k$ , we introduce
+
+a parameter  $\gamma \in (0,1)$  called sparsity ratio. If a layer has an output dimension  $N$ , then its  $k$ -WTA activation has  $k = \lfloor \gamma \cdot N \rfloor$ . Even though the sparsity ratio can be set differently for different layers, in practice we found no clear gain from introducing such a variation. Therefore, we use a fixed  $\gamma$  the only additional hyperparameter needed for the neural network.
+
+In convolutional neural networks (CNN), the output of a layer is a  $C \times H \times W$  tensor.  $C$  denotes the number of output channels;  $H$  and  $W$  indicate the feature resolution. While there are multiple choices of applying  $k$ -WTA on the tensor—for example, one can apply  $k$ -WTA individually to each channel—empirically we found that the most effective (and conceptually the simplest) way is to treat the tensor as a long  $C \cdot H \cdot W \times 1$  vector input to the  $k$ -WTA activation. Using  $k$ -WTA in this way is also backed by our theoretical understanding (see Sec. 3).
+
+The runtime cost of computing a  $k$ -WTA activation is asymptotically  $O(N)$ , because finding  $k$  largest values in a list is asymptotically equivalent to finding the  $k$ -th largest value, which has an  $O(N)$  complexity (Cormen et al., 2009). This cost is comparable to ReLU's  $O(N)$  cost on a  $N$ -length vector. Thus, replacing ReLU with  $k$ -WTA introduces no significant overhead.
+
+Remark: other WTA-type activations. Relevant to  $k$ -WTA is the local Winner-Take-All (LWTA) activation (Srivastava et al., 2013; 2014), which divides each layer's output values into local groups and applies WTA to each group individually. LWTA is similar to max-pooling (Riesenhuber & Poggio, 1999) for dividing the layer output and choosing group maximums. But unlike ReLU and max-pooling being  $C^0$  continuous, LWTA and our  $k$ -WTA are both discontinuous with respect to the input. The differences among ReLU, max-pooling, LWTA, and  $k$ -WTA are illustrated in Figure 2.
+
+LWTA is motivated toward preventing catastrophic forgetting (McCloskey & Cohen, 1989), whereas our use of  $k$ -WTA is for defending against adversarial threat. Both are discontinuous. But it remains unclear what the LWTA's discontinuity properties are and how its discontinuities affect the network training. Our theoretical analysis (Sec. 3), in contrast, sheds some light on these fundamental questions about  $k$ -WTA, rationalizing its ability for improving adversarial robustness. Indeed, our experiments confirm that  $k$ -WTA outperforms LWTA in terms of robustness (see Appendix D.3).
+
+WTA-type activation, albeit originated decades ago and widely studied in computational neuroscience (Douglas et al., 1989; Douglas & Martin, 2004), remains elusive in modern neural networks. Perhaps this is because it has not demonstrated a considerable improvement to the network's standard test accuracy, though it can offer an accuracy comparable to ReLU (Srivastava et al., 2013). Our analysis and proposed use of  $k$ -WTA and its enabled improvement on adversarial defense may suggest a renaissance of studying WTA.
+
+# 2.2 TRAINING  $k$ -WTA NETWORKS
+
+$k$ -WTA networks require no special treatment in training. Any optimization algorithm (such as stochastic gradient descent) for training ReLU networks can be directly used to train  $k$ -WTA networks.
+
+Our experiments have found that when the sparsity ratio  $\gamma$  is relatively small ( $\leq 0.2$ ), the network training converges slowly. This is not a surprise. A smaller  $\gamma$  activates fewer neurons, effectively reducing more of the layer width and in turn the network size, and the stripped "subnetwork" is much less expressive (Srivastava et al., 2013). Since different training examples activate different subnetworks, collectively they make the training harder.
+
+Nevertheless, we prefer a smaller  $\gamma$ . As we will discuss in the next section, a smaller  $\gamma$  usually leads to better robustness against finding adversarial examples. Therefore, to ease the training (when  $\gamma$  is small), we propose to use an iterative fine-tuning approach. Suppose the target sparsity ratio is  $\gamma_{1}$ . We first train the network with a larger sparsity ratio  $\gamma_{0}$  using the standard training process. Then, we iteratively fine tune the network. In each iteration, we reduce its sparsity ratio by a small  $\delta$  and train the network for two epochs. The iteration repeats until  $\gamma_{0}$  is reduced to  $\gamma_{1}$ .
+
+This incremental process introduces little training overhead, because the cost of each fine tuning is negligible in comparison to training from scratch toward  $\gamma_0$ . We also note that this process is optional. In practice we use it only when  $\gamma < 0.2$ . We show more experiments on the efficacy of the incremental training in Appendix D.2.
+
+# 3 UNDERSTANDING  $k$ -WTA DISCONTINUITY
+
+We now present our theoretical understanding of  $k$ -WTA's discontinuity behavior in the context of deep neural networks, revealing some implication toward the network's adversarial robustness.
+
+![](images/671dcc06a21e9287170f258b955e741d19148fe5e02d28b916ee297bbddd2772.jpg)  
+Figure 3: (a, b) We plot the change of 10 logits values when conducting untargeted PGD attack with 100 iterations. X-axis indicates the perturbation size  $\epsilon$  and Y-axis indicates the 10 color-coded logits values. (a) When we apply PGD attack on  $k$ -WTA ResNet18, the strong discontinuities w.r.t. to input invalidate gradient estimation, effectively defending well against the attack. (b) In contrast, for a ReLU ResNet18, PGD attack can easily find adversarial examples due to the model's smooth change w.r.t. input. (c) In the process of training  $k$ -WTA ResNet18, the loss change w.r.t. model weights is largely smooth. Thus, the training is not harmed by  $k$ -WTA's discontinuities.
+
+Activation pattern. To understand  $k$ -WTA's discontinuity, consider one layer outputting values  $x$ , passed through a  $k$ -WTA activation, and followed by the next layer whose linear weight matrix is  $W$  (see adjacent figure). Then, the value fed into the next activation can be expressed as  $W\phi_{k}(x)$ , where
+
+![](images/276b630a057fde86b29abf65dc6a42393e825b454ac735add0223385de62b416.jpg)
+
+$\phi_k(\cdot)$  is the  $k$ -WTA function defined in (1). Suppose the vector  $\pmb{x}$  has a length  $l$ . We define the  $k$ -WTA's activation pattern under the input  $\pmb{x}$  as
+
+$$
+\mathcal {A} (\boldsymbol {x}) := \left\{i \in [ l ] \mid x _ {i} \text {i s o n e o f t h e} k \text {l a r g e s t v a l u e s i n} \boldsymbol {x} \right\} \subseteq [ l ]. \tag {2}
+$$
+
+Here (and throughout this paper), we use  $[l]$  to denote the integer set  $\{1,2,\dots,l\}$ .
+
+Discontinuity. The activation pattern  $\mathcal{A}(\pmb{x})$  is a key notion for analyzing  $k$ -WTA's discontinuity behavior. Even an infinitesimal perturbation of  $\pmb{x}$  may change  $\mathcal{A}(\pmb{x})$ : some element  $i$  is removed from  $\mathcal{A}(\pmb{x})$  while another element  $j$  is added in. Corresponding to this change, in the evaluation of  $\mathsf{W}\phi_{k}(\pmb{x})$ , the contribution of W's column vector  $\mathsf{W}_i$  vanishes while another column  $\mathsf{W}_j$  suddenly takes effect. It is this abrupt change that renders the result of  $\mathsf{W}\phi_{k}(\pmb{x})$ $C^0$  discontinuous.
+
+Such a discontinuity jump can be arbitrarily large, because the column vectors  $\mathsf{W}_i$  and  $\mathsf{W}_j$  can be of any difference. Once  $\mathsf{W}$  is determined, the discontinuity jump then depends on the value of  $x_i$  and  $x_j$ . As explained in Appendix B, when the discontinuity occurs,  $x_i$  and  $x_j$  have about the same value, depending on the choice of the sparsity ratio  $\gamma$  (recall Sec. 2.1)—the smaller the  $\gamma$  is, the larger the jump will be. This relationship suggests that a smaller  $\gamma$  will make the search of adversarial examples harder. Indeed, this is confirmed through our experiments (see Appendix D.6).
+
+Piecewise linearity. Now, consider an  $n$ -layer  $k$ -WTA network, which can be expressed as  $f(\pmb{x}) = W^{(1)} \cdot \phi_k(W^{(2)} \cdot \phi_k(\dots \phi_k(W^{(n)}\pmb{x} + \pmb{b}^{(n)})) + \pmb{b}^{(2)}) + \pmb{b}^{(1)}$ , where  $W^{(i)}$  and  $\pmb{b}^{(i)}$  are the  $i$ -th layer's weight matrix and bias vector, respectively. If the activation patterns of all layers are fixed, then  $f(\pmb{x})$  is a linear function. When the activation pattern changes,  $f(\pmb{x})$  switches from one linear function to another linear function. Over the entire space of  $\pmb{x}$ ,  $f(\pmb{x})$  is piecewise linear. The specific activation patterns of all layers define a specific linear piece of the function, or a linear region (following the notion introduced by Montufar et al. (2014)). Conventional ReLU (or hard tanh) networks also represent piecewise linear functions and their linear regions are joined together at their boundaries, whereas in  $k$ -WTA networks the linear regions are disconnected (see Figure 1).
+
+Linear region density. Next, we gain some insight on the distribution of those linear regions. This is of our interest because if the linear regions are densely distributed, a small  $\Delta x$  perturbation from any data point  $x$  will likely cross the boundary of the linear region where  $x$  locates. Whenever boundary crossing occurs, the gradient becomes undefined (see Figure 3-a).
+
+For the purpose of analysis, consider an input  $\pmb{x}$  passing through a layer followed by a  $k$ -WTA activation (see adjacent figure). The output from the activation is  $\phi_k(W\pmb{x} + \pmb{b})$ . We would like to understand, when  $\pmb{x}$  is changed into  $\pmb{x}'$ , how likely the activation pattern of  $\phi_k$  will change. First, notice that
+
+![](images/e381f6548fddb5301bf09149c0d465dc6890e8571b00d91c4e335ea2763f37c6.jpg)
+
+if  $\pmb{x}'$  and  $\pmb{x}$  satisfy  $\pmb{x}' = c \cdot \pmb{x}$  with some  $c > 0$ , the activation pattern remains unchanged. Therefore, we introduce a notation  $\mathrm{d}(\pmb{x}, \pmb{x}')$  that measures the "perpendicular" distance between  $\pmb{x}$  and  $\pmb{x}'$ , one
+
+that satisfies  $\pmb{x}^{\prime} = c \cdot (\pmb{x} + \mathrm{d}(\pmb{x},\pmb{x}^{\prime})\pmb{x}_{\perp})$  for some scalar  $c$ , where  $\pmb{x}_{\perp}$  is a unit vector perpendicular to  $\pmb{x}$  and on the plane spanned by  $\pmb{x}$  and  $\pmb{x}^{\prime}$ . With this notion, and if the elements in  $W$  is initialized by sampling from  $\mathcal{N}(0,\frac{1}{l})$  and  $\pmb{b}$  is initialized as zero, we find the following property:
+
+Theorem 1 (Dense discontinuities). Given any input  $\pmb{x} \in \mathbb{R}^m$  and some  $\beta$ , and  $\forall \pmb{x}' \in \mathbb{R}^m$  such that  $\frac{\mathrm{d}^2(\pmb{x},\pmb{x}')}{\|\pmb{x}\|_2^2} \geq \beta$ , if the following condition
+
+$$
+l \geq \Omega \left(\left(\frac {m}{\gamma} \cdot \frac {1}{\beta}\right) \cdot \log \left(\frac {m}{\gamma} \cdot \frac {1}{\beta}\right)\right)
+$$
+
+is satisfied, then with a probability at least  $1 - 2^{-m}$ , we have  $\mathcal{A}(\mathsf{W}\pmb{x} + \pmb{b})\neq \mathcal{A}(\mathsf{W}\pmb{x}' + \pmb{b})$
+
+Here  $l$  is the width of the layer, and  $\gamma$  is again the sparsity ratio in  $k$ -WTA. This theorem informs us that the larger the layer width  $l$  is, the smaller  $\beta$  and thus the smaller perpendicular perturbation distance  $\mathsf{d}(\pmb{x},\pmb{x}^{\prime})$  —is needed to trigger a change of the activation pattern, that is, as the layer width increases, the piecewise linear regions become finer (see Appendix C for proof and more discussion). This property also echos a similar trend in ReLU networks, as pointed out by Raghu et al. (2017).
+
+Why is the  $k$ -WTA network trainable? While  $k$ -WTA networks are highly discontinuous as revealed by Theorem 1 and our experiments (Figure 3-a), in practice we experience no difficulty on training these networks. Our next theorem sheds some light on the reason behind the training success.
+
+Theorem 2. Consider  $N$  data points  $\pmb{x}_1, \pmb{x}_2, \dots, \pmb{x}_N \in \mathbb{R}^m$ . Suppose  $\forall i \neq j, \frac{\pmb{x}_i}{\|\pmb{x}_i\|_2} \neq \frac{\pmb{x}_j}{\|\pmb{x}_j\|_2}$ . If  $l$  is sufficiently large, then with a high probability, we have  $\forall i \neq j, \mathcal{A}(\mathsf{W}\pmb{x}_i + \pmb{b}) \cap \mathcal{A}(\mathsf{W}\pmb{x}_j + \pmb{b}) = \emptyset$ .
+
+This theorem is more formally stated in Theorem 10 in Appendix C together with a proof there. Intuitively speaking, it states that if the network is sufficiently wide, then for any  $i \neq j$ , activation pattern of input data  $\boldsymbol{x}_i$  is almost separated from that of  $\boldsymbol{x}_j$ . Thus, the weights for predicting  $\boldsymbol{x}_i$ 's and  $\boldsymbol{x}_j$ 's labels can be optimized almost independently, without changing their individual activation patterns. In practice, the activation patterns of  $\boldsymbol{x}_i$  and  $\boldsymbol{x}_j$  are not fully separated but weakly correlated. During the optimization, the activation pattern of a data point  $\boldsymbol{x}_i$  may change, but the chance is relatively low—a similar behavior has also been found in ReLU networks (Li & Liang, 2018; Du et al., 2018; Allen-Zhu et al., 2019a;b; Song & Yang, 2019).
+
+Further, notice that the training loss takes a summation over all training data points. This means a weight update would change only a small set of activation patterns (since the chance of having the pattern changed is low); the discontinuous change on the loss value, after taking the summation, will be negligible (see Figure 3-c). Thus, the discontinuities in  $k$ -WTA is not harmful to network training.
+
+# 4 EXPERIMENTAL RESULTS
+
+We evaluate the robustness of  $k$ -WTA networks under adversarial attacks. Our evaluation considers multiple training methods on different network architectures (see details below). When reporting statistics, we use  $A_{rob}$  to indicate the worst-case robustness accuracy on test data under all adversarial attacks we evaluated, and use  $A_{std}$  to indicate the accuracy on the clean test data. We use  $k$ -WTA-γ to represent  $k$ -WTA activation with sparsity ratio  $\gamma$ .
+
+# 4.1 ROBUSTNESS UNDER WHITE-BOX ATTACKS
+
+The rationale behind  $k$ -WTA activation is to destroy network gradients—information needed in white-box attacks. We therefore evaluate  $k$ -WTA networks under multiple recently proposed white-box attack methods, including Projected Gradient Descent (PGD) (Madry et al., 2017), Deepfool (Moosavi-Dezfooli et al., 2016), C&W attack (Carlini & Wagner, 2017), and Momentum Iterative Method (MIM) (Dong et al., 2018). Since  $k$ -WTA activation can be used in almost any training method, be it adversarial training or not, we also consider multiple training methods, including natural (non-adversarial) training, adversarial training (AT) (Madry et al., 2017), TRADES (Zhang et al., 2019) and free adversarial training (FAT) (Shafahi et al., 2019b).
+
+In addition, we evaluate the robustness under transfer-based Black-box (BB) attacks (Papernot et al., 2017). The black-box threat model requires no knowledge about network architecture and parameters. Thus, we use a pre-trained VGG19 network (Simonyan & Zisserman, 2014) as the source model to generate adversarial examples using PGD. As demonstrated by Su et al. (2018b), VGG networks have the strongest transferability among different architectures.
+
+Table 1: Adversarial robustness on CIFAR-10 and SVHN datasets.  $A_{rob}$  in the last column denotes the empirical worst-case robustness among different attacks (columns) for each network optimized by different training methods (row). The bold numbers indicate the best  $A_{rob}$  robustness achieved on ReLU and  $k$ -WTA networks by each training method.  
+CIFAR-10  
+
+<table><tr><td>Training</td><td>Activation</td><td>\(A_{std}\)</td><td>PGD</td><td>C&amp;W</td><td>Deepfool</td><td>MIM</td><td>BB</td><td>\(A_{rob}\)</td></tr><tr><td rowspan="3">Natural</td><td>ReLU</td><td>92.9%</td><td>0.0%</td><td>0.0%</td><td>1.5%</td><td>0.0%</td><td>18.9%</td><td>0.0%</td></tr><tr><td>k-WTA-0.1</td><td>89.3%</td><td>13.3%</td><td>27.9%</td><td>55.6%</td><td>13.1%</td><td>62.6%</td><td>13.1%</td></tr><tr><td>k-WTA-0.2</td><td>91.7%</td><td>4.2%</td><td>6.2%</td><td>47.8%</td><td>3.9%</td><td>66.8%</td><td>4.2%</td></tr><tr><td rowspan="3">AT</td><td>ReLU</td><td>83.5%</td><td>46.3%</td><td>43.6%</td><td>46.8%</td><td>45.9%</td><td>71.0%</td><td>43.6%</td></tr><tr><td>k-WTA-0.1</td><td>78.9%</td><td>51.4%</td><td>64.4%</td><td>70.4%</td><td>50.7%</td><td>73.4%</td><td>50.7%</td></tr><tr><td>k-WTA-0.2</td><td>81.4%</td><td>48.4%</td><td>52.7%</td><td>66.1%</td><td>47.4%</td><td>73.5%</td><td>47.4%</td></tr><tr><td rowspan="3">TRADES</td><td>ReLU</td><td>79.7%</td><td>49.8%</td><td>52.3%</td><td>57.6%</td><td>49.9%</td><td>70.6%</td><td>49.8%</td></tr><tr><td>k-WTA-0.1</td><td>76.6%</td><td>55.0%</td><td>62.2%</td><td>66.0%</td><td>57.5%</td><td>72.3%</td><td>55.0%</td></tr><tr><td>k-WTA-0.2</td><td>80.4%</td><td>51.5%</td><td>57.7%</td><td>63.9%</td><td>53.4%</td><td>74.7%</td><td>51.5%</td></tr><tr><td rowspan="3">FAT</td><td>ReLU</td><td>82.6%</td><td>42.7%</td><td>44.4%</td><td>49.7%</td><td>41.6%</td><td>73.4%</td><td>41.6%</td></tr><tr><td>k-WTA-0.1</td><td>78.4%</td><td>51.7%</td><td>66.3%</td><td>72.4%</td><td>49.1%</td><td>72.3%</td><td>49.1%</td></tr><tr><td>k-WTA-0.2</td><td>82.8%</td><td>48.4%</td><td>60.5%</td><td>67.2%</td><td>46.7%</td><td>76.8%</td><td>46.7%</td></tr></table>
+
+SVHN  
+
+<table><tr><td>Training</td><td>Activation</td><td>\(A_{std}\)</td><td>PGD</td><td>C&amp;W</td><td>Deepfool</td><td>MIM</td><td>BB</td><td>\(A_{rob}\)</td></tr><tr><td rowspan="3">Natural</td><td>ReLU</td><td>95.1%</td><td>0.0%</td><td>0.0%</td><td>2.5%</td><td>0.0%</td><td>14.7%</td><td>0.0%</td></tr><tr><td>k-WTA-0.1</td><td>92.6%</td><td>10.2%</td><td>19.5%</td><td>88.7%</td><td>11.6%</td><td>51.4%</td><td>10.2%</td></tr><tr><td>k-WTA-0.2</td><td>93.8%</td><td>4.3%</td><td>8.0%</td><td>86.8%</td><td>8.3%</td><td>56.7%</td><td>4.3%</td></tr><tr><td rowspan="3">AT</td><td>ReLU</td><td>84.2%</td><td>44.5%</td><td>42.7%</td><td>70.3%</td><td>48.4%</td><td>77.7%</td><td>42.7%</td></tr><tr><td>k-WTA-0.1</td><td>79.9%</td><td>62.2%</td><td>65.7%</td><td>71.5%</td><td>56.9%</td><td>76.1%</td><td>56.9%</td></tr><tr><td>k-WTA-0.2</td><td>82.4%</td><td>53.2%</td><td>63.6%</td><td>77.4%</td><td>52.3%</td><td>74.2%</td><td>52.3%</td></tr><tr><td rowspan="3">TRADES</td><td>ReLU</td><td>84.7%</td><td>47.4%</td><td>51.6%</td><td>76.9%</td><td>49.6%</td><td>76.5%</td><td>47.4%</td></tr><tr><td>k-WTA-0.1</td><td>81.6%</td><td>61.3%</td><td>77.4%</td><td>79.4%</td><td>58.3%</td><td>78.1%</td><td>58.3%</td></tr><tr><td>k-WTA-0.2</td><td>85.4%</td><td>56.7%</td><td>59.2%</td><td>71.6%</td><td>54.5%</td><td>79.3%</td><td>54.5%</td></tr><tr><td rowspan="3">FAT</td><td>ReLU</td><td>85.9%</td><td>40.8%</td><td>46.2%</td><td>76.1%</td><td>39.9%</td><td>76.9%</td><td>40.8%</td></tr><tr><td>k-WTA-0.1</td><td>85.5%</td><td>57.7%</td><td>70.0%</td><td>77.0%</td><td>62.8%</td><td>75.6%</td><td>57.7%</td></tr><tr><td>k-WTA-0.2</td><td>86.8%</td><td>54.3%</td><td>64.3%</td><td>74.7%</td><td>55.2%</td><td>74.4%</td><td>54.3%</td></tr></table>
+
+In each setup, we compare the robust accuracy of  $k$ -WTA networks with standard ReLU networks on three datasets, CIFAR-10, SVHN, and MNIST. Results on the former two are reported in Table 1, while the latter is reported in Appendix D.4. We use ResNet-18 for CIFAR-10 and SVHN. The perturbation range is 0.031 (CIFAR-10) and 0.047 (SVHN) for pixels ranging in [0, 1]. More detailed training and attacking settings are reported in Appendix D.1.
+
+The main takeaway from these experiments (in Table 1) is that  $k$ -WTA is able to universally improve the white-box robustness, regardless of the training methods. The  $k$ -WTA robustness under black-box attacks is not always significantly better than ReLU networks. But black-box attacks, due to the lack of network information, are generally much harder than white-box attacks. In this sense, white-box attacks make the networks more vulnerable, and  $k$ -WTA is able to improve a network's worst-case robustness. This improvement is not tied to any specific training method, achieved with no significant overhead, just by a simple replacement of ReLU with  $k$ -WTA.
+
+Athalye et al. (2018) showed that gradient-based defenses may render the network more vulnerable under black-box attacks than under gradient-based white-box attacks. However, we have not observed this behavior in  $k$ -WTA networks. Even under the strongest black-box attack, i.e., by generating adversarial examples from an independently trained copy of the target network, gradient-based attacks are still stronger (with higher successful rate) than black-box attacks (see Appendix D.3).
+
+Additional experiments include: 1) tests under transfer attacks across two independently trained  $k$ -WTA networks and across  $k$ -WTA and ReLU networks, 2) evaluation of  $k$ -WTA performance on different network architectures, and 3) comparison of  $k$ -WTA performance with the LWTA (recall Sec. 2.1) performance. See Appendix D.3 for details.
+
+![](images/2dfd3af56b1f878704412c1ff60a47d14e57bd522e42b10eedb86229f9a6205c.jpg)  
+Figure 4: Robustness changing w.r.t.  $\gamma$  on CIFAR. When  $\gamma$  decreases, the standard test accuracy (left) starts to drop after a certain point. The robust accuracy (right) first increases then decreases.
+
+![](images/ef15c54f1238fadf18aee7ef78bf0ea7bb3279098a2e47b8d3f597bbb3a68022.jpg)
+
+![](images/5baa7561a33bb62f7bb3b57068bf2dffc722d9dc779bedee41c395212752a17a.jpg)  
+(a) ResNet18-0.1
+
+![](images/87404352bc58b0bfdb9edd6673a7c9ba75e92b1703e19365bb473ff1a4bcc17c.jpg)  
+(b)ResNet18-0.1+adv
+
+![](images/b6d1453ead0e72de6e2892deeb418fe5a9ec97867f7f6303cbda0535609b2974.jpg)  
+Figure 5: Gradient-based attack's loss landscapes in  $k$ -WTA (a, b) and conventional ReLU models (c, d). (a,b)  $k$ -WTA Models have much more non-convex and non-smooth landscapes. Also, the model optimized by adversarial training (b) has a lower absolute value of loss.
+
+![](images/4b098b69f9d8a823f7ebf2b0b8df7a8c0c12c28e7fee3d8af652d5ec6ecab221.jpg)  
+(c) ResNet18 (Vanilla)  
+(d) ResNet18+adv
+
+# 4.2 VARYING SPARSITY RATIO  $\gamma$  AND MODEL ARCHITECTURE.
+
+We further evaluate our method on various network architectures with different sparsity ratios  $\gamma$ . Figure 4 shows the standard test accuracies and robust accuracies against PGD attacks while  $\gamma$  decreases. To test on different network architectures, we apply  $k$ -WTA to ResNet18, DenseNet121 and Wide ResNet (WRN-22-12). In each case, starting from  $\gamma = 0.2$ , we decrease  $\gamma$  using incremental fine-tuning. We then evaluate the robust accuracy on CIFAR dataset, taking 20-iteration PGD attacks with a perturbation range  $\epsilon = 0.31$  for pixels ranging in  $[0, 1]$ .
+
+We find that when  $\gamma$  is larger than  $\sim 0.1$ , reducing  $\gamma$  has little effect on the standard accuracy, but increases the robust accuracy. When  $\gamma$  is smaller than  $\sim 0.1$ , reducing  $\gamma$  drastically lowers both the standard and robust accuracies. The peaks in the  $A_{rob}$  curves (Figure 4-right) are consistent with our theoretical understanding: Theorem 1 suggests that when  $l$  is fixed, a smaller  $\gamma$  tends to sparsify the linear region boundaries, exposing more gradients to the attacker. Meanwhile, as also discussed in Sec. 3, a smaller  $\gamma$  leads to a larger discontinuity jump and thus tends to improve the robustness.
+
+# 4.3 LOSS LANDSCAPE IN GRADIENT-BASED ATTACKS
+
+We now empirically unveil why  $k$ -WTA is able to improve the network's robustness (in addition to our theoretical analysis in Sec. 3). Here we visualize the attacker's loss landscape in gradient-based attacks in order to reveal the landscape change caused by  $k$ -WTA. Similar to the analysis in Tramér et al. (2017), we plot the attack loss of a model with respect to its input on points  $x' = x + \epsilon_1 g_1 + \epsilon_2 g_2$  where  $x$  is a test sample from CIFAR test set,  $g_1$  is the direction of the loss gradient with respect to the input,  $g_2$  is another random direction,  $\epsilon_1$  and  $\epsilon_2$  sweep in the range of  $[-0.04, 0.04]$ , with 50 samples each. This results in a 3D landscape plot with 2500 data points (Figure 5).
+
+As shown in Figure 5,  $k$ -WTA models (with  $\gamma = 0.1$ ) have a highly non-convex and non-smooth loss landscape. Thus, the estimated gradient is hardly useful for adversarial searches. This explains why  $k$ -WTA models can effectively resist gradient-based attacks. In contrast, ReLU models have a much smoother loss surface, from which adversarial examples can be easily found using gradient descent.
+
+Inspecting the range of loss values in Figure 5, we find that adversarial training tends to compress the loss landscape's dynamic range in both the gradient direction and the other random direction, making the dynamic range smaller than that of the models without adversarial training. This phenomenon has been observed in ReLU networks (Madry et al., 2017; Tramér et al., 2017). Interestingly,  $k$ -WTA models manifest a similar behavior (Figure 5-a,b). Moreover, we find that in  $k$ -WTA models a larger  $\gamma$  leads to a smoother loss surface than a smaller  $\gamma$  (see Appendix D.6 for more details).
+
+# 5 CONCLUSION
+
+This paper proposes to replace widely used activation functions with the  $k$ -WTA activation for improving the neural network's robustness against adversarial attacks. This is the only change we advocate. The underlying idea is to embrace the discontinuities introduced by  $k$ -WTA functions to make the search for adversarial examples more challenging. Our method comes almost for free, harmless to network training, and readily useful in the current paradigm of neural networks.
+
+Acknowledgments. This work was supported in part by the National Science Foundation (CAREER-1453101, 1816041, 1910839, 1703925, 1421161, 1714818, 1617955, 1740833), Simons Foundation (#491119 to Alexandr Andoni), Google Research Award, a Google PhD Fellowship, a Snap Research Fellowship, a Columbia SEAS CKGSB Fellowship, and SoftBank Group.
+
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+Zhao Song and Xin Yang. Quadratic suffices for over-parametrization via matrix Chernoff bound. arXiv preprint arXiv:1906.03593, 2019.  
+Rupesh K Srivastava, Jonathan Masci, Sohrob Kazerounian, Faustino Gomez, and Jürgen Schmidhuber. Compete to compute. In Advances in Neural Information Processing Systems 26, pp. 2310-2318. 2013. URL http://papers.nips.cc/paper/5059-compete-to-compute.pdf.  
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+
+# Supplementary Document Enhancing Adversarial Defense by  $k$ -Winners-Take-All
+
+# A OTHER RELATED WORK
+
+In this section, we briefly review the key ideas of attacking neural network models and existing defense methods based on adversarial training.
+
+Attack methods. Recent years have seen adversarial attack studied extensively. The proposed attack methods fall under two general categories, white-box and black-box attacks.
+
+The white-box threat model assumes that the attacker knows the model's structure and parameters fully. This means that the attacker can exploit the model's gradient (with respect to the input) to find adversarial examples. A baseline of such attacks is the Fast Gradient Sign Method (FGSM) (Goodfellow et al., 2014), which constructs the adversarial example  $\pmb{x}^{\prime}$  of a given labeled data  $(\pmb{x},\pmb{y})$  using a gradient-based rule:
+
+$$
+\boldsymbol {x} ^ {\prime} = \boldsymbol {x} + \epsilon \operatorname {s i g n} \left(\nabla_ {\boldsymbol {x}} L (f (\boldsymbol {x}), y)\right), \tag {3}
+$$
+
+where  $f(x)$  denotes the neural network model's output,  $L(\cdot)$  is the loss function provided  $f(x)$  and input label  $y$ , and  $\epsilon$  is the perturbation range for the allowed adversarial example.
+
+Extending FGSM, Projected Gradient Descent (PGD) (Kurakin et al., 2016) utilizes local first-order gradient of the network in a multi-step fashion, and is considered the "strongest" first-order adversary (Madry et al., 2017). In each step of PGD, the adversarial example is updated by a FGSM rule, namely,
+
+$$
+\boldsymbol {x} _ {n + 1} ^ {\prime} = \Pi_ {\boldsymbol {x} ^ {\prime} \in \Delta_ {\epsilon}} \boldsymbol {x} _ {n} ^ {\prime} + \epsilon \operatorname {s i g n} \left(\nabla_ {\boldsymbol {x}} L \left(f \left(\boldsymbol {x} _ {n} ^ {\prime}\right), y\right)\right), \tag {4}
+$$
+
+where  $\pmb{x}_n'$  is the adversarial examples after  $n$  steps and  $\Pi_{\pmb{x} \in \Delta_\epsilon}(\pmb{x}_n')$  projects  $\pmb{x}_n'$  back into an allowed perturbation range  $\Delta_\epsilon$  (such as an  $\epsilon$  ball of  $\pmb{x}$  under certain distance measure). Other attacks include Deepfool (Moosavi-Dezfooli et al., 2016), C&W (Carlini & Wagner, 2017) and momentum-based attack (Dong et al., 2018). Those methods are all using first-order gradient information to construct adversarial samples.
+
+The black-box threat model is a strict subset of the white-box threat model. It assumes that the attacker has no information about the model's architecture or parameters. Some black-box attack model allows the attacker to query the victim neural network to gather (or reverse-engineer) information. By far the most successful black-box attack is transfer attack (Papernot et al., 2017; Tramère et al., 2017). The idea is to first construct adversarial examples on an adversarially trained network and then attack the black-box network model use these samples. There also exist some gradient-free black-box attack methods, such as boundary attack (Brendel et al., 2017; Chen & Jordan, 2019), one-pixel attack (Su et al., 2019) and local search attack (Narodytska & Kasiviswanathan, 2016). Those methods rely on repeatedly evaluating the model and are not as effective as gradient-based white-box attacks.
+
+Adversarial training. Adversarial training (Goodfellow et al., 2014; Madry et al., 2017; Kurakin et al., 2016; Huang et al., 2015) is by far the most successful method against adversarial attacks. It trains the network model with adversarial images generated during the training time. Madry et al. (2017) showed that adversarial training in essence solves the following min-max optimization problem:
+
+$$
+\min  _ {f} \mathbb {E} \left\{\max  _ {\boldsymbol {x} ^ {\prime} \in \Delta_ {\epsilon}} L \left(f \left(\boldsymbol {x} ^ {\prime}\right), y\right) \right\}, \tag {5}
+$$
+
+where  $\Delta_{\epsilon}$  is the set of allowed perturbations of training samples, and  $y$  denotes the true label of each training sample. Recent works that achieve state-of-the-art adversarial robustness rely on adversarial training (Zhang et al., 2019; Xie et al., 2018b). However, adversarial training is notoriously slow because it requires finding adversarial samples on-the-fly at each training epoch. Its prohibitive cost makes adversarial training difficult to scale to large datasets such as ImageNet (Deng et al., 2009) unless enormous computation resources are available. Recently, Shafahi et al. (2019b) revise the adversarial training algorithm to make it has similar training time as regular training, while keep the standard and robust accuracy comparable to standard adversarial training.
+
+Regularization. Another type of defense is based on regularizing the neural network, and many works of this type are combined with adversarial training. For example, feature denoising (Xie et al., 2018b) adds several denoise blocks to the network structure and trains the network with adversarial training. Zhang et al. (Zhang et al., 2019) explicitly added a regularization term to balance the trade-off between standard accuracy and robustness, obtaining state-of-the-art robust accuracy on CIFAR.
+
+Some other regularization-based methods require no adversarial training. For example, Ross & Doshi-Velez (2017) proposed to regularize the gradient of the model with respect to its input; Zheng et al. (2016) generated adversarial samples by adding random Gaussian noise to input data. However, these methods are shown to be brittle under stronger iterative gradient-based attacks such as PGD (Zhang et al., 2019). In contrast, as demonstrated in our experiments, our method without using adversarial training is able to greatly improve robustness under PGD and other attacks.
+
+# B DISCONTINUITY JUMP OF  $\mathsf{W}\phi_k(x)$
+
+Consider a gradual and smooth change of the vector  $\pmb{x}$ . For the ease of illustration, let us assume all the values in  $\pmb{x}$  are distinct. Because every element in  $\pmb{x}$  changes smoothly, when the activation pattern  $\mathcal{A}(\pmb{x})$  changes, the  $k$ -th largest and  $k + 1$ -th largest value in  $\pmb{x}$  must swap: the previously  $k$ -th largest value is removed from the activation pattern, while the previously  $k + 1$ -th largest value is added in the activation pattern. Let  $i$  and  $j$  denote the indices of the two values, that is,  $x_{i}$  is previously the  $k$ -th largest and  $x_{j}$  is previously the  $k + 1$ -th largest. When this swap happens,  $x_{i}$  and  $x_{j}$  must be infinitesimally close to each other, and we use  $x^{*}$  to indicate their common value.
+
+This swap affects the computation of  $\mathsf{W}\phi_{k}(\pmb {x})$ . Before the swap happens,  $x_{i}$  is in the activation pattern but  $x_{j}$  is not, therefore  $\mathsf{W}_i$  takes effect but  $\mathsf{W}_j$  does not. After the swap,  $\mathsf{W}_j$  takes effect while  $\mathsf{W}_j$  is suppressed. Therefore, the discontinuity jump due to this swap is  $(\mathsf{W}_j - \mathsf{W}_i)x^*$ .
+
+When  $\mathsf{W}$  is determined, the magnitude of the jump depends on  $x^{*}$ . Recall that  $x^{*}$  is the  $k$ -th largest value in  $\pmb{x}$  when the swap happens. Thus, it depends on  $k$  and in turn the sparsity ratio  $\gamma$ : the smaller the  $\gamma$  is, the smaller  $k$  is effectively used (for a fixed vector length). As a result, the  $k$ -th largest value becomes larger—when  $k = 1$ , the largest value of  $\pmb{x}$  is used as  $x^{*}$ .
+
+# C THEORETICAL PROOFS
+
+In this section, we will prove Theorem 1 and Theorem 2. The formal version of the two theorems are Theorem 9 and Theorem 10 respectively.
+
+Notation. We use  $[n]$  to denote the set  $\{1,2,\dots ,n\}$ . We use  $\mathbf{1}(\mathcal{E})$  to indicate an indicator variable. If the event  $\mathcal{E}$  happens, the value of  $\mathbf{1}(\mathcal{E})$  is 1. Otherwise the value of  $\mathbf{1}(\mathcal{E})$  is 0. For a weight matrix  $W$ , we use  $W_{i}$  to denote the  $i$ -th row of  $W$ . For a bias vector  $b$ , we use  $b_{i}$  to denote the  $i$ -th entry of  $b$ . In this section, we show some behaviors of the  $k$ -WTA activation function. Recall that an  $n$ -layer neural network  $f(x)$  with  $k$ -WTA activation function can be seen as the following:
+
+$$
+f (x) = W ^ {(1)} \cdot \phi_ {k} \left(W ^ {(2)} \cdot \phi_ {k} \left(\dots \phi_ {k} \left(W ^ {(n)} x + b ^ {(n)}\right)\right) + b ^ {(2)}\right) + b ^ {(1)}
+$$
+
+where  $W^{(i)}$  is the weight matrix,  $b^{(i)}$  is the bias vector of the  $i$ -th layer, and  $\phi(\cdot)$  is the  $k$ -WTA activation function, i.e., for an arbitrary vector  $y$ ,  $\phi_k(y)$  is defined as the following:
+
+$$
+\phi_ {k} (y) _ {j} = \left\{ \begin{array}{l l} y _ {j}, & \text {i f y _ {j} i s o n e o f t h e t o p - k l a r g e s t v a l u e s ,} \\ 0, & \text {o t h e r w i s e .} \end{array} \right.
+$$
+
+For simplicity of the notation, if  $k$  is clear in the context, we will just use  $\phi(y)$  for short. Notice that if there is a tie in the above definition, we assume the entry with smaller index has larger value. For a vector  $y \in \mathbb{R}^l$ , we define the activation pattern  $\mathcal{A}(y) \subseteq [l]$  as
+
+$$
+\mathcal {A} (y) = \{i \in [ l ] \mid y _ {i} \text {i s o n e o f t h e t o p - k l a r g e s t v a l u e s} \}.
+$$
+
+Notice that if the activation pattern  $\mathcal{A}(y)$  is different from  $\mathcal{A}(y')$ , then  $W \cdot \phi(y)$  and  $W \cdot \phi(y')$  will be in different linear regions. Actually,  $W \cdot \phi(y)$  may even represent a discontinuous function. In the next section, we will show that when the network is much wider, the function may be more discontinuous with respect to the input.
+
+# C.1 DISCONTINUITY WITH RESPECT TO THE INPUT
+
+We only consider the activation pattern of the output of one layer. We consider the behavior of the network after the initialization of the weight matrix and the bias vector. By initialization, the entries of the weight matrix  $W$  are i.i.d. random Gaussian variables, and the bias vector is zero. We can show that if the weight matrix is sufficiently wide, then for any vector  $x$ , with high probability, for all vector  $x'$  satisfying that the "perpendicular" distance between  $x$  and  $x'$  is larger than a small threshold, the activation patterns of  $Wx$  and  $Wx'$  are different.
+
+Notice that the scaling of  $W$  does not change the activation pattern of  $Wx$  for any  $x$ , we can thus assume that each entry of  $W$  is a random variable with standard Gaussian distribution  $N(0,1)$ .
+
+Before we prove Theorem 9, let us prove several useful lemmas. The following several lemmas do not depend on the randomness of the weight matrix.
+
+Lemma 1 (Inputs with the same activation pattern form a convex set). Given an arbitrary weight matrix  $W \in \mathbb{R}^{l \times m}$  and an arbitrary bias vector  $b \in \mathbb{R}^l$ , for any  $x \in \mathbb{R}^m$ , the set of all the vectors  $x' \in \mathbb{R}^m$  satisfying  $A(Wx' + b) = A(Wx + b)$  is convex, i.e., the set
+
+$$
+\{x ^ {\prime} \in \mathbb {R} ^ {m} \mid \mathcal {A} (W x + b) = \mathcal {A} (W x ^ {\prime} + b) \}
+$$
+
+is convex.
+
+Proof. If  $\mathcal{A}(Wx' + b) = \mathcal{A}(Wx + b)$ , then  $x'$  should satisfy:
+
+$$
+\forall i \in \mathcal {A} (W x + b), j \in [ l ] \backslash \mathcal {A} (W x + b), W _ {i} x ^ {\prime} + b _ {i} \geq (\text {o r} >) W _ {j} x ^ {\prime} + b _ {j}.
+$$
+
+Notice that the inequality  $W_{i}x^{\prime} + b_{i}\geq (\mathrm{or} > )W_{j}x^{\prime} + b_{j}$  denotes a half hyperplane  $(W_{i} - W_{j})x^{\prime} + (b_{i} - b_{j})\geq (\mathrm{or} > )0$ . Thus, the set  $\{x^{\prime}\in \mathbb{R}^{m}\mid \mathcal{A}(Wx + b) = \mathcal{A}(Wx^{\prime} + b)\}$  is convex since it is an intersection of half hyperplanes.
+
+Lemma 2 (Different patterns of input points with small angle imply different patterns of input points with large angle). Let  $\alpha \in (0,1)$ . Given an arbitrary weight matrix  $W \in \mathbb{R}^{l \times m}$ , a bias vector  $b = 0$  and a vector  $x \in \mathbb{R}^m$  with  $\|x\|_2 = 1$ , if every vector  $x' \in \mathbb{R}^m$  with  $\|x'\|_2 = 1$  and  $\langle x, x' \rangle = \alpha$  satisfies  $\mathcal{A}(Wx + b) \neq \mathcal{A}(Wx' + b)$ , then for any  $x'' \in \mathbb{R}^m$  with  $\|x''\|_2 = 1$  and  $\langle x, x'' \rangle < \alpha$ , it satisfies  $\mathcal{A}(Wx + b) \neq \mathcal{A}(Wx'' + b)$ .
+
+Proof. We draw a line between  $x$  and  $x''$ . There must be a point  $x^* \in \mathbb{R}^m$  on the line and  $\langle x, x' \rangle = \alpha$ , where  $x' = x^* / \|x^*\|_2$ . Since  $b = 0$ , we have  $\mathcal{A}(Wx^* + b) = \mathcal{A}(Wx' + b) \neq \mathcal{A}(Wx + b)$ . Since  $x^*$  is on the line between  $x$  and  $x''$ , we have  $\mathcal{A}(Wx'' + b) \neq \mathcal{A}(Wx + b)$  by convexity (see Lemma 1).
+
+Lemma 3 (A sufficient condition for different patterns). Consider two vectors  $y \in \mathbb{R}^l$  and  $y' \in \mathbb{R}^l$ . If  $\exists i \in \mathcal{A}(y), j \in [l] \setminus \mathcal{A}(y)$  such that  $y_i' < y_j'$ , then  $\mathcal{A}(y) \neq \mathcal{A}(y')$ .
+
+Proof. Suppose  $\mathcal{A}(y) = \mathcal{A}(y')$ . We have  $i \in \mathcal{A}(y')$ . It means that  $y_i'$  is one of the top- $k$  largest values among all entries of  $y'$ . Thus  $y_j'$  is also one of the top- $k$  largest values, and  $j$  should be in  $\mathcal{A}(y')$  which leads to a contradiction.
+
+In the remaining parts, we will assume that each entry of the weight matrix  $W \in \mathbb{R}^{l \times m}$  is a standard random Gaussian variable.
+
+Lemma 4 (Upper bound of the entires of  $W$ ). Consider a matrix  $W \in \mathbb{R}^{l \times m}$  where each entry is a random variable with standard Gaussian distribution  $N(0,1)$ . With probability at least  $0.99$ ,  $\forall i \in [l]$ ,  $\| W_i \|_2 \leq 10\sqrt{ml}$ .
+
+Proof. Consider a fixed  $i \in [l]$ . We have  $\mathbb{E}[\| W_i\|_2^2] = m$ . By Markov's inequality, we have  $\operatorname*{Pr}[\| W_i\|_2^2 > 100ml] \leq 0.01 / l$ . By taking union bound over all  $i \in [l]$ , with probability at least 0.99, we have  $\forall i \in [l], \| W_i\|_2 \leq 10\sqrt{ml}$ .
+
+Lemma 5 (Two vectors may have different activation patterns with a good probability). Consider a matrix  $W \in \mathbb{R}^{l \times m}$  where each entry is a random variable with standard Gaussian distribution  $N(0,1)$ . Let  $\gamma \in (0,0.48)$  be the sparsity ratio of the activation, i.e.,  $\gamma = k / l$ . For any two vectors  $x, x' \in \mathbb{R}^m$  with  $\| x \|_2 = \| x' \|_2 = 1$  and  $\langle x, x' \rangle = \alpha$  for some arbitrary  $\alpha \in (0.5,1)$ , with probability at least  $1 - 2^{-\Theta((1/\alpha^2 - 1)\gamma l)}$ ,  $\mathcal{A}(Wx) \neq \mathcal{A}(Wx')$  and  $\exists i \in \mathcal{A}(Wx)$ ,  $j \in [l] \setminus \mathcal{A}(Wx)$  such that
+
+$$
+W _ {i} x ^ {\prime} <   W _ {j} x ^ {\prime} - \frac {\sqrt {1 - \alpha^ {2}}}{2 4 \alpha} \cdot \sqrt {2 \pi}.
+$$
+
+Proof. Consider arbitrary two vectors  $x, x' \in \mathbb{R}^m$  with  $\|x\|_2 = \|x'\|_2 = 1$  and  $\langle x, x' \rangle = \alpha$ . We can find an orthogonal matrix  $Q \in \mathbb{R}^{m \times m}$  such that  $\tilde{x} := Qx = (1,0,0,\dots,0)^\top \in \mathbb{R}^m$  and  $\tilde{x}' := Qx' = (\alpha, \sqrt{1 - \alpha^2}, 0,0,\dots,0)^\top \in \mathbb{R}^m$ . Let  $\tilde{W} = WQ^\top$ . Then we have  $\tilde{W}\tilde{x} = Wx$  and  $\tilde{W}\tilde{x}' = Wx'$ . Thus, we only need to analyze the activation patterns of  $\tilde{W}\tilde{x}$  and  $\tilde{W}\tilde{x}'$ . Since  $Q^\top$  is an orthogonal matrix and each entry of  $W$  is an i.i.d. random variable with standard Gaussian distribution  $N(0,1)$ ,  $\tilde{W} = WQ^\top$  is also a random matrix where each entry is an i.i.d. random variable with standard Gaussian distribution  $N(0,1)$ . Let the entries in the first column of  $\tilde{W}$  be  $X_1, X_2, \dots, X_l$  and let the entries in the second column of  $\tilde{W}$  be  $Y_1, Y_2, \dots, Y_l$ . Then we have
+
+$$
+W x = \tilde {W} \tilde {x} = \left( \begin{array}{c} X _ {1} \\ X _ {2} \\ \dots \\ X _ {l} \end{array} \right), \quad W x ^ {\prime} = \tilde {W} \tilde {x} ^ {\prime} = \left( \begin{array}{c} \alpha X _ {1} + \sqrt {1 - \alpha^ {2}} Y _ {1} \\ \alpha X _ {2} + \sqrt {1 - \alpha^ {2}} Y _ {2} \\ \dots \\ \alpha X _ {l} + \sqrt {1 - \alpha^ {2}} Y _ {l} \end{array} \right). \tag {6}
+$$
+
+We set  $\varepsilon = \sqrt{1 - \alpha^2} / (96\alpha)$  and define  $R_1' < R_1 < R_2 < R_2'$  as follows:
+
+$$
+\Pr_ {X \sim N (0, 1)} [ X \geq R _ {2} ^ {\prime} ] = (1 - 2 \varepsilon) \gamma , \tag {7}
+$$
+
+$$
+\Pr_ {X \sim N (0, 1)} [ X \geq R _ {2} ] = (1 - \varepsilon) \gamma , \tag {8}
+$$
+
+$$
+\Pr_ {X \sim N (0, 1)} [ X \geq R _ {1} ] = (1 + \varepsilon) \gamma , \tag {9}
+$$
+
+$$
+\Pr_ {X \sim N (0, 1)} [ X \geq R _ {1} ^ {\prime} ] = (1 + 2 \varepsilon) \gamma . \tag {10}
+$$
+
+Since  $\gamma < 0.48$  and  $\varepsilon \leq 0.02$ , we have  $(1 + 2\varepsilon)\gamma < 0.5$ . It implies  $0 < R_1' < R_1 < R_2 < R_2'$ .
+
+Claim 3.
+
+$$
+R _ {2} ^ {\prime} - R _ {1} ^ {\prime} \leq 8 \varepsilon \sqrt {2 \pi}.
+$$
+
+Proof. By Equation (7) and Equation (10),
+
+$$
+\operatorname * {P r} _ {X \sim N (0, 1)} \left[ R _ {1} ^ {\prime} \leq X \leq R _ {2} ^ {\prime} \right] = 4 \varepsilon \gamma .
+$$
+
+Due to the density function of standard Gaussian distribution, we have
+
+$$
+\frac {1}{\sqrt {2 \pi}} \int_ {R _ {1} ^ {\prime}} ^ {R _ {2} ^ {\prime}} e ^ {- t ^ {2} / 2} \mathrm {d} t = \Pr_ {X \sim N (0, 1)} [ R _ {1} ^ {\prime} \leq X \leq R _ {2} ^ {\prime} ] = 4 \varepsilon \gamma .
+$$
+
+Since  $R_2' \geq R_1' \geq 0$ , we have  $\forall t \in [R_1', R_2']$ ,  $e^{-t^2/2} \geq e^{-R_2'^2/2}$ . Thus,
+
+$$
+\frac {1}{\sqrt {2 \pi}} \cdot e ^ {- R _ {2} ^ {\prime 2} / 2} (R _ {2} ^ {\prime} - R _ {1} ^ {\prime}) = \frac {1}{\sqrt {2 \pi}} \cdot e ^ {- R _ {2} ^ {\prime 2} / 2} \int_ {R _ {1} ^ {\prime}} ^ {R _ {2} ^ {\prime}} 1 \mathrm {d} t \leq \frac {1}{\sqrt {2 \pi}} \int_ {R _ {1} ^ {\prime}} ^ {R _ {2} ^ {\prime}} e ^ {- t ^ {2} / 2} \mathrm {d} t = 4 \varepsilon \gamma .
+$$
+
+By the tail bound of Gaussian distribution, we have
+
+$$
+\operatorname * {P r} _ {X \sim N (0, 1)} [ X \geq R _ {2} ^ {\prime} ] \leq e ^ {- R _ {2} ^ {\prime 2} / 2}.
+$$
+
+By combining with Equation (7), we have
+
+$$
+\begin{array}{l} (1 - 2 \varepsilon) \gamma \cdot \frac {1}{\sqrt {2 \pi}} (R _ {2} ^ {\prime} - R _ {1} ^ {\prime}) \\ = \operatorname * {P r} _ {X \sim N (0, 1)} [ X \geq R _ {2} ^ {\prime} ] \cdot \frac {1}{\sqrt {2 \pi}} (R _ {2} ^ {\prime} - R _ {1} ^ {\prime}) \\ \leq e ^ {- R _ {2} ^ {\prime 2} / 2} \cdot \frac {1}{\sqrt {2 \pi}} (R _ {2} ^ {\prime} - R _ {1} ^ {\prime}) \\ \leq 4 \varepsilon \gamma , \\ \end{array}
+$$
+
+which implies
+
+$$
+R _ {2} ^ {\prime} - R _ {1} ^ {\prime} \leq \frac {4 \varepsilon}{1 - 2 \varepsilon} \sqrt {2 \pi} \leq 8 \varepsilon \sqrt {2 \pi},
+$$
+
+where the last inequality follows from  $1 - 2\varepsilon \geq 0.5$ .
+
+![](images/51d6e06102f06eea729c473681a386c94e1e994dceaf83600d78f7809dbfbb6d.jpg)
+
+Claim 4.
+
+$$
+\Pr_ {X _ {1}, X _ {2}, \dots , X _ {l}} \left[ \sum_ {i = 1} ^ {l} \mathbf {1} \left(X _ {i} \geq R _ {2}\right) \geq (1 - \varepsilon / 2) \gamma l \right] \leq e ^ {- \varepsilon^ {2} \gamma l / 2 4} \tag {11}
+$$
+
+$$
+\Pr_ {X _ {1}, X _ {2}, \dots , X _ {l}} \left[ \sum_ {i = 1} ^ {l} \mathbf {1} \left(X _ {i} \geq R _ {1}\right) \leq (1 + \varepsilon / 2) \gamma l \right] \leq e ^ {- \varepsilon^ {2} \gamma l / 1 8} \tag {12}
+$$
+
+$$
+\Pr_ {X _ {1}, X _ {2}, \dots , X _ {l}} \left[ \sum_ {i = 1} ^ {l} \mathbf {1} \left(R _ {2} ^ {\prime} \geq X _ {i} \geq R _ {2}\right) \leq \varepsilon \gamma l / 2 \right] \leq e ^ {- \varepsilon \gamma l / 8} \tag {13}
+$$
+
+$$
+\Pr_ {X _ {1}, X _ {2}, \dots , X _ {l}} \left[ \sum_ {i = 1} ^ {l} \mathbf {1} \left(R _ {1} \geq X _ {i} \geq R _ {1} ^ {\prime}\right) \leq \varepsilon \gamma l / 2 \right] \leq e ^ {- \varepsilon \gamma l / 8} \tag {14}
+$$
+
+Proof. For  $i \in [l]$ , we have  $\mathbb{E}[\mathbf{1}(X_i \geq R_2)] = \operatorname*{Pr}[X_i \geq R_2] = (1 - \varepsilon)\gamma$  by Equation (8). By Chernoff bound, we have
+
+$$
+\Pr \left[ \sum_ {i = 1} ^ {l} \mathbf {1} \left(X _ {i} \geq R _ {2}\right) \geq (1 + \varepsilon / 2) \cdot (1 - \varepsilon) \gamma l \right] \leq e ^ {- (\varepsilon / 2) ^ {2} (1 - 2 \varepsilon) \gamma l / 3}.
+$$
+
+Since  $\varepsilon \leq 0.02$
+
+$$
+\Pr \left[ \sum_ {i = 1} ^ {l} \mathbf {1} \left(X _ {i} \geq R _ {2}\right) \geq (1 - \varepsilon / 2) \gamma l \right] \leq e ^ {- \varepsilon^ {2} \gamma l / 2 4}.
+$$
+
+We have  $\mathbb{E}[\mathbf{1}(X_i\geq R_1)] = \operatorname *{Pr}[X_i\geq R_1] = (1 + \varepsilon)\gamma$  by Equation (9). By Chernoff bound, we have
+
+$$
+\Pr \left[ \sum_ {i = 1} ^ {l} \mathbf {1} (X _ {i} \geq R _ {1}) \leq (1 - \varepsilon / 3) \cdot (1 + \varepsilon) \gamma l \right] \leq e ^ {- (\varepsilon / 3) ^ {2} (1 + \varepsilon) \gamma l / 2}.
+$$
+
+Thus,
+
+$$
+\Pr \left[ \sum_ {i = 1} ^ {l} \mathbf {1} \left(X _ {i} \geq R _ {i}\right) \leq (1 + \varepsilon / 2) \gamma l \right] \leq e ^ {- \varepsilon^ {2} \gamma l / 1 8}
+$$
+
+We have  $\mathbb{E}\left[\mathbf{1}(R_2' \geq X_i \geq R_2)\right] = \operatorname*{Pr}[R_2' \geq X_i \geq R_2] = \varepsilon \gamma$  by Equation (7) and Equation (8). By Chernoff bound, we have
+
+$$
+\Pr \left[ \sum_ {i = 1} ^ {l} \mathbf {1} \left(R _ {2} ^ {\prime} \geq X _ {i} \geq R _ {2}\right) \leq 1 / 2 \cdot \varepsilon \gamma l \right] \leq e ^ {- \varepsilon \gamma l / 8}
+$$
+
+Similarly, we have  $\mathbb{E}[\mathbf{1}(R_1\geq X_i\geq R_1^{\prime})] = \operatorname *{Pr}[R_1\geq X_i\geq R_1^{\prime}] = \varepsilon \gamma$  by Equation (9) and Equation (10). By chernoff bound, we have
+
+$$
+\Pr_ {X _ {1}, X _ {2}, \dots , X _ {l}} \left[ \sum_ {i = 1} ^ {l} \mathbf {1} \left(R _ {1} \geq X _ {i} \geq R _ {1} ^ {\prime}\right) \leq 1 / 2 \cdot \varepsilon \gamma l \right] \leq e ^ {- \varepsilon \gamma l / 8}
+$$
+
+![](images/7f52ec503b13073297a0c3f1fcb28c4932962cae74cd747913664352f9fae25f.jpg)
+
+Equation (11) says that, with high probability,  $\forall i \in [l]$  with  $X_{i} \geq R_{2}$ , it has  $i \in \mathcal{A}(Wx)$ . Equation (12) says that, with high probability,  $\forall i \in [l]$  with  $X_{i} \leq R_{1}$ , it has  $i \notin \mathcal{A}(Wx)$ . Equation (14) (Equation (13)) says that, with high probability, there are many  $i \in [l]$  such that  $W_{i}x \in [R_{1}', R_{1}]$  ( $W_{i}x \in [R_{2}, R_{2}']$ ).
+
+Let  $\mathcal{E} = \mathcal{E}_1\wedge \mathcal{E}_2\wedge \mathcal{E}_3\wedge \mathcal{E}_4$  , where
+
+$\bullet$ $\mathcal{E}_1\colon \sum_{i = 1}^l\mathbf{1}(X_i\geq R_2)\leq (1 - \varepsilon /2)\gamma l,$  
+$\mathcal{E}_2\colon \sum_{i = 1}^l\mathbf{1}(X_i\geq R_1)\geq (1 + \varepsilon /2)\gamma l,$
+
+$\bullet$ $\mathcal{E}_3\colon \sum_{i = 1}^l\mathbf{1}(R_1\geq X_i\geq R_1')\geq \varepsilon \gamma l / 2,$  
+$\bullet$ $\mathcal{E}_4\colon \sum_{i = 1}^l{\bf 1}(R_2'\geq X_i\geq R_2)\geq \varepsilon \gamma l / 2.$
+
+According to Equation (11), Equation (12), Equation (13) and Equation (14), the probability that  $\mathcal{E}$  happens is at least
+
+$$
+1 - 4 e ^ {- \varepsilon^ {2} \gamma l / 2 4} \tag {15}
+$$
+
+by union bound over  $\bar{\mathcal{E}}_1,\bar{\mathcal{E}}_2,\bar{\mathcal{E}}_3,\bar{\mathcal{E}}_4$
+
+Claim 5. Condition on  $\mathcal{E}$ , the probability that  $\exists i \in [l]$  with  $X_{i} \in [R_{2}, R_{2}^{\prime}]$  such that  $Y_{i} < -\alpha / \sqrt{1 - \alpha^{2}} \cdot 16\varepsilon \sqrt{2\pi}$  is at least
+
+$$
+1 - \left(1 6 \varepsilon \cdot \frac {\alpha}{\sqrt {1 - \alpha^ {2}}} + \frac {1}{2}\right) ^ {\varepsilon \gamma l / 2}.
+$$
+
+Proof. For a fixed  $i \in [l]$ ,
+
+$$
+\begin{array}{l} \operatorname * {P r} \left[ Y _ {i} \geq - \alpha / \sqrt {1 - \alpha^ {2}} \cdot 1 6 \varepsilon \sqrt {2 \pi} \right] = \int_ {- \alpha / \sqrt {1 - \alpha^ {2}} \cdot 1 6 \varepsilon \sqrt {2 \pi}} ^ {0} \frac {1}{\sqrt {2 \pi}} e ^ {- t ^ {2} / 2} \mathrm {d} t + \frac {1}{2} \\ \leq \frac {1}{\sqrt {2 \pi}} \cdot \alpha / \sqrt {1 - \alpha^ {2}} \cdot 1 6 \varepsilon \sqrt {2 \pi} + \frac {1}{2} \\ = 1 6 \varepsilon \cdot \frac {\alpha}{\sqrt {1 - \alpha^ {2}}} + \frac {1}{2}. \\ \end{array}
+$$
+
+Thus, according to event  $\mathcal{E}_4$ , we have
+
+$$
+\Pr \left[ \forall i \text {w i t h} X _ {i} \in \left[ R _ {2}, R _ {2} ^ {\prime} \right], Y _ {i} \geq - \alpha / \sqrt {1 - \alpha^ {2}} \cdot 1 6 \varepsilon \sqrt {2 \pi} \mid \mathcal {E} \right] \leq \left(1 6 \varepsilon \cdot \frac {\alpha}{\sqrt {1 - \alpha^ {2}}} + \frac {1}{2}\right) ^ {\varepsilon \gamma l / 2}.
+$$
+
+![](images/75c6c4e3de7221614193fdd156c3a7253b9d3785bde2c35a0cbffb5982fa6d96.jpg)
+
+Claim 6. Condition on  $\mathcal{E}$ , the probability that  $\exists i \in [l]$  with  $X_{i} \in [R_{1}', R_{1}]$  such that  $Y_{i} \geq 0$  is at least  $1 - (1/2)^{\varepsilon \gamma l/2}$ .
+
+Proof. For a fixed  $i \in [l]$ ,  $\operatorname{Pr}[Y_i \leq 0] = 1/2$ . Thus, according to event  $\mathcal{E}_3$ , we have
+
+$$
+\Pr \left[ \forall i \text {w i t h} X _ {i} \in \left[ R _ {1} ^ {\prime}, R _ {1} \right], Y _ {i} \leq 0 \mid \mathcal {E} \right] \leq (1 / 2) ^ {\varepsilon \gamma l / 2}.
+$$
+
+![](images/38639ce3235b382a171f33829073b8c0ec7c8e72ad40f489e91ca9443db17802.jpg)
+
+Condition on that  $\mathcal{E}$  happens. Because of  $\mathcal{E}_1$ , if  $X_{i} \geq R_{2}$ ,  $X_{i}$  must be one of the top- $k$  largest values. Due to Equation (6), we have  $X_{i} = W_{i}x$ . Thus, if  $X_{i} \geq R_{2}$ ,  $i \in \mathcal{A}(Wx)$ . By Claim 5, with probability at least
+
+$$
+1 - \left(1 6 \varepsilon \cdot \frac {\alpha}{\sqrt {1 - \alpha^ {2}}} + \frac {1}{2}\right) ^ {\varepsilon \gamma l / 2}, \tag {16}
+$$
+
+there is  $i\in \mathcal{A}(Wx)$  such that
+
+$$
+\begin{array}{l} W _ {i} x ^ {\prime} = \alpha X _ {i} + \sqrt {1 - \alpha^ {2}} Y _ {i} \\ \leq \alpha X _ {i} + \sqrt {1 - \alpha^ {2}} \cdot \left(- \frac {\alpha}{\sqrt {1 - \alpha^ {2}}} \cdot 1 6 \varepsilon \sqrt {2 \pi}\right) \\ = \alpha \left(X _ {i} - 1 6 \varepsilon \sqrt {2 \pi}\right) \\ \leq \alpha \left(R _ {2} ^ {\prime} - 1 6 \varepsilon \sqrt {2 \pi}\right), \tag {17} \\ \end{array}
+$$
+
+where the first step follows from Equation (6), the second step follows from  $Y_{i} \leq -\alpha / \sqrt{1 - \alpha^{2}} \cdot 16\varepsilon \sqrt{2\pi}$ , and the last step follows from  $X_{i} \in [R_{2}, R_{2}^{\prime}]$ .
+
+Because of  $\mathcal{E}_2$  if  $X_{j}\leq R_{1}$ $X_{j}$  should not be one of the top-  $k$  largest values. Due to Equation (6), we have  $X_{j} = W_{j}x$ . Thus, if  $X_{j}\leq R_{1},j\notin \mathcal{A}(Wx)$ . By Claim C.1, with probability at least
+
+$$
+1 - (1 / 2) ^ {\varepsilon \gamma l / 2}, \tag {18}
+$$
+
+there is  $j \notin \mathcal{A}(Wx)$  such that
+
+$$
+W _ {j} x ^ {\prime} = \alpha X _ {j} + \sqrt {1 - \alpha^ {2}} Y _ {j} \geq \alpha X _ {j} \geq \alpha R _ {1} ^ {\prime}, \tag {19}
+$$
+
+where the first step follows from Equation (6), the second step follows from  $Y_{j} \geq 0$ , and the last step follows from  $X_{j} \in [R_{1}', R_{1}]$ .
+
+By Equation (19) and Equation (17),  $\exists i\in \mathcal{A}(Wx),j\in [l]\setminus \mathcal{A}(Wx)$
+
+$$
+\begin{array}{l} W _ {i} x ^ {\prime} \leq \alpha (R _ {2} ^ {\prime} - 1 6 \varepsilon \sqrt {2 \pi}) \leq \alpha (R _ {1} ^ {\prime} - 8 \varepsilon \sqrt {2 \pi}) \leq W _ {j} x ^ {\prime} - 8 \alpha \varepsilon \sqrt {2 \pi} \\ \leq W _ {j} x ^ {\prime} - 4 \varepsilon \sqrt {2 \pi} = W _ {j} x ^ {\prime} - \frac {\sqrt {1 - \alpha^ {2}}}{2 4 \alpha} \cdot \sqrt {2 \pi}, \\ \end{array}
+$$
+
+where the second step follows from Claim 3, the forth step follows from  $\alpha \geq 0.5$ , and the last step follows from  $\varepsilon = \sqrt{1 - \alpha^2} / (96\alpha)$ . By Lemma 3, we can conclude  $\mathcal{A}(Wx) \neq \mathcal{A}(Wx')$ . By Equation (15), Equation (16), Equation (18), and union bound, the overall probability is at least
+
+$$
+\begin{array}{l} 1 - \left(4 e ^ {- \varepsilon^ {2} \gamma l / 2 4} + \left(1 6 \varepsilon \cdot \frac {\alpha}{\sqrt {1 - \alpha^ {2}}} + \frac {1}{2}\right) ^ {\varepsilon \gamma l / 2} + \left(\frac {1}{2}\right) ^ {\varepsilon \gamma l / 2}\right) \\ \geq 1 - \left(4 e ^ {- \varepsilon^ {2} \gamma l / 2 4} + \left(\frac {2}{3}\right) ^ {\varepsilon \gamma l / 2} + \left(\frac {1}{2}\right) ^ {\varepsilon \gamma l / 2}\right) \\ \geq 1 - 6 \cdot \left(\frac {2}{3}\right) ^ {\varepsilon^ {2} \gamma l / 2 4} \\ \geq 1 - 2 ^ {- \Theta \left(\left(\frac {1}{\alpha^ {2}} - 1\right) \gamma l\right)}, \\ \end{array}
+$$
+
+where the first and the last step follows from  $\varepsilon = \sqrt{1 - \alpha^2} /(96\alpha)$
+
+![](images/0cb77b5997f83c2cc9675c0b4f6390f36293cb4a361743e3bc0e1ab4aa7702fe.jpg)
+
+Next, we will use a tool called  $\varepsilon$ -net.
+
+Definition 7 ( $\varepsilon$ -Net). For a given set  $\mathcal{S}$ , if there is a set  $\mathcal{N} \subseteq \mathcal{S}$  such that  $\forall x \in \mathcal{S}$  there exists a vector  $y \in \mathcal{N}$  such that  $\|x - y\|_2 \leq \varepsilon$ , then  $\mathcal{N}$  is an  $\varepsilon$ -net of  $\mathcal{S}$ .
+
+There is a standard upper bound of the size of an  $\varepsilon$ -net of a unit norm ball.
+
+Lemma 6 (Wojtaszczyk (1996) II.E, 10). Given a matrix  $U \in \mathbb{R}^{m \times d}$ , let  $\mathcal{S} = \{Uy \mid \|Uy\|_2 = 1\}$ . For  $\varepsilon \in (0,1)$ , there is an  $\varepsilon$ -net  $\mathcal{N}$  of  $\mathcal{S}$  with  $|\mathcal{N}| \leq (1 + 1/\varepsilon)^d$ .
+
+Now we can extend above lemma to the following.
+
+Lemma 7 ( $\varepsilon$ -Net for the set of points with a certain angle). Given a vector  $x \in \mathbb{R}^m$  with  $\|x\|_2 = 1$  and a parameter  $\alpha \in (-1,1)$ , let  $\mathcal{S} = \{x' \in \mathbb{R}^m \mid \|x'\|_2 = 1, \langle x, x' \rangle = \alpha\}$ . For  $\varepsilon \in (0,1)$ , there is an  $\varepsilon$ -net  $\mathcal{N}$  of  $\mathcal{S}$  with  $|\mathcal{N}| \leq (1 + 1/\varepsilon)^{m-1}$ .
+
+Proof. Let  $U \in \mathbb{R}^{m \times (m - 1)}$  have orthonormal columns and  $Ux = 0$ . Then  $S$  can be represented as
+
+$$
+\mathcal {S} = \left\{\alpha \cdot x + \sqrt {1 - \alpha^ {2}} \cdot U y \mid y \in \mathbb {R} ^ {m - 1}, \| U y \| _ {2} = 1 \right\}.
+$$
+
+Let
+
+$$
+\mathcal {S} ^ {\prime} = \left\{U y \mid y \in \mathbb {R} ^ {m - 1}, \| U y \| _ {2} = 1 \right\}.
+$$
+
+According to Lemma 6, there is an  $\varepsilon$ -net  $\mathcal{N}'$  of  $S'$  with size  $|\mathcal{N}'| \leq (1 + 1 / \varepsilon)^{m - 1}$ . We construct  $\mathcal{N}$  as following:
+
+$$
+\mathcal {N} = \{\alpha \cdot x + \sqrt {1 - \alpha^ {2}} \cdot z \mid z \in \mathcal {N} ^ {\prime} \}.
+$$
+
+It is obvious that  $|\mathcal{N}| = |\mathcal{N}'| \leq (1 + 1 / \varepsilon)^{m - 1}$ . Next, we will show that  $\mathcal{N}$  is indeed an  $\varepsilon$ -net of  $S$ . Let  $x'$  be an arbitrary vector from  $S$ . Let  $x' = \alpha \cdot x + \sqrt{1 - \alpha^2} \cdot z$  for some  $z \in S'$ . There is a vector  $(\alpha \cdot x + \sqrt{1 - \alpha^2} \cdot z') \in \mathcal{N}$  such that  $z' \in \mathcal{N}'$  and  $\| z - z' \|_2 \leq \varepsilon$ . Thus, we have
+
+$$
+\left\| x ^ {\prime} - \left(\alpha \cdot x + \sqrt {1 - \alpha^ {2}} \cdot z ^ {\prime}\right) \right\| _ {2} = \sqrt {1 - \alpha^ {2}} \| z - z ^ {\prime} \| _ {2} \leq \varepsilon .
+$$
+
+![](images/e6d7b8d5f83d4002e995224f99810029c83087d1a00a86dbcf5829201609b751.jpg)
+
+Theorem 8 (Rotating a vector a little bit may change the activation pattern). Consider a weight matrix  $W \in \mathbb{R}^{l \times m}$  where each entry is an i.i.d. sample drawn from the Gaussian distribution  $N(0,1 / l)$ . Let  $\gamma \in (0,0.48)$  be the sparsity ratio of the activation function, i.e.,  $\gamma = k / l$ . With probability at least 0.99, it has  $\forall i \in [l]$ ,  $\| W_i\|_2 \leq 10\sqrt{m}$ . Condition on that  $\forall i \in [l]$ ,  $\| W_i\|_2 \leq 10\sqrt{m}$  happens, then, for any  $x \in \mathbb{R}^m$  and  $\alpha \in (0.5,1)$ , if
+
+$$
+l \geq C \cdot \left(\frac {m + \log (1 / \delta)}{\gamma} \cdot \frac {1}{1 - \alpha^ {2}}\right) \cdot \log \left(\frac {m + \log (1 / \delta)}{\gamma} \cdot \frac {1}{1 - \alpha^ {2}}\right)
+$$
+
+for a sufficiently large constant  $C$ , with probability at least  $1 - \delta \cdot 2^{-m}$ ,  $\forall x' \in \mathbb{R}^m$  with  $\frac{\langle x, x' \rangle}{\|x\|_2 \|x'\|_2} \leq \alpha$ ,  $\mathcal{A}(Wx) \neq \mathcal{A}(Wx')$ .
+
+Proof. Notice that the scale of  $W$  does not affect the activation pattern of  $Wx$  for any  $x \in \mathbb{R}^m$ . Thus, we assume that each entry of  $W$  is a standard Gaussian random variable in the remaining proof, and we will instead condition on  $\forall i \in [l]$ ,  $\| W_i\|_2 \leq 10\sqrt{ml}$ . The scale of  $x$  or  $x'$  will not affect  $\frac{\langle x,x' \rangle}{\|x\|_2\|x'\|_2}$ . It will not affect the activation pattern either. Thus, we assume  $\| x\|_2 = \| x'\|_2 = 1$ .
+
+By Lemma 4, with probability at least 0.99, we have  $\forall i\in [l],\| W_i\| _2\leq 10\sqrt{ml}$
+
+Let
+
+$$
+\mathcal {S} = \left\{y \in \mathbb {R} ^ {m} \mid \| y \| _ {2} = 1, \langle x, y \rangle = \alpha \right\}.
+$$
+
+Set
+
+$$
+\varepsilon = \frac {\sqrt {2 \pi (1 - \alpha^ {2})}}{7 2 0 \alpha \sqrt {m l}}.
+$$
+
+By Lemma 7, there is an  $\varepsilon$ -net  $\mathcal{N}$  of  $S$  such that
+
+$$
+| \mathcal {N} | \leq \left(1 + \frac {7 2 0 \alpha \sqrt {m l}}{\sqrt {2 \pi (1 - \alpha^ {2})}}\right) ^ {m}.
+$$
+
+By Lemma 5, for any  $y \in \mathcal{N}$ , with probability at least
+
+$$
+1 - 2 ^ {- \Theta ((1 / \alpha^ {2} - 1) \gamma l)},
+$$
+
+$\exists i \in \mathcal{A}(Wx), j \in [l] \setminus \mathcal{A}(Wx)$  such that
+
+$$
+W _ {i} y <   W _ {j} y - \frac {\sqrt {1 - \alpha^ {2}}}{2 4 \alpha} \cdot \sqrt {2 \pi}.
+$$
+
+By taking union bound over all  $y \in \mathcal{N}$ , with probability at least
+
+$$
+\begin{array}{l} 1 - \left| \mathcal {N} \right| \cdot 2 ^ {- \Theta \left(\left(1 / \alpha^ {2} - 1\right) \gamma l\right)} \\ \geq 1 - \left(1 + \frac {7 2 0 \alpha \sqrt {m l}}{\sqrt {2 \pi (1 - \alpha^ {2})}}\right) ^ {m} 2 ^ {- \Theta \left(\left(\frac {1}{\alpha^ {2}} - 1\right) \gamma l\right)} \\ \geq 1 - \left(1 0 0 0 \cdot \frac {\sqrt {m l}}{\sqrt {1 - \alpha^ {2}}}\right) ^ {m} 2 ^ {- \Theta \left(\left(\frac {1}{\alpha^ {2}} - 1\right) \gamma l\right)} \\ \geq 1 - \left(1 0 0 0 \cdot \frac {\sqrt {m l}}{\sqrt {1 - \alpha^ {2}}}\right) ^ {m} 2 ^ {- C ^ {\prime} \cdot \left(\frac {1}{\alpha^ {2}} - 1\right) \gamma \cdot \frac {m + \log (1 / \delta)}{\gamma} \cdot \frac {\alpha^ {2}}{1 - \alpha^ {2}} \cdot \log \left(\frac {m l}{1 - \alpha^ {2}}\right)} // C ^ {\prime} \text {i s a s u f f i c i e n t l y l a r g e c o n s t a n t} \\ = 1 - \left(1 0 0 0 \cdot \frac {\sqrt {m l}}{\sqrt {1 - \alpha^ {2}}}\right) ^ {m} 2 ^ {- C ^ {\prime} \cdot (m + \log (1 / \delta)) \cdot \log \left(\frac {m l}{1 - \alpha^ {2}}\right)} \\ \geq 1 - \delta \cdot 2 ^ {- m}, \\ \end{array}
+$$
+
+the following event  $\mathcal{E}'$  happens:  $\forall y \in \mathcal{N}, \exists i \in \mathcal{A}(Wx), j \in [l] \setminus \mathcal{A}(Wx)$  such that
+
+$$
+W _ {i} y <   W _ {j} y - \frac {\sqrt {1 - \alpha^ {2}}}{2 4 \alpha} \cdot \sqrt {2 \pi}.
+$$
+
+In the remaining of the proof, we will condition on the event  $\mathcal{E}'$ . Consider  $y' \in S$ . Since  $\mathcal{N}$  is an  $\varepsilon$ -net of  $S$ , we can always find a  $y \in \mathcal{N}$  such that
+
+$$
+\| y - y ^ {\prime} \| _ {2} \leq \varepsilon = \frac {\sqrt {2 \pi (1 - \alpha^ {2})}}{7 2 0 \alpha \sqrt {m l}}.
+$$
+
+Since event  $\mathcal{E}'$  happens, we can find  $i\in \mathcal{A}(Wx)$  and  $j\in [l]\setminus \mathcal{A}(Wx)$  such that
+
+$$
+W _ {i} y <   W _ {j} y - \frac {\sqrt {1 - \alpha^ {2}}}{2 4 \alpha} \cdot \sqrt {2 \pi}.
+$$
+
+Then, we have
+
+$$
+\begin{array}{l} W _ {i} y ^ {\prime} = W _ {i} y + W _ {i} \left(y ^ {\prime} - y\right) \\ <   W _ {j} y - \frac {\sqrt {1 - \alpha^ {2}}}{2 4 \alpha} \cdot \sqrt {2 \pi} + \| W _ {i} \| _ {2} \| y ^ {\prime} - y \| _ {2} \\ \leq W _ {j} y - \frac {\sqrt {1 - \alpha^ {2}}}{2 4 \alpha} \cdot \sqrt {2 \pi} + 1 0 \sqrt {m l} \cdot \frac {\sqrt {2 \pi (1 - \alpha^ {2})}}{7 2 0 \alpha \sqrt {m l}} \\ = W _ {j} y - \frac {\sqrt {1 - \alpha^ {2}}}{3 6 \alpha} \cdot \sqrt {2 \pi} \\ = W _ {j} y ^ {\prime} + W _ {j} (y - y ^ {\prime}) - \frac {\sqrt {1 - \alpha^ {2}}}{3 6 \alpha} \cdot \sqrt {2 \pi} \\ \leq W _ {j} y ^ {\prime} + \| W _ {j} \| _ {2} \| y - y ^ {\prime} \| _ {2} - \frac {\sqrt {1 - \alpha^ {2}}}{3 6 \alpha} \cdot \sqrt {2 \pi} \\ \leq W _ {j} y ^ {\prime} + 1 0 \sqrt {m l} \cdot \frac {\sqrt {2 \pi (1 - \alpha^ {2})}}{7 2 0 \alpha \sqrt {m l}} - \frac {\sqrt {1 - \alpha^ {2}}}{3 6 \alpha} \cdot \sqrt {2 \pi} \\ \leq W _ {j} y ^ {\prime} - \frac {\sqrt {1 - \alpha^ {2}}}{7 2 \alpha} \cdot \sqrt {2 \pi}, \\ \end{array}
+$$
+
+where the second step follows from  $W_{i}y < W_{j}y - \sqrt{1 - \alpha^{2}} / (24\alpha) \cdot \sqrt{2\pi}$  and  $W_{i}(y' - y) \leq \| W_{i}\|_{2}\| y' - y\|_{2}$ , the third step follows from  $\| W_{i}\|_{2} \leq 10\sqrt{ml}$  and  $\| y' - y\|_{2} \leq \sqrt{2\pi(1 - \alpha^{2})} / (720\alpha\sqrt{ml})$ , the sixth step follows from  $W_{j}(y - y') \leq \| W_{j}\|_{2}\| y - y'\|_{2}$ , and the seventh step follows from  $\| W_{i}\|_{2} \leq 10\sqrt{ml}$  and  $\| y' - y\|_{2} \leq \sqrt{2\pi(1 - \alpha^{2})} / (720\alpha\sqrt{ml})$ .
+
+By Lemma 3, we know that  $\mathcal{A}(Wx) \neq \mathcal{A}(Wy')$ . Thus,  $\forall y' \in \mathbb{R}^m$  with  $\|y'\|_2 = 1$  and  $\langle x, y' \rangle = \alpha$ , we have  $\mathcal{A}(Wx) \neq \mathcal{A}(Wy')$  conditioned on  $\mathcal{E}'$ . By Lemma 2, we can conclude that  $\forall x' \in \mathbb{R}^m$  with  $\|x'\|_2 = 1$  and  $\langle x, x' \rangle \leq \alpha$ , we have  $\mathcal{A}(Wx) \neq \mathcal{A}(Wx')$  conditioned on  $\mathcal{E}'$ .
+
+Theorem 9 (A formal version of Theorem 1). Consider a weight matrix  $W \in \mathbb{R}^{l \times m}$  where each entry is an i.i.d. sample drawn from the Gaussian distribution  $N(0,1 / l)$ . Let  $\gamma \in (0,0.48)$  be the sparsity ratio of the activation function, i.e.,  $\gamma = k / l$ . With probability at least 0.99, it has  $\forall i \in [l], \| W_i\|_2 \leq 10\sqrt{m}$ . Condition on that  $\forall i \in [l], \| W_i\|_2 \leq 10\sqrt{m}$  happens, then, for any  $x \in \mathbb{R}^m$ , if
+
+$$
+l \geq C \cdot \left(\frac {m + \log (1 / \delta)}{\gamma} \cdot \frac {1}{\beta}\right) \cdot \log \left(\frac {m + \log (1 / \delta)}{\gamma} \cdot \frac {1}{\beta}\right)
+$$
+
+for some  $\beta \in (0,1)$  and a sufficiently large constant  $C$ , with probability at least  $1 - \delta \cdot 2^{-m}$ ,  $\forall x' \in \mathbb{R}^m$  with  $\| \Delta x \|_2^2 / \| x \|_2^2 \geq \beta$ ,  $\mathcal{A}(Wx) \neq \mathcal{A}(Wx')$ , where  $x' = c \cdot (x + \Delta x)$  for some scalar  $c$  and  $\Delta x$  is perpendicular to  $x$ .
+
+Proof. If  $\langle x,x^{\prime}\rangle \leq 0$ , then the statement follows from Theorem 8 directly. In the following, we consider the case  $\langle x,x^{\prime}\rangle >0$ . If  $\| \Delta x\| _2 / \| x\| _2^2\geq \beta$
+
+$$
+\begin{array}{l} \frac {\langle x , x ^ {\prime} \rangle^ {2}}{\| x \| _ {2} ^ {2} \| x ^ {\prime} \| _ {2} ^ {2}} \\ = \frac {c ^ {2} \| x \| _ {2} ^ {\frac {4}{2}}}{\| x \| _ {2} ^ {2} \left(c ^ {2} \left(\| x \| _ {2} ^ {2} + \| \Delta x \| _ {2} ^ {2}\right)\right)} = \frac {\| x \| _ {2} ^ {2}}{\| x \| _ {2} ^ {2} + \| \Delta x \| _ {2} ^ {2}} \\ \leq \frac {\| x \| _ {2} ^ {2}}{\| x \| _ {2} ^ {2} + \beta \| x \| _ {2} ^ {2}} \leq \frac {1}{1 + \beta}. \\ \end{array}
+$$
+
+Thus, we have the bounds:
+
+$$
+\frac {1}{1 - \frac {\langle x , x ^ {\prime} \rangle^ {2}}{\| x \| _ {2} ^ {2} \| x ^ {\prime} \| _ {2} ^ {2}}} \leq \frac {1}{\beta} + 1 \leq O \left(\frac {1}{\beta}\right).
+$$
+
+By Theorem 8, we conclude the proof.
+
+![](images/856fd26b53771cc01dcc702f6002ebff9b6b92d850cb34c25955c1522ab87d3b.jpg)
+
+Example 1. Suppose that the training data contains  $N$  points  $x_{1}, x_{2}, \dots, x_{N} \in \mathbb{R}^{m}$  ( $m \geq \Omega(\log N)$ ), where each entry of  $x_{i}$  for  $i \in [N]$  is an i.i.d. Bernoulli random variable, i.e., each entry is 1 with some probability  $p \in (100\log(N)/m, 0.5)$  and 0 otherwise. Consider a weight matrix  $W \in \mathbb{R}^{l \times m}$  where each entry is an i.i.d. sample drawn from the Gaussian distribution  $N(0,1/l)$ . Let  $\gamma \in (0,0.48)$  be the sparsity ratio of the activation function, i.e.,  $\gamma = k/l$ . If  $l \geq \Omega(m/\gamma \cdot \log(m/\gamma))$ , then with probability at least 0.9,  $\forall i, j \in [N]$ , the activation pattern of  $Wx_{i}$  and  $Wx_{j}$  are different, i.e.,  $\mathcal{A}(Wx_{i}) \neq \mathcal{A}(Wx_{j})$ .
+
+Proof. Firstly, let us bound  $\| x_i\| _2$ . We have  $\mathbb{E}[\| x_i\|_2^2 ] = \mathbb{E}\left[\sum_{t = 1}^{m}x_{i,t}\right] = pm$ . By Bernstein inequality, we have
+
+$$
+\operatorname * {P r} \left[ \left| \sum_ {t = 1} ^ {m} x _ {i, t} - p m \right| > \frac {1}{1 0} p m \right] \leq 2 e ^ {- \frac {(p m / 1 0) ^ {2} / 2}{p m + \frac {1}{3} \cdot \frac {1}{1 0} p m}} \leq 0. 0 1 / N.
+$$
+
+Thus, by taking union bound over all  $i \in [N]$ , with probability at least 0.99,  $\forall i \in [N], \sqrt{0.9pm} \leq \| x_i\|_2 \leq \sqrt{1.1pm}$ .
+
+Next we consider  $\langle x_i, x_j \rangle$ . Notice that  $\mathbb{E}[\langle x_i, x_j \rangle] = \mathbb{E}\left[\sum_{t=1}^{m} x_{i,t} x_{j,t}\right] = p^2 m$ . There are two cases.
+
+Case 1  $(p^2 m > 20\log N)$ . By Bernstein inequality, we have
+
+$$
+\operatorname * {P r} \left[ \left| \langle x _ {i}, x _ {j} \rangle - p ^ {2} m \right| > \frac {1}{2} p ^ {2} m \right] \leq 2 e ^ {- \frac {(p ^ {2} m / 2) ^ {2} / 2}{p ^ {2} m + \frac {1}{3} \frac {1}{2} p ^ {2} m}} = 2 e ^ {- \frac {3}{2 s} p ^ {2} m} \leq 0. 0 1 / N ^ {2}.
+$$
+
+By taking union bound over all pairs of  $i, j$ , with probability at least 0.99,  $\forall i \neq j$ ,  $\langle x_i, x_j \rangle \leq \frac{3}{2} p^2 m$ . Since  $\|x_i\|_2, \|x_j\|_2 \geq \sqrt{0.9pm}$ , we have
+
+$$
+\frac {\left\langle x _ {i} , x _ {j} \right\rangle}{\| x _ {i} \| _ {2} \| x _ {j} \| _ {2}} \leq \frac {3 p ^ {2} m / 2}{0 . 9 p m} = \frac {5}{3} p \leq \frac {5}{6}.
+$$
+
+Case 2 ( $p^2 m \leq 20\log N$ ). By Bernstein inequality, we have
+
+$$
+\operatorname * {P r} \left[ \left| \langle x _ {i}, x _ {j} \rangle - p ^ {2} m \right| > 1 0 \log N \right] \leq 2 e ^ {- \frac {(1 0 \log N) ^ {2} / 2}{p ^ {2} m + \frac {1}{3} \cdot 1 0 \log N}} \leq 0. 0 1 / N ^ {2}.
+$$
+
+By taking union bound over all pairs of  $i, j$ , with probability at least  $0.99$ ,  $\forall i \neq j$ ,  $\langle x_i, x_j \rangle \leq 10 \log N$ . Since  $\|x_i\|_2, \|x_j\|_2 \geq \sqrt{0.9pm} \geq \sqrt{90 \log N}$ , we have
+
+$$
+\frac {\left\langle x _ {i} , x _ {j} \right\rangle}{\| x _ {i} \| _ {2} \| x _ {j} \| _ {2}} \leq \frac {1 0 \log N}{9 0 \log N} = \frac {1}{9}.
+$$
+
+Thus, with probability at least 0.98, we have  $\forall i \neq j$ ,  $\langle x_i, x_j \rangle / (\|x_i\|_2 \|x_j\|_2) \leq 5/6$ . By Theorem 8, with probability at least 0.99,  $\forall q \in [l]$ ,  $\|W_q\|_2 \leq 10\sqrt{m}$ . Condition on this event, and since  $\forall i \neq j$  we have  $\langle x_i, x_j \rangle / (\|x_i\|_2 \|x_j\|_2) \leq 5/6$ , by Theorem 8 again and union bound over all  $i \in [N]$ , with probability at least 0.99,  $\forall i \neq j$ ,  $\mathcal{A}(Wx_i) \neq \mathcal{A}(Wx_j)$ .
+
+# C.2 DISJOINTNESS OF ACTIVATION PATTERNS OF DIFFERENT INPUT POINTS
+
+Let  $X_{1}, X_{2}, \dots, X_{m}$  be i.i.d. random variables drawn from the standard Gaussian distribution  $N(0,1)$ . Let  $Z = \sum_{i=1}^{m} X_{i}^{2}$ . We use the notation  $\chi_{m}^{2}$  to denote the distribution of  $Z$ . If  $m$  is clear in the context, we just use  $\chi^{2}$  for short.
+
+Lemma 8 (A property of  $\chi^2$  distribution). Let  $Z$  be a random variable with  $\chi_m^2 m (m \geq 2)$  distribution. Given arbitrary  $\varepsilon, \eta \in (0,1)$ , if  $R$  is sufficiently large then
+
+$$
+\operatorname * {P r} [ Z \geq (1 + \varepsilon) R ] / \operatorname * {P r} [ (1 + \varepsilon) R \geq Z \geq R ] \leq \eta .
+$$
+
+Proof. Let  $R$  be a sufficiently large number such that:
+
+$e^{\varepsilon R / 2}\geq \frac{4}{\varepsilon}.$  
+$e^{\varepsilon R / 8}\geq R^{m / 2 - 1}.$  
+$e^{\varepsilon R / 4}\geq \frac{16}{9}\cdot \frac{1}{\eta}.$
+
+Let  $\xi = \varepsilon /4$ . By the density function of  $\chi^2$  distribution, we have
+
+$$
+\Pr [ R \leq Z \leq (1 + \varepsilon) R ] = \frac {1}{2 ^ {m / 2} \Gamma (m / 2)} \int_ {R} ^ {(1 + \varepsilon) R} t ^ {m / 2 - 1} e ^ {- t / 2} \mathrm {d} t,
+$$
+
+and
+
+$$
+\Pr [ Z \geq (1 + \varepsilon) R ] = \frac {1}{2 ^ {m / 2} \Gamma (m / 2)} \int_ {(1 + \varepsilon) R} ^ {\infty} t ^ {m / 2 - 1} e ^ {- t / 2} \mathrm {d} t,
+$$
+
+where  $\Gamma (\cdot)$  is the Gamma function, and for integer  $m / 2$ ,  $\Gamma (m / 2) = (m / 2 - 1)(m / 2 - 2)\dots 2\cdot 1 = (m / 2 - 1)!$ . By our choice of  $R$ , we have
+
+$$
+\begin{array}{l} \Pr [ R \leq Z \leq (1 + \varepsilon) R ] \geq \frac {1}{2 ^ {m / 2} \Gamma (m / 2)} \int_ {R} ^ {(1 + \varepsilon) R} e ^ {- t / 2} d t \\ = \frac {1}{2 ^ {m / 2} \Gamma (m / 2)} \cdot 2 \left(e ^ {- R / 2} - e ^ {- (1 + \varepsilon) R / 2}\right) \\ \geq \frac {1}{2 ^ {m / 2} \Gamma (m / 2)} \cdot 2 (1 - \xi) \cdot e ^ {- R / 2}, \\ \end{array}
+$$
+
+where the first step follows from  $\forall t\geq R,t^{m / 2 - 1}\geq 1$  , and the third step follows from
+
+$$
+\frac {e ^ {- (1 + \varepsilon) R / 2}}{e ^ {- R / 2}} = e ^ {- \varepsilon R / 2} \leq \xi .
+$$
+
+We also have:
+
+$$
+\begin{array}{l} \Pr [ Z \geq (1 + \varepsilon) R ] \leq \frac {1}{2 ^ {m / 2} \Gamma (m / 2)} \int_ {(1 + \varepsilon) R} ^ {+ \infty} e ^ {- (1 - \xi) t / 2} d t \\ = \frac {1}{2 ^ {m / 2} \Gamma (m / 2)} \cdot \frac {2}{1 - \xi} \cdot e ^ {- (1 - \xi) (1 + \varepsilon) R / 2} \\ \leq \frac {1}{2 ^ {m / 2} \Gamma (m / 2)} \cdot \frac {2}{1 - \xi} \cdot e ^ {- (1 + \varepsilon / 2) R / 2}, \\ \end{array}
+$$
+
+where the first step follos from  $\forall t\geq R,t^{m / 2 - 1}\leq e^{\xi t / 2}$ , and the third step follows from  $(1 - \xi)(1 + \varepsilon)\geq (1 + \varepsilon /2)$ .
+
+Thus, we have
+
+$$
+\frac {\operatorname * {P r} [ Z \geq (1 + \varepsilon) R ]}{\operatorname * {P r} [ (1 + \varepsilon) R \geq Z \geq R ]} \leq \frac {1}{(1 - \xi) ^ {2}} e ^ {- \varepsilon R / 4} \leq \frac {1 6}{9} e ^ {- \varepsilon R / 4} \leq \eta .
+$$
+
+□
+
+Lemma 9. Consider  $x, y, z \in \mathbb{R}^m$ . If  $\frac{\langle x, y \rangle}{\|x\|_2 \|y\|_2} \leq \alpha$ ,  $\frac{\langle x, z \rangle}{\|x\|_2 \|z\|_2} \geq \beta$  for some  $\alpha, \beta \geq 0$ , then  $\frac{\langle y, z \rangle}{\|y\|_2 \|z\|_2} \leq \alpha + \sqrt{1 - \beta^2}$ . Furthermore, if  $\beta = \frac{2 + \alpha + \sqrt{2 - \alpha^2}}{4}$ , then  $\frac{\langle y, z \rangle}{\|y\|_2 \|z\|_2} \leq (1 - \varepsilon_\alpha)\beta$ , where  $\varepsilon_\alpha \in (0, 1)$  only depends on  $\alpha$ .
+
+Proof. Without loss of generality, we suppose  $\| x\| _2 = \| y\| _2 = \| z\| _2 = 1$ . We can decompose  $y$  as  $ax + y^{\prime}$  where  $y^\prime$  is perpendicular to  $x$ . We can decompose  $z$  as  $b_{1}x + b_{2}y^{\prime} / \| y^{\prime}\|_{2} + z^{\prime}$  where  $z^{\prime}$  is perpendicular to both  $x$  and  $y^\prime$ . Then we have:
+
+$$
+\langle y, z \rangle = a b _ {1} + b _ {2} \| y ^ {\prime} \| _ {2} \leq \alpha + \sqrt {1 - \beta^ {2}},
+$$
+
+where the last inequality follows from  $0 \leq b_{1} \leq 1, a \leq \alpha$ , and  $b_{2} \leq \sqrt{1 - b_{1}^{2}} \leq \sqrt{1 - \beta^{2}}, 0 \leq \|y'\|_{2} \leq 1$ .
+
+By solving  $\beta \geq \alpha +\sqrt{1 - \beta^2}$ , we can get  $\beta \geq \frac{\alpha + \sqrt{2 - \alpha^2}}{2}$ . Thus, if we set
+
+$$
+\beta = \frac {1 + \frac {\alpha + \sqrt {2 - \alpha^ {2}}}{2}}{2},
+$$
+
+$\beta$  should be strictly larger than  $\alpha + \sqrt{1 - \beta^2}$ , and the gap only depends on  $\alpha$ .
+
+Lemma 10. Give  $x \in \mathbb{R}^m$ , let  $y \in \mathbb{R}^m$  be a random vector, where each entry of  $y$  is an i.i.d. sample drawn from the standard Gaussian distribution  $N(0,1)$ . Given  $\beta \in (0.5,1)$ ,  $\operatorname*{Pr}[\langle x,y\rangle /(\|x\|_2\|y\|_2) \geq \beta] \geq 1/(1 + 1/\sqrt{2(1 - \beta)})^m$ .
+
+Proof. Without loss of generality, we can assume  $\| x \|_2 = 1$ . Let  $y' = y / \| y \|_2$ . Since each entry of  $y$  is an i.i.d. Gaussian variable,  $y'$  is a random vector drawn uniformly from a unit sphere. Notice that if  $\langle x, y' \rangle \geq \beta$ , then  $\| x - y' \|_2 \leq \sqrt{2(1 - \beta)}$ . Let  $C = \{z \in \mathbb{R}^m \mid \| z \|_2 = 1, \| z - x \|_2 \leq \sqrt{2(1 - \beta)}\}$  be a cap, and let  $S = \{z \in \mathbb{R}^m \mid \| z \|_2 = 1\}$  be the unit sphere. Then we have
+
+$$
+\Pr [ \langle x, y ^ {\prime} \rangle \geq \beta ] = \operatorname {a r e a} (C) / \operatorname {a r e a} (\mathcal {S}).
+$$
+
+According to Lemma 6, there is an  $\sqrt{2(1 - \beta)}$ -net  $\mathcal{N}$  with  $|\mathcal{N}| \leq (1 + 1 / \sqrt{2(1 - \beta)})^m$ . If we put a cap centered at each point in  $\mathcal{N}$ , then the whole unit sphere will be covered. Thus, we can conclude
+
+$$
+\Pr [ \langle x, y ^ {\prime} \rangle \geq \beta ] \geq 1 / (1 + 1 / \sqrt {2 (1 - \beta)}) ^ {m}.
+$$
+
+![](images/89346a83486ccb391e568f6f4c8aa000a282ed798160105af3def9d6e86ee0f1.jpg)
+
+Theorem 10 (A formal version of Theorem 2). Consider  $N$  data points  $x_{1}, x_{2}, \dots, x_{N} \in \mathbb{R}^{m}$  and a weight matrix  $W \in \mathbb{R}^{l \times m}$  where each entry of  $W$  is an i.i.d. sample drawn from the Gaussian distribution  $N(0,1 / l)$ . Suppose  $\forall i \neq j \in [N], \langle x_{i}, x_{j} \rangle / (\| x_{i} \|_{2} \| x_{j} \|_{2}) \leq \alpha$  for some  $\alpha \in (0.5,1)$ . Fix  $k \geq 1$  and  $\delta \in (0,1)$ , if  $l$  is sufficiently large, then with probability at least  $1 - \delta$ ,
+
+$$
+\forall i, j \in [ N ], \mathcal {A} (W x _ {i}) \cap \mathcal {A} (W x _ {j}) = \emptyset .
+$$
+
+Proof. Notice that the scale of  $W$  and  $x_{1}, x_{2}, \dots, x_{N}$  do not affect either  $\langle x_{i}, x_{j} \rangle / (\| x_{i} \|_{2} \| x_{j} \|_{2})$  or the activation pattern. Thus, we can assume  $\| x_{1} \|_{2} = \| x_{2} \|_{2} = \dots = \| x_{N} \|_{2} = 1$  and each entry of  $W$  is an i.i.d. standard Gaussian random variable.
+
+Let  $\beta = \frac{2 + \alpha + \sqrt{2 - \alpha^2}}{4}$  and  $\varepsilon_{\alpha}$  be the same as mentioned in Lemma 9. Set  $\varepsilon$  and  $\beta'$  as
+
+$$
+\varepsilon = \frac {\frac {1}{\beta} - 1}{2}, \quad \beta^ {\prime} = (1 + \varepsilon) \beta .
+$$
+
+Now, set
+
+$$
+\eta = \frac {\delta / 1 0 0}{1 0 0 k \log (N / \delta) \cdot (1 + 2 / \sqrt {2 (1 - \beta^ {\prime})}) ^ {m}},
+$$
+
+and let  $R$  satisfies
+
+$$
+\operatorname * {P r} _ {Z \sim \chi_ {m} ^ {2}} [ Z \geq (1 + \varepsilon) ^ {2} R ^ {2} ] = \frac {\delta / 1 0 0}{l}.
+$$
+
+According to Lemma 8, if  $l$  is sufficiently large, then  $R$  is sufficiently large such that
+
+$$
+\operatorname * {P r} _ {Z \sim \chi_ {m} ^ {2}} [ Z \geq (1 + \varepsilon) ^ {2} R ^ {2} ] / \operatorname * {P r} _ {Z \sim \chi_ {m} ^ {2}} [ (1 + \varepsilon) ^ {2} R ^ {2} \geq Z \geq R ^ {2} ] \leq \eta .
+$$
+
+Notice that for  $t \in [l]$ ,  $\| W_t\|_2^2$  is a random variable with  $\chi_m^2$  distribution. Thus,  $\operatorname*{Pr}[\| W_t\|_2 \geq (1 + \varepsilon)R] = \frac{\delta / 100}{l}$ . By taking union bound over all  $t \in [l]$ , with probability at least  $1 - \delta / 100$ ,  $\forall t \in [l]$ ,  $\| W_t\|_2 \leq (1 + \varepsilon)R$ . In the remaining of the proof, we will condition on that  $\forall t \in [l]$ ,  $\| W_t\|_2 \leq (1 + \varepsilon)R$ . Consider  $i, j \in [N]$ ,  $t \in [l]$ , if  $W_tx_i > \beta'R$ , then we have
+
+$$
+\frac {W _ {t} x _ {i}}{\| W _ {t} \| _ {2}} > \frac {\beta^ {\prime} R}{(1 + \varepsilon) R} \geq \beta^ {\prime} / (1 + \varepsilon) = \beta .
+$$
+
+Due to Lemma 9, we have
+
+$$
+\frac {W _ {t} x _ {j}}{\| W _ {t} \| _ {2}} <   (1 - \varepsilon_ {\alpha}) \beta .
+$$
+
+Thus,
+
+$$
+W _ {t} x _ {j} <   (1 - \varepsilon_ {\alpha}) \beta \| W _ {t} \| _ {2} \leq (1 - \varepsilon_ {\alpha}) \beta (1 + \varepsilon) R \leq (1 - \varepsilon_ {\alpha}) \beta^ {\prime} R. \tag {20}
+$$
+
+Notice that for  $i \in [N], t \in [l]$ , we have
+
+$$
+\begin{array}{l} \Pr \left[ W _ {t} x _ {i} > \beta^ {\prime} R \right] \geq \Pr \left[ \| W _ {t} \| _ {2} \geq R \right] \Pr \left[ \frac {W _ {t} x _ {i}}{\| W _ {t} \| _ {2}} \geq \beta^ {\prime} \right] \\ \geq \frac {\delta / 1 0 0}{l} \cdot \frac {1}{\eta} \cdot \frac {1}{(1 + 1 / \sqrt {2 (1 - \beta^ {\prime})}) ^ {m}} \\ \geq \frac {1}{l} \cdot 1 0 0 k \log (N / \delta). \\ \end{array}
+$$
+
+By Chernoff bound, with probability at least  $1 - \delta /(100N)$
+
+$$
+\sum_ {t = 1} ^ {l} \mathbf {1} \left(W _ {t} x _ {i} > \beta^ {\prime} R\right) \geq k.
+$$
+
+By taking union bound over  $i \in [N]$ , with probability at least  $1 - \delta / 100, \forall i \in [N]$
+
+$$
+\sum_ {t = 1} ^ {l} \mathbf {1} \left(W _ {t} x _ {i} > \beta^ {\prime} R\right) \geq k.
+$$
+
+This implies that  $\forall i\in [N]$ , if  $t\in \mathcal{A}(Wx_i)$ , then  $W_{t}x_{i} > \beta^{\prime}R$ . Due to Equation (20),  $\forall j\in [N]$ , we have  $W_{t}x_{j} < \beta^{\prime}R$  which implies that  $t\notin \mathcal{A}(Wx_j)$ . Thus, with probability at least  $1 - \delta /50\geq 1 - \delta$  probability,  $\forall i\neq j,\mathcal{A}(Wx_i)\cap \mathcal{A}(Wx_j) = \emptyset$
+
+Remark 1. Consider any  $x_{1}, x_{2}, \dots, x_{N} \in \mathbb{R}^{m}$  with  $\| x_{1}\|_{2} = \| x_{2}\|_{2} = \dots = \| x_{N}\|_{2} = 1$ . If  $\forall i \neq j \in [N], \langle x_{i}, x_{j} \rangle \leq \alpha$  for some  $\alpha \in (0.5, 1)$ , then  $|N| \leq (1 + 2 / \sqrt{2(1 - \alpha)})^{m}$ .
+
+Proof. Since  $\langle x_i, x_j \rangle \leq \alpha$ ,  $\|x_i - x_j\|_2^2 = \|x_i\|_2^2 + \|x_j\|_2^2 - 2\langle x_i, x_j \rangle \geq 2 - 2\alpha$ . Let  $S$  be the unit sphere, i.e.,  $S = \{x \in \mathbb{R}^m \mid \|x\|_2 = 1\}$ . Due to Lemma 6, there is a  $(\sqrt{2(1 - \alpha)}/2)$ -net  $\mathcal{N}$  of  $S$  with size at most  $|\mathcal{N}| \leq (1 + 2/\sqrt{2(1 - \alpha)})^m$ . Consider  $x_i, x_j$ , and  $y \in \mathcal{N}$ . By triangle inequality, if  $\|x_i - y\|_2 < \sqrt{2(1 - \alpha)}/2$ , then  $\|x_j - y\|_2 > \sqrt{2(1 - \alpha)}/2$  due to  $\|x_i - x_j\|_2 \geq \sqrt{2(1 - \alpha)}$ . Since  $\mathcal{N}$  is a net of  $S$ , for each  $x_i$ , we can find a  $y \in \mathcal{N}$  such that  $\|x_i - y\|_2 < \sqrt{2(1 - \alpha)}/2$ . Thus, we can conclude  $N \leq |\mathcal{N}| \leq (1 + 2/\sqrt{2(1 - \alpha)})^m$ .
+
+Theorem 11. Consider  $N$  data points  $x_{1}, x_{2}, \dots, x_{N} \in \mathbb{R}^{m}$  with their corresponding labels  $z_{1}, z_{2}, \dots, z_{N} \in \mathbb{R}$  and a weight matrix  $W \in \mathbb{R}^{l \times m}$  where each entry of  $W$  is an i.i.d. sample drawn from the Gaussian distribution  $N(0,1 / l)$ . Suppose  $\forall i \neq j \in [N], \langle x_{i}, x_{j} \rangle / (\| x_{i} \|_{2} \| x_{j} \|_{2}) \leq \alpha$  for some  $\alpha \in (0.5,1)$ . Fix  $k \geq 1$  and  $\delta \in (0,1)$ , if  $l$  is sufficiently large, then with probability at least  $1 - \delta$ , there exists a vector  $v \in \mathbb{R}^{l}$  such that
+
+$$
+\forall i \in [ N ], \langle v, \phi_ {k} (W x _ {i}) \rangle = z _ {i}.
+$$
+
+Proof. Due to Theorem 10, with probability at least  $1 - \delta$ ,  $\forall i \neq j$ ,  $\mathcal{A}(Wx_i) \cap \mathcal{A}(Wx_j) = \emptyset$ . Let  $t_1, t_2, \dots, t_N \in [l]$  such that  $t_i \in \mathcal{A}(Wx_i)$ . Then  $t_i \notin \mathcal{A}(Wx_j)$  for  $j \neq i$ .
+
+For each entry  $v_{t}$ , if  $t = t_{i}$  for some  $i \in [N]$ , then set  $v_{t} = z_{i} / (W_{t}x_{i})$ . Then for  $i \in [N]$ , we have
+
+$$
+\langle v, \phi_ {k} (W x _ {i}) \rangle = \sum_ {t \in \mathcal {A} (W x _ {i})} v _ {t} \cdot W _ {t} x _ {i} = z _ {i} / \left(W _ {t _ {i}} x _ {i}\right) \cdot W _ {t _ {i}} x _ {i} = z _ {i}.
+$$
+
+![](images/75ee7fc4d28b57109c380a7dffa3edb077642e3e6a8ff171a28447a22bff99e3.jpg)
+
+# D ADDITIONAL EXPERIMENTAL RESULTS
+
+This section presents details of our experiment settings and additional results for evaluating and empirically understanding the robustness of  $k$ -WTA networks.
+
+# D.1 EXPERIMENT SETTINGS
+
+First, we describe the details of setting up the experiments described in Sec. 4. To compare  $k$ -WTA networks with their ReLU counterparts, we replace all ReLU activations in a network with  $k$ -WTA activation, while retaining all other modules (such as BatchNorm, Convolution, and pooling). To test on different network architectures, including ResNet18, DenseNet121, and Wide ResNet, we use the standard implementations that are publicly available<sup>3</sup>. All experiments are conducted using PyTorch framework.
+
+Training setups. We follow the same training procedure on CIFAR-10 and SVHN datasets. All the ReLU networks are trained with stochastic gradient descent (SGD) method with momentum=0.9. We use a learning rate 0.1 from the first to 50-th epoch and 0.01 from 50-th to 80-th epoch. To compare with ReLU networks, the k-WTA networks are trained in the same way as ReLU networks. All networks are trained with a batch size of 256.
+
+For  $k$ -WTA networks with a sparsity ratio  $\gamma = 0.1$ , when adversarial training is not used, we train them incrementally (recall in Sec. 2.2). starting with  $\gamma = 0.2$  for 50 epochs with SGD (using learning rate=0.1, momentum=0.9) and then decreasing  $\gamma$  by 0.005 every 2 epochs until  $\gamma$  reaches 0.1.
+
+When adversarial training is enabled, we use untargeted PGD attack with 8 iterations to construct adversarial examples. To train networks with TRADES (Zhang et al., 2019), we use the implementation of its original paper<sup>4</sup> with the parameter  $1 / \lambda = 6$ , a value that reportedly leads to the best robustness according to the paper. To train networks with the free adversarial training method (Shafahi et al., 2019b), we implement the training algorithm by following the original paper. We set the parameter  $m = 8$  as suggested in the paper.
+
+Attack setups. All attacks are evaluated under the  $\ell_{\infty}$  metric, with perturbation size  $\epsilon = 0.031$  (CIFAR-10) and 0.047 (SVHN) for pixels ranging in [0, 1]. We use Foolbox (Rauber et al., 2017), a third-party toolbox for evaluating adversarial robustness.
+
+We use the following setups for generating adversarial examples in various attack methods: For PGD attack, we use 40 iterations with random start, the step size is 0.003. For C&W attack, we set the binary search step to 5, maximum number of iterations to 20, learning rate to 0.01, and initial constant to 0.01. For Deepfool, we use 20 steps and 10 sub-samples in its configuration. For momentum attack, we set the step size to 0.003 and number of iterations to 20. All other parameters are set by Foolbox to be its default values.
+
+# D.2 EFFICACY OF INIncrementAL TRAINING
+
+We now report additional experiments to demonstrate the efficacy of the incremental fine-tuning method (described in Sec. 2.2). As shown in Figure 6 and described its caption, models trained with incremental fine-tuning (denoted as w/ FT in the plots' legend) performs better in terms of both standard accuracy (denoted as std in the plots' legend) and robust accuracy (denoted as Rob in the plots) when the  $k$ -WTA sparsity  $\gamma < 0.2$ , suggesting that fine-tuning is worthwhile when  $\gamma$  is small.
+
+# D.3 ADDITIONAL RESULTS ON CIFAR-10
+
+Tests on different network architectures. We evaluate the robustness of  $k$ -WTA on different network architectures, including ResNet-18, DenseNet-121 and WideResNet-22-10. The results are reported in Table 2, where similar to the notation used in Table 1 of the main text,  $A_{rob}$  is calculated as the worst-case robustness, i.e., under the most effective attack among PGD, C&W, Deepfool and MIM. The training and attacking settings are same as other experiments described Sec. D.1.
+
+As shown in Table 2, while the standard and robustness accuracies,  $A_{std}$  and  $A_{rob}$ , vary on different network architectures,  $k$ -WTA networks consistently improves the worst-case robustness  $A_{rob}$  over ReLU networks, no matter what the network architecture and training method are used.
+
+![](images/75ed7f5a95a1ed070a1c2c55588744947e2fd6e848c4060c15b4c9fdef30be54.jpg)  
+Figure 6: Efficacy of incremental training. We sweep through a range of sparsity ratios, and evaluate the standard and robust accuracies of two network structures (left: ResNet18 and right: Wide ResNet). We compare the performance differences between the regular training (i.e., training without incremental fine-tuning) and the training with incremental fine-tuning.
+
+![](images/66405e0e0d5ad5e882dd04604593c8e48e7e2f285cbb0984010d5cc60fd95f3a.jpg)
+
+![](images/20bbc1dab0250dbc5e02e38c572fdb3f83490e0d440d2d0c0e43d9e8e92c5ebb.jpg)
+
+Table 2: Additional CIFAR-10 results.  
+
+<table><tr><td>Training</td><td>Model</td><td>Activation</td><td>\(A_{std}\)</td><td>\(A_{rob}\)</td></tr><tr><td rowspan="5">Natural</td><td rowspan="5">ResNet-18</td><td>ReLU</td><td>92.9%</td><td>0.0%</td></tr><tr><td>LWTA-0.1</td><td>82.8%</td><td>3.7%</td></tr><tr><td>LWTA-0.2</td><td>84.6%</td><td>0.9%</td></tr><tr><td>k-WTA-0.1</td><td>89.3%</td><td>13.1%</td></tr><tr><td>k-WTA-0.2</td><td>91.7%</td><td>4.2%</td></tr><tr><td rowspan="5">AT</td><td rowspan="5">ResNet-18</td><td>ReLU</td><td>83.5%</td><td>43.6%</td></tr><tr><td>LWTA-0.1</td><td>71.4%</td><td>46.6%</td></tr><tr><td>LWTA-0.2</td><td>78.7%</td><td>43.1%</td></tr><tr><td>k-WTA-0.1</td><td>78.9%</td><td>50.7%</td></tr><tr><td>k-WTA-0.2</td><td>81.4%</td><td>47.4%</td></tr><tr><td rowspan="5">Natural</td><td rowspan="5">DenseNet-121</td><td>ReLU</td><td>93.6%</td><td>0.0%</td></tr><tr><td>LWTA-0.1</td><td>86.1%</td><td>4.6%</td></tr><tr><td>LWTA-0.2</td><td>88.5%</td><td>1.4%</td></tr><tr><td>k-WTA-0.1</td><td>90.5%</td><td>12.3%</td></tr><tr><td>k-WTA-0.2</td><td>93.3%</td><td>6.2%</td></tr><tr><td rowspan="5">AT</td><td rowspan="5">DenseNet-121</td><td>ReLU</td><td>84.2%</td><td>46.3%</td></tr><tr><td>LWTA-0.1</td><td>74.0%</td><td>49.1%</td></tr><tr><td>LWTA-0.2</td><td>80.2%</td><td>44.9%</td></tr><tr><td>k-WTA-0.1</td><td>81.6%</td><td>52.4%</td></tr><tr><td>k-WTA-0.2</td><td>83.4%</td><td>49.6%</td></tr><tr><td rowspan="5">Natural</td><td rowspan="5">WideResNet-22-10</td><td>ReLU</td><td>93.4%</td><td>0.0%</td></tr><tr><td>LWTA-0.1</td><td>83.7%</td><td>4.2%</td></tr><tr><td>LWTA-0.2</td><td>86.1%</td><td>2.8%</td></tr><tr><td>k-WTA-0.1</td><td>88.6%</td><td>18.3%</td></tr><tr><td>k-WTA-0.2</td><td>92.7%</td><td>7.4%</td></tr><tr><td rowspan="5">AT</td><td rowspan="5">WideResNet-22-10</td><td>ReLU</td><td>83.3%</td><td>43.1%</td></tr><tr><td>LWTA-0.1</td><td>74.2%</td><td>47.5%</td></tr><tr><td>LWTA-0.2</td><td>79.8%</td><td>44.7%</td></tr><tr><td>k-WTA-0.1</td><td>78.9%</td><td>50.4%</td></tr><tr><td>k-WTA-0.2</td><td>82.4%</td><td>47.1%</td></tr></table>
+
+Comparison with LWTA. We in addition compare  $k$ -WTA to LWTA activation (Srivastava et al., 2013; 2014). For fair comparisons, we use the same sparsity ratio  $\gamma$  in both  $k$ -WTA and LWTA. As shown in Table 2, on all network architectures and training methods we tested,  $k$ -WTA networks consistently have better robustness performance than LWTA networks (in terms of both  $A_{std}$  and  $A_{rob}$ ). These results suggest that  $k$ -WTA is more suitable than LWTA for defending against adversarial attacks.
+
+Transfer attack. Since a  $k$ -WTA network is architecturally similar to its ReLU counterpart—with the only difference being the activation—we evaluate their robustness under (black-box) transfer
+
+Table 3: Transferability test on CIFAR-10.  
+
+<table><tr><td rowspan="2">Target Model</td><td colspan="4">Source Model</td></tr><tr><td>ReLU</td><td>k-WTA-0.1</td><td>ReLU (AT)</td><td>k-WTA-0.1 (AT)</td></tr><tr><td>ReLU</td><td>4.8%</td><td>75.5%</td><td>59.4%</td><td>84.7%</td></tr><tr><td>k-WTA-0.1</td><td>61.2%</td><td>71.2%</td><td>67.8%</td><td>86.4%</td></tr><tr><td>ReLU (AT)</td><td>62.7%</td><td>80.9%</td><td>61.6%</td><td>78.6%</td></tr><tr><td>k-WTA-0.1 (AT)</td><td>79.2%</td><td>78.6%</td><td>69.2%</td><td>67.2%</td></tr></table>
+
+Table 4: White-box attack results on MNIST dataset.  
+
+<table><tr><td>Activation</td><td>Training</td><td>Astd</td><td>Arob</td><td>Activation</td><td>Training</td><td>Astd</td><td>Arob</td></tr><tr><td rowspan="4">ReLU</td><td>Natural</td><td>99.4%</td><td>0.0%</td><td rowspan="4">k-WTA-0.1</td><td>Natural</td><td>99.3%</td><td>62.2%</td></tr><tr><td>AT</td><td>99.2%</td><td>95.0%</td><td>AT</td><td>99.2%</td><td>96.4%</td></tr><tr><td>TRADES</td><td>99.2%</td><td>96.0%</td><td>TRADES</td><td>99.0%</td><td>96.9%</td></tr><tr><td>FAT</td><td>98.2%</td><td>94.7%</td><td>FAT</td><td>98.1%</td><td>96.0%</td></tr></table>
+
+attacks across  $k$ -WTA and ReLU networks. To this end, we build a ReLU and a  $k$ -WTA-0.1 network on ResNet-18, and train both networks with natural (non-adversarial) training as well as adversarial training. This gives us four different models denoted (in Table 3) as ReLU,  $k$ -WTA-0.1, ReLU (AT), and  $k$ -WTA-0.1 (AT). We then launch transfer attacks across each pair of models. We also consider by-far the strongest black-box attack (according to Papernot et al. (2017)): for the same model, for example, a  $k$ -WTA-0.1 network optimized by adversarial training, we train two independent versions, each with a different random initialization, and apply the transfer attacks across the two versions.
+
+The results are reported in Table 3, where each row corresponds to a target (attacked) model, and each column corresponds to a source model from which the adversarial examples are generated. On the diagonal line of Table 3, each entry corresponds to the robustness under aforementioned transfer attacks across the two versions of the same models.
+
+The results suggest that 1) it is more difficult to transfer attack  $k$ -WTA networks than ReLU networks using adversarial examples from other models, and 2) it is also more difficult to use adversarial examples of a  $k$ -WTA network to attack other models. In a sense, the adversarial examples of a  $k$ -WTA network tend to be "disjoint" from the adversarial examples of a ReLU network, despite their architectural similarity. Inspecting the diagonal entries of Table 3, we also find that  $k$ -WTA networks are more robust than their ReLU counterparts under the strongest black-box attack (Papernot et al., 2017) (i.e., transfer attacks across two different versions of the same model).
+
+# D.4 MNIST RESULTS
+
+On MNIST dataset, we conduct experiments with an adversarial perturbation size  $\epsilon = 0.3$  for pixels ranging in [0, 1]. We use Stochastic Gradient Descent (SGD) with learning rate  $= 0.01$  and momentum  $= 0.9$  to train a 3-layer CNN. The training takes 20 epochs for all the methods we evaluate. The robust accuracy are evaluated under PGD attacks that take 20 iterations with random initialization and a step size of 0.03.
+
+The results are summarized in Table 4. Again,  $k$ -WTA activation consistently improves robustness under all different training methods. Even with natural (non-adversarial) training, the resulting  $k$ -WTA network still has  $62.2\%$  robust accuracy, significantly outperforming ReLU network.
+
+# D.5 ROBUSTNESS WITH RESPECT TO NATURAL PERTURBATIONS
+
+We also evaluate the robustness of  $k$ -WTA networks under (non-adversarial) natural perturbations. We evaluated various types of perturbations, including adding Gaussian noise to the input image (std=0.05/0.1), random translation (maximum 5 pixel), random rotation (maximum 10 degrees) and color jittering (i.e., randomly changing the brightness, contrast and saturation of an image, with a maximum perturbation of 0.4), following Hendrycks & Dietterich (2019). The results are summarized in Table 5, in which all models are ResNet18.
+
+We found that under all the tested perturbations, the accuracy drops in  $k$ -WTA networks are no worse than those in ReLU networks. Note that in these tests, all our models are trained with standard data
+
+Table 5: Robustness under natural perturbations.  
+
+<table><tr><td>Model</td><td>Clean</td><td>GN(0.05)</td><td>GN (0.1)</td><td>Translation</td><td>Rotation</td><td>ColorJitter</td></tr><tr><td>ReLU</td><td>92.9%</td><td>69.7%</td><td>27.0%</td><td>92.3%</td><td>88.9%</td><td>90.7%</td></tr><tr><td>k-WTA-0.1</td><td>89.3%</td><td>80.2%</td><td>50.9%</td><td>89.1%</td><td>86.8%</td><td>87.4%</td></tr><tr><td>k-WTA-0.2</td><td>91.7%</td><td>77.9%</td><td>39.6%</td><td>91.2%</td><td>88.9%</td><td>89.4%</td></tr><tr><td>ReLU (AT)</td><td>83.5%</td><td>80.3%</td><td>73.9%</td><td>80.5%</td><td>79.8%</td><td>74.9%</td></tr><tr><td>k-WTA-0.1 (AT)</td><td>78.9%</td><td>77.8%</td><td>69.8%</td><td>75.9%</td><td>74.7%</td><td>68.7%</td></tr><tr><td>k-WTA-0.2 (AT)</td><td>81.4%</td><td>80.6%</td><td>70.9%</td><td>79.8%</td><td>78.6%</td><td>74.7%</td></tr></table>
+
+augmentations (e.g., random crop and random flip); they are not specifically trained to avert the tested perturbations.
+
+We also highlight an interesting finding here. Adding Gaussian noise leads to a large accuracy drop (e.g., from  $92.9\%$  to  $69.7\% / 27.0\%$  as shown in the table) on naturally trained ReLU network, but in  $k$ -WTA networks (especially  $k$ -WTA-0.1), the corresponding accuracy drop is much smaller (e.g., from  $89.3\%$  to  $80.2\% / 50.9\%$ ). We conjecture that the dense discontinuities in  $k$ -WTA networks (recall Figure 5) effectively add noise to the input distribution, thus making the model more robust against input noise.
+
+# D.6 LOSS LANDSCAPE VISUALIZATION
+
+In addition to the experiments shown in Figure 5 and Sec. 4.3 of the main text, we further we visualize the loss landscapes of  $k$ -WTA networks when different sparsity ratios  $\gamma$  are used. The plots are shown in Figure 7, produced in the same way as Figure 5 described in Sec. 4.3.
+
+As analyzed in Sec. 3, a larger  $\gamma$  tends to smooth the loss surface of the  $k$ -WTA network with respect to the input, while a smaller  $\gamma$  renders the loss surface more discontinuous and "spiky". In addition, adversarial training tends to reduce the range of the loss values—a similar phenomenon in ReLU networks has already been reported Madry et al. (2017); Tramér et al. (2017)—but that does not mean that the loss surface becomes smoother; the loss surface remains spiky.
+
+![](images/ac0622d9f424c3b61188a91cedfced3edb019e7c102ab9874378a2b05d6049d1.jpg)  
+ResNet18-0.3
+
+![](images/dbae821c7a45a552378325a9dd3f22d52fb794150294344ca7d4e30df5b6fb60.jpg)  
+ResNet18-0.2
+
+![](images/6878f43eae24eb1b7a8cf46b27a5628e05415f685cec3f766fdcb2ceab1b123a.jpg)  
+ResNet18-0.1
+
+![](images/b58c8383fd568d3f12c0963328aa3ae2f78890edb7cdb6d7e8f55d6511aec2c7.jpg)  
+DenseNet121-0.2
+
+![](images/c0842066060101a2c28904dfe6754e5d439a755b1f6970cbe979610060cc5a42.jpg)  
+DenseNet121-0.1
+
+![](images/f7f72d1ee0a78f82d8e8176a9eaf76aee96b6b12aaa969c3b4135ea3ba60d0a7.jpg)  
+DenseNet121-0.1+adv  
+Figure 7: Visualization of ResNet18 and DenseNet121 with different  $\gamma$  values. The last one (DenseNet121-0.1+adv) is the result using adversarial training. The others are optimized using natural (non-adversarial) training.
\ No newline at end of file
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+# ESTIMATING COUNTERFACTUAL TREATMENT OUTCOMES OVER TIME THROUGH ADVERSARIALLY BALANCED REPRESENTATIONS
+
+# Ioana Bica
+
+Department of Engineering Science  
+University of Oxford, Oxford, UK  
+The Alan Turing Institute, London, UK  
+ioana.bica@eng.ox.ac.uk
+
+# Ahmed M. Alaa
+
+Department of Electrical Engineering  
+University of California, Los Angeles, USA  
+ahmedmalaa@ucla.edu
+
+# James Jordon
+
+Department of Engineering Science  
+University of Oxford, Oxford, UK  
+james.jordon@wolfson.ox.ac.uk
+
+# Mihaela van der Schaar
+
+University of Cambridge, Cambridge, UK  
+University of California, Los Angeles, USA  
+The Alan Turing Institute, London, UK  
+mv472@cam.ac.uk
+
+# ABSTRACT
+
+Identifying when to give treatments to patients and how to select among multiple treatments over time are important medical problems with a few existing solutions. In this paper, we introduce the Counterfactual Recurrent Network (CRN), a novel sequence-to-sequence model that leverages the increasingly available patient observational data to estimate treatment effects over time and answer such medical questions. To handle the bias from time-varying confounders, covariates affecting the treatment assignment policy in the observational data, CRN uses domain adversarial training to build balancing representations of the patient history. At each timestep, CRN constructs a treatment invariant representation which removes the association between patient history and treatment assignments and thus can be reliably used for making counterfactual predictions. On a simulated model of tumour growth, with varying degree of time-dependent confounding, we show how our model achieves lower error in estimating counterfactuals and in choosing the correct treatment and timing of treatment than current state-of-the-art methods.
+
+# 1 INTRODUCTION
+
+As clinical decision-makers are often faced with the problem of choosing between treatment alternatives for patients, reliably estimating their effects is paramount. While clinical trials represent the gold standard for causal inference, they are expensive, have a few patients and narrow inclusion criteria (Booth & Tannock, 2014). Leveraging the increasingly available observational data about patients, such as electronic health records, represents a more viable alternative for estimating treatment effects.
+
+A large number of methods have been proposed for performing causal inference using observational data in the static setting (Johansson et al., 2016; Shalit et al., 2017; Alaa & van der Schaar, 2017; Li & Fu, 2017; Yoon et al., 2018; Alaa & van der Schaar, 2018; Yao et al., 2018) and only a few methods address the longitudinal setting (Xu et al., 2016; Roy et al., 2016; Soleimani et al., 2017; Schulam & Saria, 2017; Lim et al., 2018). However, estimating the effects of treatments over time poses unique opportunities such as understanding how diseases evolve under different treatment plans, how individual patients respond to medication over time, but also which are optimal timings for assigning treatments, thus providing new tools to improve clinical decision support systems.
+
+The biggest challenge when estimating the effects of time-dependent treatments from observational data involves correctly handling the time-dependent confounders: patient covariates that are affected by past treatments which then influence future treatments and outcomes (Platt et al., 2009). For instance, consider that treatment A is given when a certain patient covariate (e.g. white blood cell
+
+![](images/34a1806388509f7f8cf4819039a4d3774b594923bcb6ed10d2e2d64eaa414178.jpg)  
+(a) Decide treatment plan
+
+![](images/d1033ca3e0782d9074df9c2de408c493f81021879d98137621766c3a15f9a386.jpg)  
+(b) Decide optimal time of treatment
+
+![](images/555c6edf6cd5a2d27311f570edc549634d4e8819e90c75b275a6979ba174c55d.jpg)  
+(c) Decide when to stop treatment  
+Figure 1: Applicability of CRN in cancer treatment planning. We illustrate 3 patients with different covariate and treatment histories  $\bar{\mathbf{H}}_t$ . For a current time  $t$ , CRN can predict counterfactual trajectories (the coloured dashed branches) for planned treatments in the future. Through the counterfactual predictions, we can decide which treatment plan results in the best patient outcome (in this case, the lowest tumour volume). This way, CRN can be used to perform all of the following: choose optimal treatments (a), find timing when treatment is most effective (b) decide when to stop treatment (c).
+
+count) has been outside of normal range values for several consecutive timesteps. Suppose also that this patient covariate was itself affected by the past administration of treatment B. If these patients are more likely to die, without adjusting for the time-dependent confounding (e.g. the changes in the white blood cell count over time), we will incorrectly conclude that treatment A is harmful to patients. Moreover, estimating the effect of a different sequence of treatments on the patient outcome would require not only adjusting for the bias at the current step (in treatment A), but also for the bias introduced by the previous application of treatment B.
+
+Existing methods for causal inference in the static setting cannot be applied in this longitudinal setting since they are designed to handle the cross-sectional set-up, where the treatment and outcome depend only on a static value of the patient covariates. If we consider again the above example, these methods would not be able to model how the changes in patient covariates over time affect the assignment of treatments and they would also not be able to estimate the effect of a sequence of treatments on the patient outcome (e.g. sequential application of treatment A followed by treatment B). Different models that can handle these temporal dependencies in the observational data and varying-length patient histories are needed for estimating treatment effects over time.
+
+Time-dependent confounders are present in observational data because doctors follow policies: the history of the patients' covariates and the patients' response to past treatments are used to decide future treatments (Mansournia et al., 2012). The direct use of supervised learning methods will be biased by the treatment policies present in the observational data and will not be able to correctly estimate counterfactuals for different treatment assignment policies.
+
+Standard methods for adjusting for time-varying confounding and estimating the effects of time-varying exposures are based on ideas from epidemiology. The most widely used among these are Marginal Structural Models (MSMs) (Robins et al., 2000; Mansournia et al., 2012) which use the inverse probability of treatment weighting (IPTW) to adjust for the time-dependent confounding bias. Through IPTW, MSMs create a pseudo-population where the probability of treatment does not depend on the time-varying confounders. However, MSMs are not robust to model misspecification in computing the IPTWs. MSMs can also give high-variance estimates due to extreme weights; computing the IPTW involves dividing by probability of assigning a treatment conditional on patient history which can be numerically unstable if the probability is small.
+
+We introduce the Counterfactual Recurrent Network (CRN), a novel sequence-to-sequence architecture for estimating treatment effects over time. CRN leverages recent advances in representation learning (Bengio et al., 2012) and domain adversarial training (Ganin et al., 2016) to overcome the problems of existing methods for causal inference over time. Our main contributions are as follows.
+
+Treatment invariant representations over time. CRN constructs treatment invariant representations at each timestep in order to break the association between patient history and treatment assignment and thus removes the bias from time-dependent confounders. For this, CRN uses domain adversarial training (Ganin et al., 2016; Li et al., 2018; Sebag et al., 2019) to trade-off between build-
+
+ing this balancing representation and predicting patient outcomes. We show that these representations remove the bias from time-varying confounders and can be reliably used for estimating counterfactual outcomes. This represents the first work that introduces ideas from domain adaptation to the area of estimating treatment effects over time. In addition, by building balancing representations, we propose a novel way of removing the bias introduced by time-varying confounders.
+
+Counterfactual estimation of future outcomes. To estimate counterfactual outcomes for treatment plans (and not just single treatments), we integrate the domain adversarial training procedure as part of a sequence-to-sequence architecture. CRN consists of an encoder network which builds treatment invariant representations of the patient history that are used to initialize the decoder. The decoder network estimates outcomes under an intended sequence of future treatments, while also updating the balanced representation. By performing counterfactual estimation of future treatment outcomes, CRN can be used to answer critical medical questions such as deciding when to give treatments to patients, when to start and stop treatment regimes, and also how to select from multiple treatments over time. We illustrate in Figure 1 the applicability of our method in choosing optimal cancer treatments.
+
+In our experiments, we evaluate CRN in a realistic set-up using a model of tumour growth (Geng et al., 2017). We show that CRN achieves better performance in predicting counterfactual outcomes, but also in choosing the right treatment and timing of treatment than current state-of-the-art methods.
+
+# 2 RELATED WORK
+
+We focus on methods for estimating treatment effects over time and for building balancing representations for causal inference. A more in-depth review of related work is in Appendix A.
+
+Treatment effects over time. Standard methods for estimating the effects of time-varying exposures were first developed in the epidemiology literature and include the g-computation formula, Structural Nested Models and Marginal Structural Models (MSMs) (Robins, 1986; 1994; Robins et al., 2000; Robins & Hernán, 2008). Originally, these methods have used predictors performing logistic/linear regression which makes them unsuitable for handling complex time-dependencies (Hernán et al., 2001; Mansournia et al., 2012; Mortimer et al., 2005). To address these limitations, methods that use Bayesian non-parametrics or recurrent neural networks as part of these frameworks have been proposed. (Xu et al., 2016; Roy et al., 2016; Lim et al., 2018).
+
+To begin with, Xu et al. (2016) use Gaussian processes to model discrete patient outcomes as a generalized mixed-effects model and uses the  $g$ -computation method to handle time-varying confounders. Soleimani et al. (2017) extend the approach in Xu et al. (2016) to the continuous time-setting and model treatment responses using linear time-invariant dynamical systems. Roy et al. (2016) use Dirichlet and Gaussian processes to model the observational data and estimate the IPTW in Marginal Structural Models. Schulam & Saria (2017) build upon work from Lok et al. (2008); Arjas & Parner (2004) and use marked point processes and Gaussian processes to learn causal effects in continuous-time data. These Bayesian non-parametric methods make strong assumptions about model structure and consequently cannot handle well heterogeneous treatment effects arising from baseline variables (Soleimani et al., 2017; Schulam & Saria, 2017) and multiple treatment outcomes (Xu et al., 2016; Schulam & Saria, 2017).
+
+The work most related to ours is the one of Lim et al. (2018) which improves on the standard MSMs by using recurrent neural networks to estimate the inverse probability of treatment weights (IPTWs). Lim et al. (2018) introduces Recurrent Marginal Structural Networks (RMSNs) which also use a sequence-to-sequence deep learning architecture to forecast treatment responses in a similar fashion to our model. However, RMSNs require training additional RNNs to estimate the propensity weights and does not overcome the fundamental problems with IPTWs, such as the high-variance of the weights. Conversely, CRN takes advantage of the recent advances in machine learning, in particular, representation learning to propose a novel way of handling time-varying confounders.
+
+Balancing representations for treatment effect estimation. Balancing the distribution of control and treated groups has been used for counterfactual estimation in the static setting. The methods proposed in the static setting for balancing representations are based on using discrepancy measures in the representation space between treated and untreated patients, which do not generalize to multiple treatments (Johansson et al., 2016; Shalit et al., 2017; Li & Fu, 2017; Yao et al., 2018). Moreover, due to the sequential assignment of treatments in the longitudinal setting, and due to the change of
+
+patient covariates over time according to previous treatments, the methods for the static setting are not directly applicable to the time-varying setting (Hernán et al., 2000; Mansournia et al., 2012).
+
+# 3 PROBLEM FORMULATION
+
+Consider an observational dataset  $\mathcal{D} = \{\{\mathbf{x}_t^{(i)},\mathbf{a}_t^{(i)},\mathbf{y}_{t + 1}^{(i)}\}_{t = 1}^{T^{(i)}}\cup \{\mathbf{v}^{(i)}\} \}_{i = 1}^N$  consisting of information about  $N$  independent patients. For each patient  $(i)$ , we observe time-dependent covariates  $\mathbf{X}_t^{(i)}\in \mathcal{X}_t$ , treatment received  $\mathbf{A}_t^{(i)}\in \{A_1,\dots A_K\} = \mathcal{A}$  and outcomes  $\mathbf{Y}_{t + 1}^{(i)}\in \mathcal{Y}_{t + 1}$  for  $T^{(i)}$  discrete timesteps. The patient can also have baseline covariates  $\mathbf{V}^{(i)}\in \mathcal{V}$  such as gender and genetic information. Note that the outcome  $\mathbf{Y}_{t + 1}^{(i)}$  will be part of the observed covariates  $\mathbf{X}_{t + 1}^{(i)}$ . For simplicity, the patient superscript  $(i)$  will be omitted unless explicitly needed.
+
+We adopt the potential outcomes framework proposed by (Neyman, 1923; Rubin, 1978) and extended by (Robins & Hernán, 2008) to account for time-varying treatments. Let  $\mathbf{Y}[\bar{\mathbf{a}}]$  be the potential outcomes, either factual or counterfactual, for each possible course of treatment  $\bar{\mathbf{a}}$ . Let  $\bar{\mathbf{H}}_t = (\bar{\mathbf{X}}_t, \bar{\mathbf{A}}_{t-1}, \mathbf{V})$  represent the history of the patient covariates  $\bar{\mathbf{X}}_t = (\mathbf{X}_1, \dots, \mathbf{X}_t)$ , treatment assignments  $\bar{\mathbf{A}}_t = (\bar{\mathbf{A}}_1, \dots, \bar{\mathbf{A}}_t)$  and static features  $\mathbf{V}$ . We want to estimate:
+
+$$
+\mathbb {E} \left(\mathbf {Y} _ {t + \tau} [ \bar {\mathbf {a}} (t, t + \tau - 1) ] | \bar {\mathbf {H}} _ {t}\right), \tag {1}
+$$
+
+where  $\bar{\mathbf{a}}(t, t + \tau - 1) = [\mathbf{a}_t, \dots, \mathbf{a}_{t + \tau - 1}]$  represents a possible sequence of treatments from timestep  $t$  just until before the potential outcome  $\mathbf{Y}_{t + \tau}$  is observed. We make the standard assumptions (Robins et al., 2000; Lim et al., 2018) needed to identify the treatment effects: consistency, positivity and no hidden confounders (sequential strong ignorability). See Appendix B for more more details.
+
+# 4 COUNTERFACTUAL RECURRENT NETWORK
+
+The observational data can be used to train a supervised learning model to forecast:  $\mathbb{E}(\mathbf{Y}_{t + \tau}|\bar{\mathbf{A}} (t,t + \tau -1) = \bar{\mathbf{a}} (t,t + \tau -1),\bar{\mathbf{H}}_t)$ . However, without adjusting for the bias introduced by time-varying confounders, this model cannot be reliably used for making causal predictions (Robins et al., 2000; Robins & Hernan, 2008; Schulam & Saria, 2017). The Counterfactual Recurrent Network (CRN) removes this bias through domain adversarial training and estimates the counterfactual outcomes  $\mathbb{E}(\mathbf{Y}_{t + \tau}[\bar{\mathbf{a}} (t,t + \tau -1)]|\bar{\mathbf{H}}_t)$ , for any intended future treatment assignment  $\bar{\mathbf{a}} (t,t + \tau -1)$ .
+
+Balancing representations. The history  $\bar{\mathbf{H}}_t = (\bar{\mathbf{X}}_t, \bar{\mathbf{A}}_{t-1}, \mathbf{V})$  of the patient contains the time-varying confounders  $\bar{\mathbf{X}}_t$  which bias the treatment assignment  $\mathbf{A}_t \in \{A_1, \dots, A_K\}$  in the observational dataset. Inverse probability of treatment weighting, as performed by MSMs, creates a pseudopopulation where the probability of treatment  $\mathbf{A}_t$  does not depend on the time-varying confounders (Robins et al., 2000). In this paper, we propose instead building a representation of the history  $\bar{\mathbf{H}}_t$  that is not predictive of the treatment  $\mathbf{A}_t$ . This way, we remove the association between history, containing the time-varying confounders  $\bar{\mathbf{X}}_t$ , and current treatment  $\mathbf{A}_t$ . Robins (1999) shows that in this case, the estimation of counterfactual treatment outcomes is unbiased. See Appendix C for details and for an example of a causal graph with time-dependent confounders.
+
+Let  $\Phi$  be the representation function that maps the patient history  $\bar{\mathbf{H}}_t$  to a representation space  $\mathcal{R}$ . To obtain unbiased treatment effects,  $\Phi$  needs to construct treatment invariant representations such that  $P(\Phi (\bar{\mathbf{H}}_t)\mid \mathbf{A}_t = A_1) = \dots = P(\Phi (\bar{\mathbf{H}}_t)\mid \mathbf{A}_t = A_K)$ . To achieve this and to estimate counterfactual outcomes under a planned sequence of treatments, we integrate the domain adversarial training framework proposed by Ganin et al. (2016) and extended by Sebag et al. (2019) to the multi-domain learning setting, into a sequence-to-sequence architecture. In our case, the different treatments at each timestep are considered the different domains. Note that the novelty here comes from the use of domain adversarial training to handle the bias from the time-dependent confounders, rather than the use of sequence-to-sequence models, which have already been applied to forecast treatment responses (Lim et al., 2018). Figure 2 illustrates our model architecture.
+
+Encoder. The encoder network uses an RNN, with LSTM unit (Hochreiter & Schmidhuber, 1997), to process the history of treatments  $\bar{\mathbf{A}}_{t-1}$ , covariates  $\bar{\mathbf{X}}_t$  and baseline features  $\mathbf{V}$  to build a treatment invariant representation  $\Phi(\bar{\mathbf{H}}_t)$ , but also to predict one-step-ahead outcomes  $\mathbf{Y}_{t+1}$ . To achieve this, the encoder network aims to maximize the loss of the treatment classifier  $G_a$  and minimize the loss
+
+![](images/bfc98686609eac692794de93b498def9efd5e650e21bedd215cc86e7870db5b0.jpg)  
+Figure 2: CRN architecture. Encoder builds representation  $\Phi(\bar{\mathbf{H}}_t)$  that maximizes loss of treatment classifier  $G_{a}$  and minimizes loss of outcome predictor  $G_{y}$ .  $\Phi(\bar{\mathbf{H}}_t)$  is used to initialize the decoder, which continues to update it to predict counterfactual outcomes of a sequence of future treatments.
+
+of the outcome predictor network  $G_{y}$ . This way, the balanced representation  $\Phi(\bar{\mathbf{H}}_t)$  is not predictive of the assigned treatment  $\mathbf{A}_t$ , but is discriminative enough to estimate the outcome  $\mathbf{Y}_{t+1}$ . To train this model using gradient descent, we use the Gradient Reversal Layer (Ganin et al., 2016).
+
+Decoder. The decoder network uses the balanced representation computed by the encoder to initialize the state of an RNN that predicts the counterfactual outcomes for a sequence of future treatments. During training, the decoder uses as input the outcomes from the observational data  $(\mathbf{Y}_{t + 1},\dots \mathbf{Y}_{t + \tau -1})$  the static patient features  $\mathbf{V}$  and the intended sequence of treatments  $\bar{\mathbf{a}} (t,t + \tau -1)$ . The decoder is trained in a similar way to the encoder to update the balanced representation and to estimate the outcomes. During testing, we do not have access to ground-truth outcomes; thus, the outcomes predicted by the decoder  $(\hat{\mathbf{Y}}_{t + 1},\dots \hat{\mathbf{Y}}_{t + \tau -1})$  are auto-regressively used instead as inputs. By running the decoder with different treatment settings, and by auto-regressively feeding back the outcomes, we can determine when to start and end different treatments, which is the optimal time to give the treatment and which treatments to give over time to obtain the best patient outcomes.
+
+The representation  $\Phi(\bar{\mathbf{H}}_t)$  is built by applying a fully connected layer, with Exponential Linear Unit (ELU) activation to the output of the LSTM. The treatment classifier  $G_{a}$  and the predictor network  $G_{y}$  consist of a hidden layer each, also with ELU activation. The output layer of  $G_{a}$  uses softmax activation, while the output layer of  $G_{y}$  uses linear activation for continuous predictions. For categorical outcomes, softmax activation can be used. We follow an approach similar to Lim et al. (2018) and we split the encoder and decoder training into separate steps. See Appendix E for details.
+
+The encoder and decoder networks use variational dropout (Gal & Ghahramani, 2016) such that the CRN can also give uncertainty intervals for the treatment outcomes. This is particularity important in the estimation of treatment effects, since the model predictions should only be used when they have high confidence. Our model can also be modified to allow for irregular samplings of observations by using a PhasedLSTM (Neil et al., 2016).
+
+# 5 ADVERSARIALLY BALANCED REPRESENTATION OVER TIME
+
+At each timestep  $t$ , let the  $K$  different possible treatments  $\mathbf{A}_t \in \{A_1, \dots, A_K\}$  represent our domains. As described in Section 4, to remove the bias from time-dependent confounders, we build a representation of history  $\bar{\mathbf{H}}_t$  that is invariant across treatments:  $P(\Phi(\bar{\mathbf{H}}_t) \mid A_1) = \dots = P(\Phi(\bar{\mathbf{H}}_t) \mid A_K)$ .
+
+This requirement can be enforced by minimizing the distance in the distribution of  $\Phi(\bar{\mathbf{H}}_t)$  between any two pairs of treatments. Kifer et al. (2004); Ben-David et al. (2007), propose measuring the disparity between distributions based on their separability by a discriminatively-trained classifier. Let the symmetric hypothesis class  $\mathcal{H}$  consist of the set of symmetric multiclass classifiers, such as neural network architectures. The  $\mathcal{H}$ -divergence between all pairs of two distributions is defined in terms of the capacity of the hypothesis class  $\mathcal{H}$  to discriminate between examples from the multiple distributions. Empirically, minimizing the  $\mathcal{H}$ -divergence involves building a representation where examples from the multiple domains are as indistinguishable as possible (Ben-David et al., 2007; Li et al., 2018; Sebag et al., 2019). Ganin et al. (2016) use this idea to propose an adversarial framework
+
+for domain adaptation involving building a representation which achieves maximum error on a domain classifier and minimum error on an outcome predictor. Similarly, in our case, we use domain adversarial training to build a representation of the patient history  $\Phi(\bar{\mathbf{H}}_t)$  that is both invariant to the treatment given at timestep  $t$ ,  $\mathbf{A}_t$  and that achieves low error in estimating the outcome  $\mathbf{Y}_{t+1}$ .
+
+Let  $G_{a}(\Phi(\bar{\mathbf{H}}_{t}); \theta_{a})$  be the treatment classifier with parameters  $\theta_{a}$  and let  $G_{a}^{j}(\Phi(\bar{\mathbf{H}}_{t}); \theta_{a})$  be the output corresponding to treatment  $A_{j}$ . Let  $G_{y}(\Phi(\bar{\mathbf{H}}_{t}); \theta_{y})$  be the predictor network with parameters  $\theta_{y}$ . The representation function  $\Phi$  is parameterized by the parameters  $\theta_{r}$  in the RNN:  $\Phi(\bar{\mathbf{H}}_{t}; \theta_{r})$ . Figure 3 shows the adversarial training procedure used.
+
+For timestep  $t$  and patient  $(i)$ , let  $\mathcal{L}_{t,a}^{(i)}(\theta_r,\theta_a)$  be the treatment (domain) loss and let  $\mathcal{L}_{t,y}^{(i)}(\theta_r,\theta_y)$  the outcome loss, defined as follows:
+
+$$
+\mathcal {L} _ {t, a} ^ {(i)} \left(\theta_ {r}, \theta_ {a}\right) = - \sum_ {j = 1} ^ {K} \mathbb {I} _ {\left\{\mathbf {a} _ {t} ^ {(i)} = a _ {j} \right\}} \log \left(G _ {a} ^ {j} \left(\Phi \left(\bar {\mathbf {H}} _ {t}; \theta_ {r}\right); \theta_ {a}\right)\right) \tag {2}
+$$
+
+$$
+\mathcal {L} _ {t, y} ^ {(i)} \left(\theta_ {r}, \theta_ {y}\right) = \left\| \mathbf {Y} _ {t + 1} ^ {(i)} - \left(G _ {y} \left(\Phi \left(\bar {\mathbf {H}} _ {t}; \theta_ {r}\right), \theta_ {y}\right)\right) \right\| ^ {2}. \tag {3}
+$$
+
+If the outcome is binary, the cross-entropy loss can be used instead for  $\mathcal{L}_{t,y}$ . To build treatment invariant representations and to also estimate patient outcomes, we aim to maximize treatment loss and minimize outcome loss.
+
+Thus, the overall loss  $\mathcal{L}_{t,y}^{(i)}$  at timestep  $t$  is given by:
+
+![](images/2743d9c81c9ac3b3c3f2b1a66e0e4ff7c5d2c8a08bf98852b20eff158e314a50.jpg)  
+Figure 3: Training procedure for building balancing representation.
+
+$$
+\mathcal {L} _ {t} ^ {(i)} \left(\theta_ {r}, \theta_ {y}, \theta_ {a}\right) = \sum_ {i = 1} ^ {N} \mathcal {L} _ {t, y} ^ {(i)} \left(\theta_ {r}, \theta_ {y}\right) - \lambda \mathcal {L} _ {t, a} ^ {(i)} \left(\theta_ {r}, \theta_ {a}\right), \tag {4}
+$$
+
+where the hyperparameter  $\lambda$  controls this trade-off between domain discrimination and outcome prediction. We use the standard procedure for training domain adversarial networks from Ganin et al. (2016) and we start off with an initial value for  $\lambda$  and use an exponentially increasing schedule during training. To train the model using backpropagation, we use the Gradient Reversal Layer (GRL) (Ganin et al., 2016). For more details about the training procedure, see Appendix E.
+
+By using the objective  $\mathcal{L}_t^{(i)}(\theta_r,\theta_y,\theta_a)$ , we reach the saddle point  $(\hat{\theta}_r,\hat{\theta}_y,\hat{\theta}_a)$  that achieves the equilibrium between domain discrimination and outcome estimation.
+
+$$
+\left(\hat {\theta} _ {r}, \hat {\theta} _ {y}\right) = \arg \min  _ {\theta_ {r}, \theta_ {y}} \mathcal {L} _ {t} ^ {(i)} \left(\theta_ {r}, \theta_ {y}, \hat {\theta} _ {a}\right) \quad \hat {\theta} _ {a} = \arg \max  _ {\theta_ {a}} \mathcal {L} _ {t} ^ {(i)} \left(\hat {\theta} _ {r}, \hat {\theta} _ {y}, \theta_ {a}\right). \tag {5}
+$$
+
+The result stated in Theorem 1 proves that the treatment (domain) loss part of our objective (from equation 2) aims to remove the time-dependent confounding bias.
+
+Theorem 1. Let  $t \in \{1,2,\ldots\}$ . For each  $j = 1,\dots,K$ , let  $P_{j}$  denote the distribution of  $\bar{\mathbf{H}}_t$  conditional on  $\mathbf{A}_t = A_j$  and let  $P_{j}^{\Phi}$  denote the distribution of  $\Phi(\bar{\mathbf{H}}_t)$  conditional on  $\mathbf{A}_t = A_j$ . Let  $G_a^j$  denote the output of  $G_a$  corresponding to treatment  $A_j$ . Then the minimax game defined by
+
+$$
+\min  _ {\Phi} \max  _ {G _ {a}} \sum_ {j = 1} ^ {K} \mathbb {E} _ {\bar {\mathbf {H}} _ {t} \sim P _ {j}} \left[ \log \left(G _ {a} ^ {j} \left(\Phi \left(\bar {\mathbf {H}} _ {t}\right); \theta_ {a}\right)\right) \right] \quad \text {s u b j e c t t o} \sum_ {j = 1} ^ {K} G _ {a} ^ {j} \left(\Phi \left(\bar {\mathbf {H}} _ {t}\right)\right) = 1 \tag {6}
+$$
+
+has a global minimum which is attained if and only if  $P_1^\Phi = P_2^\Phi = \ldots = P_K^\Phi$ , i.e. when the learned representations are invariant across all treatments.
+
+Proof. This result is a restatement of the one in Li et al. (2018). For details, see the Appendix D.
+
+A good representation allows us to obtain a low error in estimating counterfactuals for all treatments, while at the same time to minimize the  $\mathcal{H}$ -divergence between induced marginal distributions of all the domains. We use an algorithm that directly minimizes a combination of the  $\mathcal{H}$ -divergence and the empirical training margin.
+
+# 6 EXPERIMENTS
+
+In real datasets, counterfactual outcomes and the degree of time-dependent confounding are not known (Schulam & Saria, 2017; Lim et al., 2018). To validate the  $\mathrm{CRN}^1$ , we evaluate it on a Pharmacokinetic-Pharmacodynamic model of tumour growth (Geng et al., 2017), which uses a state-of-the-art bio-mathematical model to simulate the combined effects of chemotherapy and radiotherapy in lung cancer patients. The same model was used by Lim et al. (2018) to evaluate RMSNs.
+
+Model of tumour growth The volume of tumour  $t$  days after diagnosis is modelled as follows:
+
+$$
+V (t + 1) = \left(1 + \underbrace {\rho \log \left(\frac {K}{V (t)}\right)} _ {\text {T u m o r g r o w t h}} - \underbrace {\beta_ {c} C (t)} _ {\text {C h e m o t h e r a p y}} - \underbrace {\left(\alpha_ {r} d (t) + \beta_ {r} d (t) ^ {2}\right)} _ {\text {R a d i o t h e r a p y}} + \underbrace {e _ {t}} _ {\text {N o i s e}}\right) V (t) \tag {7}
+$$
+
+where  $K, \rho, \beta_{c}, \alpha_{r}, \beta_{r}, e_{t}$  are sampled as described in Geng et al. (2017). To incorporate heterogeneity in patient responses, the prior means for  $\beta_{c}$  and  $\alpha_{r}$  are adjusted to create patient subgroups, which are used as baseline features. The chemotherapy concentration  $C(t)$  and radiotherapy dose  $d(t)$  are modelled as described in Appendix F. Time-varying confounding is introduced by modelling chemotherapy and radiotherapy assignment as Bernoulli random variables, with probabilities  $p_{c}$  and  $p_{r}$  depending on the tumour diameter:  $p_{c}(t) = \sigma\left(\frac{\gamma_{c}}{D_{\max}}(\bar{D}(t) - \delta_{c})\right)$  and  $p_{r}(t) = \sigma\left(\frac{\gamma_{r}}{D_{\max}}(\bar{D}(t) - \delta_{r})\right)$  where  $\bar{D}(t)$  is the average diameter over the last 15 days,  $D_{\max} = 13\mathrm{cm}$ ,  $\sigma(\cdot)$  is the sigmoid and  $\delta_{c} = \delta_{r} = D_{\max}/2$ . The amount of time-dependent confounding is controlled through  $\gamma_{c}, \gamma_{r}$ ; the higher  $\gamma_{\star}$  is, the more important the history is in assigning treatments. At each timestep, there are four treatment options: no treatment, chemotherapy, radiotherapy, combined chemotherapy and radiotherapy. For details about data simulation, see Appendix F.
+
+Benchmarks We used the following benchmarks for performance comparison: Marginal Structural Models (MSMs) (Robins et al., 2000), which use logistic regression for estimating the IPTWs and linear regression for prediction (see Appendix G for details). We also compare against the Recurrent Marginal Structural Networks (RMSNs) Lim et al. (2018), which is the current state-of-the-art model in estimating treatment responses. RMSNs use RNNs to estimate the IPTWs and the patient outcomes (details in Appendix H). To show that standard supervised learning models do not handle the time-varying confounders we compare against an RNN and a linear regression model, which receive as input treatments and covariates to predict the outcome (see Appendix I for details). Our model architecture follows the description in Sections 4 and 5, with full training details and hyperparameter optimization in Appendix J. To show the importance of adversarial training, we also benchmark against CRN ( $\lambda = 0$ ) a model with the same architecture, but with  $\lambda = 0$ , i.e our model architecture without adversarial training.
+
+# 6.1 EVALUATE MODELS ON COUNTERFACTUAL PREDICTIONS
+
+Previous methods focused on evaluating the error only for factual outcomes (observed patient outcomes) (Lim et al., 2018). However, to build decision support systems, we need to evaluate how well the models estimate the counterfactual outcomes, i.e. patient outcomes under alternative treatment options. The parameters  $\gamma_{c}$  and  $\gamma_{r}$  control the treatment assignment policy, i.e. the degree of time-dependent confounding present in the data. We evaluate the benchmarks under different degrees of time-dependent confounding by setting  $\gamma = \gamma_{c} = \gamma_{r}$ . For each  $\gamma$  we simulate a 10000 patients for training, 1000 for validation (hyperparameter tuning) and 1000 for out-of-sample testing. For the patients in the test set, for each time  $t$ , we also simulate counterfactuals  $\mathbf{Y}_{t+1}$ , represented by tumour volume  $V(t+1)$ , under all possible treatment options.
+
+![](images/1ce5f53474e0f7699b82a5c00ac45e3d532ab4af491582e520440fbb203d8659.jpg)  
+(a) One-step ahead prediction
+
+![](images/46ecd9c562acb43c0a09a0df3ad5d419c2db355a627bf3e61bef8a2e952f5945.jpg)  
+(b) Five-step ahead prediction  
+Figure 4: Results for prediction of patient counterfactuals.
+
+Figure 4 (a) shows the normalized root mean squared error (RMSE) for one-step ahead estimation of counterfactuals with varying degree of time-dependent confounding  $\gamma$ . The RMSE is normalized by the maximum tumour volume:  $V_{max} = 1150\mathrm{cm}^3$ . The linear and MSM models provide a baseline for performance as they achieve the highest RMSE. While the use of IPTW in MSMs helps when  $\gamma$  increases, using linear modelling has severe limitations. When there is no time-dependent confounding, the machine learning methods achieve similar performance, close to  $0.6\%$  RMSE. As the bias in the dataset increases, the harder it becomes for the RNN and the CRN ( $\lambda = 0$ ) to generalize to estimate outcomes of treatments not matching the training policy. When  $\gamma = 10$ , CRN improves by  $48.1\%$  on the same model architecture without domain adversarial training CRN ( $\lambda = 0$ ).
+
+Our proposed model achieves the lowest RMSE across all values of  $\gamma$ . Compared to RMSNs, CRN improves by  $\sim 17\%$  when  $\gamma > 6$ . To highlight the gains of our method even for smaller  $\gamma$ , Figure 4 (b) shows the RMSE for five-step ahead prediction (with counterfactuals generated as described in Section 6.2 and Appendix L). RMSNs also use a decoder for sequence prediction. However, RMSNs require training additional RNNs to estimate the IPTW, which are used to weight each sample during the decoder training. For  $\tau$ -step ahead prediction, IPTW involves multiplying  $\tau$  weights which can result in high variance. The results in Figure 4 (b) show the problems with using IPTW to handle the time-dependent confounding bias. See Appendix K for more results on multi-step ahead prediction.
+
+Balancing representation: To evaluate whether the CRN has indeed learnt treatment invariant representations, for  $\gamma = 5$ , we illustrate in Figure 5 the T-SNE embeddings of the balancing representations  $\Phi(\bar{\mathbf{H}}_t)$  built by the CRN encoder for test patients. We color each point by the treatment  $\mathbf{A}_t \in \{\text{no treatment, chemotherapy, radiotherapy, combined chemotherapy and radiotherapy}\}$  received at timestep  $t$  to highlight the invariance of  $\Phi(\bar{\mathbf{H}}_t)$  across the different treatments. In Figure 5(b), we show  $\Phi(\bar{\mathbf{H}}_t)$  only for chemotherapy and radiotherapy for better understanding.
+
+![](images/f77eeb050f5a4050ff93cdba90399e5559d5c7780e9788889949e4fc1ba8614b.jpg)  
+Figure 5: TSNE embedding of the balancing representation  $\Phi(\bar{\mathbf{H}}_t)$  learnt by the CRN encoder at different timesteps  $t$ . Notice that  $\Phi(\bar{\mathbf{H}}_t)$  is not predictive of the treatment  $\mathbf{A}_t$  given at timestep  $t$ .
+
+![](images/55c62645a5ec6ceae81e97cc6438bbfad8b3163c007c0f88a48f0e0dd0bb714c.jpg)
+
+![](images/4294ce88bb0ae7f54a1b027b7d4a162f7f9bbd14094e901c53aa6b487a406afd.jpg)
+
+![](images/8ef10f66eba25e171192f204cb9547533a2cf755bcee6b9896d71b58107f3a65.jpg)
+
+![](images/cb7ea7629cbf94353270cdcc1b40178ca959dfe99221210cbf58988375d8c247.jpg)
+
+![](images/d5ae45edfccc00ee302eb89d5ec3eb0a77e7d3fb9b17e6d31e842f6126c4cf51.jpg)
+
+# 6.2 EVALUATE RECOMMENDING THE RIGHT TREATMENT AND TIMING OF TREATMENT
+
+Evaluating the models just in terms of the RMSE on counterfactual estimation is also not enough for assessing their reliability when used as part of decision support systems. In this section we assess how well the models can select the correct treatment and timing of treatment for several forecasting horizons  $\tau$ . We generate test sets consisting of 1000 patients where for each horizon  $\tau$  and for each time  $t$  in a patient's trajectory, there are  $\tau$  options for giving chemotherapy at one of  $t,\ldots t + \tau -1$  and  $\tau$  options for giving radiotherapy at one of  $t,\ldots t + \tau -1$ . At the rest of the future timesteps, no treatment is applied. These  $2\tau$  treatment plans are assessed in terms of the tumour volume outcome  $\mathbf{Y}_{t + \tau}$ . We select the treatment (chemotherapy or radiotherapy) that achieves lowest  $\mathbf{Y}_{t + \tau}$ , and within the correct treatment the timing with lowest  $\mathbf{Y}_{t + \tau}$ . We also compute the normalized RMSE for
+
+Table 1: Results for recommending the correct treatment and timing of treatment.  
+
+<table><tr><td colspan="2"></td><td colspan="3">γc=5,γr=5</td><td colspan="3">γc=5,γr=0</td><td colspan="3">γc=0,γr=5</td></tr><tr><td></td><td>τ</td><td>CRN</td><td>RMSN</td><td>MSM</td><td>CRN</td><td>RMSN</td><td>MSM</td><td>CRN</td><td>RMSN</td><td>MSM</td></tr><tr><td rowspan="5">Normalized RMSE</td><td>3</td><td>2.43%</td><td>3.16%</td><td>6.75%</td><td>1.08%</td><td>1.35%</td><td>3.68%</td><td>1.54%</td><td>1.59%</td><td>3.23%</td></tr><tr><td>4</td><td>2.83%</td><td>3.95%</td><td>7.65%</td><td>1.21%</td><td>1.81%</td><td>3.84%</td><td>1.81%</td><td>2.25%</td><td>3.52%</td></tr><tr><td>5</td><td>3.18%</td><td>4.37%</td><td>7.95%</td><td>1.33%</td><td>2.13%</td><td>3.91%</td><td>2.03%</td><td>2.71%</td><td>3.63%</td></tr><tr><td>6</td><td>3.51%</td><td>5.61%</td><td>8.19%</td><td>1.42%</td><td>2.41%</td><td>3.97%</td><td>2.23%</td><td>2.73%</td><td>3.71%</td></tr><tr><td>7</td><td>3.93%</td><td>6.21%</td><td>8.52%</td><td>1.53%</td><td>2.43%</td><td>4.04%</td><td>2.43%</td><td>2.88%</td><td>3.79%</td></tr><tr><td rowspan="5">Treatment Accuracy</td><td>3</td><td>83.1%</td><td>75.3%</td><td>73.9%</td><td>83.2%</td><td>78.6%</td><td>77.1%</td><td>92.9%</td><td>87.3%</td><td>74.9%</td></tr><tr><td>4</td><td>82.5%</td><td>74.1%</td><td>68.5%</td><td>81.3%</td><td>77.7%</td><td>73.9%</td><td>85.7%</td><td>83.8%</td><td>74.1%</td></tr><tr><td>5</td><td>73.5%</td><td>72.7%</td><td>63.2%</td><td>78.3%</td><td>77.2%</td><td>72.3%</td><td>83.8%</td><td>82.1%</td><td>72.8%</td></tr><tr><td>6</td><td>69.4%</td><td>66.7%</td><td>62.7%</td><td>79.5%</td><td>76.3%</td><td>71.8%</td><td>78.6%</td><td>69.7%</td><td>64.5%</td></tr><tr><td>7</td><td>71.2%</td><td>68.8%</td><td>62.4%</td><td>72.7%</td><td>71.8%</td><td>71.6%</td><td>71.9%</td><td>69.3%</td><td>61.2%</td></tr><tr><td rowspan="5">Treatment Timing Accuracy</td><td>3</td><td>79.6%</td><td>78.1%</td><td>67.6%</td><td>80.5%</td><td>76.8%</td><td>77.5%</td><td>79.8%</td><td>75.7%</td><td>60.6%</td></tr><tr><td>4</td><td>73.9%</td><td>70.3%</td><td>63.1%</td><td>79.0%</td><td>77.2%</td><td>73.4%</td><td>75.4%</td><td>71.4%</td><td>58.2%</td></tr><tr><td>5</td><td>69.8%</td><td>68.6%</td><td>62.4%</td><td>78.3%</td><td>73.3%</td><td>63.6%</td><td>66.9%</td><td>31.3%</td><td>29.5%</td></tr><tr><td>6</td><td>66.9%</td><td>66.2%</td><td>62.6%</td><td>73.5%</td><td>72.1%</td><td>63.9%</td><td>65.8%</td><td>24.2%</td><td>15.5%</td></tr><tr><td>7</td><td>64.5%</td><td>63.6%</td><td>62.2%</td><td>70.6%</td><td>57.4%</td><td>44.2%</td><td>63.9%</td><td>25.6%</td><td>12.5%</td></tr></table>
+
+predicting  $\mathbf{Y}_{t + \tau}$ . See Appendix L for more details about the test set. The models are evaluated for 3 settings of  $\gamma_{c}$  and  $\gamma_{r}$ .
+
+Table 1 shows the results for this evaluation set-up. The treatment accuracy denotes the percentage of patients for which the correct treatment was selected, while the treatment timing accuracy is the percentage for which the correct timing was selected. Note that when  $\gamma_{c} = 0$  and  $\gamma_{r} = 5$ , RMSN and MSM select the wrong treatment timing for projection horizons  $\tau > 4$ . CRN performs similarly among the different policies present in the observational data and achieve the lowest RMSE and highest accuracy in selecting the correct treatment and timing of treatment.
+
+In Appendix M we also show the applicability of the CRN in more complex medical scenarios involving real data. We provide experimental results based on the Medical Information Mart for Intensive Care (MIMIC III) database (Johnson et al., 2016) consisting of electronic health records from patients in the ICU.
+
+# 7 CONCLUSION
+
+Despite its wide applicability, the problem of causal inference for time-dependent treatments has been relatively less studied compared to problem of causal inference in the static setting. Both new methods and theory are necessary to be able to harness the full potential of observational data for learning individualized effects of complex treatment scenarios. Further work in this direction is needed for proposing alternative methods for handling time-dependent confounders, for modelling combinations of treatments assigned over time or for estimating the individualized effects of time-dependent treatments with associated dosage.
+
+In this paper, we introduced the Counterfactual Recurrent Network (CRN), a model that estimates individualized effects of treatments over time using a novel way of handling the bias from time-dependent confounders through adversarial training. Using a model of tumour growth, we validated CRN in realistic medical scenarios and we showed improvements over existing state-of-the-art methods. We also showed the applicability of the CRN a real dataset consiting of patient electronic health records. The counterfactual predictions of CRN have the potential to be used as part of clinical decision support systems to address relevant medical challenges involving selecting the best treatments for patients over time, identify optimal treatment timings but also when the treatment is no longer needed. In future work, we will aim to build better balancing representations and to provide theoretical guarantees for the expected error on the counterfactuals.
+
+# ACKNOWLEDGMENTS
+
+We would like to thank the reviewers for their valuable feedback. The research presented in this paper was supported by The Alan Turing Institute, under the EPSRC grant EP/N510129/1 and by the US Office of Naval Research (ONR).
+
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+
+# APPENDIX
+
+# A EXTENDED RELATED WORK
+
+Causal inference in the static setting: A large number of methods have been proposed to learn treatment effects from observational data in the static setting. In this case, it is needed to adjust for the selection bias; bias caused by the fact that, in the observational dataset, the treatment assignments depend on the patient features. Several ways of handling the selection bias involve using propensity matching (Austin, 2011; Imai & Ratkovic, 2014; Abadie & Imbens, 2016), building representations where treated and un-treated populations had similar distributions (Johansson et al., 2016; Shalit et al., 2017; Li & Fu, 2017; Yao et al., 2018) or performing propensity-aware hyperparameter tuning (Alaa & van der Schaar, 2017; 2018). However, these methods for the static setting cannot be extended directly to time-varying treatments (Hernán et al., 2000; Schisterman et al., 2009).
+
+Learning optimal policies: A related problem to ours involves learning the optimal treatment policies from logged data (Swaminathan & Joachims, 2015a;b; Atan et al., 2018). That is, learning the treatment option that would give the best reward. Note the difference to the causal inference setting considered in this paper, where the aim is to learn the counterfactual patient outcomes under all possible treatment options. Learning all of the counterfactual outcomes is a harder problem and can also be used for finding the optimal treatment.
+
+A method for learning optimal policies, proposed by Atan et al. (2018) uses domain adversarial training to build a representation that is invariant to the following two domains: observational data and simulated randomized clinical trial data, where the treatments have equal probabilities. Atan et al. (2018) only considers the static setting and aims to choose the optimal treatment instead of estimating all of the counterfactual outcomes. In our paper the aim is to eliminate the bias from the time-dependent confounders and reliably estimate all of the potential outcomes; thus, at each timestep  $t$  we build a representation that is invariant to the treatment.
+
+Off-policy evaluation in reinforcement learning: In reinforcement learning, a similar problem to ours is off-policy evaluation, which uses retrospective observational data, also known as logged bandit feedback (Hoiles & Van Der Schaar, 2016; Păduraru et al., 2013; Doroudi et al., 2017). In this case, the retrospective observational data consists of sequences of states, actions and rewards which were generated by an agent operating under an unknown policy. The off-policy evaluation methods aim to use this data to estimate the expected reward of a target policy. These methods use algorithms based on importance sampling (Precup, 2000; Thomas et al., 2015; Guo et al., 2017), action-value function approximation (model based) (Hallak et al., 2015) or doubly robust combination of both approaches (Jiang & Li, 2015). Nevertheless, these methods focus on obtaining average rewards of policies, while in our case the aim is to estimate individualized patient outcomes for future treatments.
+
+# B ASSUMPTIONS
+
+The standard assumptions needed for identifying the treatment effects are (Robins & Hernán, 2008; Lim et al., 2018; Schulam & Saria, 2017):
+
+Assumption 1: Consistency. If  $\mathbf{A}_t = \mathbf{a}_t$  for a given patient, then the potential outcome for treatment  $\mathbf{a}_t$  is the same as the observed (factual) outcome:  $\mathbf{Y}_{t + 1}[\mathbf{a}_t] = \mathbf{Y}_{t + 1}$ .
+
+Assumption 2: Positivity (Overlap) (Imai & Van Dyk, 2004): If  $P(\bar{\mathbf{A}}_{t - 1} = \bar{\mathbf{a}}_{t - 1},\bar{\mathbf{X}}_t = \bar{\mathbf{x}}_t)\neq 0$  then  $P(\mathbf{A}_t = \mathbf{a}_t\mid \bar{\mathbf{A}}_{t - 1} = \bar{\mathbf{a}}_{t - 1},\bar{\mathbf{X}}_t = \bar{\mathbf{x}}_t) > 0$  for all  $\bar{\mathbf{a}}_t$ .
+
+Assumption 3: Sequential strong ignorability.  $\mathbf{Y}_{t + 1}[\mathbf{a}_t]\perp \perp \mathbf{A}_t\mid \bar{\mathbf{A}}_{t - 1},\bar{\mathbf{X}}_t,\forall \mathbf{a}_t\in \mathcal{A},\forall t.$
+
+Assumption 2 means that, for each timestep, each treatment has non-zero probability of being assigned. Assumption 3 means that there are no hidden confounders, that is, all of covariates affecting both the treatment assignment and the outcomes are present in the the observational dataset. Note that while assumption 3 is standard across all methods for estimating treatment effects, it is not testable in practice (Robins et al., 2000; Pearl et al., 2009).
+
+# C TIME-DEPENDENT CONFOUNDING
+
+Figure 6 illustrates the causal graphs for a time-varying exposures with 2-steps (Robins et al., 2000). In Figure 6 (a), the covariate  $X$  is a time-dependent confounder because it affects the treatment assignments and at the same time, its value is changed by past treatments (Mansournia et al., 2017), as illustrated by the red arrows. Thus, the treatment probabilities at each time  $t$  depend on the history of covariate  $X$  and past treatments. Note that  $U_{0}$  and  $U_{1}$  are hidden variables which only affect the covariates, i.e. they do not have arrows into the treatments. Thus, the no hidden confounders assumption (Assumption 3) is satisfied.
+
+Figure 6 (a) and (b) illustrate the two cases when there is no bias from time-dependent confounding. In Figure 6 (a) the treatment probabilities are independent, while in Figure 6 (b) they depend on past treatments.
+
+![](images/9618593638adaab44aee5185b820e5f8fa51de8d1805385f7056de2277862a91.jpg)  
+(a) Bias from time-dependent confounders.
+
+![](images/c932eb5fd72054d948aaeccecf425f505c3649ca2d2f62d3105063d46f3672a4.jpg)  
+(a) Option 1 for removing bias from time-dependent confounders.
+
+![](images/c335f4615b59f8d9f212c1cd37a106e4d047ac9e437c518448adac989d865a13.jpg)  
+(a) Option 2 for removing bias from time-dependent confounders.  
+Figure 6: Causal graphs for 2-step time-varying exposures (Robins et al., 2000).  $X_0$ ,  $X_1$  are patient covariates,  $A_0$ ,  $A_1$  are treatments,  $U_0$ ,  $U_1$  are unobserved variable and  $Y_2$  is the outcome.
+
+Marginal Structural Models Robins et al. (2000). To remove the association between time-dependent confounders and time-varying treatments, Marginal Structural Models propose using inverse probability of treatment weighting (IPTW). Without loss of generality, consider the use of MSMs with univariate treatments, baseline variables and outcomes. The outcome after  $t$  timesteps is parametrized as follows:  $\mathbf{E}[Y_{t + 1} \mid \mathbf{a}_1, \dots, \mathbf{a}_t, V] = g(\mathbf{a}_1, \dots, \mathbf{a}_n, V; \theta)$ , where  $g(\cdot)$  is usually a linear function with parameters  $\theta$ . To remove the bias from the time-dependent confounders present in the observational dataset, in the regression model  $g(\cdot)$  MSMs weights each patients using either stabilized weights:
+
+$$
+S W (t) = \prod_ {l = 1} ^ {t} \frac {f \left(\mathbf {A} _ {l} \mid \bar {\mathbf {A}} _ {l - 1}\right)}{f \left(\mathbf {A} _ {l} \mid \bar {\mathbf {X}} _ {l} , \bar {\mathbf {A}} _ {l - 1} , \mathbf {V}\right)} \tag {8}
+$$
+
+or unstabilized weights:
+
+$$
+W (t) = \prod_ {l = 1} ^ {t} \frac {1}{f \left(\mathbf {A} _ {l} \mid \bar {\mathbf {X}} _ {l} , \bar {\mathbf {A}} _ {l - 1} , \mathbf {V}\right)}, \tag {9}
+$$
+
+where  $f(\cdot)$  represents the conditional probability mass function for discrete treatments.
+
+Inverse probability of treatment weighting (IPTW) creates a pseudo-population where each member consists of themselves and  $W - 1$  (or  $SW - 1$ ) copies added though weighting. In this pseudopopulation, Robins Robins (1999) shows that  $\bar{\mathbf{X}}_t$  does not predict treatment  $\mathbf{A}_t$ , thus removing the bias from time-dependent confounders.
+
+When using unstabilized weights  $W$ , the causal graph in the pseudo-population is the one in Figure 6 (a) where  $P(\mathbf{A}_t \mid \bar{\mathbf{X}}_t, \bar{\mathbf{A}}_{t-1}, V) = P(\mathbf{A}_t)$ . On the other hand, when using stabilized weights  $SW$ , causal graph in the pseudo-population is the one in Figure 6 (b) where  $P(\mathbf{A}_t \mid \bar{\mathbf{X}}_t, \bar{\mathbf{A}}_{t-1}, V) = P(\mathbf{A}_t \mid \bar{\mathbf{A}}_{t-1})$ .
+
+Counterfactual Recurrent Networks. Instead of using IPTW, we proposed building a representation of  $\bar{\mathbf{X}}_t,\bar{\mathbf{A}}_{t - 1},V$  that is not predictive of treatment  $\mathbf{A}_t$ . At timestep  $t$ , we have  $k$  different possible treatments  $\mathbf{A}_t\in \{A_1,\dots A_K\}$ . We build a representation of the history and covariates and treatments that has the same distribution across the different possible treatments:  $P(\Phi (\bar{\mathbf{X}}_t,\bar{\mathbf{A}}_{t - 1},\mathbf{V})\mid \mathbf{A}_t = A_1) = \dots = P(\Phi (\bar{\mathbf{X}}_t,\bar{\mathbf{A}}_{t - 1},\mathbf{V})\mid \mathbf{A}_t = A_K)$ . By breaking the association between past exposure and current treatments  $\mathbf{A}_t$ , we satisfy the causal graph in Figure 6 (a) and thus we remove the bias from time-dependent confounders.
+
+# D PROOF OF THEOREM 1
+
+We first prove the following proposition.
+
+Proposition 1. For fixed  $\Phi$ , let  $x' = \Phi(\bar{\mathbf{h}}_t)$ . Then the optimal prediction probabilities of  $G_a$  are given by
+
+$$
+G _ {a} ^ {j ^ {*}} \left(x ^ {\prime}\right) = \frac {P _ {j} ^ {\Phi} \left(x ^ {\prime}\right)}{\sum_ {i = 1} ^ {K} P _ {i} ^ {\Phi} \left(x ^ {\prime}\right)}. \tag {10}
+$$
+
+Proof. For fixed  $\Phi$ , the optimal prediction probabilities are given by
+
+$$
+G _ {a} ^ {*} = \arg \max  _ {G _ {a}} \sum_ {j = 1} ^ {K} \int_ {x ^ {\prime}} \log \left(G _ {a} ^ {j} \left(x ^ {\prime}\right)\right) P _ {j} ^ {\Phi} \left(x ^ {\prime}\right) d x ^ {\prime} \quad \text {s u b j e c t} G _ {a} ^ {j} \left(x ^ {\prime}\right) = 1. \tag {11}
+$$
+
+Maximising the value function pointwise and applying Lagrange multiplies, we get
+
+$$
+G _ {a} ^ {*} = \arg \max  _ {G _ {a}} \sum_ {j = 1} ^ {K} \log \left(G _ {a} ^ {j} \left(x ^ {\prime}\right)\right) P _ {j} ^ {\Phi} \left(x ^ {\prime}\right) + \lambda \left(\sum_ {j = 1} ^ {K} G _ {a} ^ {j} \left(x ^ {\prime}\right) - 1\right). \tag {12}
+$$
+
+Setting the derivative (w.r.t.  $G_{a}^{j^{*}}(x^{\prime})$ ) to 0 and solving for  $G_{a}^{j^{*}}(x^{\prime})$  we get
+
+$$
+G _ {a} ^ {j ^ {*}} \left(x ^ {\prime}\right) = - \frac {P _ {j} ^ {\Phi} \left(x ^ {\prime}\right)}{\lambda} \tag {13}
+$$
+
+where  $\lambda$  can now be solved for using the constraint to be  $\lambda = -\sum_{i=1}^{K} P_i^\Phi(x')$ . This gives the result.
+
+Proof. (of Theorem 1) By substituting the expression from Proposition 1 into the minimax game defined in Eq. 6, the objective for  $\Phi$  becomes
+
+$$
+\min  _ {\Phi} \sum_ {j = 1} ^ {K} \mathbb {E} _ {x ^ {\prime} \sim P _ {j} ^ {\Phi}} \left[ \log \left(\frac {P _ {j} ^ {\Phi} \left(x ^ {\prime}\right)}{\sum_ {i = 1} ^ {K} P _ {i} ^ {\Phi} \left(x ^ {\prime}\right)}\right) \right]. \tag {14}
+$$
+
+We then note that
+
+$$
+\begin{array}{l} \sum_ {j = 1} ^ {K} \mathbb {E} _ {x ^ {\prime} \sim P _ {j} ^ {\Phi}} \left[ \log \left(\frac {P _ {j} ^ {\Phi} \left(x ^ {\prime}\right)}{\sum_ {i = 1} ^ {K} P _ {i} ^ {\Phi} \left(x ^ {\prime}\right)}\right) \right] + K \log K = \sum_ {j = 1} ^ {K} \left(\mathbb {E} _ {x ^ {\prime} \sim P _ {j} ^ {\Phi}} \left[ \log \left(\frac {P _ {j} ^ {\Phi} \left(x ^ {\prime}\right)}{\sum_ {i = 1} ^ {K} P _ {i} ^ {\Phi} \left(x ^ {\prime}\right)}\right) \right] + \log K\right) (15) \\ = \sum_ {j = 1} ^ {K} \mathbb {E} _ {x ^ {\prime} \sim P _ {j} ^ {\Phi}} \left[ \log \left(\frac {P _ {j} ^ {\Phi} \left(x ^ {\prime}\right)}{\frac {1}{K} \sum_ {i = 1} ^ {K} P _ {i} ^ {\Phi} \left(x ^ {\prime}\right)}\right) \right] (16) \\ = \sum_ {j = 1} ^ {K} K L \left(P _ {j} ^ {\Phi} \left(x ^ {\prime}\right) \Bigg | \Bigg | \frac {1}{K} \sum_ {i = 1} ^ {K} P _ {i} ^ {\Phi} \left(x ^ {\prime}\right)\right) (17) \\ = K \cdot J S D \left(P _ {1} ^ {\Phi}, \dots , P _ {K} ^ {\Phi}\right) (18) \\ \end{array}
+$$
+
+where  $KL(\cdot ||\cdot)$  is the Kullback-Leibler divergence and  $JSD(\cdot ,\dots,\cdot)$  is the multi-distribution Jensen-Shannon Divergence (Li et al., 2018). Since  $K$  log  $K$  is a constant and the multi-distribution JSD is non-negative and 0 if and only if all distributions are equal, we have that  $P_{1}^{\Phi} = \ldots = P_{K}^{\Phi}$ .
+
+# E TRAINING PROCEDURE FOR CRN
+
+Let  $\mathcal{D} = \left\{\{\mathbf{x}_t^{(i)},\mathbf{a}_t^{(i)},\mathbf{y}_{t + 1}^{(i)}\}_{t = 1}^{T^{(i)}}\cup \{\mathbf{v}^{(i)}\} \right\}_{i = 1}^N$  be an observational dataset consisting of information about  $N$  independent patients that we use to train CRN. The encoder and decoder networks part of CRN are trained into two separate steps.
+
+To begin with, the encoder is trained to built treatment invariant representations of the patient history and to perform one-step ahead prediction. After the encoder is optimized, we use it to compute the balancing representation  $\mathbf{br}_t^{(i)}$  for each timestep in the trajectory of patient  $(i)$ . To train the decoder, we modify the training dataset as follows. For each patient  $(i)$ , we split their trajectory into shorter sequences of the  $\tau_{\mathrm{max}}$  timesteps of the form:
+
+$$
+\left\{\mathbf {b r} _ {l} ^ {(i)} \cup \left\{\mathbf {y} _ {l + t} ^ {(i)}, \mathbf {a} _ {l + t} ^ {(i)}, \mathbf {y} _ {l + t + 1} ^ {(i)} \right\} _ {t = 1} ^ {\tau_ {m a x}} \cup \mathbf {v} ^ {(i)} \right\}, \tag {19}
+$$
+
+for  $l = 1, \ldots, T^{(i)} - \tau_{\max}$ . Thus, each patient contributes with  $T^{(i)} - \tau_{\max}$  examples in the dataset for training the decoder. The different sequences obtained for all patents are randomly grouped into minibatches and used for training.
+
+The pseudocode in Algorithm 1 shows the training procedure used for the encoder and decoder networks part of CRN. The model was implemented in TensorFlow and trained on an NVIDIA Tesla K80 GPU. The Adam optimizer (Kingma & Ba, 2014) was used for training and both the encoder and the decoder are trained for 100 epochs.
+
+Algorithm 1 Pseudo-code for training CRN
+
+Input: Training data:  $\mathcal{D} = \left\{\{\mathbf{x}_t^{(i)},\mathbf{a}_t^{(i)},\mathbf{y}_{t + 1}^{(i)}\}_{t = 1}^{T^{(i)}}\cup \mathbf{v}^{(i)}\right\}_{i = 1}^N$
+
+(1) Encoder optimization: parameters  $\theta_{E,r},\theta_{E,a},\theta_{E,y}$
+
+Learning rate:  $\mu$
+
+for  $p = 1,\ldots ,$  max epochs do
+
+$$
+\lambda_ {p} = \frac {2}{1 + \exp (- 1 0 \cdot p)} - 1
+$$
+
+for Batch  $\mathcal{B} = \left\{\{\mathbf{x}_t^{(i)},\mathbf{a}_t^{(i)},\mathbf{y}_{t + 1}^{(i)}\}_{t = 0}^{T^{(i)}}\cup \mathbf{v}^{(i)}\right\}_{i = 1}^{| \mathcal{B}|}$  in epochd o
+
+Compute  $\mathcal{L}_{E,a}^{\mathcal{B}}(\theta_{E,r},\theta_{E,a}) = \frac{1}{|\mathcal{B}|}\sum_{i\in \mathcal{B}}\sum_{t = 1}^{T^{(i)}}\mathcal{L}_{t,a}^{(i)}(\theta_{E,r},\theta_{E,a})$
+
+Compute  $\mathcal{L}_{E,y}^{\mathcal{B}}(\theta_{E,r},\theta_{E,y}) = \frac{1}{|\mathcal{B}|}\sum_{i\in \mathcal{B}}\sum_{t = 1}^{T^{(i)}}\mathcal{L}_{t,y}^{(i)}(\theta_{E,r},\theta_{E,y})$
+
+$$
+\theta_ {E, r} \leftarrow \theta_ {E, r} - \mu \left(\frac {\partial \mathcal {L} _ {E , y} ^ {B} \left(\theta_ {E , r} , \theta_ {E , y}\right)}{\partial \theta_ {E , r}} - \lambda_ {p} \frac {\partial \mathcal {L} _ {E , a} ^ {B} \left(\theta_ {E , r} , \theta_ {E , a}\right)}{\partial \theta_ {E , r}}\right)
+$$
+
+$$
+\theta_ {E, y} \leftarrow \theta_ {E, y} - \mu \frac {\partial \mathcal {L} _ {E , y} ^ {\mathcal {B}} \left(\theta_ {E , r} , \theta_ {E , y}\right)}{\partial \theta_ {E , y}}
+$$
+
+$$
+\theta_ {E, a} \leftarrow \theta_ {E, a} - \mu \frac {\partial \mathcal {L} _ {E , a} ^ {\mathcal {B}} \left(\theta_ {E , r} , \theta_ {E , a}\right)}{\partial \theta_ {E , a}}
+$$
+
+end for
+
+end for
+
+(2) Compute the encoder balanced representation and use it to initialize the decoder hidden state.
+
+for  $i = 1,\dots,N$  do
+
+for  $t = 1,\dots ,T^{(i)}$  do
+
+$$
+\mathbf {b r} _ {t} ^ {(i)} = \operatorname {e n c o d e r} \left(\bar {\mathbf {x}} _ {t} ^ {(i)}, \bar {\mathbf {a}} _ {t - 1} ^ {(i)}, \mathbf {v} ^ {(i)}; \theta_ {E, r}\right)
+$$
+
+end for
+
+end for
+
+(3) Split dataset in sequences of  $\tau_{\mathrm{max}}$  timesteps:
+
+$$
+\left\{\left\{\mathbf {b r} _ {l} ^ {(i)} \cup \{\mathbf {y} _ {l + t} ^ {(i)}, \mathbf {a} _ {l + t} ^ {(i)}, \mathbf {y} _ {l + t + 1} ^ {(i)} \} _ {t = 1} ^ {\tau_ {m a x}} \cup \mathbf {v} ^ {(i)} \right\} _ {l = 1} ^ {T ^ {(i)} - \tau_ {\max}} \right\} _ {i = 1} ^ {N}
+$$
+
+(4) Optimize decoder: parameters  $\theta_{D,r},\theta_{D,a},\theta_{D,y}$
+
+Learning rate:  $\mu$
+
+for  $p = 1, \ldots,$  max epochs do
+
+$$
+\lambda_ {p} = \frac {2}{1 + \exp (- 1 0 \cdot p)} - 1
+$$
+
+for Batch  $\mathcal{B} = \left\{\mathbf{br}_l^{(i)}\cup \{\mathbf{y}_{l + t}^{(i)},\mathbf{a}_{l + t}^{(i)},\mathbf{y}_{l + t + 1}^{(i)}\}_{t = 0}^{\tau_{max}}\cup \{\mathbf{v}^{(i)}\}\right\}_{i = 1}^{|B|}$  in epoch of do
+
+Compute  $\mathcal{L}_{D,a}^{\mathcal{B}}(\theta_{D,r},\theta_{D,a}) = \frac{1}{|\mathcal{B}|}\sum_{i\in \mathcal{B}}\sum_{t = 1}^{\tau_{max}}\mathcal{L}_{t,a}^{(i)}(\theta_{D,r},\theta_{D,a})$
+
+Compute  $\mathcal{L}_{D,y}^{\mathcal{B}}(\theta_{D,r},\theta_{D,y}) = \frac{1}{|\mathcal{B}|}\sum_{i\in \mathcal{B}}\sum_{t = 1}^{\tau_{max}}\mathcal{L}_{t,y}^{(i)}(\theta_{D,r},\theta_{D,y})$
+
+$$
+\theta_ {D, r} \leftarrow \theta_ {D, r} - \mu \left(\frac {\partial \mathcal {L} _ {D , y} ^ {\mathcal {B}} \left(\theta_ {D , r} , \theta_ {D , y}\right)}{\partial \theta_ {D , r}} - \lambda_ {p} \frac {\partial \mathcal {L} _ {D , a} ^ {\mathcal {B}} \left(\theta_ {D , r} , \theta_ {D , a}\right)}{\partial \theta_ {D , r}}\right)
+$$
+
+$$
+\theta_ {D, y} \leftarrow \theta_ {D, y} - \mu \frac {\partial \mathcal {L} _ {D , y} ^ {\mathcal {B}} \left(\theta_ {D , r} , \theta_ {D , y}\right)}{\partial \theta_ {D , y}}
+$$
+
+$$
+\theta_ {D, a} \leftarrow \theta_ {D, a} - \mu \frac {\partial \mathcal {L} _ {D , a} ^ {\mathcal {B}} \left(\theta_ {D , r} , \theta_ {D , a}\right)}{\partial \theta_ {D , a}}
+$$
+
+end for
+
+end for
+
+Output: Trained CRN encoder (parameters  $\theta_{E,r},\theta_{E,a},\theta_{E,y}$  ) and trained CRN decoder (parameters  $\theta_{D,r},\theta_{D,a},\theta_{D,y}$ .)
+
+# F PHARMACOKINETIC-PHARMACODYNAMIC MODEL OF TUMOUR GROWTH
+
+To evaluate the CRN on counterfactual estimation, we need access to the data generation mechanism to build a test set that consists of patient outcomes under all possible treatment options. For this purpose, we use the state-of-the-art pharmacokinetic-pharmacodynamic (PK-PD) model of tumour growth proposed by Geng et al. (2017) and also used by Lim et al. (2018) for evaluating RMSMs. The PK-PD model characterizes patients suffering from non-small cell lung cancer and models the evolution of their tumour under the combined effects of chemotherapy and radiotherapy. In addition, the model includes different distributions of tumour sizes based on the cancer stage at diagnosis.
+
+Model of tumour growth The volume of tumour  $t$  days after diagnosis is modelled as follows:
+
+$$
+V (t + 1) = \left(1 + \underbrace {\rho \log \left(\frac {K}{V (t)}\right)} _ {\text {T u m o r g r o w t h}} - \underbrace {\beta_ {c} C (t)} _ {\text {C h e m o t h e r a p y}} - \underbrace {\left(\alpha_ {r} d (t) + \beta_ {r} d (t) ^ {2}\right)} _ {\text {R a d i o t h e r a p y}} + \underbrace {e _ {t}} _ {\text {N o i s e}}\right) V (t) \tag {20}
+$$
+
+where the parameters  $K, \rho, \beta_c, \alpha_r, \beta_r$  are sampled from the prior distributions described in (Geng et al., 2017) and  $e_t \sim \mathcal{N}(0, 0.01^2)$  is a noise term that accounts for randomness in the tumour growth.
+
+To incorporate heterogeneity among patient responses, due to, for instance, gender or genetic factors Bartsch et al. (2007), the prior means for  $\beta_{c}$  and  $\alpha_{r}$  are adjusted to create three patient subgroups  $S^{(i)}\in \{1,2,3\}$  as described in Lim et al. (2018). This way, we incorporate in the model of tumour growth specific characteristics that affect the patient's individualized response to treatments. Thus, the prior mean  $\mu_{\beta_c}$  of  $\beta_{c}$  and the prior mean  $\mu_{\alpha_r}$  of  $\alpha_{r}$  are augmented as follows.
+
+$$
+\mu_ {\beta_ {c}} ^ {\prime} (i) = \left\{ \begin{array}{l} 1. 1 \mu_ {\beta_ {c}}, \text {i f} \mathrm {S} ^ {(i)} = 3 \\ \mu_ {\beta_ {c}}, \text {o t h e r w i s e} \end{array} \right. \quad \mu_ {\alpha_ {r}} ^ {\prime} (i) = \left\{ \begin{array}{l} 1. 1 \mu_ {\alpha_ {r}}, \text {i f} \mathrm {S} ^ {(i)} = 1 \\ \mu_ {\alpha_ {r}}, \text {o t h e r w i s e} \end{array} \right. \tag {21}
+$$
+
+where  $\mu_{\beta_c}$  and  $\mu_{\alpha_r}$  are the mean parameters from Geng et al. (2017) and  $\mu_{\beta_c}^{\prime}(i)$  and  $\mu_{\alpha_r}^{\prime}(i)$  are the parameters used in the data simulation. The patient subgroup  $S^{(i)} \in \{1, 2, 3\}$  is used as baseline features.
+
+The chemotherapy drug concentration follows an exponential decay with half life of 1 day:
+
+$$
+C (t) = \tilde {C} (t) + C (t - 1) / 2, \tag {22}
+$$
+
+where  $\tilde{C}(t) = 5.0mg/m^3$  of Vinblastine if chemotherapy is given at time  $t$ .  $d(t) = 2.0Gy$  fractions of radiotherapy if the radiotherapy treatment is applied at timestep  $t$ .
+
+Time-varying confounding is introduced by modelling chemotherapy and radiotherapy assignment as Bernoulli random variables, with probabilities  $p_c$  and  $p_r$  depending on the tumour diameter:
+
+$$
+p _ {c} (t) = \sigma \left(\frac {\gamma_ {c}}{D _ {\max }} (\bar {D} (t) - \delta_ {c})\right) \quad p _ {r} (t) = \sigma \left(\frac {\gamma_ {r}}{D _ {\max }} (\bar {D} (t) - \delta_ {r})\right), \tag {23}
+$$
+
+where  $\bar{D}(t)$  is the average tumour diameter over the last 15 days,  $D_{\mathrm{max}} = 13\mathrm{cm}$  is the maximum tumour diameter and  $\sigma(\cdot)$  is the sigmoid activation function. The parameters  $\delta_c$  and  $\delta_r$  are set to  $\delta_c = \delta_r = D_{\mathrm{max}} / 2$  such that there is 0.5 probability of receiving treatment when tumour is half of its maximum size.  $\gamma_c, \gamma_r$  control the amount of time-dependent confounding; the higher  $\gamma_\star$  is, the more important the history of tumour diameter is in assigning treatments. Thus, at each timestep, there are four treatment options: no treatment  $(A_1)$ , chemotherapy  $(A_2)$ , radiotherapy  $(A_3)$ , combined chemotherapy and radiotherapy  $(A_4)$ .
+
+Since the work most relevant to ours is the one of Lim et al. (2018) we used the same data simulation and same settings for  $\gamma = \gamma_{c} = \gamma_{r}$  as in their case. When  $\gamma = 0$ , there is no time-dependent confounding and the treatments are randomly assigned. By increasing  $\gamma$  we increase the influence of the volume size history (encoded in  $\bar{D}(t)$ ) on the treatment probability. For example, assume  $\bar{D}(t) = \frac{3D_{max}}{4}$ . From equation (7), the probability of chemotherapy in this case is  $p_{c}(t) = \sigma\left(\frac{\gamma_{c}}{D_{max}}(\bar{D}(t) - \frac{D_{max}}{2})\right) = \sigma(0.25\gamma_{c})$ , where  $\sigma(\cdot)$  is the sigmoid function. When  $\gamma = 1$ ,  $p_{c}(t) = 0.56$ , when  $\gamma = 5$ ,  $p_{c}(t) = 0.77$  and when  $\gamma = 10$ ,  $p_{c}(t) = 0.92$  in this example.  $\gamma$  can be increased further to increase the bias. However, the values used in the experiments evaluate the model on a wide range of settings for the time-dependent confounding bias.
+
+# G MARGINAL STRUCTURAL MODELS
+
+Marginal Structural Models (Robins et al., 2000; Hernán et al., 2001) have been widely used in epidemiology and as part of follow up studies. In our case, we would like to estimate the effects of a sequence of treatments in the future given the current patient history:
+
+$$
+\mathbb {E} \left(\mathbf {Y} _ {t + \tau} \mid \bar {\mathbf {A}} (t, t + \tau - 1) = \bar {\mathbf {a}} (t, t + \tau - 1), \bar {\mathbf {H}} _ {t}\right) = g (\tau , a (t, t + \tau - 1), \bar {\mathbf {H}} _ {t}), \tag {24}
+$$
+
+where  $g$  is a generic function and  $\bar{\mathbf{a}}(t, t + \tau - 1) = [\mathbf{a}_t, \dots, \mathbf{a}_{t + \tau - 1}]$  represents a possible sequence of treatments from timestep  $t$  just until before the potential outcome  $\mathbf{Y}_{t + \tau}$  is observed. After removing the bias form time-dependent confounders,  $\mathbb{E}(\mathbf{Y}_{t + \tau} \mid \bar{\mathbf{A}}(t, t + \tau - 1) = \bar{\mathbf{a}}(t, t + \tau - 1), \bar{\mathbf{H}}_t) = \mathbb{E}(\mathbf{Y}_{t + \tau}[\bar{\mathbf{a}}(t, t + \tau - 1)]$ .
+
+Note that for implementing MSMs, we encode the treatments at timestep  $t$  in the model of tumour growth as  $\mathbf{A}_t = [A_{t,c}, A_{t,d}]$  to indicate the binary application of chemotherapy and radiotherapy. In order to remove the time-dependent confounding bias and estimate future outcomes, we use the stabilized weights of MSMs to weight each patient in the dataset:
+
+$$
+S W (t, \tau) = \prod_ {n = t} ^ {t + \tau} \frac {f \left(\mathbf {A} _ {n} \mid \bar {\mathbf {A}} _ {n - 1}\right)}{f \left(\mathbf {A} _ {n} \mid \bar {\mathbf {A}} _ {n - 1} , \bar {\mathbf {X}} _ {n} , \mathbf {V}\right)} = \prod_ {n = t} ^ {t + \tau} \frac {\prod_ {k \in \{c , d \}} f \left(A _ {n , k} \mid \bar {\mathbf {A}} _ {n - 1}\right)}{\prod_ {k \in \{c , d \}} f \left(A _ {n , k} \mid \bar {\mathbf {A}} _ {n - 1} , \bar {\mathbf {X}} _ {n} , \mathbf {V}\right)}, \tag {25}
+$$
+
+where  $f(\cdot)$  represents the conditional probability mass function for discrete treatments.
+
+We adopt the implementation in (Hernán et al., 2001; Howe et al., 2012; Lim et al., 2018) for MSMs and use logistic regression for estimating the propensity weights as follows:
+
+$$
+f \left(A _ {t, k} \mid \bar {\mathbf {A}} _ {t - 1}\right) = \sigma \left(\sum_ {j = 1} ^ {k} \omega_ {k} \left(\sum_ {i = 1} ^ {t - 1} A _ {t, j}\right)\right) \tag {26}
+$$
+
+$$
+f \left(A _ {t, k} \mid \bar {\mathbf {H}} _ {t}\right) = \sigma \Big (\sum_ {k \in \{c, d \}} \phi_ {k} \left(\sum_ {i = 1} ^ {t - 1} A _ {t, k}\right) + \mathbf {w} _ {1} \mathbf {X} _ {t} + \mathbf {w} _ {2} \mathbf {X} _ {t - 1} + \mathbf {w} _ {3} \mathbf {V} \Big) \tag {27}
+$$
+
+where  $\omega_{\star}, \phi_{\star}$  and  $\mathbf{w}_{\star}$  are regression coefficients,  $k \in \{c, d\}$  indicates the chemotherapy or radiotherapy treatments and  $\sigma(\cdot)$  is the sigmoid function.
+
+For predicting the outcome, the following regression model is used, where each individual patient is weighted by its propensity score:
+
+$$
+g (\tau , a (t, t + \tau - 1), \bar {\mathbf {H}} _ {t}) = \sum_ {k \in \{c, d \}} \beta_ {k} \left(\sum_ {n = t} ^ {t + \tau - 1} A _ {n, k}\right) + \mathbf {l} _ {1} \mathbf {X} _ {t} + \mathbf {l} _ {2} \mathbf {X} _ {t - 1} + \mathbf {l} _ {3} \mathbf {V} \tag {28}
+$$
+
+where  $\beta_{\star}$  and  $\mathbf{l}_{\star}$  are regression coefficients.
+
+MSMs do not require hyperparameter tuning so we use the patients from both the train and validation sets for training.
+
+# H RECURRENT MARGINAL STRUCTURAL NETWORKS
+
+MSMs are very sensitive to model mis-specification in computing the propensity weights and estimating the outcomes. Recurrent Marginal Structural Models (RMSNs) (Lim et al., 2018) overcome this problem by using recurrent neural networks to estimate the propensity scores and to build the outcome model. RNNs are more robust to changes in the treatment assignment policy. RMSNs were implemented as described in Lim et al. (2018)<sup>2</sup>.
+
+For implementing RMSNs, we also encode the treatments at timestep  $t$  in the model of tumour growth as  $\mathbf{A}_t = [A_{t,c}, A_{t,d}]$  to indicate the binary application of chemotherapy and radiotherapy. The propensity weights are estimated using recurrent neural networks as follows:
+
+$$
+f \left(A _ {t, k} \mid \bar {\mathbf {A}} _ {t - 1}\right) = \operatorname {R N N} _ {S W _ {n}} \left(\bar {\mathbf {A}} _ {t - 1}\right) \quad f \left(A _ {t, k} \mid \bar {\mathbf {X}} _ {t}, \bar {\mathbf {A}} _ {t - 1}\right) = \operatorname {R N N} _ {S W _ {d}} \left(\bar {\mathbf {A}} _ {t - 1}, \bar {\mathbf {X}} _ {t}, \mathbf {V}\right) \tag {29}
+$$
+
+For predicting one-step-ahed outcome, R-MSNs use an encoder network:
+
+$$
+g (1, a (t, t), \bar {\mathbf {H}} _ {t}) = \operatorname {R N N} _ {E} \left(\mathbf {a} _ {t}, \bar {\mathbf {A}} _ {t - 1}, \bar {\mathbf {X}} _ {t}, \mathbf {V}\right), \tag {30}
+$$
+
+where in the loss function, each patient is weighted by their stabilized IPTW.
+
+For estimating the treatment responses for a sequence of treatments in the future, RMSNs use a decoder network:
+
+$$
+g (\tau , a (t, t + \tau - 1), \bar {\mathbf {H}} _ {t}) = \operatorname {R N N} _ {D} \left(\mathbf {a} _ {t}, \dots , \mathbf {a} _ {t + \tau - 1}, \bar {\mathbf {A}} _ {t - 1}, \bar {\mathbf {X}} _ {t}, \mathbf {V}\right). \tag {31}
+$$
+
+See Lim et al. (2018) for more details about the R-MSNs model architecture and training procedure of the propensity weights, encoder and decoder networks. Tables 2 and 3 show the hyperparameter search ranges used to optimize this model for evaluation in our paper. The hyperparameters were selected in the same way as proposed by Lim et al. (2018), based on the error on the factual outcomes in the validation dataset. All of the models are trained using Adam optimizer for 100 epochs.
+
+Table 2: Hyperparameter search range for propensity networks and encoder (same as in Lim et al. (2018)). C is the size of the input.  
+
+<table><tr><td>Hyperparameter</td><td>Search range</td></tr><tr><td>Iterations of Hyperparameter Search</td><td>50</td></tr><tr><td>Learning rate</td><td>0.01, 0.005, 0.001</td></tr><tr><td>Minibatch size</td><td>64, 128, 256</td></tr><tr><td>RNN state size</td><td>0.5C, 1C, 2C, 3C, 4C</td></tr><tr><td>Dropout rate</td><td>0.1, 0.2, 0.3, 0.4, 0.5</td></tr><tr><td>Max Gradient Norm</td><td>0.5, 1.0, 2.0</td></tr></table>
+
+Table 3: Hyperparameter search range for decoder (same as in Lim et al. (2018)). C is the input size.  
+
+<table><tr><td>Hyperparameter</td><td>Search range</td></tr><tr><td>Iterations of Hyperparameter Search</td><td>20</td></tr><tr><td>Learning rate</td><td>0.01, 0.001, 0.0001</td></tr><tr><td>Minibatch size</td><td>256, 512, 1024</td></tr><tr><td>RNN state size</td><td>1C, 2C, 4C, 8C, 16C</td></tr><tr><td>Dropout Rate</td><td>0.1, 0.2, 0.3, 0.4, 0.5</td></tr><tr><td>Max Gradient Norm</td><td>0.5, 1.0, 2.0, 4.0</td></tr></table>
+
+# I BASELINE RNN AND LINEAR MODEL
+
+For the baseline linear model, we fit the same regression model used for Marginal Structural Networks, but without using the IPTW. The baseline RNN uses an LSTM unit and, at each timestep, receives as input the current treatment, the patient covariates and the patient static features to perform one-step-ahead prediction. To have a model of similar capacity to the CRN (similar number of parameters), we add a fully connected layer on top of the output of the LSTM unit in order to obtain the outcomes. Table 4 shows the hyperparameter search range used to optimize this model. The hyperparameters were selecting according to the error on the factual outcomes in the validation set. We train the baseline RNN using the Adam optimizer for 100 epochs.
+
+Table 4: Hyperparameter search range for baseline RNN model. C is the size of the input.  
+
+<table><tr><td>Hyperparameter</td><td>Search range</td></tr><tr><td>Iterations of Hyperparameter Search</td><td>50</td></tr><tr><td>Learning rate</td><td>0.01, 0.001, 0.0001</td></tr><tr><td>Minibatch size</td><td>64, 128, 256</td></tr><tr><td>RNN hidden units</td><td>0.5C, 1C, 2C, 3C, 4C</td></tr><tr><td>FC hidden units</td><td>0.5C, 1C, 2C, 3C, 4C</td></tr><tr><td>RNN dropout probability</td><td>0.1, 0.2, 0.3, 0.4, 0.5</td></tr></table>
+
+# J HYPERPARAMETER OPTIMIZATION FOR CRN
+
+As described in Appendix C, the dataset for training the decoder are used by splitting the sequences of the patients in the training set. This creates a larger dataset for training (where each patient  $(i)$  contributes  $T^{(i)} - \tau_{\mathrm{max}}$  times to the dataset) which requires a different hyperparameter search range. Moreover, the balancing representations computed by the encoder are used to initialize the state of the RNN for the decoder. Thus, the decoder RNN size is equal to the size of the balancing representation size of the encoder. Table 5 shows the hyperparameter search ranges for the encoder and decoder networks in CRN. We selected hyperparameters based on the error of the model on the factual outcomes in the validation dataset. All models are trained for 100 epochs.
+
+In addition, Tables 6 and 7 illustrate the optimal hyperparameters chosen.
+
+Table 5: Hyperparameter search range for CRN encoder. C is the size of the input and R is the size of the balancing representation.  
+
+<table><tr><td>Hyperparameter</td><td>Search range encoder</td><td>Search range decoder</td></tr><tr><td>Iterations of Hyperparameter Search</td><td>50</td><td>30</td></tr><tr><td>Learning rate</td><td>0.01, 0.001, 0.0001</td><td>0.01, 0.001, 0.0001</td></tr><tr><td>Minibatch size</td><td>64, 128, 256</td><td>256, 512, 1024</td></tr><tr><td>RNN hidden units</td><td>0.5C, 1C, 2C, 3C, 4C</td><td>Balancing representation size of encoder</td></tr><tr><td>Balancing representation size</td><td>0.5C, 1C, 2C, 3C, 4C</td><td>0.5C, 1C, 2C, 3C, 4C</td></tr><tr><td>FC hidden units</td><td>0.5R, 1R, 2R, 3R, 4R</td><td>0.5R, 1R, 2R, 3R, 4R</td></tr><tr><td>RNN dropout probability</td><td>0.1, 0.2, 0.3, 0.4, 0.5</td><td>0.1, 0.2, 0.3, 0.4, 0.5</td></tr></table>
+
+Table 6: Optimal hyperparameters for the CRN encoder when different degrees of time-dependent confounding are applied in the model of tumour growth. The parameters  $\gamma_{c}$  and  $\gamma_{r}$  measure the degree of time-dependent confounding applied. When  $\gamma_{c}$  and  $\gamma_{r}$  are set to the same value, we denote this with  $\gamma_{\star}$ .  
+
+<table><tr><td></td><td>γ☆=0</td><td>γ☆=1</td><td>γ☆=2</td><td>γ☆=3</td><td>γ☆=4</td><td>γ☆=5</td></tr><tr><td>Learning rate</td><td>0.001</td><td>0.1</td><td>0.001</td><td>0.01</td><td>0.01</td><td>0.001</td></tr><tr><td>Minibatch size</td><td>64</td><td>64</td><td>64</td><td>128</td><td>64</td><td>128</td></tr><tr><td>RNN hidden units</td><td>12</td><td>18</td><td>24</td><td>18</td><td>24</td><td>24</td></tr><tr><td>Balancing representation size</td><td>18</td><td>18</td><td>12</td><td>18</td><td>6</td><td>12</td></tr><tr><td>FC hidden units</td><td>18</td><td>18</td><td>36</td><td>54</td><td>24</td><td>48</td></tr><tr><td>RNN dropout probability</td><td>0.1</td><td>0.1</td><td>0.1</td><td>0.2</td><td>0.2</td><td>0.1</td></tr><tr><td></td><td>γ☆=6</td><td>γ☆=7</td><td>γ☆=8</td><td>γ☆=9</td><td>γ☆=10</td><td></td></tr><tr><td>Learning rate</td><td>0.001</td><td>0.001</td><td>0.01</td><td>0.001</td><td>0.01</td><td></td></tr><tr><td>Minibatch size</td><td>64</td><td>64</td><td>128</td><td>128</td><td>128</td><td></td></tr><tr><td>RNN hidden units</td><td>24</td><td>18</td><td>12</td><td>24</td><td>24</td><td></td></tr><tr><td>Balancing representation size</td><td>12</td><td>18</td><td>24</td><td>18</td><td>12</td><td></td></tr><tr><td>FC hidden units</td><td>48</td><td>72</td><td>12</td><td>36</td><td>12</td><td></td></tr><tr><td>RNN dropout probability</td><td>0.1</td><td>0.2</td><td>0.1</td><td>0.1</td><td>0.1</td><td></td></tr><tr><td></td><td colspan="2">γc=0,γr=5</td><td colspan="2">γc=5,γr=0</td><td></td><td></td></tr><tr><td>Learning rate</td><td colspan="2">0.01</td><td colspan="2">0.001</td><td></td><td></td></tr><tr><td>Minibatch size</td><td colspan="2">128</td><td colspan="2">64</td><td></td><td></td></tr><tr><td>RNN hidden units</td><td colspan="2">12</td><td colspan="2">12</td><td></td><td></td></tr><tr><td>Balancing representation size</td><td colspan="2">18</td><td colspan="2">24</td><td></td><td></td></tr><tr><td>FC hidden units</td><td colspan="2">36</td><td colspan="2">96</td><td></td><td></td></tr><tr><td>RNN dropout probability</td><td colspan="2">0.1</td><td colspan="2">0.1</td><td></td><td></td></tr></table>
+
+Table 7: Optimal hyperparameters for the CRN decoder when different degrees of time-dependent confounding are applied in the model of tumour growth. The parameters  $\gamma_{c}$  and  $\gamma_{r}$  measure the degree of time-dependent confounding applied. When  $\gamma_{c}$  and  $\gamma_{r}$  are set to the same value, we denote this with  $\gamma_{\star}$  
+
+<table><tr><td></td><td>γ* = 1</td><td>γ* = 2</td><td>γ* = 3</td><td>γ* = 4</td><td>γ* = 5</td></tr><tr><td>Learning rate</td><td>0.001</td><td>0.001</td><td>0.001</td><td>0.001</td><td>0.001</td></tr><tr><td>Minibatch size</td><td>1024</td><td>1024</td><td>512</td><td>1024</td><td>1024</td></tr><tr><td>RNN hidden units</td><td>18</td><td>12</td><td>18</td><td>6</td><td>12</td></tr><tr><td>Balancing representation size</td><td>18</td><td>18</td><td>6</td><td>18</td><td>3</td></tr><tr><td>FC hidden units</td><td>18</td><td>36</td><td>18</td><td>72</td><td>6</td></tr><tr><td>RNN dropout probability</td><td>0.1</td><td>0.2</td><td>0.3</td><td>0.1</td><td>0.1</td></tr><tr><td></td><td>γc=0, γr=5</td><td>γc=5, γr=0</td><td></td><td></td><td></td></tr><tr><td>Learning rate</td><td>0.01</td><td>0.001</td><td></td><td></td><td></td></tr><tr><td>Minibatch size</td><td>512</td><td>1024</td><td></td><td></td><td></td></tr><tr><td>RNN hidden units</td><td>18</td><td>24</td><td></td><td></td><td></td></tr><tr><td>Balancing representation size</td><td>18</td><td>12</td><td></td><td></td><td></td></tr><tr><td>FC hidden units</td><td>36</td><td>24</td><td></td><td></td><td></td></tr><tr><td>RNN dropout probability</td><td>0.1</td><td>0.03</td><td></td><td></td><td></td></tr></table>
+
+# K FULL RESULTS FOR COUNTERFACTUAL PREDICTION
+
+# K.1 MULTI-STEP AHEAD PREDICTION OF COUNTERFACTUALS
+
+Figure 7 shows the normalized RMSE for multiple step-ahead prediction of counterfactuals. The RMSE is normalized by the maximum tumour volume:  $V_{max} = 1150\mathrm{cm}^3$ . The counterfactuals in this case are generated as described in Section 6.3 and Appendix I. We notice that performance gains of CRN compared to RMSN increase with the number of future timesteps for which the counterfactuals are estimated.
+
+![](images/41dea6ae802a8a0136af177b77355929a2c976d60ae54b7379b7b08205df089f.jpg)  
+(a) Two-step ahead prediction
+
+![](images/2335111aacfe06aee5e8258cfc69d4ca8f7577fa709ed016a369251038b963e2.jpg)  
+(b) Three-step ahead prediction
+
+![](images/c9e7d81eefc78a90ffeffc45895352be91f28c6ba07d837914118ebc433c72c3.jpg)  
+(c) Four-step ahead prediction
+
+![](images/66c25b8a7a075c9fca9cc0141b02952c7376adf63c1bb4dda37e0ac751219240.jpg)  
+(d) Five-step ahead prediction
+
+![](images/3ad3008b52ceaccd3092017bec3fcc9cb51c5e931cf288597d8c981d3466910b.jpg)  
+(e) Six-step ahead prediction  
+Figure 7: Results for prediction of patient counterfactuals for multiple steps ahead.
+
+# K.2 DETAILED RESULTS FOR THE COUNTERFACTUAL PREDICTIONS
+
+Tables 8 and 9 show detailed results for the counterfactual predictions.
+
+Table 8: Normalized RMSE for one-step-ahead prediction of counterfactuals. The parameter  $\gamma$  measures the degree of time-dependent confounding applied.  
+
+<table><tr><td></td><td>γ = 0</td><td>γ = 1</td><td>γ = 2</td><td>γ = 3</td><td>γ = 4</td><td>γ = 5</td></tr><tr><td>Linear (no IPTW)</td><td>0.99%</td><td>1.08%</td><td>1.36%</td><td>1.68%</td><td>2.11%</td><td>2.77%</td></tr><tr><td>MSM</td><td>0.99%</td><td>1.08%</td><td>1.34%</td><td>1.63%</td><td>2.02%</td><td>2.61%</td></tr><tr><td>RNN</td><td>0.70%</td><td>0.70%</td><td>0.84%</td><td>1.05%</td><td>1.24%</td><td>1.69%</td></tr><tr><td>CRN (λ = 0)</td><td>0.66%</td><td>0.77%</td><td>0.92%</td><td>0.95%</td><td>1.24%</td><td>1.54%</td></tr><tr><td>RMSN</td><td>0.60%</td><td>0.61%</td><td>0.72%</td><td>0.81%</td><td>0.94%</td><td>1.23%</td></tr><tr><td>CRN</td><td>0.56%</td><td>0.57%</td><td>0.62%</td><td>0.67%</td><td>0.87%</td><td>1.20%</td></tr><tr><td></td><td>γ = 6</td><td>γ = 7</td><td>γ = 8</td><td>γ = 9</td><td>γ = 10</td><td></td></tr><tr><td>Linear (no IPTW)</td><td>3.55%</td><td>4.15%</td><td>4.80%</td><td>5.09%</td><td>5.22%</td><td></td></tr><tr><td>MSM</td><td>3.30%</td><td>3.79%</td><td>4.30%</td><td>4.47%</td><td>4.47%</td><td></td></tr><tr><td>RNN</td><td>2.03%</td><td>2.52%</td><td>2.88%</td><td>3.79%</td><td>4.01%</td><td></td></tr><tr><td>CRN (λ = 0)</td><td>1.98%</td><td>2.42%</td><td>2.73%</td><td>3.17%</td><td>3.57%</td><td></td></tr><tr><td>RMSN</td><td>1.70%</td><td>2.18%</td><td>2.37%</td><td>2.77%</td><td>2.83%</td><td></td></tr><tr><td>CRN</td><td>1.48%</td><td>1.56%</td><td>2.05%</td><td>2.36%</td><td>2.41%</td><td></td></tr></table>
+
+Table 9: Normalized RMSE for  $\tau$ -step-ahead prediction of counterfactuals. The parameter  $\gamma$  measures the degree of time-dependent confounding applied.  
+
+<table><tr><td></td><td></td><td>γ = 1</td><td>γ = 2</td><td>γ = 3</td><td>γ = 4</td><td>γ = 5</td></tr><tr><td rowspan="2">τ = 2</td><td>RMSN</td><td>0.90%</td><td>1.15%</td><td>1.53%</td><td>2.14%</td><td>2.91%</td></tr><tr><td>CRN</td><td>0.84%</td><td>0.96%</td><td>1.21%</td><td>1.46%</td><td>2.45%</td></tr><tr><td rowspan="2">τ = 3</td><td>RMSN</td><td>0.97%</td><td>1.36%</td><td>1.87%</td><td>2.44%</td><td>3.47%</td></tr><tr><td>CRN</td><td>0.86%</td><td>0.96%</td><td>1.47%</td><td>1.51%</td><td>2.84%</td></tr><tr><td rowspan="2">τ = 4</td><td>RMSN</td><td>1.24%</td><td>1.79%</td><td>2.60%</td><td>3.33%</td><td>3.88%</td></tr><tr><td>CRN</td><td>0.91%</td><td>1.08%</td><td>1.74%</td><td>1.76%</td><td>2.82%</td></tr><tr><td rowspan="2">τ = 5</td><td>RMSN</td><td>1.51%</td><td>2.13%</td><td>3.06%</td><td>4.07%</td><td>4.58%</td></tr><tr><td>CRN</td><td>0.85%</td><td>1.10%</td><td>1.73%</td><td>2.00%</td><td>3.43%</td></tr><tr><td rowspan="2">τ = 6</td><td>RMSN</td><td>2.10%</td><td>2.89%</td><td>3.06%</td><td>4.16%</td><td>6.32%</td></tr><tr><td>CRN</td><td>1.16%</td><td>1.52%</td><td>2.29%</td><td>2.66%</td><td>4.91%</td></tr></table>
+
+# L TEST SET GENERATION FOR EVALUATING TIMING OF TREATMENT
+
+In order to evaluate how well the models select the correct treatment and timing of treatment we simulate counterfactual outcomes as follows. We generate 1000 test samples using the model of tumour growth described in Section 6. Let  $\bar{\mathbf{H}}_t$  be the current history of the patient and let  $\tau$  be a future time horizon. For each timestep in the future, we have 4 treatment options at: no treatment  $(A_0)$ , chemotherapy  $(A_1)$ , radiotherapy  $(A_2)$ , chemotherapy and radiotherapy.  $(A_3)$ .
+
+Using the model of tumour growth where the outcome  $\mathbf{Y}_{t + \tau}$  is given by the volume of the tumour, we generate the following  $2\tau$  counterfactuals:
+
+Chemotherapy application
+
+$$
+\mathbf {Y} _ {t + \tau} \quad | \quad \mathbf {a} _ {t} = A _ {1}, \mathbf {a} _ {t + 1} = A _ {0}, \dots \mathbf {a} _ {t + \tau - 1} = A _ {0}, \bar {\mathbf {H}} _ {t} \tag {32}
+$$
+
+$$
+\mathbf {Y} _ {t + \tau} \quad | \quad \mathbf {a} _ {t} = A _ {0}, \mathbf {a} _ {t + 1} = A _ {1}, \dots \mathbf {a} _ {t + \tau - 1} = A _ {0}, \bar {\mathbf {H}} _ {t} \tag {33}
+$$
+
+$$
+\mathbf {Y} _ {t + \tau} \quad | \quad \mathbf {a} _ {t} = A _ {0}, \mathbf {a} _ {t + 1} = A _ {0}, \dots \mathbf {a} _ {t + \tau - 1} = A _ {1}, \bar {\mathbf {H}} _ {t} \tag {34}
+$$
+
+Radiotherapy application
+
+$$
+\mathbf {Y} _ {t + \tau} \quad | \quad \mathbf {a} _ {t} = A _ {2}, \mathbf {a} _ {t + 1} = A _ {0}, \dots \mathbf {a} _ {t + \tau - 1} = A _ {0}, \bar {\mathbf {H}} _ {t} \tag {35}
+$$
+
+$$
+\mathbf {Y} _ {t + \tau} \quad | \quad \mathbf {a} _ {t} = A _ {0}, \mathbf {a} _ {t + 1} = A _ {2}, \dots \mathbf {a} _ {t + \tau - 1} = A _ {0}, \bar {\mathbf {H}} _ {t} \tag {36}
+$$
+
+.
+
+$$
+\mathbf {Y} _ {t + \tau} \quad | \quad \mathbf {a} _ {t} = A _ {0}, \mathbf {a} _ {t + 1} = A _ {0}, \dots \mathbf {a} _ {t + \tau - 1} = A _ {2}, \bar {\mathbf {H}} _ {t} \tag {37}
+$$
+
+We perform this for each patient in the test set and at each time  $t$  in the history. For instance, for a patient with 50 timesteps in the model of tumour growth and for time horizon  $\tau = 3$ , we generate  $2 \cdot 3 \cdot 50 = 300$  counterfactuals.
+
+Using the true generated counterfactual data, we select the treatment that has the lowest  $\mathbf{Y}_{t + \tau}$  among the  $\tau$  options generated for each treatment. Then, we select the time of applying treatment (among  $t,t + 1,\ldots t + \tau -1$ ) that resulted in the lowest  $\mathbf{Y}_{t + \tau}$ . For each model, we generate the counterfactuals under the same treatment plans and patient histories. Then, we perform the selection of treatment and timing of treatment in the same way and we compare these with the true ones. Note that in order to account for numerical instability (two outcomes  $Y_{t + \tau}$  having very similar values), we consider two outcomes the same if they are within  $\epsilon = 0.001$  of each other.
+
+# M RESULTS ON FACTUAL PREDICTION ON MIMIC III
+
+Using the Medical Information Mart for Intensive Care (MIMIC III) (Johnson et al., 2016) database consisting of electronic health records from patients in the ICU, we also show how the CRN can be used on a real medical dataset. From MIMIC III we extracted the patients on antibiotics, with trajectories up to 30 timesteps, thus obtaining a dataset with 3487 patients. For each patient, we extracted 25 patient covariates including lab tests and vital signs measured over time, as well as static patient features such as age and gender.
+
+We used a binary treatment at each timestep indicating whether the patient was administered antibiotics or not. Note that for the longitudinal covariates we used aggregate value for each day since the ICU admission. The reason for this is because antibiotic treatment is decided daily for the patient. We split the dataset into 2826/313/348 patients for training, validation and testing respectively. We performed hyperparameter optimization on the validation patient set, using the search ranges in Table 5 and we again selected hyperparameters based on the error on the factual outcomes.
+
+We estimate the individualized effect of antibiotics assigned over time on the patient's white blood cell count. A high white blood cell count is associated with severe illness and poor outcome for ICU patients (Waheed et al., 2003). Antibiotic administration in the ICU aims to reduce the white blood cell count. However, the effectiveness of the antibiotics treatment in reducing the white blood cell count is highly dependent on the time they are administered with respect to the history of the patient covariates. In this context we again have time-dependent confounders: the patient features change over time and are affected by the previous administration of antibiotics. Moreover, the history of the patient features also determines antibiotics administration and affects future patient outcomes (De Bus et al., 2018; Ali et al., 2019).
+
+In Table 10 we report the root mean squared error for factual prediction of the patients' white blood cell count for multiple prediction horizons  $\tau$ . Note that for this dataset we do not have access to counterfactual data, which is why we report error on factual predictions.
+
+Table 10: RMSE for  $\tau$  -step-ahead prediction of factual outcomes on MIMIC III.  
+
+<table><tr><td></td><td>τ = 1</td><td>τ = 2</td><td>τ = 3</td><td>τ = 4</td></tr><tr><td>RMSN</td><td>2.84</td><td>3.87</td><td>4.46</td><td>4.79</td></tr><tr><td>CRN</td><td>2.68</td><td>3.54</td><td>4.07</td><td>4.67</td></tr></table>
+
+We notice that CRN also achieves better performance than RMSN in estimating factual outcomes in a real-world dataset containing electronic health records. In this context, where counterfactual data is not available, domain expert knowledge is required to validate the model's counterfactual predictions under other antibiotic treatment alternatives. This further medical validation is outside the scope of this paper.
\ No newline at end of file
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+# ESTIMATING GRADIENTS FOR DISCRETE RANDOM VARIABLES BY SAMPLING WITHOUT REPLACEMENT
+
+Wouter Kool
+
+University of Amsterdam ORTEC
+
+w.w.m.kool@uva.nl
+
+Herke van Hoof
+
+University of Amsterdam h.c.vanhoof@uva.nl
+
+Max Welling
+
+University of Amsterdam CIFAR
+
+m.welling@uva.nl
+
+# ABSTRACT
+
+We derive an unbiased estimator for expectations over discrete random variables based on sampling without replacement, which reduces variance as it avoids duplicate samples. We show that our estimator can be derived as the Rao-Blackwellization of three different estimators. Combining our estimator with REINFORCE, we obtain a policy gradient estimator and we reduce its variance using a built-in control variate which is obtained without additional model evaluations. The resulting estimator is closely related to other gradient estimators. Experiments with a toy problem, a categorical Variational Auto-Encoder and a structured prediction problem show that our estimator is the only estimator that is consistently among the best estimators in both high and low entropy settings.
+
+# 1 INTRODUCTION
+
+Put replacement in your basement! We derive the unordered set estimator<sup>1</sup>: an unbiased (gradient) estimator for expectations over discrete random variables based on (unordered sets of) samples without replacement. In particular, we consider the problem of estimating (the gradient of) the expectation of  $f(x)$  where  $x$  has a discrete distribution  $p$  over the domain  $D$ , i.e.
+
+$$
+\mathbb {E} _ {\boldsymbol {x} \sim p (\boldsymbol {x})} [ f (\boldsymbol {x}) ] = \sum_ {\boldsymbol {x} \in D} p (\boldsymbol {x}) f (\boldsymbol {x}). \tag {1}
+$$
+
+This expectation comes up in reinforcement learning, discrete latent variable modelling (e.g. for compression), structured prediction (e.g. for translation), hard attention and many other tasks that use models with discrete operations in their computational graphs (see e.g. Jang et al. (2016)). In general,  $x$  has structure (such as a sequence), but we can treat it as a 'flat' distribution, omitting the bold notation, so  $x$  has a categorical distribution over  $D$  given by  $p(x), x \in D$ . Typically, the distribution has parameters  $\theta$ , which are learnt through gradient descent. This requires estimating the gradient  $\nabla_{\theta} \mathbb{E}_{x \sim p_{\theta}(x)}[f(x)]$ , using a set of samples  $S$ . A gradient estimate  $e(S)$  is unbiased if
+
+$$
+\mathbb {E} _ {S} [ e (S) ] = \nabla_ {\boldsymbol {\theta}} \mathbb {E} _ {x \sim p _ {\boldsymbol {\theta}} (x)} [ f (x) ]. \tag {2}
+$$
+
+The samples  $S$  can be sampled independently or using alternatives such as stratified sampling which reduce variance to increase the speed of learning. In this paper, we derive an unbiased gradient estimator that reduces variance by avoiding duplicate samples, i.e. by sampling  $S$  without replacement. This is challenging as samples without replacement are dependent and have marginal distributions that are different from  $p(x)$ . We further reduce the variance by deriving a built-in control variate, which maintains the unbiasedness and does not require additional samples.
+
+Related work. Many algorithms for estimating gradients for discrete distributions have been proposed. A general and widely used estimator is REINFORCE (Williams, 1992). Biased gradients based on a continuous relaxations of the discrete distribution (known as Gumbel-Softmax or Concrete) were jointly introduced by Jang et al. (2016) and Maddison et al. (2016). These can be combined with the straight through estimator (Bengio et al., 2013) if the model requires discrete samples or be used to construct control variates for REINFORCE, as in REBAR (Tucker et al., 2017) or
+
+RELAX (Grathwohl et al., 2018). Many other methods use control variates and other techniques to reduce the variance of REINFORCE (Paisley et al., 2012; Ranganath et al., 2014; Gregor et al., 2014; Mnih & Gregor, 2014; Gu et al., 2016; Mnih & Rezende, 2016).
+
+Some works rely on explicit summation of the expectation, either for the marginal distribution (Titsias & Lázaro-Gredilla, 2015) or globally summing some categories while sampling from the remainder (Liang et al., 2018; Liu et al., 2019). Other approaches use a finite difference approximation to the gradient (Lorberbom et al., 2018; 2019). Yin et al. (2019) introduced ARSM, which uses multiple model evaluations where the number adapts automatically to the uncertainty.
+
+In the structured prediction setting, there are many algorithms for optimizing a quantity under a sequence of discrete decisions, using (weak) supervision, multiple samples (or deterministic model evaluations), or a combination both (Ranzato et al., 2016; Shen et al., 2016; He et al., 2016; Norouzi et al., 2016; Bahdanau et al., 2017; Edunov et al., 2018; Leblond et al., 2018; Negrinho et al., 2018). Most of these algorithms are biased and rely on pretraining using maximum likelihood or gradually transitioning from supervised to reinforcement learning. Using Gumbel-Softmax based approaches in a sequential setting is difficult as the bias accumulates because of mixing errors (Gu et al., 2018).
+
+# 2 PRELIMINARIES
+
+Throughout this paper, we will denote with  $B^{k}$  an ordered sample without replacement of size  $k$  and with  $S^{k}$  an unordered sample (of size  $k$ ) from the categorical distribution  $p$ .
+
+Restricted distribution. When sampling without replacement, we remove the set  $C \subset D$  already sampled from the domain and we denote with  $p^{D \setminus C}$  the distribution restricted to the domain  $D \setminus C$ :
+
+$$
+p ^ {D \backslash C} (x) = \frac {p (x)}{1 - \sum_ {c \in C} p (c)}, \quad x \in D \backslash C. \tag {3}
+$$
+
+Ordered sample without replacement  $B^{k}$ . Let  $B^{k} = (b_{1},\dots,b_{k}), b_{i} \in D$  be an ordered sample without replacement, which is generated from the distribution  $p$  as follows: first, sample  $b_{1} \sim p$ , then sample  $b_{2} \sim p^{D \setminus \{b_{1}\}}$ ,  $b_{3} \sim p^{D \setminus \{b_{1},b_{2}\}}$ , etc. i.e. elements are sampled one by one without replacement. Using this procedure,  $B^{k}$  can be seen as a (partial) ranking according to the Plackett-Luce model (Plackett, 1975; Luce, 1959) and the probability of obtaining the vector  $B^{k}$  is
+
+$$
+p \left(B ^ {k}\right) = \prod_ {i = 1} ^ {k} p ^ {D \backslash B ^ {i - 1}} \left(b _ {i}\right) = \prod_ {i = 1} ^ {k} \frac {p \left(b _ {i}\right)}{1 - \sum_ {j <   i} p \left(b _ {j}\right)}. \tag {4}
+$$
+
+We can also restrict  $B^k$  to the domain  $D \setminus C$ , which means that  $b_i \notin C$  for  $i = 1, \dots, k$ :
+
+$$
+p ^ {D \backslash C} (B ^ {k}) = \prod_ {i = 1} ^ {k} \frac {p ^ {D \backslash C} \left(b _ {i}\right)}{1 - \sum_ {j <   i} p ^ {D \backslash C} \left(b _ {j}\right)} = \prod_ {i = 1} ^ {k} \frac {p \left(b _ {i}\right)}{1 - \sum_ {c \in C} p (c) - \sum_ {j <   i} p \left(b _ {j}\right)}. \tag {5}
+$$
+
+Unordered sample without replacement. Let  $S^k \subseteq D$  be an unordered sample without replacement from the distribution  $p$ , which can be generated simply by generating an ordered sample and discarding the order. We denote elements in the sample with  $s \in S^k$  (so without index) and we write  $\mathcal{B}(S^k)$  as the set of all  $k!$  permutations (orderings)  $B^k$  that correspond to (could have generated)  $S^k$ . It follows that the probability for sampling  $S^k$  is given by:
+
+$$
+p \left(S ^ {k}\right) = \sum_ {B ^ {k} \in \mathcal {B} \left(S ^ {k}\right)} p \left(B ^ {k}\right) = \sum_ {B ^ {k} \in \mathcal {B} \left(S ^ {k}\right)} \prod_ {i = 1} ^ {k} \frac {p \left(b _ {i}\right)}{1 - \sum_ {j <   i} p \left(b _ {j}\right)} = \left(\prod_ {s \in S ^ {k}} p (s)\right) \cdot \sum_ {B ^ {k} \in \mathcal {B} \left(S ^ {k}\right)} \prod_ {i = 1} ^ {k} \frac {1}{1 - \sum_ {j <   i} p \left(b _ {j}\right)}. \tag {6}
+$$
+
+The last step follows since  $B^{k} \in \mathcal{B}(S^{k})$  is an ordering of  $S^{k}$ , such that  $\prod_{i=1}^{k} p(b_{i}) = \prod_{s \in S} p(s)$ . Naive computation of  $p(S^{k})$  is  $O(k!)$ , but in Appendix B we show how to compute it efficiently.
+
+When sampling from the distribution restricted to  $D \setminus C$ , we sample  $S^k \subseteq D \setminus C$  with probability:
+
+$$
+p ^ {D \backslash C} (S ^ {k}) = \left(\prod_ {s \in S ^ {k}} p (s)\right) \cdot \sum_ {B ^ {k} \in \mathcal {B} (S ^ {k})} \prod_ {i = 1} ^ {k} \frac {1}{1 - \sum_ {c \in C} p (c) - \sum_ {j <   i} p \left(b _ {j}\right)}. \tag {7}
+$$
+
+The Gumbel-Top- $k$  trick. As an alternative to sequential sampling, we can also sample  $B^k$  and  $S^k$  by taking the top  $k$  of Gumbel variables (Yellott, 1977; Vieira, 2014; Kim et al., 2016). Following notation from Kool et al. (2019c), we define the perturbed log-probability  $g_{\phi_i} = \phi_i + g_i$ , where  $\phi_i = \log p(i)$  and  $g_i \sim \mathrm{Gumbel}(0)$ . Then let  $b_1 = \arg \max_{i \in D} g_{\phi_i}$ ,  $b_2 = \arg \max_{i \in D \setminus \{b_1\}} g_{\phi_i}$ , etc., so  $B^k$  is the top  $k$  of the perturbed log-probabilities in decreasing order. The probability of obtaining  $B_k$  using this procedure is given by equation 4, so this provides an alternative sampling method which is effectively a (non-differentiable) reparameterization of sampling without replacement. For a differentiable reparameterization, see Grover et al. (2019).
+
+It follows that taking the top  $k$  perturbed log-probabilities without order, we obtain the unordered sample set  $S^k$ . This way of sampling underlies the efficient computation of  $p(S^k)$  in Appendix B.
+
+# 3 METHODOLOGY
+
+In this section, we derive the unordered set policy gradient estimator: a low-variance, unbiased estimator of  $\nabla_{\theta}\mathbb{E}_{p_{\theta}(x)}[f(x)]$  based on an unordered sample without replacement  $S^k$ . First, we derive the generic (non-gradient) estimator for  $\mathbb{E}[f(x)]$  as the Rao-Blackwellized version of a single sample Monte Carlo estimator (and two other estimators!). Then we combine this estimator with REINFORCE (Williams, 1992) and we show how to reduce its variance using a built-in baseline.
+
+# 3.1 RAO-BLACKWELLIZATION OF THE SINGLE SAMPLE ESTIMATOR
+
+A very crude but simple estimator for  $\mathbb{E}[f(x)]$  based on the ordered sample  $B^{k}$  is to only use the first element  $b_{1}$ , which by definition is a sample from the distribution  $p$ . We define this estimator as the single sample estimator, which is unbiased, since
+
+$$
+\mathbb {E} _ {B ^ {k} \sim p (B ^ {k})} [ f (b _ {1}) ] = \mathbb {E} _ {b _ {1} \sim p (b _ {1})} [ f (b _ {1}) ] = \mathbb {E} _ {x \sim p (x)} [ f (x) ]. \tag {8}
+$$
+
+Discarding all but one sample, the single sample estimator is inefficient, but we can use Rao-Blackwellization (Casella & Robert, 1996) to significantly improve it. To this end, we consider the distribution  $B^{k}|S^{k}$ , which is, knowing the unordered sample  $S^{k}$ , the conditional distribution over ordered samples  $B^{k}\in \mathcal{B}(S^{k})$  that could have generated  $S^k$ . Using  $B^{k}|S^{k}$ , we rewrite  $\mathbb{E}[f(b_1)]$  as
+
+$$
+\mathbb {E} _ {B ^ {k} \sim p (B ^ {k})} [ f (b _ {1}) ] = \mathbb {E} _ {S ^ {k} \sim p (S ^ {k})} \left[ \mathbb {E} _ {B ^ {k} \sim p (B ^ {k} | S ^ {k})} \left[ f (b _ {1}) \right] \right] = \mathbb {E} _ {S ^ {k} \sim p (S ^ {k})} \left[ \mathbb {E} _ {b _ {1} \sim p (b _ {1} | S ^ {k})} \left[ f (b _ {1}) \right] \right].
+$$
+
+The Rao-Blackwellized version of the single sample estimator computes the inner conditional expectation exactly. Since  $B^{k}$  is an ordering of  $S^k$ , we have  $b_{1}\in S^{k}$  and we can compute this as
+
+$$
+\mathbb {E} _ {b _ {1} \sim p \left(b _ {1} \mid S ^ {k}\right)} [ f (b _ {1}) ] = \sum_ {s \in S ^ {k}} P \left(b _ {1} = s \mid S ^ {k}\right) f (s) \tag {9}
+$$
+
+where, in a slight abuse of notation,  $P(b_{1} = s|S^{k})$  is the probability that the first sampled element  $b_{1}$  takes the value  $s$ , given that the complete set of  $k$  samples is  $S^{k}$ . Using Bayes' Theorem we find
+
+$$
+P \left(b _ {1} = s \mid S ^ {k}\right) = \frac {p \left(S ^ {k} \mid b _ {1} = s\right) P \left(b _ {1} = s\right)}{p \left(S ^ {k}\right)} = \frac {p ^ {D \backslash \{s \}} \left(S ^ {k} \backslash \{s \}\right) p (s)}{p \left(S ^ {k}\right)}. \tag {10}
+$$
+
+The step  $p(S^k | b_1 = s) = p^{D \setminus \{s\}}(S^k \setminus \{s\})$  comes from analyzing sequential sampling without replacement: given that the first element sampled is  $s$ , the remaining elements have a distribution restricted to  $D \setminus \{s\}$ , so sampling  $S^k$  (including  $s$ ) given the first element  $s$  is equivalent to sampling the remainder  $S^k \setminus \{s\}$  from the restricted distribution, which has probability  $p^{D \setminus \{s\}}(S^k \setminus \{s\})$  (see equation 7).
+
+The unordered set estimator. For notational convenience, we introduce the leave-one-out ratio.
+
+Definition 1. The leave-one-out ratio of  $s$  w.r.t. the set  $S$  is given by  $R(S^k, s) = \frac{p^{D \setminus \{s\}}(S^k \setminus \{s\})}{p(S^k)}$ .
+
+Rewriting equation 10 as  $P(b_{1} = s|S^{k}) = p(s)R(S^{k},s)$  shows that the probability of sampling  $s$  first, given  $S^k$ , is simply the unconditional probability multiplied by the leave-one-out ratio. We now define the unordered set estimator as the Rao-Blackwellized version of the single-sample estimator.
+
+Theorem 1. The unordered set estimator, given by
+
+$$
+e ^ {U S} \left(S ^ {k}\right) = \sum_ {s \in S ^ {k}} p (s) R \left(S ^ {k}, s\right) f (s) \tag {11}
+$$
+
+is the Rao-Blackwellized version of the (unbiased!) single sample estimator.
+
+Proof. Using  $P(b_{1} = s|S^{k}) = p(s)R(S^{k},s)$  in equation 9 we have
+
+$$
+\mathbb {E} _ {b _ {1} \sim p \left(b _ {1} \mid S ^ {k}\right)} [ f (b _ {1}) ] = \sum_ {s \in S ^ {k}} P (b _ {1} = s \mid S ^ {k}) f (s) = \sum_ {s \in S ^ {k}} p (s) R \left(S ^ {k}, s\right) f (s). \tag {12}
+$$
+
+![](images/ea7a693592c230f0b32b3c61197316cac71cefd017d290f103abd79f986e3f8c.jpg)
+
+The implication of this theorem is that the unordered set estimator, in explicit form given by equation 11, is an unbiased estimator of  $\mathbb{E}[f(x)]$  since it is the Rao-Blackwellized version of the unbiased single sample estimator. Also, as expected by taking multiple samples, it has variance equal or lower than the single sample estimator by the Rao-Blackwell Theorem (Lehmann & Scheffé, 1950).
+
+# 3.2 RAO-BLACKWELLIZATION OF OTHER ESTIMATORS
+
+The unordered set estimator is also the result of Rao-Blackwellizing two other unbiased estimators: the stochastic sum-and-sample estimator and the importance-weighted estimator.
+
+The sum-and-sample estimator. We define as sum-and-sample estimator any estimator that relies on the identity that for any  $C \subset D$
+
+$$
+\mathbb {E} _ {x \sim p (x)} [ f (x) ] = \mathbb {E} _ {x \sim p ^ {D \backslash C} (x)} \left[ \sum_ {c \in C} p (c) f (c) + \left(1 - \sum_ {c \in C} p (c)\right) f (x) \right]. \tag {13}
+$$
+
+For the derivation, see Appendix C.1 or Liang et al. (2018); Liu et al. (2019). In general, a sum-and-sample estimator with a budget of  $k > 1$  evaluations sums expectation terms for a set of categories  $C$  (s.t.  $|C| < k$ ) explicitly (e.g. selected by their value  $f$  (Liang et al., 2018) or probability  $p$  (Liu et al., 2019)), and uses  $k - |C|$  (down-weighted) samples from  $D \setminus C$  to estimate the remaining terms. As is noted by Liu et al. (2019), selecting  $C$  such that  $\frac{1 - \sum_{c \in C} p(c)}{k - |C|}$  is minimized guarantees to reduce variance compared to a standard minibatch of  $k$  samples (which is equivalent to setting  $C = \emptyset$ ). See also Fearnhead & Clifford (2003) for a discussion on selecting  $C$  optimally. The ability to optimize  $C$  depends on whether  $p(c)$  can be computed efficiently a-priori (before sampling). This is difficult in high-dimensional settings, e.g. sequence models which compute the probability incrementally while ancestral sampling. An alternative is to select  $C$  stochastically (as equation 13 holds for any  $C$ ), and we choose  $C = B^{k-1}$  to define the stochastic sum-and-sample estimator:
+
+$$
+e ^ {\text {S S A S}} \left(B ^ {k}\right) = \sum_ {j = 1} ^ {k - 1} p \left(b _ {j}\right) f \left(b _ {j}\right) + \left(1 - \sum_ {j = 1} ^ {k - 1} p \left(b _ {j}\right)\right) f \left(b _ {k}\right). \tag {14}
+$$
+
+For simplicity, we consider the version that sums  $k - 1$  terms here, but the following results also hold for a version that sums  $k - m$  terms and uses  $m$  samples (without replacement) (see Appendix C.3). Sampling without replacement, it holds that  $b_{k}|B^{k - 1}\sim p^{D\setminus B^{k - 1}}$ , so the unbiasedness follows from equation 13 by separating the expectation over  $B^k$  into expectations over  $B^{k - 1}$  and  $b_{k}|B^{k - 1}$ :
+
+$$
+\mathbb {E} _ {B ^ {k - 1} \sim p (B ^ {k - 1})} \left[ \mathbb {E} _ {b _ {k} \sim p (b _ {k} | B ^ {k - 1})} \left[ e ^ {\mathrm {S S A S}} (B ^ {k}) \right] \right] = \mathbb {E} _ {B ^ {k - 1} \sim p (B ^ {k - 1})} \left[ \mathbb {E} [ f (x) ] \right] = \mathbb {E} [ f (x) ].
+$$
+
+In general, a sum-and-sample estimator reduces variance if the probability mass is concentrated on the summed categories. As typically high probability categories are sampled first, the stochastic sum-and-sample estimator sums high probability categories, similar to the estimator by Liu et al. (2019) which we refer to as the deterministic sum-and-sample estimator. As we show in Appendix C.2, Rao-Blackwellizing the stochastic sum-and-sample estimator also results in the unordered set estimator. This even holds for a version that uses  $m$  samples and  $k - m$  summed terms (see Appendix C.3), which means that the unordered set estimator has equal or lower variance than the optimal (in terms of  $m$ ) stochastic sum-and-sample estimator, but conveniently does not need to choose  $m$ .
+
+The importance-weighted estimator. The importance-weighted estimator (Vieira, 2017) is
+
+$$
+e ^ {\mathrm {I W}} \left(S ^ {k}, \kappa\right) = \sum_ {s \in S ^ {k}} \frac {p (s)}{q (s , \kappa)} f (s). \tag {15}
+$$
+
+This estimator is based on the idea of priority sampling (Duffield et al., 2007). It does not use the order of the sample, but assumes sampling using the Gumbel-Top- $k$  trick and requires access to  $\kappa$ , the  $(k + 1)$ -th largest perturbed log-probability, which can be seen as the 'threshold' since  $g_{\phi_s} > \kappa \forall s \in S^k$ .  $q(s,a) = P(g_{\phi_s} > a)$  can be interpreted as the inclusion probability of  $s \in S^k$  (assuming a fixed threshold  $a$  instead of a fixed sample size  $k$ ). For details and a proof of unbiasedness, see Vieira (2017) or Kool et al. (2019c). As the estimator has high variance, Kool et al. (2019c) resort to normalizing the importance weights, resulting in biased estimates. Instead, we use Rao-Blackwellization to eliminate stochasticity by  $\kappa$ . Again, the result is the unordered set estimator (see Appendix D.1), which thus has equal or lower variance.
+
+# 3.3 THE UNORDERED SET POLICY GRADIENT ESTIMATOR
+
+Writing  $p_{\theta}$  to indicate the dependency on the model parameters  $\theta$ , we can combine the unordered set estimator with REINFORCE (Williams, 1992) to obtain the unordered set policy gradient estimator.
+
+Corollary 1. The unordered set policy gradient estimator, given by
+
+$$
+e ^ {U S P G} \left(S ^ {k}\right) = \sum_ {s \in S ^ {k}} p _ {\boldsymbol {\theta}} (s) R \left(S ^ {k}, s\right) \nabla_ {\boldsymbol {\theta}} \log p _ {\boldsymbol {\theta}} (s) f (s) = \sum_ {s \in S ^ {k}} \nabla_ {\boldsymbol {\theta}} p _ {\boldsymbol {\theta}} (s) R \left(S ^ {k}, s\right) f (s), \tag {16}
+$$
+
+is an unbiased estimate of the policy gradient.
+
+Proof. Using REINFORCE (Williams, 1992) combined with the unordered set estimator we find:
+
+$$
+\nabla_ {\boldsymbol {\theta}} \mathbb {E} _ {p _ {\boldsymbol {\theta}} (x)} [ f (x) ] = \mathbb {E} _ {p _ {\boldsymbol {\theta}} (x)} [ \nabla_ {\boldsymbol {\theta}} \log p _ {\boldsymbol {\theta}} (x) f (x) ] = \mathbb {E} _ {S ^ {k} \sim p _ {\boldsymbol {\theta}} (S ^ {k})} \left[ \sum_ {s \in S ^ {k}} p _ {\boldsymbol {\theta}} (s) R (S ^ {k}, s) \nabla_ {\boldsymbol {\theta}} \log p _ {\boldsymbol {\theta}} (s) f (s) \right].
+$$
+
+Variance reduction using a built-in control variate. The variance of REINFORCE can be reduced by subtracting a baseline from  $f$ . When taking multiple samples (with replacement), a simple and effective baseline is to take the mean of other (independent!) samples (Mnih & Rezende, 2016). Sampling without replacement, we can use the same idea to construct a baseline based on the other samples, but we have to correct for the fact that the samples are not independent.
+
+Theorem 2. The unordered set policy gradient estimator with baseline, given by
+
+$$
+e ^ {U S P G B L} \left(S ^ {k}\right) = \sum_ {s \in S ^ {k}} \nabla_ {\boldsymbol {\theta}} p _ {\boldsymbol {\theta}} (s) R \left(S ^ {k}, s\right) \left(f (s) - \sum_ {s ^ {\prime} \in S ^ {k}} p _ {\boldsymbol {\theta}} \left(s ^ {\prime}\right) R ^ {D \backslash \{s \}} \left(S ^ {k}, s ^ {\prime}\right) f \left(s ^ {\prime}\right)\right), \tag {17}
+$$
+
+where
+
+$$
+R ^ {D \backslash \{s \}} \left(S ^ {k}, s ^ {\prime}\right) = \frac {p _ {\boldsymbol {\theta}} ^ {D \backslash \{s , s ^ {\prime} \}} \left(S ^ {k} \backslash \{s , s ^ {\prime} \}\right)}{p _ {\boldsymbol {\theta}} ^ {D \backslash \{s \}} \left(S ^ {k} \backslash \{s \}\right)} \tag {18}
+$$
+
+is the second order leave-one-out ratio, is an unbiased estimate of the policy gradient.
+
+Proof. See Appendix E.1.
+
+![](images/21760681bc0b9772bcdcb5aff0ecea983aefba57c6f09343a5b3912b61525981.jpg)
+
+This theorem shows how to include a built-in baseline based on dependent samples (without replacement), without introducing bias. By having a built-in baseline, the value  $f(s)$  for sample  $s$  is compared against an estimate of its expectation  $\mathbb{E}[f(s)]$ , based on the other samples. The difference is an estimate of the advantage (Sutton & Barto, 2018), which is positive if the sample  $s$  is 'better' than average, causing  $p_{\theta}(s)$  to be increased (reinforced) through the sign of the gradient, and vice versa. By sampling without replacement, the unordered set estimator forces the estimator to compare different alternatives, and reinforces the best among them.
+
+Including the pathwise derivative. So far, we have only considered the scenario where  $f$  does not depend on  $\theta$ . If  $f$  does depend on  $\theta$ , for example in a VAE (Kingma & Welling, 2014; Rezende et al., 2014), then we use the notation  $f_{\theta}$  and we can write the gradient (Schulman et al., 2015) as
+
+$$
+\nabla_ {\boldsymbol {\theta}} \mathbb {E} _ {p _ {\boldsymbol {\theta}} (x)} [ f _ {\boldsymbol {\theta}} (x) ] = \mathbb {E} _ {p _ {\boldsymbol {\theta}} (x)} [ \nabla_ {\boldsymbol {\theta}} \log p _ {\boldsymbol {\theta}} (x) f _ {\boldsymbol {\theta}} (x) + \nabla_ {\boldsymbol {\theta}} f _ {\boldsymbol {\theta}} (x) ]. \tag {19}
+$$
+
+The additional second ('pathwise') term can be estimated (using the same samples) with the standard unordered set estimator. This results in the full unordered set policy gradient estimator:
+
+$$
+\begin{array}{l} e ^ {\mathrm {F U S P G}} \left(S ^ {k}\right) = \sum_ {s \in S ^ {k}} \nabla_ {\boldsymbol {\theta}} p _ {\boldsymbol {\theta}} (s) R \left(S ^ {k}, s\right) f _ {\boldsymbol {\theta}} (s) + \sum_ {s \in S ^ {k}} p _ {\boldsymbol {\theta}} (s) R \left(S ^ {k}, s\right) \nabla_ {\boldsymbol {\theta}} f _ {\boldsymbol {\theta}} (s) \\ = \sum_ {s \in S ^ {k}} R \left(S ^ {k}, s\right) \nabla_ {\boldsymbol {\theta}} \left(p _ {\boldsymbol {\theta}} (s) f _ {\boldsymbol {\theta}} (s)\right) \tag {20} \\ \end{array}
+$$
+
+Equation 20 is straightforward to implement using an automatic differentiation library. We can also include the baseline (as in equation 17) but we must make sure to call STOP_GRADIENT (DETACH in PyTorch) on the baseline (but not on  $f_{\theta}(s)!$ ). Importantly, we should never track gradients through the leave-one-out ratio  $R(S^k, s)$  which means it can be efficiently computed in pure inference mode.
+
+Scope & limitations. We can use the unordered set estimator for any discrete distribution from which we can sample without replacement, by treating it as a univariate categorical distribution over its domain. This includes sequence models, from which we can sample using Stochastic Beam Search (Kool et al., 2019c), as well as multivariate categorical distributions which can also be treated as sequence models (see Section 4.2). In the presence of continuous variables or a stochastic function  $f$ , we may separate this stochasticity from the stochasticity over the discrete distribution, as in Lorberbom et al. (2019). The computation of the leave-one-out ratios adds some overhead, although they can be computed efficiently, even for large  $k$  (see Appendix B). For a moderately sized model, the costs of model evaluation and backpropagation dominate the cost of computing the estimator.
+
+# 3.4 RELATIONTOOTHERMULTI-SAMPLEESTIMATORS
+
+Relation to Murthy's estimator. We found out that the 'vanilla' unordered set estimator (equation 11) is actually a special case of the estimator by Murthy (1957), known in statistics literature for estimation of a population total  $\Theta = \sum_{i\in D}y_i$ . Using  $y_{i} = p(i)f(i)$ , we have  $\Theta = \mathbb{E}[f(i)]$ , so Murthy's estimator can be used to estimate expectations (see equation 11). Murthy derives the estimator by 'unordering' a convex combination of Raj (1956) estimators, which, using  $y_{i} = p(i)f(i)$ , are stochastic sum-and-sample estimators in our analogy.
+
+Murthy (1957) also provides an unbiased estimator of the variance, which may be interesting for future applications. Since Murthy's estimator can be used with arbitrary sampling distribution, it is straightforward to derive importance-sampling versions of our estimators. In particular, we can sample  $S$  without replacement using  $q(x) > 0$ ,  $x \in D$ , and use equations 11, 16, 17 and 20, as long as we compute the leave-one-out ratio  $R(S^k, s)$  using  $q$ .
+
+While part of our derivation coincides with Murthy (1957), we are not aware of previous work using this estimator to estimate expectations. Additionally, we discuss practical computation of  $p(S)$  (Appendix B), we show the relation to the importance-weighted estimator, and we provide the extension to estimating policy gradients, especially including a built-in baseline without adding bias.
+
+Relation to the empirical risk estimator. The empirical risk loss (Edunov et al., 2018) estimates the expectation in equation 1 by summing only a subset  $S$  of the domain, using normalized probabilities  $\hat{p}_{\boldsymbol{\theta}}(s) = \frac{p_{\boldsymbol{\theta}}(s)}{\sum_{s' \in S} p_{\boldsymbol{\theta}}(s)}$ . Using this loss, the (biased) estimate of the gradient is given by
+
+$$
+e ^ {\text {R I S K}} \left(S ^ {k}\right) = \sum_ {s \in S ^ {k}} \nabla_ {\boldsymbol {\theta}} \left(\frac {p _ {\boldsymbol {\theta}} (s)}{\sum_ {s ^ {\prime} \in S ^ {k}} p _ {\boldsymbol {\theta}} \left(s ^ {\prime}\right)}\right) f (s). \tag {21}
+$$
+
+The risk estimator is similar to the unordered set policy gradient estimator, with two important differences: 1) the individual terms are normalized by the total probability mass rather than the leave-one-out ratio and 2) the gradient w.r.t. the normalization factor is taken into account. As a result, samples 'compete' for probability mass and only the best can be reinforced. This has the same effect as using a built-in baseline, which we prove in the following theorem.
+
+Theorem 3. By taking the gradient w.r.t. the normalization factor into account, the risk estimator has a built-in baseline, which means it can be written as
+
+$$
+e ^ {R I S K} \left(S ^ {k}\right) = \sum_ {s \in S ^ {k}} \nabla_ {\boldsymbol {\theta}} p _ {\boldsymbol {\theta}} (s) \frac {1}{\sum_ {s ^ {\prime \prime} \in S ^ {k}} p _ {\boldsymbol {\theta}} \left(s ^ {\prime \prime}\right)} \left(f (s) - \sum_ {s ^ {\prime} \in S ^ {k}} p _ {\boldsymbol {\theta}} \left(s ^ {\prime}\right) \frac {1}{\sum_ {s ^ {\prime \prime} \in S ^ {k}} p _ {\boldsymbol {\theta}} \left(s ^ {\prime \prime}\right)} f \left(s ^ {\prime}\right)\right). \tag {22}
+$$
+
+Proof. See Appendix F.1
+
+![](images/61b8a2d5cf42f708cdf3750d9c494431935736ae9a67aae37f5c3d981542e284.jpg)
+
+This theorem highlights the similarity between the biased risk estimator and our unbiased estimator (equation 17), and suggests that their only difference is the weighting of terms. Unfortunately, the implementation by Edunov et al. (2018) has more sources of bias (e.g. length normalization), which are not compatible with our estimator. However, we believe that our analysis helps analyze the bias of the risk estimator and is a step towards developing unbiased estimators for structured prediction.
+
+Relation to VIMCO. VIMCO (Mnih & Rezende, 2016) is an estimator that uses  $k$  samples (with replacement) to optimize an objective of the form  $\log \frac{1}{k} \sum_{i} f(x_i)$ , which is a multi-sample stochastic lower bound in the context of variational inference. VIMCO reduces the variance by using a local baseline for each of the  $k$  samples, based on the other  $k - 1$  samples. While we do not have a log term, as our goal is to optimize general  $\mathbb{E}[f(x)]$ , we adopt the idea of forming a baseline based on the other samples, and we define REINFORCE with replacement (with built-in baseline) as the estimator that computes the gradient estimate using samples with replacement  $X^k = (x_1, \dots, x_k)$  as
+
+$$
+e ^ {\mathrm {R F W R}} \left(X ^ {k}\right) = \frac {1}{k} \sum_ {i = 1} ^ {k} \nabla_ {\boldsymbol {\theta}} \log p _ {\boldsymbol {\theta}} \left(x _ {i}\right) \left(f \left(x _ {i}\right) - \frac {1}{k - 1} \sum_ {j \neq i} f \left(x _ {j}\right)\right). \tag {23}
+$$
+
+This estimator is unbiased, as  $\mathbb{E}_{x_i,x_j}[\nabla_\theta \log p_\theta (x_i)f(x_j)] = 0$  for  $i\neq j$  (see also Kool et al. (2019b)). We think of the unordered set estimator as the without-replacement version of this estimator, which weights terms by  $p_{\theta}(s)R(S^{k},s)$  instead of  $\frac{1}{k}$ . This puts more weight on higher probability elements to compensate for sampling without replacement. If probabilities are small and (close to) uniform, there are (almost) no duplicate samples and the weights will be close to  $\frac{1}{k}$ , so the gradient estimate of the with- and without-replacement versions are similar.
+
+Relation to ARSM. ARSM (Yin et al., 2019) also uses multiple evaluations ('pseudo-samples') of  $p_{\theta}$  and  $f$ . This can be seen as similar to sampling without replacement, and the estimator also has a built-in control variate. Compared to ARSM, our estimator allows direct control over the computational cost (through the sample size  $k$ ) and has wider applicability, for example it also applies to multivariate categorical variables with different numbers of categories per dimension.
+
+Relation to stratified/systematic sampling. Our estimator aims to reduce variance by changing the sampling distribution for multiple samples by sampling without replacement. There are alternatives, such as using stratified or systematic sampling (see, e.g. Douc & Cappé (2005)). Both partition the domain  $D$  into  $k$  strata and take a single sample from each stratum, where systematic sampling uses common random numbers for each stratum. In applications involving high-dimensional or structured domains, it is unclear how to partition the domain and how to sample from each partition. Additionally, as samples are not independent, it is non-trivial to include a built-in baseline, which we find is a key component that makes our estimator perform well.
+
+![](images/5c1b93fe9be5d9d6387b71f57a63b6ff4b227b87a87648f1fade709e032168f6.jpg)  
+(a) High entropy  $(\eta = 0)$
+
+![](images/999d1dd32e191f203149113933ce73c0c9c8897444ef154e7cf49f2ea8b82d35.jpg)  
+(b) Low entropy  $(\eta = -4)$  
+Figure 1: Bernoulli gradient variance (on log scale) as a function of the number of model evaluations (including baseline evaluations, so the sum-and-sample estimators with sampled baselines use twice as many evaluations). Note that for some estimators, the variance is 0 (log variance  $-\infty$  ) for  $k = 8$
+
+# 4 EXPERIMENTS
+
+# 4.1 BERNOULLI TOY EXPERIMENT
+
+We use the code by Liu et al. (2019) to reproduce their Bernoulli toy experiment. Given a vector  $\mathbf{p} = (0.6, 0.51, 0.48)$  the goal is to minimize the loss  $\mathcal{L}(\eta) = \mathbb{E}_{x_1, x_2, x_3 \sim \mathrm{Bern}(\sigma(\eta))}\left[\sum_{i=1}^{3}(x_i - p_i)^2\right]$ . Here  $x_1, x_2, x_3$  are i.i.d. from the Bernoulli  $(\sigma(\eta))$  distribution, parameterized by a scalar  $\eta \in \mathbb{R}$ , where  $\sigma(\eta) = (1 + \exp(-\eta))^{-1}$  is the sigmoid function. We compare different estimators, with and without baseline (either 'built-in' or using additional samples, referred to as REINFORCE+ in Liu et al. (2019)). We report the (log-)variance of the scalar gradient  $\frac{\partial\mathcal{L}}{\partial\eta}$  as a function of the number of model evaluations, which is twice as high when using a sampled baseline (for each term).
+
+As can be seen in Figure 1, the unordered set estimator is the only estimator that has consistently the lowest (or comparable) variance in both the high  $(\eta = 0)$  and low entropy  $(\eta = -4)$  regimes and for different number of samples/model evaluations. This suggests that it combines the advantages of the other estimators. We also ran the actual optimization experiment, where with as few as  $k = 3$  samples the trajectory was indistinguishable from using the exact gradient (see Liu et al. (2019)).
+
+# 4.2 CATEGORICAL VARIATIONAL AUTO-ENCODER
+
+We use the code from Yin et al. (2019) to train a categorical Variational Auto-Encoder (VAE) with 20 dimensional latent space, with 10 categories per dimension (details in Appendix G.1). To use our estimator, we treat this as a single factorized distribution with  $10^{20}$  categories from which we can sample without replacement using Stochastic Beam Search (Kool et al., 2019c), sequentially sampling each dimension as if it were a sequence model. We also perform experiments with  $10^{2}$  latent space, which provides a lower entropy setting, to highlight the advantage of our estimator.
+
+Measuring the variance. In Table 1, we report the variance of different gradient estimators with  $k = 4$  samples, evaluated on a trained model. The unordered set estimator has the lowest variance in both the small and large domain (low and high entropy) setting, being on-par with the best of the (stochastic<sup>3</sup>) sum-and-sample estimator and REINFORCE with replacement<sup>4</sup>. This confirms the toy experiment, suggesting that the unordered set estimator provides the best of both estimators. In Appendix G.2 we repeat the same experiment at different stages of training, with similar results.
+
+<table><tr><td colspan="2">Domain</td><td>ARSM</td><td>RELAX</td><td>REINFORCE (no bl)</td><td>Sum &amp; sample (sample bl)</td><td>REINF. w.r. (built-in bl)</td><td>Unordered (built-in bl)</td></tr><tr><td>Small</td><td>\( 10^{2} \)</td><td>13.45</td><td>11.67</td><td>11.52</td><td>7.49</td><td>6.29</td><td>6.65</td></tr><tr><td>Large</td><td>\( 10^{20} \)</td><td>15.55</td><td>15.86</td><td>13.81</td><td>8.48</td><td>13.77</td><td>8.44</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>7.06</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>7.05</td></tr></table>
+
+![](images/61a99cb6bfaf397c1e24288b989b633b993a0db45a9af0fcbadc53f5dc6785f8.jpg)  
+(a) Small domain (latent space size  $10^{2}$ )
+
+![](images/5bdd6bd8453c2c99be34fd4dddfaf0dfd4380b6e3b82dab1b4ccfe4525f42592.jpg)  
+(b) Large domain (latent space size  $10^{20}$ )  
+Figure 2: VAE smoothed training curves (-ELBO) of two independent runs when training with different estimators with  $k = 1, 4$  or 8 (thicker lines) samples (ARSM has a variable number). Some lines coincide, so we sort the legend by the lowest -ELBO achieved and report this value.
+
+ELBO optimization. We use different estimators to optimize the ELBO (details in Appendix G.1). Additionally to the baselines by Yin et al. (2019) we compare against REINFORCE with replacement and the stochastic sum-and-sample estimator. In Figure 2 we observe that our estimator performs on par with REINFORCE with replacement (and built-in baseline, equation 23) and outperforms other estimators in at least one of the settings. There are a lot of other factors, e.g. exploration that may explain why we do not get a strictly better result despite the lower variance. We note some overfitting (see validation curves in Appendix G.2), but since our goal is to show improved optimization, and to keep results directly comparable to Yin et al. (2019), we consider regularization a separate issue outside the scope of this paper. These results are using MNIST binarized by a threshold of 0.5. In Appendix G.2 we report results using the standard binarized MNIST dataset from Salakhutdinov & Murray (2008).
+
+# 4.3 STRUCTURED PREDICTION FOR THE TRAVELLING SALESMAN PROBLEM
+
+To show the wide applicability of our estimator, we consider the structured prediction task of predicting routes (sequences) for the Travelling Salesman Problem (TSP) (Vinyals et al., 2015; Bello et al., 2016; Kool et al., 2019a). We use the code by Kool et al.  $(2019\mathrm{a})^5$  to reproduce their TSP experiment with 20 nodes. For details, see Appendix H.
+
+We implement REINFORCE with replacement (and built-in baseline) as well as the stochastic sum-and-sample estimator and our estimator, using Stochastic Beam Search (Kool et al., 2019c) for sampling. Also, we include results using the biased normalized importance-weighted policy gradient estimator with built-in baseline (derived in Kool et al. (2019b), see Appendix D.2). Additionally, we compare against REINFORCE with greedy rollout baseline (Rennie et al., 2017) used by Kool et al. (2019c) and a batch-average baseline. For reference, we also include the biased risk estimator, either 'sampling' using stochastic or deterministic beam search (as in Edunov et al. (2018)).
+
+In Figure 3a, we compare training progress (measured on the validation set) as a function of the number of training steps, where we divide the batch size by  $k$  to keep the total number of samples equal. Our estimator outperforms REINFORCE with replacement, the stochastic sum-and-sample estimator and the strong greedy rollout baseline (which uses additional baseline model evaluations) and performs on-par with the biased risk estimator. In Figure 3b, we plot the same results against the number of instances, which shows that, compared to the single sample estimators, we can train with less data and less computational cost (as we only need to run the encoder once for each instance).
+
+![](images/613f09c1ec2079170966787a6eaa7350676778d06161d25a118cdaa50e59f992.jpg)  
+(a) Performance vs. training steps
+
+![](images/186532f52b07f05a4d78510dc80e613b8f4e96b1b96c443c79f39232453af6ad.jpg)  
+(b) Performance vs. number of instances  
+Figure 3: TSP validation set optimality gap measured during training. Raw results are light, smoothed results are darker (2 random seeds). We compare our estimator against different unbiased and biased (dotted) multi-sample estimators and against single-sample REINFORCE, with batch-average or greedy rollout baseline.
+
+# 5 DISCUSSION
+
+We introduced the unordered set estimator, a low-variance, unbiased gradient estimator based on sampling without replacement, which can be used as an alternative to the popular biased Gumbel-Softmax estimator (Jang et al., 2016; Maddison et al., 2016). Our estimator is the result of Rao-Blackwellizing three existing estimators, which guarantees equal or lower variance, and is closely related to a number of other estimators. It has wide applicability, is parameter free (except for the sample size  $k$ ) and has competitive performance to the best of alternatives in both high and low entropy regimes.
+
+In our experiments, we found that REINFORCE with replacement, with multiple samples and a built-in baseline as inspired by VIMCO (Mnih & Rezende, 2016), is a simple yet strong estimator which has performance similar to our estimator in the high entropy setting. We are not aware of any recent work on gradient estimators for discrete distributions that has considered this estimator as baseline, while it may be often preferred given its simplicity. In future work, we want to investigate if we can apply our estimator to estimate gradients 'locally' (Titsias & Lázaro-Gredilla, 2015), as locally we have a smaller domain and expect more duplicate samples.
+
+# ACKNOWLEDGMENTS
+
+This research was funded by ORTEC. We would like to thank anonymous reviewers for their feedback that helped improve the paper.
+
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+
+# A NOTATION
+
+Throughout this appendix we will use the following notation from Maddison et al. (2014):
+
+$$
+e _ {\phi} (g) = \exp (- g + \phi)
+$$
+
+$$
+F _ {\phi} (g) = \exp (- \exp (- g + \phi))
+$$
+
+$$
+f _ {\phi} (g) = e _ {\phi} (g) F _ {\phi} (g).
+$$
+
+This means that  $F_{\phi}(g)$  is the CDF and  $f_{\phi}(g)$  the PDF of the Gumbel(φ) distribution. Additionally we will use the identities by Maddison et al. (2014):
+
+$$
+F _ {\phi} (g) F _ {\gamma} (g) = F _ {\log \left(\exp (\phi) + \exp (\gamma)\right)} (g) \tag {24}
+$$
+
+$$
+\int_ {g = a} ^ {b} e _ {\gamma} (g) F _ {\phi} (g) \partial g = \left(F _ {\phi} (b) - F _ {\phi} (a)\right) \frac {\exp (\gamma)}{\exp (\phi)}. \tag {25}
+$$
+
+Also, we will use the following notation, definitions and identities (see Kool et al. (2019c)):
+
+$$
+\phi_ {i} = \log p (i) \tag {26}
+$$
+
+$$
+\phi_ {S} = \log \sum_ {i \in S} p (i) = \log \sum_ {i \in S} \exp \phi_ {i} \tag {27}
+$$
+
+$$
+\phi_ {D \backslash S} = \log \sum_ {i \in D \backslash S} p (i) = \log \left(1 - \sum_ {i \in S} p (i)\right) = \log \left(1 - \exp \left(\phi_ {S}\right)\right) \tag {28}
+$$
+
+$$
+G _ {\phi_ {i}} \sim \operatorname {G u m b e l} \left(\phi_ {i}\right) \tag {29}
+$$
+
+$$
+G _ {\phi_ {S}} = \max  _ {i \in S} G _ {\phi_ {i}} \sim \operatorname {G u m b e l} \left(\phi_ {S}\right) \tag {30}
+$$
+
+For a proof of equation 30, see Maddison et al. (2014).
+
+# B COMPUTATION OF  $p(S^k)$ $p^{D\setminus C}(S\setminus C)$  AND  $R(S^k,s)$
+
+We can sample the set  $S^k$  from the Plackett-Luce distribution using the Gumbel-Top- $k$  trick by drawing Gumbel variables  $G_{\phi_i} \sim \mathrm{Gumbel}(\phi_i)$  for each element and returning the indices of the  $k$  largest Gumbels. If we ignore the ordering, this means we will obtain the set  $S^k$  if  $\min_{i \in S^k} G_{\phi_i} > \max_{i \in D \setminus S^k} G_{\phi_i}$ . Omitting the superscript  $k$  for clarity, we can use the Gumbel-Max trick, i.e. that  $G_{\phi_{D \setminus S}} = \max_{i \notin S} G_{\phi_i} \sim \mathrm{Gumbel}(\phi_{D \setminus S})$  (equation 30) and marginalize over  $G_{\phi_{D \setminus S}}$ :
+
+$$
+\begin{array}{l} p (S) = P \left(\min  _ {i \in S} G _ {\phi_ {i}} > G _ {\phi_ {D \backslash S}}\right) \\ = P \left(G _ {\phi_ {i}} > G _ {\phi_ {D \backslash S}}, i \in S\right) \\ = \int_ {g _ {\phi_ {D} \backslash S} = - \infty} ^ {\infty} f _ {\phi_ {D \backslash S}} (g _ {\phi_ {D \backslash S}}) P (G _ {\phi_ {i}} > g _ {\phi_ {D \backslash S}}, i \in S) \partial g _ {\phi_ {D \backslash S}} \\ = \int_ {g _ {\phi_ {D} \backslash S}} ^ {\infty} f _ {\phi_ {D} \backslash S} \left(g _ {\phi_ {D} \backslash S}\right) \prod_ {i \in S} \left(1 - F _ {\phi_ {i}} \left(g _ {\phi_ {D} \backslash S}\right)\right) \partial g _ {\phi_ {D} \backslash S} (31) \\ = \int_ {u = 0} ^ {1} \prod_ {i \in S} \left(1 - F _ {\phi_ {i}} \left(F _ {\phi_ {D \backslash S}} ^ {- 1} (u)\right)\right) \partial u (32) \\ \end{array}
+$$
+
+Here we have used a change of variables  $u = F_{\phi_{D \setminus S}}(g_{\phi_{D \setminus S}})$ . This expression can be efficiently numerically integrated (although another change of variables may be required for numerical stability depending on the values of  $\phi$ ).
+
+Exact computation in  $O(2^k)$ . The integral in equation 31 can be computed exactly using the identity
+
+$$
+\prod_ {i \in S} (a _ {i} - b _ {i}) = \sum_ {C \subseteq S} (- 1) ^ {| C |} \prod_ {i \in C} b _ {i} \prod_ {i \in S \backslash C} a _ {i}
+$$
+
+which gives
+
+$$
+\begin{array}{l} p (S) = \int_ {g _ {\phi_ {D \setminus S}} = - \infty} ^ {\infty} f _ {\phi_ {D \setminus S}} (g _ {\phi_ {D \setminus S}}) \prod_ {i \in S} \left(1 - F _ {\phi_ {i}} (g _ {\phi_ {D \setminus S}})\right) \partial g _ {\phi_ {D \setminus S}} \\ = \sum_ {C \subseteq S} (- 1) ^ {| C |} \int_ {g _ {\phi_ {D \setminus S}} = - \infty} ^ {\infty} f _ {\phi_ {D \setminus S}} (g _ {\phi_ {D \setminus S}}) \prod_ {i \in C} F _ {\phi_ {i}} (g _ {\phi_ {D \setminus S}}) \prod_ {i \in S \setminus C} 1 \partial g _ {\phi_ {D \setminus S}} \\ = \sum_ {C \subseteq S} (- 1) ^ {| C |} \int_ {g _ {\phi_ {D \setminus S}} = - \infty} ^ {\infty} e _ {\phi_ {D \setminus S}} (g _ {\phi_ {D \setminus S}}) F _ {\phi_ {D \setminus S}} (g _ {\phi_ {D \setminus S}}) F _ {\phi_ {C}} (g _ {\phi_ {D \setminus S}}) \partial g _ {\phi_ {D \setminus S}} \\ = \sum_ {C \subseteq S} (- 1) ^ {| C |} \int_ {g _ {\phi_ {D \backslash S}}} ^ {\infty} e _ {\phi_ {D \backslash S}} (g _ {\phi_ {D \backslash S}}) F _ {\phi_ {(D \backslash S) \cup C}} (g _ {\phi_ {D \backslash S}}) \partial g _ {\phi_ {D \backslash S}} \\ = \sum_ {C \subseteq S} (- 1) ^ {| C |} (1 - 0) \frac {\exp \left(\phi_ {D \backslash S}\right)}{\exp \left(\phi_ {(D \backslash S) \cup C}\right)} \\ = \sum_ {C \subseteq S} (- 1) ^ {| C |} \frac {1 - \sum_ {i \in S} p (i)}{1 - \sum_ {i \in S \backslash C} p (i)}. \tag {33} \\ \end{array}
+$$
+
+Computation of  $p^{D \setminus C}(S \setminus C)$ . When using the Gumbel-Top- $k$  trick over the restricted domain  $D \setminus C$ , we do not need to renormalize the log-probabilities  $\phi_s, s \in D \setminus C$  since the Gumbel-Top- $k$  trick applies to unnormalized log-probabilities. Also, assuming  $C \subseteq S^k$ , it holds that  $(D \setminus C) \setminus (S \setminus C) = D \setminus S$ . This means that we can compute  $p^{D \setminus C}(S \setminus C)$  similar to equation 31:
+
+$$
+\begin{array}{l} p ^ {D \backslash C} (S \backslash C) = P \big (\min  _ {i \in S \backslash C} G _ {\phi_ {i}} > G _ {\phi_ {(D \backslash C) \backslash (S \backslash C)}} \big) \\ = P \left(\min  _ {i \in S \backslash C} G _ {\phi_ {i}} > G _ {\phi_ {D \backslash S}}\right) \\ = \int_ {g _ {\phi_ {D} \backslash S}} ^ {\infty} f _ {\phi_ {D \backslash S}} \left(g _ {\phi_ {D \backslash S}}\right) \prod_ {i \in S \backslash C} \left(1 - F _ {\phi_ {i}} \left(g _ {\phi_ {D \backslash S}}\right)\right) \partial g _ {\phi_ {D \backslash S}}. \tag {34} \\ \end{array}
+$$
+
+Computation of  $R(S^k, s)$ . Note that, using equation 10, it holds that
+
+$$
+\sum_ {s \in S ^ {k}} \frac {p ^ {D \setminus \{s \}} (S ^ {k} \setminus \{s \}) p (s)}{p (S ^ {k})} = \sum_ {s \in S ^ {k}} P (b _ {1} = s | S ^ {k}) = 1
+$$
+
+from which it follows that
+
+$$
+p (S ^ {k}) = \sum_ {s \in S ^ {k}} p ^ {D \backslash \{s \}} (S ^ {k} \backslash \{s \}) p (s)
+$$
+
+such that
+
+$$
+R \left(S ^ {k}, s\right) = \frac {p ^ {D \backslash \{s \}} \left(S ^ {k} \backslash \{s \}\right)}{p \left(S ^ {k}\right)} = \frac {p ^ {D \backslash \{s \}} \left(S ^ {k} \backslash \{s \}\right)}{\sum_ {s ^ {\prime} \in S ^ {k}} p ^ {D \backslash \{s ^ {\prime} \}} \left(S ^ {k} \backslash \{s ^ {\prime} \}\right) p \left(s ^ {\prime}\right)}. \tag {35}
+$$
+
+This means that, to compute the leave-one-out ratio for all  $s \in S^k$ , we only need to compute  $p^{D \setminus \{s\}}(S^k \setminus \{s\})$  for  $s \in S^k$ . When using the numerical integration or summation in  $O(2^k)$ , we can reuse computation, whereas using the naive method, the cost is  $O(k \cdot (k - 1)!) = O(k!)$ , making the total computational cost comparable to computing just  $p(S^k)$ , and the same holds when computing the 'second-order' leave one out ratios for the built-in baseline (equation 17).
+
+Details of numerical integration. For computation of the leave-one-out ratio (equation 35) for large  $k$  we can use the numerical integration, where we need to compute equation 34 with  $C = \{s\}$ . For this purpose, we rewrite the integral as
+
+$$
+\begin{array}{l} p ^ {D \backslash C} (S \backslash C) = \int_ {g _ {\phi_ {D \backslash S}} = - \infty} ^ {\infty} f _ {\phi_ {D \backslash S}} \left(g _ {\phi_ {D \backslash S}}\right) \prod_ {i \in S \backslash C} \left(1 - F _ {\phi_ {i}} \left(g _ {\phi_ {D \backslash S}}\right)\right) \partial g _ {\phi_ {D \backslash S}} \\ = \int_ {u = 0} ^ {1} \prod_ {i \in S \backslash C} \left(1 - F _ {\phi_ {i}} \left(F _ {\phi_ {D \backslash S}} ^ {- 1} (u)\right)\right) \partial u \\ = \int_ {u = 0} ^ {1} \prod_ {i \in S \backslash C} \left(1 - u ^ {\exp \left(\phi_ {i} - \phi_ {D \backslash S}\right)}\right) \partial u \\ = \exp (b) \cdot \int_ {v = 0} ^ {1} v ^ {\exp (b) - 1} \prod_ {i \in S \backslash C} \left(1 - v ^ {\exp \left(\phi_ {i} - \phi_ {D \backslash S} + b\right)}\right) \partial v \\ = \exp (a + \phi_ {D \backslash S}) \cdot \int_ {v = 0} ^ {1} v ^ {\exp (a + \phi_ {D \backslash S}) - 1} \prod_ {i \in S \backslash C} \left(1 - v ^ {\exp (\phi_ {i} + a)}\right) \partial v. \\ \end{array}
+$$
+
+Here we have used change of variables  $v = u^{exp(-b)}$  and  $a = b - \phi_{D\backslash S}$ . This form allows to compute the integrands efficiently, as
+
+$$
+\prod_ {i \in S \backslash C} \left(1 - v ^ {\exp (\phi_ {i} + a)}\right) = \frac {\prod_ {i \in S} \left(1 - v ^ {\exp (\phi_ {i} + a)}\right)}{\prod_ {i \in C} \left(1 - v ^ {\exp (\phi_ {i} + a)}\right)}
+$$
+
+where the numerator only needs to be computed once, and, since  $C = \{s\}$  when computing equation 35, the denominator only consists of a single term.
+
+The choice of  $a$  may depend on the setting, but we found that  $a = 5$  is a good default option which leads to an integral that is generally smooth and can be accurately approximated using the trapezoid rule. We compute the integrands in logarithmic space and sum the terms using the stable LOGSUMEXP trick. In our code we provide an implementation which also computes all second-order leave-one-out ratios efficiently.
+
+# C THE SUM-AND-SAMPLE ESTIMATOR
+
+# C.1 UNBIASEDNESS OF THE SUM-AND-SAMPLE ESTIMATOR
+
+We show that the sum-and-sample estimator is unbiased for any set  $C \subset D$  (see also Liang et al. (2018); Liu et al. (2019)):
+
+$$
+\begin{array}{l} \mathbb {E} _ {x \sim p ^ {D \backslash C} (x)} \left[ \sum_ {c \in C} p (c) f (c) + \left(1 - \sum_ {x \in C} p (c)\right) f (x) \right] \\ = \sum_ {c \in C} p (c) f (c) + \left(1 - \sum_ {c \in C} p (c)\right) \mathbb {E} _ {x \sim p ^ {D \setminus C} (x)} [ f (x) ] \\ = \sum_ {c \in C} p (c) f (c) + \left(1 - \sum_ {c \in C} p (c)\right) \sum_ {x \in D \backslash C} \frac {p (x)}{1 - \sum_ {c \in C} p (c)} f (x) \\ = \sum_ {c \in C} p (c) f (c) + \sum_ {x \in D \backslash C} p (x) f (x) \\ = \sum_ {x \in D} p (x) f (x) \\ = \mathbb {E} _ {x \sim p (x)} [ f (x) ] \\ \end{array}
+$$
+
+# C.2 RAO-BLACKWELLIZATION OF THE STOCHASTIC SUM-AND-SAMPLE ESTIMATOR
+
+In this section we give the proof that Rao-Blackwellizing the stochastic sum-and-sample estimator results in the unordered set estimator.
+
+Theorem 4. Rao-Blackwellizing the stochastic sum-and-sample estimator results in the unordered set estimator, i.e.
+
+$$
+\mathbb {E} _ {B ^ {k} \sim p \left(B ^ {k} \mid S ^ {k}\right)} \left[ \sum_ {j = 1} ^ {k - 1} p \left(b _ {j}\right) f \left(b _ {j}\right) + \left(1 - \sum_ {j = 1} ^ {k - 1} p \left(b _ {j}\right)\right) f \left(b _ {k}\right) \right] = \sum_ {s \in S ^ {k}} p (s) R \left(S ^ {k}, s\right) f (s). \tag {36}
+$$
+
+Proof. To give the proof, we first prove three Lemmas.
+
+# Lemma 1.
+
+$$
+P \left(b _ {k} = s \mid S ^ {k}\right) = \frac {p \left(S ^ {k} \setminus \{s \}\right)}{p \left(S ^ {k}\right)} \frac {p (s)}{1 - \sum_ {s ^ {\prime} \in S ^ {k} \setminus \{s \}} p \left(s ^ {\prime}\right)} \tag {37}
+$$
+
+Proof. Similar to the derivation of  $P(b_{1} = s|S^{k})$  (equation 10 in the main paper), we can write:
+
+$$
+\begin{array}{l} P (b _ {k} = s | S ^ {k}) = \frac {P (S ^ {k} \cap b _ {k} = s)}{p (S ^ {k})} \\ = \frac {p \left(S ^ {k} \backslash \{s \}\right) p ^ {D \backslash \left(S ^ {k} \backslash \{s \}\right)} (s)}{p \left(S ^ {k}\right)} \\ = \frac {p (S ^ {k} \setminus \{s \})}{p (S ^ {k})} \frac {p (s)}{1 - \sum_ {s ^ {\prime} \in S ^ {k} \setminus \{s \}} p (s ^ {\prime})}. \\ \end{array}
+$$
+
+The step from the first to the second row comes from analyzing the event  $S^k \cap b_k = s$  using sequential sampling: to sample  $S^k$  (including  $s$ ) with  $s$  being the  $k$ -th element means that we should first sample  $S^k \setminus \{s\}$  (in any order), and then sample  $s$  from the distribution restricted to  $D \setminus (S^k \setminus \{s\})$ .
+
+# Lemma 2.
+
+$$
+p (S) + p \left(S \backslash \{s \}\right) \frac {1 - \sum_ {s ^ {\prime} \in S} p \left(s ^ {\prime}\right)}{1 - \sum_ {s ^ {\prime} \in S \backslash \{s \}} p \left(s ^ {\prime}\right)} = p ^ {D \backslash \{s \}} \left(S \backslash \{s \}\right) \tag {38}
+$$
+
+Dividing equation 33 by  $1 - \sum_{s' \in S} p(s')$  on both sides, we obtain
+
+Proof.
+
+$$
+\begin{array}{l} \frac {p (S)}{1 - \sum_ {s ^ {\prime} \in S} p (s ^ {\prime})} \\ = \sum_ {C \subseteq S} (- 1) ^ {| C |} \frac {1}{1 - \sum_ {s ^ {\prime} \in S \backslash C} p (s ^ {\prime})} \\ = \sum_ {C \subseteq S \backslash \{s \}} \left((- 1) ^ {| C |} \frac {1}{1 - \sum_ {s ^ {\prime} \in S \backslash C} p (s ^ {\prime})} + (- 1) ^ {| C \cup \{s \} |} \frac {1}{1 - \sum_ {s ^ {\prime} \in S \backslash (C \cup \{s \})} p (s ^ {\prime})}\right) \\ = \sum_ {C \subseteq S \backslash \{s \}} (- 1) ^ {| C |} \frac {1}{1 - \sum_ {s ^ {\prime} \in S \backslash C} p (s ^ {\prime})} + \sum_ {C \subseteq S \backslash \{s \}} (- 1) ^ {| C \cup \{s \} |} \frac {1}{1 - \sum_ {s ^ {\prime} \in S \backslash (C \cup \{s \})} p (s ^ {\prime})} \\ = \sum_ {C \subseteq S \backslash \{s \}} (- 1) ^ {| C |} \frac {1}{1 - p (s) - \sum_ {s ^ {\prime} \in (S \backslash \{s \}) \backslash C} p (s ^ {\prime})} - \sum_ {C \subseteq S \backslash \{s \}} (- 1) ^ {| C |} \frac {1}{1 - \sum_ {s ^ {\prime} \in (S \backslash \{s \}) \backslash C} p (s ^ {\prime})} \\ = \frac {1}{1 - p (s)} \sum_ {C \subseteq S \backslash \{s \}} (- 1) ^ {| C |} \frac {1}{1 - \sum_ {s ^ {\prime} \in (S \backslash \{s \}) \backslash C} \frac {p (s ^ {\prime})}{1 - p (s)}} - \frac {p (S \backslash \{s \})}{1 - \sum_ {s ^ {\prime} \in S \backslash \{s \}} p (s ^ {\prime})} \\ = \frac {1}{1 - p (s)} \frac {p ^ {D \setminus \{s \}} (S \setminus \{s \})}{1 - \sum_ {s ^ {\prime} \in S \setminus \{s \}} \frac {p (s ^ {\prime})}{1 - p (s)}} - \frac {p (S \setminus \{s \})}{1 - \sum_ {s ^ {\prime} \in S \setminus \{s \}} p (s ^ {\prime})} \\ = \frac {p ^ {D \backslash \{s \}} (S \backslash \{s \})}{1 - p (s) - \sum_ {s ^ {\prime} \in S \backslash \{s \}} p \left(s ^ {\prime}\right)} - \frac {p (S \backslash \{s \})}{1 - \sum_ {s ^ {\prime} \in S \backslash \{s \}} p \left(s ^ {\prime}\right)} \\ = \frac {p ^ {D \backslash \{s \}} (S \backslash \{s \})}{1 - \sum_ {s ^ {\prime} \in S} p \left(s ^ {\prime}\right)} - \frac {p (S \backslash \{s \})}{1 - \sum_ {s ^ {\prime} \in S \backslash \{s \}} p \left(s ^ {\prime}\right)}. \\ \end{array}
+$$
+
+Multiplying by  $1 - \sum_{s' \in S} p(s')$  and rearranging terms proves Lemma 2.
+
+![](images/048769f471b1756b018b53bb81daf929a8c2775900c6dd1eda56079d9f0d1ef0.jpg)
+
+Lemma 3.
+
+$$
+p (s) + \left(1 - \sum_ {s ^ {\prime} \in S ^ {k}} p \left(s ^ {\prime}\right)\right) P \left(b _ {k} = s \mid S ^ {k}\right) = p (s) R \left(S ^ {k}, s\right) \tag {39}
+$$
+
+Proof. First using Lemma 1 and then Lemma 2 we find
+
+$$
+\begin{array}{l} p (s) + \left(1 - \sum_ {s ^ {\prime} \in S ^ {k}} p \left(s ^ {\prime}\right)\right) P \left(b _ {k} = s \mid S ^ {k}\right) \\ = p (s) + \left(1 - \sum_ {s ^ {\prime} \in S ^ {k}} p \left(s ^ {\prime}\right)\right) \frac {p \left(S ^ {k} \backslash \{s \}\right)}{p \left(S ^ {k}\right)} \frac {p (s)}{1 - \sum_ {s ^ {\prime} \in S ^ {k} \backslash \{s \}} p \left(s ^ {\prime}\right)} \\ = \frac {p (s)}{p \left(S ^ {k}\right)} \left(p \left(S ^ {k}\right) + \frac {1 - \sum_ {s ^ {\prime} \in S ^ {k}} p \left(s ^ {\prime}\right)}{1 - \sum_ {s ^ {\prime} \in S ^ {k} \setminus \{s \}} p \left(s ^ {\prime}\right)} p \left(S ^ {k} \setminus \{s \}\right)\right) \\ = \frac {p (s)}{p \left(S ^ {k}\right)} p ^ {D \backslash \{s \}} \left(S ^ {k} \backslash \{s \}\right) \\ = p (s) R \left(S ^ {k}, s\right). \\ \end{array}
+$$
+
+Now we can complete the proof of Theorem 4 by adding  $p(b_k)f(b_k) - p(b_k)f(b_k) = 0$  to the estimator, moving the terms independent of  $B^k$  outside the expectation and using Lemma 3:
+
+$$
+\begin{array}{l} \mathbb {E} _ {B ^ {k} \sim p \left(B ^ {k} \mid S ^ {k}\right)} \left[ \sum_ {j = 1} ^ {k - 1} p (b _ {j}) f (b _ {j}) + \left(1 - \sum_ {j = 1} ^ {k - 1} p (b _ {j})\right) f (b _ {k}) \right] \\ = \mathbb {E} _ {B ^ {k} \sim p \left(B ^ {k} \mid S ^ {k}\right)} \left[ \sum_ {j = 1} ^ {k} p (b _ {j}) f (b _ {j}) + \left(1 - \sum_ {j = 1} ^ {k} p (b _ {j})\right) f (b _ {k}) \right] \\ = \sum_ {s \in S ^ {k}} p (s) f (s) + \mathbb {E} _ {B ^ {k} \sim p \left(B ^ {k} \mid S ^ {k}\right)} \left[ \left(1 - \sum_ {s ^ {\prime} \in S ^ {k}} p \left(s ^ {\prime}\right)\right) f \left(b _ {k}\right) \right] \\ = \sum_ {s \in S ^ {k}} p (s) f (s) + \sum_ {s \in S ^ {k}} \left(1 - \sum_ {s ^ {\prime} \in S ^ {k}} p \left(s ^ {\prime}\right)\right) P \left(b _ {k} = s \mid S ^ {k}\right) f (s) \\ = \sum_ {s \in S ^ {k}} \left(p (s) + \left(1 - \sum_ {s ^ {\prime} \in S ^ {k}} p \left(s ^ {\prime}\right)\right) P \left(b _ {k} = s \mid S ^ {k}\right)\right) f (s) \\ = \sum_ {s \in S ^ {k}} p (s) R \left(S ^ {k}, s\right) f (s). \\ \end{array}
+$$
+
+![](images/b978c5552860b4fdc0c5e787f5a401527a20a0c0304e94ad52a2d0ad46b66f01.jpg)
+
+# C.3 THE STOCHASTIC SUM-AND-SAMPLE ESTIMATOR WITH MULTIPLE SAMPLES
+
+As was discussed in Liu et al. (2019), one can trade off the number of summed terms and number of sampled terms to maximize the achieved variance reduction. As a generalization of Theorem 4 (the stochastic sum-and-sample estimator with  $k - 1$  summed terms), we introduce here the stochastic sum-and-sample estimator that sums  $k - m$  terms and samples  $m > 1$  terms without replacement. To estimate the sampled term, we use the unordered set estimator on the  $m$  samples without replacement, on the domain restricted to  $D \setminus B^{k - m}$ . In general, we denote the unordered set estimator restricted to the domain  $D \setminus C$  by
+
+$$
+e ^ {\mathrm {U S}, D \backslash C} \left(S ^ {k}\right) = \sum_ {s \in S ^ {k} \backslash C} p (s) R ^ {D \backslash C} \left(S ^ {k}, s\right) f (s) \tag {40}
+$$
+
+where  $R^{D \setminus C}(S^k, s)$  is the leave-one-out ratio restricted to the domain  $D \setminus C$ , similar to the second order leave-one-out ratio in equation 18:
+
+$$
+R ^ {D \backslash C} \left(S ^ {k}, s\right) = \frac {p _ {\boldsymbol {\theta}} ^ {(D \backslash C) \backslash \{s \}} \left(\left(S ^ {k} \backslash C\right) \backslash \{s \}\right)}{p _ {\boldsymbol {\theta}} ^ {D \backslash C} \left(S ^ {k} \backslash C\right)}. \tag {41}
+$$
+
+While we can also constrain  $S^k \subseteq (D \setminus C)$ , this definition is consistent with equation 18 and allows simplified notation.
+
+Theorem 5. Rao-Blackwellizing the stochastic sum-and-sample estimator with  $m > 1$  samples results in the unordered set estimator, i.e.
+
+$$
+\mathbb {E} _ {B ^ {k} \sim p \left(B ^ {k} \mid S ^ {k}\right)} \left[ \sum_ {j = 1} ^ {k - m} p (b _ {j}) f (b _ {j}) + \left(1 - \sum_ {j = 1} ^ {k - m} p (b _ {j})\right) e ^ {U S, D \backslash B ^ {k - m}} \left(S ^ {k}\right) \right] = \sum_ {s \in S ^ {k}} p (s) R \left(S ^ {k}, s\right) f (s). \tag {42}
+$$
+
+Proof. Recall that for the unordered set estimator, it holds that
+
+$$
+e ^ {\mathrm {U S}} \left(S ^ {k}\right) = \mathbb {E} _ {b _ {1} \sim p \left(b _ {1} \mid S ^ {k}\right)} [ f (b _ {1}) ] = \mathbb {E} _ {x \sim p (x)} [ f (x) | x \in S ^ {k} ] \tag {43}
+$$
+
+which for the restricted equivalent (with restricted distribution  $p^{D \setminus C}$ ) translates into
+
+$$
+e ^ {\mathrm {U S}, D \backslash C} (S ^ {k}) = \mathbb {E} _ {x \sim p ^ {D \backslash C} (x)} [ f (x) | x \in S ^ {k} ] = \mathbb {E} _ {x \sim p (x)} [ f (x) | x \in S ^ {k}, x \notin C ]. \tag {44}
+$$
+
+Now we consider the distribution  $b_{k - m + 1}|S^k$ ,  $B^{k - m}$ : the distribution of the first element sampled (without replacement) after sampling  $B^{k - m}$ , given (conditionally on the event) that the set of  $k$  samples is  $S^k$ , so we have  $b_{k - m + 1} \in S^k$  and  $b_{k - m + 1} \notin B^{k - m}$ . This means that its conditional expectation of  $f(b_{k - m + 1})$  is the restricted unordered set estimator for  $C = B^{k - m}$  since
+
+$$
+\begin{array}{l} e ^ {\mathrm {U S}, D \backslash B ^ {k - m}} (S ^ {k}) = \mathbb {E} _ {x \sim p (x)} [ f (x) | x \in S ^ {k}, x \notin B ^ {k - m} ] \\ = \mathbb {E} _ {b _ {k - m + 1} \sim p \left(b _ {k - m + 1} \mid S ^ {k}, B ^ {k - m}\right)} \left[ f \left(b _ {k - m + 1}\right) \right]. \tag {45} \\ \end{array}
+$$
+
+Observing that the definition (equation 42) of the stochastic sum-and-sample estimator does not depend on the actual order of the  $m$  samples, and using equation 45, we can reduce the multisample estimator to the stochastic sum-and-sample estimator with  $k' = k - m + 1$ , such that the result follows from equation 36.
+
+$$
+\begin{array}{l} \mathbb {E} _ {B ^ {k} \sim p (B ^ {k} | S ^ {k})} \left[ \sum_ {j = 1} ^ {k - m} p (b _ {j}) f (b _ {j}) + \left(1 - \sum_ {j = 1} ^ {k - m} p (b _ {j})\right) e ^ {\mathrm {U S}, D \setminus B ^ {k - m}} (S ^ {k}) \right] \\ = \mathbb {E} _ {B ^ {k - m} \sim p (B ^ {k - m} | S ^ {k})} \left[ \sum_ {j = 1} ^ {k - m} p (b _ {j}) f (b _ {j}) + \left(1 - \sum_ {j = 1} ^ {k - m} p (b _ {j})\right) e ^ {\mathrm {U S}, D \backslash B ^ {k - m}} (S ^ {k}) \right] \\ = \mathbb {E} _ {B ^ {k - m} \sim p (B ^ {k - m} | S ^ {k})} \left[ \sum_ {j = 1} ^ {k - m} p (b _ {j}) f (b _ {j}) + \left(1 - \sum_ {j = 1} ^ {k - m} p (b _ {j})\right) \mathbb {E} _ {b _ {k - m + 1} \sim p (b _ {k - m + 1} | S ^ {k}, B ^ {k - m})} \left[ f (b _ {k - m + 1}) \right] \right] \\ = \mathbb {E} _ {B ^ {k - m + 1} \sim p (B ^ {k - m + 1} | S ^ {k})} \left[ \sum_ {j = 1} ^ {k - m} p (b _ {j}) f (b _ {j}) + \left(1 - \sum_ {j = 1} ^ {k - m} p (b _ {j})\right) f (b _ {k - m + 1}) \right] \\ = \mathbb {E} _ {S ^ {k - m + 1} \mid S ^ {k}} \left[ \mathbb {E} _ {B ^ {k - m + 1} \sim p (B ^ {k - m + 1} \mid S ^ {k - m + 1})} \left[ \sum_ {j = 1} ^ {k - m} p (b _ {j}) f (b _ {j}) + \left(1 - \sum_ {j = 1} ^ {k - m} p (b _ {j})\right) f (b _ {k - m + 1}) \right] \right] \\ = \mathbb {E} _ {S ^ {k - m + 1} \mid S ^ {k}} \left[ \sum_ {s \in S ^ {k}} p (s) R \left(S ^ {k}, s\right) f (s) \right] \\ = \sum_ {s \in S ^ {k}} p (s) R \left(S ^ {k}, s\right) f (s). \tag {46} \\ \end{array}
+$$
+
+# D THE IMPORTANCE-WEIGHTED ESTIMATOR
+
+# D.1 RAO-Blackwellization OF THE IMPORTANCE-WEIGHTED ESTIMATOR
+
+In this section we give the proof that Rao-Blackwellizing the importance-weighted estimator results in the unordered set estimator.
+
+Theorem 6. Rao-Blackwellizing the importance-weighted estimator results in the unordered set estimator, i.e.:
+
+$$
+\mathbb {E} _ {\kappa \sim p (\kappa | S ^ {k})} \left[ \sum_ {s \in S ^ {k}} \frac {p (s)}{1 - F _ {\phi_ {s}} (\kappa)} f (s) \right] = \sum_ {s \in S ^ {k}} p (s) R \left(S ^ {k}, s\right) f (s). \tag {47}
+$$
+
+Here we have slightly rewritten the definition of the importance-weighted estimator, using that  $q(s,a) = P(g_{\phi_s} > a) = 1 - F_{\phi_s}(a)$ , where  $F_{\phi_s}$  is the CDF of the Gumbel distribution (see Appendix A).
+
+Proof. We first prove the following Lemma:
+
+# Lemma 4.
+
+$$
+\mathbb {E} _ {\kappa \sim p (\kappa \mid S ^ {k})} \left[ \frac {1}{1 - F _ {\phi_ {s}} (\kappa)} \right] = R \left(S ^ {k}, s\right) \tag {48}
+$$
+
+Proof. Conditioning on  $S^k$ , we know that the elements in  $S^k$  have the  $k$  largest perturbed log-probabilities, so  $\kappa$ , the  $(k + 1)$ -th largest perturbed log-probability is the largest perturbed log-probability in  $D \setminus S^k$ , and satisfies  $\kappa = \max_{s \in D \setminus S^k} g_{\phi_s} = g_{\phi_{D \setminus S^k}} \sim \mathrm{Gumbel}(\phi_{D \setminus S^k})$ . Computing  $p(\kappa | S^k)$  using Bayes' Theorem, we have
+
+$$
+p (\kappa | S ^ {k}) = \frac {p \left(S ^ {k} \mid \kappa\right) p (\kappa)}{p \left(S ^ {k}\right)} = \frac {\prod_ {s \in S ^ {k}} \left(1 - F _ {\phi_ {s}} (\kappa)\right) f _ {\phi_ {D \backslash S ^ {k}}} (\kappa)}{p \left(S ^ {k}\right)} \tag {49}
+$$
+
+which allows us to compute (using equation 34 with  $C = \{s\}$  and  $g_{\phi_D\backslash S} = \kappa$ )
+
+$$
+\begin{array}{l} \mathbb {E} _ {\kappa \sim p (\kappa | S ^ {k})} \left[ \frac {1}{1 - F _ {\phi_ {s}} (\kappa)} \right] \\ = \int_ {\kappa = - \infty} ^ {\infty} p (\kappa | S ^ {k}) \frac {1}{1 - F _ {\phi_ {s}} (\kappa)} \partial \kappa \\ = \int_ {\kappa = - \infty} ^ {\infty} \frac {\prod_ {s \in S ^ {k}} (1 - F _ {\phi_ {s}} (\kappa)) f _ {\phi_ {D \setminus S ^ {k}}} (\kappa)}{p (S ^ {k})} \frac {1}{1 - F _ {\phi_ {s}} (\kappa)} \partial \kappa \\ = \frac {1}{p (S ^ {k})} \int_ {\kappa = - \infty} ^ {\infty} \prod_ {s \in S ^ {k} \backslash \{s \}} \left(1 - F _ {\phi_ {s}} (\kappa)\right) f _ {\phi_ {D \backslash S ^ {k}}} (\kappa) \partial \kappa \\ = \frac {1}{p (S ^ {k})} p ^ {D \backslash \{s \}} (S \backslash \{s \}) \\ = R \left(S ^ {k}, s\right). \\ \end{array}
+$$
+
+![](images/abf8f51fb1ddfb5ccabfa600e2ac8cb61196e32008b4c229c1552cfd42b09ed9.jpg)
+
+Using Lemma 4 we find
+
+$$
+\begin{array}{l} \mathbb {E} _ {\kappa \sim p (\kappa | S ^ {k})} \left[ \sum_ {s \in S ^ {k}} \frac {p (s)}{1 - F _ {\phi_ {s}} (\kappa)} f (s) \right] \\ = \sum_ {s \in S ^ {k}} p (s) \mathbb {E} _ {\kappa \sim p (\kappa | S ^ {k})} \left[ \frac {1}{1 - F _ {\phi_ {s}} (\kappa)} \right] f (s) \\ = \sum_ {s \in S ^ {k}} p (s) R (S ^ {k}, s) f (s). \\ \end{array}
+$$
+
+![](images/23a9d1f0c25b76e1007ed61f2eb571ac9e56fb27c247de525697da1e1f41a5ec.jpg)
+
+# D.2 THE IMPORTANCE-WEIGHTED POLICY GRADIENT ESTIMATOR WITH BUILT-IN BASELINE
+
+For self-containment we include this section, which is adapted from our unpublished workshop paper (Kool et al., 2019b). The importance-weighted policy gradient estimator combines REINFORCE (Williams, 1992) with the importance-weighted estimator (Duffield et al., 2007; Vieira, 2017) in equation 15 which results in an unbiased estimator of the policy gradient  $\nabla_{\theta}\mathbb{E}_{p_{\theta}(x)}[f_{\theta}(x)]$ :
+
+$$
+e ^ {\mathrm {I W P G}} \left(S ^ {k}, \kappa\right) = \sum_ {s \in S ^ {k}} \frac {p _ {\boldsymbol {\theta}} (s)}{q _ {\boldsymbol {\theta} , \kappa} (s)} \nabla_ {\boldsymbol {\theta}} \log p _ {\boldsymbol {\theta}} (s) f (s) = \sum_ {s \in S ^ {k}} \frac {\nabla_ {\boldsymbol {\theta}} p _ {\boldsymbol {\theta}} (s)}{q _ {\boldsymbol {\theta} , \kappa} (s)} f (s) \tag {50}
+$$
+
+Recall that  $\kappa$  is the  $(k + 1)$ -th largest perturbed log-probability (see Section 3.2). We compute a lower variance but biased variant by normalizing the importance weights using the normalization  $W(S^{k}) = \sum_{s\in S^{k}}\frac{p_{\theta}(s)}{q_{\theta,\kappa}(s)}$ .
+
+As we show in Kool et al. (2019b), we can include a 'baseline'  $B(S^k) = \sum_{s \in S^k} \frac{p_\theta(s)}{q_{\theta, \kappa}(s)} f(s)$  and correct for the bias (since it depends on the complete sample  $S^k$ ) by weighting individual terms of the estimator by  $1 - p_\theta(s) + \frac{p_\theta(s)}{q_{\theta, \kappa}(s)}$ :
+
+$$
+e ^ {\mathrm {I W P G B L}} \left(S ^ {k}, \kappa\right) = \sum_ {s \in S ^ {k}} \frac {\nabla_ {\boldsymbol {\theta}} p _ {\boldsymbol {\theta}} (s)}{q _ {\boldsymbol {\theta} , \kappa} (s)} \left(f (s) \left(1 - p _ {\boldsymbol {\theta}} (s) + \frac {p _ {\boldsymbol {\theta}} (s)}{q _ {\boldsymbol {\theta} , \kappa} (s)}\right) - B \left(S ^ {k}\right)\right) \tag {51}
+$$
+
+For the normalized version, we use the normalization  $W(S^k) = \sum_{s\in S^k}\frac{p_\theta(s)}{q_{\theta,\kappa}(s)}$  for the baseline, and  $W_{i}(S^{k}) = W(S^{k}) - \frac{p_{\theta}(s)}{q_{\theta,\kappa}(s)} +p_{\theta}(s)$  to normalize the individual terms:
+
+$$
+\nabla_ {\boldsymbol {\theta}} \mathbb {E} _ {y \sim p _ {\boldsymbol {\theta}} (y)} [ f (y) ] \approx \sum_ {s \in S ^ {k}} \frac {1}{W _ {i} \left(S ^ {k}\right)} \cdot \frac {\nabla_ {\boldsymbol {\theta}} p _ {\boldsymbol {\theta}} (s)}{q _ {\boldsymbol {\theta} , \kappa} (s)} \left(f (s) - \frac {B \left(S ^ {k}\right)}{W \left(S ^ {k}\right)}\right) \tag {52}
+$$
+
+It seems odd to normalize the terms in the outer sum by  $\frac{1}{W_i(S^k)}$  instead of  $\frac{1}{W(S^k)}$ , but equation 52 can be rewritten into a form similar to equation 17, i.e. with a different baseline for each sample, but this form is more convenient for implementation (Kool et al., 2019b).
+
+# E THE UNORDERED SET POLICY GRADIENT ESTIMATOR
+
+# E.1 PROOF OF UNBIASEDNESS OF THE UNORDERED SET POLICY GRADIENT ESTIMATOR WITH BASELINE
+
+To prove the unbiasedness of result we need to prove that the control variate has expectation 0:
+
+Lemma 5.
+
+$$
+\mathbb {E} _ {S ^ {k} \sim p _ {\boldsymbol {\theta}} \left(S ^ {k}\right)} \left[ \sum_ {s \in S ^ {k}} \nabla_ {\boldsymbol {\theta}} p _ {\boldsymbol {\theta}} (s) R \left(S ^ {k}, s\right) \sum_ {s ^ {\prime} \in S ^ {k}} p _ {\boldsymbol {\theta}} \left(s ^ {\prime}\right) R ^ {D \backslash \{s \}} \left(S ^ {k}, s ^ {\prime}\right) f \left(s ^ {\prime}\right) \right] = 0. \tag {53}
+$$
+
+Proof. Similar to equation 10, we apply Bayes' Theorem conditionally on  $b_{1} = s$  to derive for  $s' \neq s$
+
+$$
+\begin{array}{l} P (b _ {2} = s ^ {\prime} | S ^ {k}, b _ {1} = s) = \frac {P (S ^ {k} | b _ {2} = s ^ {\prime} , b _ {1} = s) P (b _ {2} = s ^ {\prime} | b _ {1} = s ^ {\prime})}{P (S ^ {k} | b _ {1} = s)} \\ = \frac {p _ {\boldsymbol {\theta}} ^ {D \setminus \{s , s ^ {\prime} \}} (S ^ {k} \setminus \{s , s ^ {\prime} \}) p _ {\boldsymbol {\theta}} ^ {D \setminus \{s \}} (s ^ {\prime})}{p _ {\boldsymbol {\theta}} ^ {D \setminus \{s \}} (S ^ {k} \setminus \{s \})} \\ = \frac {p _ {\boldsymbol {\theta}} \left(s ^ {\prime}\right)}{1 - p _ {\boldsymbol {\theta}} (s)} R ^ {D \backslash \{s \}} \left(S ^ {k}, s ^ {\prime}\right). \tag {54} \\ \end{array}
+$$
+
+For  $s' = s$  we have  $R^{D \setminus \{s\}}(S^k, s') = 1$  by definition, so using equation 54 we can show that
+
+$$
+\begin{array}{l} \sum_ {s ^ {\prime} \in S ^ {k}} p _ {\boldsymbol {\theta}} (s ^ {\prime}) R ^ {D \backslash \{s \}} (S ^ {k}, s ^ {\prime}) f (s ^ {\prime}) \\ = p _ {\boldsymbol {\theta}} (s) f (s) + \sum_ {s ^ {\prime} \in S ^ {k} \backslash \{s \}} p _ {\boldsymbol {\theta}} (s ^ {\prime}) R ^ {D \backslash \{s \}} \left(S ^ {k}, s ^ {\prime}\right) f (s ^ {\prime}) \\ = p _ {\boldsymbol {\theta}} (s) f (s) + \left(1 - p _ {\boldsymbol {\theta}} (s)\right) \sum_ {s ^ {\prime} \in S ^ {k} \backslash \{s \}} \frac {p _ {\boldsymbol {\theta}} \left(s ^ {\prime}\right)}{1 - p _ {\boldsymbol {\theta}} (s)} R ^ {D \backslash \{s \}} \left(S ^ {k}, s ^ {\prime}\right) f \left(s ^ {\prime}\right) \\ = p _ {\boldsymbol {\theta}} (s) f (s) + \left(1 - p _ {\boldsymbol {\theta}} (s)\right) \sum_ {s ^ {\prime} \in S ^ {k} \backslash \{s \}} P (b _ {2} = s ^ {\prime} | S ^ {k}, b _ {1} = s) f (s ^ {\prime}) \\ = p _ {\boldsymbol {\theta}} (s) f (s) + \left(1 - p _ {\boldsymbol {\theta}} (s)\right) \mathbb {E} _ {b _ {2} \sim p _ {\boldsymbol {\theta}} \left(b _ {2} \mid S ^ {k}, b _ {1} = s\right)} [ f (b _ {2}) ] \\ = \mathbb {E} _ {b _ {2} \sim p _ {\boldsymbol {\theta}} \left(b _ {2} \mid S ^ {k}, b _ {1} = s\right)} \left[ p _ {\boldsymbol {\theta}} \left(b _ {1}\right) f \left(b _ {1}\right) + \left(1 - p _ {\boldsymbol {\theta}} \left(b _ {1}\right)\right) f \left(b _ {2}\right) \right]. \\ \end{array}
+$$
+
+Now we can show that the control variate is actually the result of Rao-Blackwellization:
+
+$$
+\begin{array}{l} \mathbb {E} _ {S ^ {k} \sim p _ {\boldsymbol {\theta}} (S ^ {k})} \left[ \sum_ {s \in S ^ {k}} \nabla_ {\boldsymbol {\theta}} p _ {\boldsymbol {\theta}} (s) R (S ^ {k}, s) \sum_ {s ^ {\prime} \in S ^ {k}} p _ {\boldsymbol {\theta}} (s ^ {\prime}) R ^ {D \backslash \{s \}} (S ^ {k}, s ^ {\prime}) f (s ^ {\prime}) \right] \\ = \mathbb {E} _ {S ^ {k} \sim p _ {\boldsymbol {\theta}} (S ^ {k})} \left[ \sum_ {s \in S ^ {k}} p _ {\boldsymbol {\theta}} (s) R (S ^ {k}, s) \nabla_ {\boldsymbol {\theta}} \log p _ {\boldsymbol {\theta}} (s) \sum_ {s ^ {\prime} \in S ^ {k}} p _ {\boldsymbol {\theta}} \left(s ^ {\prime}\right) R ^ {D \backslash \{s \}} \left(S ^ {k}, s ^ {\prime}\right) f \left(s ^ {\prime}\right) \right] \\ = \mathbb {E} _ {S ^ {k} \sim p _ {\boldsymbol {\theta}} (S ^ {k})} \left[ \sum_ {s \in S ^ {k}} P \left(b _ {1} = s \mid S ^ {k}\right) \nabla_ {\boldsymbol {\theta}} \log p _ {\boldsymbol {\theta}} (s) \sum_ {s ^ {\prime} \in S ^ {k}} p _ {\boldsymbol {\theta}} \left(s ^ {\prime}\right) R ^ {D \backslash \{s \}} \left(S ^ {k}, s ^ {\prime}\right) f \left(s ^ {\prime}\right) \right] \\ = \mathbb {E} _ {S ^ {k} \sim p _ {\boldsymbol {\theta}} (S ^ {k})} \left[ \mathbb {E} _ {b _ {1} \sim p _ {\boldsymbol {\theta}} (b _ {1} | S ^ {k})} \left[ \nabla_ {\boldsymbol {\theta}} \log p _ {\boldsymbol {\theta}} (b _ {1}) \sum_ {s ^ {\prime} \in S ^ {k}} p _ {\boldsymbol {\theta}} \left(s ^ {\prime}\right) R ^ {D \backslash \{b _ {1} \}} \left(S ^ {k}, s ^ {\prime}\right) f \left(s ^ {\prime}\right) \right] \right] \\ = \mathbb {E} _ {S ^ {k} \sim p _ {\boldsymbol {\theta}} (S ^ {k})} \left[ \mathbb {E} _ {b _ {1} \sim p _ {\boldsymbol {\theta}} (b _ {1} | S ^ {k})} \left[ \nabla_ {\boldsymbol {\theta}} \log p _ {\boldsymbol {\theta}} (b _ {1}) \mathbb {E} _ {b _ {2} \sim p _ {\boldsymbol {\theta}} (b _ {2} | S ^ {k}, b _ {1})} \left[ p _ {\boldsymbol {\theta}} (b _ {1}) f (b _ {1}) + (1 - p _ {\boldsymbol {\theta}} (b _ {1})) f (b _ {2}) \right] \right] \right] \\ = \mathbb {E} _ {S ^ {k} \sim p _ {\boldsymbol {\theta}} (S ^ {k})} \left[ \mathbb {E} _ {B ^ {k} \sim p _ {\boldsymbol {\theta}} (B ^ {k} | S ^ {k})} \left[ \nabla_ {\boldsymbol {\theta}} \log p _ {\boldsymbol {\theta}} (b _ {1}) \left(p _ {\boldsymbol {\theta}} (b _ {1}) f (b _ {1}) + (1 - p _ {\boldsymbol {\theta}} (b _ {1})) f (b _ {2})\right) \right] \right] \\ = \mathbb {E} _ {B ^ {k} \sim p _ {\boldsymbol {\theta}} \left(B ^ {k}\right)} \left[ \nabla_ {\boldsymbol {\theta}} \log p _ {\boldsymbol {\theta}} \left(b _ {1}\right) \left(p _ {\boldsymbol {\theta}} \left(b _ {1}\right) f \left(b _ {1}\right) + \left(1 - p _ {\boldsymbol {\theta}} \left(b _ {1}\right)\right) f \left(b _ {2}\right)\right) \right] \\ \end{array}
+$$
+
+This expression depends only on  $b_{1}$  and  $b_{2}$  and we recognize the stochastic sum-and-sample estimator for  $k = 2$  used as 'baseline'. As a special case of equation 13 for  $C = \{b_{1}\}$ , we have
+
+$$
+\mathbb {E} _ {b _ {2} \sim p _ {\boldsymbol {\theta}} (b _ {2} | b _ {1})} \left[ \left(p _ {\boldsymbol {\theta}} (b _ {1}) f (b _ {1}) + \left(1 - p _ {\boldsymbol {\theta}} (b _ {1})\right) f (b _ {2})\right) \right] = \mathbb {E} _ {i \sim p _ {\boldsymbol {\theta}} (i)} [ f (i) ]. \tag {55}
+$$
+
+Using this, and the fact that  $\mathbb{E}_{b_1\sim p_\theta (b_1)}[\nabla_\theta \log p_\theta (b_1)] = \nabla_\theta \mathbb{E}_{b_1\sim p_\theta (b_1)}[1] = \nabla_\theta 1 = 0$  we find
+
+$$
+\begin{array}{l} \mathbb {E} _ {S ^ {k} \sim p _ {\boldsymbol {\theta}} (S ^ {k})} \left[ \sum_ {s \in S ^ {k}} \nabla_ {\boldsymbol {\theta}} p _ {\boldsymbol {\theta}} (s) R (S ^ {k}, s) \sum_ {s ^ {\prime} \in S ^ {k}} p _ {\boldsymbol {\theta}} \left(s ^ {\prime}\right) R ^ {D \backslash \{s \}} \left(S ^ {k}, s ^ {\prime}\right) f \left(s ^ {\prime}\right) \right] \\ = \mathbb {E} _ {B ^ {k} \sim p _ {\boldsymbol {\theta}} \left(B ^ {k}\right)} \left[ \nabla_ {\boldsymbol {\theta}} \log p _ {\boldsymbol {\theta}} \left(b _ {1}\right) \left(p _ {\boldsymbol {\theta}} \left(b _ {1}\right) f \left(b _ {1}\right) + \left(1 - p _ {\boldsymbol {\theta}} \left(b _ {1}\right)\right) f \left(b _ {2}\right)\right) \right] \\ = \mathbb {E} _ {b _ {1} \sim p _ {\boldsymbol {\theta}} (b _ {1})} \left[ \nabla_ {\boldsymbol {\theta}} \log p _ {\boldsymbol {\theta}} (b _ {1}) \mathbb {E} _ {b _ {2} \sim p _ {\boldsymbol {\theta}} (b _ {2} | b _ {1})} \left[ \left(p _ {\boldsymbol {\theta}} (b _ {1}) f (b _ {1}) + \left(1 - p _ {\boldsymbol {\theta}} (b _ {1})\right) f (b _ {2})\right) \right] \right] \\ = \mathbb {E} _ {b _ {1} \sim p _ {\theta} (b _ {1})} \left[ \nabla_ {\theta} \log p _ {\theta} (b _ {1}) \mathbb {E} _ {x \sim p _ {\theta} (x)} [ f (x) ] \right] \\ = \mathbb {E} _ {b _ {1} \sim p _ {\boldsymbol {\theta}} (b _ {1})} \left[ \nabla_ {\boldsymbol {\theta}} \log p _ {\boldsymbol {\theta}} (b _ {1}) \right] \mathbb {E} _ {x \sim p _ {\boldsymbol {\theta}} (x)} [ f (x) ] \\ = 0 \cdot \mathbb {E} _ {x \sim p _ {\boldsymbol {\theta}} (x)} [ f (x) ] \\ = 0 \\ \end{array}
+$$
+
+# F THE RISK ESTIMATOR
+
+# F.1 PROOF OF BUILT-IN BASELINE
+
+We show that the RISK estimator, taking gradients through the normalization factor actually has a built-in baseline. We first use the log-derivative trick to rewrite the gradient of the ratio as the ratio times the logarithm of the gradient, and then swap the summation variables in the double sum that arises:
+
+$$
+\begin{array}{l} e ^ {\mathrm {R I S K}} (S) = \sum_ {s \in S} \nabla_ {\pmb {\theta}} \left(\frac {p _ {\pmb {\theta}} (s)}{\sum_ {s ^ {\prime} \in S} p _ {\pmb {\theta}} (s ^ {\prime})}\right) f (s) \\ = \sum_ {s \in S} \frac {p _ {\pmb {\theta}} (s)}{\sum_ {s ^ {\prime} \in S} p _ {\pmb {\theta}} (s ^ {\prime})} \nabla_ {\pmb {\theta}} \log \left(\frac {p _ {\pmb {\theta}} (s)}{\sum_ {s ^ {\prime} \in S} p _ {\pmb {\theta}} (s ^ {\prime})}\right) f (s) \\ = \sum_ {s \in S} \frac {p _ {\boldsymbol {\theta}} (s)}{\sum_ {s ^ {\prime} \in S} p _ {\boldsymbol {\theta}} \left(s ^ {\prime}\right)} \left(\nabla_ {\boldsymbol {\theta}} \log p _ {\boldsymbol {\theta}} (s) - \nabla_ {\boldsymbol {\theta}} \log \sum_ {s ^ {\prime} \in S} p _ {\boldsymbol {\theta}} \left(s ^ {\prime}\right)\right) f (s) \\ = \sum_ {s \in S} \frac {p _ {\boldsymbol {\theta}} (s)}{\sum_ {s ^ {\prime} \in S} p _ {\boldsymbol {\theta}} (s ^ {\prime})} \left(\frac {\nabla_ {\boldsymbol {\theta}} p _ {\boldsymbol {\theta}} (s)}{p _ {\boldsymbol {\theta}} (s)} - \frac {\sum_ {s ^ {\prime} \in S} \nabla_ {\boldsymbol {\theta}} p _ {\boldsymbol {\theta}} (s ^ {\prime})}{\sum_ {s ^ {\prime} \in S} p _ {\boldsymbol {\theta}} (s ^ {\prime})}\right) f (s) \\ = \sum_ {s \in S} \frac {\nabla_ {\boldsymbol {\theta}} p _ {\boldsymbol {\theta}} (s) f (s)}{\sum_ {s ^ {\prime} \in S} p _ {\boldsymbol {\theta}} \left(s ^ {\prime}\right)} - \frac {\sum_ {s , s ^ {\prime} \in S} p _ {\boldsymbol {\theta}} (s) \nabla_ {\boldsymbol {\theta}} p _ {\boldsymbol {\theta}} \left(s ^ {\prime}\right) f (s)}{\left(\sum_ {s ^ {\prime} \in S} p _ {\boldsymbol {\theta}} \left(s ^ {\prime}\right)\right) ^ {2}} \\ = \sum_ {s \in S} \frac {\nabla_ {\boldsymbol {\theta}} p _ {\boldsymbol {\theta}} (s) f (s)}{\sum_ {s ^ {\prime} \in S} p _ {\boldsymbol {\theta}} (s ^ {\prime})} - \frac {\sum_ {s , s ^ {\prime} \in S} p _ {\boldsymbol {\theta}} (s ^ {\prime}) \nabla_ {\boldsymbol {\theta}} p _ {\boldsymbol {\theta}} (s) f (s ^ {\prime})}{\left(\sum_ {s ^ {\prime} \in S} p _ {\boldsymbol {\theta}} (s ^ {\prime})\right) ^ {2}} \\ = \sum_ {s \in S} \frac {\nabla_ {\boldsymbol {\theta}} p _ {\boldsymbol {\theta}} (s)}{\sum_ {s ^ {\prime} \in S} p _ {\boldsymbol {\theta}} \left(s ^ {\prime}\right)} \left(f (s) - \frac {\sum_ {s ^ {\prime} \in S} p _ {\boldsymbol {\theta}} \left(s ^ {\prime}\right) f \left(s ^ {\prime}\right)}{\sum_ {s ^ {\prime} \in S} p _ {\boldsymbol {\theta}} \left(s ^ {\prime}\right)}\right) \\ = \sum_ {s \in S} \frac {\nabla_ {\pmb {\theta}} p _ {\pmb {\theta}} (s)}{\sum_ {s ^ {\prime \prime} \in S} p _ {\pmb {\theta}} (s ^ {\prime \prime})} \left(f (s) - \sum_ {s ^ {\prime} \in S} \frac {p _ {\pmb {\theta}} (s ^ {\prime})}{\sum_ {s ^ {\prime \prime} \in S} p _ {\pmb {\theta}} (s ^ {\prime \prime})} f (s ^ {\prime})\right). \\ \end{array}
+$$
+
+# G CATEGORICAL VARIATIONAL AUTO-ENCODER
+
+# G.1 EXPERIMENTAL DETAILS
+
+We use the code<sup>6</sup> by Yin et al. (2019) to reproduce their categorical VAE experiment, of which we include details here for self-containment. The dataset is MNIST, statically binarized by thresholding at 0.5 (although we include results using the standard binarized dataset by Salakhutdinov & Murray (2008); Larochelle & Murray (2011) in Section G.2). The latent representation  $z$  is  $K = 20$  dimensional with  $C = 10$  categories per dimension with a uniform prior  $p(z_k = c) = 1 / C, k = 1, \dots, K$ . The encoder is parameterized by  $\phi$  as  $q_{\phi}(z|x) = \prod_k q_{\phi}(z_k|x)$  and has two fully connected hidden layers with 512 and 256 hidden nodes respectively, with LeakyReLU ( $\alpha = 0.1$ ) activations. The decoder, parameterized by  $\theta$ , is given by  $p_{\theta}(x|z) = \prod_i p_{\theta}(x_i|z)$ , where  $x_i \in \{0,1\}$  are the pixel values, and has fully connected hidden layers with 256 and 512 nodes and LeakyReLU activation.
+
+ELBO optimization. The evidence lower bound (ELBO) that we optimize is given by
+
+$$
+\begin{array}{l} \mathcal {L} (\phi , \boldsymbol {\theta}) = \mathbb {E} _ {\boldsymbol {z} \sim q _ {\phi} (\boldsymbol {z} | \boldsymbol {x})} \left[ \ln p _ {\boldsymbol {\theta}} (\boldsymbol {x} | \boldsymbol {z}) + \ln p (\boldsymbol {z}) - \ln q _ {\phi} (\boldsymbol {z} | \boldsymbol {x}) \right] (56) \\ = \mathbb {E} _ {\boldsymbol {z} \sim q _ {\phi} (\boldsymbol {z} | \boldsymbol {x})} \left[ \ln p _ {\boldsymbol {\theta}} (\boldsymbol {x} | \boldsymbol {z}) \right] - K L \left(q _ {\phi} (\boldsymbol {z} | \boldsymbol {x}) | | p (\boldsymbol {z})\right). (57) \\ \end{array}
+$$
+
+For the decoder parameters  $\theta$ , since  $q_{\phi}(z|x)$  does not depend on  $\theta$ , it follows that
+
+$$
+\nabla_ {\boldsymbol {\theta}} \mathcal {L} (\boldsymbol {\phi}, \boldsymbol {\theta}) = \mathbb {E} _ {\boldsymbol {z} \sim q _ {\boldsymbol {\phi}} (\boldsymbol {z} | \boldsymbol {x})} \left[ \nabla_ {\boldsymbol {\theta}} \ln p _ {\boldsymbol {\theta}} (\boldsymbol {x} | \boldsymbol {z}) \right]. \tag {58}
+$$
+
+For the encoder parameters  $\phi$ , we can write  $\nabla_{\phi}\mathcal{L}(\phi,\theta)$  using equation 57 and equation 19 as
+
+$$
+\nabla_ {\phi} \mathcal {L} (\phi , \boldsymbol {\theta}) = \mathbb {E} _ {\boldsymbol {z} \sim q _ {\phi} (\boldsymbol {z} | \boldsymbol {x})} \left[ \nabla_ {\phi} \ln q _ {\phi} (\boldsymbol {z} | \boldsymbol {x}) \ln p _ {\boldsymbol {\theta}} (\boldsymbol {x} | \boldsymbol {z}) \right] - \nabla_ {\phi} K L \left(q _ {\phi} (\boldsymbol {z} | \boldsymbol {x}) \| p (\boldsymbol {z})\right). \tag {59}
+$$
+
+This assumes we can compute the KL divergence analytically. Alternatively, we can use a sample estimate for the KL divergence, and use equation 56 with equation 19 to obtain
+
+$$
+\begin{array}{l} \nabla_ {\phi} \mathcal {L} (\phi , \boldsymbol {\theta}) = \mathbb {E} _ {\boldsymbol {z} \sim q _ {\phi} (\boldsymbol {z} | \boldsymbol {x})} \left[ \nabla_ {\phi} \ln q _ {\phi} (\boldsymbol {z} | \boldsymbol {x}) \left(\ln p _ {\boldsymbol {\theta}} (\boldsymbol {x} | \boldsymbol {z}) + \ln p (\boldsymbol {z}) - \ln q _ {\phi} (\boldsymbol {z} | \boldsymbol {x})\right) + \nabla_ {\phi} \ln q _ {\phi} (\boldsymbol {z} | \boldsymbol {x}) \right] (60) \\ = \mathbb {E} _ {\boldsymbol {z} \sim q _ {\phi} (\boldsymbol {z} | \boldsymbol {x})} \left[ \nabla_ {\phi} \ln q _ {\phi} (\boldsymbol {z} | \boldsymbol {x}) (\ln p _ {\boldsymbol {\theta}} (\boldsymbol {x} | \boldsymbol {z}) - \ln q _ {\phi} (\boldsymbol {z} | \boldsymbol {x})) \right]. (61) \\ \end{array}
+$$
+
+Here we have left out the term  $\mathbb{E}_{\boldsymbol{z} \sim q_{\phi}(\boldsymbol{z}|\boldsymbol{x})}[\nabla_{\phi}\ln q_{\phi}(\boldsymbol{z}|\boldsymbol{x})] = 0$ , similar to Roeder et al. (2017), and, assuming a uniform (i.e. constant) prior  $\ln p(\boldsymbol{z})$ , the term  $\mathbb{E}_{\boldsymbol{z} \sim q_{\phi}(\boldsymbol{z}|\boldsymbol{x})}[\nabla_{\phi}\ln q_{\phi}(\boldsymbol{z}|\boldsymbol{x})\ln p(\boldsymbol{z})] = 0$ . With a built-in baseline, this second term cancels out automatically, even if it is implemented. Despite the similarity of the equation 56 and equation 57, their gradient estimates (equation 60 and equation 59) are structurally dissimilar and care should be taken to implement the REINFORCE estimator (or related estimators such as ARSM and the unordered set estimator) correctly using automatic differentiation software. Using Gumbel-Softmax and RELAX, we take gradients 'directly' through the objective in equation 57.
+
+We optimize the ELBO using the analytic KL for 1000 epochs using the Adam (Kingma & Ba, 2015) optimizer. We use a learning rate of  $10^{-3}$  for all estimators except Gumbel-Softmax and RELAX, which use a learning rate of  $10^{-4}$  as we found they diverged with a higher learning rate. For ARSM, as an exception we use the sample KL, and a learning rate of  $3 \cdot 10^{-4}$ , as suggested by the authors. All reported ELBO values are computed using the analytic KL. Our code is publicly available<sup>7</sup>.
+
+# G.2 ADDITIONAL RESULTS
+
+Gradient variance during training. We also evaluate gradient variance of different estimators during different stages of training. We measure the variance of different estimators with  $k = 4$  samples during training with REINFORCE with replacement, such that all estimators are computed for the same model parameters. The results during training, given in Figure 4, are similar to the results for the trained model in Table 1, except for at the beginning of training, although the rankings of different estimator are mostly the same.
+
+Negative ELBO on validation set. Figure 5 shows the -ELBO evaluated during training on the validation set. For the large latent space, we see validation error quickly increase (after reaching a minimum) which is likely because of overfitting (due to improved optimization), a phenomenon observed before (Tucker et al., 2017; Grathwohl et al., 2018). Note that before the overfitting starts, both REINFORCE without replacement and the unordered set estimator achieve a validation error similar to the other estimators, such that in a practical setting, one can use early stopping.
+
+Results using standard binarized MNIST dataset. Instead of using the MNIST dataset binarized by thresholding values at 0.5 (as in the code and paper by Yin et al. (2019)) we also experiment with the standard (fixed) binarized dataset by Salakhutdinov & Murray (2008); Larochelle & Murray (2011), for which we plot train and validation curves for two runs on the small and large domain in Figure 6. This gives more realistic (higher) -ELBO scores, although we still observe the effect of overfitting. As this is a bit more unstable setting, one of the runs using REINFORCE with replacement diverged, but in general the relative performance of estimators is similar to using the dataset with 0.5 threshold.
+
+![](images/8ceb9602a57b820e6b501c99b444db884425873ad3e17774779cc9e4689432ed.jpg)  
+(a) Small domain (latent space size  $10^{2}$ )
+
+![](images/0231a9ff6cc290ed7727571e846d3c7d08da5bd2cd09ee043a6a050f364c0ecb.jpg)  
+(b) Large domain (latent space size  $10^{20}$ )  
+Figure 4: Gradient log variance of different unbiased estimators with  $k = 4$  samples, estimated every 100 (out of 1000) epochs while training using REINFORCE with replacement. Each estimator is computed 1000 times with different latent samples for a fixed minibatch (the first 100 records of training data). We report (the logarithm of) the sum of the variances per parameter (trace of the covariance matrix). Some lines coincide, so we sort the legend by the last measurement and report its value.
+
+![](images/34857ae82e41c89fa687b48ba5f32b4c530cf6cc6d62b579eb90c7d639b55ef0.jpg)  
+(a) Small domain (latent space size  $10^{2}$ )  
+Figure 5: Smoothed validation -ELBO curves during training of two independent runs when with different estimators with  $k = 1, 4$  or 8 (thicker lines) samples (ARSM has a variable number). Some lines coincide, so we sort the legend by the lowest -ELBO achieved and report this value.
+
+![](images/d16a5bd2acae7bc8bb92e1c25155f5ae420445e07d8ef57d60d45c764ccfb70e.jpg)  
+(b) Large domain (latent space size  $10^{20}$ )
+
+![](images/643660664aec70b7b9ead5960896877a4cc6dac10ce9fa14f10429e2d371c805.jpg)  
+(a) Training -ELBO, small domain  $(10^{2})$
+
+![](images/e62b8e4e7f1a2a5e54c51cdbac553cb932773993f503f0e64ba5f3e594a66173.jpg)  
+(b) Training -ELBO, large domain  $(10^{20})$
+
+![](images/715d6cbd85035a2cacb03a5e4c0bfdeb9ad9fc4ab9312876f3c7edbc05083855.jpg)  
+(c) Validation -ELBO, small domain  $(10^{2})$
+
+![](images/4009c4b03fa5622acdcf570819d026e964f814eeff129288a39231a9ae598ce4.jpg)  
+(d) Validation -ELBO, large domain  $(10^{20})$  
+Figure 6: Smoothed training and validation -ELBO curves during training on the standard binarized MNIST dataset (Salakhutdinov & Murray, 2008; Larochelle & Murray, 2011) of two independent runs when with different estimators with  $k = 1, 4$  or 8 (thicker lines) samples (ARSM has a variable number). Some lines coincide, so we sort the legend by the lowest -ELBO achieved and report this value.
+
+# H TRAVELLING SALESMAN PROBLEM
+
+The Travelling Salesman Problem (TSP) is a discrete optimization problem that consists of finding the order in which to visit a set of locations, given as  $x, y$  coordinates, to minimize the total length of the tour, starting and ending at the same location. As a tour can be considered a sequence of locations, this problem can be set up as a sequence modelling problem, that can be either addressed using supervised (Vinyals et al., 2015) or reinforcement learning (Bello et al., 2016; Kool et al., 2019a).
+
+Kool et al. (2019a) introduced the Attention Model, which is an encoder-decoder model which considers a TSP instances as a fully connected graph. The encoder computes embeddings for all nodes (locations) and the decoder produces a tour, which is sequence of nodes, selecting one note at the time using an attention mechanism, and uses this autoregressively as input to select the next node. In Kool et al. (2019a), this model is trained using REINFORCE, with a greedy rollout used as baseline to reduce variance.
+
+We use the code by Kool et al. (2019a) to train the exact same Attention Model (for details we refer to Kool et al. (2019a)), and minimize the expected length of a tour predicted by the model, using different gradient estimators. We did not do any hyperparameter optimization and used the exact same training details, using the Adam optimizer (Kingma & Ba, 2015) with a learning rate of  $10^{-4}$  (no decay) for 100 epochs for all estimators. For the baselines, we used the same batch size of 512, but for estimators that use  $k = 4$  samples, we used a batch size of  $\frac{512}{4} = 128$  to compensate for the additional samples (this makes multi-sample methods actually faster since the encoder still needs to be evaluated only once).
\ No newline at end of file
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+# EXPLANATION BY PROGRESSIVE EXAGGERATION
+
+Sumedha Singla
+
+Department of Computer Science
+
+University of Pittsburgh
+
+Brian Pollack, Junxiang Chen
+
+Department of Biomedical Informatics
+
+University of Pittsburgh
+
+Kayhan Batmanghelich
+
+Department of Biomedical Informatics
+
+Department of Computer Science
+
+Intelligent Systems Program
+
+University of Pittsburgh
+
+# ABSTRACT
+
+As machine learning methods see greater adoption and implementation in high stakes applications such as medical image diagnosis, the need for model interpretability and explanation has become more critical. Classical approaches that assess feature importance (e.g., saliency maps) do not explain how and why a particular region of an image is relevant to the prediction. We propose a method that explains the outcome of a classification black-box by gradually exaggerating the semantic effect of a given class. Given a query input to a classifier, our method produces a progressive set of plausible variations of that query, which gradually changes the posterior probability from its original class to its negation. These counter-factly generated samples preserve features unrelated to the classification decision, such that a user can employ our method as a "tuning knob" to traverse a data manifold while crossing the decision boundary. Our method is model agnostic and only requires the output value and gradient of the predictor with respect to its input.
+
+# 1 INTRODUCTION
+
+With the explosive adoption of deep learning for real-world applications, explanation and model interpretability have received substantial attention from the research community (Kim, 2015; Doshi-Velez & Kim, 2017; Molnar, 2019; Guidotti et al., 2019). Explaining an outcome of a model in high stake applications, such as medical diagnosis from radiology images, is of paramount importance to detect hidden biases in data (Cramer et al., 2018), evaluate the fairness of the model (Doshi-Velez & Kim, 2017), and build trust in the system (Glass et al., 2008). For example, consider evaluating a computer-aided diagnosis of Alzheimer's disease from medical images. The physician should be able to assess whether or not the model pays attention to age-related or disease-related variations in an image in order to trust the system. Given a query, our model provides an explanation that gradually exaggerates the semantic effect of one class, which is equivalent to traversing the decision boundary from side to another.
+
+Although not always clear, there are subtle differences between interpretability and explanation (Turner, 2016). While the former mainly focuses on building or approximating models that are locally or globally interpretable (Ribeiro et al., 2016), the latter aims at explaining a predictor a posteriori. The explanation approach does not compromise the prediction performance. However, a rigorous definition for what is a good explanation is elusive. Some researchers focused on providing feature importance (e.g., in the form of a heatmap (Selvaraju et al., 2017)) that influence the outcome of the predictor. In some applications (e.g., diagnosis with medical images) the causal changes are spread out across a large number of features (i.e., large portions of the image are impacted by a disease). Therefore, a heatmap may not be informative or useful, as almost all image features are highlighted. Furthermore, those methods do not explain why a predictor returns an outcome. Others have introduced local occlusion or perturbations to the input (Zhou et al., 2014; Fong & Vedaldi, 2017) by assessing which manipulations have the largest impact on the predictors. There is also
+
+recent interest in generating counterfactual inputs that would change the black box classification decision with respect to the query inputs (Goyal et al., 2019; Liu et al., 2019). Local perturbations of a query are not guaranteed to generate realistic or plausible inputs, which diminishes the usefulness of the explanation, especially for end users (e.g., physicians). We argue that the explanation should depend not only on the predictor function but also on the data. Therefore, it is reasonable to train a model that learns from data as well as the black-box classifier (e.g., (Chang et al., 2019; Dabkowski & Gal, 2017; Fong & Vedaldi, 2017)).
+
+Our proposed method falls into the local explanation paradigm. Our approach is model agnostic and only requires access to the predictor values and its gradient with respect to the input. Given a query input to a black-box, we aim at explaining the outcome by providing plausible and progressive variations to the query that can result in a change to the output. The plausibility property ensures that perturbation is natural-looking. A user can employ our method as a "tuning knob" to progressively transform inputs, traverse the decision boundary from one side to the other, and gain understanding about how the predictor makes a decision. We introduce three principles for an explanation function that can be used beyond our application of interest. We evaluate our method on a set of benchmarks as well as real medical imaging data. Our experiments show that the counterfactually generated samples are realistic-looking and in the real medical application, satisfy the external evaluation. We also show that the method can be used to detect bias in training of the predictor.
+
+# 2 METHOD
+
+Consider a black box classifier that maps an input space  $\mathcal{X}$  (e.g., images) to an output space  $\mathcal{Y}$  (e.g., labels). In this paper, we consider binary classification problems where  $\mathcal{Y} = \{-1, + 1\}$ . To model the black-box, we use  $f(\mathbf{x}) = \mathbb{P}(y|\mathbf{x})$  to denote the posterior probability of the classification. We assume that  $f$  is a differentiable function and we have access to its value as well as its gradient with respect to the input  $\nabla_{\mathbf{x}}f(\mathbf{x})$ .
+
+![](images/3b07d2cc09fc751a0a58138b03f199f0520dcb5bdda8fdda38744c959f472e61.jpg)  
+(a)  
+Figure 1: (a) The schematic of the method:  $f$  is the black-box function producing the posterior probability.  $\delta$  is the required change in black-box's output  $f(\mathbf{x})$ .  $\mathcal{I}_f(\mathbf{x},\delta)$  is an explainer function for  $f$ , which shifts the value of  $f(\mathbf{x})$  by  $\delta$ . The  $E(\cdot)$  is an encoder that maps the data manifold  $\mathcal{M}_x$  to the embedding manifold  $\mathcal{M}_z$ .  $\mathbf{x}_{\delta}$  is an abbreviation for  $\mathcal{I}_f(\mathbf{x},\delta)$ . (b) The architecture of our model:  $E$  is the encoder,  $G_{f}^{\delta}$  denotes the conditional generator  $G(\cdot ,c_{f}(x,\delta))$ ,  $f$  is the black-box and  $D$  is the discriminator. The circles denote loss functions.
+
+![](images/1b3bdb1d66716c93ffa59c4395d85194c0ccb655a70d5408a71b11e62d275cd5.jpg)  
+(b)
+
+We view the (visual) explanation of the black-box as a generative process that produces an input for the black-box that slightly perturbs current prediction  $(f(\mathbf{x}) + \delta)$  while remaining plausible and realistic. By repeating this process towards each end of the binary classification spectrum, we can traverse the prediction space from one end to the other and exaggerate the underlying effect. We conceptualize the traversal from one side of the decision boundary to the other as walking across a data manifold,  $\mathcal{M}_x$ . We assume the walk has a fixed step size and each step of the walk makes  $\delta$  change to the posterior probability of the classifier,  $f$ . Since the output of  $f$  is bounded between
+
+[0, 1], we can take at-most  $\left\lfloor \frac{1}{\delta} \right\rfloor$  steps. Each positive (negative) step increases (decreases) the posterior probability of the previous step. We assume that there is a low-dimensional embedding space  $(\mathcal{M}_z)$  that encodes the walk. An encoder,  $E: \mathcal{M}_x \to \mathcal{M}_z$ , maps an input,  $\mathbf{x}$ , from the data manifold,  $\mathcal{M}_x$ , to the embedding space. A generator,  $G: \mathcal{M}_z \to \mathcal{M}_y$ , takes both the embedding coordinate and the number of steps and maps it back to the data manifold (see Figure1).
+
+We use  $\mathcal{I}_f(\cdot ,\cdot)$  to denote the explainer function. Formally,  $\mathcal{I}_f(\mathbf{x},\delta):(X,\mathbb{R})\to X$  is a function that takes two arguments: a query image  $\mathbf{x}$  and the desired perturbation  $\delta$ . This function generates a perturbed image which is then passed through function  $f$ . The difference between the outputs of  $f$  given the original image and the perturbed image should be the desired change i.e.,  $f(\mathbf{x}_{\delta}) - f(\mathbf{x}) = \delta$ . We use  $\mathbf{x}_{\delta}$  to denote  $\mathcal{I}_f(\mathbf{x},\delta)$ . This formulation enables us to use  $\delta$  as a knob to exaggerate the visual explanations of the query sample while it is crossing the decision boundary given by function  $f$ . Our proposed interpretability function  $\mathcal{I}_f$  should satisfy the following properties:
+
+1. Data Consistency: perturbed samples generated by  $\mathcal{I}_f$  should lie on the data manifold,  $\mathcal{M}_x$ , to be consistent with real data. In other words, the generated samples should look realistic when compared to other samples.  
+2. Compatibility with  $f$ : changing the second argument in  $\mathcal{I}_f(\mathbf{x},\cdot)$  should produce the desired outcome from classifier  $f$ , i.e.,  $f(\mathcal{I}_f(\mathbf{x},\delta)) \approx f(\mathbf{x}) + \delta$ .  
+3. Self Consistency: Applying reverse perturbation should bring  $\mathbf{x}$  back to its original form i.e.,  $\mathcal{I}_f(\mathcal{I}_f(\mathbf{x},\delta), - \delta) = \mathbf{x}$ . Also, applying setting  $\delta$  to zero should return the query, i.e.,  $\mathcal{I}_f(\mathbf{x},0) = \mathbf{x}$ .
+
+Each criterion is enforced via a loss function which are discussed in the following sections.
+
+# 2.1 DATA CONSISTENCY
+
+We adopt the Generative Adversarial Networks (GANs) framework for our model (Goodfellow et al., 2014). The GANs implicitly model the underlying data distribution by setting up a min-max game between generative  $(G)$  and discriminative  $(D)$  networks:
+
+$$
+\mathcal {L} _ {\mathrm {G A N}} (D, G) = \mathbb {E} _ {\mathbf {x}, c \sim P (\mathbf {x})} \left[ \log \left(D (\mathbf {x})\right) \right] + \mathbb {E} _ {\mathbf {z} \sim P _ {\mathbf {z}}} \left[ \log \left(1 - D (G (\mathbf {z}))\right) \right],
+$$
+
+where  $\mathbf{z}$  and  $P_{\mathbf{z}}$  are the noise distribution and the corresponding canonical distribution. There has been significant progress toward improving GANs stability as well as sample quality (Brock et al., 2019; Karras et al., 2019). The advantage of GANs is that they produce realistic-looking samples without an explicit likelihood assumption about the underlying probability distribution. This property is appealing for our application.
+
+Furthermore, we need to provide the desired amount of perturbation to the black-box,  $f$ . Hence, we use a Conditional GAN (cGAN) that allows the incorporation of a context as a condition to the GAN (Mirza & Osindero, 2014; Miyato & Koyama, 2018). To define the condition, we fix the step size,  $\delta$ , and descritize the walk which effectively cuts the posterior probability range of the predictor (i.e., [0, 1]) into  $\lfloor \frac{1}{\delta} \rfloor$  equally-sized bins. Hence, one can view the perturbation from  $f(\mathbf{x})$  to  $f(\mathbf{x}) + \delta$  as changing the bin index from the current value  $c_{f}(\mathbf{x}, 0)$  to  $c_{f}(\mathbf{x}, \delta)$  where  $c_{f}(\mathbf{x}, \delta)$  returns the bin index of  $f(\mathbf{x}) + \delta$ . We use  $c_{f}(\mathbf{x}, \delta)$  as a condition to the cGAN.
+
+The cGAN optimizes the following loss function:
+
+$$
+\mathcal {L} _ {\mathrm {c G A N}} (D, G) = \mathbb {E} _ {\mathbf {x}, c \sim P (\mathbf {x}, c)} [ \log (D (\mathbf {x}, c)) ] + \mathbb {E} _ {\mathbf {z} \sim P _ {\mathbf {z}}, c \sim P _ {c}} [ \log (1 - D (G (\mathbf {z}, c), c)) ], \tag {1}
+$$
+
+where  $c$  denotes a condition. Instead of generating random samples from  $P_{\mathbf{z}}$ , we use the output of an encoder,  $E(\mathbf{x})$ , as input to the generator. Finally, the explainer function is defined as:
+
+$$
+\mathcal {I} _ {f} (\mathbf {x}, \delta) = G (E (\mathbf {x}), c _ {f} (\mathbf {x}, \delta)). \tag {2}
+$$
+
+Our architecture is based on Projection GAN (Miyato & Koyama, 2018), a modification of cGAN. An advantage of the Projection GAN is that it scales well with the number of classes allowing  $\delta \rightarrow 0$ . The Projection GAN imposes the following structure on the discriminator loss function:
+
+$$
+\mathcal {L} _ {\mathrm {c G A N}} (D, \hat {G}) (\mathbf {x}, \mathbf {c}) = \log \frac {p _ {\text {d a t a}} (\mathbf {c} | \mathbf {x})}{q (\mathbf {c} | \mathbf {x})} + \log \frac {p _ {\text {d a t a}} (\mathbf {x})}{q (\mathbf {x})} := r (c | \mathbf {x}) + \psi (\phi (\hat {G} (\mathbf {z}))), \tag {3}
+$$
+
+where  $\mathcal{L}_{\mathrm{cGAN}}(D,\hat{G})$  indicates the loss function in Eq. 1 when  $\hat{G}$  is fixed,  $\phi (\cdot)$  and  $\psi (\cdot)$  are networks producing vector (feature) and scalar outputs respectively. The  $r(c|\mathbf{x})$  is a conditional ratio function which will be discussed in Section 2.2.
+
+# 2.2 COMPATIBILITY WITH THE BLACK BOX
+
+In our model, the condition  $c$  is an ordered variable i.e.,  $c_{f}(\mathbf{x},\delta_{1}) < c_{f}(\mathbf{x},\delta_{2})$  when  $\delta_1 < \delta_2$ . Therefore, we adapt the first term in Eq. 3 to account for ordinal multi-class regression by transforming  $c_{f}(\mathbf{x},\delta)$  into  $\lfloor \frac{1}{\delta}\rfloor -1$  binary classification terms (Frank & Hall, 2001):
+
+$$
+r (c = k | \mathbf {x}) := \sum_ {i <   k} \mathbf {v} _ {i} ^ {T} \phi (\mathbf {x}), \tag {4}
+$$
+
+where  $\phi (\cdot)$  is the feature network in Eq. 3 and  $\mathbf{v}_i$ 's are parameters. We also need to ensure that plugging  $\mathbf{x}_{\delta}$  into  $f(\cdot)$  yields  $f(\mathbf{x}) + \delta$  (i.e., compatible with  $f$ ). This condition is enforce by a KullbackLeibler (KL) divergence loss term. Adding the KL loss and the conditional ratio function we arrive at the following loss:
+
+$$
+\mathcal {L} _ {f} (D, G) := r (c | \mathbf {x}) + D _ {\mathrm {K L}} \left(f (\mathbf {x}) + \delta \| f \left(\mathcal {I} _ {f} (\mathbf {x}, \delta)\right)\right).
+$$
+
+While the first term is a function of both  $G$  and  $D$ , the second term influences only the generator  $G$ .
+
+# 2.3 SELF CONSISTENCY
+
+We use a reconstruction loss term to enforce encoder-decoder consistency and satisfy the identity constraint of  $\mathbf{x} = \mathcal{I}_f(\mathbf{x},0)$
+
+$$
+\mathcal {L} _ {\operatorname {r e c}} (G) = \left\| \mathbf {x} - G \left(E (\mathbf {x}), c _ {f} (\mathbf {x}, 0)\right) \right\| _ {1}, \tag {5}
+$$
+
+We also require that the perturbation is reversible (i.e.,  $\mathcal{I}_f(\mathcal{I}_f(\mathbf{x},\delta), - \delta) = \mathbf{x})$ . We use a cycle-consistency (Zhu et al., 2017) loss to reconstruct the input from its corresponding perturbed image,
+
+$$
+\mathcal {L} _ {\text {c y c}} (G) = \left\| \mathbf {x} - G \left(E \left(\mathbf {x} _ {\delta}\right), c _ {f} (\mathbf {x}, 0)\right) \right\| _ {1}. \tag {6}
+$$
+
+Note that the conditions for the generators in Eq. 5 and 6 are the same. However, in the former, we are reconstructing the input  $\mathbf{x}$  from its latent space, but in the latter, we perturb  $\mathbf{x}_{\delta}$  from the bin index  $c_{f}(\mathbf{x},\delta)$  back to original bin index  $c_{f}(\mathbf{x},0)$ .
+
+# 2.4 OBJECTIVE FUNCTIONS
+
+We adapted the hinge version of the adversarial loss for  $\mathcal{L}_{\mathrm{cGAN}}(G,D)$
+
+$$
+\begin{array}{l} \mathcal {L} _ {\mathrm {c G A N}} (D) = - \mathbb {E} _ {\mathbf {x} \sim p _ {\mathrm {d a t a}}} \left[ \min (0, - 1 + D (\mathbf {x}, c _ {f} (\mathbf {x}, 0))) \right] \\ - \mathbb {E} _ {\mathbf {x} \sim p _ {\mathrm {d a t a}}, c _ {f} (\mathbf {x}, \delta) \in [ 0, \frac {1}{\delta} ]} \left[ \min  (0, - 1 - D (G (E (\mathbf {x}), c _ {f} (\mathbf {x}, \delta)), c _ {f} (\mathbf {x}, \delta))) \right] \tag {7} \\ \end{array}
+$$
+
+$$
+\mathcal {L} _ {\mathrm {c G A N}} (G, E) = - \mathbb {E} _ {\mathbf {x} \sim p _ {\text {d a t a}}, c _ {f} (\mathbf {x}, \delta) \in [ 0, \frac {1}{\delta} ]} \left[ D \left(G \left(E (\mathbf {x}), c _ {f} (\mathbf {x}, \delta)\right), c _ {f} (\mathbf {x}, \delta)\right) \right] \tag {8}
+$$
+
+The overall objective function is
+
+$$
+\min  _ {E, G} \max  _ {D} \lambda_ {\mathrm {c G A N}} \mathcal {L} _ {\mathrm {c G A N}} (D, G) + \lambda_ {f} \mathcal {L} _ {f} (D, G) + \lambda_ {\mathrm {r e c}} \mathcal {L} _ {\mathrm {r e c}} (G) + \lambda_ {\mathrm {r e c}} \mathcal {L} _ {\mathrm {c y c}} (G) \tag {9}
+$$
+
+where  $\lambda_{\mathrm{cGAN}}, \lambda_f, \lambda_{\mathrm{rec}}$  are the hyper-parameters that balance the importance of the loss terms.
+
+# 3 RELATED WORK
+
+Our work broadly relates to literature in interpretation methods that are designed to provide a visual explanation of the decisions made by a black-box function  $f$ , for a given query sample  $\mathbf{x}$ .
+
+Perturbation-based methods: These methods provide interpretation by showing what minimal changes are required in  $\mathbf{x}$  to induce a desirable output of  $f$ . Some methods employed image manipulation via the removal of image patches (Zhou et al., 2014) or the occlusion of image regions (Zhou et al., 2014) to change the classification score. Recently, the use of influence function, as proposed by (Koh & Liang, 2017) are applied as a form of data perturbation to modify a classifier's response. The authors in (Fong & Vedaldi, 2017) proposed the use of optimal perturbation, defined as removing the smallest possible image region in  $\mathbf{x}$  that results in the maximum drop in classification score. In another approach, (Chang et al., 2019) proposed a generative process to find and fill the image
+
+regions that correspond to the largest change in the decision output of a classifier. To switch the decision of a classifier, (Goyal et al., 2019) suggested generating counterfactuals by replacing the regions of  $\mathbf{x}$  with patches from images with a different class label. All of the aforementioned works perform pixel- or patch-level manipulation to  $\mathbf{x}$ , which may not result in natural-looking images. In contrast, our model enforces that the perturbed data be consistent with the unperturbed data to ensure that the perturbation is plausible. Furthermore, our method can be applied to general data and is not restricted to the imaging domain.
+
+Saliency map-based methods: Saliency maps explains the decision of  $f$  on  $\mathbf{x}$  by highlighting the relevant regions of  $x$ . Some earlier work in this direction (Simonyan et al., 2013; Springenberg et al., 2015; Bach et al., 2015) focuses on computing the gradient of the target class with respect to  $\mathbf{x}$  and considers the image regions with large gradients as most informative. Building on this work, the class activation map (CAM) (Zhou et al., 2016) and its generalized version Grad-CAM Selvaraju et al. (2017) and other variants such as LPR (Bach et al., 2015) use a linear or non-linear combination of the activation layers to derive relevance score for every pixel in an image. These gradient-based methods are not model-agnostic and require access to intermediate layers. Recently, Adebayo et al. (2018) have shown that some saliency methods are independent both of the model and of the data generating process. We used their propose evaluation to validate our interpretation model. The saliency maps are also prone to adversarial attacks as shown by Ghorbani et al. (2019) and Kindermans et al. (2017). Furthermore, if the causal effect of a class is distributed across an image, which is the case in radiology images, the saliency approaches highlight large sections of the image, which greatly reduce the usefulness of the interpretation.
+
+Generative explanation-based methods: These are interpretation models that uses a generative process to produce visual explanations. The contrastive explanations method (CEM) (Dhurandhar et al., 2018) generates explanations that show minimum regions in  $\mathbf{x}$  which must be present/absent for a particular classification decision. In another work, (Liu et al., 2019; Joshi et al., 2019; Samangouei et al., 2018) generates explanations that highlight what features should be changed in  $\mathbf{x}$  so that the classifier confidence in the prediction is strengthened (prototype) or weakened (counterfactual). Our approach is aligned with these latter lines of work, although our method and model architecture is different. Our method allows for the gradual change of the class effect, and our consistency criteria result in high-quality feasible perturbation in  $\mathbf{x}$ . We rigorously evaluate our method on real medical imaging applications, in addition to the curated computer vision datasets.
+
+# 4 EXPERIMENTS
+
+We set up four experiments to evaluate our method. First, we assess if our method satisfies the three criteria of the explainer function introduced in Section 2. We report both qualitative and quantitative results. Second, we apply our method on a medical image diagnosis task. We use external domain knowledge about the disease to perform a quantitative evaluation of the explanation. Third, we train two classifiers on biased and unbiased data and examine the performance of our method in identifying the bias. While our method does not produce a saliency map, in our last experiment, we use the two counterfactual samples on the boundary  $[0,1]$  to generate a saliency map and compare it with the other methods. In Appendix A, we show further experiments to evaluate our model in human experiments, to demonstrate its compatibility with a multi-label classifier and, an ablation study, to show the relative importance of each of the three criteria of the explainer function.
+
+Our experiments are conducted on the CelebA (Liu et al., 2015) and CheXpert (Irvin et al., 2019) datasets. CelebA contains 200K celebrity face images, each with forty attribute labels. We considered binary classifier trained on the "smiling" and "young" attributes. CheXpert is a medical dataset containing 224K chest x-ray images from 65K patients and has labels for fourteen radio-graphic observations. We considered Cardiomegaly as the target class for generating explanations. All images are re-sized to  $128 \times 128$  before processing.
+
+# 4.1 EVALUATING THE CRITERIA OF THE EXPLainer
+
+Figure 2 reports the qualitative results on three datasets. Given a query image  $\mathbf{x}$  at inference time, our model generates a series of images  $\mathbf{x}_{\delta}$  as visual explanations, which gradually increase the posterior probability  $f(\mathbf{x}_{\delta})$  (top label). We show results for three prediction tasks: smiling or not-smiling,
+
+young or old, and Cardiomegaly or healthy. The values on the top of each figure report the  $f(\mathbf{x}_{\delta})$ 's. For Cardiomegaly, we show the outlines of the heart as well as its normalized size (values inside the parenthesis), which is indicative of the disease.
+
+![](images/64d149a768cbfb35ac2e8449e1109ff5c5ec59408b37f9090018a263231138e2.jpg)  
+Figure 2: Visual explanations generated for three prediction tasks: smiling/not-smiling face (first two rows), young/old face (middle two rows) and Cardiomegaly/healthy chest x-ray (bottom two rows). The first column shows the query image, followed by the corresponding generated explanations. The values above each image are the output of the classifier  $f$ . For Cardiomegaly, we show the segmentation of the heart (yellow edge) and report normalized heart size (values in parenthesis), which is indicative of the disease.
+
+Data Consistency: The generated explanations are synthesized variations of the query image. To quantitatively compare their visual quality, we consider Fréchet Inception Distance (FID) (Heusel et al., 2017). We compared our results against the counterfactual explanations produced by xGEM (Joshi et al., 2018). The details of the xGEM model are given in appendix A.2. We divided the real and fake (i.e., generated explanations) images into two groups (on either boundary of  $f(\mathbf{x}) \in [0,1]$ ) and reported the FID for each group and the overall score. Our method significantly outperforms xGEM, producing crisp and more realistic-looking images. xGEM is based on variational autoencoder (VAE) which are known to produce blurry images (see Figure 7).
+
+Compatibility with the black-box  $f$ : To quantify whether the generation process is aligned with the desire perturbation  $\delta$ , we plotted the expected outcome  $f(\mathbf{x}) + \delta$  against the actual response of the classifier for the generated explanations,  $f(\mathbf{x}_{\delta})$ . Figure 3 shows how our model performs when generating a series of explanations starting from a wide range of initial query images. The performance is almost perfect for Young/Old, but less so for more challenging classification problems such as Smiling or Cardiomegaly. The plot also validates that we are producing perturb images covering the entire classification range, [0, 1]. Appendix A.3 shows additional result from CelebA dataset.
+
+<table><tr><td rowspan="2">Target Class</td><td colspan="2">CelebA:Smiling</td><td colspan="2">CelebA:Young</td><td colspan="2">Xray:Cardiomegaly</td></tr><tr><td>xGEM</td><td>Ours</td><td>xGEM</td><td>Ours</td><td>xGEM</td><td>Ours</td></tr><tr><td>Present (f(xδ) ∈ [0.9, 1])</td><td>111.0</td><td>46.9</td><td>115.2</td><td>67.6</td><td>368.6</td><td>82.9</td></tr><tr><td>Absent (f(xδ) ∈ [0, 0.1])</td><td>112.9</td><td>56.3</td><td>170.3</td><td>74.4</td><td>394.6</td><td>84.8</td></tr><tr><td>Overall (f(xδ) ∈ [0, 1])</td><td>106.3</td><td>35.8</td><td>117.9</td><td>53.4</td><td>326.3</td><td>58.1</td></tr></table>
+
+Table 1: The Fréchet Inception Distance (FID) score, measuring the quality of the generated explanations for the three prediction tasks. Lower FID corresponds to better image quality. Top (bottom) row corresponds the top (bottom)  $10\%$  of the decision interval.
+
+![](images/66430c252e9433bf703ae6811c4ad376c4a41c34de27876d45400ba1d0721a6c.jpg)  
+Figure 3: Plot of the expected outcome from the classifier,  $f(\mathbf{x}) + \delta$ , against the actual response of the classifier on generated explanations,  $f(\mathbf{x}_{\delta})$ . The monotonically increasing trend shows a positive correlation between  $f(\mathbf{x}) + \delta$  and  $f(\mathbf{x}_{\delta})$ , and thus the generated explanations are consistent with the expected condition.
+
+Identity preservation: The generated explanations should differ only in semantic features associated with the target class, while retaining the identity of the query image. We extracted the latent embedding for real images  $(E(\mathbf{x}))$  and their corresponding explanations  $(E(\mathbf{x}_{\delta}))$ , for different values of  $\delta$ . We calculated latent space closeness as the percentage of the times,  $\mathbf{x}_{\delta}$  is closest to the query image  $\mathbf{x}$  as compared to other generated explanations
+
+$$
+\forall \delta , \mathbf {x} \in \mathcal {X}, \quad | | E (\mathbf {x}) - E \left(\mathbf {x} _ {\delta}\right) | | _ {2} <   \min  _ {\mathbf {m} \in \mathcal {I} _ {f} \left(\mathcal {X} - \{\mathbf {x} \}, \delta\right)} | | E \left(\mathbf {x} _ {\delta}\right) - E (\mathbf {m}) | | _ {2}, \tag {10}
+$$
+
+where,  $\mathbf{m} \in \mathcal{I}_f(\mathcal{X} - \{\mathbf{x}\}, \delta)$  is the set of explanations generated for all the real images excluding the query image  $\mathbf{x}$ . Another, popular approach to quantify identity of two face images, is to perform face verification. We used state-of-the-art face recognition model trained on VGGFace2 dataset (Cao et al., 2018) as feature extractor for both real images and their corresponding fake explanations. For face verification, we calculated the closeness between real and fake image as cosine distance between their feature vectors. The faces were considered as verified i.e., fake explanation have same identity as real image, if the distance is below 0.5. Table 2 summarizes the results.
+
+Our method achieved high performance on localized attribute "smiling", which alters a relatively small region of the face image as compare to attribute "age" which affects the entire face. Medical images like chest x-ray have very fine grain details which are difficult to preserve in the generative process of GAN. Our explainer function preserves the high level features like shape and size of the lung, but it struggles to retain the low level features like anatomy of the breast and shape of the collar bones. Also, it should be noted that both the datasets have multiple images for same person, but we ignore this information in our analysis and treat each image as a different identity. We compared our performance against xGEM (Joshi et al., 2018). VAE explicitly minimizes for latent space closeness. The generated explanation by xGEM were blurry version of the query image. Hence, although they were close to query image in latent space, but they didn't preserve the identity of the individual as shown in face verification task and is evident in Figure 7 in appendix A.2. In comparison, our model achieved good performance on both the tasks.
+
+# 4.2 COUNTERFACTUAL EVALUATION ON MEDICAL DATA
+
+Cardiomegaly refers to an abnormal enlargement of the heart (Brakohiapa et al., 2017). To understand the explanations derived for Cardiomegaly target class, we overlaid the heart segmentation
+
+<table><tr><td></td><td colspan="2">CelebA:Smiling</td><td colspan="2">CelebA:Young</td><td colspan="2">Xray:Cardiomegaly</td></tr><tr><td></td><td>xGEM</td><td>Ours</td><td>xGEM</td><td>Ours</td><td>xGEM</td><td>Ours</td></tr><tr><td>Latent Space Closeness</td><td>88.2</td><td>88.0</td><td>89.5</td><td>81.6</td><td>2.2</td><td>27.9</td></tr><tr><td>Face Verification Accuracy</td><td>0.0</td><td>85.3</td><td>0.0</td><td>72.2</td><td>-</td><td>-</td></tr></table>
+
+Table 2: Identity preserving performance on three prediction tasks.
+
+over the x-ray image and visualize the gradual change in heart size. The heart segmentation is shown as outlines in Figure 2, with their corresponding heart size (top values in parentheses). The heart segmentation is derive by training a UNet (Ronneberger et al., 2015) model on the segmentation in chest radiograph (SCR) dataset (van Ginneken et al., 2006). We registered  $\mathbf{x}$  with its associated  $\mathbf{x}_{\delta}$  and applied the resulting transformation to the heart masks of  $\mathbf{x}$  to derive the heart masks for  $\mathbf{x}_{\delta}$ .
+
+For population-level analysis, we plotted the average heart size of  $\mathbf{x}_{\delta}$  vs the condition used for generation  $(f(\mathbf{x}) + \delta)$  in Figure 4 (a). The plot shows a positive correlation between the heart size and the response of the classifier  $f(\mathbf{x})$ , which agrees with the definition of Cardiomegaly. To better understand the results, we divided the population into two groups, the first group  $(\mathbf{x}^h; f(\mathbf{x}^h) < 0.1)$  consists of real images of healthy x-rays, and the second group  $(\mathbf{x}^c; f(\mathbf{x}^c) > 0.9)$  contains real images of abnormal x-rays positive for Cardiomegaly. For  $\mathbf{x}^h$  we generated counterfactual as  $\mathbf{x}_{\delta}^c$  such that  $f(\mathbf{x}_{\delta}^{c}) > 0.9$ . Similarly, counterfactuals for  $\mathbf{x}^c$  are derived as  $\mathbf{x}_{\delta}^h$  such that  $f(\mathbf{x}_{\delta}^h) < 0.1$ . In Figure 4 (b), we show the distribution of heart size in the four groups. We reported the dependent t-test statistics for paired samples  $(\mathbf{x}^h$  and  $\mathbf{x}_{\delta}^{c})$ ,  $(\mathbf{x}^{c}$  and  $\mathbf{x}_{\delta}^{h})$ . A significant p-value  $\ll 0.001$  rejected the null hypothesis (i.e., that the two groups have similar distributions). We also reported the independent two-sample t-test statistics for healthy  $(\mathbf{x}^h$  and  $\mathbf{x}_{\delta}^{h}$ , p-value  $>0.01$ ) and abnormal  $(\mathbf{x}^{c}$  and  $\mathbf{x}_{\delta}^{c}$ , p-value  $< 0.01$ ) populations. Given higher p-values, we cannot reject the null hypothesis of identical average distributions with high confidence. Our model derived explanations successfully captured the change in heart size while generating counterfactual explanations.
+
+![](images/dc3af051e57a269068e872c705b23a2ac705dd97659b3a20286c1016fbe5ee3f.jpg)  
+Figure 4: Cardiomegaly disease is associated with large heart size. In (a) we show the positive correlation between the heart size and the response of the classifier  $f(\mathbf{x})$ . (b) Comparison of the distribution of the heart size in the four groups. (c) Plot to show the drop in accuracy of the classifier as we perturb the most relevant pixels (relevance calculated from saliency map) in the image.
+
+![](images/7240b2d155efc75e055118a1d6b6fd8bb879bd388859e567fcd7367dcf26cb3a.jpg)
+
+![](images/ef4aa0062b647ab312b3bce7a5d59f18c7145c569eed5eb9b9f42aa6b659b9aa.jpg)
+
+# 4.3 SALIENCY MAP
+
+Saliency maps show the importance of each pixel of an image in the context of classification. Our method is not designed to produce saliency maps as a continuous score for every feature of the input. We extract an approximate saliency map by quantifying the regions that changed the most when comparing explanations at the opposing ends of the classification spectrum. For each query image, we generated two visual explanations corresponding to the two extremes of the decision boundary  $(f(\mathbf{x}_{\delta}) = 0$  and  $f(\mathbf{x}_{\delta}) = 1)$ . The absolute difference between these explanations is our saliency map. Figure 5 shows the saliency map obtained from our method and its comparison with popular gradient based methods. We restricted the saliency maps obtained from different methods to have positive values and normalize them to range [0,1]. Subjective, the saliency maps produced by our method are very localized and are comparable to the other methods.
+
+We adapted the metric introduced in (Samek et al., 2016) to compare the different saliency maps. In an iterative procedure, we progressively replace a percentage of the most relevant pixels in an image (as given by the saliency map) with random values sampled from a uniform distribution. We observe the corresponding change in the classification performance as shown in Figure 4 (c). All the methods experienced a drop in the accuracy of the classifier with increase in the fraction of perturb pixels. The saliency maps produced by our model is significantly better than random maps and are comparable to the other saliency map methods. It should be noted that, there are many ways to quantify important regions in a image, using the series of explanations generated by our method. We didn't optimize to find the best saliency map and showed results for one such method.
+
+![](images/e316eb1bd5da5365eb2a7039beb3e5d6c6f799530468e2aa7d2e28332eff0921.jpg)  
+Figure 5: Our comparison with popular gradient-based saliency map producing methods on the prediction task of identifying smiling faces in CelebA dataset.
+
+# 4.4 BIAS DETECTION
+
+Our model can discover confounding bias in the data used for training the black-box classifier. Confounding bias provides an alternative explanation for an association between the data and the target label. For example, a classifier trained to predict the presence of a disease may make decisions based on hidden attributes like gender, race, or age. In a simulated experiment, we trained two classifiers to identify smiling vs not-smiling images in the CelebA dataset. The first classifier  $f_{\mathrm{Biased}}$  is trained on a biased dataset, confounded with gender such that all smiling images are of male faces. We train a second classifier  $f_{\mathrm{No-biased}}$  on an unbiased dataset, with data uniformly distributed with respect to gender. Note that we evaluate both the classifiers on the same validation set. Additionally, we assume access to a proxy Oracle classifier  $f_{\mathrm{Gender}}$  that perfectly classifies the confounding attribute i.e., gender. As shown in Cohen et al. (2018), if the training data for the GAN is biased, then the inference would reflect that bias. In Figure 6, we compare the explanations generated for the two classifiers. The visual explanations for the biased classifier change gender as it increases the amount of smile. We adapted the confounding metric proposed in Joshi et al. (2018) to summarize our results in Table 3. Given the data  $\mathcal{D} = \{(x_i, y_i, a_i), x_i \in \mathcal{X}, y_i, a_i \in \mathcal{Y}\}$ , we quantify that a classifier is confounded by an attribute  $a$  if the generated explanation  $\hat{x}_{\delta}$  has a different attribute  $a$  as compared to query image  $\mathbf{x}$ , when processed through the Oracle classifier  $f_{\mathrm{Gender}}$ . The metric is formally defined as  $\mathbb{E}_{\mathcal{D}}[1(g^{*}(\mathbf{x}_{\delta}) \neq a)] / |\mathcal{D}|$ . For a biased classifier, the Oracle function predicted the female class for the majority of the images, while the unbiased classifier is consistent with the true distribution of the validation set for gender. Thus, we the fraction of generated explanations that changed the confounding attribute "gender" was found to be high for the biased classifier.
+
+<table><tr><td colspan="3">Target Label</td></tr><tr><td>Black-box classifier</td><td>Smiling</td><td>Not-Smiling</td></tr><tr><td rowspan="3">fBiased</td><td>Male: 0.52</td><td>Male: 0.18</td></tr><tr><td>Female: 0.48</td><td>Female: 0.82</td></tr><tr><td>Overall: 0.12</td><td>Overall: 0.35</td></tr><tr><td rowspan="3">fNo-biased</td><td>Male: 0.48</td><td>Male: 0.47</td></tr><tr><td>Female: 0.52</td><td>Female: 0.53</td></tr><tr><td>Overall: 0.07</td><td>Overall: 0.08</td></tr></table>
+
+Table 3: Confounding metric for biased detection. For target label "Smiling" and "Not-Smiling", the explanations are generated using condition  $f(x) + \delta > 0.9$  and  $f(x) + \delta < 0.1$  respectively. The Male and Female values quantify the fraction of the generated explanations classifier as male or female, respectively by oracle classifier  $f_{\text{Gender}}$ . The overall value quantifies the fraction of the generated explanations who have different gender as compared to the query image. A small overall value shows least bias.
+
+![](images/344572f82f71937a5e5418fe11d9e1a9003b42ca91984ecb4570ea8cd295b726.jpg)  
+Figure 6: The visual explanations for two classifiers, both trained to classify "Smiling" attribute on CelebA dataset. For each example, the top row shows results from "Biased" classifier whose data distribution is confounded with "Gender". The bottom row shows explanations from "No-Biased" classifier with uniform data distribution w.r.t gender. The top label indicates output of the classifier and the bottom label is the output of an oracle classifier for the con-founding attribute gender. The visual explanations for the "Biased" classifier changes the gender as it adds smile on the face.
+
+# 5 CONCLUSION
+
+In this paper, we proposed a novel interpretation method that explains the decision of a black-box classifier by producing natural-looking, gradual perturbations of the query image, resulting in an equivalent change in the output of the classifier. We evaluated our model on two very different datasets, including a medical imaging dataset. Our model produces high-quality explanations while preserving the identity of the query image. Our analysis shows that our explanations are consistent with the definition of the target disease without explicitly using that information. Our method can also be used to generate a saliency map in a model agnostic setting. In addition to the interpretability advantages, our proposed method can also identify plausible confounding biases in a classifier.
+
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+
+# A APPENDIX
+
+# A.1 IMPLEMENTATION DETAILS
+
+The architecture for the generator and discriminator is adapted from Miyato & Koyama (2018). The image encoding learned by encoder  $E(\mathbf{x})$  is fed into the generator. The condition  $c_{f}(\mathbf{x},\delta)$  is passed to each resnet block in the generator, using conditional batch normalization. The generator has five resnet blocks, where each block consists of BN-ReLU-Conv3-BN-ReLU-Conv3. BN is batch normalization, ReLU is activation function, and Conv3 is the convolution filter. The encoder function uses the same structure but downsamples the image. The discriminator function has five resnet blocks, each of which has the form ReLU-Conv3-ReLU-Conv3.
+
+# A.2 XGEM IMPLEMENTATION
+
+We refers to Joshi et al. (2019) for the implementation of xGEM. First, VAE is trained to generate face images. The VAE used is available at:https://github.com/LynnHo/VAE-Tensorflow. All settings and architectures were set to default values. The original code generates an image of dimension 64x64. We extended the given network to produce an image with dimensions 128x128. The pretrained VAE is then extended to incorporate the cross-entropy loss for flipping the label of the query image. The model evaluates the cross-entropy loss by passing the generated image through the classifier. Figure 7 shows the qualitative difference between the explanations generated by our proposed method and xGEM.
+
+# A.3 EXTENDED RESULTS FOR EVALUATING THE CRITERIA OF THE EXPLainer
+
+Here, we provide results for four more prediction tasks on celebA dataset: no-beard or beard, heavy makeup or light makeup, black hair or not back hair, and bangs or no-bangs. Figure 8 shows the qualitative results, an extended version of results in Figure 2. We evaluated the results from these prediction tasks for compatibility with black-box  $f$  (see Figure 9), data consistency and self consistency (see Table 4).
+
+<table><tr><td rowspan="2">Prediction Task</td><td colspan="3">Data Consistency (FID)</td><td colspan="2">Self Consistency</td></tr><tr><td>Present</td><td>Absent</td><td>Overall</td><td>LSC</td><td>FVA</td></tr><tr><td>Smiling vs Not-smiling</td><td>46.9</td><td>56.3</td><td>35.8</td><td>88.0</td><td>85.3</td></tr><tr><td>Young vs Old</td><td>67.5</td><td>74.4</td><td>53.4</td><td>81.6</td><td>72.2</td></tr><tr><td>No beard vs Beard</td><td>79.2</td><td>72.3</td><td>45.4</td><td>89.6</td><td>83.3</td></tr><tr><td>Heavy makeup vs Light makeup</td><td>64.9</td><td>98.2</td><td>39.2</td><td>89.2</td><td>75.3</td></tr><tr><td>Black hair vs Not black hair</td><td>55.8</td><td>72.8</td><td>34.8</td><td>79.4</td><td>81.6</td></tr><tr><td>Bangs vs No bangs</td><td>54.1</td><td>57.8</td><td>40.6</td><td>76.5</td><td>87.3</td></tr></table>
+
+Table 4: Our model results for six prediction tasks on CelebA dataset. FID (Frechet Inception Distance) score measures the quality of the generated explanations. Lower FID is better. LSC (Latent Space Closeness) quantifies the fraction of the population where generated explanation is nearest to the query image than any other generated explanation in embedding space. FVA (Face verification accuracy) measures percentage of the times the query image and generated explanation have same face identity as per model trained on VGGFace2. Higher LSC and FVA is better.
+
+# A.4 HUMAN EVALUATION
+
+We used Amazon Mechanical Turk (AMT) to conduct human experiments to demonstrate that the progressive exaggeration produced by our model is visually perceivable to humans. We presented AMT workers with three tasks. In the first task, we evaluated if humans can detect the relative order between two explanations produced for a given image. We ask the AMT workers, "Given two images of the same person, in which image is the person younger (or smiling more)?" (see Figure 10). We experimented with 200 query images and generated two pairs of explanations for each query image (i.e., 400 hits). The first pair (easy) imposed the two images are samples from opposite ends of the explanation spectrum (counterfactuals), while the second pair (hard) makes no such assumption.
+
+In the second task, we evaluated if humans can identify the target class for which our model has provided the explanations. We ask the AMT workers, "What is changing in the images? (age, smile, hair-style or beard)". We experimented with 100 query images from each of the four attributes (i.e., 400 hits). In the third task, we demonstrate that our model can help the user to identify problems like possible bias in the black-box training. Here, we used the same setting as in the second task but also showed explanations generated for a biased classifier. We ask the AMT workers, "What is changing in the images? (smile or smile and gender)" (see Figure 10). We generated explanations for 200 query images each, from a biased-classifier  $(f_{\mathrm{Biased}})$  explainer from Section 4.4 and an unbiased classifier  $(f_{\mathrm{No - biased}})$  explainer (i.e., 400 hits). In all the three tasks, we collected eight votes for each task, evaluated against the ground truth, and used the majority vote for calculating accuracy.
+
+We summarize our results in Table 5. In the first task, the annotators achieved high accuracy for the easy pair when there was a significant difference among the two explanation images, as compared to the hard pair when the two explanations can have very subtle differences. Overall, the annotators were successful in identifying the relative order between the two explanation images.
+
+In the second task, the annotators were generally successful in correctly identifying the target class. The target class "bangs" proved to be the most difficult to identify, which was expected. The generated images for "bangs" were qualitatively, the most subtle. For the third task, the correct answer was always the target class i.e., "smile". In the case of biased classifier explainer, the annotators selected "Smile and Gender"  $12.5\%$  of the times. The gradual progression made by the explainer for a biased classifier was very subtle and was changing large regions of the face as compared to the unbiased explainer. The difference is much more visible when we compare the explanation generated for the same query image for a biased and no-biased classifier, as in Figure 6. But in a realistic scenario, the no-biased classifier would not be available to compare against. Nevertheless, the annotators detected bias at roughly the same level of accuracy as our classifier (Table 3). Future work could improve upon bias detection.
+
+<table><tr><td rowspan="2">Annotation Task</td><td colspan="3">Overall</td><td colspan="2">Sub categories</td></tr><tr><td>Accuracy</td><td>κ-statistic</td><td>Category</td><td>Accuracy</td><td>κ-statistic</td></tr><tr><td rowspan="2">Task-1 (Age)</td><td rowspan="2">83.5%</td><td rowspan="2">0.41 (Moderate)</td><td>Hard</td><td>73%</td><td>0.31 (Fair)</td></tr><tr><td>Easy</td><td>94%</td><td>0.51 (Moderate)</td></tr><tr><td rowspan="2">Task-1 (Smile)</td><td rowspan="2">77.5%</td><td rowspan="2">0.28 (Fair)</td><td>Hard</td><td>66%</td><td>0.23 (Fair)</td></tr><tr><td>Easy</td><td>89.5%</td><td>0.32 (Fair)</td></tr><tr><td rowspan="4">Task-2 (Identify Target Class)</td><td rowspan="4">77%</td><td rowspan="4">0.35 (Fair)</td><td>Age</td><td>72%</td><td>-</td></tr><tr><td>Smile</td><td>99%</td><td>-</td></tr><tr><td>Bangs</td><td>50%</td><td>-</td></tr><tr><td>Beard</td><td>87%</td><td>-</td></tr><tr><td rowspan="2">Task-3 (Bias Detection)</td><td rowspan="2">93.75%</td><td rowspan="2">0.14 (Slight)</td><td>fBiased</td><td>87.5%</td><td>0.09 (Slight)</td></tr><tr><td>fNo-biased</td><td>100%</td><td>0.02 (Slight)</td></tr></table>
+
+Table 5: Summarizing the results of human evaluation. The  $\kappa$ -statistics measure inter-rater agreement for qualitative classification of items into some mutually exclusive categories. One possible interpretation of  $\kappa$  as given in Viera et al. (2005) is  $< 0.0$ : Poor,  $0.01 - 0.2$ : Slight,  $0.21 - 0.40$ : Fair,  $0.41 - 0.60$ : Moderate,  $0.61 - 0.80$ : Substantial and  $0.81 - 1.00$ : Almost perfect agreement.
+
+# A.5 EVALUATING CLASS DISCRIMINATION
+
+In multi-label settings, multiple labels can be true for a given image. In this test, we evaluated the sensitivity of our generated explanations to the class being explained. We consider a classifier trained to identify multiple attributes: young, smiling, black-hair, no-beard and bangs in face images from CelebA dataset. We used our model to generate explanations while considering one of the attributes as the target. Ideally, an explanation model trained to explain a target attribute should produce explanations consistent with the query image on all the attributes beside the target. Figure 11 plots the fraction of the generated explanations, that have flipped in source attribute as compared to the query image. Each column represents one source attribute. Each row is one run of our method to explain a given target attribute.
+
+# A.6 ABLATION STUDY
+
+Our proposed model has three types of loss functions: adversarial loss from cGAN, KL loss, and reconstruction loss. The three losses enforce the three properties of our proposed explainer function: data consistency, compatibility with  $f$ , and self-consistency, respectively. In the ablation study, we quantify the importance of each of these components by training different models, which differ in one hyper-parameter while rest are equivalent ( $\lambda_{\mathrm{cGAN}} = 1$ ,  $\lambda_f = 1$  and  $\lambda_{\mathrm{rec}} = 100$ ). For data consistency, we evaluate Fréchet Inception Distance (FID). FID score measures the visual quality of the generated explanations by comparing them with the real images. We show results for two
+
+groups. In the first group, we consider real and fake images where the classifier has high confidence in presence of the target label i.e.,  $f(\mathbf{x}_{\delta}), f(\mathbf{x}) \in [0.9, 1.0]$ . In second group, the target label is absent i.e.,  $f(\mathbf{x}_{\delta}), f(\mathbf{x}) \in [0.0, 0.1)$ . We also report an overall score by considering all the real and generated explanations together. For compatability with  $f$  we plotted the desired output of the classifier i.e.,  $f(\mathbf{x}) + \delta$  against the actual output of the classifier  $f(\mathbf{x}_{\delta})$  for the generated explanations. For self consistency, we calculated the Latent Space Closeness (LSC) measure and Face verification accuracy (FVA). LSC quantifies the fraction of the population in which the generated explanation is nearest to the query image than any other generated explanation in embedding space. FVA measures the percentage of the instances in which the query image and generated explanation have the same face identity as per the model trained on VGGFace2. For the ablation study, we consider the prediction task of young vs old on the CelebA dataset. Figure 12 shows the results for compatibility with  $f$ . Table 6 summarizes the results for data consistency and self-consistency.
+
+<table><tr><td colspan="3">Configuration</td><td colspan="3">Data Consistency (FID)</td><td colspan="2">Self Consistency</td></tr><tr><td>λcGAN</td><td>λf</td><td>λrec</td><td>Present</td><td>Absent</td><td>Overall</td><td>LSC</td><td>FVA</td></tr><tr><td>0</td><td>1</td><td>100</td><td>69.7</td><td>105.7</td><td>67.2</td><td>96.1</td><td>99.8</td></tr><tr><td>1</td><td>1</td><td>100</td><td>67.5</td><td>74.4</td><td>53.4</td><td>81.6</td><td>72.2</td></tr><tr><td>10</td><td>1</td><td>100</td><td>89.4</td><td>105.2</td><td>63.0</td><td>68.0</td><td>82.7</td></tr><tr><td>100</td><td>1</td><td>100</td><td>71.6</td><td>80.6</td><td>44.26</td><td>75.3</td><td>18.0</td></tr><tr><td>1</td><td>0</td><td>100</td><td>66.2</td><td>66.2</td><td>44.9</td><td>77.2</td><td>99.4</td></tr><tr><td>1</td><td>1</td><td>100</td><td>67.5</td><td>74.4</td><td>53.4</td><td>81.6</td><td>72.2</td></tr><tr><td>1</td><td>10</td><td>100</td><td>95.5</td><td>90.4</td><td>62.4</td><td>71.83</td><td>96.8</td></tr><tr><td>1</td><td>100</td><td>100</td><td>77.4</td><td>73.1</td><td>71.2</td><td>55.4</td><td>42.23</td></tr><tr><td>1</td><td>1</td><td>0</td><td>116.2</td><td>118.9</td><td>72.2</td><td>16.6</td><td>0.0</td></tr><tr><td>1</td><td>1</td><td>1</td><td>63.0</td><td>78.6</td><td>61.6</td><td>32.2</td><td>5.5</td></tr><tr><td>1</td><td>1</td><td>10</td><td>87.6</td><td>83.6</td><td>65.7</td><td>71.5</td><td>88.8</td></tr><tr><td>1</td><td>1</td><td>100</td><td>67.5</td><td>74.4</td><td>53.4</td><td>81.6</td><td>72.2</td></tr></table>
+
+Table 6: Our model with ablation on prediction task of young vs old on CelebA dataset. FID (Fréchet Inception Distance) score measures the quality of the generated explanations. Lower FID is better. LSC (Latent Space Closeness) quantifies the fraction of the population where generated explanation is nearest to the query image than any other generated explanation in embedding space. FVA (Face verification accuracy) measures percentage of the times the query image and generated explanation have same face identity as per model trained on VGGFace2. Higher LSC and FVA is better.
+
+![](images/d18a9cb00b55b89741345ed3ddef4f7c61bbe9f1efe908481fe3d15a10abbc54.jpg)  
+Figure 7: Visual explanations generated for three prediction tasks on CelebA dataset. The first column shows the query image, followed by the corresponding generated explanations.
+
+![](images/8eda7eb146d6551ed96fd49d8dad89da2adba615e776ec255810b7881f0d658e.jpg)  
+Figure 8: Visual explanations generated for six prediction tasks on CelebA dataset. The first column shows the query image, followed by the corresponding generated explanations. The values above each image are the output of the classifier  $f$ .
+
+![](images/dba69c7353ec47553dcc40a889cdd6ae5e873b49d23a88bdda0ba845f33aa408.jpg)  
+Figure 9: Plot of the expected outcome from the classifier,  $f(\mathbf{x}) + \delta$ , against the actual response of the classifier on generated explanations,  $f(\mathbf{x}_{\delta})$ . The monotonically increasing trend shows a positive correlation between  $f(\mathbf{x}) + \delta$  and  $f(\mathbf{x}_{\delta})$ , and thus the generated explanations are consistent with the expected condition.
+
+![](images/68805c80cadf9d77a13cfcfbcf47a8b4d9f52ff493a7c51ed7b2061edc2abb89.jpg)  
+Figure 10: The interface for the human evaluation done using Amazon Mechanical Turk (AMT). Task-1 evaluated if humans can detect the relative order between two explanations. Task-2 evaluated if humans can identify the target class for which our model has provided the explanations. Task-3 demonstrated that our model can help the user to identify problems like possible bias in the black-box training.
+
+![](images/c6f7a1cf0d77c2522ebe8088fe3fd88ce9babca66dd92d46b1738cb535c0a59d.jpg)  
+Figure 11: Each cell is the fraction of the generated explanations, that have flipped in source attribute as compared to the query image. The x-axis is source attribute and y-axis is the target attribute for which explanation is generated. Note: This is not a confusion matrix.
+
+![](images/6b6e51c5690cfb2138165f4c29b2fd67b59462e2b59ce334f7311849d168cb7d.jpg)  
+Figure 12: Ablation study to show the effect of KL loss term. Plot of the expected outcome from the classifier,  $f(\mathbf{x}) + \delta$ , against the actual response of the classifier on generated explanations,  $f(\mathbf{x}_{\delta})$ .
\ No newline at end of file
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+# FINITE DEPTH AND WIDTH CORRECTIONS TO THE NEURAL TANGENT KERNEL
+
+# Boris Hanin
+
+Department of Mathematics
+
+Texas A&M University
+
+College Station, TX 77843, USA
+
+bhanin@math.tamu.edu
+
+# Mihai Nica
+
+Department of Mathematics
+
+University of Toronto
+
+Toronto, Canada
+
+mnica@math.utoronto.ca
+
+# ABSTRACT
+
+We prove the precise scaling, at finite depth and width, for the mean and variance of the neural tangent kernel (NTK) in a randomly initialized ReLU network. The standard deviation is exponential in the ratio of network depth to width. Thus, even in the limit of infinite overparameterization, the NTK is not deterministic if depth and width simultaneously tend to infinity. Moreover, we prove that for such deep and wide networks, the NTK has a non-trivial evolution during training by showing that the mean of its first SGD update is also exponential in the ratio of network depth to width. This is sharp contrast to the regime where depth is fixed and network width is very large. Our results suggest that, unlike relatively shallow and wide networks, deep and wide ReLU networks are capable of learning data-dependent features even in the so-called lazy training regime.
+
+# 1 INTRODUCTION
+
+Modern neural networks are typically overparameterized: they have many more parameters than the size of the datasets on which they are trained. That some setting of parameters in such networks can interpolate the data is therefore not surprising. But it is a priori unexpected that not only can such interpolating parameter values can be found by stochastic gradient descent (SGD) on the highly non-convex empirical risk but also that the resulting network function generalizes to unseen data. In an overparameterized neural network  $N(x)$  the individual parameters can be difficult to interpret, and one way to understand training is to rewrite the SGD updates
+
+$$
+\Delta \theta_ {p} = - \lambda \frac {\partial \mathcal {L}}{\partial \theta_ {p}}, p = 1, \ldots , P
+$$
+
+of trainable parameters  $\theta = \{\theta_p\}_{p=1}^P$  with a loss  $\mathcal{L}$  and learning rate  $\lambda$  as kernel gradient descent updates for the values  $N(x)$  of the function computed by the network:
+
+$$
+\Delta N (x) = - \lambda \left\langle K _ {N} (x, \cdot), \nabla \mathcal {L} (\cdot) \right\rangle = - \frac {\lambda}{| \mathcal {B} |} \sum_ {j = 1} ^ {| \mathcal {B} |} K _ {N} (x, x _ {j}) \frac {\partial \mathcal {L}}{\partial \mathcal {N}} (x _ {j}, y _ {j}). \tag {1}
+$$
+
+Here  $\mathcal{B} = \{(x_1, y_1), \ldots, (x_{|\mathcal{B}|}, y_{|\mathcal{B}|})\}$  is the current batch, the inner product is the empirical  $\ell_2$  inner product over  $\mathcal{B}$ , and  $K_N$  is the neural tangent kernel (NTK):
+
+$$
+{K _ {N} (x, x ^ {\prime})} = {\sum_ {p = 1} ^ {P} \frac {\partial N}{\partial \theta_ {p}} (x) \frac {\partial N}{\partial \theta_ {p}} (x ^ {\prime}).}
+$$
+
+Relation (1) is valid to first order in  $\lambda$ . It translates between two ways of thinking about the difficulty of neural network optimization:
+
+(i) The parameter space view where the loss  $\mathcal{L}$ , a complicated function of  $\theta \in \mathbb{R}^{\# \text{parameters}}$ , is minimized using gradient descent with respect to a simple (Euclidean) metric;
+
+(ii) The function space view where the loss  $\mathcal{L}$ , which is a simple function of the network mapping  $x\mapsto N(x)$ , is minimized over the manifold  $M_N$  of all functions representable by the architecture of  $N$  using gradient descent with respect to a potentially complicated Riemannian metric  $K_{N}$  on  $M_N$ .
+
+A remarkable observation of Jacot et al. (2018) is that  $K_N$  simplifies dramatically when the network depth  $d$  is fixed and its width  $n$  tends to infinity. In this setting, by the universal approximation theorem (Cybenko, 1989; Hornik et al., 1989), the manifold  $M_N$  fills out any (reasonable) ambient linear space of functions. The results in Jacot et al. (2018) then show that the kernel  $K_N$  in this limit is frozen throughout training to the infinite width limit of its average  $\mathbb{E}[K_N]$  at initialization, which depends on the depth and non-linearity of  $N$  but not on the dataset.
+
+This mapping between parameter space SGD and kernel gradient descent for a fixed kernel can be viewed as two separate statements. First, at initialization, the distribution of  $K_{N}$  converges in the infinite width limit to the delta function on the infinite width limit of its mean  $\mathbb{E}[K_N]$ . Second, the infinite width limit of SGD dynamics in function space is kernel gradient descent for this limiting mean kernel for any fixed number of SGD iterations. As long as the loss  $\mathcal{L}$  is well-behaved with respect to the network outputs  $N(x)$  and  $\mathbb{E}[K_N]$  is non-degenerate in the subspace of function space given by values on inputs from the dataset, SGD for infinitely wide networks will converge with probability 1 to a minimum of the loss. Further, kernel method-based theorems show that even in this infinitely overparameterized regime neural networks will have non-vacuous guarantees on generalization (Wei et al., 2018). However, as (Wei et al., 2018) shows, the regularized neural networks at finite width can have better sample complexity the corresponding infinite width kernel method.
+
+But replacing neural network training by gradient descent for a fixed kernel in function space is also not completely satisfactory for several reasons. First, it suggests that no feature learning occurs during training for infinitely wide networks in the sense that the kernel  $\mathbb{E}[K_N]$  (and hence its associated feature map) is data-independent. In fact, empirically, networks with finite but large width trained with initially large learning rates often outperform NTK predictions at infinite width (Arora et al., 2019). One interpretation is that, at finite width,  $K_N$  evolves through training, learning data-dependent features not captured by the infinite width limit of its mean at initialization. In part for such reasons, it is important to study both empirically and theoretically finite width corrections to  $K_N$ . Another interpretation is that the specific NTK scaling of weights at initialization (Chizat & Bach, 2018b;a; Mei et al., 2019; 2018; Rotskoff & Vanden-Eijnden, 2018a;b) and the implicit small learning rate limit (Li et al., 2019) obscure important aspects of SGD dynamics. Second, even in the infinite width limit, although  $K_N$  is deterministic, it has no simple analytical formula for deep networks, since it is defined via a layer by layer recursion. In particular, the exact dependence, even in the infinite width limit, of  $K_N$  on network depth is not well understood.
+
+Moreover, the joint statistical effects of depth and width on  $K_N$  in finite size networks remain unclear, and the purpose of this article is to shed light on the simultaneous effects of depth and width on  $K_N$  for finite but large widths  $n$  and any depth  $d$ . Our results apply to fully connected ReLU networks at initialization for which our main contributions are:
+
+1. In contrast to the regime in which the depth  $d$  is fixed but the width  $n$  is large,  $K_{N}$  is not approximately deterministic at initialization so long as  $d / n$  is bounded away from 0. Specifically, for a fixed input  $x$  the normalized on-diagonal second moment of  $K_{N}$  satisfies
+
+$$
+\begin{array}{l} \frac {\mathbb {E} \left[ K _ {N} (x , x) ^ {2} \right]}{\mathbb {E} \left[ K _ {N} (x , x) \right] ^ {2}} \simeq \exp (5 d / n) \left(1 + O (d / n ^ {2})\right). \end{array}
+$$
+
+Thus, when  $d / n$  is bounded away from 0, even when both  $n, d$  are large, the standard deviation of  $K_{N}(x,x)$  is at least as large as its mean, showing that its distribution at initialization is not close to a delta function. See Theorem 1.
+
+2. Moreover, when  $\mathcal{L}$  is the square loss, the average of the SGD update  $\Delta K_N(x,x)$  to  $K_{N}(x,x)$  from a batch of size one containing  $x$  satisfies
+
+$$
+\frac {\mathbb {E} \left[ \Delta K _ {N} (x , x) \right]}{\mathbb {E} \left[ K _ {N} (x , x) \right]} \simeq \frac {d ^ {2}}{n n _ {0}} \exp (5 d / n) \left(1 + O (d / n ^ {2})\right),
+$$
+
+where  $n_0$  is the input dimension. Therefore, if  $d^2 / nn_0 > 0$ , the NTK will have the potential to evolve in a data-dependent way. Moreover, if  $n_0$  is comparable to  $n$  and  $d / n > 0$  then it is possible that this evolution will have a well-defined expansion in  $d / n$ . See Theorem 2.
+
+In both statements above,  $\simeq$  means is bounded above and below by universal constants. We emphasize that our results hold at finite  $d, n$  and the implicit constants in both  $\simeq$  and in the error terms  $O(d / n^2)$  are independent of  $d, n$ . Moreover, our precise results, stated in §2 below, hold for networks with variable layer widths. We have denoted network width by  $n$  only for the sake of exposition. The appropriate generalization of  $d / n$  to networks with varying layer widths is the parameter
+
+$$
+\beta := \sum_ {i = 1} ^ {d} \frac {1}{n _ {j}},
+$$
+
+which in light of the estimates in (1) and (2) plays the role of an inverse temperature.
+
+# 1.1 PRIOR WORK
+
+A number of articles (Bietti & Mairal, 2019; Dyer & Gur-Ari, 2019; Lee et al., 2019; Yang, 2019) have followed up on the original NTK work Jacot et al. (2018). Related in spirit to our results is the work Dyer & Gur-Ari (2019), which uses Feynman diagrams to study finite width corrections to general correlations functions (and in particular the NTK). The most complete results obtained by Dyer & Gur-Ari (2019) are for deep linear networks but a number of estimates hold general non-linear networks as well. The results there, like in essentially all previous work, fix the depth  $d$  and let the layer widths  $n$  tend to infinity. In contrast, our results (as well as those of Hanin (2018); Hanin & Nica (2018); Hanin & Rolnick (2018)), do not treat  $d$  as a constant, suggesting that the  $1/n$  expansions (e.g. in Dyer & Gur-Ari (2019)) can be promoted to  $d/n$  expansions. Also, the sum-over-path approach to studying correlation functions in randomly initialized ReLU nets was previously taken up for the forward pass by Hanin & Rolnick (2018) and for the backward pass by Hanin (2018) and Hanin & Nica (2018). We also point the reader to Theorems 3.1 and 3.2 in Arora et al. (2019), which provide quantitative rates of convergence for both the neural tangent kernel and the resulting full optimization trajectory of neural networks at large but finite width (and fixed depth).
+
+# 2 FORMAL STATEMENT OF RESULTS
+
+Consider a ReLU network  $N$  with input dimension  $n_0$ , hidden layer widths  $n_1, \ldots, n_{d-1}$ , and output dimension  $n_d = 1$ . We will assume that the output layer of  $N$  is linear and initialize the biases in  $N$  to zero. Therefore, for any input  $x \in \mathbb{R}^{n_0}$ , the network  $N$  computes  $N(x) = x^{(d)}$  given by
+
+$$
+x ^ {(0)} = x, \quad y ^ {(i)} := \widehat {W} ^ {(i)} x ^ {(i - 1)}, \quad x ^ {(i)} := \operatorname {R e L U} \left(y ^ {(i)}\right), \quad i = 1, \dots , d, \tag {2}
+$$
+
+where for  $i = 1,\dots ,d - 1$
+
+$$
+\widehat {W} ^ {(d)} := \left(1 / n _ {i - 1}\right) ^ {- 1 / 2} W ^ {(i)}, \quad \widehat {W} ^ {(i)} := \left(2 / n _ {i - 1}\right) ^ {- 1 / 2} W ^ {(i)}, \quad W _ {\alpha , \beta} ^ {(i)} \sim \mu i. i. d., \tag {3}
+$$
+
+and  $\mu$  is a fixed probability measure on  $\mathbb{R}$  that we assume has a density with respect to Lebesgue measure and satisfies:
+
+$$
+\mu \text {i s} \quad \operatorname {V a r} [ \mu ] = 1, \quad \int_ {- \infty} ^ {\infty} x ^ {4} d \mu (x) = \mu_ {4} <   \infty . \tag {4}
+$$
+
+The three assumptions in (4) hold for virtually all standard network initialization schemes with the exception of orthogonal weight initialization. But believe our results extend hold also for this case but not do take up this issue. The on-diagonal NTK is
+
+$$
+K _ {N} (x, x) := \sum_ {j = 1} ^ {d} \sum_ {\alpha = 1} ^ {n _ {j - 1}} \sum_ {\beta = 1} ^ {n _ {j}} \left(\frac {\partial N}{\partial W _ {\alpha , \beta} ^ {(j)}} (x)\right) ^ {2} + \sum_ {j = 1} ^ {d} \sum_ {\beta = 1} ^ {n _ {j}} \left(\frac {\partial N}{\partial b _ {\beta} ^ {(j)}} (x)\right) ^ {2}, \tag {5}
+$$
+
+and we emphasize that although we have initialized the biases to zero, they are not removed from the list of trainable parameters. Our first result is the following:
+
+Theorem 1 (Mean and Variance of NKT on Diagonal at Init). We have
+
+$$
+\mathbb {E} [ K _ {\boldsymbol {N}} (x, x) ] = d \left(\frac {1}{2} + \frac {\| x \| _ {2} ^ {2}}{n _ {0}}\right).
+$$
+
+Moreover, we have that  $\mathbb{E}[K_N(x,x)^2 ]$  is bounded above and below by universal constants times
+
+$$
+\exp \left(5 \beta\right) \left(\frac {d ^ {2} \| x \| _ {2} ^ {4}}{n _ {0} ^ {2}} + \frac {d \| x \| _ {2} ^ {2}}{n _ {0}} \sum_ {j = 1} ^ {d} e ^ {- 5 \sum_ {i = 1} ^ {j} \frac {1}{n _ {i}}} + \sum_ {\stackrel {i, j = 1} {i \leq j}} ^ {d} e ^ {- 5 \sum_ {i = 1} ^ {j} \frac {1}{n _ {i}}}\right), \quad \beta = \sum_ {i = 1} ^ {d} \frac {1}{n _ {i}}
+$$
+
+times a multiplicative error  $\left(1 + O\left(\sum_{i = 1}^{d}\frac{1}{n_i^2}\right)\right)$ . In particular, if all the hidden layer widths are equal (i.e.  $n_i = n$ , for  $i = 1,\ldots ,d - 1$ ), we have
+
+$$
+\frac {\mathbb {E} \left[ K _ {N} (x , x) ^ {2} \right]}{\mathbb {E} \left[ K _ {N} (x , x) \right] ^ {2}} \simeq \exp \left(5 \beta\right) \left(1 + O \left(\beta / n\right)\right), \quad \beta = d / n,
+$$
+
+where  $f \simeq g$  means  $f$  is bounded above and below by universal constants times  $g$ .
+
+This result shows that in the deep and wide double scaling limit
+
+$$
+n _ {i}, d \rightarrow \infty , \quad 0 <   \lim  _ {n _ {i}, d \rightarrow \infty} \sum_ {i = 1} ^ {d} \frac {1}{n _ {i}} <   \infty ,
+$$
+
+the NTK does not converge to a constant in probability. This is contrast to the wide and shallow regime where  $n_i \to \infty$  and  $d < \infty$  is fixed.
+
+Our next result shows that when  $\mathcal{L}$  is the square loss  $K_{N}(x,x)$  is not frozen during training. To state it, fix an input  $x\in \mathbb{R}^{n_0}$  to  $\pmb{N}$  and define  $\Delta K_N(x,x)$  to be the update from one step of SGD with a batch of size 1 containing  $x$  (and learning rate  $\lambda$ ).
+
+Theorem 2 (Mean of Time Derivative of NTK on Diagonal at Init). We have that  $\mathbb{E}\left[\lambda^{-1}\Delta K_N(x,x)\right]$  is bounded above and below by universal constants times
+
+$$
+\left[\frac{\|x\|_{2}^{4}}{n_{0}^{2}}\sum_{\substack{i_{1},i_{2} = 1\\ i_{i} <   i_{2}}}^{d}\sum_{\ell = i_{1}}^{i_{2} - 1}\frac{e^{-5 / n_{\ell} - 6\sum_{i = i_{1}}^{\ell}\frac{1}{n_{i}}}}{n_{\ell}} +\frac{\|x\|_{2}^{2}}{n_{0}}\sum_{\substack{i_{i},i_{2} = 1\\ i_{1} <   i_{2}}}^{d}e^{-5\sum_{i = 1}^{i_{1}}\frac{1}{n_{i}}}\sum_{\ell = i_{1}}^{i_{2} - 1}\frac{e^{-6\sum_{i = i_{1} + 1}^{\ell - 1}\frac{1}{n_{i}}}}{n_{\ell}}\right]\exp \left(5\beta\right)
+$$
+
+times a multiplicative error of size  $\left(1 + O\left(\sum_{i = 1}^{d}\frac{1}{n_i^2}\right)\right)$ , where  $\beta = \sum_{i = 1}^{d}1 / n_{i}$ , as in Theorem I. In particular, if all the hidden layer widths are equal (i.e.  $n_i = n$ , for  $i = 1,\ldots ,d - 1$ ), we find
+
+$$
+\frac {\mathbb {E} \left[ \Delta K _ {N} (x , x) \right]}{\mathbb {E} \left[ K _ {N} (x , x) \right]} \simeq \frac {d \beta}{n _ {0}} \exp \left(5 \beta\right) \left(1 + O \left(\beta / n\right)\right), \quad \beta = d / n.
+$$
+
+Observe that when  $d$  is fixed and  $n_i = n \to \infty$ , the pre-factor in front of  $\exp(5\beta)$  scales like  $1/n$ . This is in keeping with the results from Dyer & Gur-Ari (2019) and Jacot et al. (2018). Moreover, it shows that if  $d, n, n_0$  grow in any way so that  $d\beta / n_0 = d^2 / nn_0 \to 0$ , the update  $\Delta K_N(x, x)$  to  $K_N(x, x)$  from the batch  $\{x\}$  at initialization will have mean 0. It is unclear whether this will be true also for larger batches and when the arguments of  $K_N$  are not equal. In contrast, if  $n_i \simeq n$  and  $\beta = d/n$  is bounded away from  $0, \infty$ , and the  $n_0$  is proportional to  $d$ , the average update  $\mathbb{E}[\Delta K_N(x, x)]$  has the same order of magnitude as  $\mathbb{E}[K_N(x)]$ .
+
+# 2.1 ORGANIZATION FOR THE REST OF THE ARTICLE
+
+The remainder of this article is structured as follows. First, we give an outline of the proofs of Theorems 1 and 2 in §3 and particularly in §3.1, which gives an in-depth but informal explanation of our strategy for computing moments of  $K_N$  and its time derivative. Next, in the Appendix Section §A, we introduce some notation about paths and edges in the computation graph of  $N$ . This
+
+notation will be used in the proofs of Theorems 1 and 2 presented in the Appendix Section §B- $\S$ D. The computations in §B explain how to handle the contribution to  $K_{N}$  and  $\Delta K_{N}$  coming only from the weights of the network. They are the most technical and we give them in full detail. Then, the discussion in §C and §D show how to adapt the method developed in §B to treat the contribution of biases and mixed bias-weight terms in  $K_{N}, K_{N}^{2}$  and  $\Delta K_{N}$ . Since the arguments are simpler in these cases, we omit some details and focus only on highlighting the salient differences.
+
+# 3 OVERVIEW OF PROOF OF THEOREMS 1 AND 2
+
+The proofs of Theorems 1 and 2 are so similar that we will prove them at the same time. In this section and in §3.1 we present an overview of our argument. Then, we carry out the details in Appendix Sections §B-§D below. Fix an input  $x \in \mathbb{R}^{n_0}$  to  $N$ . Recall from (5) that
+
+$$
+K _ {N} (x, x) = K _ {\mathrm {w}} + K _ {\mathrm {b}},
+$$
+
+where we've set
+
+$$
+K _ {\mathrm {w}} := \sum_ {\text {w e i g h t s} w} \left(\frac {\partial N}{\partial w} (x)\right) ^ {2}, \quad K _ {\mathrm {b}} := \sum_ {\text {b i a s e s} b} \left(\frac {\partial N}{\partial b} (x)\right) ^ {2} \tag {6}
+$$
+
+and have suppressed the dependence on  $x, N$ . Similarly, we have
+
+$$
+- \frac {1}{2 \lambda} \Delta K _ {\boldsymbol {N}} (x, x) = \Delta_ {\mathrm {w w}} + 2 \Delta_ {\mathrm {w b}} + \Delta_ {\mathrm {b b}},
+$$
+
+where we have introduced
+
+$$
+\Delta_ {\mathrm {w w}} := \sum_ {\mathrm {w e i g h t s} w, w ^ {\prime}} \frac {\partial \boldsymbol {N}}{\partial w} (x) \frac {\partial^ {2} \boldsymbol {N}}{\partial w \partial w ^ {\prime}} (x) \frac {\partial \boldsymbol {N}}{\partial w ^ {\prime}} (x) (\boldsymbol {N} (x) - \boldsymbol {N} _ {*} (x))
+$$
+
+$$
+\Delta_ {\mathrm {w b}} := \sum_ {\text {w e i g h t} w, \text {b i a s} b} \frac {\partial \boldsymbol {N}}{\partial w} (x) \frac {\partial^ {2} \boldsymbol {N}}{\partial w \partial b} (x) \frac {\partial \boldsymbol {N}}{\partial b} (x) (\boldsymbol {N} (x) - \boldsymbol {N} _ {*} (x))
+$$
+
+$$
+\Delta_ {\mathrm {b b}} := \sum_ {\text {b i a s e s} b, b ^ {\prime}} \frac {\partial \boldsymbol {N}}{\partial b} (x) \frac {\partial^ {2} \boldsymbol {N}}{\partial b \partial b} (x) \frac {\partial \boldsymbol {N}}{\partial b ^ {\prime}} (x) \left(\boldsymbol {N} (x) - \boldsymbol {N} _ {*} (x)\right)
+$$
+
+and have used that the loss on the batch  $\{x\}$  is given by  $\mathcal{L}(x) = \frac{1}{2} (N(x) - N_{*}(x))^{2}$  for some target value  $N_{*}(x)$ . To prove Theorem 1 we must estimate the following quantities:
+
+$$
+\mathbb {E} \left[ K _ {\mathrm {w}} \right], \quad \mathbb {E} \left[ K _ {\mathrm {b}} \right], \quad \mathbb {E} \left[ K _ {\mathrm {w}} ^ {2} \right], \quad \mathbb {E} \left[ K _ {\mathrm {w}} K _ {\mathrm {b}} \right], \quad \mathbb {E} \left[ K _ {\mathrm {b}} ^ {2} \right].
+$$
+
+To prove Theorem 2, we must control in addition
+
+$$
+\mathbb {E} [ \Delta_ {\mathrm {w w}} ], \quad \mathbb {E} [ \Delta_ {\mathrm {w b}} ], \quad \mathbb {E} [ \Delta_ {\mathrm {b b}} ].
+$$
+
+The most technically involved computations will turn out to be those involving only weights: namely, the terms  $\mathbb{E}[K_{\mathrm{w}}],\mathbb{E}[K_{\mathrm{w}}^2 ],\mathbb{E}[\Delta_{\mathrm{ww}}]$ . These terms are controlled by writing each as a sum over certain paths  $\gamma$  that traverse the network from the input to the output layers. The corresponding results for terms involving the bias will then turn out to be very similar but with paths that start somewhere in the middle of network (corresponding to which bias term was used to differentiate the network output). The main result about the pure weight contributions to  $K_N$  is the following
+
+Proposition 3 (Pure weight moments for  $K_N, \Delta K_N$ ). We have
+
+$$
+\mathbb {E} \left[ K _ {\mathrm {w}} \right] = \frac {d}{n _ {0}} \| x \| _ {2} ^ {2}.
+$$
+
+Moreover,
+
+$$
+\mathbb {E} [ K _ {\mathrm {w}} ^ {2} ] \simeq \frac {d ^ {2}}{n _ {0} ^ {2}} \| x \| _ {2} ^ {4} \exp {(5 \beta)} \left(1 + O \left(\sum_ {i = 1} ^ {d} \frac {1}{n _ {i} ^ {2}}\right)\right), \quad \beta := \sum_ {i = 1} ^ {d} \frac {1}{n _ {i}}.
+$$
+
+Finally,
+
+$$
+\mathbb {E} [ \Delta_ {\mathrm {w w}} ] \simeq \frac {\| x \| _ {2} ^ {4}}{n _ {0} ^ {2}} \left[ \sum_ {\substack {i _ {1}, i _ {2} = 1 \\ i _ {i} <   i _ {2}}} ^ {d} \sum_ {\ell = i _ {1}} ^ {i _ {2} - 1} \frac {1}{n _ {\ell}} e ^ {- 5 / n _ {\ell} - 6} \sum_ {i = i _ {1}} ^ {\ell - 1} \frac {1}{n _ {i}} \right] \exp {(5 \beta)} \left(1 + O \left(\sum_ {i = 1} ^ {d} \frac {1}{n _ {i} ^ {2}}\right)\right).
+$$
+
+We prove Proposition 3 in §B below. The proof already contains all the ideas necessary to treat the remaining moments. In §C and §D we explain how to modify the proof of Proposition 3 to prove the following two Propositions:
+
+Proposition 4 (Pure bias moments for  $K_N, \Delta K_N$ ). We have
+
+$$
+\mathbb {E} [ K _ {\mathrm {b}} ] = \frac {d}{2}.
+$$
+
+Moreover,
+
+$$
+\mathbb {E} [ K _ {\mathrm {b}} ^ {2} ] \simeq \left[ \sum_ {i, j = 1 \atop i \leq j} ^ {d} e ^ {- 5 \sum_ {\ell = 1} ^ {j} \frac {1}{n _ {i}}} \right] \exp \left(5 \sum_ {i = 1} ^ {d} \frac {1}{n _ {i}}\right) \left(1 + O \left(\sum_ {i = 1} ^ {d} \frac {1}{n _ {i} ^ {2}}\right)\right).
+$$
+
+Finally, with probability 1, we have  $\Delta_{\mathrm{bb}} = 0$
+
+Proposition 5 (Mixed bias-weight moments for  $K_N, \Delta K_N$ ). We have
+
+$$
+\mathbb {E} [ K _ {\mathrm {b}} K _ {\mathrm {w}} ] \simeq \frac {d \| x \| _ {2} ^ {2}}{n _ {0}} \left[ \sum_ {j = 1} ^ {d} e ^ {- 5 \sum_ {i = 1} ^ {j} \frac {1}{n _ {i}}} \right] \exp \left(5 \sum_ {i = 1} ^ {d} \frac {1}{n _ {i}}\right) \left(1 + O \left(\sum_ {i = 1} ^ {d} \frac {1}{n _ {i} ^ {2}}\right)\right).
+$$
+
+Further,  $\mathbb{E}[\Delta_{\mathrm{wb}}]$  is bounded above and below by universal constants times
+
+$$
+\frac{\|x\|_{2}^{2}}{n_{0}}\exp \left(5\sum_{i = 1}^{d}\frac{1}{n_{i}}\right)\left[\sum_{\substack{i,j = 1\\ j <   i}}^{d}e^{-5\sum_{\alpha = 1}^{j}\frac{1}{n_{\alpha}}}\sum_{\ell = j}^{i - 1}\frac{1}{n_{\ell}} e^{-6\sum_{\alpha = j + 1}^{\ell -1}\frac{1}{n_{\alpha}}}\right]\left(1 + O\left(\sum_{i = 1}^{d}\frac{1}{n_{i}^{2}}\right)\right).
+$$
+
+The statements in Theorems 1 and 2 that hold for general  $n_i$  now follow directly from Propositions 3-5. The asymptotics when  $n_i \simeq n$  follow from some routine algebra.
+
+# 3.1 IDEA OF PROOF OF PROPOSITIONS 3-5
+
+Before turning to the details of the proof of Propositions 3-5 below, we give an intuitive explanation of the key steps in our sum-over-path analysis of the moments of  $K_{\mathrm{w}}$ ,  $K_{\mathrm{b}}$ ,  $\Delta_{\mathrm{ww}}$ ,  $\Delta_{\mathrm{wb}}$ ,  $\Delta_{\mathrm{bb}}$ . Since the proofs of all three Propositions follow a similar structure and Proposition 3 is the most complicated, we will focus on explaining how to obtain the first 2 moments of  $K_{\mathrm{w}}$ . Since the biases are initialized to zero and  $K_{\mathrm{w}}$  involves only derivatives with respect to the weights, for the purposes of analyzing  $K_{\mathrm{w}}$  the biases play no role. Without the biases, the output of the neural network,  $N(x)$  can be express as a weighted sum over paths in the computational graph of the network:
+
+$$
+\boldsymbol {N} (\boldsymbol {x}) = \sum_ {a = 1} ^ {n _ {0}} x _ {a} \sum_ {\gamma \in \Gamma_ {a} ^ {1}} \mathrm {w t} (\gamma),
+$$
+
+where  $\Gamma_{a}^{1}$  is the collection of paths in  $N$  starting at neuron  $a$  and the weight of a path  $\mathrm{wt}(\gamma)$  is defined in (13) in the Appendix and includes both the product of the weights along  $\gamma$  and the condition that every neuron in  $\gamma$  is open at  $x$ . The path  $\gamma$  begins at some neuron in the input layer of  $N$  and passes through a neuron in every subsequent layer until ending up at the unique neuron in the output layer (see (10)). Being a product over edge weights in a given path, the derivative of  $\mathrm{wt}(\gamma)$  with respect to a weight  $W_{e}$  on an edge  $e$  of the computational graph of  $N$  is:
+
+$$
+\frac {\partial \operatorname {w t} (\gamma)}{\partial W _ {e}} = \frac {\operatorname {w t} (\gamma)}{W _ {e}} \mathbf {1} _ {\{e \in \gamma \}}. \tag {7}
+$$
+
+There is a subtle point here that  $\mathrm{wt}(\gamma)$  also involves indicator functions of the events that neurons along  $\gamma$  are open at  $x$ . However, with probability 1, the derivative with respect to  $W_{e}$  of these indicator functions is identically 0 at  $x$ . The details are in Lemma 11.
+
+Because  $K_{\mathrm{w}}$  is a sum of derivatives squared (see (6)), ignoring the dependence on the network input  $x$ , the kernel  $K_{\mathrm{w}}$  roughly takes the form
+
+$$
+K _ {\mathrm {w}} \sim \sum_ {\gamma_ {1}, \gamma_ {2}} \sum_ {e \in \gamma_ {1} \cap \gamma_ {2}} \frac {\prod_ {k = 1} ^ {2} \mathrm {w t} (\gamma_ {k})}{W _ {e} ^ {2}},
+$$
+
+where the sum is over collections  $(\gamma_{1},\gamma_{2})$  of two paths in the computation graph of  $N$  and edges  $e$  in the computational graph of  $N$  that lie on both (see Lemma 6 for the precise statement). When computing the mean,  $\mathbb{E}[K_{\mathrm{w}}]$ , by the mean zero assumption of the weights  $W_{e}$  (see (4)), the only contribution is when every edge in the computational graph of  $N$  is traversed by an even number of paths. Since there are exactly two paths, the only contribution is when the two paths are identical, dramatically simplifying the problem. This gives rise to the simple formula for  $\mathbb{E}[K_{\mathrm{w}}]$  (see (23)). The expression
+
+$$
+K_{\mathrm{w}}^{2}\sim \sum_{\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4}}\sum_{\substack{e_{1}\in \gamma_{1}\cap \gamma_{2}\\ e_{2}\in \gamma_{3}\cap \gamma_{4}}}\frac{\prod_{k = 1}^{4}\mathrm{wt}(\gamma_{k})}{W_{e_{1}}^{2}W_{e_{2}}^{2}},
+$$
+
+for  $K_{\mathrm{w}}^{2}$  is more complex. It involves sums over four paths in the computational graph of  $N$  as in the second statement of Lemma 6. Again recalling that the moments of the weights have mean 0, the only collections of paths that contribute to  $\mathbb{E}[K_{\mathrm{w}}^{2}]$  are those in which every edge in the computational graph of  $N$  is covered an even number of times:
+
+$$
+\mathbb {E} \left[ K _ {\mathrm {w}} ^ {2} \right] \sim \sum_ {\substack {\gamma_ {1}, \gamma_ {2}, \gamma_ {3}, \gamma_ {4} \\ \text {even}}} \sum_ {\substack {e _ {1} \in \gamma_ {1} \cap \gamma_ {2} \\ e _ {2} \in \gamma_ {3} \cap \gamma_ {4}}} \mathbb {E} \left[ \frac {\prod_ {k = 1} ^ {4} \operatorname{wt} (\gamma_ {k})}{W _ {e _ {1}} ^ {2} W _ {e _ {2}} ^ {2}} \right] \tag{8}
+$$
+
+However, there are now several ways the four paths can interact to give such a configuration. It is the combinatorics of these interactions, together with the stipulation that the marked edges  $e_1, e_2$  belong to particular pairs of paths, which complicates the analysis of  $\mathbb{E}[K_{\mathrm{w}}^2]$ . We estimate this expectation in several steps:
+
+1. Obtain an exact formula for the expectation in (8):
+
+$$
+\mathbb {E} \left[ \frac {\prod_ {k = 1} ^ {4} \mathrm {w t} \left(\gamma_ {k}\right)}{W _ {e _ {1}} ^ {2} W _ {e _ {2}} ^ {2}} \right] = F (\Gamma , e _ {1}, e _ {2}),
+$$
+
+where  $F(\Gamma, e_1, e_2)$  is the product over the layers  $\ell = 1, \ldots, d$  in  $N$  of the "cost" of the interactions of  $\gamma_1, \ldots, \gamma_4$  between layers  $\ell - 1$  and  $\ell$ . The precise formula is in Lemma 7.
+
+2. Observe the dependence of  $F(\Gamma, e_1, e_2)$  on  $e_1, e_2$  is only up to a multiplicative constant:
+
+$$
+F (\Gamma , e _ {1}, e _ {2}) \simeq F _ {*} (\Gamma).
+$$
+
+The precise relation is (24). This shows that, up to universal constants,
+
+$$
+\mathbb{E}[K_{\mathrm{w}}^{2}]\simeq \sum_{\substack{\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4}\\ \text{even}}}F_{*}(\Gamma)\# \left\{\ell_{1},\ell_{2}\in [d]\big|\begin{array}{c}\gamma_{1},\gamma_{2}\text{togethe at layer}\ell_{1}\\ \gamma_{3},\gamma_{4}\text{togethe at layer}\ell_{2} \end{array} \right\} .
+$$
+
+This is captured precisely by the terms  $I_{j},II_{j}$  defined in (27),(28).
+
+3. Notice that  $F_{*}(\Gamma)$  depends only on the un-ordered multiset of edges  $E = E^{\Gamma} \in \Sigma_{even}^{4}$  determined by  $\Gamma$  (see (17) for a precise definition). We therefore change variables in the sum from the previous step to find
+
+$$
+\mathbb{E}[K_{\mathrm{w}}^{2}]\simeq \sum_{E\in \Sigma_{even}^{4}}F_{*}(E)\mathrm{Jacobian}(E,e_{1},e_{2}),
+$$
+
+where  $\operatorname{Jacobian}(E, e_1, e_2)$  counts how many collections of four paths  $\Gamma \in \Gamma_{even}^4$  that have the same  $E^\Gamma$  also have paths  $\gamma_1, \gamma_2$  pass through  $e_1$  and paths  $\gamma_3, \gamma_4$  pass through  $e_2$ . Lemma 8 gives a precise expression for this Jacobian. It turns out, as explained just below Lemma 8, that
+
+$$
+\operatorname {J a c o b i a n} (E, e _ {1}, e _ {2}) \simeq 6 ^ {\# \text {l o o p s} (E)},
+$$
+
+where a loop in  $E$  occurs when the four paths interact. More precisely, a loop occurs whenever all four paths pass through the same neuron in some layer (see Figures 1 and 2).
+
+4. Change variables from unordered multisets of edges  $E \in \Sigma_{\text{even}}^4$  in which every edge is covered an even number of times to pairs of paths  $V \in \Gamma^2$ . The Jacobian turns out to be  $2^{-\# \text{loops}(E)}$  (Lemma 9), giving
+
+$$
+\mathbb {E} \left[ K _ {\mathrm {w}} ^ {2} \right] \simeq \sum_ {V \in \Gamma^ {2}} F _ {*} (V) 3 ^ {\# \text {l o o p s} (V)}.
+$$
+
+5. Just like  $F_{*}(V)$ , the term  $3^{\# \mathrm{loops}(V)}$  is again a product over layers  $\ell$  in the computational graph of  $N$  of the "cost" of interactions between our four paths. Aggregating these two terms into a single functional  $\widehat{F}_{*}(E)$  and factoring out the  $1 / n_{\ell}$  terms in  $F_{*}(V)$  we find that:
+
+$$
+\mathbb {E} [ K _ {\mathrm {w}} ^ {2} ] \simeq \frac {1}{n _ {0} ^ {2}} \mathcal {E} \left[ \widehat {F} _ {*} (V) \right],
+$$
+
+where the  $1 / n_{\ell}$  terms cause the sum to become an average over collections  $V$  of two independent paths in the computational graph of  $N$ , with each path sampling neurons uniformly at random in every layer. The precise result, including the dependence on the input  $x$ , is in (42).
+
+6. Finally, we use Proposition 10 to obtain for this expectation estimates above and below that match up multiplicative constants.
+
+![](images/133ab8837042f39a112b9b5a1b3b3962a4446936e992ec716942a4acb087b377.jpg)  
+Figure 1: Cartoon of the four paths  $\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4}$  between layers  $\ell_1$  and  $\ell_2$  in the case where there is no interaction. Paths stay with there original partners  $\gamma_{1}$  with  $\gamma_{2}$  and  $\gamma_{3}$  with  $\gamma_{4}$  at all intermediate layers.
+
+![](images/1e8d042286c1c28b524249015f7535f819b3c2cfb3c1dd43aace3fd4861998a8.jpg)  
+Figure 2: Cartoon of the four paths  $\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4}$  between layers  $\ell_1$  and  $\ell_2$  in the case where there is exactly one "loop" interaction between the marked layers. Paths swap away from their original partners exactly once at some intermediate layer after  $\ell_1$ , and then swap back to their original partners before  $\ell_2$ .
+
+# 3.2 CONCLUSION
+
+Taken together Theorems 1 and 2 show that in fully connected ReLU nets that are both deep and wide the neural tangent kernel  $K_{N}$  is genuinely stochastic and enjoys a non-trivial evolution during training. This suggests that in the overparameterized limit  $n, d \to \infty$  with  $d / n \in (0, \infty)$ , the kernel  $K_{N}$  may learn data-dependent features. Moreover, our results show that the fluctuations of both  $K_{N}$  and its time derivative are exponential in the inverse temperature  $\beta = d / n$ .
+
+It would be interesting to obtain an exact description of its statistics at initialization and to describe the law of its trajectory during training. Assuming this trajectory turns out to be data-dependent,
+
+our results suggest that the double descent curve Belkin et al. (2018; 2019); Spigler et al. (2018) that trades off complexity vs. generalization error may display significantly different behaviors depending on the mode of network overparameterization.
+
+However, it is also important to point out that the results in Hanin (2018); Hanin & Nica (2018); Hanin & Rolnick (2018) show that, at least for fully connected ReLU nets, gradient-based training is not numerically stable unless  $d / n$  is relatively small (but not necessarily zero). Thus, we conjecture that there may exist a "weak feature learning" NTK regime in which network depth and width are both large but  $0 < d / n \ll 1$ . In such a regime, the network will be stable enough to train but flexible enough to learn data-dependent features. In the language of Chizat & Bach (2018b) one might say this regime displays weak lazy training in which the model can still be described by a stochastic positive definite kernel whose fluctuations can interact with data.
+
+Finally, it is an interesting question to what extent our results hold for non-linearities other than ReLU and for network architectures other than fully connected (e.g. convolutional and residual). Even in fully connected networks, the input/output Jacobian already displays different spectral statistics depending on the non-linearity (Pennington et al., 2017). Moreover, typical ConvNets, are significantly wider than they are deep, and we leave it to future work to adapt the techniques from the present article to these more general settings (the neural tangent kernel for finite depth convolutional networks is studied in part in (Arora et al., 2019)).
+
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+Grant M Rotskoff and Eric Vanden-Eijnden. Neural networks as interacting particle systems: Asymptotic convexity of the loss landscape and universal scaling of the approximation error. arXiv preprint arXiv:1805.00915, 2018b.  
+Stefano Spigler, Mario Geiger, Stephane d'Ascoli, Levent Sagun, Giulio Biroli, and Matthieu Wyart. A jamming transition from under-to over-parametrization affects loss landscape and generalization. arXiv preprint arXiv:1810.09665, 2018.  
+Colin Wei, Jason Lee, Qiang Liu, and Tengyu Ma. Regularization matters: Generalization and optimization of neural nets v.s. their induced kernel. arXiv preprint arXiv:1810.05369, 2018.  
+Greg Yang. Scaling limits of wide neural networks with weight sharing: Gaussian process behavior, gradient independence, and neural tangent kernel derivation. arXiv preprint arXiv:1902.04760, 2019.
+
+# A NOTATION
+
+In this section, we introduce some notation, adapted in large part from Hanin & Nica (2018), that will be used in the proofs of Theorems 1 and 2. For  $n \in \mathbb{N}$ , we will write
+
+$$
+[ n ] := \{1, \dots , n \}.
+$$
+
+It will also be convenient to denote
+
+$$
+[ n ] _ {e v e n} ^ {k} := \{a \in [ n ] ^ {k} \mid \text {e v e r y e n t r y i n a p p a r s e a n e v e n n u m b e r o f t i m e s} \}.
+$$
+
+Given a ReLU network  $N$  with input dimension  $n_0$ , hidden layer widths  $n_1, \ldots, n_{d-1}$ , and output dimension  $n_d = 1$ , its computational graph is a directed multipartite graph whose vertex set is the disjoint union  $[n_0] \coprod \cdots \coprod [n_d]$  and in which edges are all possible ways of connecting vertices from  $[n_{i-1}]$  with vertices from  $[n_i]$  for  $i = 1, \ldots, d$ . The vertices are the neurons in  $N$ , and we will write for  $\ell \in \{0, \ldots, d\}$  and  $\alpha \in [n_\ell]$
+
+$$
+z (\ell , \alpha) := \text {n e u r o n n u m b e r} \alpha \text {i n l a y e r} \ell . \tag {9}
+$$
+
+Definition 1 (Path in the computational graph of  $N$ ). Given  $0 \leq \ell_1 < \ell_2 \leq d$  and  $\alpha_1 \in [n_{\ell_1}]$ ,  $\alpha_2 \in [n_{\ell_2}]$ , a path  $\gamma$  in the computational graph of  $N$  from neuron  $z(\ell_1, \alpha_1)$  to neuron  $z(\ell_2, \alpha_2)$  is a collection of neurons in layers  $\ell_1, \ldots, \ell_2$ :
+
+$$
+\gamma = (\gamma (\ell_ {1}), \dots , \gamma (\ell_ {2})), \quad \gamma (j) \in [ n _ {j} ], \quad \gamma (\ell_ {1}) = \alpha_ {1}, \gamma (\ell_ {2}) = \alpha_ {2}. \tag {10}
+$$
+
+Further, we will write
+
+$$
+Z ^ {k} = \left\{\left(z _ {1}, \dots , z _ {k}\right) \mid z _ {j} \text {a r e n e u r o n s i n} N \right\}.
+$$
+
+Given a collection of neurons
+
+$$
+Z = (z (\ell_ {1}, \alpha_ {1}), \dots , z (\ell_ {k}, \alpha_ {k})) \in Z ^ {k}
+$$
+
+we denote by
+
+$$
+\Gamma_ {Z} ^ {k} := \left\{(\gamma_ {1}, \ldots , \gamma_ {k}) \mid \begin{array}{c} \gamma_ {j} \text {i s a p a t h s t a r t i n g a t n e u r o n} z (\ell_ {j}, \alpha_ {j}) \\ \text {e n d i n g a t t h e o u t p u t n e u r o n} z (d, 1) \end{array} \right\}
+$$
+
+Note that with this notation, we have  $\gamma_{i}\in \Gamma^{1}_{z(\ell_{i},\alpha_{i})}$  for each  $i = 1,\ldots ,k$ . For  $\Gamma \in \Gamma_Z^k$ , we also set
+
+$$
+\Gamma (\ell) = \{\alpha \in [ n _ {\ell} ] \mid \exists j \in [ k ] \mathrm {s . t .} \gamma_ {j} (\ell) = k \}.
+$$
+
+Correspondingly, we will write
+
+$$
+| \Gamma (\ell) | := \# \text {d i s t i n c t e l e m e n t s i n} \Gamma (\ell). \tag {11}
+$$
+
+If each edge  $e$  in the computational graph of  $N$  is assigned a weight  $\widehat{W}_e$ , then associated to a path  $\gamma$  is a collection of weights:
+
+$$
+\widehat {W} _ {\gamma} ^ {(i)} := \widehat {W} _ {(\gamma (i - 1), \gamma (i))}. \tag {12}
+$$
+
+Definition 2 (Weight of a path in the computational graph of  $N$ ). Fix  $0 \leq \ell \leq d$ , and let  $\gamma$  be a path in the computation graph of  $N$  starting at layer  $\ell$  and ending at the output. The weight of a this path at a given input  $x$  to  $N$  is
+
+$$
+\operatorname {w t} (\gamma) := \widehat {W} _ {\gamma} ^ {(d)} \prod_ {j = \ell + 1} ^ {d - 1} \widehat {W} _ {\gamma} ^ {(j)} \mathbf {1} _ {\{\gamma \text {o p e n a t} x \}}, \tag {13}
+$$
+
+where
+
+$$
+\mathbf {1} _ {\{\gamma \text {o p e n a t} x \}} = \prod_ {i = \ell} ^ {d} \xi_ {\gamma} ^ {(i)} (x), \qquad \xi_ {\gamma} ^ {(\ell)} (x) := \mathbf {1} _ {\{y _ {\gamma} ^ {(\ell)} > 0 \}},
+$$
+
+is the event that all neurons along  $\gamma$  are open for the input  $x$ . Here  $y^{(\ell)}$  is as in (2).
+
+Next, for an edge  $e \in [n_{i-1}] \times [n_i]$  in the computational graph of  $N$  we will write
+
+$$
+\ell (e) = i \tag {14}
+$$
+
+for the layer of  $e$ . In the course of proving Theorems 1 and 2, it will be useful to associate to every  $\Gamma \in \Gamma^k(\vec{n})$  an unordered multi-set of edges  $E^\Gamma$ .
+
+Definition 3 (Unordered multiset of edges and their endpoints). For  $n, n', \ell \in \mathbb{N}$  set
+
+$$
+\Sigma^ {k} (n, n ^ {\prime}) = \left\{\left(\alpha_ {1}, \beta_ {1}\right), \dots , \left(\alpha_ {k}, \beta_ {k}\right) \mid \left(\alpha_ {j}, \beta_ {j}\right) \in [ n ] \times [ n ^ {\prime} ] \right\}
+$$
+
+to be the unordered multiset of edges in the complete directed bi-paritite graph  $K_{n,n'}$  oriented from  $[n]$  to  $[n']$ . For every  $E \in \Sigma^k(n, n')$  define its left and right endpoints to be
+
+$$
+L (E) := \{\alpha \in [ n ] \mid \exists j = 1, \dots , k s. t. \alpha = \alpha_ {j} \} \tag {15}
+$$
+
+$$
+R (E) := \left\{\beta \in [ n ^ {\prime} ] \mid \exists j = 1, \dots , k s. t. \beta = \beta_ {j} \right\}, \tag {16}
+$$
+
+where  $L(E),R(E)$  are unordered multi-sets.
+
+Using this notation, for any collection  $Z = (z(\ell_1, \alpha_1), \ldots, z(\ell_k, \alpha_k))$  of neurons and  $\Gamma = (\gamma_1, \ldots, \gamma_k) \in \Gamma_Z^k$ , define for each  $\ell \in [d]$  the associated unordered multiset
+
+$$
+E ^ {\Gamma} (\ell) := \left\{(\alpha , \beta) \in [ n _ {\ell - 1}, n _ {\ell} ] \mid \exists j = 1, \dots , k \text {s . t .} \gamma_ {j} (\ell - 1) = \alpha , \gamma_ {j} (\ell) = \beta \right\}
+$$
+
+of edges between layers  $\ell - 1$  and  $\ell$  that are present in  $\Gamma$ . Similarly, we will write
+
+$$
+\Sigma_ {Z} ^ {k} := \left\{\left(E (0), \dots , E (d)\right) \in \Sigma^ {k} \left(n _ {0}, n _ {1}\right) \times \dots \times \Sigma^ {k} \left(n _ {d - 1}, n _ {d}\right) \mid \exists \Gamma \in \Gamma_ {Z} ^ {k} \text {s . t .} E (\ell) = E ^ {\Gamma} (\ell), \ell \in [ d ] \right\} \tag {17}
+$$
+
+for the set of all possible edge multisets realized by paths in  $\Gamma_Z^k$ . On a number of occasions, we will also write
+
+$$
+\Sigma_ {Z, e v e n} ^ {k} := \{E \in \Sigma_ {Z} ^ {k} \mid \text {e v e r y e d g e i n E a p p a r s e a n e v e n n u m b e r o f t i m e s} \}
+$$
+
+and correspondingly
+
+$$
+\Gamma_ {Z, e v e n} ^ {k} := \{\Gamma \in \Gamma_ {Z} ^ {k} \mid E ^ {\Gamma} \in \Sigma_ {Z, e v e n} ^ {k} \}.
+$$
+
+We will moreover say that for a path  $\gamma$  an edge  $e = (\alpha, \beta) \in [n_{i-1}] \times [n_i]$  in the computational graph of  $N$  belongs to  $\gamma$  (written  $e \in \gamma$ ) if
+
+$$
+\gamma (i - 1) = \alpha , \quad \gamma (i) = \beta . \tag {18}
+$$
+
+Finally, for an edge  $e = (\alpha, \beta) \in [n_{i-1}] \times [n_i]$  in the computational graph of  $N$ , we set
+
+$$
+W _ {e} = W _ {\alpha , \beta} ^ {(i)}, \quad \widehat {W} _ {e} = \widehat {W} _ {\alpha , \beta} ^ {(i)}
+$$
+
+for the normalized and unnormalized weights on the edge corresponding to  $e$  (see (3)).
+
+# B PROOF OF PROPOSITION 3
+
+We begin with the well-known formula for the output of a ReLU net  $N$  with biases set to 0 and a linear final layer with one neuron:
+
+$$
+N (x) = \sum_ {a = 1} ^ {n _ {0}} x _ {a} \sum_ {\gamma \in \Gamma_ {a} ^ {1}} \operatorname {w t} (\gamma). \tag {19}
+$$
+
+The weight of a path  $\mathrm{wt}(\gamma)$  was defined in (13) and includes both the product of the weights along  $\gamma$  and the condition that every neuron in  $\gamma$  is open at  $x$ . As explained in §A, the inner sum in (19) is over paths  $\gamma$  in the computational graph of  $N$  that start at neuron  $a$  in the input layer and end at the output neuron and the random variables  $\widehat{W}_{\gamma}^{(i)}$  are the normalized weights on the edge of  $\gamma$  between layer  $i - 1$  and layer  $i$  (see (12)). Differentiating this formula gives sum-over-path expressions for the derivatives of  $N$  with respect to both  $x$  and its trainable parameters. For the NTK and its first SGD update, the result is the following:
+
+Lemma 6 (weight contribution to  $K_N$  and  $\Delta K_N$  as a sum-over-paths). With probability 1,
+
+$$
+K_{\mathrm{w}} = \sum_{a\in [n_{0}]^{2}}\prod_{k = 1}^{2}x_{a_{k}}\sum_{\substack{\Gamma \in \Gamma_{a}^{2}\\ \Gamma = (\gamma_{1},\gamma_{2})}}\sum_{e\in \gamma_{1}\cap \gamma_{2}}\frac{\prod_{k = 1}^{2}\mathrm{wt}(\gamma_{k})}{W_{e}^{2}},
+$$
+
+where the sum is over collections  $\Gamma$  of two paths in the computation graph of  $N$  and edges  $e$  that lie on both paths. Similarly, almost surely,
+
+$$
+K_{\mathrm{w}}^{2} = \sum_{a\in [n_{0}]^{4}}\prod_{k = 1}^{4}x_{a_{k}}\sum_{\substack{\Gamma \in \Gamma^ {4}_{a}(\vec{n})\\ \Gamma = (\gamma_{1},\ldots ,\gamma_{4})}}\sum_{\substack{e_{1}\in \gamma_{1}\cap \gamma_{2}\\ e_{2}\in \gamma_{3},\gamma_{4}}}\frac{\prod_{k = 1}^{4}\mathrm{wt}(\gamma_{k})}{W^{2}_{e_{1}}W^{2}_{e_{2}}},
+$$
+
+and
+
+$$
+\Delta_{\mathrm{ww}} = \sum_{a\in [n_{0}]^{4}}\prod_{k = 1}^{4}x_{a_{k}}\sum_{\substack{\Gamma \in \Gamma^ {4}_{a}(\vec{n})\\ \Gamma = (\gamma_{1},\ldots ,\gamma_{4})}}\sum_{\substack{e_{1}\in \gamma_{1}\cap \gamma_{2}\\ e_{2}\in \gamma_{2},\gamma_{3}\\ e_{1}\neq e_{2}}}\frac{\prod_{k = 1}^{4}\mathrm{wt}(\gamma_{k})}{W_{e_{1}}^{2}W_{e_{2}}^{2}}
+$$
+
+plus a term that has mean 0.
+
+The notation  $[n_0]^k$ ,  $\Gamma_a^k$ ,  $e \in \gamma$ , etc is defined in §A. We prove Lemma 6 in §B.1 below. Let us emphasize that the expressions for  $K_{\mathrm{w}}^{2}$  and  $\Delta_{\mathrm{ww}}$  are almost identical. The main difference is that in the expression for  $\Delta_{\mathrm{ww}}$ , the second path  $\gamma_{2}$  must contain both  $e_1$  and  $e_2$  while  $\gamma_{4}$  has no restrictions. Hence, while for  $K_{\mathrm{w}}^{2}$  the contribution from a collection of four paths  $\Gamma = (\gamma_1, \gamma_2, \gamma_3, \gamma_4)$  is the same as from the collection  $\Gamma' = (\gamma_2, \gamma_1, \gamma_4, \gamma_3)$ , for  $\Delta_{\mathrm{ww}}$  the contributions are different. This seemingly small discrepancy, as we shall see, causes the normalized expectation  $\mathbb{E}[\Delta_{\mathrm{ww}}] / \mathbb{E}[K_{\mathrm{w}}]$  to converge to zero when  $d < \infty$  is fixed and  $n_i \to \infty$  (see the  $1/n_{\ell}$  factors in the statement of Theorem 2). In contrast, in the same regime, the normalized second moment  $\mathbb{E}[K_{\mathrm{w}}^2] / \mathbb{E}[K_{\mathrm{w}}]^2$  remains bounded away from zero as in the statement of Theorem 1. Both statements are consistent with prior results in the literature Dyer & Gur-Ari (2019); Jacot et al. (2018). Taking expectations in Lemma 6 yields the following result.
+
+Lemma 7 (Expectation of  $K_{\mathrm{w}}$ ,  $K_{\mathrm{w}}^{2}$ ,  $\Delta_{\mathrm{ww}}$  as sums over 2, 4 paths). We have,
+
+$$
+\mathbb {E} \left[ K _ {\mathrm {w}} \right] = \sum_ {a \in \left[ n _ {0} \right] _ {e v e n} ^ {2}} \prod_ {k = 1} ^ {2} x _ {a _ {k}} \sum_ {\substack {\Gamma_ {a} \in \Gamma_ {e v e n} ^ {2} \\ \Gamma = (\gamma_ {1}, \gamma_ {2})}} \sum_ {e \in \gamma_ {1} \cap \gamma_ {2}} H (\Gamma , e) \tag{20}
+$$
+
+where
+
+$$
+H (\Gamma , e) = \mathbf {1} _ {\{\gamma_ {1} = \gamma_ {2} \}} \prod_ {i = 1} ^ {d} \frac {1}{n _ {i - 1}}
+$$
+
+Similarly,
+
+$$
+\mathbb {E} \left[ K _ {\mathrm {w}} ^ {2} \right] = \sum_ {a \in \left[ n _ {0} \right] _ {e v e n} ^ {4}} \prod_ {k = 1} ^ {4} x _ {a _ {k}} \sum_ {\substack {\Gamma \in \Gamma_ {a, e v e n} ^ {4} \\ \Gamma = \left(\gamma_ {1}, \dots , \gamma_ {4}\right)}} \sum_ {\substack {e _ {1} \in \gamma_ {1} \cap \gamma_ {2} \\ e _ {2} \in \gamma_ {3} \cap \gamma_ {4}}} F (\Gamma , e _ {1}, e _ {2}), \tag{21}
+$$
+
+where
+
+$$
+F (\Gamma , e _ {1}, e _ {2}) = \frac {1}{2} \prod_ {i = 1} ^ {d} \frac {2 ^ {2 - | \Gamma (i) |}}{n _ {i - 1} ^ {2}} \prod_ {i \neq \ell (e _ {1}), \ell (e _ {2})} \mu_ {4} ^ {\mathbf {1} _ {\{| \Gamma (i - 1) | = | \Gamma (i) | = 1 \}}}.
+$$
+
+Finally,
+
+$$
+\mathbb {E} \left[ \Delta_ {\mathrm {w w}} \right] = \sum_ {a \in [ n _ {0} ] _ {e v e n} ^ {4}} \prod_ {k = 1} ^ {4} x _ {a _ {k}} \sum_ {\substack {\Gamma \in \Gamma_ {a, e v e n} ^ {4} \\ \Gamma = (\gamma_ {1}, \dots , \gamma_ {4})}} \sum_ {\substack {e _ {1} \in \gamma_ {1} \cap \gamma_ {2} \\ e _ {2} \in \gamma_ {2}, \gamma_ {3} \\ e _ {1} \neq e _ {2}}} F (\Gamma , e _ {1}, e _ {2}). \tag{22}
+$$
+
+Lemma 7 is proved in §B.2. The expression (20) is simple to evaluate due to the delta function in  $H(\Gamma, e)$ . We obtain:
+
+$$
+\mathbb {E} \left[ K _ {\mathrm {w}} \right] = \sum_ {a = 1} ^ {n _ {0}} x _ {a} ^ {2} \sum_ {\gamma \in \Gamma^ {1} (\vec {n})} \sum_ {e \in \gamma} \prod_ {i = 1} ^ {d} \frac {1}{n _ {i - 1}} = d \prod_ {i = 1} ^ {d} \frac {1}{n _ {i - 1}} \prod_ {i = 1} ^ {d} n _ {i} \| x \| _ {2} ^ {2} = \frac {d}{n _ {0}} \| x \| _ {2} ^ {2}, \tag {23}
+$$
+
+where in the second-to-last equality we used that the number of paths in the computational graph of  $N$  from a given neuron in the input to the output neuron equals  $\prod_{i=1,\dots,d} n_i$  and in the last equality we used that  $n_d = 1$ . This proves the first equality in Theorem 1.
+
+It therefore remains to evaluate (21) and (22). Since they are so similar, we will continue to discuss them in parallel. To start, notice that the expression  $F(\Gamma, e_1, e_2)$  appearing in (21) and (22) satisfies
+
+$$
+\frac {1}{2 \mu_ {4} ^ {2}} F _ {*} (\Gamma) \leq F (\Gamma , e _ {1}, e _ {2}) \leq \frac {1}{2} F _ {*} (\Gamma),
+$$
+
+where
+
+$$
+F _ {*} (\Gamma) := \prod_ {i = 1} ^ {d} \frac {2 ^ {2 - | \Gamma (i) |}}{n _ {i - 1} ^ {2}} \mu_ {4} ^ {\mathbf {1} _ {\{| \Gamma (i - 1) | = | \Gamma (i) | = 1 \}}}. \tag {24}
+$$
+
+For the remainder of the proof we will write
+
+$$
+f \simeq g \iff \exists \text {c o n s t a n t s} C, c > 0 \text {d e p e n d i n g o n l y o n} \mu \text {s . t .} \quad c g \leq f \leq C g.
+$$
+
+Thus, in particular,
+
+$$
+F (\Gamma , e _ {1}, e _ {2}) \simeq F _ {*} (\Gamma).
+$$
+
+The advantage of  $F_{*}(\Gamma)$  is that it does not depend on  $e_1, e_2$ . Observe that for every  $a = (\alpha_1, \alpha_2, \alpha_3, \alpha_4) \in [n_0]_{even}^4$ , we have that either  $\alpha_1 = \alpha_2$ ,  $\alpha_1 = \alpha_3$ , or  $\alpha_1 = \alpha_4$ . Thus, by symmetry, the sum over  $\Gamma_{even}^4(\vec{n})$  in (21) and (22) takes only four distinct values, represented by the following possibilities:
+
+$$
+a _ {j} \in [ n _ {0} ] _ {e v e n} ^ {4} := \left\{ \begin{array}{l l} (1, 1, 1, 1), & j = 1 \\ (1, 2, 1, 2), & j = 2 \\ (1, 1, 2, 2), & j = 3 \\ (1, 2, 2, 1), & j = 4 \end{array} \right.,
+$$
+
+keeping track of which paths  $\gamma_1, \ldots, \gamma_4$  begin at the same neuron in the input layer to  $N$ . Hence, since
+
+$$
+\sum_{\substack{a = (a_{1},\ldots ,a_{4})\in [n_{0}]_{even}^{4}\\ a_{1} = a_{2},  a_{3} = a_{4},  a_{1}\neq a_{3}}}\prod_{k = 1}^{4}x_{a_{k}} = \| x\|_{2}^{4} - \| x\|_{4}^{4}
+$$
+
+we find
+
+$$
+\mathbb {E} \left[ K _ {\mathrm {w}} ^ {2} \right] \simeq \| x \| _ {4} ^ {4} I _ {1} + \left(\| x \| _ {2} ^ {4} - \| x \| _ {4} ^ {4}\right) \left(I _ {2} + I _ {3} + I _ {4}\right), \tag {25}
+$$
+
+and similarly,
+
+$$
+\mathbb {E} \left[ \Delta_ {\mathrm {w w}} \right] \simeq \| x \| _ {4} ^ {4} I I _ {1} + \left(\| x \| _ {2} ^ {4} - \| x \| _ {4} ^ {4}\right) \left(I I _ {2} + I I _ {3} + I I _ {4}\right), \tag {26}
+$$
+
+where
+
+$$
+I _ {j} = \sum_ {\substack {\Gamma \in \Gamma_ {a _ {j}, e v e n} ^ {4} (\vec {n}) \\ \Gamma = (\gamma_ {1}, \dots , \gamma_ {4})}} F _ {*} (\Gamma) \# \left\{\text {edges } e _ {1}, e _ {2} \mid e _ {1} \in \gamma_ {1} \cap \gamma_ {2}, e _ {2} \in \gamma_ {3} \cap \gamma_ {4} \right\} \tag{27}
+$$
+
+$$
+I I _ {j} = \sum_ {\substack {\Gamma \in \Gamma_ {a _ {j}, e v e n} ^ {4} (\vec {n}) \\ \Gamma = (\gamma_ {1}, \dots , \gamma_ {4})}} F _ {*} (\Gamma) \# \left\{\text {edges} e _ {1}, e _ {2} \mid e _ {1} \in \gamma_ {1} \cap \gamma_ {2}, e _ {2} \in \gamma_ {2}, \gamma_ {3}, e _ {1} \neq e _ {2} \right\}. \tag{28}
+$$
+
+To evaluate  $I_{j}$ ,  $II_{j}$  let us write
+
+$$
+T _ {i} ^ {\alpha , \beta} (\Gamma) := \mathbf {1} _ {\left\{ \begin{array}{c} \gamma_ {\alpha} (i - 1) = \gamma_ {\beta} (i - 1) \\ \gamma_ {\alpha} (i) = \gamma_ {\beta} (i) \end{array} \right\}}, \qquad \Gamma = (\gamma_ {1}, \dots , \gamma_ {4}), \qquad \alpha , \beta = 1, \dots , 4 \tag {29}
+$$
+
+for the indicator function of the event that paths  $\gamma_{\alpha},\gamma_{\beta}$  pass through the same edge between layers  $i - 1,i$  in the computational graph of  $N$ . Observe that
+
+$$
+\# \left\{\text {e d g e s} e _ {1}, e _ {2} \mid e _ {1} \in \gamma_ {1} \cap \gamma_ {2}, e _ {2} \in \gamma_ {3} \cap \gamma_ {4} \right\} = \sum_ {i _ {1}, i _ {2} = 1} ^ {d} T _ {i _ {1}} ^ {1, 2} T _ {i _ {2}} ^ {3, 4}
+$$
+
+and
+
+$$
+\# \left\{\text{edges} e_{1},e_{2}\mid e_{1}\in \gamma_{1}\cap \gamma_{2},  e_{2}\in \gamma_{2},\gamma_{3},  e_{1}\neq e_{2}\right\} = \sum_{\substack{i_{1},i_{2} = 1\\ i_{1}\neq i_{2}}}^{d}T_{i_{1}}^{1,2}T_{i_{2}}^{2,3}.
+$$
+
+Thus, we have
+
+$$
+I_{j} = \sum_{i_{1},i_{2} = 1}^{d}I_{j,i_{1},i_{2}},\qquad II_{j} = \sum_{\substack{i_{1},i_{2} = 1\\ i_{1}\neq i_{2}}}^{d}II_{j,i_{1},i_{2}},
+$$
+
+where
+
+$$
+I_{j,i_{1},i_{2}} = \sum_{\substack{\Gamma \in \Gamma^{4}_{a_{j},even}\\ \Gamma = (\gamma_{1},\ldots ,\gamma_{4})}}F_{*}(\Gamma)T^{1,2}_{i_{1}}T^{3,4}_{i_{2}},\qquad II_{j,i_{1},i_{2}} = \sum_{\substack{\Gamma \in \Gamma^{4}_{a_{j},even}\\ \Gamma = (\gamma_{1},\ldots ,\gamma_{4})}}F_{*}(\Gamma)T^{1,2}_{i_{1}}T^{2,3}_{i_{2}}.
+$$
+
+To simplify  $I_{j,i_1,i_2}$  and  $II_{j,i_1,i_2}$  observe that  $F_{*}(\Gamma)$  depends only on  $\Gamma$  only via the unordered edge multi-set (i.e. only which edges are covered matters; not their labelling)
+
+$$
+E ^ {\Gamma} = \left(E ^ {\Gamma} (1), \dots , E ^ {\Gamma} (d)\right) \in \Sigma_ {e v e n} ^ {4}
+$$
+
+defined in Definition 3. Hence, we find that for  $j = 1,2,3,4$ ,  $i_1,i_2 = 1,\ldots ,d$
+
+$$
+I _ {j, i _ {1}, i _ {2}} = \sum_ {E \in \Sigma_ {a _ {j}, e v e n} ^ {4} (\vec {n})} F _ {*} (E) \# \left\{\Gamma \in \Gamma_ {a _ {j}, e v e n} ^ {4} (\vec {n}) \mid_ {\gamma_ {1} (i _ {t} - 1) = \gamma_ {2} (i _ {t} - 1), \gamma_ {1} (i _ {t}) = \gamma_ {2} (i _ {t})} ^ {E ^ {\Gamma} = E, \Gamma (0) = a _ {j}, t = 1, 2} \right\} \tag {30}
+$$
+
+$$
+I I _ {j, i _ {1}, i _ {2}} = \sum_ {E \in \Sigma_ {a _ {j}, e v e n} ^ {4} (\vec {n})} F _ {*} (E) \# \left\{\Gamma \in \Gamma_ {a _ {j}, e v e n} ^ {4} (\vec {n}) \left| \begin{array}{c} E ^ {\Gamma} = E, \Gamma (0) = a _ {j} \\ \gamma_ {1} (i _ {1} - 1) = \gamma_ {2} (i _ {1} - 1), \gamma_ {1} (i _ {1}) = \gamma_ {2} (i _ {1}) \\ \gamma_ {2} (i _ {2} - 1) = \gamma_ {3} (i _ {2} - 1), \gamma_ {2} (i _ {2}) = \gamma_ {3} (i _ {2}) \end{array} \right. \right\} \tag {31}
+$$
+
+The counts in  $I_{j,*i_1,i_2}$  and  $II_{j,*i_1,i_2}$  have a convenient representation in terms of
+
+$$
+C \left(E, i _ {1}, i _ {2}\right) := \mathbf {1} _ {\{\exists \ell = \min  \left(i _ {1}, i _ {2}\right), \dots , \max  \left(i _ {1}, i _ {2} - 1\right) \text {s . t .} | R (E (\ell)) | = 1 \}} \tag {32}
+$$
+
+$$
+\widehat {C} (E, i _ {1}, i _ {2}) := \mathbf {1} _ {\{\exists \ell = 0, \dots , \min  \left(i _ {1}, i _ {2}\right) - 1 \text {s . t .} | R (E (\ell)) | = 1 \}}. \tag {33}
+$$
+
+Informally, the event  $\widehat{C}(E, i_1, i_2)$  indicates the presence of a "collision" of the four paths in  $\Gamma$  before the earlier of the layers  $i_1, i_2$ , while  $C(E, i_1, i_2)$  gives a "collision" between layers  $i_1, i_2$ ; see Section 3.1 for the intuition behind calling these collisions. We also write
+
+$$
+\begin{array}{l} A(E,i_{1},i_{2}):= \mathbf{1}_{\left\{ \begin{array}{c}|L(E(i_{1}))| = |R(E(i_{1}))| = 1\\ |L(E(i_{2}))| = |R(E(i_{2}))| = 1 \end{array} \right\}} + \frac{1}{6}\mathbf{1}_{\left\{ \begin{array}{c}|L(E(i_{1}))| = |R(E(i_{1}))| = 1,|R(E(i_{2}))| = 2\text{or}\\ |L(E(i_{2}))| = |R(E(i_{2}))| = 1,|R(E(i_{1}))| = 2 \end{array} \right\}} \\ + \frac {1}{6} \mathbf {1} _ {\left\{ \begin{array}{l} | R (E (i _ {1})) | = | R (E (i _ {2})) | = 2 \\ \exists \min  (i _ {1}, i _ {2}) \leq \ell <   \max  (i _ {1}, i _ {2}) \\ \text {s . t .} | R (\bar {E} (\ell)) | = 1 \end{array} \right\}} + \frac {1}{3 6} \mathbf {1} _ {\left\{ \begin{array}{c} E (i _ {1}), E (i _ {2}) \in U \\ \exists \min  (i _ {1}, i _ {2}) \leq \ell <   \max  (i _ {1}, i _ {2}) \\ \text {s . t .} | R (\bar {E} (\ell)) | = 1 \end{array} \right\}}. \tag {34} \\ \end{array}
+$$
+
+Finally, for  $E \in \Sigma_{a,even}^{4}(\vec{n})$ , we will define
+
+$$
+\# \operatorname {l o o p s} (E) = \# \left\{i \in [ d ] \mid | L (E (i)) | = 1, | R (E (i)) | = 2 \right\}. \tag {35}
+$$
+
+That is, a loop is created at layer  $i$  if the four edges in  $E$  all begin at occupy the same vertex in layer  $i - 1$  but occupy two different vertices in layer  $i$ . We have the following Lemma.
+
+Lemma 8 (Evaluation of Counting Terms in (30) and (31)). Suppose  $E \in \Sigma_{a_j, \text{even}}^4$  for some  $j = 1, 2, 3, 4$ . For each  $i_1, i_2 \in \{1, \ldots, d\}$ ,
+
+$$
+\# \left\{\Gamma = (\gamma_{1},\ldots ,\gamma_{4})\in \Gamma^{4}_{a_{j},e v e n}\big| \begin{array}{c}E^{\Gamma} = E,\Gamma (0) = a_{j},t = 1,2\\ \gamma_{1}(i_{t} - 1) = \gamma_{2}(i_{t} - 1),\gamma_{1}(i_{t}) = \gamma_{2}(i_{t}) \end{array} \right\}
+$$
+
+equals
+
+$$
+6 ^ {\# \text {l o o p s} (E)} A \left(E, i _ {1}, i _ {2}\right) \cdot \left\{ \begin{array}{l l} 1, & j = 1, 2 \\ \widehat {C} \left(E, i _ {1}, i _ {2}\right), & j = 3, 4 \end{array} . \right. \tag {36}
+$$
+
+Similarly,
+
+$$
+\# \left\{\Gamma \in \Gamma_ {a _ {j}, e v e n} ^ {4} \left| \begin{array}{c} E ^ {\Gamma} = E, \Gamma (0) = a _ {j}, t = 1, 2 \\ \gamma_ {1} (i _ {1} - 1) = \gamma_ {2} (i _ {1} - 1), \gamma_ {1} (i _ {1}) = \gamma_ {2} (i _ {1}) \\ \gamma_ {2} (i _ {2} - 1) = \gamma_ {3} (i _ {2} - 1), \gamma_ {2} (i _ {2}) = \gamma_ {3} (i _ {2}) \end{array} \right. \right\}
+$$
+
+equals
+
+$$
+6 ^ {\# \text {l o o p s} (E)} A \left(E, i _ {1}, i _ {2}\right) C \left(E, i _ {1}, i _ {2}\right) \cdot \left\{ \begin{array}{l l} 1, & j = 1, 2 \\ \widehat {C} \left(E, i _ {1}, i _ {2}\right), & j = 3, 4 \end{array} . \right. \tag {37}
+$$
+
+We prove Lemma 8 in §B.3 below. Assuming it for now, observe that
+
+$$
+\frac {1}{3 6} \leq A (E, i _ {1}, i _ {2}) \leq 1
+$$
+
+and that the conditions  $L(E(1)) = a_{j}$  are the same for  $j = 2,3,4$  since the equality it is in the sense of unordered multi-sets. Thus, we find that  $\mathbb{E}[K_{\mathrm{w}}^2]$  is bounded above/below by a constant times
+
+$$
+\| x \| _ {4} ^ {4} \sum_ {i _ {1}, i _ {2} = 1} ^ {d} \sum_ {E \in \Sigma_ {a _ {1}, e v e n} ^ {4}} F _ {*} (E) + (\| x \| _ {2} ^ {4} - \| x \| _ {4} ^ {4}) \sum_ {E \in \Sigma_ {a _ {2}, e v e n} ^ {4}} F _ {*} (E) (1 + 2 \widehat {C} (E, i _ {1}, i _ {2})). \tag {38}
+$$
+
+Similarly,  $\mathbb{E}[\Delta_{\mathrm{ww}}]$  is bounded above/below by a constant times
+
+$$
+\begin{array}{l} \sum_ { \begin{array}{l} i _ {1}, i _ {2} = 1 \\ i _ {1} \neq i _ {2} \end{array} } ^ {d} \left[ \| x \| _ {4} ^ {4} \sum_ {E \in \Sigma_ {a _ {1}, e v e n} ^ {4}} F _ {*} (E) 6 ^ {\# \text {l o o p s} (E)} C (E, i _ {1}, i _ {2}) \right. \tag {39} \\ \left. + \left(\| x \| _ {2} ^ {4} - \| x \| _ {4} ^ {4}\right) \sum_ {E \in \Sigma_ {a _ {2}, e v e n} ^ {4}} F _ {*} (E) 6 ^ {\# \text {l o o p s} (E)} C (E, i _ {1}, i _ {2}) \left(1 + 2 \widehat {C} (E, i _ {2}, i _ {2})\right) \right]. \\ \end{array}
+$$
+
+Observe that every unordered multi-set four edge multiset  $E \in \Sigma_{even}^{4}$  can be obtained by starting from some  $V \in \Gamma^{2}$ , considering its unordered edge multi-set  $E^{V}$  and doubling all its edges. This map from  $\Gamma^{2}$  to  $\Sigma_{even}^{4}$  is surjective but not injective. The sizes of the fibers is computed by the following Lemma.
+
+Lemma 9. Fix  $E \in \Sigma_{even}^{4}$ . The number of  $V \in \Gamma_Z^2$  so that  $E = 2 \cdot E^V$  is  $2^{\# loops(V) + \mathbf{1}_{\{|V(0)| = 2\}}},$  where as in (35),
+
+$$
+\# \text {l o o p s} (V) = \# \{i \in [ d ] \mid | V (i - 1) | = 1, | V (i) | = 2 \}.
+$$
+
+Lemma 9 is proved in §B.4. Using it and that  $0 \leq \widehat{C}(E, i_1, i_2) \leq 1$ , the relation (38) shows that  $\mathbb{E}[K_{\mathrm{w}}^2]$  is bounded above/below by a constant times
+
+$$
+d ^ {2} \sum_ {V \in \Gamma^ {2}} F _ {*} (V) 3 ^ {\# \operatorname {l o o p s} (V)} \left(\| x \| _ {4} ^ {4} \mathbf {1} _ {\{| V (0) | = 1 \}} + \left(\| x \| _ {2} ^ {4} - \| x \| _ {4} ^ {4}\right) \mathbf {1} _ {\{| V (0) | = 2 \}}\right). \tag {40}
+$$
+
+Similarly,  $\mathbb{E}[\Delta_{\mathrm{ww}}]$  is bounded above/below by a constant times
+
+$$
+\sum_{\substack{i_{1},i_{2} = 1\\ i_{1}\neq i_{2}}}^{d}\sum_{V\in \Gamma^{2}}F_{*}(V)3^{\# \mathrm{loops}(V)}C(V,i_{1},i_{2})\left(\| x\|_{4}^{4}\mathbf{1}_{\{|V(0)| = 1\}} + (\| x\|_{2}^{4} - \| x\|_{4}^{4})\mathbf{1}_{\{|V(0)| = 2\}}\right), \tag{41}
+$$
+
+where, in analogy to (32), we have
+
+$$
+C (V, i _ {1}, i _ {2}) := \mathbf {1} _ {\{\exists \ell = i _ {1}, \dots , i _ {2} - 1 \text {s . t .} | V (\ell) | = 1 \}}.
+$$
+
+Let us introduce
+
+$$
+\begin{array}{l} \widehat {F} _ {*} (V) := F _ {*} (V) \cdot 3 ^ {\# \text {l o o p s} (V)} \prod_ {i = 0} ^ {d} n _ {i} ^ {2} \\ = 2 ^ {\# \{i \in [ d ] \mid | V (i) | = 1 \}} 3 ^ {\# \operatorname {l o o p s} (V)} \mu_ {4} ^ {\# \{i \in [ d ] \mid | V (i - 1) | = | V (i) | = 1 \}}. \\ \end{array}
+$$
+
+Since the number of  $V$  in  $\Gamma^2 (\vec{n})$  with specified  $V(0)$  equals  $\prod_{i = 1}^{d}n_i^2$ , we find that so that for each  $x\neq 0$ , we have
+
+$$
+\frac {\mathbb {E} \left[ K _ {\mathrm {w}} ^ {2} \right]}{\| x \| _ {2} ^ {4}} \simeq \frac {d ^ {2}}{n _ {0} ^ {2}} \mathcal {E} _ {x} \left[ \widehat {F} _ {*} (V) \right], \tag {42}
+$$
+
+and similarly,
+
+$$
+\frac{\mathbb{E}[\Delta_{\mathrm{ww}}]}{\|x\|_{2}^{4}}\simeq \frac{1}{n_{0}^{2}}\sum_{\substack{i_{1},i_{2} = 1\\ i_{1}\neq i_{2}}}^{d}\mathcal{E}_{x}\left[\widehat{F}_{*}(V)C(V,i_{1},i_{2})\right].
+$$
+
+Here,  $\mathcal{E}_x$  is the expectation with respect to the probability measure on  $V = (v_{1}, v_{2}) \in \Gamma^{2}$  obtained by taking  $v_{1}, v_{2}$  independent, each drawn from the products of the measure  $\left(x_1^2 / \| x\|_2^2, \ldots, x_{n_0}^2 / \| x\|_2^2\right)$  on  $[n_0]$  and the uniform measure on  $[n_i], i = 1, \ldots, d$ .
+
+We are now in a position to complete the proof of Theorems 1 and 2. To do this, we will evaluate the expectations  $\mathcal{E}_x$  above to leading order in  $\sum_{i}1 / n_{i}$  with the help of the following elementary result which is proven as Lemma 18 in Hanin & Nica (2018).
+
+Proposition 10. Let  $A_0, A_1, \ldots, A_d$  be independent events with probabilities  $p_0, \ldots, p_d$  and  $B_0, \ldots, B_d$  be independent events with probabilities  $q_0, \ldots, q_d$  such that
+
+$$
+A _ {j} \cap B _ {j} = \emptyset , \quad \forall j = 0, \dots , d.
+$$
+
+Denote by  $X_{i}$  the indicator that the event  $A_{i}$  happens,  $X_{i} \coloneqq \mathbf{1}_{\{A_{i}\}}$ , and by  $Y_{i}$  the indicator that  $B_{i}$  happens,  $Y_{i} = \mathbf{1}_{\{B_{i}\}}$ . Further, fix for every  $i \in 1, \ldots, d$  some  $\alpha_{i} \geq 1$ ,  $K_{i} \geq 1$  as well as  $\gamma_{i} > 0$ . Define
+
+$$
+Z = \prod_ {i = 1} ^ {d} \alpha_ {i} ^ {X _ {i}} \gamma_ {i} ^ {X _ {i - 1} X _ {i}} K _ {i} ^ {Y _ {i}}.
+$$
+
+Then, if  $\gamma_{i}\geq 1$  for every  $i$ , we have:
+
+$$
+\mathbb {E} [ Z ] \leq \prod_ {i = 1} ^ {d} (1 + p _ {i} (\alpha_ {i} - 1) + q _ {i} (K _ {i} - 1) + p _ {i} p _ {i - 1} \alpha_ {i} \alpha_ {i - 1} \gamma_ {i - 1} (\gamma_ {i} - 1)), \tag {43}
+$$
+
+where by convention  $\alpha_0 = \gamma_0 = 1$ . In contrast, if  $\gamma_i \leq 1$  for every  $i$ , we have:
+
+$$
+\mathbb {E} [ Z ] \geq \prod_ {i = 1} ^ {d} (1 + p _ {i} (\alpha_ {i} - 1) + p _ {i} p _ {i - 1} \alpha_ {i - 1} \alpha_ {i} (\gamma_ {i} - 1)) \tag {44}
+$$
+
+We first apply Proposition 10 to the estimates above for  $\mathbb{E}[K_{\mathrm{w}}^2]$ . To do this, recall that
+
+$$
+3 ^ {\# \operatorname {l o o p s} (V)} = \prod_ {i = 1} ^ {d} 3 ^ {\mathbf {1} _ {\{| V (i - 1) | = 1, | V (i) | = 2 \}}}.
+$$
+
+Since  $|V(d)| = 1$ , we may also write
+
+$$
+3 ^ {\# \mathrm {l o o p s} (V)} = \frac {1}{3} \prod_ {i = 1} ^ {d} 3 ^ {\mathbf {1} _ {\{| V (i - 1) | = 2, | V (i) | = 1 \}}} = \frac {1}{3} \prod_ {i = 1} ^ {d} \left(\frac {1}{3}\right) ^ {\mathbf {1} _ {\{| V (i - 1) | = | V (i) | = 1 \}}} 3 ^ {\mathbf {1} _ {\{| V (i) | = 1 \}}}.
+$$
+
+Putting this together with (42) and noting that
+
+$$
+\prod_ {i = 1} ^ {d} 2 ^ {2 - | V (i) |} = \prod_ {i = 1} ^ {d} 2 ^ {\mathbf {1} _ {\{| V (i) | = 1 \}}},
+$$
+
+we find that
+
+$$
+\mathbb {E} [ K _ {\mathrm {w}} ^ {2} ] / \| x \| _ {2} ^ {4} \simeq \frac {1}{n _ {0} ^ {2}} \mathcal {E} _ {x} \left[ \prod_ {i = 1} ^ {d} \left(\frac {\mu_ {4}}{3}\right) ^ {\mathbf {1} _ {\{| V (i - 1) | = | V (i) | = 1 \}}} 6 ^ {\mathbf {1} _ {\{| V (i) | = 1 \}}} \right].
+$$
+
+Since the contribution for each layer in the product is bounded above and below by constants, we have that  $\mathbb{E}[K_{\mathrm{w}}^2 ] / \| x\| _2^4$  is bounded below by a constant times
+
+$$
+\frac {d ^ {2}}{n _ {0} ^ {2}} \mathcal {E} _ {x} \left[ \prod_ {i = 2} ^ {d - 1} \left(1 \wedge \frac {\mu_ {4}}{3}\right) ^ {\mathbf {1} \{| V (i - 1) | = | V (i) | = 1 \}} 6 ^ {\mathbf {1} \{| V (i) | = 1 \}} \right] \tag {45}
+$$
+
+and above by a constant times
+
+$$
+\frac {d ^ {2}}{n _ {0} ^ {2}} \mathcal {E} _ {x} \left[ \prod_ {i = 2} ^ {d - 1} \left(1 \vee \frac {\mu_ {4}}{3}\right) ^ {\mathbf {1} _ {\{| V (i - 1) | = | V (i) | = 1 \}}} 6 ^ {\mathbf {1} _ {\{| V (i) | = 1 \}}} \right]. \tag {46}
+$$
+
+Here, note that the initial condition given by  $x$  and the terminal condition that all paths end at one neuron in the final layer are irrelevant. The expression (45) is there precisely  $\mathbb{E}[Z_{d - 1} / n_0^2]$  from Proposition 10 where  $X_{i}$  is the event that  $|V(i)| = 1$ ,  $Y_{i} = \emptyset$ ,  $\alpha_{i} = 6$ ,  $\gamma_{i} = 1 \wedge \frac{\mu_{4}}{3} \leq 1$ , and  $K_{i} = 1$ . Thus, since for  $i = 1, \ldots, d - 1$ , the probability of  $X_{i}$  is  $1 / n_{i} + O(1 / n_{i}^{2})$ , we find that
+
+$$
+\mathbb {E} [ K _ {\mathrm {w}} ^ {2} ] / \| x \| _ {2} ^ {4} \geq \frac {d ^ {2}}{n _ {0} ^ {2}} \prod_ {i = 2} ^ {d - 1} \left(1 + \frac {5}{n _ {i}} + O \left(\frac {1}{n _ {i} ^ {2}} + \frac {1}{n _ {i - 1} ^ {2}}\right)\right) \geq \frac {d ^ {2}}{n _ {0} ^ {2}} \exp \left(5 \sum_ {i = 2} ^ {d - 1} \frac {1}{n _ {i}} + O \left(\sum_ {i = 2} ^ {d - 1} \frac {1}{n _ {i} ^ {2}}\right)\right),
+$$
+
+where in the last inequality we used that  $1 + x \geq e^{x - x^2 / 2}$  for  $x \geq 0$ . Since  $e^{-1 / n_1 + 1 / n_d} \simeq 1$ , we conclude
+
+$$
+\mathbb {E} [ K _ {\mathrm {w}} ^ {2} ] / \| x \| _ {2} ^ {4} \geq \frac {d ^ {2}}{n _ {0} ^ {2}} \exp {(5 \beta)} \left(1 + O \left(\beta^ {- 1} \sum_ {i = 1} ^ {d} \frac {1}{n _ {i} ^ {2}}\right)\right), \quad \beta = \sum_ {i = 1} ^ {d} \frac {1}{n _ {i}}.
+$$
+
+When combined with (23) this gives the lower bound in Proposition 3. The matching upper bound is obtained from (46) in the same way using the opposite inequality from Proposition 10.
+
+To complete the proof of Proposition 3, we prove the analogous bounds for  $\mathbb{E}[\Delta_{\mathrm{ww}}]$  in a similar fashion. Namely, we fix  $1 \leq i_1 < i_2 \leq d$  and write
+
+$$
+C(V,i_{1},i_{2}) = \sum_{\ell = i_{1}}^{i_{2} - 1}\mathbf{1}_{A_{\ell}},\qquad A_{\ell}:= \left\{ \begin{array}{c}|V(i)| = 2,  i = i_{1},\ldots ,\ell -1\\ \text{and} |V(\ell)| = 1 \end{array} \right\} .
+$$
+
+The set  $A_{\ell}$  is the event that the first collision between layers  $i_1, i_2$  occurs at layer  $\ell$ . We then have
+
+$$
+\mathcal {E} _ {x} \left[ \widehat {F} _ {*} (V) C (V, i _ {1}, i _ {2}) \right] = \sum_ {\ell = i _ {1}} ^ {i _ {2} - 1} \mathcal {E} _ {x} \left[ \widehat {F} _ {*} (V) \mathbf {1} _ {\{A _ {\ell} \}} \right],
+$$
+
+On the event  $A_{\ell}$ , notice that  $\widehat{F}_{*}(V)$  only depends on the layers  $1 \leq i \leq i_1$  and layers  $\ell < i \leq d$  because the event  $A_{\ell}$  fixes what happens in layers  $i_1 < i \leq \ell$ . Mimicking the estimates (45), (46) and the application of Proposition 10 and using independence, we get that:
+
+$$
+\mathcal{E}_{x}\Big[\widehat{F}_{*}(V)1\{A_{\ell}\} \Big] \simeq \exp \left(\sum_{\substack{i = 1\\ i\notin [i_{1},\ell)}}^{d}\frac{1}{n_{i}}\right)\mathcal{E}_{x}\left(\mathbf{1}_{\{A_{\ell}\}}\right)
+$$
+
+Finally, we compute:
+
+$$
+\mathcal {E} _ {x} \left(\mathbf {1} _ {\{A _ {\ell} \}}\right) = \mathbb {P} (A _ {\ell}) = \frac {1}{n _ {\ell}} \prod_ {i = i _ {1}} ^ {\ell - 1} \left(1 - \frac {1}{n _ {i}}\right) \simeq \frac {1}{n _ {\ell}} \exp \left(- \sum_ {i = i _ {1}} ^ {\ell - 1} \frac {1}{n _ {i}}\right),
+$$
+
+Combining this we obtain that  $\mathbb{E}[\Delta_{\mathrm{ww}}] / \| x\| _2^4$  is bounded above and below by constants times
+
+$$
+\frac{1}{n_{0}^{2}}\left[\sum_{\substack{i_{1},i_{2} = 1\\ i_{i} <   i_{2}}}^{d}\sum_{\ell = i_{1}}^{i_{2} - 1}\frac{1}{n_{\ell}} e^{-5 / n_{\ell} - 6\sum_{i = i_{1}}^{\ell -1}\frac{1}{n_{i}}}\right]\exp \left(5\sum_{i = 1}^{d}\frac{1}{n_{i}}\right)\left(1 + O\left(\sum_{i = 1}^{d}\frac{1}{n_{i}^{2}}\right)\right).
+$$
+
+This completes the proof of Proposition 3, modulo the proofs of Lemmas 6-9, which we supply below.
+
+# B.1 PROOF OF LEMMA 6
+
+Fix an input  $x \in \mathbb{R}^{n_0}$  to  $N$ . We will continue to write as in (2)  $y^{(i)}$  for the vector of pre-activations as layer  $i$  corresponding to  $x$ . We need the following simple Lemma.
+
+Lemma 11. With probability 1, either there exists  $i$  so that  $y^{(i)} = 0$  or, for every  $i \in [d], j \in [n_i]$  we have  $y_j^{(i)} \neq 0$ .
+
+Proof. The argument is similar to Lemma 8 in Hanin & Rolnick (2019). Namely, fix  $i \in [d], j \in [n_i]$ . If  $y^{(\ell)} \neq 0$  for every  $\ell$ , then there exists at least one path  $\gamma$  in the computational graph of the map  $x \mapsto y_j^{(i)}$  so that,  $y_{\gamma}^{(\ell)} > 0$  for each  $\ell = 1, \ldots, i - 1$ . For event that  $y_j^{(i)} = 0$  is therefore contained in the union over all non-empty subsets  $\Gamma$  of the collection of all paths in the computational graph of  $x \mapsto y_j^{(i)}$  of the event that
+
+$$
+\sum_ {\gamma \in \Gamma} \prod_ {\ell = 1} ^ {i} \widehat {W} _ {\gamma} ^ {(\ell)} = 0.
+$$
+
+For each fixed  $\Gamma$  this event defines a co-dimension 1 set in the space of all the weights. Hence, since the joint distribution of the weights has a density with respect to Lebesgue measure (see just before (4)), the union of this (finite number) of events has measure 0. This shows that on the even that  $y^{(\ell)}\neq 0$  for every  $\ell ,y_j^{(i)}\neq 0$  with probability 1. Taking the union over  $i,j$  completes the proof.
+
+Lemma 11 shows that for our fixed  $x$ , with probability 1, the derivative of each  $\xi_j^{(i)}$  in (19) vanishes. Hence, almost surely, for any edge  $e$  in the computational graph of  $N$ :
+
+$$
+\frac {\partial N}{\partial W _ {e} ^ {(j)}} (x) = \sum_ {a = 1} ^ {n _ {0}} x _ {a} \sum_ {\substack {\gamma \in \Gamma_ {a} ^ {1} \\ e \in \gamma}} \frac {\operatorname{wt} (\gamma)}{W _ {e}}. \tag{47}
+$$
+
+This proves the formulas for  $K_N, K_N^2$ . To derive the result for  $\Delta K_N$ , we write
+
+$$
+\Delta K _ {N} = - \lambda \sum_ {\text {e d g e s} e} \left(\frac {\partial}{\partial W _ {e}} K _ {N}\right) \frac {\partial \mathcal {L}}{\partial W _ {e}},
+$$
+
+where the loss  $\mathcal{L}$  on a single batch containing only  $x$  is  $\frac{1}{2} (N(x) - N_{*}(x))^{2}$ . We therefore find
+
+$$
+\Delta K _ {N} = - 2 \lambda \sum_ {\text {e d g e s} e _ {1}, e _ {2}} \frac {\partial N}{\partial W _ {e _ {1}}} \frac {\partial^ {2} N}{\partial W _ {e _ {1}} \partial W _ {e _ {2}}} \frac {\partial N}{\partial W _ {e _ {2}}} (N (x) - N _ {*} (x)).
+$$
+
+Using (47) and again applying Lemma 11, we find that with probability 1
+
+$$
+\frac{\partial^{2}\boldsymbol{N}}{\partial W_{e_{1}}\partial W_{e_{2}}} = \sum_{a = 1}^{n_{0}}x_{a}\sum_{\substack{\gamma \in \Gamma_{a}^{1}\\ e_{1},e_{2}\in \gamma ,e_{1}\neq e_{2}}}\frac{\mathrm{wt}(\gamma_{1})\mathrm{wt}(\gamma_{2})}{W_{e_{1}}W_{e_{2}}}.
+$$
+
+Thus, almost surely
+
+$$
+\begin{array}{l} -\frac{1}{2\lambda}\Delta K_{N} = \sum_{a\in [n_{0}]^{4}}\prod_{k = 1}^{4}x_{a_{k}}\sum_{\Gamma_{a}\in \Gamma^{4}(\vec{n})}\sum_{\substack{e_{1}\in \gamma_{1},\gamma_{2}\\ e_{2}\in \gamma_{2},\gamma_{3}\\ e_{1}\neq e_{2}}}\frac{\prod_{k = 1}^{4}\mathrm{wt}(\gamma_{k})}{W_{e_{1}}^{2}W_{e_{2}}^{2}} \\ - N _ {*} (x) \sum_ {a \in [ n _ {0} ] ^ {3}} \prod_ {k = 1} ^ {3} x _ {a _ {k}} \sum_ {\Gamma \in \Gamma_ {a} ^ {3} (\vec {n})} \sum_ {\substack {e _ {1} \in \gamma_ {1}, \gamma_ {2} \\ e _ {2} \in \gamma_ {2}, \gamma_ {3} \\ e _ {1} \neq e _ {2}}} \frac {\prod_ {k = 1} ^ {4} \mathrm {w t} (\gamma_ {k})}{W _ {e _ {1}} ^ {2} W _ {e _ {2}} ^ {2}}. \\ \end{array}
+$$
+
+To complete the proof of Lemma 6 it therefore remains to check that this last term has mean 0. To do this, recall that the output layer of  $N$  is assumed to be linear and that the distribution of each weight is symmetric around 0 (and hence has vanishing odd moments). Thus, the expectation over the weights in layer  $d$  has either 1 or 3 weights in it and so vanishes.
+
+# B.2 PROOF OF LEMMA 7
+
+Lemma 7 is almost a corollary of Theorem 3 in Hanin (2018) and Proposition 2 in Hanin & Nica (2018). The difference is that, in Hanin (2018); Hanin & Nica (2018), the biases in  $N$  were assumed to have a non-degenerate distribution, whereas here we've set them to zero. The nondegeneracy assumption is not really necessary, so we repeat here the proof from Hanin (2018) with the necessary modifications.
+
+If  $x = 0$ , then  $\mathcal{N}(x) = 0$  for any configuration of weights since the network biases all vanish. Will therefore suppose that  $x \neq 0$ . Let us first show (20). We have from Lemma 6 that
+
+$$
+\mathbb {E} \left[ K _ {\boldsymbol {N}} (x, x) \right] = \sum_ {a \in [ n _ {0} ] ^ {2}} x _ {a _ {1}} x _ {a _ {2}} \sum_ {\Gamma \in \Gamma_ {a, e v e n} ^ {2}} \sum_ {e \in \gamma_ {1} \cap \gamma_ {2}} \mathbb {E} \left[ \frac {\prod_ {k = 1} ^ {2} \operatorname {w t} (\gamma_ {k})}{W _ {e} ^ {2}} \right]. \tag {48}
+$$
+
+To compute the inner expectation, write  $\pmb{F}_j$  for the sigma algebra generated by the weight in layers up to and including  $j$ . Let us also define the events:
+
+$$
+S _ {j} := \{x ^ {(j)} \neq 0 \},
+$$
+
+where we recall from (2) that  $x^{(j)}$  are the post-activations in layer  $j$ . Supposing first that  $e$  is not in layer  $d$ , the expectation becomes
+
+$$
+\mathbb {E} \left[ \frac {\prod_ {i = 1} ^ {d - 1} \widehat {W} _ {\gamma_ {1}} ^ {(i)} \widehat {W} _ {\gamma_ {2}} ^ {(i)} \mathbf {1} _ {\{y _ {\gamma_ {1}} ^ {(i)} > 0 \}} \mathbf {1} _ {\{y _ {\gamma_ {2}} ^ {(i)} > 0 \}}}{W _ {e} ^ {2}} \mathbb {E} \left[ \widehat {W} _ {\gamma_ {1}} ^ {(d)} \widehat {W} _ {\gamma_ {2}} ^ {(d)} \Bigg | \mathcal {F} _ {d - 1} \right] \right].
+$$
+
+We have
+
+$$
+\mathbb{E}\left[\widehat{W}_{\gamma_{1}}^{(d)}\widehat{W}_{\gamma_{2}}^{(d)}\bigg|  \mathcal{F}_{d - 1}\right] = \frac{1}{n_{d - 1}}\mathbf{1}_{\left\{ \begin{array}{c}\gamma_{1}(d - 1) = \gamma_{2}(d - 1)\\ \gamma_{1}(d) = \gamma_{2}(d) \end{array} \right\}}
+$$
+
+Thus, the expectation in (48) becomes  $\frac{1}{n_{d-1}}\mathbf{1}\left\{\begin{array}{c}\gamma_1(d-1)=\gamma_2(d-1) \\ \gamma_1(d)=\gamma_2(d)\end{array}\right\}$  times
+
+$$
+\mathbb {E} \left[ \frac {\prod_ {k = 1} ^ {2} \prod_ {i = 1} ^ {d - 2} \widehat {W} _ {\gamma_ {k}} ^ {(i)} \mathbf {1} _ {\{y _ {\gamma_ {k}} ^ {(i)} > 0 \}}}{W _ {e} ^ {2}} \mathbb {E} \left[ \prod_ {k = 1} ^ {2} \widehat {W} _ {\gamma_ {k}} ^ {(d - 1)} \mathbf {1} _ {\{y _ {\gamma_ {k}} ^ {(d - 1)} > 0 \}} \Bigg | \mathcal {F} _ {d - 2} \right] \right].
+$$
+
+Note that given  $\mathcal{F}_{d - 2}$ , the pre-activations  $y_{j}^{(d - 1)}$  of different neurons in layer  $d - 1$  are independent. Hence,
+
+$$
+\begin{array}{r l} \mathbb {E} \left[ \prod_ {k = 1} ^ {2} \widehat {W} _ {\gamma_ {k}} ^ {(d - 1)} \mathbf {1} _ {\{y _ {\gamma_ {k}} ^ {(d - 1)} > 0 \}} \Bigg | \mathcal {F} _ {d - 2} \right] & = \left\{ \begin{array}{l l} \prod_ {k = 1} ^ {2} \mathbb {E} \left[ \widehat {W} _ {\gamma_ {k}} ^ {(d - 1)} \mathbf {1} _ {\{y _ {\gamma_ {k}} ^ {(d - 1)} > 0 \}} \Bigg | \mathcal {F} _ {d - 2} \right], & \gamma_ {1} (d - 1) \neq \gamma_ {2} (d - 1) \\ \mathbb {E} \left[ \mathbf {1} _ {\{y _ {\gamma_ {1}} ^ {(d - 1)} > 0 \}} \prod_ {k = 1} ^ {2} \widehat {W} _ {\gamma_ {k}} ^ {(d - 1)} \Bigg | \mathcal {F} _ {d - 2} \right], & \gamma_ {1} (d - 1) = \gamma_ {2} (d - 1) \end{array} . \right. \end{array}
+$$
+
+Recall that by assumption, the weight matrix  $\widehat{W}^{(d - 1)}$  in layer  $d - 1$  is equal in distribution to  $-\widehat{W}^{(d - 1)}$ . This replacement leaves the product  $\prod_{k = 1}^{2}\widehat{W}_{\gamma_k}^{(d - 1)}$  unchanged but changes  $\mathbf{1}_{\{y_{\gamma_1}^{(d - 1)} > 0\}}$  to  $\mathbf{1}_{\{y_{\gamma_1}^{(d - 1)} \leq 0\}}$ . On the event  $S_{d - 1}$  (which occurs whenever  $y_{\gamma_k}^{(d - 2)} > 0$ ) we have that  $y_{\gamma_1}^{(d - 1)} \neq 0$  with probability 1 since we assumed that the distribution of each weight has a density relative to Lebesgue measure. Hence, symmetrizing over  $\pm \widehat{W}^{(d)}$ , we find that
+
+$$
+\mathbb {E} \left[ \prod_ {k = 1} ^ {2} \widehat {W} _ {\gamma_ {k}} ^ {(d - 1)} \mathbf {1} _ {\{y _ {\gamma_ {k}} ^ {(d - 1)} > 0 \}} \Bigg | \mathcal {F} _ {d - 2} \right] = \frac {1}{n _ {d - 2}} \mathbf {1} _ {\left\{ \begin{array}{l} \gamma_ {1} (d - 1) = \gamma_ {2} (d - 1) \\ \gamma_ {1} (d - 2) = \gamma_ {2} (d - 2) \end{array} \right\}}.
+$$
+
+Similarly, if  $e$  is in layer  $i$ , then we automatically find that  $\gamma_1(i - 1) = \gamma_2(i - 1)$  and  $\gamma_1(i) = \gamma_2(i)$ , giving an expectation of  $1 / n_{i - 1}\mathbf{1}_{\left\{ \begin{array}{c}\gamma_1(i) = \gamma_2(i)\\ \gamma_1(i - 1) = \gamma_2(i - 1) \end{array} \right\}}$ . Proceeding in this way yields
+
+$$
+\begin{array}{l} \mathbb {E} [ K _ {\boldsymbol {N}} (x, x) ] = \sum_ {a \in [ n _ {0} ] ^ {2}} x _ {a _ {1}} x _ {a _ {2}} \prod_ {i = 1} ^ {d} \frac {1}{n _ {i - 1}} \sum_ {\Gamma \in \Gamma_ {a, e v e n} ^ {2}} \sum_ {\vec {n} \in \gamma_ {1} \cap \gamma_ {2}} \delta_ {\gamma_ {1} = \gamma_ {2}} \\ = \sum_ {a \in [ n _ {0} ] ^ {2}} x _ {a _ {1}} x _ {a _ {2}} \sum_ {\Gamma \in \Gamma_ {a, e v e n} ^ {2} (\vec {n})} \delta_ {\gamma_ {1} = \gamma_ {2}} \prod_ {i = 1} ^ {d} \frac {1}{n _ {i - 1}}, \\ \end{array}
+$$
+
+which is precisely (20). The proofs of (21) and (22) are similar. We have
+
+$$
+\mathbb {E} [ K _ {\boldsymbol {N}} (x, x) ^ {2} ] = \sum_ {a \in [ n _ {0} ] ^ {4}} ^ {n _ {0}} \prod_ {k = 1} ^ {4} x _ {a _ {k}} \sum_ {\Gamma \in \Gamma_ {a} ^ {4}} \sum_ { \begin{array}{c} e _ {1} \in \gamma_ {1}, \gamma_ {2} \\ e _ {2} \in \gamma_ {3}, \gamma_ {4} \end{array} } \mathbb {E} \left[ \frac {\prod_ {k = 1} ^ {4} \mathrm {w t} (\gamma_ {k})}{W _ {e _ {1}} ^ {2} W _ {e _ {2}} ^ {2}} \right].
+$$
+
+As before let us first assume that edges  $e_1, e_2$  are not in layer  $d$ . Then,
+
+$$
+\mathbb {E} \left[ \prod_ {k = 1} ^ {4} \operatorname {w t} \left(\gamma_ {k}\right) \right] = \mathbb {E} \left[ \prod_ {k = 1} ^ {4} \prod_ {i = 1} ^ {d - 1} \widehat {W} _ {\gamma_ {k}} ^ {(i)} \mathbf {1} _ {\left\{y _ {\gamma_ {k}} ^ {(i)} > 0 \right\}} \mathbb {E} \left[ \prod_ {k = 1} ^ {4} \widehat {W} _ {\gamma_ {k}} ^ {(d)} \mid \mathcal {F} _ {d - 1} \right] \right].
+$$
+
+The inner expectation is
+
+$$
+\mathbf{1}_{\left\{ \begin{array}{c}\text{each weight appears an}\\ \text{even number of times} \end{array} \right\}}\cdot \frac{1}{n_{d - 1}^{2}}\mu_{4}^{\mathbf{1}_{\{| \Gamma (d - 1)| = |\Gamma (d)| = 1\}}.
+$$
+
+In contrast, if  $d = \ell(e_1)$  or  $d = \ell(e_2)$ , then the inner expectation is
+
+$$
+1 _ {\left\{ \begin{array}{l} \text {e a c h w e i g h t a p p e a r s a n} \\ \text {e v e n n u m b e r o f t i m e s} \end{array} \right\}} \frac {1}{n _ {d - 1} ^ {2}}.
+$$
+
+Again symmetrizing with respect to  $\pm \widehat{W}^{(d)}$  and using that the pre-activation of different neurons are independent given the activations in the previous layer we find that, on the event  $\{y_{\gamma_k}^{(d - 2)} > 0\}$ ,
+
+$$
+\mathbb{E}\left[\prod_{k = 1}^{4}\widehat{W}_{\gamma_{k}}^{(d - 1)}\mathbf{1}_{\{y_{\gamma_{k}}^{(d - 1)} > 0\}}\bigg|  \mathcal{F}_{d - 2}\right]\\ =   \mathbf{1}_{\left\{\text{each weight appears an}\atop \text{even number of times}\right\}}\frac{2^{2 - |\Gamma(d - 1)|}}{n_{d - 1}^{2}}\mu_{4}^{\mathbf{1}_{L}},
+$$
+
+where  $L$  is the event that  $|\Gamma(d - 1)| = |\Gamma(d)| = 1$  and  $e_1, e_2$  are not in layer  $d - 1$ . Proceeding in this way one layer at a time completes the proofs of (21) and (22).
+
+# B.3 PROOF OF LEMMA 8
+
+Fix  $j = 1, \ldots, 4$ , edges  $e_1, e_2$  with  $\ell(e_1) \leq \ell(e_2)$  in the computational graph of  $N$  and  $E \in \Sigma_{a_j, \text{even}}^4$ . The key idea is to decompose  $E$  into loops. To do this, define
+
+$$
+i _ {0} = - 1, \quad i _ {k} (E) := \min  \left\{i > i _ {i - 1} \mid | L (E (i)) | = 1, | R (E (i)) | = 2 \right\}, k \geq 1, \dots , \# \text {l o o p s} (E).
+$$
+
+For each  $i = 1,\ldots ,d$  there exists unique  $k = 1,\dots ,\# \mathrm{loops}(E)$  so that
+
+$$
+i _ {k - 1} (E) \leq i <   i _ {k} (E).
+$$
+
+We will say that two layers  $i, j = 1, \ldots, d$  belong to the same loop of  $E$  if exists  $k = 1, \ldots, \# \mathrm{loops}(E)$  so that
+
+$$
+i _ {k - 1} (E) \leq i, j <   i _ {k} (E).
+$$
+
+We proceed layer by layer to count the number of  $\Gamma \in \Gamma_{a_j,even}^4$  satisfying  $\Gamma (0) = a_{j}$  and  $E^{\Gamma} = E$ . To do this, suppose we are given  $\Gamma (i - 1)\in [n_{i - 1}]^4$  and we have  $L(E(i)) = 2$ . Then  $\Gamma (i - 1)$  is some permutation of  $(\alpha_{1},\alpha_{1},\alpha_{2},\alpha_{2})$  with  $\alpha_{1}\neq \alpha_{2}$ . Moreover, for  $j = 1,2$  there is a unique edge (with multiplicity 2) in  $E(i)$  whose left endpoint is  $\alpha_{j}$ . Therefore,  $\Gamma (i - 1)$  determines  $\Gamma (i)$  when  $L(E(i)) = 2$ . In contrast, suppose  $L(E(i)) = 1$ . If  $R(E(i)) = 1$ , then  $E(i)$  consists of a single edge with multiplicity 4, which again determines  $\Gamma (i - 1),\Gamma (i)$ . In short,  $\Gamma (i)$  determines  $\Gamma (j)$  for all  $j$  belonging to the same loop of  $E$  as  $i$ . Therefore, the initial condition  $\Gamma (0) = a_{j}$  determines  $\Gamma (i)$  for all  $i\leq i_1$  and the conditions  $e_1\in \gamma_1,e_2\in \gamma_2$  determine  $\Gamma$  in the loops of  $E$  containing the layers of  $e_1,e_2$ .
+
+Finally, suppose  $L(E(i)) = 1$  and  $R(E(i)) = 2$  (i.e.  $i = i_k(E)$  for some  $k = 1, \ldots, d$ ) and that  $e_1, e_2$  are not contained in the same loop of  $E$  layer  $i$ . Then all  $\binom{4}{2} = 6$  choices of  $\Gamma(i)$  satisfy  $\Gamma(i) = R(E(i))$ , accounting for the factor of  $6^{\# \mathrm{loops}(E)}$ . The concludes the proof in the case  $j = 1$ . The only difference in the cases  $j = 2, 3, 4$  is that if  $\gamma_1(0) \neq \gamma_2(0)$  (and hence  $\gamma_3(0) \neq \gamma_4(0)$ ), then since  $\ell(e_1) \leq \ell(e_2)$  in order to satisfy  $e_1 \in \gamma_1, \gamma_2$  we must have that  $i_1(E) < \ell(e_1)$ .
+
+# B.4 PROOF OF LEMMA 9
+
+The proof of Lemma 9 is essentially identical to the proof of Lemma 8. In fact it is slightly simpler since there are no distinguished edges  $e_1, e_2$  to consider. We omit the details.
+
+# C PROOF OF PROPOSITION 4
+
+In this section, we seek to estimate  $\mathbb{E}[K_{\mathrm{b}}], \mathbb{E}[K_{\mathrm{b}}^2], \mathbb{E}[\Delta_{\mathrm{bb}}]$ . The approach is essentially identical to but somewhat simpler than our proof of Proposition 3 in §B. We will therefore focus here on explaining the salient differences. Our starting point is the following analog of Lemma 6, which gives a sum-over-paths expression for the bias contribution  $K_{\mathrm{b}}$  to the neural tangent kernel. To state it, let us define, for any collection  $Z = (z_1, \ldots, z_k) \in Z^k$  of  $k$  neurons in  $N$
+
+$$
+\mathbf {1} _ {\{y _ {Z} > 0 \}} := \prod_ {j = 1} ^ {k} \mathbf {1} _ {\{y _ {z _ {j}} > 0 \}},
+$$
+
+to be the event that the pre-activations of the neurons  $z_{k}$  are positive.
+
+Lemma 12 ( $K_{\mathrm{b}}$  as a sum over paths). With probability 1,
+
+$$
+K _ {\mathrm {b}} = \sum_ {Z \in Z ^ {1}} \mathbf {1} _ {\{Z > 0 \}} \sum_ {\Gamma \in \Gamma_ {(Z, Z)} ^ {2}} \prod_ {k = 1} ^ {2} \operatorname {w t} \left(\gamma_ {k}\right), \tag {49}
+$$
+
+where  $Z^1$ ,  $\Gamma_{(Z,Z)}^2$ ,  $\mathrm{wt}(\gamma)$  are defined in §A. Further, almost surely,
+
+$$
+\Delta_ {\mathrm {b b}} = 0. \tag {50}
+$$
+
+The proof of this result is a small modification of the proof of Lemma 6 and hence is omitted. Taking expectations, we therefore obtain the following analog to Lemma 7.
+
+Lemma 13 (Expectation of  $K_{\mathrm{b}}$ ,  $K_{\mathrm{b}}^{2}$  as a sum over paths). We have
+
+$$
+\mathbb {E} \left[ K _ {\mathrm {b}} \right] = \frac {1}{2} \sum_ {Z \in Z ^ {1}} \sum_ {\substack {\Gamma \in \Gamma_ {(Z, Z), e v e n} ^ {2} \\ \Gamma = (\gamma_ {1}, \gamma_ {2})}} H (\Gamma), \quad H (\Gamma) = \mathbf {1} _ {\{\gamma_ {1} = \gamma_ {2} \}} \prod_ {i = \ell (Z) + 1} ^ {d} \frac {1}{n _ {i - 1}}. \tag{51}
+$$
+
+Moreover,
+
+$$
+\mathbb{E}[K_{\mathrm{b}}^{2}] = \frac{1}{2}\sum_{\substack{Z = (z_{1},z_{2})\in Z^{2} \\ \ell (z_{1})\leq \ell (z_{2})}}\sum_{\Gamma \in \Gamma^{4}_{(Z,Z),even}}\widehat{H} (\Gamma),
+$$
+
+where for  $\Gamma = (\gamma_1,\dots ,\gamma_4)\in \Gamma^{4}_{(Z,Z),even}$  we have
+
+$$
+\widehat {H} (\Gamma) = \prod_ {i ^ {\prime} = \ell \left(z _ {1}\right) + 1} ^ {\ell \left(z _ {2}\right)} \mathbf {1} _ {\left\{ \begin{array}{l} \gamma_ {1} \left(i ^ {\prime}\right) = \gamma_ {2} \left(i ^ {\prime}\right) \\ \gamma_ {1} \left(i ^ {\prime} - 1\right) = \gamma_ {2} \left(i ^ {\prime} - 1\right) \end{array} \right\}} \frac {1}{n _ {i ^ {\prime} - 1}} \prod_ {i = \ell \left(z _ {2}\right) + 1} ^ {d} \frac {2 ^ {| \Gamma (i) | - 2}}{n _ {i - 1} ^ {2}} \mu_ {4} ^ {\mathbf {1} _ {\{| \Gamma (i) | = | \Gamma (i - 1) | \}}}. \tag {52}
+$$
+
+The proof is identical to the argument used in §B.2 to establish Lemma 7, so we omit the details. The relation (51) is easy to simplify:
+
+$$
+\mathbb {E} [ K _ {\mathrm {b}} ] = \frac {1}{2} \sum_ {Z \in Z ^ {1}} \sum_ {\gamma \in \Gamma_ {Z}} \prod_ {i = \ell (Z) + 1} ^ {d} \frac {1}{n _ {i - 1}} = \frac {1}{2} \sum_ {z \in Z ^ {1}} \frac {1}{n _ {\ell (z)}} = \frac {d}{2},
+$$
+
+where we used that the number paths from a neuron in layer  $\ell$  to the output of  $N$  equals  $\prod_{i = \ell +1}^{d}n_{i}$ . This proves the first statement in Proposition 4. Next, let us explain how to simplify  $\mathbb{E}[K_{\mathrm{b}}^2]$ . The key computation is the following
+
+Lemma 14. Fix two neurons  $z_{1},z_{2}$  with  $\ell (z_1)\leq \ell (z_2)$  and write  $Z = (z_{1},z_{1},z_{2},z_{2})$  . Then,
+
+$$
+\sum_ {\Gamma \in \Gamma_ {Z, e v e n} ^ {4}} \widehat {H} (\Gamma) \simeq \frac {1}{n _ {\ell \left(z _ {1}\right)} n _ {\ell \left(z _ {2}\right)}} \exp \left(5 \sum_ {i = \ell \left(z _ {2}\right) + 1} ^ {d} \frac {1}{n _ {i}}\right) \left(1 + O \left(\sum_ {i = \ell \left(z _ {2}\right) + 1} ^ {d} \frac {1}{n _ {i} ^ {2}}\right)\right). \tag {53}
+$$
+
+Proof. The proof of Lemma 14 is a simplified version of the computation of  $\mathbb{E}[K_{\mathrm{w}}^2]$  (starting around (24) and ending at the end of the proof of Proposition 3). Specifically, note that for  $\Gamma = (\gamma_1,\ldots ,\gamma_4)\in \Gamma_{Z,even}^4$  with  $\ell (z_1)\leq \ell (z_2)$ , the delta functions  $\mathbf{1}_{\{\gamma_1(i') = \gamma_2(i')\}}\mathbf{1}_{\{\gamma_1(i' - 1) = \gamma_2(i' - 1)\}}$  in the definition (52) of  $\widehat{H} (\Gamma)$  ensures that  $\gamma_{1},\gamma_{2}$  go through the same neuron in layer  $\ell (z_{2})$ . To condition on the index of this neuron, we recall that we denote by  $z(j,\beta)$  neuron number  $\beta$  in layer  $j$ . We have
+
+$$
+\begin{array}{l} \sum_{\Gamma \in \Gamma^{4}_{Z,even}}\widehat{H} (\Gamma) = \sum_{\beta = 1}^{n_{\ell (z_{2})}}\sum_{\substack{\gamma_{1},\gamma_{2}:z_{1}\to z(\ell (z_{2}),\beta)}}\prod_{i^{\prime} = \ell (z_{1}) + 1}^{\ell (z_{2})}\mathbf{1}_{\left\{ \begin{array}{c}\gamma_{1}(i^{\prime}) = \gamma_{2}(i^{\prime})\\ \gamma_{1}(i^{\prime} - 1) = \gamma_{2}(i^{\prime} - 1) \end{array} \right\}}\frac{1}{n_{i^{\prime} - 1}}\sum_{\Gamma \in \Gamma^{4}_{Z^{\prime},even}}H(\Gamma) \\ = \frac {1}{n _ {\ell \left(z _ {1}\right)} - 1} \sum_ {\beta = 1} ^ {n _ {\ell \left(z _ {2}\right)}} \sum_ {\Gamma \in \Gamma_ {Z ^ {\prime}, e v e n} ^ {4}} H (\Gamma), \tag {54} \\ \end{array}
+$$
+
+where  $Z^{\prime} = (z(\ell (z_{2}),\beta),z(\ell (z_{2}),\beta),z_{2},z_{2})$  and
+
+$$
+H (\Gamma) = \prod_ {i = \ell (z _ {2}) + 1} ^ {d} \frac {2 ^ {| \Gamma (i) | - 2}}{n _ {i - 1} ^ {2}} \mu_ {4} ^ {\mathbf {1} _ {\{| \Gamma (i) | = | \Gamma (i - 1) | \}}}.
+$$
+
+Since the inner sum in (54) is independent of  $\beta$  by symmetry, we find
+
+$$
+\sum_ {\Gamma \in \Gamma_ {Z, e v e n} ^ {4}} \hat {H} (\Gamma) = \frac {n _ {\ell \left(z _ {2}\right)}}{n _ {\ell \left(z _ {1}\right) - 1}} \sum_ {\Gamma \in \Gamma_ {Z ^ {\prime \prime}, e v e n} ^ {4}} H (\Gamma), \tag {55}
+$$
+
+where  $Z'' = (1, 1, z_2, z_2)$ . The inner sum in (55) is now precisely one of the terms  $I_j$  from (27) without counting terms involving edges  $e_1, e_2$ , except that the paths start at neuron 1 in layer  $\ell(z_2)$ . The changes of variables from  $\Gamma \in \Gamma_{even}^4$  to  $E \in \Sigma_{even}^4$  to  $V \in \Gamma^2$  that we used to estimate the  $I_j$ 's are no far simpler. In particular, Lemma 8 still holds but without any of the  $A(E, i_1, i_2)$ ,  $C(E, i_1, i_2)$ ,  $\widehat{C}(E, i_1, i_2)$  terms. Thus, we find that
+
+$$
+\sum_ {\Gamma \in \Gamma_ {Z ^ {\prime \prime}} ^ {4, e v e n}} H (\Gamma) \simeq \sum_ {E \in \Sigma_ {Z ^ {\prime \prime \prime}} ^ {4, e v e n}} H (E) 6 ^ {\# \mathrm {l o o p s} (E)} \simeq \sum_ {V \in \Gamma_ {Z ^ {\prime \prime \prime}} ^ {2}} H (V) 3 ^ {\# \mathrm {l o o p s} (V)},
+$$
+
+where for the second estimate we applied Lemma 9 and have written  $Z''' = (1, z_2)$ . Thus, as in the derivation of (42), we find that
+
+$$
+\sum_ {\Gamma \in \Gamma_ {Z ^ {\prime \prime}} ^ {4, e v e n}} H (\Gamma) \simeq \frac {1}{n _ {\ell (z _ {2})} ^ {2}} \mathcal {E} \left[ H _ {*} (V) \right],
+$$
+
+where
+
+$$
+H _ {*} (V) = 2 ^ {\# \{i \in [ d ] \mid | V (i) | = 1 \}} 3 ^ {\# \text {l o o p s} (V)} \mu_ {4} ^ {\# \{i \in [ d ] \mid | V (i - 1) | = | V (i) | = 1 \}}
+$$
+
+and  $\mathcal{E}$  is the expectation over pairs of paths starting from neurons  $1, z_{2}$  in layer  $\ell(z_{2})$  to the output of the network for which neurons in subsequent layers are chosen independently and uniformly among all neurons in that layer. This is precisely the expectation we evaluated in the end of the proof for Proposition 3. Thus, applying Proposition 10 exactly as in that case, we find that
+
+$$
+\sum_ {\Gamma \in \Gamma_ {Z} ^ {4, e v e n}} F (\Gamma) \simeq \frac {1}{n _ {\ell (z _ {2})} ^ {2}} \exp \left(5 \sum_ {i = \ell (z _ {2}) + 1} ^ {d} \frac {1}{n _ {i}} + O \left(\sum_ {i = \ell (z _ {2}) + 1} ^ {d} \frac {1}{n _ {i} ^ {2}}\right)\right).
+$$
+
+Putting this together with (54) completes the proof of Lemma 14.
+
+![](images/0b2b246ce5a648dfced3e54cc3ad2ee1cd4c5ee799397937d51716d6a4850e74.jpg)
+
+Lemma 14 combined with Lemma 13 yields
+
+$$
+\mathbb{E}[K_{\mathrm{b}}^{2}]\simeq \sum_{\substack{i,j = 1\\ i\leq j}}^{d}\exp \left(5\sum_{i = j + 1}^{d}\frac{1}{n_{i}}\right)\left(1 + O\left(\sum_{i = 1}^{d}\frac{1}{n_{i}^{2}}\right)\right),
+$$
+
+as claimed in the statement Proposition 4.
+
+![](images/daef91af343710c63d289d78d8954304ac0f906eac4172d39fcdc5dd9e1236ea.jpg)
+
+# D PROOF OF PROPOSITION 5
+
+We begin by computing  $\mathbb{E}[K_{\mathrm{b}}K_{\mathrm{w}}]$ . We will use a hybrid of the procedures for computing  $\mathbb{E}[K_{\mathrm{b}}^2]$  and  $\mathbb{E}[K_{\mathrm{w}}^2]$ . Recall from Lemmas 6 and 12 that
+
+$$
+K_{\mathrm{b}} = \sum_{Z\in Z^{1}}\mathbf{1}_{\{y_{Z} > 0\}}\sum_{\Gamma \in \Gamma^{2}_{(Z,Z)}}\prod_{k = 1}^{2}\mathrm{wt}(\gamma_{k}),\qquad K_{\mathrm{w}} = \sum_{a\in [n_{0}]^{2}}\prod_{k = 1}^{2}x_{a_{k}}\sum_{\substack{\Gamma \in \Gamma^{2}_{a}\\ \Gamma = (\gamma_{1},\gamma_{2})}}\sum_{e\in \gamma_{1},\gamma_{2}}\frac{\prod_{k = 1}^{2}\mathrm{wt}(\gamma_{k})}{W^{2}_{e}}.
+$$
+
+Therefore, the expectation of the product  $\mathbb{E}[K_{\mathrm{b}}K_{\mathrm{w}}]$  has the following form
+
+$$
+\sum_ {a \in [ n _ {0} ] ^ {2}} \prod_ {k = 1} ^ {2} x _ {a _ {k}} \sum_ {Z \in Z ^ {1}} \sum_ {\Gamma \in \Gamma_ {(Z, Z, a)}} ^ {4} \sum_ {e \in \gamma_ {3}, \gamma_ {4}} \mathbb {E} \left[ \mathbf {1} _ {\{y _ {Z} > 0 \}} \frac {\prod_ {k = 1} ^ {4} \mathrm {w t} (\gamma_ {k})}{W _ {e} ^ {2}} \right].
+$$
+
+Here, for a neuron  $Z$  and  $a = (a_1, a_2) \in [n_0]^2$  we've denoted by  $\Gamma_{(Z,Z,a)}^4$  the set of four tuples  $(\gamma_1, \ldots, \gamma_4)$  of paths in the computational graph of  $N$  where  $\gamma_1, \gamma_2$  start from  $Z$  and  $\gamma_3, \gamma_4$  start at neurons  $a_1, a_2$  respectively. The analog of Lemmas 7 and 13 (with essentially the same proof), gives that the expectation in the previous line equals
+
+$$
+\frac {\| x \| ^ {2}}{2} \prod_ {i = 1} ^ {\ell (Z)} \mathbf {1} _ {\{\gamma_ {3} (i) = \gamma_ {4} (i) \}} \frac {1}{n _ {i - 1}} \prod_ {i = \ell (Z) + 1} ^ {d} \frac {2 ^ {2 - | \Gamma (i) |}}{n _ {i - 1} ^ {2}} \mu_ {4} ^ {\mathbf {1} _ {\{| \Gamma (i - 1) | = | \Gamma (i) | = 1, i \neq \ell (e) \}}},
+$$
+
+which, up to a multiplicative constant equals
+
+$$
+G _ {Z} (\Gamma) := \| x \| ^ {2} \prod_ {i = 1} ^ {\ell (Z)} \mathbf {1} _ {\left\{\gamma_ {3} (i) = \gamma_ {4} (i) \right\}} \frac {1}{n _ {i - 1}} \prod_ {i = \ell (Z) + 1} ^ {d} \frac {2 ^ {2 - | \Gamma (i) |}}{n _ {i - 1} ^ {2}} \mu_ {4} ^ {\mathbf {1} _ {\{| \Gamma (i - 1) | = | \Gamma (i) | = 1 \}}}, \tag {56}
+$$
+
+which is independent of  $e$ . Thus, we find
+
+$$
+\mathbb {E} [ K _ {\mathrm {b}} K _ {\mathrm {w}} ] \simeq \| x \| ^ {2} \sum_ {i = 1} ^ {d} \sum_ {Z \in Z ^ {1}} \sum_ {\Gamma \in \Gamma_ {(Z, Z, 1, 1), e v e n} ^ {4}} G _ {Z} (\Gamma) T _ {3, 4} ^ {i} (\Gamma),
+$$
+
+where if  $\Gamma = (\gamma_1,\dots ,\gamma_4)$  we recall that  $T_{3,4}^{i}(\Gamma)$  is the indicator function of the event that paths  $\gamma_3,\gamma_4$  pass through the same edge in the computational graph of  $N$  at layer  $i$  (see (29)).
+
+As before, note that the delta functions  $\mathbf{1}_{\{\gamma_3(i) = \gamma_4(i)\}}$  ensure that  $\gamma_3,\gamma_4$  pass through the same neuron in layer  $\ell (Z)$ . Thus, we may condition on the common neuron through which  $\gamma_3,\gamma_4$  must pass at layer  $\ell (Z)$  to obtain that  $\mathbb{E}[K_{\mathrm{b}}K_{\mathrm{w}}]$  is bounded above and below by a constant times
+
+$$
+\| x \| ^ {2} \sum_ {i = 1} ^ {d} \sum_ {Z \in Z ^ {1}} \sum_ {\Gamma \in \Gamma_ {(Z, Z, 1, 1), e v e n} ^ {4}} G _ {Z} (\Gamma) T _ {3, 4} ^ {i} (\Gamma) = \frac {\| x \| ^ {2}}{n _ {0}} \sum_ {i = 1} ^ {d} \sum_ {Z \in Z ^ {1}} \sum_ {\beta = 1} ^ {n _ {\ell (Z)}} \sum_ {\Gamma \in \Gamma_ {Z ^ {\prime}, e v e n} ^ {4}} \widehat {G} _ {Z} (\Gamma) T _ {3, 4} ^ {i} (\Gamma),
+$$
+
+where  $Z^{\prime} = (Z,Z,z(\ell (Z),\beta),z(\ell (Z),\beta))$  and we have set
+
+$$
+\widehat {G} _ {Z} (\Gamma) = \prod_ {i = \ell (Z) + 1} ^ {d} \frac {2 ^ {2 - | \Gamma (i) |}}{n _ {i - 1} ^ {2}} \mu_ {4} ^ {\mathbf {1} _ {\{| \Gamma (i - 1) | = | \Gamma (i) | = 1 \}}}.
+$$
+
+Notice that  $T_{3,4}^{i} = 1$  if  $i \leq \ell(Z)$ . Moreover, for  $i \geq \ell(Z) + 1$ , the same argument as in the proof of Lemma 8 shows that the number of  $\Gamma \in \Gamma_{Z',\text{even}}^{4}$  for which  $\gamma_{3}, \gamma_{4}$  pass through the same edge at layer  $i$  and correspond to the same unordered multiset of edges  $E$  equals
+
+$$
+6 ^ {\# \operatorname {l o o p s} (E) - 1 _ {\{| R (E (i)) | \cdot | L (E (i)) | \neq 1 \}}} \simeq 6 ^ {\# \operatorname {l o o p s} (E)}.
+$$
+
+As in the proof of Proposition 3, observe that  $\widehat{G}_Z(\Gamma)T_{3,4}^i (\Gamma)$  depends only on the unordered multiset of edges  $E^{\Gamma}$  in  $\Gamma$ . Thus, we find that
+
+$$
+\mathbb {E} \left[ K _ {\mathrm {b}} K _ {\mathrm {w}} \right] \simeq \frac {d \| x \| ^ {2}}{n _ {0}} \sum_ {Z \in Z ^ {1}} n _ {\ell (Z)} \sum_ {E \in \Sigma_ {Z ^ {\prime}, e v e n} ^ {4}} G _ {Z} (E) 6 ^ {\# \text {l o o p s} (E)}.
+$$
+
+Applying Proposition 10 as in the end of the proof of Propositions 3 and 4 we conclude
+
+$$
+\sum_ {E \in \Sigma_ {Z ^ {\prime}, e v e n} ^ {4}} G _ {Z} (E) 6 ^ {\# \mathrm {l o o p s} (E)} = \frac {1}{n _ {\ell (Z)} ^ {2}} \exp \left(5 \sum_ {i = \ell (Z) + 1} ^ {d} \frac {1}{n _ {i}}\right) \left(1 + O \left(\sum_ {i = 1} ^ {d} \frac {1}{n _ {i} ^ {2}}\right)\right).
+$$
+
+Hence,
+
+$$
+\mathbb {E} \left[ K _ {\mathrm {b}} K _ {\mathrm {w}} \right] \simeq \frac {d \| x \| ^ {2}}{n _ {0}} \left[ \sum_ {j = 1} ^ {d} \exp \left(- 5 \sum_ {i = 1} ^ {j} \frac {1}{n _ {i}}\right) \right] \exp \left(5 \sum_ {i = 1} ^ {d} \frac {1}{n _ {i}}\right) \left(1 + O \left(\sum_ {i = 1} ^ {d} \frac {1}{n _ {i} ^ {2}}\right)\right).
+$$
+
+To complete the proof of Proposition 5 it remains to evaluate  $\mathbb{E}[\Delta_{\mathrm{wb}}]$ . To do this, we note that, as in the proof of Lemma 6, we have
+
+$$
+\Delta_{\mathrm{wb}}  =   \sum_{\substack{a\in [n_{0}]^{2}\\ a = (a_{1},a_{2})}}\prod_{k = 1}^{2}x_{a_{k}}\sum_{Z\in Z^{1}}{\bf 1}_{\{y_{Z} > 0\}}\sum_{\Gamma \in \Gamma^{4}_{(Z,Z,a_{1},a_{2})}}\sum_{e\in \gamma_{2},\gamma_{3}}\frac{\prod_{k = 1}^{4}\mathrm{wt}(\gamma_{k})}{\widehat{W}_{e}^{2}}
+$$
+
+plus a term that has mean 0. Therefore, as in Lemma 7, we find
+
+$$
+\mathbb {E} [ \Delta_ {\mathrm {w b}} ] \simeq \frac {\| x \| _ {2} ^ {2}}{n _ {0}} \sum_ {Z \in Z ^ {1}} \sum_ {\Gamma \in \Gamma_ {(Z, Z, 1, 1), e v e n} ^ {4}} P (\Gamma) \# \{\text {e d g e s} e \text {b e l o n g i n g t o b o t h} \gamma_ {2}, \gamma_ {3} \},
+$$
+
+where
+
+$$
+P(\Gamma) = \prod_{i = 1}^{\ell (Z)}\frac{1}{n_{i - 1}}\mathbf{1}_{\left\{ \begin{array}{c}\gamma_{3}(i - 1) = \gamma_{4}(i - 1)\\ \gamma_{3}(i) = \gamma_{4}(i) \end{array} \right\}}\prod_{i = \ell (Z) + 1}^{d}\frac{2^{2 - |\Gamma(i)|}}{n_{i - 1}^{2}}\mu_{4}^{\mathbf{1}_{\{| \Gamma (i - 1)| = |\Gamma (i)| = 1\}}}.
+$$
+
+Thus, we have
+
+$$
+\mathbb {E} [ \Delta_ {\mathrm {w b}} ] \simeq \frac {\| x \| _ {2} ^ {2}}{n _ {0}} \sum_ {i = 1} ^ {d} \sum_ {Z \in Z ^ {1}} \sum_ {E \in \Sigma_ {Z ^ {\prime}, e v e n} ^ {4}} P (E) T _ {2, 3} ^ {i} (E) \# \left\{\Gamma \in \Gamma_ {Z ^ {\prime}, e v e n} ^ {4} \big | _ {\gamma_ {2} (i - 1) = \gamma_ {3} (i - 1)} ^ {E ^ {\Gamma} = E \gamma_ {2} (i) = \gamma_ {3} (i)} \right\},
+$$
+
+where  $T_{2,3}^{i}(E)$  is as in (29), the sum is over unordered edge multiset sets  $E$  (see (17)), and we've set
+
+$$
+Z ^ {\prime} = (Z, Z, z (0, 1), z (0, 1)).
+$$
+
+As in Lemma 8, the counting term satisfies
+
+$$
+\# \left\{\Gamma \in \Gamma_ {Z ^ {\prime}, e v e n} ^ {4} \Big | _ {\gamma_ {2} (i - 1) = \gamma_ {3} (i - 1)} ^ {E ^ {\Gamma} = E \gamma_ {2} (i) = \gamma_ {3} (i)} \right\} \simeq \mathbf {1} _ {\{\ell (Z) <   i \}} C (E, \ell (Z), i) 6 ^ {\# \mathrm {l o o p s} (E)},
+$$
+
+where  $C(E, i, j)$  was defined in (32) and is the event that there exists a collision between layers  $i, j$  (i.e. there exists  $\ell = i, \dots, j - 1$  so that  $|R(E(\ell))| = 1$ ). Proceeding now as in the derivation of  $\mathbb{E}[\Delta_{\mathrm{ww}}]$  at the end of the proof of Proposition 3, we find
+
+$$
+\mathbb{E}[\Delta_{\mathrm{wb}}]\simeq \frac{\|x\|_{2}^{2}}{n_{0}}\exp \left(5\sum_{i = 1}^{d}\frac{1}{n_{i}}\right)\left[ \sum_{\substack{i,j = 1\\ j <   i}}^{d}e^{-5\sum_{\alpha = 1}^{j}\frac{1}{n_{\alpha}}}\sum_{\ell = j}^{i - 1}\frac{1}{n_{\ell}} e^{-6\sum_{\alpha = j + 1}^{\ell -1}\frac{1}{n_{\alpha}}} \right]\left(1 + O\left(\sum_{i = 1}^{d}\frac{1}{n_{i}^{2}}\right)\right).
+$$
+
+This completes the proof of Proposition 5.
\ No newline at end of file
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+# FREELB: ENHANCED ADVERSARIAL TRAINING FOR NATURAL LANGUAGE UNDERSTANDING
+
+Chen Zhu $^{1}$ , Yu Cheng $^{2}$ , Zhe Gan $^{2}$ , Siqi Sun $^{2}$ , Tom Goldstein $^{1}$ , Jingjing Liu $^{2}$
+
+<sup>1</sup>University of Maryland, College Park <sup>2</sup>Microsoft Dynamics 365 AI Research
+
+{chenzhu,tomg}@cs.umd.edu, {yu.cheng,zhe.gan,siqi.sun,jingjl}@microsoft.com
+
+# ABSTRACT
+
+Adversarial training, which minimizes the maximal risk for label-preserving input perturbations, has proved to be effective for improving the generalization of language models. In this work, we propose a novel adversarial training algorithm, FreeLB, that promotes higher invariance in the embedding space, by adding adversarial perturbations to word embeddings and minimizing the resultant adversarial risk inside different regions around input samples. To validate the effectiveness of the proposed approach, we apply it to Transformer-based models for natural language understanding and commonsense reasoning tasks. Experiments on the GLUE benchmark show that when applied only to the finetuning stage, it is able to improve the overall test scores of BERT-base model from 78.3 to 79.4, and RoBERTa-large model from 88.5 to 88.8. In addition, the proposed approach achieves state-of-the-art single-model test accuracies of  $85.44\%$  and  $67.75\%$  on ARC-Easy and ARC-Challenge. Experiments on CommonsenseQA benchmark further demonstrate that FreeLB can be generalized and boost the performance of RoBERTa-large model on other tasks as well. $^1$
+
+# 1 INTRODUCTION
+
+Adversarial training is a method for creating robust neural networks. During adversarial training, mini-batches of training samples are contaminated with adversarial perturbations (alterations that are small and yet cause misclassification), and then used to update network parameters until the resulting model learns to resist such attacks. Adversarial training was originally proposed as a means to enhance the security of machine learning systems (Goodfellow et al., 2015), especially for safety-critical systems like self-driving cars (Xiao et al., 2018) and copyright detection (Saadatpanah et al., 2019).
+
+In this paper, we turn our focus away from the security benefits of adversarial training, and instead study its effects on generalization. While adversarial training boosts the robustness, it is widely accepted by computer vision researchers that it is at odds with generalization, with classification accuracy on non-corrupted images dropping as much as  $10\%$  on CIFAR-10, and  $15\%$  on ImageNet (Madry et al., 2018; Xie et al., 2019). Surprisingly, people observe the opposite result for language models (Miyato et al., 2017; Cheng et al., 2019), showing that adversarial training can improve both generalization and robustness.
+
+We will show that adversarial training significantly improves performance of state-of-the-art models for many language understanding tasks. In particular, we propose a novel adversarial training algorithm, called FreeLB (Free Large-Batch), which adds adversarial perturbations to word embeddings and minimizes the resultant adversarial loss around input samples. The method leverages recently proposed "free" training strategies (Shafahi et al., 2019; Zhang et al., 2019) to enrich the training data with diversified adversarial samples under different norm constraints at no extra cost than PGD-based (Projected Gradient Descent) adversarial training (Madry et al., 2018), which enables us to perform such diversified adversarial training on large-scale state-of-the-art models. We observe improved invariance in the embedding space for models trained with FreeLB, which is positively correlated with generalization.
+
+We perform comprehensive experiments to evaluate the performance of a variety of adversarial training algorithms on state-of-the-art language understanding models and tasks. In the comparisons with standard PGD (Madry et al., 2018), FreeAT (Shafahi et al., 2019) and YOPO (Zhang et al., 2019), FreeLB stands out to be the best for the datasets and models we evaluated. With FreeLB, we achieve state-of-the-art results on several important language understanding benchmarks. On the GLUE benchmark, FreeLB pushes the performance of the BERT-base model from 78.3 to 79.4. The overall score of the RoBERTa-large models on the GLUE benchmark is also lifted from 88.5 to 88.8, achieving best results on most of its sub-tasks. Experiments also show that FreeLB can boost the performance of RoBERTa-large on question answering tasks, such as the ARC and Common-senseQA benchmarks. We also provide a comprehensive ablation study and analysis to demonstrate the effectiveness of our training process.
+
+# 2 RELATED WORK
+
+# 2.1 ADVERSARIAL TRAINING
+
+To improve the robustness of neural networks against adversarial examples, many defense strategies and models have been proposed, in which PGD-based adversarial training (Madry et al., 2018) is widely considered to be the most effective, since it largely avoids the obfuscated gradient problem (Athalye et al., 2018). It formulates a class of adversarial training algorithms (Kurakin et al., 2017) into solving a minimax problem on the cross-entropy loss, which can be achieved reliably through multiple projected gradient ascent steps followed by a SGD (Stochastic Gradient Descent) step.
+
+Despite being verified by Athalye et al. (2018) to avoid obfuscated gradients, Qin et al. (2019) shows that PGD-based adversarial training still leads to highly convolved and non-linear loss surfaces when  $K$  is small, which could be readily broken under stronger adversaries. Thus, to be effective, the cost of PGD-based adversarial training is much higher than conventional training. To mitigate this cost, Shafahi et al. (2019) proposed a "free" adversarial training algorithm that simultaneously updates both model parameters and adversarial perturbations on a single backward pass. Using a similar formulation, Zhang et al. (2019) effectively reduce the total number of full forward and backward propagations for obtaining adversarial examples by restricting most of its adversarial updates in the first layer.
+
+# 2.2 ADVERSARIAL EXAMPLES IN NATURAL LANGUAGES
+
+Adversarial examples have been explored primarily in the image domain, and received many attention in text domain recently. Previous works on text adversaries have focused on heuristics for creating adversarial examples in the black-box setting, or on specific tasks. Jia & Liang (2017) propose to add distracting sentences to the input document in order to induce mis-classification. Zhao et al. (2018) generate text adversaries by projecting the input data to a latent space using GANs, and searching for adversaries close to the original instance. Belinkov & Bisk (2018) manipulate every word in a sentence with synthetic or natural noise in machine translation systems. Iyyer et al. (2018) propose a neural paraphrase model based on back-translated data to produce paraphrases that have different sentence structures. Different from previous work, ours is not to produce actual adversarial examples, but only take the benefit of adversarial training for natural language understanding.
+
+We are not the first to observe that robust language models may perform better on clean test data. Miyato et al. (2017) extend adversarial and virtual adversarial training (Miyato et al., 2019) to the text domain to improve the performance on semi-supervised classification tasks. Ebrahimi et al. (2018) propose a character/word replacement for crafting attacks, and show employing adversarial examples in training renders the models more robust. Ribeiro et al. (2018) show that adversarial attacks can be used as a valuable tool for debugging NLP models. Cheng et al. (2019) also find that crafting adversarial examples can help neural machine translation significantly. Notably, these studies have focused on simple models or text generation tasks. Our work explores how to efficiently use the gradients obtained in adversarial training to boost the performance of state-of-the-art transformer-based models.
+
+# 3 ADVERSARIAL TRAINING FOR LANGUAGE UNDERSTANDING
+
+Pre-trained large-scale language models, such as BERT (Devlin et al., 2019), RoBERTa (Liu et al., 2019b), ALBERT (Lan et al., 2020) and T5 (Raffel et al., 2019), have proven to be highly effective for downstream tasks. We aim to further improve the generalization of these pre-trained language models on the downstream language understanding tasks by enhancing their robustness in the embedding space during finetuning on these tasks. We achieve this goal by creating "virtual" adversarial examples in the embedding space, and then perform parameter updates on these adversarial embeddings. Creating actual adversarial examples for language is difficult; even with state-of-the-art language models as guidance (e.g., (Cheng et al., 2019)), it remains unclear how to construct label-preserving adversarial examples via word/character replacement without human evaluations, because the meaning of each word/character depends on the context (Ribeiro et al., 2018). Since we are only interested in the effects of adversarial training, rather than producing actual adversarial examples, we add norm-bounded adversarial perturbations to the embeddings of the input sentences using a gradient-based method. Note that our embedding-based adversary is strictly stronger than a more conventional text-based adversary, as our adversary can make manipulations on word embeddings that are not possible in the text domain.
+
+For models that incorporate various input representations, including word or subword embeddings, segment embeddings and position embeddings, our adversaries only modify the concatenated word or sub-word embeddings, leaving other components of the sentence representation unchanged.  ${}^{2}$  Denote the sequence of one-hot representations of the input subwords as  $Z = \left\lbrack  {{z}_{1},{z}_{2},\ldots ,{z}_{n}}\right\rbrack$  ,the embedding matrix as  $V$  ,and the language model (encoder) as a function  $y = {f}_{\theta }\left( X\right)$  ,where  $X = V\mathbf{Z}$  is the subword embeddings,  $y$  is the output of the model (e.g.,class probabilities for classification models), and  $\theta$  denotes all the learnable parameters including the embedding matrix  $V$  . We add adversarial perturbations  $\delta$  to the embeddings such that the prediction becomes  ${y}^{\prime } = {f}_{\theta }\left( {X + \delta }\right)$  . To preserve the semantics, we constrain the norm of  $\delta$  to be small,and assume the model's prediction should not change after the perturbation. This formulation is analogous to Miyato et al. (2017), with the difference that we do not require  $X$  to be normalized.
+
+# 3.1 PGD FOR ADVERSARIAL TRAINING
+
+Standard adversarial training seeks to find optimal parameters  $\theta^{*}$  to minimize the maximum risk for any  $\delta$  within a norm ball as:
+
+$$
+\min  _ {\boldsymbol {\theta}} \mathbb {E} _ {(\boldsymbol {Z}, y) \sim \mathcal {D}} \left[ \max  _ {\| \boldsymbol {\delta} \| \leq \epsilon} L \left(f _ {\boldsymbol {\theta}} (\boldsymbol {X} + \boldsymbol {\delta}), y\right) \right], \tag {1}
+$$
+
+where  $\mathcal{D}$  is the data distribution,  $y$  is the label, and  $L$  is some loss function. We use the Frobenius norm to constrain  $\delta$ . For neural networks, the outer "min" is non-convex, and the inner "max" is non-concave. Nonetheless, Madry et al. (2018) demonstrated that this saddle-point problem can be solved reliably with SGD for the outer minimization and PGD (a standard method for large-scale constrained optimization, see (Combettes & Pesquet, 2011) and (Goldstein et al., 2014)), for the inner maximization. In particular, for the constraint  $\| \delta \|_F \leq \epsilon$ , with an additional assumption that the loss function is locally linear, PGD takes the following step (with step size  $\alpha$ ) in each iteration:
+
+$$
+\boldsymbol {\delta} _ {t + 1} = \Pi_ {\| \boldsymbol {\delta} \| _ {F} \leq \epsilon} \left(\boldsymbol {\delta} _ {t} + \alpha g (\boldsymbol {\delta} _ {t}) / \| g (\boldsymbol {\delta} _ {t}) \| _ {F}\right), \tag {2}
+$$
+
+where  $g(\delta_t) = \nabla_{\delta} L(f_\theta(X + \delta_t), y)$  is the gradient of the loss with respect to  $\delta$ , and  $\Pi_{\|\delta\|_F \leq \epsilon}$  performs a projection onto the  $\epsilon$ -ball. To achieve high-level robustness, multi-step adversarial examples are needed during training, which is computationally expensive. The  $K$ -step PGD ( $K$ -PGD) requires  $K$  forward-backward passes through the network, while the standard SGD update requires only one. As a result, the adversary generation step in adversarial training increases run-time by an order of magnitude—a catastrophic amount when training large state-of-the-art language models.
+
+# 3.2 LARGE-BATCH ADVERSARIAL TRAINING FOR FREE
+
+In the inner ascent steps of PGD, the gradients of the parameters can be obtained with almost no overhead when computing the gradients of the inputs. From this observation, FreeAT (Shafahi et al.,
+
+Algorithm 1 "Free" Large-Batch Adversarial Training (FreeLB-  $K$  
+Require: Training samples  $X = \{(Z,y)\}$ , perturbation bound  $\epsilon$ , learning rate  $\tau$ , ascent steps  $K$ , ascent step size  $\alpha$   
+1: Initialize  $\theta$   
+2: for epoch = 1...Nep do  
+3: for minibatch  $B \subset X$  do  
+4:  $\delta_0 \gets \frac{1}{\sqrt{N_\delta}} U(-\epsilon, \epsilon)$   
+5:  $g_0 \gets 0$   
+6: for  $t = 1\ldots K$  do  
+7: Accumulate gradient of parameters  $\theta$   
+8:  $g_t \gets g_{t-1} + \frac{1}{K} \mathbb{E}_{(Z,y) \in B} [\nabla_\theta L(f_\theta(X + \delta_{t-1}), y)]$   
+9: Update the perturbation  $\delta$  via gradient ascend  
+10:  $g_{adv} \gets \nabla_\delta L(f_\theta(X + \delta_{t-1}), y)$   
+11:  $\delta_t \gets \Pi_{\|\delta\|_F \leq \epsilon} (\delta_{t-1} + \alpha \cdot g_{adv} / \|g_{adv}\|_F)$   
+12: end for  
+13:  $\theta \gets \theta - \tau g_K$   
+14: end for  
+15: end for
+
+2019) and YOPO (Zhang et al., 2019) have been proposed to accelerate adversarial training. They achieve comparable robustness and generalization as standard PGD-trained models using only the same or a slightly larger number of forward-backward passes as natural training (i.e., SGD on clean samples). FreeAT takes one descent step on the parameters together with each of the  $K$  ascent steps on the perturbation. As a result, FreeAT may suffer from the "stale gradient" problem ( Dutta et al., 2018), where in every step  $t$ ,  $\delta_{t}$  does not necessarily maximize the model with parameter  $\theta_{t}$  since its update is based on  $\nabla_{\delta}L(f_{\theta_{t-1}}(X + \delta_{t-1}), y)$ , and vice versa,  $\theta_{t}$  does not necessarily minimize the adversarial risk with adversary  $\delta_{t}$  since its update is based on  $\nabla_{\theta}L(f_{\theta_{t-1}}(X + \delta_{t-1}), y)$ . Such a problem may be more significant when the step size is large.
+
+Different from FreeAT, YoPO accumulates the gradient of the parameters from each of the ascent steps, and updates the parameters only once after the  $K$  inner ascent steps. YoPO also advocates that after each back-propagation, one should take the gradient of the first hidden layer as a constant and perform several additional updates on the adversary using the product of this constant and the Jacobian of the first layer of the network to obtain strong adversaries. However, when the first hidden layer is a linear layer as in their implementation, such an operation is equivalent to taking a larger step size on the adversary. The analysis backing the extra update steps also assumes a twice continuously differentiable loss, which does not hold for ReLU-based neural networks they experimented with, and thus the reasons for the success of such an algorithm remains obscure. We give empirical comparisons between YoPO and our approach in Sec. 4.3.
+
+To obtain better solutions for the inner max and avoid fundamental limitations on the function class, we propose FreeLB, which performs multiple PGD iterations to craft adversarial examples, and simultaneously accumulates the "free" parameter gradients  $\nabla_{\theta}L$  in each iteration. After that, it updates the model parameter  $\pmb{\theta}$  all at once with the accumulated gradients. The overall procedure is shown in Algorithm 1, in which  $\pmb{X} + \pmb{\delta}_{t}$  is an approximation to the local maximum within the intersection of two balls  $\mathcal{I}_t = \mathcal{B}_{\pmb{X} + \pmb{\delta}_0}(\alpha t)\cap \mathcal{B}_{\pmb{X}}(\epsilon)$ . By taking a descent step along the averaged gradients at  $X + \delta_0,\dots,X + \delta_{K - 1}$ , we approximately optimize the following objective:
+
+$$
+\min  _ {\boldsymbol {\theta}} \mathbb {E} _ {(\boldsymbol {Z}, y) \sim \mathcal {D}} \left[ \frac {1}{K} \sum_ {t = 0} ^ {K - 1} \max  _ {\delta_ {t} \in \mathcal {I} _ {t}} L \left(f _ {\boldsymbol {\theta}} \left(\boldsymbol {X} + \boldsymbol {\delta} _ {t}\right), y\right) \right], \tag {3}
+$$
+
+which is equivalent to replacing the original batch  $\mathbf{X}$  with a  $K$ -times larger virtual batch, consisting of samples whose embeddings are  $\mathbf{X} + \delta_0, \dots, \mathbf{X} + \delta_{K-1}$ . Compared with PGD-based adversarial training (Eq. 1), which minimizes the maximum risk at a single estimated point in the vicinity of each training sample, FreeLB minimizes the maximum risk at each ascent step at almost no overhead.
+
+Intuitively, FreeLB could be a learning method with lower generalization error than PGD. Sokolic et al. (2017) have proved that the generalization error of a learning method invariant to a set of  $T$  transformations may be up to  $\sqrt{T}$  smaller than a non-invariant learning method. According to
+
+their theory, FreeLB could have a more significant improvement over natural training, since FreeLB enforces the invariance to  $K$  adversaries from a set of up to  $K$  different norm constraints, $^3$  while PGD only enforces invariance to a single norm constraint  $\epsilon$ .
+
+Empirically, FreeLB does lead to higher robustness and invariance than PGD in the embedding space, in the sense that the maximum increase of loss in the vicinity of  $X$  for models trained with FreeLB is smaller than that with PGD. See Sec. 4.3 for details. In theory, such improved robustness can lead to better generalization (Xu & Mannor, 2012), which is consistent with our experiments. Qin et al. (2019) also demonstrated that PGD-based method leads to highly convolved and non-linear loss surfaces in the vicinity of input samples when  $K$  is small, indicating a lack of robustness.
+
+# 3.3 WHEN ADVERSARIAL TRAINING MEETS DROPOUT
+
+Usually, adversarial training is not used together with dropout (Srivastava et al., 2014). However, for some language models like RoBERTa (Liu et al., 2019b), dropout is used during the finetuning stage. In practice, when dropout is turned on, each ascent step of Algorithm 1 is optimizing  $\delta$  for a different network. Specifically, denote the dropout mask as  $m$  with each entry  $m_i \sim \text{Bernoulli}(p)$ . Similar to our analysis for FreeAT, the ascent step from  $\delta_{t-1}$  to  $\delta_t$  is based on  $\nabla_\delta L(f_{\theta(m_{t-1})}(X + \delta_{t-1}), y)$ , so  $\delta_t$  is sub-optimal for  $L(f_{\theta(m_t)}(X + \delta), y)$ . Here  $\theta(m)$  is the effective parameters under dropout mask  $m$ .
+
+The more plausible solution is to use the same  $m$  in each step. When applying dropout to any network, the objective for  $\theta$  is to minimize the expectation of loss under different networks determined by the dropout masks, which is achieved by minimizing the Monte Carlo estimation of the expected loss. In our case, the objective becomes:
+
+$$
+\min  _ {\boldsymbol {\theta}} \mathbb {E} _ {(\boldsymbol {Z}, \boldsymbol {y}) \sim \mathcal {D}, \boldsymbol {m} \sim \mathcal {M}} \left[ \frac {1}{K} \sum_ {t = 0} ^ {K - 1} \max  _ {\delta_ {t} \in \mathcal {I} _ {t}} L \left(f _ {\boldsymbol {\theta} (\boldsymbol {m})} \left(\boldsymbol {X} + \boldsymbol {\delta} _ {t}\right), y\right) \right], \tag {4}
+$$
+
+where the 1-sample Monte Carlo estimation should be  $\frac{1}{K}\sum_{t=0}^{K-1}\max_{\delta_t \in \mathcal{I}_t} L(f_{\theta(\boldsymbol{m}_0)}(\boldsymbol{X} + \delta_t), y)$  and can be minimized by using FreeLB with dropout mask  $\boldsymbol{m}_0$  in each ascent step. This is similar to applying Variational Dropout to RNNs as used in Gal & Ghahramani (2016).
+
+# 4 EXPERIMENTS
+
+In this section, we provide comprehensive analysis on FreeLB through extensive experiments on three Natural Language Understanding benchmarks: GLUE (Wang et al., 2019), ARC (Clark et al., 2018) and CommonsenseQA (Talmor et al., 2019). We also compare the robustness and generalization of FreeLB with other adversarial training algorithms to demonstrate its strength. Additional experimental details are provided in the Appendix.
+
+# 4.1 DATASETS
+
+GLUE Benchmark. The GLUE benchmark is a collection of 9 natural language understanding tasks, namely Corpus of Linguistic Acceptability (CoLA; Warstadt et al. (2018)), Stanford Sentiment Treebank (SST; Socher et al. (2013)), Microsoft Research Paraphrase Corpus (MRPC; Dolan & Brockett (2005)), Semantic Textual Similarity Benchmark (STS; Agirre et al. (2007)), Quora Question Pairs (QQP; Iyer et al. (2017)), Multi-Genre NLI (MNLI; Williams et al. (2018)), Question NLI (QNLI; Rajpurkar et al. (2016)), Recognizing Textual Entailment (RTE; Dagan et al. (2006); Bar Haim et al. (2006); Giampiccolo et al. (2007); Bentivogli et al. (2009)) and Winograd NLI (WNLI; Levesque et al. (2011)). 8 of the tasks are formulated as classification problems and only STS-B is formulated as regression, but FreeLB applies to all of them. For BERT-base, we use the HuggingFace implementation<sup>4</sup>, and follow the single-task finetuning procedure as in Devlin et al. (2019). For RoBERTa, we use the fairseq implementation<sup>5</sup>. Same as Liu et al. (2019b), we also use
+
+<table><tr><td>Method</td><td>MNLI (Acc)</td><td>QNLI (Acc)</td><td>QQP (Acc)</td><td>RTE (Acc)</td><td>SST-2 (Acc)</td><td>MRPC (Acc)</td><td>CoLA (Mcc)</td><td>STS-B (Pearson)</td></tr><tr><td>Reported</td><td>90.2</td><td>94.7</td><td>92.2</td><td>86.6</td><td>96.4</td><td>90.9</td><td>68.0</td><td>92.4</td></tr><tr><td>ReImp</td><td>-</td><td>-</td><td>-</td><td>85.61 (1.7)</td><td>96.56 (.3)</td><td>90.69 (.5)</td><td>67.57 (1.3)</td><td>92.20 (.2)</td></tr><tr><td>PGD</td><td>90.53 (.2)</td><td>94.87 (.2)</td><td>92.49 (.07)</td><td>87.41 (.9)</td><td>96.44 (.1)</td><td>90.93 (.2)</td><td>69.67 (1.2)</td><td>92.43 (7.)</td></tr><tr><td>FreeAT</td><td>90.02 (.2)</td><td>94.66 (.2)</td><td>92.48 (.08)</td><td>86.69 (15.)</td><td>96.10 (.2)</td><td>90.69 (.4)</td><td>68.80 (1.3)</td><td>92.40 (.3)</td></tr><tr><td>FreeLB</td><td>90.61 (.1)</td><td>94.98 (.2)</td><td>92.60 (.03)</td><td>88.13 (1.2)</td><td>96.79 (.2)</td><td>91.42 (.7)</td><td>71.12 (.9)</td><td>92.67 (.08)</td></tr></table>
+
+Table 1: Results (median and variance) on the dev sets of GLUE based on the RoBERTa-large model, from 5 runs with the same hyperparameter but different random seeds. ReImp is our reimplementation of RoBERTa-large. The training process can be very unstable even with the vanilla version. Here, both PGD on STS-B and FreeAT on RTE demonstrates such instability, with one unconverged instance out of five.  
+
+<table><tr><td>Model</td><td>Score</td><td>CoLA 8.5k</td><td>SST-2 67k</td><td>MRPC 3.7k</td><td>STS-B 7k</td><td>QQP 364k</td><td>MNLI-m/mm 393k</td><td>QNLI 108k</td><td>RTE 2.5k</td><td>WNLI 634</td><td>AX</td></tr><tr><td>BERT-base1</td><td>78.3</td><td>52.1</td><td>93.5</td><td>88.9/84.8</td><td>87.1/85.8</td><td>71.2/89.2</td><td>84.6/83.4</td><td>90.5</td><td>66.4</td><td>65.1</td><td>34.2</td></tr><tr><td>FreeLB-BERT</td><td>79.4</td><td>54.5</td><td>93.6</td><td>88.1/83.5</td><td>87.7/86.7</td><td>72.7/89.6</td><td>85.7/84.6</td><td>91.8</td><td>70.1</td><td>65.1</td><td>36.9</td></tr><tr><td>MT-DNN2</td><td>87.6</td><td>68.4</td><td>96.5</td><td>92.7/90.3</td><td>91.1/90.7</td><td>73.7/89.9</td><td>87.9/87.4</td><td>96.0</td><td>86.3</td><td>89.0</td><td>42.8</td></tr><tr><td>XLNet-Large3</td><td>88.4</td><td>67.8</td><td>96.8</td><td>93.0/90.7</td><td>91.6/91.1</td><td>74.2/90.3</td><td>90.2/89.8</td><td>98.6</td><td>86.3</td><td>90.4</td><td>47.5</td></tr><tr><td>RoBERTa4</td><td>88.5</td><td>67.8</td><td>96.7</td><td>92.3/89.8</td><td>92.2/91.9</td><td>74.3/90.2</td><td>90.8/90.2</td><td>98.9</td><td>88.2</td><td>89.0</td><td>48.7</td></tr><tr><td>FreeLB-RoB</td><td>88.8</td><td>68.0</td><td>96.8</td><td>93.1/90.8</td><td>92.4/92.2</td><td>74.8/90.3</td><td>91.1/90.7</td><td>98.8</td><td>88.7</td><td>89.0</td><td>50.1</td></tr><tr><td>Human</td><td>87.1</td><td>66.4</td><td>97.8</td><td>86.3/80.8</td><td>92.7/92.6</td><td>59.5/80.4</td><td>92.0/92.8</td><td>91.2</td><td>93.6</td><td>95.9</td><td>-</td></tr></table>
+
+Table 2: Results on GLUE from the evaluation server, as of Sep 25, 2019. Metrics are the same as the leaderboard. Number under each task's name is the size of the training set. FreeLB-BERT is the single-model results of BERT-base finetuned with FreeLB, and FreeLB-RoB is the ensemble of 7 RoBERTa-Large models for each task. References:  ${}^{1}$  : (Devlin et al.,2019);  ${}^{2}$  : (Liu et al.,2019a);  ${}^{3}$  : (Yang et al.,2019);  ${}^{4}$  : (Liu et al., 2019b).
+
+single-task finetuning for all dev set results, and start with MNLI-finetuned models on RTE, MRPC and STS-B for the test submissions.
+
+ARC Benchmark. The ARC dataset (Clark et al., 2018) is a collection of multi-choice science questions from grade-school level exams. It is further divided into ARC-Challenge set with 2,590 question answer (QA) pairs and ARC-Easy set with 5,197 QA pairs. Questions in ARC-Challenge are more difficult and cannot be handled by simply using a retrieval and co-occurrence based algorithm (Clark et al., 2018). A typical question is:
+
+Which property of a mineral can be determined just by looking at it?
+
+(A) luster [correct] (B) mass (C) weight (D) hardness.
+
+CommonsenseQA Benchmark. The CommonsenseQA dataset (Talmor et al., 2019) consists of 12,102 natural language questions that require human commonsense reasoning ability to answer. A typical question is:
+
+Where can I stand on a river to see water falling without getting wet?
+
+(A) waterfall, (B) bridge [correct], (C) valley, (D) stream, (E) bottom.
+
+Each question has five candidate answers from ConceptNet (Speer et al., 2017). To make the question more difficult to solve, most answers have the same relation in ConceptNet to the key concept in the question. As shown in the above example, most answers can be connected to "river" by "At-Location" relation in ConceptNet. For a fair comparison with the reported results in papers and leaderboard<sup>6</sup>, we use the official random split 1.11.
+
+# 4.2 EXPERIMENTAL RESULTS
+
+GLUE We summarize results on the dev sets of GLUE in Table 1, comparing the proposed FreeLB against other adversarial training algorithms (PGD (Madry et al., 2018) and FreeAT (Shafahi et al., 2019)). We use the same step size  $\alpha$  and number of steps  $m$  for PGD, FreeAT and FreeLB. FreeLB is consistently better than the two baselines. Comparisons and detailed discussions about
+
+<table><tr><td rowspan="2"></td><td colspan="2">ARC-Easy</td><td colspan="2">ARC-Challenge</td><td colspan="2">ARC-Merge</td><td colspan="2">CQA</td></tr><tr><td>Dev</td><td>Test</td><td>Dev</td><td>Test</td><td>Dev</td><td>Test</td><td>Dev</td><td>Test</td></tr><tr><td>RoBERTa (Reported)</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>78.43</td><td>72.1</td></tr><tr><td>RoBERTa (ReImp)</td><td>84.39</td><td>84.13</td><td>64.54</td><td>64.44</td><td>77.83</td><td>77.62</td><td>77.56</td><td>-</td></tr><tr><td>FreeLB-RoBERTa</td><td>84.91</td><td>84.81</td><td>65.89</td><td>65.36</td><td>78.37</td><td>78.39</td><td>78.81</td><td>72.2</td></tr><tr><td>AristoRoBERTaV7 (MTL)</td><td>-</td><td>85.02</td><td>-</td><td>66.47</td><td>-</td><td>78.89</td><td>-</td><td>-</td></tr><tr><td>XLNet + RoBERTa (MTL+Ens)</td><td>-</td><td>-</td><td>-</td><td>67.06</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>FreeLB-RoBERTa (MTL)</td><td>84.91</td><td>85.44</td><td>70.23</td><td>67.75</td><td>79.86</td><td>79.60</td><td>-</td><td>-</td></tr></table>
+
+Table 3: Results on ARC and CommonsenseQA (CQA). ARC-Merge is the combination of ARC-Easy and ARC-Challenge, "MTL" stands for multi-task learning and "Ens" stands for ensemble. Results of XLNet + RoBERTa (MTL+Ens) and AristoRoBERTaV7 (MTL) are from the ARC leaderboards. Test (E) denotes the test set results with ensembles. For CQA, we report the highest dev and test accuracies among all models. The models with 78.81/72.19 dev/test accuracy (as in the table) have 71.84/78.64 test/dev accuracies respectively.
+
+YOPO (Zhang et al., 2019) are provided in Sec. 4.3. We have also submitted our results to the evaluation server, results provided in Table 2. FreeLB lifts the performance of the BERT-base model from 78.3 to 79.4, and RoBERTa-large model from 88.5 to 88.8 on overall scores.
+
+ARC For ARC, a corpus of 14 million related science documents (from ARC Corpus, Wikipedia and other sources) is provided. For each QA pair, we first use a retrieval model to select top 10 related documents. Then, given these retrieved documents<sup>7</sup>, we use RoBERTa-large model to encode  $\langle s \rangle$  Retrieved Documents  $\langle /s \rangle$  Question + Answer  $\langle /s \rangle$ , where  $\langle s \rangle$  and  $\langle /s \rangle$  are special tokens for RoBERTa model<sup>8</sup>. We then apply a fully-connected layer to the representation of the [CLS] token to compute the final logit, and use standard cross-entropy loss for model training.
+
+Results are summarized in Table 3. Following Sun et al. (2018), we first finetune the RoBERTa model on the RACE dataset (Lai et al., 2017). The finetuned RoBERTa model achieves  $85.70\%$  and  $85.24\%$  accuracy on the development and test set of RACE, respectively. Based on this, we further finetune the model on both ARC-Easy and ARC-Challenge datasets with the same hyper-parameter searching strategy (for 5 epochs), which achieves  $84.13\% /64.44\%$  test accuracy on ARC-Easy/ARC-Challenge. And by adding FreeLB finetuning, we can reach  $84.81\% /65.36\%$ , a significant boost on ARC benchmark, demonstrating the effectiveness of FreeLB.
+
+To further improve the results, we apply a multi-task learning (MTL) strategy using additional datasets. We first finetune the model on RACE (Lai et al., 2017), and then finetune on a joint dataset of ARC-Easy, ARC-Challenge, OpenbookQA (Mihaylov et al., 2018) and Regents Living Environment<sup>9</sup>. Based on this, we further finetune our model on ARC-Easy and ARC-Challenge with FreeLB. After finetuning, our single model achieves  $67.75\%$  test accuracy on ARC-Challenge and  $85.44\%$  on ARC-Easy, both outperforming the best submission on the official leaderboard<sup>10</sup>.
+
+CommonsenseQA Similar to the training strategy in Liu et al. (2019b), we construct five inputs for each question by concatenating the question and each answer separately, then encode each input with the representation of the [CLS] token. A final score is calculated by applying the representation of [CLS] to a fully-connected layer. Following the fairseq repository<sup>11</sup>, the input is formatted as: " $\langle s \rangle Q$ : Where can I stand on a river to see water falling without getting wet?  $\langle /s \rangle A$ : waterfall  $\langle /s \rangle$ ". where 'Q:' and 'A:' are the prefix for question and answer, respectively.
+
+Results are summarized in Table 3. We obtained a dev-set accuracy of  $77.56\%$  with the RoBERTa-large model. When using FreeLB finetuning, we achieved  $78.81\%$ , a  $1.25\%$  absolute gain. Compared with the results reported from fairseq repository, which obtains  $78.43\%$  accuracy on the dev-set, FreeLB still achieves better performance. Our submission to the CommonsenseQA leaderboard achieves  $72.2\%$  single-model test set accuracy, and the result of a 20-model ensemble is  $73.1\%$ , which achieves No.1 among all the submissions without making use of ConceptNet.
+
+<table><tr><td>Methods</td><td>Vanilla</td><td>FreeLB-3*</td><td>FreeLB-3</td><td>YOPO-3-2</td><td>YOPO-3-3</td></tr><tr><td>RTE</td><td>85.61 (1.67)</td><td>87.14 (1.29)</td><td>88.13 (1.21)</td><td>87.05 (1.36)</td><td>87.05 (0.20)</td></tr><tr><td>CoLA</td><td>67.57 (1.30)</td><td>69.31 (1.16)</td><td>71.12 (0.90)</td><td>70.40 (0.91)</td><td>69.91 (1.16)</td></tr><tr><td>MRPC</td><td>90.69 (0.54)</td><td>90.93 (0.66)</td><td>91.42 (0.72)</td><td>90.44 (0.62)</td><td>90.69 (0.37)</td></tr></table>
+
+Table 4: The median and standard deviation of the scores on the dev sets of RTE, CoLA and MRPC from the GLUE benchmark, computed from 5 runs with the same hyper-parameters except for the random seeds. We use FreeLB- $m$  to denote FreeLB with  $m$  ascent steps, and FreeLB- $3^*$  to denote the version without reusing the dropout mask.  
+
+<table><tr><td>Methods</td><td colspan="3">RTE</td><td colspan="3">CoLA</td><td colspan="3">MRPC</td></tr><tr><td></td><td>M-Inc (10-4)</td><td>M-Inc (R) (10-4)</td><td>N-Loss (10-4)</td><td>M-Inc (10-4)</td><td>M-Inc (R) (10-4)</td><td>N-Loss (10-4)</td><td>M-Inc (10-3)</td><td>M-Inc (R) (10-3)</td><td>N-Loss (10-3)</td></tr><tr><td>Vanilla</td><td>5.1</td><td>5.3</td><td>4.5</td><td>6.1</td><td>5.7</td><td>5.2</td><td>10.2</td><td>10.2</td><td>1.9</td></tr><tr><td>PGD</td><td>4.7</td><td>4.9</td><td>6.2</td><td>128.2</td><td>130.1</td><td>436.1</td><td>5.7</td><td>5.7</td><td>5.4</td></tr><tr><td>FreeLB</td><td>3.0</td><td>2.6</td><td>4.1</td><td>1.4</td><td>1.3</td><td>7.2</td><td>3.6</td><td>3.6</td><td>2.7</td></tr></table>
+
+Table 5: Median of the maximum increase in loss in the vicinity of the dev set samples for RoBERTa-Large model finetuned with different methods. Vanilla models are naturally trained RoBERTa's. M-Inc: Max Inc, M-Inc (R): Max Inc (R). Nat Loss (N-Loss) is the loss value on clean samples. Notice we require all clean samples here to be correctly classified by all models, which results in 227, 850 and 355 samples for RTE, CoLA and MRPC, respectively. We also give the variance in the Appendix.
+
+# 4.3 ABLATION STUDY AND ANALYSIS
+
+In this sub-section, we first show the importance of reusing dropout mask, then conduct a thorough ablation study on FreeLB over the GLUE benchmark to analyze the robustness and generalization strength of different approaches. We observe that it is unnecessary to perform shallow-layer updates on the adversary as YoPO for our case, and FreeLB results in improved robustness and generalization compared with PGD.
+
+Importance of Reusing Mask Table 4 (columns 2 to 4) compares the results of FreeLB with and without reusing the same dropout mask in each ascent step, as proposed in Sec. 3.3. With reusing, FreeLB can achieve a larger improvement over the naturally trained models. Thus, we enable mask reusing for all experiments involving RoBERTa.
+
+Comparing the Robustness Table 5 provides the comparisons of the maximum increment of loss in the vicinity of each sample, defined as:
+
+$$
+\Delta L _ {\max } (\boldsymbol {X}, \epsilon) = \max  _ {\| \delta \| \leq \epsilon} L \left(f _ {\boldsymbol {\theta}} (\boldsymbol {X} + \delta), y\right) - L \left(f _ {\boldsymbol {\theta}} (\boldsymbol {X}), y\right), \tag {5}
+$$
+
+which reflects the robustness and invariance of the model in the embedding space. In practice, we use PGD steps as in Eq. 2 to find the value of  $\Delta L_{\mathrm{max}}(X,\epsilon)$ . We found that when using a step size of  $5\cdot 10^{-3}$  and  $\epsilon = 0.01\| X\| _F$ , the PGD iterations converge to almost the same value, starting from 100 different random initializations of  $\delta$  for the RoBERTa models, trained with or without FreeLB. This indicates that PGD reliably finds  $\Delta L_{\mathrm{max}}$  for these models. Therefore, we compute  $\Delta L_{\mathrm{max}}(X,\epsilon)$  for each  $X$  via a 2000-step PGD.
+
+Samples with small margins exist even for models with perfect accuracy, which could give a false sense of vulnerability of the model. To rule out the outlier effect and make  $\Delta L_{\mathrm{max}}(X,\epsilon)$  comparable across different samples, we only consider samples that all the evaluated models can correctly classify, and search for an  $\epsilon$  for each sample such that the reference model can correctly classify all samples within the  $\epsilon$  ball. $^{12}$  However, such choice of per-sample  $\epsilon$  favors the reference model by design. To make fair comparisons, Table 5 provides the median of  $\Delta L_{\mathrm{max}}(X,\epsilon)$  with per-sample  $\epsilon$  from models trained by FreeLB (Max Inc) and PGD (Mac Inc (R)), respectively.
+
+Across all three datasets and different reference models, FreeLB has the smallest median increment even when starting from a larger natural loss than vanilla models. This demonstrates that FreeLB is more robust and invariant in most cases. Such results are also consistent with the models' dev set performance (the performances for Vanilla/PGD/FreeLB models on RTE, CoLA and MRPC are 86.69/87.41/89.21, 69.91/70.84/71.40, 91.67/91.17/91.17, respectively).
+
+Comparing with YOPO The original implementation of YOPO (Zhang et al., 2019) chooses the first convolutional layer of the ResNets as  $f_{0}$  for updating the adversary in the "s-loop". As a result, each step of the "s-loop" should be using exactly the same value to update the adversary, and YOPO- $m-n$  degenerates into FreeLB with a  $n$ -times large step size. To avoid that, we choose the layers up to the output of the first Transformer block as  $f_{0}$  when implementing YOPO. To make the total amount of update on the adversary equal, we take the hyper-parameters for FreeLB- $m$  and only change the step size  $\alpha$  into  $\alpha / n$  for YOPO- $m-n$ . Table 4 shows that FreeLB performs consistently better than YOPO on all three datasets. Accidentally, we also give the results comparing with YOPO- $m-n$  without changing the step size  $\alpha$  for YOPO in Table 8. The gap between two approaches seem to shrink, which may be caused by using a larger total step size for the YOPO adversaries. We leave exhaustive hyperparameter search for both models as our future work.
+
+# 5 CONCLUSION
+
+In this work, we have developed an adversarial training approach, FreeLB, to improve natural language understanding. The proposed approach adds perturbations to continuous word embeddings using a gradient method, and minimizes the resultant adversarial risk in an efficient way. FreeLB is able to boost Transformer-based model (BERT and RoBERTa) on several datasets and achieve new state of the art on GLUE and ARC benchmarks. Empirical study demonstrates that our method results in both higher robustness in the embedding space than natural training and better generalization ability. Such observation is also consistent with recent findings in Computer Vision. However, adversarial training still takes significant overhead compared with vanilla SGD. How to accelerate this process while improving generalization is an interesting future direction.
+
+Acknowledgements: Goldstein and Zhu were supported in part by the DARPA GARD, DARPA QED for RML, and AFOSR MURI programs.
+
+# REFERENCES
+
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+
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+
+# A ADDITIONAL EXPERIMENTAL DETAILS
+
+# A.1 PROBLEM FORMULATIONS
+
+For tasks with ranking loss like ARC, CommonsenseQA, WNLI and QNLI, add the perturbation to the concatenation of the embeddings of all question/answer pairs.
+
+Additional tricks are required to achieve high performance on WNLI and QNLI for the GLUE benchmark. We use the same tricks as Liu et al. (2019b). For WNLI, we use the same WSC data provided by Liu et al. (2019b) for training. For testing, Liu et al. (2019b) also provided the test set with span annotations, but the order is different from the GLUE dataset. We re-order their test set by matching. For the QNLI, we follow Liu et al. (2019b) and formulate the problem as pairwise ranking problem, which is the same for CommonsenseQA. We find the matching pairs for both training set and testing set by matching the queries in the dev set. We predict "entailment" if the candidate has the higher score, and "not_ entailment" otherwise.
+
+# A.2 HYPER-PARAMETERS
+
+<table><tr><td></td><td>MNLI</td><td>QNLI</td><td>QQP</td><td>RTE</td><td>SST-2</td><td>MRPC</td><td>CoLA</td><td>STS-B</td></tr><tr><td>ε</td><td>2E-1</td><td>1.5E-1</td><td>4.5E-1</td><td>1.5E-1</td><td>6E-1</td><td>4E-1</td><td>2E-1</td><td>3E-1</td></tr><tr><td>α</td><td>1E-1</td><td>1E-1</td><td>1.5E-1</td><td>3E-2</td><td>1E-1</td><td>4E-2</td><td>2.5E-2</td><td>1E-1</td></tr><tr><td>m</td><td>2</td><td>2</td><td>2</td><td>3</td><td>2</td><td>3</td><td>3</td><td>3</td></tr></table>
+
+As other adversarial training methods, introduces three additional hyper-parameters: step size  $\alpha$ , maximum perturbation  $\epsilon$ , number of steps  $m$ . For all other hyper-parameters such as learning rate and number of iterations, we either search in the same interval as RoBERTa (on CommonsenseQA, ARC, and WNLI), or use exactly the same setting as RoBERTa (except for MRPC, where we find using a learning rate of  $5 \times 10^{-6}$  gives better results).[13] We list the best combinations for  $\alpha$ ,  $\epsilon$  and  $m$  for each of the GLUE tasks in Table 6. For WSC/WNLI, the best combination is  $\epsilon = 1e - 2$ ,  $\alpha = 5e - 3$ ,  $m = 2$ . Notice even when  $m\alpha < \epsilon$ , the maximum perturbation could still reach  $\epsilon$  due to the random initialization.
+
+# B VARIANCE OF MAXIMUM INIncrement OF LOSS
+
+Table 6: Additional hyper-parameters on GLUE tasks.  
+
+<table><tr><td>Methods</td><td colspan="3">RTE</td><td colspan="3">CoLA</td><td colspan="3">MRPC</td></tr><tr><td></td><td>M-Inc (10-4)</td><td>M-Inc (R) (10-4)</td><td>N-Loss (10-4)</td><td>M-Inc (10-4)</td><td>M-Inc (R) (10-4)</td><td>N-Loss (10-4)</td><td>M-Inc (10-3)</td><td>M-Inc (R) (10-3)</td><td>N-Loss (10-3)</td></tr><tr><td>Vanilla</td><td>5.1(15087)</td><td>5.3(14346)</td><td>4.5(920)</td><td>6.1(13118)</td><td>5.7(14122)</td><td>5.2(447)</td><td>10.2(929)</td><td>10.2(955)</td><td>1.9(76)</td></tr><tr><td>PGD</td><td>4.7(10138)</td><td>4.9(1828)</td><td>6.2(752)</td><td>128.2(1300)</td><td>130.1(893)</td><td>436.1(1276)</td><td>5.7(321)</td><td>5.7(155)</td><td>5.4(54)</td></tr><tr><td>FreeLB</td><td>3.0(1614)</td><td>2.6(11114)</td><td>4.1(595)</td><td>1.4(1392)</td><td>1.3(6231)</td><td>7.2(615)</td><td>3.6(167)</td><td>3.6(601)</td><td>2.7(54)</td></tr></table>
+
+Table 7: Median and Standard Deviation of the maximum increase in loss in the vicinity of the dev set samples for RoBERTa-Large model finetuned with different methods. Vanilla models are naturally trained RoBERTa's. Nat Loss is the loss value on clean samples. Notice we require all clean samples here to be correctly classified by all models, which results in 227, 850 and 355 samples for RTE, CoLA and MRPC.
+
+Table 7 provides the complete results for the increment of loss in the interval, with median and standard deviation.
+
+# C ADDITIONAL RESULTS FOR ABLATION STUDIES
+
+Here we provide some additional results for comparison with YOPO as complementary results to Table 4. We will release the complete results for comparing with YOPO and without variational dropout on each of the GLUE tasks in our next revision. From the current results, there is no need
+
+<table><tr><td>Methods</td><td>Vanilla</td><td>FreeLB-3</td><td>YOPO-3-2</td><td>YOPO-3-3</td></tr><tr><td>STS-B</td><td>92.20 (.2)</td><td>92.67 (.08)</td><td>92.60 (.17)</td><td>92.60 (0.20)</td></tr><tr><td>SST-2</td><td>96.56 (.3)</td><td>96.79 (.2)</td><td>96.44 (.2)</td><td>96.33 (.1)</td></tr><tr><td>QNLI</td><td>-</td><td>94.98 (.2)</td><td>94.96 (.1)</td><td>-</td></tr><tr><td>QQP</td><td>-</td><td>92.60 (.03)</td><td>92.55 (.05)*</td><td>92.50 (.02)*</td></tr><tr><td>MNLI</td><td>-</td><td>90.61 (.1)</td><td>90.59 (.2)</td><td>90.45 (.2)</td></tr></table>
+
+Table 8: The median and standard deviation of the scores on the dev sets of STS-B, SST-2, QNLI, QQP and MNLI from the GLUE benchmark, each computed from 5 runs with the same hyper-parameters except for the random seeds (except for the results with YOPO on QQP, which are from 4 runs). Also note here we use a step size of  $\alpha$  for the adversary of YOPO- $m$ - $n$ , so YOPO effectively uses a step size of  $n\alpha$ . We use FreeLB- $m$  to denote FreeLB with  $m$  ascent steps, and YOPO- $3$ - $n$  to denote YOPO with  $n$  shallow-layer ascents.
+
+in using extra shallow-layer updates thatYOPO advocates, since this consistently deteriorates the performance while introducing extra computations.
\ No newline at end of file
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+# GENERALIZATION OF TWO-LAYER NEURAL NETWORKS: AN ASYMPTOTIC VIEWPOINT
+
+Jimmy Ba $^{1,2}$ , Murat A. Erdogdu $^{1,2}$ , Taiji Suzuki $^{3,4}$ , Denny Wu $^{1,2,4}$ , Tianzong Zhang $^{2,5}$
+
+University of Toronto<sup>1</sup>, Vector Institute<sup>2</sup>, University of Tokyo<sup>3</sup>, RIKEN AIP<sup>4</sup>, Tsinghua University<sup>5</sup>
+
+{jba,erdogdu,dennywu}@cs.toronto.edu,
+
+taiji@mist.i.u-tokyo.ac.jp, ztz16@mails.tsinghua.edu.cn
+
+# ABSTRACT
+
+This paper investigates the generalization properties of two-layer neural networks in high-dimensions, i.e. when the number of samples  $n$ , features  $d$ , and neurons  $h$  tend to infinity at the same rate. Specifically, we derive the exact population risk of the unregularized least squares regression problem with two-layer neural networks when either the first or the second layer is trained using a gradient flow under different initialization setups. When only the second layer coefficients are optimized, we recover the double descent phenomenon: a cusp in the population risk appears at  $h \approx n$  and further overparameterization decreases the risk. In contrast, when the first layer weights are optimized, we highlight how different scales of initialization lead to different inductive bias, and show that the resulting risk is independent of overparameterization. Our theoretical and experimental results suggest that previously studied model setups that provably give rise to double descent might not translate to optimizing two-layer neural networks.
+
+# 1 INTRODUCTION
+
+In modern neural networks, the number of parameters can easily exceed the number of training samples, yet in many circumstances, there is little sign of overfitting even in the absence of explicit regularization (Zhang et al., 2016). This phenomenon is usually explained by the interplay between the model architecture and the optimization method. Existing works have analyzed the implicit regularization of gradient descent on simple models (Gunasekar et al., 2018; Ji and Telgarsky, 2018), and provided generalization guarantees (Arora et al., 2018; Bartlett et al., 2017; Dziugaite and Roy, 2017) that align with the empirical observations.
+
+Recently, a series of works highlighted the implicit regularization of interpolators in the overparameterized regime (Belkin et al., 2018; Spigler et al., 2018; Geiger et al., 2018; Advani and Saxe, 2017). Specifically, a second decrease in the population risk is observed when the model is further overparameterized beyond the interpolation limit, i.e. when the model achieves zero training error. This phenomenon is known as double descent, and can be precisely quantified for certain linear models (Hastie et al., 2019; Mei and Montanari, 2019; Belkin et al., 2019; Bartlett et al., 2019; Xu and Hsu, 2019). Among the recent works, Hastie et al. (2019) and Mei and Montanari (2019) explicitly derived the population risk of linear regression and random features regression models in high dimensions using tools from random matrix theory.
+
+However, there is still a gap between the practical benefit of overparameterization and the recently proved double descent phenomenon, which is typically established under models that exhibits the following structure: the trained model solves a linear inverse problem, and the "cusp" in the risk arises from the instability of the inverse at the interpolation threshold. Moreover, given a dataset or fixed  $n$ ,  $d$ , the number of parameters in the linear regression model is also fixed, i.e. the level of overparameterization cannot be altered. It is therefore unclear if the trend persists in the optimization of more complex models, for instance in two-layer neural networks where overparameterization can be controlled simply by adding more neurons.
+
+In this work, we analyze the generalization properties of two-layer neural networks in the unregularized least squares regression setting and examine the presence/absence of the double descent phenomenon. We consider the proportional asymptotic limit where the number of samples  $n$ , input
+
+features  $d$ , and neurons  $h$  tend to infinity at the same rate, under which overparameterization corresponds to increasing the limit of  $h / n$  (network "width"). This regime is particularly interesting because even though  $n \to \infty$ , the empirical risk is not equivalent to the population risk. In addition, the joint scaling of  $n, d, h$  is parallel to the practical choice of model architectures, where it is common to train a larger network when the number of samples and input features are larger. Following Hastie et al. (2019), we assume unit Gaussian input and noisy linear observations, and analytically derive the population risk of the solution of gradient flow on either the first or the second layer parameters when the flow is initialized close to zero.
+
+Our findings can be summarized as follows (see Figure 1):
+
+- When only the second layer is optimized, we derive the risk in its bias-variance decomposition and demonstrate the presence of the double descent phenomenon.  
+- When the first layer is optimized, we compare two solutions of gradient flow from different scales of initialization, which we term as vanishing and non-vanishing initialization, and show in both cases the population risk is independent to overparameterization.  
+- For the vanishing initialization, we show that the risk of the gradient flow solution is asymptotically close to that of a rank-1 model. For non-vanishing initialization, we show that the gradient flow solution is well-approximated by a kernel model and derive the risk.
+
+![](images/3a26c1792d5cd8c480c98a9d417891d49a82035243ca8b64842c4453d02d822d.jpg)  
+Figure 1: Illustration of the double descent risk curve in two-layer linear networks (SNR = 16). Brighter color indicates larger  $\gamma_{1} = d / n$ . Double descent is observed when the second layer coefficients are optimized (main figure), but not when the first layer weights are optimized (subfigure).
+
+# 1.1 RELATED WORKS
+
+Global Convergence of Two-layer Networks. A plethora of recent works have explored the global convergence of shallow neural networks. Mei et al. (2018; 2019); Chizat and Bach (2018a); Rotskoff and Vanden-Eijnden (2018); Sirignano and Spiliopoulos (2018); Nitanda and Suzuki (2017) studied the mean-field limit where the number of neurons  $h \to \infty$  and the second layer scaled by  $1 / h$ , and established correspondence between the main-particle limit of gradient descent and Wasserstein gradient flow to demonstrate global convergence. On the other hand, Jacot et al. (2018); Du et al. (2018); Oymak and Soltanolkotabi (2019); Allen-Zhu et al. (2018b); Song and Yang (2019) considered a different scaling and showed that gradient descent on overparameterized models converges to global minimizer at a linear rate; key to these results is an observation that optimization via gradient descent is asymptotically equivalent to kernel regression with respect to the neural tangent kernel.
+
+Active vs. Lazy Training. Following Chizat and Bach (2018b), we refer to the two aforementioned scalings as the active and lazy (kernel) regime. It has been observed that different regimes lead to contrasting inductive biases. Williams et al. (2019); Woodworth et al. (2019); Li et al. (2017) showed that for certain two-layer network or overparameterized linear model, the scale of initialization controls the implicit regularization of gradient descent (from sparse to smooth solution). In the student-teacher setup (Tian, 2017; Zhong et al., 2017), Ghorbani et al. (2019b;a) showed that kernel models in high dimensions perform no better than low-degree polynomials on the input or fully-trained two-layer network. Additionally, Suzuki (2018); Allen-Zhu and Li (2019); Yehudai and Shamir (2019); Wei et al. (2018) demonstrated that neural network outperforms linear estimators (including kernel method) in learning various target functions. The difference between fixed bases and adaptive bases mirrors the difference in optimizing the first or second layer in our setup.
+
+Generalization of Overparameterized Models. It is often observed that overparameterization does not result in overfitting (Neyshabur et al., 2014). In the lazy regime, generalization guarantees can be derived from the distance traveled by the parameters (Neyshabur et al., 2018; Nagarajan and Kolter, 2019), which becomes small if the model is sufficiently overparameterized (Arora et al., 2019b; Li and Liang, 2018; Allen-Zhu et al., 2018a; Cao and Gu, 2019). Compared to these guarantees that usually require significant overparameterization, our result relies on stronger data assumptions, but consequently we obtain the exact population risk instead of a vacuous upper-bound. Beyond the kernel regime, Advani and Saxe (2017); Goldt et al. (2019) analyzed the generalization dynamics of overparameterized models in the student-teacher setup.
+
+Double Descent. The term double descent refers to the phenomenon that the population risk of an empirical risk minimizer manifests a "cusp" at the interpolation threshold, and further overparameterization decreases the risk. First observed in Krogh and Hertz (1992), the phenomenon has been recently connected to the benefit of overparameterization (Belkin et al., 2018; Geiger et al., 2018; Spigler et al., 2018; Advani and Saxe, 2017), and can be precisely characterized for certain simple models (Hastie et al., 2019; Belkin et al., 2019; Bartlett et al., 2019; Xu and Hsu, 2019). Our work is inspired by Hastie et al. (2019) which uses random matrix theory to derive the asymptotic risk for linear and random feature models. Concurrent to our work, Mei and Montanari (2019) analyzed the random features model and derived its population risk for which double descent occurs both in bias and variance. This aligns with our results on optimizing the second layer in Section 4 although we do not derive the bias component explicitly. Compared to Hastie et al. (2019); Mei and Montanari (2019), the focus of this work is to highlight the different generalization property of models obtained from optimizing different layers of the network and from different initialization.
+
+Random Matrix Theory. High-dimensional models, including kernel models and neural networks, can be analyzed by studying the properties of random matrices. El Karoui et al. (2010); Cheng and Singer (2013); Fan and Montanari (2019) studied the spectral properties of kernel matrix via decomposing the nonlinearity with Taylor series or Hermite polynomials, which in turn explains the generalization of high-dimensional kernel ridgeless interpolators (Liang and Rakhlin, 2018). In addition, similar tools have been used to study two-layer neural networks (Louart et al., 2018; Pennington and Worah, 2017) and related quantities such as the Fisher information matrix (Karakida et al., 2018; Pennington and Worah, 2018).
+
+# 2 PRELIMINARIES: TWO-LAYER NEURAL NETWORK
+
+Consider the following bias-free two-layer neural network  $f: \mathbb{R}^d \to \mathbb{R}$  with  $h$  hidden units
+
+$$
+f (\boldsymbol {x}) = \sum_ {i = 1} ^ {h} a _ {i} \phi (\langle \boldsymbol {x}, \boldsymbol {w} _ {i} \rangle), \tag {1}
+$$
+
+where  $\pmb{x} \in \mathbb{R}^d$  is the input,  $\pmb{w}_i \in \mathbb{R}^d$  is the weights corresponding to neuron  $i$ ,  $a_i \in \mathbb{R}$  is the  $i$ -th coefficient of the second layer, and  $\phi : \mathbb{R} \to \mathbb{R}$  is a Lipschitz continuous activation function with bounded Gaussian moments, i.e.  $\mathbb{E}[\phi(G)^k] < \infty$ ,  $\forall k \in \mathbb{Z}_+$  for  $G \sim \mathcal{N}(0,1)$ . For concise notation, we write  $W = [\pmb{w}_1, \dots, \pmb{w}_h] \in \mathbb{R}^{d \times h}$  for the weight matrix,  $\pmb{a} = [a_1, \dots, a_h] \in \mathbb{R}^h$  for the coefficient vector,  $X = [\pmb{x}_1, \dots, \pmb{x}_n] \in \mathbb{R}^{d \times n}$  for the data matrix,  $\pmb{y} \in \mathbb{R}^n$  for the corresponding vector of labels, and  $\Phi = \phi(W^\top X) \in \mathbb{R}^{h \times n}$  for the feature matrix at the first layer. We omit arguments of  $f$  when they are clear from the context.
+
+We consider a student-teacher setup, in which data is generated by a teacher model  $F: \mathbb{R}^d \to \mathbb{R}$  with additive noise, and the student model aims to minimize the squared loss:
+
+$$
+\left(\boldsymbol {x} _ {i}, \varepsilon_ {i}\right) \stackrel {\text {i . i . d .}} {\sim} P _ {\boldsymbol {x}} \times P _ {\varepsilon}, \quad y _ {i} = F \left(\boldsymbol {x} _ {i}\right) + \varepsilon_ {i}, \quad L (X; f) = \frac {1}{2 n} \sum_ {i = 1} ^ {n} \left(y _ {i} - f \left(\boldsymbol {x} _ {i}\right)\right) ^ {2}, \tag {2}
+$$
+
+where  $\mathbb{E}[\pmb{x}_i] = 0$ ,  $\mathrm{Cov}(\pmb{x}_i) = \Sigma$ ,  $\mathbb{E}[\varepsilon_i] = 0$ ,  $\mathrm{Var}(\varepsilon_i) = \sigma^2$ . We are interested in the population risk  $R(f) = \mathbb{E}_{P_x}[(F(\pmb{x}) - f(\pmb{x}))^2]$ . Our analysis will be made under the proportional asymptotics:
+
+$$
+n, d, h \to \infty ; \quad d / n \to \gamma_ {1}, h / n \to \gamma_ {2}; \quad \gamma_ {1}, \gamma_ {2} \in (0, \infty),
+$$
+
+in which overparameterization corresponds to increasing  $\gamma_{2}$ . Thus the characteristics of double descent considered in this work are: 1) large population risk as  $\gamma_{2} \rightarrow 1$ ; 2) decrease in the risk for  $\gamma_{2} > 1$ . While the empirical risk can be minimized in various ways, we analyze the solution of gradient flow, in which we update either the first layer  $W$  or the second layer  $\pmb{a}$ :
+
+$$
+\mathrm {d} W (t) = - \nabla_ {W} L (X; f) \mathrm {d} t \quad \text {o r} \quad \mathrm {d} \boldsymbol {a} (t) = - \nabla_ {\boldsymbol {a}} L (X; f) \mathrm {d} t, \tag {3}
+$$
+
+from small initialization. The rest of the paper is organized as follows. In Section 3, we start with a simple example of two-layer linear network as warm-up. In Section 4, we consider optimizing the second layer coefficients (flow over  $\mathbf{a}$ ) of a non-linear two-layer neural network under fixed Gaussian first layer, which is a random feature model. Section 5 considers optimizing the first layer weights (flow over  $W$ ) of such network under fixed Rademacher second layer. We defer all proofs and details on experiments to appendix.
+
+# 3 WARM-UP: LINEAR NETWORK
+
+We begin with a simple linear model with  $\phi (\pmb {x}) = \pmb{x}$ , i.e.  $\Phi = W^{\top}X$ . We remark that although the model is linear, the solution obtained by gradient flow on the two-layer model can be different than that from directly solving the linear regression problem on input features.
+
+Training the Second Layer. Following Hastie et al. (2019), we fix the first layer parameters to be randomly drawn from a unit Gaussian and optimize the coefficients  $\mathbf{a}$  by minimizing  $\left\| \mathbf{a}^{\top}\Phi - \mathbf{y}\right\|_{2}^{2}$ . The following lemma characterizes the solution of the gradient flow.
+
+Lemma 1 (Least squares solution). Given data matrix  $X$ , response vector  $\mathbf{y}$  and model  $f(\mathbf{x}) = \langle \phi (\mathbf{x}^{\top}W),\mathbf{a}\rangle$  with fixed first layer coefficients  $W$ , gradient flow on the coefficients  $\mathbf{a}$  starting from zero initialization converges to  $\hat{\mathbf{a}} = \Phi^{\dagger}\mathbf{y}$ , where  $\dagger$  stands for the Moore-Penrose inverse.
+
+We make two assumptions on the data and the teacher model to simplify the computation.
+
+(A1) Gaussian Features:  $x_{i} \sim \mathcal{N}(0, I_{d})$ ; (A2) Linear Teacher:  $F(\pmb{x}) = \langle \pmb{x}, \beta \rangle$ ,  $\| \beta \| = r$ .
+
+Denote the linear student network as  $f(\pmb{x}) = \langle \pmb{x},\hat{\beta}\rangle$ , where  $\hat{\beta} = W\hat{\pmb{a}}$  and  $\hat{\pmb{a}}$  is the least-square solution defined by Lemma 1. We write the population risk in its bias-variance decomposition.
+
+$$
+R = \mathbb {E} _ {\boldsymbol {x} \sim P _ {\boldsymbol {x}}} [ \| \hat {\boldsymbol {\beta}} - \boldsymbol {\beta} \| _ {\Sigma} ^ {2} | X, W ] = \underbrace {\| \mathbb {E} [ \hat {\boldsymbol {\beta}} | X , W ] - \boldsymbol {\beta} \| _ {2} ^ {2}} _ {B = \text {b i a s}} + \underbrace {\operatorname {t r} \left(\operatorname {C o v} (\hat {\boldsymbol {\beta}} | X , W)\right)} _ {V = \text {v a r i a n c e}}, \tag {4}
+$$
+
+where  $\| \pmb {x}\|_{\Sigma}^{2} = \pmb{x}^{\top}\pmb {\Sigma}\pmb{x}$  . We compute the bias and the variance separately to obtain the risk.
+
+Theorem 2. Given (A1)(A2) and let  $\boldsymbol{w}_i \stackrel{\mathrm{i.i.d.}}{\sim} \mathcal{N}(0, d^{-1}I_d)$ , at  $n, d, h \to \infty$  we have
+
+$$
+R _ {(\gamma_ {1} <   1)} \rightarrow \left\{\begin{array}{l l}\frac {\gamma_ {1} - \gamma_ {2}}{\gamma_ {1} g _ {2}} r ^ {2} + \frac {\gamma_ {2}}{g _ {2}} \sigma^ {2},&\gamma_ {2} <   \gamma_ {1},\\\frac {\gamma_ {1}}{g _ {1}} \sigma^ {2},&\gamma_ {2} > \gamma_ {1},\end{array}\quad R _ {(\gamma_ {1} > 1)} \rightarrow \left\{\begin{array}{l l}\frac {\gamma_ {1} - \gamma_ {2}}{\gamma_ {1} g _ {2}} r ^ {2} + \frac {\gamma_ {2}}{g _ {2}} \sigma^ {2},&\gamma_ {2} <   1,\\\frac {\gamma_ {2} g _ {1}}{\gamma_ {1} g _ {2}} r ^ {2} + \frac {g _ {1} + g _ {2}}{g _ {1} g _ {2}} \sigma^ {2},&\gamma_ {2} > 1.\end{array}\right.\right. \tag {5}
+$$
+
+where  $d / n\to \gamma_1$ $h / n\rightarrow \gamma_{2}$ $g_{1} = |\gamma_{1} - 1|$  , and  $g_{2} = |\gamma_{2} - 1|$
+
+We observe that when  $d > n$  (i.e.  $\gamma_1 > 1$ ), we obtain the double descent risk curve, i.e., the population risk achieves its maximum at  $\gamma_2 \to 1$  and further overparameterization  $(\gamma_2 > 1)$  reduces both the bias and the variance. Conversely when  $n > d$  and  $h > d$  (i.e.  $\gamma_1 < \min(1, \gamma_2)$ ), the population risk becomes constant and equals to that of the minimum-norm solution  $\hat{\beta}_{\mathrm{min}} = X^{\dagger} \mathbf{y}$  on the input features.
+
+Training the First Layer. When the first layer of a linear network is optimized via gradient flow and the second layer is fixed, the following holds for zero-initialization of  $W$ .
+
+Proposition 3. Given  $W(0) = 0$  and fixed  $a^{init} \neq 0$ , at any time  $t > 0$  of the gradient flow on  $W$ ,  $W(t)$  is rank-1. Further,  $\hat{\beta} = \widehat{W}\mathbf{a}$  converges to the least squares solution of  $\mathbf{y} = X^{\top}\hat{\beta}$ , the population risk of which is given in (Hastie et al., 2019, Thm. 1 & 3)) as
+
+$$
+R _ {(\gamma_ {1} <   1)} \rightarrow \frac {\gamma_ {1}}{1 - \gamma_ {1}} \sigma^ {2}; \quad R _ {(\gamma_ {1} > 1)} \rightarrow \frac {\gamma_ {1} - 1}{\gamma_ {1}} r ^ {2} + \frac {1}{\gamma_ {1} - 1} \sigma^ {2}. \tag {6}
+$$
+
+In this case, overparameterization by increasing  $\gamma_{2}$  does not influence the population risk. In addition, since the obtained two-layer linear model is equivalent to the minimum-norm solution on the input features  $\hat{\beta}_{\mathrm{min}}$ , optimizing the first layer always results in smaller or equal population risk compared to optimizing the second layer.
+
+In this simple scenario for two-layer linear networks, double descent is observed only when the second layer is optimized, which reduces the objective to least squares regression on the intermediate features. On the other hand, training the first layer from zero-initialization always yields the same solution that is independent to overparameterization. One natural question to ask is: does this phenomenon generalize to nonlinear two-layer neural networks? The following sections answer this question in the affirmative under certain conditions.
+
+![](images/c77d7cc11c82d9c0141f902c862721743f4ac4287f95644a7b060225bc36052b.jpg)  
+(a) risk  $(\gamma_{1} < 1)$ .  
+Figure 2: Population risk of two-layer neural networks with optimized second layer under (A1)(A2). Brighter color indicates larger  $\gamma_{1}$ . (a) risk of linear network with  $r^2 / \sigma^2 = 16$  and  $\gamma_{1} < 1$ . ( $\gamma_{1} > 1$  is shown in Figure 1) (b) variance of network with ReLU activation. Black line corresponds to  $\gamma_{1} \to \infty$  predicted by Corollary 5. (c) bias of network with ReLU activation. Black line corresponds to  $\gamma_{1} \to \infty$  for linear network, which is empirically observed as an upper-bound. Note that as  $\gamma_{2} \to 1$  both bias and variance becomes unbounded.
+
+![](images/110d994ad59c8346ed8ab275d8289dd81478287b2f2fed0ddeff34dcb0fb0630.jpg)  
+(b) variance (ReLU).
+
+![](images/dfd5535bc6dc1983a912bfd72d914d5f461e29faefcda32cd564c3c08eefe7a5.jpg)  
+(c) bias (ReLU).
+
+# 4 NONLINEAR MODEL: OPTIMIZING THE SECOND LAYER
+
+In this section, we analyze the case when the second layer  $\pmb{a}$  is learned under fixed  $W$  and a nonlinear activation function  $\phi$  (a random feature model). We first observe that by Lemma 1, the gradient flow finds the solution  $\hat{\pmb{a}} = \Phi^{\dagger}\pmb{y}$ . We again consider the following bias-variance decomposition.
+
+$$
+\begin{array}{l} R = \mathbb {E} _ {\boldsymbol {x} \sim P _ {\boldsymbol {x}}} \left[ \left\| \phi (\boldsymbol {x} ^ {\top} W) \hat {\boldsymbol {a}} - F (\boldsymbol {x}) \right\| _ {2} ^ {2} | X, W \right] \tag {7} \\ = \underbrace {\mathbb {E} _ {\boldsymbol {x}} \left[ \left\| \mathbb {E} [ \phi (\boldsymbol {x} ^ {\top} W) \hat {\boldsymbol {a}} | X , W ] - F (\boldsymbol {x}) \right\| _ {2} ^ {2} \right]} _ {B = \text {b i a s}} + \underbrace {\mathbb {E} _ {\boldsymbol {x}} \left[ \left\| \phi (\boldsymbol {x} ^ {\top} W) \hat {\boldsymbol {a}} - \mathbb {E} [ \phi (\boldsymbol {x} ^ {\top} W) \hat {\boldsymbol {a}} ] \right\| _ {2} ^ {2} | X , W \right]} _ {V = \text {v a r i a n c e}}. \\ \end{array}
+$$
+
+We highlight that the variance term does not depend on the target function. The following result characterizes the variance of the random feature model.
+
+Theorem 4. Given (A1) and  $\boldsymbol{w}_i \stackrel{\mathrm{i.i.d.}}{\sim} \mathcal{N}(0, d^{-1}I_d)$ , when  $n, d, h \to \infty$ , we have
+
+$$
+V = \left\{ \begin{array}{l l} \sigma^ {2} \frac {\gamma_ {2}}{1 - \gamma_ {2}}, & \gamma_ {2} <   1, \\ \sigma^ {2} \lim  _ {\xi \to 0} - \left[ \left. \gamma_ {2} \frac {\partial}{\partial x} m _ {1} (\xi , c _ {1} x, c _ {2} x) \right| _ {x = 0} + \left. \frac {\partial}{\partial x} m _ {2} (\xi , c _ {1} x, c _ {2} x) \right| _ {x = 0} + \frac {\gamma_ {2} - 1}{\xi^ {2}} \right] & \gamma_ {2} > 1. \end{array} \right.
+$$
+
+in which  $m_{1}(\xi, \rho, \tau)$ ,  $m_{2}(\xi, \rho, \tau)$  is the unique solution in  $\{|m_1|, |m_2| < 1 / \Im \xi\}$  of
+
+$$
+m _ {1} ^ {- 1} = - \xi - \rho - \gamma_ {1} ^ {- 1} \gamma_ {2} \tau^ {2} m _ {1} - c _ {1} m _ {2} + \frac {\tau^ {2} \gamma_ {1} ^ {- 1} \gamma_ {2} m _ {1} ^ {2} (c _ {2} m _ {2} - \tau) - 2 \tau c _ {2} m _ {1} m _ {2} + c _ {2} ^ {2} m _ {1} m _ {2} ^ {2}}{m _ {1} (c _ {2} m _ {2} - \tau) - \gamma_ {1} \gamma_ {2} ^ {- 1}}, \tag {8}
+$$
+
+$$
+m _ {2} ^ {- 1} = - \xi - r \gamma_ {2} m _ {1} + \frac {\gamma_ {2} c _ {2} m _ {1} ^ {2} \left(c _ {2} m _ {2} - \tau\right)}{m _ {1} \left(c _ {2} m _ {2} - \tau\right) - \gamma_ {1} \gamma_ {2} ^ {- 1}}, \tag {9}
+$$
+
+where variables  $\xi, \rho, \tau$  satisfies  $\Im \xi > 0$  or  $\xi < 0$ ,  $\rho > \tau > 0$ , and constants  $c_1, c_2$  defined as,
+
+$$
+c _ {1} = \mathbb {E} [ \phi (G) ^ {2} ] - \mathbb {E} [ \phi (G) ] ^ {2}, \quad c _ {2} = \mathbb {E} [ G \phi (G) ] ^ {2}, \tag {10}
+$$
+
+for  $G\sim \mathcal{N}(0,1)$  and  $\Im \xi$  denoting the imaginary part of  $\xi$
+
+Remark.  $c_{1} \geq c_{2}$  and the equality holds iff  $\phi$  is linear.
+
+Corollary 5. If we let  $\gamma_1 \to \infty$ , the variance is equal to the lowest value of the variance of the linear model  $V_{(\gamma_1 \to \infty)} = \sigma^2 \min \{\gamma_2, 1\} / |1 - \gamma_2|$ .
+
+The proof of Theorem 4 largely follows from Hastie et al. (2019) with techniques similar to Cheng and Singer (2013), but with modifications in order to handle unnormalized and uncentered activation functions. The above theorem holds irrespective of the underlying teacher model, and is consistent with the double descent risk curve as it suggests that for all  $\gamma_{1}$ , variance of the random feature model
+
+peaks at  $h = n$  then drops as  $\gamma_{2}$  further increases. Note that as  $\gamma_{1} \to \infty$ , a linear and nonlinear network would have the same asymptotic variance.
+
+Since double descent is observed in the variance term, we do not derive the bias for all  $\gamma_{1},\gamma_{2}$ . Instead, we show that for linear teacher, the bias also becomes unbounded as  $\gamma_{2}\rightarrow 1$
+
+Proposition 6. Given (A1)(A2) and  $\boldsymbol{w}_i \stackrel{\mathrm{i.i.d.}}{\sim} \mathcal{N}(0, I_d)$ , then  $B \to \infty$  as  $\gamma_2 \to 1$ . Furthermore,  $B$  is finite when  $\gamma_2 > 1$ .
+
+Thus we have shown that a "cusp" in the population risk appears at  $h = n$ , which aligns with the double descent phenomenon. Empirically as  $\gamma_1 \to \infty$  the nonlinear model also shares the same asymptotic bias with the linear model, as shown in Figure 2. We note that (Mei and Montanari, 2019, Thm. 1 & 3) analytically solved the risk of random feature model for a larger class of target functions than ours and confirmed that double descent appears in both the bias and the variance.
+
+# 5 NONLINEAR MODEL: OPTIMIZING THE FIRST LAYER
+
+Having observed the double descent phenomenon in optimizing the second layer, in the sequel we consider a two-layer neural network with fixed second layer coefficients initialized from a Rademacher distribution  $a_{i}\sim \mathrm{Unif}\{-1 / \sqrt{h},1\sqrt{h}\}$ , and the first layer  $W$  is optimized with the following update
+
+$$
+\frac {\partial W (t)}{\partial t} = - \frac {\partial L (X ; W)}{\partial W} = \frac {1}{n} \sum_ {i = 1} ^ {n} \left[ y _ {i} - \boldsymbol {a} ^ {\top} \phi \left(W ^ {\top} \boldsymbol {x} _ {i}\right) \right] \boldsymbol {x} _ {i} \left[ \phi^ {\prime} \left(\boldsymbol {x} _ {i} ^ {\top} W\right) \circ \boldsymbol {a} \right], \tag {11}
+$$
+
+which potentially has different stationary solutions with no explicit form, depending on the initialization. We denote the solution of this flow at time  $t$  started from designated initialization by  $W^{\mathrm{init}}(t)$ , its stationary solution by  $\widehat{W}$ , and the corresponding network by  $\hat{f}$ .
+
+Remark. Although we let  $n \to \infty$ , this dynamics does not correspond to the population gradient flow considered in Tian (2017). For instance when  $x \sim \mathcal{N}(0, I / d)$ , the spectrum of the data covariance is Marčenko-Pastur, whereas the population covariance is identity.
+
+As we cannot characterize the gradient flow solution from all possible initializations, we consider two specific scales of initialization:
+
+$\mathbf{V}$  anishing:  $\pmb {w}_i^{\mathrm{Van}}(0)\sim \mathcal{N}(0,I_d / dh^{1 + \epsilon})$  Non-vanishing:  $w_{i}^{\mathrm{NV}}(0)\sim \mathcal{N}(0,I_{d} / d^{-\epsilon})$
+
+Note that neither of the two initializations correspond to the "mean-field" regime (e.g. analyzed in Mei et al. (2018)) due to the  $1 / \sqrt{h}$  scaling of the second layer. In other words, as  $h$  increases, we expect the distance traveled by each parameter to decrease under both initializations. The difference, however, is the "relative" amount the parameters traveled compared to their initialized magnitude, which leads to solutions with contrasting properties. As we will see, under (A1)(A2) and vanishing initialization we have  $\| W(0) - \widehat{W}\|_{F} / \| W(0)\|_{F}\gg 1$ , i.e. the contribution of initialization vanishes at the end of training, whereas for non-vanishing initialization the inequality is in the opposite direction, i.e.  $\widehat{W}$  "barely moves" and resembles the initialization  $W(0)$ .
+
+# 5.1 VANISHING INITIALIZATION
+
+As  $d, h \to \infty$ , the vanishing initialization becomes arbitrarily close to zero-initialization. We thus expect the gradient flow under vanishing initialization to "resemble" that of starting from exactly zero if the flow converges sufficiently fast and the gradient being Lipschitz. The Lipschitz condition (Lemma 18) can be established under the following assumption on the activation.
+
+(A3):  $\phi$  is smooth, Lipschitz and monotone with  $\phi'(0) \neq 0$ ;  $|\phi'(\pm x) - \phi'(\pm \infty)| = O(e^{-x})$ .
+
+The above assumption requires that the derivative of the nonlinearity  $\phi$  saturates beyond a O(1) region, which holds true for the commonly-used smooth activations such as sigmoid and SoftPlus. In addition, the choice of scaling ensures that the gradient flow converges sufficiently fast. We thus have the following characterization of the population risk:
+
+Theorem 7. Given (A1-3). Let  $T = O(\log \log h)$  and  $\hat{f}(\cdot) = f^{\text{van}}(\cdot, W(T))$ , then as  $n, d, h \to \infty$ , the gradient flow reaches a o(1) first-order stationary point at time  $T$ , i.e.  $\| \partial W(T) / \partial t \|_F \in o(1)$ , at which point the population risk is given as
+
+$$
+R (\hat {f}) \rightarrow \max  \left\{0, \frac {\gamma_ {1} - 1}{\gamma_ {1}} \right\} r ^ {2} + \frac {\min  \left\{\gamma_ {1} , 1 \right\}}{| 1 - \gamma_ {1} |} \sigma^ {2}. \tag {12}
+$$
+
+The expression above is the same as the risk of the least squares solution on input  $\hat{\beta} = X^{\dagger}\pmb{y}$ ; therefore the risk is independent to overparameterization (increasing  $\gamma_{2}$ ). The intuition is that when the weights are initialized sufficiently small and travel infinitesimally, then the activation can be linearized around 0 and thus the model is equivalent to a two-layer linear network. Note that this result does not apply to the non-smooth ReLU activation. Instead, in Appendix E we heuristically show that under the additional assumption that the data is symmetric, the risk of ReLU network is also independent to  $\gamma_{2}$ .
+
+# 5.2 NON-VANISHING INITIALIZATION
+
+When initialization is sufficiently large, the amount each parameter travels to minimize the empirical risk becomes asymptotically negligible compared to the magnitude of initialization. In this case we establish under (A1-3) that (11) is asymptotically equivalent to the kernel gradient flow on the tangent kernel:  $k(\pmb{x}, \pmb{y}) = \langle \nabla_{W^{\mathrm{init}}} f(\pmb{x}), \nabla_{W^{\mathrm{init}}} f(\pmb{y}) \rangle$ . The converged parameters under this linearized dynamics have the following closed-form:
+
+$$
+\operatorname {v e c} \left(W ^ {*}\right) \approx \operatorname {v e c} \left(W ^ {\text {i n i t}}\right) + \Delta ; \quad \Delta = J ^ {\dagger} (\boldsymbol {y} - f ^ {\text {i n i t}} (X)); \quad J _ {[ i, j ]} = \nabla_ {\operatorname {v e c} \left(W ^ {\text {i n i t}}\right) _ {j}} f ^ {\text {i n i t}} \left(\boldsymbol {x} _ {i}\right), \tag {13}
+$$
+
+where  $J \in \mathbb{R}^{n \times (d \times h)}$  is the Jacobian matrix w.r.t. to the model parameters. We remark that in contrast to most NTK-type global convergence results that require the width of the model to grow faster than the number of data points (e.g. Du et al. (2018)), our result is not built upon the overparameterization on width, but instead an anti-concentration that relies on the scale of initialization. Consequently the above initialization is larger than the scale that is commonly used in practice.
+
+One may naturally expect the double descent phenomenon to appear in this kernel solution, as  $\Delta$  exhibits the form of a least squares solution which contains a pseudo-inverse. However, we show that this is not the case under the same assumptions in Section 4; in fact, the risk is also independent to  $\gamma_{2}$ .
+
+An obstacle in computing the risk of the kernel model is the potentially non-zero  $f^{\mathrm{init}}(X)$ . We thus adopt the "doubling-trick" from Chizat and Bach (2018b) to ensure  $f^{\mathrm{init}}(\cdot) = 0$ , i.e. we assume the following symmetric property on the initialized weights:
+
+$$
+\text {(A 4) S y m m e t r i c I n i t i z a t i o n :} \forall i \in [ 1, h ], \exists ! j \in [ 1, h ] \text {s . t .} a _ {i} w _ {i} ^ {\mathrm {i n i t}} = - a _ {j} w _ {j} ^ {\mathrm {i n i t}}.
+$$
+
+Theorem 8. Given (A1-4) and let  $n, d, h \to \infty$ , the stationary solution  $\hat{f}$  has the following risk
+
+$$
+\begin{array}{l} R (\hat {f}) \rightarrow \left(\frac {\gamma_ {1} - 1}{2 \gamma_ {1}} + \frac {\gamma_ {1} (\gamma_ {1} + \gamma_ {1} m + m - 2) + 1}{2 \gamma_ {1} \sqrt {\gamma_ {1} (\gamma_ {1} + m (\gamma_ {1} (m + 2) + 2) - 2) + 1}}\right) r ^ {2} \\ + \left(\frac {\gamma_ {1} + \gamma_ {1} m + 1}{4 \sqrt {\gamma_ {1} (\gamma_ {1} + m (\gamma_ {1} (m + 2) + 2) - 2) + 1}} - \frac {1}{4}\right) \sigma^ {2}, \tag {14} \\ \end{array}
+$$
+
+where  $m = b_1^2 /b_0^2$ ,  $b_{0}^{2} = \mathbb{E}[\phi^{\prime}(G)]^{2}$ , and  $b_{1}^{2} = \mathbb{E}[\phi^{\prime}(G)^{2}] - b_{0}^{2}$ ,  $G\sim \mathcal{N}(0,1)$ .
+
+Note that the population risk is again independent of  $\gamma_{2}$ , and thus double descent does not appear for this initialization. Roughly speaking, the reason that the risk does not become unbounded at some point is that in the asymptotic limit the pseudo-inverse  $(JJ^{\top})^{\dagger}$  is stable due to the nonlinearity and  $dh\gg n$ . We make two additional observations:
+
+- the stability of the inverse at  $\gamma_{1} \to 1$  depends on the lowest eigenvalue of the tangent kernel matrix (smaller  $\lambda_{\mathrm{min}}(K)$  entails larger variance), which is determined by the nonlinearity;  
+- While our result only holds for network with zero initial output, for non-symmetric (i.i.d.) initialization we also observe that the risk is independent to  $\gamma_{2}$ , but the bias is higher than that in symmetric initialization as shown in Figure 8. We comment that the non-zero  $f^{\mathrm{init}}(X)$  in (13) behaves as zero-mean Gaussian due to central limit theorem; therefore, in the kernel regime the function output at initialization is equivalent to additive noise to the labels  $\mathbf{y}$ , and we thus expect the magnitude of  $f^{\mathrm{init}}(X)$  to negatively influence the model generalization.
+
+![](images/2ced30a4b685b98c704a2ce095b07385b2e3d0c18e51ae25f2eff8b662fffb8d.jpg)  
+(a) bias.
+
+![](images/9fff623a7b566efcde89643850a50806c8b29cfe5c4d87e4e94f926a79cc94ba.jpg)  
+(b) variance.  
+Figure 3: Bias and variance of two-layer sigmoid network with optimized first layer under (A1)(A2). Individual dotted lines correspond to different  $\gamma_{2}$  (from 0.2 to 2) which is independent to the risk. The bias and variance for both initializations is well-aligned with Theorem 7 and Theorem 8.
+
+# 5.3 COMPARING THE INITIALIZATIONS
+
+Figure 3 shows the agreement between theoretical prediction and experimental results. Although in both cases the risk is independent of overparameterization  $(\gamma_{2})$ , the two initializations lead to models with contrasting properties, as demonstrated by the following comparison on the risk.
+
+Corollary 9. For any  $\gamma_1 \in (0, \infty)$  and nonlinearity  $\phi$ ,  $B(\hat{f}^{\mathrm{Van}}) \leq B(\hat{f}^{\mathrm{NV}}) \leq 1$ . On the other hand, for all  $m > 0$ ,  $V(\hat{f}^{\mathrm{NV}}) = O(1)$ , whereas  $V(\hat{f}^{\mathrm{Van}})$  can be arbitrarily large as  $\gamma_1 \to 1$ .
+
+Remark.  $m \geq 0$  for all smooth activations  $\phi$ , and the equality holds if  $\phi$  is linear.
+
+Intuitively, small initialization enables the model "evolve" more during optimization and better align with the data and target function. This potentially results in a lower bias, at the expense of overfitting more to the noise (high variance). In contrast, with sufficiently large initialization the final model becomes close to the initialized model, and thus we may expect it to be less "aligned" to the target (high bias) but is more stable (lower variance).
+
+In illustrate the different inductive bias of the two initializations, we plot the trajectory of neurons in Appendix A Figure 4. Observe that for vanishing initialization the neurons stay close to one another throughout the trajectory, which results in a low-rank weight matrix, as predicted by Theorem 7. In contrast, for non-vanishing initialization the neurons stay close to initialization (therefore full-rank), which validates the kernel approximation. Last but not least, although the derived risk is only for learning a linear target function, we empirically observe that when the teacher is also a two-layer network, the population risk follows the same trend, i.e. double descent occurs when only the second layer is optimized, as shown in Figure 7.
+
+# 6 DISCUSSION AND FUTURE WORKS
+
+We derived the exact population risk of high-dimensional two-layer neural networks in learning a linear target function over Gaussian data with additive label noise, and showed that optimizing the first or the second layer via gradient flow results in solutions with contrasting properties. Specifically, double descent is present when the second layer coefficients are optimized, but not when the first layer weights are optimized under certain initializations. Moreover, we highlight that the scale of initialization leads to different inductive bias in optimizing the first layer.
+
+It should be noted that our analysis only applies to the unregularized objective: it has been shown that explicit regularization (such as  $\ell_2$  penalty) stabilizes the singularity at  $\gamma_{2}\rightarrow 1$  (Mei and Montanari, 2019), and algorithmic regularization (Li et al., 2019a; Dong et al., 2019) also provides robustness against noisy observations. We further remark that our findings do not directly contradict the experimental double descent phenomenon, nor the practical benefit of overparameterization. In particular, the interpolation limit could occur at  $\gamma_{2}\to 0$  which is beyond the regime we consider (such as Figure 4 in (Belkin et al., 2018)). Thus what we conclude is that under the studied proportional
+
+asymptotics, the mechanism that provably gives rise to double descent from previous works on least squares regression might not translate to neural networks trained with gradient descent.  
+To simplify the computation, we rely on a set of strong assumptions similar to those in Hastie et al. (2019), some of which we believe can be relaxed in future works, such as isotropic Gaussian input and linear target. Importantly, the two specific scales of initialization studied in Section 5 are by no means exhaustive, and thus one would expect that under a different initialization of the first layer, or the mean-field  $1 / h$  scaling of the second layer, the risk of the trained model can be very different. Changing the loss function may also alter the generalization behavior of the network. Another challenging problem is to extend the current analysis to beyond two layers. Last but not least, our result characterizes gradient flow which resembles gradient descent with small stepsize, and thus it would be interesting to study the effect of learning rate schedule, which has a known impact on generalization (Smith and Le, 2017; Li et al., 2019b).
+
+# ACKNOWLEDGEMENT
+
+We thank Xiuyuan Cheng, Xuechen Li, Yiping Lu, Atsushi Nitanda, Shengyang Sun and anonymous reviewers for helpful comments and feedback. JB and DW were partially funded by LG Electronics and NSERC. JB and MAE were supported by the CIFAR AI Chairs program. TS was partially supported by JSPS Kakenhi (26280009, 15H05707 and 18H03201), Japan Digital Design and JST-CREST.
+
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+
+# A ADDITIONAL FIGURES AND PLOTS
+
+![](images/8d35ce2bc2929f934b743ea7df87ee1b5ff82fe88609fcbf8bf50985aba56e75.jpg)  
+(a) vanishing initialization.
+
+![](images/a6cd3934637e2c504a6e81c2299bad6e4f1dacae4c5695b195b9085bcce53b68.jpg)  
+(b) non-vanishing initialization.
+
+![](images/42398fe5effef73a1c9c3a2f077744c9f4697964b798b10dc5c99804826217fb.jpg)  
+Figure 4: trajectory of neurons from initialization (dark blue) to optimum (orange) on the first two dimensions (two-layer SoftPlus student and linear teacher;  $\mathrm{SNR} = 1 / 4$ ). For vanishing initialization the neurons stay close to one another throughout the trajectory, whereas for non-vanishing initialization the neurons stay close to initialization.  
+(a)  $\gamma_{1} < 1$
+
+![](images/a93143991a2c7a032067c3bc714679cd9b7fc29db81376895d5c5e6622414a97.jpg)  
+(b)  $\gamma_{1} > 1$  
+Figure 5: Population risk of two-layer linear network with fixed random 1st layer with  $\mathrm{SNR} = 2^{5 / 16}$  under Gaussian input and linear teacher. Brighter color indicates larger  $\gamma_{1}$ .
+
+SUMMARY OF THE PRESENCE / ABSENCE OF DOUBLE DESCENT  
+
+<table><tr><td>Singularity in</td><td>2nd Layer Trained (RF)</td><td>Vanishing Init.</td><td>Non-vanishing Init.</td></tr><tr><td>Bias</td><td>γ1: No; γ2: Yes</td><td>γ1: No; γ2: No</td><td>γ1: No; γ2: No</td></tr><tr><td>Variance</td><td>γ1: No; γ2: Yes</td><td>γ1: Yes; γ2: No</td><td>γ1: No; γ2: No</td></tr></table>
+
+![](images/08821bed02f0d99c73a325103699b21603446b0a95d503e1b85e8cfe44cd56be.jpg)  
+(a) bias.
+
+![](images/873d97653cfde00d3439e5b4f0f818a4b3e9d56d4f9620c6e915def04f68d190.jpg)  
+(b) variance.  
+Figure 6: Bias and variance of two-layer SoftPlus network with optimized first layer under (A1)(A2). Individual dotted lines correspond to different  $\gamma_{2}$  (from 0.2 to 2) which is independent to the risk. The bias and variance for both initializations is well-aligned with Theorem 7 and Theorem 8, respectively.
+
+![](images/258ef17e3383bede540f4035c9a38c13017348535598ae86957569f1e137d622.jpg)  
+(a) trained 2nd layer.
+
+![](images/5942be3264182106e6a2624f13262ce602d9253c4747c4529876c5e084c4b883.jpg)  
+(b) trained 1st layer.  
+Figure 7: Population risk (scaled by  $1 / d$ ) of two-layer ReLU network trained to fit a two-layer ReLU teacher model with  $h = d$  neurons. Brighter color corresponds to larger  $\gamma_{1}$ . Similar to the linear teacher case, double descent is observed when the second layer is optimized (a) but not when the first layer is optimized (b).
+
+![](images/f1150f48f917089f6d356f3069520890de01e24e16842a4fa1b0958e5edaf97b.jpg)  
+(a) SoftPlus activation.  
+Figure 8: Bias of (a) SoftPlus and (b) sigmoid two-layer network with optimized first layer under (A1)(A2). Note that bias under i.i.d. initialization is also independent to overparameterization  $(\gamma_{2})$ , but is higher than the bias under symmetric initialization ("doubling trick") and not always upper-bounded by the null risk  $r^2$ .
+
+![](images/575d08e31468603f23a38035bf902bcb693210cb8fa70dd8fffabdb430281c4b.jpg)  
+(b) sigmoid activation.
+
+# B BACKGROUND
+
+# B.1 ROTATIONAL INVARIANCE
+
+The rotational invariance of Gaussian distribution is crucial in our analysis throughout this paper. A basic observation is that for a random Gaussian matrix  $X$  and any fixed unitary matrix  $U$ , the distribution of  $X$  and  $UX$  are the same.
+
+Lemma 10 (Rotational Invariance). Denote  $A(X) \in \mathbb{R}^{d \times d}$  a matrix function of  $X \in \mathbb{R}^{d \times n}$ . If  $A(X)$  satisfies that  $A(UX) = UA(X)U^T$  for all unitary  $U$ , then
+
+$$
+\mathbb {E} _ {X} \left[ \boldsymbol {\beta} ^ {T} A (X) \boldsymbol {\beta} \right] = \frac {1}{d} \boldsymbol {\beta} ^ {T} \boldsymbol {\beta} \mathbb {E} _ {X} [ \operatorname {t r} (A (X)) ]. \tag {15}
+$$
+
+for any fixed nonzero  $\beta \in \mathbb{R}^d$  and random matrix  $X$  with each entry i.i.d.  $X_{ij}\sim \mathcal{N}(0,\sigma^2)$
+
+Proof. Choose a set of Unitary matrices  $\{U_i\}_{i=1}^d$  such that  $U_i^\top \beta = \|\beta\| e_i$ , where  $e_i$  is the  $i$ -th canonical vector in  $\mathbb{R}^d$ . Since  $U_i X \sim X$ , we have  $\mathbb{E}[A(X)] = \mathbb{E}[A(U_i X)] = U_i \mathbb{E}[A(X)] U_i^\top$  and hence
+
+$$
+\mathbb {E} \left[ \boldsymbol {\beta} ^ {T} A (X) \boldsymbol {\beta} \right] = \frac {1}{d} \sum_ {i = 1} ^ {d} \mathbb {E} \left[ \boldsymbol {\beta} ^ {T} U _ {i} A (X) U _ {i} ^ {\top} \boldsymbol {\beta} \right] = \frac {1}{d} \sum_ {i = 1} ^ {d} \boldsymbol {e} _ {i} ^ {T} \mathbb {E} [ A (X) ] \boldsymbol {e} _ {i} = \frac {\boldsymbol {\beta} ^ {T} \boldsymbol {\beta}}{d} \mathbb {E} [ \operatorname {t r} (A (X)) ]. \tag {16}
+$$
+
+Note that the property also holds for matrix function  $A(X)$  that satisfies  $A(XU) = UA(X)U^T$ , and can be extended to matrix function  $A$  that takes multiple matrices as input.
+
+For brevity we will refer to rotational invariance instead of equation (15).
+
+# B.2 MARCENKO-PASTUR LAW
+
+For a real symmetric random matrix  $A \in \mathbb{R}^{p \times p}$ , define its empirical spectral density as
+
+$$
+\mu_ {A} (d \lambda) = \frac {1}{p} \sum_ {i = 1} ^ {p} \delta_ {\lambda_ {i} (A)} (\lambda) d \lambda ,
+$$
+
+where  $\delta_{a}(x) = \delta (x - a)$  is the Dirac delta function. Assume  $A = W_{p}\sim W_{p}(I,n)$  is a Wishart matrix, i.e.  $W_{p} = X^{\top}X / n$  and  $X\in \mathbb{R}^{n\times p}$  is random Gaussian matrix with each column i.i.d.  $X_{i}\sim \mathcal{N}(\mathbf{0},I)$ . Marčenko and Pastur (1967) showed that as  $n,p\to \infty$  and  $p / n = \gamma \in (0,\infty)$ , the empirical spectral density  $\mu_{W_p}(d\lambda)$  converges weakly to a limiting density  $\mu_{\mathrm{MP}(\gamma)}(\lambda)$ :
+
+$$
+\mu_ {\mathrm {M P} (\gamma)} (d \lambda) = [ 1 - \gamma^ {- 1} ] _ {+} \delta_ {0} (\lambda) d \lambda + \frac {1}{2 \pi \gamma \lambda} \sqrt {\left((1 + \sqrt {\gamma}) ^ {2} - \lambda\right) \left(\lambda - (1 - \sqrt {\gamma}) ^ {2}\right)} d \lambda . \tag {17}
+$$
+
+We say  $\mu_{\mathrm{MP}(\gamma)}$  is the density of the Marčenko-Pastur distribution with support  $S = [(1 - \sqrt{\gamma})^2, (1 + \sqrt{\gamma})^2]$  (for  $0 < \gamma < 1$ ) or  $S = \{0\} \cup [(1 - \sqrt{\gamma})^2, (1 + \sqrt{\gamma})^2]$  (for  $\gamma \geq 1$ ). Note that this implies that the smallest non-zero eigenvalue of  $A$  is bounded away from 0 a.s. for  $\gamma \neq 1$ .
+
+The explicit form of Marčenko-Pastur distribution allows us to investigate the asymptotic properties of random matrices. Generally speaking, by Pormanteau theorem one can translate any bounded continuous function on the empirical spectral density to the one on Marčenko-Pastur distribution, i.e. for any bounded  $f(\lambda) \in C(S)$ , as  $n, p \to \infty$  with  $p / n = \gamma$ , almost surely
+
+$$
+\int_ {S} f (\lambda) \cdot \mu_ {W} (d \lambda) \rightarrow \int f _ {S} (\lambda) \cdot \mu_ {M P} (d \lambda). \tag {18}
+$$
+
+One implication is the following trace concentration on the inverse Wishart matrix for  $\gamma < 1$ ,
+
+$$
+\operatorname {t r} \left(X ^ {\top} X\right) = \operatorname {t r} \left(\frac {1}{p} W _ {p} ^ {- 1}\right) = \frac {1}{p} \sum_ {i = 1} ^ {p} \frac {1}{\lambda_ {i} \left(W _ {p}\right)} = \int_ {S} \frac {1}{\lambda} \mu_ {W _ {p}} (d \lambda) \rightarrow \int_ {S} \frac {1}{\lambda} \mu_ {\mathrm {M P} (\gamma)} (d \lambda) = \frac {1}{1 - \gamma}. \tag {19}
+$$
+
+We remark that instability of the trace of the invert Wishart matrix as  $\gamma \rightarrow 1$  plays an important role in the double descent phenomenon.
+
+# B.3 ORTHOGONAL POLYNOMIALS
+
+Orthogonal polynomials are useful in the analysis of nonlinear random matrices. Suppose  $\phi$  is a function in  $L^2 (\mathbb{R},\mu_G)$ , where  $G\sim \mathcal{N}(0,1)$ ,  $\mu_G(dx) = (\sqrt{2\pi})^{-1}e^{-x^2 /2}dx$  is the Gaussian measure. For  $n\geq 0$ , define the Hermite polynomials
+
+$$
+H _ {n} (x) = (- 1) ^ {n} e ^ {- x ^ {2} / 2} \frac {\partial^ {n}}{\partial x ^ {n}} e ^ {- x ^ {2} / 2}, \tag {20}
+$$
+
+Note that orthogonality can be easily verified:
+
+$$
+\mathbb {E} [ H _ {j} (G) H _ {k} (G) ] = \int_ {\mathbb {R}} H _ {j} (x) H _ {k} (x) \mu_ {G} (d x) = j! \cdot \delta_ {j k}. \tag {21}
+$$
+
+Since  $\{H_i(x)\}_{i = 0}^{\infty}$  forms a set of orthogonal basis in  $L^2 (\mathbb{R},\mu_G)$ , the function  $\phi (x)$  can be expanded under the Hermite basis as
+
+$$
+\phi (x) = \sum_ {i = 0} ^ {\infty} c _ {i} H _ {i} (x) = \sum_ {i = 0} ^ {\infty} \left(\frac {1}{k !} \int_ {\mathbb {R}} \phi (a) H _ {i} (a) \mu_ {G} (d a)\right) H _ {i} (x). \tag {22}
+$$
+
+Following Cheng and Singer (2013), we mainly focus on the first few terms of the Hermite expansion. One can check that  $H_0(x) = 1$ ,  $H_1(x) = x$ , and  $\phi(x)$  can be expanded as
+
+$$
+\phi (x) = c _ {0} + c _ {1} x + \phi_ {\perp} (x), \tag {23}
+$$
+
+where  $c_{0} = \mathbb{E}[\phi (G)]$ ,  $c_{1} = \mathbb{E}[G\phi (G)]$ , and terms in the RHS are orthogonal to one another in  $L^2 (\mathbb{R},\mu_G)$ . Taking square and expectation over both sides yields
+
+$$
+\mathbb {E} [ \phi (G) ^ {2} ] = \mathbb {E} [ \phi (G) ] ^ {2} + \mathbb {E} [ G \phi (G) ] ^ {2} + \mathbb {E} [ \phi_ {\perp} (G) ^ {2} ], \tag {24}
+$$
+
+which indicates  $\mathbb{E}[\phi (G)^2 ] - \mathbb{E}[\phi (G)]^2 -\mathbb{E}[G\phi (G)]^2 = \mathbb{E}[\phi_{\perp}(G)^2 ]\geq 0$  . Note that the equality holds if and only if  $\mathbb{E}[\phi_{\perp}(G)^2 ] = 0$  , i.e.  $\phi (x) = c_0 + c_1x$  is linear. One application of this decomposition is to "linearize" a non-linear matrix in high dimensions, which will be useful for the following sections.
+
+# C PROOF OF MAIN RESULTS
+
+# C.1 PROOF OF LEMMA 1
+
+Given features  $X \in \mathbb{R}^{d \times n}$ , labels  $\pmb{y} \in \mathbb{R}^d$  and model parameters  $\pmb{\theta}$ , the gradient flow of  $\pmb{\theta}$  on the squared loss  $\left\| \pmb{y} - X^\top \pmb{\theta} \right\|_2^2$  can be written as
+
+$$
+\frac {\partial \boldsymbol {\theta} (t)}{\partial t} = \frac {1}{n} X \left(y - X ^ {\top} \boldsymbol {\theta} (t)\right). \tag {25}
+$$
+
+Thus with initialization  $\theta_0$ , the solution of this ODE at time  $t$  can be written in explicit form
+
+$$
+\boldsymbol {\theta} (t) = e ^ {- \frac {t}{n} X X ^ {\top}} \boldsymbol {\theta} _ {0} + \left(X X ^ {\top}\right) ^ {\dagger} \left(I - e ^ {- \frac {t}{n} X X ^ {\top}}\right) X \boldsymbol {y}. \tag {26}
+$$
+
+Since  $\theta_0 = 0$ , taking  $t \to \infty$  yields the desired result.
+
+# C.2 PROOF OF THEOREM 2
+
+We compute the bias and variance for different cases of  $\gamma_{1},\gamma_{2}$ . We first discuss the case where the random feature  $\Phi_X$  is not full rank (Case I). Otherwise when  $\Phi_X$  is full rank, we discuss whether it is full column rank (Case II) or full row rank (Case III).
+
+Case I:  $W^{\top}X$  is not full rank, i.e.  $\gamma_1 < 1, \gamma_2 > \gamma_1$ . In this case  $\operatorname{rank}(\Phi_X) = d < \min(n, h)$ , and thus by taking the Moore-Penrose inverse we obtain
+
+$$
+\hat {\boldsymbol {\beta}} = W \left(X ^ {\top} W\right) ^ {\dagger} \boldsymbol {y} = \left(X X ^ {\top}\right) ^ {- 1} X \boldsymbol {y}. \tag {27}
+$$
+
+It is clear that the mean and variance is identical to the underparameterized regime in Hastie et al. (2019), i.e. when  $n, d, h \to \infty$
+
+$$
+B \rightarrow 0; \quad V \rightarrow \frac {\gamma_ {1}}{1 - \gamma_ {1}} \sigma^ {2}. \tag {28}
+$$
+
+Case II:  $W^{\top}X$  has full column rank, i.e.  $\gamma_{2} < 1, \gamma_{1} > \gamma_{2}$ . By Lemma 1, the solution of the second layer coefficients is
+
+$$
+\hat {\boldsymbol {\beta}} = W \left(W ^ {\top} X X ^ {\top} W\right) ^ {- 1} W ^ {\top} X \boldsymbol {y}. \tag {29}
+$$
+
+Denote  $W = U\Sigma V^{\top}$  the singular value decomposition. We perform the block decomposition:
+
+$$
+\Sigma = \left[ \begin{array}{l} \Sigma_ {0} \\ 0 \end{array} \right], X = \left[ \begin{array}{l} X _ {0} \\ X _ {1} \end{array} \right], \tag {30}
+$$
+
+where  $\Sigma_0\in \mathbb{R}^{h\times h}$ ,  $X_0\in \mathbb{R}^{h\times n}$ ,  $X_{1}\in \mathbb{R}^{(d - h)\times n}$ , and notice that  $X_0,X_1$  are independent. By a concentration of measure argument (e.g. Tao (2012); Hastie et al. (2019)) one can show that the quantity below tightly concentrates at its expectation. For the variance we have
+
+$$
+\begin{array}{l} V = \operatorname {t r} \left(W \left(W ^ {\top} X X ^ {\top} W\right) ^ {- 1} W ^ {\top} X ^ {\top} \sigma^ {2} X W \left(\left(W ^ {\top} X X ^ {\top} W\right) ^ {- 1} W ^ {\top}\right) \right. \\ = \sigma^ {2} \operatorname {t r} \left(W ^ {\top} W \left(W ^ {\top} X X ^ {\top} W\right) ^ {- 1}\right) = \sigma^ {2} \operatorname {t r} \left(\Sigma^ {\top} \Sigma \left(\Sigma^ {\top} U ^ {\top} X X ^ {\top} U \Sigma\right) ^ {- 1}\right) \\ \sim \sigma^ {2} \operatorname {t r} \left(\Sigma^ {\top} \Sigma \left(\Sigma^ {\top} X X ^ {\top} \Sigma\right) ^ {- 1}\right) = \sigma^ {2} \operatorname {t r} \left(\left(X _ {0} X _ {0} ^ {\top}\right) ^ {- 1}\right)\rightarrow \sigma^ {2} \frac {\gamma_ {2}}{1 - \gamma_ {2}}, \tag {31} \\ \end{array}
+$$
+
+where the last equality follows from Appendix B.2. Similarly for the bias term we have
+
+$$
+\begin{array}{l} B = \left\| W (W ^ {\top} X X ^ {\top} W) ^ {- 1} W ^ {\top} X X ^ {\top} \beta - \beta \right\| _ {2} ^ {2} \\ = \boldsymbol {\beta} ^ {\top} \left(W \left(W ^ {\top} X X ^ {\top} W\right) ^ {- 1} W ^ {\top} X X ^ {\top} - I _ {d}\right) ^ {\top} \left(W \left(W ^ {\top} X X ^ {\top} W\right) ^ {- 1} W ^ {\top} X X ^ {\top} - I _ {d}\right) \boldsymbol {\beta} \\ \stackrel {(i)} {=} \frac {r ^ {2}}{d} \operatorname {t r} \left(\left(W (W ^ {\top} X X ^ {\top} W) ^ {- 1} W ^ {\top} X X ^ {\top} - I _ {d}\right) ^ {\top} \left(W (W ^ {\top} X X ^ {\top} W) ^ {- 1} W ^ {\top} X X ^ {\top} - I _ {d}\right)\right) \\ = \frac {r ^ {2}}{d} \mathrm {t r} \left(\left(U \Sigma V ^ {\top} (V \Sigma^ {\top} U ^ {\top} X X ^ {\top} U \Sigma V ^ {\top}) ^ {- 1} V \Sigma^ {\top} U ^ {\top} X X ^ {\top} - I _ {d}\right) ^ {\top} \left(\dots\right)\right) \\ = \frac {r ^ {2}}{d} \operatorname {t r} \left(\left(U \Sigma (\Sigma^ {\top} X X ^ {\top} \Sigma) ^ {- 1} \Sigma^ {\top} X X ^ {\top} U ^ {\top} - U U ^ {\top}\right) ^ {\top} (\dots)\right) \\ = \frac {r ^ {2}}{d} \operatorname {t r} \left(\left(\Sigma \left(\Sigma^ {\top} X X ^ {\top} \Sigma\right) ^ {- 1} \Sigma^ {\top} X X ^ {\top} - I _ {d}\right) ^ {\top} (\dots)\right), \tag {32} \\ \end{array}
+$$
+
+where symmetric arguments are omitted as  $(\cdot, \cdot)$ , and (i) follows from the rotational invariance argument introduced in Lemma 10 and that  $\beta^{\top} \beta = r^{2}$ . By the block decomposition (30),
+
+$$
+\Sigma \left(\Sigma^ {\top} X X ^ {\top} \Sigma\right) ^ {- 1} \Sigma^ {\top} X X ^ {\top} - I _ {d} = \left[ \begin{array}{c c} 0 & \left(X _ {0} X _ {0} ^ {\top}\right) ^ {- 1} X _ {0} X _ {1} ^ {\top} \\ 0 & - I _ {d - h} \end{array} \right]. \tag {33}
+$$
+
+Therefore the bias term simplifies to
+
+$$
+\begin{array}{l} B = \frac {r ^ {2}}{d} \operatorname {t r} \left(\left(\Sigma (\Sigma^ {\top} X X ^ {\top} \Sigma) ^ {- 1} \Sigma^ {\top} X X ^ {\top} - I _ {d}\right) ^ {\top} \left(\Sigma (\Sigma^ {\top} X X ^ {\top} \Sigma) ^ {- 1} \Sigma^ {\top} X X ^ {\top} - I _ {d}\right)\right) \\ = \frac {r ^ {2}}{d} \left(\operatorname {t r} \left((X _ {0} X _ {0} ^ {\top}) ^ {- 1} X _ {0} X _ {1} ^ {\top} X _ {1} X _ {0} ^ {\top} (X _ {0} X _ {0} ^ {\top}) ^ {- 1}\right) + (d - h)\right) \\ \rightarrow \frac {r ^ {2}}{d} \left(\frac {(d - h) h}{n - h - 1} + d - h\right)\rightarrow \frac {\gamma_ {1} - \gamma_ {2}}{\gamma_ {1} (1 - \gamma_ {2})} r ^ {2}. \tag {34} \\ \end{array}
+$$
+
+Thus we have obtained that as  $n, d, h \to \infty$
+
+$$
+B \rightarrow \frac {\gamma_ {1} - \gamma_ {2}}{\gamma_ {1} (1 - \gamma_ {2})} r ^ {2}. \tag {35}
+$$
+
+Case III:  $W^{\top}X$  has full row rank, i.e.  $\gamma_1 > 1, \gamma_2 > 1$ . Similarly, the least squares solution is
+
+$$
+\hat {\boldsymbol {\beta}} = W W ^ {\top} X \left(X ^ {\top} W W ^ {\top} X\right) ^ {- 1} \boldsymbol {y}, \tag {36}
+$$
+
+Simplifying the variance:
+
+$$
+\begin{array}{l} V = \operatorname {t r} \left(W W ^ {\top} X \left(X ^ {\top} W W ^ {\top} X\right) ^ {- 1} \sigma^ {2} \left(X ^ {\top} W W ^ {\top} X\right) ^ {- 1} X ^ {\top} W W ^ {\top}\right) \\ = \sigma^ {2} \operatorname {t r} \left(W W ^ {\top} U \Sigma V ^ {\top} \left(V \Sigma^ {\top} U ^ {\top} W W ^ {\top} U \Sigma V ^ {\top}\right) ^ {- 2} V \Sigma^ {\top} U ^ {\top} W W ^ {\top}\right) \\ \sim \sigma^ {2} \operatorname {t r} \left(W W ^ {\top} \Sigma \left(\Sigma^ {\top} W W ^ {\top} \Sigma\right) ^ {- 2} \Sigma^ {\top} W W ^ {\top}\right). \tag {37} \\ \end{array}
+$$
+
+where we applied the SVD of  $X = U\Sigma V^{\top}$  and the rotational invariance argument. Using a similar block decomposition on  $W$ :
+
+$$
+\Sigma = \left[ \begin{array}{l} \Sigma_ {0} \\ 0 \end{array} \right], W = \left[ \begin{array}{l} W _ {0} \\ W _ {1} \end{array} \right], \tag {38}
+$$
+
+where  $\Sigma_0\in \mathbb{R}^{n\times n}$ ,  $W_{0}\in \mathbb{R}^{n\times h}$ ,  $W_{1}\in \mathbb{R}^{(d - n)\times h}$ , and  $W_{0},W_{1}$  independent. We thus simplify  $V$  as
+
+$$
+\begin{array}{l} V = \sigma^ {2} \operatorname {t r} \left(W W ^ {\top} \Sigma \left(\Sigma^ {\top} W W ^ {\top} \Sigma\right) ^ {- 2} \Sigma^ {\top} W W ^ {\top}\right) \\ = \sigma^ {2} \mathrm {t r} \left(\left[ \begin{array}{c c} W _ {0} W _ {0} ^ {\top} \Sigma_ {0} (\Sigma_ {0} ^ {\top} W _ {0} W _ {0} ^ {\top} \Sigma_ {0}) ^ {- 2} \Sigma_ {0} W _ {0} W _ {0} ^ {\top} & \dots \\ \dots & W _ {1} W _ {0} ^ {\top} \Sigma_ {0} (\Sigma_ {0} ^ {\top} W _ {0} W _ {0} ^ {\top} \Sigma_ {0}) ^ {- 2} \Sigma_ {0} W _ {0} W _ {1} ^ {\top} \end{array} \right]\right) \\ = \sigma^ {2} \left(\operatorname {t r} \left(\Sigma_ {0} ^ {- T} \Sigma_ {0} ^ {- 1}\right) + \operatorname {t r} \left(W _ {1} W _ {0} ^ {\top} \left(W _ {0} W _ {0} ^ {\top}\right) ^ {- 1} \Sigma_ {0} ^ {- T} \Sigma_ {0} ^ {- 1} \left(W _ {0} W _ {0} ^ {\top}\right) ^ {- 1} W _ {0} W _ {1} ^ {\top}\right)\right) \\ = \sigma^ {2} \operatorname {t r} \left(\left(X ^ {\top} X\right) ^ {- 1}\right) + \sigma^ {2} \operatorname {t r} \left(W _ {1} ^ {\top} W _ {1} W _ {0} ^ {\top} \left(W _ {0} W _ {0} ^ {\top}\right) ^ {- 1} \left(\Sigma_ {0} ^ {\top} \Sigma_ {0}\right) ^ {- 1} \left(W _ {0} W _ {0} ^ {\top}\right) ^ {- 1} W _ {0}\right). \tag {39} \\ \end{array}
+$$
+
+Hence we obtain the following expression on the variance
+
+$$
+V \rightarrow \sigma^ {2} \frac {1}{\gamma_ {1} - 1} + \sigma^ {2} (d - n) \mathbb {E} _ {W, X} V \operatorname {t r} \left(\left(W _ {0} W _ {0} ^ {\top}\right) ^ {- 1} \left(\Sigma_ {0} ^ {\top} \Sigma_ {0}\right) ^ {- 1}\right)\rightarrow \sigma^ {2} \left(\frac {1}{\gamma_ {1} - 1} + \frac {1}{\gamma_ {2} - 1}\right). \tag {40}
+$$
+
+We omit the derivation of bias, which follows a similar derivation:
+
+$$
+B \rightarrow \frac {\gamma_ {2} \left(\gamma_ {1} - 1\right)}{\gamma_ {1} \left(\gamma_ {2} - 1\right)} r ^ {2}. \tag {41}
+$$
+
+Combining Case I, II, III yields theorem 2.
+
+# C.3 PROOF OF PROPOSITION 3
+
+Given the squared loss, one can derive the dynamics of  $W$  with fixed second layer  $a$  w.r.t the loss:
+
+$$
+\frac {\partial W (t)}{\partial t} = - \frac {1}{n} X (\boldsymbol {y} - X ^ {\top} W (t) \boldsymbol {a}) \boldsymbol {a} ^ {\top}. \tag {42}
+$$
+
+Note that the update of  $W$  can be written as a linear combination of  $\pmb{a}$ . Since  $W(0) = 0$ , we can write  $W(t) = \hat{\pmb{w}}(t)\pmb{a}^{\top}$  for some  $\hat{\pmb{w}}$ . The corresponding flow on  $\hat{\pmb{w}}$  is
+
+$$
+\frac {\partial \hat {\boldsymbol {w}} (t)}{\partial t} = - \frac {1}{n} X (\boldsymbol {y} - X ^ {\top} \hat {\boldsymbol {w}} (t) \| \boldsymbol {a} \| _ {2} ^ {2}), \tag {43}
+$$
+
+which gives the following solution
+
+$$
+\hat {\boldsymbol {w}} ^ {*} = \frac {1}{\left\| \boldsymbol {a} \right\| _ {2} ^ {2}} X ^ {\dagger} \boldsymbol {y} \Rightarrow \hat {\boldsymbol {\beta}} = W ^ {*} \boldsymbol {a} = X ^ {\dagger} \boldsymbol {y}. \tag {44}
+$$
+
+Thus gradient flow on the first layer leads to the minimum-norm solution on the input features.
+
+# C.4 PROOF OF THEOREM 4
+
+Following the bias-variance decomposition (7), the variance term can be written as  $(\sigma^2$  omitted)
+
+$$
+\begin{array}{l} V = \mathbb {E} _ {\boldsymbol {x}, \varepsilon} \left[ \| \boldsymbol {a} ^ {\top} \phi (W ^ {\top} \boldsymbol {x}) - \mathbb {E} _ {\varepsilon} \boldsymbol {a} ^ {\top} \phi (W ^ {\top} \boldsymbol {x}) \| _ {2} ^ {2} \right] \\ = \mathbb {E} _ {\boldsymbol {x}} \left[ \left\| \left[ \phi (W ^ {\top} X) \right] ^ {\dagger} \phi (W ^ {\top} \boldsymbol {x}) \right\| _ {2} ^ {2} \right] \\ = \operatorname {t r} \left(\left[ \phi (X ^ {\top} W) \right] ^ {\dagger} \left[ \phi (W ^ {\top} X) \right] ^ {\dagger} \mathbb {E} _ {\boldsymbol {x}} \left[ \phi (W ^ {\top} \boldsymbol {x}) \phi (W ^ {\top} \boldsymbol {x}) ^ {\top} \right]\right) \\ = \operatorname {t r} \left(\left[ \phi \left(X ^ {\top} W\right) \right] ^ {\dagger} \left[ \phi \left(W ^ {\top} X\right) \right] ^ {\dagger} K _ {W}\right), \tag {45} \\ \end{array}
+$$
+
+where we define the expected non-linear Gram matrix  $K_W \in \mathbb{R}^{h \times h}$  as
+
+$$
+K _ {W} = \mathbb {E} _ {\boldsymbol {x}} \left[ \phi \left(W ^ {\top} \boldsymbol {x}\right) \phi \left(W ^ {\top} \boldsymbol {x}\right) ^ {\top} \right]. \tag {46}
+$$
+
+and for each entry we have  $(K_W)_{[i,j]} = \mathbb{E}_{\pmb{x}}\Big[\phi (\pmb {w}_i^\top \pmb {x})\phi (\pmb {w}_j^\top \pmb {x})\Big]$ .
+
+Random matrix in the form of covariance matrix of nonlinear features has been studied in many works Hastie et al. (2019); Mei and Montanari (2019); Liao and Couillet (2018); Louart et al. (2018); Pennington and Worah (2017). We note that our setup for the variance term is very similar to that for nonlinear features in Hastie et al. (2019) with modifications mentioned below.
+
+In contrast to the linear network in Section C.2, the Gram matrix of a nonlinear activation is almost surely full-rank, as specified in the following lemma from Pennington and Worah (2017):
+
+Lemma 11. Suppose  $\phi$  is not linear. Then the smallest singular value of  $\Phi = \phi(W^{\top}X) \in \mathbb{R}^{h \times n}$  is of order  $O(\sqrt{n})$ . To be precise, consider the empirical spectral density  $\mu_{\Phi \Phi^{\top} / n}(d\lambda)$ , when  $n, d, h \to \infty$  with  $d / n \to \gamma_1$  and  $h / n \to \gamma_2$ ,  $\mu_{\Phi \Phi^{\top} / n}(d\lambda)$  converges weakly to
+
+$$
+\mu_ {\Phi \Phi^ {\top} / n} (d \lambda) \rightarrow [ 1 - \gamma_ {2} ^ {- 1} ] _ {+} \delta_ {0} (\lambda) d \lambda + \mu^ {+} (d \lambda),
+$$
+
+where  $\mu^{+}(d\lambda)$  has non-negative support  $[\rho, \infty)$  with  $\rho > 0$ .
+
+We therefore consider two scenarios:  $\Phi$  is full column rank (Case I) or full row rank (Case II).
+
+Case 1.  $h <   n$  . In this case (45) simplifies into
+
+$$
+V = \operatorname {t r} \left(\left(\phi (W ^ {\top} X) \phi (X ^ {\top} W)\right) ^ {- 1} K _ {W}\right) = \lim  _ {\xi \to 0 ^ {-}} \operatorname {t r} \left(\left(\Phi \Phi^ {\top} - \xi I\right) ^ {- 1} K _ {W}\right) = \lim  _ {\xi \to 0 ^ {-}} V _ {\xi},
+$$
+
+where the continuity and boundness of  $V_{\xi}$  at  $\xi = 0^{-}$  is guaranteed by Lemma 11 when  $\gamma_2 = h / n\neq 1$  A ridge  $\xi$  is added make use of (Louart et al., 2018, Theorem 1), which derived the asymptotic equivalent of the resolvent  $\left(\Phi \Phi^{\top} - \xi I\right)^{-1}$ . It follows that as  $n,d,h\to \infty$
+
+$$
+\begin{array}{l} \operatorname {t r} \left(h ^ {- 1} \left(\Phi \Phi^ {\top} - \xi I\right) ^ {- 1} \left(\frac {n}{h} \frac {K _ {W}}{1 + h ^ {- 1} \operatorname {t r} \left(h \left(\Phi \Phi^ {\top} - \xi I\right) ^ {- 1} K _ {W}\right)}\right) - h ^ {- 1} I\right) \\ \leq \frac {1}{h} \left\| \left(\Phi \Phi^ {\top} - \xi I\right) ^ {- 1} - \left(\frac {n}{h} \frac {K _ {W}}{1 + h ^ {- 1} \operatorname {t r} \left(h \left(\Phi \Phi^ {\top} - \xi I\right) ^ {- 1} K _ {W}\right)}\right) ^ {- 1} \right\| _ {F} \\ \cdot \left. \left\| \frac {n}{h} \frac {K _ {W}}{1 + h ^ {- 1} \operatorname {t r} \left(h \left(\Phi \Phi^ {\top} - \xi I\right) ^ {- 1} K _ {W}\right)} \right\| _ {2} \right. \\ \leq \frac {1}{h} O \left(n ^ {- 1 / 2 + \epsilon}\right) O \left(n ^ {1 / 2}\right)\rightarrow 0. \tag {47} \\ \end{array}
+$$
+
+where we have used the inequality  $\operatorname{tr}(AB) \leq \|A\|_F \|B\|_2$ , and equivalently by taking  $\xi \to 0$  we get
+
+$$
+\lim  _ {\xi \rightarrow 0} \lim  _ {n, d, h \rightarrow \infty} \frac {n}{h} \frac {\operatorname {t r} \left(\left(\Phi \Phi^ {\top} - \xi I\right) ^ {- 1} K _ {W}\right)}{1 + \operatorname {t r} \left(\left(\Phi \Phi^ {\top} - \xi I\right) ^ {- 1} K _ {W}\right)} = 1. \tag {48}
+$$
+
+Therefore  $\lim_{\xi \to 0}\lim_{n,d,h\to \infty}n / h\cdot \mathrm{tr}\left(\left(\Phi \Phi^{\top} - \xi I\right)^{-1}K_{W}\right) = \gamma_{2} / (1 - \gamma_{2})$ . Note that  $\partial V_{\xi} / \partial \xi =$ $\mathrm{tr}\left(\xi (\Phi \Phi^{\top} - \xi I)^{-2}K_{W}\right)$  is bounded around the neighbourhood of  $\xi = 0$ , hence following the same argument as (Hastie et al., 2019, Theorem 4), we exchange the limit of  $\xi \rightarrow 0$  and  $n,d,h\rightarrow 0$ :
+
+$$
+V \rightarrow \frac {\gamma_ {2}}{1 - \gamma_ {2}}. \tag {49}
+$$
+
+Case 2.  $h > n$ . Techniques used in the current proof are largely borrowed from Hastie et al. (2019); Cheng and Singer (2013), and we include the full proof for completeness. It should be noted that compared to Hastie et al. (2019) we handle the non-zero expectation of the nonlinearity under Gaussian distribution, i.e. the off-diagonal entries of the kernel matrix is no longer zero-centered. For simplicity we mainly adhere to the notations in Hastie et al. (2019).
+
+We briefly summarizes the procedure for deriving  $V$ . Instead of calculating the variance directly, we analyze a modified quantity  $V_{\xi}$  and then take  $\xi \rightarrow 0$ , which can be connected to the trace of the resolvent of matrix  $\tilde{A}$  defined in (56); this translates the calculation of  $V_{\xi}$  into the calculation of the Stieltjes transform of  $\tilde{A}$  (57), (61), (62).
+
+# C.5 DERIVING THE VARIANCE  $V$  FOR  $h > n$
+
+Step 1. An equivalent expression. For notational simplicity we omit the magnitude  $\sigma$ :
+
+$$
+V = \operatorname {t r} \left(\phi \left(W ^ {\top} X\right) \left(\phi \left(X ^ {\top} W\right) \phi \left(W ^ {\top} X\right)\right) ^ {- 2} \phi \left(X ^ {\top} W\right) K _ {W}\right). \tag {50}
+$$
+
+and due to the same continuity argument as in Case 1 we have
+
+$$
+V = \lim _ {\xi \to 0} \frac {1}{n} \Big [ \mathrm {t r} \left(S (S ^ {\top} S - \xi I _ {n}) ^ {- 2} S ^ {\top} K _ {W}\right) \Big ] = \lim _ {\xi \to 0} V _ {\xi}.
+$$
+
+where  $S = \phi (W^{\top}X) / \sqrt{n} = \Phi /\sqrt{n},\xi \in \mathbb{C}$  and  $\Im \xi >0$  or  $\xi <  0$
+
+We decompose the normalized feature matrix  $S = \phi(W^{\top}X) / \sqrt{n}$  as  $S = U\Sigma V^{\top}$ , where  $\Sigma = \mathrm{diag}_{h\times n}(\phi_1,\dots ,\phi_n)\in \mathbb{R}^{h\times n}$  is a tall diagonal matrix, and  $U = [\pmb{u}_1,\dots ,\pmb{u}_h]\in \mathbb{R}^{h\times h}$  is the set of orthogonal eigenvectors of  $SS^{\top} = \phi (W^{\top}X)\phi (X^{\top}W) / n$ , and  $V\in \mathbb{R}^{n\times n}$  is the set of orthogonal eigenvectors of  $S^{\top}S = \phi (X^{\top}W)\phi (W^{\top}X) / n$ . Now the variance can be written as
+
+$$
+V _ {\xi} = \frac {1}{n} \operatorname {t r} \left(U \Sigma \left(\Sigma^ {\top} \Sigma + \xi I _ {n}\right) ^ {- 2} \Sigma U ^ {\top} K _ {W}\right). \tag {51}
+$$
+
+By the same argument as in Lemma 13 of Hastie et al. (2019) one can show that when  $n, d, h \to \infty$
+
+$$
+V _ {\xi} = \frac {1}{n} \left[ \operatorname {t r} \left(U \Sigma \left(\Sigma^ {\top} \Sigma + \xi I _ {n}\right) ^ {- 2} \Sigma U ^ {\top} K _ {W}\right)\right]\rightarrow \frac {1}{n} \left[ \operatorname {t r} \left(U \Sigma \left(\Sigma^ {\top} \Sigma + \xi I _ {n}\right) ^ {- 2} \Sigma U ^ {\top} \tilde {K} _ {W}\right)\right], \tag {52}
+$$
+
+in which  $\tilde{K}_W$  is the approximation of  $K_W$  defined in Lemma 16. Writing the trace explicitly (denote eigenvalues  $\lambda_i = \phi_i^2$ , and  $\phi_{n+1} = \dots = \phi_h = 0$ ), we have
+
+$$
+V _ {\xi} \rightarrow \frac {1}{n} \operatorname {t r} \left(U \Sigma \left(\Sigma^ {\top} \Sigma + \xi I _ {n}\right) ^ {- 2} \Sigma U ^ {\top} \tilde {K} _ {W}\right) = \gamma_ {2} \frac {1}{h} \sum_ {i = 1} ^ {h} \frac {\lambda_ {i}}{\left(\lambda_ {i} + \xi\right) ^ {2}} \boldsymbol {u} _ {i} ^ {\top} \tilde {K} _ {W} \boldsymbol {u} _ {i}. \tag {53}
+$$
+
+Since the positive support of spectrum  $\lambda$  is lower bounded and the density at 0 is  $1 - \gamma_2^{-1}$ , we have
+
+$$
+V _ {\xi} \rightarrow \gamma_ {2} \frac {1}{h} \lim  _ {h, d, n \rightarrow \infty} \sum_ {i = 1} ^ {h} \frac {\lambda_ {i}}{(\lambda_ {i} + \xi) ^ {2}} \boldsymbol {u} _ {i} ^ {\top} \tilde {K} _ {W} \boldsymbol {u} _ {i} = \gamma_ {2} \int \frac {\lambda}{(\lambda + \xi) ^ {2}} \mu_ {\infty} (d \lambda) = \gamma_ {2} \int_ {\lambda > \rho} \frac {\lambda}{(\lambda + \xi) ^ {2}} \mu_ {\infty} ^ {+} (d \lambda), \tag {54}
+$$
+
+where we define  $\mu_n(x) = \frac{1}{h}\sum_{i=1}^h \delta_{\lambda_i}(x)\pmb{u}_i^\top \tilde{K}_W\pmb{u}_i$  and its positive part  $\mu_n^+(x)$ . Hence we have
+
+$$
+V = \lim  _ {\xi \rightarrow 0} V _ {\xi} = \lim  _ {\xi \rightarrow 0} \gamma_ {2} \int_ {\lambda > \rho} \frac {\lambda}{(\lambda + \xi) ^ {2}} \mu_ {\infty} ^ {+} (d \lambda) = \gamma_ {2} \int_ {\lambda > \rho} \frac {1}{\lambda} \mu_ {\infty} ^ {+} (d \lambda). \tag {55}
+$$
+
+We define the following matrix  $\tilde{A}_n(\rho, \varsigma, \tau) \in \mathbb{R}^{N \times N}$  where  $N = n + h$ :
+
+$$
+\tilde {A} _ {n} (\rho , \varsigma , \tau) = \left[ \begin{array}{c c} \rho I _ {h} + \varsigma \mathbf {1} _ {h} \mathbf {1} _ {h} ^ {\top} + \tau Q & S \\ S ^ {\top} & 0 _ {n} \end{array} \right]. \tag {56}
+$$
+
+And denote the Stieltjes transform of  $\tilde{A}_n$  as
+
+$$
+\tilde {m} _ {n} (\xi , \rho , \varsigma , \tau) = \frac {1}{n} \operatorname {t r} \left(\left(\tilde {A} _ {n} (\rho , \varsigma , \tau) - \xi I _ {N}\right) ^ {- 1}\right). \tag {57}
+$$
+
+Then following the definition of  $\tilde{K}_W$  one can show that
+
+$$
+\tilde {m} _ {n} (\xi , r x, s x, t x) = \frac {1}{n} \operatorname {t r} \left(\left[ \begin{array}{c c} \tilde {K} _ {W} x - \xi I _ {h} & S \\ S ^ {\top} & - I _ {n} \end{array} \right] ^ {- 1}\right), \tag {58}
+$$
+
+and taking matrix derivative gives
+
+$$
+\begin{array}{l} \left. - \frac {\partial}{\partial x} \tilde {m} _ {n} (\xi , r x, s x, t x) \right| _ {x = 0} = \frac {1}{n} \operatorname {t r} \left(\left[ \begin{array}{c c} - \xi I _ {h} & S \\ S ^ {\top} & - I _ {n} \end{array} \right] ^ {- 1} \left[ \begin{array}{c c} \tilde {K} _ {W} & 0 \\ 0 & 0 \end{array} \right] \left[ \begin{array}{c c} - \xi I _ {h} & S \\ S ^ {\top} & - I _ {n} \end{array} \right] ^ {- 1}\right) \\ = \frac {1}{n} \mathrm {t r} \left(\left[ \begin{array}{c c} \xi^ {2} I _ {h} + S S ^ {\top} & 0 \\ 0 & I _ {n} + S ^ {\top} S \end{array} \right] ^ {- 1} \left[ \begin{array}{c c} \tilde {K} _ {W} & 0 \\ 0 & 0 \end{array} \right]\right) \\ = \frac {1}{n} \operatorname {t r} \left(U \left(\Sigma \Sigma^ {\top} + \xi^ {2} I _ {h}\right) ^ {- 1} U ^ {\top} \tilde {K} _ {W}\right) = \frac {1}{n} \sum_ {i = 1} ^ {h} \frac {1}{\lambda + \xi^ {2}} \boldsymbol {u} _ {i} ^ {\top} \tilde {K} _ {W} \boldsymbol {u} _ {i}. \tag {59} \\ \end{array}
+$$
+
+Denote the limit  $\tilde{m} (\xi ,\rho ,\varsigma ,\tau) = \lim_{n,h,d\to \infty}\tilde{m}_n(\xi ,\rho ,\varsigma ,\tau)$ , and the derivative is given as
+
+$$
+\begin{array}{l} \left. - \frac {\partial}{\partial x} \tilde {m} (\xi , r x, s x, t x) \right| _ {x = 0} = \lim  _ {h, d, n \rightarrow \infty} \frac {1}{n} \sum_ {i = 1} ^ {h} \frac {1}{\lambda + \xi^ {2}} \boldsymbol {u} _ {i} ^ {\top} \tilde {K} _ {W} \boldsymbol {u} _ {i} \\ = \gamma_ {2} \int_ {\lambda \geq 0} \frac {1}{\lambda + \xi^ {2}} \mu_ {\infty} (d \lambda) = \frac {\gamma_ {2} - 1}{\xi^ {2}} + \gamma_ {2} \int_ {\lambda > \rho} \frac {1}{\lambda + \xi^ {2}} \mu_ {\infty} ^ {+} (d \lambda). \tag {60} \\ \end{array}
+$$
+
+For simplicity we define the following function on  $\xi$ :
+
+$$
+q (\xi) = - \frac {\partial}{\partial x} \tilde {m} (\xi , r x, s x, t x) \bigg | _ {x = 0}, \tag {61}
+$$
+
+also denote  $q_{+}(\xi) = q(\xi) - \frac{\gamma_{2} - 1}{\xi^{2}} = \gamma_{2}\int_{\lambda >\rho}\frac{1}{\lambda + \xi^{2}}\mu_{\infty}^{+}(d\lambda)$ . Comparing with (55) yields:
+
+$$
+V = \lim  _ {\xi \rightarrow 0} q _ {+} (\xi). \tag {62}
+$$
+
+Step 2. Calculating  $q(\xi)$  and  $m_{n}(\xi, \rho, \varsigma, \tau)$  This subsection aims to calculate  $\tilde{m}(\xi, \rho, \varsigma, \tau)$  and  $q(\xi) = -\tilde{m}_{x}'(\xi, rx, sx, tx)|_{x=0}$ , from which the variance can be computed from (57)(61)(62).
+
+We define  $A_{n}$  by subtracting the off-diagonal entries of the upper-left block of  $\tilde{A}_n$ :
+
+$$
+A _ {n} (\rho , \tau) = \left[ \begin{array}{c c} \rho I _ {h} + \tau Q & S \\ S ^ {\top} & 0 _ {n} \end{array} \right], \tag {63}
+$$
+
+where  $S = \tilde{S} - a_0 I_{p \times n}$ , i.e.  $S_{ik} = \phi(\boldsymbol{w}_i^\top \boldsymbol{x}_k) - a_0 = \varphi(\boldsymbol{w}_i^\top \boldsymbol{x}_k)$ ,  $a_0 = \mathbb{E}[\phi(x)]$ . The Stieltjes transform of  $A_n$  given by  $m_n(\xi, \rho, \tau) = \frac{1}{n} \mathrm{tr}\left((A_n(\rho, \tau) - \xi I_N)^{-1}\right)$ . The following Lemma shows that  $\tilde{m}_n$  and  $m_n$  have the same limit:
+
+Lemma 12. when  $n\to \infty$  and for  $\Im \xi >0$  or  $\xi <  0$  , we have  $m_{n}(\xi ,\rho ,\tau)\rightarrow \tilde{m}_{n}(\xi ,\rho ,\varsigma ,\tau)$
+
+Proof. By definition of  $\bar{A}_n$  and  $A_{n}$
+
+$$
+\tilde {A} _ {n} (\rho , \varsigma , \tau) - A _ {n} (\rho , \tau) = \left[ \begin{array}{l l} 1 _ {h} & 0 _ {h} \\ 0 _ {n} & 1 _ {n} \end{array} \right] \left[ \begin{array}{l l} \xi & a _ {0} \\ a _ {0} & 0 \end{array} \right] \left[ \begin{array}{l l} 1 _ {h} & 0 _ {h} \\ 0 _ {n} & 1 _ {n} \end{array} \right] ^ {\top} \tag {64}
+$$
+
+which is a rank-2 matrix. By theorem A.43 from Bai and Silverstein (2010), which characterizes the effect of finite-rank perturbation on the e.s.d. of random matrices:
+
+$$
+\sup  _ {x} \left| F ^ {\prime A _ {n}} (x) - F ^ {\prime A _ {n}} (x) \right| \leq O \left(n ^ {- 1}\right), \tag {65}
+$$
+
+where  $F^M$  is the empirical spectral distribution of  $M \in \mathbb{R}^{n \times n}$ . The claim follows from the Stieltjes continuity theorem (e.g. Section 2.4 in Tao (2012)).
+
+To calculate the Stieltjes transform  $m_{n}$ , we take advantage of the block structure of  $A_{n}$ :
+
+$$
+m _ {1, n} (\xi , \rho , \tau) = \frac {1}{p} \operatorname {t r} \left(\left(A _ {n} (\rho , \tau) - \xi I _ {N}\right) _ {[ 1.. p, 1.. p ]} ^ {- 1}\right), \tag {66}
+$$
+
+$$
+m _ {2, n} (\xi , \rho , \tau) = \frac {1}{n} \operatorname {t r} \left(\left(A _ {n} (\rho , \tau) - \xi I _ {N}\right) _ {[ p + 1.. p + n, p + 1.. p + n ]} ^ {- 1}\right). \tag {67}
+$$
+
+One can observe that  $m_{n}(\xi ,\rho ,\tau) = \gamma_{2}m_{1,n}(\xi ,\rho ,\tau) + m_{2,n}(\xi ,\rho ,\tau)$
+
+In the following equations we omit the subscript  $n$ , as well as dependency on  $\rho, \varsigma, \tau$ . Following Hastie et al. (2019), we rewrite the matrix  $A_{n} = A = \left[ \begin{array}{cc}A_{*} & \boldsymbol{a}\\ \boldsymbol{a}^{\top} & 0 \end{array} \right]$ , where  $A^{*}$  is a  $(N - 1)\times (N - 1)$  matrix with last column and row of  $A$  removed and  $\boldsymbol{s}$  the activation vector:
+
+$$
+A _ {*} = \left[ \begin{array}{c c} \rho I _ {h} + \tau Q & S _ {*} \\ S _ {*} ^ {\top} & 0 _ {n - 1} \end{array} \right]; \quad \boldsymbol {a} ^ {\top} = \left[ \phi \left(W ^ {\top} \boldsymbol {x} _ {n}\right) ^ {\top} \quad \mathbf {0} _ {n - 1} ^ {\top} \right] = \left[ \begin{array}{l l} \boldsymbol {s} ^ {\top} & \mathbf {0} _ {n - 1} ^ {\top} \end{array} \right]. \tag {68}
+$$
+
+Hence by the block matrix inverse formula
+
+$$
+\left(A - \xi I _ {N}\right) ^ {- 1} = \left[ \begin{array}{l l} * & * \\ * & \left[ - \xi - \boldsymbol {a} ^ {\top} \left(A _ {*} - \xi I _ {N - 1}\right) ^ {- 1} \boldsymbol {a} \right] ^ {- 1} \end{array} \right]. \tag {69}
+$$
+
+Plugging this back in the Stieltjes transform we have
+
+$$
+\begin{array}{l} m _ {2, n} (\xi , \rho , \tau) = \frac {1}{n} \operatorname {t r} \left((A _ {n} (\rho , \varsigma , \tau) - \xi I _ {N}) _ {[ h + 1, N ]} ^ {- 1}\right) = \mathbb {E} _ {\boldsymbol {a}} \left[ (A _ {n} (\rho , \tau) - \xi I _ {N}) ^ {- 1} \right] _ {N N} \\ = \mathbb {E} _ {\boldsymbol {a}} \left[ \left(- \xi - \boldsymbol {a} ^ {\top} \left(A ^ {*} - \xi I _ {N - 1}\right) ^ {- 1} \boldsymbol {a}\right) ^ {- 1} \right]. \tag {70} \\ \end{array}
+$$
+
+To obtain an asymptotic description of  $m_{2,n}$ , we perform the orthonormal decomposition on the nonlinearity  $\varphi$  introduced in Cheng and Singer (2013):  $\varphi(x) = a_1x + \varphi_\perp(x)$ , where  $a_1 = \mathbb{E}_{x \sim \mathcal{N}(0,1)}[x\varphi(x)]$ . For each  $\boldsymbol{w}_i(1 \leq i \leq h)$ , we perform the following orthonormal decomposition (along the direction of  $\boldsymbol{x}_n$  and the direction of  $\tilde{\boldsymbol{w}}_i$  perpendicular to  $\boldsymbol{x}_n$ ):
+
+$$
+\boldsymbol {w} _ {i} = \underbrace {\boldsymbol {w} _ {i} ^ {\top} \boldsymbol {x} _ {n}} _ {\eta_ {i}} \frac {\boldsymbol {x} _ {n}}{\| \boldsymbol {x} _ {n} \|} + \tilde {\boldsymbol {w}} _ {i} = \eta_ {i} \frac {\boldsymbol {x} _ {n}}{\| \boldsymbol {x} _ {n} \|} + \tilde {\boldsymbol {w}} _ {i}. \tag {71}
+$$
+
+We can thus simplify the activation vector  $\mathbf{a}$  as
+
+$$
+\begin{array}{l} \boldsymbol {a} ^ {\top} = \left[ \begin{array}{l l l l} \frac {1}{\sqrt {n}} \varphi (\| \boldsymbol {x} _ {n} \| \eta_ {1}) & \dots & \frac {1}{\sqrt {n}} \varphi (\| \boldsymbol {x} _ {n} \| \eta_ {h}) & \underbrace {0 \quad \cdots \quad 0} _ {n - 1} \end{array} \right] \\ = \underbrace {\left[ \begin{array}{c c c} \frac {1}{\sqrt {n}} a _ {1} \| \boldsymbol {x} _ {n} \| \eta_ {1} & \cdots & \frac {1}{\sqrt {n}} a _ {1} \| \boldsymbol {x} _ {n} \| \eta_ {h} \\ & & \underbrace {0 \quad \cdots \quad 0} _ {n - 1} \end{array} \right]} _ {\boldsymbol {a} _ {1} ^ {\top} = [ \boldsymbol {s} _ {1} ^ {\top}, 0 ^ {\top} ]} \\ + \underbrace {\left[ \frac {1}{\sqrt {n}} \varphi_ {\perp} \left(\| \boldsymbol {x} _ {n} \| \eta_ {1}\right) \quad \cdots \quad \frac {1}{\sqrt {n}} \varphi_ {\perp} \left(\| \boldsymbol {x} _ {n} \| \eta_ {h}\right) \quad \underbrace {0 \quad \cdots \quad 0} _ {n - 1} \right]} _ {\boldsymbol {a} _ {2} ^ {\top} = [ \boldsymbol {s} _ {2} ^ {\top}, 0 ^ {\top} ]}. \tag {72} \\ \end{array}
+$$
+
+and for  $1 \leq i \neq j \leq h, 1 \leq k \leq n - 1$ ,
+
+$$
+Q _ {i j} = \left(\eta_ {i} \frac {\boldsymbol {x} _ {n}}{\| \boldsymbol {x} _ {n} \|} + \tilde {\boldsymbol {w}} _ {i}\right) ^ {\top} \left(\eta_ {j} \frac {\boldsymbol {x} _ {n}}{\| \boldsymbol {x} _ {n} \|} + \tilde {\boldsymbol {w}} _ {j}\right) = \eta_ {i} \eta_ {j} + \underbrace {\tilde {\boldsymbol {w}} _ {i} ^ {\top} \tilde {\boldsymbol {w}} _ {j}} _ {\tilde {Q} _ {i j}}. \tag {73}
+$$
+
+Similarly we decompose the activation function in  $S$ ,
+
+$$
+\begin{array}{l} S _ {i k} = \frac {1}{\sqrt {n}} \varphi \left(\eta_ {i} \frac {\boldsymbol {x} _ {n} ^ {\top} \boldsymbol {x} _ {k}}{\| \boldsymbol {x} _ {n} \|} + \tilde {\boldsymbol {w}} _ {i} ^ {\top} \boldsymbol {x} _ {k}\right) = \frac {1}{\sqrt {n}} a _ {1} \eta_ {i} \frac {\boldsymbol {x} _ {n} ^ {\top} \boldsymbol {x} _ {k}}{\| \boldsymbol {x} _ {n} \|} + \frac {1}{\sqrt {n}} a _ {1} \tilde {\boldsymbol {w}} _ {i} ^ {\top} \boldsymbol {x} _ {k} + \frac {1}{\sqrt {n}} \varphi_ {\perp} \left(\eta_ {i} \frac {\boldsymbol {x} _ {n} ^ {\top} \boldsymbol {x} _ {k}}{\| \boldsymbol {x} _ {n} \|} + \tilde {\boldsymbol {w}} _ {i} ^ {\top} \boldsymbol {x} _ {k}\right) \\ = \underbrace {\frac {1}{\sqrt {n}} \varphi \left(\tilde {\boldsymbol {w}} _ {i} ^ {\top} \boldsymbol {x} _ {k}\right)} _ {\tilde {S} _ {i k}} + \underbrace {\frac {1}{\sqrt {n}} a _ {1} \eta_ {i} \frac {\boldsymbol {x} _ {n} ^ {\top} \boldsymbol {x} _ {k}}{\| \boldsymbol {x} _ {n} \|}} _ {a _ {1} \eta_ {i} u _ {k}} + \underbrace {\frac {1}{\sqrt {n}} \left[ \varphi_ {\perp} \left(\eta_ {i} \frac {\boldsymbol {x} _ {n} ^ {\top} \boldsymbol {x} _ {k}}{\| \boldsymbol {x} _ {n} \|} + \tilde {\boldsymbol {w}} _ {i} ^ {\top} \boldsymbol {x} _ {k}\right) - \varphi_ {\perp} \left(\tilde {\boldsymbol {w}} _ {i} ^ {\top} \boldsymbol {x} _ {k}\right) \right]} _ {E _ {i k}}. \tag {74} \\ \end{array}
+$$
+
+We thus have an equivalent expression of matrix  $A_{*}$
+
+$$
+\begin{array}{l} A _ {*} = \left[ \begin{array}{c c} \rho I _ {h} + \tau \tilde {Q} & \tilde {S} _ {*} \\ \tilde {S} _ {*} ^ {\top} & 0 _ {n - 1} \end{array} \right] + \left[ \begin{array}{c c} t \boldsymbol {\eta} \boldsymbol {\eta} ^ {\top} & a _ {1} \boldsymbol {\eta} \boldsymbol {u} ^ {\top} \\ a _ {1} \boldsymbol {u} \boldsymbol {\eta} ^ {\top} & 0 _ {n - 1} \end{array} \right] + \left[ \begin{array}{c c} E _ {0} & E _ {1} \\ E _ {1} ^ {\top} & 0 _ {n - 1} \end{array} \right] \\ = \underbrace {\left[ \begin{array}{c c} \rho I _ {h} + \tau \tilde {Q} & \tilde {S} _ {*} \\ \tilde {S} _ {*} ^ {\top} & 0 _ {n - 1} \end{array} \right]} _ {\tilde {A} _ {*}} + \underbrace {\left[ \begin{array}{c c} \boldsymbol {\eta} & \mathbf {0} _ {h} \\ \mathbf {0} _ {n - 1} & \boldsymbol {u} \end{array} \right]} _ {U} \underbrace {\left[ \begin{array}{c c} \tau & a _ {1} \\ a _ {1} & 0 \end{array} \right]} _ {C} \underbrace {\left[ \begin{array}{c c} \boldsymbol {\eta} & \mathbf {0} _ {h} \\ \mathbf {0} _ {n - 1} & \boldsymbol {u} \end{array} \right]} _ {U ^ {\top}} ^ {\top} + \underbrace {\left[ \begin{array}{c c} E _ {0} & E _ {1} \\ E _ {1} ^ {\top} & 0 _ {n - 1} \end{array} \right]} _ {E} \\ = \tilde {A} _ {*} + U C U ^ {\top} + E. \tag {75} \\ \end{array}
+$$
+
+By argument similar to (Hastie et al., 2019, B.1.2),  $E$  diminishes to 0 as  $n \to \infty$  with respect to the Frobenius norm, therefore by the Woodbury's identity and the expression of  $m_{2,n}$  in (70)
+
+$$
+\begin{array}{l} m _ {2, n} (\xi , \rho , \tau) = \mathbb {E} _ {\boldsymbol {a}} \left[ \left(- \xi - \boldsymbol {a} ^ {\top} \left(A _ {*} - \xi I _ {N - 1}\right) ^ {- 1} \boldsymbol {a}\right) ^ {- 1} \right] \\ \rightarrow \mathbb {E} _ {\boldsymbol {a}} \left[\left(- \xi - \boldsymbol {a} ^ {\top} \left(\tilde {A} _ {*} - \xi I _ {N - 1} + U C U ^ {\top}\right) ^ {- 1} \boldsymbol {a}\right) ^ {- 1} \right] \\ = \mathbb {E} _ {\boldsymbol {a}} \left[ \left(- \xi - \underbrace {\boldsymbol {a} ^ {\top} \left(\tilde {A} _ {*} - \xi I _ {N - 1}\right) ^ {- 1} \boldsymbol {a}} _ {u} + \right. \right. \\ \underbrace {\boldsymbol {a} ^ {\top} \left(\tilde {A} _ {*} - \xi I _ {N - 1}\right) ^ {- 1} U} _ {\boldsymbol {v} ^ {\top}} \underbrace {\left(C ^ {- 1} + U ^ {\top} \left(\tilde {A} _ {*} - \xi I _ {N - 1}\right) ^ {- 1} U\right) ^ {- 1}} _ {S} \underbrace {\left. U \left(\tilde {A} _ {*} - \xi I _ {N - 1}\right) ^ {- 1} \boldsymbol {a} \right) ^ {- 1} ]}. \tag {76} \\ \end{array}
+$$
+
+We bound each term  $u, v, S$  to compute  $m_{2,n}$ . For  $u$
+
+$$
+\mathbb {E} _ {\boldsymbol {a}} u = \mathbb {E} _ {\boldsymbol {a}} \left[ \boldsymbol {a} ^ {\top} \left(\tilde {A} _ {*} - \xi I _ {N - 1}\right) ^ {- 1} \boldsymbol {a} \right] = \operatorname {t r} \left(\mathbb {E} _ {\boldsymbol {s}} [ \boldsymbol {s} \boldsymbol {s} ^ {\top} ] \left(\tilde {A} _ {*} - \xi I _ {N - 1}\right) _ {[ 1.. h, 1.. h ]} ^ {- 1}\right) = b \gamma_ {2} m _ {1, n} (\xi , \rho , \tau), \tag {77}
+$$
+
+where  $b = \mathbb{E}_{x\sim \mathcal{N}(0,1)}[\varphi (x)^2 ] = \mathbb{E}_{x\sim \mathcal{N}(0,1)}[(\phi (x) - \mathbb{E}\phi (x))^2 ] = r$ . From a standard concentration of measure argument we have that as  $n,h,d\to \infty$
+
+$$
+u \rightarrow \mathbb {E} _ {\boldsymbol {a}} u = r \gamma_ {2} m _ {1, n} (\xi , \rho , \tau). \tag {78}
+$$
+
+And for  $\pmb{v}$  (note that  $U$  is dependent on  $\pmb{a}$ )
+
+$$
+\begin{array}{l} \mathbb {E} _ {\boldsymbol {a}} \boldsymbol {v} ^ {\top} = \mathbb {E} _ {\boldsymbol {a}} \left[ \boldsymbol {a} ^ {\top} (\tilde {A} _ {*} - \xi I _ {N - 1}) ^ {- 1} U \right] = \mathbb {E} _ {\boldsymbol {a}} \left[ [ \boldsymbol {s} ^ {\top}, 0 _ {n - 1} ^ {\top} ] (\tilde {A} _ {*} - \xi I _ {N - 1}) ^ {- 1} \left[ \begin{array}{c c} \boldsymbol {\eta} & \boldsymbol {0} _ {h} \\ \boldsymbol {0} _ {n - 1} & \boldsymbol {u} \end{array} \right] \right] \\ = \left[ \mathbb {E} _ {\boldsymbol {s}} \left[ \boldsymbol {s} ^ {\top} \left(\tilde {A} _ {*} - \xi I _ {N - 1}\right) _ {[ 1.. h, 1.. h ]} ^ {- 1} \boldsymbol {\eta} \right] \quad 0 \right] = \left[ \underbrace {\operatorname {t r} \left(\left(\tilde {A} _ {*} - \xi I _ {N - 1}\right) _ {[ 1.. h , 1.. h ]} ^ {- 1} \mathbb {E} _ {\boldsymbol {s}} \left[ \boldsymbol {\eta} \boldsymbol {s} ^ {\top} \right]\right)} _ {v} \quad 0 \right]. \tag {79} \\ \end{array}
+$$
+
+where  $v = \operatorname{tr}\left((\tilde{A}_* - \xi I_{N-1})_{[1..h,1..h]}^{-1}\mathbb{E}_s[\pmb{\eta}\pmb{s}^\top]\right) = a_1\sqrt{\gamma_2^2 / \gamma_1} m_{1,n}(\xi,\rho,\tau)$ . We thus have
+
+$$
+\boldsymbol {v} \rightarrow \mathbb {E} _ {\boldsymbol {a}} \boldsymbol {v} = \left[ a _ {1} \sqrt {\gamma_ {2} ^ {2} / \gamma_ {1}} m _ {1, n} (\xi , \rho , \tau) \quad 0 \right]. \tag {80}
+$$
+
+as  $n, d, p \to \infty$ . And finally for  $S$ ,
+
+$$
+\begin{array}{l} \mathbb {E} _ {\boldsymbol {a}} S ^ {- 1} = \mathbb {E} _ {\boldsymbol {a}} \left[ C ^ {- 1} + U ^ {\top} \left(\tilde {A} _ {*} - \xi I _ {N - 1}\right) ^ {- 1} U \right] \\ = \left[ \begin{array}{c c} \tau & a _ {1} \\ a _ {1} & 0 \end{array} \right] ^ {- 1} + \mathbb {E} _ {\boldsymbol {a}} \left[ \left[ \begin{array}{c c} \boldsymbol {\eta} & \mathbf {0} _ {h} \\ \mathbf {0} _ {n - 1} & \boldsymbol {u} \end{array} \right] ^ {\top} (\tilde {A} _ {*} - \xi I _ {N - 1}) ^ {- 1} \left[ \begin{array}{c c} \boldsymbol {\eta} & \mathbf {0} _ {h} \\ \mathbf {0} _ {n - 1} & \boldsymbol {u} \end{array} \right] \right] \\ = \left[ \begin{array}{c c} 0 & 1 / a _ {1} \\ 1 / a _ {1} & - \tau / a _ {1} ^ {2} \end{array} \right] + \mathbb {E} _ {\boldsymbol {a}} \left[ \begin{array}{c c} \boldsymbol {\eta} ^ {\top} (\tilde {A} _ {*} - \xi I _ {N - 1}) _ {[ 1.. h ]} ^ {- 1} \boldsymbol {\eta} & 0 \\ 0 & \boldsymbol {u} ^ {\top} (\tilde {A} _ {*} - \xi I _ {N - 1}) _ {[ h + 1.. h + n - 1 ]} ^ {- 1} \boldsymbol {u} \end{array} \right] \\ = \left[ \begin{array}{c c} 0 & 1 / a _ {1} \\ 1 / a _ {1} & - \tau / a _ {1} ^ {2} \end{array} \right] + \left[ \begin{array}{c c} \operatorname {t r} \left((\tilde {A} _ {*} - \xi I _ {N - 1}) _ {[ 1.. h ]} ^ {- 1} \mathbb {E} _ {\boldsymbol {a}} [ \boldsymbol {\eta} \boldsymbol {\eta} ^ {\top} ]\right) & 0 \\ 0 & \operatorname {t r} \left((\tilde {A} _ {*} - \xi I _ {N - 1}) _ {[ h + 1.. h + n - 1 ]} ^ {- 1} \mathbb {E} _ {\boldsymbol {a}} [ \boldsymbol {u u} ^ {\top} ]\right) \end{array} \right] \\ = \left[ \begin{array}{c c} 0 & 1 / a _ {1} \\ 1 / a _ {1} & - \tau / a _ {1} ^ {2} \end{array} \right] + \left[ \begin{array}{c c} \gamma_ {2} / \gamma_ {1} m _ {1, n} (\xi , \rho , \tau) & 0 \\ 0 & m _ {2, n} (\xi , \rho , \tau) \end{array} \right] \\ = \left[ \begin{array}{c c} \gamma_ {1} ^ {- 1} \gamma_ {2} m _ {1, n} (\xi , \rho , \tau) & 1 / a _ {1} \\ 1 / a _ {1} & m _ {2, n} (\xi , \rho , \tau) - \tau / a ^ {2} \end{array} \right]. \tag {81} \\ \end{array}
+$$
+
+And hence as  $n, d, h \to \infty$ ,
+
+$$
+S ^ {- 1} \rightarrow \mathbb {E} _ {\boldsymbol {a}} S ^ {- 1} = \left[\begin{array}{c c}\gamma_ {1} ^ {- 1} \gamma_ {2} m _ {1, n} (\xi , \rho , \tau)&1 / a _ {1}\\1 / a _ {1}&m _ {2, n} (\xi , \rho , \tau) - \tau / a ^ {2}\end{array}\right]. \tag {82}
+$$
+
+Therefore by combining (78), (80), (82), we arrive at the following expression on  $m_{2,n}$
+
+$$
+\begin{array}{l} m _ {2, n} (\xi , \rho , \tau) \rightarrow \mathbb {E} _ {\boldsymbol {a}} \left[\left(- \xi - u + \boldsymbol {v} ^ {\top} S \boldsymbol {v}\right) ^ {- 1} \right]\rightarrow \left(- \xi - u + \boldsymbol {v} ^ {\top} S \boldsymbol {v}\right) ^ {- 1} \\ \rightarrow \Big (- \xi - r \gamma_ {2} m _ {1, n} + \left(a _ {1} \sqrt {\gamma_ {2} ^ {2} / \gamma_ {1}} m _ {1, n}\right) ^ {2} \left[\begin{array}{c c}\gamma_ {1} ^ {- 1} \gamma_ {2} m _ {1, n} (\xi , \rho , \tau)&1 / a _ {1}\\1 / a _ {1}&m _ {2, n} (\xi , \rho , \tau) - \tau / a ^ {2}\end{array}\right] _ {[ 1, 1 ]} ^ {- 1} \Big) ^ {- 1} \\ = \left(- \xi - r \gamma_ {2} m _ {1, n} + \frac {\gamma_ {2} a _ {1} ^ {2} m _ {1 , n} ^ {2} \left(a _ {1} ^ {2} m _ {2 , n} - \tau\right)}{m _ {1 , n} \left(a _ {1} ^ {2} m _ {2 , n} - \tau\right) - \gamma_ {1} \gamma_ {2} ^ {- 1}}\right) ^ {- 1}. \tag {83} \\ \end{array}
+$$
+
+Similarly we can calculate  $m_{1,n}(\xi, \rho, \tau)$  as
+
+$$
+\begin{array}{l} m _ {1, n} (\xi , \rho , \tau) \rightarrow \\ \left(- \xi - \rho - \gamma_ {1} ^ {- 1} \gamma_ {2} \tau^ {2} m _ {1, n} - r m _ {2, n} + \frac {\tau^ {2} \gamma_ {1} ^ {- 1} \gamma_ {2} m _ {1 , n} ^ {2} \left(a _ {1} ^ {2} m _ {2} - \tau\right) - 2 \tau a _ {1} ^ {2} m _ {1 , n} m _ {2 , n} + a _ {1} ^ {4} m _ {1 , n} m _ {2 , n} ^ {2}}{m _ {1 , n} \left(a _ {1} ^ {2} m _ {2 , n} - \tau\right) - \gamma_ {1} \gamma_ {2} ^ {- 1}}\right) ^ {- 1}. \tag {84} \\ \end{array}
+$$
+
+Uniqueness in (84)(83) follows from (Hastie et al., 2019, Sec B.1) and is omitted.
+
+# C.6 PROOF OF COROLLARY 5
+
+In this section we take the limit  $\gamma_1\to \infty$  . In this case (8), (9) simplify to
+
+$$
+m _ {2} = \left(- \xi - r \gamma_ {2} m _ {1}\right) ^ {- 1}, \tag {85}
+$$
+
+$$
+m _ {1} = \left(- \xi - \rho - \gamma_ {2} \tau^ {2} m _ {1} - r m _ {2}\right) ^ {- 1}. \tag {86}
+$$
+
+Recall that  $m(\xi, \rho, \tau) = \gamma_2 m_1(\xi, \rho, \tau) + m_2(\xi, \rho, \tau)$ . By taking the derivative we have
+
+$$
+\begin{array}{l} - q (\xi) = \left. \frac {\partial}{\partial x} m (\xi , r x, t x) \right| _ {x = 0} = \left. r \frac {\partial}{\partial \rho} m (\xi , \rho , 0) \right| _ {\rho = 0} + \left. t \frac {\partial}{\partial \tau} m (\xi , 0, \tau) \right| _ {\tau = 0} \\ = r \gamma_ {2} \frac {\partial}{\partial \rho} m _ {1} (\xi , \rho , 0) \Big | _ {\rho = 0} + r \frac {\partial}{\partial \rho} m _ {2} (\xi , \rho , 0) \Big | _ {\rho = 0} + t \gamma_ {2} \frac {\partial}{\partial \tau} m _ {1} (\xi , 0, \tau) \Big | _ {\tau = 0} + t \frac {\partial}{\partial \tau} m _ {2} (\xi , 0, \tau) \Big | _ {\tau = 0}. \tag {87} \\ \end{array}
+$$
+
+Observe that (85), (86) constitutes a set of implicit functions. Thus differentiating the two functions with respect to  $\tau, \rho$  and then substitute by  $\rho = \tau = 0$  gives
+
+$$
+q (\xi) = \frac {\left(r \left(\gamma_ {2} - 1\right) - \xi^ {2}\right) \left(\sqrt {\left(r \left(\gamma_ {2} - 1\right) + \xi^ {2}\right) ^ {2} - 4 r \gamma_ {2} \xi^ {2}} + r \left(\gamma_ {2} - 1\right) + \xi^ {2}\right)}{2 \xi^ {2} \sqrt {\left(r \left(\gamma_ {2} - 1\right) + \xi^ {2}\right) ^ {2} - 4 r \gamma_ {2} \xi^ {2}}}. \tag {88}
+$$
+
+Hence by (62) we obtain the asymptotic variance:
+
+$$
+V _ {(\gamma_ {1} \rightarrow \infty)} = \lim  _ {\xi \rightarrow 0} q _ {+} (\xi) = \lim  _ {\xi \rightarrow 0} \left(q (\xi) - \frac {\gamma_ {2} - 1}{\xi^ {2}}\right) = \frac {1}{\gamma_ {2} - 1}. \tag {89}
+$$
+
+Combining the case where  $\gamma_{2} < 1$  in Theorem 4 completes the proof.
+
+![](images/2a709d7317a36c41b1e568f59eadb81048ab61c35a012876e922958e2784eefa.jpg)
+
+Remark. For  $\phi(x) = \mathrm{ReLU}(x)$ ,  $c_1 = 1/2 - 1/(2\pi)$ ,  $c_2 = 1/4$ . For  $\phi(x) = \mathrm{SoftPlus}(x) = \log(1 + e^x)$ , numerical integration yields  $c_1 \approx 0.2715$ ,  $c_2 = 1/4$ .
+
+# C.7 PROOF OF COROLLARY 6
+
+# C.7.1 UNBOUNDEDBIASWHEN  $h = n$
+
+From the bias-variance decomposition (7), the bias  $B$  is written as  $B = \frac{r^2}{d} \operatorname{tr}(Q_1 + Q_2 + I_d)$ , where
+
+$$
+Q _ {1} = X \left[ \phi \left(W ^ {\top} X\right) \right] ^ {\dagger} K _ {W} \left[ \phi \left(X ^ {\top} W\right) \right] ^ {\dagger} X ^ {\top}; \quad Q _ {2} = X \left[ \phi \left(W ^ {\top} X\right) \right] ^ {\dagger} W ^ {\top}. \tag {90}
+$$
+
+When  $h = n$ , due to the nonlinearity of  $\phi$ , we have  $\phi(W^{\top}X)$  is full rank a.s., and hence  $[\phi(X^{\top}W)]^{\dagger} = [\phi(X^{\top}W)]^{-1}$ . We have the following bound for  $Q_{1}$
+
+$$
+\begin{array}{l} \frac {1}{d} \operatorname {t r} \left(Q _ {1}\right) = \frac {1}{d} \operatorname {t r} \left(X [ \phi (W ^ {\top} X) ] ^ {\dagger} K _ {W} [ \phi (X ^ {\top} W) ] ^ {\dagger} X ^ {\top}\right) = \frac {1}{d} \operatorname {t r} \left(K _ {W} [ \phi (X ^ {\top} W) ] ^ {- 1} X ^ {\top} X [ \phi (W ^ {\top} X) ] ^ {- 1}\right) \\ \geq \frac {1}{d} \lambda_ {\min} (K _ {W}) \mathrm {t r} \left([ \phi (X ^ {\top} W) ] ^ {- 1} X ^ {\top} X [ \phi (W ^ {\top} X) ] ^ {- 1}\right) = \frac {\lambda_ {\min} (K _ {W})}{n} \mathrm {t r} \left((S S ^ {\top}) ^ {- 1} \cdot \frac {1}{d} X ^ {\top} X\right). \\ \end{array}
+$$
+
+Since  $W$  and  $X / \sqrt{d}$  are  $\mathbb{R}^{d\times n} = \mathbb{R}^{d\times p}$  follows the same distribution where each entry i.i.d.  $\mathcal{N}(0,1 / d)$ ,
+
+$$
+\begin{array}{l} \frac {1}{d \lambda_ {\min } (K _ {W})} \operatorname {t r} \left(Q _ {1}\right) \geq \frac {1}{n} \operatorname {t r} \left((S S ^ {\top}) ^ {- 1} \cdot \frac {1}{d} X ^ {\top} X\right) \sim \frac {1}{n} \operatorname {t r} \left(\left(S ^ {\top} S\right) ^ {- 1} \cdot W ^ {\top} W\right) \\ = \frac {1}{n} \operatorname {t r} \left(\left(S ^ {\top} S\right) ^ {- 1} \cdot (I + Q)\right) = \lim  _ {\xi \rightarrow 0} - \frac {\partial}{\partial x} \tilde {m} _ {n} (\xi , x, 0, x) \rightarrow \infty , \tag {91} \\ \end{array}
+$$
+
+where in Section C.5 we have showed (91) is unbounded when  $n \to \infty$ . Moreover, by (154) and Weyl's theorem we have  $\lambda_{\min}(K_W) = O(1)$ , and thus  $d^{-1} \operatorname{tr}(Q_1)$  is unbounded. For  $d^{-1} \operatorname{tr}(Q_2)$ ,
+
+$$
+\frac {1}{d} \operatorname {t r} \left(Q _ {2}\right) = \frac {1}{d} \operatorname {t r} \left(W ^ {\top} X [ \phi (W ^ {\top} X) ] ^ {- 1}\right) \leq \frac {1}{d} \lambda_ {\max } (\phi (W ^ {\top} X) ^ {- 1}) \operatorname {t r} \left(W ^ {\top} X\right) = O (1). \tag {92}
+$$
+
+To sum up, for  $n\to \infty$  and  $\gamma_{2}\rightarrow 1$  we have  $B\rightarrow \infty$
+
+# C.7.2 BOUNDEDBIASWHEN  $\gamma_{2} > 1$
+
+Since  $h > n$ , the two terms in the expression of the bias can be written as
+
+$$
+Q _ {1} = X \left(\phi \left(X ^ {\top} W\right) \phi \left(W ^ {\top} X\right)\right) ^ {- 1} \phi \left(X ^ {\top} W\right) K _ {W} \phi \left(W ^ {\top} X\right) \left(\phi \left(X ^ {\top} W\right) \phi \left(W ^ {\top} X\right)\right) ^ {- 1} X ^ {\top}, \tag {93}
+$$
+
+$$
+Q _ {2} = X \left(\phi \left(X ^ {\top} W\right) \phi \left(W ^ {\top} X\right)\right) ^ {- 1} \phi \left(X ^ {\top} W\right) W ^ {\top}. \tag {94}
+$$
+
+Therefore we have
+
+$$
+\frac {2}{d} \operatorname {t r} \left(Q _ {1}\right) = 2 \operatorname {t r} \left(\frac {X ^ {\top} X}{d} \left(\phi (X ^ {\top} W) \phi (W ^ {\top} X)\right) ^ {- 1} \phi (X ^ {\top} W) K _ {W} \phi (W ^ {\top} X) \left(\phi (X ^ {\top} W) \phi (W ^ {\top} X)\right) ^ {- 1}\right)
+$$
+
+$$
+\begin{array}{l} \leq 2 \lambda_ {\max } \left(\frac {X ^ {\top} X}{d}\right) \operatorname {t r} \left(\left(\phi (X ^ {\top} W) \phi (W ^ {\top} X)\right) ^ {- 1} \phi (X ^ {\top} W) K _ {W} \phi (W ^ {\top} X) \left(\phi (X ^ {\top} W) \phi (W ^ {\top} X)\right) ^ {- 1}\right) \\ = O (1) \cdot \operatorname {t r} \left(\phi \left(W ^ {\top} X\right) \left(\phi \left(X ^ {\top} W\right) \phi \left(W ^ {\top} X\right)\right) ^ {- 2} \phi \left(X ^ {\top} W\right) K _ {W}\right) \\ \leq O (1) \cdot \lambda_ {\max } \left(\phi \left(W ^ {\top} X\right) \left(\phi \left(X ^ {\top} W\right) \phi \left(W ^ {\top} X\right)\right) ^ {- 2} \phi \left(X ^ {\top} W\right)\right) \cdot \operatorname {t r} \left(K _ {W}\right) \\ = O (1) \cdot \sigma_ {\min } ^ {- 2} \left(\phi \left(X ^ {\top} W\right)\right) \operatorname {t r} \left(K _ {W}\right) = O (1) \cdot O \left(n ^ {- 1}\right) \cdot O (n) = O (1), \tag {95} \\ \end{array}
+$$
+
+in which we used Lemma 11 and  $\operatorname{tr}(AB) \leq \lambda_{\max}(A)\operatorname{tr}(B)$  for positive semi-definite  $A, B$ . Similarly
+
+$$
+\begin{array}{l} \frac {2}{d} \operatorname {t r} \left(Q _ {2}\right) = 2 \operatorname {t r} \left(\left(\phi \left(X ^ {\top} W\right) \phi \left(W ^ {\top} X\right)\right) ^ {- 1} \phi \left(X ^ {\top} W\right) \cdot \frac {1}{d} W ^ {\top} X\right) \\ \leq \operatorname {t r} \left(\left(\left(\phi (X ^ {\top} W) \phi (W ^ {\top} X)\right) ^ {- 1} \phi (X ^ {\top} W)\right) (\dots) ^ {\top}\right) + \operatorname {t r} \left(d ^ {- 2} X ^ {\top} W W ^ {\top} X\right) \\ \leq n \cdot \sigma_ {\min } \left(\phi \left(X ^ {\top} W\right)\right) ^ {- 2} + \frac {1}{d} \lambda_ {\max } \left(\frac {1}{d} X X ^ {\top}\right) \operatorname {t r} \left(W W ^ {\top}\right) = O (1), \tag {96} \\ \end{array}
+$$
+
+where we applied  $\operatorname{tr}(AB) \leq (\operatorname{tr}\left(A^\top A\right) + \operatorname{tr}\left(B^\top B\right)) / 2$ . We therefore conclude that  $B$  is bounded when  $h > n$  and  $h, n \to \infty$ .
+
+Remark. Concurrent to this work, Mei and Montanari (2019) provides a complete characterization of the bias term and confirms our observations above.
+
+# C.8 PROOF OF THEOREM 7
+
+For simplicity we assume  $n, d, h$  to be even and let  $d_0 = d/2$ ,  $n_0 = n/2$  and  $h_0 = h/2$ . Since the second layer is fixed  $a_i \sim \mathrm{Unif}\{-1/\sqrt{h}, 1/\sqrt{h}\}$ , we let  $a_i = 1/\sqrt{h}$  and  $a_{i+h_0} = -1/\sqrt{h}$  for all  $1 \leq i \leq h_0$ . We therefore write  $\pmb{a}^{\top} = h^{-1/2}[\mathbf{1}_{h_0}, -\mathbf{1}_{h_0}]^{\top}$  and  $W = [W_+, W_-]$ :
+
+$$
+f (\boldsymbol {x}; W _ {-}, W _ {+}) = \boldsymbol {a} ^ {\top} \phi (W ^ {\top} \boldsymbol {x}) = \frac {1}{\sqrt {h}} \mathbf {1} ^ {\top} \phi \left(W _ {+} ^ {\top} \boldsymbol {x}\right) - \frac {1}{\sqrt {h}} \mathbf {1} ^ {\top} \phi \left(W _ {-} ^ {\top} \boldsymbol {x}\right). \tag {97}
+$$
+
+The empirical risk can thus be written as
+
+$$
+L (X; W _ {+}, W _ {-}) = \frac {1}{n} \sum_ {\boldsymbol {x} \in X} L (\boldsymbol {x}; W _ {+}, W _ {-}) = \frac {1}{n} \sum_ {\boldsymbol {x} \in X} \left[ y - \frac {1}{\sqrt {h}} \mathbf {1} ^ {\top} \phi \left(W _ {+} ^ {\top} \boldsymbol {x}\right) + \frac {1}{\sqrt {h}} \mathbf {1} ^ {\top} \phi \left(W _ {-} ^ {\top} \boldsymbol {x}\right) \right] ^ {2}. \tag {98}
+$$
+
+# C.8.1 DEFINING GRADIENT FLOWS
+
+In this section we define three gradient flows and show that the three flows are similar in some sense. GF-Original is the original gradient flow (11), i.e.
+
+$$
+\begin{array}{l} \frac {\partial W _ {+} ^ {O}}{\partial t} = \frac {1}{2 n _ {0}} \sum_ {i = 1} ^ {2 n _ {0}} \left[ \frac {1}{\sqrt {h}} \left(y _ {i} - \frac {1}{\sqrt {h}} \mathbf {1} ^ {\top} \phi \left(W _ {+} ^ {O \top} \boldsymbol {x} _ {i}\right) + \frac {1}{\sqrt {h}} \mathbf {1} ^ {\top} \phi \left(W _ {-} ^ {O \top} \boldsymbol {x} _ {i}\right)\right) \boldsymbol {x} _ {i} \phi^ {\prime} \left(\boldsymbol {x} _ {i} ^ {\top} W _ {+} ^ {O}\right) \right] (99) \\ = \frac {1}{2 n _ {0}} \left[ X \left(\boldsymbol {y} - \boldsymbol {y} ^ {O} (t)\right) \frac {1}{\sqrt {h}} \mathbf {1} ^ {\top} \circ \phi^ {\prime} \left(X W _ {+} ^ {O}\right) \right], (100) \\ \end{array}
+$$
+
+starting from the vanishing initialization  $\boldsymbol{w}_i^O(0) \sim \mathcal{N}(\mathbf{0}, I / dh^{1 + \epsilon})$ . Note that the gradient for the negative part  $W_-^O$  can be similarly defined.
+
+We now define the flow under the same objective but from exact zero initialization  $\pmb{w}_i^D (0) = \mathbf{0}$  termed GF-Double. Due to zero initialization, a basic observation is that the solution  $[W_{+}^{D},W_{-}^{D}]$  is
+
+at most rank-2, and more precisely, the parameters in the flow takes the form of  $W_{\pm}^{D}(t) = \pmb{w}_{\pm}^{D}(t)\mathbf{1}^{\top}$  where  $\pmb{w}_{\pm}^{D}(t)$  admits the following dynamics:
+
+$$
+\frac {\partial \boldsymbol {w} _ {+} ^ {D}}{\partial t} = \boldsymbol {g} _ {+} ^ {D} \left(\boldsymbol {w} _ {+} ^ {D}\right) = \frac {1}{2 n _ {0}} \sum_ {i = 1} ^ {2 n _ {0}} \left[ \frac {1}{\sqrt {h}} \left(y _ {i} - \sqrt {h} \phi \left(\boldsymbol {w} _ {+} ^ {D \top} \boldsymbol {x} _ {i}\right) + \sqrt {h} \phi \left(\boldsymbol {w} _ {-} ^ {D \top} \boldsymbol {x} _ {i}\right)\right) \phi^ {\prime} \left(\boldsymbol {w} _ {+} ^ {D \top} \boldsymbol {x} _ {i}\right) \boldsymbol {x} _ {i} \right]. \tag {101}
+$$
+
+Lastly, we define the  $GF$ -Single with solution denoted as  $\boldsymbol{w}_{\pm} = \boldsymbol{w}_{\pm}^{S}(t)$ :
+
+$$
+\frac {\partial \boldsymbol {w} _ {+} ^ {S}}{\partial t} = \boldsymbol {g} _ {+} ^ {S} \left(\boldsymbol {w} _ {+} ^ {S}\right) = \frac {1}{2 n _ {0}} \sum_ {i = 1} ^ {2 n _ {0}} \left[ \frac {1}{\sqrt {h}} \left(y _ {i} - \sqrt {h} \phi^ {\prime} (0) \boldsymbol {w} _ {+} ^ {S \top} \boldsymbol {x} _ {i} + \sqrt {h} \phi^ {\prime} (0) \boldsymbol {w} _ {-} ^ {1 \top} \boldsymbol {x} _ {i}\right) \phi^ {\prime} (0) \boldsymbol {x} _ {i} \right]. \tag {102}
+$$
+
+from zero initialization  $\pmb{w}_{\pm}^{D}(0) = \mathbf{0}$ . This can be seen as replacing the nonlinearity  $\phi$  with its first-order Taylor expansion at the origin.
+
+# C.8.2 FROM GF-DOUBLE TO GF-SINGLE
+
+Step 1. Solution of GF-single. Among the three flows defined above, only GF-single an explicit form at any time  $t$ . Specifically, the solution can be written as:
+
+$$
+\boldsymbol {w} _ {+} ^ {S} (t) = - \boldsymbol {w} _ {-} ^ {S} (t) = \frac {1}{2 \sqrt {h}} \left(I - e ^ {- \frac {\phi^ {\prime} (0) ^ {2}}{n _ {0}} X X ^ {\top} t}\right) \left(X X ^ {\top}\right) ^ {- 1} X \boldsymbol {y}, \tag {103}
+$$
+
+when  $d < n$ , or otherwise
+
+$$
+\boldsymbol {w} _ {+} ^ {S} (t) = - \boldsymbol {w} _ {-} ^ {S} (t) = \frac {1}{2 \sqrt {h}} X \left(I - e ^ {- \frac {\phi^ {\prime} (0) ^ {2}}{n _ {0}} X ^ {\top} X t}\right) \left(X ^ {\top} X\right) ^ {- 1} \boldsymbol {y}. \tag {104}
+$$
+
+when  $d > n$ . For simplicity we elaborate the proof only for  $d < n$ . We provide a condition under which the difference between the trajectories can be controlled, and we first that this condition holds for the linearized flow GF-single for all  $t$  in Lemma 17.
+
+$$
+\textbf {C o n d i t i o n A :} \| \boldsymbol {w} (t) \| _ {2} = O \left(\frac {1}{\sqrt {d}}\right); \| X ^ {\top} \boldsymbol {w} (t) \| _ {\infty} = O \left(\frac {\operatorname {p o l y} \log d}{\sqrt {d}}\right).
+$$
+
+Step 2. Bounding the Difference in Gradient Flow Trajectory. Due to low rank property of GF-single and GF-double, in this subsection we slightly abuse the notation and define
+
+$$
+f (\boldsymbol {x}; \boldsymbol {w} _ {\pm}) = f (\boldsymbol {x}; \boldsymbol {w} _ {+} \mathbf {1} ^ {\top}, \boldsymbol {w} _ {-} \mathbf {1} ^ {\top}) = \sqrt {h} \phi \left(\boldsymbol {w} _ {+} ^ {\top} \boldsymbol {x}\right) - \sqrt {h} \phi \left(\boldsymbol {w} _ {-} ^ {\top} \boldsymbol {x}\right). \tag {105}
+$$
+
+We now show that the difference between the two trajectories defined above is asymptotically vanishing for  $\pmb{w}_{+}$  ( $\pmb{w}_{-}$  follows the same argument). Compare the two trajectories up to time  $T$
+
+$$
+\begin{array}{l} \left\| \boldsymbol {w} _ {+} ^ {D} (T) - \boldsymbol {w} _ {+} ^ {S} (T) \right\| _ {2} = \left\| \int_ {0} ^ {T} \boldsymbol {g} _ {+} ^ {D} (\boldsymbol {w} _ {+} ^ {D} (t)) - \boldsymbol {g} _ {+} ^ {S} (\boldsymbol {w} _ {+} ^ {S} (t)) d t \right\| _ {2} \\ \leq \left\| \int_ {0} ^ {T} \boldsymbol {g} _ {+} ^ {S} (\boldsymbol {w} _ {+} ^ {S} (t)) - \boldsymbol {g} _ {+} ^ {S} (\boldsymbol {w} _ {+} ^ {D} (t)) \mathrm {d} t \right\| _ {2} + \left\| \int_ {0} ^ {T} \boldsymbol {g} _ {+} ^ {D} (\boldsymbol {w} _ {+} ^ {D} (t)) - \boldsymbol {g} _ {+} ^ {S} (\boldsymbol {w} _ {+} ^ {D} (t)) \mathrm {d} t \right\| _ {2} \\ = \left\| \int_ {0} ^ {T} \frac {1}{2 n _ {0}} \sum_ {i = 1} ^ {2 n _ {0}} \left[ \left(- \phi^ {\prime} (0) \left(\boldsymbol {w} _ {+} ^ {D} (t) - \boldsymbol {w} _ {+} ^ {S} (t)\right) ^ {\top} \boldsymbol {x} _ {i} + \phi^ {\prime} (0) \left(\boldsymbol {w} _ {-} ^ {D} (t) - \boldsymbol {w} _ {-} ^ {S} (t)\right) ^ {\top} \boldsymbol {x} _ {i}\right) \phi^ {\prime} (0) \boldsymbol {x} _ {i} \right] \mathrm {d} t \right\| _ {2} + E _ {+} \\ = O (1) \int_ {0} ^ {T} \left(\left\| \boldsymbol {w} _ {+} ^ {D} (t) - \boldsymbol {w} _ {+} ^ {S} (t) \right\| _ {2} + \left\| \boldsymbol {w} _ {-} ^ {D} (t) - \boldsymbol {w} _ {-} ^ {S} (t) \right\| _ {2}\right) d t + E _ {+}, \tag {106} \\ \end{array}
+$$
+
+where we have defined the error term as
+
+$$
+E _ {+} = \left\| \int_ {0} ^ {T} \boldsymbol {g} _ {+} ^ {D} \left(\boldsymbol {w} _ {+} ^ {D} (s)\right) - \boldsymbol {g} _ {+} ^ {S} \left(\boldsymbol {w} _ {+} ^ {D} (s)\right) \mathrm {d} t \right\| _ {2}. \tag {107}
+$$
+
+To bound the error term, we note that at  $t = 0$  Condition A holds for GF-double. Assume that for some  $0 \leq t \leq T$ , Condition A also holds for GF-double, we have
+
+$$
+E _ {+} = \left\| \int_ {0} ^ {T} \boldsymbol {g} _ {+} ^ {D} \left(\boldsymbol {w} _ {+} ^ {D} (t)\right) - \boldsymbol {g} _ {+} ^ {S} \left(\boldsymbol {w} _ {+} ^ {D} (t)\right) \mathrm {d} t \right\| _ {2}
+$$
+
+$$
+\leq \left\| \int_ {0} ^ {T} \frac {1}{n} \sum_ {i = 1} ^ {n} \left[ \frac {1}{\sqrt {h}} \left(y _ {i} - \sqrt {h} \phi \left(\boldsymbol {w} _ {+} ^ {D \top} \boldsymbol {x} _ {i}\right) + \sqrt {h} \phi \left(\boldsymbol {w} _ {-} ^ {D \top} \boldsymbol {x} _ {i}\right)\right) \left(\phi^ {\prime} \left(\boldsymbol {w} _ {+} ^ {D \top} \boldsymbol {x} _ {i}\right) - \phi^ {\prime} (0)\right) \boldsymbol {x} _ {i} \right] \mathrm {d} t \right\| _ {2}
+$$
+
+$$
++ \left\| \int_ {0} ^ {T} \frac {1}{n} \sum_ {i = 1} ^ {n} \left[ \left(- \phi (\boldsymbol {w} _ {+} ^ {D \top} \boldsymbol {x} _ {i}) + \phi^ {\prime} (0) \boldsymbol {w} _ {+} ^ {D \top} \boldsymbol {x} _ {i} + \phi (0) + \phi (\boldsymbol {w} _ {-} ^ {D \top} \boldsymbol {x} _ {i}) - \phi^ {\prime} (0) \boldsymbol {w} _ {-} ^ {D \top} - \phi (0)\right) \phi^ {\prime} (0) \boldsymbol {x} _ {i} \right] d t \right\| _ {2}
+$$
+
+$$
+\stackrel {(i)} {\leq} \left\| \int_ {0} ^ {T} \frac {1}{n \sqrt {h}} X (\boldsymbol {y} - \boldsymbol {y} ^ {D} (t)) \circ \left(\phi^ {\prime} \left(X ^ {\top} \boldsymbol {w} _ {+} ^ {D}\right) - \phi^ {\prime} (0) \mathbf {1}\right) d t \right\| _ {2}
+$$
+
+$$
++ \left\| \int_ {0} ^ {T} \frac {1}{n} \sum_ {i = 1} ^ {n} \left[ \left(O \left(\boldsymbol {w} _ {+} ^ {D \top} \boldsymbol {x} _ {i}\right) ^ {2} - O \left(\boldsymbol {w} _ {-} ^ {D \top} \boldsymbol {x} _ {i}\right) ^ {2}\right) \boldsymbol {x} _ {i} \right] \mathrm {d} t \right\| _ {2}
+$$
+
+$$
+\stackrel {(i)} {\leq} \frac {1}{n \sqrt {h}} \int_ {0} ^ {T} \| X \| _ {2} \left\| \boldsymbol {y} - \boldsymbol {y} ^ {D} (t) \right\| _ {\infty} \left\| O \left(\phi^ {\prime} \left(X ^ {\top} \boldsymbol {w} _ {+} ^ {D}\right)\right) \right\| _ {2} d t + O \left(n ^ {- 1}\right) \int_ {0} ^ {T} \| X \| _ {2} \sqrt {n} \left\| X ^ {\top} \boldsymbol {w} _ {\pm} ^ {D} \right\| _ {\infty} ^ {2} d t
+$$
+
+$$
+\leq \frac {1}{n \sqrt {h}} \int_ {0} ^ {T} O (\sqrt {d}) O \left(\log^ {c} d\right) O (1) \mathrm {d} t + \frac {1}{\sqrt {n}} \int_ {0} ^ {T} O (\sqrt {d}) O \left(\frac {\log^ {c} d}{d}\right) \mathrm {d} t = O \left(\frac {\log^ {c} d}{d}\right) T. \tag {108}
+$$
+
+where (i) follows from the error of Taylor expansion on  $\phi$  and (ii) from Condition A. Therefore, by Equation (106) and Gronwall's inequality,
+
+$$
+\left\| \boldsymbol {w} _ {+} ^ {D} (T) - \boldsymbol {w} _ {+} ^ {S} (T) \right\| _ {2} \leq C _ {1} \cdot \frac {\log^ {c} h}{h} e ^ {C _ {2} T} = O \left(\frac {\text {p o l y} \log h}{h}\right)\rightarrow 0. \tag {109}
+$$
+
+for  $T \in O(\log \log h)$ . This shows that up to time  $T$ , the difference between the trajectories of GF-single and GF-double vanishes under the assumption that Condition A holds for  $\boldsymbol{w}^D$  up to  $T$ . Importantly, note that  $\boldsymbol{w}^S(0) = \boldsymbol{w}^D(0) = 0$ , and that Condition A holds for  $\boldsymbol{w}^S$  for all  $T > 0$ . Therefore, by a standard contradiction argument (e.g. (Du et al., 2018, Lemma 3.4)), one can show that this closeness between the two trajectories implies that Condition A also holds for  $\boldsymbol{w}^D$  up to time  $T$ . With Equation (108) we bound the difference in population risk between the two flows.
+
+Step 3. Bounding the Difference in Risk. Now we consider the difference of the population risk  $R^S$  and  $R^D$  of two models with parameters  $\boldsymbol{w}_{\pm}^{S}$  and  $\boldsymbol{w}_{\pm}^{D}$ .
+
+$$
+\begin{array}{l} \left| 2 \left(R ^ {S} - R ^ {D}\right) \right| = \left| \mathbb {E} _ {\boldsymbol {x}} \left(\boldsymbol {x} ^ {\top} \boldsymbol {\beta} - f (\boldsymbol {x}; \boldsymbol {w} _ {\pm} ^ {S})\right) ^ {2} - \mathbb {E} _ {\boldsymbol {x}} \left(\boldsymbol {x} ^ {\top} \boldsymbol {\beta} - f (\boldsymbol {x}; \boldsymbol {w} _ {\pm} ^ {D})\right) ^ {2} \right| \\ \stackrel {(i)} {\leq} \sqrt {\mathbb {E} _ {\boldsymbol {x}} [ f (\boldsymbol {x} ; \boldsymbol {w} _ {\pm} ^ {S}) - f (\boldsymbol {x} ; \boldsymbol {w} _ {\pm} ^ {D}) ] ^ {2} \mathbb {E} _ {\boldsymbol {x}} [ \boldsymbol {\beta} ^ {\top} \boldsymbol {x} - f (\boldsymbol {x} ; \boldsymbol {w} _ {\pm} ^ {S}) + \boldsymbol {\beta} ^ {\top} \boldsymbol {x} - f (\boldsymbol {x} ; \boldsymbol {w} _ {\pm} ^ {D}) ] ^ {2}} \\ \leq \sqrt {h \mathbb {E} _ {\boldsymbol {x}} [ | \phi (\boldsymbol {w} _ {+} ^ {S \top} \boldsymbol {x}) - \phi (\boldsymbol {w} _ {+} ^ {D \top} \boldsymbol {x}) | + | \phi (\boldsymbol {w} _ {-} ^ {S \top} \boldsymbol {x}) - \phi (\boldsymbol {w} _ {-} ^ {D \top} \boldsymbol {x}) | ] ^ {2}} \\ \cdot \sqrt {\mathbb {E} _ {\boldsymbol {x}} [ | \boldsymbol {\beta} ^ {\top} \boldsymbol {x} - f (\boldsymbol {x} ; \boldsymbol {w} _ {\pm} ^ {S}) | + | \boldsymbol {\beta} ^ {\top} \boldsymbol {x} - f (\boldsymbol {x} ; \boldsymbol {w} _ {\pm} ^ {D}) | ] ^ {2}} \\ \stackrel {(i i)} {\leq} 2 \sqrt {h \mathbb {E} _ {\boldsymbol {x}} [ | \phi (\boldsymbol {w} _ {+} ^ {S \top} \boldsymbol {x}) - \phi (\boldsymbol {w} _ {+} ^ {D \top} \boldsymbol {x}) | ^ {2} + | \phi (\boldsymbol {w} _ {-} ^ {S \top} \boldsymbol {x}) - \phi (\boldsymbol {w} _ {-} ^ {D \top} \boldsymbol {x}) | ^ {2} ]} \\ \cdot \sqrt {\mathbb {E} _ {\boldsymbol {x}} [ | \boldsymbol {\beta} ^ {\top} \boldsymbol {x} - f (\boldsymbol {x} ; \boldsymbol {w} _ {\pm} ^ {S}) | ^ {2} + | \boldsymbol {\beta} ^ {\top} \boldsymbol {x} - f (\boldsymbol {x} ; \boldsymbol {w} _ {\pm} ^ {D}) | ^ {2} ]} \\ \stackrel {(i i i)} {\leq} 2 \sqrt {h \operatorname {L i p} (\phi) \mathbb {E} _ {\boldsymbol {x}} [ | \boldsymbol {w} _ {+} ^ {S \top} \boldsymbol {x} - \boldsymbol {w} _ {+} ^ {D \top} \boldsymbol {x} | ^ {2} + | \boldsymbol {w} _ {-} ^ {S \top} \boldsymbol {x} - \boldsymbol {w} _ {-} ^ {D \top} \boldsymbol {x} | ^ {2} ]} \cdot \sqrt {R ^ {S} + R ^ {D}} \\ \leq O (\sqrt {h}) \sqrt {\left\| \boldsymbol {w} _ {+} ^ {S} - \boldsymbol {w} _ {+} ^ {D} \right\| _ {2} ^ {2} + \left\| \boldsymbol {w} _ {-} ^ {S} - \boldsymbol {w} _ {-} ^ {D} \right\| _ {2} ^ {2}} \leq O \left(\frac {\text {p o l y} \log h}{\sqrt {h}}\right), \tag {110} \\ \end{array}
+$$
+
+where (i) is due to Cauchy-Schwarz inequality on norm, (ii) from Jenson's inequality on squares and Young's inequality, and (iii) from the Lipschitz assumption on the activation, and the observation
+
+that both  $R^S$  and  $R^D$  are finite for  $\gamma_1 \neq 1$  due to the justified Condition A above. Therefore the difference between  $R^S$  and  $R^D$  vanishes for  $T = O(\log \log h)$ .
+
+Step 4. Bounding the Difference from Stationarity We compute the difference in risk between the model at some finite time  $t$  and the stationary point i.e.  $t = \infty$ ,
+
+$$
+\begin{array}{l} \left| R ^ {S} (t) - R ^ {S} (\infty) \right| \leq C \sqrt {h} \cdot \left\| \boldsymbol {w} _ {+} ^ {S} (t) - \boldsymbol {w} _ {+} ^ {S} (\infty) \right\| = C \sqrt {h} \left\| \frac {1}{2 \sqrt {h}} e ^ {- \frac {\phi^ {\prime} (0) ^ {2}}{n _ {0}} X X ^ {\top} t} (X X ^ {\top}) ^ {- 1} X \boldsymbol {y} \right\| _ {2} \\ = C \left\| e ^ {- \frac {\phi^ {\prime} (0) ^ {2}}{n _ {0}} X X ^ {\top} t} \left(X X ^ {\top}\right) ^ {- 1} X X ^ {\top} \boldsymbol {\beta} \right\| _ {2} \leq C \exp \left(- \phi^ {\prime} (0) ^ {2} \left\| \frac {1}{n} X X ^ {\top} \right\| _ {2} t\right) \| \boldsymbol {\beta} \| _ {2} = C _ {3} e ^ {- C _ {4} t}. \tag {111} \\ \end{array}
+$$
+
+for constants  $C_3, C_4 > 0$ . Combining (110) and (111) yields for  $T = \log \log h$
+
+$$
+\begin{array}{l} \left| R ^ {D} (T) - R ^ {S} (\infty) \right| \leq \left| R ^ {S} (T) - R ^ {D} (T) \right| + \left| R ^ {S} (T) - R ^ {S} (\infty) \right| \\ = O \left(\frac {\text {p o l y} \log h}{\sqrt {h}}\right) + O \left(\frac {1}{\text {p o l y} \log h}\right)\rightarrow 0. \tag {112} \\ \end{array}
+$$
+
+The result in (112) for case  $d > n$  follows a similar proof and is omitted.
+
+# C.8.3 FROM GF-ORIGINAL TO GF-DOUBLE
+
+In this section we compare GF-original with GF-double. Note that the two flows differ only at initialization: vanishing initialization  $\pmb{w}_i^O (0)\sim \mathcal{N}(\mathbf{0},I / dh^{1 + \epsilon})$  v.s. zero initialization  $\pmb{w}_i^D (0) = \pmb{0}$ . Note that at initialization  $\| \pmb {y} - f(\pmb {X})\| _2 = O(\sqrt{n})$ , and gradient flow decreases the empirical risk; therefore the condition for Lemma 18 is satisfied, and by the Lipschitz condition on the empirical gradient we have
+
+$$
+\begin{array}{l} \left\| W ^ {O} (T) - W ^ {D} (T) \right\| _ {F} ^ {2} \\ = \left\| W ^ {O} (0) - W ^ {D} (0) \right\| _ {F} ^ {2} + \int_ {0} ^ {T} \frac {\partial \left\| W ^ {O} (t) - W ^ {D} (t) \right\| _ {F} ^ {2}}{\partial t} d t \\ = \left\| W ^ {O} (0) \right\| _ {F} ^ {2} + \int_ {0} ^ {T} \left\| W ^ {O} (t) - W ^ {D} (t)) \right\| _ {F} \left\| \frac {\partial L (X ; W ^ {O} (t))}{\partial W} - \frac {\partial L (X ; W ^ {D} (t))}{\partial W} \right\| _ {F} d t \\ \leq \left\| W ^ {O} (0) \right\| _ {F} ^ {2} + \int_ {0} ^ {T} \left\| W ^ {O} (t) - W ^ {D} (t) \right\| _ {F} ^ {2} + \left\| \frac {\partial L (X ; W ^ {O} (t))}{\partial W} - \frac {\partial L (X ; W ^ {D} (t))}{\partial W} \right\| _ {F} ^ {2} d t \\ \leq \left\| W ^ {O} (0) \right\| _ {F} ^ {2} + \left(1 + L _ {f}\right) \int_ {0} ^ {T} \left\| W ^ {O} (t) - W ^ {D} (t) \right\| _ {F} ^ {2} d t, \tag {113} \\ \end{array}
+$$
+
+And hence by Gronwall's lemma one obtains:
+
+$$
+\left\| W ^ {O} (T) - W ^ {D} (T) \right\| _ {F} \leq \left\| W ^ {O} (0) \right\| _ {F} e ^ {(1 + L _ {f}) T / 2} = O \left(d ^ {- (1 + \epsilon) / 2}\right) e ^ {C T}. \tag {114}
+$$
+
+Therefore the difference in the function output can be bounded as
+
+$$
+\begin{array}{l} \left\| f (\boldsymbol {x}, W ^ {D} (T)) - f (\boldsymbol {x}, W ^ {O} (T)) \right\| _ {2} = \left\| \phi (\boldsymbol {x} ^ {\top} W ^ {D} (T)) \boldsymbol {a} - \phi (\boldsymbol {x} ^ {\top} W ^ {O} (T)) \boldsymbol {a} \right\| _ {2} \\ \leq \left\| \phi (\boldsymbol {x} ^ {\top} W ^ {D} (T)) - \phi (\boldsymbol {x} ^ {\top} W ^ {O} (T)) \right\| _ {2} \| \boldsymbol {a} \| _ {2} \leq L _ {\phi} \| \boldsymbol {x} \| _ {2} \left\| W ^ {D} (T) - W ^ {O} (T) \right\| _ {F} \| \boldsymbol {a} \| _ {2} \\ = O (\sqrt {d}) \cdot \left\| W ^ {D} (T) - W ^ {O} (T) \right\| _ {F} = O \left(d ^ {- \epsilon / 2}\right) e ^ {C T}. \tag {115} \\ \end{array}
+$$
+
+Taking  $T = \log \log h$ , together with the same argument in Step 1 yields
+
+$$
+\left| R ^ {O} (T) - R ^ {D} (T) \right| = O \left(\frac {\text {p o l y} \log h}{d ^ {\epsilon / 2}}\right)\rightarrow 0. \tag {116}
+$$
+
+# C.8.4 PUTTING THINGS TOGETHER
+
+By (112) and (116), we know that for  $T = \log \log h$
+
+$$
+\left| R ^ {O} (T) - R ^ {S} (\infty) \right| \leq \left| R ^ {O} (T) - R ^ {D} (T) \right| + \left| R ^ {D} (T) - R ^ {S} (\infty) \right|\rightarrow 0. \tag {117}
+$$
+
+Finally the proof is completed by observing that  $R^S(\infty)$  is the risk of the minimum-norm solution on the input discussed in Section 3. In addition, note that at  $T = \log \log h$ , from (109)(114) one obtains that  $\left\| W^O(t) - W^S(t) \right\|_F \to 0$ , and therefore by Lemma 18 we have  $\left\| \partial L(X; W^O(t)) / \partial W \right\|_F \to 0$ , i.e. the flow on the original objective reaches a  $o(1)$  first-order stationary point.
+
+# C.9 PROOF OF THEOREM 8
+
+Denote  $\omega = \operatorname{vec}(W) = \operatorname{vec}([W_{+}, W_{-}])$ , and  $\omega_0 = \operatorname{vec}(W^{\mathrm{init}})$ . Define
+
+$$
+K (t) = \frac {\partial \boldsymbol {f} (X ; \boldsymbol {\omega} (t))}{\partial \boldsymbol {\omega} (t)} ^ {\top} \frac {\partial \boldsymbol {f} (X ; \boldsymbol {\omega} (t))}{\partial \boldsymbol {\omega} (t)}, \tag {118}
+$$
+
+which is the kernel matrix of the neural tangent kernel Jacot et al. (2018); Du et al. (2018). In the following sections we show that under the non-vanishing initialization, the trained two-layer network is well-approximated by the regression model on the NTK, for which we derive the population risk.
+
+# C.9.1 THE KERNEL LINEARIZATION
+
+Write  $\pmb{y}_{NN}(t) = f_{NN}(X,t)\in \mathbb{R}^n$  and its evolution:
+
+$$
+\mathrm {d} \boldsymbol {y} _ {N N} (t) = \frac {1}{n} K (t) (\boldsymbol {y} - \boldsymbol {y} _ {N N} (t)) \mathrm {d} t, \tag {119}
+$$
+
+and the corresponding linearized flow:
+
+$$
+\mathrm {d} \boldsymbol {y} _ {N T K} (t) = \frac {1}{n} K (0) (\boldsymbol {y} - \boldsymbol {y} _ {N T K} (t)) \mathrm {d} t, \tag {120}
+$$
+
+Previous works (e.g. Du et al. (2018); Oymak and Soltanolkotabi (2019)) have proved (non-asymptotically) that the two trajectories (119) and (120) are close if the model is overparameterized, i.e.  $h = \mathrm{poly}(n)$ , under no assumptions on the teacher model. In our asymptotic setup (together with assumptions (A1-3)), we argue that similar conclusion holds without significant overparameterization.
+
+We first show the global convergence of the training of two-layer neural network  $f_{NN}$ . We employ an argument similar to (Du et al., 2018, Theo. 3.2) by first identifying the condition under which training converges at linear rate:
+
+$$
+\text {C o n d i t i o n} \quad \lambda_ {\min } (K (t)) = O (d).
+$$
+
+From Corollary 15 we know that at initialization the lowest eigenvalue of the NTK matrix satisfies  $\lambda_{\mathrm{min}}(K(0)) = O(d)$ , and thus the kernel regression on the NTK enjoys linear convergence. By Lemma 19, we know that for  $\| W(t) - W(0)\| _2 = O(d^{1 - \epsilon})$ , the order of  $\lambda_{\mathrm{min}}(K(t)) = O(d)$  remains unchanged. Therefore, if we assume that up to time  $T$  the weights satisfy  $\| \pmb {w}_i(t) - \pmb {w}_i(0)\| _2 = O(d^{-1 / 2})$ , then Condition B is satisfied for  $\pmb{y}_{NN}$  and from (Chizat and Bach, 2018b, Lemma B1) we have the following linear convergence
+
+$$
+\left\| \boldsymbol {y} _ {N N} (t) - \boldsymbol {y} \right\| _ {2} \leq C _ {1} \left\| \boldsymbol {y} _ {N N} (0) - \boldsymbol {y} \right\| _ {2} e ^ {- C _ {2} t}. \tag {121}
+$$
+
+Since  $\| \pmb{y}_{NN}(0) - \pmb{y}\| _2 = O(\sqrt{n})$  at initialization, setting  $T = O(\log d)$  ensures that the training loss  $\frac{1}{n}\| \pmb{y}_{NN}(T) - \pmb{y}\| _2^2\to 0$  as  $n\to \infty$ . Consequently it is easy to check that  $\left\| \frac{\partial L(X;W(T))}{\partial W(T)}\right\| _2\to 0$  and thus at time T the gradient flow reaches an o(1) first order stationary point.
+
+We now verify that each weight vector  $\pmb{w}_i$  travels at most  $O(d^{-1/2})$  from initialization. The norm of gradient for  $\pmb{w}_i$  can be bounded as:
+
+$$
+\begin{array}{l} \left\| \frac {\partial L (X ; \boldsymbol {w} _ {i} (t))}{\partial \boldsymbol {w} _ {i} (t)} \right\| _ {2} = \left\| \frac {1}{n} X \left[ \frac {1}{\sqrt {h}} (\boldsymbol {y} - \boldsymbol {y} _ {N N} (t)) \circ \phi^ {\prime} (X ^ {\top} \boldsymbol {w} _ {i} (t)) \right] \right\| _ {2} \\ \stackrel {(i)} {\leq} O (1) \frac {1}{d ^ {1 . 5}} \| X \| _ {2} \| \boldsymbol {y} - \boldsymbol {y} _ {N N} (t) \| _ {2} \leq O (1) d ^ {- 1} \| \boldsymbol {y} - \boldsymbol {y} _ {N N} (0) \| _ {2} e ^ {- t}, \tag {122} \\ \end{array}
+$$
+
+where (i) follows from the boundedness of  $\phi'$ . Integrating the gradient yields  $\| \pmb{w}_i(t) - \pmb{w}_i(0) \|_2 = O(d^{-1/2})$ , i.e.  $\| W(t) - W(0) \|_2 = O(1)$ . Thus the distance traveled by  $W$  indeed satisfies the norm
+
+assumption above, and following the same argument as (Du et al., 2018, Lemma 3.4) we conclude that Condition B holds true for the gradient flow of  $f_{NN}$  for  $t > 0$ .
+
+Next we show that for any input  $\hat{\pmb{x}}$  with  $\| \hat{\pmb{x}}\| _2 = O(\sqrt{d})$ , the prediction of the neural network is uniformly close to the prediction of the prediction of the kernel model, a result similar to (Arora et al., 2019a, Lemma F.1). Following the notation of Arora et al. (2019a), we write the time derivative of the prediction as
+
+$$
+\frac {\mathrm {d}}{\mathrm {d} t} f _ {N N} (\hat {\boldsymbol {x}}, t) = \frac {1}{n} \boldsymbol {u} _ {N N} (\hat {\boldsymbol {x}}, t) ^ {\top} (\boldsymbol {y} - \boldsymbol {y} _ {N N} (t)); \frac {\mathrm {d}}{\mathrm {d} t} f _ {N T K} (\hat {\boldsymbol {x}}, t) = \frac {1}{n} \boldsymbol {u} _ {N T K} (\hat {\boldsymbol {x}}, t) ^ {\top} (\boldsymbol {y} - \boldsymbol {y} _ {N T K} (t)),
+$$
+
+where  $\pmb{u}_{NN}(\pmb{x},t) = \frac{\partial\pmb{f}(X;\pmb{\omega}(t))}{\partial\pmb{\omega}(t)}^{\top}\frac{\partial f(\pmb{x};\pmb{\omega}(t))}{\partial\pmb{\omega}(t)}\in \mathbb{R}^{n}$  and  $\pmb{u}_{NTK}(\pmb{x},t)$  similarly defined on the initialized weights  $\omega (0)$ . We bound the difference between the predictions on  $\hat{\pmb{x}}$  up to terminal time T as
+
+$$
+\begin{array}{l} | f _ {N N} (\hat {\boldsymbol {x}}, t) - f _ {N T K} (\hat {\boldsymbol {x}}, t) | = | \int_ {0} ^ {T} \left[ \frac {\mathrm {d}}{\mathrm {d} t} f _ {N N} (\hat {\boldsymbol {x}}, t) - \frac {\mathrm {d}}{\mathrm {d} t} f _ {N T K} (\hat {\boldsymbol {x}}, t) \right] \mathrm {d} t | \\ = \frac {1}{n} \left| \int_ {0} ^ {T} \left[ \boldsymbol {u} _ {N N} (\hat {\boldsymbol {x}}, t) ^ {\top} (\boldsymbol {y} - \boldsymbol {y} _ {N N} (t)) - \boldsymbol {u} _ {N T K} (\hat {\boldsymbol {x}}, t) ^ {\top} (\boldsymbol {y} - \boldsymbol {y} _ {N T K} (t)) \right] d t \right| \\ \leq \frac {1}{n} \left| \int_ {0} ^ {T} \boldsymbol {u} _ {N T K} (\hat {\boldsymbol {x}}, t) ^ {\top} \left(\boldsymbol {y} _ {N N} (t) - \boldsymbol {y} _ {N T K} (t)\right) \mathrm {d} t \right| \\ + \frac {1}{n} \left| \int_ {0} ^ {T} \left(\boldsymbol {u} _ {N N} (\hat {\boldsymbol {x}}, t) - \boldsymbol {u} _ {N T K} (\hat {\boldsymbol {x}}, t)\right) ^ {\top} \left(\boldsymbol {y} - \boldsymbol {y} _ {N N} (t)\right) \mathrm {d} t \right| \\ \leq \frac {1}{n} \| \boldsymbol {u} _ {N T K} (\hat {\boldsymbol {x}}, t) \| _ {2} \int_ {0} ^ {T} \| \boldsymbol {y} _ {N N} (t) - \boldsymbol {y} _ {N T K} (t) \| _ {2} d t \\ + \frac {1}{n} \max  _ {t <   T} \| \boldsymbol {u} _ {N N} (\hat {\boldsymbol {x}}, t) - \boldsymbol {u} _ {N T K} (\hat {\boldsymbol {x}}, t) \| _ {2} \int_ {0} ^ {T} \| \boldsymbol {y} - \boldsymbol {y} _ {N N} (t) \| _ {2} \mathrm {d} t. \tag {123} \\ \end{array}
+$$
+
+For the first term have
+
+$$
+\left\| \boldsymbol {y} _ {N N} (T) - \boldsymbol {y} _ {N T K} (T) \right\| _ {2} \leq \frac {1}{n} \int_ {0} ^ {T} \left\| K (t) (\boldsymbol {y} - \boldsymbol {y} _ {N N} (t)) - K (0) (\boldsymbol {y} - \boldsymbol {y} _ {N T K} (t)) \right\| _ {2} d t
+$$
+
+$$
+\leq \frac {1}{n} \max  _ {0 <   t <   T} \| K (t) - K (0) \| _ {2} \int_ {0} ^ {T} \| \boldsymbol {y} - \boldsymbol {y} _ {N N} (t) \| _ {2} d t + \frac {1}{n} \| K (0) \| _ {2} \int_ {0} ^ {T} \| \boldsymbol {y} _ {N N} (t) - \boldsymbol {y} _ {N T K} (t) \| _ {2} d t
+$$
+
+$$
+\leq \frac {1}{n} O \left(d ^ {1 / 2 - \epsilon^ {\prime}}\right) O (\sqrt {d}) + O (1) \int_ {0} ^ {T} \| \boldsymbol {y} _ {N N} (t) - \boldsymbol {y} _ {N T K} (t) \| _ {2} \mathrm {d} t \stackrel {(i i)} {\leq} O \left(d ^ {- \epsilon^ {\prime}}\right), \tag {124}
+$$
+
+where (i) is due to Corollary 15, Lemma 19 for some  $\epsilon' > 0$  and the linear convergence of  $\mathbf{y}_{NN}$ , and (ii) is due to Gronwall's inequality. Note that the log factor in  $T$  is omitted. Similarly, for the second term we have
+
+$$
+\frac {1}{n} \max  _ {t <   T} \| \boldsymbol {u} _ {N N} (\hat {\boldsymbol {x}}, t) - \boldsymbol {u} _ {N T K} (\hat {\boldsymbol {x}}, t) \| _ {2} \int_ {0} ^ {T} \| \boldsymbol {y} - \boldsymbol {y} _ {N N} (t) \| _ {2} d t \stackrel {(i)} {\leq} \frac {1}{n} O \left(d ^ {1 / 2 - \epsilon^ {\prime}}\right) O (\sqrt {d}) = O \left(d ^ {- \epsilon^ {\prime}}\right), \tag {125}
+$$
+
+where we used Lemma 19 and the linear convergence of  $\pmb{y}_{NN}$  in (i). Combining the two cases yields
+
+$$
+\left| f _ {N N} (\hat {\boldsymbol {x}}, t) - f _ {N T K} (\hat {\boldsymbol {x}}, t) \right| \leq \frac {1}{n} \| \boldsymbol {u} _ {N T K} (\hat {\boldsymbol {x}}, t) \| _ {2} O \left(d ^ {- \epsilon^ {\prime}}\right) + O \left(d ^ {- \epsilon^ {\prime}}\right) \stackrel {(i)} {=} O \left(d ^ {- \epsilon^ {\prime}}\right), \tag {126}
+$$
+
+where we utilized Corollary 14 in (i). Thus we know that the difference between the population risk of  $f_{NN}$  and  $f_{NTK}$  is also asymptotically vanishing (note that the derivation above is independent of the target function as long as  $\| \pmb{y} \|_2 = O(\sqrt{n})$ ). Therefore, in the following subsection we compute the risk of the linearized (kernel) model  $f_{NTK}$ .
+
+# C.9.2 COMPUTING THE KERNEL RISK
+
+Given input  $X \in \mathbb{R}^{d \times n}$  and label  $\boldsymbol{y} = \boldsymbol{\beta}^{\top} X + \varepsilon$ , gradient flow on the tangent kernel solves the following equation of the parameters  $\omega$ :
+
+$$
+\boldsymbol {y} = \boldsymbol {f} (X; \omega) = \frac {\partial \boldsymbol {f} (X ; \omega_ {0})}{\partial \omega} ^ {\top} (\omega - \omega_ {0}), \tag {127}
+$$
+
+where  $\partial f(X;\omega_0) / \partial \omega$  is a  $dh\times n$  matrix with column  $\partial f(\pmb {x}_i;\pmb {\omega}_0) / \partial \pmb {\omega}$ . Note that for  $n\to \infty$  and  $\gamma_{1},\gamma_{2}\in (0,\infty)$ ,  $dh > n$  trivially holds, and by Corollary 15 we know that solution is given by
+
+$$
+\boldsymbol {\omega} _ {1} = \boldsymbol {\omega} _ {0} + \frac {\partial \boldsymbol {f} (X ; \boldsymbol {\omega} _ {0})}{\partial \boldsymbol {\omega}} \left(\frac {\partial \boldsymbol {f} (X ; \boldsymbol {\omega} _ {0})}{\partial \boldsymbol {\omega}} ^ {\top} \frac {\partial \boldsymbol {f} (X ; \boldsymbol {\omega} _ {0})}{\partial \boldsymbol {\omega}}\right) ^ {- 1} (X ^ {\top} \boldsymbol {\beta} + \varepsilon). \tag {128}
+$$
+
+And the population risk can be written as (note that there is a factor of 2 due to the "doubling trick" at initialization to ensure  $f^{\mathrm{init}}(\cdot) = 0$ ):
+
+$$
+\begin{array}{l} 2 R = \mathbb {E} _ {\boldsymbol {x}, \varepsilon} \left[ \left(\boldsymbol {x} ^ {\top} \boldsymbol {\beta} - f (x; \omega_ {1})\right) ^ {2} \right] \\ = \mathbb {E} _ {\boldsymbol {x}, \varepsilon} \left[ \left(\boldsymbol {x} ^ {\top} \beta - \frac {\partial f (\boldsymbol {x} ; \boldsymbol {\omega} _ {0})}{\partial \boldsymbol {\omega}} ^ {\top} \left(\boldsymbol {\omega} _ {1} - \boldsymbol {\omega} _ {0}\right)\right) ^ {2} \right] \\ = \mathbb {E} _ {\boldsymbol {x}, \varepsilon} \left[ \left(\boldsymbol {x} ^ {\top} \boldsymbol {\beta} - \frac {\partial f (\boldsymbol {x} ; \omega_ {0})}{\partial \omega} ^ {\top} \frac {\partial \boldsymbol {f} (X ; \omega_ {0})}{\partial \omega} \left(\frac {\partial \boldsymbol {f} (X ; \omega_ {0})}{\partial \omega} ^ {\top} \frac {\partial \boldsymbol {f} (X ; \omega_ {0})}{\partial \omega}\right) ^ {- 1} (X ^ {\top} \boldsymbol {\beta} + \varepsilon)\right) ^ {2} \right] \\ = \underbrace {\mathbb {E} _ {\boldsymbol {x}} \left[ \left(\boldsymbol {x} ^ {\top} \boldsymbol {\beta} - \hat {\boldsymbol {u}} ^ {\top} \hat {K} _ {X} ^ {- 1} X ^ {\top} \boldsymbol {\beta}\right) ^ {2} \right]} _ {2 B} + \underbrace {\mathbb {E} _ {\boldsymbol {x}} \left[ \hat {\boldsymbol {u}} \hat {K} _ {X} ^ {- 1} \hat {K} _ {X} ^ {- 1} \hat {\boldsymbol {u}} ^ {\top} \right] \sigma^ {2}} _ {2 V}, \tag {129} \\ \end{array}
+$$
+
+where a bias-variance decomposition is made here, and for simplicity we define
+
+$$
+\hat {\boldsymbol {u}} = \frac {\partial \boldsymbol {f} (X ; \omega_ {0}) ^ {\top}}{\partial \boldsymbol {\omega}} \frac {\partial f (\boldsymbol {x} ; \omega_ {0})}{\partial \boldsymbol {\omega}}, \quad \hat {K} _ {X} = \frac {\partial \boldsymbol {f} (X ; \omega_ {0}) ^ {\top}}{\partial \boldsymbol {\omega}} \frac {\partial \boldsymbol {f} (X ; \omega_ {0})}{\partial \boldsymbol {\omega}}. \tag {130}
+$$
+
+# C.9.3 APPROXIMATING THE KERNEL MATRIX
+
+In this section we drop the negligible  $\epsilon$  in the initialization. Following Cheng and Singer (2013) we utilize the orthonormal decomposition of  $\phi'(x)$  in  $L^2(\mathbb{R}, \mu_G)$ . Denote  $b_0 = \mathbb{E}[\phi'(G)]$ , and  $b_1^2 = \mathbb{E}[\phi'(G)^2] - b_0^2$ . We have the orthogonal decomposition of  $\phi'$
+
+$$
+\phi^ {\prime} (x) = b _ {0} + \phi_ {\perp} ^ {\prime} (x), \tag {131}
+$$
+
+where  $\mathbb{E}[\phi_{\perp}'(G)] = 0$ . We develop the following lemmas to approximate the kernel matrix.
+
+Lemma 13 (Approximation of  $(\hat{K}_X)_{ij}$ ). There exist constants  $c, c' > 0$  such that for  $i \neq j$  with probability  $1 - e^{-cn\varepsilon^2}$  we have
+
+$$
+\left| \frac {1}{d} \frac {\partial f \left(\boldsymbol {x} _ {i} ; \boldsymbol {\omega} _ {0}\right)}{\partial \boldsymbol {\omega}} ^ {\top} \frac {\partial f \left(\boldsymbol {x} _ {j} ; \boldsymbol {\omega} _ {0}\right)}{\partial \boldsymbol {\omega}} - \frac {1}{d} b _ {0} ^ {2} \boldsymbol {x} _ {i} ^ {\top} \boldsymbol {x} _ {j} \right| <   \varepsilon^ {2}, \tag {132}
+$$
+
+and with probability  $1 - e^{-c'n\varepsilon^2}$ ,
+
+$$
+\left| \frac {1}{d} \frac {\partial f \left(\boldsymbol {x} _ {i} ; \boldsymbol {\omega} _ {0}\right)}{\partial \boldsymbol {\omega}} ^ {\top} \frac {\partial f \left(\boldsymbol {x} _ {i} ; \boldsymbol {\omega} _ {0}\right)}{\partial \boldsymbol {\omega}} - \left(b _ {0} ^ {2} + b _ {1} ^ {2}\right) \right| <   \varepsilon . \tag {133}
+$$
+
+Proof. When  $i \neq j$  (i.e. Equation (132)), we have
+
+$$
+\begin{array}{l} \frac {1}{d} [ \hat {K} _ {X} ] _ {i j} = \frac {1}{d} \frac {\partial f (\boldsymbol {x} _ {i} ; \boldsymbol {\omega} _ {0})}{\partial \boldsymbol {\omega}} ^ {\top} \frac {\partial f (\boldsymbol {x} _ {j} ; \boldsymbol {\omega} _ {0})}{\partial \boldsymbol {\omega}} \\ = \frac {1}{d} \sum_ {k = 1} ^ {h} \frac {\partial f (\boldsymbol {x} _ {i} ; \boldsymbol {\omega} _ {0})}{\partial \boldsymbol {w} _ {k}} ^ {\top} \frac {\partial f (\boldsymbol {x} _ {j} ; \boldsymbol {\omega} _ {0})}{\partial \boldsymbol {w} _ {k}} = \frac {1}{d h} \sum_ {k = 1} ^ {h} \boldsymbol {x} _ {i} ^ {\top} \boldsymbol {x} _ {j} \phi^ {\prime} (\boldsymbol {w} _ {k} ^ {\top} \boldsymbol {x} _ {i}) \phi^ {\prime} (\boldsymbol {w} _ {k} ^ {\top} \boldsymbol {x} _ {j}) \\ \rightarrow \frac {1}{d} \boldsymbol {x} _ {i} ^ {\top} \boldsymbol {x} _ {j} \mathbb {E} _ {\boldsymbol {w}} \left[ \phi^ {\prime} \left(\boldsymbol {w} ^ {\top} \boldsymbol {x} _ {i}\right) \phi^ {\prime} \left(\boldsymbol {w} ^ {\top} \boldsymbol {x} _ {j}\right)\right] = \frac {1}{d} H (\boldsymbol {x} _ {i}, \boldsymbol {x} _ {j}). \tag {134} \\ \end{array}
+$$
+
+The matrix  $H(\pmb{x}_i, \pmb{x}_j) = \pmb{x}_i^\top \pmb{x}_j \mathbb{E}_{\pmb{w}} \left[ \phi'(\pmb{w}^\top \pmb{x}_i) \phi'(\pmb{w}^\top \pmb{x}_j) \right]$  can be seen as the expected tangent kernel of nonlinear activation function studied in Du et al. (2018); Arora et al. (2019b). Moreover, due to the assumed boundedness of  $\phi'(x)$  (A3), by Hoeffding's inequality we have
+
+$$
+\Pr \left| \frac {1}{d} \frac {\partial f \left(\boldsymbol {x} _ {i} ; \boldsymbol {\omega} _ {0}\right)}{\partial \boldsymbol {\omega}} ^ {\top} \frac {\partial f \left(\boldsymbol {x} _ {j} ; \boldsymbol {\omega} _ {0}\right)}{\partial \boldsymbol {\omega}} - \frac {1}{d} H \left(\boldsymbol {x} _ {i}, \boldsymbol {x} _ {j}\right) \right| <   \frac {1}{d} \boldsymbol {x} _ {i} ^ {\top} \boldsymbol {x} _ {j} \varepsilon > 1 - e ^ {- c _ {1} h \varepsilon^ {2}}. \tag {135}
+$$
+
+In addition, by the concentration of  $\pmb{x}_i^\top \pmb{x}_j$  and  $\| \pmb{x}_i\|_2^2$ , i.e.  $\operatorname{Pr} \pmb{x}_i^\top \pmb{x}_j / d > \varepsilon < 1 - e^{-c_2d\varepsilon^2}$  and  $\operatorname{Pr} |x_i^\top x_i / d - 1| < \varepsilon > 1 - e^{-c_3d\varepsilon^2}$ , the orthonormal decomposition  $\phi'(x) = b_0x + \phi_\perp'(x)$  leads to the following linear approximation of the matrix  $H$
+
+$$
+\frac {1}{d} H \left(\boldsymbol {x} _ {i}, \boldsymbol {x} _ {j}\right) = b _ {0} ^ {2} \frac {1}{d} \boldsymbol {x} _ {i} ^ {\top} \boldsymbol {x} _ {j} + O \left(\left(\boldsymbol {x} _ {i} ^ {\top} \boldsymbol {x} _ {j} / d\right) ^ {2}\right). \tag {136}
+$$
+
+and by taking  $\varepsilon = \pmb{x}_i^\top \pmb{x}_j / d$  under the joint event we can show that
+
+$$
+\left| \frac {1}{d} \frac {\partial f \left(\boldsymbol {x} _ {i} ; \boldsymbol {\omega} _ {0}\right)}{\partial \boldsymbol {\omega}} ^ {\top} \frac {\partial f \left(\boldsymbol {x} _ {j} ; \boldsymbol {\omega} _ {0}\right)}{\partial \boldsymbol {\omega}} - \frac {1}{d} b _ {0} ^ {2} \boldsymbol {x} _ {i} ^ {\top} \boldsymbol {x} _ {j} \right| <   \varepsilon^ {2} \tag {137}
+$$
+
+with probability  $1 - e^{-cd\varepsilon^2}$ . The same argument follows for the case where  $i = j$ .
+
+![](images/0697c60760fac48c5333ecfb1dbea99682f80ba205a9ad541b5ced06bd550f66.jpg)
+
+Corollary 14 (Approximation of  $\hat{\pmb{u}}$ ). For large enough  $l > 0$ , with probability  $1 - de^{-c\log^d d}$
+
+$$
+\frac {1}{d} \| \hat {\boldsymbol {u}} - \tilde {\boldsymbol {u}} \| _ {2} = \left\| \frac {1}{d} \frac {\partial f (\boldsymbol {x} ; \boldsymbol {\omega} _ {0})}{\partial \boldsymbol {\omega}} ^ {\top} \frac {\partial \boldsymbol {f} (X ; \boldsymbol {\omega} _ {0})}{\partial \boldsymbol {\omega}} - \frac {1}{d} \tilde {\boldsymbol {u}} \right\| _ {2} <   \frac {\log^ {l} d}{d}, \tag {138}
+$$
+
+where  $\tilde{\pmb{u}} = b_0^2\pmb{x}^\top X$
+
+Proof. Taking  $\varepsilon = \log^l d / d$  together with Lemma 13 yields the desired result.
+
+![](images/31707fac27cb8535deddf3917d76a38e09a8639a9a047f775803e22bdd439bb7.jpg)
+
+Corollary 15 (Approximation of  $\hat{K}_X$ ). With probability  $1 - de^{-c\log^d d}$
+
+$$
+\frac {1}{d} \left\| \hat {K} _ {X} - \tilde {K} _ {X} \right\| _ {F} = \left\| \frac {1}{d} \frac {\partial \boldsymbol {f} (X ; \boldsymbol {\omega} _ {0})}{\partial \boldsymbol {\omega}} ^ {\top} \frac {\partial \boldsymbol {f} (X ; \boldsymbol {\omega} _ {0})}{\partial \boldsymbol {\omega}} - \frac {1}{d} \tilde {K} _ {X} \right\| _ {F} <   \log^ {l} d, \tag {139}
+$$
+
+where  $\tilde{K}_X = b_0^2 X^\top X + b_1^2 dI$ .
+
+Proof. Also by directly applying Lemma 13.
+
+![](images/e7e2b5e933501cd7c8240d6722af36eb25a1aeca7a979c55d1e8d803642f2b1f.jpg)
+
+Remark. For initialization larger or equal to  $\pmb{w}_i(0) \sim N(0, I_d / d)$ , the above approximation does not depend on the scale of initialization.
+
+Remark. For  $\phi(x) = \mathrm{SoftPlus}(x)$ ,  $b_0^2 = 1/4$ ,  $b_1^2 = 0.043379$ . For  $\phi(x) = \mathrm{sigmoid}(x) = (1 + e^{-x})^{-1}$ ,  $b_0^2 = 0.042692$ ,  $b_1^2 = 0.002144$ . Note that  $b_1 \geq 0$  for all smooth activations  $\phi$ , and the equality holds (i.e.  $b_1 = 0$ ) if and only if  $\phi$  is linear. We comment that smaller  $b_1$  entails larger variance as  $\gamma_1 \to 1$ , and vice versa, as shown in the following section.
+
+# C.9.4 THE BIAS TERM
+
+With these approximation above we proceed to calculating (129)
+
+$$
+2 B = \mathbb {E} _ {\boldsymbol {x}} \left[ \left(\boldsymbol {x} ^ {\top} \boldsymbol {\beta} - \hat {\boldsymbol {u}} ^ {\top} \hat {K} _ {X} ^ {- 1} X ^ {\top} \boldsymbol {\beta}\right) ^ {2} \right]. \tag {140}
+$$
+
+We first bound the error in substituting  $\hat{\pmb{u}}$  with  $\tilde{\pmb{u}}$ :
+
+$$
+\begin{array}{l} \left\| \hat {\boldsymbol {u}} ^ {\top} \hat {K} _ {X} ^ {- 1} X ^ {\top} \boldsymbol {\beta} - \tilde {\boldsymbol {u}} ^ {\top} \hat {K} _ {X} ^ {- 1} X ^ {\top} \boldsymbol {\beta} \right\| _ {2} \leq \| \hat {\boldsymbol {u}} - \tilde {\boldsymbol {u}} \| _ {2} \left\| \hat {K} _ {X} ^ {- 1} \right\| _ {2} \| X \| _ {2} \| \boldsymbol {\beta} \| _ {2} \\ \stackrel {(i)} {=} O \left(\log^ {l} d \cdot d ^ {- 1} \cdot \sqrt {d} \cdot 1\right) = O \left(\frac {\log^ {l} d}{\sqrt {d}}\right), \tag {141} \\ \end{array}
+$$
+
+where (i) is due to the fact that  $\left\| \hat{K}_X^{-1}\right\|_2 = \lambda_{\min}^{-1}(\hat{K}_X) = O(1 / d)$  and  $\| X\|_2 = O(\sqrt{d})$ . Therefore we have as  $n, d, h \to \infty$
+
+$$
+\begin{array}{l} 2 B = \mathbb {E} _ {\boldsymbol {x}} \left[\left(\boldsymbol {x} ^ {\top} \boldsymbol {\beta} - \hat {\boldsymbol {u}} ^ {\top} \hat {K} _ {X} ^ {- 1} X ^ {\top} \boldsymbol {\beta}\right) ^ {2} \right]\rightarrow \mathbb {E} _ {\boldsymbol {x}} \left[\left(\boldsymbol {x} ^ {\top} \boldsymbol {\beta} - \tilde {\boldsymbol {u}} ^ {\top} \hat {K} _ {X} ^ {- 1} X ^ {\top} \boldsymbol {\beta}\right) ^ {2} \right] \\ = \mathbb {E} _ {\boldsymbol {x}} \left[ \left(\boldsymbol {x} ^ {\top} \boldsymbol {\beta} - b _ {0} ^ {2} \boldsymbol {x} ^ {\top} X \hat {K} _ {X} ^ {- 1} X ^ {\top} \boldsymbol {\beta}\right) ^ {2} \right]. \tag {142} \\ \end{array}
+$$
+
+By taking expectation over  $\pmb{x}$  and the rotational invariance argument similar to Hastie et al. (2019),
+
+$$
+\begin{array}{l} \mathbb {E} _ {\boldsymbol {x}} \left[ \left(\boldsymbol {x} ^ {\top} \boldsymbol {\beta} - b _ {0} ^ {2} \boldsymbol {x} ^ {\top} X \hat {K} _ {X} ^ {- 1} X ^ {\top} \boldsymbol {\beta}\right) ^ {2} \right] = \mathbb {E} _ {\boldsymbol {x}} \left[ \boldsymbol {\beta} ^ {\top} \left(I - b _ {0} ^ {2} X \hat {K} _ {X} ^ {- 1} X ^ {\top}\right) ^ {2} \boldsymbol {\beta} \right] \\ = \frac {\beta^ {\top} \beta}{d} \operatorname {t r} \left(\left(I - b _ {0} ^ {2} X \hat {K} _ {X} ^ {- 1} X ^ {\top}\right) \left(I - b _ {0} ^ {2} X \hat {K} _ {X} ^ {- 1} X ^ {\top}\right)\right). \tag {143} \\ \end{array}
+$$
+
+In addition, we bound the error in substituting  $\hat{K}_X$  by  $\tilde{K}_X$  defined in (139):
+
+$$
+\begin{array}{l} \left| \frac {1}{d} \operatorname {t r} \left(X \hat {K} _ {X} ^ {- 1} X ^ {\top} - X \tilde {K} _ {X} ^ {- 1} X ^ {\top}\right) \right| = \left| \frac {1}{d} \operatorname {t r} \left(X ^ {\top} X \hat {K} _ {X} ^ {- 1} (\hat {K} _ {X} - \tilde {K} _ {X}) \tilde {K} _ {X} ^ {- 1}\right) \right| \\ <   \frac {1}{d} \left\| X ^ {\top} X \right\| _ {2} \left\| \hat {K} _ {X} ^ {- 1} \right\| _ {2} \left\| \hat {K} _ {X} - \tilde {K} _ {X} \right\| _ {F} \left\| \tilde {K} _ {X} ^ {- 1} \right\| _ {2} \\ = O \left(d ^ {- 1} \cdot d \cdot d ^ {- 1} \cdot d \log^ {l} d \cdot d ^ {- 1}\right) = O \left(\frac {\log^ {l} d}{d}\right), \tag {144} \\ \end{array}
+$$
+
+and similarly,
+
+$$
+\begin{array}{l} \left| \frac {1}{d} \operatorname {t r} \left(X \hat {K} _ {X} ^ {- 1} X ^ {\top} X \hat {K} _ {X} ^ {- 1} X ^ {\top} - X \tilde {K} _ {X} ^ {- 1} X ^ {\top} X \tilde {K} _ {X} ^ {- 1} X ^ {\top}\right) \right| \\ = \left| \frac {1}{d} \mathrm {t r} \left(X ^ {\top} X (\hat {K} _ {X} ^ {- 1} - \tilde {K} _ {X} ^ {- 1}) X ^ {\top} X (\hat {K} _ {X} ^ {- 1} + \tilde {K} _ {X} ^ {- 1})\right) \right| \\ <   \frac {1}{d} \left\| X ^ {\top} X \right\| _ {2} \left\| \hat {K} _ {X} ^ {- 1} \right\| _ {2} \left\| \hat {K} _ {X} - \tilde {K} _ {X} \right\| _ {F} \left\| \tilde {K} _ {X} ^ {- 1} \right\| _ {2} \left\| X ^ {\top} X \right\| _ {2} \left\| \hat {K} _ {X} ^ {- 1} + \tilde {K} _ {X} ^ {- 1} \right\| _ {2} \\ = O \left(d ^ {- 1} \cdot d \cdot d ^ {- 1} \cdot d \log^ {l} d \cdot d ^ {- 1} \cdot d \cdot d ^ {- 1}\right) = O \left(\frac {\log^ {l} d}{d}\right). \tag {145} \\ \end{array}
+$$
+
+Combining these two formulas in (143) yields
+
+$$
+\begin{array}{l} 2 B \to \mathbb {E} _ {\pmb {x}} \left[ \left(\pmb {x} ^ {\top} \pmb {\beta} - b _ {0} ^ {2} \pmb {x} ^ {\top} X \hat {K} _ {X} ^ {- 1} X ^ {\top} \pmb {\beta}\right) ^ {2} \right] \\ \rightarrow \frac {\boldsymbol {\beta} ^ {\top} \boldsymbol {\beta}}{d} \operatorname {t r} \left(\left(I - b _ {0} ^ {2} X \tilde {K} _ {X} ^ {- 1} X ^ {\top}\right)\left(I - b _ {0} ^ {2} X \tilde {K} _ {X} ^ {- 1} X ^ {\top}\right)\right) \\ = \frac {\boldsymbol {\beta} ^ {\top} \boldsymbol {\beta}}{d} \operatorname {t r} \left(\left(I - b _ {0} ^ {2} X \left(b _ {0} ^ {2} X ^ {\top} X + b _ {1} ^ {2} d I\right) ^ {- 1} X ^ {\top}\right) ^ {2}\right). \tag {146} \\ \end{array}
+$$
+
+Utilizing the Marčenko-Pastur law from Section B.2 we obtain
+
+$$
+B = \beta^ {\top} \beta \left(\frac {\gamma_ {1} - 1}{2 \gamma_ {1}} + \frac {\gamma_ {1} \left(\gamma_ {1} + \gamma_ {1} m + m - 2\right) + 1}{2 \gamma_ {1} \sqrt {\gamma_ {1} \left(\gamma_ {1} + m \left(\gamma_ {1} (m + 2) + 2\right) - 2\right) + 1}}\right), \tag {147}
+$$
+
+where  $m = b_0^{-2}b_1^2$
+
+# C.9.5 THE VARIANCE TERM
+
+Similarly, for the variance we utilize the approximation
+
+$$
+2 V = \mathbb {E} _ {\boldsymbol {x}} \left[ \hat {\boldsymbol {u}} \hat {K} _ {X} ^ {- 1} \hat {K} _ {X} ^ {- 1} \hat {\boldsymbol {u}} ^ {\top} \right] \sigma^ {2} \tag {148}
+$$
+
+Specifically, we bound the approximation error
+
+$$
+\begin{array}{l} \left| \hat {\boldsymbol {u}} \hat {K} _ {X} ^ {- 1} \hat {K} _ {X} ^ {- 1} \hat {\boldsymbol {u}} ^ {\top} - \tilde {\boldsymbol {u}} \hat {K} _ {X} ^ {- 1} \hat {K} _ {X} ^ {- 1} \tilde {\boldsymbol {u}} ^ {\top} \right| \leq \| \hat {\boldsymbol {u}} - \tilde {\boldsymbol {u}} \| _ {2} \left\| \hat {K} _ {X} ^ {- 1} \right\| _ {2} \| \hat {\boldsymbol {u}} + \tilde {\boldsymbol {u}} \| _ {2} \left\| \hat {K} _ {X} ^ {- 1} \right\| _ {2} \\ = O \left(\log^ {l} d \cdot \frac {1}{d} \cdot d \cdot \frac {1}{d}\right) = O \left(\frac {\log^ {l} d}{d}\right), \tag {149} \\ \end{array}
+$$
+
+and similarly
+
+$$
+\begin{array}{l} \left| \mathbb {E} _ {\boldsymbol {x}} \left[ \tilde {\boldsymbol {u}} \hat {K} _ {X} ^ {- 1} \hat {K} _ {X} ^ {- 1} \tilde {\boldsymbol {u}} ^ {\top} - \tilde {\boldsymbol {u}} \tilde {K} _ {X} ^ {- 1} \tilde {K} _ {X} ^ {- 1} \tilde {\boldsymbol {u}} ^ {\top} \right] \right| \\ = \operatorname {t r} \left(\left(\hat {K} _ {X} ^ {- 1} - \tilde {K} _ {X} ^ {- 1}\right) \left(\hat {K} _ {X} ^ {- 1} + \tilde {K} _ {X} ^ {- 1}\right) \mathbb {E} _ {\boldsymbol {x}} \tilde {\boldsymbol {u}} \tilde {\boldsymbol {u}} ^ {\top}\right) \\ = \operatorname {t r} \left(\hat {K} _ {X} ^ {- 1} \left(\hat {K} _ {X} - \tilde {K} _ {X}\right) \tilde {K} _ {X} ^ {- 1} \left(\hat {K} _ {X} ^ {- 1} + \tilde {K} _ {X} ^ {- 1}\right) X ^ {T} X\right) \\ \leq \left\| \hat {K} _ {X} ^ {- 1} \right\| _ {2} \left\| \hat {K} _ {X} - \tilde {K} _ {X} \right\| _ {F} \left\| \tilde {K} _ {X} ^ {- 1} \right\| _ {2} \left\| \hat {K} _ {X} ^ {- 1} + \tilde {K} _ {X} ^ {- 1} \right\| _ {2} \left\| X ^ {T} X \right\| _ {2} \\ = O \left(d ^ {- 1} \cdot d \log^ {l} d \cdot d ^ {- 1} \cdot d ^ {- 1} \cdot d\right) = O \left(\frac {\log^ {l} d}{d}\right), \tag {150} \\ \end{array}
+$$
+
+By combining the two approximations above we know that as  $n, d, p \to \infty$
+
+$$
+\left| 2 V - \mathbb {E} _ {\boldsymbol {x}} \left[ \tilde {\boldsymbol {u}} \tilde {K} _ {X} ^ {- 1} \tilde {K} _ {X} ^ {- 1} \tilde {\boldsymbol {u}} ^ {\top} \right] \sigma^ {2} \right| = O \left(\frac {\log^ {l} d}{d}\right)\rightarrow 0. \tag {151}
+$$
+
+Therefore the variance is given as
+
+$$
+\begin{array}{l} 2 V \rightarrow \sigma^ {2} \mathbb {E} _ {\boldsymbol {x}} \left[ \tilde {\boldsymbol {u}} \tilde {K} _ {X} ^ {- 1} \tilde {K} _ {X} ^ {- 1} \tilde {\boldsymbol {u}} ^ {\top} \right] \\ = \sigma^ {2} \mathbb {E} _ {\boldsymbol {x}} \left[ b _ {0} ^ {4} \boldsymbol {x} ^ {T} X \left(b _ {0} ^ {2} X ^ {\top} X + b _ {1} ^ {2} d I\right) ^ {- 2} X ^ {T} \boldsymbol {x} \right] \\ = \sigma^ {2} \frac {1}{d} \operatorname {t r} \left(\frac {1}{d} X ^ {T} X \cdot \left(\frac {1}{d} X ^ {T} X + b _ {0} ^ {- 2} b _ {1} ^ {2} I\right) ^ {- 2}\right) \\ = \sigma^ {2} \left(- \frac {1}{2} + \frac {\gamma_ {1} + \gamma_ {1} m + 1}{2 \sqrt {\gamma_ {1} (\gamma_ {1} + m (\gamma_ {1} (m + 2) + 2) - 2) + 1}}\right), \tag {152} \\ \end{array}
+$$
+
+where  $m$  is defined in the derivation of the bias term.
+
+# C.9.6 PUTTING THINGS TOGETHER
+
+Recall the population risk is the sum of the bias and variance
+
+$$
+\begin{array}{l} R \rightarrow r ^ {2} \left(\frac {\gamma_ {1} - 1}{2 \gamma_ {1}} + \frac {\gamma_ {1} (\gamma_ {1} + \gamma_ {1} m + m - 2) + 1}{2 \gamma_ {1} \sqrt {\gamma_ {1} (\gamma_ {1} + m (\gamma_ {1} (m + 2) + 2) - 2) + 1}}\right) \\ + \sigma^ {2} \left(- \frac {1}{4} + \frac {\gamma_ {1} + \gamma_ {1} m + 1}{4 \sqrt {\gamma_ {1} (\gamma_ {1} + m (\gamma_ {1} (m + 2) + 2) - 2) + 1}}\right). \tag {153} \\ \end{array}
+$$
+
+Observe that the population risk is independent of  $\gamma_{2}$ , i.e. double descent does not occur when the network is overparameterized via changing the width. In addition, the bias is monotonically increasing and upper-bounded by the null risk  $r^2$  and lower-bounded by the bias of the least squares solution on the input features  $\hat{\beta} = X^{\dagger}\mathbf{y}$ , whereas the variance remains bounded for all  $\gamma_{1} \in (0,\infty)$  as long as  $m > 0$ , i.e.  $\phi$  is nonlinear.
+
+![](images/adeec3371f903336fb10e7d87fd077f3473eb0308ee728201f3d9e0d33825719.jpg)
+
+# D USEFUL LEMMAS
+
+Lemma 16. Given  $K_W$  from (46), define
+
+$$
+\tilde {K} _ {W} = r I _ {h} + s \mathbf {1} _ {h} \mathbf {1} _ {h} ^ {\top} + t Q, \tag {154}
+$$
+
+where  $Q \in \mathbb{R}^{h \times h}$  with  $Q_{i \neq j} = \pmb{w}_i^\top \pmb{w}_j$  and  $Q_{i,i} = 0$ , and  $r = \mathbb{E}[\phi(G)^2] - \mathbb{E}[\phi(G)]^2$ ,  $s = \mathbb{E}[\phi(G)]^2$ ,  $t = \mathbb{E}[G\phi(G)]^2$ . Then as  $n, d, h \to \infty$ ,  $\left\| K_W - \tilde{K}_W \right\|_F \leq \log^c d$  a.s..
+
+Proof. Consider the event where  $\mathcal{A}_{\epsilon} = \left\{\left|\left|\boldsymbol{w}_{i}\right|_{2} - 1\right| < \epsilon, \left|\boldsymbol{w}_{i}^{\top}\boldsymbol{w}_{j}\right| < \epsilon\right\}$ . Under event  $\mathcal{A}_{\epsilon}$ , for the diagonal term of the kernel matrix we have
+
+$$
+\begin{array}{l} \left| \left[ K _ {W} \right] _ {i i} - \left[ \tilde {K} _ {W} \right] _ {i i} \right| = \left| \mathbb {E} _ {\boldsymbol {x}} \left[ \phi \left(\boldsymbol {w} _ {i} ^ {\top} \boldsymbol {x}\right) \phi \left(\boldsymbol {w} _ {i} ^ {\top} \boldsymbol {x}\right) \right] - \mathbb {E} [ \phi (G) ^ {2} ] \right| \\ = \left| \mathbb {E} \left[ \phi \left(\left\| \boldsymbol {w} _ {i} \right\| _ {2} G\right) ^ {2} \right] - \mathbb {E} \left[ \phi (G) ^ {2} \right] \right| = O (\epsilon). \tag {155} \\ \end{array}
+$$
+
+And for off-diagonal term, by the decomposition introduced in Section B.3 we have
+
+$$
+\left[ K _ {W} \right] _ {i j} = \mathbb {E} _ {\boldsymbol {x}} \left[ \phi \left(\boldsymbol {w} _ {i} ^ {\top} \boldsymbol {x}\right) \phi \left(\boldsymbol {w} _ {j} ^ {\top} \boldsymbol {x}\right) \right] = \| \boldsymbol {w} _ {i} \| _ {2} \| \boldsymbol {w} _ {j} \| _ {2} + \mathbb {E} _ {\boldsymbol {x}} \left[ \phi_ {\perp} \left(\boldsymbol {w} _ {i} ^ {\top} x\right) \phi_ {\perp} \left(\boldsymbol {w} _ {j} ^ {\top} x\right) \right], \tag {156}
+$$
+
+hence  $|[K_W]_{ij} - [\tilde{K}_W]_{ij}| < (\boldsymbol{w}_i^\top \boldsymbol{w}_j)^2$ . Notice that  $\mathcal{A}_{\epsilon}$  holds a.s. for  $\epsilon = \log^c d / \sqrt{d}$  and large enough  $c > 0$ ; we therefore have  $\left\| K_W - \tilde{K}_W \right\|_F \leq \log^c d$ .
+
+Lemma 17. Let  $\hat{\beta}(t)$  be the solution to the gradient flow at time  $t$  defined in (103)(104). Then as  $n, d \to \infty$  and  $\gamma_1 \neq 1$  the following holds.:
+
+$$
+\| \hat {\boldsymbol {\beta}} (t) \| _ {2} = O _ {P} (1); \quad \| X ^ {\top} \hat {\boldsymbol {\beta}} (t) \| _ {\infty} = O _ {P} (\text {p o l y} \log d).
+$$
+
+Proof. We consider  $d < n$  for simplicity, and result for the other case follows in similar fashion.
+
+In this case  $\hat{\beta}(t) = \left(I - \exp\left(-\frac{t}{n}XX^{\top}\right)\right)(XX^{\top})^{-1}X\pmb{y}$ . From (Hastie et al., 2019, Corollary 1) we know that  $\|\hat{\beta}(\infty)\|_2 = \left\|(XX^{\top})^{-1}X\pmb{y}\right\|_2 = O(1)$  for  $\gamma_1 < 1$ . Note that  $\left\|I - \exp\left(-\frac{t}{n}XX^{\top}\right)\right\|_2 = O(1)$  for  $t \geq 0$ ; it follows that  $\|\hat{\beta}(t)\|_2 = O(1)$ .
+
+For the second part, we utilize the SVD  $X = U\Sigma V^{\top}$ , where  $\Sigma = [\hat{\Sigma};0]$ ,  $\hat{\Sigma} \in \mathbb{R}^{d\times d}$  and  $\hat{\Sigma}_{ii} = \lambda_i$ . We have  $\left(I - \exp \left(-\frac{t}{n} XX^{\top}\right)\right) = U\bar{\Sigma} U^{\top}$  where  $\bar{\Sigma}_{i,i} = 1 - \exp (-t\lambda_i^2 /n)$  if  $i\leq d$  and 0 otherwise.
+
+$$
+\begin{array}{l} \left\| X ^ {\top} \hat {\boldsymbol {\beta}} (t) \right\| _ {\infty} = \left\| X ^ {\top} \left(I - \exp \left(- \frac {t}{n} X X ^ {\top}\right)\right) \left(X X ^ {\top}\right) ^ {- 1} X \boldsymbol {y} \right\| _ {\infty} \\ \leq \left\| V \Sigma^ {\top} U ^ {\top} U \bar {\Sigma} U ^ {\top} U \hat {\Sigma} ^ {- 2} U ^ {\top} U \Sigma V ^ {\top} \right\| _ {\infty} \| \boldsymbol {y} \| _ {\infty} \\ \leq \left\| V \Sigma^ {\top} \bar {\Sigma} \hat {\Sigma} ^ {- 2} \Sigma V ^ {\top} \right\| _ {\infty} \left(\| X ^ {\top} \boldsymbol {\beta} \| _ {\infty} + \| \varepsilon \| _ {\infty}\right) \\ \leq \| V \| _ {\infty} \left\| \Sigma^ {\top} \bar {\Sigma} \hat {\Sigma} ^ {- 2} \Sigma \right\| _ {\infty} \| V ^ {\top} \| _ {\infty} \left(\| X ^ {\top} \beta \| _ {\infty} + \| \varepsilon \| _ {\infty}\right) \overset {(i)} {\leq} O _ {P} (\text {p o l y} \log d), \tag {157} \\ \end{array}
+$$
+
+where (i) follows from the concentration of the Gaussian maxima, and the fact that the law of  $V$  is the Haar measure on  $SO(n)$ , and thus for any unit vector  $z$  independent to  $V$ ,  $Vz$  is uniform on sphere and  $\| Vz\|_{\infty} = O(\log d / \sqrt{d})$ . We wherefore have
+
+$$
+\| V \| _ {\infty} = \sup  _ {z} \frac {\| V z \| _ {\infty}}{\| z \| _ {\infty}} = O \left(\frac {\log d}{\sqrt {d}}\right) \frac {\| z \| _ {2}}{\| z \| _ {\infty}} = O (\log d), \tag {158}
+$$
+
+Note that this result also implies that  $\| \pmb {y} - X^{\top}\pmb {\beta}(t)\|_{\infty} = O_P(\mathrm{poly}\log d)$
+
+□
+
+Lemma 18. For weight matrices  $W, W'$  satisfying  $\| \pmb{y} - f(\pmb{X}) \|_2 = O(\sqrt{n})$ , where  $f(X) = \phi(XW)\pmb{a}$  with fixed  $a_i \sim \mathrm{Unif}\{-1/\sqrt{h}, 1/\sqrt{h}\}$ , given (A1)-(A3), the gradient of the empirical risk defined in (11) is Lipschitz w.r.t.  $W$  in the Frobenius norm, i.e.
+
+$$
+\left\| \frac {\partial L (X ; W)}{\partial W} - \frac {\partial L (X ; W ^ {\prime})}{\partial W} \right\| _ {F} \leq L \| W - W ^ {\prime} \| _ {F}. \tag {159}
+$$
+
+Proof. Denote  $\pmb{y}_1 = \phi(X^\top W_1)\pmb{a}$  and  $\pmb{y}_2 = \phi(X^\top W_2)\pmb{a}$  for  $W_1, W_2$  satisfying the assumption above (which can be seen as a condition on the magnitude of training loss), we have
+
+$$
+\begin{array}{l} \left\| \frac {\partial L (W _ {1})}{\partial W _ {1}} - \frac {\partial L (W _ {2})}{\partial W _ {2}} \right\| _ {F} \\ = \left\| \frac {1}{n} X \left[ (\boldsymbol {y} - \boldsymbol {y} _ {1}) \boldsymbol {a} ^ {\top} \circ \phi^ {\prime} (X ^ {\top} W _ {1}) \right] - \frac {1}{n} X \left[ (\boldsymbol {y} - \boldsymbol {y} _ {2}) \boldsymbol {a} ^ {\top} \circ \phi^ {\prime} (X ^ {\top} W _ {2}) \right] \right\| _ {F} \\ \leq \frac {1}{n} \| X \| _ {2} \left\| (\boldsymbol {y} - \boldsymbol {y} _ {1}) \boldsymbol {a} ^ {\top} \circ \phi^ {\prime} (X ^ {\top} W _ {1}) - (\boldsymbol {y} - \boldsymbol {y} _ {2}) \boldsymbol {a} ^ {\top} \circ \phi^ {\prime} (X ^ {\top} W _ {2}) \right\| _ {F} \\ \leq O \left(\frac {1}{\sqrt {d}}\right) \left\| \left(\boldsymbol {y} _ {2} - \boldsymbol {y} _ {1}\right) \boldsymbol {a} ^ {\top} \circ \phi^ {\prime} \left(X ^ {\top} W _ {1}\right) \right\| _ {F} \\ + O \left(\frac {1}{\sqrt {d}}\right) \left\| \left(\boldsymbol {y} - \boldsymbol {y} _ {2}\right) \boldsymbol {a} ^ {\top} \circ \left(\phi^ {\prime} \left(X ^ {\top} W _ {1}\right) - \phi^ {\prime} \left(X ^ {\top} W _ {2}\right)\right) \right\| _ {F}. \tag {160} \\ \end{array}
+$$
+
+We upper bound the two terms separately:
+
+$$
+\begin{array}{l} \left\| \left(\boldsymbol {y} _ {2} - \boldsymbol {y} _ {1}\right) \boldsymbol {a} ^ {\top} \circ \phi^ {\prime} \left(X ^ {\top} W _ {1}\right) \right\| _ {2} \overset {(i)} {\leq} \max  \left\{\phi^ {\prime} \left(X ^ {\top} W _ {1}\right) _ {i j} \right\} \| \boldsymbol {y} _ {2} - \boldsymbol {y} _ {1} \| _ {2} \| \boldsymbol {a} \| _ {2} \\ \stackrel {(i i)} {\leq} O (1) \left\| \phi^ {\prime} \left(X ^ {\top} W _ {1}\right) \boldsymbol {a} - \phi^ {\prime} \left(X ^ {\top} W _ {2}\right) \boldsymbol {a} \right\| _ {F} \\ \stackrel {(i i i)} {\leq} O (1) \| X \| _ {2} \| W _ {1} - W _ {2} \| _ {F} = O (\sqrt {d}) \| W _ {1} - W _ {2} \| _ {F}, \tag {161} \\ \end{array}
+$$
+
+where we applied the inequality  $\| A\circ B\| _F\leq \max \{|A_{ij}|\}\| B\| _F$  in (i), boundedness of  $\phi^\prime$  in (ii) and Lipschitzity of  $\phi$  in (iii). Similarly, for the second term
+
+$$
+\left\| \left(\boldsymbol {y} - \boldsymbol {y} _ {2}\right) \boldsymbol {a} ^ {\top} \circ \left(\phi^ {\prime} \left(X ^ {\top} W _ {1}\right) - \phi^ {\prime} \left(X ^ {\top} W _ {2}\right)\right) \right\| _ {F}
+$$
+
+$$
+\leq \max  \left\{\left| a _ {i} \right| \right\} \| \boldsymbol {y} - \boldsymbol {y} _ {2} \| _ {2} \left\| \phi^ {\prime} \left(X ^ {\top} W _ {1}\right) - \phi^ {\prime} \left(X ^ {\top} W _ {2}\right) \right\| _ {F}
+$$
+
+$$
+\stackrel {(i)} {\leq} O (1) \| X \| _ {2} \| W _ {1} - W _ {2} \| _ {F} = O (\sqrt {d}) \| W _ {1} - W _ {2} \| _ {F}, \tag {162}
+$$
+
+where we used the assumption on the training loss and the Lipschitzity of  $\phi^{\prime}$  in (i). Combining the two terms yields the desired result.
+
+![](images/deb02751861df30ff8f8c7e4b63dea0ba98b45055d164a562bcb2970fcb03678.jpg)
+
+Lemma 19. Under assumptions (A1-3) and the non-vanishing initialization, given that  $\| \pmb{w}_i(t) - \pmb{w}_i(0)\|_2 = O(d^{-1/2})$  for all  $i$ , then we have  $\| K(t) - K(0)\|_2 = O(d^{1/2 - \epsilon'})$  for some positive  $\epsilon' \in \Theta(1)$ .
+
+Proof. Recall the definition of the NTK:
+
+$$
+K _ {i j} (t) = \frac {\partial f (\boldsymbol {x} _ {i} ; \boldsymbol {\omega} (t))}{\partial \boldsymbol {\omega} (t)} ^ {\top} \frac {\partial f (\boldsymbol {x} _ {j} ; \boldsymbol {\omega} (t))}{\partial \boldsymbol {\omega} (t)} = \boldsymbol {x} _ {i} ^ {\top} \boldsymbol {x} _ {j} \frac {1}{h} \sum_ {k = 1} ^ {h} \phi^ {\prime} \left(\boldsymbol {w} _ {k} (t) ^ {\top} \boldsymbol {x} _ {i}\right) \phi^ {\prime} \left(\boldsymbol {w} _ {k} (t) ^ {\top} \boldsymbol {x} _ {j}\right), \tag {163}
+$$
+
+or equivalently the matrix form
+
+$$
+K (t) = X ^ {\top} X \circ \frac {1}{h} \left[ \phi^ {\prime} \left(X ^ {\top} W (t)\right) \phi^ {\prime} \left(W (t) ^ {\top} X\right) \right]. \tag {164}
+$$
+
+At initialization,  $\pmb{x}_i \sim N(0, I_d)$  and  $\pmb{w}_k(0) \sim N(0, d^\epsilon I_d)$ . Thus for fixed  $\pmb{x}_i$ , by Gaussian anti-concentration we have  $\operatorname*{Pr} | \pmb{x}_i^\top \pmb{w}_k | < \log d \leq O(1 / d^{1/2 + \epsilon_1})$  for some  $\epsilon_1 > 0$ . In addition, note that  $\| \pmb{w}_k(t) - \pmb{w}_k(0) \|_2 = O(d^{-1/2})$  for all  $k$ , and therefore for  $i, j, k$  such that
+
+$\left|\pmb{x}_i^\top \pmb{w}_k(0)\right| > O(\log d)$  and  $\left|\pmb{x}_j^\top \pmb{w}_k(0)\right| > O(\log d)$ , we know that  $\left|\phi^{\prime}(\pmb{x}_i^\top \pmb{w}_k(t))\phi^{\prime}(\pmb{x}_j^\top \pmb{w}_k(t)) - \right.$ $\phi^{\prime}(\pmb{x}_i^\top \pmb{w}_k(0))\phi^{\prime}(\pmb{x}_j^\top \pmb{w}_k(0))| = O(d^{-2})$
+
+Given fixed  $\pmb{x}_i$ , define  $y_k = 1\{|\pmb{x}_i^\top \pmb{w}_k| < \log d\}$  as the indicator variable that the  $k$ -th neuron does not saturate. We know that  $\mathbb{E}[y_k] = O(1 / d^{1/2 + \epsilon_1})$ , and  $\mathrm{Var}[y_k] = \mathbb{E}[y_k^2] - \mathbb{E}[y_k]^2 = O(1 / d^{1/2 + \epsilon_1})$ . By Bernstein's inequality
+
+$$
+\Pr \left| \frac {1}{h} \sum_ {k = 1} ^ {h} y _ {k} - \mathbb {E} \left[ y _ {k} \right] \right| > \varepsilon \leq 2 \exp \left(- \frac {h \varepsilon^ {2}}{2 \sigma^ {2} + 2 \varepsilon / 3}\right). \tag {165}
+$$
+
+Setting  $\varepsilon = \sqrt{\frac{c\log h}{h^{1 + \epsilon_2}}}$ , we know that with probability at least  $1 - h^{-c}$
+
+$$
+\frac {1}{h} \sum_ {k = 1} ^ {h} y _ {k} \leq \varepsilon + \mathbb {E} \left[ y _ {k} \right] = O \left(\frac {\operatorname {p o l y} \log h}{h ^ {1 / 2 + \epsilon_ {3}}}\right). \tag {166}
+$$
+
+Therefore, given  $\pmb{x}_i$  and  $\pmb{x}_j$ , for large enough  $c_1$  with probability at least  $1 - h^{-3}$  we have
+
+$$
+\left| \sum_ {k = 1} ^ {h} \phi^ {\prime} \left(\boldsymbol {w} _ {k} (t) ^ {\top} \boldsymbol {x} _ {i}\right) \phi^ {\prime} \left(\boldsymbol {w} _ {k} (t) ^ {\top} \boldsymbol {x} _ {j}\right) - \sum_ {k = 1} ^ {h} \phi^ {\prime} \left(\boldsymbol {w} _ {k} (0) ^ {\top} \boldsymbol {x} _ {i}\right) \phi^ {\prime} \left(\boldsymbol {w} _ {k} (0) ^ {\top} \boldsymbol {x} _ {j}\right) \right| = O \left(h ^ {1 / 2 - \epsilon_ {4}}\right), \tag {167}
+$$
+
+in which we utilized the boundedness of  $\phi'$ . Taking union bound over  $d^2$  elements in the random feature matrix yields
+
+$$
+\begin{array}{l} \left\| K (t) - K (0) \right\| _ {2} \\ = \left\| X ^ {\top} X \circ \frac {1}{h} \left[ \phi^ {\prime} (X ^ {\top} W (t)) \phi^ {\prime} (W (t) ^ {\top} X) - \phi^ {\prime} (X ^ {\top} W (0)) \phi^ {\prime} (W (0) ^ {\top} X) \right] \right\| \\ \leq \frac {1}{h} \left\| X ^ {\top} X \right\| _ {2} \max  \left\{\left| \phi^ {\prime} (X ^ {\top} W (t)) \phi^ {\prime} (W (t) ^ {\top} X) - \phi^ {\prime} (X ^ {\top} W (0)) \phi^ {\prime} (W (0) ^ {\top} X) \right| _ {i j} \right\} \\ \leq \frac {1}{h} O (d) O \left(d ^ {1 / 2 - \epsilon^ {\prime}}\right) = O \left(d ^ {1 / 2 - \epsilon^ {\prime}}\right). \tag {168} \\ \end{array}
+$$
+
+Using the exact same argument, one can derive that  $\| \pmb{u}_{NN}(\hat{\pmb{x}}) - \pmb{u}_{NTK}(\hat{\pmb{x}})\| _2 = O(d^{1 / 2 - \epsilon '})$ , the proof of which we omit.
+
+![](images/77bc883657bd117e38cf8e02da99fbeb93d6ef1cb6e3fcc1e9a4b1453c9a9abb.jpg)
+
+# E ADDITIONAL RESULTS
+
+# E.1 RISK OF ReLU NETWORK UNDER SYMMETRIC DATA
+
+If the dataset is symmetric, that is
+
+$$
+\forall i \in [ 1, n ], \exists ! j \in [ 1, n ] \text {s . t .} x _ {i} + x _ {j} = 0, \tag {A5}
+$$
+
+then population risk of the gradient flow solution can be given explicitly for certain nonlinearities:
+
+Proposition 20. Given (A1-3)(A5), if the nonlinearity satisfies  $\phi^{\prime}(\pmb {x}) + \phi^{\prime}(-\pmb {x}) = C$  for constant  $C$  then as  $n,d,h\to \infty$
+
+$$
+R _ {(\gamma_ {1} <   0. 5)} (\hat {f}) \rightarrow \frac {2 \gamma_ {1}}{1 - 2 \gamma_ {1}} \sigma^ {2}; R _ {(\gamma_ {1} \geq 0. 5)} (\hat {f}) = \left(1 - \frac {1}{2 \gamma_ {1}}\right) r ^ {2} + \frac {1}{2 \gamma_ {1} - 1} \sigma^ {2}. \tag {169}
+$$
+
+Note that the requirement on the nonlinearity holds for ReLU and SoftPlus. This expression is again independent of  $\gamma_{2}$  and aligns with the experimental results in Figure 9 (we only plot the bias component for verification). In addition, the bias is upper-bounded by the null risk for all  $\gamma_{1}$ . We remark that the symmetry assumption does not hold for i.i.d. samples from symmetric distributions, and Figure 9 demonstrates that the additional condition alters the risk.
+
+![](images/3a7e3fa84141210e56572b8508e99ce3afef787c69981f3b0d4f15b409fc12f2.jpg)  
+(a) bias of vanishing initialization.  
+Figure 9: Bias of two-layer ReLU networks with optimized first layer under Gaussian data and linear teacher. Individual dotted lines correspond to different  $\gamma_{2}$  (from 0.2 to 2) which is independent to the risk. (a) Vanishing initialization. The bias under symmetric data is predicted by Proposition 20. (b) Non-vanishing initialization. The red and blue lines represent models optimized from i.i.d. and symmetric initialization, respectively. The bias for symmetric initialization is predicted by Theorem 8.
+
+![](images/909d5be4c22251a0dcd4209acb516ee9ccab5cbb233d4683bd617be6902c18ce.jpg)  
+(b) bias of non-vanishing initialization.
+
+Proof. Without loss of generality assume  $X = [X_0, -X_0]$ . Then by (100) we have
+
+$$
+\frac {\partial \boldsymbol {w} _ {+}}{\partial t} = \frac {1}{2 n _ {0}} \sum_ {i = 1} ^ {2 n _ {0}} \left[ \left(y _ {i} - h _ {0} \phi \left(\boldsymbol {w} _ {+} ^ {\top} \boldsymbol {x} _ {i}\right) + h _ {0} \phi \left(\boldsymbol {w} _ {-} ^ {\top} \boldsymbol {x} _ {i}\right)\right) \phi^ {\prime} \left(\boldsymbol {w} _ {+} ^ {\top} \boldsymbol {x} _ {i}\right) \boldsymbol {x} _ {i} \right], \tag {170}
+$$
+
+and the flow for  $w_{-}$  follows from symmetry. In this case one can show that from exact zero initialization, for nonlinearity satisfying  $\phi(x) - \phi(-x) = x$ , such as ReLU and SoftPlus,
+
+$$
+\frac {\partial \left(\boldsymbol {w} _ {+}\right)}{\partial t} + \frac {\partial \left(\boldsymbol {w} _ {-}\right)}{\partial t} = \frac {1}{2 n _ {0}} \sum_ {i = 1} ^ {2 n _ {0}} \left[ \left(y _ {i} - h _ {0} \phi \left(\boldsymbol {w} _ {+} ^ {\top} \boldsymbol {x} _ {i}\right) + h _ {0} \phi \left(\boldsymbol {w} _ {-} ^ {\top} \boldsymbol {x} _ {i}\right)\right) \left(\phi^ {\prime} \left(\boldsymbol {w} _ {+} ^ {\top} \boldsymbol {x} _ {i}\right) - \phi^ {\prime} \left(\boldsymbol {w} _ {-} ^ {\top} \boldsymbol {x} _ {i}\right)\right) \boldsymbol {x} _ {i} \right] = 0. \tag {171}
+$$
+
+And therefore the gradient flow of  $\pmb{w}_{+}$  is
+
+$$
+\begin{array}{l} \frac {\partial \boldsymbol {w} _ {+}}{\partial t} = \frac {1}{2 n _ {0}} \sum_ {i = 1} ^ {2 n _ {0}} \left[ \left(y _ {i} - h _ {0} \phi (\boldsymbol {w} _ {+} ^ {\top} \boldsymbol {x} _ {i}) + h _ {0} \phi (- \boldsymbol {w} _ {+} ^ {\top} \boldsymbol {x} _ {i})\right) \phi^ {\prime} (\boldsymbol {w} _ {+} ^ {\top} \boldsymbol {x} _ {i}) \boldsymbol {x} _ {i} \right] \\ = \frac {1}{2 n _ {0}} \sum_ {i = 1} ^ {n _ {0}} \left[ \left(y _ {i} - h \phi \left(\boldsymbol {w} _ {+} ^ {\top} \boldsymbol {x} _ {i}\right) + h _ {0} \phi \left(- \boldsymbol {w} _ {+} ^ {\top} \boldsymbol {x} _ {i}\right)\right) \left(\phi^ {\prime} \left(\boldsymbol {w} _ {+} ^ {\top} \boldsymbol {x} _ {i}\right) + \phi^ {\prime} \left(- \boldsymbol {w} _ {+} ^ {\top} \boldsymbol {x} _ {i}\right)\right) \boldsymbol {x} _ {i} \right] \\ = \frac {1}{2 n _ {0}} \sum_ {i = 1} ^ {n _ {0}} \left[ \left(y _ {i} - h _ {0} \boldsymbol {w} _ {+} ^ {\top} \boldsymbol {x} _ {i}\right) \boldsymbol {x} _ {i} \right] = \frac {1}{2 n _ {0}} X _ {0} \boldsymbol {y} _ {0} - \frac {1}{2 n _ {0}} h _ {0} X _ {0} X _ {0} ^ {\top} \boldsymbol {w} _ {+}. \tag {172} \\ \end{array}
+$$
+
+The flow of  $\boldsymbol{w}_{-}$  follows from symmetry. Solving for the stationary points (i.e. gradient becomes zero), it the clear that
+
+$$
+\boldsymbol {w} _ {+} ^ {(t = \infty)} = - \boldsymbol {w} _ {-} ^ {(t = \infty)} = \left\{ \begin{array}{l l} \frac {1}{h _ {0}} \left(X X ^ {\top}\right) ^ {- 1} X \boldsymbol {y}, & \gamma_ {1} <   0. 5, \\ \frac {1}{h _ {0}} X \left(X ^ {\top} X\right) ^ {- 1} \boldsymbol {y}, & \gamma_ {1} > 0. 5. \end{array} \right. \tag {173}
+$$
+
+And hence the asymptotic risk is
+
+$$
+R _ {(\gamma_ {1} <   0. 5)} \rightarrow \frac {2 \gamma_ {1}}{1 - 2 \gamma_ {1}} \sigma^ {2}; \quad R _ {(\gamma_ {1} \geq 0. 5)} = \left(1 - \frac {1}{2 \gamma_ {1}}\right) r ^ {2} + \frac {1}{2 \gamma_ {1} - 1} \sigma^ {2}. \tag {174}
+$$
+
+The same conclusion holds for vanishing initialization if we assume that the trajectory stays close to that of exact zero initialization. Note that although the prediction aligns well with the experimental results, the argument in Theorem 7 does not directly apply due to the undefined derivative of ReLU at the origin, and thus this result is not rigorously justified.
+
+# F EXPERIMENT SETUP
+
+Optimizing the Second Layer. We compute the minimum-norm solution by directly solving the pseudo-inverse. We set  $n = 1000$  and vary  $\gamma_1, \gamma_2$  from 0.1 to 3. The linear teacher model  $F(\pmb{x}) = \pmb{x}^\top \pmb{\beta}$  is fixed as  $\beta = -\mathbf{1}_d / \sqrt{d}$ . For each  $(\gamma_1, \gamma_2)$  we average across 50 random draws of data.
+
+**Optimizing the First Layer.** For both initializations, we use gradient descent with small step size  $(\eta = 0.1)$  and train the model for minimally 25000 steps and till  $\| \nabla_W f(X,W) \|_F^2 < 10^{-6}$ . We fix  $n = 320$  and vary  $\gamma_1, \gamma_2$  from 0.1 to 3 with the same linear teacher model  $\beta = -\mathbf{1}_d / \sqrt{d}$ . The risk is averaged across 20 models trained from different initializations.
\ No newline at end of file
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+# GEOM-GCN: GEOMETRIC GRAPH CONVOLUTIONAL NETWORKS
+
+Hongbin Pei $^{1,5,*}$
+
+peihb15@mails.jlu.edu.cn
+
+Bingzhe Wei2
+
+bwei6@illinois.edu
+
+Kevin Chen-Chuan Chang $^{2,3}$
+
+kcchang@illinois.edu
+
+Yu Lei
+
+csylei@comp.polyu.edu.hk
+
+Bo Yang $^{1,5,\dagger}$
+
+ybo@jlu.edu.cn
+
+<sup>1</sup>College of Computer Science and Technology, Jilin University, China  
+$^{2}$ Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, USA  
+$^{3}$ Department of Computer Science, University of Illinois at Urbana-Champaign, USA  
+$^{4}$ Department of Computing, Hong Kong Polytechnic University, Hong Kong  
+<sup>5</sup>Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, China
+
+# ABSTRACT
+
+Message-passing neural networks (MPNNs) have been successfully applied to representation learning on graphs in a variety of real-world applications. However, two fundamental weaknesses of MPNNs' aggregators limit their ability to represent graph-structured data: losing the structural information of nodes in neighborhoods and lacking the ability to capture long-range dependencies in disassortative graphs. Few studies have noticed the weaknesses from different perspectives. From the observations on classical neural network and network geometry, we propose a novel geometric aggregation scheme for graph neural networks to overcome the two weaknesses. The behind basic idea is the aggregation on a graph can benefit from a continuous space underlying the graph. The proposed aggregation scheme is permutation-invariant and consists of three modules, node embedding, structural neighborhood, and bi-level aggregation. We also present an implementation of the scheme in graph convolutional networks, termed Geom-GCN, to perform transductive learning on graphs. Experimental results show the proposed Geom-GCN achieved state-of-the-art performance on a wide range of open datasets of graphs.
+
+# 1 INTRODUCTION
+
+Message-passing neural networks (MPNNs), such as GNN (Scarselli et al., 2008), ChebNet (Defferrard et al., 2016), GG-NN (Li et al., 2016), GCN (Kipf & Welling, 2017), are powerful for learning on graphs with various applications ranging from brain networks to online social network (Gilmer et al., 2017; Wang et al., 2019). In a layer of MPNNs, each node sends its feature representation, a "message", to the nodes in its neighborhood; and then updates its feature representation by aggregating all "messages" received from the neighborhood. The neighborhood is often defined as the set of adjacent nodes in graph. By adopting permutation-invariant aggregation functions (e.g., summation, maximum, and mean), MPNNs are able to learn representations which are invariant to isomorphic graphs, i.e., graphs that are topologically identical.
+
+Although existing MPNNs have been successfully applied in a wide variety of scenarios, two fundamental weaknesses of MPNNs' aggregators limit their ability to represent graph-structured data. Firstly, the aggregators lose the structural information of nodes in neighborhoods. Permutation invariance is an essential requirement for any graph learning method. To meet it, existing MPNNs adopt permutation-invariant aggregation functions which treat all "messages" from neighborhood as
+
+a set. For instance, GCN simply sums the normalized "messages" from all one-hop neighbors (Kipf & Welling, 2017). Such aggregation loses the structural information of nodes in neighborhood because it does not distinguish the "messages" from different nodes. Therefore, after such aggregation, we cannot know which node contributes what to the final aggregated output.
+
+Without modeling such structural information, as shown in (Kondor et al., 2018) and (Xu et al., 2019), the existing MPNNs cannot discriminate between certain non-isomorphic graphs. In those cases, MPNNs may map non-isomorphic graphs to the same feature representations, which is obviously not desirable for graph representation learning. Unlike MPNNs, classical convolutional neural networks (CNNs) avoid this problem by using aggregators (i.e., convolutional filters) with a structural receiving file defined on grids, i.e., a Euclidean space, and are hence able to distinguish each input unit. As shown by our experiments, such structural information often contains clues regarding topology patterns in graph (e.g., hierarchy), and should be extracted and used to learn more discriminating representations for graph-structured data.
+
+Secondly, the aggregators lack the ability to capture long-range dependencies in disassortative graphs. In MPNNs, the neighborhood is defined as the set of all neighbors one hop away (e.g., GCN), or all neighbors up to  $r$  hops away (e.g., ChebNet). In other words, only messages from nearby nodes are aggregated. The MPNNs with such aggregation are inclined to learn similar representations for proximal nodes in a graph. This implies that they are probably desirable methods for assortative graphs (e.g., citation networks (Kipf & Welling, 2017) and community networks (Chen et al., 2019)) where node homophily holds (i.e., similar nodes are more likely to be proximal, and vice versa), but may be inappropriate to the disassortative graphs (Newman, 2002) where node homophily does not hold. For example, Ribeiro et al. (2017) shows disassortative graphs where nodes of the same class exhibit high structural similarity but are far apart from each other. In such cases, the representation ability of MPNNs may be limited significantly, since they cannot capture the important features from distant but informative nodes.
+
+A straightforward strategy to address this limitation is to use a multi-layered architecture so as to receive "messages" from distant nodes. For instance, due to the localized nature of convolutional filters in classical CNNs, a single convolutional layer is similarly limited in its representational ability. CNNs typically use multiple layers connected in a hierarchical manner to learn complex and global representations. However, unlike CNNs, it is difficult for multi-layer MPNNs to learn good representations for disassortative graphs because of two reasons. On one hand, relevant messages from distant nodes are mixed indistinguishably with a large number of irrelevant messages from proximal nodes in multi-layer MPNNs, which implies that the relevant information will be "washed out" and cannot be extracted effectively. On the other hand, the representations of different nodes would become very similar in multi-layer MPNNs, and every node's representation actually carries the information about the entire graph (Xu et al., 2018).
+
+In this paper, we overcome the aforementioned weaknesses of graph neural networks starting from two basic observations: i) Classical neural networks effectively address the similar limitations thanks to the stationarity, locality, and compositionality in a continuous space (Bronstein et al., 2017); ii) The notion of network geometry bridges the gap between continuous space and graph (Hoff et al., 2002; Muscoloni et al., 2017). Network geometry aims to understand networks by revealing the latent continuous space underlying them, which assumes that nodes are sampled discretely from a latent continuous space and edges are established according to their distance. In the latent space, complicated topology patterns in graphs can be preserved and presented as intuitive geometry, such as subgraph (Narayanan et al., 2016), community (Ni et al., 2019), and hierarchy (Nickel & Kiela, 2017; 2018). Inspired by those two observations, we raise an enlightening question about the aggregation scheme in graph neural network.
+
+- Can the aggregation on a graph benefit from a continuous latent space, such as using geometry in the space to build structural neighborhoods and capture long-range dependencies in the graph?
+
+To answer the above question, we propose a novel aggregation scheme for graph neural networks, termed the geometric aggregation scheme. In the scheme, we map a graph to a continuous latent space via node embedding, and then use the geometric relationships defined in the latent space to build structural neighborhoods for aggregation. Also, we design a bi-level aggregator operating on the structural neighborhoods to update the feature representations of nodes in graph neural networks, which are able to guarantee permutation invariance for graph-structured data. Compared with exist
+
+ing MPNNs, the scheme extracts more structural information of the graph and can aggregate feature representations from distant nodes via mapping them to neighborhoods defined in the latent space.
+
+We then present an implementation of the geometric aggregation scheme in graph convolutional networks, which we call Geom-GCN, to perform transductive learning, node classification, on graphs. We design particular geometric relationships to build the structural neighborhood in Euclidean and hyperbolic embedding space respectively. We choose different embedding methods to map the graph to a suitable latent space for different applications, where suitable topology patterns of graph are preserved. Finally, we empirically validate and analyze Geom-GCN on a wide range of open datasets of graphs, and Geom-GCN achieved the state-of-the-art results.
+
+In summary, the contribution of this paper is three-fold: i) We propose a novel geometric aggregation scheme for graph neural network, which operates in both graph and latent space, to overcome the aforementioned two weaknesses; ii) We present an implementation of the scheme, Geom-GCN, for transductive learning in graph; iii) We validate and analyze Geom-GCN via extensive comparisons with state-of-the-art methods on several challenging benchmarks.
+
+# 2 GEOMETRIC AGGREGATION SCHEME
+
+In this section, we start by presenting the geometric aggregation scheme, and then outline its advantages and limitations compared to existing works. As shown in Fig. 1, the aggregation scheme consists of three modules, node embedding (panel A1 and A2), structural neighborhood (panel B1 and B2), and bi-level aggregation (panel C). We will elaborate on them in the following.
+
+![](images/61612b810dd4006ad4ef7dd82ac9c9e70d7bc05eda37ad1ef43bce79eade401d.jpg)
+
+![](images/a726d9b9d26a313b6f97062a5049fc8eeac0de673fa2cfd64d9b98a5b16cca5b.jpg)
+
+![](images/47dcf1ef0307476ade7fb4a84472febc7c9161033ec888d6207a221f476b37e5.jpg)
+
+![](images/20fa48b84e041cb1a2a2cde9ad5153bc0255f28c8609f690b928f4b2bf5fd6bb.jpg)  
+Figure 1: An illustration of the geometric aggregation scheme. A1-A2 The original graph is mapped to a latent continuous space. B1-B2 The structural neighborhood. All adjacent nodes lie in a small region around a center node in B1 for visualization. In B2, the neighborhood in the graph contains all adjacent nodes in graph; the neighborhood in the latent space contains the nodes within the dashed circle whose radius is  $\rho$ . The relational operator  $\tau$  is illustrated by a colorful  $3 \times 3$  grid where each unit is corresponding to a geometric relationship to the red target node. C Bi-level aggregation on the structural neighborhood. Dashed and solid arrows denote the low-level and high-level aggregation, respectively. Blue and green arrows denote the aggregation on the neighborhood in the graph and the latent space, respectively.
+
+![](images/20e7c0297d05525431bc43a42412fda81ccaae3ff079601d6569c915b839e695.jpg)
+
+A. Node embedding. This is a fundamental module which maps the nodes in a graph to a latent continuous space. Let  $\mathcal{G} = (V, E)$  be a graph, where each node  $v \in V$  has a feature vector  $\boldsymbol{x}_v$  and each edge  $e \in E$  connects two nodes. Let  $f: v \to \boldsymbol{z}_v$  be a mapping function from a node in graph to a representation vector. Here,  $\boldsymbol{z}_v \in \mathbb{R}^d$  can also be considered as the position of node  $v$  in a latent continuous space, and  $d$  is the number of dimensions of the space. During the mapping, the structure and properties of graph are preserved and presented as the geometry in the latent space. For instance, hierarchical pattern in graph is presented as the distance to the original in embedding hyperbolic space (Nickel & Kiela, 2017). One can employ various embedding methods to infer the latent space (Cai et al., 2018; Wang et al., 2018).
+
+B. Structural neighborhood. Based on the graph and the latent space, we then build a structural neighborhood,  $\mathcal{N}(v) = (\{N_g(v), N_s(v)\}, \tau)$ , for the next aggregation. The structural neighborhood consists of a set of neighborhood  $\{N_g(v), N_s(v)\}$ , and a relational operator on neighborhoods  $\tau$ .
+
+The neighborhood in the graph,  $N_{g}(v) = \{u|u\in V,(u,v)\in E\}$ , is the set of adjacent nodes of  $v$ . The neighborhood in the latent space,  $N_{s}(v) = \{u|u\in V,d(\pmb{z}_{u},\pmb{z}_{v}) < \rho \}$ , is the set of nodes from which the distance to  $v$  is less than a pre-given parameter  $\rho$ . The distance function  $d(\cdot ,\cdot)$  depends on the particular metric in the space. Compared with  $N_{g}(v)$ ,  $N_{s}(v)$  may contain nodes which are far from  $v$  in the graph, but have a certain similarity with  $v$ , and hence are mapped together with  $v$  in the latent space though preserving the similarity. By aggregating on such neighborhood  $N_{s}(v)$ , the long-range dependencies in disassortative graphs can be captured.
+
+The relational operator  $\tau$  is a function defined in the latent space. It inputs an ordered position pair  $(z_v, z_u)$  of nodes  $v$  and  $u$ , and outputs a discrete variable  $r$  which indicates the geometric relationship from  $v$  to  $u$  in the latent space. For  $u, v \in V$ ,
+
+$$
+\tau : \left(\boldsymbol {z} _ {v}, \boldsymbol {z} _ {u}\right)\rightarrow r \in R,
+$$
+
+where  $R$  is the set of the geometric relationships. According to the particular latent space and application,  $r$  can be specified as an arbitrary geometric relationship of interest. A requirement on  $\tau$  is that it should guarantee that each ordered position pair has only one geometric relationship. For example,  $\tau$  is illustrated in Fig. 1B by a colorful  $3 \times 3$  grid in a 2-dimensional Euclidean space, in which each unit is corresponding to a geometric relationship to node  $v$ .
+
+C. Bi-level aggregation. With the structural neighborhood  $\mathcal{N}(v)$ , we propose a novel bi-level aggregation scheme for graph neural network to update the hidden features of nodes. The bi-level aggregation consists of two aggregation functions and operates in a neural network layer. It can extract effectively structural information of nodes in neighborhoods as well as guarantee permutation invariance for graph. Let  $h_v^l$  be the hidden features of node  $v$  at the  $l$ -th layer, and  $h_v^0 = x_v$  be the node features. The  $l$ -th layer updates  $h_v^l$  for every  $v \in V$  by the following.
+
+$\pmb{e}_{(i,r)}^{v,l + 1} = p(\{\pmb{h}_u^l | u \in N_i(v), \tau(\pmb{z}_v, \pmb{z}_u) = r\}), \forall i \in \{g,s\}, \forall r \in R$  (Low-level aggregation)
+
+$\pmb{m}_{v}^{l + 1} = q_{i\in \{g,s\},r\in R}((\pmb{e}_{(i,r)}^{v,l + 1},(i,r)))$  (High-level aggregation) (1)
+
+$\pmb{h}_{v}^{l + 1} = \sigma (\pmb{W}_{l}\cdot \pmb{m}_{v}^{l + 1})$  (Non-linear transform)
+
+In the low-level, the hidden features of nodes that are in the same neighborhood  $i$  and have the same geometric relationship  $r$  are aggregated to a virtual node via the aggregation function  $p$ . The features of the virtual node are  $e_{(i,r)}^{v,l+1}$ , and the virtual node is indexed by  $(i,r)$  which is corresponding to the combination of a neighborhood  $i$  and a relationship  $r$ . It is required to adopt a permutation-invariant function for  $p$ , such as an  $L_p$ -norm (the choice of  $p = 1,2$ , or  $\infty$  results in average, energy, or max pooling). The low level aggregation is illustrated by dashed arrows in Fig. 1C.
+
+In the high-level, the features of virtual nodes are further aggregated by function  $q$ . The inputs of function  $q$  contain both the features of virtual nodes  $e_{(i,r)}^{v,l+1}$  and the identity of virtual nodes  $(i,r)$ . That is,  $q$  can be a function that takes an ordered object as input, e.g., concatenation, to distinguish the features of different virtual nodes, thereby extracting the structural information in the neighborhoods explicitly. The output of high-level aggregation is a vector  $m_v^{l+1}$ . Then new hidden features of  $v$ ,  $h_v^{(l+1)}$ , are given by a non-linear transform, wherein  $W_l$  is a learnable weight matrix on the  $l$ -th layer shared by all nodes, and  $\sigma(\cdot)$  is a non-linear activation function, e.g., a ReLU.
+
+Permutation invariance is an essential requirement for aggregators in graph neural networks. Thus, we then prove that the proposed bi-level aggregation, Eq. 1, is able to guarantee invariance for any permutation of nodes. We firstly give a definition for permutation-invariant mapping of graph.
+
+Definition 1. Let a bijective function  $\psi : V \to V$  be a permutation for nodes, which renames  $v \in V$  as  $\psi(v) \in V$ . Let  $V'$  and  $E'$  be the node and edge set after a permutation  $\psi$ , respectively. A mapping of graph,  $\phi(\mathcal{G})$ , is permutation-invariant if, given any permutation  $\psi$ , we have  $\phi(\mathcal{G}) = \phi(\mathcal{G}')$ ,  $\mathcal{G}' = (V', E')$ .
+
+Lemma 1. For a composite function  $\phi_1\circ \phi_2(\mathcal{G})$ , if  $\phi_2(\mathcal{G})$  is permutation-invariant, the entire composite function  $\phi_1\circ \phi_2(\mathcal{G})$  is permutation-invariant.
+
+Proof. Let  $\mathcal{G}'$  be an isomorphic graph of  $\mathcal{G}$  after a permutation  $\psi$ , as defined in Definition 1. If  $\phi_2(\mathcal{G})$  is permutation-invariant, we have  $\phi_2(\mathcal{G}) = \phi_2(\mathcal{G}')$ . Therefore, the entire composite function  $\phi_1 \circ \phi_2(\mathcal{G})$  is permutation-invariant because  $\phi_1 \circ \phi_2(\mathcal{G}) = \phi_1 \circ \phi_2(\mathcal{G}')$ .
+
+Theorem 1. Given a graph  $\mathcal{G} = (V, E)$  and its structural neighborhood  $\mathcal{N}(v), \forall v \in V$ , the bi-level aggregation, Eq. 1, is a permutation-invariant mapping of graph.
+
+Proof. The bi-level aggregation, Eq. 1, is a composite function, where the low-level aggregation is the input of the high-level aggregation. Thus, Eq. 1 is permutation-invariant if the low-level aggregation is permutation-invariant according to Lemma 1.
+
+We then prove that the low-level aggregation is permutation-invariant. The low-level aggregation consists of  $2 \times |R|$  sub-aggregations, each of which is corresponding to the nodes in a neighborhood  $i$  and with a relationship  $r$  to  $v$ . Firstly, the input of each sub-aggregations is permutation-invariant because both  $i \in \{g, s\}$  and  $r \in R$  are determined by the given structural neighborhood  $\mathcal{N}(v), \forall v \in V$ , which is constant for any permutation. Secondly, Eq. 1 adopts a permutation-invariant aggregation function  $p$  for the sub-aggregations. Thus the low-level aggregation is permutation-invariant.
+
+# 2.1 COMPARISONS TO RELATED WORK
+
+We now discuss how the proposed geometric aggregation scheme overcomes the two aforementioned weaknesses, i.e., how it effectively models the structural information and captures the long-range dependencies, in comparison to some closely related works.
+
+To overcome the first weakness of MPNNs, i.e., losing the structural information of nodes in neighborhoods, the proposed scheme explicitly models the structural information by exploiting the geometric relationship between nodes in latent space and then extracting the information effectively by using the bi-level aggregations. In contrast, several existing works attempt to learn some implicit structure-like information to distinguish different neighbors when aggregating features. For example, GAT (Velickovic et al., 2017), LGCL (Gao et al., 2018) and GG-NN (Li et al., 2016) learn weights on "messages" from different neighbors by using attention mechanisms and node and/or edge attributes. CCN (Kondor et al., 2018) utilizes a covariance architecture to learn structure-aware representations. The major difference between these works and ours is that we offer an explicit and interpretable way to model the structural information of nodes in neighborhood, with the assistance of the geometry in a latent space. We note that our work is orthogonal with existing methods and thus can be readily incorporated to further improve their performance. In particular, we exploit geometric relationships from the aspect of graph topology, while other methods focus on that of feature representation—the two aspects are complementary.
+
+For the second weakness of MPNNs, i.e., lacking the ability to capture long-range dependencies, the proposed scheme models the long-range dependencies in disassortative graphs in two different ways. First of all, the distant (but similar) nodes in the graph can be mapped into a latent-space-based neighborhood of the target node, and then their useful feature representations can be used for aggregations. This way depends on an appropriate embedding method, which is able to preserve the similarities between the distant nodes and the target node. On the other hand, the structural information enables the method to distinguish different nodes in a graph-based neighborhood (as mentioned above). The informative nodes may have some special geometric relationships to the target node (e.g., a particular angle or distance), whose relevant features hence will be passed to the target node with much higher weights, compared to the uninformative nodes. As a result, the long-range dependencies are captured indirectly through the whole message propagation process in all graph-based neighborhoods. In literature, a recent method JK-Nets (Xu et al., 2018) captures the long-range dependencies by skipping connections during feature aggregations.
+
+# 2.1.1 CASE STUDY ON DistinguISHING NON-ISOMORPHIC GRAPHS
+
+In literature, Kondor et al. (2018) and Xu et al. (2019) construct several non-isomorphic example graphs that cannot be distinguished by the aggregators (e.g., mean and maximum) in existing MPNNs. We present a case study to illustrate how to distinguish the non-isomorphic example graphs once the structural neighborhood is applied. We take two non-isomorphic graphs in (Xu et al., 2019) as an example, where each node has the same feature  $a$  and after any mapping  $f(a)$  remains the same across all nodes, as shown in Fig. 2 (left). Then the aggregator, e.g., mean or maximum, over  $f(a)$  remains  $f(a)$ , and hence the final representations of the nodes are the same. That is, mean and maximum aggregators fail to distinguish the two different graphs.
+
+![](images/838bb533e51a3ef133fcdb2c1c2047c3f0cc8c3a4fab533a839203fffea09209.jpg)  
+Figure 2: An illustration to distinguish non-isomorphic graphs by proposed structural neighborhood.
+
+In contrast, the two graphs become distinguishable once we apply a structural neighborhood in aggregation. With the structural neighborhood, the nodes have different geometric relationships to the center node  $V_{1}$  in the structural neighborhood, as shown in Fig. 2 (right). Taking aggregation for  $V_{1}$  as an example, we can adopt different mapping function  $f_{r}, r \in R$  to the neighbors with different geometric relationship  $r$  to  $V_{1}$ . Then, the aggregator in two graph have different inputs,  $\{f_{2}(a), f_{8}(a)\}$  in the left graph and  $\{f_{2}(a), f_{7}(a), f_{9}(a)\}$  in the right graph. Finally, the aggregator (mean or maximum) will output different representations for the node  $V_{1}$  in the two graphs, thereby distinguishing the topological difference between the two graphs.
+
+# 3 GEOM-GCN: AN IMPLEMENTATION OF THE SCHEME
+
+In this section, we present Geom-GCN, a specific implementation of the geometric aggregation scheme in graph convolutional networks, to perform transductive learning in graphs. To implement the general aggregation scheme, one needs to specify its three modules: node embedding, structural neighborhood, and bi-level aggregation function.
+
+Node embedding is the fundamental. As shown in our experiments, a common embedding method which only preserves the connection and distance pattern in a graph can already benefit the aggregation. For particular applications, one can specify embedding methods to create suitable latent spaces where particular topology patterns (e.g., hierarchy) are preserved. We employ three embedding methods, Isomap (Tenenbaum et al., 2000), Poincare embedding (Nickel & Kiela, 2017), and struc2vec (Ribeiro et al., 2017), which result in three Geom-GCN variants: Geom-GCN-I, GeomGCN-P, and Geom-GCN-S. Isomap is a widely used isometry embedding method, by which distance patterns (lengths of shortest paths) are preserved explicitly in the latent space. Poincare embedding and struc2vec can create particular latent spaces that preserve hierarchies and local structures in a graph, respectively. We use an embedding space of dimension 2 for ease of explanation.
+
+The structural neighborhood  $\mathcal{N}(v) = (\{N_g(v), N_s(v)\}, \tau)$  of node  $v$  includes its neighborhoods in both the graph and latent space. The neighborhood-in-graph  $N_g(v)$  consists of the set of  $v$ 's adjacent nodes in the graph, and the neighborhood-in-latent-space  $N_s(v)$  those nodes whose distances to  $v$  are less than a parameter  $\rho$  in the latent space. We determine  $\rho$  by increasing  $\rho$  from zero until the average cardinality of  $N_s(v)$  equals to that of  $N_g(v), \forall v \in V$ -i.e., when the average neighborhood sizes in the graph and latent spaces are the same. We use Euclidean distance in the Euclidean space. In the hyperbolic space, we approximate the geodesic distance between two nodes via their Euclidean distance in the local tangent plane.
+
+Here we simply implement the geometric operator  $\tau$  as four relationships of the relative positions between two nodes in a 2-D Euclidean or hyperbolic space. Particularly, the relationship set  $R = \{\text{upper left, upper right, lower left, lower right}\}$ , and a  $\tau(z_v, z_u)$  is given by Table 1. Note that, we adopt the rectangular coordinate system in the Euclidean space and angular coordinate in the hyperbolic space. By this way, the relationship "upper" indicates the node nearer to the origin and thus lie in a higher level in a hierarchical graph. One can design a more sophisticated operator  $\tau$ , such as borrowing the structure of descriptors in manifold geometry (Kokkinos et al., 2012; Monti et al., 2017), thereby preserving more and richer structural information in neighborhood.
+
+Table 1: The relationship operator  
+
+<table><tr><td>τ(zv, zu)</td><td>zv[0] &gt; zu[0]</td><td>zv[0] ≤ zu[0]</td></tr><tr><td>zv[1] ≤ zu[1]</td><td>upper left</td><td>upper right</td></tr><tr><td>zv[1] &gt; zu[1]</td><td>lower left</td><td>lower right</td></tr></table>
+
+Finally, to implement the bi-level aggregation, we adopt the same summation of normalized hidden features as GCN (Kipf & Welling, 2017) as the aggregation function  $p$  in the low-level aggregation,
+
+$$
+\boldsymbol {e} _ {(i, r)} ^ {v, l + 1} = \sum_ {u \in N _ {i} (v)} \delta (\tau (\boldsymbol {z} _ {v}, \boldsymbol {z} _ {u}), r) (\deg (v) \deg (u)) ^ {\frac {1}{2}} \boldsymbol {h} _ {u} ^ {l}, \forall i \in \{g, s \}, \forall r \in R,
+$$
+
+where  $\deg(v)$  is the degree of node  $v$  in graph, and  $\delta(\cdot, \cdot)$  is a Kronecker delta function that only allows the nodes with relationship  $r$  to  $v$  to be included. The features of all virtual nodes  $e_{(i,r)}^{v,l+1}$  are further aggregated in the high-level aggregation. The aggregation function  $q$  is a concatenation || for all layers except the final layer, which uses mean for its aggregation function. Then, the overall bi-level aggregation of Geom-GCN is given by
+
+$$
+\boldsymbol{h}_{v}^{l + 1} = \sigma (\boldsymbol{W}_{l}\cdot \underset {i\in \{g,s\}}{\|}\underset {r\in R}{\|}\boldsymbol{e}_{(i,r)}^{v,l + 1})
+$$
+
+where we use ReLU as the non-linear activation function  $\sigma (\cdot)$  and  $W_{l}$  is the weight matrix to estimate by backpropagation.
+
+# 4 EXPERIMENTS
+
+We validate Geom-GCN by comparing Geom-GCN's performance with the performance of Graph Convolutional Networks (GCN) (Kipf & Welling (2017)) and Graph Attention Networks (GAT) (Velickovic et al. (2017)). Two state-of-the-art graph neural networks, on transductive node-label classification tasks on a wide variety of open graph datasets.
+
+# 4.1 DATASETS
+
+We utilize nine open graph datasets to validate the proposed Geom-GCN. An overview summary of characteristics of the datasets is given in Table 2.
+
+Table 2: Datasets statistics  
+
+<table><tr><td>Dataset</td><td>Cora</td><td>Cite.</td><td>Pubm.</td><td>Cham.</td><td>Squi.</td><td>Actor</td><td>Corn.</td><td>Texa.</td><td>Wisc.</td></tr><tr><td># Nodes</td><td>2708</td><td>3327</td><td>19717</td><td>2277</td><td>5201</td><td>7600</td><td>183</td><td>183</td><td>251</td></tr><tr><td># Edges</td><td>5429</td><td>4732</td><td>44338</td><td>36101</td><td>217073</td><td>33544</td><td>295</td><td>309</td><td>499</td></tr><tr><td># Features</td><td>1433</td><td>3703</td><td>500</td><td>2325</td><td>2089</td><td>931</td><td>1703</td><td>1703</td><td>1703</td></tr><tr><td># Classes</td><td>7</td><td>6</td><td>3</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td></tr></table>
+
+Citation networks. Cora, CiteSeer, and Pubmed are standard citation network benchmark datasets (Sen et al., 2008; Namata et al., 2012). In these networks, nodes represent papers, and edges denote citations of one paper by another. Node features are the bag-of-words representation of papers, and node label is the academic topic of a paper.
+
+WebKB. WebKB<sup>1</sup> is a webpage dataset collected from computer science departments of various universities by Carnegie Mellon University. We use the three subdatasets of it, Cornell, Texas, and Wisconsin, where nodes represent web pages, and edges are hyperlinks between them. Node features are the bag-of-words representation of web pages. The web pages are manually classified into the five categories, student, project, course, staff, and faculty.
+
+Actor co-occurrence network. This dataset is the actor-only induced subgraph of the film-director-actor-writer network (Tang et al., 2009). Each nodes correspond to an actor, and the edge between two nodes denotes co-occurrence on the same Wikipedia page. Node features correspond to some keywords in the Wikipedia pages. We classify the nodes into five categories in term of words of actor's Wikipedia.
+
+Wikipedia network. Chameleon and squirrel are two page-page networks on specific topics in Wikipedia (Rozemberczki et al., 2019). In those datasets, nodes represent web pages and edges are mutual links between pages. And node features correspond to several informative nouns in the Wikipedia pages. We classify the nodes into five categories in term of the number of the average monthly traffic of the web page.
+
+# 4.2 EXPERIMENTAL SETUP
+
+As mentioned in Section 3, we construct three Geom-GCN variants by using three embedding methods, Isomap (Geom-GCN-I), Poincare (Geom-GCN-P), and struc2vec (Geom-GCN-S). We specify the dimension of embedding space as two, and use the relationship operator  $\tau$  defined in Table 1, and apply mean and concatenation as the low- and high- level aggregation function, respectively.
+
+With the structural neighborhood, we perform a hyper-parameter search for all models on validation set. For fairness, the size of search space for each method is the same. The searching hyperparameters include number of hidden unit, initial learning rate, weight decay, and dropout. We fix the number of layer to 2 and use Adam optimizer (Kingma & Ba, 2014) for all models. We use ReLU as the activation function for Geom-GCN and GCN, and ELU for GAT.
+
+The final hyper-parameter setting is dropout of  $p = 0.5$ , initial learning rate of 0.05, patience of 100 epochs, weight decay of  $5E - 6$  (WebKB datasets) or  $5E - 5$  (the other all datasets). In GCN, the number of hidden unit is 16 (Cora), 16 (CiteSeer), 64 (Pubmed), 32 (WebKB), 48 (Wikipedia), and 32 (Actor). In Geom-GCN, the number of hidden unit is 8 times as many as the number in GCN since Geom-GCN has 8 virtual nodes. For each attention head in GAT, the number of hidden unit is 8 (Citation networks), 32 (WebKB), 48 (Wikipedia), and 32 (Actor). GAT has 8 attention heads in layer one and 8 (Pubmed) or 1 (the all other datasets) attention heads in layer two.
+
+For all graph datasets, we randomly split nodes of each class into  $60\%$ ,  $20\%$ , and  $20\%$  for training, validation and testing. With the hyper-parameter setting, we report the average performance of all models on the test sets over 10 random splits.
+
+# 4.3 RESULTS AND ANALYSIS
+
+Results are summarized in Table 3. The reported numbers denote the mean classification accuracy in percent. In general, Geom-GCN achieves state-of-the-art performance. The best performing method is highlighted. From the results, Isomap embedding (Geom-GCN-I) which only preserves the connection and distance pattern in graph can already benefit the aggregation. We can also specify an embedding method to create a suitable latent space for a particular application (e.g., disassortative graph or hierarchical graph), by doing which a significant performance improvement is achieved (e.g., Geom-GCN-P).
+
+Table 3: Mean Classification Accuracy (Percent)  
+
+<table><tr><td>Dataset</td><td>Cora</td><td>Cite.</td><td>Pubm.</td><td>Cham.</td><td>Squi.</td><td>Actor</td><td>Corn.</td><td>Texa.</td><td>Wisc.</td></tr><tr><td>GCN</td><td>85.77</td><td>73.68</td><td>88.13</td><td>28.18</td><td>23.96</td><td>26.86</td><td>52.70</td><td>52.16</td><td>45.88</td></tr><tr><td>GAT</td><td>86.37</td><td>74.32</td><td>87.62</td><td>42.93</td><td>30.03</td><td>28.45</td><td>54.32</td><td>58.38</td><td>49.41</td></tr><tr><td>Geom-GCN-I</td><td>85.19</td><td>77.99</td><td>90.05</td><td>60.31</td><td>33.32</td><td>29.09</td><td>56.76</td><td>57.58</td><td>58.24</td></tr><tr><td>Geom-GCN-P</td><td>84.93</td><td>75.14</td><td>88.09</td><td>60.90</td><td>38.14</td><td>31.63</td><td>60.81</td><td>67.57</td><td>64.12</td></tr><tr><td>Geom-GCN-S</td><td>85.27</td><td>74.71</td><td>84.75</td><td>59.96</td><td>36.24</td><td>30.30</td><td>55.68</td><td>59.73</td><td>56.67</td></tr></table>
+
+# 4.3.1 ABLATION STUDY ON CONTRIBUTIONS FROM TWO NEIGHBORHOODS
+
+The proposed Geom-GCN aggregates "message" from two neighborhoods which are defined in graph and latent space respectively. In this section, we present an ablation study to evaluate the contribution from each neighborhood though constructing new Geom-GCN variants with only one neighborhood. For the variants with only neighborhood in graph, we use "g" as a suffix of their name (e.g., Geom-GCN-I-g), and use suffix "s" to denote the variants with only neighborhood in latent space (e.g., Geom-GCN-I-s). Here we set GCN as a baseline so that the contribution can be measured via the performance improvement comparing with GCN. The results are summarized in Table 4, where positive improvement is denoted by an up arrow  $\uparrow$  and negative improvement by a down arrow  $\downarrow$ . The best performing method is highlighted.
+
+We also design an index denoted by  $\beta$  to measure the homophily in a graph,
+
+$$
+\beta = \frac {1}{| V |} \sum_ {v \in V} \frac {\text {N u m b e r o f v ' s n e i g h b o r s w h o h a v e t h e s a m e l a b e l a s v}}{\text {N u m b e r o f v ' s n e i g h b o r s}}.
+$$
+
+A large  $\beta$  value implies that the homophily, in term of node label, is strong in a graph, i.e., similar nodes tend to connect together. From Table 4, one can see that assortative graphs (e.g., citation networks) have a much larger  $\beta$  than disassortative graphs (e.g., WebKB networks).
+
+Table 4 exhibits three interesting patterns: i) Neighborhoods in graph and latent space both benefit the aggregation in most cases; ii) Neighborhoods in latent space have larger contributions in disassortative graphs (with a small  $\beta$ ) than assortative ones, which implies relevant information from disconnected nodes is captured effectively by the neighborhoods in latent space; iii) To our surprise, several variants with only one neighborhood (in Table 4) achieve better performances than the variants with two neighborhoods (in Tabel 3). We think the reason is that Geom-GCN with two neighborhoods aggregate more irrelevant "messages" than Geom-GCN with only one neighborhood, and the irrelevant "messages" adversely affect the performance. Thus, we believe an attention mechanism can alleviate this issue- which we will study as future work.
+
+Table 4: Mean Classification Accuracy (Percent)  
+
+<table><tr><td>Dataset
+β</td><td>Cora
+0.83</td><td>Cite.
+0.71</td><td>Pumb.
+0.79</td><td>Cham.
+0.25</td><td>Squi.
+0.22</td><td>Actor
+0.24</td><td>Corn.
+0.11</td><td>Texa.
+0.06</td><td>Wisc.
+0.16</td></tr><tr><td>Geom-GCN-I-g</td><td>86.26
+↑0.48</td><td>80.64
+↑6.96</td><td>90.72
+↑2.59</td><td>68.00
+↑39.82</td><td>46.01
+↑22.05</td><td>31.96
+↑4.04</td><td>65.40
+↑12.70</td><td>72.51
+↑21.35</td><td>68.23
+↑22.35</td></tr><tr><td>Geom-GCN-I-s</td><td>77.34
+↓8.34</td><td>72.22
+↓1.46</td><td>85.02
+↓3.11</td><td>61.64
+↑33.46</td><td>37.98
+↑14.02</td><td>30.59
+↑2.67</td><td>62.16
+↑9.46</td><td>60.54
+↑8.38</td><td>64.90
+↑19.01</td></tr><tr><td>Geom-GCN-P-g</td><td>86.30
+↑0.52</td><td>75.45
+↑1.76</td><td>88.40
+↑0.27</td><td>63.07
+↑34.89</td><td>38.41
+↑14.45</td><td>31.55
+↑3.63</td><td>64.05
+↑11.35</td><td>73.05
+↑21.89</td><td>69.41
+↑23.53</td></tr><tr><td>Geom-GCN-P-s</td><td>73.14
+↓12.63</td><td>71.65
+↓2.04</td><td>86.95
+↓1.18</td><td>43.20
+↑15.02</td><td>30.47
+↑6.51</td><td>34.59
+↑6.67</td><td>75.40
+↑22.70</td><td>73.51
+↑21.35</td><td>80.39
+↑34.51</td></tr><tr><td>Geom-GCN-S-g</td><td>87.00
+↑1.23</td><td>75.73
+↑2.04</td><td>88.44
+↑0.31</td><td>67.04
+↑38.86</td><td>44.92
+↑20.96</td><td>31.27
+↑3.35</td><td>67.02
+↑14.32</td><td>71.62
+↑19.46</td><td>69.41
+↑23.52</td></tr><tr><td>Geom-GCN-S-s</td><td>66.92
+↓18.85</td><td>66.03
+↓7.65</td><td>79.41
+↓8.72</td><td>49.21
+↑21.03</td><td>31.27
+↑7.31</td><td>30.32
+↑2.40</td><td>62.43
+↑9.73</td><td>63.24
+↑11.08</td><td>64.51
+↑18.63</td></tr></table>
+
+# 4.3.2 ANALYSIS OF EMBEDDING SPACE COMBINATION
+
+The structural neighborhood in Geom-GCN is very flexible, where one can combine arbitrary embedding space. To study which combination of embedding spaces is desirable, we construct new Geom-GCN variants by adopting neighborhoods built by different embedding space. For the variants adopted Isomap and poincare embedding space to build neighborhood in graph and in latent space respectively, we use Geom-GCN-IP to denote it. The naming rule is the same for other combinations. The performances of all variants are summarized in Table 5. One can observe that several combinations achieve better performance than Geom-GCN with neighborhoods built by only one embedding space (in Table 3); and there are also many combinations that have bad performance. Thus, we think it's significant future work to design an end-to-end framework that can automatically determine the right embedding spaces for Geom-GCN.
+
+Table 5: Mean Classification Accuracy (Percent)  
+
+<table><tr><td>Dataset</td><td>Cora</td><td>Cite.</td><td>Pubm.</td><td>Cham.</td><td>Squi.</td><td>Actor</td><td>Corn.</td><td>Texa.</td><td>Wisc.</td></tr><tr><td>Geom-GCN-IP</td><td>85.13</td><td>79.41</td><td>90.49</td><td>65.77</td><td>45.49</td><td>31.94</td><td>60.00</td><td>66.49</td><td>62.75</td></tr><tr><td>Geom-GCN-PI</td><td>85.09</td><td>75.08</td><td>85.64</td><td>59.19</td><td>32.65</td><td>29.16</td><td>58.11</td><td>58.11</td><td>58.63</td></tr><tr><td>Geom-GCN-IS</td><td>84.51</td><td>77.83</td><td>88.66</td><td>58.40</td><td>35.29</td><td>29.41</td><td>54.32</td><td>57.57</td><td>57.65</td></tr><tr><td>Geom-GCN-SI</td><td>85.31</td><td>75.50</td><td>85.52</td><td>62.13</td><td>32.57</td><td>28.97</td><td>57.30</td><td>60.00</td><td>55.10</td></tr><tr><td>Geom-GCN-PS</td><td>85.65</td><td>74.84</td><td>84.96</td><td>56.34</td><td>28.27</td><td>29.53</td><td>58.11</td><td>62.43</td><td>60.59</td></tr><tr><td>Geom-GCN-SP</td><td>85.43</td><td>75.71</td><td>88.00</td><td>65.81</td><td>44.53</td><td>31.16</td><td>58.38</td><td>67.84</td><td>65.10</td></tr></table>
+
+# 4.3.3 ANALYSIS OF TIME COMPLEXITY
+
+Time complexity is very important for graph neural networks because real-world graphs are always very large. In this subsection, we firstly present the theoretical time complexity of Geom-GCN and then compare the real running time of GCN, GAT, and Geom-GCN.
+
+To update the representations of one node, the time complexity of Geom-GCN is  $O(n \times m \times 2|R|)$  where  $n$  is the size of input representations,  $m$  is the number of hidden unit in non-linear transform
+
+for each virtual node (i.e.,  $(i, r)$ ), and  $2|R|$  is the number of virtual nodes. Geom-GCN has  $2|R|$  times complexity than GCN whose time complexity is  $O(n \times m)$ .
+
+We also compare the real running time (500 epochs) of GCN, GAT, and Geom-GCN on all datasets with the hyper-parameters described in Section 4.2. Results are shown in Fig. 3 (a). One can see that GCN is the fastest, and GAT and Geom-GCN are on the same level. An important future work is to develop accelerating technology so as to solve the scalability of Geom-GCN.
+
+![](images/fbf77a485ca1fda79f57e1522f825e83b7ae8fec4da97210638b8ce549e95b03.jpg)  
+(a)
+
+![](images/cb5454f444d8f05ca9ca720cd78632efaa4998d076df69ca42ddb565531869c9.jpg)  
+(b)  
+Figure 3: (a) Running time comparison. GCN, GAT, and Geom-GCN both run 500 epochs, and  $y$  axis is the log seconds. GCN is the fastest, and GAT and Geom-GCN are on the same level. (b) A visualization for the feature representations of Cora obtained from Geom-GCN-P in a 2-D space. Node colors denote node labels. There are two obvious patterns, nodes with the same label exhibit a spatial clustering and all nodes distribute radially. The radial pattern indicates graph's hierarchy learned by Poincaré embedding.
+
+# 4.3.4 VISUALIZATION
+
+To study what patterns are learned in the feature representations of node by Geom-GCN, we visualize the feature representations extracted by the last layer of Geom-GCN-P on Cora dataset by mapping it into a 2-D space though t-SNE (Maaten & Hinton, 2008), as shown in Fig. 3 (b). In the figure, the nodes with the same label exhibit spatial clustering, which could shows the discriminative power of Geom-GCN. That all nodes distribute radially in the figure indicates the proposed model learn graph's hierarchy by Poincare embedding.
+
+# 4.4 CONCLUSION AND FUTURE WORK
+
+We tackle the two major weaknesses of existing message-passing neural networks over graphs—losses of discriminative structures and long-range dependencies. As our key insight, we bridge a discrete graph to a continuous geometric space via graph embedding. That is, we exploit the principle of convolution: spatial aggregation over a meaningful space—and our approach thus extracts or “recovers” the lost information (discriminative structures and long-range dependencies) in an embedding space from a graph. We proposed a general geometric aggregation scheme and instantiated it with several specific Geom-GCN implementations, and our experiments validated clear advantages over the state-of-the-art. As future work, we will explore techniques for choosing a right embedding method—depending not only on input graphs but also on target applications, such as epidemic dynamic prediction on social contact network (Yang et al., 2017; Pei et al., 2018).
+
+# ACKNOWLEDGMENTS
+
+We thank the reviewers for their valuable feedback. This work was supported in part by National Natural Science Foundation of China under grant 61876069, 61572226 and 61902145, National Science Foundation IIS 16-19302 and IIS 16-33755, Jilin Province Key Scientific and Technological Research and Development project under grants 20180201067GX and 20180201044GX, University science and technology research plan project of Jilin Province under grants JJKH20190156KJ, Zhejiang University ZJU Research 083650, Futurewei Technologies HF2017060011 and 094013, UIUC OVCR CCIL Planning Grant 434S34, UIUC CSBS Small Grant 434C8U, Advanced Digital Sciences Center Faculty Grant, and China Scholarships Council under scholarship 201806170202. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the author(s) and do not necessarily reflect the views of the funding agencies.
+
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+# GRADIENTLESS DESCENT: HIGH-DIMENSIONAL ZEROTH-ORDER OPTIMIZATION
+
+Daniel Golovin, John Karro, Greg Kochanski, Chansoo Lee Xingyou Song, Qiuyi (Richard) Zhang*
+
+Google Brain
+
+{dgg,karro,gpk,chansoo,xingyousong,qiuyiz}@google.com
+
+# ABSTRACT
+
+Zeroth-order optimization is the process of minimizing an objective  $f(x)$ , given oracle access to evaluations at adaptively chosen inputs  $x$ . In this paper, we present two simple yet powerful GradientLess Descent (GLD) algorithms that do not rely on an underlying gradient estimate and are numerically stable. We analyze our algorithm from a novel geometric perspective and present a novel analysis that shows convergence within an  $\epsilon$ -ball of the optimum in  $O(kQ\log(n)\log(R/\epsilon))$  evaluations, for any monotone transform of a smooth and strongly convex objective with latent dimension  $k < n$ , where the input dimension is  $n$ ,  $R$  is the diameter of the input space and  $Q$  is the condition number. Our rates are the first of its kind to be both 1) poly-logarithmically dependent on dimensionality and 2) invariant under monotone transformations. We further leverage our geometric perspective to show that our analysis is optimal. Both monotone invariance and its ability to utilize a low latent dimensionality are key to the empirical success of our algorithms, as demonstrated on BBOB and MuJoCo benchmarks.
+
+# 1 INTRODUCTION
+
+We consider the problem of zeroth-order optimization (also known as gradient-free optimization, or bandit optimization), where our goal is to minimize an objective function  $f: \mathbb{R}^n \to \mathbb{R}$  with as few evaluations of  $f(x)$  as possible. For many practical and interesting objective functions, gradients are difficult to compute and there is still a need for zeroth-order optimization in applications such as reinforcement learning (Mania et al., 2018; Salimans et al., 2017; Choromanski et al., 2018), attacking neural networks (Chen et al., 2017; Papernot et al., 2017), hyperparameter tuning of deep networks (Snoek et al., 2012), and network control (Liu et al., 2017).
+
+The standard approach to zeroth-order optimization is, ironically, to estimate the gradients from function values and apply a first-order optimization algorithm (Flaxman et al., 2005). Nesterov & Spokoiny (2011) analyze this class of algorithms as gradient descent on a Gaussian smoothing of the objective and gives an accelerated  $O(n\sqrt{Q}\log ((LR^2 +F) / \epsilon))$  iteration complexity for an  $L$ -Lipschitz convex function with condition number  $Q$  and  $R = \| x_0 - x^*\|$  and  $F = f(x_0) - f(x^*)$ . They propose a two-point evaluation scheme that constructs gradient estimates from the difference between function values at two points that are close to each other. This scheme was extended by (Duchi et al., 2015) for stochastic settings, by (Ghadimi & Lan, 2013) for nonconvex settings, and by (Shamir, 2017) for non-smooth and non-Euclidean norm settings. Since then, first-order techniques such as variance reduction (Liu et al., 2018), conditional gradients (Balasubramanian & Ghadimi, 2018), and diagonal preconditioning (Mania et al., 2018) have been successfully adopted in this setting. This class of algorithms are also known as stochastic search, random search, or (natural) evolutionary strategies and have been augmented with a variety of heuristics, such as the popular CMA-ES (Auger & Hansen, 2005).
+
+These algorithms, however, suffer from high variance due to non-robust local minima or highly non-smooth objectives, which are common in the fields of deep learning and reinforcement learn
+
+ing. Mania et al. (2018) notes that gradient variance increases as training progresses due to higher variance in the objective functions, since often parameters must be tuned precisely to achieve reasonable models. Therefore, some attention has shifted into direct search algorithms that usually finds a descent direction  $u$  and moves to  $x + \delta u$ , where the step size is not scaled by the function difference.
+
+The first approaches for direct search were based on deterministic approaches with a positive spanning set and date back to the 1950s (Brooks, 1958). Only recently have theoretical bounds surfaced, with Gratton et al. (2015) giving an iteration complexity that is a large polynomial of  $n$  and Dodangeh & Vicente (2016) giving an improved  $O(n^{2}L^{2} / \epsilon)$ . Stochastic approaches tend to have better complexities: Stich et al. (2013) uses line search to give a  $O(nQ\log (F / \epsilon))$  iteration complexity for convex functions with condition number  $Q$  and most recently, Gorbunov et al. (2019) uses importance sampling to give a  $O(n\bar{Q}\log (F / \epsilon))$  complexity for convex functions with average condition number  $\bar{Q}$ , assuming access to sampling probabilities. Stich et al. (2013) notes that direct search algorithms are invariant under monotone transforms of the objective, a property that might explain their robustness in high-variance settings.
+
+In general, zeroth order optimization suffers an at least linear dependence on input dimension  $n$  and recent works have tried to address this limitation when  $n$  is large but  $f(x)$  admits a low-dimensional structure. Some papers assume that  $f(x)$  depends only on  $k$  coordinates and Wang et al. (2017) applies Lasso to find the important set of coordinates, whereas Balasubramanian & Ghadimi (2018) simply change the step size to achieve an  $O(k(\log(n)/\epsilon)^2)$  iteration complexity. Other papers assume more generally that  $f(x) = g(\mathbf{P_A}x)$  only depends on a  $k$ -dimensional subspace given by the range of  $\mathbf{P_A}$  and Djolonga et al. (2013) apply low-rank approximation to find the low-dimensional subspace while Wang et al. (2013) use random embeddings. Hazan et al. (2017) assume that  $f(x)$  is a sparse collection of  $k$ -degree monomials on the Boolean hypercube and apply sparse recovery to achieve a  $O(n^k)$  runtime bound. We will show that under the case that  $f(x) = g(\mathbf{P_A}x)$ , our algorithm will inherently pick up any low-dimensional structure in  $f(x)$  and achieve a convergence rate that depends on  $k\log(n)$ . This initial convergence rate survives, even if we perturb  $f(x) = g(\mathbf{P_A}x) + h(x)$ , so long as  $h(x)$  is sufficiently small.
+
+We will not cover the whole variety of black-box optimization methods, such as Bayesian optimization or genetic algorithms. In general, these methods attempt to solve a broader problem (e.g. multiple optima), have weaker theoretical guarantees and may require substantial computation at each step: e.g. Bayesian optimization generally has theoretical iteration complexities that grow exponentially in dimension, and CMA-ES lacks provable complexity bounds beyond convex quadratic functions. In addition to the slow runtime and weaker guarantees, Bayesian optimization assumes the success of an inner optimization loop of the acquisition function. This inner optimization is often implemented with many iterations of a simpler zeroth-order methods, justifying the need to understand gradient-less descent algorithms within its own context.
+
+# 1.1 OUR CONTRIBUTIONS
+
+In this paper, we present GradientLess Descent (GLD), a class of truly gradient-free algorithms (also known as direct search algorithms) that are parameter free and provably fast. Our algorithms are based on a simple intuition: for well-conditioned functions, if we start from a point and take a small step in a randomly chosen direction, there is a significant probability that we will reduce the objective function value. We present a novel analysis that relies on facts in high dimensional geometry and can thus be viewed as a geometric analysis of gradient-free algorithms, recovering the standard convergence rates and step sizes. Specifically, we show that if the step size is on the order of  $O\left(\frac{1}{\sqrt{n}}\right)$ , we can guarantee an expected decrease of  $1 - \Omega\left(\frac{1}{n}\right)$  in the optimality gap, based on geometric properties of the sublevel sets of a smooth and strongly convex function.
+
+Our results are invariant under monotone transformations of the objective function, thus our convergence results also hold for a large class of non-convex functions that are a subclass of quasi-convex functions. Specifically, note that monotone transformations of convex functions are not necessarily convex. However, a monotone transformation of a convex function is always quasi-convex. The maximization of quasi-concave utility functions, which is equivalent to the minimization of quasi-convex functions, is an important topic of study in economics (e.g. Arrow & Enthoven (1961)).
+
+Table 1: Comparison of zeroth order optimization for well-conditioned convex functions where  $R = \| x_0 - x^*\|$  and  $F = f(x_0) - f(x^*)$ . 'Monotone' column indicates the invariance under monotone transformations (Definition 4). 'k-Sparse' and 'k-Affine' columns indicate that iteration complexity is poly $(k, \log(n))$  when  $f(x)$  depends only on a k-sparse subset of coordinates or on a rank- $k$  affine subspace.  
+
+<table><tr><td>Algorithm</td><td>Iteration Complexity</td><td>Monotone</td><td>k-Sparse</td><td>k-Affine</td></tr><tr><td>Nesterov &amp; Spokoiny (2011)</td><td>n log((R2+F)/ε)</td><td>No</td><td>No</td><td>No</td></tr><tr><td>Balasubramanian &amp; Ghadimi (2018)</td><td>n log(F/ε)</td><td>No</td><td>Yes</td><td>No</td></tr><tr><td>Stich et al. (2013)</td><td>n log(F/ε)</td><td>Yes</td><td>No</td><td>No</td></tr><tr><td>Gorbunov et al. (2019)</td><td>n log(F/ε)</td><td>Yes</td><td>No</td><td>No</td></tr><tr><td>This paper (GLD)</td><td>n log(R/ε)</td><td>Yes</td><td>Yes</td><td>Yes</td></tr></table>
+
+Intuition suggests that the step-size dependence on dimensionality can be improved when  $f(x)$  admits a low-dimensional structure. With a careful choice of sampling distribution we can show that if  $f(x) = g(\mathbf{P}_{\mathbf{A}}x)$ , where  $\mathbf{P}_{\mathbf{A}}$  is a rank  $k$  matrix, then our step size can be on the order of  $O\left(\frac{1}{\sqrt{k}}\right)$  as our optimization behavior is preserved under projections. We call this property affine-invariance and show that the number of function evaluations needed for convergence depends logarithmically on  $n$ . Unlike most previous algorithms in the high-dimensional setting, no expensive sparse recovery or subspace finding methods are needed. Furthermore, by novel perturbation arguments, we show that our fast convergence rates are robust and holds even under the more realistic assumption when  $f(x) = g(\mathbf{P}_{\mathbf{A}}x) + h(x)$  with  $h(x)$  being sufficiently small.
+
+Theorem 1 (Convergence of GLD: Informal Restatement of Theorem 7 and Theorem 14). Let  $f(x)$  be any monotone transform of a convex function with condition number  $Q$  and  $R = \| x_0 - x^*\|$ . Let  $y$  be a sample from an appropriate distribution centered at  $x$ . Then, with constant probability,
+
+$$
+f (y) - f (x ^ {*}) \leq \left(f (x) - f (x ^ {*})\right) \left(1 - \frac {1}{5 n Q}\right)
+$$
+
+Therefore, we can find  $x_{T}$  such that  $\| x_{T} - x^{*}\| \leq \epsilon$  after  $T = \widetilde{O}(nQ\log(R / \epsilon))$  function evaluations. Furthermore, for functions  $f(x) = g(\mathbf{P}_{\mathbf{A}}x) + h(x)$  with rank  $k$  matrix  $\mathbf{P}_{\mathbf{A}}$  and sufficiently small  $h(x)$ , we only require  $\widetilde{O}(kQ\log(n)\log(R / \epsilon))$  evaluations.
+
+Another advantage of our non-standard geometric analysis is that it allows us to deduce that our rates are optimal with a matching lower bound (up to logarithmic factors), presenting theoretical evidence that gradient-free inherently requires  $\Omega(nQ)$  function evaluations to converge. While gradient-estimation algorithms can achieve a better theoretical iteration complexity of  $O(n\sqrt{Q})$ , they lack the monotone and affine invariance properties. Empirically, we see that invariance properties are important to successful optimization, as validated by experiments on synthetic BBOB and MuJoCo benchmarks that show the competitiveness of GLD against standard optimization procedures.
+
+# 2 PRELIMINARIES
+
+We first define a few notations for the rest of the paper. Let  $\mathcal{X}$  be a compact subset of  $\mathbb{R}^n$  and let  $\| \cdot \|$  denote the Euclidean norm. The diameter of  $\mathcal{X}$ , denoted  $\| \mathcal{X} \| = \max_{x, x' \in \mathcal{X}} \| x - x' \|$ , is the maximum distance between elements in  $\mathcal{X}$ . Let  $f: \mathcal{X} \to \mathbb{R}$  be a real-valued function which attains its minimum at  $x^*$ . We use  $f(\mathcal{X}) = \{f(x): x \in \mathcal{X}\}$  to denote the image of  $f$  on a subset  $\mathcal{X}$  of  $\mathbb{R}^n$ , and  $\mathcal{B}(c, r) = \{x \in \mathbb{R}^n: \| c - x \| \leq r\}$  to denote the ball of radius  $r$  centered at  $c$ .
+
+Definition 2. The level set of  $f$  at point  $x \in \mathcal{X}$  is  $\mathcal{L}_c(f) = \{y \in \mathcal{X} : f(y) = f(x)\}$ . The sub-level set of  $f$  at point  $x \in \mathcal{X}$  is  $\mathcal{L}_c^\downarrow(f) = \{y \in \mathcal{X} : f(y) \leq f(x)\}$ . When the function  $f$  is clear from the context, we omit it.
+
+Definition 3. We say that  $f$  is  $\alpha$ -strongly convex for  $\alpha > 0$  if  $f(y) \geq f(x) + \langle \nabla f(x), y - x \rangle + \frac{\alpha}{2} \| y - x \|^2$  for all  $x, y \in \mathcal{X}$  and  $\beta$ -smooth for  $\beta > 0$  if  $f(y) \leq f(x) + \langle \nabla f(x), y - x \rangle + \frac{\beta}{2} \| y - x \|^2$  for all  $x, y \in \mathcal{X}$ .
+
+Definition 4. We say that  $g \circ f$  is a monotone transformation of  $f$  if  $g: f(\mathcal{X}) \to \mathbb{R}$  is a monotonically (and strictly) increasing function.
+
+Monotone transformations preserve the level sets of a function in the sense that  $\mathcal{L}_x(f) = \mathcal{L}_x(g\circ f)$ . Because our algorithms depend only on the level set properties, our results generalize to any monotone transformation of a strongly convex and strongly smooth function. This leads to our extended notion of condition number.
+
+Definition 5. A function  $f$  has condition number  $Q \geq 1$  if it is the minimum ratio  $\beta / \alpha$  over all functions  $g$  such that  $f$  is a monotone transformation of  $g$  and  $g$  is  $\alpha$ -strongly convex and  $\beta$  smooth.
+
+When we work with low rank extensions of  $f$ , we only care about the condition number of  $f$  within a rank  $k$  subspace. Indeed, if  $f$  only varies along a rank  $k$  subspace, then it has a strong convexity value of 0, making its condition number undefined. If  $f$  is  $\alpha$ -strongly convex and  $\beta$ -smooth, then its Hessian matrix always has eigenvalues bounded between  $\alpha$  and  $\beta$ . Therefore, we need a notion of a projected condition number. Let  $\mathbf{A} \in \mathbb{R}^{d \times k}$  be some orthonormal matrix and let  $\mathbf{P}_{\mathbf{A}} = \mathbf{A}\mathbf{A}^{\top}$  be the projection matrix onto the column space of  $\mathbf{A}$ .
+
+Definition 6. For some orthonormal  $\mathbf{A} \in \mathbb{R}^{d \times k}$  with  $d > k$ , a function  $f$  has condition number restricted to  $\mathbf{A}$ .  $Q(\mathbf{A}) \geq 1$ , if it is the minimum ratio  $\beta / \alpha$  over all functions  $g$  such that  $f$  is a monotone transformation of  $g$  and  $h(y) = g(\mathbf{A}y)$  is  $\alpha$ -strongly convex and  $\beta$  smooth.
+
+# 3 ANALYSIS OF DESCENT STEPS
+
+The GLD template can be summarized as follows: given a sampling distribution  $\mathcal{D}$ , we start at  $x_0$  and in iteration  $t$ , we choose a scalar radii  $r_t$  and we sample  $y_t$  from a distribution  $r_t\mathcal{D}$  centered around  $x_t$ , where  $r_t$  provides the scaling of  $\mathcal{D}$ . Then, if  $f(y_t) < f(x_t)$ , we update  $x_{t+1} = y_t$ ; otherwise, we set  $x_{t+1} = x_t$ . The analysis of GLD follows from the main observation that the sub-level set of a monotone transformation of a strongly convex and strongly smooth function contains a ball of sufficiently large radius tangent to the level set (Lemma 15). In this section, we show that this property, combined with facts of high-dimensional geometry, implies that moving in a random direction from any point has a good chance of significantly improving the objective.
+
+As we mentioned before, the key to fast convergence is the careful choice of step sizes, which we describe in Theorem 7. The intuition here is that we would like to take as large steps as possible while keeping the probability of improving the objective function reasonably high, so by insights in high-dimensional geometry, we choose a step size of  $\Theta(1/\sqrt{n})$ . Also, we show that if  $f(x)$  admits a latent rank- $k$  structure, then this step size can be increased to  $\Theta(1/\sqrt{k})$  and is therefore only dependent on the latent dimensionality of  $f(x)$ , allowing for fast high-dimensional optimization. Lastly, our geometric understanding allows us to show that our convergence rates are optimal with a matching lower bound. Without loss of generality, this section assumes that  $f(x)$  is strongly convex and smooth with condition number  $Q$ .
+
+# 3.1 STEP SIZE
+
+Theorem 7. For any  $x$  such that  $\frac{3}{5Q}\| x - x^{*}\| \in [C_1,C_2]$ , we can find integers  $0\leq k_{1},k_{2} < \log \frac{C_{2}}{C_{1}}$  such that if  $r = 2^{k_1}C_1$  or  $r = 2^{-k_2}C_2$ , then a random sample  $y$  from uniform distribution over  $B_{x} = \mathcal{B}(x,\frac{r}{\sqrt{n}})$  satisfies
+
+$$
+f (y) - f \left(x ^ {*}\right) \leq \left(f (x) - f \left(x ^ {*}\right)\right) \left(1 - \frac {1}{5 n Q}\right)
+$$
+
+with probability at least  $\frac{1}{4}$ .
+
+Proving the above theorem requires the following lemma about the intersection of balls in high dimensions and it is proved in the appendix.
+
+Lemma 8. Let  $B_{1}$  and  $B_{2}$  be two balls in  $\mathbb{R}^n$  of radii  $r_1$  and  $r_2$  respectively. Let  $\ell$  be the distance between the centers. If  $r_1 \in \left[\frac{\ell}{2\sqrt{n}}, \frac{\ell}{\sqrt{n}}\right]$  and  $r_2 \geq \ell - \frac{\ell}{4n}$ , then
+
+$$
+\operatorname {v o l} \left(B _ {1} \cap B _ {2}\right) \geq c _ {n} \operatorname {v o l} \left(B _ {1}\right),
+$$
+
+where  $c_{n}$  is a dimension-dependent constant that is lower bounded by  $\frac{1}{4}$  at  $n = 1$ .
+
+# 3.2 GAUSSIAN SAMPLING AND LOW RANK STRUCTURE
+
+A direct application of Lemma 8 seems to imply that uniform sampling of a high-dimensional ball is necessary. Upon further inspection, this can be easily replaced with a much simpler Gaussian sampling procedure that concentrates the mass close to the surface to the ball. This procedure lends itself to better analysis when  $f(x)$  admits a latent low-dimensional structure since any affine projection of a Gaussian is still Gaussian.
+
+Lemma 9. Let  $B_{1}$  and  $B_{2}$  be two balls in  $\mathbb{R}^n$  of radii  $r_1$  and  $r_2$  respectively. Let  $\ell$  be the distance between the centers. If  $r_1 \in [\frac{\ell}{2\sqrt{n}}, \frac{\ell}{\sqrt{n}}]$  and  $r_2 \geq \ell - \frac{\ell}{n}$  and  $X = (X_1, \dots, X_n)$  are independent Gaussians with mean centered at the center of  $B_{1}$  and variance  $\frac{r_1^2}{n}$ , then
+
+$$
+\Pr \left[ X \in B _ {2} \right] > c,
+$$
+
+where  $c$  is a dimension-independent constant.
+
+Assume that there exists some rank  $k$  projection matrix  $\mathbf{P}_{\mathbf{A}}$  such that  $f(x) = g(\mathbf{P}_{\mathbf{A}}x)$ , where  $k$  is much smaller than  $n$ . Because Gaussians projected on a  $k$ -dimensional subspace are still Gaussians, we show that our algorithm has a dimension dependence on  $k$ . We let  $Q_{g}(\mathbf{A})$  be the condition number of  $g$  restricted to the subspace  $\mathbf{A}$  that drives the dominant changes in  $f(x)$ .
+
+Theorem 10. Let  $f(x) = g(\mathbf{P}_{\mathbf{A}}x)$  for some unknown rank  $k$  matrix  $\mathbf{P}_{\mathbf{A}}$  with  $k < n$  and suppose  $\frac{3}{5Q} \| \mathbf{P}_{\mathbf{A}}(x - x^{*}) \| \in [C_{1}, C_{2}]$  for some numbers  $C_1, C_2 \in \mathbb{R}^+$ . Then, there exist integers  $0 \leq k_{1}, k_{2} < \log \frac{C_{2}}{C_{1}}$  such that if  $r = 2^{k_{1}}C_{1}$  or  $r = 2^{-k_{2}}C_{2}$ , then a random sample  $y$  from a Gaussian distribution  $\mathcal{N}(x, \frac{r^2}{k}\mathbf{I})$  satisfies
+
+$$
+f (y) - f \left(x ^ {*}\right) \leq \left(f (x) - f \left(x ^ {*}\right)\right) \left(1 - \frac {1}{5 k Q _ {g} (\mathbf {A})}\right)
+$$
+
+with constant probability.
+
+Note that the speed-up in progress is due to the fact that we can now tolerate the larger sampling radius of  $\Omega(1/\sqrt{k})$ , while maintaining a high probability of making progress. If  $k$  is unknown, we can simply use binary search to find the correct radius with an extra factor of  $\log(n)$  in our runtime.
+
+The low-rank assumption is too restrictive to be realistic; however, our fast rates still hold, at least for the early stages of the optimization, even if we assume that  $f(x) = g(\mathbf{P_A}x) + h(x)$  and  $|h(x)| \leq \delta$  is a full-rank function that is bounded by  $\delta$ . In this setting, we can show that convergence remains fast, at least until the optimality gap approaches  $\delta$ .
+
+Theorem 11. Let  $f(x) = g(\mathbf{P}_{\mathbf{A}}x) + h(x)$  for some unknown rank  $k$  matrix  $\mathbf{P}_{\mathbf{A}}$  with  $k < n$  where  $g, h$  are convex and  $|h| \leq \delta$ . Suppose  $\frac{3}{5Q} \| \mathbf{P}_{\mathbf{A}}x - z^{*} \| \in [C_{1}, C_{2}]$  for some numbers  $C_{1}, C_{2} \in \mathbb{R}^{+}$  where  $z^{*}$  minimizes  $g(z)$ . Then, there exist integers  $0 \leq k_{1}, k_{2} < \log \frac{C_{2}}{C_{1}}$  such that if  $r = 2^{k_{1}}C_{1}$  or  $r = 2^{-k_{2}}C_{2}$ , then a random sample  $y$  from a Gaussian distribution  $\mathcal{N}(x, \frac{r^2}{k}\mathbf{I})$  satisfies
+
+$$
+f (y) - f (x ^ {*}) \leq \left(f (x) - f (x ^ {*})\right) \left(1 - \frac {1}{1 0 k Q _ {g} (\mathbf {A})}\right)
+$$
+
+with constant probability whenever  $f(x) - f(x^{*}) \geq 60\delta kQ_{g}(\mathbf{A})$ .
+
+# 3.3 LOWER BOUNDS
+
+We show that our upper bounds given in the previous section are tight up to logarithmic factors for any symmetric sampling distribution  $\mathcal{D}$ . These lower bounds are easily derived from our geometric perspective as we show that a sampling distribution with a large radius gives an extremely low probability of intersection with the desired sub-level set. Therefore, while gradient-approximation algorithms can be accelerated to achieve a runtime that depends on the square-root of the condition number  $Q$ , gradient-less methods that rely on random sampling are likely unable to be accelerated according to our lower bound. However, we emphasize that monotone invariance allows these results to apply to a broader class of objective functions, beyond smooth and convex, so the results can be useful in practice despite the seemingly worse theoretical bounds.
+
+Algorithm 1: Gradientless Descent with Binary Search (GLD-Search)  
+Input: function:  $f:\mathbb{R}^n\to \mathbb{R},T\in \mathbb{Z}_+$  : number of iterations,  $x_0$  starting point,  $\mathcal{D}$  sampling distribution,  $R$  : maximum search radius,  $r$  : minimum search radius  
+1 Set  $K = \log (R / r)$   
+2 for  $t = 0,\dots ,T$  do  
+3 Ball Sampling Trial i:  
+4 for  $k = 0,\ldots ,K$  do  
+5 Set  $r_{i,k} = 2^{-k}R$   
+6 Sample  $v_{i,k}\sim r_{i,k}\mathcal{D}$   
+7 end  
+8 Update:  $x_{t + 1} = \arg \min_k\left\{f(y)\bigg|y = x_t,y = x_t + v_{i,k}\right\}$   
+9 end  
+10 return  $x_{t}$
+
+Theorem 12. Let  $y = x + v$ , where  $v$  is a random sample from  $r\mathcal{D}$  for some radius  $r > 0$  and  $\mathcal{D}$  is standard Gaussian or any rotationally symmetric distribution. Then, there exist a region  $X$  with positive measure such that for any  $x \in X$ ,
+
+$$
+f (y) - f \left(x ^ {*}\right) \geq \left(f (x) - f \left(x ^ {*}\right)\right) \left(1 - \frac {\sqrt {5 \log (n Q)}}{n Q}\right)
+$$
+
+with probability at least  $1 - \frac{1}{poly(nQ)}$ .
+
+# 4 GRADIENTLESS ALGORITHMS
+
+In this section, we present two algorithms that follow the same Gradientless Descent (GLD) template: GLD-Search and GLD-Fast, with the latter being an optimized version of the former when an upper bound on the condition number of a function is known. For both algorithms, since they are monotone-invariant, we appeal to the previous section to derive fast convergence rates for any monotone transform of convex  $f(x)$  with good condition number. We show the efficacy of both algorithms experimentally in the Experiments section.
+
+# 4.1 GRADIENTLESS DESCENT WITH BINARY SEARCH
+
+Although the sampling distribution  $\mathcal{D}$  is fixed, we have a choice of radii for each iteration of the algorithm. We can apply a binary search procedure to ensure progress. The most straightforward version of our algorithm is thus with a naive binary sweep across an interval in  $[r,R]$  that is unchanged throughout the algorithm. This allows us to give convergence guarantees without previous knowledge of the condition number at a cost of an extra factor of  $\log (n / \epsilon)$ .
+
+Theorem 13. Let  $x_0$  be any starting point and  $f$  a blackbox function with condition number  $Q$ . Running Algorithm 1 with  $r = \frac{\epsilon}{\sqrt{n}}$ ,  $R = \| \mathcal{X} \|$  and  $\mathcal{D} = \mathcal{N}(0, \mathbf{I})$  as a standard Gaussian returns a point  $x_T$  such that  $\| x_T - x^* \| \leq 2Q^{3/2} \epsilon$  after  $O(nQ \log (n \| \mathcal{X} \| / \epsilon)^2)$  function evaluations with high probability.
+
+Furthermore, if  $f(x) = g(\mathbf{P}_{\mathbf{A}}x)$  admits a low-rank structure with  $\mathbf{P}_{\mathbf{A}}$  a rank  $k$  matrix, then we only require  $O(kQ_{g}(\mathbf{A})\log (n\| \mathcal{X}\| /\epsilon)^{2})$  function evaluations to guarantee  $\| \mathbf{P}_{\mathbf{A}}(x_T - x^*)\| \leq \epsilon$ . This holds analogously even if  $f(x) = g(\mathbf{P}_{\mathbf{A}}x) + h(x)$  is almost low-rank where  $|h|\leq \delta$  and  $\epsilon >60\delta kQ_{g}(\mathbf{A})$ .
+
+# 4.2 GRADIENTLESS DESCENT WITH FAST BINARY SEARCH
+
+GLD-Search (Algorithm 1) uses a naive lower and upper bound for the search radius  $\| x_t - x^*\|$ , which incurs an extra factor of  $\log(1/\epsilon)$  in the runtime bound. In GLD-Fast, we remove this extra factor dependence on  $\log(1/\epsilon)$  by drastically reducing the range of the binary search. This is done by exploiting the assumption that  $f$  has a good condition number upper bound  $\hat{Q}$  and by slowly halving the diameter of the search space every few iterations since we expect  $x_t \to x^*$  as  $t \to \infty$ .
+
+Algorithm 2: Gradientless Descent with Fast Binary Search (GLD-Fast)  
+Input: function  $f:\mathbb{R}^n\to \mathbb{R},T\in \mathbb{Z}_+$  : number of iterations,  $x_0$  starting point,  $\mathcal{D}$  sampling distribution,  $R$  diameter of search space,  $Q$  condition number bound   
+1 Set  $K = \log (4\sqrt{Q})$ $H = nQ\log (Q)$    
+2 for  $t = 1,\dots ,T$  do   
+3 Set  $R = R / 2$  when  $T\equiv 0$  mod  $H$  (every  $H$  iterations).   
+4 Ball Sampling Trial i:   
+5 for  $k = -K,\ldots ,0,\ldots ,K$  do   
+6 Set  $r_{i,k} = 2^{-k}R.$    
+7 Sample  $v_{i,k}\sim r_{i,k}\mathcal{D}$    
+8 end   
+9 Update:  $x_{t + 1} = \arg \min_i\left\{f(y)\bigg|y = x_t,y = x_t + v_i\right\}$    
+10 end   
+11 return  $x_{t}$
+
+Theorem 14. Let  $x_0$  be any starting point and  $f$  a blackbox function with condition number upper bounded by  $Q$ . Running Algorithm 2 with suitable parameters returns a point  $x_T$  such that  $f(x_T) - f(x^*) \leq \epsilon$  after  $O(nQ\log^2(Q)\log(\|\mathcal{X}\|/\epsilon))$  function evaluations with high probability.
+
+Furthermore, if  $f(x) = g(\mathbf{P}_{\mathbf{A}}x)$  admits a low-rank structure with  $\mathbf{P}_{\mathbf{A}}$  a rank  $k$  matrix, then we only require  $O(kQ_{g}(\mathbf{A})\log (n)\log^{2}(Q_{g}(\mathbf{A}))\log (\| \mathcal{X}\| /\epsilon))$  function evaluations to guarantee  $\| \mathbf{P}_{\mathbf{A}}(x_T - x^*)\| \leq \epsilon$ . This holds analogously even if  $f(x) = g(\mathbf{P}_{\mathbf{A}}x) + h(x)$  is almost low-rank where  $|h|\leq \delta$  and  $\epsilon >60\delta kQ_g(\mathbf{A})$ .
+
+# 5 EXPERIMENTS
+
+We tested GLD algorithms on a simple class of objective functions and compare it to Accelerated Random Search (ARS) by Nesterov & Spokoiny (2011), which has linear convergence guarantees on strongly convex and strongly smooth functions. To our knowledge, ARS makes the weakest assumption among the zeroth-order algorithms that have linear convergence guarantees and perform only a constant order of operations per iteration. Our main conclusion is that GLD-Fast is comparable to ARS and tends to achieve a reasonably low error much faster than ARS in high dimensions  $(\geq 50)$ . In low dimensions, GLD-Search is competitive with GLD-Fast and ARS though it requires no information about the function.
+
+We let  $H_{\alpha, \beta, n} \in \mathbb{R}^{n \times n}$  be a diagonal matrix with its  $i$ -th diagonal equal to  $\alpha + (\beta - \alpha)\frac{i-1}{n-1}$ . In simple words, its diagonal elements form an evenly space sequence of numbers from  $\alpha$  to  $\beta$ . Our objective function is then  $f_{\alpha, \beta, n}: \mathbb{R}^n \to \mathbb{R}$  as  $f_{\alpha, \beta, n}(x) = \frac{1}{2} x^\top H_{\alpha, \beta, n}x$ , which is  $\alpha$ -strongly convex and  $\beta$ -strongly smooth. We always use the same starting point  $x = \frac{1}{\sqrt{n}} (1, \ldots, 1)$ , which requires  $\| X \| = \sqrt{Q}$  for our algorithms. We plot the optimality gap  $f(b_t) - f(x^*)$  against the number of function evaluations, where  $b_t$  is the best point observed so far after  $t$  evaluations. Although all tested algorithms are stochastic, they have a low variance on the objective functions that we use; hence we average the results over 10 runs and omit the error bars in the plots.
+
+We ran experiments on  $f_{1,8,n}$  with imperfect curvature information  $\hat{\alpha}$  and  $\hat{\beta}$  (see Figure 3 in appendix). GLD-Search is independent of the condition number. GLD-Fast takes only one parameter, which is the upper bound on the condition number; if approximation factor is  $z$ , then we pass  $8z$  as the upper bound. ARS requires both strong convexity and smoothness parameters. We test three different distributions of the approximation error; when the approximation factor is  $z$ , then ARS-alpha gets  $(\alpha / z, \beta)$ , ARS-beta gets  $(\alpha, z\beta)$ , and ARS-even gets  $(\alpha / \sqrt{z}, \sqrt{z}\beta)$  as input. GLD-Fast is more robust and faster than ARS when the condition number is over-approximated. When the condition number is underestimated, GLD-Fast still steadily converges.
+
+![](images/479eeeb0baaac5fc8b5503e6c520f03b3c81379f248b4cb93a0e18ca80496f96.jpg)  
+Figure 1: The average optimality gap on a quadratic objective function that is strongly convex and smooth objective (top); and its monotone transformation (bottom). Further experiments on nonconvex BBOB functions show similar behavior and are in the appendix.
+
+# 5.1 MONOTONE TRANSFORMATIONS
+
+In Figure 1, we ran experiments on  $f_{1,8,n}$  for different settings of dimensionality  $n$ , and its monotone transformation with  $g(y) = -\exp(-\sqrt{y})$ . For this experiment, we assume a perfect oracle for the strong convexity and smoothness parameters of  $f$ . The convergence of GLD is totally unaffected by the monotone transformation. For the low-dimension cases of a transformed function (bottom half of the figure), we note that there are inflection points in the convergence curve of ARS. This means that ARS initially struggles to gain momentum and then struggles to stop the momentum when it gets close to the optimum. Another observation is that unlike ARS that needs to build up momentum, GLD-Fast starts from a large radius and therefore achieves a reasonably low error much faster than ARS, especially in higher dimensions.
+
+# 5.2 BBOB BENCHMARKS
+
+To show that practicality of GLD on practical and non-convex settings, we also test GLD algorithms on a variety of BlackBox Optimization Benchmarking (BBOB) functions (Hansen et al., 2009). For each function, the optima is known and we use the log optimality gap as a measure of competence. Because each function can exhibit varying forms of non-smoothness and convexity, all algorithms are ran with a smoothness constant of 10 and a strong convexity constant of 0.1. All other setup details are same as before, such as using a fixed starting point.
+
+The plots, given in Appendix C, underscore the superior performance of GLD algorithms on various BBOB functions, demonstrating that GLD can successfully optimize a diverse set of functions even without explicit knowledge of condition number. We note that BBOB functions are far from convex and smooth, many exhibiting high conditioning, multi-modal valleys, and weak global structure. Due to our radius search produce, our algorithm appears more robust to non-ideal settings with non-convexity and ill conditioning. As expected, we note that GLD-Fast tend to outperform GLD-Search, especially as the dimension increases, matching our theoretical understanding of GLD.
+
+# 5.3 MUJOCO CONTROL BENCHMARKS AND AFFINE TRANSFORMATIONS
+
+We also ran experiments on the Mujoco benchmarks with varying architectures, both linear and nonlinear. This demonstrates the viability of our approach even in the non-convex, high dimensional setting. We note that however, unlike e.g. ES which uses all queries to form a gradient direction, our algorithm removes queries which produce less reward than using the current arg-max, which can be an information handicap. Nevertheless, we see that our algorithm still achieves competitive performance on the maximum reward. We used a horizon of 1000 for all experiments.
+
+We further tested the affine invariance of GLD on the policy parameters from using Gaussian ball sampling, under the HalfCheetah benchmark by projecting the state  $s$  of the MDP with linear policy to a higher dimensional state  $Ws$ , using a matrix multiplication with an orthonormal  $W$ . Specifically, in this setting, for a linear policy parametrized by matrix  $K$ , the objective function is thus  $J(KW)$  where  $\pi_K(Ws) = KWs$ . Note that when projecting into a high dimension, there is a slowdown factor of  $\log \frac{d_{new}}{d_{old}}$  where  $d_{new}, d_{old}$  are the new high dimension and previous base dimension, respectively, due to the binary search in our algorithm on a higher dimensional space. For our HalfCheetah case, we projected the 17 base dimension to a 200-length dimension, which suggests that the slowdown factor is a factor  $\log \frac{200}{17} \approx 3.5$ . This can be shown in our plots in the appendix (Figure 15).
+
+Table 2: Final rewards by GLD with linear (L) and deep (H41) policies on Mujoco Benchmarks show that GLD is competitive. We apply an affine projection on HalfCheetah to test affine invariance. We use the reward threshold found from (Mania et al., 2018) with Reacher's threshold (Schulman et al., 2017) for a reasonable baseline.  
+
+<table><tr><td>Env.</td><td>Arch</td><td>Rew. at (104, 105, Max) Queries</td><td>Rew. Thresh.</td></tr><tr><td>HalfCheetah-v1</td><td>L</td><td>3799, 3903, 4064</td><td>3430</td></tr><tr><td>HalfCheetah-v1, Proj-200</td><td>L</td><td>1594, 3509, 4342</td><td>3430</td></tr><tr><td>HalfCheetah-v1</td><td>H41</td><td>2741, 3074, 3392</td><td>3430</td></tr><tr><td>Hopper-v1</td><td>L</td><td>1017, 3359, 3375</td><td>3120</td></tr><tr><td>Hopper-v1</td><td>H41</td><td>2708, 3370, 3566</td><td>3120</td></tr><tr><td>Reacher-v1</td><td>L</td><td>-70, -5, -4</td><td>-10</td></tr><tr><td>Reacher-v1</td><td>H41</td><td>-231, -17, -15</td><td>-10</td></tr><tr><td>Swimmer-v1</td><td>L</td><td>365, 369, 369</td><td>325</td></tr><tr><td>Swimmer-v1</td><td>H41</td><td>353, 369, 369</td><td>325</td></tr><tr><td>Walker2d-v1</td><td>L</td><td>1027, 2201, 2201</td><td>4390</td></tr><tr><td>Walker2d-v1</td><td>H41</td><td>1630, 1963, 2146</td><td>4390</td></tr></table>
+
+# 6 CONCLUSION
+
+We introduced GLD, a robust zeroth-order optimization algorithm that is simple, efficient, and we show strong theoretical convergence bounds via our novel geometric analysis. As demonstrated by our experiments on BBOB and MuJoCo benchmarks, GLD performs very robustly even in the non-convex setting and its monotone and affine invariance properties give theoretical insight on its practical efficiency.
+
+GLD is very flexible and allows easy modifications. For example, it could use momentum terms to keep moving in the same direction that improved the objective, or sample from adaptively chosen ellipsoids similarly to adaptive gradient methods. (Duchi et al., 2011; McMahan & Streeter, 2010). Just as one may decay or adaptively vary learning rates for gradient descent, one might use a similar change the distribution from which the ball-sampling radii are chosen, perhaps shrinking the minimum radius as the algorithm progresses, or concentrating more probability mass on smaller radii.
+
+Likewise, GLD could be combined with random restarts or other restart policies developed for gradient descent. Analogously to adaptive per-coordinate learning rates Duchi et al. (2011); McMahan & Streeter (2010), one could adaptively change the shape of the balls being sampled into ellipsoids with various length-scale factors. Arbitrary combinations of the above variants are also possible.
+
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+Yurii Nesterov and Vladimir Spokoiny. Random gradient-free minimization of convex functions. Technical report, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE), 2011.  
+Nicolas Papernot, Patrick McDaniel, Ian Goodfellow, Somesh Jha, Z Berkay Celik, and Ananthram Swami. Practical black-box attacks against machine learning. In Proceedings of the 2017 ACM on Asia conference on computer and communications security, pp. 506-519. ACM, 2017.  
+Tim Salimans, Jonathan Ho, Xi Chen, Szymon Sidor, and Ilya Sutskever. Evolution strategies as a scalable alternative to reinforcement learning. arXiv preprint arXiv:1703.03864, 2017.  
+John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal policy optimization algorithms. CoRR, abs/1707.06347, 2017. URL http://arxiv.org/abs/1707.06347.  
+Ohad Shamir. An optimal algorithm for bandit and zero-order convex optimization with two-point feedback. Journal of Machine Learning Research, 18(52):1-11, 2017.  
+Jasper Snoek, Hugo Larochelle, and Ryan P Adams. Practical bayesian optimization of machine learning algorithms. In Advances in neural information processing systems, pp. 2951-2959, 2012.  
+Sebastian U Stich, Christian L Muller, and Bernd Gartner. Optimization of convex functions with random pursuit. SIAM Journal on Optimization, 23(2):1284-1309, 2013.  
+Yining Wang, Simon Du, Sivaraman Balakrishnan, and Aarti Singh. Stochastic zeroth-order optimization in high dimensions. arXiv preprint arXiv:1710.10551, 2017.  
+Ziyu Wang, Masrour Zoghi, Frank Hutter, David Matheson, and Nando De Freitas. Bayesian optimization in high dimensions via random embeddings. In Twenty-Third International Joint Conference on Artificial Intelligence, 2013.
+
+# A PROOFS OF SECTION 3
+
+Lemma 15. If  $h$  has condition number  $Q$ , then for all  $x \in \mathcal{X}$ , there is a ball of radius  $Q^{-1} \| x - x^{*} \|$  that is tangent at  $x$  and inside the sublevel set  $\mathcal{L}_{x}^{\downarrow}(h)$ .
+
+Proof. Write  $h = g \circ f$  such that  $f$  is  $\alpha$ -strongly convex and  $\beta$ -smooth for some  $\beta = Q\alpha$  and  $g$  is monotonically increasing. From the smoothness assumption, we have for any  $s$ ,
+
+$$
+\begin{array}{l} f \left(x - \frac {1}{\beta} \nabla f (x) + s\right) \\ \leq f (x) + \left\langle \nabla f (x), s - \frac {1}{\beta} \nabla f (x) \right\rangle + \frac {\beta}{2} \left\| s - \frac {1}{\beta} \nabla f (x) \right\| ^ {2} \\ = f (x) + \frac {\beta}{2} \left(\| s \| ^ {2} - \frac {1}{\beta^ {2}} \| \nabla f (x) \| ^ {2}\right). \\ \end{array}
+$$
+
+Consider the ball  $B = \mathcal{B}(x - \frac{1}{\beta}\nabla f(x), \frac{1}{\beta}\| \nabla f(x)\|)$ . For any  $y \in B$ , the above inequality implies  $f(y) \leq f(x)$ . Hence, when we apply  $g$  on both sides, we still have  $h(y) \leq h(x)$  for all  $y \in B$ . Therefore,  $B \subseteq \mathcal{L}_{h(y)}^{\downarrow}$ .
+
+By strong convexity,  $\| \nabla f(x)\| \geq \alpha \| x - x^{*}\|$  . It follows that the radius of  $B$  is at least  $\frac{\alpha}{\beta}\| x - x^*\|$
+
+Proof of Lemma 8. Without loss of generality, consider the unit distance case where  $\ell = 1$ . Furthermore, it suffices to prove for the smallest possible radius  $r_2 = 1 - \frac{1}{4n}$ .
+
+Since  $|r_1 - r_2| \leq \ell \leq r_1 + r_2$ , the intersection  $B_1 \cap B_2$  is composed of two hyperspherical caps glued end to end. We lower bound vol  $(B_1 \cap B_2)$  by the volume of the cap  $C_1$  of  $B_1$  that is contained in the intersection. Consider the triangle with sides  $r_1, r_2$  and  $\ell$ . From classic geometry, the height of  $C_1$  is
+
+$$
+c _ {1} = \frac {1}{2} \left(1 + r _ {1} ^ {2} - r _ {2} ^ {2}\right) > 0. \tag {1}
+$$
+
+The volume of a spherical cap is Li (2011),
+
+$$
+\operatorname {v o l} \left(C _ {1}\right) = \frac {1}{2} \operatorname {v o l} \left(B _ {1}\right) I _ {1 - \frac {c _ {1} ^ {2}}{r _ {1} ^ {2}}} (\frac {n + 1}{2}, \frac {1}{2}).
+$$
+
+where  $I$  is the regularized incomplete beta function defined as
+
+$$
+I _ {x} (a, b) = \frac {\int_ {0} ^ {x} t ^ {a - 1} (1 - t) ^ {b - 1} d t}{\int_ {0} ^ {1} t ^ {a - 1} (1 - t) ^ {b - 1} d t}
+$$
+
+where  $x \in [0,1]$  and  $a, b \in (0,\infty)$ . Note that for any fixed  $a$  and  $b$ ,  $I_x(a,b)$  is increasing in  $x$ . Hence, in order to obtain a lower bound on  $\operatorname{vol}(C_1)$ , we want to lower bound  $1 - \frac{c_1^2}{r_1^2}$  or equivalently, upper bound  $\frac{c_1^2}{r_1^2}$ .
+
+Write  $r_1 = \frac{\alpha}{2\sqrt{n}}$  for some  $\alpha \in [1,2]$ . From Eq. (1),
+
+$$
+c _ {1} = \frac {1}{4 n} + \frac {\alpha^ {2}}{8 n} - \frac {1}{3 2 n ^ {2}}.
+$$
+
+Hence,
+
+$$
+\frac {c _ {1}}{r _ {1}} = \frac {1}{1 6 \sqrt {n}} \left(\frac {8}{\alpha} + 4 \alpha - \frac {1}{n}\right)
+$$
+
+Since  $g(\alpha) = \frac{8}{\alpha} + 4\alpha$  is convex in  $[1,2]$ ,  $g(\alpha) \leq \max(g(1), g(2)) = 12$ . It follows that  $\frac{c_1}{r_1} \leq \frac{1}{16\sqrt{n}} \left(12 - \frac{1}{n}\right) \leq \frac{3}{4\sqrt{n}}$ . So,  $1 - \frac{c_1^2}{r_1^2} \geq 1 - \frac{9}{16n}$ . To complete the proof, note that  $V_n := I_{1 - \frac{9}{16n}} \left(\frac{n + 1}{2}, \frac{1}{2}\right)$  is increasing in  $n$ , and  $V_1 = \frac{1}{4}$ . As  $n$  goes to infinity, this value converges to 1 as  $B_1 \subset B_2$ .
+
+Proof of Lemma 7. Let  $\nu = \frac{1}{5nQ}$ . Let  $q = (1 - \nu)x + \nu x^{*}$ . Let  $B_{q} = \mathcal{B}(c_{q},r_{q})$  be a ball that has  $q$  on its surface, lies inside  $\mathcal{L}_q^\downarrow$ , and has radius  $r_q = Q^{-1}\| x - x^*\|$ . Lemma 15 guarantees its existence.
+
+Suppose that
+
+$$
+\operatorname {v o l} \left(B _ {x} \cap B _ {q}\right) \geq \frac {1}{4} \operatorname {v o l} \left(B _ {x}\right) \tag {2}
+$$
+
+and that a random sample  $y$  from  $B_{x}$  belongs to  $B_{q}$ , which happens with probability at least  $\frac{1}{4}$ . Then, our guarantee follows by
+
+$$
+\begin{array}{l} f (y) - f \left(x ^ {*}\right) \leq f (q) - f \left(x ^ {*}\right) \\ \leq (1 - \nu) f (x) + \nu f \left(x ^ {*}\right) - f \left(x ^ {*}\right) \\ \leq (1 - \nu) (f (x) - f (x ^ {*})) \\ \end{array}
+$$
+
+where the first line follows from Lemma 15 and second line from convexity of  $f$ .
+
+Therefore, it now suffices to prove Eq. 2. To do so, we will apply Lemma 8 after showing that the radius of  $B_{x}$  and  $B_{q}$  are in the proper ranges. Let  $\ell = \| x - c_{q}\|$  and note that
+
+$$
+\begin{array}{l} \ell \leq \| x - q \| + r _ {q} (3) \\ \leq \nu \| x - x ^ {*} \| + r _ {q} = \nu \| x - x ^ {*} \| + Q ^ {- 1} \| q - x ^ {*} \| \\ \leq (\nu + Q ^ {- 1} (1 - \nu)) \| x - x ^ {*} \| (4) \\ \leq \frac {6}{5 Q} \left\| b _ {x} - x ^ {*} \right\|. \\ \end{array}
+$$
+
+Since  $x$  is outside of  $B_q$ , we also have
+
+$$
+\begin{array}{l} \ell \geq r _ {q} = Q ^ {- 1} \| q - x ^ {*} \| = Q ^ {- 1} (1 - \nu) \| x - x ^ {*} \| \\ \geq \frac {4}{5 Q} \| b _ {x} - x ^ {*} \|. \tag {5} \\ \end{array}
+$$
+
+It follows that
+
+$$
+\frac {\ell}{2} \leq \frac {3}{5 Q} \| b _ {x} - x ^ {*} \| \leq \ell .
+$$
+
+In the  $\log_2$  space, our choice of  $k_{1}$  is equivalent to starting from  $\log_2C_1$  and sweeping through the range  $[\log_2C_1,\log_2C_2]$  at the interval of size 1. This is guaranteed to find a point between  $\frac{\ell}{2}$  and  $\ell$ , which is also an interval of size 1. Therefore, there exists a  $k_{1}$  satisfying the theorem statement, and similarly, we can prove the existence of  $k_{2}$ .
+
+Finally, it remains to show that  $r_q \geq (1 - 1/(4n))\ell$ . From Eq. (3), it suffices to show that  $\|x - q\| \leq \frac{\ell}{4n}$  or equivalently  $\nu \|x - x^*\| \leq \frac{\ell}{4n}$ . From Eq. (4),
+
+$$
+\| x - q \| = \nu \| x - x ^ {*} \| \leq \nu Q (1 - \nu) ^ {- 1} \ell .
+$$
+
+For any  $Q, n \geq 1, 1 - \nu \geq \frac{4}{5}$ . So,
+
+$$
+\nu Q (1 - \nu) ^ {- 1} = \frac {1}{5 n} (1 - \nu) ^ {- 1} \leq \frac {1}{4 n} \tag {6}
+$$
+
+and the proof is complete.
+
+![](images/e9bdfa686ea8b6f04d548c7a35ec41b3fe60193ace1a2161c49195c66a16c878.jpg)
+
+Proof of Lemma 9. Without loss of generality, let  $\ell = 1$  and  $B_{2}$  is centered at the origin with radius  $r_2$  and  $B_{1}$  is centered at  $e_1 = (1,0,\dots,0)$ . Then, we simply want to show that
+
+$$
+\mathbf {P r} \left[ (1 + X _ {1}) ^ {2} + \sum_ {i = 2} ^ {n} X _ {i} ^ {2} \leq r _ {2} ^ {2} \right] > c _ {n}
+$$
+
+By Markov's inequality, we see that  $\sum_{i=2}^{n} X_i^2 \leq 2r_1^2 = 2/n$  with probability at most  $1/2$ . And since  $X_1$  is independent and  $r_2 \geq 1 - 1/n$ , it suffices to show that
+
+$$
+\Pr \left[ (1 + X _ {1}) ^ {2} \leq 1 - 4 / n \right] > \Omega (1)
+$$
+
+Since  $X_{1}$  has standard deviation at least  $r_1 / \sqrt{n} \geq 1 / (2n)$ , we see that the probability of deviating at least a few standard deviations below is at least a constant.
+
+Proof of Theorem 10. We can consider the projection of all points onto the column space of  $A$  and since the Gaussian sampling process is preserved, our proof follows from applying Theorem 7 restricted onto the  $k$ -dimensional subspace and using Lemma 9 in place of Lemma 8.
+
+Proof of Theorem 11. By the boundedness of  $h$ , since  $f(x) - f(x^{*}) \geq 60\delta kQ_{g}(\mathbf{A})$ , we see that  $g(\mathbf{P}_{\mathbf{A}}x) - g(x^{*}) \geq 60\delta kQ_{g}(\mathbf{A}) - 2\delta > 0$ . By Lemma 9, we see that if we sample from a Gaussian distribution  $y \sim \mathcal{N}(x,\frac{r^2}{k}\mathbf{I})$ , then if  $z^{*}$  is the minimum of  $g(x)$  restricted to the column space of  $\mathbf{A}$ , then
+
+$$
+g (\mathbf {P _ {A}} y) - g (z ^ {*}) \leq (g (\mathbf {P _ {A}} x) - g (z ^ {*})) \left(1 - \frac {1}{5 k Q _ {g} (\mathbf {A})}\right)
+$$
+
+with constant probability. By boundedness on  $h$ , we know that  $h(y) \leq h(x) + 2\delta$ . Furthermore, this also implies that  $g(\mathbf{P}_{\mathbf{A}}x^{*}) \leq g(z^{*}) + 2\delta$ . Therefore, we know that the decrease is at least
+
+$$
+\begin{array}{l} f (y) - f \left(x ^ {*}\right) = g \left(\mathbf {P} _ {\mathbf {A}} y\right) - g \left(\mathbf {P} _ {\mathbf {A}} x ^ {*}\right) + h (y) - h \left(x ^ {*}\right) \\ \leq g \left(\mathbf {P} _ {\mathbf {A}} y\right) - g \left(z ^ {*}\right) + 2 \delta \\ \leq \left(g \left(\mathbf {P} _ {\mathbf {A}} x\right) - g \left(z ^ {*}\right)\right) \left(1 - \frac {1}{5 k Q _ {g} (\mathbf {A})}\right) + 2 \delta \\ \leq \left(g (\mathbf {P} _ {\mathbf {A}} x) - g (\mathbf {P} _ {\mathbf {A}} x ^ {*}) + 2 \delta\right) \left(1 - \frac {1}{5 k Q _ {g} (\mathbf {A})}\right) + 2 \delta \\ \leq \left(f (x) - f \left(x ^ {*}\right) + 4 \delta\right) \left(1 - \frac {1}{5 k Q _ {g} (\mathbf {A})}\right) + 2 \delta \\ \leq \left(f (x) - f \left(x ^ {*}\right)\right) \left(1 - \frac {1}{5 k Q _ {g} (\mathbf {A})}\right) + 6 \delta \\ \end{array}
+$$
+
+Since  $f(x) - f(x^{*}) \geq 10\delta kQ_{g}(A)$ , we conclude that  $(f(x) - f(x^{*}))\left(1 - \frac{1}{5kQ_{g}(\mathbf{A})}\right) + 6\delta \leq (f(x) - f(x^{*}))\left(1 - \frac{1}{10kQ_{g}(\mathbf{A})}\right)$  and our proof is complete.
+
+Proof of Theorem 12. Our main proof strategy is to show that progress can only be made with a radius size of  $O(\sqrt{\log(nQ)}/(nQ))$ ; larger radii cannot find descent directions with high probability. Consider a simple ellipsoid function  $f(x) = x^{\top}Dx$ , where  $D$  is a diagonal matrix and  $D_{11} \leq D_{22} \leq \ldots \leq D_{nn}$ , where WLOG we let  $D_{11} = 1$  and  $D_{ii} = Q$  for  $i > 1$ . The optima is  $x^{*} = 0$  with  $f(x^{*}) = 0$ .
+
+Consider the region  $X = \{x = (x_{1}, x_{2}, \dots, x_{n}) | 1 \geq x_{1} \geq 0.9, |x_{i}| \leq 0.1 / (Q\sqrt{n})\}$ . Then, if we let  $v \sim N(0, I)$  be a standard Gaussian vector, then for some radius  $r$ , we see that the probability of finding a descent direction is:
+
+$$
+\begin{array}{l} \Pr [ f (x + r v) \leq f (x) ] = \Pr \left[ \left(x _ {1} + r v _ {1}\right) ^ {2} + \sum_ {i > 1} D _ {i i} \left(x _ {i} + r v _ {i}\right) ^ {2} \leq x _ {1} ^ {2} + \sum_ {i > 1} D _ {i i} x _ {i} ^ {2} \right] \\ = \Pr \left[ 2 r x _ {1} v _ {1} + r ^ {2} v _ {1} ^ {2} + Q \sum_ {i > 1} \left(2 r x _ {i} v _ {i} + r ^ {2} v _ {i} ^ {2}\right) \leq 0 \right] \\ \leq \Pr \left[ 2 r x _ {1} v _ {1} \leq - Q \sum_ {i > 1} 2 r x _ {i} v _ {i} - Q r ^ {2} \sum_ {i > 1} v _ {i} ^ {2} \right] \\ = \mathbf {P r} \left[ v _ {1} X _ {1} \leq - Q \sum_ {i > 1} x _ {i} v _ {i} - \frac {1}{2} Q r \sum_ {i > 1} v _ {i} ^ {2} \right] \\ \end{array}
+$$
+
+By standard concentration bounds for sub-exponential variables, we have
+
+$$
+\mathbf {P r} \left[ \left| \frac {1}{n - 1} \sum_ {i > 1} v _ {i} ^ {2} - 1 \right| \geq t \right] \leq 2 e ^ {- (n - 1) t ^ {2} / 8}
+$$
+
+Therefore, with exponentially high probability,  $\sum_{i > 1}X_i^2\geq n / 2$ . Also, since  $|x_{i}|\leq 0.1 / (Q\sqrt{n})$ , Chernoff bounds give:
+
+$$
+\Pr \left[ \left| \sum_ {i > 1} x _ {i} v _ {i} \right| \geq t \right] \leq 2 e ^ {- 5 0 (Q t) ^ {2}}
+$$
+
+Therefore, with probability at least  $1 - 1 / (nQ)^3$ ,  $\left|\sum_{i > 1}v_iX_i\right|\leq \sqrt{\log(nQ)} /Q$
+
+If  $Qrn \geq \Omega(\sqrt{\log(nQ)})$ , then we have
+
+$$
+- Q \sum_ {i > 1} v _ {i} X _ {i} - \frac {1}{2} Q r \sum_ {i > 1} X _ {i} ^ {2} \leq - \Omega (\sqrt {\log (n Q)})
+$$
+
+We conclude that the probability of descent is upper bounded by  $\mathbf{Pr}\left[v_1X_1\leq -\Omega (\sqrt{\log(nQ)})\right]$ . This probability is exactly  $\Phi (-l)$ , where  $\Phi$  is the cumulative density of a standard normal and  $l = \Omega (\sqrt{\log(nQ)})$ . By a naive upper bound, we see that
+
+$$
+\begin{array}{l} \Phi (- l) = \frac {1}{\sqrt {2 \pi}} \int_ {l} ^ {\infty} e ^ {- x ^ {2} / 2} d x \\ \leq \frac {C}{l} \int_ {l} ^ {\infty} x e ^ {- x ^ {2} / 2} d x \\ = \frac {C}{l} e ^ {- l ^ {2} / 2} \\ \end{array}
+$$
+
+Since  $l = \Omega(\sqrt{\log(nQ)})$ , we conclude that with probability at least  $1 - 1/poly(nQ)$ , we have  $f(y) - f(x^{*}) \geq f(x) - f(x^{*})$ .
+
+Otherwise, we are in the case that  $Qrn \leq O(\sqrt{\log(nQ)})$ . Arguing similarly as before, with high probability, our objective function and each coordinate can change by at most  $O(\sqrt{\log(nQ)}/(Qn))$ .
+
+Next, we extend our proof to any symmetric distribution  $\mathcal{D}$ . Since  $\mathcal{D}$  is rotationally symmetric, if we parametrize  $v = (r,\theta)$  is polar-coordinates, then the p.d.f. of any scaling of  $\mathcal{D}$  must take the form  $p(v) = p_r(r)u(\theta)$ , where  $u(\theta)$  induces the uniform distribution over the unit sphere. Therefore, if  $Y$  is a random variable that follows  $\mathcal{D}$ , then we may write  $Y = Rv / \| v\|$ , where  $R$  is a random scalar with p.d.f  $p_r(r)$  and  $v$  is a standard Gaussian vector and  $R,X$  are independent.
+
+As previously argued,  $\| v \| \in [0.5n, 1.5n]$  with exponentially high probability. Therefore, if  $R \geq \Omega(\sqrt{\log(nQ)} / Q)$ , the same arguments will imply that  $Y$  is a descent direction with polynomially small probability. Thus, when  $Y$  is a descent direction, it must be that  $R \leq \Omega(\sqrt{\log(nQ)} / Q)$  and as argued previously, our lower bound follows similarly.
+
+![](images/f848a949fdb40d5073dcf9ade3afc7ebe208349eaceebcda00f7f20aa046e85e.jpg)
+
+# B PROOFS OF SECTION 4
+
+Proof of Theorem 13. By the Gaussian version of Theorem 7 (full rank version of Theorem 10), as long as our binary search sweeps between minimum search radius  $r \leq \frac{3}{5Q\sqrt{n}} \| x - x^{*} \|$  and
+
+maximum search radius of the diameter of the whole space  $R = \| \mathcal{X}\|$ , the objective value will decrease multiplicatively by  $1 - \frac{1}{5nQ}$  in each iteration with constant probability. Therefore, if  $\| x_{t} - x^{*}\| \geq 2Q\epsilon$  and we set  $r = \frac{\epsilon}{\sqrt{n}}$  and  $R = \| \mathcal{X}\|$ , then with high probability, we expect  $f(x_{T}) - f(x^{*}) \leq \beta Q^{2}\epsilon^{2}$  after  $T = O(nQ\log (\| \mathcal{X}\| /(Q\epsilon)))$  iterations, where we note that  $F = f(x_0) - f(x^*) \leq \beta \| \mathcal{X}\|^2$  by smoothness.
+
+Otherwise, if there exists some  $x_{t}$  such that  $\| x_{t} - x^{*}\| \leq 2Q\epsilon$ , then  $f(x_{T}) - f(x^{*}) \leq f(x_{t}) - f(x^{*}) \leq 4\beta Q^{2}\epsilon^{2}$ . Therefore, by strong convexity, we conclude that in either case,  $\| x_{T} - x^{*}\| \leq 2Q^{3 / 2}\epsilon$ . Finally note that each iteration uses a binary search that requires  $O(\log (R / r)) = O(\log (n\| \mathcal{X}\| /\epsilon))$  function evaluations.
+
+Therefore, by combining these bounds, we derive our result. The low-rank result follows from applying Theorem 10 and Theorem 11 instead.
+
+Proof of Theorem 14. Let  $H = O(nQ\log (Q))$  be the number of iterations between successive radius halving and we initialize  $R = \| \mathcal{X}\|$  and half  $R$  every  $H$  iterations. We call the iterations between two halving steps an epoch. We claim that  $\| x_{i} - x_{0}\| \leq R$  for all iterations and proceed with induction on the epoch number. The base case is trivial.
+
+Assume that  $\| x_{i} - x_{0}\| \leq R$  for all iterations in the previous epoch and let iteration  $i_{s}$  be the start of the epoch and iteration  $i_{s} + H$  be the end of the epoch. Then, since  $\| x_{i_s} - x^*\| \leq R$ , we see that  $f(x_{i_s}) - f(x^*)\leq \beta R^2$  by smoothness. If  $\frac{R}{4\sqrt{Q}}\leq \| x_i - x^*\| \leq 4\sqrt{Q} R$  for all  $i$  in the previous epoch, then by the Gaussian version of Theorem 7 (Theorem 10), since we do a binary sweep from  $\frac{R}{4Q}$  to  $4\sqrt{Q} R$ , we can choose  $\mathcal{D}$  accordingly so that we are guaranteed that our objective value will decrease multiplicatively by  $1 - \frac{1}{5nQ}$  with constant probability at a cost of  $O(\log (Q))$  function evaluations per iteration. This implies that with high probability, after  $O(nQ\log (Q))$  iterations, we conclude
+
+$$
+f (x _ {i _ {s} + H}) - f (x ^ {*}) \leq \frac {1}{4 Q} (f (x _ {i _ {s}}) - f (x ^ {*})) \leq \frac {\alpha}{4} \| x _ {i _ {s}} - x ^ {*} \| ^ {2} \leq \frac {\alpha}{4} R ^ {2}
+$$
+
+Otherwise, there exists some  $1 \leq j \leq H$  such that  $\| x_{i_s + j} - x^* \| \geq 4\sqrt{Q} R$  or  $\| x_{i_s + j} - x^* \| \leq \frac{R}{4\sqrt{Q}}$ . If it is the former, then by strong convexity,  $f(x_{i_s + j}) - f(x^*) \geq \alpha \| x_{i_s + j} - x^* \|^2 \geq 2\beta R^2$ , which contradicts the fact that  $f(x_{i_s}) - f(x^*) \leq \beta R^2$  by smoothness. If it is the latter, then by smoothness, we reach the same conclusion:
+
+$$
+f \left(x _ {i _ {s} + H}\right) - f \left(x ^ {*}\right) \leq f \left(x _ {i _ {s} + j}\right) - f \left(x ^ {*}\right) \leq \beta \| x _ {i _ {s} + j} - x ^ {*} \| ^ {2} \leq \frac {\alpha}{4} R ^ {2}
+$$
+
+Therefore, by strong convexity, we have
+
+$$
+\left\| x _ {i _ {s} + H} - x ^ {*} \right\| \leq \sqrt {\frac {f \left(x _ {i _ {s} + H}\right) - f \left(x ^ {*}\right)}{\alpha}} \leq \frac {R}{2}
+$$
+
+And our induction is complete. Therefore, we conclude that after  $\log (\| \mathcal{X}\| /\epsilon)$  epochs, we have  $\| x_{T} - x^{*}\| \leq \epsilon$ . Each epoch has  $H$  iterations, each with  $O(\log (Q))$  function evaluations and so our result follows.
+
+The low-rank result follows from applying Theorem 10 and Theorem 11 instead. However, note that since we do not know the latent dimension  $k$ , we must extend the binary search to incur an extra  $\log (n)$  factor in the binary search cost.
+
+# C FIGURES
+
+![](images/008ccaf611aa1b5fef5cb063e9619080f866286667597301004a3afb52d5e808.jpg)  
+Figure 2: The average optimality gap by the condition number of the objective function.
+
+![](images/b51f57f7d5ae0b3b17aa12315d1939f08737c687a373a16575297315e2aea5bd.jpg)
+
+![](images/8998a917094abc39b9de5ab1cc8b9d15babb9d028b5b264530ab9893305fb0f5.jpg)
+
+![](images/04363a554f54096cc2c469417a56422a84385e5d5cb7c3e5871a1c43d5b7c9b9.jpg)  
+GLD-Search  
+GLD-Fast  
+ARS
+
+![](images/84f2ebc719573f0c324afab9f011e8c00c0fe2a58e8dfcbe863ccc851c1cdc0c.jpg)  
+Figure 3: The average optimality gap by the accuracy of the condition number estimate, where approximation factor is the ratio of estimated to true condition number. The dimension  $n = 20$ .
+
+![](images/ee016e7bf132c3c4bc78f90fe6be7f6b56ee1ffe904fb9bd34e85e982eddf61a.jpg)
+
+![](images/bc0a39c32a4f6cf726c235c2bff33241de9625b048f0cbcb7535aba3cf2b2233.jpg)
+
+![](images/995cd17bf4b8abeeffd7b76e31a26636000b9a6249739406c65f8a8fec97afcd.jpg)  
+- GLD-Search
+- GLD-Fast
+- ARS-alpha
+- ARS-beta
+- ARS-even
+
+# C.1 BBOB FUNCTION PLOTS
+
+![](images/a9ea2bf2014eb25dbcdd3df1fe66c9e14edf832319c950179b6441b2ba296770.jpg)  
+Figure 4: Convergence plot for the BBOB Rastrigin Function.
+
+![](images/7ef1b814679adac14465fe371248701ad41c6bd18c764f78be07dba3673df00b.jpg)  
+Figure 5: Convergence plot for the BBOB BentCigar Function.
+
+![](images/83c8d4898d14b5d188770372b37d85694af25a5723f5adbe35f1342ac5f66fad.jpg)  
+Figure 6: Convergence plot for the BBOB BuecheRastrigin Function.  
+Convergence Plots for BuecheRastrigin Function
+
+![](images/14478faeda08e85646775d36cd142f09366337165c17e295bfa6ab9c341fe062.jpg)  
+Figure 7: Convergence plot for the BBOB DifferentPowers Function.  
+Convergence Plots for DifferentPowers Function
+
+![](images/2f356461464bd804a4e3fd555aea7cb996b0f67c897918577af4ba77841f8f33.jpg)  
+Figure 8: Convergence plot for the BBOB Discuss Function.  
+Convergence Plots for Discus Function
+
+![](images/d7fc73bbd452737bec566757082fae8acea4557c9b4d4ee91d43fec0476f6c27.jpg)  
+Figure 9: Convergence plot for the BBOB Ellipsoidal Function.  
+Convergence Plots for Ellipsoidal Function
+
+![](images/f19b229fc49c9573c28a89b37d6bb4cdacb229d5948377a9669e2e0264c7c22c.jpg)  
+Figure 10: Convergence plot for the BBOB Katsuura Function.
+
+![](images/019d74f4cfadd0683b749b93219ba9ea897939f1ae844a89f2963705ba5887b7.jpg)  
+Convergence Plots for Katsuura Function
+
+![](images/b375c788ff852b53145f451c6cb2597e78eb4f922d517a9a6e51b2bfca37ddf9.jpg)
+
+![](images/3913ab8f4736ee247edb10d6037ffc5c78bd4928a36c4249f72560d28ae07bac.jpg)
+
+![](images/058521f017eda206c6ae80c9b8dad8bb8844424054807ce52aa01ab73e33f832.jpg)
+
+![](images/4cf51f48975adaa13f747143d25f5e55c3cf63df11f34cde7dab7da5015e6818.jpg)  
+Figure 11: Convergence plot for the BBOB SchaffersF7 Function.
+
+![](images/edbf1720af8a919c51eca7989a2fcd3ae8c168eb01bfe9212cab6ee55d288b45.jpg)  
+Convergence Plots for SchaffersF7 Function
+
+![](images/1868bca6c60af7d8a5ebaf6971d1f87a3a1828ed39fe0e8c58f787b6e18207df.jpg)  
+Figure 12: Convergence plot for the BBOB Ill-Conditioned SchaffersF7 Function.
+
+![](images/275e542e35407d7d24ec310373c9c81838b8b8320d1203d473fffb52aef51b1b.jpg)
+
+![](images/f8a0bab85bbe142ea3eeb5259bacf6663e367902c8f0a8ba306e7950b52fdf28.jpg)
+
+![](images/d09d888ef111544916231e060f976f897cbbb90011e388ade85542cc8dc09747.jpg)
+
+![](images/c24095c0fe94678996b0844808c4e4f6e6097eb4272af7452f9b9d014662bb59.jpg)  
+Convergence Plots for SchaffersF7IIIConditioned Function
+
+![](images/3680c1e1cfc127bc52336038e66a61c6c50924fa25e1b0def17264902be4cbb1.jpg)  
+Figure 13: Convergence plot for the BBOB SharpRidge Function.
+
+![](images/c15cbaafdae03ef54a1279fd51a5413df88a81eb72d9d595c916afcc490f24b8.jpg)
+
+![](images/2f3d7091d5d9964bade7115ecd295b127c9dcbf6f61292b4807d99c4d2ce31a1.jpg)
+
+![](images/673faf7d4023a312ee7d023c950af2ecaf3b3e4ab82e882740f80f90d22bd1a0.jpg)
+
+![](images/ffac4b59a645ceafe2434e4aaa1e853a2c4c8d73d22711bf1c761a4ac026d11a.jpg)  
+Convergence Plots for SharpRidge Function
+
+![](images/95b8b22640fd071d0ccc717330a4cdd4ffcfd211a524e5b6b7c53a37f2d397df.jpg)
+
+![](images/358fd934081879e9c59ee39d06df61b4463657f5f4f8a2a07cf72a1df74262d1.jpg)
+
+![](images/d7b2fdcfb2c8bcdd0f16846d99e2633efa68d6dea1f141fa4e9315427c153109.jpg)
+
+![](images/0f32233bcfc7b8ba0cf5b306895fa9c689cc85ef53a1e8c1528d6fc4e6330f17.jpg)  
+Figure 14: Convergence plot for the BBOB Weierstass Function.
+
+# C.2 MUJOCO CONTROL PLOTS
+
+![](images/5dcb5eb909f5be725ac412f7ee13de5ee7abe244aeb3d24ba456aa9dabd7f9f3.jpg)  
+Figure 15: Plot of maximum reward so far found by algorithm. Main line is the median trajectory across 3 runs.
+
+![](images/76be28398f39af10f879686c98efc8473516ca3bdb8327df7148dace241b336e.jpg)  
+Figure 16: Example of rewards found by all samples by algorithm.
\ No newline at end of file
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+# GRAPH NEURAL NETWORKS EXPONENTIALLY LOSE EXPRESSIVE POWER FOR NODE CLASSIFICATION
+
+Kenta Oono $^{1,2}$ , Taiji Suzuki $^{1,3}$
+
+{kenta_oono, taiji}@mist.i.u-tokyo.ac.jp
+
+<sup>1</sup>The University of Tokyo <sup>2</sup>Preferred Networks, Inc.
+
+$^{3}$ RIKEN Center for Advanced Intelligence Project (AIP)
+
+# ABSTRACT
+
+Graph Neural Networks (graph NNs) are a promising deep learning approach for analyzing graph-structured data. However, it is known that they do not improve (or sometimes worsen) their predictive performance as we pile up many layers and add non-linearity. To tackle this problem, we investigate the expressive power of graph NNs via their asymptotic behaviors as the layer size tends to infinity. Our strategy is to generalize the forward propagation of a Graph Convolutional Network (GCN), which is a popular graph NN variant, as a specific dynamical system. In the case of a GCN, we show that when its weights satisfy the conditions determined by the spectra of the (augmented) normalized Laplacian, its output exponentially approaches the set of signals that carry information of the connected components and node degrees only for distinguishing nodes. Our theory enables us to relate the expressive power of GCNs with the topological information of the underlying graphs inherent in the graph spectra. To demonstrate this, we characterize the asymptotic behavior of GCNs on the Erdős - Rényi graph. We show that when the Erdős - Rényi graph is sufficiently dense and large, a broad range of GCNs on it suffers from the "information loss" in the limit of infinite layers with high probability. Based on the theory, we provide a principled guideline for weight normalization of graph NNs. We experimentally confirm that the proposed weight scaling enhances the predictive performance of GCNs in real data<sup>1</sup>.
+
+# 1 INTRODUCTION
+
+Motivated by the success of Deep Learning (DL), several attempts have been made to apply DL models to non-Euclidean data, particularly, graph-structured data such as chemical compounds, social networks, and polygons. Recently, Graph Neural Networks (graph NNs) (Duvenaud et al., 2015; Li et al., 2016; Gilmer et al., 2017; Hamilton et al., 2017; Kipf & Welling, 2017; Nguyen et al., 2017; Schlichtkrull et al., 2018; Battaglia et al., 2018; Xu et al., 2019; Wu et al., 2019a) have emerged as a promising approach. However, despite their practical popularity, theoretical research of graph NNs has not been explored extensively.
+
+The characterization of DL model expressive power, i.e., to identify what function classes DL models can (approximately) represent, is a fundamental question in theoretical research of DL. Many studies have been conducted for Fully Connected Neural Networks (FNNs) (Cybenko, 1989; Hornik, 1991; Hornik et al., 1989; Barron, 1993; Mhaskar, 1993; Sonoda & Murata, 2017; Yarotsky, 2017) and Convolutional Neural Networks (CNNs) (Petersen & Voigtlaender, 2018; Zhou, 2018; Oono & Suzuki, 2019). For such models, we have theoretical and empirical justification that deep and nonlinear architectures can enhance representation power (Telgarsky, 2016; Chen et al., 2018b; Zhou & Feng, 2018). However, for graph NNs, several papers have reported that node representations go indistinguishable (known as over-smoothing) and prediction performances severely degrade when we stack many layers (Kipf & Welling, 2017; Wu et al., 2019b; Li et al., 2018). Besides, Wu et al. (2019a) reported that graph NNs achieved comparable performance even if they removed intermediate non-linear functions. These studies posed a question about the current architecture and made us aware of the need for the theoretical analysis of the graph NN expressive power.
+
+In this paper, we investigate the expressive power of graph NNs by analyzing their asymptotic behaviors as the layer size goes to infinity. Our theory gives new theoretical conditions under which neither layer stacking nor non-linearity contributes to improving expressive power. We consider a specific dynamics that includes a transition defining a Markov process and the forward propagation of a Graph Convolutional Network (GCN) (Kipf & Welling, 2017), which is one of the most popular graph NN variants, as special cases. We prove that under certain conditions, the dynamics exponentially approaches a subspace that is invariant under the dynamics. In the case of GCN, the invariant space is a set of signals that correspond to the lowest frequency of graph spectra and that have "no information" other than connected components and node degrees for a node classification task whose goal is to predict the nodes' properties in a graph. The rate of the distance between the output and the invariant space is  $O((s\lambda)^L)$  where  $s$  is the maximum singular values of weights,  $\lambda$  is typically a quantity determined by the spectra of the (augmented) normalized Laplacian, and  $L$  is the layer size. See Sections 3.3 (general case) and 4 (GCN case) for precise statements.
+
+We can interpret our theorem as the generalization of the well-known property that if a finite and discrete Markov process is irreducible and aperiodic, it exponentially converges to a unique equilibrium and the eigenvalues of its transition matrix determine the convergence rate (see, e.g., Chung & Graham (1997)). Different from the Markov process case, which is linear, the existence of intermediate non-linear functions complicates the analysis. We overcame this problem by leveraging the combination of the ReLU activation function (Krizhevsky et al., 2012) and the positivity of eigenvectors of the Laplacian associated with the smallest positive eigenvalues.
+
+Our theory enables us to investigate asymptotic behaviors of graph NNs via the spectral distribution of the underlying graphs. To demonstrate this, we take GCNs defined on the Erdős - Rényi graph  $G_{N,p}$ , which has  $N$  nodes and each edge appears independently with probability  $p$ , for an example. We prove that if  $\frac{\log N}{pN} = o(1)$  as a function of  $N$ , any GCN whose weights have maximum singular values at most  $C\sqrt{\frac{Np}{\log(N / \varepsilon)}}$  approaches the "information-less" invariant space with probability at least  $1 - \varepsilon$ , where  $C$  is a universal constant. Intuitively, if the graph on which we define graph NNs is sufficiently dense, graph-convolution operations mix signals on nodes fast and hence the feature maps lose information for distinguishing nodes quickly.
+
+Our contributions are as follows:
+
+- We relate asymptotic behaviors of graph NNs with the topological information of underlying graphs via the spectral distribution of the (augmented) normalized Laplacian.  
+- We prove that if the weights of a GCN satisfy conditions determined by the graph spectra, the output of the GCN carries no information other than the node degrees and connected components for discriminating nodes when the layer size goes to infinity (Theorems 1, 2).  
+- We apply our theory to Erdős – Rényi graphs as an example and show that when the graph is sufficiently dense and large, many GCNs suffer from the information loss (Theorem 3).  
+- We propose a principled guideline for weight normalization of graph NNs and empirically confirm it using real data.
+
+# 2 RELATED WORK
+
+MPNN-type Graph NNs. Since many graph NN variants have been proposed, there are several unified formulations of graph NNs (Gilmer et al., 2017; Battaglia et al., 2018). Our approach is the closest to the formulation of Message Passing Neural Network (MPNN) (Gilmer et al., 2017), which unified graph NNs in terms of the update and readout operations. Many graph NNs fall into this formulation such as Duvenaud et al. (2015), Li et al. (2016), and Velicković et al. (2018). Among others, GCN (Kipf & Welling, 2017) is an important application of our theory because it is one of the most widely used graph NNs. In addition, GCNs are interesting from a theoretical research perspective because, in addition to an MPNN-type graph NN, we can interpret GCNs as a simplification of spectral-type graph NNs (Henaff et al., 2015; Defferrard et al., 2016), that make use of the graph Laplacian.
+
+Our approach, which considers the asymptotic behaviors graph NNs as the layer size goes to infinity, is similar to Scarselli et al. (2009), one of the earliest works about graph NNs. They obtained node
+
+representations by iterating message passing between nodes until convergence. Their formulation is general in that we can use any local aggregation operation as long as it is a contraction map. Our theory differs from theirs in that we proved that the output of a graph NN approaches a certain space even if the local aggregation function is not necessarily a contraction map.
+
+Expressive Power of Graph NNs. Several studies have focused on theoretical analysis and the improvement of graph NN expressive power. For example, Xu et al. (2019) proved that graph NNs are no more powerful than the Weisfeiler - Lehman (WL) isomorphism test (Weisfeiler & A.A., 1968) and proposed a Graph Isomorphism Network (GIN), that is approximately as powerful as the WL test. Although they experimentally showed that GIN has improved accuracy in supervised learning tasks, their analysis was restricted to the graph isomorphism problem. Xu et al. (2018) analyzed the non-asymptotic properties of GCNs through the lens of random walk theory. They proved the limitations of GCNs in expander-like graphs and proposed a Jumping Knowledge Network (JK-Net) to address the issue. To handle the non-linearity, they linearized networks by a randomization assumption (Choromanska et al., 2015). We take a different strategy and make use of the interpretation of ReLU as a projection onto a cone. Recently, NT & Maehara (2019) showed that a GCN approximately works as a low-pass filter plus an MLP in a certain setting. Although they analyzed finite-depth GCNs, our theory has similar spirits with theirs because our "information-less" space corresponds to the lowest frequency of a graph Laplacian. Another point is that they imposed assumptions that input signals consist of low-frequent true signals and high-frequent noise, whereas we need not such an assumption.
+
+Role of Deep and Non-linear Structures. For ordinal DL models such as FNNs and CNNs, we have both theoretical and empirical justification of deep and non-linear architectures for enhancing of the expressive power (e.g., Telgarsky (2016); Petersen & Voigtlaender (2018); Oono & Suzuki (2019)). In contrast, several studies have witnessed severe performance degradation when stacking many layers on graph NNs (Kipf & Welling, 2017; Wu et al., 2019b). Li et al. (2018) reported that feature vectors on nodes in a graph go indistinguishable as we increase layers in several tasks. They named this phenomenon over-smoothing. Regarding non-linearity, Wu et al. (2019a) empirically showed that graph NNs achieve comparable performance even if we omit intermediate non-linearity. These observations gave us questions about the current models of deep graph NNs in terms of their expressive power. Several studies gave theoretical explanations of the over-smoothing phenomena for linear GNNs (Li et al., 2018; Zhang, 2019; Zhao & Akoglu, 2020). We can think of our theory as an extension of their results to non-linear GNNs.
+
+# 3 PROBLEM SETTING AND MAIN RESULT
+
+# 3.1 NOTATION
+
+Let  $\mathbb{N}_{+}$  be the set of positive integers. For  $N\in \mathbb{N}_{+}$ , we denote  $[N]\coloneqq \{1,\ldots ,N\}$ . For a vector  $v\in \mathbb{R}^N$ , we write  $v\geq 0$  if and only if  $v_{n}\geq 0$  for all  $n\in [N]$ . Similarly, for a matrix  $X\in \mathbb{R}^{N\times C}$ , we write  $X\geq 0$  if and only if  $X_{nc}\geq 0$  for all  $n\in [N]$  and  $c\in [C]$ . We say such a vector and matrix is non-negative.  $\langle \cdot ,\cdot \rangle$  denotes the inner product of vectors or matrices, depending on the context:  $\langle u,v\rangle \coloneqq u^\top v$  for  $u,v\in \mathbb{R}^N$  and  $\langle X,Y\rangle \coloneqq \mathrm{tr}(X^T Y)$  for  $X,Y\in \mathbb{R}^{N\times C}$ .  $\mathbf{1}_P$  equals to 1 if the proposition  $P$  is true else 0. For vectors  $v\in \mathbb{R}^N$  and  $w\in \mathbb{R}^C$ ,  $v\otimes w\in \mathbb{R}^{N\times C}$  denotes the Kronecker product of  $v$  and  $w$  defined by  $(v\otimes w)_{nc}\coloneqq v_nw_c$ . For  $X\in \mathbb{R}^{N\times C}$ ,  $\| X\|_{\mathrm{F}}\coloneqq \langle X,X\rangle^{\frac{1}{2}}$  denotes the Frobenius norm of  $X$ . For a vector  $v\in \mathbb{R}^N$ ,  $\mathrm{diag}(v)\coloneqq (v_n\delta_{nm})_{n,m\in [N]}\in \mathbb{R}^{N\times N}$  denotes the diagonalization of  $v$ .  $I_N\in \mathbb{R}^{N\times N}$  denotes the identity matrix of size  $N$ . For a linear operator  $P:\mathbb{R}^N\to \mathbb{R}^M$  and a subset  $V\subset \mathbb{R}^N$ , we denote the restriction of  $P$  to  $V$  by  $P|_V$ .
+
+# 3.2 DYNAMICAL SYSTEM
+
+Although we are mainly interested in GCNs, we develop our theory more generally using dynamical systems. We will specialize to the GCNs in Section 4.
+
+For  $N,C,H_{l}\in \mathbb{N}_{+}$  ( $l\in \mathbb{N}_{+}$ ), let  $P\in \mathbb{R}^{N\times N}$  be a symmetric matrix and  $W_{lh}\in \mathbb{R}^{C\times C}$  for  $l\in \mathbb{N}_{+}$  and  $h\in [H_l]$ . We define  $f_{l}:\mathbb{R}^{N\times C}\to \mathbb{R}^{N\times C}$  by  $f_{l}(X)\coloneqq \mathrm{MLP}_{l}(PX)$ . Here,  $\mathrm{MLP}_l:\mathbb{R}^{N\times C}\to \mathbb{R}^{N\times C}$  is the  $l$ -th multi-layer perceptron common to all nodes (Xu et al., 2019) and is defined by  $\mathrm{MLP}_l(X)\coloneqq \sigma (\dots \sigma (\sigma (X)W_{l1})W_{l2}\dots W_{lH_l})$ , where  $\sigma :\mathbb{R}^{N\times C}\to \mathbb{R}^{N\times C}$
+
+is an element-wise ReLU function (Krizhevsky et al., 2012) defined by  $\sigma(X)_{nc} \coloneqq \max(X_{nc}, 0)$  for  $n \in [N], c \in [C]$ . We consider the dynamics  $X^{(l+1)} \coloneqq f_l(X^{(l)})$  with some initial value  $X^{(0)} \in \mathbb{R}^{N \times C}$ . We are interested in the asymptotic behavior of  $X^{(l)}$  as  $l \to \infty$ .
+
+For  $M \leq N$ , let  $U$  be a  $M$ -dimensional subspace of  $\mathbb{R}^N$ . We assume that  $U$  and  $P$  satisfy the following properties that generalize the situation where  $U$  is the eigenspace associated with the smallest eigenvalue of a (normalized) graph Laplacian  $\Delta$  (that is, zero) and  $P$  is a polynomial of  $\Delta$ .
+
+Assumption 1.  $U$  has an orthonormal basis  $(e_m)_{m\in [M]}$  that consists of non-negative vectors.
+
+Assumption 2.  $U$  is invariant under  $P$ , i.e., if  $u \in U$ , then  $Pu \in U$ .
+
+We endow  $\mathbb{R}^N$  with the ordinal inner product and denote the orthogonal complement of  $U$  by  $U^\perp \coloneqq \{u\in \mathbb{R}^N\mid \langle u,v\rangle = 0,\forall v\in U\}$ . By the symmetry of  $P$ , we can show that  $U^{\perp}$  is invariant under  $P$ , too (Appendix E.1, Proposition 2). Therefore, we can regard  $P$  as a linear mapping  $P|_{U^{\perp}}: U^{\perp}\to U^{\perp}$ . We denote the operator norm of  $P|_{U^{\perp}}$  by  $\lambda$ . When  $U$  is the eigenspace associated with the smallest eigenvalue of  $\Delta$  and  $P$  is  $g(\Delta)$  where  $g$  is a polynomial, then,  $\lambda$  corresponds to  $\lambda = \sup_{\mu}|g(\mu)|$  where sup ranges over all eigenvalues except the smallest one.
+
+# 3.3 MAIN RESULT
+
+We define the subspace  $\mathcal{M}$  of  $\mathbb{R}^{N\times C}$  by  $\mathcal{M}:= U\otimes \mathbb{R}^C = \{\sum_{m = 1}^M e_m\otimes w_m\mid w_m\in \mathbb{R}^C\}$  where  $(e_m)_{m\in [M]}$  is the orthonormal basis of  $U$  appeared in Assumption 1. For  $X\in \mathbb{R}^{N\times C}$ , we denote the distance between  $X$  and  $\mathcal{M}$  by  $d_{\mathcal{M}}(X)\coloneqq \inf \left\{\| X - Y\|_{\mathrm{F}}\mid Y\in \mathcal{M}\right\}$ . We denote the maximum singular value of  $W_{lh}$  by  $s_{lh}$  and set  $s_l\coloneqq \prod_{h = 1}^{H_l}s_{lh}$ . With these preparations, we introduce the main theorem of the paper.
+
+Theorem 1. Under Assumptions 1 and 2, we have  $d_{\mathcal{M}}(f_l(X)) \leq s_l \lambda d_{\mathcal{M}}(X)$  for any  $X \in \mathbb{R}^{N \times C}$ .
+
+The proof key is that the non-linear operation  $\sigma$  decreases the distance  $d_{\mathcal{M}}$ , that is,  $d_{\mathcal{M}}(\sigma(X)) \leq d_{\mathcal{M}}(X)$ . We use the non-negativity of  $e_m$  to prove this claim. See Appendix A for the complete proof. We also discuss the strictness of Theorem 1 in Appendix E.3.
+
+By setting  $d_{\mathcal{M}}(X) = 0$ , this theorem implies that  $\mathcal{M}$  is invariant under  $f_{l}$ . In addition, if the maximum value of singular values are small,  $X^{(l)}$  asymptotically approaches  $\mathcal{M}$  in the sense of Johnson (1973) for any initial value  $X^{(0)}$ . That is, the followings hold under Assumptions 1 and 2.
+
+Corollary 1.  $\mathcal{M}$  is invariant under  $f_{l}$  for any  $l\in \mathbb{N}_{+}$ , that is, if  $X\in \mathcal{M}$ , then we have  $f_{l}(X)\in \mathcal{M}$ .
+
+Corollary 2. Let  $s \coloneqq \sup_{l \in \mathbb{N}_+} s_l$ . We have  $d_{\mathcal{M}}(X^{(l)}) = O((s\lambda)^l)$ . In particular, if  $s\lambda < 1$ , then  $X_l$  exponentially approaches  $\mathcal{M}$  as  $l \to \infty$  for any initial value  $X^{(0)}$ .
+
+Suppose the operator norm of  $P|_{U}: U \to U$  is no larger than  $\lambda$ , then, under the assumption of  $s\lambda < 1$ ,  $X^{(l)}$  converges to 0, the trivial fixed point (see Appendix E.2, Proposition 3). Therefore, we are mainly interested in the case where the operator norm of  $P|_{U}$  is strictly larger than  $\lambda$  (see Proposition 1). Finally, we restate Theorem 1 specialized to the situation where  $U$  is the direct sum of eigenspaces associated with the largest  $M$  eigenvalues of  $P$ . Note that the eigenvalues of  $P$  is real since  $P$  is symmetric.
+
+Corollary 3. Let  $\lambda_1 \leq \dots \leq \lambda_N$  be the eigenvalue of  $P$ , sorted in ascending order. Suppose the multiplicity of the largest eigenvalue  $\lambda_N$  is  $M(\leq N)$ , i.e.,  $\lambda_{N - M} < \lambda_{N - M + 1} = \dots = \lambda_N$ . We define  $\lambda := \max_{n \in [N - M]} |\lambda_n|$ . We denote  $U$  by the eigenspace associated with  $\lambda_N$  and assume that  $U$  satisfies Assumption 1. Then, we have  $d_{\mathcal{M}}(X^{(l + 1)}) \leq s_l \lambda d_{\mathcal{M}}(X^{(l)})$ .
+
+Remark 1. It is known that any Markov process on finite states converges to a unique distribution (equilibrium) if it is irreducible and aperiodic (see e.g., Norris (1998)). Theorem 1 includes this proposition as a special case with  $M = 1$ ,  $C = 1$ , and  $W_{l} = 1$  for all  $l \in \mathbb{N}_{+}$ . This is essentially the direct consequence of Perron-Frobenius' theorem (see e.g., Meyer (2000)). See Appendix F.
+
+# 4 APPLICATION TO GCN
+
+We formulate a GCN (Kipf & Welling, 2017) without readout operations (Gilmer et al., 2017) using the dynamical system in the previous section and derive a sufficient condition in terms of the spectra of underlying graphs in which layer stacking nor non-linearity are not helpful for node classification.
+
+Let  $G = (V, E)$  be an undirected graph where  $V$  is a set of nodes and  $E$  is a set of edges. We denote the number of nodes in  $G$  by  $N = |V|$  and identify  $V$  with  $[N]$  by fixing an order of  $V$ . We associate a  $C$  dimensional signal to each node.  $X$  in the previous section corresponds to concatenation of the signals. GCNs iteratively update signals on  $V$  using the connection information and weights.
+
+Let  $A \coloneqq (\mathbf{1}_{\{(i,j) \in E\}})_{i,j \in [N]} \in \mathbb{R}^{N \times N}$  be the adjacency matrix and  $D \coloneqq \mathrm{diag}(\deg(i)_{i \in [N]}) \in \mathbb{R}^{N \times N}$  be the degree matrix of  $G$  where  $\deg(i) \coloneqq |\{j \in V | (i,j) \in E\}|$  is the degree of node  $i$ . Let  $\tilde{A} \coloneqq A + I_N$ ,  $\tilde{D} \coloneqq D + I_N$  be the adjacent and degree matrix of graph  $G$  augmented with self-loops. We define the augmented normalized Laplacian (Wu et al., 2019a) of  $G$  by  $\tilde{\Delta} \coloneqq I_N - \tilde{D}^{-\frac{1}{2}}\tilde{A}\tilde{D}^{-\frac{1}{2}}$  and set  $P \coloneqq I_N - \tilde{\Delta}$ . Let  $L, C \in \mathbb{N}_+$  be the layer and channel sizes, respectively. For weights  $W_l \in \mathbb{R}^{C \times C}$  ( $l \in [L]$ ), we define a GCN² associated with  $G$  by  $f = f_L \circ \dots \circ f_1$  where  $f_l: \mathbb{R}^{N \times C} \to \mathbb{R}^{N \times C}$  is defined by  $f_l(X) \coloneqq \sigma(PXW_l)$ . We are interested in the asymptotic behavior of the output  $X^{(L)}$  of the GCN as  $L \to \infty$ .
+
+Suppose  $G$  has  $M$  connected components and let  $V = V_{1} \sqcup \dots \sqcup V_{M}$  be the decomposition of the node set  $V$  into connected components. We denote an indicator vector of the  $m$ -th connected component by  $u_{m} := (\mathbf{1}_{\{n \in V_{m}\}})_{n \in [N]} \in \mathbb{R}^{N}$ . The following proposition shows that GCN satisfies the assumption of Corollay 3 (see Appendix B for proof).
+
+Proposition 1. Let  $\lambda_1 \leq \dots \leq \lambda_N$  be the eigenvalue of  $P$  sorted in ascending order. Then, we have  $-1 < \lambda_1$ ,  $\lambda_{N - M} < 1$ , and  $\lambda_{N - M + 1} = \dots = \lambda_N = 1$ . In particular, we have  $\lambda := \max_{n=1,\ldots,N-M} |\lambda_n| < 1$ . Further,  $e_m := \tilde{D}^{\frac{1}{2}} u_m$  for  $m \in [M]$  are the basis of the eigenspace associated with the eigenvalue 1.
+
+Theorem 2. For any initial value  $X^{(0)}$ , the output of  $l$ -th layer  $X^{(l)}$  satisfies  $d_{\mathcal{M}}(X^{(l)}) \leq (s\lambda)^l d_{\mathcal{M}}(X^{(0)})$ . In particular,  $d_{\mathcal{M}}(X^{(l)})$  exponentially converges to 0 when  $s\lambda < 1$ .
+
+In the context of node classification tasks, we can interpret this corollary as the "information loss" of GCNs in the limit of infinite layers. For any  $X \in \mathcal{M}$ , if two nodes  $i, j \in V$  are in a same connected component and their degrees are identical, then, the column vectors of  $X$  that correspond to nodes  $i$  and  $j$  are identical. It means that we cannot distinguish these nodes using  $X$ . In this sense,  $\mathcal{M}$  only has information about connected components and node degrees and we can interpret this theorem as the exponential information loss of GCNs in terms of the layer size. Similarly to the discussion in the previous section,  $X^{(l)}$  converges to the trivial fixed point 0 when  $s < 1$  (remember  $\lambda_N = 1$ ). An interesting point is that even if  $s \geq 1$ ,  $X^{(l)}$  can suffer from this information loss when  $s < \lambda^{-1}$ .
+
+We note that the rate  $s\lambda$  in Theorem 2 depends on the spectra of the augmented normalized Laplacian, which is determined by the topology of the underlying graph  $G$ . Hence, our result explicitly relates the topological information of graphs and asymptotic behaviors of graph NNs.
+
+Remark 2. The old preprint (version 2) of Luan et al. (2019) formulated a theorem that explains the over-smoothing of non-linear GNNs. Specifically, it claimed that if a graph does not have a bipartite component and the input distribution is continuous, the rank of the output of a GCN converges to the number of connected components of the underlying graph as the layer size goes to infinity almost surely. However, it is not true in general as we give a counterexample in Appendix C.
+
+# 5 ASYMPTOTIC BEHAVIOR OF GCN ON ERDOS - RÉNYI GRAPH
+
+Theorem 2 gives us a way to characterize the asymptotic behaviors of GCNs via the spectral distributions of the underlying graphs. To demonstrate this, we consider an Erdős - Rényi graph  $G_{N,p}$  (Erdős & Rényi, 1959; Gilbert, 1959), which is a random graph that has  $N$  nodes and whose edges between two distinct nodes appear independently with probability  $p \in [0,1]$ , as an example. First, consider a (non-random) graph  $G$  with  $M$  connected components. Let  $0 = \tilde{\mu}_1 = \dots = \tilde{\mu}_M < \tilde{\mu}_{M+1} \leq \dots \leq \tilde{\mu}_N < 2$  be eigenvalues of the augmented normalized Laplacian of  $G$  (see, Proposition 1) and set  $\lambda := \min_{m=M+1,\dots,N} |1 - \tilde{\mu}_m| (< 1)$ . By Theorem 2, the output of GCN "loses information" as the layer size goes to infinity when the largest singular values of weights are strictly smaller than  $\lambda^{-1}$ . Therefore, the closer the positive eigenvalues  $\mu_m$  are to 1, the broader range of GCNs satisfies the assumption of Theorem 2.
+
+![](images/5122dd52ea471be69a88ea3f17ddbe67e69e8806d0c0355d91764e437ce0552c.jpg)  
+Figure 1: Visualization of vector field  $V(X) \coloneqq f(X) - X$  induced by the one-step transition. Color maps indicate the absolute value  $|V(X)|$  at the point  $X$ . Dotted lines are the subspace  $\mathcal{M}$ . Left: Case 1. Right: Case 2. Best view in color.
+
+![](images/a464812635882d6eb31e4f4489b877744adc4189dfea035064a1cc03a5e57910.jpg)
+
+For an Erdős – Rényi graph  $G_{N,p}$ , Chung & Radcliffe (2011) showed that when  $\frac{\log N}{Np} = o(1)$ , the eigenvalues of the (usual) normalized Laplacian except for the smallest one converge to 1 with high probability (see Theorem 2 therein). We can interpret this theorem as the convergence of Erdős–Rényi graphs to the complete graph in terms of graph spectra. We can prove that the augmented normalized Laplacian behaves similarly (Lemma 6). By combining this fact with the discussion in the previous paragraph, we obtain the asymptotic behavior of GCNs on the Erdős – Rényi graph. See Appendix D for the complete proof.
+
+Theorem 3. Consider a GCN on the Erdős-Rényi graph  $G_{N,p}$  such that  $\frac{\log N}{Np} = o(1)$  as a function of  $N$ . For any  $\varepsilon > 0$ , if the supremum  $s$  of the maximum singular values of weights in the GCN satisfies  $s < s_0 \coloneqq \frac{1}{7}\sqrt{\frac{Np - p + 1}{\log(4N / \varepsilon)}}$ , then, for sufficiently large  $N$ , the GCN satisfies the condition of Theorem 2 with probability at least  $1 - \varepsilon$ .
+
+Theorem 3 requires that an underlying graph is not extremely sparse. For example, suppose the node size is  $N = 20,000$ , which is the approximately the maximum node size of datasets we use in experiments, and the edge probability is  $p = \log N / N$ . Then, each node has the order of  $Np \approx 4.3$  adjacent nodes.
+
+Under the condition of Theorem 3, the upper bound  $s_0 \to \infty$  as  $N \to \infty$ . It means that if the graph is sufficiently large and not extremely sparse, most GCNs suffer from the information loss. For the dependence on the edge probability  $p$ ,  $s_0$  is an increasing function of  $p$ , which means the denser a graph is, the more quickly graph convolution operations mix signals on nodes and move them close to each other.
+
+Theorem 3 implies that graph NNs perform poorly on dense NNs. More aggressively, we can hypothesize that the sparsity of practically available graphs is one of the reasons for the success of graph NNs in node classification tasks. To confirm this hypothesis, we artificially add edges to citation networks to make them dense in the experiments and observe the failure of graph NNs as expected (see Section 6.3).
+
+# 6 EXPERIMENT
+
+# 6.1 SYNTHESIS DATA: ONE-STEP TRANSITION
+
+We numerically investigate how the transition  $f(X) \coloneqq \sigma(PXW)$  changes inputs using the vector field  $V(X) \coloneqq f(X) - X^4$ . For this purpose, we set  $N = 2$ ,  $M = 1$ , and  $C = 1$ . Let  $\lambda_1 \leq \lambda_2$  be the eigenvalues of  $P$ . We choose  $W$  as  $|\lambda_2|^{-1} \leq W < |\lambda_1|^{-1}$  so that Theorem 1 is applicable but is not reduced to the trivial situation (see, Appendix E.2). We choose the eigenvector  $e \in \mathbb{R}^2$
+
+![](images/1ff69019752f257b7a51ffed8e40be7c57b40eb808689aa35cc83f9a41acf532.jpg)  
+Figure 2: The actual distances to the invariant space  $\mathcal{M}$  and their upper bounds. Solid lines are the log relative distance defined by  $y(l) = \log (d_{\mathcal{M}}(X^{(l)}) / d_{\mathcal{M}}(X^{(0)}))$  and dotted lines are upper bound  $y(l) = l\log (s\lambda)$ , where  $X^{(0)}$  is the input signal and  $X^{(l)}$  is the output of the  $l$ -th layer.
+
+![](images/08d047c8dc64542afab2cafaebe0f42e91ed2de00468149a6552eb88b6713b55.jpg)
+
+![](images/a094b95832f7000a98488ab823c29a9db9bc42f1349e4904c6bd3822983d0497.jpg)  
+Figure 3: Node prediction results on Noisy Cora. Left: Effect of the maximum singular values on weights on model performance. The horizontal dotted line indicates the chance rate  $(30.2\%)$ . The error bar is the standard deviation of 3 trials. Right: Transition of maximum singular values during training. See Appendix I.3 for results using other datasets. Best view in color.
+
+![](images/f1d63f0134d96760667c336d6268742210c486563d813af338a21bae43a745cc.jpg)
+
+associated with  $\lambda_{2}$  in two ways as described below. See Appendix H.1 for the concrete values of  $P$ ,  $e$ , and  $W$ . Figure 1 shows the visualization of  $V$ . First, we choose the non-negative eigenvector  $e$  so that it satisfies Assumption 1 (Case 1). We see that the transition function  $f$  uniformly decreases the distance from  $\mathcal{M}$ . This is consistent with the consequence of Theorem 1. Next, we choose the eigenvector  $e = [e_1 e_2]^\top$  such that the signs of  $e_1$  and  $e_2$  differ (Case 2), which violates Assumption 1. We see that  $\mathcal{M}$  is not invariant under  $f$  and  $f$  does not uniformly decrease the distance from  $\mathcal{M}$ . Therefore, we cannot remove the non-negativity assumption from Theorem 1.
+
+# 6.2 SYNTHESIS DATA: DISTANCE TO INVARIANT SPACE
+
+We evaluate the distance to the invariant space  $\mathcal{M}$  using synthesis data. We randomly generate an Erdős–Rényi graph, a GCN on it, and an input signal  $X^{(0)}$ . We compute the distance between the  $l$ -th intermediate output  $X^{(l)}$  and the invariant space  $\mathcal{M}$  for various edge probability  $p$  and maximum singular value  $s$ . Figure 2 plots the logarithm of the relative distance  $y(l) = \log \frac{d_{\mathcal{M}}(X^{(l)})}{d_{\mathcal{M}}(X^{(0)})}$  with respect to the layer index  $l$ . From Theorem 1, we know that it is upper bounded by  $y(l) = l\log (s\lambda)$ . We see that this bound well approximates the actual value when  $s\lambda$  is small. On the other hand, it is loose for large  $s\lambda$ . We leave tighter bounds for  $d_{\mathcal{M}}$  in such a case for future research.
+
+# 6.3 REAL DATA: EFFECT OF MAXIMUM SINGULAR VALUES ON PERFORMANCE
+
+Theorem 2 implies that if  $s$  is smaller than the threshold  $\lambda^{-1}$ , we cannot expect deep GCN to achieve good prediction accuracy. Conversely, if we can successfully train the model,  $s$  should avoid the region  $s \leq \lambda^{-1}$ . We empirically confirm these hypotheses using real datasets.
+
+We use Cora, CiteSeer, and PubMed (Sen et al., 2008), which are standard citation network datasets. The task is to classify the genre of papers using word occurrences and citation relationships. We regard each paper as a node and citation relationship as an edge. Due to space constraints, we focus on Cora in the main article. See Appendix H.3 and I.3 for the other datasets. The discussion in Section 5 implies that Theorem 2 can support a wide range of GCNs when the underlying graph is relatively dense. However, the citation networks are too sparse to examine the aforementioned hypotheses — Theorem 2 gives a non-trivial result only when  $1 \leq s < \lambda^{-1} \approx 1 + 3.62 \times 10^{-3}$ . To circumvent this, we make noisy versions of citation networks by randomly adding edges to graphs. Through this manipulation, we can increase the value of  $\lambda^{-1}$  to 1.11.
+
+Figure 3 (left) shows the accuracy for the test dataset in terms of the maximum singular values and the number of graph convolution layers. We can observe that when GCNs whose maximum singular value  $s$  is out of the region  $s < \lambda^{-1}$  outperform those inside the region in almost all configurations. Furthermore, the accuracy of GCNs with  $s = 10$  are better than those without normalization (unnormized). Figure 3 (right) shows the transition of the maximum singular values of the weights during training when we use a three-layered GCN. We can observe that the maximum singular value  $s$  does not shrink to the region  $s \leq \lambda^{-1}$ . In addition, when the layer size is small and predictive accuracy is high, GCNs gradually increase  $s$  from the initial value and avoid the region. In conclusion, the experiment results are consistent with the theorems.
+
+# 6.4 REAL DATA: EFFECT OF SIGNAL COMPONENT PERPENDICULAR TO INVARIANT SPACE
+
+We can decompose the output  $X$  of a model as  $X = X_0 + X_1$  ( $X_0 \in \mathcal{M}$ ,  $X_1 \in \mathcal{M}^\perp$ ). According to the theory,  $X_0$  has limited information for node classification. We hypothesize that the model emphasizes the perpendicular component  $X_1$  to perform good predictions. To quantitatively evaluate it, we define the relative magnitude of the perpendicular component of the output  $X$  by  $t(X) := X_1 / X_0$ . Figure 4 compares this quantity and the prediction accuracy on the noisy version of Cora (see Appendix I.4 for other datasets). We observe that these two quantities are correlated ( $R = 0.545$ ). If we remove GCNs have only one layer (corresponding to right points in the figure), the correlation coefficient is 0.827. This result does not contradict to the hypothesis above<sup>5</sup>.
+
+![](images/55c3e92c56fa816b9de12d5190f7b26d1494e7815d5079942ac8e1332ff6f620.jpg)  
+Figure 4:  $\log t(X)$  and prediction accuracy on Noisy Cora.
+
+# 7 DISCUSSION
+
+Applicability to Graph NNs on Sparse Graphs. We have theoretically and empirically shown that when the underlying graph is sufficiently dense and large, the threshold  $\lambda^{-1}$  is large (Theorem 2 and Section 6.3), which means many graph CNNs are eligible. However, real-world graphs are not often dense, which means that Theorem 2 is applicable to very limited GCNs. In addition, Coja-Oghlan (2007) theoretically proved that if the expected average degree of  $G_{N,p}$  is bounded, the smallest positive eigenvalue of the normalized Laplacian of  $G_{N,p}$  is  $o(1)$  with high probability. The asymptotic behaviors of graph NNs on sparse graphs are left for future research.
+
+**Remedy for Over-smoothing.** Based on our theory, we can propose several techniques for mitigating the over-smoothing phenomena. One idea is to (randomly) sample edges in an underlying graph. The sparsity of practically available graphs could be a factor in the success of graph NNs. Assuming this hypothesis is correct, there is a possibility that we can relive the effect of over-smoothing by sparsification. Since we can never restore the information in pruned edges if we remove them permanently, random edge sampling could work better as FastGCN (Chen et al., 2018a) and GraphSAGE (Hamilton et al., 2017) do. Another idea is to scale node representations (i.e., intermediate or final output of graph NNs) appropriately so that they keep away from the invariant space  $\mathcal{M}$ . Our proposed weight scaling mechanism takes this strategy. Recently, Zhao & Akoglu (2020) has pro
+
+posed PairNorm to alleviate the over-smoothing phenomena. Although the scaling target is different – they rescaled signals whereas we normalized weights – theirs and ours have similar spirits.
+
+Graph NNs with Large Weights. Our theory suggests that the maximum singular values of weights in a GCN should not be smaller than a threshold  $\lambda^{-1}$  because it suffers from information loss for node classification. On the other hand, if the scale of weights are very large, the model complexity of the function class represented by graph NNs increases, which may cause large generalization errors. Therefore, from a statistical learning theory perspective, we conjecture that the graph NNs with too-large weights perform poorly, too. A trade-off should exist between the expressive power and model complexity and there should be a "sweet spot" on the weight scale that balances the two.
+
+Relation to Double Descent Phenomena. Belkin et al. (2019) pointed out that modern deep models often have double descent risk curves: when a model is under-parameterized, a classical bias-variance trade-off occurs. However, once the model has a large capacity and perfectly fits the training data, the test error decreases as we increase the number of parameters. To the best of our knowledge, no literature reported the double descent phenomena for graph NNs (it is consistent with the picture of the classical U-shaped risk curve in the previous paragraph). It is known that double descent phenomena do not occur in some situations, especially depending on regularization types. For example, while Belkin et al. (2019) employed the interpolating hypothesis with the minimum norm, Mei & Montanari (2019) found that the double descent was alleviated or disappeared when they used Ridge-type regularization techniques. Therefore, one can hypothesize the over-smoothing is a cause or consequence of regularization that is more like a Ridge-type rather than minimum-norm inductive bias.
+
+Limitations in Graph NN Architectures. Our analysis is limited to graph NNs with the ReLU activation function because we implicitly use the property that ReLU is a projection onto the cone  $\{X\geq 0\}$  (Appendix A, Lemma 3). This fact enables the ReLU function to get along with the nonnegativity of eigenvectors associated with the largest eigenvalues. Therefore, it is far from trivial to extend our results to other activation functions such as the sigmoid function or Leaky ReLU (Maas et al., 2013). Another point is that our formulation considers the update operation (Gilmer et al., 2017) of graph NNs only and does not take readout operations into account. In particular, we cannot directly apply our theory to graph classification tasks in which each sample is a graph.
+
+Over-smoothing of Residual GNNs. Considering the correspondence of graph NNs and Markov processes (see Appendix F), one can imagine that residual links do not contribute to alleviating the over-smoothing phenomena because adding residual connections to a graph NN corresponds to converting a Markov process to its lazy version. When a Markov process converges to a stable distribution, the corresponding lazy process also converges eventually under certain conditions. It implies that residual links might not be helpful. However, Li et al. (2019) reported that graph NNs with as many as 56 layers performed well if they added residual connections. Considering that, the situation could be more complicated than our intuitions. The analysis of the role of residual connections in graph NNs is a promising direction for future research.
+
+# 8 CONCLUSION
+
+In this paper, to understand the empirically observed phenomena that deep non-linear graph NNs do not perform well, we analyzed their asymptotic behaviors by interpreting them as a dynamical system that includes GCN and Markov process as special cases. We gave theoretical conditions under which GCNs suffer from the information loss in the limit of infinite layers. Our theory directly related the expressive power of graph NNs and topological information of the underlying graphs via spectra of the Laplacian. It enabled us to leverage spectral and random graph theory to analyze the expressive power of graph NNs. To demonstrate this, we considered GCN on the Erdős - Rényi graph as an example and showed that when the underlying graph is sufficiently dense and large, a wide range of GCNs on the graph suffer from information loss. Based on the theory, we gave a principled guideline for how to determine the scale of weights of graph NNs and empirically showed that the weight normalization implied by our theory performed well in real datasets. One promising direction of research is to analyze the optimization and statistical properties such as the generalization power (Verma & Zhang, 2019) of graph NNs via spectral and random graph theories.
+
+# ACKNOWLEDGMENTS
+
+We thank Katsuhiko Ishiguro for providing a part of code for the experiments, Kohei Hayashi and Haru Negami Oono for giving us feedback and comments on the draft, Keyulu Xu and anonymous reviewers for fruitful discussions via OpenReview, and Ryuta Osawa for pointing out errors and suggesting improvements of the paper. TS was partially supported by JSPS KAKENHI (15H05707, 18K19793, and 18H03201), Japan Digital Design, and JST CREST.
+
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+
+# A PROOF OF THEOREM 1
+
+As we wrote in the main article, it is enough to show the following lemmas (definition of miscellaneous variables are as in Section 3.2). Remember that  $\lambda = \sup_{n\in [N - M]}\left|\lambda_n\right|$  and  $s_{lh}$  is the maximum singular value of  $W_{lh}$
+
+Lemma 1. For any  $X\in \mathbb{R}^{N\times C}$ , we have  $d_{\mathcal{M}}(PX)\leq \lambda d_{\mathcal{M}}(X)$ .
+
+Lemma 2. For any  $X\in \mathbb{R}^{N\times C}$ , we have  $d_{\mathcal{M}}(XW_{lh})\leq s_{lh}d_{\mathcal{M}}(X)$ .
+
+Lemma 3. For any  $X\in \mathbb{R}^{N\times C}$ , we have  $d_{\mathcal{M}}(\sigma (X))\leq d_{\mathcal{M}}(X)$ .
+
+Proof of Lemma 1. Since  $P$  is a symmetric linear operator on  $U^{\perp}$ , we can choose the orthonormal basis  $(e_m)_{m = M + 1,\dots ,N}$  of  $U^{\perp}$  consisting of the eigenvalue of  $P|_{U^{\perp}}$ . Let  $\lambda_{m}$  be the eigenvalue of  $P$  to which  $e_m$  is associated  $(m = M + 1,\ldots ,N)$ . Note that since the operator norm of  $P|_{U^{\perp}}$  is  $\lambda$ , we have  $|\lambda_{m}| \leq \lambda$  for all  $m = M + 1,\ldots ,N$ . Since  $(e_m)_{m\in [N]}$  forms the orthonormal basis of
+
+$\mathbb{R}^N$ , we can uniquely write  $X \in \mathbb{R}^{N \times C}$  as  $X = \sum_{m=1}^{N} e_m \otimes w_m$  for some  $w_m \in \mathbb{R}^C$ . Then, we have  $d_{\mathcal{M}}^2(X) = \sum_{m=M+1}^{N} \|w_m\|^2$  where  $\| \cdot \|$  is the 2-norm of a vector. On the other hand, we have
+
+$$
+\begin{array}{l} P X = \sum_ {m = 1} ^ {N} P e _ {m} \otimes w _ {m} \\ = \sum_ {m = 1} ^ {M} P e _ {m} \otimes w _ {m} + \sum_ {m = M + 1} ^ {N} P e _ {m} \otimes w _ {m} \\ = \sum_ {m = 1} ^ {M} P e _ {m} \otimes w _ {m} + \sum_ {m = M + 1} ^ {N} e _ {m} \otimes (\lambda_ {m} w _ {m}) \\ \end{array}
+$$
+
+Since  $U$  is invariant under  $P$ , for any  $m \in [M]$ , we can write  $Pe_m$  as a linear combination of  $e_n(n \in [M])$ . Therefore, we have  $d_{\mathcal{M}}^2 (PX) = \sum_{m = M + 1}^{N} \| \lambda_m w_m\|^2$ . Then, we obtain the desired inequality as follows:
+
+$$
+\begin{array}{l} d _ {\mathcal {M}} ^ {2} (P X) = \sum_ {m = M + 1} ^ {N} \| \lambda_ {m} w _ {m} \| ^ {2} \\ \leq \lambda^ {2} \sum_ {m = M + 1} ^ {N} \| w _ {m} \| ^ {2} \\ \leq \lambda^ {2} \sum_ {m = M + 1} ^ {N} \| w _ {m} \| ^ {2} \\ = \lambda^ {2} d _ {\mathcal {M}} ^ {2} (X). \\ \end{array}
+$$
+
+![](images/ec6455e5c20a6e98e67b83b9ca3d00bfa65c107aaec817ad607a4d9fdbe505b5.jpg)
+
+Proof of Lemma 2. Using the same decomposition of  $X$  as the proof in Lemma 1, we have
+
+$$
+\begin{array}{l} X W _ {l h} = \sum_ {m = 1} ^ {N} e _ {m} \otimes \left(W _ {l h} ^ {\top} w _ {m}\right) \\ = \sum_ {m = 1} ^ {M} e _ {m} \otimes \left(W _ {l h} ^ {\top} w _ {m}\right) + \sum_ {m = M + 1} ^ {N} e _ {m} \otimes \left(W _ {l h} ^ {\top} w _ {m}\right). \\ \end{array}
+$$
+
+Therefore, we have
+
+$$
+\begin{array}{l} d _ {\mathcal {M}} ^ {2} (X W _ {l h}) = \sum_ {m = M + 1} ^ {N} \| W _ {l h} ^ {\top} w _ {m} \| ^ {2} \\ \leq s _ {l h} ^ {2} \sum_ {m = M + 1} ^ {N} \| w _ {m} \| ^ {2} \\ = s _ {l h} ^ {2} d _ {\mathcal {M}} ^ {2} (X). \\ \end{array}
+$$
+
+![](images/378304be44a8f6d9a0e47b21e30ae3f36098fd34460c105ddd592b6f222c0082.jpg)
+
+Proof of Lemma 3. We choose  $(e_m)_{m = N - M + 1,\dots ,N}$  as in the proof of Lemma 1. We denote  $X = (X_{nc})_{n\in [N],c\in [C]}$  and  $e_n = (e_{mn})_{m\in [N]}$ , respectively. Let  $(e_c')_{c\in [C]}$  be the standard basis of  $\mathbb{R}^C$ . Then,  $(e_n\otimes e_c')_{n\in [N],c\in [C]}$  is the orthonormal basis of  $\mathbb{R}^{N\times C}$ , endowed with the standard inner product as a Euclid space. Therefore, we can decompose  $X$  as  $X = \sum_{n = 1}^{N}\sum_{c = 1}^{C}a_{nc}e_n\otimes e_c'$  where  $a_{nc} = \langle X,e_n\otimes e_c'\rangle = \sum_{m = 1}^{N}X_{mc}e_{mn}$ . Then, we have  $d_{\mathcal{M}}^2 (X) = \sum_{n = M + 1}^{N}\| \sum_{c = 1}^{C}a_{nc}e_c'\|^2$
+
+which is further transformed as
+
+$$
+\begin{array}{l} d _ {\mathcal {M}} ^ {2} (X) = \sum_ {n = M + 1} ^ {N} \left\| \sum_ {c = 1} ^ {C} a _ {n c} e _ {c} ^ {\prime} \right\| ^ {2} \\ = \sum_ {n = M + 1} ^ {N} \sum_ {c = 1} ^ {C} a _ {n c} ^ {2} \\ = \sum_ {c = 1} ^ {C} \left(\sum_ {n = 1} ^ {N} a _ {n c} ^ {2} - \sum_ {n = 1} ^ {M} a _ {n c} ^ {2}\right) \\ = \sum_ {c = 1} ^ {C} \left(\left\| X _ {\cdot c} \right\| ^ {2} - \sum_ {n = 1} ^ {M} \langle X _ {\cdot c}, e _ {n} \rangle^ {2}\right), \\ \end{array}
+$$
+
+where  $X_{c}$  is the  $c$ -th column vector of  $X$ . Similarly, we have
+
+$$
+d _ {\mathcal {M}} ^ {2} (\sigma (X)) = \sum_ {c = 1} ^ {C} \left(\| X _ {. c} ^ {+} \| ^ {2} - \sum_ {n = 1} ^ {M} \langle X _ {. c} ^ {+}, e _ {n} \rangle^ {2}\right),
+$$
+
+where we denote  $\sigma(X) = (X_{nc}^{+})_{n \in [N], c \in [C]}$  in shorthand. Therefore, the inequality follows from the following lemma.
+
+Lemma 4. Let  $x \in \mathbb{R}^N$  and  $v_1, \ldots, v_M \in \mathbb{R}^N$  be orthonormal vectors (i.e.,  $\langle v_m, v_n \rangle = \delta_{mn}$ ) satisfying  $v_m \geq 0$  for all  $m \in [M]$ . Then, we have  $\| x\|^2 - \sum_{m=1}^{M} \langle x, v_m \rangle^2 \geq \| x^+ \|^2 - \sum_{m=1}^{M} \langle x^+, v_m \rangle^2$  where  $x^+ := \max(x, 0)$  for  $x \in \mathbb{R}$ .
+
+Proof. The value  $\| y \|^2 - \sum_{m=1}^{M} \langle y, u_m \rangle^2$  is invariant under simultaneous coordinate permutation of  $y$  and  $u_m$ 's. Therefore, we can assume without loss of generality that the coordinate of  $x$  are sorted:  $x_1 \leq \ldots \leq x_L < 0 \leq x_{L+1} \leq \ldots \leq x_N$  for some  $L \leq N$ . Then, we have
+
+$$
+\left\| x \right\| ^ {2} - \left\| x ^ {+} \right\| ^ {2} = \sum_ {n = 1} ^ {L} x _ {n} ^ {2}. \tag {1}
+$$
+
+When  $L = 0$ , the sum in the right hand side is treated as 0. On the other hand, writing as  $v_{m} = (v_{nm})_{n\in [N]}$ , direct calculation shows
+
+$$
+\sum_ {m = 1} ^ {M} \langle x, v _ {m} \rangle^ {2} - \langle x ^ {+}, v _ {m} \rangle^ {2} = \sum_ {m = 1} ^ {M} \left(\left(\sum_ {n = 1} ^ {L} x _ {n} v _ {n m}\right) ^ {2} - 2 \sum_ {n = 1} ^ {L} \sum_ {l = L + 1} ^ {N} x _ {n} x _ {l} v _ {n m} v _ {l m}\right). \tag {2}
+$$
+
+Let  $I_{m} \coloneqq \{n \in [N] \mid v_{nm} > 0\}$  be the support of  $v_{m}$  for  $m \in [M]$ . We note that if  $m \neq m' \in [M]$ , we have  $I_{m} \cap I_{m'} = \emptyset$  since if there existed  $n \in I_{m} \cap I_{m'}$ , we have
+
+$$
+0 = \left\langle v _ {m}, v _ {m ^ {\prime}} \right\rangle \geq v _ {n m} v _ {n m ^ {\prime}} > 0,
+$$
+
+which is contradictory. Therefore,
+
+$$
+\begin{array}{l} \sum_ {m = 1} ^ {N} \left(\sum_ {n = 1} ^ {L} x _ {n} v _ {n m}\right) ^ {2} = \sum_ {m = 1} ^ {N} \left(\sum_ {n \in I _ {m} \cap [ L ]} x _ {n} v _ {n m}\right) ^ {2} \\ \leq \sum_ {m = 1} ^ {N} \left(\sum_ {n \in I _ {m} \cap [ L ]} x _ {n} ^ {2}\right) \left(\sum_ {n \in I _ {m} \cap [ L ]} v _ {n m} ^ {2}\right) \quad (\because \text {C a u c h y - S c h w a r z i n e q u a l i t y}) \\ \leq \sum_ {m = 1} ^ {N} \left(\sum_ {n \in I _ {m} \cap [ L ]} x _ {n} ^ {2}\right) \quad (\because \| v _ {m} \| ^ {2} = 1) \\ \leq \sum_ {n = 1} ^ {L} x _ {n} ^ {2}. \tag {3} \\ \end{array}
+$$
+
+We used the fact that  $I_{m}$ 's are disjoint and  $v_{nm} = 0$  if  $n \notin \cup_{m} I_{m}$  in the first equality above. Further, we have  $x_{n}x_{l}v_{nm}v_{lm} \leq 0$  for  $1 \leq n \leq L$  and  $L + 1 \leq l \leq N$  by the definition of  $L$  and non-negativity of  $v_{m}$ . By combining (1), (2), and (3), we have
+
+$$
+\sum_ {m = 1} ^ {M} \langle x, v _ {m} \rangle^ {2} - \langle x ^ {+}, v _ {m} \rangle^ {2} \leq \sum_ {n = 1} ^ {L} x _ {n} ^ {2} = \| x \| ^ {2} - \| x ^ {+} \| ^ {2}.
+$$
+
+![](images/ef473adc5317e1fe7fc9530f14f375b7058bdce9cc15c5306bcbb4d14066ad0b.jpg)
+
+Proof of Theorem 1. By Lemma 1, 2, and 3, we have
+
+$$
+d_{\mathcal{M}}(f_{l}(X)) = d_{\mathcal{M}}(\underbrace{\sigma(\cdots\sigma(\sigma(PX)W_{l1})W_{l2}\cdots W_{lH_{l}})}_{H\text{times}})
+$$
+
+$$
+\begin{array}{l} \leq d_{\mathcal{M}}(\underbrace{\sigma(\cdots\sigma(\sigma(PX)W_{l1})W_{l2}\cdots)}_{H - 1\text{times}}W_{lH_{l}}) \\ \leq s _ {l H _ {l} - 1} d _ {\mathcal {M}} (\underbrace {\sigma (\cdots \sigma (\sigma (P X) W _ {l 1}) W _ {l 2} \cdots) W _ {l H _ {l} - 1}))} _ {H - 1 \text {t i m e s}} \\ \end{array}
+$$
+
+.
+
+$$
+\begin{array}{l} \leq \left(\prod_ {h = 1} ^ {H _ {l}} s _ {l h}\right) d _ {\mathcal {M}} (P X) \\ \leq s _ {l} d _ {\mathcal {M}} (P X) \\ \leq s _ {l} \lambda d _ {\mathcal {M}} (X). \\ \end{array}
+$$
+
+![](images/3c1a70d689cb7569682936fc9d69e1e63a3c4dcf3fb1ce8a3ca933d30b645dff.jpg)
+
+# B PROOF OF PROPOSITION 1
+
+Proof. Let  $\tilde{\mu}_1 \leq \dots \leq \tilde{\mu}_N$  be the eigenvalue of the augmented normalized Laplacian  $\tilde{\Delta}$ , sorted in ascending order. Since  $P = I_N - \tilde{\Delta}$ , it is enough to show  $\tilde{\mu}_1 = \dots = \tilde{\mu}_M = 0$ ,  $\tilde{\mu}_{M + 1} > 0$ , and  $\tilde{\mu}_N < 2$ . For the first two, the statements are equivalent to that  $\tilde{\Delta}$  is positive semi-definite and that the multiplicity of the eigenvalue 0 is same as the number of connected components  $^6$ . This is well-known for Laplacian or its normalized version (see, e.g., Chung & Graham (1997)) and the proof for  $\tilde{\Delta}$  is similar. By direct calculation, we have
+
+$$
+x ^ {\top} \tilde {\Delta} x = \frac {1}{2} \sum_ {i, j = 1} ^ {N} a _ {i j} \left(\frac {x _ {i}}{\sqrt {d _ {i} + 1}} - \frac {x _ {j}}{\sqrt {d _ {j} + 1}}\right) ^ {2}
+$$
+
+for any  $x = [x_{1}\dots x_{N}]^{\top}\in \mathbb{R}^{N}$  . Therefore,  $\tilde{\Delta}$  is positive semi-definite and hence  $\tilde{\mu}_1\geq 0$
+
+Suppose temporally that  $G$  is connected. If  $x \in \mathbb{R}^N$  is an eigenvector associated to 0, then, by the aforementioned calculation,  $\frac{x_i}{\sqrt{d_i + 1}}$  and  $\frac{x_j}{\sqrt{d_j + 1}}$  must be same for all pairs  $(i,j)$  such that  $a_{ij} > 0$ . However, since  $G$  is connected,  $\frac{x_i}{\sqrt{d_i + 1}}$  must be the same value for all  $i \in [N]$ . That means the multiplicity of the eigenvalue 0 is 1 and any eigenvector associated to 0 must be proportional to  $\tilde{D}^{\frac{1}{2}}\mathbf{1}$ . Now, suppose  $G$  has  $M$  connected components  $V_1, \ldots, V_M$ . Let  $\tilde{\Delta}_m$  be the augmented normalized Laplacians corresponding to each connected component  $V_m$  for  $m \in [M]$ . By the aforementioned discussion,  $\tilde{\Delta}_m$  has the eigenvalue 0 with multiplicity 1. Since  $\tilde{\Delta}$  is the direct sum of  $\tilde{\Delta}_m' s$ , the eigenvalue of  $\tilde{\Delta}$  is the union of those for  $\tilde{\Delta}_m$ 's. Therefore,  $\tilde{\Delta}$  has the eigenvalue 0 with multiplicity  $M$  and  $e_m = \tilde{D}^{\frac{1}{2}}\mathbf{1}_m$ 's are the orthogonal basis of the eigenspace.
+
+Finally, we prove  $\tilde{\mu}_N < 2$ . Let  $\mu_N$  be the largest eigenvalue of the normalized Laplacian  $\Delta = D^{-\frac{1}{2}}(D - A)D^{-\frac{1}{2}}$ , where  $D^{-\frac{1}{2}} \in \mathbb{R}^{N \times N}$  is the diagonal matrix defined by
+
+$$
+D _ {i i} ^ {- \frac {1}{2}} = \left\{ \begin{array}{l l} \deg (i) ^ {- \frac {1}{2}} & (\text {i f} \deg (i) \neq 0) \\ 0 & (\text {i f} \deg (i) = 0) \end{array} \right..
+$$
+
+Note that  $D^{-\frac{1}{2}}D^{\frac{1}{2}}$  nor  $D^{\frac{1}{2}}D^{-\frac{1}{2}}$  are not equal to the identity matrix  $I_{N}$  in general. However, we have
+
+$$
+L = D ^ {\frac {1}{2}} D ^ {- \frac {1}{2}} L D ^ {- \frac {1}{2}} D ^ {\frac {1}{2}} \tag {4}
+$$
+
+where  $L = D - A$  is the (unnormized) Laplacian. Therefore, we have
+
+$$
+\begin{array}{l} \tilde {\mu} _ {N} = \max _ {x \neq 0} \frac {x ^ {\top} \tilde {\Delta} x}{\| x \|} \\ = \max  _ {x \neq 0} \frac {x ^ {\top} \tilde {D} ^ {- \frac {1}{2}} L \tilde {D} ^ {- \frac {1}{2}} x}{\| x \|} \\ = \max  _ {x \neq 0} \frac {x ^ {\top} \tilde {D} ^ {- \frac {1}{2}} D ^ {\frac {1}{2}} D ^ {- \frac {1}{2}} L D ^ {- \frac {1}{2}} D ^ {\frac {1}{2}} \tilde {D} ^ {- \frac {1}{2}} x}{\| x \|} \quad (\because (4)) \\ = \max  _ {x \neq 0} \frac {\left(D ^ {\frac {1}{2}} \tilde {D} ^ {- \frac {1}{2}} x\right) ^ {\top} \Delta \left(D ^ {\frac {1}{2}} \tilde {D} ^ {- \frac {1}{2}} x\right)}{\| x \|} \\ = \max_{\substack{x\neq 0\\ D^{\frac{1}{2}}\tilde{D}^{-\frac{1}{2}}x\neq 0}}\frac{(D^{\frac{1}{2}}\tilde{D}^{-\frac{1}{2}}x)^{\top}\Delta(D^{\frac{1}{2}}\tilde{D}^{-\frac{1}{2}}x)}{\|x\|} \\ = \max_{\substack{x\neq 0\\ D^{\frac{1}{2}}\tilde{D}^{-\frac{1}{2}}x\neq 0}}\frac{(D^{\frac{1}{2}}\tilde{D}^{-\frac{1}{2}}x)^{\top}\Delta(D^{\frac{1}{2}}\tilde{D}^{-\frac{1}{2}}x)}{\|D^{\frac{1}{2}}\tilde{D}^{-\frac{1}{2}}x\|}\frac{\|D^{\frac{1}{2}}\tilde{D}^{-\frac{1}{2}}x\|}{\|x\|} \\ \leq \max_{\substack{x\neq 0\\ D^{\frac{1}{2}}\tilde{D}^{-\frac{1}{2}}x\neq 0}}\frac{(D^{\frac{1}{2}}\tilde{D}^{-\frac{1}{2}}x)^{\top}\Delta(D^{\frac{1}{2}}\tilde{D}^{-\frac{1}{2}}x)}{\|D^{\frac{1}{2}}\tilde{D}^{-\frac{1}{2}}x\|}\max_{\substack{x\neq 0\\ D^{\frac{1}{2}}\tilde{D}^{-\frac{1}{2}}x\neq 0}}\frac{\|D^{\frac{1}{2}}\tilde{D}^{-\frac{1}{2}}x\|}{\|x\|} \\ \leq \max  _ {y \neq 0} \frac {y ^ {\top} \Delta y}{\| y \|} \max  _ {x \neq 0} \frac {\| D ^ {\frac {1}{2}} \tilde {D} ^ {- \frac {1}{2}} x \|}{\| x \|} \\ = \mu_ {N} \max _ {n \in [ N ]} \left(\frac {\deg (i)}{\deg (i) + 1}\right) ^ {\frac {1}{2}} \\ \leq \mu_ {N}. \\ \end{array}
+$$
+
+Therefore, we have  $\tilde{\mu}_N \leq \mu_N^7$ . Since  $\max_{i \in [N]} \left( \frac{\deg(i)}{\deg(i) + 1} \right)^{\frac{1}{2}} < 1$ , the equality  $\tilde{\mu}_N = \mu_N$  holds if and only if  $\mu_N = 0$ , that is,  $G$  has  $N$  connected components. On the other hand, it is known that  $\mu_N \leq 2$  and the equality holds if and only if  $G$  has non-trivial bipartite graph as a connected component (see, e.g., Chung & Graham (1997)). Therefore,  $\tilde{\mu}_N = \mu_N$  and  $\mu_N = 2$  does not hold simultaneously and we obtain  $\mu_N < 2$ .
+
+# C COUNTEREXAMPLE OF PREVIOUS STUDY ON OVER-SMOOTHING FOR NON-LINEAR GNNS
+
+We restate Theorem 1 of the preprint (version2) of Luan et al. (2019) $^{8}$ . Let  $G$  be a simple undirected graph with  $N$  nodes and  $k$  connected components such that it does not have a bipartite component. Let  $L = \tilde{D}^{-1/2} \tilde{A} \tilde{D}^{-1/2} \in \mathbb{R}^{N \times N}$  be the augmented normalized Laplacian of  $G$ . Let  $F \in \mathbb{N}_{+}$  and
+
+$W_{n}\in \mathbb{R}^{F\times F}$  be the weight of the  $n$  -th layer for  $n\in \mathbb{N}_{+}$ . For the input  $X\in \mathbb{R}^{N\times F}$ , we define the output  $Y_{n}\in \mathbb{R}^{N\times F}$  of the  $n$  -th layer of a GCN by  $Y_{n} = \sigma (L\dots \sigma (LXW_{0})\dots W_{n})$  where  $\sigma$  is the ReLU function. We assume the input  $X$  is drawn from a continuous distribution on  $\mathbb{R}^{N\times F}$ . Then, the theorem claims that we have  $\lim_{n\to \infty}\mathrm{rank}(Y_n) = k$  almost surely with respect to the distribution of  $X$ .
+
+We construct a counterexample. Consider a graph  $G$  consisting of  $N = 4$  nodes whose adjacency matrix is
+
+$$
+A = \left[ \begin{array}{c c c c} 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 0 \\ 1 & 0 & 0 & 1 \end{array} \right].
+$$
+
+Note that  $G$  is connected (i.e.,  $k = 1$ ) and is not bipartite. We make a GCN with  $F = 3$  channels and whose weight matrices are  $W_{n} = I_{3}$  (the identity matrix of size 3) for all  $n \in \mathbb{N}$ . For the distribution of the input  $X$ , we consider an absolutely continuous distribution with respect to the Lebesgue measure on  $\mathbb{R}^{4 \times 3}$  such that  $P(X \geq 0) > 0$  (here,  $X \geq 0$  means the element-wise comparison). For example, the standard Gaussian distribution satisfies the condition.
+
+Since  $L \geq 0$ , we have  $Y_{n} = L^{n}X$  if  $X \geq 0$ . Let  $L = P^{\top}\Lambda P$  be the diagonalization of  $L$  where  $P \in O(4)$  is an orthogonal matrix of size 4. Since  $\mathrm{rank}(L) = 3$ , we have  $\mathrm{rank}(\Lambda^n) = 3$  for any  $n$  (we can assume that  $\Lambda_{44} = 0$  without loss of generality). Therefore, under the condition  $X \geq 0$ , we have
+
+$$
+\begin{array}{l} \operatorname {r a n k} \left(Y _ {n}\right) = 3 \iff \operatorname {r a n k} \left(P ^ {\top} \Lambda^ {n} P X\right) = 3 \\ \Longleftrightarrow X \in \left\{P ^ {- 1} \left[ \begin{array}{c c} B & v \end{array} \right] ^ {\top} \mid B \in \mathbb {R} ^ {3 \times 3} \text {i s i n v e r t i b l e}, v \in \mathbb {R} ^ {3} \right\}. \\ \end{array}
+$$
+
+Note that the last condition is independent of  $n$ . Since the set of invertible matrices is dense in the set of all matrices of the same size (with respect to the standard topology of the Euclidean space), we have  $P(\{\mathrm{rank}(Y_n) = 3 \text{ for all } n \in \mathbb{N}\}) > 0$ . Therefore, we have  $\lim_{n \to \infty} \mathrm{rank}(Y_n) = 3$  with a non-zero probability.
+
+# D PROOF OF THEOREM 3
+
+We follow the proof of Theorem 2 of Chung & Radcliffe (2011). The idea is to relate the spectral distribution of the normalized Laplacian with that of its expected version. Since we can compute the latter one explicitly for the Erdős-Rényi graph, we can derive the convergence of spectra. We employ this technique and derive similar conclusions for the augmented normalized Laplacian.
+
+First, we consider general random graphs not restricted to Erdős-Rényi graphs. Let  $N \in \mathbb{N}_{+}$ , and  $P = (p_{ij})_{i,j \in [N]}$  be a non-negative symmetric matrix (meaning that  $p_{ij} \geq 0$  for any  $i, j \in [N]$ ). Let  $G$  be an undirected random graph with  $N$  nodes such that an edge between  $i$  and  $j$  is independently present with probability  $p_{ij}$ . Let  $A$  and  $D$  be the adjacency and the degree matrices of  $G$ , respectively (that is,  $A_{ij} \sim \mathrm{Ber}(p_{ij})$ , i.i.d.). Define the expected node degree of node  $i$  by  $t_i := \sum_{j=1}^{N} p_{ij}$ . Let  $\tilde{A} := A + I_N$ ,  $\tilde{D} := D + I_N$  and define  $\bar{A} := \mathbb{E}[\tilde{A}] = P + I_N$  and  $\bar{D} := \mathbb{E}[\tilde{D}] = \mathrm{diag}(t_1, \ldots, t_N) + I_N$  correspondingly. We define the augmented normalized Laplacian  $\tilde{\Delta}$  of  $G$  by  $\tilde{\Delta} := I_N - \tilde{D}^{-\frac{1}{2}}\tilde{A}\tilde{D}^{-\frac{1}{2}}$  and its expected version by  $\bar{\Delta} := I_N - \bar{D}^{-\frac{1}{2}}\bar{A}\bar{D}^{-\frac{1}{2}}$ . For a symmetric matrix  $X \in \mathbb{R}^N$ , we define its eigenvalues, sorted in ascending order by  $\lambda_1(X) \leq \dots \leq \lambda_N(X)$  and its operator norm by  $\|X\| = \max_{n \in [N]} |\lambda_n(X)|$ .
+
+Lemma 5 (Ref. Chung & Radcliffe (2011) Theorem 2). Let  $\delta \coloneqq \min_{n\in [N]}t_n$  be the minimum expected degree of  $G$ . Set  $k(\varepsilon)\coloneqq 3(1 + \log (4 / \varepsilon))$ . Then, for any  $\varepsilon >0$ , if  $\delta +1 > k(\varepsilon)\log N$ , we have
+
+$$
+\max _ {n \in [ N ]} \left| \lambda_ {n} (\tilde {\Delta}) - \lambda_ {n} (\bar {\Delta}) \right| \leq 4 \sqrt {\frac {3 \log (4 N / \varepsilon)}{\delta + 1}}
+$$
+
+with probability at least  $1 - \varepsilon$ .
+
+Proof. By Weyl's theorem, we have  $\max_{n\in [N]}\left|\lambda_n(\tilde{\Delta}) - \lambda_n(\bar{\Delta})\right|\leq \| \tilde{\Delta} -\bar{\Delta}\|$ . Therefore, it is enough to bound  $\| \tilde{\Delta} -\bar{\Delta}\|$ . Let  $C\coloneqq I_N - \bar{D}^{-\frac{1}{2}}\tilde{A}\bar{D}$ . By the triangular inequality, we have  $\| \tilde{\Delta} -\bar{\Delta}\| \leq \| \tilde{\Delta} -C\| +\| C - \bar{\Delta}\|$ . We will bound these terms respectively.
+
+First, we bound  $\| C - \bar{\Delta} \|$ . Direct calculation shows  $C - \bar{\Delta} = -\bar{D}^{-\frac{1}{2}}(A - P)\bar{D}^{-\frac{1}{2}}$ . Let  $E^{ij} \in \mathbb{R}^{N \times N}$  be a matrix defined by
+
+$$
+(E ^ {i j}) _ {k l} = \left\{ \begin{array}{l l} 1 & \text {i f (i = k a n d i = l) o r (i = l a n d j = k) ,} \\ 0 & \text {o t h e r w i s e .} \end{array} \right.
+$$
+
+We define the random variable  $Y_{ij}$  by
+
+$$
+Y _ {i j} := \frac {A _ {i j} - p _ {i j}}{\sqrt {t _ {i} + 1} \sqrt {t _ {j} + 1}} E ^ {i j}.
+$$
+
+Then,  $Y_{ij}$ 's are independent and we have  $C - \bar{\Delta} = \sum_{i,j=1}^{N} Y_{ij}$ . To apply Theorem 5 of Chung & Radcliffe (2011) to  $Y_{ij}$ 's, we bound  $\| Y_{ij} - \mathbb{E}[Y_{ij}] \|$  and  $\| \sum_{i,j=1}^{N} \mathbb{E}[Y_{ij}^2] \|$ . First, we have
+
+$$
+\| Y _ {i j} - \mathbb {E} [ Y _ {i j} ] \| = \| Y _ {i j} \| \leq \frac {\| E ^ {i j} \|}{\sqrt {t _ {i} + 1} \sqrt {t _ {j} + 1}} \leq (\delta + 1) ^ {- 1}.
+$$
+
+Since
+
+$$
+\mathbb {E} [ Y _ {i j} ^ {2} ] = \frac {p _ {i j} - p _ {i j} ^ {2}}{(t _ {i} + 1) (t _ {j} + 1)} \left\{ \begin{array}{l l} E ^ {i i} + E ^ {j j} & (\text {i f} i \neq j), \\ E ^ {i i} & (\text {i f} i = j), \end{array} \right.
+$$
+
+we have
+
+$$
+\begin{array}{l} \left\| \sum_ {i, j = 1} ^ {N} \mathbb {E} [ Y _ {i j} ^ {2} ] \right\| = \left\| \sum_ {i, j = 1} ^ {N} \frac {p _ {i j} - p _ {i j} ^ {2}}{(t _ {i} + 1) (t _ {j} + 1)} E ^ {i i} \right\| \\ = \max  _ {i \in [ N ]} \left(\sum_ {j = 1} ^ {N} \frac {p _ {i j} - p _ {i j} ^ {2}}{(t _ {i} + 1) (t _ {j} + 1)}\right) \\ \leq \max  _ {i \in [ N ]} \left(\sum_ {j = 1} ^ {N} \frac {p _ {i j}}{\left(t _ {i} + 1\right) \left(t _ {j} + 1\right)}\right) \\ \leq (\delta + 1) ^ {- 1}. \\ \end{array}
+$$
+
+By letting  $a \gets \sqrt{\frac{3\log(4N / \varepsilon)}{\delta + 1}}$ ,  $M \gets (\delta + 1)^{-1}$ ,  $v^2 \gets (\delta + 1)^{-1}$  and applying Theorem 5 of Chung & Radcliffe (2011), we have
+
+$$
+\begin{array}{l} \Pr (\| C - \bar {\Delta} \| > a) \leq 2 N \exp \left(- \frac {a ^ {2}}{2 (\delta + 1) ^ {- 1} + 2 (\delta + 1) ^ {- 1} a / 3}\right) \\ \leq 2 N \exp \left(- \frac {3 \log (4 N / \varepsilon)}{2 (1 + a / 3)}\right). \\ \end{array}
+$$
+
+By the definition of  $k(\varepsilon)$ , we have  $a < 1$  if  $\delta + 1 > k(\varepsilon) \log n$ . For such  $\delta$ , we have
+
+$$
+\begin{array}{l} \Pr (\| C - \bar {\Delta} \| > a) \leq 2 N \exp \left(- \frac {3 \log (4 N / \varepsilon)}{2 (1 + a / 3)}\right) \\ \leq 2 N \exp (- \log (4 N / \varepsilon)) \quad (\because a <   1) \\ = \frac {\varepsilon}{2}. \tag {5} \\ \end{array}
+$$
+
+Next, we bound  $\|\tilde{\Delta} - C\|$ . First, since  $a < 1$ , by Chernoff bound (see, e.g. Angluin & Valiant (1979); Hagerup & Rüb (1990)), we have
+
+$$
+\begin{array}{l} \Pr \left(\left| d _ {i} - t _ {i} \right| > a \left(t _ {i} + 1\right)\right) \leq 2 \exp \left(- \frac {a ^ {2} \left(t _ {i} + 1\right)}{3}\right) \\ \leq 2 \exp \left(- \frac {a ^ {2} (\delta + 1)}{3}\right) \\ = \frac {\varepsilon}{2 N}. \\ \end{array}
+$$
+
+Therefore, if  $|d_i - t_i| \leq a(t_i + 1)$ , then we have
+
+$$
+\begin{array}{l} \left| \sqrt {\frac {d _ {i} + 1}{t _ {i} + 1}} - 1 \right| \leq \left| \frac {d _ {i} + 1}{t _ {i} + 1} - 1 \right| (\because | \sqrt {x} - 1 | \leq | x - 1 | \text {f o r} x \geq 0) \\ = \left| \frac {d _ {i} - t _ {i}}{t _ {i} + 1} \right| \\ \leq a. \\ \end{array}
+$$
+
+Therefore, by union bound, we have
+
+$$
+\left\| \bar {D} ^ {- \frac {1}{2}} \tilde {D} ^ {\frac {1}{2}} - I _ {N} \right\| = \max  _ {i \in [ N ]} \left| \sqrt {\frac {d _ {i} + 1}{t _ {i} + 1}} - 1 \right| \leq a
+$$
+
+with probability at least  $1 - \varepsilon /2$ . Further, since the eigenvalue of the augmented normalized Laplacian is in [0, 2] by the proof of Proposition 1, we have  $\| I_N - \tilde{\Delta}\| \leq 1$ . By combining them, we have
+
+$$
+\begin{array}{l} \left\| \tilde {\Delta} - C \right\| = \left\| \left(\bar {D} ^ {- \frac {1}{2}} \tilde {D} ^ {\frac {1}{2}} - I _ {N}\right) \left(I _ {N} - \tilde {\Delta}\right) \tilde {D} ^ {\frac {1}{2}} \bar {D} ^ {- \frac {1}{2}} + \left(I _ {N} - \tilde {\Delta}\right) \left(I - \tilde {D} ^ {\frac {1}{2}} \bar {D} ^ {- \frac {1}{2}}\right) \right\| \\ \leq \left\| \left(\bar {D} ^ {- \frac {1}{2}} \tilde {D} ^ {\frac {1}{2}} - I _ {N} \right\| \left\| \tilde {D} ^ {\frac {1}{2}} \bar {D} ^ {- \frac {1}{2}} \right\| + \left\| I - \tilde {D} ^ {\frac {1}{2}} \bar {D} ^ {- \frac {1}{2}} \right\| \right. \\ \leq a (a + 1) + a. \tag {6} \\ \end{array}
+$$
+
+From (5) and (6), we have
+
+$$
+\begin{array}{l} \| \tilde {\Delta} - \bar {\Delta} \| \leq \| \tilde {\Delta} - C \| + \| C - \bar {\Delta} \| \\ \leq a + a (a + 1) + a \\ \leq a ^ {2} + 3 a \\ \leq 4 a \quad (\because a <   1) \\ \end{array}
+$$
+
+with probability at least  $1 - \varepsilon$  by union bound.
+
+Let  $N \in \mathbb{N}_{+}$  and  $p > 0$ . In the case of the Erdős-Rényi graph  $G_{N,p}$ , we should set  $P = p(J_{N} - I_{N})$  where  $J_{N} \in \mathbb{R}^{N \times N}$  are the all-one matrix. Then, we have  $\bar{A} = pJ_{N} + (1 - p)I_{N}$ ,  $\bar{D} = (Np - p + 1)I_{N}$ , and  $\bar{\Delta} = \frac{p}{Np - p + 1}(NI_{N} - J_{N})$ . Since the eigenvalue of  $J_{N}$  is  $N$  (with multiplicity 1) and 0 (with multiplicity  $N - 1$ ), the eigenvalue of  $\bar{\Delta}$  is 0 (with multiplicity 1) and  $\frac{Np}{Np - p + 1}$  (with multiplicity  $N - 1$ ). For  $G_{N,p}$ ,  $\delta$  is the expected average degree  $(N - 1)p$ . Hence, we have the following lemma from Lemma 5:
+
+Lemma 6. Let  $\tilde{\Delta}$  be its augmented normalized Laplacian of the Erdős-Rényi graph  $G_{N,p}$ . For any  $\varepsilon > 0$ , if  $\frac{Np - p + 1}{\log N} > k(\varepsilon) \coloneqq 3(1 + \log (4 / \varepsilon))$ , then, with probability at least  $1 - \varepsilon$ , we have
+
+$$
+\max _ {i = 2, \dots , N} \left| \lambda_ {i} (\tilde {\Delta}) - \frac {N p}{N p - p + 1} \right| \leq 4 \sqrt {\frac {3 \log (4 N / \varepsilon)}{N p - p + 1}}.
+$$
+
+Corollary 4. Consider GCN on  $G_{N,p}$ . Let  $W_{l}$  be the weight of the  $l$ -th layer of GCN and  $s_{l}$  be the maximum singular value of  $W_{l}$  for  $l \in \mathbb{N}_{+}$ . Set  $s := \sup_{l \in \mathbb{N}_{+}}$ . Let  $\varepsilon > 0$ . We define  $k(\varepsilon) := 3(1 + \log (4 / \varepsilon))$  and  $l(N,p,\varepsilon) = \frac{1 - p}{Np - p + 1} + 4\sqrt{\frac{3\log(4N / \varepsilon)}{Np - p + 1}}$ . If  $\frac{Np - p + 1}{\log N} > k(\varepsilon)$  and  $s \leq l(N,\varepsilon)^{-1}$ , then, GCN on  $G_{N,p}$  satisfies the assumption of Theorem 2 with probability at least  $1 - \varepsilon$ .
+
+Proof of Theorem 3. Since  $\frac{\log N}{Np} = o(1)$ , for fixed  $\varepsilon$ , we have
+
+$$
+\frac {N p - p + 1}{\log N} > \frac {N p}{\log N} > k (\varepsilon)
+$$
+
+for sufficiently large  $N$ . Further,  $Np \to \infty$  as  $N \to \infty$  when  $\frac{\log N}{Np} = o(1)$ . Therefore, we have
+
+$$
+\frac {(1 - p) ^ {2}}{N p - p + 1} \leq \frac {1}{N p} \leq (7 - 4 \sqrt {3}) ^ {2} \log \left(\frac {4 N}{\varepsilon}\right)
+$$
+
+for sufficiently large  $N$ . Hence.
+
+$$
+\frac {1 - p}{N p - p + 1} \leq (7 - 4 \sqrt {3}) \sqrt {\frac {\log (4 N / \varepsilon)}{N p - p + 1}}.
+$$
+
+Therefore, we have  $l(N,p,\varepsilon) \leq 7\sqrt{\frac{\log(4N / \varepsilon)}{Np - p + 1}}$ . Therefore, if  $s \leq \frac{1}{7}\sqrt{\frac{Np - p + 1}{\log(4N / \varepsilon)}}$ , then we have  $s \leq l(N,p,\varepsilon)^{-1}$ .
+
+# E MISCELLANEOUS PROPOSITIONS
+
+# E.1 INVARIANCE OF ORTHOGONAL COMPLEMENT SPACE
+
+Proposition 2. Let  $P \in \mathbb{R}^{N \times N}$  be a symmetric matrix, treated as a linear operator  $P: \mathbb{R}^N \to \mathbb{R}^N$ . If a subspace  $U \subset \mathbb{R}^N$  is invariant under  $P$  (i.e., if  $u \in U$ , then  $Pu \in U$ ), then,  $U^\perp$  is invariant under  $P$ , too.
+
+Proof. For any  $u \in U^{\perp}$  and  $v \in U$ , by symmetry of  $P$ , we have
+
+$$
+\langle P u, v \rangle = (P u) ^ {\top} v = u ^ {\top} P ^ {\top} v = u ^ {\top} P v = \langle u, P v \rangle .
+$$
+
+Since  $U$  is an invariant space of  $P$ , we have  $Pv \in U$ . Hence, we have  $\langle u, Pv \rangle = 0$  because  $u \in U^{\perp}$ . We obtain  $Pu \in U^{\perp}$  by the definition of  $U^{\perp}$ .
+
+# E.2 CONVERGENCE TO TRIVIAL FIXED POINT
+
+Let  $P \in \mathbb{R}^{N \times N}$  be a symmetric matrix,  $W_{l} \in \mathbb{R}^{C \times C}$ ,  $s_{l}$  be the maximum singular value of  $W_{l}$  for  $l \in \mathbb{N}_{+}$ . We define  $f_{l}: \mathbb{R}^{N \times C} \to \mathbb{R}^{N \times C}$  by  $f_{l}(X) := \sigma(PXW_{l})$  where  $\sigma$  is the element-wise ReLU function.
+
+Proposition 3. Suppose further that the operator norm of  $P$  is no larger than  $\lambda$ , then we have  $\| f_{l}(X)\|_{\mathrm{F}}\leq s_{l}\lambda \| X\|_{\mathrm{F}}$  for any  $l\in \mathbb{N}_{+}$ . In particular, let  $s\coloneqq \sup_{l\in \mathbb{N}_{+}}s_{l}$ . If  $s\lambda < 1$ , then,  $X_{l}$  exponentially approaches 0 as  $l\to \infty$ .
+
+Proof. Since  $\lambda$  is the operator norm of  $P|_{U^{\perp}}$ , the assumption implies that the operator norm of  $P$  itself is no larger than  $\lambda$ . Therefore, we have  $\| PXW_l\|_{\mathrm{F}} \leq \lambda \| XW_l\|_{\mathrm{F}} \leq s_l\lambda \| X\|_{\mathrm{F}}$ . On the other hand, since  $\sigma(x)^2 \leq x^2$  for any  $x \in \mathbb{R}$ , we have  $\|\sigma(X)\|_{\mathrm{F}} \leq \|X\|_F$  for any  $X \in \mathbb{R}^{N \times C}$ . Combining the two, we have  $\|f_l(X)\|_{\mathrm{F}} \leq \|PXW_l\|_{\mathrm{F}} \leq s_l\lambda \|X\|_{\mathrm{F}}$ .
+
+# E.3 STRICTNESS OF THEOREM 1
+
+Theorem 1 implies that if  $s\lambda \leq 1$ , then, one-step transition  $f_{l}$  does not increase the distance to  $\mathcal{M}$ . In this section, we first prove that this theorem is strict in the sense that, there exists a situation in which  $s_l\lambda > 1$  holds and the distance  $d_{\mathcal{M}}$  increases by one-step transition  $f_{l}$  at some point  $X$ .
+
+Set  $N\gets 2,C\gets 1$  , and  $M\gets 1$  in Section 3.2. For  $\mu ,\lambda >0$  , we set
+
+$$
+P \leftarrow \left[ \begin{array}{c c} \mu & 0 \\ 0 & \lambda \end{array} \right], e \leftarrow \left[ \begin{array}{c} 1 \\ 0 \end{array} \right], U \leftarrow \left\{\left[ \begin{array}{c} x \\ y \end{array} \right] \mid y = 0 \right\}.
+$$
+
+Then, by definition, we can check that the 3-tuple  $(P,e,U)$  satisfies the Assumptions 1 and 2. Set  $\mathcal{M} \coloneqq U \otimes \mathbb{R} = U$  and choose  $W \in \mathbb{R}$  so that  $W > \lambda^{-1}$ . Finally define  $f: \mathbb{R}^{N \times C} \to \mathbb{R}^{N \times C}$  by  $f(X) \coloneqq \sigma(PXW)$  where  $\sigma$  is the element-wise ReLU function.
+
+Proposition 4. We have  $d_{\mathcal{M}}(f(X)) > d_{\mathcal{M}}(X)$  for any  $X = [x_1 \quad x_2]^\top \in \mathbb{R}^2$  such that  $x_2 > 0$ .
+
+Proof. By definition, we have  $d_{\mathcal{M}}(X) = |x_2|$ . On the other hand, direct calculation shows that  $f_l(X) = [(W\mu X_1)^+ (W\lambda X_2)^+]^\top$  and  $d_{\mathcal{M}}(f_l(X)) = (W\lambda X_2)^+$  where  $x^{+} := \max(x,0)$  for  $x \in \mathbb{R}$ . Since  $W > \lambda^{-1}$  and  $x_2 > 0$ , we have  $d_{\mathcal{M}}(f_l(X)) > d_{\mathcal{M}}(X)$ .
+
+Next, we prove the non-strictness of Theorem 1 in the sense that there exists a situation in which  $s_l \lambda > 1$  holds and the distance  $d_{\mathcal{M}}$  uniformly decreases by  $f_l$ . Again, we set  $\text{Set } N \gets 2, C \gets 1$ , and  $M \gets 1$ . Let  $\lambda \in (1, 2)$  and set
+
+$$
+P \leftarrow \frac {\lambda}{2} \left[ \begin{array}{c c} 1 & - 1 \\ - 1 & 1 \end{array} \right], e \leftarrow \frac {1}{\sqrt {2}} \left[ \begin{array}{c} 1 \\ 1 \end{array} \right], U \leftarrow \left\{\left[ \begin{array}{c} x \\ y \end{array} \right] \mid x = y \right\}
+$$
+
+Then, we can directly show that 3-tuple  $(P,e,U)$  satisfies the Assumptions 1 and 2. Set  $W\gets 1$
+
+Proposition 5. We have  $W\lambda > 1$  and  $d_{\mathcal{M}}(f_l(X)) < d_{\mathcal{M}}(X)$  for all  $X \in \mathbb{R}^2$ .
+
+Proof. First, note that  $e' := \frac{1}{\sqrt{2}} [1 - 1]^\top$  is the eigenvector of  $P$  associated to  $\lambda$ :  $Pe' = \lambda e'$ . For  $X = ae + be'$  ( $a, b \in \mathbb{Z}$ ), the distance between  $X$  and  $\mathcal{M}$  is  $d_{\mathcal{M}}(X) = |b|$ . On the other hand, by direct computation, we have
+
+$$
+f (X) = \sigma (P X W) = \left\{ \begin{array}{l l} \left[ \begin{array}{c c} 0 & \frac {\lambda b}{\sqrt {2}} \end{array} \right] ^ {\top} & (\text {i f} b \geq 0), \\ \left[ \begin{array}{c c} \frac {- \lambda b}{\sqrt {2}} & 0 \end{array} \right] ^ {\top} & (\text {i f} b <   0). \end{array} \right.
+$$
+
+Therefore, the distance between  $f(X)$  and  $\mathcal{M}$  is  $d_{\mathcal{M}}(f(X)) = \lambda |b| / 2$ . Since  $\lambda < 2$ , we have  $d_{\mathcal{M}}(f(X)) < d_{\mathcal{M}}(X)$  for any  $X \in \mathbb{R}^2$ .
+
+We have shown that the non-negativity of  $e$  (Assumption 1) is not a redundant condition in Section 6.1.
+
+# F RELATIONTO MARKOV PROCESS
+
+It is known that any Markov process on finite states converges to a unique distribution (equilibrium) if it is irreducible and aperiodic (see, e.g., Norris (1998)). As we see in this section, this theorem is the special case of Corollary 3.
+
+Let  $S \coloneqq \{1, \dots, N\}$  be a finite discrete state space. Consider a Markov process on  $S$  characterized by a symmetric transition matrix  $P = (p_{ij})_{i,j \in [N]} \in \mathbb{R}^{N \times N}$  such that  $P \geq 0$  and  $P\mathbf{1} = \mathbf{1}$  where  $\mathbf{1}$  is the all-one vector. We interpret  $p_{ij}$  as the transition probability from a state  $i$  to  $j$ . We associate  $P$  with a graph  $G_P = (V_P, E_P)$  by  $V_P = [N]$  and  $(i,j) \in E_P$  if and only if  $p_{ij} > 0$ . Since  $P$  is symmetric, we can regard  $G_P$  as an undirected graph. We assume  $P$  is irreducible and aperiodic<sup>10</sup>. Perron-Frobenius' theorem (see, e.g., Meyer (2000)) implies that  $P$  satisfy the assumption of Corollary 3 with  $M = 1$ .
+
+Proposition 6 (Perron-Frobenius). Let the eigenvalues of  $P$  be  $\lambda_1 \leq \dots \leq \lambda_N$ . Then, we have  $-1 < \lambda_1$ ,  $\lambda_{N-1} < 1$ , and  $\lambda_N = 1$ . Further, there exists a unique vector  $e \in \mathbb{R}^N$  such that  $e \geq 0$ ,  $\| e \| = 1$ , and  $e$  is the eigenvector for the eigenvalue 1.
+
+Corollary 5. Let  $\lambda \coloneqq \max_{n = 1,\dots ,N - 1}|\lambda_n|(< 1)$  and  $\mathcal{M}\coloneqq \{e\otimes w\mid w\in \mathbb{R}^C\}$ . If  $s\lambda < 1$ , then, for any initial value  $X_{1}$ ,  $X_{l}$  exponentially approaches  $\mathcal{M}$  as  $l\to \infty$ .
+
+If we set  $C = 1$  and  $W_{l} = 1$  for all  $l \in \mathbb{N}_{+}$ , then, we can inductively show that  $X_{l} \geq 0$  for any  $l \geq 2$ . Therefore, we can interpret  $X_{l}$  as a measure on  $S$ . Suppose further that we take the initial value  $X_{1}$  as  $X_{1} \geq 0$  and  $X_{1}^{\top} \mathbf{1} = 1$  so that we can interpret  $X_{1}$  as a probability distribution on  $S$ . Then, we can inductively show that  $X_{l} \geq 0$ ,  $X_{l}^{\top} \mathbf{1} = 1$  (i.e.,  $X_{l}$  is a probability distribution on
+
+$S$ ), and  $X_{l + 1} = \sigma (PX_lW_l) = PX_l$  for all  $l\in \mathbb{N}_{+}$ . In conclusion, the corollary is reduced to the fact that if a finite and discrete Markov process is irreducible and aperiodic, any initial probability distribution converges exponentially to an equilibrium. In addition, the rate  $\lambda$  corresponds to the mixing time of the Markov process.
+
+# G GCN DEFINED BY NORMALIZED LAPLACIAN
+
+In Section 4, we defined  $P$  using the augmented normalized Laplacian  $\tilde{\Delta}$  by  $P = I_N - \tilde{\Delta}$ . We can alternatively use the usual normalized Laplacian  $\Delta$  instead of the augmented one to define  $P$  and want to apply the theory developed in Section 3.2. We write the normalized Laplacian version as  $P_{\Delta} := I_N - \Delta$ . The only obstacle is that the smallest eigenvalue  $\lambda_1$  of  $P_{\Delta}$  can be equal to  $-1$ , while that of  $P$  is strictly larger than  $-1$  (see, Proposition 1). This corresponds to that fact the largest eigenvalue of  $\tilde{\Delta}$  is strictly smaller than 2, while that for  $\Delta$  can be 2. It is known that the largest eigenvalue of  $\Delta$  is 2 if and only if the graph has a non-trivial bipartite connected component (see, e.g., Chung & Graham (1997)). Therefore, we can develop a theory using the normalized Laplacian instead of the augmented one in parallel for such a graph  $G$ .
+
+In Section 5, we characterized the asymptotic behavior of GCN defined by the augmented normalized Laplacian via its spectral distribution (Lemma 6 of Appendix D). We can derive a similar claim for GCN defined via the normalized Laplacian using the original theorem for the normalized Laplacian in Chung & Radcliffe (2011) (Theorem 7 therein). The normalized Laplacian version of GCN is advantageous over the one made from the augmented one because we know its spectral distribution for broader range of random graphs. For example, Chung & Radcliffe (2011) proved the convergence of the spectral distribution of the normalized Laplacian for Chung-Lu's model (Chung & Lu, 2002), which includes power law graphs as a special case (see, Theorem 4 of Chung & Radcliffe (2011)).
+
+# H DETAILS OF EXPERIMENT SETTINGS
+
+# H.1 EXPERIMENT OF SECTION 6.1
+
+We set the eigenvalue of  $P$  to  $\lambda_1 = 0.5$  and  $\lambda_2 = 1.0$  and randomly generated  $P$  until the eigenvector  $e$  associated to  $\lambda_2$  satisfies the condition of each case described in the main article. We set  $W = 1.2$  and used the following values for each case as  $P$  and  $e$ .
+
+# H.1.1 CASE 1
+
+$$
+P = \left[ \begin{array}{c c} 0. 7 4 6 9 9 1 5 & 0. 2 4 9 9 8 1 9 \\ 0. 2 4 9 9 8 1 9 & 0. 7 5 3 0 0 8 5 \end{array} \right], \quad e = \left[ \begin{array}{c} 0. 7 0 2 8 3 9 2 \\ - 0. 7 1 1 3 4 8 7 6 \end{array} \right].
+$$
+
+# H.1.2 CASE 2
+
+$$
+P = \left[ \begin{array}{c c} 0. 6 8 9 9 5 7 4 & - 0. 2 4 2 6 8 2 7 \\ - 0. 2 4 2 6 8 2 7 & 0. 8 1 0 0 4 2 6 \end{array} \right], \quad e = \left[ \begin{array}{c} 0. 6 1 6 3 7 2 3 4 \\ - 0. 7 8 7 4 5 4 8 5 \end{array} \right].
+$$
+
+# H.2 EXPERIMENT OF SECTION 6.2
+
+We randomly generated an Erdős– Rényi graph  $G_{N,p}$  with  $N = 1000$  and randomly generated a one-of-  $K$  hot vector for each node and embed it to a  $C$ -dimensional vector using a random matrix whose elements were randomly sampled from the standard Gaussian distribution. Here,  $K = 10$  and  $C = 32$ . We treated the resulting single as the input signal  $X^{(0)} \in \mathbb{R}^{N \times C}$ . We constructed a GCN with  $L = 10$  layers and  $C$  channels. All parameters were i.i.d. sampled from the Gaussian distribution whose standard deviation is same as the one used in LeCun et al. (2012)<sup>11</sup> and multiplied a scalar to each weight matrix so that the largest singular value equals to a specified value  $s$ . We used three configurations  $(p,s) = (0.1,0.1), (0.5,1.0), (0.5,10.0)$ .  $\lambda$  of the generated GCNs are 0.063, 0.197, 0.194, respectively. See Appendix 6.2 for the results of other configurations of  $(p,s)$ .
+
+Table 1: Dataset specifications. The threshold  $\lambda^{-1}$  in the table indicates the upper bound of Corollary 2.  
+
+<table><tr><td></td><td>#Node</td><td>#Edge</td><td>#Class</td><td>Chance Rate</td><td>Threshold λ-1</td></tr><tr><td>Cora</td><td>2708</td><td>5429</td><td>6</td><td>30.2%</td><td>1 + 3.62 × 10-3</td></tr><tr><td>CiteSeer</td><td>3312</td><td>4732</td><td>7</td><td>21.1%</td><td>1 + 1.25 × 10-3</td></tr><tr><td>PubMed</td><td>19717</td><td>44338</td><td>3</td><td>39.9%</td><td>1 + 9.57 × 10-3</td></tr></table>
+
+Table 2: Dataset specifications for noisy citation networks. The threshold  $\lambda^{-1}$  in the table indicates the upper bound of Corollary 2.  
+
+<table><tr><td></td><td>Original Dataset</td><td>#Edge Added</td><td>Threshold λ-1</td></tr><tr><td>Noisy Cora 2500</td><td>Cora</td><td>2495</td><td>1.11</td></tr><tr><td>Noisy Cora 5000</td><td>Cora</td><td>4988</td><td>1.15</td></tr><tr><td>Noisy CiteSeer</td><td>CiteSeer</td><td>4991</td><td>1.13</td></tr><tr><td>Noisy PubMed</td><td>PubMed</td><td>24993</td><td>1.17</td></tr></table>
+
+# H.3 EXPERIMENT OF SECTION 6.3
+
+# H.3.1 DATASET
+
+We used the Cora (McCallum et al., 2000; Sen et al., 2008), CiteSeer (Giles et al., 1998; Sen et al., 2008), and PubMed(Sen et al., 2008) datasets for experiments. We obtained the preprocessed dataset from the code repository of Kipf & Welling  $(2017)^{12}$ . Table 1 summarizes specifications of datasets and their noisy version (explained in the next section).
+
+The Cora dataset is a citation network dataset consisting of 2708 papers and 5429 links. Each paper is represented as the occurrence of 1433 unique words and is associated to one of 7 genres (Case Based, Genetic Algorithms, Neural Networks, Probabilistic Methods, Reinforcement Learning, Rule Learning, Theory). The graph made from the citation links has 78 connected components and the smallest positive eigenvalue of the augmented Normalized Laplacian is approximately  $\tilde{\mu} = 3.62 \times 10^{-3}$ . Therefore, the upper bound of Theorem 2 is  $\lambda^{-1} = (1 - \tilde{\mu})^{-1} \approx 1 + 3.62 \times 10^{-3}$ . 818 out of 2708 samples are labelled as "Probabilistic Methods", which is the largest proportion. Therefore, the chance rate is  $818 / 2708 = 30.2\%$ .
+
+The CiteSeer dataset is a citation network dataset consisting of 3312 papers and 4732 links. Each paper is represented as the occurrence of 3703 unique words and is associated to one of 6 genres (Agents, AI, DB, IR, ML, HCI). The graph made from the citation links has 438 connected components and the smallest positive eigenvalue of the augmented Normalized Laplacian is approximately  $\tilde{\mu} = 1.25 \times 10^{-3}$ . Therefore, the upper bound of Theorem 2 is  $\lambda^{-1} = (1 - \tilde{\mu})^{-1} \approx 1 + 1.25 \times 10^{-3}$ . 701 out of 2708 samples are labelled as "IR", which is the largest proportion. Therefore, the chance rate is  $701 / 3312 = 21.1\%$ .
+
+The PubMed dataset is a citation network dataset consisting of 19717 papers and 44338 links. Each paper is represented as the occurrence of 500 unique words and is associated to one of 3 genres ("Diabetes Mellitus, Experimental", "Diabetes Mellitus Type 1", "Diabetes Mellitus Type 2"). The graph made from the citation links has 438 connected components and the smallest positive eigenvalue of the augmented Normalized Laplacian is approximately  $\tilde{\mu} = 9.48 \times 10^{-3}$ . Therefore, the upper bound of Theorem 2 is  $\lambda^{-1} = (1 - \tilde{\mu})^{-1} \approx 1 + 9.57 \times 10^{-3}$ . 7875 out of 19717 samples are labelled as "Diabetes Mellitus Type 2", which is the largest proportion. Therefore, the chance rate is  $7875 / 19717 = 39.9\%$ .
+
+# H.3.2 NOISY CITATION NETWORKS
+
+We summarize the properties of noisy citation networks in Table 2.
+
+![](images/eb55c8329ff668c57bf284d14f6a518ab711a922ebf09337ee3e1b88593c519c.jpg)
+
+![](images/f4a2ce2d6fbe4340eefbe96508c30810430fe65c664a65b8f75a17434507be23.jpg)
+
+![](images/f4e277adc81ff49a67e594fedf183fd2cf37b18b8c78106ed08b0cf384bf67c8.jpg)  
+Figure 5: Spectral distribution of Laplacian for the citation network datasets. Left: normalized Laplacian. Right: augmented normalized Laplacian. Top: Cora and Noisy Cora (2500, 5000). Bottom: CiteSeer and Noisy CiteSeer.
+
+![](images/813178b5607f0b179000465d0198aba9b1495b0ea865c936d5db00a6fc27d34d.jpg)
+
+We created two datasets from the Cora dataset: Noisy Cora 2500 and Noisy Cora 5000. Noisy Cora 2500 is made from the Cora dataset by uniformly randomly adding 2500 edges, respectively. Since some random edges are overlapped with existing edges, the number of newly-added edges is 2495 in total. We only changed the underlying graph from the Cora dataset and did not change word occurrences (feature vectors) and genres (labels). The underlying graph of the Noisy Cora dataset has two connected components and the smallest positive eigenvalue is  $\tilde{\mu} \approx 9.62 \times 10^{-2}$ . Therefore, the threshold of the maximum singular values of in Theorem 2 has been increased to  $\lambda^{-1} = (1 - \tilde{\mu})^{-1} \approx 1.11$ . Similarly, Noisy Cora 5000 was made by adding 5000 edges uniformly randomly. The number of newly added edges is 4988 and the graph is connected (i.e., it has only 1 connected component).  $\tilde{\mu}$  and  $\lambda$  are  $\tilde{\mu} \approx 1.32 \times 10^{-1}$  and  $\lambda = (1 - \tilde{\mu})^{-1} \approx 1.15$ , respectively.
+
+We made the noisy version of CiteSeer (Noisy CiteSeer) and PubMed (Noisy PubMed), in the similar way, by adding 5000 and 25000 edges uniformly randomly to the datasets. Since some random edges were overlapped with existing edges, 4991 and 24993 edges are newly added, respectively. This manipulation reduced the number of connected components of the graph to 3.  $\tilde{\mu}$  is approximately  $1.11\times 10^{-1}$  (Noisy CiteSeer) and  $1.43\times 10^{-1}$  (Noisy PubMed) and  $\lambda^{-1} = (1 - \tilde{\mu})^{-1}$  is approximately 1.13 (Noisy CiteSeer) and 1.17 (Noisy PubMed), respectively. Figure 5 (right) shows the spectral distribution of the augmented normalized Laplacian For comparison, we show in Figure 5 (left) the spectral distribution of the normalized Laplacian for these datasets $^{13}$ .
+
+# H.3.3 MODEL ARCHITECTURE
+
+We used a GCN consisting of a single node embedding layer, one to nine graph convolution layers, and a readout operation (Gilmer et al., 2017), which is a linear transformation common to all nodes in our case. We applied softmax function to the output of GCN. The output dimension of GCN is same as the number of classes (i.e., seven for Noisy Cora 2500/5000, six for Noisy CiteSeer, and three for Noisy PubMed). We treated the number of units in each graph convolution layer as a hyperparameter. Optionally, we specified the maximum singular values  $s$  of graph convolution
+
+Table 3: Hyperparameters of the experiment in Section 6.3.  $X \sim$  LogUnif[10a, 10b] denotes the random variable  $\log_{10}X$  obeys the uniform distribution over  $[a,b]$ . "Learning rate" corresponds to  $\alpha$  when "Optimization algorithm" is Adam (Kingma & Ba, 2015).  
+
+<table><tr><td>Name</td><td>Value</td></tr><tr><td>Unit size</td><td>{10,20,...,500}</td></tr><tr><td>Epoch</td><td>{10,20,...,100}</td></tr><tr><td>Optimization algorithm</td><td>{SGD, MomentumSGD, Adam}</td></tr><tr><td>Learning rate</td><td>LogUnif[10-5, 10-2]</td></tr></table>
+
+layers. The choice of  $s$  is either 0.5 (smaller than 1),  $s_1$  (in the interval  $\{1 \leq s < \lambda^{-1}\}$ ), 3 and 10 (larger than  $\lambda^{-1}$ ). We used  $s_1 = 1.05$  for Noisy Cora 2500, Noisy CiteSeer, and Noisy PubMed, and  $s_1 = 1.1$  for Noisy Cora 5000 and Noisy CiteSeer so that  $s_1$  is not close to the edges of the the interval  $\{1 \leq s < \lambda^{-1}\}$ .
+
+# H.3.4 PERFORMANCE EVALUATION PROCEDURE
+
+We split all nodes in a graph (either Noisy Cora 2500/5000 or Noisy CiteSeer) into training, validation, and test sets. Data split is the same as the one done by Kipf & Welling (2017). This is a transductive learning (Pan & Yang, 2010) setting because we can use node properties of the validation and test data during training. We trained the model three times for each choice of hyperparameters using the training set and defined the objective function as the average accuracy on the validation set. We chose the combination of hyperparameters that achieves the best value of objective function. We evaluate the accuracy of the test dataset three times using the chosen combination of hyperparameters and computed their average and the standard deviation.
+
+# H.3.5 TRAINING
+
+At initialization, we sampled parameters from the i.i.d. Gaussian distribution. If the scale of maximum singular values  $s$  was specified, we subsequently scaled weight matrices of graph convolution layers so that their maximum singular values were normalized to  $s$ . The loss function was defined as the sum of the cross entropy loss for all training nodes. We train the model using the one of gradient-based optimization methods described in Table 3.
+
+# H.3.6 HYPERPRAMETERS
+
+Table 3 shows the set of hyperparameters from which we chose. Since we compute the representations of all nodes at once at each iteration, each epoch consists of 1 iteration. We employ Tree-structured Parzen Estimator (Bergstra et al., 2011) for hyperparameter optimization.
+
+# H.3.7 IMPLEMENTATION
+
+We used Chainer Chemistry $^{14}$ , which is an extension library for the deep learning framework Chainer (Tokui et al., 2015; 2019), to implement GCNs and Optuna (Akiba et al., 2019) for hyperparameter tuning. We conducted experiments in a signal machine which has 2 Intel(R) Xeon(R) Gold 6136 CPU@3.00GHz (24 cores), 192 GB memory (DDR4), and 3 GPGPUs (NVIDIA Tesla V100). Our implementation achieved  $68.1\%$  with Dropout (Srivastava et al., 2014) (2 graph convolution layers) and  $64.2\%$  without Dropout (1 graph convolution layer) on the test dataset. These are slightly worse than the accuracy reported in Kipf & Welling (2017), but are still comparable with it.
+
+# H.4 EXPERIMENT OF SECTION 6.4
+
+The experiment settings are almost same as the experiment in Section 6.3. The only difference is that we did not train the node embedding layer, which we put before convolution layers of a GCN, while we did in Section 6.3. This is because we wanted to see the the effect of convolution operations
+
+on the perpendicular component of signals, while we interested in the prediction accuracy in real training settings in the previous experiment.
+
+# I ADDITIONAL EXPERIMENT RESULTS
+
+# I.1 EXPERIMENT OF SECTION 6.1
+
+We show the vector field  $V$  for various  $W$  in Figure 6 (Case 1) and Figure 7 (Case 2). Parameters other than  $W$  are same as experiments in Section 6.1 (detail values are available in Appendix H.1).
+
+# I.2 EXPERIMENT OF SECTION 6.2
+
+Figure 8 shows the relative log distance of signals and their upper bound for various edge probability  $p$  and the maximum singular value  $s$ . Note that we generate a new graph for each configuration of  $(p, s)$ . Therefore, different configurations may have different graphs and hence different  $\lambda$  even they have a same edge probability  $p$  in common.
+
+# I.3 EXPERIMENT OF SECTION 6.3
+
+# I.3.1 PREDICTIVE ACCURACY
+
+Figure 9 shows the comparison of predictive performance in terms the maximum singular value and layer size when the dataset is Noisy Cora 5000 (left) and Noisy Citeseer (right), respectively. Concrete values are available in Table 4.
+
+# I.3.2 TRANSITION OF MAXIMUM SINGULAR VALUES
+
+Figure 10 - 13 show the transition of weight of graph convolution layers during training when the dataset is Noisy Cora 2500, Noisy Cora 5000, and Noisy CiteSeer, respectively. We note that the result of 3-layered GCN from the Noisy Cora 2500 is identical to Figure 3 (right) of the main article.
+
+# I.4 EXPERIMENT OF SECTION 6.4
+
+Figure 14 shows the logarithm of relative perpendicular component and prediction accuracy on Noisy Cora, Noisy CiteSeer, and Noisy PubMed datasets. We use Pearson R as a correlation coefficient. If GCNs have only one layer, it has more large relative perpendicular components (corresponding to right points in the figures) than GCNs which have other number of layers. The correlation between the logarithm of relative perpendicular components and prediction accuracies are  $0.827(p = 6.890 \times 10^{-6})$  for Noisy Cora,  $0.524(p = 1.771 \times 10^{-2})$  for Noisy CiteSeer, and  $0.679(p = 1.002 \times 10^{-3})$  for Noisy PubMed, if we treat the one-layer case as outliers and remove them.
+
+![](images/0ed143f59debf54ddd3c0eab5391c62c21bd6d25f3aea19046fa736902653007.jpg)
+
+![](images/60881755d9a793bc42c8b9dae0187669e075a788dd08cd399b3b1eb0c581397c.jpg)
+
+![](images/680dd1f41a4cd8bc0103e5c8f64b5e36c7b62dc4278104ad3ce1ace22b19e9be.jpg)
+
+![](images/4edf35554b99298850e0b67b1bba45afd42379b0f0cb3c0f8bf131c3d1d77715.jpg)
+
+![](images/2107b7aa730df3d034a5c6a30f117e75eae880789e5dee78c6df5d7a24e13d4c.jpg)  
+Figure 6: Vector field  $V$  for various  $W$  for Case 1. Top left:  $W = 0.5$ . Top right:  $W = 1.0$ . Middle left:  $W = 1.2$  (same as Figure 1 in the main article). Middle right:  $W = 1.5$ . Bottom left:  $W = 2.0$ . Bottom right:  $W = 4.0$ . Best view in color.
+
+![](images/4f9538c16f46ca886cc683963b57e4726d396fe5f4546ac344da36127af2afc7.jpg)
+
+![](images/eb7dc7abd9deb802b144c3aff974b6ffa8afe3c0a1acdd194ae2eecc955c62c3.jpg)
+
+![](images/0e088e58bab77852a1cb01583a2aaf5c15640b9b8caea6d0ea1fdfa405c3f9a8.jpg)
+
+![](images/18448379027fc8d0d563c7145c1d086146c4318501c7eb238954f46ee1be3c68.jpg)
+
+![](images/2f5e4ae7b76ff6835924b4f447455e0533faec633f3d6f959da7c6c6c0693617.jpg)
+
+![](images/2eee00948e6e48187d3575b3246188e246a58152b73a86d0a0c038dec1d827ac.jpg)  
+Figure 7: Vector field  $V$  for various  $W$  for Case 2. Top left:  $W = 0.5$ . Top right:  $W = 1.0$ . Middle left:  $W = 1.2$  (same as Figure 1 in the main article). Middle right:  $W = 1.5$ . Bottom left:  $W = 2.0$ . Bottom right:  $W = 4.0$ . Best view in color.
+
+![](images/ebeeaff02617a24704674475dbb3b1141b4ca440611d96b87e4aae5c3f9c1f44.jpg)
+
+Table 4: Comparison of performance in terms of maximum singular value of weights and layer size. "U" in the right most column indicates the accuracy of GCN without weight normalization.  
+Noisy Cora 2500  
+
+<table><tr><td rowspan="2">Depth</td><td colspan="5">Maximum Singular Value</td></tr><tr><td>1</td><td>1.05</td><td>3</td><td>10</td><td>U</td></tr><tr><td>1</td><td>0.389 ± 0.101</td><td>0.429 ± 0.090</td><td>0.552 ± 0.014</td><td>0.632 ± 0.007</td><td>0.587 ± 0.008</td></tr><tr><td>3</td><td>0.273 ± 0.051</td><td>0.309 ± 0.017</td><td>0.580 ± 0.058</td><td>0.661 ± 0.003</td><td>0.494 ± 0.041</td></tr><tr><td>5</td><td>0.319 ± 0.000</td><td>0.267 ± 0.059</td><td>0.462 ± 0.065</td><td>0.602 ± 0.004</td><td>0.326 ± 0.029</td></tr><tr><td>7</td><td>0.261 ± 0.076</td><td>0.262 ± 0.080</td><td>0.407 ± 0.021</td><td>0.501 ± 0.017</td><td>0.279 ± 0.129</td></tr><tr><td>9</td><td>0.261 ± 0.080</td><td>0.319 ± 0.000</td><td>0.284 ± 0.109</td><td>0.443 ± 0.014</td><td>0.319 ± 0.000</td></tr></table>
+
+Noisy Cora 5000  
+
+<table><tr><td rowspan="2">Depth</td><td colspan="5">Maximum Singular Value</td></tr><tr><td>1</td><td>1.1</td><td>3</td><td>10</td><td>U</td></tr><tr><td>1</td><td>0.301 ± 0.080</td><td>0.333 ± 0.099</td><td>0.557 ± 0.004</td><td>0.561 ± 0.019</td><td>0.555 ± 0.016</td></tr><tr><td>3</td><td>0.245 ± 0.066</td><td>0.247 ± 0.076</td><td>0.370 ± 0.041</td><td>0.587 ± 0.009</td><td>0.286 ± 0.066</td></tr><tr><td>5</td><td>0.274 ± 0.048</td><td>0.237 ± 0.070</td><td>0.257 ± 0.076</td><td>0.535 ± 0.031</td><td>0.319 ± 0.000</td></tr><tr><td>7</td><td>0.263 ± 0.080</td><td>0.297 ± 0.031</td><td>0.260 ± 0.074</td><td>0.339 ± 0.060</td><td>0.319 ± 0.000</td></tr><tr><td>9</td><td>0.262 ± 0.081</td><td>0.258 ± 0.064</td><td>0.262 ± 0.080</td><td>0.261 ± 0.082</td><td>0.318 ± 0.002</td></tr></table>
+
+Noisy CiteSeer  
+
+<table><tr><td rowspan="2">Depth</td><td colspan="5">Maximum Singular Value</td></tr><tr><td>0.5</td><td>1.1</td><td>3</td><td>10</td><td>U</td></tr><tr><td>1</td><td>0.461 ± 0.018</td><td>0.467 ± 0.012</td><td>0.490 ± 0.016</td><td>0.494 ± 0.006</td><td>0.495 ± 0.009</td></tr><tr><td>3</td><td>0.438 ± 0.027</td><td>0.436 ± 0.010</td><td>0.450 ± 0.019</td><td>0.462 ± 0.007</td><td>0.417 ± 0.061</td></tr><tr><td>5</td><td>0.285 ± 0.008</td><td>0.371 ± 0.016</td><td>0.373 ± 0.011</td><td>0.425 ± 0.007</td><td>0.380 ± 0.024</td></tr><tr><td>7</td><td>0.213 ± 0.006</td><td>0.282 ± 0.011</td><td>0.309 ± 0.012</td><td>0.385 ± 0.007</td><td>0.308 ± 0.012</td></tr><tr><td>9</td><td>0.182 ± 0.005</td><td>0.242 ± 0.030</td><td>0.303 ± 0.021</td><td>0.325 ± 0.003</td><td>0.229 ± 0.033</td></tr></table>
+
+Noisy Pubmed  
+
+<table><tr><td rowspan="2">Depth</td><td colspan="5">Maximum Singular Value</td></tr><tr><td>0.5</td><td>1.1</td><td>3</td><td>10</td><td>U</td></tr><tr><td>1</td><td>0.488 ± 0.039</td><td>0.636 ± 0.006</td><td>0.641 ± 0.010</td><td>0.632 ± 0.002</td><td>0.631 ± 0.010</td></tr><tr><td>3</td><td>0.442 ± 0.027</td><td>0.426 ± 0.026</td><td>0.658 ± 0.004</td><td>0.661 ± 0.005</td><td>0.631 ± 0.013</td></tr><tr><td>5</td><td>0.431 ± 0.033</td><td>0.431 ± 0.034</td><td>0.561 ± 0.083</td><td>0.641 ± 0.004</td><td>0.424 ± 0.093</td></tr><tr><td>7</td><td>0.428 ± 0.032</td><td>0.443 ± 0.051</td><td>0.449 ± 0.035</td><td>0.619 ± 0.011</td><td>0.440 ± 0.041</td></tr><tr><td>9</td><td>0.413 ± 0.009</td><td>0.438 ± 0.039</td><td>0.539 ± 0.052</td><td>0.569 ± 0.042</td><td>0.473 ± 0.031</td></tr></table>
+
+![](images/99ab3e78cb559046859f3915dc14ca283662648b00d204f09fb1d8a447b44dc2.jpg)
+
+![](images/888d41d43d6d003cd66eaf03f24a333368810b75af490b4c852b985516ea0487.jpg)
+
+![](images/98e8569fa5bd4f1fe5d7f9d2656276c14696179c250aebfd4b1c3878faed490a.jpg)
+
+![](images/97514e6eb9e01d6b29a27603219875e52e357085cb6e7c02b686552e171552e8.jpg)
+
+![](images/6044865655f267633bbd42f1f2a7ee5beb95ee7f5df059dea2118f2ecbff4c25.jpg)
+
+![](images/e7f73992ed48b51314cfff7a4c2351b0cebdfd822bf763a4ce9eb0bfaf478d9a.jpg)
+
+![](images/a33e3335774bd3d05e0d80be4ec8441ffdc402d9a994adc5df38de074cd96ac2.jpg)  
+Figure 8: The actual distance  $d_{\mathcal{M}}$  to the invariant space  $\mathcal{M}$  and the upper bound inferred by Theorem 1. The edge probability  $p$  takes 0.01(top), 0.1, 0.9(bottom) and the maximum singular value  $s$  takes 0.1(left), 1.0, 10(right). Blue lines are the log relative distance defined by  $y(l) \coloneqq \log \frac{d_{\mathcal{M}}(X^{(l)})}{d_{\mathcal{M}}(X^{(0)})}$  and orange dotted lines are upper bound  $y(l) \coloneqq l\log (s\lambda)$ , where  $X^{(0)}$  is the input signal and  $X^{(l)}$  is the output of the  $l$ -th layer. Best view in color.
+
+![](images/21724d81cbb945ee3c02728cbc3af99a584e10428324aa008cb63555782b9575.jpg)
+
+![](images/c2a492fa65135419b265560e6a6e6e43252dad713c787dd89c936af2f625c107.jpg)
+
+![](images/2ab037c047c4a6cd25f37dc8b306b3fc5a70d794d59d0c13b900d69e1c39c21e.jpg)
+
+![](images/3a79ad5153d1263b821dacab47872deb2b19c5d3e32ea3f8763685db5342f4e3.jpg)
+
+![](images/234348492acada62c53d7743fee0b808d3e29f2b3760285bafb686fe94d45f1c.jpg)  
+Figure 9: Effect of the maximum singular values of weights on predictive performance. Horizontal dotted lines indicate the chance rates (30.2% for Noisy Cora 5000, 21.2% for Noisy CiteSeer, and 39.9% for Noisy PubMed). The error bar is the standard deviation of 3 trials. Left: Noisy Cora 5000. Right: Noisy CiteSeer. Bottom: Noisy PubMed. Best view in color.
+
+![](images/734cd8da0fcca1e6b901e1a3ba8a783efcc24b6eba5a0bcf23b3c80afbaf4f5e.jpg)  
+Noisy Cora 2500
+
+![](images/780aa9c590ba4ec34204b1532989855766e6e801ead5401bdc36c1fe479ddb0a.jpg)
+
+![](images/1d2676e655c613c63a728a0aa7afa3933653d8d524e6e38c9bed3ffe9e4d7123.jpg)
+
+![](images/2ac3772cbc7eb4a479811e6910ac76a4a9271067a4d12ecbb8cdf9659b6564c6.jpg)
+
+![](images/655e9ac332274d3fb2c840b1cec230c8eb38c55844913a82b930fc64f8ed09bd.jpg)  
+Figure 10: Transition of maximum singular values of GCN during training using Noisy Cora 2500. Top left: 1 layer. Top right: 5 layers. Bottom left: 7 layers. Bottom right: 9 layers.
+
+![](images/1ea85b8693e1586c0487530b1110bdf4dd0b9cf3ac464716efd345c8fda43568.jpg)  
+Noisy Cora 5000
+
+![](images/e9839554890906cf4e5a5e563434d2847344ace332636917e437ea2f14173b2b.jpg)
+
+![](images/89101284882f38043f132aa7f1a28850ec6baf7586b1654ff2c2ee2c795e38e9.jpg)
+
+![](images/b67294415dc61a45d2346996db71a568478a345970d323fd0a8d2b0e39b3c0b9.jpg)
+
+![](images/d5c13e006fcc87319f8e3d17eacbe35c7da1e661a8683b052e60d8afe8b23ea0.jpg)  
+Figure 11: Transition of maximum singular values of GCN during training using Noisy Cora 5000. Top left: 1 layer. Top right: 3 layers. Middle left: 5 layers. Middle right: 7 layers. Bottom: 9 layers.
+
+![](images/96369e0226467f3604c73de8b665bbbbc36b1c8b9fbf4cc186a2a46a67e93703.jpg)  
+Noisy CiteSeer
+
+![](images/a89b50100acb33c1e0d5ceb25753755906d05e86709b62043ec05abda0c924bd.jpg)
+
+![](images/4112344b26681b74ee0279f2567ba8ccaeeced0fabf02941b47f42e9f07b3bea.jpg)
+
+![](images/0307c4158e70dc11d6f07bb2f5a3d9de633c1e9000fb6960fcd1a09e431b7e2f.jpg)
+
+![](images/1656753019ed5d376b72f3b379511b5ea5abbb9071d64ddd36cdf2acae560ba3.jpg)  
+Figure 12: Transition of maximum singular values of GCN during training using Noisy CiteSeer. Top left: 1 layer. Top right: 3 layers. Middle left: 5 layers. Middle right: 7 layers. Bottom: 9 layers.
+
+![](images/8d3a8b34c042ab7d002b632e2329c343433103878e7a85c6af16bf5e9371907c.jpg)  
+Noisy PubMed
+
+![](images/a472b0f7d5a76bfc362fdc5a3244a9124894b029c8cdc20be52f4c94919ac15e.jpg)
+
+![](images/72352d4bf2de59e9450595663e9ef5865d5d8e169d07dcc0509c819a16846577.jpg)
+
+![](images/1502e5a26cd4fee06308fdf8df9792c221253182ec4d94d5f1fa75a3c8c603f3.jpg)
+
+![](images/ae49ee8fadae98b5e92e30f196e41dfe206beb1cbca2dc681183d5ec7ab2f97f.jpg)  
+Figure 13: Transition of maximum singular values of GCN during training using Noisy PubMed. Top left: 1 layer. Top right: 3 layers. Middle left: 5 layers. Middle right: 7 layers. Bottom: 9 layers.
+
+![](images/d93fc5d02448d80f63c7264d7334405e5e4ad7efa745c09050b79b0cbfe0224d.jpg)  
+Figure 14: Logarithm of relative perpendicular component and prediction accuracy. Left: Noisy CiteSeer. Right: Noisy PubMed.  $p$  in the title represents the  $p$ -value for the Pearson R coefficients.
+
+![](images/a433c8e4470500a0928f085b4b4850eece9e6752f25a647abdf3421043814952.jpg)
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+# HAMILTONIAN GENERATIVE NETWORKS
+
+Peter Toth*
+
+DeepMind
+
+petertoth@google.com
+
+Danilo J. Rezende*
+
+DeepMind
+
+danilor@google.com
+
+Andrew Jaegle
+
+DeepMind
+
+drewjaegle@google.com
+
+Sebastien Racanière
+
+DeepMind
+
+sracaniere@google.com
+
+Aleksandar Botev
+
+DeepMind
+
+botev@google.com
+
+Irina Higgins
+
+DeepMind
+
+irinah@google.com
+
+# ABSTRACT
+
+The Hamiltonian formalism plays a central role in classical and quantum physics. Hamiltonians are the main tool for modelling the continuous time evolution of systems with conserved quantities, and they come equipped with many useful properties, like time reversibility and smooth interpolation in time. These properties are important for many machine learning problems—from sequence prediction to reinforcement learning and density modelling—but are not typically provided out of the box by standard tools such as recurrent neural networks. In this paper, we introduce the Hamiltonian Generative Network (HGN), the first approach capable of consistently learning Hamiltonian dynamics from high-dimensional observations (such as images) without restrictive domain assumptions. Once trained, we can use HGN to sample new trajectories, perform rollouts both forward and backward in time and even speed up or slow down the learned dynamics. We demonstrate how a simple modification of the network architecture turns HGN into a powerful normalising flow model, called Neural Hamiltonian Flow (NHF), that uses Hamiltonian dynamics to model expressive densities. We hope that our work serves as a first practical demonstration of the value that the Hamiltonian formalism can bring to deep learning.
+
+# 1 INTRODUCTION
+
+Any system capable of a wide range of intelligent behaviours within a dynamic environment requires a good predictive model of the environment's dynamics. This is true for intelligence in both biological (Friston, 2009; 2010; Clark, 2013) and artificial (Hafner et al., 2019; Battaglia et al., 2013; Watter et al., 2015; Watters et al., 2019) systems. Predicting environmental dynamics is also of fundamental importance in physics, where Hamiltonian dynamics and the structure-preserving transformations it provides have been used to unify, categorise and discover new physical entities (Noether, 1915; Livio, 2012).
+
+Hamilton's fundamental result was a system of two first-order differential equations that, in a stroke, unified the predictions made by prior Newtonian and Lagrangian mechanics (Hamilton, 1834). After well over a century of development, it has proven to be essential for parsimonious descriptions of nearly all of physics.
+
+![](images/c05a8a13369d522f63050bb5e5dae5ed3a6797493effcd19f128a391a2725fe9.jpg)  
+Figure 1: The Hamiltonian manifold hypothesis: natural images lie on a low-dimensional manifold in pixel space, and natural image sequences (such as one produced by watching a two-body system, as shown in red) correspond to movement on the manifold according to Hamiltonian dynamics.
+
+![](images/cf62cec17edbd66f2c57950d965f688cbf65798e52d32d337dd57785476b702f.jpg)  
+Figure 2: Hamiltonian Generative Network schematic. The encoder takes a stacked sequence of images and infers the posterior over the initial state. The state is rolled out using the learnt Hamiltonian. Note that we depict Euler updates of the state for schematic simplicity, while in practice this is done using a leapfrog integrator. For each unroll step we reconstruct the image from the position  $\mathbf{q}$  state variables only and calculate the reconstruction error.
+
+Hamilton's equations provide a way to predict a system's future behavior from its current state in phase space (that is, its position and momentum for classical Newtonian systems, and its generalized position and momentum more broadly). Hamiltonian mechanics induce dynamics with several nice properties: they are smooth, they include paths along which certain physical quantities are conserved (symmetries) and their time evolution is fully reversible. These properties are also useful for machine learning systems. For example, capturing the time-reversible dynamics of the world state might be useful for agents attempting to account for how their actions led to effects in the world; recovering an abstract low-dimensional manifold with paths that conserve various properties is tightly connected to outstanding problems in representation learning (see e.g. Higgins et al. (2018) for more discussion); and the ability to conserve energy is related to expressive density modelling in generative approaches (Rezende & Mohamed, 2015). Hence, we propose a reformulation of the well-known image manifold hypothesis by extending it with a Hamiltonian assumption (illustrated in Fig. 1): natural images lie on a low-dimensional manifold embedded within a high-dimensional pixel space and natural sequences of images trace out paths on this manifold that follow the equations of an abstract Hamiltonian.
+
+Given the rich set of established tools provided by Hamiltonian descriptions of system dynamics, can we adapt these to solve outstanding machine learning problems? When it comes to adapting the Hamiltonian formalism to contemporary machine learning, two questions need to be addressed: 1) how should a system's Hamiltonian be learned from data; and 2) how should a system's abstract phase space be inferred from the high-dimensional observations typically available to machine learning systems? Note that the inferred state may need to include information about properties that play no physical role in classical mechanics but which can still affect their behavior or function, like the colour or shape of an object. The first question was recently addressed by the Hamiltonian Neural Network (HNN) (Greydanus et al., 2019) approach, which was able to learn the Hamiltonian of three simple physical systems from noisy phase space observations. However, to address the second question, HNN makes assumptions that restrict it to Newtonian systems and appear to limit its ability to scale to more challenging video datasets.
+
+In this paper we introduce the first model that answers both of these questions without relying on restrictive domain assumptions. Our model, the Hamiltonian Generative Network (HGN), is a generative model that infers the abstract state from pixels and then unrolls the learned Hamiltonian following the Hamiltonian equations (Goldstein, 1980). We demonstrate that HGN is able to reliably learn the Hamil
+
+tonian dynamics from noisy pixel observations on four simulated physical systems: a pendulum, a mass-spring and two- and three- body systems. Our approach outperforms HNN by a significant margin. After training, we demonstrate that HGN produces meaningful samples with reversible dynamics and that the speed of rollouts can be controlled by changing the time derivative of the integrator at test time. Finally, we show that a small modification of our architecture yields a flexible, normalising flow-based generative model that respects Hamiltonian dynamics. We show that this model, which we call Neural Hamiltonian Flow (NHF), inherits the beneficial properties of the Hamiltonian formalism (including volume preservation) and is capable of expressive density modelling, while offering computational benefits over standard flow-based models.
+
+# 2 RELATED WORK
+
+Most machine learning approaches to modeling dynamics use discrete time steps, which often results in an accumulation of the approximation errors when producing rollouts and, therefore, to a fast drop in accuracy. Our approach, on the other hand, does not discretise continuous dynamics and models them directly using the Hamiltonian differential equations, which leads to slower divergence for longer rollouts. The density model version of HGN (NHF) uses the Hamiltonian dynamics as normalising flows along with a numerical integrator, making our approach somewhat related to the recently published neural ODE work (Chen et al., 2018; Grathwohl et al., 2018). What makes our approach different is that Hamiltonian dynamics are both invertible and volume-preserving (as discussed in Sec. 3.3), which makes our approach computationally cheaper than the alternatives and more suitable as a model of physical systems and other processes that have these properties. Also related is recent work attempting to learn a model of physical system dynamics end-to-end from image sequences using an autoencoder (de Avila Belbute-Peres et al., 2018). Unlike our work, this model does not exploit Hamiltonian dynamics and is trained in a supervised or semi-supervised regime.
+
+# 2.1 HAMILTONIAN NEURAL NETWORK
+
+One of the most comparable approaches to ours is the Hamiltonian Neural Network (HNN) (Greydanus et al., 2019). This work, done concurrently to ours, proposes a way to learn Hamiltonian dynamics from data by training the gradients of a neural network (obtained by backpropagation) to match the time derivative of a target system in a supervised fashion. In particular, HNN learns a differentiable function  $\mathcal{H}(q,p)$  that maps a system's state (its position,  $q$ , and momentum,  $p$ ) to a scalar quantity interpreted as the system's Hamiltonian. This model is trained so that  $H(p,q)$  satisfies the Hamiltonian equation by minimizing
+
+$$
+\mathcal {L} _ {\mathrm {H N N}} = \frac {1}{2} \left[ \left(\frac {\partial \mathcal {H}}{\partial p} - \frac {d q}{d t}\right) ^ {2} + \left(\frac {\partial \mathcal {H}}{\partial q} + \frac {d p}{d t}\right) ^ {2} \right], \tag {1}
+$$
+
+where the derivatives  $\frac{\partial\mathcal{H}}{\partial q}$  and  $\frac{\partial\mathcal{H}}{\partial p}$  are computed by backpropagation. Hence, this learning procedure is most directly applicable when the true state space (in canonical coordinates) and its time derivatives are known. Accordingly, in the majority of the experiments presented by the authors, the Hamiltonian was learned from the ground truth state space directly, rather than from pixel observations.
+
+The single experiment with pixel observations required a modification of the model. First, the input to the model became a concatenated, flattened pair of images  $\boldsymbol{o}_t = [x_t, x_{t+1}]$ , which was then mapped to a low-dimensional embedding space  $\boldsymbol{z}_t = [q_t, p_t]$  using an encoder neural network. Note that the dimensionality of this embedding ( $z \in \mathbb{R}^2$  in the case of the pendulum system presented in the paper) was chosen to perfectly match the ground truth dimensionality of the phase space, which was assumed to be known a priori. This, however, is not always possible. The latent embedding was then treated as an estimate of the position and the momentum of the system depicted in the images, where the momentum was assumed to be equal to the velocity of the system – an assumption enforced by the additional constraint found necessary to encourage learning, which encouraged the time derivative of the position latent to equal the momentum latent using finite differences on the split latents:
+
+$$
+\mathcal {L} _ {\mathrm {C C}} = \left(p _ {t} - \left(q _ {t + 1} - q _ {t}\right)\right) ^ {2}. \tag {2}
+$$
+
+This assumption is appropriate in the case of the simple pendulum system presented in the paper, however it does not hold more generally. Note that our approach does not make any assumptions on the dimensionality of the learned phase space, or the form of the momenta coordinates, which makes our approach more general and allows it to perform well on a wider range of image domains as presented in Sec. 4.
+
+# 3 METHODS
+
+# 3.1 THE HAMILTONIAN FORMALISM
+
+The Hamiltonian formalism describes the continuous time evolution of a system in an abstract phase space  $s = (q, p) \in \mathbb{R}^{2n}$ , where  $q \in \mathbb{R}^n$  is a vector of position coordinates, and  $p \in \mathbb{R}^n$  is the corresponding vector of momenta. The time evolution of the system in phase space is given by the Hamiltonian equations:
+
+$$
+\frac {\partial \boldsymbol {q}}{\partial t} = \frac {\partial \mathcal {H}}{\partial \boldsymbol {p}}, \quad \frac {\partial \boldsymbol {p}}{\partial t} = - \frac {\partial \mathcal {H}}{\partial \boldsymbol {q}} \tag {3}
+$$
+
+where the Hamiltonian  $\mathcal{H}:\mathbb{R}^{2n}\to \mathbb{R}$  maps the state  $s = (q,p)$  to a scalar representing the energy of the system. The Hamiltonian specifies a vector field over the phase space that describes all possible dynamics of the system. For example, the Hamiltonian for an undamped mass-spring system is  $\mathcal{H}(q,p) = \frac{1}{2} kq^2 +\frac{p^2}{2m}$ , where  $m$  is the mass,  $q\in \mathbb{R}^1$  is its position,  $p\in \mathbb{R}^1$  is its momentum and  $k$  is the spring stiffness coefficient. The Hamiltonian can often be expressed as the sum of the kinetic  $T$  and potential  $V$  energies  $\mathcal{H} = T(\pmb {p}) + V(\pmb {q})$ , as is the case for the mass-spring example. Identifying a system's Hamiltonian is in general a very difficult problem, requiring carefully instrumented experiments and researcher insight produced by years of training. In what follows, we describe a method for modeling a system's Hamiltonian from raw observations (such as pixels) by inferring a system's state with a generative model and rolling it out with the Hamiltonian equations.
+
+# 3.2 LEARNING HAMILTONIANS WITH THE HAMILTONIAN GENERATIVE NETWORK
+
+Our goal is to build a model that can learn a Hamiltonian from observations. We assume that the data  $X = \{(\pmb{x}_0^1, \dots, \pmb{x}_T^1), \dots, (\pmb{x}_0^K, \dots, \pmb{x}_T^K)\}$  comes in the form of high-dimensional noisy observations, where each  $\pmb{x}_i = \mathrm{G}(\pmb{s}_i) = \mathrm{G}(\pmb{q}_i)$  is a non-deterministic function of the generalised position in the phase space, and the full state is a non-deterministic function of a sequence of images  $\pmb{s}_i = (\pmb{q}_i, \pmb{p}_i) = \mathrm{D}(\pmb{x}_0^i, \dots, \pmb{x}_t^i)$ , since the momentum (and hence the full state) cannot in general be recovered from a single observation. Our goal is to infer the abstract state and learn the Hamiltonian dynamics in phase space by observing  $K$  motion sequences, discretised into  $T + 1$  time steps each. In the process, we also want to learn an approximation to the generative process  $G(s)$  in order to be able to move in both directions between the high dimensional observations and the low-dimensional abstract phase space.
+
+Although the Hamiltonian formalism is general and does not depend on the form of the observations, we present our model in terms of visual observations, since many known physical Hamiltonian systems, like a mass-spring system, can be easily observed visually. In this section we introduce the Hamiltonian Generative Network (HGN), a generative model that is trained to behave according to the Hamiltonian dynamics in an abstract phase space learned from raw observations of image sequences. HGN consists of three parts (see Fig. 2): an inference network, a Hamiltonian network and a decoder network, which are discussed next.
+
+The inference network takes in a sequence of images  $(\pmb{x}_0^i,\dots \pmb{x}_T^i)$ , concatenated along the channel dimension, and outputs a posterior over the initial state  $z\sim q_{\phi}(\cdot |\pmb{x}_0,\dots \pmb{x}_T)$ , corresponding to the system's coordinates in phase space at the first frame of the sequence. We parametrise  $q_{\phi}(\pmb {z})$  as a diagonal Gaussian with a unit Gaussian prior  $p(z) = \mathcal{N}(0,\mathbb{I})$  and optimise it using the usual reparametrisation trick (Kingma & Welling, 2014). To increase the expressivity of the abstract phase space  $s_0$ , we map samples from the posterior with another function  $s_0 = f_\psi (\pmb {z})$  to obtain the system's initial state. As mentioned in Sec. 3.1, the Hamiltonian function expects the state to be of the form  $\pmb {s} = (\pmb {q},\pmb {p})$ , hence we initialise  $\pmb {s}_0\in \mathbb{R}^{2n}$  and arbitrarily assign the first half of the units to represent abstract position  $\pmb{q}$  and the other half to represent abstract momentum  $\pmb{p}$ .
+
+![](images/44633e8e674dae00fa7096cbbb15157bb2bec1bb29ed3236ed001f553d17f45d.jpg)  
+Figure 3: A: standard normalising flow, where the invertible function  $f_{i}$  is implemented by a neural network. B: Hamiltonian flows, where the initial density is transformed using the learned Hamiltonian dynamics. Note that we depict Euler updates of the state for schematic simplicity, while in practice this is done using a leapfrog integrator.
+
+![](images/7543b7708cb397d751aed86050f2a03f32a8df342cb7cc0affc5ea6b63f7041c.jpg)
+
+![](images/3e9908bd5ad52ba493a3e5640ce4f2e377e1c1cef8f200012360e14f500de5ef.jpg)  
+Figure 4: A schematic representation of NFH which can perform expressive density modelling by using the learned Hamiltonians as normalising flows. Note that we depict Euler updates of the state for schematic simplicity, while in practice this is done using a leapfrog integrator.
+
+The Hamiltonian network is parametrised as a neural network with parameters  $\gamma$  that takes in the inferred abstract state and maps it to a scalar  $\mathcal{H}_{\gamma}(s_t) \in \mathbb{R}$ . We can use this function to do rollouts in the abstract state space using the Hamiltonian equations (Eq. 3), for example by Euler integration:  $s_{t + 1} = (q_{t + 1},p_{t + 1}) = (q_t + \frac{\partial\mathcal{H}}{\partial p_t} dt,p_t - \frac{\partial\mathcal{H}}{\partial q_t} dt)$ . In this work we assume a separable Hamiltonian, so in practice we use a more sophisticated leapfrog integrator to roll out the system, since it has better theoretical properties and results in better performance in practice (see Sec. A.6 in Supplementary Materials for more details).
+
+The decoder network is a standard deconvolutional network (we use the architecture from Karras et al. (2018)) that takes in a low-dimensional representation vector and produces a high-dimensional pixel reconstruction. Given that each instantaneous image does not depend on the momentum information, we restrict the decoder to take only the position coordinates of the abstract state as input:  $p_{\theta}(\boldsymbol{x}_t) = d_{\theta}(\boldsymbol{q}_t)$ .
+
+The objective function. Given a sequence of  $T + 1$  images, HGN is trained to optimise the following objective:
+
+$$
+\mathcal {L} (\phi , \psi , \gamma , \theta ; \boldsymbol {x} _ {0}, \dots \boldsymbol {x} _ {T}) = \frac {1}{T + 1} \sum_ {t = 0} ^ {T} \left[ \mathbb {E} _ {q _ {\phi} (\boldsymbol {z} | \boldsymbol {x} _ {1}, \dots \boldsymbol {x} _ {T})} \left[ \log p _ {\psi , \gamma , \theta} \left(\boldsymbol {x} _ {t} \mid \boldsymbol {q} _ {t}\right) \right] \right] - K L \left(q _ {\phi} (\boldsymbol {z}) \mid | p (\boldsymbol {z})\right), \tag {4}
+$$
+
+which can be seen as a temporally extended variational autoencoder (VAE) (Kingma & Welling, 2014; Rezende et al., 2014) objective, consisting of a reconstruction term for each frame, and an additional term that encourages the inferred posterior to match a prior. The key difference with a standard VAE lies in how we generate rollouts – these are produced using the Hamiltonian equations of motion in learned Hamiltonian phase space.
+
+# 3.3 LEARNING HAMILTONIAN FLOWS
+
+In this section, we describe how the architecture described above can be modified to produce a model for flexible density estimation. Learning computationally feasible and accurate estimates of complex
+
+![](images/42ccf27565c6765b62e892fadc5a39c2562d31609104dd899511e119f8ed8cb8.jpg)
+
+![](images/404950a77e9636f956785b0ea7e2ea566e279975017aee55be53f3d906ba1ee7.jpg)
+
+![](images/7e2006b08fd721aff14e9dc2c2706c65028c602e1b863c7df64b541d24fd1629.jpg)  
+Figure 5: Ground truth Hamiltonians and samples from generated datasets for the ideal pendulum, mass-spring, and two- and three-body systems used to train HGN.
+
+densities is an open problem in generative modelling. A common idea to address this problem is to start with a simple prior distribution  $\pi(\boldsymbol{u})$  and then transform it into a more expressive form  $p(\boldsymbol{x})$  through a series of composable invertible transformations  $f_{i}(\boldsymbol{u})$  called normalising flows (Rezende & Mohamed, 2015) (see Fig. 3A). Sampling can then be done according to  $\boldsymbol{x} = f_{T} \circ \ldots \circ f_{1}(\boldsymbol{u})$ , where  $\boldsymbol{u} \sim \pi(\cdot)$ . Density evaluation, however, requires more expensive computations of both inverting the flows and calculating the determinants of their Jacobians. For a single flow step, this equates to the following:  $p(\boldsymbol{x}) = \pi(f^{-1}(\boldsymbol{x})) \left| \det \left( \frac{\partial f^{-1}}{\partial \boldsymbol{x}} \right) \right|$ . While a lot of work has been done recently into proposing better alternatives for flow-based generative models in machine learning (Rezende & Mohamed, 2015; Kingma et al., 2016; Papamakarios et al., 2017; Dinh et al., 2017; Huang et al., 2018; Kingma & Dhariwal, 2018; Hoogeboom et al., 2019; Chen et al., 2019; Grathwohl et al., 2018), none of the approaches manage to produce both sampling and density evaluation steps that are computationally scalable.
+
+The two requirements for normalising flows are that they are invertible and volume preserving, which are exactly the two properties that Hamiltonian dynamics possess. This can be seen by computing the determinant of the Jacobian of the infinitesimal transformation induced by the Hamiltonian  $\mathcal{H}$ . By Jacobi's formula, the derivative of the determinant at the identity is the trace and:
+
+$$
+\det  \left[ \mathbb {I} + d t \left( \begin{array}{c c} \frac {\partial^ {2} \mathcal {H}}{\partial q _ {i} \partial p _ {j}} & - \frac {\partial^ {2} \mathcal {H}}{\partial q _ {i} \partial q _ {j}} \\ \frac {\partial^ {2} \mathcal {H}}{\partial p _ {i} \partial p _ {j}} & - \frac {\partial^ {2} \mathcal {H}}{\partial p _ {i} \partial q _ {j}} \end{array} \right) \right] = 1 + d t \operatorname {T r} \left( \begin{array}{c c} \frac {\partial^ {2} \mathcal {H}}{\partial q _ {i} \partial p _ {j}} & - \frac {\partial^ {2} \mathcal {H}}{\partial q _ {i} \partial q _ {j}} \\ \frac {\partial^ {2} \mathcal {H}}{\partial p _ {i} \partial p _ {j}} &  - \frac {\partial^ {2} \mathcal {H}}{\partial p _ {i} \partial q _ {j}} \end{array} \right) + O (d t ^ {2}) = 1 + O (d t ^ {2}) (5)
+$$
+
+where  $i \neq j$  are the off-diagonal entries of the determinant of the Jacobian. Hence, in this section we describe a simple modification of HGN that allows it to act as a normalising flow. We will refer to this modification as the Neural Hamiltonian Flow (NHF) model. First, we assume that the initial state  $s_0$  is a sample from a simple prior  $s_0 \sim \pi_0(\cdot)$ . We then chain several Hamiltonians  $\mathcal{H}_i$  to transform the sample to a new state  $s_T = \mathcal{H}_T \circ \ldots \circ \mathcal{H}_1(s_0)$  which corresponds to a sample from the more expressive final density  $s_T \sim p(\pmb{x})$  (see Fig. 3B for an illustration of a single Hamiltonian flow). Note that unlike HGN, where the Hamiltonian dynamics are shared across time steps (a single Hamiltonian is learned and its parameters are shared across time steps of a rollout), in NHF each step of the flow (corresponding to a single time step of a rollout) can be parametrised by a different Hamiltonian. The inverse of such a Hamiltonian flow can be easily obtained by replacing  $dt$  by  $-dt$  in the Hamiltonian equations and reversing the order of the transformations,  $s_0 = \mathcal{H}_1^{-dt} \circ \ldots \circ \mathcal{H}_T^{-dt}(s_T)$  (we will use the appropriate  $dt$  or  $-dt$  superscript from now on to make the direction of integration of the Hamiltonian dynamics more explicit). The resulting density  $p(s_T)$  is given by the following equation:
+
+$$
+\ln p (\boldsymbol {s} _ {T}) = \ln \pi (\boldsymbol {s} _ {0}) = \ln \pi \left(\mathcal {H} _ {1} ^ {- d t} \circ \dots \circ \mathcal {H} _ {T} ^ {- d t} (\boldsymbol {s} _ {T})\right) + O (d t ^ {2})
+$$
+
+Our proposed NHF is more computationally efficient than many other flow-based approaches, because it does not require the expensive step of calculating the trace of the Jacobian. Hence, the NHF model constitutes a more structured form of a Neural ODE flow (Chen et al., 2018), but with a few notable differences: (i) The Hamiltonian ODE is volume-preserving, which makes the computation of log-likelihood cheaper than for a general ODE flow. (ii) General ODE flows are only invertible in the limit  $dt \to 0$ , whereas for some Hamiltonians we can use more complex integrators (like the symplectic leapfrog integrator described in Sec. A.6) that are both invertible and volume-preserving for any  $dt > 0$ . The structure  $s = (q,p)$  on the state-space imposed by the Hamiltonian dynamics can be constraining from the point of view of density estimation. We choose to use the trick proposed in the Hamiltonian
+
+![](images/ad65b3b7caf8bc88092c099682e8012e18bd2de287fbf7e2a2d42bf3057b0d9f.jpg)  
+Figure 6: Average pixel MSE for each step of a single train and test unroll on four physical systems. All versions of HGN outperform HNN, which often learned to reconstruct an average image. Note the different scales of the plots comparing HGN to HNN and different versions of HGN.
+
+Monte Carlo (HMC) literature (Neal et al., 2011; Salimans et al., 2015; Levy et al., 2017), which treats the momentum  $\pmb{p}$  as a latent variable (see Fig. 4). This is an elegant solution which avoids having to artificially split the density into two disjoint sets. As a result, the data density that our Hamiltonian flows are modelling becomes exclusively parametrised by  $p(\pmb{q}_T)$ , which takes the following form:  $p(\pmb{q}_T) = \int p(\pmb{q}_T, \pmb{p}_T) d\pmb{p}_T = \int \pi (\mathcal{H}_1^{-dt} \circ \dots \circ \mathcal{H}_T^{-dt} (\pmb{q}_T, \pmb{p}_T)) d\pmb{p}_T$ . This integral is intractable, but the model can still be trained via variational methods where we introduce a variational density  $f_{\psi}(\pmb{p}_T | \pmb{q}_T)$  with parameters  $\psi$  and instead optimise the following ELBO:
+
+$$
+\operatorname {E L B O} \left(\boldsymbol {q} _ {T}\right) = \mathbb {E} _ {f _ {\psi} \left(\boldsymbol {p} _ {T} \mid \boldsymbol {q} _ {T}\right)} \left[ \ln \pi \left(\mathcal {H} _ {1} ^ {- d t} \circ \dots \circ \mathcal {H} _ {T} ^ {- d t} \left(\boldsymbol {q} _ {T}, \boldsymbol {p} _ {T}\right)\right) - \ln f _ {\psi} \left(\boldsymbol {p} _ {T} \mid \boldsymbol {q} _ {T}\right) \right] \leq \ln p \left(\boldsymbol {q} _ {T}\right), \tag {6}
+$$
+
+Note that, in contrast to VAEs (Kingma & Welling, 2014; Rezende et al., 2014), the ELBO in Eq. 6 is not explicitly in the form of a reconstruction error term plus a KL term.
+
+# 4 RESULTS
+
+In order to directly compare the performance of HGN to that of its closest baseline, HNN, we generated four datasets analogous to the data used in Greydanus et al. (2019). The datasets contained observations of the time evolution of four physical systems: mass-spring, pendulum, two- and three-body (see Fig. 5). In order to generate each trajectory, we first randomly sampled an initial state, then produced a 30 step rollout following the ground truth Hamiltonian dynamics, before adding Gaussian noise with standard deviation  $\sigma^2 = 0.1$  to each phase-space coordinate, and rendering a corresponding  $64\times 64$  pixel observation. We generated 50 000 train and 10 000 test trajectories for each dataset. When sampling initial states, we start by first sampling the total energy of the system denoted as a radius  $r$  in the phase space, before sampling the initial state  $(q,p)$  uniformly on the circle of radius  $r$ . Note that our pendulum dataset is more challenging than the one described in Greydanus et al. (2019), where the pendulum had a fixed radius and was initialized at a maximum angle of  $30^\circ$  from the central axis.
+
+Mass-spring. The dynamics of a frictionless mass-spring system are modeled by the Hamiltonian  $\mathcal{H} = \frac{1}{2} k q^2 + \frac{p^2}{2m}$ , where  $k$  is the spring constant and  $m$  is the mass. We fix  $k = 2$  and  $m = 0.5$ , then sample a radius from a uniform distribution  $r \sim \mathbb{U}(0.1, 1.0)$ .
+
+Pendulum. The dynamics of a frictionless pendulum are modeled by the Hamiltonian  $\mathcal{H} = 2mgl(1 - \cos (q)) + \frac{p^2}{2ml^2}$ , where  $g$  is the gravitational constant and  $l$  is the length of the pendulum. We fix  $g = 3$ ,  $m = 0.5$ ,  $l = 1$ , then sample a radius from a uniform distribution  $r \sim \mathbb{U}(1.3,2.3)$ .
+
+Two- and three- body problems. In an n-body problem, particles interact with each other through an attractive force, like gravity. The dynamics are represented by the following Hamiltonian  $\mathcal{H} = \sum_{i}^{n}\frac{\|p_i\|^2}{2m_i} -\sum_{1\leq i < j\leq n}\frac{gm_im_j}{||q_j - q_i||}$ . We set  $m = 1$  and  $g = 1$  for both systems. For the two-body problem, we set  $r\sim \mathbb{U}(0.5,1.5)$ , and we also change the observation noise to  $\sigma^2 = 0.05$ . For the three-body problem, we set  $r\sim U(0.9,1.2)$ , and set the observation noise to  $\sigma^2 = 0.2$ .
+
+Learning the Hamiltonian We tested whether HGN and the HNN baseline could learn the dynamics of the four systems described above. To ensure that our re-implementation of HNN was correct, we replicated all the results presented in the original paper (Greydanus et al., 2019) by verifying that it could learn the dynamics of the mass-spring, pendulum and two-body systems well from the ground truth state, and the dynamics of a restricted pendulum from pixels. We also compared different modifications of HGN: a version trained and tested with an Euler rather than a leapfrog integrator (HGN Euler), a version trained with no additional function between the posterior and the prior (HGN no  $f_{\psi}$ ) and a deterministic version (HGN determ), which did not include the sampling step from the posterior  $q_{\phi}(z|x_0\dots x_T)$ .
+
+<table><tr><td rowspan="2">MODEL</td><td colspan="2">MASS-SPRING</td><td colspan="2">PENDULUM</td><td colspan="2">TWO-BODY</td><td colspan="2">THREE-BODY</td></tr><tr><td>TRAIN</td><td>TEST</td><td>TRAIN</td><td>TEST</td><td>TRAIN</td><td>TEST</td><td>TRAIN</td><td>TEST</td></tr><tr><td>HNN</td><td>50.38±1.75</td><td>104.37±52.51</td><td>69.39±0.89</td><td>64.62±0.87</td><td>80.52±1.79</td><td>78.9±1.93</td><td>61.63±3.31</td><td>57.54±2.39</td></tr><tr><td>HNN (CONV)</td><td>18.97±0.77</td><td>119.09±69.17</td><td>13.14±0.66</td><td>106.07±41.94</td><td>7.3±0.38</td><td>120.71±44.48</td><td>15.22±1.99</td><td>62.77±37.57</td></tr><tr><td>HGN (EULER)</td><td>3.67±1.09</td><td>6.2±2.69</td><td>5.43±2.53</td><td>10.93±4.32</td><td>6.62±3.93</td><td>15.06±7.01</td><td>7.51±3.49</td><td>9.4±3.92</td></tr><tr><td>HGN (DETERM)</td><td>0.23±0.23</td><td>3.07±1.06</td><td>0.79±1.24</td><td>10.68±3.19</td><td>2.34±2.3</td><td>14.47±5.24</td><td>4.1±2.05</td><td>5.17±1.96</td></tr><tr><td>HGN (NO fψ)</td><td>4.95±1.71</td><td>7.04±2.55</td><td>6.83±3.29</td><td>13.98±4.94</td><td>6.35±3.86</td><td>16.49±6.6</td><td>8.37±3.13</td><td>10.41±3.72</td></tr><tr><td>HGN (LEAPFROG)</td><td>3.84±1.07</td><td>6.23±2.03</td><td>4.9±1.86</td><td>11.72±4.14</td><td>6.36±3.29</td><td>16.47±7.15</td><td>7.88±3.55</td><td>9.8±3.72</td></tr></table>
+
+Table 1: Average pixel MSE over a 30 step unroll on the train and test data on four physical systems. All values are multiplied by  $1e + 4$ . We evaluate two versions of the Hamiltonian Neural Network (HNN) (Greydanus et al., 2019): the original architecture and a convolutional version closely matched to the architecture of HGN. We also compare four versions of our proposed Hamiltonian Generative Network (HGN): the full version, a version trained and tested with an Euler rather than a leapfrog integrator, a deterministic rather than a generative version, and a version of HGN with no extra network between the posterior and the initial state.
+
+Tbl. 1 and Fig. 6 demonstrate that HGN and its modifications learned well on all four datasets. However, when we attempted to train HNN on the four datasets described above, its Hamiltonian often collapsed to 0 and the model failed to reproduce any dynamics, defaulting to a static single image. We were unable to improve on this performance despite our best efforts, including a modification of the architecture to closely match ours (referred to as HNN Conv) (see Sec. A.3 of the appendix for details). Tbl. 1 shows that the average mean squared error (MSE) of the pixel reconstructions on both the train and test data is an order of magnitude better for HGN compared to both versions of HNN. The same holds when visualising the average per-frame MSE of a single train and test rollout for each dataset shown in Fig. 6.
+
+Note that the different versions of HGN have different trade-offs. The deterministic version produces more accurate reconstructions but it does not allow sampling. This effect is equivalent to a similar distinction between autoencoders (Hinton & Salakhutdinov, 2006) and VAEs. Using the simpler Euler integrator rather than the more involved leapfrog one might be conceptually more appealing, however it does not provide the same energy conservation and reversibility properties as the leapfrog integrator, as evidenced by the increase by an order of magnitude of the variance of the learned Hamiltonian throughout a sequence rollout as shown in Tbl. 2. The full version of HGN, on the other hand, is capable of reproducing the dynamics well, is capable of producing diverse yet plausible rollout samples (Fig. 8) and its rollouts can be reversed in time, sped up or slowed down by either changing the value or the sign of  $dt$  used in the integrator (Fig. 7).
+
+Expressive density modelling using learned Hamiltonian flows We evaluate whether NF is capable of expressive density modelling by stacking learned Hamiltonians into a series of normalising
+
+<table><tr><td rowspan="2">MODEL</td><td colspan="2">MASS-SPRING</td><td colspan="2">PENDULUM</td><td colspan="2">TWO-BODY</td><td colspan="2">THREE-BODY</td></tr><tr><td>TRAIN</td><td>TEST</td><td>TRAIN</td><td>TEST</td><td>TRAIN</td><td>TEST</td><td>TRAIN</td><td>TEST</td></tr><tr><td>HNN</td><td>N/A</td><td>N/A</td><td>N/A</td><td>N/A</td><td>N/A</td><td>N/A</td><td>0.72</td><td>1.43</td></tr><tr><td>HNN (CONV)</td><td>N/A</td><td>N/A</td><td>N/A</td><td>N/A</td><td>N/A</td><td>N/A</td><td>2.17</td><td>28.34</td></tr><tr><td>HGN (EULER)</td><td>157.28</td><td>120.16</td><td>135.53</td><td>19.53</td><td>405.83</td><td>682.02</td><td>1.18</td><td>8.36</td></tr><tr><td>HGN (DETERM)</td><td>0.08</td><td>0.09</td><td>0.03</td><td>0.02</td><td>0.03</td><td>0.05</td><td>3.08</td><td>0.22</td></tr><tr><td>HGN (NO fψ)</td><td>0.02</td><td>0.02</td><td>0.02</td><td>0.04</td><td>0.04</td><td>0.02</td><td>4.03</td><td>1.82</td></tr><tr><td>HGN (LEAPFROG)</td><td>1.38</td><td>0.68</td><td>3.15</td><td>7.13</td><td>0.15</td><td>0.45</td><td>0.31</td><td>1.65</td></tr></table>
+
+Table 2: Variance of the Hamiltonian on four physical systems over single train and test rollouts shown in Fig. 6. The numbers reported are scaled by a factor of  $1e + 4$ . High variance indicates that the energy is not conserved by the learned Hamiltonian throughout the rollout. Many HNN Hamiltonians have collapsed to 0, as indicated by N/A. HGN Hamiltonians are meaningful, and different versions of HGN conserve energy to varying degrees.
+
+![](images/865668eedc02451dceace8d477e3fcc0194084c042922e4012856948f76533d4.jpg)  
+Figure 7: Example of a train and a test sequence from the dataset of a three-body system, its inferred forward, backward, double speed and half speed rollouts in time from HGN, and a forward rollout from HNN. HNN did not learn the dynamics of the system and instead learned to reconstruct an average image.
+
+![](images/d11f390779a9e904c4740c7382eb70fcf0b6cb38231d2b167e0e47110ed17528.jpg)
+
+![](images/fe0637d371c6fd1289bd15a287cdf8fbfd5e2385d8c53b44b11a8fe10b1336b6.jpg)  
+Figure 8: Examples of sample rollouts for all four datasets from a trained HGN.
+
+![](images/66231d357982a9f2f87a4107064401bd822e3b8cce4a73285af7948577fdbbb3.jpg)
+
+flows. Fig. 9 demonstrates that NHF can transform a simple soft-uniform prior distribution  $\pi (s_0;\sigma ,\beta)$  into significantly more complex densities with arbitrary numbers of modes. The soft-uniform density,  $\pi (s_0;\sigma ,\beta)\propto f(\beta (s + \sigma \frac{1}{2}))f(-\beta (s - \sigma \frac{1}{2}))$  , where  $f$  is the sigmoid function and  $\beta$  is a constant, was chosen to make it easier to visualise the learned attractors. The model also performed well with other priors, including a Normal distribution. It is interesting to note that the trained model is very interpretable. When decomposed into the equivalents of the kinetic and potential energies, it can be seen that the learned potential energy  $V(q)$  learned to have several local minima, one for each mode of the data. As a consequence, the trajectory of the initial samples through the flow has attractors at the modes of the data. We have also compared the performance of NHF to that of the RNVP baseline
+
+![](images/9ea53db387d2812bae4e586e6eedd215e70c85fd3e0dfc13a2c9e360a639c8db.jpg)  
+Figure 9: Multimodal density learning using Hamiltonian flows. From left to right: KDE estimators of the target and learned densities; learned kinetic energy  $K(\pmb{p})$  and potential energy  $V(\pmb{q})$ ; single leapfrog step and an integrated flow. The potential energy learned multiple attractors, also clearly visible in the integrated flow plot. The basins of attraction are centred at the modes of the data.
+
+![](images/b66963e0bfc3f128bcd4b45a3f36d6fb52aaed4ec8040863bf98fcb9c0ea285f.jpg)
+
+![](images/52583bdee79040406654310eda460f1b710b7ad33dd053f5e815f8a01453523c.jpg)  
+Figure 10: Comparison between RNVP (Dinh et al., 2017) and NHF on a Gaussian Mixture Dataset. The RNVPs use alternating masks with two layers (red) or 3 layers (purple). NHF uses 1, 2 or 3 leapfrog steps (blue, yellow and green respectively).
+
+(Dinh et al., 2017). Fig. 10 shows that the two approaches are comparable in their performance, but NHF is more computationally efficient as discussed at the end of Sec. 3.3.
+
+# 5 CONCLUSIONS
+
+We have presented HGN, the first deep learning approach capable of reliably learning Hamiltonian dynamics from pixel observations. We have evaluated our approach on four classical physical systems and demonstrated that it outperformed the only relevant baseline by a large margin. Hamiltonian dynamics have a number of useful properties that can be exploited more widely by the machine learning community. For example, the Hamiltonian induces a smooth manifold and a vector field in the abstract phase space along which certain physical quantities are conserved. The time evolution along these paths is also completely reversible. These properties can have wide implications in such areas of machine learning as reinforcement learning, representation learning and generative modelling. We have demonstrated the first step towards applying the learnt Hamiltonian dynamics as normalising flows for expressive yet computationally efficient density modelling. We hope that this work serves as the first step towards a wider adoption of the rich body of physics literature around the Hamiltonian principles in the machine learning community.
+
+# ACKNOWLEDGEMENTS
+
+We thank Alvaro Sanchez for useful feedback.
+
+# REFERENCES
+
+Peter W. Battaglia, Jessica B. Hamrick, and Joshua B. Tenenbaum. Simulation as an engine of physical scene understanding. PNAS, 110, 2013.  
+Ricky T. Q. Chen, Jens Behrmann, David Duvenaud, and Jörn-Henrik Jacobsen. Residual flows for invertible generative modeling. arXiv, 2019.  
+Tian Qi Chen, Yulia Rubanova, Jesse Bettencourt, and David K Duvenaud. Neural ordinary differential equations. In NeurIPS, 2018.  
+Andy Clark. Whatever next? predictive brains, situated agents, and the future of cognitive science. Behavioral and Brain Sciences, 36(3), 2013.  
+Filipe de Avila Belbute-Peres, Kevin Smith, Kelsey Allen, Josh Tenenbaum, and J. Zico Kolter. End-to-end differentiable physics for learning and control. NeurIPS, 2018.  
+Laurent Dinh, Jascha Sohl-Dickstein, and Samy Bengio. Density estimation using real nvp. *ICLR*, 2017.  
+K. Friston. The free-energy principle: a rough guide to the brain? Trends in cognitive sciences, 13, 2009.  
+Karl Friston. The free-energy principle: a unified brain theory? Nature Reviews Neuroscience, 2010.  
+Herbert Goldstein. Classical Mechanics. Addison-Wesley Pub. Co, Reading, Mass., 1980.  
+Will Grathwohl, Ricky T. Q. Chen, Jesse Bettencourt, Ilya Sutskever, and David Duvenaud. Ffjord: Free-form continuous dynamics for scalable reversible generative models. *ICLR*, 2018.  
+Sam Greydanus, Misko Dzamba, and Jason Yosinski. Hamiltonian neural networks. NeurIPS, 2019.  
+Danijar Hafner, Timothy Lillicrap, Ian Fischer, Ruben Villegas, David Ha, Honglak Lee, and James Davidson. Learning latent dynamics for planning from pixels. PMLR, 2019.  
+William R. Hamilton. On a general method in dynamics. Philosophical Transactions of the Royal Society, II, pp. 247-308, 1834.  
+Irina Higgins, David Amos, David Pfau, Sebastien Racaniere, Loic Matthew, Danilo Rezende, and Alexander Lerchner. Towards a definition of disentangled representations. arXiv, 2018.  
+G. E. Hinton and R.R. Salakhutdinov. "reducing the dimensionality of data with neural networks. Science, 313:504-507, 2006.  
+Emiel Hoogeboom, Rianne van den Berg, and Max Welling. Emerging convolutions for generative normalizing flows. ICML, 2019.  
+Chin-Wei Huang, David Krueger, Alexandre Lacoste, and Aaron Courville. Neural autoregressive flows. ICML, 2018.  
+Eric Jones, Travis Oliphant, PEARu Peterson, et al. SciPy: Open source scientific tools for Python, 2001. URL http://www.scipy.org/.  
+Tero Karras, Timo Aila, Samuli Laine, and Jaakko Lehtinen. Progressive growing of gans for improved quality, stability, and variation. *ICLR*, 2018.  
+Diederik P. Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.  
+Diederik P. Kingma and Prafulla Dhariwal. Glow: Generative flow with invertible 1x1 convolutions. In Advances in Neural Information Processing Systems, pp. 10215-10224, 2018.  
+Diederik P. Kingma and Max Welling. Auto-encoding variational bayes. *ICLR*, 2014.  
+Diederik P. Kingma, Tim Salimans, Rafal Jozefowicz, Xi Chen, Ilya Sutskever, and Max Welling. Improving variational inference with inverse autoregressive flow. NeurIPS, 2016.
+
+Daniel Levy, Matthew D Hoffman, and Jascha Sohl-Dickstein. Generalizing hamiltonian monte carlo with neural networks. arXiv preprint arXiv:1711.09268, 2017.  
+Mario Livio. Why symmetry matters. Nature, 490, 2012.  
+Radford M Neal et al. MCMC using hamiltonian dynamics. Handbook of markov chain monte carlo, 2 (11):2, 2011.  
+Emmy Noether. The finiteness theorem for invariants of finite groups. Mathematische Annalen, 77: 89-92, 1915.  
+George Papamakarios, Theo Pavlakou, and Iain Murray. Masked autoregressive flow for density estimation. NeurIPS, 2017.  
+Danilo J Rezende and Fabio Viola. Generalized elbo with constrained optimization, geco. Workshop on Bayesian Deep Learning, NeurIPS, 2018.  
+Danilo Jimenez Rezende and Shakir Mohamed. Variational inference with normalizing flows. arXiv preprint arXiv:1505.05770, 2015.  
+Danilo Jimenez Rezende, Shakir Mohamed, and Daan Wierstra. Stochastic backpropagation and approximate inference in deep generative models. ICML, 2014.  
+Tim Salimans, Diederik Kingma, and Max Welling. Markov chain Monte Carlo and variational inference: Bridging the gap. In ICML, 2015.  
+Manuel Watter, Jost Tobias Springenberg, Joschka Boedecker, and Martin Riedmiller. Embed to control: A locally linear latent dynamics model for control from raw images. NeurIPS, 2015.  
+Nicholas Watters, Loic Matthew, Matko Bosnjak, Christopher P. Burgess, and Alexander Lerchner. *Cobra: Data-efficient model-based rl through unsupervised object discovery and curiosity-driven exploration.* arxiv, 2019.
+
+# A SUPPLEMENTARY MATERIALS
+
+# A.1 HAMILTONIAN GENERATIVE NETWORK
+
+The Hamiltonian Generative Network (HGN) consists of three major parts, an encoder, the Hamiltonian transition network and a decoder. During training the encoder starts with a sequence
+
+$$
+\left(\boldsymbol {x} _ {1}, \boldsymbol {x} _ {2}, \dots , \boldsymbol {x} _ {n}\right) \in \mathbb {R} ^ {3 2 \times 3 2 \times 3} \tag {7}
+$$
+
+of raw training images and encodes it into a probabilistic prior representation transformed with an additional network on top
+
+$$
+q = q \left(\boldsymbol {z} \mid \boldsymbol {x} _ {1}, \boldsymbol {x} _ {2}, \dots , \boldsymbol {x} _ {n}\right) \sim p = \mathcal {N} \left(\boldsymbol {s} _ {0} \mid (0, 1)\right) \tag {8}
+$$
+
+into a start state
+
+$$
+\boldsymbol {s} _ {0} \in \mathbb {R} ^ {4 \times 4 \times (2 \times 1 6)} \tag {9}
+$$
+
+consisting of a downsized spatial representation in latent space  $(4\times 4)$ , where each abstract pixel is the the concatenation of abstract position  $(q)$  and momentum  $(p)$  (each of dimension 16). The encoder network is a convolutional neural network with 8 layers, with 32 filters on the first layer, then 64 filters on each subsequent layer, while in the last layer we have 48 filters. The final encoder transformer network is a convolutional neural network with 3 layers and 64 filters on each layer. Starting from this initial embedded state, the Hamiltonian transition network generates subsequent states using a symplectic integrator approximating the Hamiltonian equations. The Hamiltonian transition network represents the Hamiltonian function
+
+$$
+\mathcal {H}: \boldsymbol {s} _ {t} \in \mathbb {R} ^ {4 \times 4 \times (2 \times 1 6)} \rightarrow \mathbb {R} \tag {10}
+$$
+
+as a function from the abstract position and momentum space to the real numbers at any time step  $t$ . The Hamiltonian transition network is a convolutional neural network of 6 layers, each consisting of 64 filters. The discrete timestep we use for the symplectic integrator update step is  $dt = 0.125$
+
+At each time step  $t$  the decoder network  $d_{\theta}$  takes only the abstract position part  $\pmb{q}_t$  of the state  $\pmb{s}_t$  and decodes it back to an output image  $\widetilde{\pmb{x}}_t \in \mathbb{R}^{32 \times 32 \times 3}$  of the same shape as the input images.
+
+$$
+d _ {\theta}: \boldsymbol {q} _ {t} \rightarrow \widetilde {\boldsymbol {x}} _ {t}. \tag {11}
+$$
+
+The decoder network is a progressive network consisting of 3 residual blocks, where each residual block resizes the current input image by a factor of 2 using the nearest neighbor method (at the end we have to upscale our latent spatial dimension of 4 to the desired output image dimension of 32 in these steps), followed by 2 blocks of a one layer convolutional neural network with 64 filters and a leaky ReLU activation function, closing by a sigmoid activation in each block. After the 3 blocks a final one layer convolutional neural network outputs the output image with the right number of channels.
+
+We use Adam optimisier (Kingma & Ba, 2014) with learning rate  $1.5\mathrm{e - }4$  . When optimising the loss, in practice we do not learn the variance of the decoder  $p_{\theta}(\boldsymbol {x}|s)$  and fix it to 1, which makes the reconstruction objective equivalent to a scaled L2 loss. Furthermore, we introduce a Lagrange multiplier in front of the KL term and optimise it using the same method as in Rezende & Viola (2018).
+
+# A.2 NEURAL HAMILTONIAN FLOW
+
+For all NHF experiments the Hamiltonian was of the form  $\mathcal{H}(q,p) = K(p) + V(q)$ . The kinetic energy term  $K$  and the potential energy term  $V$  are soft-plus MLPs with layer-sizes  $[d,128,128,1]$  where  $d$  is the dimension of the data. Soft-plus non-linearities were chosen because the optimisation of Hamiltonian flows involves second-order derivatives of the MLPs used for parametrising the Hamiltonians. This makes ReLU non-linearities unsuitable. The encoder network was parametrized as  $f_{\psi}(\pmb {p}|\pmb {q}) = \mathcal{N}(\pmb {p};\mu (\pmb {q}),\sigma (\pmb {q}))$ , where  $\mu$  and  $\sigma$  are ReLU MLPs with size  $[d,128,128,d]$ . The Hamiltonian flow  $\mathcal{H}^{dt}$ , was approximated using a leapfrog integrator (Neal et al., 2011) since it preserves volume and is invertible for any  $dt$  (see also section A.6). We found that only two leapfrog steps where sufficient for all our examples. Parameters were optimised using Adam (Kingma & Ba, 2014) (learning rate 3e-4) and Lagrange multipliers were optimised using the same method as in Rezende & Viola (2018). All
+
+shown kernel density estimate (KDE) plots used 1000 samples and isotropic Gaussian kernel bandwidth of 0.3. For the RNVP baseline (Dinh et al., 2017) we used alternating masks with two or three layers. Each RNVP layer used an affine coupling parametrized by a two-layer relu-MLP that matched those used in the leapfrog.
+
+# A.3 HAMILTONIAN NEURAL NETWORK
+
+The Hamiltonian Neural Network (HNN) (Greydanus et al., 2019) learns a differentiable function  $\mathcal{H}(\pmb{q},\pmb{p})$  that maps a system's state in phase space (its position  $\pmb{q}$  and momentum  $\pmb{p}$ ) to a scalar quantity interpreted as the system's Hamiltonian. This model is trained so that  $\mathcal{H}(\pmb{q},\pmb{p})$  satisfies the Hamiltonian equation by minimizing
+
+$$
+\mathcal {L} _ {\mathrm {H N N}} = \frac {1}{2} \left[ \left(\frac {\partial \mathcal {H}}{\partial \boldsymbol {p}} - \frac {d \boldsymbol {q}}{d t}\right) ^ {2} + \left(\frac {\partial \mathcal {H}}{\partial \boldsymbol {q}} + \frac {d \boldsymbol {p}}{d t}\right) ^ {2} \right], \tag {12}
+$$
+
+where the derivatives  $\frac{\partial\mathcal{H}}{\partial q}$  and  $\frac{\partial\mathcal{H}}{\partial p}$  are computed by backpropagation. In the original paper, these targets are either assumed to be known or are estimated by finite differences using the state at times  $t$  and  $t + 1$ . Accordingly, in the majority of the experiments presented by the authors, the Hamiltonian was learned from the ground truth state space directly, rather than from pixel observations.
+
+The original HNN model is trained in a supervised fashion on the ground truth state of a physical system and its time derivatives. As such, it is not directly comparable to our method, which learns a Hamiltonian directly from pixels. Instead, we compare to the PixelHNN variant of the HNN, which is introduced in the same paper, and which is able to learn a Hamiltonian from images and in the absence of true state or time derivative in some settings.
+
+This required a modification of the model. First, the input to the model became a concatenated, flattened pair of images  $X_{t} = (x_{t}, x_{t + 1})$ , which was then mapped to a low-dimensional embedding space  $z_{t} = (q_{t}, p_{t})$  using an encoder neural network. Note that the dimensionality of this embedding  $(z \in \mathbb{R}^2$  in the case of the pendulum system presented in the paper) is chosen to perfectly match the ground truth dimensionality of the phase space, which was assumed to be known a priori. This, however, is not always possible, as when a system has not yet been identified. The latent embedding was then treated as an estimate of the position and the momentum of the system depicted in the images, where the momentum was assumed to be equal to the velocity of the system – an assumption enforced by the additional constraint found necessary to encourage learning, which encouraged the time derivative of the position latent to equal the momentum latent using finite differences on the split latents:
+
+$$
+\mathcal {L} _ {\mathrm {C C}} = \left(\boldsymbol {p} _ {t} - \left(\boldsymbol {q} _ {t + 1} - \boldsymbol {q} _ {t}\right)\right) ^ {2}. \tag {13}
+$$
+
+This loss is motivated by the observation that in simple Newtonian systems with unit mass, the system's state is fully described by the position and its time derivative (the system's velocity). An image embedding that corresponds to the position and velocity of the system will minimize this loss. This assumption is appropriate in the case of the simple pendulum system presented in the paper, however it does not hold more generally.
+
+As mentioned earlier, PixelHNN takes as input a concatenated, flattened pair of images and maps them to an embedding space  $\mathbf{z}_t = (\mathbf{q}_t, \mathbf{p}_t)$ , which is treated as an estimate of the position and momentum of the system depicted in the images. Note that  $X_t$  always consists of two images in order to make the momentum observable. This embedding space is used as the input to an HNN, which is trained to learn the Hamiltonian of the system as before, but using the embedding instead of the true system state.
+
+To enable stable learning in this configuration, the PixelHNN uses a standard mean-squared error autoencoding loss:
+
+$$
+\mathcal {L} _ {\mathrm {A E}} = \frac {1}{N} \sum_ {i = 1} ^ {N} \left(X _ {t} ^ {i} - \hat {X} _ {t} ^ {i}\right) ^ {2}, \tag {14}
+$$
+
+where  $\hat{X}_t$  is the autoencoder output and  $X_t^i$  is the value of pixel  $i$  of  $N$  total pixels in  $X_t$ . This loss encourages the network embedding to reflect the content of the input images and to avoid the trivial solution to (12).
+
+The full PixelHNN loss is:
+
+$$
+\mathcal {L} _ {\text {P i x e l H N N}} = \mathcal {L} _ {\mathrm {A E}} + \mathcal {L} _ {\mathrm {C C}} + \lambda_ {\mathrm {H N N}} \mathcal {L} _ {\mathrm {H N N}} + \lambda_ {\mathrm {W D}} \mathcal {L} _ {\mathrm {W D}}, \tag {15}
+$$
+
+where  $\mathcal{L}_{\mathrm{HNN}}$  is computed using the finite difference estimate of the time derivative of the embedding.  $\lambda_{\mathrm{HNN}}$  is a Lagrange multiplier, which is set to 0.1, as in the original paper.  $\mathcal{L}_{\mathrm{WD}}$  is a standard L2 weight decay and its Lagrange multiplier  $\lambda_{\mathrm{WD}}$  is set to 1e-5, as in the original paper.
+
+In the experiments presented here, we reimplemented the PixelHNN architecture as described in Greydanus et al. (2019) and trained it using the full loss (15). As in the original paper, we used a PixelHNN with HNN, encoder, and decoder subnetworks each parameterized by a multi-layer perceptron (MLP). The encoder and decoder MLPs use ReLU nonlinearities. Each consists of 4 layers, with 200 units in each hidden layer and an embedding of the same size as the true position and momentum of the system depicted (2 for mass-spring and pendulum, 8 for two-body, and 12 for 3-body). The HNN MLP uses tanh nonlinearities and consists of two hidden layers with 200 units and a one-dimensional output.
+
+To ensure the difference in performance between the PixelHNN and HGN are not due primarily to architectural choices, we also compare to a variant of the PixelHNN architecture using the same convolutional encoder and decoder as used in HGN. We used identical hyperparameters to those described in section A.1. We map between the convolutional latent space used by the encoder and decoder and the vector-valued latent required by the HNN using one additional linear layer for the encoder and decoder.
+
+In the original paper, the PixelHNN model is trained using full-batch gradient descent. To make it more comparable to our approach, we train it here using stochastic gradient descent using minibatches of size 64 and around 15000 training steps. As in the original paper, we train the model using the Adam optimizer (Kingma & Ba, 2014) and a learning rate of 1e-3. As in the original paper, we produce rollouts of the model using a Runge-Kutta integrator (RK4). See Section A.6 for a description of RK4. Note that, as in the original paper, we use the more sophisticated algorithm implemented in scipy (scipy.integrate.solve_ivp) (Jones et al., 2001).
+
+# A.4 DATASETS
+
+The datasets for the experiments described in 4 were generated in a similar manner to Greydanus et al. (2019) for comparative purposes. All of the datasets simulate the exact Hamiltonian dynamics of the underlying differential equation using the default scipy initial value problem solver Jones et al. (2001). After creating a dataset of trajectories for each system, we render those into a sequence of images. The system depicted in each dataset can be visualized by rendering circular objects:
+
+- For the mass-spring the mass object is rendered as a circle and the spring and pivot are invisible.  
+- For the pendulum only the weight (the bob) is rendered as a circle, while the massless rod and pivot are invisible.  
+- For the two and three body problem we render each point mass as a circle in a different color.
+
+Additionally, we smooth out the circles such that they do not have hard edges, as can be seen in Fig. 7.
+
+# A.5 CONVERGENCE OF HGN AND HNN
+
+In order to obtain the results presented in Tbl. 1 we trained both HGN and HNN for 15000 iterations, with batch size of 16 for HGN, and 64 for the HNN. Given the dataset sizes, this means that HGN was trained for around 5 epochs and HNN was trained for around 19 epochs, which took around 16 hours. Figs. 11-12 plot the convergence rates for HGN (leapfrog) and HNN (conv) on the four datasets.
+
+![](images/fc181d0ecfd3a16c1a8d628785fad1711c6b7375355b44a45e31b6594b5ed91d.jpg)
+
+![](images/ba3c415c95f084ae531c8e7a581c177707bea69f7c925daeb73b0e3b60a79164.jpg)
+
+![](images/96102014f3e3041ae695806071ea1655c556d36d890c081144f6f65d847f51e3.jpg)  
+Figure 11: Average MSE for pixel reconstructions during training of HGN (leapfrog). The x axis indicates the training iteration number  $1e + 2$ .
+
+![](images/1b837d321a2b283b8af0ed03d1f40d77d2554b7b2f2b1653bf94684204f53aa3.jpg)
+
+# A.6 INTEGRATORS
+
+Throughout this paper, we estimate the future state of systems from inferred values of the system position and momentum by numerically integrated the Hamiltonian. We explore three methods of numerical integration: (i) Euler integration, (ii) Runge-Kutta integration and (iii) leapfrog integration.
+
+Euler integration estimates the value of a function at time  $t + dt$  by incrementing the function's value with the value accumulated by the function's derivative, assuming it stays constant in the interval  $[t, t + dt]$ . In the Hamiltonian framework, Euler integration takes the form:
+
+$$
+\boldsymbol {q} _ {t + d t} = \boldsymbol {q} _ {t} + d t \frac {\partial \mathcal {H}}{\partial \boldsymbol {p}} \Big | _ {\boldsymbol {p} = \boldsymbol {p} _ {t}} \tag {16}
+$$
+
+$$
+\boldsymbol {p} _ {t + d t} = \boldsymbol {p} _ {t} - d t \frac {\partial \mathcal {H}}{\partial \boldsymbol {q}} \Bigg | _ {\boldsymbol {q} = \boldsymbol {q} _ {t}} \tag {17}
+$$
+
+Because Euler integration estimates a function's future value by extrapolating along its first derivative, the method ignores the contribution of higher-order derivatives to the function's change in time. Accordingly, while Euler integration can reasonably estimate a function's value over short periods, its errors accumulate rapidly as it is integrated over longer periods or when it is applied multiple times. This limitation motivates the use of methods that are stable over more steps and longer integration times.
+
+One such method is four-step Runge-Kutta integration (RK4), the most widely used member of the Runge-Kutta family of integrators. Whereas Euler integration estimates the value of a function at time  $t + dt$  using only the function's derivative at time  $t$ , RK4 accumulates multiple estimates of the function's value in the interval  $[t, t + dt]$ . This integral more correctly reflects the behavior of the function in the interval, resulting in a more stable estimate of the function's value.
+
+RK4 estimates the state at time  $t + dt$  as:
+
+![](images/41a5f1a19b8669ec834b914d368c2b63ec20ac0aa905c84fd4515d90d1f4cd52.jpg)
+
+![](images/a53389c2f9ce07adacef1ed609e8f53c47a131ab60f454627448f7be86915a44.jpg)
+
+![](images/d4bedc2b00869b6cabb705d1a14e850027ccc794ead858a0f32596faefba521b.jpg)  
+Figure 12: Average MSE for pixel reconstructions during training of HNN (conv). The x axis indicates the training iteration number  $1e + 2$ .
+
+![](images/6745126af27a6b41326ef4df2002444b0180634202319247a9e7940b9dfb2f6c.jpg)
+
+$$
+\boldsymbol {x} _ {t + d t} = \boldsymbol {x} _ {t} + d t \left(\frac {1}{6} \boldsymbol {k} _ {1} + \frac {1}{3} \boldsymbol {k} _ {2} + \frac {1}{3} \boldsymbol {k} _ {3} + \frac {1}{6} \boldsymbol {k} _ {4}\right), \tag {18}
+$$
+
+where, for compactness, we write the full system state as  $\boldsymbol{x}_t = [\boldsymbol{q}_t, \boldsymbol{p}_t]$  and the Hamiltonian as  $\mathcal{H}(\boldsymbol{x})$ . In the Hamiltonian framework,  $\boldsymbol{k}_1, \boldsymbol{k}_2, \boldsymbol{k}_3, \boldsymbol{k}_4$  are obtained by evaluating the derivative at four points in the interval  $[t, t + dt]$ :
+
+$$
+\boldsymbol {k} _ {1} = \frac {d \mathcal {H} (\boldsymbol {x})}{d t} \Bigg | _ {\boldsymbol {x} = \boldsymbol {x} _ {t}} \tag {19}
+$$
+
+$$
+\boldsymbol {k} _ {2} = \frac {d \mathcal {H} (\boldsymbol {x})}{d t} \Bigg | _ {\boldsymbol {x} = \boldsymbol {x} _ {t} + \frac {d t}{2} \boldsymbol {k} _ {1}} \tag {20}
+$$
+
+$$
+\boldsymbol {k} _ {3} = \frac {d \mathcal {H} (\boldsymbol {x})}{d t} \Bigg | _ {\boldsymbol {x} = \boldsymbol {x} _ {t} + \frac {d t}{2} \boldsymbol {k} _ {2}} \tag {21}
+$$
+
+$$
+\boldsymbol {k} _ {4} = \frac {d \mathcal {H} (\boldsymbol {x})}{d t} \Bigg | _ {\boldsymbol {x} = \boldsymbol {x} _ {t} + d t \cdot \boldsymbol {k} _ {3}} \tag {22}
+$$
+
+While RK4 may produce reasonably stable estimates over short periods of time, it is not guaranteed to behave stably indefinitely. Neither RK4 nor Euler integration is guaranteed to preserve the energy of the system being integrated, and in practice both will produce estimates that drift away from the true system dynamics over timescales that are relevant for simulating real systems.
+
+Fortunately, there are well-known methods for numerical integration that preserve energy and can be applied to Hamiltonian systems, like the one we propose here. One such method is leapfrog integration, which is a special method for integrating differential equations of the form:
+
+![](images/3863d0798e83af3f4339fa01ddb3d0153910e1a36c0e139c8a3264ee2f54de83.jpg)
+
+![](images/52cfa8533137ae864dc0e6a6c04b73ca74332d433bf1e394eb0623df70c21899.jpg)
+
+![](images/61f2a6b3837406f2e5d41ae1acae37409d34e04db3fe14835ce90e4557bb10f1.jpg)  
+Figure 13: A: example of using a symplectic (leapfrog) and a non-symplectic (Euler) integrators on the Hamiltonian of a harmonic oscillator. The blue quadrilaterals depict a volume in phase space over the course of integration. While the symplectic integrator conserves the volume of this region, but the non-symplectic integrator causes it to increase in volume with each integration step. The symplectic integrator clearly introduces less divergence in the phase space than the non-symplectic alternative over the same integration window. B: an illustration of the leapfrog updates in the phase space, where  $\mathbf{q}$  is position and  $\mathbf{p}$  is momentum.
+
+$$
+\frac {d y}{d t} = F (x), \quad \frac {d x}{d t} = y. \tag {23}
+$$
+
+If we assume that the Hamiltonian equations take this form, we can integrate them using the leapfrog integrator, which in essence updates the position and momentum variables at interleaved time points in a way that resembles the updates "leapfrogging" over each other (see Fig. 13B for an illustration).
+
+In particular, the following updates can be applied to a Hamiltonian of the form  $\mathcal{H} = V(\pmb{q}) + T(\pmb{p})$ , where  $V$  is the potential energy and  $T$  is the kinetic energy of the system:
+
+$$
+\boldsymbol {p} _ {t + d t \frac {1}{2}} = \boldsymbol {p} _ {t} - \left. \frac {d t}{2} \frac {\partial V}{\partial \boldsymbol {q}} \right| _ {\boldsymbol {q} = \boldsymbol {q} _ {t}} \tag {24}
+$$
+
+$$
+\boldsymbol {q} _ {t + d t} = \boldsymbol {q} _ {t} + d t \boldsymbol {p} _ {t + \frac {d t}{2}}. \tag {25}
+$$
+
+As discussed above, leapfrog integration is more stable and accurate over long rollouts than integrators like Euler or RK4. This is because the leapfrog integrator is an example of a symplectic integrator, which means it is guaranteed to preserve the special form of the Hamiltonian even after repeated application. An example visual comparison between a symplectic (leapfrog) and non-symplectic (Euler) integrator applied over the Hamiltonian for a harmonic oscillator is shown in Fig. 13A. For a more thorough discussion of the properties of leapfrog integration, see (Neal et al., 2011).
+
+# A.7 PROOF OF THE HAMILTONIAN FLOW ELBO
+
+The Hamiltonian Flow consists of two components. Firstly, it defines a density model over the joint space  $s_{T} = (q_{T},p_{T})$  using the Hamiltonian Flow as described in section 3.3. However, we assume that the observable variable represents only  $q_{T}$  and treat  $p_{T}$  as a latent variable which we have to marginalize over. Since the integral is intractable using the introduced variational distribution we can derive the lower bound on the marginal likelihood:
+
+$$
+\begin{array}{l} \ln p (\boldsymbol {q} _ {T}) = \ln \int p (\boldsymbol {q} _ {T}, \boldsymbol {p} _ {T}) d \boldsymbol {p} _ {T} \\ = \ln \int \frac {p (\boldsymbol {q} _ {T} , \boldsymbol {p} _ {T})}{f _ {\psi} (\boldsymbol {p} _ {T} | \boldsymbol {q} _ {T})} f _ {\psi} (\boldsymbol {p} _ {T} | \boldsymbol {q} _ {T}) d \boldsymbol {p} _ {T} \\ = \ln \mathbb {E} _ {f _ {\psi} (\boldsymbol {p} _ {T} | \boldsymbol {q} _ {T})} [ \frac {p (\boldsymbol {q} _ {T} , \boldsymbol {p} _ {T}) d \boldsymbol {p} _ {T}}{f _ {\psi} (\boldsymbol {p} _ {T} | \boldsymbol {q} _ {T})} ] \\ \geq \mathbb {E} _ {f _ {\psi} (\boldsymbol {p} _ {T} | \boldsymbol {q} _ {T})} [ \ln \frac {p (\boldsymbol {q} _ {T} , \boldsymbol {p} _ {T}) d \boldsymbol {p} _ {T}}{f _ {\psi} (\boldsymbol {p} _ {T} | \boldsymbol {q} _ {T})} ] \\ = \mathbb {E} _ {f _ {\psi} (\boldsymbol {p} _ {T} | \boldsymbol {q} _ {T})} [ \ln \pi (\mathcal {H} _ {1} ^ {- d t} \circ \dots \circ \mathcal {H} _ {T} ^ {- d t} (\boldsymbol {q} _ {T}, \boldsymbol {p} _ {T})) - \ln f _ {\psi} (\boldsymbol {p} _ {T} | \boldsymbol {q} _ {T}) ] \\ = \operatorname {E L B O} \left(\boldsymbol {q} _ {T}\right) \\ \end{array}
+$$
\ No newline at end of file
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+# HARNESSING THE POWER OF INFINITELY WIDE DEEP NETS ON SMALL-DATA TASKS
+
+Sanjeev Arora
+
+Princeton University
+
+arora@cs.princeton.edu
+
+Simon S. Du
+
+Institute for Advanced Study  
+ssdu@ias.edu
+
+Zhiyuan Li
+
+Princeton University
+
+zhiyuanli@cs.princeton.edu
+
+Ruslan Salakhutdinov
+
+Carnegie Mellon University
+
+rsalakhu@cs.cmu.edu
+
+Ruosong Wang
+
+Carnegie Mellon University
+
+ruosongw@andrew.cmu.edu
+
+Dingli Yu
+
+Princeton University
+
+dingliy@cs.princeton.edu
+
+# ABSTRACT
+
+Recent research shows that the following two models are equivalent: (a) infinitely wide neural networks (NNs) trained under  $\ell_2$  loss by gradient descent with infinitesimally small learning rate (b) kernel regression with respect to so-called Neural Tangent Kernels (NTKs) (Jacot et al., 2018). An efficient algorithm to compute the NTK, as well as its convolutional counterparts, appears in Arora et al. (2019a), which allowed studying performance of infinitely wide nets on datasets like CIFAR-10. However, super-quadratic running time of kernel methods makes them best suited for small-data tasks. We report results suggesting neural tangent kernels perform strongly on low-data tasks.
+
+1. On a standard testbed of classification/regression tasks from the UCI database, NTK SVM beats the previous gold standard, Random Forests (RF), and also the corresponding finite nets.  
+2. On CIFAR-10 with 10 - 640 training samples, Convolutional NTK consistently beats ResNet-34 by  $1\% - 3\%$ .  
+3. On VOC07 testbed for few-shot image classification tasks on ImageNet with transfer learning (Goyal et al., 2019), replacing the linear SVM currently used with a Convolutional NTK SVM consistently improves performance.  
+4. Comparing the performance of NTK with the finite-width net it was derived from, NTK behavior starts at lower net widths than suggested by theoretical analysis(Arora et al., 2019a). NTK's efficacy may trace to lower variance of output.
+
+# 1 INTRODUCTION
+
+Modern neural networks (NNs) have way more parameters than training data points, which allow them to achieve near-zero training error while simultaneously — for some reason yet to be understood — have low generalization error (Zhang et al., 2016). This motivated formal study of highly overparametrized networks, including networks whose width (i.e., number of nodes in layers, or number of channels in convolutional layers) goes to infinity. A recent line of theoretical results shows that with  $\ell_2$  loss and infinitesimal learning rate, in the limit of infinite width the trajectory of training converges to kernel regression with a particular kernel, neural tangent kernel (NTK) (Jacot et al., 2018). For convolutional networks, the kernel is CNTK. See Section 2 for more discussions. Arora et al. (2019a) gave an algorithm to exactly compute the kernel corresponding to the infinite limit of various realistic NN architectures with convolutions and pooling layers, allowing them to compute performance on CIFAR-10, which revealed that the infinite networks have 6 to  $8\%$  higher error than their finite counterparts. This is still fairly good performance for a fixed kernel.
+
+Ironically, while the above-mentioned analysis, at first sight, appears to reduce the study of a complicated model — deep networks — to an older, simpler model — kernel regression — in practice the simpler model is computationally less efficient because running time of kernel regression can be quadratic in the number of data points! Thus computing using CNTK kernel on large datasets like ImageNet currently appears infeasible. Even on CIFAR-10, it seems infeasible to incorporate data augmentation.
+
+However, kernel classifiers are very efficient on small datasets. Here NTKs could conceivably be practical while at the same time bringing some of the power of deep networks to these settings. We recall that recently Olson et al. (2018) showed that multilayer neural networks can be reasonably effective on small datasets, specifically on a UCI testbed of tasks with as few as dozens of training examples. Of course, this required some hyperparameter tuning, although they noted that such tuning is also needed for the champion method, Random Forests (RF), which multilayer neural networks could not beat.
+
+It is thus natural to check if NTK — corresponding to infinitely wide fully-connected networks — performs well in such small-data tasks $^2$ . Convex objectives arising from kernels have stable solvers with minimal hyperparameter tuning. Furthermore, random initialization in deep network training seems to lead to higher variance in the output, which can hurt performance in small-data settings. Can NTK's do better? Below we will see that in the setup of Olson et al. (2018), NTK predictors indeed outperforms corresponding finite deep networks, and also slightly beats the earlier gold standard, Random Forests. This suggests NTK predictors should belong in any list of off-the-shelf machine learning methods.
+
+Following are low-data settings where we used NTKs and CNTKs:
+
+- In the testbed of 90 classification tasks from UCI database, NTK predictor achieves superior, and arguably the strongest classification performance. This is verified via several standard statistical tests, including Friedman Rank, Average Accuracy, Percentage of the Maximum Accuracy (PMA) and probability of achieving  $90\% / 95\%$  maximum accuracy (P90 and P95), performed to compare performances of different classifiers on 90 datasets from UCI database. (The authors plan to release the code, to allow off-the-shelf use of this method. It does not require GPUs.)  
+- We find the performance of NN is close to that of NTK. On every dataset from UCI database, the difference between the classification accuracy of NN and that of NTK is within  $5\%$ . On the other hand, on some datasets, the difference between classification accuracy of NN (or NTK) and that of other classifiers like RF can be as high as  $20\%$ . This indicates in low-data settings, NTK is indeed a good description of NN. Furthermore, we find NTK is more stable (smaller variance), which seems to help it achieve better accuracy on small datasets (cf. Figure 2b).  
+- CNTK is useful in computer vision tasks with small-data. On CIFAR-10, we compare CNTK with ResNet using 10 - 640 training samples and find CNTK can beat ResNet by  $1\% - 3\%$ . We further study few-shot image classification task on VOC07 dataset. The standard method is to first use a pre-trained network, e.g., ResNet-50 trained on ImageNet, to extract features and then directly apply a linear classifier on the extracted features (Goyal et al., 2019). Here we replace the linear classifier with CNTK and obtain better classification accuracy in various setups.
+
+Paper organization. Section 2 discusses related work. Section 3 reviews the derivation of NTK. Section 4 presents experiments using NN and NTK on UCI datasets. Section 5 presents experiments using CNN and CNTK on small CIFAR-10 datasets. Section 6 presents experiments using CNTK for the few-shot learning setting. Additional technical details are presented in appendix.
+
+# 2 RELATED WORK
+
+Our paper is inspired by Fernandez-Delgado et al. (2014) which conducted extensive experiments on UCI dataset. Their conclusion is random forest performs the best, which is followed by the SVM with Gaussian kernel. Therefore, RF may be considered as a reference ("gold-standard") to compare
+
+with new classifiers. Olson et al. (2018) followed this testing strategy to evaluate the performance of modern neural networks concluding that modern neural networks, even though being highly overparameterized, still give reasonable performances on these small datasets, though not as strong as RFs. Our paper follows the same testing strategy to evaluate the performance of NTK.
+
+The focus of this paper, neural tangent kernel is induced from a neural network architecture. The connection between infinitely wide neural networks and kernel methods is not new (Neal, 1996; Williams, 1997; Roux & Bengio, 2007; Hazan & Jaakkola, 2015; Lee et al., 2018; Matthews et al., 2018; Novak et al., 2019; Garriga-Alonso et al., 2019; Cho & Saul, 2009; Daniely et al., 2016; Daniely, 2017). However, these kernels correspond to neural network where only the last layer is trained. Neural tangent kernel, first proposed by Jacot et al. (2018), is fundamentally different as NTKs correspond to infinitely wide NNs with all layer being trained. Theoretically, a line of work study the optimization and generalization behavior of ultra-wide NNs (Allen-Zhu et al., 2018b;a; Arora et al., 2019b; Du et al., 2018b;a; Li & Liang, 2018; Zou et al., 2018; Yang, 2019). Recently, Arora et al. (2019a) gave non-asymptotic perturbation bound between the NN predictor trained by gradient descent and the NTK predictor. Empirically, Lee et al. (2019) verified on small scale data, NTK is a good approximation to NN. However, Arora et al. (2019a) showed on large scale dataset, NN can outperform NTK which may due to the effect of finite-width and/or optimization procedure.
+
+Generalization to architectures other than fully-connected NN and CNN are recently proposed (Yang, 2019; Du et al., 2019; Bietti & Mairal, 2019). Du et al. (2019) showed graph neural tangent kernel (GNTK) can achieve better performance than its counter part, graph neural network (GNN), on datasets with up to 5000 samples.
+
+# 3 NEURAL NETWORK AND NEURAL TANGENT KERNEL
+
+Since NTK is induced by a NN architecture, we first define a NN formally. Let  $x \in \mathbb{R}^d$  be the input, and denote  $g^{(0)}(x) = x$  and  $d_0 = d$  for notational convenience. We define an  $L$ -hidden-layer fully-connected neural network recursively:
+
+$$
+f ^ {(h)} (x) = W ^ {(h)} g ^ {(h - 1)} (x) \in \mathbb {R} ^ {d _ {h}}, \quad g ^ {(h)} (x) = \sqrt {\frac {c _ {\sigma}}{d _ {h}}} \sigma \left(f ^ {(h)} (x)\right) \in \mathbb {R} ^ {d _ {h}}, \quad h = 1, 2, \dots , L, \tag {1}
+$$
+
+where  $W^{(h)}\in \mathbb{R}^{d_h\times d_{h - 1}}$  is the weight matrix in the  $h$ -th layer  $(h\in [L])$ ,  $\sigma :\mathbb{R}\to \mathbb{R}$  is a coordinatewise activation function, and  $c_{\sigma}$  is a scaling factor. In this paper, for NN we will consider  $\sigma$  being ReLU or ELU (Clevert et al., 2015) and for NTK we will only consider kernel functions induced by NNs with ReLU activation. The last layer of the neural network is
+
+$$
+\begin{array}{l} f (w, x) = f ^ {(L + 1)} (x) = W ^ {(L + 1)} \cdot g ^ {(L)} (x) \\ = W ^ {(L + 1)} \cdot \sqrt {\frac {c _ {\sigma}}{d _ {L}}} \sigma \left(W ^ {(L)} \cdot \sqrt {\frac {c _ {\sigma}}{d _ {L - 1}}} \sigma \left(W ^ {(L - 1)} \dots \sqrt {\frac {c _ {\sigma}}{d _ {1}}} \sigma \left(W ^ {(1)} x\right)\right)\right), \tag {2} \\ \end{array}
+$$
+
+where  $W^{(L + 1)} \in \mathbb{R}^{1 \times d_L}$  is the weights in the final layer, and we let  $w = (W^{(1)}, \ldots, W^{(L + 1)})$  be all parameters in the network. All the weights are initialized to be i.i.d.  $\mathcal{N}(0,1)$  random variables. From now on, by NTK initialization we mean a neural network with parameterization defined in Equation 2 with all weighted being initialized to be i.i.d.  $\mathcal{N}(0,1)$ .
+
+When the hidden widths  $d_{1}, d_{2}, \ldots, d_{L} \to \infty$ , certain limiting behavior emerges along the gradient trajectory. Let  $x, x' \in \mathbb{R}^{d}$  be two data points, the covariance kernel of the  $h$ -th layer's outputs,  $\Sigma^{(h)}(x, x') = f^{(h)}(x) \cdot f^{(h)}(x')$ , can be recursively defined in an analytical form:
+
+$$
+\Sigma^ {(0)} (x, x ^ {\prime}) = x ^ {\top} x ^ {\prime},
+$$
+
+$$
+\Lambda^ {(h)} (x, x ^ {\prime}) = \left( \begin{array}{c c} \Sigma^ {(h - 1)} (x, x) & \Sigma^ {(h - 1)} (x, x ^ {\prime}) \\ \Sigma^ {(h - 1)} \left(x ^ {\prime}, x\right) & \Sigma^ {(h - 1)} \left(x ^ {\prime}, x ^ {\prime}\right) \end{array} \right) \in \mathbb {R} ^ {2 \times 2}, \tag {3}
+$$
+
+$$
+\Sigma^ {(h)} (x, x ^ {\prime}) = c _ {\sigma} \mathbb {E} _ {(u, v) \sim \mathcal {N} \left(0, \Lambda^ {(h)}\right)} [ \sigma (u) \sigma (v) ],
+$$
+
+<table><tr><td>Classifier</td><td>Friedman Rank</td><td>Average Accuracy</td><td>P90</td><td>P95</td><td>PMA</td></tr><tr><td>NTK</td><td>28.34</td><td>81.95% ±14.10%</td><td>88.89%</td><td>72.22%</td><td>95.72% ±5.17%</td></tr><tr><td>NN (He init)</td><td>40.97</td><td>80.88%±14.96%</td><td>81.11%</td><td>65.56%</td><td>94.34% ±7.22%</td></tr><tr><td>NN (NTK init)</td><td>38.06</td><td>81.02%±14.47%</td><td>85.56%</td><td>60.00%</td><td>94.55% ±5.89%</td></tr><tr><td>RF</td><td>33.51</td><td>81.56% ±13.90%</td><td>85.56%</td><td>67.78%</td><td>95.25% ±5.30%</td></tr><tr><td>Gaussian Kernel</td><td>35.76</td><td>81.03% ± 15.09%</td><td>85.56%</td><td>72.22%</td><td>94.56% ±8.22%</td></tr><tr><td>Polynomial Kernel</td><td>38.44</td><td>78.21% ± 20.30%</td><td>80.00%</td><td>62.22%</td><td>91.29% ±18.05%</td></tr></table>
+
+Table 1: Comparisons of different classifiers on 90 UCI datasets. P90/P95: the number of datasets a classifier achieves  $90\% /95\%$  or more of the maximum accuracy, divided by the total number of datasets. PMA: average percentage of the maximum accuracy.
+
+for  $h \in [L]$ . Crucially, this analytical form holds not only at the initialization, but also holds during the training (when gradient descent with small learning rate is used as the optimization routine).
+
+Formally, NTK is defined as the limiting gradient kernel
+
+$$
+\Theta \left(x, x ^ {\prime}\right) \triangleq \left\langle \frac {\partial f (w , x)}{\partial w}, \frac {\partial f (w , x ^ {\prime})}{\partial w} \right\rangle = \sum_ {h = 1} ^ {L + 1} \left\langle \frac {\partial f (w , x)}{\partial W ^ {(h)}}, \frac {\partial f (w , x ^ {\prime})}{\partial W ^ {(h)}} \right\rangle .
+$$
+
+Again, one can obtain a recursive formula
+
+$$
+\dot {\Sigma} ^ {(h)} (x, x ^ {\prime}) = c _ {\sigma} \mathbb {E} _ {(u, v) \sim \mathcal {N} \left(0, \Lambda^ {(h)}\right)} [ \dot {\sigma} (u) \dot {\sigma} (v) ], h = 1, \dots , L + 1 \tag {4}
+$$
+
+$$
+\Theta (x, x ^ {\prime}) = \sum_ {h = 1} ^ {L + 1} \left(\Sigma^ {(h - 1)} \left(x, x ^ {\prime}\right) \cdot \prod_ {h ^ {\prime} = h} ^ {L + 1} \dot {\Sigma} ^ {\left(h ^ {\prime}\right)} \left(x, x ^ {\prime}\right)\right), \tag {5}
+$$
+
+where we let  $\dot{\Sigma}^{(L + 1)}(x,x^{\prime}) = 1$  for convenience. It is easy to check if we fix the first  $L^{\prime}$  layers and only train the remaining  $(L + 1 - L^{\prime})$  layers, then the resulting NTK is  $\Theta (x,x^{\prime}) = \sum_{h = L^{\prime} + 1}^{L + 1}\left(\Sigma^{(h - 1)}(x,x^{\prime})\cdot \prod_{h^{\prime} = h}^{L + 1}\dot{\Sigma}^{(h^{\prime})}(x,x^{\prime})\right)$ . Note when  $L^{\prime} = L$ , then the resulting NTK is  $\Sigma^{(L)}(x,x^{\prime})$ , which is the NNGP kernel (Lee et al., 2018).  $L^{\prime}$  can be viewed as a hyperparameter of NTK classifier, and in our UCI experiment we tune  $L^{\prime}$ . Given a kernel function, one can directly use it for downstream classification tasks (Scholkopf & Smola, 2001).
+
+# 4 EXPERIMENTS ON UCI DATASETS
+
+In this section, we present our experimental results on UCI datasets which follow the setup of Fernández-Delgado et al. (2014) with extensive comparisons of classifiers, including random forest, kernel SVM, multilayer neural networks, etc. Section 4.2 discusses the performance of NTK through detailed comparisons with other classifiers tested by Fernández-Delgado et al. (2014). Section 4.2 compares NTK classifier and the corresponding NN classifier and verifies how similar their predictions are. See Table 6 in Appendix A for a summary of datasets we used. The detailed experiment setup, including the choices the datasets, training / test splitting and ranges of hyperparameters, is described in Appendix A. We note that usual methods of obtaining confidence bounds in these low-data settings are somewhat heuristic.
+
+# 4.1 OVERALL PERFORMANCE COMPARISONS
+
+Table 1 lists the performance of 6 classifiers under various metrics: the three top classifiers identified in Fernandez-Delgado et al. (2014), namely, RF, Gaussian kernel and polynomial kernel, along with our new methods NTK, NN with He initialization and NN with NTK initialization. Table 1 shows NTK is the best classifier under all metrics, followed by RF, the best classifier identified in Fernandez-Delgado et al. (2014). Now we interpret each metric in more details.
+
+Friedman Ranking and Average Accuracy. NTK is the best (Friedman Rank 28.34, Average Accuracy  $81.95\%$ ), followed by RF (Friedman Rank 33.51, Average Accuracy  $81.56\%$ ) and then followed by SVM with Gaussian kernel (Friedman Rank 35.76, Average Accuracy  $81.03\%$ ). The difference between NTK and RF is significant (-5.17 in Friedman Rank and +0.39% in Average Accuracy), just as the superiority of RF is significant compared to other classifiers as claimed in Fernandez-Delgado et al. (2014). NN (with either He initialization or NTK initialization) performs significantly better than the polynomial kernel in terms of the Average Accuracy (80.88% and 81.02% vs. 78.21%), but in terms of Friedman Rank, NN with He initialization is worse than polynomial kernel (40.97 vs. 38.44) and NN with NTK initialization is slightly better than polynomial kernel (38.06 vs. 38.44). On many datasets where most classifiers have similar performances, NN's rank is high as well, whereas on other datasets, NN is significantly better than most classifiers, including SVM with polynomial kernel. Therefore, NN enjoys higher Average Accuracy but suffers higher Friedman Rank. For example, on OZONE dataset, NN with NTK initialization is only 0.25% worse than polynomial kernel but their difference in terms of rank is 56. It is also interesting to see that NN with NTK initialization performs better than NN with He initialization (38.06 vs. 40.97 in Friedman Rank and 81.02% vs. 80.88% in Average Accuracy).
+
+P90/P95 and PMA. This measures, for a given classifier, the fraction of datasets on which it achieves more than  $90\% / 95\%$  of the maximum accuracy among all classifiers. NTK is one of the best classifiers (ties with Gaussian kernel on P95), which shows NTK can consistently achieve superior classification performance across a broad range of datasets. Lastly, we consider the Percentage of the Maximum Accuracy (PMA). NTK achieves the best average PMA followed by RF whose PMA is  $0.47\%$  below that of NTK and other classifiers are all below  $94.6\%$ . An interesting observation is that NTK, NN with NTK initialization and RF have small standard deviation  $(5.17\%, 5.89\%$  and  $5.30\%)$  whereas all other classifiers have much larger standard deviation.
+
+# 4.2 PAIRWISE COMPARISONS
+
+NTK vs. RF. In Figure 1a, we compare NTK with RF. There are 42 datasets that NTK outperforms RF and 40 datasets that RF outperforms NTK. $^{4}$  The mean difference is  $2.51\%$ , which is statistically significant by a Wilcoxon signed rank test. We see NTK and RF perform similarly when the Bayes error rate is low with NTK being slightly better. There are some exceptions. For example, on the BALANCE-SCALE dataset, NTK achieves  $98\%$  accuracy whereas RF only achieves  $84.1\%$ . When the Bayes error rate is high, either classifier can be significantly better than the other.
+
+NTK vs. Gaussian Kernel. The gap between NTK and Gaussian kernel is more significant. As shown in Figure 1b, NTK generally performs better than Gaussian kernel no matter Bayes error rate is low or high. There are 43 datasets that NTK outperforms Gaussian kernel and there are 34 datasets that Gaussian kernel outperforms NTK. The mean difference is  $2.22\%$ , which is also statistically significant by a Wilcoxon signed rank test. On BALLOONS, HEART-SWITZERLAND, PITTSBURG-BRIDGES-MATERIAL, TEACHING, TRAINS datasets, NTK is better than Gaussian kernel by at least  $11\%$  in terms of accuracy. These five datasets all have less than 200 samples, which shows that NTK can perform much better than Gaussian kernel when the number of samples is small.
+
+NTK vs. NN In this section we compare NTK with NN. The goals are (i) comparing the performance and (ii) verifying NTK is a good approximation to NN.
+
+In Figure 2a, we compare the performance between NTK and NN with He initialization. For most datasets, these two classifiers perform similarly. However, there are a few datasets that NTK performs significantly better. There are 50 datasets that NTK outperforms NN with He initialization and there are 27 datasets that NN with He initialization outperforms NTK. The mean difference is  $1.96\%$ , which is also statistically significant by a Wilcoxon signed rank test.
+
+In Figure 2b, we compare the performance between NTK and NN with NTK initialization. Recall that for NN with NTK initialization, when the width goes to infinity, the predictor is just NTK.
+
+![](images/ac2c0e5e8403617755c9fea923497840154dc4e01f889955a4e30072120ee38b.jpg)  
+(a) RF vs. NTK
+
+![](images/7abc5d133f8386f4e989ec4006a0a696056df004a128055c0ddff11f97bd8116.jpg)  
+(b) Gaussian Kernel vs. NTK
+
+![](images/0b5ef9707a0965526d0f6b01b95d4da10632823ce0a2b8195e569ce888db8912.jpg)  
+(a) NN w. He init vs. NTK  
+Figure 2: Performance Comparisons between NTK and NN.
+
+![](images/995b3193d6a4b9721e4111b455fcae236c177d3c841004b870ea57f0f16ab5c0.jpg)  
+Figure 1: Performance comparisons between NTK and other classifiers.  
+(b) NN w. NTK init vs. NTK
+
+Therefore, we expect these two predictors give similar performance. Figure 2b verifies our conjecture. There is no dataset the one classifier is significantly better than the other. We do not have the same observation in Figure 1a, 1b or 2a. Nevertheless, NTK often performs better than NN. There are 52 datasets that NTK outperforms NN with He initialization and there are 25 datasets NN with He initialization outperforms NTK. The mean difference is  $1.54\%$ , which is also statistically significant by a Wilcoxon signed rank test.
+
+# 5 EXPERIMENTS ON SMALL CIFAR-10 DATASET
+
+In this section, we study the performance of CNTK on subsampled CIFAR-10 dataset. We randomly choose  $n$  samples from CIFAR-10 training set, use them to train CNTK and ResNet-34, and test both classifiers on the whole test set. In our experiments, we vary  $n$  from 10 to 1280, and the number of convolutional layers of CNTK varies from 5 to 14. See Appendix B for detailed experiment setup.
+
+The results are reported in Table 2. It can be observed that in this setting CNTK consistently outperforms ResNet-34. The largest gap occurs at  $n = 320$  where 14-layer CNTK achieves  $36.57\%$  accuracy and ResNet achieves  $33.15\%$ . The smallest improvement occurs at  $n = 10$ :  $15.33\%$  vs.
+
+<table><tr><td>n</td><td>ResNet</td><td>5-layer CNTK</td><td>8-layer CNTK</td><td>11-layer CNTK</td><td>14-layer CNTK</td></tr><tr><td>10</td><td>14.59% ± 1.99%</td><td>15.08% ± 2.43%</td><td>15.24% ± 2.44%</td><td>15.31% ± 2.38%</td><td>15.33% ± 2.43%</td></tr><tr><td>20</td><td>17.50% ± 2.47%</td><td>18.03% ± 1.91%</td><td>18.50% ± 2.03%</td><td>18.69% ± 2.07%</td><td>18.79% ± 2.13%</td></tr><tr><td>40</td><td>19.52% ± 1.39%</td><td>20.83% ± 1.68%</td><td>21.07% ± 1.80%</td><td>21.23% ± 1.86%</td><td>21.34% ± 1.91%</td></tr><tr><td>80</td><td>23.32% ± 1.61%</td><td>24.82% ± 1.75%</td><td>25.18% ± 1.80%</td><td>25.40% ± 1.84%</td><td>25.48% ± 1.91%</td></tr><tr><td>160</td><td>28.30% ± 1.38%</td><td>29.63% ± 1.13%</td><td>30.17% ± 1.11%</td><td>30.46% ± 1.15%</td><td>30.48% ± 1.17%</td></tr><tr><td>320</td><td>33.15% ± 1.20%</td><td>35.26% ± 0.97%</td><td>36.05% ± 0.92%</td><td>36.44% ± 0.91%</td><td>36.57% ± 0.88%</td></tr><tr><td>640</td><td>41.66% ± 1.09%</td><td>41.24% ± 0.78%</td><td>42.10% ± 0.74%</td><td>42.44% ± 0.72%</td><td>42.63% ± 0.68%</td></tr><tr><td>1280</td><td>49.14% ± 1.31%</td><td>47.21% ± 0.49%</td><td>48.22% ± 0.49%</td><td>48.67% ± 0.57%</td><td>48.86% ± 0.68%</td></tr></table>
+
+Table 2: Performance of ResNet-34 and CNTK on small CIFAR-10 Dataset.  
+
+<table><tr><td>k</td><td>linear SVM</td><td>1-layer CNTK</td><td>2-layer CNTK</td><td>3-layer CNTK</td><td>4-layer CNTK</td><td>5-layer CNTK</td><td>6-layer CNTK</td></tr><tr><td>1</td><td>49.46 ± 4.71</td><td>50.04 ± 4.28</td><td>49.94 ± 4.19</td><td>49.66 ± 4.19</td><td>49.24 ± 4.20</td><td>48.66 ± 4.26</td><td>48.00 ± 4.41</td></tr><tr><td>2</td><td>64.39 ± 5.02</td><td>65.81 ± 5.28</td><td>65.89 ± 5.23</td><td>65.65 ± 5.17</td><td>65.30 ± 5.10</td><td>64.76 ± 5.07</td><td>64.12 ± 5.00</td></tr><tr><td>3</td><td>70.38 ± 3.28</td><td>71.60 ± 3.51</td><td>71.56 ± 3.65</td><td>71.32 ± 3.74</td><td>70.96 ± 3.84</td><td>70.51 ± 3.92</td><td>69.98 ± 3.95</td></tr><tr><td>4</td><td>72.43 ± 3.21</td><td>73.65 ± 3.29</td><td>73.59 ± 3.28</td><td>73.35 ± 3.23</td><td>73.00 ± 3.20</td><td>72.56 ± 3.20</td><td>72.09 ± 3.20</td></tr><tr><td>5</td><td>74.49 ± 3.08</td><td>75.87 ± 2.88</td><td>75.69 ± 2.89</td><td>75.45 ± 2.89</td><td>75.13 ± 2.85</td><td>74.68 ± 2.88</td><td>74.24 ± 2.85</td></tr><tr><td>6</td><td>76.59 ± 2.15</td><td>77.64 ± 2.37</td><td>77.57 ± 2.43</td><td>77.39 ± 2.50</td><td>77.13 ± 2.56</td><td>76.87 ± 2.60</td><td>76.52 ± 2.63</td></tr><tr><td>7</td><td>77.11 ± 1.25</td><td>78.64 ± 1.35</td><td>78.57 ± 1.39</td><td>78.39 ± 1.41</td><td>78.12 ± 1.42</td><td>77.80 ± 1.41</td><td>77.43 ± 1.41</td></tr><tr><td>8</td><td>79.57 ± 0.87</td><td>80.42 ± 0.88</td><td>80.38 ± 0.86</td><td>80.23 ± 0.86</td><td>79.99 ± 0.88</td><td>79.72 ± 0.89</td><td>79.41 ± 0.92</td></tr></table>
+
+14.59%. When the number of training data is large, i.e.,  $n = 1280$ , ResNet can outperform CNTK. It is also interesting to see that 14-layer CNTK is the best performing CNTK for all values of  $n$ , suggesting depth has a significant effect on this task.
+
+# 6 EXPERIMENTS ON FEW-SHOT LEARNING
+
+In this section, we test the ability of NTK as a drop-in classifier to replace the linear classifier in the few-shot learning setting. Using linear SVM on top of extracted features is arguably the most widely used strategy in few-shot learning as linear SVM is easy and fast to train whereas more complicated strategies like fine-tuning or training a neural network on top of the extracted feature have more randomness and may overfit due to the small size of the training set. NTK has the same benefits (easy and fast to train, no randomness) as the linear classifier but also allows some non-linearity in the design. Note since we consider image classification tasks, we use CNTK in experiments in this section.
+
+Experiment Setup. We mostly follow the settings of Goyal et al. (2019). Features are extracted from layer conv1, conv2, conv3, conv4, conv5 in ResNet-50 (He et al., 2016) trained on ImageNet (Deng et al., 2009), and we use these features for VOC07 classification task. The number of positive examples  $k$  varies from 1 to 8. For each value of  $k$ , for each class in the VOC07 dataset, we choose  $k$  positive examples and  $19k$  negative examples. For each  $k \in \{1,2,\dots,8\}$  and for each class, we randomly choose 10 independent sets with  $20k$  training samples in each set. We
+
+Table 3: Performance of linear SVM and CNTK with different number of convolutional layers on the 11-20th classes in VOC07. The cost value  $C$  is tuned on the first 10 classes. Feature extracted from conv5 in ResNet-50.  
+
+<table><tr><td>k</td><td>linear SVM</td><td>1-layer CNTK</td><td>2-layer CNTK</td><td>3-layer CNTK</td><td>4-layer CNTK</td><td>5-layer CNTK</td><td>6-layer CNTK</td></tr><tr><td>1</td><td>23.25 ± 2.89</td><td>24.54 ± 3.39</td><td>24.57 ± 3.46</td><td>24.57 ± 3.50</td><td>24.52 ± 3.51</td><td>24.43 ± 3.50</td><td>24.34 ± 3.49</td></tr><tr><td>2</td><td>32.26 ± 3.85</td><td>33.81 ± 4.25</td><td>33.77 ± 4.33</td><td>33.68 ± 4.40</td><td>33.53 ± 4.46</td><td>33.31 ± 4.51</td><td>33.04 ± 4.55</td></tr><tr><td>3</td><td>38.41 ± 2.74</td><td>40.06 ± 2.70</td><td>40.10 ± 2.47</td><td>40.02 ± 2.41</td><td>39.87 ± 2.37</td><td>39.65 ± 2.35</td><td>39.36 ± 2.29</td></tr><tr><td>4</td><td>39.88 ± 2.08</td><td>42.20 ± 1.78</td><td>41.80 ± 2.13</td><td>42.64 ± 2.00</td><td>42.51 ± 2.02</td><td>42.32 ± 2.03</td><td>42.08 ± 2.03</td></tr><tr><td>5</td><td>42.46 ± 2.70</td><td>44.71 ± 3.12</td><td>44.69 ± 2.98</td><td>44.72 ± 3.01</td><td>44.69 ± 3.04</td><td>44.57 ± 3.07</td><td>44.40 ± 3.10</td></tr><tr><td>6</td><td>45.71 ± 3.27</td><td>49.02 ± 2.63</td><td>48.63 ± 2.69</td><td>48.97 ± 2.68</td><td>48.92 ± 2.71</td><td>48.79 ± 2.74</td><td>48.63 ± 2.77</td></tr><tr><td>7</td><td>47.97 ± 3.25</td><td>50.89 ± 3.50</td><td>50.43 ± 3.48</td><td>50.50 ± 3.47</td><td>50.48 ± 3.44</td><td>50.39 ± 3.44</td><td>50.24 ± 3.41</td></tr><tr><td>8</td><td>49.81 ± 2.18</td><td>52.32 ± 2.65</td><td>52.06 ± 2.32</td><td>52.23 ± 2.38</td><td>52.31 ± 2.40</td><td>52.26 ± 2.45</td><td>52.14 ± 2.50</td></tr></table>
+
+Table 4: Performance of linear SVM and CNTK with different number of convolutional layers on the 11-20th classes in VOC07. The cost value  $C$  is tuned on the first 10 classes. Feature extracted from conv4 in ResNet-50.
+
+<table><tr><td>k</td><td>linear SVM</td><td>1-layer CNTK</td><td>2-layer CNTK</td><td>3-layer CNTK</td><td>4-layer CNTK</td><td>5-layer CNTK</td><td>6-layer CNTK</td></tr><tr><td>1</td><td>14.38 ± 1.98</td><td>15.42 ± 1.90</td><td>15.36 ± 1.93</td><td>15.30 ± 1.97</td><td>15.24 ± 2.02</td><td>15.17 ± 2.05</td><td>15.10 ± 2.08</td></tr><tr><td>2</td><td>16.67 ± 1.50</td><td>18.48 ± 1.58</td><td>18.39 ± 1.55</td><td>18.28 ± 1.52</td><td>18.18 ± 1.49</td><td>18.09 ± 1.47</td><td>17.99 ± 1.45</td></tr><tr><td>3</td><td>19.56 ± 1.04</td><td>21.79 ± 1.69</td><td>21.73 ± 1.63</td><td>21.67 ± 1.60</td><td>21.61 ± 1.55</td><td>21.54 ± 1.50</td><td>21.47 ± 1.46</td></tr><tr><td>4</td><td>20.53 ± 1.62</td><td>23.39 ± 2.13</td><td>23.36 ± 2.18</td><td>23.29 ± 2.22</td><td>23.20 ± 2.24</td><td>23.11 ± 2.26</td><td>23.03 ± 2.27</td></tr><tr><td>5</td><td>21.51 ± 1.68</td><td>25.13 ± 2.36</td><td>25.09 ± 2.37</td><td>25.03 ± 2.36</td><td>24.96 ± 2.36</td><td>24.87 ± 2.36</td><td>24.78 ± 2.37</td></tr><tr><td>6</td><td>23.51 ± 2.39</td><td>26.51 ± 2.39</td><td>26.26 ± 2.43</td><td>26.10 ± 2.47</td><td>25.97 ± 2.47</td><td>25.87 ± 2.47</td><td>25.78 ± 2.46</td></tr><tr><td>7</td><td>24.33 ± 1.59</td><td>27.98 ± 2.46</td><td>27.75 ± 2.52</td><td>28.24 ± 2.22</td><td>28.20 ± 2.22</td><td>28.14 ± 2.21</td><td>28.08 ± 2.21</td></tr><tr><td>8</td><td>25.31 ± 2.07</td><td>27.76 ± 2.87</td><td>28.52 ± 2.65</td><td>28.52 ± 2.69</td><td>28.49 ± 2.70</td><td>28.44 ± 2.72</td><td>28.38 ± 2.74</td></tr></table>
+
+Table 5: Performance of linear SVM and CNTK with different number of convolutional layers on the 11-20th classes in VOC07. The cost value  $C$  is tuned on the first 10 classes. Feature extracted from conv3. in ResNet-50
+
+report the mean and standard deviation of mAP on the test split of VOC07 dataset. This setting has been used in numbers of previous few-shot learning papers (Goyal et al., 2019; Zhang et al., 2017).
+
+We take the extracted features as the input to CNTK. We tried CNTK with 0-6 convolution layers, a global average pooling layer and a fully-connected layer. We normalize the data in the feature space, and finally use SVM to train the classifiers. Note without the convolution layer, it is equivalent to directly applying linear SVM after global average pooling and normalization. We use sklearn.svm.LinearSVC to train linear SVMs, and sklearn.svm.SVC to train kernel SVMs (for CNTK). To train SVM, we choose the cost value  $C$  from  $2^{[-19,-4]} \cup 10^{[-7,2]}$  and set the class_weight ratio to be  $2:1$  for positive/negative classes as in Goyal et al. (2019).
+
+Since the number of given samples is usually small, Goyal et al. (2019) chooses to report the performance of the best cost value  $C$ . In our experiments, we use a more standard method to perform cross-validation. We use the first 10 classes of VOC07 to tune  $C$ , and report the performance of selected  $C$  in the other 10 classes in Table 3-5. We also report the performance of the best  $C$  as in Goyal et al. (2019), in Tables 7-9 in the appendix for completeness.
+
+**Discussions.** First, we find CNTK is a strong drop-in classifier to replace the linear classifier in few-shot learning. Tables 3-9 clearly demonstrate that CNTK is consistently better than linear classifier. Note Tables 3-9 only show the prediction accuracy in an average sense. In fact, we find that on every randomly sampled training set, CNTK always gives a better performance. We conjecture that CNTK gives better performance because the non-linearity in CNTK helps prediction. This is verified by looking at the performance gain of CNTK for different feature extractors. Note Conv5 is often considered to be most useful for linear classification. There, CNTK only outperforms the linear classifier by about  $1\%$ . On the other hand, with Conv3 and Conv4 features, CNTK can outperform linear classifier by around  $2\%$  and sometimes by  $3\%-4\%$  (last two rows in Table 5). We believe this happens because Conv3 and Conv4 correspond to middle-level features, and thus non-linearity is indeed beneficial for better accuracy.
+
+We also observe that CNTK often performs the best with a single convolutional layer. This is expected since features extracted by ResNet-50 already produce a good representation. Nevertheless, we find for middle-level features Conv3 with  $k = 7$  or 8, CNTK with 3 convolutional layers give the best performance. We believe this is because with more data, utilizing the non-linearity induced by CNTK can further boost the performance.
+
+# 7 CONCLUSION
+
+The Neural Tangent Kernel, discovered by mathematical curiosity about deep networks in the limit of infinite width, is found to yield superb performance on low-data tasks, beating extensively tuned versions of classic methods such as random forests. The (fully-connected) NTK classifiers are easy to compute (no GPU required) and thus should be a good off-the-shelf classifier in many settings. We plan to release our code to allow drop-in replacement for SVMs and linear regression.
+
+Many theoretical questions arise. Do NTK SVMs correspond to some infinite net architecture (as NTK ridge regression does)? What explains generalization in small-data settings? (This understanding is imperfect even for random forests.) Finally one can derive NTK corresponding to other
+
+architectures, e.g., recurrent neural tangent kernel (RNTK) induced by recurrent neural networks. It would be an interesting future research direction to test their performance on benchmark tasks.
+
+# ACKNOWLEDGMENTS
+
+S. Arora, Z. Li and D. Yu are supported by NSF, ONR, Simons Foundation, Schmidt Foundation, Amazon Research, DARPA and SRC. S. S. Du is supported by National Science Foundation (Grant No. DMS-1638352) and theInfosys Membership. R. Salakhutdinov and R. Wang are supported in part by NSF IIS-1763562, AFRL CogDeCON FA875018C0014, and DARPA SAGAMORE HR00111990016. Part of this work was done while S. S. Du was visiting Google Brain Princeton and R. Wang was visiting Princeton University. The authors would like to thank Amazon Web Services for providing compute time for the experiments in this paper, and NVIDIA for GPU support. We thank Priya Goyal for providing experiment details of Goyal et al. (2019). We thank Xiaolong Wang for discussing the few-shot learning task.
+
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+
+# A ADDITIONAL EXPERIMENTAL DETAILS ON UCI
+
+In this section, we describe our experiment setup.
+
+Dataset Selection Since we only wish to test on small datasets, we only select UCI datasets with number of samples smaller than 5000. Furthermore, we only use datasets without explicit training / testing splitting from the link: http://persoal.citius.usc.es/manuel.fernandez.delgado/papers/jmlr/data.tar.gz, which are the pre-processed datasets by Fernandez-Delgado et al. (2014). These datasets are originally from UCI datasets (including most of the datasets before March, 2013) and 4 real-world datasets not included in the UCI repository (see Fernandez-Delgado et al. (2014) for details). The datasets are pre-processed to be classification problems and all categorical features are turned into numerical features and normalized along the samples for each feature. The reason to discard datasets with explicit splitting is that we found there is an obvious distributional shift between training and testing data. The pre-processed data provided from the link shows that training and testing data are normalized with different mean and standard deviation. For example, on the AUDIOLOGY-STD dataset, the support of the first feature is  $\{-0.761526, 1.30547\}$  in training and  $\{-1.2, 0.8\}$  in testing; on the ANNEALING dataset, the support of the first feature is  $\{-0.670274, -0.223425, 3.79822\}$  in training and  $\{-0.111111, 0, 1\}$  in testing. See Table 6 for a summary of the datasets.
+
+<table><tr><td></td><td># samples</td><td># features</td><td># classes</td></tr><tr><td>min</td><td>10</td><td>3</td><td>2</td></tr><tr><td>25%</td><td>178</td><td>8</td><td>2</td></tr><tr><td>50%</td><td>583</td><td>16</td><td>3</td></tr><tr><td>75%</td><td>1022</td><td>32</td><td>6</td></tr><tr><td>max</td><td>5000</td><td>262</td><td>100</td></tr></table>
+
+Table 6: Dataset Summary
+
+Performance Comparison Details We follow the comparison setup in Fernandez-Delgado et al. (2014) that we report 4-fold cross-validation. For hyperparameters, we tune them with the same validation methodology in Fernandez-Delgado et al. (2014): all available training samples are randomly split into one training and one test set, while imposing that each class has the same number of training and test samples. Then the parameter with best validation accuracy is selected. It is possible to give confidence bounds for this parameter tuning scheme, but they are worse than standard ones for separated training/validation/testing data. For NTK and NN classifiers we train them on these 90 datasets. For other classifiers, we use the results from Fernandez-Delgado et al. (2014).
+
+NTK Specification We calculate NTK induced fully-connected neural networks with  $L$  layers where  $L'$  bottom layers are fixed, and then use  $C$ -support vector classification implemented by sklearnnn.svm. We tune hyperparameters  $L$  from 1 to 5,  $L'$  from 0 to  $L - 1$ , and cost value  $C$  as powers of ten from -2 to 4. The number of kernels used is 15, so the total number of parameter combinations is 105. Note this number is much less than the number of hyperparameter of Gaussian Kernel reported in Fernández-Delgado et al. (2014) where they tune hyperparameters with 500 combinations.
+
+NN Specification We use fully-connected NN with  $L$  layers, 512 number of hidden nodes per layer and use gradient descent to train the neural network. We tune hyperparameters  $L$  from 1 to 5, with / without batch normalization and learning rate 0.1 or 1. We run gradient descent for 2000 epochs. We treat NN with He initialization and NTK initialization as two classifiers and report their results separately.
+
+# B ADDITIONAL EXPERIMENTAL DETAILS ON SMALL CIFAR-10 DATASETS
+
+We randomly choose  $n / 10$  samples from each class of CIFAR-10 training set and test classifiers on the whole testing set.  $n$  varies from 10 to 1280. For each  $n$ , we repeat 20 times and report the mean accuracy and its standard deviation for each classifier.
+
+The number of convolution layers of CNTK ranges from 5-14. After convolutional layers, we apply a global pooling layer and a fully connected layer. We refer readers to Arora et al. (2019a) for exact formulas of CNTK. We normalize the kernel such that each sample has unit length in feature space.
+
+We use ResNet-34 with width 64,128,256 and default hyperparameters: learning rate 0.1, momentum 0.9, weight decay 0.0005. We decay the learning rate by 10 at the epoch of 80 and 120, with 160 training epochs in total. The training batch size is the minimum of the size of the whole training dataset and 160. We report the best testing accuracy among epochs.
+
+# C ADDITIONAL RESULTS IN FEW-SHOT LEARNING
+
+Tables 7-9 show the performance of the best  $C$  as has been done in Goyal et al. (2019).  
+
+<table><tr><td>k</td><td>linear SVM</td><td>1-layer CNTK</td><td>2-layer CNTK</td><td>3-layer CNTK</td><td>4-layer CNTK</td><td>5-layer CNTK</td><td>6-layer CNTK</td></tr><tr><td>1</td><td>52.63 ± 5.35</td><td>53.16 ± 4.96</td><td>53.08 ± 4.89</td><td>52.83 ± 4.85</td><td>52.44 ± 4.85</td><td>51.88 ± 4.85</td><td>51.21 ± 4.88</td></tr><tr><td>2</td><td>65.08 ± 2.69</td><td>66.25 ± 3.09</td><td>66.29 ± 3.09</td><td>66.11 ± 3.07</td><td>65.79 ± 3.04</td><td>65.33 ± 3.00</td><td>64.77 ± 2.99</td></tr><tr><td>3</td><td>71.78 ± 1.85</td><td>72.76 ± 1.76</td><td>72.72 ± 1.73</td><td>72.49 ± 1.71</td><td>72.14 ± 1.73</td><td>71.71 ± 1.76</td><td>71.17 ± 1.78</td></tr><tr><td>4</td><td>74.40 ± 1.86</td><td>75.37 ± 1.95</td><td>75.33 ± 1.95</td><td>75.14 ± 1.96</td><td>74.85 ± 1.96</td><td>74.48 ± 1.99</td><td>74.05 ± 1.97</td></tr><tr><td>5</td><td>75.94 ± 2.51</td><td>77.01 ± 2.39</td><td>76.91 ± 2.39</td><td>76.72 ± 2.39</td><td>76.47 ± 2.37</td><td>76.13 ± 2.39</td><td>75.76 ± 2.39</td></tr><tr><td>6</td><td>76.39 ± 1.27</td><td>77.15 ± 1.47</td><td>77.14 ± 1.50</td><td>77.02 ± 1.55</td><td>76.83 ± 1.58</td><td>76.62 ± 1.60</td><td>76.35 ± 1.61</td></tr><tr><td>7</td><td>78.18 ± 0.86</td><td>79.42 ± 0.85</td><td>79.36 ± 0.87</td><td>79.22 ± 0.88</td><td>79.00 ± 0.91</td><td>78.75 ± 0.93</td><td>78.50 ± 0.94</td></tr><tr><td>8</td><td>79.78 ± 0.65</td><td>80.47 ± 0.59</td><td>80.45 ± 0.61</td><td>80.33 ± 0.63</td><td>80.14 ± 0.65</td><td>79.92 ± 0.67</td><td>79.66 ± 0.70</td></tr></table>
+
+Table 7: Performance of linear SVM and CNTK with different number of convolutional layers on all classes in VOC07 with the best  $C$ . Feature extracted from conv5 in ResNet-50  
+
+<table><tr><td>k</td><td>linear SVM</td><td>1-layer CNTK</td><td>2-layer CNTK</td><td>3-layer CNTK</td><td>4-layer CNTK</td><td>5-layer CNTK</td><td>6-layer CNTK</td></tr><tr><td>1</td><td>24.18 ± 2.64</td><td>25.15 ± 2.93</td><td>25.13 ± 3.00</td><td>25.06 ± 3.05</td><td>24.95 ± 3.10</td><td>24.80 ± 3.13</td><td>24.60 ± 3.14</td></tr><tr><td>2</td><td>31.59 ± 2.64</td><td>32.88 ± 2.91</td><td>32.81 ± 2.94</td><td>32.67 ± 2.96</td><td>32.48 ± 2.98</td><td>32.24 ± 2.99</td><td>31.94 ± 3.01</td></tr><tr><td>3</td><td>37.34 ± 2.18</td><td>38.69 ± 2.39</td><td>38.64 ± 2.26</td><td>38.53 ± 2.28</td><td>38.36 ± 2.31</td><td>38.12 ± 2.33</td><td>37.83 ± 2.35</td></tr><tr><td>4</td><td>39.67 ± 2.31</td><td>41.34 ± 2.33</td><td>41.39 ± 2.28</td><td>41.33 ± 2.24</td><td>41.21 ± 2.22</td><td>41.02 ± 2.20</td><td>40.77 ± 2.18</td></tr><tr><td>5</td><td>41.74 ± 1.20</td><td>43.48 ± 1.40</td><td>43.48 ± 1.24</td><td>43.44 ± 1.25</td><td>43.32 ± 1.27</td><td>43.12 ± 1.30</td><td>42.89 ± 1.33</td></tr><tr><td>6</td><td>44.68 ± 1.84</td><td>46.77 ± 1.82</td><td>46.42 ± 1.91</td><td>46.44 ± 1.90</td><td>46.39 ± 1.91</td><td>46.25 ± 1.93</td><td>46.08 ± 1.94</td></tr><tr><td>7</td><td>47.71 ± 1.84</td><td>50.07 ± 1.97</td><td>49.74 ± 1.95</td><td>49.63 ± 1.81</td><td>49.56 ± 1.81</td><td>49.41 ± 1.82</td><td>49.19 ± 1.81</td></tr><tr><td>8</td><td>48.91 ± 1.90</td><td>51.32 ± 1.97</td><td>50.98 ± 1.87</td><td>51.09 ± 1.84</td><td>51.09 ± 1.81</td><td>51.00 ± 1.78</td><td>50.84 ± 1.76</td></tr></table>
+
+Table 8: Performance of linear SVM and CNTK with different number of convolutional layers on all classes in VOC07 with the best  $C$ . Feature extracted from conv4 in ResNet-50  
+
+<table><tr><td>k</td><td>linear SVM</td><td>1-layer CNTK</td><td>2-layer CNTK</td><td>3-layer CNTK</td><td>4-layer CNTK</td><td>5-layer CNTK</td><td>6-layer CNTK</td></tr><tr><td>1</td><td>14.72 ± 1.49</td><td>15.01 ± 1.27</td><td>14.96 ± 1.28</td><td>14.92 ± 1.30</td><td>14.87 ± 1.32</td><td>14.81 ± 1.34</td><td>14.75 ± 1.36</td></tr><tr><td>2</td><td>16.67 ± 1.04</td><td>17.52 ± 1.05</td><td>17.45 ± 1.03</td><td>17.36 ± 1.02</td><td>17.28 ± 1.00</td><td>17.20 ± 0.99</td><td>17.11 ± 0.97</td></tr><tr><td>3</td><td>19.21 ± 1.31</td><td>20.65 ± 1.68</td><td>20.63 ± 1.65</td><td>20.58 ± 1.63</td><td>20.53 ± 1.61</td><td>20.47 ± 1.59</td><td>20.41 ± 1.57</td></tr><tr><td>4</td><td>20.87 ± 1.49</td><td>22.28 ± 1.80</td><td>22.27 ± 1.81</td><td>22.22 ± 1.83</td><td>22.16 ± 1.83</td><td>22.09 ± 1.83</td><td>22.03 ± 1.82</td></tr><tr><td>5</td><td>21.98 ± 1.17</td><td>23.45 ± 1.14</td><td>23.45 ± 1.12</td><td>23.43 ± 1.08</td><td>23.38 ± 1.07</td><td>23.32 ± 1.05</td><td>23.25 ± 1.04</td></tr><tr><td>6</td><td>23.02 ± 1.56</td><td>24.44 ± 1.61</td><td>24.29 ± 1.66</td><td>24.27 ± 1.82</td><td>24.25 ± 1.81</td><td>24.21 ± 1.80</td><td>24.18 ± 1.78</td></tr><tr><td>7</td><td>24.38 ± 1.52</td><td>25.96 ± 1.80</td><td>25.97 ± 1.70</td><td>25.98 ± 1.70</td><td>25.96 ± 1.69</td><td>25.92 ± 1.67</td><td>25.88 ± 1.66</td></tr><tr><td>8</td><td>25.28 ± 1.39</td><td>27.06 ± 1.61</td><td>27.15 ± 1.58</td><td>27.18 ± 1.56</td><td>27.17 ± 1.53</td><td>27.13 ± 1.51</td><td>27.09 ± 1.50</td></tr></table>
+
+Table 9: Performance of linear SVM and CNTK with different number of convolutional layers on all classes in VOC07 with the best  $C$ . Feature extracted from conv3 in ResNet-50
\ No newline at end of file
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+# HOPPITY: LEARNING GRAPH TRANSFORMATIONS TO DETECT AND FIX BUGS IN PROGRAMS
+
+Elizabeth Dinella*
+
+University of Pennsylvania
+
+Hanjun Dai*
+
+Google Brain
+
+Ziyang Li
+
+University of Pennsylvania
+
+Mayur Naik
+
+University of Pennsylvania
+
+Le Song
+
+Georgia Tech
+
+Ke Wang
+
+Visa Research
+
+# ABSTRACT
+
+We present a learning-based approach to detect and fix a broad range of bugs in Javascript programs. We frame the problem in terms of learning a sequence of graph transformations: given a buggy program modeled by a graph structure, our model makes a sequence of predictions including the position of bug nodes and corresponding graph edits to produce a fix. Unlike previous works built upon deep neural networks, our approach targets bugs that are more diverse and complex in nature (i.e. bugs that require adding or deleting statements to fix). We have realized our approach in a tool called HOPPITY. By training on 290,715 Javascript code change commits on Github, HOPPITY correctly detects and fixes bugs in 9,490 out of 36,361 programs in an end-to-end fashion. Given the bug location and type of the fix, HOPPITY also outperforms the baseline approach by a wide margin.
+
+# 1 INTRODUCTION
+
+The sheer size and complexity of modern codebases makes it impossible for them to be bug-free. As a result, a more reasonable and effective strategy has emerged, which aims to prevent bugs in production by applying automated tools to detect and even fix them early in the development process.
+
+This trend has gained increasing popularity in recent years. Examples include Google's Tricorder (Sadowski et al., 2015), Facebook's Getafix (Scott et al., 2019) and Zoncolan, and Microsoft's Visual Studio IntelliCode. The techniques underlying these tools can be classified into broadly two categories: logical, rule-based techniques (Sadowski et al., 2015) and statistical, data-driven techniques (Allamanis et al., 2018; Pradel & Sen, 2018; Vasic et al., 2019). The former uses manually written rules capturing undesirable code patterns and scans the entire codebase for these classes of bugs. The latter learns to detect abnormal code from a large code corpus using deep neural networks. Despite great strides, however, both kinds of tools are limited in generality because they target error patterns in specific codebases or they target specific bug types. For instance, Zoncolan's rules are designed to be specifically applicable to Facebook's codebases, and deep learning models target specialized bugs in variable naming (Allamanis et al., 2018) or binary expressions (Pradel & Sen, 2018). Moreover, the patterns are relatively syntactic, allowing them to be specified by human experts using logical rules or learnt from a corpus of programs.
+
+In this paper, we propose a novel learning-based approach for finding and fixing bugs in Javascript programs automatically. Javascript is a scripting language designed for web application development. It has been the most popular programming language on GitHub since 2014 (Github, 2019). Repairing Javascript code presents a unique challenge as bugs manifest in diverse forms due to unusual language features and the lack of tooling support. Therefore, the primary goal of our approach is generality since it must be effective against a board spectrum of programming errors, such as using wrong operators or identifiers, accessing undefined properties, mishandling variable scopes, triggering type incompatibilities, among many others. Another important novel aspect concerns our approach's ability to deal with bugs that are more complex and semantic in nature, namely, bugs that require adding or removing statements from a program, which are not considered by prior works. Finally, compared to automated program repair techniques (Le Goues et al., 2019; Scott et al., 2019; Hua et al., 2018; Chen
+
+et al., 2018) which require knowledge of bug location, this paper presents an end-to-end approach including localizing bugs, predicting the types of fixes, and generating patches.
+
+We design our model architecture in a similar vein as a Neural Turing Machine (NTM) (Graves et al., 2014). It consists of an external memory (a Graph Neural Network) for embedding a buggy program and a central controller (an LSTM) that makes a sequence of primitive actions (e.g., predicting type, generating patch, etc.) to perform a fix. The multi-step decision process is implemented by an autoregressive model. Crucially, our model differs from the standard NTM in how the memory is manipulated: apart from the common read and write operations, the controller can also expand or shrink the memory when adding or deleting nodes in the original graph.
+
+We have realized our approach in a tool called HOPPITY. By training on 290,715 Javascript code change commits collected from Github, HOPPITY correctly detects and fixes bugs in 9,490 out of 36,361 programs using a beam size of three.
+
+# 2 MOTIVATING EXAMPLES
+
+Javascript is quite different compared to traditional object-oriented languages (e.g. C++, Java, or C#). In addition to the weak, dynamic typing discipline of scripting languages, Javascript supports many peculiar features that do not exist in other languages. For example, it allows a property (i.e., a field) to be added to or removed from an object at runtime. As another example, Javascript did not support block-level scoping until recently, allowing a variable defined in a block structure such as a for loop to be exposed to the entire function in which the loop occurs. While the latest ES6 language standard incorporates block-level scoping, developers have been programming without it for decades, resulting in a large body of legacy code. Finally, Javascript's eval function, which interprets and executes a string as a code fragment, is widely regarded as a major source of bugs and vulnerabilities. All of these aspects make programming in Javascript a frustrating and error-prone experience.
+
+```javascript
+function clearEmployeeListOnLinkClick(){ document.querySelector("a").addEventListener("click", function(event){ document.querySelector("ul").InnerHTML  $\equiv$  ""; ）；   
+}
+```
+
+(a) InnerHTML should have been innerHTML.
+
+```javascript
+if (matches) { return { episode: Number(matches_groupsepisode), hosts: matches_groups.hosts.split(/([,&]  $^+$  |\sand\s))/. map(el  $\Rightarrow$  S(el).trim().s) };   
+}
+```
+
+(b) Highlighted parentheses should have been removed.
+
+```javascript
+module.exports  $=$  function (grant){ grant.initConfig({ execute:{...},copy:{...},checktextdomain:{...} wp_readme_tomarkdown:{...},makepot:{...}) ... grunt.registerTask('default',[wp_readme_tomarkdown' ,'makepot', 'execute', 'checktextdomain']) };
+```
+
+```javascript
+export default {
+  computed:
+    level()
+    return dictMapskillLevel[parseInt((this.value === 0 ? 1 : this.value)/20)];
+}
+```
+
+(c) copy function should have also been included in the highlighted list.
+
+(d) parseInt should have been removed because  $= =$  implies this.value is an integer.
+
+Figure 1: Example programs that illustrate limitations of existing approaches including both rule-based static analyzers and neural-based bug predictors.
+
+Static analyzers aim to detect common coding errors in Javascript programs by applying logical rule-based reasoning on source code. TAJS (Jensen et al., 2009) and ESLint (Zakas, 2013) are prominent examples. These tools face important challenges to be effective. We present several examples in Figure 1 to illustrate their limitations. Due to the complex nature of client-side web APIs, TAJS and ESLint choose to ignore analyzing built-in libraries for the sake of scalability. As depicted in Figure 1a, when developers mistakenly capitalize the first letter of innerHTML, a property of class Element in DOM (Document Object Model), both analyzers fail to catch the error. Javascript will then silently allow developers to set the previously non-existent property InnerHTML to the empty string. Later, when developers attempt to access the intended property, innerHTML, the program will crash and potentially cause a security vulnerability or incur a costly debugging experience. Additionally, static analyzers can never deal with functional bugs (i.e., errors that violate the program specification and yet conform to the coding rules). Figure 1b shows one such example. The goal is to split a string using regular expressions. However, the program incorrectly splits the input ' and ' into ['', ' and ', '] instead of ['', ''], which is what the developer intended. Since
+
+![](images/86d85a45fea30d6dffc8b4574ce7532f72f24a58cf42f310bce78efe5929d1a4.jpg)  
+Figure 2: Code repair as graph transformation. Each step the source code graph is edited via one of the operator module until STOP is triggered by controller.
+
+the error is simply a mismatch between the developer's implicit specification and implementation, static analyzers are incapable of catching it.
+
+The bugs that static analyzers missed in both cases are in hindsight quite obvious to human programmers. The criteria they use is very simple: any code snippet that seems to deviate from common code patterns is likely to be buggy. This is precisely the observation that our approach seeks to mimic. In particular, if a model observes a property or an unusual way of splitting strings that never appeared in the training data, it is likely to recognize those abnormal code fragments as potential bugs. The main advantage of our approach over existing neural-based bug detectors (Allamanis et al., 2018; Pradel & Sen, 2018; Vasic et al., 2019) is its generality. Unlike prior works that target specific classes of bugs (e.g., variable naming issues or binary expression bugs), we train a single model to deal with a wide range of bug types, encompassing all previously proposed ones. Compared to past approaches that leverage a graph-based neural network model (Allamanis et al., 2018), our model is capable of more sophisticated transformations such as adding or removing nodes, as shown in Figure 1c and 1d. Finally, our model not only locates but also fixes bugs, whereas program repair (Le Goues et al., 2019; Scott et al., 2019; Hua et al., 2018; Chen et al., 2018) or bug localization (Ball et al., 2003; Jose & Majumdar, 2011; Wang et al., 2019) techniques only solve a single task.
+
+# 3 MODEL
+
+We model the problem of detecting and repairing bugs in programs as a structured prediction problem on a graph-based representation of programs. Given a graph  $g_{bug}$  that represents a buggy program, we wish to predict a graph  $g_{fix}$  that represents the fixed program. Our model aims to capture the structured prediction by a sequence of up to  $T$  steps of graph transformations:
+
+$$
+p \left(g _ {f i x} \mid g _ {b u g}; \theta\right) = p \left(g _ {1} \mid g _ {b u g}; \theta\right) p \left(g _ {2} \mid g _ {1}; \theta\right) \dots p \left(g _ {f i x} \mid g _ {T - 1}; \theta\right) \tag {1}
+$$
+
+The high-level overview of the graph sequence transformation is shown in Figure 2. Different programs may need a different number of steps  $T(g_{bug})$  which is also determined by the model.
+
+We first introduce our representation module for programs in Sec 3.1. We then elucidate each step of the above transformation in Sec 3.2. Finally, we summarize and present the full model in Sec 3.3.
+
+# 3.1 PROGRAM REPRESENTATION
+
+Programs written in a high-level language have rich structure. Researchers have proposed graph-based representations to capture this structure (Allamanis et al., 2018). We start with this approach of representing programs using graphs with certain modifications for our task.
+
+As shown in the left part of Figure 2, we first parse the program's source code into an abstract syntax tree (AST) form that captures the program's syntactic structure. We then connect the leaf nodes with SuccToken edges. Unlike previous approaches, we additionally add value nodes that store the actual content of the leaf nodes, with special ValueLink edges connecting them together. The purpose of introducing this additional set of nodes is to provide a name-independent strategy for code representation and modification, which we elucidate in the next section. Hereafter, we use  $g_{fix}$ ,  $g_{bug}$  or  $g$  in general to represent either the source code or the corresponding graph structure.
+
+After representing the program as a graph, we use a Graph Neural Network (GNN) (Scarselli et al., 2008) to map the graph into a representation in a fixed dimensional vector space. Specifically, given
+
+![](images/dcb0c14a12904a66e90f4c9d6f99f1fb650ace00e17bb7a85367143f5ed74d56.jpg)  
+Figure 3: Graph edit operators with low-level primitives.
+
+a graph  $g = (V, E)$  with set of nodes  $V$  and edges  $E$ , we need a function  $f(g) \mapsto (\mathbb{R}^d, \mathbb{R}^{|V| \times d})$  to obtain the  $d$ -dimensional representation of graph  $g$  (denoted as  $\vec{g}$ ), as well as representations of individual nodes  $v \in V$  (denoted as  $\vec{v}$ ). To parameterize  $f(\cdot)$ , we employ the form in GIN (Xu et al., 2018), with our adaptation to our multigraph for program representation in the following manner:
+
+$$
+\begin{array}{l} h _ {v} ^ {(l + 1), k} = \sigma \left(\sum_ {u \in \mathcal {N} ^ {k} (v)} \mathbf {W} _ {1} ^ {l, k} h _ {u} ^ {(l)}\right), \forall k \in \{1, 2, \dots , K \} \\ h _ {v} ^ {(l + 1)} = \sigma \left(\mathbf {W} _ {2} ^ {l} \left[ h _ {v} ^ {(l + 1), 1}, h _ {v} ^ {(l + 1), 2}, \dots , h _ {v} ^ {(l + 1), K} \right] + h _ {v} ^ {(l)}\right) \tag {2} \\ \end{array}
+$$
+
+where  $\mathbf{W}_1^{l,k} \in \mathbb{R}^{d \times d}$ ,  $\mathbf{W}_2^l \in \mathbb{R}^{dK \times d}$  are model parameters and  $\sigma(\cdot)$  is  $\tanh$  in this implementation.  $K$  is the total number of edge types in this multi-graph representation. In the end, the node embedding is  $\vec{v} = h_v^{(L)}$ , where  $L$  is the total number of propagations in the GNN.  $\mathcal{N}^k(v)$  is the set of neighbors of node  $v$  that are connected by edge with type  $k$ . Following GIN, the graph representation  $\vec{g}$  is the aggregation of  $h_v^l, \forall l \in [0,1,\dots,L]$ . We use max pooling to aggregate  $h_v^l$  for each  $l$ , and then take the average of these  $L + 1$  vectors to obtain  $\vec{g}$ .
+
+Initially, we use the node type as one-hot features as a starting value for  $h_v^{(0)}$ , where the types are either obtained from the AST representation, or from the local value table as shown in Figure 2. Note that we don't use features like variable names or function names in this graph representation, as different programs may follow different naming conventions. Instead, we focus on the syntactic structure of the source code, so as to enable naming-agnostic representation across different programs.
+
+# 3.2 ONE-STEP GRAPH EDIT
+
+There are five types of operators to choose from for a single step graph edit, namely, adding a node (ADD), deleting a node (DEL), replacing a node value (REP_VAL), replacing a node type (REP_TYPE) and stop (NO_OP). When combined with multi-step edits, these operators suffice to capture a rich variety of code modifications. These operators share some common low-level primitives, such as finding the location, predicting value, etc. We first introduce the individual low-level primitives and then present how to assemble these for each type of graph edit operator.
+
+# 3.2.1 LOW-LEVEL PRIMITIVES
+
+Our low-level primitives contain location, type, and value prediction. These primitives can be combined for different operators later on. In this section, we assume the availability of a controller, represented as  $\vec{c} \in \mathbb{R}^d$ . It keeps track of the global state, including the original source code, as well as the edits made so far. We will elaborate this when we assemble different primitives together.
+
+Location The location primitive locates a specific position in the source code. While it corresponds to region selection in the original text representation, with the graph representation, we can easily treat it as a node selection step. As different programs have different numbers of nodes, we employ a pointer network (Vinyals et al., 2015) into the graph structure. Specifically, after obtaining the node embeddings  $\{\vec{v}\}_{v\in V}$ , we select the node via  $loc(\vec{c},g) = \arg \max_{v\in V}\vec{v}^{\top}\vec{c}$  for simplicity.
+
+Value The value primitive assigns a value for a leaf node in the AST. Instead of predicting the replacement value using a language generative model (Chen et al., 2018) or GNN score function (Allamanis
+
+et al., 2018), we adopt the attention mechanism to let the model to choose from either the values appearing in the current file (local value table), or a collection of global values that are common for the specific language. Let  $D_{val}$  be the global dictionary of commonly used leaf-node values in the language, where each item  $i_v \in D_{val}$  is associated with a vector representation  $\vec{i_v} \in \mathbb{R}^d$ . The local value table is denoted as  $V_{val}$  which is a subset of the nodes in current graph. Then, the value is predicted via  $val(\vec{c}, g) = \operatorname{argmax}_{t \in D_{val} \cup V_{val}} \vec{t}^\top \vec{c}$ . Again we use inner product simply for efficiency, while more expressive score functions can also be used.
+
+Type The type primitive assigns the type for non-terminal nodes in an AST. As the total possible number of types is finite and fixed for a given language, the type prediction is simply a multiclass classification problem. However, we can utilize the AST grammar checker with contextual information to prune the output space. To predict the type of a given non-terminal node, we can obtain its parent node and current children. Then, by looping over the valid production rules at the current location, we can obtain a list of all valid types. The final type is only chosen from this set.
+
+# 3.2.2 GRAPH EDIT OPERATORS
+
+The  $t$ -th round of edit starts with the current graph  $g_{t-1}$ , the corresponding graph embedding, and the 'macro-context' embedding  $\overrightarrow{c_{Mt-1}}$  that captures the edit history so far. Every type of edit operation (excluding NO_OP), requires prediction of the buggy location. So, in each round, the location primitive is invoked to determine the node to target. Then the edit type  $e$  that is feasible at this location  $v$  is predicted out of the five operators. A 'micro-context' embedding  $\overrightarrow{c_{mt}}$  is obtained from the macro embedding updated by two LSTM calls with location node embedding  $\vec{v}$  and operator embedding  $\vec{e}$ . To summarize:
+
+$$
+\overrightarrow {c _ {M _ {t}} ^ {\rightarrow}} = \operatorname {L S T M} (\overrightarrow {g _ {t - 1}} | \overrightarrow {c _ {M _ {t - 1}}}), \overrightarrow {c _ {m _ {t}}} = \operatorname {L S T M} (\overrightarrow {e _ {t}} | \operatorname {L S T M} (\overrightarrow {v _ {t}} | \overrightarrow {c _ {M _ {t}} ^ {\rightarrow}})), \tag {3}
+$$
+
+The micro-context embedding is used as the controller throughout the process of each operator. In the following content, we present these operators in detail.
+
+ADD This operation adds a new node to the graph. Unlike in Li et al. (2018) where the node and corresponding edges are added in separate stages, which would introduce extra complexity, we introduce a simple mechanism that can uniquely add a node and corresponding edges. As is shown in Figure 3, this process invokes one location primitive, one value primitive, and one type primitive. The location primitive invoked before the edit (i.e., node  $v$ ) determines the parent of the node to be added, while the location primitive called during the edit chooses the left sibling of the node. In a special case where the parent node does not have any children, then such left sibling node is set to the parent node itself. With this information, we can uniquely determine the position to insert into the AST. Finally, the corresponding edges—SuccToken, ValueLink, and AST edges—can automatically be inferred with the location of new node to be added.
+
+As this process is autoregressive, the micro-context embedding is kept updated with all the primitive calls. For this specific operator, the context is updated in the order of:  $c_{m1}^{\rightarrow} = \mathrm{LSTM}(v_{sibling}|c_m^{\rightarrow})$ ,  $c_{m2}^{\rightarrow} = \mathrm{LSTM}(val(c_{m1},g)|c_{m1}^{\rightarrow})$  and  $c_{m3}^{\rightarrow} = \mathrm{LSTM}(type(c_{m2},g)|c_{m2}^{\rightarrow})$ . In the end,  $\vec{c}_{ADD} = \vec{c}_{m3}$  summarizes the process.
+
+DEL This operator deletes a node and corresponding edges in the graph. If it is a non-terminal node in the AST, then the corresponding subtree is removed as well. The micro-context embedding is updated by the LSTM via the embedding of the node being deleted.
+
+REP_VAL This operator replaces the value of a leaf (terminal) node in the AST. This procedure requires the prediction the value. The leaf node is linked to the new value node in the internal value table via a ValueLink edge. Also, the micro-context embedding is updated by the LSTM via the embedding of the corresponding node and value.
+
+REP_TYPE This operator changes the type of a non-terminal node, which involves type primitive steps. The micro-context embedding is updated by the LSTM via the embedding of the corresponding node and type.
+
+NO_OP This op does not change the graph. It simply denotes the end of the sequence of graph edits.
+
+# 3.3 GRAPH TRANSFORMATION
+
+Our end-to-end model for graph transformation inference is shown in Alg 1. We denote the buggy graph  $g_{bug}$  as  $g_0$  for simplicity. Then, for the  $t$ -th graph edit, the following steps are performed:
+
+1. Obtain the graph representation  $\overrightarrow{g_{t-1}}$  and node embeddings. Update the macro-context embedding using  $\overrightarrow{g_{t-1}}$ ;  
+2. Choose edit location  $v_{t}$  by performing location primitive and update the context embedding;  
+3. Pick the graph edit operator  $e_t$  that is compatible with  $v_t$ ; Use both  $v_t$  and  $e_t$  to obtain the micro-context embedding.  
+4. Perform the edit, obtain the corresponding micro-context summary  $\overrightarrow{c_{et}}$  and update the macro-context embedding.  
+5. If the edit is not NO_OP, then go back to step 1; otherwise return the graph.
+
+# Algorithm 1 Transformation inference of  $p(g_{fix}|g_{bug})$
+
+1: Input  $g_{bug} \sim \mathcal{D}$  and model parameters  $\theta$ .  
+2: Obtain  $\vec{g_0}$ ,  $\{\vec{v}_{v\in g_{bug}}\} = f_0(g_{bug})$ , let  $c_{M_0}$  be null.  
+3: for  $t = 1$  to  $T$  do  
+4: Obtain  $\overrightarrow{c_{M_t}^{\prime}} = \mathrm{LSTM}(\overrightarrow{g_{t - 1}} |\overrightarrow{c_{M_{t - 1}}})$  
+5: Choose location  $v_{t}$ , then edit type  $e_{t}$  
+6: if  $e_t = \text{NO\_OP}$  then  
+7: set  $g_{T} = g_{t - 1}$  and exit the loop.  
+8: end if  
+9: Perform operator  $e_t$  with  $\overrightarrow{c_{m_t}}$ , obtained by Eq 3.  
+10: Get new graph  $g_{t}$ , update  $\overrightarrow{c_{M_t}}$  with  $\overrightarrow{c_{e_t}}$ .  
+11: end for  
+12: Return  $g_{T}$
+
+This process repeats until it reaches the maximum steps  $T$  or the NO_OP operator is selected. Note that our framework can capture the situation when the input program is bug-free. In this case, the NO_OP operator is supposed to be triggered at the first step. Also, each edit step is not limited to a single node level operation. It can be extended to modify a certain substructure (e.g., replace a tree node with one of its children). This in turn allows program repair to be performed in fewer edit steps.
+
+# 4 LEARNING
+
+Given the dataset  $\mathcal{D} = \{(g_{bug}^{(i)}, g_{fix}^{(i)})\}_{i=1}^{|D|}$  which consists of pairs of buggy code and the fixed code, the learning objective  $\max_{\theta} \mathbb{E}_{(g_{bug}, g_{fix}) \sim \mathcal{D}} p(g_{fix} | g_{bug}; \theta)$  maximizes the likelihood of fixes.
+
+Since the probability is factorized according to Eq 1 where a sequence of transformations is performed, we parse the source code using the SHIFT AST format, and utilize a JSON diff toolbox to compile the code differences into a sequence of AST edits. This serves as the fine-grained supervision mechanism for our graph transformation formulation. Thus, the MLE objective above is realized with the sum of cross entropy loss at each step of graph edits. During training, we jointly optimize the graph representation module  $\{f_t(\cdot)\}_{t=1}^T$ , each of the operator module and the controller module which is parameterized by LSTM. We use the Adam optimizer with  $\beta_1 = 0.9$ ,  $\beta_2 = 0.99$  and initial learning rate of  $10^{-3}$ . Due to the large size of each sample, we use a small batch size of 10 during training. Furthermore, to stabilize the training, we apply the gradient clip with the maximum norm of 5.
+
+# 5 INFERENCE
+
+The inference procedure involves searching for the maximum in the combinatorial space:  $\arg \max_{g_{fix}} p(g_{fix} | g_{bug}; \theta)$ . Since the search space is very large, however, we use beam-search to approximately find the fixes with highest probabilities.
+
+Specifically, we maintain a pool of partially fixed programs  $\{\tilde{g}\}$ , which starts with simply the single buggy program  $g_{bug}$ . The pool size is limited by the beam-search size  $B$ . For each  $\tilde{g}$ , we propose the top  $B$  locations to be modified, top  $B$  operators or top  $B$  primitives (location, type, value), depending on the current stage of the edit  $\tilde{g}$ . Then the total joint one-step graph transformation solutions are ranked together based on the joint log-likelihood, and the top  $B$  solutions with the largest likelihood are kept in the pool for the next round of beam search.
+
+Unlike beam search for language models where the vocabulary size is fixed, in our setting, the available choices or even the steps of inference may vary (e.g., the ADD operator has more steps of primitive calls than the DEL operator). Our implementation is based on PyTorch with customized GPU kernels to enable efficient inference on GPUs.
+
+# 6 EXPERIMENTS
+
+Dataset Our model is trained and evaluated on a corpus of nearly half a million data points. We have created a robust system to continuously collect small changes in Javascript programs from Github. Given a commit, we download the Javascript file before and after the change:  $(src_{buggy}, src_{fixed})$ . Commits can contain many types of changes such as feature additions, refactorings, bug fixes, etc. In
+
+<table><tr><td></td><td>ADD</td><td>REP_TYPE</td><td>REP_VAL</td><td>DEL</td><td>total</td></tr><tr><td>train</td><td>6,473</td><td>1,864</td><td>251,097</td><td>31,281</td><td>290,715</td></tr><tr><td>validate</td><td>790</td><td>245</td><td>31,357</td><td>3,957</td><td>36,349</td></tr><tr><td>test</td><td>796</td><td>233</td><td>31,387</td><td>3,945</td><td>36,361</td></tr></table>
+
+Table 1: Statistic of OneDiff dataset. See appendix for more information of other dataset.  
+
+<table><tr><td rowspan="2"></td><td colspan="2">Total</td><td colspan="2">Location</td><td colspan="2">Operator</td><td colspan="2">Value</td><td colspan="2">Type</td></tr><tr><td>Top-3</td><td>Top-1</td><td>Top-3</td><td>Top-1</td><td>Top-1</td><td>Top-3</td><td>Top-1</td><td>Top-3</td><td>Top-1</td><td></td></tr><tr><td>TOTAL</td><td>26.1</td><td>14.2</td><td>35.5</td><td>20.4</td><td>34.4</td><td>52.3</td><td>29.1</td><td>76.1</td><td>66.7</td><td></td></tr><tr><td>ADD</td><td>52.9</td><td>39.2</td><td>69.6</td><td>51.4</td><td>70.6</td><td>65.7</td><td>55.1</td><td>76.8</td><td>68.5</td><td></td></tr><tr><td>REP_VAL</td><td>23.4</td><td>11.9</td><td>33.3</td><td>18.5</td><td>31.7</td><td>53.0</td><td>28.8</td><td>-</td><td>-</td><td></td></tr><tr><td>REP_TYPE</td><td>71.7</td><td>52.4</td><td>73.0</td><td>52.8</td><td>79.4</td><td>-</td><td>-</td><td>74.7</td><td>61.0</td><td></td></tr><tr><td>DEL</td><td>39.6</td><td>24.8</td><td>44.0</td><td>27.5</td><td>45.8</td><td>-</td><td>-</td><td>-</td><td>-</td><td></td></tr><tr><td>Random</td><td>.08</td><td>.07</td><td>2.28</td><td>1.4</td><td>27.7</td><td>.01</td><td>.01</td><td>.27</td><td>0</td><td></td></tr></table>
+
+Table 2: Evaluation of model on the OneDiff dataset: accuracy  $(\%)$
+
+an attempt to filter our dataset to only include bug fixes, we use a heuristic based on the number of changes to the AST. Our insight is that a commit with a smaller number of AST differences is more likely to be a bug fix than a commit containing large changes. Thus for the experiments, we use three different datasets: OneDiff with precisely one edit; ZeroOneDiff with zero and one edit and ZeroOneTwoDiff with zero, one or two edits. We additionally filter out data points with ASTs larger than 500 nodes as a parameter in our system. A detailed overview of our corpus crawler is available in Appendix B.
+
+# 6.1 EVALUATION
+
+We train the model for 3 epochs on the training set until the validation loss converges. We tried different configurations of our model with different number of layers and different graph embedding methods besides the generic one in Eq 2. We report on these ablation studies in Appendix C.
+
+Table 2 shows the evaluation results of our model on a held out test set consisting of samples from our OneDiff dataset. Additional experiments on ZeroOneDiff and ZeroOneTwoDiff datasets are available in Appendix A. We also provide experimental results with respect to different configurations.
+
+Accuracy is shown for each graph edit operation type. Accuracy is measured in a complete discrete graph edit operation step. For example consider Figure 1a, in which we edit an object property name with the REP_VAL operation. If the model incorrectly predicts the edit operation to be of type DEL, then it will not go on to predict a value. In this case, the model will be penalized twice in the operation accuracy as well as the value accuracy. A prediction is considered totally correct only if the entire sequence of graph edit primitives is correct. Note that top-1 greedy prediction is not always among top-3 when beam search is used. Additionally, operator prediction is only evaluated on the top prediction as the search space only includes four operators.
+
+To demonstrate the magnitude of the search space, we compare HOPPITY to a model that selects uniformly at random, in each step of the graph edit process. The random model performs well at operation type selection since the search space only has four options (ADD, REP_VAL, REP_TYPE, DEL). However, after the operation type is predicted, the random model's accuracy drops, as there are up to 500 nodes in the buggy AST. When it predicts value, the accuracy drops even further as our vocabulary contains 5,000 values. Lastly, type prediction has slightly better accuracy than value prediction because the number of the types of AST nodes in total is smaller than our vocabulary.
+
+# 6.2 BASELINES
+
+As existing approaches cannot be applied for comparison in Table 7, we adapt the baselines to some restricted settings in this section. We report the results on the OneDiff dataset as most of the baselines target repair of a single bug. Note that for all comparisons we provide equal amounts of information to HOPPITY and the baseline without retraining our model.
+
+<table><tr><td>Type</td><td>GGNN-Rep</td><td>GGNN-Cls</td><td>HOPPITY</td></tr><tr><td>Top-1</td><td>53.2%</td><td>99.6%</td><td>90.0%</td></tr><tr><td>Top-3</td><td>85.8%</td><td>99.6%</td><td>94.8%</td></tr></table>
+
+Table 3: REP_TYPE accuracies with location+op
+
+<table><tr><td>Value</td><td>GGNN-Rep</td><td>GGNN-RNN</td><td>HOPPITY</td></tr><tr><td>Top-1</td><td>63.8%</td><td>60.3%</td><td>69.1%</td></tr><tr><td>Top-3</td><td>67.6%</td><td>63.6%</td><td>73.4%</td></tr></table>
+
+Table 4: REP_VAL accuracies with location+op.  
+
+<table><tr><td></td><td>Top-1</td><td>Top-3</td></tr><tr><td>HOPPITY</td><td>67.7%</td><td>73.3%</td></tr><tr><td>SequenceR</td><td>64.2%</td><td>68.6%</td></tr></table>
+
+Table 5: Overall OneDiff accuracy with location.  
+
+<table><tr><td>Bug Type</td><td>Amount</td><td>TAJS</td><td>HOPPITY</td></tr><tr><td>Undefined Property</td><td>7</td><td>0</td><td>1</td></tr><tr><td>Functional Bug</td><td>11</td><td>0</td><td>3</td></tr><tr><td>Refactoring</td><td>12</td><td>0</td><td>1</td></tr><tr><td>Total</td><td>30</td><td>0</td><td>5</td></tr></table>
+
+Table 6: Comparison with TAJS.
+
+GGNN: Allamanis et al. (2018), uses Gated Graph Neural Networks (GGNN) for two specific bug repair tasks: VARMISUSE, in which the model learns to select the correct variable that should be used at a given location, and VARNAMING, in which the model predicts a variable name based on its usage. We adapt these tasks to compare with HOPPITY on the REP_TYPE and REP_VAL tasks. Specifically, for REP_TYPE prediction we have
+
+- GGNN-Rep: we adopt VARMISUSE to replace with candidate node type and modify the graph structure correspondingly; we use their proposed max-margin formulation for training.  
+- GGNN-Cls: we perform multi-class classification using the target node and graph embedding.
+
+For REP_VAL prediction, we also made two versions of adaptations:
+
+- GGNN-Rep: similar to above, here the candidate set is from values in the current graphs plus the top-100 frequent values used for repair in the training set.  
+- GGNN-RNN: we adopt VARNAMING approach to predict value directly. Due to the huge vocabulary size, we use char-level language model for predicting the replacement.
+
+Table 3 and 4 show the comparison when buggy node is known. Regarding the type prediction, as the number of types is large, the likelihood formulation with classification objective outperforms the max-margin loss based one (i.e., GGNN-Rep). As in this limited case GGNN-Cls and HOPPITY are quite similar except for graph representation, the performance is expected to be comparable. As HOPPITY is not trained to predict type fix only, it performs slightly worse than GGNN-Cls. Also for the value prediction, our formulation of pointer on graph is more effective. We found when the space of decisions is large, it is hard to apply structured prediction method like GGNN-Rep in this setting. Since real-world programs are noisy, the sentences used in different programs vary greatly, making it difficult for language models to predict the exact accurate value. A possible extension is to combine the language model with the graph pointer, which we will explore in future work.
+
+SequenceR: The model proposed by Chen et al. (2018) is a translation based model that predicts a fixed sequence of tokens when given a buggy line in the source code. We compare with our model by providing location information to both approaches.
+
+Table 5 summarizes the total accuracy for fixing a single bug. In order to provide a fair comparison, we allow SequenceR to predict the same information as our model (i.e., predict op, value etc.in a sequential way), rather than an entire sequence of raw textual tokens. This experiment shows the benefit of formulating code repair with graphs over text tokens.
+
+With the above two baselines, we can see that in the restricted case our model can still yield comparable or even better performances. Given that our model can go for more edits without location information, we believe this tool is more generic and effective for code repair.
+
+TAJS: We also compare the bug detection ability of HOPPITY against TAJS (Jensen et al., 2009) which is a well-known static analysis tool for Javascript programs. Automating the comparison for our entire test set proved to be infeasible. For example, TAJS only accepts JavaScript ES5 programs, while the vast majority of current JavaScript projects use ES6 or other variants like React JSX. Another problem is that TAJS does not analyze code that is not invoked, e.g., a library function that is not called by client code. Moreover, determining the right command-line options of TAJS is non-trivial since it provides many options targeting different JavaScript runtime environments. Due to these issues, we forgo a large-scale comparison, and instead pick 30 random points in our test set to manually analyze using TAJS. Table 6 depicts the results (Appendix D provides further details).
+
+We restrict the chosen test points to satisfy a necessary condition for undefined property bugs since TAJS claims to be proficient in detecting this class of bugs. In the process, we also pick some functional bugs, as well as cases of refactoring modifications. By resolving the numerous issues that prevented us from automating the comparison, we were able to run TAJS manually. TAJS failed to detect any real bugs in the 30 test points. While functional bugs and refactoring modifications are beyond TAJS, however, TAJS also raises many unrelated false alarms due to its failures in locating NodeJS libraries, importing JSON files, or recognizing built-in global variables. These warnings are detrimental because TAJS suspends the analysis as soon as it detects what it preceives to be a bug. To further aid TAJS, we omitted parts of each program that are unrelated to the bug, in the hope of driving TAJS's analysis as deep as possible. After all these measures, TAJS managed to detect two of the undefined property bugs (Bug IDs 4 and 6 in Appendix D).
+
+In contrast, HOPPITY is able to correctly detect 5 bug locations of the 30 testing points within our top 3 predictions. Moreover, HOPPITY also produces 4 patches that are identical to the developer's fixes. Our comparison highlights HOPPITY's two important strengths compared to TAJS. First, HOPPITY relieves developers from the enormous burden of manual configuration. Second, HOPPITY achieves far better performance in detecting as well as fixing the bugs in Javascript programs.
+
+# 7 RELATED WORK
+
+Static analysis for bug detection. Static analyzers such as FindBugs, Error-Prone, and Semmle use syntactic pattern-matching and dataflow analysis to find common bugs. Typically, detecting even a single class of bugs can require dozens or even hundreds of patterns. Coverity (Bessey et al., 2010), SonarQube, and Clang Static Analyzer check for semantic inconsistencies in code based on more sophisticated path analyses. Infer (Calcagno et al., 2015) is built upon sound principles and can prove the absence of certain classes of bugs. TAJS belongs to this category as well. Due to the undecidability of the problem, however, approximations are inevitable which voids the guarantees in practice. Compared to all static analysis tools, HOPPITY offers the following advantages: (1) it targets a board range of programming errors; (2) it not only localizes bugs but also fixes them; and (3) it has significantly higher signal-to-noise ratio (i.e., detects more bugs with less false alarms).
+
+Learning-based bug detection. Allamanis et al. (2018) target variable-misuse errors and present a solution based on a gated graph neural network model to predict the correct variable name given a buggy location. Vasic et al. (2019) present a pointer network on top of a RNN which outperforms Allamanis et al. (2018) on the same task. DeepBugs (Pradel & Sen, 2018) proposes a name-based bug detection scheme. Their model is trained to predict three classes of bugs: swapped function arguments, wrong binary operator, and wrong operand in a binary operation. Compared to these models, our approach is capable of detecting and fixing a wide range of errors in Javascript. SequenceR (Chen et al., 2018) uses sequence-to-sequence model to translate a buggy code segment into correct one; Getafix (Scott et al., 2019) produces human-like bug fixes by learning from past fixes. It employs a hierarchical clustering algorithm that sorts fix patterns according to their generality. While these approaches are general against different types of bugs, they still need the bug location as input.
+
+Graph learning and optimization. Our work is closely related to the literature in graph representation learning and optimization. Our model uses a variant of GNN that is inspired by many representative works (Li et al., 2015; Xu et al., 2018; Si et al., 2018), with the adaptation of local value table and pointer mechanism. Our work is also related to auto-regressive graph modeling Johnson (2016); Li et al. (2018); Brockschmidt et al. (2018); Dai et al. (2018), but with more generic operations such as subtree deletion and attribute modifications. Some other works model the graph modification in latent space (Jin et al., 2018; Yin et al., 2018), but such frameworks lack fine-grained control over the generative process, and thus are not very suitable for performing code repair.
+
+# 8 CONCLUSION
+
+We proposed an end-to-end learning-based approach to detect and fix bugs in Javascript programs. We realized the approach in a tool HOPPITY and demonstrated that it correctly predicts 9,490 out of 36,361 code changes in real programs on Github. In the future, we plan to expand the targeted bugs to include those that are caused by the interdependence among multiple files or that require multiple steps to fix. We will also deploy HOPPITY in an IDE to further evaluate its accuracy and utility. Finally, we plan to extend our learning framework to support other languages. Due to its language-independence, we believe HOPPITY will benefit developers beyond Javascript as well.
+
+# ACKNOWLEDGMENTS
+
+We thank the reviewers for their insightful comments. This research was supported in part by NSF awards #1836936 and #1836822, ONR award #N00014-18-1-2021, and Facebook research awards.
+
+# REFERENCES
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+Daniel D Johnson. Learning graphical state transitions. 2016.  
+Manu Jose and Rupak Majumdar. Cause clue clauses: Error localization using maximum satisfiability. In Proceedings of the 32nd ACM SIGPLAN Conference on Programming Language Design and Implementation, 2011.  
+Claire Le Goues, Michael Pradel, and Abhik Roychoudhury. Automated program repair. Commun. ACM, 2019.  
+Yujia Li, Daniel Tarlow, Marc Brockschmidt, and Richard Zemel. Gated graph sequence neural networks. arXiv preprint arXiv:1511.05493, 2015.  
+Yujia Li, Oriol Vinyals, Chris Dyer, Razvan Pascanu, and Peter Battaglia. Learning deep generative models of graphs. arXiv preprint arXiv:1803.03324, 2018.  
+Michael Pradel and Koushik Sen. Deepbugs: A learning approach to name-based bug detection. Proc. ACM Program. Lang., 2(OOPSLA), 2018.
+
+Caitlin Sadowski, Jeffrey van Gogh, Ciera Jaspan, Emma Söderberg, and Collin Winter. Tricorder: Building a program analysis ecosystem. In Proceedings of the 37th International Conference on Software Engineering, 2015.  
+Franco Scarselli, Marco Gori, Ah Chung Tsoi, Markus Hagenbuchner, and Gabriele Monfardini. The graph neural network model. IEEE Transactions on Neural Networks, 20(1):61-80, 2008.  
+Andrew Scott, Johannes Bader, and Satish Chandra. Getafix: Learning to fix bugs automatically. volume 2, 2019.  
+Xujie Si, Hanjun Dai, Mukund Raghothaman, Mayur Naik, and Le Song. Learning loop invariants for program verification. In Advances in Neural Information Processing Systems, pp. 7751-7762, 2018.  
+Marko Vasic, Aditya Kanade, Petros Maniatis, David Bieber, and Rishabh Singh. Neural program repair by jointly learning to localize and repair. International Conference on Learning Representations, 2019.  
+Oriol Vinyals, Meire Fortunato, and Navdeep Jaitly. Pointer networks. In Advances in Neural Information Processing Systems, pp. 2692-2700, 2015.  
+Yu Wang, Fengjuan Gao, Linzhang Wang, and Ke Wang. Learning a static bug finder from data. arXiv preprint arXiv:1907.05579, 2019.  
+Keyulu Xu, Weihua Hu, Jure Leskovec, and Stefanie Jegelka. How powerful are graph neural networks? arXiv preprint arXiv:1810.00826, 2018.  
+Pengcheng Yin, Graham Neubig, Miltiadis Allamanis, Marc Brockschmidt, and Alexander L Gaunt. Learning to represent edits. arXiv preprint arXiv:1810.13337, 2018.  
+Nicholas C. Zakas. ESLint. https://eslint.org/, 2013.
+
+A ADDITIONAL EXPERIMENTS  
+
+<table><tr><td></td><td colspan="2">Total</td><td colspan="2">Location</td><td colspan="2">Operator</td><td colspan="2">Value</td><td colspan="2">Type</td></tr><tr><td></td><td>Top-3</td><td>Top-1</td><td>Top-3</td><td>Top-1</td><td>Top-1</td><td>Top-3</td><td>Top-1</td><td>Top-3</td><td>Top-1</td><td></td></tr><tr><td>ZeroOneTwoDiff</td><td>40.8</td><td>29.7</td><td>18.9</td><td>3.9</td><td>30.3</td><td>35.0</td><td>6.5</td><td>38.6</td><td>3.4</td><td></td></tr><tr><td>ZeroOneDiff</td><td>51.6</td><td>34.5</td><td>27.1</td><td>5.5</td><td>35.6</td><td>45.4</td><td>10.4</td><td>73.9</td><td>58.9</td><td></td></tr><tr><td>OneDiff</td><td>26.1</td><td>14.2</td><td>35.5</td><td>20.4</td><td>34.4</td><td>52.3</td><td>29.1</td><td>76.1</td><td>66.7</td><td></td></tr><tr><td>Random</td><td>.08</td><td>.07</td><td>2.28</td><td>1.4</td><td>27.7</td><td>.01</td><td>.01</td><td>.27</td><td>0</td><td></td></tr></table>
+
+Table 7: Evaluation of models on each dataset. The Random model is evaluated on the OneDiff dataset and is shown for comparison.
+
+Full experiment results In addition to the evaluation of samples with one edit Table 7, we also evaluate HOPPITY on the following datasets:
+
+ZeroOneDiff-Includes samples with labels of zero or one edit  
+- ZeroOneTwoDiff - Includes samples with labels of zero, one, or two edits.
+
+We trained models on each dataset for roughly 12 hours on a single GTX 2080Ti GPU. Accuracy on the ZeroOneDiff is the highest as predicting that an AST is not buggy does not consist of any low level primitive predictions. This makes it a much easier prediction for the model than say, an ADD operation which the parent location, left sibling, value, and type must all be predicted correctly in order to be considered accurate.
+
+![](images/3836c626968f0cb3ff2fb5c2aa33417e89679ab7d3e04c09bd38fd38fd486b50.jpg)
+
+False positive/negative study An evaluation of false positives and false negatives is available in Table 8. In this setting, we treat the problem as a classification problem on our ZeroOneDiff dataset and our model attempts to predict if a given AST is BUGGY / NOT BUGGY. If the model predicts ADD, REP_VAL, REP_TYPE, or DEL, we consider this a prediction of "BUGGY." Accordingly, if the model predicts NO_OP, we consider this to be a prediction of "NOT BUGGY."
+
+Accuracy v.s. size of graph To demonstrate the affect of AST size on HOPPITY's prediction accuracy on the OneDiff dataset, we include Figure 4. As expected, AST size and accuracy are inversely related.
+
+Table 8: Results on true/false predictions.  
+
+<table><tr><td>Beam Size (k)</td><td>Top-k Accuracy (%)</td></tr><tr><td>1</td><td>14.37%</td></tr><tr><td>2</td><td>21.10%</td></tr><tr><td>3</td><td>26.14%</td></tr><tr><td>4</td><td>30.12%</td></tr><tr><td>5</td><td>33.58%</td></tr></table>
+
+Table 9: Accuracy vs beam sizes.
+
+Accuracy v.s. beam search size In Table 9 we compare the performance with different beam sizes on the OneDiff dataset. As we can see, the top-3 accuracy with beam size 3 is significantly better than top-1 accuracy with just greedy prediction. This is expected, as in the decision process there are 'bottleneck' stages with only a few predictions (e.g., the op prediction). Thus from beam-1 to beam-3 there's huge improvement, but further beyond the performance maxed out.
+
+![](images/0a844c80d540569076716415de8ab34073dff045b4ecc94a8cabe2124eba879d.jpg)  
+Figure 4: End-to-end code repair accuracy v.s. size of AST of source code.
+
+# B DATA COLLECTION
+
+We have built a robust system to automatically collect millions of bug-fixes in Javascript programs from Github. Our system continuously crawls Github for commits containing Javascript files and creates a label consisting of the change to the AST corresponding to each such file.
+
+Our system consists of three entirely automated parallel steps:
+
+1. Collect Commits: Our system uses the GH Archive API to easily access Github event data for a specific hour in time. After obtaining all data for the hour, we filter this using the Github API to only include commits that consist of edits to Javascript files.  
+2. Download Files: As we are obtaining a list of valid commits from step 1, we begin downloading the pair:  $(src_{buggy}, src_{fixed})$  where  $src_{buggy}$  is the file prior to the commit, and  $src_{fixed}$  is the file following the commit that contains the changes made.  
+3. Create Label: For each Javascript file downloaded, we parse the source code into a JSON format of the AST. Our system uses the SHIFT AST<sup>1</sup>. Abstract Syntax Tree representations are designed to naturally and intuitively represent the structure of the source code. Because of this design goal, small changes in the source code can often lead to very large changes in the AST. We chose the SHIFT AST representation with consideration to our goal of maximizing the number of commits with only one difference between the ASTs. This component produces a pair of ASTs:  $(AST_{buggy}, AST_{fixed})$ , at which point a JSON differencing algorithm, fast-json-patch<sup>2</sup> is applied to create a label. The label includes the operation type and node edited for each difference between  $AST_{buggy}$  and  $AST_{fixed}$ .
+
+Each step of this process is parallelized in order to grow our corpus as quickly as possible. Our dataset has the advantage that it is continuously growing without human input.
+
+Our system is language independent and highly extensible and modular. For example, it can handle any language so long as it can be parsed into a JSON AST.
+
+For each label, we must download two files  $src_{buggy}$  and  $src_{fixed}$ . Additionally, if source files cannot be parsed into a SHIFT AST, a label cannot be created. For our learning corpus, we limit the dataset to only include labels with one AST difference. Additionally, in an attempt to limit graph size, we only include data points in which the  $AST_{buggy}$  and  $AST_{fixed}$  have less than 500 nodes.
+
+<table><tr><td>Total Files Downloaded:</td><td>52,719,402</td></tr><tr><td>Total Labelled Data Points:</td><td>15,225,347</td></tr><tr><td># AST differences:</td><td># data points:</td></tr><tr><td>0</td><td>3,473,391</td></tr><tr><td>1</td><td>1,863,193</td></tr><tr><td>2-10</td><td>3,247,437</td></tr><tr><td>11-20</td><td>2,117,977</td></tr><tr><td>21-50</td><td>2,047,998</td></tr><tr><td>51-100</td><td>858,981</td></tr><tr><td>101+</td><td>921,754</td></tr></table>
+
+Table 10: Data collection statistics.
+
+![](images/12f019227d5012d810a37d8595f4722164d42498f81a3f96f9882843c7f5a76c.jpg)  
+Figure 5: Distribution of number of edits in the entire crawled dataset.  
+Figure 5 plots the distribution of number of edits that are recorded in Table 10. We can see the distribution is long tail, with majority of edits as 1 or 2.
+
+# C ABLATION STUDY
+
+We tried different graph representations with corresponding graph embedding methods. The multi represents the multi-graph defined by different edge types, with the parameterization of message passing function mentioned in Eq 2; the code2inv is the parameterization used in Si et al. (2018); the single instead uses a single graph with edge types as one-hot edge features. We found that more layers does not lead to better generalization in our setting, and it becomes slower in terms of convergence. So we report the results with 4 layers in our main paper.
+
+<table><tr><td>model</td><td>maxLv</td><td>Total</td><td>Operator</td><td>Location</td><td>Value</td><td>Type</td></tr><tr><td>multi</td><td>20</td><td>7.63</td><td>30.0</td><td>13.1</td><td>22.6</td><td>54.5</td></tr><tr><td>multi</td><td>14</td><td>11.05</td><td>48.0</td><td>17.9</td><td>38.6</td><td>61.6</td></tr><tr><td>multi</td><td>4</td><td>13.33</td><td>53.4</td><td>36.2</td><td>38.6</td><td>56.4</td></tr><tr><td>code2inv</td><td>20</td><td>10.3</td><td>18.1</td><td>25.7</td><td>38.8</td><td>57.7</td></tr><tr><td>code2inv</td><td>14</td><td>8.92</td><td>40.0</td><td>18.1</td><td>36.0</td><td>55.9</td></tr><tr><td>code2inv</td><td>4</td><td>13.29</td><td>30.8</td><td>18.9</td><td>28.2</td><td>68.21</td></tr><tr><td>single</td><td>20</td><td>5.00</td><td>20.2</td><td>10.3</td><td>14.2</td><td>44.8</td></tr><tr><td>single</td><td>14</td><td>10.69</td><td>67.7</td><td>18.6</td><td>49.6</td><td>38.7</td></tr><tr><td>single</td><td>4</td><td>12.88</td><td>55.8</td><td>20.8</td><td>43.2</td><td>55.8</td></tr></table>
+
+Table 11: Ablation study with different graph embedding parameterizations and different number of layers. Full end-to-end repair accuracy as well as the accuracies for each primitives are reported. All the numbers are for top-1 prediction.
+
+D 30 RANDOM TESTING POINTS FOR TAJS BASELINE STUDY  
+
+<table><tr><td>(   )pəmətɪsɪ
+(   )ɪtɪdʒɪsɪd
+(   )bɪtɪtɪsɪsɪe
+(   )bɪtɪtɪsɪsɪe
+(   )bɪtɪtɪsɪe
+(   )bɪtɪtɪsɪe
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+(   ) bɪtɪtɪsɪe
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+(   )bɪtɪtɪsɪe
+(   )bɪtɪtɪsɪe
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+(   )bɪtɪtɪsɪe
+(   )bɪtɪtɪsɪe
+(   )bɪtɪtɪsɪe</td></tr><tr><td>(   )pəmətɪsɪ
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+(   )bɪtɪtɪsɪsɪe
+(   )bɪtɪtɪsɪe
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+(   )bɪtsɪsɪe
+(   )bɪtsɪe
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+(   )bɪtsɪe
+(   )bɪtsɪe
+(   )bɪtsɪe
+(   )bɪtsʌsɪe
+(   )mɪp·eɪnɪe
+(   )mɪp·eɪnɪe
+(   )mɪp·eɪnɪe
+(   )mɪp·eɪnɪe
+(   )mɪp·eɪnɪe
+(   )mɪp·eɪnɪe
+(   )mɪp·eɪnɪe
+(   )mɪp·eɪnɪe
+(   )mɪrɪsɪe
+(   )mɪrɪsɪe
+(   )mɪrɪsɪe
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+(   )mɪrɪsɪe
+(   )mɪrɪs¹e
+(   )mɪrɪsɪe
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+(   )mɪrɪsɪe
+(   )mɪrɪsɪe
+(   )mɪrɪsɪe
+(   )mɪrɪsɪe</td></tr><tr><td>(   )pəmətɪsɪ
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+(   )bɪtɪtɪsɪsɪe
+(   )bɪtɪtɪsɪe
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+(   )bɪtɪtɪsɪe
+（   )bɪtɪtɪsɪe
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+(   )bɪtɪtɪse
+(   )bɪtɪtɪsɪe
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+(   )bɪtɪsɪe
+(   )bɪtsɪsɪe
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+(   )bɪtsʌtⅡe
+(   )bɪtsʌtⅡe
+( 二)Bɪtsʌsɪe
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+(   )bʌsɪe
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+(   )bɑːnɪsɪe
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+(   )bɑːnɪsɪe</td></tr><tr><td>(   )pəmətɪsɪ
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+(   )bɪtɪdʒɪsɪd
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+(   )bɪtɪdʒɪsɪd
+（   )bɪtɪdʒɪsɪd
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+(   )bɪtɪdʒɪsɪd
+( 二)Bɪtsʌsɪe
+(   )bɑːnɪsɪe
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+(   )bɑːn iəsɪe
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+(   )bɑːn iəsɪe
+(   )bʌsɪe
+(   )bʌsɪe
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+(   )bʌtɪ sɪe
+(   )bʌtɪ sɪe
+(   )bʌtɪ sɪe
+(   )bʌtɪ sɪe
+(   )bʌtɪ sɪe
+(   )bʌtɪ sɪe
+(   )bʌtɪ sɪe
+(   )bʌtɪ sɪe
+(   )bʌtɪ sⅡe
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+
+<table><tr><td></td><td>\</td><td></td><td colspan="2">... = sèsèsèsèsèsèsèsèsèsèsèsèsèsèsèsèsèsèsèsèsèsèsèsèsèsèsèsèsèsèsèsèsèsèsèsèsèsèsèsèsèsèsèsèsèsèsèsèsèsèsè</td><td>Λνλελαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλαλβεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεεε</td><td></td></tr><tr><td></td><td></td><td></td><td>( ) MeiP·eWeb·S·T·C·(05L) eQoT·s·p·T</td><td></td><td>( ) MeiP·eWeb·S·T·C·(05L) eQoT·s·p·T</td><td></td></tr><tr><td></td><td>\</td><td></td><td>(···) dntēaesuaspppe·ræpTauordeTauaes</td><td>S T</td><td>( ) dntēaesuaspppe·ræpTauordeTauaes</td><td>S T</td></tr><tr><td></td><td></td><td></td><td></td><td>ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssSS</td><td></td><td></td></tr><tr><td>\</td><td></td><td></td><td>(···,···) suwred·dde</td><td>( ) wēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTē TTT</td><td>( ) wēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTēTē TTT</td><td>( )(8) L</td></tr><tr><td>ε doL</td><td>2 doL</td><td>1 doL</td><td>duoonononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononon</td><td></td><td></td><td></td></tr><tr><td>Σ doL</td><td>Σ doL</td><td>I doL</td><td>duoonononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononon</td><td></td><td></td><td></td></tr><tr><td>Σ do L</td><td>Σ do L</td><td>I do L</td><td>duoononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononon</td><td></td><td></td><td></td></tr></table>
\ No newline at end of file
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+# HOW MANY POSITION INFORMATION DO CONVOLUTIONAL NEURAL NETWORKS ENCODE?
+
+Md Amirul Islam*,1,2, Sen Jia*,1, Neil D. B. Bruce1,2
+
+$^{1}$ Ryerson University, Canada  
+$^{2}$ Vector Institute for Artificial Intelligence, Canada
+
+amirul@scs.ryerson.ca, sen.jia@ryerson.ca, bruce@ryerson.ca
+
+# ABSTRACT
+
+In contrast to fully connected networks, Convolutional Neural Networks (CNNs) achieve efficiency by learning weights associated with local filters with a finite spatial extent. An implication of this is that a filter may know what it is looking at, but not where it is positioned in the image. Information concerning absolute position is inherently useful, and it is reasonable to assume that deep CNNs may implicitly learn to encode this information if there is a means to do so. In this paper, we test this hypothesis revealing the surprising degree of absolute position information that is encoded in commonly used neural networks. A comprehensive set of experiments show the validity of this hypothesis and shed light on how and where this information is represented while offering clues to where positional information is derived from in deep CNNs.
+
+# 1 INTRODUCTION
+
+Convolutional Neural Networks (CNNs) have achieved state-of-the-art results in many computer vision tasks, e.g. object classification (Simonyan & Zisserman, 2014; He et al., 2016) and detection (Redmon et al., 2016; Ren et al., 2015), face recognition (Taigman et al., 2014), semantic segmentation (Long et al., 2015; Chen et al., 2018; Noh et al., 2015; Islam et al., 2017) and saliency detection (Cornia et al., 2018; Li et al., 2014; Jia & Bruce, 2019; Islam et al., 2018). However, CNNs have faced some criticism in the context of deep learning for the lack of interpretability (Lipton, 2018).
+
+The classic CNN model is considered to be spatially-agnostic and therefore capsule (Sabour et al., 2017) or recurrent networks (Visin et al., 2015) have been utilized to model relative spatial relationships within learned feature layers. It is unclear if CNNs capture any absolute spatial information which is important in position-dependent tasks (e.g. semantic segmentation and salient object detection). As shown in Fig. 1, the regions determined to be most salient (Jia & Bruce, 2018) tend to be near the center of an image. While detecting saliency on a cropped version of the images, the most salient region shifts even though the visual features have not been changed. This is somewhat surprising, given the limited spatial extent of CNN filters through which the image is interpreted. In this paper, we examine the role of absolute position information by performing a series of randomization tests with the hypothesis that CNNs might indeed learn to encode position information as a cue for decision making. Our experiments reveal that position information is implicitly learned from the commonly used padding operation (zero-padding). Zero-padding is widely used for keeping the same dimensionality when applying convolution. However, its hidden effect in representation learning has been long omitted. This work helps to better understand the nature of the learned features in CNNs and highlights an important observation and fruitful direction for future investigation.
+
+Previous works try to visualize learned feature maps to demystify how CNNs work. A simple idea is to compute losses and pass these backwards to the input space to generate a pattern image that can maximize the activation of a given unit (Hinton et al., 2006; Erhan et al., 2009). However, it is very difficult to model such relationships when the number of layers grows. Recent work (Zeiler & Fergus, 2014) presents a non-parametric method for visualization. A deconvolutional network (Zeiler et al., 2011) is leveraged to map learned features back to the input space and their
+
+![](images/b8f231dfd9a15c651277a5f27303c9ba5de68b06d95c0bde36652abebd08e5b0.jpg)  
+Figure 1: Sample predictions for salient regions for input images (left), and a slightly cropped version (right). Cropping results in a shift in position rightward of features relative to the centre. It is notable that this has a significant impact on output and decision of regions deemed salient despite no explicit position encoding and a modest change to position in the input.
+
+![](images/4dcdebea061fc1b39bf1861db748ad47a120a55fefa8b8b842f2a4a1320d58bb.jpg)
+
+![](images/9bd868d1c13ea882eaeee7a02152fc0048aa941b4345c8c80d879c0aa502878c.jpg)
+
+![](images/751f7c5956d019f0ecc4689009cccf30b8b8524c9a11945b147992f47ba11414.jpg)
+
+![](images/b3aad9b76adc807040569f1f483df6d98a49f7f2a68ab7518c5aea558ac8f9a4.jpg)
+
+![](images/e12efbfd21a8a4c01ab8822d941f0ca663425796b6b718acbcf56093b04c3083.jpg)
+
+results reveal what types of patterns a feature map actually learns. Another work (Selvaraju et al., 2017) proposes to combine pixel-level gradients with weighted class activation mapping to locate the region which maximizes class-specific activation. As an alternative to visualization strategies, an empirical study (Zhang et al., 2016) has shown that a simple network can achieve zero training loss on noisy labels. We share the similar idea of applying a randomization test to study the CNN learned features. However, our work differs from existing approaches in that these techniques only present interesting visualizations or understanding, but fail to shed any light on spatial relationships encoded by a CNN model.
+
+In summary, CNNs have emerged as a way of dealing with the prohibitive number of weights that would come with a fully connected end-to-end network. A trade-off resulting from this is that kernels and their learned weights only have visibility of a small subset of the image. This would seem to imply solutions where networks rely more on cues such as texture and color rather than shape (Baker et al., 2018). Nevertheless, position information provides a powerful cue for where objects might appear in an image (e.g. birds in the sky). It is conceivable that networks might rely sufficiently on such cues that they implicitly encode spatial position along with the features they represent. It is our hypothesis that deep neural networks succeed in part by learning both what and where things are. This paper tests this hypothesis, and provides convincing evidence that CNNs do indeed rely on and learn information about spatial positioning in the image to a much greater extent than one might expect.
+
+# 2 POSITION INFORMATION IN CNNS
+
+CNNs naturally try to extract fine-level high spatial-frequency details (e.g. edges, texture, lines) in the early convolutional stages while at the deepest layers of encoding the network produces the richest possible category specific features representation (Simonyan & Zisserman, 2014; He et al., 2016; Badrinarayanan et al., 2017). In this paper, we propose a hypothesis that position information is implicitly encoded within the extracted feature maps and plays an important role in classifying, detecting or segmenting objects from a visual scene. We therefore aim to prove this hypothesis by predicting position information from different CNN archetypes in an end-to-end manner. In the following sections, we first introduce the problem definition followed by a brief discussion of our proposed position encoding network.
+
+Problem Formulation: Given an input image  $\mathcal{I}_m\in \mathbb{R}^{h\times w\times 3}$ , our goal is to predict a gradient-like position information mask  $\hat{f}_p\in \mathbb{R}^{h\times w}$  where each pixel value defines the absolute coordinates of an pixel from left  $\rightarrow$  right or top  $\rightarrow$  bottom. We generate gradient-like masks  $\mathcal{G}_{pos}\in \mathbb{R}^{h\times w}$  (Sec. 2.2) for supervision in our experiments, with weights of the base CNN archetypes being fixed.
+
+# 2.1 POSITION ENCODING NETWORK
+
+Our Position Encoding Network (PosENet) (See Fig. 2) consists of two key components: a feedforward convolutional encoder network  $f_{enc}$  and a simple position encoding module, denoted as  $f_{perm}$ . The encoder network extracts features at different levels of abstraction, from shallower to deeper layers. The position encoding module takes multi-scale features from the encoder network as input and predicts the absolute position information at the end.
+
+Encoder: We use ResNet and VGG based architectures to build encoder networks ( $f_{enc}$ ) by removing the average pooling layer and the layer that assigns categories. As shown in Fig. 2, the
+
+![](images/b0f537dff18f408355a346dec7c3535537eeeab4af80ddec897efeadedccc6c1.jpg)  
+Figure 2: Illustration of PosENet architecture.  
+Figure 3: Sample images and generated gradient-like ground-truth position maps.
+
+![](images/327d39f3723f91eb52649c1cb89b074fed5db92c4d0c138e202dba914cbd4329.jpg)
+
+encoder module consists of five feature extractor blocks denoted by  $(f_{\vartheta}^{1}, f_{\vartheta}^{2}, f_{\vartheta}^{3}, f_{\vartheta}^{4}, f_{\vartheta}^{5})$ . The extracted multi-scale features from bottom to top layers of the canonical network are denoted by  $(f_{pos}^{1}, f_{pos}^{2}, f_{pos}^{3}, f_{pos}^{4}, f_{pos}^{5})$ . We summarize the key operations as follows:
+
+$$
+f _ {p o s} ^ {i} = f _ {\vartheta} ^ {i} \left(\mathbf {W} _ {\mathbf {a}} * \mathcal {I} _ {m}\right) \tag {1}
+$$
+
+where  $\mathbf{W}_{\mathrm{a}}$  denotes weights that are frozen.  $*$  denotes the convolution operation. Note that in probing the encoding network, only the position encoding module  $f_{perm}$  is trained to focus on extracting position information while the encoder network is forced to maintain their existing weights.
+
+Position Encoding Module: The position encoding module takes multi-scale features  $(f_{pos}^{1},\dots ,f_{pos}^{5})$  from  $f_{enc}$  as input and generates the desired position map  $\hat{f}_p$  thorough a transformation function  $\mathcal{T}_{pos}$ . The transformation function  $\mathcal{T}_{pos}$  first applies a bi-linear interpolation operation on the feature maps to have the same spatial dimension resulting in a feature map  $f_{pos}^{c}$ . Once we have the same spatial dimension for multi-scale features, we concatenate them together followed by a sequence of  $k\times k$  convolution operations. In our experiments, we vary the value of  $k$  between  $\{1,3,5,7\}$  and most experiments are carried out with a single convolutional layer in the position encoding module  $f_{perm}$ . The key operations can be summarized as follows:
+
+$$
+f _ {p o s} ^ {c} = \left(f _ {p o s} ^ {1} \oplus \dots \oplus f _ {p o s} ^ {5}\right) \quad \hat {f} _ {p} = \left(\mathbf {W} _ {\mathbf {p o s}} ^ {\mathbf {c}} * f _ {p o s} ^ {c}\right) \tag {2}
+$$
+
+where  $\mathbf{W}_{pos}^{c}$  is the trainable weights attached with the transformation function  $\mathcal{T}_{pos}$ .
+
+The main objective of the encoding module is to validate whether position information is implicitly learned when trained on categorical labels. Additionally, the position encoding module models the relationship between hidden position information and the gradient like ground-truth mask. The output is expected to be random if there is no position information encoded in the features maps and vice versa (ignoring any guidance from image content).
+
+# 2.2 SYNTHETIC DATA AND GROUND-TRUTH GENERATION
+
+To validate the existence of position information in a network, we implement a randomization test by assigning a normalized gradient-like  $^1$  position map as ground-truth shown in Fig. 3. We first generate gradient-like masks in Horizontal (H) and vertical (V) directions. Similarly, we apply a Gaussian filter to design another type of ground-truth map, Gaussian distribution (G). The key motivation of generating these three patterns is to validate if the model can learn absolute position on one or two axes. Additionally, We also create two types of repeated patterns, horizontal and vertical stripes, (HS, VS). Regardless of the direction, the position information in the multi-level features is likely to be modelled through a transformation by the encoding module  $f_{perm}$ . Our design of gradient ground-truth can be considered as a type of random label because there is no correlation between the input image and the ground-truth with respect to position. Since the extraction of position information is independent of the content of images, we can choose any image datasets. Meanwhile, we also build synthetic images to validate our hypothesis.
+
+# 2.3 TRAINING THE NETWORK
+
+As we implicitly aim to encode the position information from a pretrained network, we freeze the encoder network  $f_{enc}$  in all of our experiments. Our position encoding module  $f_{perm}$  generates the
+
+position map  $\hat{f}_p$  of interest. During training, for a given input image  $\mathcal{I}_m\in \mathbb{R}^{h\times w\times 3}$  and associated ground-truth position map  $\mathcal{G}_{pos}^h$ , we apply the supervisory signal on  $\hat{f}_p$  by upsampling it to the size of  $\mathcal{G}_{pos}^h$ . Then, we define a pixel-wise mean squared error loss to measure the difference between predicted and ground-truth position maps as follows:
+
+$$
+\Delta_ {\hat {f} _ {p}} = \frac {1}{2 n} \sum_ {i = 1} ^ {n} \left(x _ {i} - y _ {i}\right) ^ {2} \tag {3}
+$$
+
+where  $x \in \mathbb{R}^n$  and  $y \in \mathbb{R}^n$  ( $n$  denotes the spatial resolution) are the vectorized predicted position and ground-truth map respectively.  $x_i$  and  $y_i$  refer to a pixel of  $\hat{f}_p$  and  $\mathcal{G}_{pos}^h$  respectively.
+
+# 3 EXPERIMENTS
+
+# 3.1 DATASET AND EVALUATION METRICS
+
+Datasets: We use the DUT-S dataset (Wang et al., 2017) as our training set, which contains 10,533 images for training. Following the common training protocol used in (Zhang et al., 2017; Liu et al., 2018), we train the model on the training set of DUT-S and evaluate the existence of position information on the natural images of the PASCAL-S (Li et al., 2014) dataset. The synthetic images (white, black and Gaussian noise) are also used as described in Section 2.2. Note that we follow the common setting used in saliency detection just to make sure that there is no overlap between the training and test sets. However, any images can be used in our experiments given that the position information is relatively content independent.
+
+Evaluation Metrics: As position encoding measurement is a new direction, there is no universal metric. We use two different natural choices for metrics (Spearmen Correlation (SPC) and Mean Absolute Error (MAE)) to measure the position encoding performance. The SPC is defined as the Spearman's correlation between the ground-truth and the predicted position map. For ease of interpretation, we keep the SPC score within range [-1 1]. MAE is the average pixel-wise difference between the predicted position map and the ground-truth gradient position map.
+
+# 3.2 IMPLEMENTATION DETAILS
+
+We initialize the architecture with a network pretrained for the ImageNet classification task. The new layers in the position encoding branch are initialized with xavier initialization (Glorot & Bengio, 2010). We train the networks using stochastic gradient descent for 15 epochs with momentum of 0.9, and weight decay of  $1e - 4$ . We resize each image to a fixed size of  $224\times 224$  during training and inference. Since the spatial extent of multi-level features are different, we align all the feature maps to a size of  $28\times 28$ . We report experimental results for the following baselines that are described as follows: VGG indicates PosENet is based on the features extracted from the VGG16 model. Similarly, ResNet represents the combination of ResNet-152 and PosENet. PosENet alone denotes only the PosENet model is applied to learn position information directly from the input image. H, V, G, HS and VS represent the five different ground-truth patterns, horizontal and vertical gradients, 2D Gaussian distribution, horizontal and vertical stripes respectively.
+
+# 3.3 EXISTENCE OF POSITION INFORMATION
+
+Position Information in Pretrained Models: We first conduct experiments to validate the existence of position information encoded in a pretrained model. Following the same protocol, we train the VGG and ResNet based networks on each type of the ground-truth and report the experimental results in Table 1. We also report results when we only train PosENet without using any pretrained model to justify that the position information is not driven from prior knowledge of objects. Our experiments do not focus on achieving higher performance on the metrics but instead validate how much position information a CNN model encodes or how easily PosENet can extract this information. Note that, we only use one convolutional layer with a kernel size of  $3 \times 3$  without any padding in the PosENet for this experiment.
+
+As shown in Table 1, PosENet (VGG and ResNet) can easily extract position information from the pretrained CNN models, especially the ResNet based PosENet model. However, training PosENet
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Model</td><td colspan="2">PASCAL-S</td><td colspan="2">Black</td><td colspan="2">White</td><td colspan="2">Noise</td></tr><tr><td>SPC</td><td>MAE</td><td>SPC</td><td>MAE</td><td>SPC</td><td>MAE</td><td>SPC</td><td>MAE</td></tr><tr><td rowspan="3">H</td><td>PosENet</td><td>.012</td><td>.251</td><td>.0</td><td>.251</td><td>.0</td><td>.251</td><td>.001</td><td>.251</td></tr><tr><td>VGG</td><td>.742</td><td>.149</td><td>.751</td><td>.164</td><td>.873</td><td>.157</td><td>.591</td><td>.173</td></tr><tr><td>ResNet</td><td>.933</td><td>.084</td><td>.987</td><td>.080</td><td>.994</td><td>.078</td><td>.973</td><td>.077</td></tr><tr><td rowspan="3">V</td><td>PosENet</td><td>.131</td><td>.248</td><td>.0</td><td>.251</td><td>.0</td><td>.251</td><td>.053</td><td>.250</td></tr><tr><td>VGG</td><td>.816</td><td>.129</td><td>.846</td><td>.146</td><td>.927</td><td>.138</td><td>.771</td><td>.150</td></tr><tr><td>ResNet</td><td>.951</td><td>.083</td><td>.978</td><td>.069</td><td>.979</td><td>.072</td><td>.968</td><td>.074</td></tr><tr><td rowspan="3">G</td><td>PosENet</td><td>-.001</td><td>.233</td><td>.0</td><td>.186</td><td>.0</td><td>.186</td><td>-.034</td><td>.214</td></tr><tr><td>VGG</td><td>.814</td><td>.109</td><td>.842</td><td>.123</td><td>.898</td><td>.116</td><td>.762</td><td>.129</td></tr><tr><td>ResNet</td><td>.936</td><td>.070</td><td>.953</td><td>.068</td><td>.964</td><td>.064</td><td>.971</td><td>.055</td></tr><tr><td rowspan="3">HS</td><td>PosENet</td><td>-.001</td><td>.712</td><td>-.055</td><td>.704</td><td>.0</td><td>.704</td><td>.023</td><td>.710</td></tr><tr><td>VGG</td><td>.405</td><td>.556</td><td>.532</td><td>.583</td><td>.576</td><td>.574</td><td>.375</td><td>.573</td></tr><tr><td>ResNet</td><td>.534</td><td>.528</td><td>.566</td><td>.518</td><td>.562</td><td>.515</td><td>.471</td><td>.530</td></tr><tr><td rowspan="3">VS</td><td>PosENet</td><td>.006</td><td>.723</td><td>.081</td><td>.709</td><td>.081</td><td>.709</td><td>.018</td><td>.714</td></tr><tr><td>VGG</td><td>.374</td><td>.567</td><td>.538</td><td>.575</td><td>.437</td><td>.578</td><td>.526</td><td>.566</td></tr><tr><td>ResNet</td><td>.520</td><td>.537</td><td>.574</td><td>.523</td><td>.593</td><td>.514</td><td>.523</td><td>.545</td></tr></table>
+
+Table 1: Quantitative comparison of different networks in terms of SPC and MAE across different image types.
+
+![](images/1136d0e7de36997afc34d12e8b5c357554cdc25db26e530c2584205005009d2d.jpg)  
+Figure 4: Qualitative results of PosENet based networks corresponding to different ground-truth patterns.
+
+(PosENet) separately achieves much lower scores across different patterns and source images. This result implies that it is very difficult to extract position information from the input image alone. PosENet can extract position information consistent with the ground-truth position map only when coupled with a deep encoder network. As mentioned prior, the generated ground-truth map can be considered as a type of randomization test given that the correlation with input has been ignored (Zhang et al., 2016). Nevertheless, the high performance on the test sets across different ground-truth patterns reveals that the model is not blindly overfitting to the noise and instead is extracting true position information. However, we observe low performance on the repeated patterns (HS and VS) compared to other patterns due to the model complexity and specifically the lack of correlation between ground-truth and absolute position (last two rows of Table 1). The  $H$  pattern can be seen as one quarter of a sine wave whereas the striped patterns (HS and VS) can be considered as repeated periods of a sine wave which requires a deeper comprehension.
+
+The qualitative results for several architectures across different patterns are shown in Fig. 4. We can see the correlation between the predicted and the ground-truth position maps corresponding to  $\mathbf{H}$ ,  $\mathbf{G}$  and  $\mathbf{HS}$  patterns, which further reveals the existence of position information in these networks. The quantitative and qualitative results strongly validate our hypothesis that position information is implicitly encoded in every architecture without any explicit supervision towards this objective.
+
+Moreover, PosENet alone shows no capacity to output a gradient map based on the synthetic data. We further explore the effect of image semantics in Sec. 4.1. It is interesting to note the performance gap among different architectures specifically the ResNet based models achieve higher performance than the VGG16 based models. The reason behind this could be the use of different convolutional
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Layers</td><td colspan="2">PosENet</td><td colspan="2">VGG</td></tr><tr><td>SPC</td><td>MAE</td><td>SPC</td><td>MAE</td></tr><tr><td rowspan="3">H</td><td>1 Layer</td><td>.012</td><td>.251</td><td>.742</td><td>.149</td></tr><tr><td>2 Layers</td><td>.056</td><td>.250</td><td>.797</td><td>.128</td></tr><tr><td>3 Layers</td><td>.055</td><td>.250</td><td>.830</td><td>.117</td></tr><tr><td rowspan="3">G</td><td>1 Layer</td><td>-.001</td><td>.233</td><td>.814</td><td>.109</td></tr><tr><td>2 Layers</td><td>.067</td><td>.187</td><td>.828</td><td>.105</td></tr><tr><td>3 Layers</td><td>.126</td><td>.186</td><td>.835</td><td>.104</td></tr><tr><td rowspan="3">HS</td><td>1 Layer</td><td>-.001</td><td>.712</td><td>.405</td><td>.556</td></tr><tr><td>2 Layers</td><td>-.006</td><td>.628</td><td>.483</td><td>.538</td></tr><tr><td>3 Layers</td><td>.003</td><td>.628</td><td>.491</td><td>.540</td></tr></table>
+
+(a)  
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Kernel</td><td colspan="2">PosENet</td><td colspan="2">VGG</td></tr><tr><td>SPC</td><td>MAE</td><td>SPC</td><td>MAE</td></tr><tr><td rowspan="3">H</td><td>1 × 1</td><td>.013</td><td>.251</td><td>.542</td><td>.196</td></tr><tr><td>3 × 3</td><td>.012</td><td>.251</td><td>.742</td><td>.149</td></tr><tr><td>7 × 7</td><td>.060</td><td>.250</td><td>.828</td><td>.120</td></tr><tr><td rowspan="3">G</td><td>1 × 1</td><td>.017</td><td>.188</td><td>.724</td><td>.127</td></tr><tr><td>3 × 3</td><td>-.001</td><td>.233</td><td>.814</td><td>.109</td></tr><tr><td>7 × 7</td><td>.068</td><td>.187</td><td>.816</td><td>.111</td></tr><tr><td rowspan="3">HS</td><td>1 × 1</td><td>-.004</td><td>.628</td><td>.317</td><td>.576</td></tr><tr><td>3 × 3</td><td>-.001</td><td>.723</td><td>.405</td><td>.556</td></tr><tr><td>7 × 7</td><td>.002</td><td>.628</td><td>.487</td><td>.532</td></tr></table>
+
+(b)
+
+Table 2: Quantitative comparison on the PASCAL-S dataset in terms of SPC and MAE with varying (a) number of layers and (b) kernel sizes. Note that (a) the kernel size is fixed to  $3 \times 3$  but different numbers of layers are used in the PosENet. (b) Number of layers is fixed to one but we use different kernel sizes in the PosENet.
+
+kernels in the architecture or the degree of prior knowledge of the semantic content. We show an ablation study in the next experiment for further investigation. For the rest of this paper, we only focus on the natural images, PASCAL-S dataset, and three representative patterns,  $\mathbf{H}$ ,  $\mathbf{G}$  and  $\mathbf{HS}$ .
+
+# 3.4 ANALYZING POSENET
+
+In this section, we conduct ablation studies to examine the role of the proposed position encoding network by highlighting two key design choices. (1) the role of varying kernel size in the position encoding module and (2) stack length of convolutional layers we add to extract position information from the multi-level features.
+
+Impact of Stacked Layers: Experimental results in Table 1 show the existence of position information learned from an object classification task. In this experiment, we change the design of PosENet to examine if it is possible to extract hidden position information more accurately. The PosENet used in the prior experiment (Table 1) has only one convolutional layer with a kernel size of  $3 \times 3$ . Here, we apply a stack of convolutional layers of varying length to the PosENet and report the experimental results in Table 2 (a). Even though the stack size is varied, we aim to retain a relatively simple PosENet to only allow efficient readout of positional information. As shown in Table 2, we keep the kernel size fixed at  $3 \times 3$  while stacking multiple layers. Applying more layers in the PosENet can improve the readout of position information for all the networks. One reason could be that stacking multiple convolutional filters allows the network to have a larger effective receptive field, for example two  $3 \times 3$  convolution layers are spatially equal to one  $5 \times 5$  convolution layer (Simonyan & Zisserman, 2014). An alternative possibility is that positional information may be represented in a manner that requires more than first order inference (e.g. a linear readout).
+
+Impact of varying Kernel Sizes: We further validate PosENet by using only one convolutional layer with different kernel sizes and report the experimental results in Table 2 (b). From Table 2 (b), we can see that the larger kernel sizes are likely to capture more position information compared to smaller sizes. This finding implies that the position information may be distributed spatially within layers and in feature space as a larger receptive field can better resolve position information.
+
+We further show the visual impact of varying number of layers and kernel sizes to learn position information in Fig. 5.
+
+# 3.5 WHERE IS THE POSITION INFORMATION STORED?
+
+Our previous experiments reveal that the position information is encoded in a pretrained CNN model. It is also interesting to see whether position information is equally distributed across the layers. In this experiment, we train PosENet on each of the extracted features,  $f_{pos}^{1}, f_{pos}^{2}, f_{pos}^{3}, f_{pos}^{4}, f_{pos}^{5}$  separately using VGG16 to examine which layer encodes more position information. Similar to Sec. 3.3, we only apply one  $3 \times 3$  kernel in  $F_{perm}$  to obtain the position map.
+
+![](images/e13dfc9d57309f05d2fdb9758082dbbeed9643135da59abbfa1c6be9f40d1823.jpg)  
+Figure 5: The effect of more Layers (Top row) and varying Kernel Size (bottom row) applied in the PoseNet. Order (left  $\rightarrow$  right): GT (G), PosENet (L=1, KS=1), PosENet (L=2, KS=3), PosENet (L=3, KS=7), VGG (L=1, KS=1), VGG (L=2, KS=3), VGG (L=3, KS=7).
+
+<table><tr><td></td><td>Method</td><td>f1pos</td><td>f2pos</td><td>f3pos</td><td>f4pos</td><td>f5pos</td><td>SPC</td><td>MAE</td></tr><tr><td rowspan="6">H</td><td rowspan="6">VGG</td><td>✓</td><td></td><td></td><td></td><td></td><td>.101</td><td>.249</td></tr><tr><td></td><td>✓</td><td></td><td></td><td></td><td>.344</td><td>.225</td></tr><tr><td></td><td></td><td>✓</td><td></td><td></td><td>.472</td><td>.203</td></tr><tr><td></td><td></td><td></td><td>✓</td><td></td><td>.610</td><td>.181</td></tr><tr><td></td><td></td><td></td><td></td><td>✓</td><td>.657</td><td>.177</td></tr><tr><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>.742</td><td>.149</td></tr><tr><td rowspan="6">G</td><td rowspan="6">VGG</td><td>✓</td><td></td><td></td><td></td><td></td><td>.241</td><td>.182</td></tr><tr><td></td><td>✓</td><td></td><td></td><td></td><td>.404</td><td>.168</td></tr><tr><td></td><td></td><td>✓</td><td></td><td></td><td>.588</td><td>.146</td></tr><tr><td></td><td></td><td></td><td>✓</td><td></td><td>.653</td><td>.138</td></tr><tr><td></td><td></td><td></td><td></td><td>✓</td><td>.693</td><td>.135</td></tr><tr><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>✓</td><td>.814</td><td>.109</td></tr></table>
+
+Table 3: Performance of VGG on natural images with a varying extent of the reach of different feed-forward blocks.
+
+As shown in Table 3, the VGG based PosENet with top  $f_{pos}^{5}$  features achieves higher performance compared to the bottom  $f_{pos}^{1}$  features. This may partially a result of more feature maps being extracted from deeper as opposed to shallower layers, 512 vs 64 respectively. However, it is likely indicative of stronger encoding of the positional information in the deepest layers of the network where this information is shared by high-level semantics. We further investigate this effect for VGG16 where the top two layers ( $f_{pos}^{4}$  and  $f_{pos}^{5}$ ) have the same number of features. More interestingly,  $f_{pos}^{5}$  achieves better results than  $f_{pos}^{4}$ . This comparison suggests that the deeper feature contains more position information, which validates the common belief that top level visual features are associated with global features.
+
+# 4 WHERE DOES POSITION INFORMATION COME FROM?
+
+We believe that the padding near the border delivers position information to learn. Zero-padding is widely used in convolutional layers to maintain the same spatial dimensions for the input and output, with a number of zeros added at the beginning and at the end of both axes, horizontal and vertical. To validate this, we remove all the padding mechanisms implemented within VGG16 but still initialize the model with the ImageNet pretrained weights. Note that we perform this experiment only using VGG based PosENet since removing padding on ResNet models will lead to inconsistent sizes of skip connections. We first test the effect of zero-padding used in VGG, no padding used in PosENet. As we can see from Table 4, the VGG16 model without zero-padding achieves much lower performance than the default setting (padding=1) on the natural images. Similarly, we introduce position information to the PosENet by applying zero-padding. PosENet with padding=1 (concatenating one zero around the frame) achieves higher performance than the original (padding=0). When we set padding=2, the role of position information is more obvious. This also validates our experiment in Section 3.3, that shows PosENet is unable to extract noticeable position information because no padding was applied, and the information is encoded from a pretrained CNN model. This is why we did not apply zero-padding in PosENet in our previous experiments. Moreover, we aim to explore
+
+<table><tr><td rowspan="2">Model</td><td colspan="2">H</td><td colspan="2">G</td><td colspan="2">HS</td></tr><tr><td>SPC</td><td>MAE</td><td>SPC</td><td>MAE</td><td>SPC</td><td>MAE</td></tr><tr><td>PosENet</td><td>.012</td><td>.251</td><td>-.001</td><td>.233</td><td>-.001</td><td>.712</td></tr><tr><td>PosENet with padding=1</td><td>.274</td><td>.239</td><td>.205</td><td>.184</td><td>.148</td><td>.608</td></tr><tr><td>PosENet with padding=2</td><td>.397</td><td>.223</td><td>.380</td><td>.177</td><td>.214</td><td>.595</td></tr><tr><td>VGG16</td><td>.742</td><td>.149</td><td>.814</td><td>.109</td><td>.405</td><td>.556</td></tr><tr><td>VGG16 w/o. padding</td><td>.381</td><td>.223</td><td>.359</td><td>.174</td><td>.011</td><td>.628</td></tr></table>
+
+Table 4: Quantitative comparison subject to padding in the convolution layers used in PosENet and VGG (w/o and with zero padding) on natural images.
+
+![](images/fdf64c076e3a883c31fad898248d2dcdae94753f2f34cf753d61493a6bf1a56e.jpg)  
+Figure 6: The effect of zero-padding on Gaussian pattern. Left to right: GT (G), Pad=0 (.286, .186), Pad=1 (.227, .180), Pad=2 (.473, .169), VGG Pad=1 (.928, .085), VGG Pad=0(.405, .170).
+
+how much position information is encoded in the pretrained model instead of directly combining with the PosENet. Fig. 6 illustrates the impact of zero-padding on encoding position information subject to padding using a Gaussian pattern.
+
+# 4.1 CASE STUDY
+
+Recall that the position information is considered to be content independent but our results in Table 1 show that semantics within an image may affect the position map. To visualize the impact of semantics, we compute the content loss heat map using the following equation:
+
+$$
+\mathcal {L} = \frac {\left| \left(\mathcal {G} _ {p o s} ^ {h} - \hat {f} _ {p} ^ {h}\right) \right| + \left| \left(\mathcal {G} _ {p o s} ^ {v} - \hat {f} _ {p} ^ {v}\right) \right| + \left| \left(\mathcal {G} _ {p o s} ^ {g} - \hat {f} _ {p} ^ {g}\right) \right|}{3} \tag {4}
+$$
+
+where  $\hat{f}_p^h, \hat{f}_p^v$ , and  $\hat{f}_p^g$  are the predicted position maps from horizontal, vertical and Gaussian patterns respectively.
+
+As shown in Figure 7, the heatmaps of PosENet have larger content loss around the corners. While the loss maps of VGG and ResNet correlate more with the semantic content. Especially for ResNet, the deeper understanding of semantic content leads to a stronger interference in generating a smooth gradient. The highest losses are from the face, person, cat, airplane and vase respectively (from left to right). This visualization can be an alternative method to show which regions a model focuses on, especially in the case of ResNet.
+
+![](images/c1a03556d9543c79192798d649808b488cb11e1332e9490d39602f7f2bb307fb.jpg)  
+Figure 7: Error heat maps of PosENet (1<sup>st</sup> row), VGG (2<sup>nd</sup> row) and ResNet (3<sup>rd</sup> row).
+
+# 4.2 ZERO-PADDING DRIVEN POSITION INFORMATION
+
+Saliency Detection: We further validate our findings in the position-dependent tasks (semantic segmentation and salient object detection (SOD)). First, we train the VGG network with and without zero-padding from scratch to validate if the position information delivered by zero-padding is critical for detecting salient regions. For these experiments, we use the publicly available MSRA dataset (Cheng et al., 2015) as our SOD training set and evaluate on three other datasets (ECSSD, PASCAL-S, and DUT-OMRON). From Table 5 (a), we can see that VGG without padding achieves much worse results on both of the metrics (F-measure and MAE) which further validates our findings that zero-padding is the key source of position information.
+
+Semantic Segmentation: We also validate the impact of zero-padding on the semantic segmentation task. We train the VGG16 network with and without zero padding on the training set of PASCAL VOC 2012 dataset and evaluate on the validation set. Similar to SOD, the model with zero padding significantly outperforms the model with no padding.
+
+<table><tr><td rowspan="2">Model</td><td colspan="2">ECSSD</td><td colspan="2">PASCAL-S</td><td colspan="2">DUT-OMRON</td></tr><tr><td>Fm</td><td>MAE</td><td>Fm</td><td>MAE</td><td>Fm</td><td>MAE</td></tr><tr><td>VGG w/o padding</td><td>.36</td><td>.48</td><td>.32</td><td>.48</td><td>.25</td><td>.48</td></tr><tr><td>VGG</td><td>.78</td><td>.17</td><td>.66</td><td>.21</td><td>.63</td><td>.18</td></tr></table>
+
+(a)  
+
+<table><tr><td>Model</td><td>mIoU (%)</td></tr><tr><td>VGG w/o padding</td><td>12.3</td></tr><tr><td>VGG</td><td>23.1</td></tr></table>
+
+We believe that CNN models pretrained on these two tasks can learn more position information than classification task. To validate this hypothesis, we take the VGG model pretrained on ImageNet as our baseline. Meanwhile, we train two VGG models on the tasks of semantic segmentation and saliency detection from scratch, denoted as VGG-SS and VGG-SOD respectively. Then we finetune these three VGG models following the protocol used in Section 3.3. From Table 6, we can see that the VGG-SS and VGG-SOD models outperform VGG by a large margin. These experiments further reveal that the zero-padding strategy plays an important role in a position-dependent task, an observation that has been long-ignored in neural network solutions to vision problems.
+
+(b)  
+Table 5: VGG models with and w/o zero-padding for (a) SOD and (b) semantic segmentation.  
+
+<table><tr><td rowspan="2"></td><td rowspan="2">Model</td><td colspan="2">PASCAL-S</td><td colspan="2">BLACK</td><td colspan="2">WHITE</td><td colspan="2">NOISE</td></tr><tr><td>SPC</td><td>MAE</td><td>SPC</td><td>MAE</td><td>SPC</td><td>MAE</td><td>SPC</td><td>MAE</td></tr><tr><td rowspan="3">H</td><td>VGG</td><td>.742</td><td>.149</td><td>.751</td><td>.164</td><td>.873</td><td>.157</td><td>.591</td><td>.173</td></tr><tr><td>VGG-SOD</td><td>.969</td><td>.055</td><td>.857</td><td>.099</td><td>.938</td><td>.087</td><td>.965</td><td>.060</td></tr><tr><td>VGG-SS</td><td>.982</td><td>.038</td><td>.990</td><td>.030</td><td>.985</td><td>.032</td><td>.985</td><td>.033</td></tr><tr><td rowspan="3">G</td><td>VGG</td><td>.814</td><td>.109</td><td>.842</td><td>.123</td><td>.898</td><td>.116</td><td>.762</td><td>.129</td></tr><tr><td>VGG-SOD</td><td>.948</td><td>.067</td><td>.904</td><td>.086</td><td>.907</td><td>.085</td><td>.912</td><td>.077</td></tr><tr><td>VGG-SS</td><td>.971</td><td>.055</td><td>.984</td><td>.050</td><td>.989</td><td>.046</td><td>.982</td><td>.051</td></tr><tr><td rowspan="3">HS</td><td>VGG</td><td>.405</td><td>.556</td><td>.532</td><td>.583</td><td>.576</td><td>.574</td><td>.375</td><td>.573</td></tr><tr><td>VGG-SOD</td><td>.667</td><td>.476</td><td>.699</td><td>.506</td><td>.709</td><td>.482</td><td>.668</td><td>.489</td></tr><tr><td>VGG-SS</td><td>.810</td><td>.430</td><td>.802</td><td>.426</td><td>.810</td><td>.426</td><td>.789</td><td>.428</td></tr></table>
+
+Table 6: Comparison of VGG models pretrained for classification, SOD, and semantic segmentation.
+
+# 5 CONCLUSION
+
+In this paper we explore the hypothesis that absolute position information is implicitly encoded in convolutional neural networks. Experiments reveal that positional information is available to a strong degree. More detailed experiments show that larger receptive fields or non-linear readout of positional information further augments the readout of absolute position, which is already very strong from a trivial single layer  $3 \times 3$  PosENet. Experiments also reveal that this recovery is possible when no semantic cues are present and interference from semantic information suggests joint encoding of what (semantic features) and where (absolute position). Results point to zero padding and borders as an anchor from which spatial information is derived and eventually propagated over the whole image as spatial abstraction occurs. These results demonstrate a fundamental property of CNNs that was unknown to date, and for which much further exploration is warranted.
+
+# REFERENCES
+
+Vijay Badrinarayanan, Alex Kendall, and Roberto Cipolla. Segnet: A deep convolutional encoder-decoder architecture for scene segmentation. TPAMI, 2017.  
+Nicholas Baker, Hongjing Lu, Gennady Erlikhman, and Philip J. Kellman. Deep convolutional networks do not classify based on global object shape. PLOS Computational Biology, 2018.  
+Liang-Chieh Chen, George Papandreou, Iasonas Kokkinos, Kevin Murphy, and Alan L Yuille. DeepLab: Semantic image segmentation with deep convolutional nets, atrous convolution, and fully connected crfs. TPAMI, 2018.  
+M. Cheng, N. J. Mitra, X. Huang, P. H. S. Torr, and S. Hu. Global contrast based salient region detection. TPAMI, 2015.  
+M. Cornia, L. Baraldi, G. Serra, and R. Cucchiara. Predicting human eye fixations via an LSTM-based saliency attentive model. TIP, 2018.  
+Dumitru Erhan, Yoshua Bengio, Aaron Courville, and Pascal Vincent. Visualizing higher-layer features of a deep network. University of Montreal, 2009.  
+Xavier Glorot and Yoshua Bengio. Understanding the difficulty of training deep feedforward neural networks. In AISTATS, 2010.  
+Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In CVPR, 2016.  
+Geoffrey Hinton, Simon Osindero, Max Welling, and Yee-Whye Teh. Unsupervised discovery of nonlinear structure using contrastive backpropagation. Cognitive science, 2006.  
+Md Amirul Islam, Mrigank Rochan, Neil DB Bruce, and Yang Wang. Gated feedback refinement network for dense image labeling. In CVPR, 2017.  
+Md Amirul Islam, Mahmoud Kalash, and Neil DB Bruce. Revisiting salient object detection: Simultaneous detection, ranking, and subitizing of multiple salient objects. In CVPR, 2018.  
+Sen Jia and Neil DB Bruce. Eml-net: An expandable multi-layer network for saliency prediction. arXiv preprint arXiv:1805.01047, 2018.  
+Sen Jia and Neil DB Bruce. Richer and deeper supervision network for salient object detection. arXiv preprint arXiv:1901.02425, 2019.  
+Y. Li, X. Hou, C. Koch, J. M. Rehg, and A. L. Yuille. The secrets of salient object segmentation. In CVPR, 2014.  
+Zachary C Lipton. The mythos of model interpretability. Queue, 2018.  
+Nian Liu, Junwei Han, and Ming-Hsuan Yang. Picanet: Learning pixel-wise contextual attention for saliency detection. In CVPR, 2018.  
+Jonathan Long, Evan Shelhamer, and Trevor Darrell. Fully convolutional networks for semantic segmentation. In CVPR, 2015.  
+Hyeonwoo Noh, Seunghoon Hong, and Bohyung Han. Learning deconvolution network for semantic segmentation. In ICCV, 2015.  
+Joseph Redmon, Santosh Divvala, Ross Girshick, and Ali Farhadi. You only look once: Unified, real-time object detection. In CVPR, 2016.  
+Shaoqing Ren, Kaiming He, Ross Girshick, and Jian Sun. Faster r-cnn: Towards real-time object detection with region proposal networks. In NIPS, 2015.  
+Sara Sabour, Nicholas Frosst, and Geoffrey E Hinton. Dynamic routing between capsules. In NIPS, 2017.
+
+Ramprasaath R Selvaraju, Michael Cogswell, Abhishek Das, Ramakrishna Vedantam, Devi Parikh, and Dhruv Batra. Grad-cam: Visual explanations from deep networks via gradient-based localization. In ICCV, 2017.  
+Karen Simonyan and Andrew Zisserman. Very deep convolutional networks for large-scale image recognition. arXiv preprint arXiv:1409.1556, 2014.  
+Yaniv Taigman, Ming Yang, Marc'Aurelio Ranzato, and Lior Wolf. Deepface: Closing the gap to human-level performance in face verification. In CVPR, 2014.  
+F Visin, K Kastner, K Cho, M Matteucci, A Courville, and Y Bengio. A recurrent neural network based alternative to convolutional networks. arXiv preprint arXiv:1505.00393, 2015.  
+Lijun Wang, Huchuan Lu, Yifan Wang, Mengyang Feng, Dong Wang, Baocai Yin, and Xiang Ruan. Learning to detect salient objects with image-level supervision. In CVPR, 2017.  
+M. D. Zeiler, G. W. Taylor, and R. Fergus. Adaptive deconvolutional networks for mid and high level feature learning. In ICCV, 2011.  
+Matthew D Zeiler and Rob Fergus. Visualizing and understanding convolutional networks. In ECCV, 2014.  
+Chiyuan Zhang, Samy Bengio, Moritz Hardt, Benjamin Recht, and Oriol Vinyals. Understanding deep learning requires rethinking generalization. arXiv preprint arXiv:1611.03530, 2016.  
+Pingping Zhang, Dong Wang, Huchuan Lu, Hongyu Wang, and Xiang Ruan. Amulet: Aggregating multi-level convolutional features for salient object detection. In ICCV, 2017.
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+# IMPROVING GENERALIZATION IN META REINFORCEMENT LEARNING USING LEARNED OBJECTIVES
+
+Louis Kirsch, Sjoerd van Steenkiste, Jürgen Schmidhuber
+
+The Swiss AI Lab IDSIA, USI, SUPSI
+
+{louis, sjoerd, juergen}@idsia.ch
+
+# ABSTRACT
+
+Biological evolution has distilled the experiences of many learners into the general learning algorithms of humans. Our novel meta reinforcement learning algorithm MetaGenRL is inspired by this process. MetaGenRL distills the experiences of many complex agents to meta-learn a low-complexity neural objective function that decides how future individuals will learn. Unlike recent meta-RL algorithms, MetaGenRL can generalize to new environments that are entirely different from those used for meta-training. In some cases, it even outperforms human-engineered RL algorithms. MetaGenRL uses off-policy second-order gradients during meta-training that greatly increase its sample efficiency.
+
+# 1 INTRODUCTION
+
+The process of evolution has equipped humans with incredibly general learning algorithms. They enable us to solve a wide range of problems, even in the absence of a large number of related prior experiences. The algorithms that give rise to these capabilities are the result of distilling the collective experiences of many learners throughout the course of natural evolution. By essentially learning from learning experiences in this way, the resulting knowledge can be compactly encoded in the genetic code of an individual to give rise to the general learning capabilities that we observe today.
+
+In contrast, Reinforcement Learning (RL) in artificial agents rarely proceeds in this way. The learning rules that are used to train agents are the result of years of human engineering and design, (e.g. Williams (1992); Wierstra et al. (2008); Mnih et al. (2013); Lillicrap et al. (2016); Schulman et al. (2015a)). Correspondingly, artificial agents are inherently limited by the ability of the designer to incorporate the right inductive biases in order to learn from previous experiences.
+
+Several works have proposed an alternative framework based on meta reinforcement learning (Schmidhuber, 1994; Wang et al., 2016; Duan et al., 2016; Finn et al., 2017; Houthooft et al., 2018; Clune, 2019). Meta-RL distinguishes between learning to act in the environment (the reinforcement learning problem) and learning to learn (the meta-learning problem). Hence, learning itself is now a learning problem, which in principle allows one to leverage prior learning experiences to meta-learn general learning rules that surpass human-engineered alternatives. However, while prior work found that learning rules could be meta-learned that generalize to slightly different environments or goals (Finn et al., 2017; Plappert et al., 2018; Houthooft et al., 2018), generalization to entirely different environments remains an open problem.
+
+In this paper we present MetaGenRL<sup>1</sup>, a novel meta reinforcement learning algorithm that meta-learns learning rules that generalize to entirely different environments. MetaGenRL is inspired by the process of natural evolution as it distills the experiences of many agents into the parameters of an objective function that decides how future individuals will learn. Similar to Evolved Policy Gradients (EPG; Houthooft et al. (2018)), it meta-learns low complexity neural objective functions that can be used to train complex agents with many parameters. However, unlike EPG, it is able to meta-learn using second-order gradients, which offers several advantages as we will demonstrate.
+
+We evaluate MetaGenRL on a variety of continuous control tasks and compare to  $\mathbf{R}\mathbf{L}^2$  (Wang et al., 2016; Duan et al., 2016) and EPG in addition to several human engineered learning algorithms.
+
+Compared to  $\mathrm{RL}^2$  we find that MetaGenRL does not overfit and is able to train randomly initialized agents using meta-learned learning rules on entirely different environments. Compared to EPG we find that MetaGenRL is more sample efficient, and outperforms significantly under a fixed budget of environment interactions. The results of an ablation study and additional analysis provide further insight into the benefits of our approach.
+
+# 2 PRELIMINARIES
+
+Notation We consider the standard MDP Reinforcement Learning setting defined by a tuple  $e = (S, A, P, \rho_0, r, \gamma, T)$  consisting of states  $S$ , actions  $A$ , the transition probability distribution  $P: S \times A \times S \to \mathbb{R}_+$ , an initial state distribution  $\rho_0: S \to \mathbb{R}_+$ , the reward function  $r: S \times A \to [-R_{max}, R_{max}]$ , a discount factor  $\gamma$ , and the episode length  $T$ . The objective for the probabilistic policy  $\pi_\phi: S \times A \to \mathbb{R}_+$  parameterized by  $\phi$  is to maximize the expected discounted return:
+
+$$
+\mathbb {E} _ {\tau} \left[ \sum_ {t = 0} ^ {T - 1} \gamma^ {t} r _ {t} \right], \text {w h e r e} s _ {0} \sim \rho_ {0} (s _ {0}), a _ {t} \sim \pi_ {\phi} (a _ {t} | s _ {t}), s _ {t + 1} \sim P (s _ {t + 1} | s _ {t}, a _ {t}), r _ {t} = r (s _ {t}, a _ {t}), \tag {1}
+$$
+
+with  $\tau = (s_0, a_0, r_0, s_1, \dots, s_{T-1}, a_{T-1}, r_{T-1})$ .
+
+Human Engineered Gradient Estimators A popular gradient-based approach to maximizing Equation 1 is REINFORCE (Williams, 1992). It directly differentiates Equation 1 with respect to  $\phi$  using the likelihood ratio trick to derive gradient estimates of the form:
+
+$$
+\nabla_ {\phi} \mathbb {E} _ {\tau} \left[ L _ {R E I N F} \left(\tau , \pi_ {\phi}\right) \right] := \mathbb {E} _ {\tau} \left[ \nabla_ {\phi} \sum_ {t = 0} ^ {T - 1} \log \pi_ {\phi} \left(a _ {t} \mid s _ {t}\right) \cdot \sum_ {t ^ {\prime} = t} ^ {T - 1} \gamma^ {t ^ {\prime} - t} r _ {t ^ {\prime}}\right). \tag {2}
+$$
+
+Although this basic estimator is rarely used in practice, it has become a building block for an entire class of policy-gradient algorithms of this form. For example, a popular extension from Schulman et al. (2015b) combines REINFORCE with a Generalized Advantage Estimate (GAE) to yield the following policy gradient estimator:
+
+$$
+\nabla_ {\phi} \mathbb {E} _ {\tau} \left[ L _ {G A E} \left(\tau , \pi_ {\phi}, V\right) \right] := \mathbb {E} _ {\tau} \left[ \nabla_ {\phi} \sum_ {t = 0} ^ {T - 1} \log \pi_ {\phi} \left(a _ {t} \mid s _ {t}\right) \cdot A (\tau , V, t) \right]. \tag {3}
+$$
+
+where  $A(\tau, V, t)$  is the GAE and  $V: S \to \mathbb{R}$  is a value function estimate. Several recent other extensions include TRPO (Schulman et al., 2015a), which discourages bad policy updates using trust regions and iterative off-policy updates, or PPO (Schulman et al., 2017), which offers similar benefits using only first order approximations.
+
+Parametrized Objective Functions In this work we note that many of these human engineered policy gradient estimators can be viewed as specific implementations of a general objective function  $L$  that is differentiated with respect to the policy parameters:
+
+$$
+\nabla_ {\phi} \mathbb {E} _ {\tau} [ L (\tau , \pi_ {\phi}, V) ]. \tag {4}
+$$
+
+Hence, it becomes natural to consider a generic parametrization of  $L$  that, for various choices of parameters  $\alpha$ , recovers some of these estimators. In this paper, we will consider neural objective functions where  $L_{\alpha}$  is implemented by a neural network. Our goal is then to optimize the parameters  $\alpha$  of this neural network in order to give rise to a new learning algorithm that best maximizes Equation 1 on an entire class of (different) environments.
+
+# 3 META-LEARNING NEURAL OBJECTIVES
+
+In this work we propose MetaGenRL, a novel meta reinforcement learning algorithm that meta-learns neural objective functions of the form  $L_{\alpha}(\tau ,\pi_{\phi},V)$ . MetaGenRL makes use of value functions and second-order gradients, which makes it more sample efficient compared to prior work (Duan et al., 2016; Wang et al., 2016; Houthooft et al., 2018). More so, as we will demonstrate, MetaGenRL meta-learns objective functions that generalize to vastly different environments.
+
+![](images/044d0f429797a73e4c2ba646a796544f5cfb843d9f28e894fd6777b531f52de3.jpg)  
+Figure 1: A schematic of MetaGenRL. On the left a population of agents  $(i\in 1,\dots ,N)$ , where each member consists of a critic  $Q_{\theta}^{(i)}$  and a policy  $\pi_{\phi}^{(i)}$  that interact with a particular environment  $e^{(i)}$  and store collected data in a corresponding replay buffer  $B^{(i)}$ . On the right a meta-learned neural objective function  $L_{\alpha}$  that is shared across the population. Learning (dotted arrows) proceeds as follows: Each policy is updated by differentiating  $L_{\alpha}$ , while the critic is updated using the usual TD-error (not shown).  $L_{\alpha}$  is meta-learned by computing second-order gradients that can be obtained by differentiating through the critic.
+
+Our key insight is that a differentiable critic  $Q_{\theta}: S \times A \to \mathbb{R}$  can be used to measure the effect of locally changing the objective function parameters  $\alpha$  based on the quality of the corresponding policy gradients. This enables a population of agents to use and improve a single parameterized objective function  $L_{\alpha}$  through interacting with a set of (potentially different) environments. During evaluation (meta-test time), the meta-learned objective function can then be used to train a randomly initialized RL agent in a new environment.
+
+# 3.1 FROM DDPG TO GRADIENT-BASED META-LEARNING OF NEURAL OBJECTIVES
+
+We will formally introduce MetaGenRL as an extension of the DDPG actor-critic framework (Silver et al., 2014; Lillicrap et al., 2016). In DDPG, a parameterized critic of the form  $Q_{\theta}: S \times A \to \mathbb{R}$  transforms the non-differentiable RL reward maximization problem into a myopic value maximization problem for any  $s_t \in S$ . This is done by alternating between optimization of the critic  $Q_{\theta}$  and the (here deterministic) policy  $\pi_{\phi}$ . The critic is trained to minimize the TD-error by following:
+
+$$
+\nabla_ {\theta} \sum_ {\left(s _ {t}, a _ {t}, r _ {t}, s _ {t + 1}\right)} \left(Q _ {\theta} \left(s _ {t}, a _ {t}\right) - y _ {t}\right) ^ {2}, \text {w h e r e} y _ {t} = r _ {t} + \gamma \cdot Q _ {\theta} \left(s _ {t + 1}, \pi_ {\phi} \left(s _ {t + 1}\right)\right), \tag {5}
+$$
+
+and the dependence of  $y_{t}$  on the parameter vector  $\theta$  is ignored. The policy  $\pi_{\phi}$  is improved to increase the expected return from arbitrary states by following the gradient  $\nabla_{\phi}\sum_{s_t}Q_\theta (s_t,\pi_\phi (s_t))$ . Both gradients can be computed entirely off-policy by sampling trajectories from a replay buffer.
+
+MetaGenRL builds on this idea of differentiating the critic  $Q_{\theta}$  with respect to the policy parameters. It incorporates a parameterized objective function  $L_{\alpha}$  that is used to improve the policy (i.e. by following the gradient  $\nabla_{\phi}L_{\alpha}$ ), which adds one extra level of indirection: The critic  $Q_{\theta}$  improves  $L_{\alpha}$ , while  $L_{\alpha}$  improves the policy  $\pi_{\phi}$ . By first differentiating with respect to the objective function parameters  $\alpha$ , and then with respect to the policy parameters  $\phi$ , the critic can be used to measure the effect of updating  $\pi_{\phi}$  using  $L_{\alpha}$  on the estimated return<sup>2</sup>:
+
+$$
+\nabla_ {\alpha} Q _ {\theta} \left(s _ {t}, \pi_ {\phi^ {\prime}} \left(s _ {t}\right)\right), \text {w h e r e} \phi^ {\prime} = \phi - \nabla_ {\phi} L _ {\alpha} (\tau , x (\phi), V). \tag {6}
+$$
+
+This constitutes a type of second order gradient  $\nabla_{\alpha}\nabla_{\phi}$  that can be used to meta-train  $L_{\alpha}$  to provide better updates to the policy parameters in the future. In practice we will use batching to optimize Equation 6 over multiple trajectories  $\tau$ .
+
+Similarly to the policy-gradient estimators from Section 2, the objective function  $L_{\alpha}(\tau ,x(\phi),V)$  receives as inputs an episode trajectory  $\tau = (s_{0:T - 1},a_{0:T - 1},r_{0:T - 1})$ , the value function estimates
+
+Algorithm 1 MetaGenRL: Meta-Training  
+Require:  $p(e)$  a distribution of environments  $P\Leftarrow \{(e_1\sim p(e),\phi_1,\theta_1,B_1\leftarrow \emptyset),\ldots \}$  Randomly initialize objective function  $L_{\alpha}$  while  $L_{\alpha}$  has not converged do for  $e,\phi ,\theta ,B\in P$  do For each agent  $i$  in parallel if extend replay buffer  $B$  then Extend  $B$  using  $\pi_{\phi}$  in  $e$  Sample trajectories from  $B$  Update critic  $Q_{\theta}$  using TD-error Update policy by following  $\nabla_{\phi}L_{\alpha}$  Compute objective function gradient  $\Delta_{i}$  for agent  $i$  according to Equation 6 Sum gradients  $\sum_{i}\Delta_{i}$  to update  $L_{\alpha}$
+
+$V$ , and an auxiliary input  $x(\phi)$  (previously  $\pi_{\phi}$ ) that can be differentiated with respect to the policy parameters. The latter is critical to be able to differentiate with respect to  $\phi$  and in the simplest case it consists of the action as predicted by the policy. While Equation 6 is used for meta-learning  $L_{\alpha}$ , the objective function  $L_{\alpha}$  itself is used for policy learning by following  $\nabla_{\phi}L_{\alpha}(\tau, x(\phi), V)$ . See Figure 1 for an overview. MetaGenRL consists of two phases: During meta-training, we alternate between critic updates, objective function updates, and policy updates to meta-learn an objective function  $L_{\alpha}$  as described in Algorithm 1. During meta-testing in Algorithm 2, we take the learned objective function  $L_{\alpha}$  and keep it fixed while training a randomly initialized policy in a new environment to assess its performance.
+
+We note that the inputs to  $L_{\alpha}$  are sampled from a replay buffer rather than solely using on-policy data. If  $L_{\alpha}$  were to represent a REINFORCE-type objective then it would mean that differentiating  $L_{\alpha}$  yields biased policy gradient estimates. In our experiments we will find that the gradients from  $L_{\alpha}$  work much better in comparison to a biased off-policy REINFORCE algorithm, and to an importance-sampled unbiased REINFORCE algorithm, while also improving over the popular on-policy REINFORCE and PPO algorithms.
+
+# 3.2 PARAMETRIZING THE OBJECTIVE FUNCTION
+
+We will implement  $L_{\alpha}$  using an LSTM (Gers et al., 2000; Hochreiter & Schmidhuber, 1997) that iterates over  $\tau$  in reverse order and depends on the current policy action  $\pi_{\phi}(s_t)$  (see Figure 2). At every time-step  $L_{\alpha}$  receives the reward  $r_t$ , taken action  $a_t$ , predicted action by the current policy  $\pi_{\phi}(s_t)$ , the time  $t$ , and value function estimates  $V_t, V_{t+1}^3$ . At each step the LSTM outputs the objective value  $l_t$ , all of which are summed to yield a single scalar output value that can be differentiated with respect to  $\phi$ . In order to accommodate varying action dimensionalities across different environments, both  $\pi_{\phi}(s_t)$  and  $a_t$  are first convolved and then averaged to obtain an action embedding that does not depend on the action dimensionality. Additional details, including suggestions for more expressive alternatives are available in Appendix B.
+
+By presenting the trajectory in reverse order to the LSTM (and  $L_{\alpha}$  correspondingly), it is able to assign credit to an action  $a_{t}$  based on its future impact on the reward, similar to policy gradient estimators. More so, as a general function approximator using these inputs, the LSTM is in principle able to learn different variance and bias reduction techniques, akin to advantage estimates, generalized advantage estimates, or importance weights<sup>4</sup>. Due to these properties, we expect the class of objective functions that is supported to somewhat relate to a REINFORCE (Williams, 1992) estimator that uses generalized advantage estimation (Schulman et al., 2015b).
+
+<table><tr><td>Algorithm 2 MetaGenRL: Meta-Testing</td></tr><tr><td>Require: A test environment e, and an objective function Lα
+Randomly initialize πφ, Vθ, B ← ∅
+while πφ has not converged do
+if extend replay buffer B then
+Extend B using πφ in e
+Sample trajectories from B
+Update Vθ using TD-error
+Update policy by following ∇φLα</td></tr></table>
+
+![](images/e6dbad9116e3fd51452c8baba84632692c943d216dcb12a171862b04fcb66d35.jpg)  
+Figure 2: An overview of  $L_{\alpha}(\tau, x(\phi), V)$ .
+
+# 3.3 GENERALITY AND EFFICIENCY OF METAGENRL
+
+MetaGenRL offers a general framework for meta-learning objective functions that can represent a wide range of learning algorithms. In particular, it is only required that both  $\pi_{\phi}$  and  $L_{\alpha}$  can be differentiated w.r.t. to the policy parameters  $\phi$ . In the present work, we use this flexibility to leverage population-based meta-optimization, increase sample efficiency through off-policy second-order gradients, and to improve the generalization capabilities of meta-learned objective functions.
+
+Population-Based A general objective function should be applicable to a wide range of environments and agent parameters. To this extent MetaGenRL is able to leverage the collective experience of multiple agents to perform meta-learning by using a single objective function  $L_{\alpha}$  shared among a population of agents that each act in their own (potentially different) environment. Each agent locally computes Equation 6 over a batch of trajectories, and the resulting gradients are combined to update  $L_{\alpha}$ . Thus, the relevant learning experience of each individual agent is compressed into the objective function that is available to the entire population at any given time.
+
+Sample Efficiency An alternative to learning neural objective functions using a population of agents is through evolution as in EPG (Houthooft et al., 2018). However, we expect meta-learning using second-order gradients as in MetaGenRL to be much more sample efficient. This is due to off-policy training of the objective function  $L_{\alpha}$  and its subsequent off-policy use to improve the policy. Indeed, unlike in evolution there is no need to train multiple randomly initialized agents in their entirety in order to evaluate the objective function, thus speeding up credit assignment. Rather, at any point in time, any information that is deemed useful for future environment interactions can directly be incorporated into the objective function. Finally, using the formulation in Equation 6 one can measure the effects of improving the policy using  $L_{\alpha}$  for multiple steps by increasing the corresponding number of gradient steps before applying  $Q_{\theta}$ , which we will explore in Section 5.2.3.
+
+Meta-Generalization The focus of this work is to learn general learning rules that during test-time can be applied to vastly different environments. A strict separation between the policy and the learning rule, the functional form of the latter, and training across many environments all contribute to this. Regarding the former, a clear separation between the policy and the learning rule as in MetaGenRL is expected to be advantageous for two reasons. Firstly, it allows us to specify the number of parameters of the learning rule independent of the policy and critic parameters. For example, our implementation of  $L_{\alpha}$  uses only  $15K$  parameters for the objective function compared to  $384K$  parameters for the policy and critic. Hence, we are able to only use a short description length for the learning rule. A second advantage that is gained is that the meta-learner is unable to directly change the policy and must, therefore, learn to make use of the objective function. This makes it difficult for the meta-learner to overfit to the training environments.
+
+# 4 RELATED WORK
+
+Among the earliest pursuits in meta-learning are meta-hierarchies of genetic algorithms (Schmidhuber, 1987) and learning update rules in supervised learning (Bengio et al., 1990). While the former introduced a general framework of entire meta-hierarchies, it relied on discrete non-differentiable programs. The latter introduced local update rules that included free parameters, which could be
+
+learned using gradients in a supervised setting. Schmidhuber (1993) introduced a differentiable self-referential RNN that could address and modify its own weights, albeit difficult to learn.
+
+Hochreiter et al. (2001) introduced differentiable meta-learning using RNNs to scale to larger problem instances. By giving an RNN access to its prediction error, it could implement its own meta-learning algorithm, where the weights are the meta-learned parameters, and the hidden states the subject of learning. This was later extended to the RL setting (Wang et al., 2016; Duan et al., 2016; Santoro et al., 2016; Mishra et al., 2018) (here referred to as  $\mathrm{RL}^2$ ). As we show empirically in our paper, meta-learning with  $\mathrm{RL}^2$  does not generalize well. It lacks a clear separation between policy and objective function, which makes it easy to overfit on training environments. This is exacerbated by the imbalance of  $O(n^{2})$  meta-learned parameters to learn  $O(n)$  activations, unlike in MetaGenRL.
+
+Many other recent meta-learning algorithms learn a policy parameter initialization that is later finetuned using a fixed reinforcement learning algorithm (Finn et al., 2017; Schulman et al., 2017; Grant et al., 2018; Yoon et al., 2018). Different from MetaGenRL, these approaches use second order gradients on the same policy parameter vector instead of using a separate objective function. Albeit in principle general (Finn & Levine, 2018), the mixing of policy and learning algorithm leads to a complicated way of expressing general update rules. Similar to RL², adaptation to related tasks is possible, but generalization is difficult (Houthooft et al., 2018).
+
+Objective functions have been learned prior to MetaGenRL. Houthooft et al. (2018) evolve an objective function that is later used to train an agent. Unlike MetaGenRL, this approach is extremely costly in terms of the number of environment interactions required to evaluate and update the objective function. Most recently, Bechtle et al. (2019) introduced learned loss functions for reinforcement learning that also make use of second-order gradients, but use a policy gradient estimator instead of a Q-function. Similar to other work, their focus is only on narrow task distributions. Learned objective functions have also been used for learning unsupervised representations (Metz et al., 2019), DDPG-like meta-gradients for hyperparameter search (Xu et al., 2018), and learning from human demonstrations (Yu et al., 2018). Concurrent to our work, Alet et al. (2020) uses techniques from architecture search to search for viable artificial curiosity objectives that are composed of primitive objective functions.
+
+Li & Malik (2016; 2017) and Andrychowicz et al. (2016) conduct meta-learning by learning optimizers that update parameters  $\phi$  by modulating the gradient of some fixed objective function  $L$ :  $\Delta \phi = f_{\alpha}(\nabla_{\phi}L)$  where  $\alpha$  is learned. They differ from MetaGenRL in that they only modulate the gradient of a fixed objective function  $L$  instead of learning  $L$  itself.
+
+Another connection exists to meta-learned intrinsic reward functions (Schmidhuber, 1991a; Dayan & Hinton, 1993; Wiering & Schmidhuber, 1996; Singh et al., 2004; Niekum et al., 2011; Zheng et al., 2018; Jaderberg et al., 2019). Choosing  $\nabla_{\phi}L_{\alpha} = \tilde{\nabla}_{\phi}\sum_{t=1}^{T}\bar{r}_{t}(\tau)$ , where  $\bar{r}_{t}$  is a meta-learned reward and  $\tilde{\nabla}_{\theta}$  is a gradient estimator (such as a value based or policy gradient based estimator) reveals that meta-learning objective functions include meta-learning the gradient estimator  $\tilde{\nabla}$  itself as long as it is expressible by a gradient  $\nabla_{\theta}$  on an objective  $L_{\alpha}$ . In contrast, for intrinsic reward functions, the gradient estimator  $\tilde{\nabla}$  is normally fixed.
+
+Finally, we note that positive transfer between different tasks (reward functions) as well as environments (e.g. different Atari games) has been shown previously in the context of transfer learning (Kistler et al., 1997; Parisotto et al., 2015; Rusu et al., 2016; 2019; Nichol et al., 2018) and meta-critic learning across tasks (Sung et al., 2017). In contrast to this work, the approaches that have shown to be successful in this domain rely entirely on human-engineered learning algorithms.
+
+# 5 EXPERIMENTS
+
+We investigate the learning and generalization capabilities of MetaGenRL on several continuous control benchmarks including HalfCheetah (Cheetah) and Hopper from MuJoCo (Todorov et al., 2012), and LunarLanderContinuous (Lunar) from OpenAI gym (Brockman et al., 2016). These environments differ significantly in terms of the properties of the underlying system that is to be controlled, and in terms of the dynamics that have to be learned to complete the environment. Hence, by training meta-RL algorithms on one environment and testing on other environments they provide a reasonable measure of out-of-distribution generalization.
+
+Table 1: Mean return across multiple seeds (MetaGenRL: 6 meta-train  $\times$  2 meta-test seeds,  $\mathbf{R}\mathbf{L}^{2}$  : 6 meta-train  $\times$  2 meta-test seeds, EPG: 3 meta-train  $\times$  2 meta-test seeds) obtained by training randomly initialized agents during meta-test time on previously seen environments (cyan) and on unseen environments (brown). Boldface highlights best meta-learned algorithm. Mean returns (6 seeds) of several human-engineered algorithms are also listed.  
+
+<table><tr><td colspan="2">Training \Testing</td><td>Cheetah</td><td>Hopper</td><td>Lunar</td></tr><tr><td rowspan="3">Cheetah &amp; Hopper</td><td>MetaGenRL</td><td>2185</td><td>2439</td><td>18</td></tr><tr><td>EPG</td><td>-571</td><td>20</td><td>-540</td></tr><tr><td>\( RL^2 \)</td><td>5180</td><td>289</td><td>-479</td></tr><tr><td rowspan="3">Lunar &amp; Cheetah</td><td>MetaGenRL</td><td>2552</td><td>2363</td><td>258</td></tr><tr><td>EPG</td><td>-701</td><td>8</td><td>-707</td></tr><tr><td>\( RL^2 \)</td><td>2218</td><td>5</td><td>283</td></tr><tr><td>Lunar &amp; Hopper &amp; Walker &amp; Ant</td><td>MetaGenRL (40 agents)</td><td>3106</td><td>2869</td><td>201</td></tr><tr><td>Cheetah &amp; Lunar &amp; Walker &amp; Ant</td><td></td><td>3331</td><td>2452</td><td>-71</td></tr><tr><td>Cheetah &amp; Hopper &amp; Walker &amp; Ant</td><td></td><td>2541</td><td>2345</td><td>-148</td></tr><tr><td rowspan="4"></td><td>PPO</td><td>1455</td><td>1894</td><td>187</td></tr><tr><td>DDPG / TD3</td><td>8315</td><td>2718</td><td>288</td></tr><tr><td>off-policy REINFORCE (GAE)</td><td>-88</td><td>1804</td><td>168</td></tr><tr><td>on-policy REINFORCE (GAE)</td><td>38</td><td>565</td><td>120</td></tr></table>
+
+In our experiments, we will mainly compare to EPG and to RL² to evaluate the efficacy of our approach. We will also compare to several fixed model-free RL algorithms to measure how well the algorithms meta-learned by MetaGenRL compare to these handcrafted alternatives. Unless otherwise mentioned, we will meta-train MetaGenRL using 20 agents that are distributed equally over the indicated training environments⁵. Meta-learning uses clipped double-Q learning, delayed policy & objective updates, and target policy smoothing from TD3 (Fujimoto et al., 2018). We will allow for  $600K$  environment interactions per agent during meta-training and then meta-test the objective function for  $1M$  interactions. Further details are available in Appendix B.
+
+# 5.1 COMPARISON TO PRIOR WORK
+
+**Evaluating on previously seen environments** We meta-train MetaGenRL on Lunar and compare its ability to train a randomly initialized agent at test-time (i.e. using the learned objective function and keeping it fixed) to DDPG, PPO, and on- and off-policy REINFORCE (both using GAE) across multiple seeds. Figure 3a shows that MetaGenRL markedly outperforms both the REINFORCE baselines and PPO. Compared to DDPG, which finds the optimal policy, MetaGenRL performs only slightly worse on average although the presence of outliers increases its variance. In particular, we find that some meta-test agents get 'stuck' for some time before reaching the optimal policy (see Section A.2 for additional analysis). Indeed, when evaluating only the best meta-learned objective function that was obtained during meta-training (MetaGenRL (best objective func) in Figure 3a) we are able to observe a strong reduction in variance and even better performance.
+
+We also report results (Figure 3a) when meta-training MetaGenRL on both Lunar and Cheetah, and compare to EPG and  $\mathrm{RL}^2$  that were meta-trained on these same environments. For MetaGenRL we were able to obtain similar performance to meta-training on only Lunar in this case. In contrast, for EPG it can be observed that even one billion environment interactions is insufficient to find a good objective function (in Figure 3a quickly dropping below -300). Finally, we find that  $\mathrm{RL}^2$  reaches the optimal policy after 100 million meta-training iterations, and that its performance is unaffected by additional steps during testing on Lunar. We note that  $\mathrm{RL}^2$  does not separate the policy and the learning rule and indeed in a similar 'within distribution' evaluation,  $\mathrm{RL}^2$  was found successful (Wang et al., 2016; Duan et al., 2016).
+
+![](images/026c2b9bffa1f5f9d0cf8d25e0f71be82283679f46ff81a167cbfb7035e9b2b2.jpg)  
+(a) Previously seen Lunar environment.
+
+![](images/de1919804d968a8ece41747c6e59f75bd60d4c2c4469e25aa5559362936e875d.jpg)  
+(b) Unseen Hopper environment.  
+Figure 3: Comparing the test-time training behavior of the meta-learned objective functions by MetaGenRL to other (meta) reinforcement learning algorithms. We train randomly initialized agents on (a) environments that were encountered during training, and (b) on significantly different environments that were unseen. Training environments are denoted by  $\dagger$  in the legend. All runs are shown with mean and standard deviation computed over multiple random seeds (MetaGenRL: 6 meta-train  $\times 2$  meta-test seeds,  $\mathrm{RL}^2$ : 6 meta-train  $\times 2$  meta-test seeds, EPG: 3 meta-train  $\times 2$  meta-test seeds, and 6 seeds for all others).
+
+Table 1 provides a similar comparison for two other environments. Here we find that in general MetaGenRL is able to outperform the REINFORCE baselines and PPO, and in most cases (except for Cheetah) performs similar to DDPG<sup>7</sup>. We also find that MetaGenRL consistently outperforms EPG, and often RL<sup>2</sup>. For an analysis of meta-training on more than two environments we refer to Appendix A.
+
+Generalization to vastly different environments We evaluate the same objective functions learned by MetaGenRL, EPG and the recurrent dynamics by  $\mathbf{R}\mathbf{L}^2$  on Hopper, which is significantly different compared to the meta-training environments. Figure 3b shows that the learned objective function by MetaGenRL continues to outperform both PPO and our implementations of REINFORCE, while the best performing configuration is even able to outperform DDPG.
+
+When comparing to related meta-RL approaches, we find that MetaGenRL is significantly better in this case. The performance of EPG remains poor, which was expected given what was observed on previously seen environments. On the other hand, we now find that the  $\mathrm{RL}^2$  baseline fails completely (resulting in a flat low-reward evaluation), suggesting that the learned learning rule that was previously found to be successful is in fact entirely overfitted to the environments that were seen during meta-training. We were able to observe similar results when using different train and test environment splits as reported in Table 1, and in Appendix A.
+
+# 5.2 ANALYSIS
+
+# 5.2.1 META-TRAINING PROGRESSION OF OBJECTIVE FUNCTIONS
+
+Previously we focused on test-time training randomly initialized agents using an objective function that was meta-trained for a total of  $600K$  steps (corresponding to a total of  $12M$  environment interactions across the entire population). We will now investigate the quality of the objective functions during meta-training.
+
+Figure 4 displays the result of evaluating an objective function on Hopper at different intervals during meta-training on Cheetah and Lunar. Initially (28K steps) it can be seen that due to lack of meta-training there is only a marginal improvement in the return obtained during test time. However, after only meta-training for 86K steps we find (perhaps surprisingly) that the meta-trained
+
+![](images/93d60d9cdc36d645b7a499b31ffaf85e2acfe900d1d81f52ca747b4f8801a24b.jpg)  
+Figure 4: Meta-training with 20 agents on Cheetah and Lunar. We test the objective function at five stages of meta-training by using it to train three randomly initialized agents on Hopper.
+
+![](images/c25c858d73c2b638a54582b6f7141dcb2e25335a0b89a87a9a4031706c10e15a.jpg)  
+(a) Meta-training on Lunar & Cheetah
+
+![](images/644a3b9b4f445fe2c504dbca0caff9e8ea1f3b4cf2029c7d9236a08858071ce6.jpg)  
+Figure 5: We meta-train MetaGenRL using several alternative parametrizations of  $L_{\alpha}$  on a) Lunar and Cheetah, and b) present results of testing on Cheetah. During meta-training a representative example of a single agent population is shown with shaded regions denoting standard deviation across the population. Meta-test results are reported as per usual across 6 meta-train × 2 meta-test seeds.
+
+![](images/2409f6b3fd7b3b8114f7f047f0633f0431a5bd6b3192a784a39e3ffa5439bf6a.jpg)  
+(b) Meta-testing on Cheetah
+
+objective function is already able to make consistent progress in optimizing a randomly initialized agent during test-time. On the other hand, we observe large variances at test-time during this phase of meta-training. Throughout the remaining stages of meta-training we then observe an increase in convergence speed, more stable updates, and a lower variance across seeds.
+
+# 5.2.2 ABLATION STUDY
+
+We conduct an ablation study of the neural objective function that was described in Section 3.2. In particular, we assess the dependence of  $L_{\alpha}$  on the value estimates  $V_{t}, V_{t+1}$  and on the time component that could to some extent be learned. Other ablations, including limiting access to the action chosen or to the received reward, are expected to be disastrous for generalization to any other environment (or reward function) and therefore not explored.
+
+**Dependence on  $t$ ** We use a parameterized objective function of the form  $L_{\alpha}(a_t, r_t, V_t, \pi_{\phi}(s_t) | t \in [0, \dots, T - 1])$  as in Figure 2 except that it does not receive information about the time-step  $t$  at each step. Although information about the current time-step is required in order to learn (for example), a generalized advantage estimate (Schulman et al., 2015b), the LSTM could in principle learn such time tracking on its own, and we expect only minor effects on meta-training and during meta-testing. Indeed in Figure 5b it can be seen that the neural objective function performs well without access to  $t$ , although it converges slower on Cheetah during meta-training (Figure 5a).
+
+Dependence on  $V$  We use a parameterized objective function of the form  $L_{\alpha}(a_t,r_t,t,\pi_\phi (s_t)|t\in$ $0,\dots ,T - 1)$  as in Figure 2 except that it does not receive any information about the value estimates at time-step  $t$  . There exist reinforcement learning algorithms that work without value function estimates (eg. Williams (1992); Schmidhuber & Zhao (1998)), although in the absence of an alternative baseline these often have a large variance. Similar results are observed for this ablation in Figure 5a
+
+![](images/c790831b3fb8e1d63bbb85f274bb37c103527db2a430c438a67972cc55810286.jpg)  
+Figure 6: We meta-train MetaGenRL on the LunarLander and HalfCheetah environments using one, three, and five inner gradient steps on  $\phi$ . Meta-test results are reported across 3 meta-train  $\times$  2 meta-test seeds.
+
+![](images/60a065aa8f6f2d7cee910e6d9886c253b76adfee8ce99f4223f6b5696d73e1ed.jpg)
+
+during meta-training where a possibly large variance appears to affect meta-training. Correspondingly during test-time (Figure 5b) we do not find any meaningful training progress to take place. In contrast, we find that we can remove the dependence on one of the value function estimates, i.e. remove  $V_{t + 1}$  but keep  $V_{t}$ , which during some runs even increases performance.
+
+# 5.2.3 MULTIPLE GRADIENT STEPS
+
+We analyze the effect of making multiple gradient updates to the policy using  $L_{\alpha}$  before applying the critic to compute second-order gradients with respect to the objective function parameters as in Equation 6. While in previous experiments we have only considered applying a single update, multiple gradient updates might better capture long term effects of the objective function. At the same time, moving further away from the current policy parameters could reduce the overall quality of the second-order gradients. Indeed, in Figure 6 it can be observed that using 3 gradient steps already slightly increases the variance during test-time training on Hopper and Cheetah after metatraining on LunarLander and Cheetah. Similarly, we find that further increasing the number of gradient steps to 5 harms performance.
+
+# 6 CONCLUSION
+
+We have presented MetaGenRL, a novel off-policy gradient-based meta reinforcement learning algorithm that leverages a population of DDPG-like agents to meta-learn general objective functions. Unlike related methods the meta-learned objective functions do not only generalize in narrow task distributions but show similar performance on entirely different tasks while markedly outperforming REINFORCE and PPO. We have argued that this generality is due to MetaGenRL's explicit separation of the policy and learning rule, the functional form of the latter, and training across multiple agents and environments. Furthermore, the use of second order gradients increases MetaGenRL's sample efficiency by several orders of magnitude compared to EPG (Houthooft et al., 2018).
+
+In future work, we aim to further improve the learning capabilities of the meta-learned objective functions, including better leveraging knowledge from prior experiences. Indeed, in our current implementation, the objective function is unable to observe the environment or the hidden state of the (recurrent) policy. These extensions are especially interesting as they may allow more complicated curiosity-based (Schmidhuber, 1991b; 1990; Houthooft et al., 2016; Pathak et al., 2017) or model-based (Schmidhuber, 1990; Weber et al., 2017; Ha & Schmidhuber, 2018) algorithms to be learned. To this extent, it will be important to develop introspection methods that analyze the learned objective function and to scale MetaGenRL to make use of many more environments and agents.
+
+# ACKNOWLEDGEMENTS
+
+We thank Paulo Rauber, Imanol Schlag, and the anonymous reviewers for their feedback. This work was supported by the ERC Advanced Grant (no: 742870) and computational resources by the Swiss National Supercomputing Centre (CSCS, project: s978). We also thank NVIDIA Corporation for donating a DGX-1 as part of the Pioneers of AI Research Award and to IBM for donating a Minsky machine.
+
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+
+# A ADDITIONAL RESULTS
+
+# A.1 ALL TRAINING AND TEST REGIMES
+
+In the main text, we have shown several combinations of meta-training, and testing environments. We will now show results for all combinations, including the respective human engineered baselines.
+
+![](images/725c4b25866354c4b75fc7fa2d6a3d271080473da01d0b9f7331fdce70c80a30.jpg)  
+(a) Within distribution generalization.
+
+![](images/117a6e764425db8f629555c713e60287aabb691c15575bc37045b7862864f329.jpg)  
+(b) Out of distribution generalization.  
+Figure 7: Comparing the test-time training behavior of the meta-learned objective functions by MetaGenRL to other (meta) reinforcement learning algorithms on Hopper. We consider within distribution testing (a), and out of distribution testing (b) by varying the meta-training environments (denoted by  $\dagger$ ) for the meta-RL approaches. All runs are shown with mean and standard deviation computed over multiple random seeds (MetaGenRL: 6 meta-train  $\times$  2 meta-test seeds,  $\mathrm{RL}^2$ : 6 meta-train  $\times$  2 meta-test seeds, EPG: 3 meta-train  $\times$  2 meta-test seeds, and 6 seeds for all others).
+
+Hopper On Hopper (Figure 7) we find that MetaGenRL works well, both in terms of generalization to previously seen environments, and to unseen environments. The PPO, REINFORCE,  $\mathsf{RL}^2$ , and EPG baselines are outperformed significantly. Regarding  $\mathsf{RL}^2$  we observe that it is only able to obtain reward when Hopper was included during meta-training, although its performance is generally poor. Regarding EPG, we observe some learning progress during meta-testing on Hopper after meta-training on Cheetah and Hopper (Figure 7a), although it drops back down quickly as test-time training proceeds. In contrast, when meta-testing on Hopper after meta-training on Cheetah and Lunar (Figure 7b) no test-time training progress is observed at all.
+
+Cheetah Similar results are observed in Figure 8 for Cheetah, where MetaGenRL outperforms PPO and REINFORCE significantly. On the other hand, it can be seen that DDPG notably outperforms MetaGenRL on this environment. It will be interesting to further study these differences in the future to improve the expressibility of our approach. Regarding  $\mathrm{RL}^2$  and EPG only within distribution generalization results are available due to Cheetah having larger observations and / or action spaces compared to Hopper and Lunar. We observe that  $\mathrm{RL}^2$  performs similar to our earlier findings on Hopper but significantly improves in terms of within-distribution generalization (likely due to greater overfitting, as was consistently observed for other splits). EPG shows initially more promise on within distribution generalization (Figure 8a), but ends up like before.
+
+Lunar On Lunar (Figure 9) we find that MetaGenRL is only marginally better compared to the REINFORCE and PPO baselines in terms of within distribution generalization and worse in terms of out of distribution generalization. Analyzing this result reveals that although many of the runs train rather well, some get stuck during the early stages of training without or only delayed recovering. These outliers lead to a seemingly very large variance for MetaGenRL in Figure 9b. We will provide a more detailed analysis of this result in Section A.2. If we focus on the best performing objective function then we observe competitive performance to DDPG (Figure 9a). Nonetheless, we notice that the objective function trained on Hopper generalizes worse to Lunar, despite our earlier result that objective functions trained on Lunar do in fact generalize well to Hopper. MetaGenRL is still able to outperform both  $\mathrm{RL}^2$  and EPG in terms of out of distribution generalization. We do note
+
+![](images/430ab91b678f2c099dfe7638cff87b0addf0b32c5b1a909f1cb1143f40f64610.jpg)  
+(a) Within distribution generalization.
+
+![](images/10585e8142b6d28f96bdfaa1da4c90657dd29eb9616dca80dd9d3ca74989ad8c.jpg)  
+(b) Out of distribution generalization.
+
+![](images/f1e195c46664158c9ef831fd39797648a38060a0a4efdb0700fc3eda33cf9a87.jpg)  
+Figure 8: Comparing the test-time training behavior of the meta-learned objective functions by MetaGenRL to other (meta) reinforcement learning algorithms on Cheetah. We consider within distribution testing (a), and out of distribution testing (b) by varying the meta-training environments (denoted by  $\dagger$ ) for the meta-RL approaches. All runs are shown with mean and standard deviation computed over multiple random seeds (MetaGenRL: 6 meta-train  $\times$  2 meta-test seeds, RL $^2$ : 6 meta-train  $\times$  2 meta-test seeds, EPG: 3 meta-train  $\times$  2 meta-test seeds, and 6 seeds for all others).  
+(a) Within distribution generalization.  
+Figure 9: Comparing the test-time training behavior of the meta-learned objective functions by MetaGenRL to other (meta) reinforcement learning algorithms on Lunar. We consider within distribution testing (a), and out of distribution testing (b) by varying the meta-training environments (denoted by  $\dagger$ ) for the meta-RL approaches. All runs are shown with mean and standard deviation computed over multiple random seeds (MetaGenRL: 6 meta-train  $\times$  2 meta-test seeds,  $\mathrm{RL}^2$ : 6 meta-train  $\times$  2 meta-test seeds, EPG: 3 meta-train  $\times$  2 meta-test seeds, and 6 seeds for all others).
+
+![](images/c7a1662c49d1aa52d96a6a496e3f6381b1b5aebc1dee0dff7ac565b93013217c.jpg)  
+(b) Out of distribution generalization.
+
+that EPG is able to meta-learn objective functions that are able to improve to some extent during test time.
+
+Comparing final scores An overview of the final scores that were obtained for MetaGenRL in comparison to the human engineered baselines is shown in Table 2. It can be seen that MetaGenRL outperforms PPO and off-/on-policy REINFORCE in most configurations while DDPG with TD3 tricks remains stronger on two of the three environments. Note that DDPG is currently not among the representable algorithms by MetaGenRL.
+
+# A.2 STABILITY OF LEARNED OBJECTIVE FUNCTIONS
+
+In the results presented in Figure 9 on Lunar we observed a seemingly large variance for MetaGenRL that was due to outliers. Indeed, when analyzing the individual runs meta-trained on Lunar and tested on Lunar we found that that one of the runs converged to a local optimum early on during
+
+Table 2: Agent mean return across multiple seeds (MetaGenRL: 6 meta-train  $\times$  2 meta-test seeds, and 6 seeds for all others) for meta-test training on previously seen environments (cyan) and on unseen (different) environments (brown) compared to human engineered baselines.  
+
+<table><tr><td></td><td>Training (below) / Test (right)</td><td>Cheetah</td><td>Hopper</td><td>Lunar</td></tr><tr><td rowspan="5">MetaGenRL (20 agents)</td><td>Cheetah &amp; Hopper</td><td>2185</td><td>2433</td><td>18</td></tr><tr><td>Cheetah &amp; Lunar</td><td>2551</td><td>2363</td><td>258</td></tr><tr><td>Hopper &amp; Lunar</td><td>4160</td><td>2966</td><td>146</td></tr><tr><td>Hopper</td><td>3646</td><td>2937</td><td>-62</td></tr><tr><td>Lunar</td><td>4366</td><td>2717</td><td>244</td></tr><tr><td rowspan="3">MetaGenRL (40 agents)</td><td>Lunar &amp; Hopper &amp; Walker &amp; Ant</td><td>3106</td><td>2869</td><td>201</td></tr><tr><td>Cheetah &amp; Lunar &amp; Walker &amp; Ant</td><td>3331</td><td>2452</td><td>-71</td></tr><tr><td>Cheetah &amp; Hopper &amp; Walker &amp; Ant</td><td>2541</td><td>2345</td><td>-148</td></tr><tr><td>PPO</td><td>-</td><td>1455</td><td>1894</td><td>187</td></tr><tr><td>DDPG / TD3</td><td>-</td><td>8315</td><td>2718</td><td>288</td></tr><tr><td>off-policy REINFORCE (GAE)</td><td>-</td><td>-88</td><td>1804</td><td>168</td></tr><tr><td>on-policy REINFORCE (GAE)</td><td>-</td><td>38</td><td>565</td><td>120</td></tr></table>
+
+![](images/1c775ba2b1d290aadafc5db3921085c9c57578c3b36689b4e8273f428dd608be.jpg)  
+Figure 10: Meta-training with 20 agents on LunarLander. We meta-test the objective function at different stages in training on the same environment.
+
+training and was unable to recover from this afterwards. On the other hand, we also observed that runs can be 'stuck' for a long time to then make very fast learning progress. It suggests that the objective function may sometimes experience difficulties in providing meaningful updates to the policy parameters during the early stages of training.
+
+We have further analyzed this issue by evaluating one of the objective functions at several intervals throughout meta-training in Figure 10. From the meta-training curve (bottom) it can be seen that meta-training in Lunar converges very early. This means that from then on, updates to the objective function will be based on mostly converged policies. As the test-time plots show, these additional updates appear to negatively affect test-time performance. We hypothesize that the objective function essentially ' forgets' about the early stages of training a randomly initialized agent, by only incorporating information about good performing agents. A possible solution to this problem would be to keep older policies in the meta-training agent population or use early stopping.
+
+Finally, if we exclude four random seeds (of 12), we indeed find a significant reduction in the variance (and increase in the mean) of the results observed for MetaGenRL (see Figure 11).
+
+# A.3 ABLATION OF AGENT POPULATION SIZE AND UNIQUE ENVIRONMENTS
+
+In our experiments we have used a population of 20 agents during meta-training to ensure diversity in the conditions under which the objective function needs to optimize. The size of this population is a crucial parameter for a stable meta-optimization. Indeed, in Figure 12 it can be seen that meta-training becomes increasingly unstable as the number of agents in the population decreases.
+
+Using a similar argument, one would expect to gain from increasing the number of distinct environments (or agents) during meta-training. In order to verify this, we have evaluated two additional
+
+![](images/b119296b5c1982419a13cd9770528666ca54cfaf92637d1341da99ce5f905af3.jpg)  
+Figure 11: The left plot shows all 12 random seeds on the meta-test environment Lunar while the right has the 4 worst random seeds removed. The variance is now reduced significantly.
+
+![](images/4f458c57d3e819482df29fcd0f7f020f2d23c4af9219fea30b512d5291673b86.jpg)
+
+![](images/e09afeaee7bcfb43f924e97c4f15d239622c7e51a72cce52202fd7debb8b34bb.jpg)  
+Figure 12: Stable meta-training requires a large population size of at least 20 agents. Metatraining performance is shown for a single run with the mean and standard deviation across the agent population.
+
+![](images/20bb3e76d3b2a62b1192ad4f056e362f9eda4f722f9d5d9277d4ac3b33149312.jpg)  
+Figure 13: Meta-training on Cheetah, Lunar, Walker, and Ant with 20 or 40 agents; meta-testing on the out-of-distribution Hopper environment. We compare to previous MetaGenRL configurations.
+
+settings: Meta-training on Cheetah & Lunar & Walker & Ant with 20 and 40 agents respectively. Figure 13 shows the result of meta-testing on Hopper for these experiments (also see the final results reported for 40 agents in Table 2). Unexpectedly, we find that increasing the number of distinct environments does not yield a significant improvement and, in fact, sometimes even decrease performance. One possibility is that this is due to the simple form of the objective function under consideration, which has no access to the environment observations to efficiently distinguish between them. Another possibility is that MetaGenRL's hyperparameters require additional tuning in order to be compatible with these setups.
+
+# B EXPERIMENT DETAILS
+
+In the following we describe all experimental details regarding the architectures used, meta-training, hyperparameters, and baselines. The code to reproduce our experiments is available at http://louiskirsch.com/code/metagenrl.
+
+# B.1 NEURAL OBJECTIVE FUNCTION ARCHITECTURE
+
+Neural Architecture In this work we use an LSTM to implement the objective function (Figure 2). The LSTM runs backwards in time over the state, action, and reward tuples that were encountered during the trajectory  $\tau$  under consideration. At each step  $t$  the LSTM receives as input the reward  $r_t$ , value estimates of the current and previous state  $V_t$ ,  $V_{t+1}$ , the current timestep  $t$  and finally the action that was taken at the current timestep  $a_t$  in addition to the action as determined by the current policy  $\pi_\phi(s_t)$ . The actions are first processed by one dimensional convolutional layers striding over the action dimension followed by a reduction to the mean. This allows for different action sizes between environments. Let  $A^{(B)} \in \mathbb{R}^{1 \times D}$  be the action from the replay buffer,  $A^{(\pi)} \in \mathbb{R}^{1 \times D}$  be the action predicted by the policy, and  $W \in \mathbb{R}^{2 \times N}$  a learnable matrix corresponding to  $N$  outgoing units, then the actions are transformed by
+
+$$
+\frac {1}{D} \sum_ {i = 1} ^ {D} ([ A ^ {(B)}, A ^ {(\pi)} ] ^ {T} W) _ {i}, \tag {7}
+$$
+
+where  $[a, b]$  is a concatenation of  $a$  and  $b$  along the first axis. This corresponds to a convolution with kernel size 1 and stride 1. Further transformations with non-linearities can be added after applying  $W$ , if necessary. We found it helpful (but not strictly necessary) to use ReLU activations for half of the units and square activations for the other half.
+
+At each time-step the LSTM outputs a scalar value  $l_{t}$  (bounded between  $-\eta$  and  $\eta$  using a scaled tanh activation), which are summed to obtain the value of the neural objective function. Differentiating this value with respect to the policy parameters  $\phi$  then yields gradients that can be used to improve  $\pi_{\phi}$ . We only allow gradients to flow backwards through  $\pi_{\phi}(s_{t})$  to  $\phi$ . This implementation is closely related to the functional form of a REINFORCE (Williams, 1992) estimator using the generalized advantage estimation (Schulman et al., 2015b).
+
+All feed-forward networks (critic and policy) use ReLU activations and layer normalization (Ba et al., 2016). The LSTM uses tanh activations for cell and hidden state transformations, sigmoid activations for the gates. The input time  $t$  is normalized between 0 at the beginning of the episode and 1 at the final transition. Any other hyper-parameters can be seen in Table 3.
+
+Extensibility The expressability of the objective function can be further increased through several means. One possibility is to add the entire sequence of state observations  $o_{1:T}$  to its inputs, or by introducing a bi-directional LSTM. Secondly, additional information about the policy (such as the hidden state of a recurrent policy) can be provided to  $L$ . Although not explored in this work, this would in principle allow one to learn an objective that encourages certain representations to emerge, e.g. a predictive representation about future observations, akin to a world model (Schmidhuber, 1990; Ha & Schmidhuber, 2018; Weber et al., 2017). In turn, these could create pressure to adapt the policy's actions to explore unknown dynamics in the environment (Schmidhuber, 1991b; 1990; Houthooft et al., 2016; Pathak et al., 2017).
+
+# B.2 META-TRAINING
+
+Annealing with DDPG At the beginning of meta-training (learning  $L_{\alpha}$ ), the objective function is randomly initialized and thus does not make sensible updates to the policies. This can lead to irreversibly breaking the policies early during training. Our current implementation circumvents this issue by linearly annealing  $\nabla_{\phi}L_{\alpha}$  the first 10k timesteps ( $\sim 2\%$  of all timesteps) with DDPG  $\nabla_{\phi}Q_{\theta}(s_t,\pi_{\phi}(s_t))$ . Preliminary experiments suggested that an exponential learning rate schedule on the gradient of  $\nabla_{\phi}L_{\alpha}$  for the first 10k steps can replace the annealing with DDPG. The learning rate anneals exponentially between a learning rate of zero and 1e-3. However, in some rare cases this may still lead to unsuccessful training runs, and thus we have omitted this approach from the present work.
+
+Standard training During training, the critic is updated twice as many times as the policy and objective function, similar to TD3 (Fujimoto et al., 2018). One gradient update with data sampled from the replay buffer is applied for every timestep collected from the environment. The gradient with respect to  $\phi$  in Equation 6 is combined with  $\phi$  using a fixed learning rate in the standard way, all other parameter updates use Adam (Kingma & Ba, 2015) with the default parameters. Any other hyper-parameters can be seen in Table 3 and Table 4.
+
+Using additional gradient steps In our experiments (Section 5.2.3) we analyzed the effect of applying multiple gradient updates to the policy using  $L_{\alpha}$  before applying the critic to compute second-order gradients with respect to the objective function parameters. For two updates, this gives
+
+$$
+\nabla_ {\alpha} Q _ {\theta} \left(s _ {t}, \pi_ {\phi^ {\dagger}} \left(s _ {t}\right)\right) \text {w i t h} \phi^ {\dagger} = \phi^ {\prime} - \nabla_ {\phi^ {\prime}} L _ {\alpha} \left(\tau_ {1}, x \left(\phi^ {\prime}\right), V\right) \tag {8}
+$$
+
+$$
+\text {a n d} \phi^ {\prime} = \phi - \nabla_ {\phi} L _ {\alpha} (\tau_ {2}, x (\phi), V)
+$$
+
+and can be extended to more than two correspondingly. Additionally, we use disjoint mini batches of data  $\tau$ :  $\tau_{1}, \tau_{2}$ . When updating the policy using  $\nabla_{\phi} L_{\alpha}$  we continue to use only a single gradient step.
+
+# B.3 BASELINES
+
+$\mathbf{R}\mathbf{L}^{2}$  The implementation for  $\mathrm{RL^2}$  mimics the paper by Duan et al. (Duan et al., 2016). However, we were unable to achieve good results with TRPO (Schulman et al., 2015a) on the MuJoCo environments and thus used PPO (Schulman et al., 2017) instead. The PPO hyperparameters and implementation are taken from rlib (Liang et al., 2018). Our implementation uses an LSTM with 64 units and does not reset the state of the LSTM for two episodes in sequence. Resetting after additional episodes were given did not improve training results. Different action and observation dimensionalities across environments were handled by using an environment wrapper that pads both with zeros appropriately.
+
+EPG We use the official EPG code base https://github.com/openai/EPG from the original paper (Houthooft et al., 2018). The hyperparameters are taken from the paper,  $V = 64$  noise vectors, an update frequency of  $M = 64$ , and 128 updates for every inner loop, resulting in an inner loop length of 8196 steps. During meta-test training, we run with the same update frequency for a total of 1 million steps.
+
+PPO & On-Policy REINFORCE with GAE We use the tuned implementations from https: //spinningup.openui.com/en/latest/spinningup/bench.html which include a GAE (Schulman et al., 2015b) baseline.
+
+Off-Policy Reinforce with GAE The implementation is equivalent to MetaGenRL except that the objective function is fixed to be the REINFORCE estimator with a GAE (Schulman et al., 2015b) baseline. Thus, experience is sampled from a replay buffer. We have also experimented with an importance weighted unbiased estimator but this resulted in poor performance.
+
+DDPG Our implementation is based on https://spinningup.openai.com/en/latest/spinningup/bench.html and uses the same TD3 tricks (Fujimoto et al., 2018) and hyperparameters (where applicable) that MetaGenRL uses.
+
+Table 3: Architecture hyperparameters  
+
+<table><tr><td>Parameter</td><td>Value</td></tr><tr><td>Critic number of layers</td><td>3</td></tr><tr><td>Critic number of units</td><td>350</td></tr><tr><td>Policy number of layers</td><td>3</td></tr><tr><td>Policy number of units</td><td>350</td></tr><tr><td>Objective function LSTM units</td><td>32</td></tr><tr><td>Objective function action conv layers</td><td>3</td></tr><tr><td>Objective function action conv filters</td><td>32</td></tr><tr><td>Error bound η</td><td>1000</td></tr></table>
+
+Table 4: Training hyperparameters  
+
+<table><tr><td>Parameter</td><td>Value</td></tr><tr><td>Truncated episode length</td><td>20</td></tr><tr><td>Global norm gradient clipping</td><td>1.0</td></tr><tr><td>Critic learning rate λ1</td><td>1e-3</td></tr><tr><td>Policy learning rate λ2</td><td>1e-3</td></tr><tr><td>Second order learning rate λ3</td><td>1e-3</td></tr><tr><td>Obj. func. learning rate λ4</td><td>1e-3</td></tr><tr><td>Critic noise</td><td>0.2</td></tr><tr><td>Critic noise clip</td><td>0.5</td></tr><tr><td>Target network update speed</td><td>0.005</td></tr><tr><td>Discount factor</td><td>0.99</td></tr><tr><td>Batch size</td><td>100</td></tr><tr><td>Random exploration timesteps</td><td>10000</td></tr><tr><td>Policy gaussian noise std</td><td>0.1</td></tr><tr><td>Timesteps per agent</td><td>1M</td></tr></table>
\ No newline at end of file
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+# INDUCTIVE MATRIX COMPLETION BASED ON GRAPH NEURAL NETWORKS
+
+Muhan Zhang*
+
+Washington University in St. Louis
+
+muhan@wustl.edu
+
+*Now at Facebook
+
+Yixin Chen
+
+Washington University in St. Louis
+
+chen@cse.wustl.edu
+
+# ABSTRACT
+
+We propose an inductive matrix completion model without using side information. By factorizing the (rating) matrix into the product of low-dimensional latent embeddings of rows (users) and columns (items), a majority of existing matrix completion methods are transductive, since the learned embeddings cannot generalize to unseen rows/columns or to new matrices. To make matrix completion inductive, most previous works use content (side information), such as user's age or movie's genre, to make predictions. However, high-quality content is not always available, and can be hard to extract. Under the extreme setting where not any side information is available other than the matrix to complete, can we still learn an inductive matrix completion model? In this paper, we propose an Inductive Graph-based Matrix Completion (IGMC) model to address this problem. IGMC trains a graph neural network (GNN) based purely on 1-hop subgraphs around (user, item) pairs generated from the rating matrix and maps these subgraphs to their corresponding ratings. It achieves highly competitive performance with state-of-the-art transductive baselines. In addition, IGMC is inductive - it can generalize to users/items unseen during the training (given that their interactions exist), and can even transfer to new tasks. Our transfer learning experiments show that a model trained out of the MovieLens dataset can be directly used to predict Douban movie ratings with surprisingly good performance. Our work demonstrates that: 1) it is possible to train inductive matrix completion models without using side information while achieving similar or better performances than state-of-the-art transductive methods; 2) local graph patterns around a (user, item) pair are effective predictors of the rating this user gives to the item; and 3) Long-range dependencies might not be necessary for modeling recommender systems.
+
+# 1 INTRODUCTION
+
+Matrix completion (Candès & Recht, 2009) is one common formulation of recommender systems, where rows and columns of a matrix represent users and items, respectively, and predicting users' interest in items corresponds to filling in the missing entries of the rating matrix. By assuming a low-rank rating matrix, many of the most popular matrix completion algorithms use factorization techniques that decompose a rating  $r_{ij}$  into  $\mathbf{w}_i^\top \mathbf{h}_j$ , the inner product of user  $i$ 's and item  $j$ 's latent feature vectors  $\mathbf{w}_i$  and  $\mathbf{h}_j$ , respectively, which have achieved great successes (Adomavicius & Tuzhilin, 2005; Schafer et al., 2007; Koren et al., 2009; Bobadilla et al., 2013)
+
+However, matrix factorization is intrinsically transductive, meaning that the learned latent features (embeddings) for users/items are not generalizable to users/items unseen during the training. When the rating matrix has changed values or has new rows/columns added, it often requires a complete retraining to get the new embeddings. To make matrix completion inductive, Inductive Matrix Completion (IMC) has been proposed, which leverages content (side information) of users and items (Jain & Dhillon, 2013; Xu et al., 2013). In IMC, a rating is decomposed by  $r_{ij} = \mathbf{x}_i^\top \mathbf{Q}\mathbf{y}_j$ , where  $\mathbf{x}_i$  and  $\mathbf{y}_j$  are content feature vectors of user  $i$  and item  $j$ , respectively, and  $\mathbf{Q}$  is a learnable matrix modeling the feature interactions. To accurately predict missing entries, IMC methods have strong constraints on the content quality, which often leads to inferior performance when high-quality content is not available. Other content-based recommender systems (Lops et al., 2011) face similar problems. In
+
+![](images/60e7217cfefdc7997f21f0870fbc7d0a44bcffcbb29f1d7e3401e48db861b9de.jpg)  
+Figure 1: We extract a local enclosing subgraph around each rating (dark green), and train a GNN to map subgraphs to ratings. Each enclosing subgraph is induced by the user and item associated with the target rating as well as their  $h$ -hop neighbors (here  $h = 1$ ). From the subgraphs, a GNN can learn mixed graph patterns (such as average ratings, paths, etc.) useful for rating prediction. We illustrate some possible patterns in the GNN box. We use the trained GNN to complete the missing entries.
+
+some extreme settings, there is not any content available for user, such as a website where users are completely anonymous. In these cases, inductive matrix completion seems impossible.
+
+Recently, Hartford et al. (2018) propose exchangeable matrix layers, which apply permutation equivariant operations on the rating matrix to reconstruct missing entries. The resulting models are inductive and do not rely on side information. However, these models take the whole rating matrix  $\mathbf{R}$  as input and output another reconstructed matrix, which poses scalability challenges for practical datasets with millions of users/items. In this paper, we propose a novel inductive matrix completion method that does not use any content. Further, our method does not need to take the whole rating matrix as input, and can infer ratings for individual user-item pairs. The key that frees us from using content or whole rating matrix is local graph pattern. If for each observed rating we add an edge between the corresponding user and item, we can build a bipartite graph from the rating matrix. Subsequently, predicting unknown ratings converts equivalently to predicting labeled links in this bipartite graph. This transforms matrix completion into a link prediction problem (Liben-Nowell & Kleinberg, 2007), where graph patterns play a major role in determining link existences.
+
+A major class of link prediction methods are heuristic methods, which predict links based on some heuristic scores. For example, the common neighbors heuristic count the common neighbors between two nodes to predict links, while the Katz index (Katz, 1953) uses a weighted sum of all the walks between two nodes. See (Liben-Nowell & Kleinberg, 2007) for an overview. These heuristics can be seen as some predefined graph structure features calculated based on the local or global graph patterns around links, which have achieved great successes due to their simplicity and effectiveness.
+
+However, these traditional link prediction heuristics only work for simple graphs where nodes and edges both only have a single type. Can we find some heuristics for labeled link prediction in bipartite graph? Intuitively, such heuristics should exist. For example, if a user  $u_{0}$  likes an item  $v_{0}$ , we may expect to see very often that  $v_{0}$  is also liked by some other user  $u_{1}$  who shares a similar taste to  $u_{0}$ . By similar taste, we mean  $u_{1}$  and  $u_{0}$  have together both liked some other item  $v_{1}$ . In the bipartite graph, such a pattern is realized as a "like" path ( $u_{0} \to_{\text{like}} v_{1} \to_{\text{liked by}} u_{1} \to_{\text{like}} v_{0}$ ). If there are many such paths between  $u_{0}$  and  $v_{0}$ , we may infer that  $u_{0}$  is highly likely to like  $v_{0}$ . Thus, we may count the number of such paths as an indicator of how likely  $u_{0}$  likes  $v_{0}$ . In fact, many neighborhood-based recommender systems (Desrosiers & Karypis, 2011) rely on similar heuristics.
+
+Of course we can try to manually define many such intuitive heuristics and test their effectiveness. In this work, however, we take a different approach that automatically learns suitable heuristics from the given bipartite graph. To do so, we first extract an  $h$ -hop enclosing subgraph for each training user-item pair  $(u, v)$ , which is defined to be the subgraph induced from the bipartite graph by nodes  $u, v$  and their neighbors within  $h$  hops. Such local subgraphs contain rich graph pattern information about the rating that  $u$  may give to  $v$ . For example, all the  $(u_0 \to_{\mathrm{like}} v_1 \to_{\mathrm{liked}} u_1 \to_{\mathrm{like}} v_0)$  paths are included in the 1-hop enclosing subgraph around  $(u_0, v_0)$ . By feeding these enclosing subgraphs to a graph neural network (GNN), we train a graph regression model that maps each subgraph to the rating that its center user gives to its center item. Figure 1 illustrates the overall framework.
+
+Due to the superior graph learning ability, a GNN can learn highly expressive graph structure features useful for inferring the ratings without restricting the features to predefined heuristics. Given a trained GNN, we can also apply it to unseen users/items without retraining. Our resulting algorithm is inductive, and is named Inductive Graph-based Matrix Completion (IGMC). Note that IGMC does not address the extreme cold-start problem, as it still requires an unseen user-item pair's enclosing subgraph (i.e., the user and item should at least have some interactions with neighbors so that the enclosing subgraph is not empty). This scenario is very common in practice. For example, a newly registered YouTube user may quickly watch some videos without completing their personal information. In this case, if we cannot retrain user embeddings frequently, IGMC can be of great value by still making recommendations based purely on this user's interaction history with videos.
+
+We compare IGMC with state-of-the-art matrix completion algorithms on five benchmark datasets. Without using any content, IGMC achieves the smallest RMSEs on four of them, even beating many transductive baselines augmented by side information. Our model is also equipped with excellent transfer learning ability. We show that an IGMC model trained on the MovieLens-100K dataset can be directly used to predict Douban movie ratings and even outperforms some baselines trained specifically on Douban. We also analyze IGMC's behavior on sparse rating matrices. We show that IGMC is more robust than transductive methods on sparse matrices. Under an extremely sparse case (only  $0.1\%$  of MovieLens-1M training ratings are kept), IGMC can still achieve less than 0.95 RMSE, beating a state-of-the-art transductive method GC-MC by more than 0.1 RMSE. Finally, our visualization confirms that local enclosing subgraphs are indeed strong predictors of ratings.
+
+# 2 RELATED WORK
+
+Graph neural networks Graph neural networks (GNNs) are a new type of neural networks for learning over graphs (Scarselli et al., 2009; Bruna et al., 2013; Duvenaud et al., 2015; Li et al., 2015; Kipf & Welling, 2016; Niepert et al., 2016; Dai et al., 2016). There are two types of GNNs: Node-level GNNs use message passing layers to iteratively pass messages between each node and its neighbors in order to extract a feature vector for each node encoding its local substructure. Graph-level GNNs additionally use a pooling layer such as summing which aggregates node feature vectors into a graph representation to enable graph-level tasks such as graph classification/regression. Due to the superior graph representation learning ability, GNNs have achieved state-of-the-art performance on semi-supervised node classification (Kipf & Welling, 2016), network embedding (Hamilton et al., 2017), graph classification (Zhang et al., 2018), and link prediction (Zhang & Chen, 2018), etc.
+
+GNNs for matrix completion The matrix completion problem has been studied using GNNs. Monti et al. (2017) develop a multi-graph CNN (MGCNN) model to extract user and item latent features from their respective nearest-neighbor networks. Berg et al. (2017) propose graph convolutional matrix completion (GC-MC) which directly applies a GNN to the user-item bipartite graph to extract user and item latent features using a GNN. The SpectralCF model of (Zheng et al., 2018) uses a spectral-GNN on the bipartite graph to learn node embeddings. Although using GNNs for matrix completion, all these models are still transductive - MGCNN and SpectralCF require graph Laplacians which do not generalize to new graphs, while GC-MC uses one-hot encoding of node IDs as initial node features, thus cannot generalize to unseen users/items. A recent inductive graph-based recommender system, PinSage (Ying et al., 2018a), uses node content as initial node features (instead of the one-hot encoding in GC-MC), and is successfully used in recommending related pins in Pinterest. Although being inductive, PinSage relies heavily on the rich visual and text content associated with the pins which is not often accessible in other recommendation tasks. In comparison, our IGMC model is inductive and does not rely on any content. All previous approaches use node-level GNNs to learn embeddings for nodes, while our IGMC uses a graph-level GNN to learn representations for subgraphs. We will discuss this crucial difference in more details in Section 4.
+
+Another related previous work is (Hartford et al., 2018), which defines exchangeable matrix layers to perform permutation-equivariant operations on matrices to achieve inductive matrix completion without using content. In particular, the operation updates each matrix entry by a weighted sum of itself, entries of its row, entries of its column, and all other entries of the matrix, where parameters for each of the four components are shared across all entries. It can also be regarded as a GNN with the exceptions that 1) the message passing is performed on edges (final edge features are pooled into node features), and 2) all edges (including those not connected to the center edge) pass messages
+
+Algorithm 1ENCLOSING SUBGRAPH EXTRACTION  
+1: input:  $h$ , target user-item pair  $(u, v)$ , the bipartite graph  $G$   
+2: output: the  $h$ -hop enclosing subgraph  $G_{u,v}^{h}$  for  $(u, v)$   
+3:  $U = U_{fringe} = \{u\}$ ,  $V = V_{fringe} = \{v\}$   
+4: for  $i = 1, 2, \ldots, h$  do  
+5:  $U_{fringe}' = \{u_i : u_i \sim V_{fringe}\} \setminus U$   
+6:  $V_{fringe}' = \{v_i : v_i \sim U_{fringe}\} \setminus V$   
+7:  $U_{fringe} = U_{fringe}'$ ,  $V_{fringe} = V_{fringe}'$   
+8:  $U = U \cup U_{fringe}$ ,  $V = V \cup V_{fringe}$   
+9: Let  $G_{u,v}^{h}$  be the vertex-induced subgraph from  $G$  using vertices  $U, V$   
+10: Remove edge  $(u, v)$  from  $G_{u,v}^{h}$ .  
+11: end for  
+12: return  $G_{u,v}^{h}$   
+Note:  $\{u_i : u_i \sim V_{fringe}\}$  is the set of nodes that are adjacent to at least one node in  $V_{fringe}$  with any edge type.
+
+to the center edge in each round. One limitation of (Hartford et al., 2018) is that it takes the entire rating matrix as input, which might raise concerns for large matrices. In comparison, our IGMC takes only local subgraphs as input which avoids the issue and enables predicting individual ratings.
+
+Link prediction based on graph patterns Learning supervised heuristics (graph patterns) has been studied for link prediction in simple graphs. Zhang & Chen (2017) propose Weisfeiler-Lehman Neural Machine (WLNM), which learns graph structure features using a fully-connected neural network on the subgraphs' adjacency matrices. Later, they improve this work by replacing the fully-connected neural network with a GNN and achieves state-of-the-art link prediction results (Zhang & Chen, 2018). Our work generalizes this line of research from predicting link existence in simple graphs to predicting values of links in bipartite graphs (i.e., matrix completion). In (Chen et al., 2005; Zhou et al., 2007), traditional link prediction heuristics are adapted to bipartite graphs which show promising performance for recommender systems. Our work differs in that we do not use any predefined heuristics, but learn general graph structure features using a GNN. Another similar work to ours is (Li & Chen, 2013), where graph kernels are used to learn graph structure features. However, graph kernels require quadratic time and space complexity to compute and store the kernel matrices thus are unsuitable for modern recommender systems.
+
+# 3 INDUCTIVE GRAPH-BASED MATRIX COMPLETION (IGMC)
+
+We now present our Inductive Graph-based Matrix Completion (IGMC) framework. We use  $G$  to denote the undirected bipartite graph constructed from the given rating matrix  $\mathbf{R}$ . In  $G$ , a node is either a user (denoted by  $u$ , corresponding to a row in  $\mathbf{R}$ ) or an item (denoted by  $v$ , corresponding to a column in  $\mathbf{R}$ ). Edges can exist between user and item, but cannot exist between two users or two items. Each edge  $(u, v)$  has a value  $r = \mathbf{R}_{u,v}$ , corresponding to the rating that  $u$  gives to  $v$ . We use  $\mathcal{R}$  to denote the set of all possible ratings (e.g.,  $\mathcal{R} = \{1, 2, 3, 4, 5\}$  in MovieLens), and use  $\mathcal{N}_r(u)$  to denote the set of  $u$ 's neighbors that connect to  $u$  with edge type  $r$ .
+
+# 3.1 ENCLOSING SUBGRAPH EXTRACTION
+
+The first part of IGMC is enclosing subgraph extraction. For each observed rating  $\mathbf{R}_{u,v}$ , we extract an  $h$ -hop enclosing subgraph around  $(u,v)$  from  $G$ . Algorithm 1 describes the BFS procedure for extracting  $h$ -hop enclosing subgraphs. We will feed these enclosing subgraphs to a GNN and regress on their ratings. Then, for each testing  $(u,v)$  pair, we again extract its  $h$ -hop enclosing subgraph from  $G$ , and use the trained GNN model to predict its rating. Note that after extracting a training enclosing subgraph for  $(u,v)$ , we should remove the edge  $(u,v)$  because it is the target to predict.
+
+# 3.2 NODE LABELING
+
+The second part of IGMC is node labeling. Before we feed an enclosing subgraph to the GNN, we first apply a node labeling to it, which gives an integer label to every node in the subgraph. The
+
+purpose is to use different labels to mark nodes' different roles in a subgraph. Ideally, our node labeling should be able to: 1) distinguish the target user and target item between which the target rating is located, and 2) differentiate user-type nodes from item-type nodes. Otherwise, the GNN cannot tell between which user and item to predict the rating, and might lose node-type information. To satisfy these conditions, we propose a node labeling as follows: We first give label 0 and 1 to the target user and target item, respectively. Then, we determine other nodes' labels according to at which hop they are included in the subgraph in Algorithm 1. If a user-type node is included at the  $i^{\text{th}}$  hop, we will give it a label  $2i$ . If an item-type node is included at the  $i^{\text{th}}$  hop, we will give it  $2i + 1$ . Such a node labeling can sufficiently discriminate: 1) target nodes from "context" nodes, 2) users from items (users always have even labels), and 3) nodes of different distances to the target rating.
+
+Note that this is not the only possible way of node labeling, but we empirically verified its excellent performance. The one-hot encoding of these node labels will be treated as the initial node features  $\mathbf{x}^0$  of the subgraph when fed to the GNN. Note that our node labels are determined completely inside each enclosing subgraph, thus are independent of the global bipartite graph. Given a new enclosing subgraph, we can as well predict its rating even if all of its nodes are from a different bipartite graph, because IGMC purely relies on graph patterns within local enclosing subgraphs without leveraging any global information specific to the bipartite graph. Our node labeling is also different from using the global node IDs as in GC-MC (Berg et al., 2017). Using one-hot encoding of global IDs is essentially transforming the first message passing layer's parameters into latent node embedding associated with each particular ID (equivalent to an embedding lookup table). Such a model cannot generalize to nodes whose IDs are out of range, thus is transductive.
+
+# 3.3 GRAPH NEURAL NETWORK ARCHITECTURE
+
+The third part of IGMC is to train a graph neural network (GNN) model predicting ratings from the enclosing subgraphs. In previous node-based approaches such as GC-MC, a node-level GNN is applied to the entire bipartite graph to extract node embeddings. Then, the node embeddings of  $u$  and  $v$  are input to an inner-product or bilinear operator to reconstruct the rating on  $(u, v)$ . In contrast, IGMC applies a graph-level GNN to the enclosing subgraph around  $(u, v)$  and maps the subgraph to the rating. There are thus two components in our GNN: 1) message passing layers that extract a feature vector for each node in the subgraph, and 2) a pooling layer to summarize a subgraph representation from node features.
+
+To learn the rich graph patterns introduced by the different edge types, we adopt the relational graph convolutional operator (R-GCN) (Schlichtkrull et al., 2018) as our GNN's message passing layers, which has the following form:
+
+$$
+\mathbf {x} _ {i} ^ {l + 1} = \mathbf {W} _ {0} ^ {l} \mathbf {x} _ {i} ^ {l} + \sum_ {r \in \mathcal {R}} \sum_ {j \in \mathcal {N} _ {r} (i)} \frac {1}{| \mathcal {N} _ {r} (i) |} \mathbf {W} _ {r} ^ {l} \mathbf {x} _ {j} ^ {l}, \tag {1}
+$$
+
+where  $\mathbf{x}_i^l$  denotes node  $i$ 's feature vector at layer  $l$ ,  $\mathbf{W}_0^l$  and  $\{\mathbf{W}_r^l | r \in \mathcal{R}\}$  are learnable parameter matrices. Since neighbors  $j$  connected to  $i$  with different edge types  $r$  are processed by different parameter matrices  $\mathbf{W}_r^l$ , we are able to learn a large amount of enriched graph patterns inside the edge types, such as the average rating the target user gives to items, the average rating the target item receives, and by which paths the two target nodes are connected, etc. We stack  $L$  message passing layers with tanh activations between two layers. Following (Zhang et al., 2018; Xu et al., 2018), node  $i$ 's feature vectors from different layers are concatenated as its final representation  $\mathbf{h}_i$ :
+
+$$
+\mathbf {h} _ {i} = \operatorname {c o n c a t} \left(\mathbf {x} _ {i} ^ {1}, \mathbf {x} _ {i} ^ {2}, \dots , \mathbf {x} _ {i} ^ {L}\right). \tag {2}
+$$
+
+Next, we pool the node representations into a graph-level feature vector. There are many choices such as summing, averaging, SortPooling (Zhang et al., 2018), DiffPooling (Ying et al., 2018b), etc. In this work, however, we use a different pooling layer which concatenates the final representations of only the target user and item as the graph representation:
+
+$$
+\mathbf {g} = \operatorname {c o n c a t} \left(\mathbf {h} _ {u}, \mathbf {h} _ {v}\right), \tag {3}
+$$
+
+where we use  $\mathbf{h}_u$  and  $\mathbf{h}_v$  to denote the final representations of the target user and target item, respectively. Our particular choice is due to the extra importance that these two target nodes carry
+
+compared to other context nodes. Although being very simple, we empirically verified its better performance than summing and other advanced pooling layers for our matrix completion tasks.
+
+After getting the final graph representation, we use an MLP to output the predicted rating:
+
+$$
+\hat {r} = \mathbf {w} ^ {\top} \sigma (\mathbf {W} \mathbf {g}), \tag {4}
+$$
+
+where  $\mathbf{W}$  and  $\mathbf{w}$  are parameters of the MLP which map the graph representation  $\mathbf{g}$  to a scalar rating  $\hat{r}$ , and  $\sigma$  is an activation function (we take ReLU in this paper).
+
+# 3.4 MODEL TRAINING
+
+Loss function We minimize the mean squared error (MSE) between the predictions and the ground truth ratings:
+
+$$
+\mathcal {L} = \frac {1}{| \{(u , v) | \boldsymbol {\Omega} _ {u , v} = 1 \} |} \sum_ {(u, v): \boldsymbol {\Omega} _ {u, v} = 1} \left(R _ {u, v} - \hat {R} _ {u, v}\right) ^ {2}, \tag {5}
+$$
+
+where we use  $R_{u,v}$  and  $\dot{R}_{u,v}$  to denote the true rating and predicted rating of  $(u,v)$ , respectively, and  $\Omega$  is a 0/1 mask matrix indicating the observed entries of the rating matrix  $\mathbf{R}$ .
+
+Adjacent rating regularization The R-GCN layer (1) used in our GNN has different parameters  $\mathbf{W}_r$  for different rating types. One drawback here is that it fails to take the magnitude of ratings into consideration. For instance, a rating of 4 and a rating of 5 in MovieLens both indicate that the user likes the movie, while a rating of 1 indicates that the user does not like the movie. Ideally, we expect our model to be aware of the fact that a rating of 4 is more similar to 5 than 1 is. In R-GCN, however, ratings 1, 4 and 5 are only treated as three independent edge types – the magnitude and order information of the ratings is completely lost. To fix that, we propose an adjacent rating regularization (ARR) technique, which encourages ratings adjacent to each other to have similar parameter matrices. Assume the ratings in  $\mathcal{R}$  exhibit an ordering  $r_1, r_2, \ldots, r_{|\mathcal{R}|}$  which indicates increasingly higher preference that users have for items. Then, the ARR regularizer is:
+
+$$
+\mathcal {L} _ {\mathrm {A R R}} = \sum_ {i = 1, 2, \dots , | \mathcal {R} | - 1} \| \mathbf {W} _ {r _ {i + 1}} - \mathbf {W} _ {r _ {i}} \| _ {F} ^ {2}, \tag {6}
+$$
+
+where  $\| \cdot \| _F$  denotes the Frobenius norm of a matrix. The above regularizer restrains the parameter matrices of adjacent ratings from having too much differences, which not only takes into consideration of the ratings' order, but also helps the optimization of those infrequent ratings by transferring knowledge from their adjacent ratings. The final loss function is given by:
+
+$$
+\mathcal {L} _ {\text {f i n a l}} = \mathcal {L} + \lambda \mathcal {L} _ {\mathrm {A R R}}, \tag {7}
+$$
+
+where  $\lambda$  trades-off the importance of the MSE loss and the ARR regularizer. There are many other ways to model rating magnitude and order, which are left for future work.
+
+# 4 GRAPH-LEVEL GNN VS. NODE-LEVEL GNN
+
+![](images/cd16f3522340d528a0f6059e02934b537bd40cf3819c143d8dc5568ae8370e74.jpg)
+
+![](images/43214fa2f0abfeaa3e81e12c95a7929bfc868047ab949f3e3a2a213ff23f55bc.jpg)
+
+Compared to previous graph matrix completion approaches such as PinSage and GC-MC, one important difference of IGMC is that it uses a graph-level GNN to map the enclosing subgraph around the target user and item to their rating (left figure (a)), instead of using a node-level GNN on the bipartite graph  $G$  to learn target user's and item's embeddings and use the node embeddings to predict the rating (left figure (b)). One drawback of the latter node-based approach is that the learned node embeddings are essentially encoding the two rooted sub
+
+trees around the two nodes independently, which fails to model the interactions and correspondences between the nodes of the two trees. For example, from the two subtrees of the left figure (b) we do not really know whether the two target nodes are just isolated from each other like in (b) or actually densely connected like in (a); these two cases look identical to a node-based approach.
+
+Table 1: Statistics of each dataset.  
+
+<table><tr><td>Dataset</td><td>Users</td><td>Items</td><td>Ratings</td><td>Density</td><td>Rating types</td></tr><tr><td>Flickr</td><td>3,000</td><td>3,000</td><td>26,173</td><td>0.0029</td><td>0.5, 1, 1.5, ..., 5</td></tr><tr><td>Douban</td><td>3,000</td><td>3,000</td><td>136,891</td><td>0.0152</td><td>1, 2, 3, 4, 5</td></tr><tr><td>YahooMusic</td><td>3,000</td><td>3,000</td><td>5,335</td><td>0.0006</td><td>1, 2, 3, ..., 100</td></tr><tr><td>ML-100K</td><td>943</td><td>1,682</td><td>100,000</td><td>0.0630</td><td>1, 2, 3, 4, 5</td></tr><tr><td>ML-1M</td><td>6,040</td><td>3,706</td><td>1,000,209</td><td>0.0447</td><td>1, 2, 3, 4, 5</td></tr></table>
+
+In comparison, a graph-level GNN can discriminate the two cases through a sufficient number of message passing rounds. Since the learning is confined to the subgraph, stacking multiple graph convolution layers will learn more and more refined local structural features which can adequately discriminate up to all subgraphs that the Weisfeiler-Lehman algorithm can discriminate (Xu et al., 2018). However, for node-based approaches, since there is no subgraph boundary, stacking multiple graph convolutions will only extend the convolution range to unrelated distant nodes and oversmooth the node embeddings (Li et al., 2018). This is reflected in that previous node-based approaches mainly use only one or two message passing layers (Berg et al., 2017; Ying et al., 2018a).
+
+Nevertheless, using a graph-level GNN on every target rating's enclosing subgraph has higher complexity than using a node-level GNN on the entire bipartite graph. Suppose the bipartite graph has  $|E|$  edges. Then performing one round of message passing using a node-level GNN has  $\mathcal{O}(|E|)$  complexity. Assume the maximum number of edges in all enclosing subgraphs is  $K$ . Performing one round of message passing for all enclosing subgraphs then has  $\mathcal{O}(K|E|)$  complexity. In practice, we can use subsampling to restrict  $K$  to a small number to reduce IGMC's complexity.
+
+# 5 EXPERIMENTS
+
+We conduct experiments on five common matrix completion datasets: Flixster (Jamali & Ester, 2010), Douban (Ma et al., 2011), YahooMusic (Dror et al., 2011), MovieLens-100K and MovieLens-1M (Miller et al., 2003). For ML-100K, we train and evaluate on the canonical u1.base/u1.test train/test split. For ML-1M, we randomly split it into  $90\%$  and  $10\%$  train/test sets. For Flixster, Douban and YahooMusic we use the preprocessed subsets and splits provided by (Monti et al., 2017). Dataset statistics are summarized in Table 1. We implemented IGMC using pytorch-geometric (Fey & Lenssen, 2019). We tuned model hyperparameters based on cross validation results on ML-100K, and used them across all datasets. The final architecture uses 4 R-GCN layers with 32, 32, 32, 32 hidden dimensions. Basis decomposition with 4 bases is used to reduce the number of parameters in  $\mathbf{W}_r$  (Schlichtkrull et al., 2018). The final MLP has 128 hidden units and a dropout rate of 0.5. We use 1-hop enclosing subgraphs for all datasets, and find them sufficiently good. We find using 2 or more hops can slightly increase the performance but take much longer training time. For each enclosing subgraph, we randomly drop out its adjacency matrix entries with a probability of 0.2 during the training. We set the  $\lambda$  in (7) to 0.001. We train our model using the Adam optimizer (Kingma & Ba, 2014) with a batch size of 50 and an initial learning rate of 0.001, and multiply the learning rate by 0.1 every 20 epochs for ML-1M, and every 50 epochs for all other datasets. Our code is publicly available at https://github.com/muhanzhang/IGMC.
+
+# 5.1 FLIXSTER, DOUBAN AND YAHOOMUSIC
+
+For these three datasets, we compare our IGMC with GRALS (Rao et al., 2015), sRGCNN (Monti et al., 2017), GC-MC (Berg et al., 2017), F-EAE (Hartford et al., 2018), and PinSage (Ying et al., 2018a). Among them, GRALS is a graph regularized matrix completion algorithm. GC-MC and sRGCNN are transductive node-level-GNN-based matrix completion methods. F-EAE uses exchangeable matrix layers to perform inductive matrix completion without using content. PinSage is an inductive node-level-GNN-based model using content, which is originally used to predict related pins and is adapted to predicting ratings here. We further implemented an inductive GC-MC model (IGC-MC) which replaces the one-hot encoding of node IDs with the content features to make it
+
+Table 2: RMSE test results on Flixster, Douban and YahooMusic.  
+
+<table><tr><td>Model</td><td>Inductive</td><td>Content</td><td>Flixster</td><td>Douban</td><td>YahooMusic</td></tr><tr><td>GRALS</td><td>no</td><td>yes</td><td>1.245</td><td>0.833</td><td>38.0</td></tr><tr><td>sRGCNN</td><td>no</td><td>yes</td><td>0.926</td><td>0.801</td><td>22.4</td></tr><tr><td>GC-MC</td><td>no</td><td>yes</td><td>0.917</td><td>0.734</td><td>20.5</td></tr><tr><td>IGC-MC</td><td>yes</td><td>yes</td><td>0.999±0.062</td><td>0.990±0.082</td><td>21.3±0.989</td></tr><tr><td>F-EAE</td><td>yes</td><td>no</td><td>0.908</td><td>0.738</td><td>20.0</td></tr><tr><td>PinSage</td><td>yes</td><td>yes</td><td>0.954±0.005</td><td>0.739±0.002</td><td>22.9±0.629</td></tr><tr><td>IGMC (ours)</td><td>yes</td><td>no</td><td>0.872±0.001</td><td>0.721±0.001</td><td>19.1±0.138</td></tr></table>
+
+Table 3: RMSE test results on MovieLens-100K (left) and MovieLens-1M (right).  
+
+<table><tr><td>Model</td><td>Inductive</td><td>Content</td><td>ML-100K</td><td>Model</td><td>Inductive</td><td>Content</td><td>ML-1M</td></tr><tr><td>MC</td><td>no</td><td>no</td><td>0.973</td><td>PMF</td><td>no</td><td>no</td><td>0.883</td></tr><tr><td>IMC</td><td>no</td><td>yes</td><td>1.653</td><td>I-RBM</td><td>no</td><td>no</td><td>0.854</td></tr><tr><td>GMC</td><td>no</td><td>yes</td><td>0.996</td><td>NNMF</td><td>no</td><td>no</td><td>0.843</td></tr><tr><td>GRALS</td><td>no</td><td>yes</td><td>0.945</td><td>I-AutoRec</td><td>no</td><td>no</td><td>0.831</td></tr><tr><td>sRGCNN</td><td>no</td><td>yes</td><td>0.929</td><td>CF-NADE</td><td>no</td><td>no</td><td>0.829</td></tr><tr><td>GC-MC</td><td>no</td><td>yes</td><td>0.905</td><td>GC-MC</td><td>no</td><td>no</td><td>0.832</td></tr><tr><td>IGC-MC</td><td>yes</td><td>yes</td><td>1.142</td><td>IGC-MC</td><td>yes</td><td>yes</td><td>1.259</td></tr><tr><td>F-EAE</td><td>yes</td><td>no</td><td>0.920</td><td>F-EAE</td><td>yes</td><td>no</td><td>0.860</td></tr><tr><td>PinSage</td><td>yes</td><td>yes</td><td>0.951</td><td>PinSage</td><td>yes</td><td>yes</td><td>0.906</td></tr><tr><td>IGMC</td><td>yes</td><td>no</td><td>0.905</td><td>IGMC</td><td>yes</td><td>no</td><td>0.857</td></tr></table>
+
+inductive. The content in these datasets are presented in the form of user and item graphs. We summarize whether each algorithm is inductive and whether it uses content in Table 2.
+
+We train our model for 40 epochs, and save the model parameters every 10 epochs. The final predictions are given by averaging the predictions from epochs 10, 20, 30 and 40. We repeat the experiment five times and report the average RMSEs. The baseline results are taken from (Hartford et al., 2018). Table 2 shows the results. Our model achieves the smallest RMSEs on all three datasets without using any content, significantly outperforming all the compared baselines, regardless of whether they are transductive or inductive. Further, except F-EAE, all the baselines have used content information to assist the matrix completion. This further highlights IGMC's great performance advantages without relying on content.
+
+# 5.2 ML-100K AND ML-1M
+
+We further conduct experiments on MovieLens datasets. Side information is present for both users (age, gender, occupation, etc.) and movies (genres). For ML-100K, we compare against matrix completion (MC) (Candès & Recht, 2009), inductive matrix completion (IMC) (Jain & Dhillon, 2013), geometric matrix completion (GMC) (Kalofolias et al., 2014), as well as GRALS, sRGCNN, GC-MC, F-EAE and PinSage. We train IGMC for 80 epochs and report the ensemble performance of epochs 50, 60, 70 and 80. For ML-1M, besides the baselines GC-MC, F-EAE and PinSage, we further include state-of-the-art algorithms including PMF (Mnih & Salakhutdinov, 2008), I-RBM (Salakhutdinov et al., 2007), NNMF (Dziugaite & Roy, 2015), I-AutoRec (Sedhain et al., 2015) and CF-NADE (Zheng et al., 2016). We train IGMC for 40 epochs and report the ensemble performance of epochs 25, 30, 35 and 40. The experiments are repeated five times and the average results are reported in Table 3 (standard deviations are less than 0.001). As we can see, IGMC achieves the best performance on ML-100K, in parallel with GC-MC despite that IGMC is an inductive model, while GC-MC is transductive and additionally uses content information. For ML-1M, IGMC cannot catch up with state-of-the-art transductive models such as CF-NADE and GC-MC, but outperforms other inductive models. We will analyze this dataset further in Section 5.3.
+
+Table 4: RMSE of transferring the model trained on ML-100K to Flixster, Douban and YahooMusic.  
+
+<table><tr><td>Model</td><td>Inductive</td><td>Content</td><td>Flixster</td><td>Douban</td><td>YahooMusic</td></tr><tr><td>IGC-MC</td><td>yes</td><td>no</td><td>1.290</td><td>1.144</td><td>25.7</td></tr><tr><td>F-EAE</td><td>yes</td><td>no</td><td>0.987</td><td>0.766</td><td>23.3</td></tr><tr><td>IGMC (ours)</td><td>yes</td><td>no</td><td>0.906</td><td>0.759</td><td>20.1</td></tr></table>
+
+# 5.3 SPARSE RATING MATRIX ANALYSIS
+
+![](images/973223c14cab0b10900a8a0bfb7f3b65e6c097d6fa77dda08bca6b0efdfc69a3.jpg)  
+Figure 2: ML-1M results under different sparsity ratios.
+
+To gain insight into when inductive graph-based matrix completion is more suitable than transductive methods, we compare IGMC with GC-MC on ML-1M under different sparsity levels of the rating matrix. We sequentially increase the sparsity level by randomly keeping only 0.2, 0.1, 0.05, 0.01, and 0.001 of the original training ratings. Then, we train both models on the sparsified rating matrices, and evaluate on the original test set. Figure 2 shows the results. As we can see, although IGMC falls behind GC-MC initially with full ratings, it starts to perform better after the sparsity ratio is less than  $20\%$ . The advantage becomes even greater under extremely sparse cases. This seems to indicate that IGMC is a better choice than transductive methods
+
+when there is not a large amount of training data, which is particularly suitable for the initial rating collection phase of a recommender system. It also suggests that transductive matrix completion relies more on the dense user-item interactions than inductive graph-based matrix completion does.
+
+# 5.4 TRANSFER LEARNING
+
+A great advantage of an inductive model is its potential for transferring to other tasks. We conduct a transfer learning experiment by applying the IGMC model trained on ML-100K to Flixster, Douban and YahooMusic. Among the three datasets, only Douban has exactly the same rating types as ML-100K (1,2,3,4,5). Thus for Flixster and YahooMusic, we bin their edge types into groups 1 to 5 before feeding into the ML-100K model, and multiply the YahooMusic predictions by 20 to account for the different scales. Despite all the compromises, the transferred IGMC model achieves excellent performance (Table 4). We also show the transfer learning results of other two inductive models, IGC-MC and F-EAE. Note that an inductive model using content features (such as PinSage) is not transferrable, due to the different feature spaces between MovieLens and the target datasets. Thus for IGC-MC, we replace its content features with node degrees. As we can see, IGMC outperforms the other two models by large margins in terms of transfer learning ability. Furthermore, the transferred IGMC even outperforms a wide range of baselines trained especially on each dataset (Table 2).
+
+# 5.5 ABLATION STUDIES
+
+To understand the individual contributions of some components in IGMC, we conduct several ablation studies. In particular, we are interested in: 1) whether the proposed pooling layer in Equation (3) helps; 2) whether the proposed adjacent rating regularization (ARR) helps; and 3) whether incorporating content can further improve IGMC's performance. We present the results in the appendix.
+
+# 5.6 VISUALIZATION
+
+Finally, we visualize 10 testing enclosing subgraphs with the highest and lowest predicted ratings for Flixster, Douban, YahooMusic, and ML-100K, respectively, in Figure 3. As we can see, there are substantially different patterns between high-score and low-score subgraphs, which is why IGMC can predict ratings merely from these subgraphs. For example, high-score subgraphs typically show
+
+both high user average rating and high item average rating, while low-score subgraphs often have mixed ratings from non-target users and have low user average rating.
+
+![](images/7ed25c3a82aaa410c6b5156451e81fb273ff2abf87aa90678aa53dc338162fdd.jpg)  
+Figure 3: Testing enclosing subgraphs for Flixster, Douban, YahooMusic, and ML-100K. Top 5 and bottom 5 are subgraphs with the highest and lowest predicted ratings, respectively. In each subgraph, red nodes in the left are users, blue nodes in the right are items; the bottom red and blue nodes are the target user and item; the predicted rating and true rating (in parenthesis) are shown underneath. We visualize edge ratings using the color map shown in the right. Higher ratings are redder.
+
+# 6 CONCLUSION
+
+In this paper, we have proposed Inductive Graph-based Matrix Completion (IGMC). Instead of learning transductive latent features, IGMC learns local graph patterns related to ratings inductively based on graph neural networks. Compared to previous inductive matrix completion methods, IGMC does not rely on content (side information) of users/items. We show that IGMC has highly competitive performance compared to state-of-the-art baselines. In addition, IGMC is transferrable to new tasks without any retraining, a property much desired in those recommendation tasks having few training data. We hope IGMC can provide a new idea to matrix completion and recommender systems.
+
+# ACKNOWLEDGMENTS
+
+The work is supported in part by the National Science Foundation under award numbers III-1526012 and SCH-1622678, and by the National Institute of Health under award number 1R21HS024581.
+
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+
+Muhan Zhang and Yixin Chen. Link prediction based on graph neural networks. In Advances in Neural Information Processing Systems, pp. 5165-5175, 2018.  
+Muhan Zhang, Zhicheng Cui, Marion Neumann, and Yixin Chen. An end-to-end deep learning architecture for graph classification. In AAAI, pp. 4438-4445, 2018.  
+Lei Zheng, Chun-Ta Lu, Fei Jiang, Jiawei Zhang, and Philip S Yu. Spectral collaborative filtering. In Proceedings of the 12th ACM Conference on Recommender Systems, pp. 311-319. ACM, 2018.  
+Yin Zheng, Bangsheng Tang, Wenkui Ding, and Hanning Zhou. A neural autoregressive approach to collaborative filtering. arXiv preprint arXiv:1605.09477, 2016.  
+Tao Zhou, Jie Ren, Matúš Medo, and Yi-Cheng Zhang. Bipartite network projection and personal recommendation. Physical Review E, 76(4):046115, 2007.
+
+# A ABLATION STUDIES
+
+In this section, we conduct ablation experiments to study the individual contributions of some components in IGMC. In particular, we are interested in: 1) whether the proposed pooling layer in Equation (3) helps; 2) whether the proposed adjacent rating regularization (ARR) in Equation (6) helps; and 3) whether incorporating content can further improve IGMC's performance. Therefore, we compare the original IGMC with: 1) IGMC with SumPooling replacing the proposed pooling layer, 2) IGMC without ARR by setting  $\lambda$  in (7) to 0, and 3) IGMC with content by concatenating target user and item's content feature vectors to the graph representation  $\mathbf{g}$  before feeding into the MLP (4), which is similar to (Berg et al., 2017). The results are shown in Table 5.
+
+Table 5: RMSE test results of ablation experiments.  
+
+<table><tr><td>Model</td><td>Flixster</td><td>Douban</td><td>YahooMusic</td><td>ML-100K</td></tr><tr><td>IGMC (original)</td><td>0.872±0.001</td><td>0.721±0.001</td><td>19.1±0.138</td><td>0.905±0.001</td></tr><tr><td>IGMC (SumPooling)</td><td>0.879±0.002</td><td>0.729±0.003</td><td>22.9±1.102</td><td>0.933±0.001</td></tr><tr><td>IGMC (no ARR)</td><td>0.872±0.001</td><td>0.721±0.001</td><td>19.2±0.140</td><td>0.908±0.001</td></tr><tr><td>IGMC (content)</td><td>0.886±0.002</td><td>0.726±0.001</td><td>19.1±0.069</td><td>0.906±0.001</td></tr></table>
+
+From Table 5, we have the following observations. Firstly, using the proposed pooling layer shows a huge improvement over a standard SumPooling. This might be because SumPooling assigns equal importance to all nodes in a subgraph, which fails to distinguish the target user and item from the context nodes. This indicates that a pooling layer able to highlight the target user and item is important for IGMC.
+
+Secondly, we can see that disabling ARR results in a 0.003 performance drop on ML-100K, while seeming to have no influence on the other three datasets. One possible explanation is that ARR is more useful for modeling large and dense enclosing subgraphs, since ARR enhances the modeling power by modeling the extra relationships between edge types in terms of their magnitude differences. Another possible reason is that we only tuned ARR's  $\lambda$  on ML-100K and used  $\lambda = 0.001$  uniformly on all datasets. This is partly verified by that when increasing  $\lambda$  to 0.1 for YahooMusic, we can further decrease IGMC's RMSE to  $18.98 \pm 0.140$ . We did not fully verify this, and leave better ways of modeling rating magnitude for future study.
+
+Thirdly, we observe that incorporating content does not improve IGMC's performance on ML-100K, and often hurts the performance on the other datasets. For Flixster, Douban and YahooMusic, this phenomenon can be explained by that the content of these three datasets are user/item's respective graphs, which has little to no gains to a model that has already exploited graph structure information between users and items very well. In addition, the content feature vectors are presented in 3000-dimensional adjacency vectors, which might pose new problems due to their size and sparsity. For ML-100K, the lost of some performance when adding content actually contradicts with our initial experiments. In our initial experiments when the RMSE on ML-100K without content was around 0.910, we observed that adding content was indeed helpful and reduced the RMSE to
+
+0.907. However, after we introduced ARR and redid the hyperparameter tuning, the RMSE of ML-100K without content became 0.905, and adding content no longer helped. We hypothesize that the benefits of content reduce with the better modeling of graph structure features.
+
+The way we incorporate content might be another reason for why content is not useful in IGMC. In our experiments we only concatenate the target user and item's content vectors with the final graph representation output by the GNN, similar to (Berg et al., 2017). However, this method fails to model the interactions between content and graph structures in the early graph convolution stage. On the other hand, as Berg et al. (2017) and Zhang & Chen (2018) found, directly concatenating content with initial node features (the one-hot encoding vectors) as the input to GNN often led to worse performance due to information flow bottlenecks. We leave exploring better ways to combine content and graph structures for future work.
\ No newline at end of file
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+# INFLUENCE-BASED MULTI-AGENT EXPLORATION
+
+Tonghan Wang*, Jianhao Wang*, Yi Wu & Chongjie Zhang
+
+Institute for Interdisciplinary Information Sciences
+
+Tsinghua University
+
+Beijing, China
+
+wangth18@mails.tsinghua.edu.cn, wjh720.eric@gmail.com
+
+jxwuyi@openai.com, chongjie@tsinghua.edu.cn
+
+# ABSTRACT
+
+Intrinsically motivated reinforcement learning aims to address the exploration challenge for sparse-reward tasks. However, the study of exploration methods in transition-dependent multi-agent settings is largely absent from the literature. We aim to take a step towards solving this problem. We present two exploration methods: exploration via information-theoretic influence (EITI) and exploration via decision-theoretic influence (EDTI), by exploiting the role of interaction in coordinated behaviors of agents. EITI uses mutual information to capture the interdependence between the transition dynamics of agents. EDTI uses a novel intrinsic reward, called Value of Interaction (VoI), to characterize and quantify the influence of one agent's behavior on expected returns of other agents. By optimizing EITI or EDTI objective as a regularizer, agents are encouraged to coordinate their exploration and learn policies to optimize the team performance. We show how to optimize these regularizers so that they can be easily integrated with policy gradient reinforcement learning. The resulting update rule draws a connection between coordinated exploration and intrinsic reward distribution. Finally, we empirically demonstrate the significant strength of our methods in a variety of multi-agent scenarios.
+
+# 1 INTRODUCTION
+
+Reinforcement learning algorithms aim to learn a policy that maximizes the accumulative reward from an environment. Many advances of deep reinforcement learning rely on a dense shaped reward function, such as distance to the goal (Mirowski et al., 2016; Wu et al., 2018), scores in games (Mnih et al., 2015) or expert-designed rewards (Wu & Tian, 2016; OpenAI, 2018), but they tend to struggle in many real-world scenarios with sparse rewards (Burda et al., 2019). Therefore, many recent works propose to introduce additional intrinsic incentives to boost exploration, including pseudocounts (Bellemare et al., 2016; Tang et al., 2017; Ostrovski et al., 2017), model-learning improvements (Burda et al., 2019; Pathak et al., 2017; Burda et al., 2018), and information gain (Florensa et al., 2017; Gupta et al., 2018; Youngseok Kim, 2019). These works result in significant progress in many challenging tasks such as Montezuma Revenge (Burda et al., 2018), robotic manipulation (Pathak et al., 2018; Riedmiller et al., 2018), and Super Mario games (Burda et al., 2019; Pathak et al., 2017).
+
+Notably, most of the existing breakthroughs on sparse-reward environments have been focusing on single-agent scenarios and leave the exploration problem largely unstudied for multi-agent settings – it is common in real-world applications that multiple agents are required to solve a task in a coordinated fashion (Cao et al., 2012; Nowé et al., 2012; Zhang & Lesser, 2011). This problem has recently attracted attention and several exploration strategies have been proposed for transition-independent cooperative multi-agent settings (Dimakopoulou & Van Roy, 2018; Dimakopoulou et al., 2018; Bargiacchi et al., 2018; Iqbal & Sha, 2019b). Nevertheless, how to explore effectively in more general scenarios with complex reward and transition dependency among cooperative agents remains an open research problem.
+
+This paper aims to take a step towards this goal. Our basic idea is to coordinate agents' exploration by taking into account their interactions during their learning processes. Configurations where interaction happens (interaction points) lie at critical junctions in the state-action space, through these critical configurations can transit to potentially important under-explored regions. To exploit this idea, we propose exploration strategies where agents start with decentralized exploration driven by their individual curiosity, and are also encouraged to visit interaction points to influence the exploration processes of other agents and help them get more extrinsic and intrinsic rewards. Based on how to quantify influence among agents, we propose two exploration methods. Exploration via information-theoretic influence (EITI) uses mutual information (MI) to capture the interdependence between the transition dynamics of agents. Exploration via decision-theoretic influence (EDTI) goes further and uses a novel measure called value of interaction (VoI) to disentangle the effect of one agent's state-action pair on the expected (intrinsic) value of other agents. By optimizing MI or VoI as a regularizer to the value function, agents are encouraged to explore state-action pairs where they can exert influences on other agents for learning sophisticated multi-agent cooperation strategies.
+
+To efficiently optimize MI and VoI, we propose augmented policy gradient formulations so that the gradients can be estimated purely from trajectories. The resulting update rule draws a connection between coordinated exploration and the distribution of individual intrinsic rewards among team members, which further explains why our methods are able to facilitate multi-agent exploration.
+
+We demonstrate the effectiveness of our methods on a variety of sparse-reward cooperative multiagent tasks. Empirical results show that both EITI and EDTI allow for the discovery of influential states and EDTI further filter out interactions that have no effects on the performance. Our results also imply that these influential states are implicitly discovered as subgoals in search space that guide and coordinate exploration. The video of experiments is available at https://sites.google.com/view/influence-based-mae/.
+
+# 2 SETTINGS
+
+In our work, we consider a fully cooperative multi-agent task that can be modelled by a factored multi-agent MDP  $G = \langle N, S, A, T, r, h, n \rangle$ , where  $N \equiv \{1, 2, \dots, n\}$  is the finite set of agents,  $S \equiv \times_{i \in N} S_i$  is the finite set of joint states and  $S_i$  is the state set of agent  $i$ . At each timestep, each agent selects an action  $a_i \in A_i$  at state  $s$ , forming a joint action  $a \in A \equiv \times_{i \in N} A_i$ , resulting in a shared extrinsic reward  $r(s, a)$  for each agent and the next state  $s'$  according to the transition function  $T(s'|s, a)$ .
+
+The objective of the task is that each agent learns a policy  $\pi_i(a_i|s_i)$ , jointly maximizing team performance. The joint policy  $\pi = \langle \pi_1, \dots, \pi_n \rangle$  induces an action-value function,  $Q^{ext,\pi}(s, \boldsymbol{a}) = \mathbb{E}_{\tau}[\sum_{t=0}^{h} r^t | s^0 = s, \boldsymbol{a}^0 = \boldsymbol{a}, \pi]$ , and a value function  $V^{ext,\pi}(s) = \max_{\boldsymbol{a}} Q^{ext,\pi}(s, \boldsymbol{a})$ , where  $\tau$  is the episode trajectory and  $h$  is the horizon.
+
+We adopt a centralized training and decentralized execution paradigm, which has been widely used in multi-agent deep reinforcement learning (Foerster et al., 2016; Lowe et al., 2017; Foerster et al., 2018; Rashid et al., 2018). During training, agents are granted access to the states, actions, (intrinsic) rewards, and value functions of other agents, while decentralized execution only requires individual states.
+
+# 3 INFLUENCE-BASED COORDINATED MULTI-AGENT EXPLORATION
+
+Efficient exploration is critical for reinforcement learning, particularly in sparse-reward tasks. Intrinsic motivation (Oudeyer & Kaplan, 2009) is a crucial mechanism for behaviour learning since it provides the driver of exploration. Therefore, to trade off exploration and exploitation, it is common for an RL agent to maximize an objective of the expected extrinsic reward augmented by the expected intrinsic reward. Curiosity is one of the extensively-studied intrinsic rewards to encourage an agent to explore according to its uncertainty about the environment, which can be measured by model prediction error (Burda et al., 2019; Pathak et al., 2017; Burda et al., 2018) or state visitation count (Bellemare et al., 2016; Tang et al., 2017; Ostrovski et al., 2017).
+
+While such an intrinsic motivation as curiosity drives effective individual exploration, it is often not sufficient enough for learning in collaborative multi-agent settings, because it does not take
+
+into account agent interactions. To encourage interactions, we propose an influence value aims to quantify one agent's influence on the exploration processes of other agents. Maximizing this value will encourage agents to visit interaction points more often through which the agent team can reach configurations that are rarely visited by decentralized exploration. In next sections, we will provide two ways to formulate the influence value with such properties, leading to two exploration strategies.
+
+Thus, for each agent  $i$ , our overall optimization objective is:
+
+$$
+J _ {\theta_ {i}} \left[ \pi_ {i} \mid \pi_ {- i}, p _ {0} \right] \equiv V ^ {e x t, \boldsymbol {\pi}} (\boldsymbol {s} _ {0}) + V _ {i} ^ {i n t, \boldsymbol {\pi}} (\boldsymbol {s} _ {0}) + \beta \cdot I _ {- i \mid i} ^ {\boldsymbol {\pi}}, \tag {1}
+$$
+
+where  $p_0(s_0)$  is the initial state distribution,  $\pi_{-i}$  is the joint policy excluding that of agent  $i$ , and  $V_i^{int,\pi}(s)$  is the intrinsic value function of agent  $i$ ,  $I_{-i|i}^{\pi}$  is the influence value,  $\beta > 0$  is a weighting term. In this paper, we use the following notations:
+
+$$
+\tilde {r} _ {i} (\boldsymbol {s}, \boldsymbol {a}) = r (\boldsymbol {s}, \boldsymbol {a}) + u _ {i} \left(s _ {i}, a _ {i}\right), \tag {2}
+$$
+
+$$
+V _ {i} ^ {\pi} (\boldsymbol {s}) = V ^ {\text {e x t}, \pi} (\boldsymbol {s}) + V _ {i} ^ {\text {i n t}, \pi} (\boldsymbol {s}), \tag {3}
+$$
+
+$$
+Q _ {i} ^ {\pi} (\boldsymbol {s}, \boldsymbol {a}) = \tilde {r} _ {i} (\boldsymbol {s}, \boldsymbol {a}) + \sum_ {\boldsymbol {s} ^ {\prime}} T \left(\boldsymbol {s} ^ {\prime} \mid \boldsymbol {s}, \boldsymbol {a}\right) V _ {i} ^ {\pi} \left(\boldsymbol {s} ^ {\prime}\right), \tag {4}
+$$
+
+where  $u_{i}(s_{i},a_{i})$  is a curiosity-derived intrinsic reward,  $\tilde{r}_i(\pmb {s},\pmb {a})$  is a sum of intrinsic and extrinsic rewards,  $V_{i}^{\pi}(\pmb {s})$  and  $Q_{i}^{\pi}(\pmb {s},\pmb {a})$  here contain both intrinsic and extrinsic rewards.
+
+# 3.1 EXPLORATION VIA INFORMATION-THEORETIC INFLUENCE
+
+One critical problem in our learning framework presented above is to define the influence value  $I$ . For simplicity, we start with a two-agent case. The first method we propose is to use mutual information between agents' trajectories to measure one agent's influence on other agents' learning processes. Such mutual information can be defined as information gain of one agent's state transition given the other's state and action. Without loss of generality, we define it from the perspective of agent 1:
+
+$$
+M I _ {2 | 1} ^ {\pi} \left(S _ {2} ^ {\prime}; S _ {1}, A _ {1} \mid S _ {2}, A _ {2}\right) = \sum_ {\boldsymbol {s}, \boldsymbol {a}, s _ {2} ^ {\prime} \in (S, A, S _ {2})} p ^ {\pi} (\boldsymbol {s}, \boldsymbol {a}, s _ {2} ^ {\prime}) \left[ \log p ^ {\pi} \left(s _ {2} ^ {\prime} \mid \boldsymbol {s}, \boldsymbol {a}\right) - \log p ^ {\pi} \left(s _ {2} ^ {\prime} \mid s _ {2}, a _ {2}\right) \right], \tag {5}
+$$
+
+where  $s = (s_1, s_2)$  is the joint state,  $\pmb{a} = (a_1, a_2)$  is the joint action, and  $S_i$  and  $A_i$  are the random variables of state and action of agent  $i$  subject to the distribution induced by the joint policy  $\pi$ . So we define  $I_{2|1}^{\pi}$  as  $M I_{2|1}^{\pi}(S_2'; S_1, A_1 | S_2, A_2)$  that captures transition interactions between agents. Optimizing this objective encourages agent 1 to visited critical points where it can influence the transition probability of agent 2. We call such an exploration method exploration via information-theoretic influence (EITI).
+
+Optimizing  $MI_{2|1}^{\pi}$  with respect to the policy parameters  $\theta_{1}$  of agent 1 is a little bit challenging, because it is an expectation with respect to a distribution that depends on  $\theta_{1}$ . The gradient consists of two terms:
+
+$$
+\begin{array}{l} \nabla_ {\theta_ {1}} M I ^ {\pi} \left(S _ {2} ^ {\prime}; S _ {1}, A _ {1} \mid S _ {2}, A _ {2}\right) = \sum_ {\boldsymbol {s}, \boldsymbol {a}, s _ {2} ^ {\prime} \in (S, A, S _ {2})} \nabla_ {\theta_ {1}} \left(p ^ {\pi} \left(\boldsymbol {s}, \boldsymbol {a}, s _ {2} ^ {\prime}\right)\right) \log \frac {p \left(s _ {2} ^ {\prime} \mid \boldsymbol {s} , \boldsymbol {a}\right)}{p ^ {\pi} \left(s _ {2} ^ {\prime} \mid s _ {2} , a _ {2}\right)} \tag {6} \\ + \sum_ {\boldsymbol {s}, \boldsymbol {a}, s _ {2} ^ {\prime} \in (S, A, S _ {2})} p ^ {\boldsymbol {\pi}} (\boldsymbol {s}, \boldsymbol {a}, s _ {2} ^ {\prime}) \nabla_ {\theta_ {1}} \log \frac {p \left(s _ {2} ^ {\prime} \mid \boldsymbol {s} , \boldsymbol {a}\right)}{p ^ {\boldsymbol {\pi}} \left(s _ {2} ^ {\prime} \mid s _ {2} , a _ {2}\right)}. \\ \end{array}
+$$
+
+While the second term is an expectation over the trajectory and can be shown to be zero (see Appendix B.1), it is unwieldy to deal with the first term because it requires the gradient of the stationary distribution, which depends on the policies and the dynamics of the environment. Fortunately, the gradient can still be estimated purely from sampled trajectories by drawing inspiration from the proof of the policy gradient theorem (Sutton et al., 2000).
+
+The resulting policy gradient update is:
+
+$$
+\nabla_ {\theta_ {1}} J _ {\theta_ {1}} (t) = \left(\hat {R} _ {1} ^ {t} - \hat {V} _ {1} ^ {\pi} \left(s _ {t}\right)\right) \nabla_ {\theta_ {1}} \log \pi_ {\theta_ {1}} \left(a _ {1} ^ {t} \mid s _ {1} ^ {t}\right) \tag {7}
+$$
+
+where  $\hat{V}_1^\pi (s_t)$  is an augmented value function of  $\hat{R}_1^t = \sum_{t' = t}^{h}\hat{r}_1^{t'}$  and
+
+$$
+\hat {r} _ {1} ^ {t} = r ^ {t} + u _ {1} ^ {t} + \beta \log \frac {p \left(s _ {2} ^ {t + 1} \mid s _ {1} ^ {t} , s _ {2} ^ {t} , a _ {1} ^ {t} , a _ {2} ^ {t}\right)}{p \left(s _ {2} ^ {t + 1} \mid s _ {2} ^ {t} , a _ {2} ^ {t}\right)}. \tag {8}
+$$
+
+The third term, which we call EITI reward, is 0 when the agents are transition-independent, i.e., when  $p(s_2^{t + 1}|s_1^t,s_2^t,a_1^t,a_2^t) = p(s_2^{t + 1}|s_2^t,a_2^t)$ , and is positive when  $s_1^t,a_1^t$  increase the probability of agent 2 translating to  $s_2^{t + 1}$ . Therefore, the EITI reward is an intrinsic motivation that encourages agent 1 to visit more frequently the state-action pairs where it can influence the trajectory of agent 2. The estimation of  $p(s_{2}^{t + 1}|s_{1}^{t},s_{2}^{t},a_{1}^{t},a_{2}^{t})$  and  $p(s_2^{t + 1}|s_2^t,a_2^t)$  are discussed in Appendix C. We assume that agents know the states and actions of other agents, but this information is only available during centralized training. When execution, agents only have access to their local observations.
+
+# 3.2 EXPLORATION VIA DECISION-THEORETIC INFLUENCE
+
+Mutual information characterizes the influence of one agent's trajectory on that of the other and captures interactions between the transition functions of the agents. However, it does not provide the value of these interactions to identify interactions related to more internal and external rewards  $(\tilde{r})$ . To address this issue, we propose exploration via decision-theoretic influence (EDTI) based on a decision-theoretic measure of  $I$ , called Value of Interaction (VoI), which disentangles both transition and reward influences. VoI is defined as the expected difference between the action-value function of one agent (e.g., agent 2) and its counterfactual action-value function without considering the state and action of the other agent (e.g., agent 1):
+
+$$
+V o I _ {2 | 1} ^ {\pi} \left(S _ {2} ^ {\prime}; S _ {1}, A _ {1} \mid S _ {2}, A _ {2}\right) = \sum_ {\boldsymbol {s}, \boldsymbol {a}, s _ {2} ^ {\prime} \in (S, A, S _ {2})} p ^ {\pi} (\boldsymbol {s}, \boldsymbol {a}, s _ {2} ^ {\prime}) \left[ Q _ {2} ^ {\pi} (\boldsymbol {s}, \boldsymbol {a}, s _ {2} ^ {\prime}) - Q _ {2 | 1} ^ {\pi , *} \left(s _ {2}, a _ {2}, s _ {2} ^ {\prime}\right) \right], \tag {9}
+$$
+
+where  $Q_2^\pi (s,a,s_2^{\prime})$  is the expected rewards (including intrinsic rewards) of agent 2 defined as:
+
+$$
+Q _ {2} ^ {\pi} (\boldsymbol {s}, \boldsymbol {a}, s _ {2} ^ {\prime}) = \tilde {r} _ {2} (\boldsymbol {s}, \boldsymbol {a}) + \gamma \sum_ {s _ {1} ^ {\prime}} p \left(s _ {1} ^ {\prime} \mid \boldsymbol {s}, \boldsymbol {a}, s _ {2} ^ {\prime}\right) V _ {2} ^ {\pi} \left(\boldsymbol {s} ^ {\prime}\right), \tag {10}
+$$
+
+and the counterfactual action-value function  $Q_{2}^{\pi, *}$  (also includes intrinsic and extrinsic rewards) can be obtained by marginalizing out the state and action of agent 1:
+
+$$
+Q _ {2 | 1} ^ {\pi , *} \left(s _ {2}, a _ {2}, s _ {2} ^ {\prime}\right) = \sum_ {s _ {1} ^ {*}, a _ {1} ^ {*}} p ^ {\pi} \left(s _ {1} ^ {*}, a _ {1} ^ {*} \mid s _ {2}, a _ {2}\right) \left[ \tilde {r} _ {2} \left(s _ {1} ^ {*}, s _ {2}, a _ {1} ^ {*}, a _ {2}\right) + \gamma \sum_ {s _ {1} ^ {\prime}} p \left(s _ {1} ^ {\prime} \mid s _ {1} ^ {*}, s _ {2}, a _ {1} ^ {*}, a _ {2}, s _ {2} ^ {\prime}\right) V _ {2} ^ {\pi} \left(\boldsymbol {s} ^ {\prime}\right) \right]. \tag {11}
+$$
+
+Note that the definition of VoI is analogous to that of MI and the difference lies in that  $\log p(\cdot)$  measures the amount of information while  $Q$  measures the action value. Although VoI can be obtained by learning  $Q_{2}^{\pi}(s, a)$  and  $Q_{2}^{\pi}(s_{2}, a_{2})$  and calculating the difference, we propose to explicitly marginalize out  $s_{1}^{*}$  and  $a_{1}^{*}$  utilizing the estimated model transition probability  $p^{\pi}(s_{2}'|s_{2}, a_{2})$  and  $p(s_{2}'|s, a)$  to get a more accurate value estimate (Feinberg et al., 2018). The performance of these two formulations are compared in the experiments.
+
+Value functions  $Q$  and  $V$  used in VoI contains both expected external rewards and internal rewards, which will not only encourage coordinated exploration by the influence between intrinsic rewards but also filter out meaningless interactions which can not lead to extrinsic reward after intrinsic reward diminishes. To facilitate the optimization of VoI, we rewrite it as an expectation over state-action trajectories.
+
+$$
+\left. \right. V o I _ {2 | 1} ^ {\pi} \left(S _ {2} ^ {\prime}; S _ {1}, A _ {1} \mid S _ {2}, A _ {2}\right) = \mathbb {E} _ {\tau} \left[ \tilde {r} _ {2} (\boldsymbol {s}, \boldsymbol {a}) - \tilde {r} _ {2} ^ {\pi} \left(s _ {2}, a _ {2}\right) + \gamma \left(1 - \frac {p ^ {\pi} \left(s _ {2} ^ {\prime} \mid s _ {2} , a _ {2}\right)}{p \left(s _ {2} ^ {\prime} \mid s , \boldsymbol {a}\right)}\right) V _ {2} ^ {\pi} \left(\boldsymbol {s} ^ {\prime}\right)\right], \tag {12}
+$$
+
+where  $\tilde{r}_2^\pi (s_2,a_2)$  is the counterfactual immediate reward. The detailed proof is deferred to Appendix B.2. From this definition, we can intuitively see how VoI reflects the value of interactions.  $\tilde{r}_2(s,\pmb {a})-$ $\tilde{r}_2^{\pi}(s_2,a_2)$  and  $1 - p^{\pi}(s_2'|s_2,a_2) / p(s_2'|s,\pmb {a})$  measure the influence of agent 1 on the immediate reward and the transition function of agent 2, and  $V_{2}^{\pi}(s^{\prime})$  serves as a scale factor in terms of future value. Only when agent 1 and agent 2 are both transition- and reward-independent, i.e., when  $p^\pi (s_2'|s_2,a_2) = p(s_2'|s,\pmb {a})$  and  $r_2^\pi (s_2,a_2) = r_2(s,\pmb {a})$  will VoI equal to 0. In particular, maximizing
+
+VoI with respect to policy parameters  $\theta_{1}$  will lead agent 1 to meaningful interaction points, where  $V_{2}^{\pi}(s')$  is high and  $s_1$ ,  $a_1$  can increase the probability that  $s'$  is reached.
+
+In this learning framework, agents initially explore the environment individually driven by its own curiosity, during which process they will discover potentially valuable interaction points where they can influence the transition function and (intrinsic) rewarding structure of each other. VoI highlights these points and encourages agents to visit these configurations more frequently. As intrinsic reward diminishes, VoI can gradually distinguish those interaction points which are necessary to get extrinsic rewards.
+
+# 3.2.1 POLICY OPTIMIZATION WITH VOI
+
+We want to optimize  $J_{\theta_i}$  with respect to the policy parameters  $\theta_i$ , where the most cumbersome term is  $\nabla_{\theta_i} V o I_{-i|i}$ . For brevity, we can consider a two-agent case, e.g., optimizing  $V o I_{2|1}$  with respect to the policy parameters  $\theta_1$ . Directly computing the gradient  $\nabla_{\theta_1} V o I_{2|1}$  is not stable, because  $V o I_{2|1}$  contains policy-dependent functions  $\tilde{r}_2^\pi(s_2, a_2), p^\pi(s_2'|s_2, a_2)$ , and  $V_2^\pi(s')$  (see Eq. 12). To stabilize training, we use target functions to approximate these policy-dependent functions, which is a commonly used technique in deep RL (Mnih et al., 2015). With this approximation, we denote
+
+$$
+g _ {2} (\boldsymbol {s}, \boldsymbol {a}) = \tilde {r} _ {2} (\boldsymbol {s}, \boldsymbol {a}) - \tilde {r} _ {2} ^ {-} (s _ {2}, a _ {2}) + \gamma \sum_ {\boldsymbol {s} ^ {\prime}} T (\boldsymbol {s} ^ {\prime} | \boldsymbol {s}, \boldsymbol {a}) \left(1 - \frac {p ^ {-} \left(s _ {2} ^ {\prime} \mid s _ {2} , a _ {2}\right)}{p \left(s _ {2} ^ {\prime} \mid \boldsymbol {s} , \boldsymbol {a}\right)}\right) V _ {2} ^ {-} \left(s _ {1} ^ {\prime}, s _ {2} ^ {\prime}\right). \tag {13}
+$$
+
+where  $r_2^-, p^-$ , and  $V_2^-$  are corresponding target functions. As these target functions are only periodically updated during the learning, their gradients over  $\theta_{1}$  can be approximately ignored. Therefore, from Eq. 12, we have
+
+$$
+\nabla_ {\theta_ {1}} \operatorname {V o} I _ {2 | 1} ^ {\pi} \left(S _ {2} ^ {\prime}; S _ {1}, A _ {1} \mid S _ {2}, A _ {2}\right) \approx \sum_ {\boldsymbol {s}, \boldsymbol {a} \in (S, A)} \left(\nabla_ {\theta_ {1}} p ^ {\pi} (\boldsymbol {s}, \boldsymbol {a})\right) g _ {2} (\boldsymbol {s}, \boldsymbol {a}). \tag {14}
+$$
+
+Similar to the calculation of  $\nabla_{\theta_i}MI$ , we get the gradient at every step (see Appendix B.3 for proof):
+
+$$
+\nabla_ {\theta_ {1}} J _ {\theta_ {1}} (t) \approx \left(\hat {R} _ {1} ^ {t} - \hat {V} _ {1} ^ {\pi} \left(s _ {t}\right)\right) \nabla_ {\theta_ {1}} \log \pi_ {\theta_ {1}} \left(a _ {1} ^ {t} \mid s _ {1} ^ {t}\right), \tag {15}
+$$
+
+where  $\hat{V}_1^\pi (s_t)$  is an augmented value function regressed towards  $\hat{R}_1^t = \sum_{t' = t}^{h}\hat{r}_1^{t'}$  and
+
+$$
+\hat {r} _ {1} ^ {t} = r ^ {t} + u _ {1} ^ {t} + \beta \left[ u _ {2} ^ {t} + \gamma \left(1 - \frac {p ^ {-} \left(s _ {2} ^ {t + 1} \mid s _ {2} ^ {t} , a _ {2} ^ {t}\right)}{p \left(s _ {2} ^ {t + 1} \mid s _ {1} ^ {t} , s _ {2} ^ {t} , a _ {1} ^ {t} , a _ {2} ^ {t}\right)}\right) V _ {2} ^ {-} \left(s _ {1} ^ {t + 1}, s _ {2} ^ {t + 1}\right) \right]. \tag {16}
+$$
+
+We call  $u_2^t + \gamma \left(1 - \frac{p^- (s_2^{t+1} | s_2^t, a_2^t)}{p(s_2^{t+1} | s_1^t, s_2^t, a_1^t, a_2^t)} \right) V_2^- (s_1^{t+1}, s_2^{t+1})$  the EDTI reward.
+
+# 3.3 DISCUSSIONS
+
+Scale to Large Settings: For cases with more than two agents, the VoI of agent  $i$  on other agents can be defined similarly to Eq. 9, which is annotated with  $VoI_{-i|i}^{\pi}(S_{-i}'; S_i, A_i|S_{-i}, A_{-i})$ , where  $S_{-i}$  and  $A_{-i}$  are the state and action sets of all agents other than agent  $i$ . In practice, agents interaction can often be decomposed to pairwise interaction so  $VoI_{-i|i}^{\pi}(S_{-i}'; S_i, A_i|S_{-i}, A_{-i})$  is well approximated by the sum of values of pairwise value of interaction:
+
+$$
+V o I _ {- i | i} ^ {\pi} \left(S _ {- i} ^ {\prime}; S _ {i}, A _ {i} \mid S _ {- i}, A _ {- i}\right) \approx \sum_ {j \in N, j \neq i} V o I _ {j | i} ^ {\pi} \left(S _ {j} ^ {\prime}; S _ {i}, A _ {i} \mid S _ {- i}, A _ {- i}\right). \tag {17}
+$$
+
+Relationship between EITI and EDTI: EITI and EDTI gradient updates are obtained by information- and decision-theoretical influence respectively. Therefore, it is nontrivial to derive that part of the EDTI reward is a lower bound of the EITI reward:
+
+$$
+1 - \frac {p \left(s _ {- i} ^ {\prime} \mid s _ {- i} , a _ {- i}\right)}{p \left(s _ {- i} ^ {\prime} \mid \boldsymbol {s} , \boldsymbol {a}\right)} \leq \log \frac {p \left(s _ {- i} ^ {\prime} \mid \boldsymbol {s} , \boldsymbol {a}\right)}{p \left(s _ {- i} ^ {\prime} \mid s _ {- i} , a _ {- i}\right)}, \quad \forall \boldsymbol {s}, \boldsymbol {a}, s _ {- i} ^ {\prime} \tag {18}
+$$
+
+which easily follows given that  $\log x\geq 1 - 1 / x$  for  $\forall x > 0$ . This draws a connection between EITI and EDTI reward.
+
+Table 1: Baseline algorithms. The third column is the reward used to train the value function of PPO.  $u_{i}$  and  $u_{cen}$  are curiosity about individual state  $s_{i}$  and global state  $s$ ,  $T_{1} = \log(p(s_{-i}'|s, a)/p(s_{-i}'|s_{-i}, a_{-i}))$ ,  $T_{2} = 1 - p(s_{-i}'|s_{-i}, a_{-i}) / p(s_{-i}'|s, a)$ , and  $\Delta Q_{-i}(s, a) = Q_{-i}(s, a) - Q_{-i}(s_{-i}, a_{-i})$ . Social influence (Jaques et al., 2018) and COMA (Foerster et al., 2018) are augmented with curiosity.  
+
+<table><tr><td></td><td>Alg.</td><td>Reward</td><td>Description</td></tr><tr><td>Ours</td><td>EITIEDTI</td><td>r+ui+βT1r+ui+β(u-i+γT2V-i)</td><td>Influence-theoretic influenceDecision-theoretic influence</td></tr><tr><td rowspan="4">Other Exploration Methods</td><td rowspan="4">randomcendeccen_control</td><td>r</td><td>Pure PPO</td></tr><tr><td>r+ucen</td><td>Decentralized PPO with cen curiosity</td></tr><tr><td>r+ui</td><td>Decentralized PPO with dec curiosity</td></tr><tr><td>r+ucen</td><td>Centralized PPO with cen curiosity</td></tr><tr><td rowspan="4">Ablations</td><td rowspan="4">r_influenceplusVshared_criticQ-Q</td><td>r+ui+βu-i</td><td>Disentangle reward interaction</td></tr><tr><td>r+ui+βV-i</td><td>Use other agents&#x27; value functions</td></tr><tr><td>r+ucen</td><td>PPO with shared V and cen curiosity</td></tr><tr><td>r+ui+βΔQ_i(s,a)</td><td>EDTI without explicit counterfactual</td></tr><tr><td rowspan="3">Related Works</td><td rowspan="3">socialCOMAMulti</td><td>—</td><td>By Jaques et al. (2018)</td></tr><tr><td>—</td><td>By Foerster et al. (2018)</td></tr><tr><td>—</td><td>By Iqbal &amp; Sha (2019b)</td></tr></table>
+
+Comparing EDTI to Centralized Methods: Different from a centralized method which directly includes value functions of other agents in the optimization objective, (i.e., by setting total reward  $\hat{r}_i = r + u_i + \beta (u_{-i} + \gamma V_{-i})$ , which is called plusV henceforth), the EDTI reward for agent  $i$  disentangles its contributions to values of another agents using a counterfactual formulation. This difference is important for quantifying influence because the value of another agent does not just contain the contributions from agent  $i$ , but also those of itself and third-party agents. Therefore, EDTI is a kind of intrinsic reward assignment. Our experiments in the next section will compare the performance of plusV against our methods, which verify the importance of the intrinsic reward assignment.
+
+# 4 EXPERIMENTAL RESULTS
+
+Our experiments aim to answer the following questions: (1) Can EITI and EDTI rewards capture interaction points? If they can, how do these points change throughout exploration? (2) Can exploiting these interaction points facilitate exploration and learning performance? (3) Can EDTI filter out interaction points that are not related to environmental rewards? (4) What if only reward influence between agents are disentangled? We evaluate our approach on a set of multi-agent tasks with sparse rewards based on a discrete version of multi-agent particle world environment (Lowe et al., 2017). PPO (Schulman et al., 2017) is used as the underlying algorithm. For evaluation, all experiments are carried out with 5 different random seeds and results are shown with  $95\%$  confidence interval. Demonstrative videos<sup>1</sup> are available online.
+
+Baselines We compare our methods with various baselines shown in Table 1. In particular, we carry out the following ablation studies: i) r_influence disentangles immediate reward influence between agents, (derivation of the associated augmented reward can be found in Appendix B.4. Reward influence in long term is not considered because it inevitably involves transition interactions) ii) PlusV as described in Sec. 3.3. iii) Shared_critic uses decentralized PPO agents with shared centralized value function and thus is a cooperative version of MADDPG (Lowe et al., 2017) augmented with intrinsic reward of curiosity. iv) Q-Q is similar to EDTI but without explicit counterfactual formulation, as described in Sec. 3.2. We also note that EITI is an ablation of EDTI which considers transition interactions. PlusV, shared_critic, Q-Q, and cen_control have access to global or other agents' value functions during training. When execution, all the methods except cen_control only require local state.
+
+![](images/c0052723a5c3781999207b94fbfa9af1b45b8f92ebae657c63c4fec2279eaeaf.jpg)  
+Figure 1: Didactic examples. Left: task Pass. Two agents starting at the upper-left corner are only rewarded when both of them reach the other room through the door, which will open only when at least one of the switches is occupied by one or more agents. Middle: Secret-Room. An extension of Pass with 4 rooms and switches. When the switch 1 is occupied, all the three doors turn open. And the three switches on the right only control the door of its room. The agents need to reach the upper right room to achieve any reward. Right: comparison of our methods with ablations on Pass.
+
+![](images/6a01c4fc3b24351f81f47d4adeb728bcb9adb40b1bcb2b5548249407796ef880.jpg)
+
+![](images/f6f39af8db092ec1e3dfd7b9275d8c465d505b193113df0d36babd0f6366cf79.jpg)
+
+# 4.1 DIDACTIC EXAMPLES
+
+We present two didactic examples of multi-agent cooperation tasks with sparse reward to explain how EITI and EDTI work. The first didactic example consists of a  $30 \times 30$  maze with two rooms and a door with two switches (Fig. 1 left). In the optimal strategy, one agent should first step on switch 1 to help the other agent pass the door, and then the agent that has already reached the right half should further go to switch 2 to bring the remaining agent in. There are two pairs of interaction points in this task: (switch 1, door) and (switch 2, door), i.e., transition probability of the agent near door is determined by whether another agent is on one of the switch.
+
+Fig. 1-right and Fig. 2-top show the learning curves of our methods and all the baselines, among which EITI, EDTI, r.influence, Multi, and centralized control can learn the winning strategy and ours learn much more efficiently. Fig. 2-bottom gives a possible explanation why our methods work. EITI and EDTI rewards successfully highlight the interaction points (before 100 and 2100 updates, respectively). Agents are encouraged to explore these configurations more frequently and thus have better chance to learn the goal strategy. EDTI reward considers the value function of the other agent, so it converges slower than the EITI reward. In contrast, directly adding the other agent's intrinsic rewards and value functions is noisy (see "plusV reward") and confuses the agent because these contain the effect of the other agent's exploration. As for centralized control, global curiosity encourages agents to try all possible configurations, so it can find environmental rewards in most tasks. However, visiting all configurations without bias renders it inefficient - external rewards begin to dominate the behaviors of agents after 7000 updates even with the help of centralized learning algorithm. Our methods use the same information as centralized exploration but take advantages of agents' interactions to accelerate exploration.
+
+In order to evaluate whether EDTI can help filter out noisy interaction points and accelerate exploration, we conduct experiments in a second didactic task (see Fig. 1 middle). It is also a navigation task in a  $25 \times 25$  maze where agents are rewarded for being in a goal room. However, in this experiment, we consider a case where there are four rooms and the upper right one is attached to reward. This task contains 6 pairs of interaction points (switch 1 with each of the doors, each switch with the door of the same room), but only two of them are related to external rewards, i.e., (switch 1, door 1) and (switch 2, door 1). As Fig. 3-right shows, EITI agents treat three doors equally even after 7400 updates (see Fig. 3 right, 7400 updates, top row). In comparison, although EDTI reward suffers from noise in the beginning, it clearly highlights two pairs of valuable interaction points (see Fig. 3 right, 7400 updates, bottom row) as intrinsic reward diminishes. This can explain why EDTI outperforms EITI (Fig. 3 left).
+
+![](images/0c0f70de8d7005bc16b4856461fe41ce7867056c7722570347a65c33ef88121a.jpg)  
+Figure 2: Development of performance of our methods compared to baselines and intrinsic reward terms of EITI, EDTI, and plusV over the training period of 9000 PPO updates segmented into three phases. "Team Reward" shows averaged team reward gained in a episode, with a maximum of 1000. It shows that only EITI, EDTI, and centralized control and Multi can learn the strategy during this stage. "EITI reward", "EDTI reward", and "plusV reward" demonstrate the evolving of corresponding intrinsic rewards.
+
+![](images/3afd1ee057debc74757440ec1d55e6e961b5828566e5d9c3d5d1a0dffa6f8068.jpg)  
+Figure 3: Left: performance comparison between EDTI and EITI on Secret-Room over 7400 PPO updates. Right: EITI and EDTI terms of two agents after 100, 2900, and 7400 updates.
+
+![](images/f8db0a24527c68ba0dad413607743c8263c7bee18bfc733953c46d674b908baa.jpg)
+
+![](images/09f343ef12692c5842ce06e5850991cfef331d81b58e62ce4e85058be4cd48d4.jpg)
+
+![](images/959e647609b55a9bd2a1cb13b5eb33a1e1510f482c4e79b23f651c3df47705a9.jpg)
+
+![](images/888fb64542bae2fce3fd06712d6182162f2aa4bea0ae1d388a2985fdccaf8f8c.jpg)  
+Figure 4: Comparison of our methods against ablations for Push-Box, Island, and Large-Island. Comparison with baselines is shown in Fig. 8 in Appendix D.
+
+![](images/c6389f0a18ba638cf3a2fe5334ac26507d2d49ee5eef3135b1016a79270b9881.jpg)
+
+![](images/e65bef5b6e4a4fc7b04be9ec59f356ab93e01123818cc8b6b10450bb6f2ce750.jpg)
+
+# 4.2 EXPLORATION IN COMPLEX TASKS
+
+Next, we evaluate the performance of our methods on more complex tasks. To this end, we use three sparse reward cooperative multi-agent tasks depicted in Fig. 7 of Appendix D and analyzed below. Details of implementation and experiment settings are also described in Appendix D.
+
+Push-Box: A  $15 \times 15$  room is populated with 2 agents and 1 box. Agents need to push the box to the wall in 300 environment steps to get a reward of 1000. However, the box is so heavy that only when two agents push it in the same direction at the same time can it be moved a grid. Agents need to coordinate their positions and actions for multiple steps to earn a reward. The purpose of this task is to demonstrate that EITI and EDTI can explore long-term cooperative strategy.
+
+Island: This task is a modified version of the classic Stag Hunt game (Peysakhovich & Lerer, 2018) where two agents roam a  $10 \times 10$  island populated with 9 treasures and a random walking beast for 300 environment steps. Agents can collect a treasure by stepping on it to get a team reward of 10 or, by attacking the beast within their attack range, capture it for a reward of 300. The beast would also attack the agents when they are too close. The beast and agent have a maximum energy of 8 and 5 respectively, which will be subtracted by 1 every time attacked. Therefore, an agent is too weak to beat the beast alone and they have to cooperate. In order to learn optimal strategy in this task, one method has to keep exploring after sub-optimal external rewards are found.
+
+Large-Island: Similar to Island but with more agents (4), more treasures (16), and a beast with more energy (16) and a higher reward (600) for being caught. This task aims to demonstrate feasibility of our methods in cases with more than 2 agents.
+
+Push-Box requires agents to take coordinated actions at certain positions for multiple steps to get rewarded. Therefore, this task is particularly challenging and all the baselines struggle to earn any reward (Fig. 4 left and Fig. 8 left). Our methods are considerably more successful because interaction happens when the box is moved – agents remain unmoved when they push the box alone but will move by a grid if push it together. In this way, EITI and EDTI agents are rewarded intrinsically to move the box and thus are able to quickly find the optimal policy.
+
+In the Island task, collecting treasures is a easily-attainable local optimal. However, efficient treasures collecting requires the agents to spread on the island. This leads to a situation where attempting to attack the beast seems a bad choice since it is highly possible that agents will be exposed to the beast's attack alone. They have to give up profitable spreading strategy and take the risk of being killed to discover that if they attack the beast collectively for several timesteps, they will get much more rewards. Our methods help solve this challenge by giving agents intrinsic incentives to appear together in the attack range of the beast, where they have indirect interactions (health is part of the state and it decreases slower when the two are attacked alternatively). Fig. 9 in Appendix D demonstrates that our methods learn to catch the beast quickly, and thus have better performance (Fig. 8 right).
+
+Finally, outperformance of our methods on Large-Island proves that they can successfully handle cases with more than two agents.
+
+In summary, both of our methods are able to facilitate effective exploration on all the tasks by exploiting interactions. EITI outperforms EDTI in scenarios where all interaction points align with extrinsic rewards. On other tasks, EDTI performs better than EITI due to its ability to filter out interaction points that can not lead to more values.
+
+We also study EDTI with only intrinsic rewards, discussion and results are included in Appendix A.
+
+# 5 RELATED WORKS
+
+Single-agent exploration achieves conspicuous success recently. Provably efficient methods are proposed, such as upper confidence bound (UCB) (Jaksch et al., 2010; Azar et al., 2017; Jin et al., 2018) and posterior sampling for reinforcement learning (PSRL) (Strens, 2000; Osband et al., 2013; Osband & Van Roy, 2016; Agrawal & Jia, 2017). Given that these methods do not scale well to large or continuous settings, another line of research has been focusing on curiosity-driven exploration (Schmidhuber, 1991; Chentanez et al., 2005; Oudeyer et al., 2007; Barto, 2013; Bellemare et al., 2016; Pathak et al., 2017; Ostrovski et al., 2017), and have shown impressive results (Burda
+
+et al., 2019; 2018; Hyoungseok Kim, 2019). In addition, methods based on variational information maximization (Houthooft et al., 2016; Barron et al., 2018) and mutual information (Rubin et al., 2012; Still & Precup, 2012; Salge et al., 2014; Mohamed & Rezende, 2015; Hyoungseok Kim, 2019) have been proposed for single-agent intrinsically motivated exploration.
+
+Although multi-agent reinforcement learning (MARL) has been making significant progresses in recent years (Foerster et al., 2018; Lowe et al., 2017; Wen et al., 2019; Iqbal & Sha, 2019a; Sunehag et al., 2018; Son et al., 2019; Rashid et al., 2018), less attention has been drawn to multi-agent exploration. Dimakopoulou & Van Roy (2018) and Dimakopoulou et al. (2018) propose posterior sampling methods for exploration of concurrent reinforcement learning in coverage problems, Bargiacchi et al. (2018) presents a multi-agent upper confidence exploration method for repeated single-stage problems, and Iqbal & Sha (2019b) investigates methods to combine several decentralized curiosity-driven exploration strategies. All these works focus on transition-independent settings. Another Bayesian exploration approach has been proposed for learning in stateless repeated games (Chalkiadakis & Boutilier, 2003). In contrast, this paper focuses on more general multi-agent sequential decision making problems with complex reward dependencies and transition interactions among agents.
+
+In the literature of MARL, COMA (Foerster et al., 2018) shares some similarity with our decision-theoretic EDTI approach in that both of them use the idea of counterfactual formulations. However, they are quite different in terms of definition and optimization: (1) conceptually, EDTI measures the influence of one agent on the value functions of other agents, while COMA quantifies individual contribution to the team value; (2) EDTI is defined on counterfactual Q-value over state-action pairs of other agents given its own state-action pair, while COMA uses the counterfactual Q-value just over its own action without considering state information, which is critical for exploration; (3) we explicitly derive the gradients for optimizing EDTI influence for coordinated exploration in the policy gradient framework, which provides more accurate feedback, while COMA uses the counterfactual Q value as a critic. Another line of relevant works (Oliehoek et al., 2012; de Castro et al., 2019) propose influence-based abstraction to predict influence sources to help local decision making of agents. In contrast, this paper presents two novel approaches that quantify and maximize the influence between agents for enabling coordinated multi-agent exploration.
+
+In addition, some previous MARL work has also studied intrinsic rewards. One notably relevant work is Jaques et al. (2018), which models the social influence of one agent on other agents' policies. In contrast, EITI measures the influence of one agent on the transition dynamics of other agents. Accompanying this distinction, EITI includes states of agents in the calculation of influence while social influence dos not. Apart from that, the optimization methods are also different – we directly derive the gradients of mutual information and incorporate its optimization in the policy gradient framework, while Jaques et al. (2018) adds social influence reward to the immediate environmental reward for training policies. Hughes et al. (2018) proposes an inequality aversion reward for learning in intertemporal social dilemmas. Strouse et al. (2018) uses mutual information between goal and states or actions as an intrinsic reward to train the agent to share or hide their intentions.
+
+# 6 CLOSING REMARKS
+
+In this paper, we study the multi-agent exploration problem and propose two influence-based methods that exploits the interaction structure. These methods are based on two interaction measures, MI and Value of Interaction (VoI), which respectively measure the amount and value of one agent's influence on the other agents' exploration processes. These two measures can be best regraded as exploration bonus distribution. We also propose an optimization method in the policy gradient framework, which enables agents to achieve coordinated exploration in a decentralized manner and optimize team performance.
+
+# REFERENCES
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+
+# APPENDIX
+
+# A INTRINSIC EDTI
+
+Value of interaction (VoI) captures both transition and reward influence among agents, and it facilitates coordinated exploration by encouraging interactions. VoI contains influence of both intrinsic and extrinsic rewards. Since single-agent literature has studied purely curiosity-driven learning and gets cutting-edge performance (Burda et al., 2019), it is interesting to investigate the performance of VoI given only intrinsic rewards.
+
+Intuitively, intrinsic VoI distributes individual curiosity among team members and facilitates exploration by encouraging agents to help each other to reach under-explored states. Specifically, we use the following objective:
+
+$$
+J _ {\theta_ {i}} [ \pi_ {i} | \pi_ {- i}, p _ {0} ] \equiv V ^ {e x t, \boldsymbol {\pi}} (\boldsymbol {s} _ {0}) + V _ {i} ^ {i n t, \boldsymbol {\pi}} (\boldsymbol {s} _ {0}) + \beta \cdot V o I _ {- i | i} ^ {i n t, \boldsymbol {\pi}}. \tag {19}
+$$
+
+The corresponding augmented reward is:
+
+$$
+\hat {r} _ {1} ^ {t} = r _ {t} + u _ {1} ^ {t} + \beta \left[ u _ {2} ^ {t} + \gamma \left(1 - \frac {p ^ {-} \left(s _ {2} ^ {t + 1} \mid s _ {2} ^ {t} , a _ {2} ^ {t}\right)}{p \left(s _ {2} ^ {t + 1} \mid s _ {1} ^ {t} , s _ {2} ^ {t} , a _ {1} ^ {t} , a _ {2} ^ {t}\right)}\right) V _ {2} ^ {\text {i n t}, -} \left(s _ {1} ^ {t + 1}, s _ {2} ^ {t + 1}\right) \right] \tag {20}
+$$
+
+We use this method (intrinsic EDTI) to train the agents on Pass, Secret-Room, Push-Box, and Island and show the results in Fig. 5.
+
+# B MATHEMATICAL DETAILS
+
+# B.1 GRADIENT OF MUTUAL INFORMATION
+
+To encourage agents to exert influence on transitions of other agents, we optimize mutual information between agent's trajectories. In particular, in the following, we show that term 2 in Eq. 6 is always zero.
+
+$$
+\begin{array}{l} T 2 = \sum_ {\boldsymbol {s}, \boldsymbol {a}, s _ {2} ^ {\prime} \in (S, A, S _ {2})} p ^ {\pi} (\boldsymbol {s}, \boldsymbol {a}, s _ {2} ^ {\prime}) \nabla_ {\theta_ {1}} \log \frac {p \left(s _ {2} ^ {\prime} \mid \boldsymbol {s} , \boldsymbol {a}\right)}{p ^ {\pi} \left(s _ {2} ^ {\prime} \mid s _ {2} , a _ {2}\right)} (21) \\ = - \sum_ {\boldsymbol {s}, \boldsymbol {a}, s _ {2} ^ {\prime}} p ^ {\pi} (\boldsymbol {s}, \boldsymbol {a}, s _ {2} ^ {\prime}) \nabla_ {\theta_ {1}} \log p ^ {\pi} \left(s _ {2} ^ {\prime} \mid s _ {2}, a _ {2}\right) (22) \\ = - \sum_ {\boldsymbol {s}, \boldsymbol {a}, s _ {2} ^ {\prime}} p ^ {\pi} (\boldsymbol {s}, \boldsymbol {a}, s _ {2} ^ {\prime}) \frac {\nabla_ {\theta_ {1}} \left(p ^ {\pi} \left(s _ {2} ^ {\prime} \mid s _ {2} , a _ {2}\right)\right)}{p ^ {\pi} \left(s _ {2} ^ {\prime} \mid s _ {2} , a _ {2}\right)} (23) \\ = - \sum_ {\boldsymbol {s}, \boldsymbol {a}, s _ {2} ^ {\prime}} \frac {p ^ {\pi} \left(\boldsymbol {s} , \boldsymbol {a} , s _ {2} ^ {\prime}\right)}{p ^ {\pi} \left(s _ {2} ^ {\prime} \mid s _ {2} , a _ {2}\right)} \nabla_ {\theta_ {1}} \left(\sum_ {s _ {1} ^ {*}, a _ {1} ^ {*}} p \left(s _ {2} ^ {\prime} \mid s _ {2}, a _ {2}, s _ {1} ^ {*}, a _ {1} ^ {*}\right) p \left(s _ {1} ^ {*} \mid s _ {2}, a _ {2}\right) \pi_ {\theta_ {1}} \left(a _ {1} ^ {*} \mid s _ {1} ^ {*}\right)\right) (24) \\ = - \sum_ {\boldsymbol {s}, \boldsymbol {a}, s _ {2} ^ {\prime}} \frac {p ^ {\pi} (\boldsymbol {s} , \boldsymbol {a} , s _ {2} ^ {\prime})}{p ^ {\pi} \left(s _ {2} ^ {\prime} \mid s _ {2} , a _ {2}\right)} \sum_ {s _ {1} ^ {*}, a _ {1} ^ {*}} p \left(s _ {2} ^ {\prime} \mid s _ {2}, a _ {2}, s _ {1} ^ {*}, a _ {1} ^ {*}\right) p \left(s _ {1} ^ {*} \mid s _ {2}, a _ {2}\right) \nabla_ {\theta_ {1}} \pi_ {\theta_ {1}} \left(a _ {1} ^ {*} \mid s _ {1} ^ {*}\right) (25) \\ = - \sum_ {s _ {2}, a _ {2}, s _ {2} ^ {\prime}} \frac {p ^ {\pi} \left(s _ {2} , a _ {2} , s _ {2} ^ {\prime}\right)}{p ^ {\pi} \left(s _ {2} ^ {\prime} \mid s _ {2} , a _ {2}\right)} \sum_ {s _ {1} ^ {*}, a _ {1} ^ {*}} p \left(s _ {2} ^ {\prime} \mid s _ {2}, a _ {2}, s _ {1} ^ {*}, a _ {1} ^ {*}\right) p \left(s _ {1} ^ {*} \mid s _ {2}, a _ {2}\right) \nabla_ {\theta_ {1}} \pi_ {\theta_ {1}} \left(a _ {1} ^ {*} \mid s _ {1} ^ {*}\right) (26) \\ = - \sum_ {s _ {2}, a _ {2}, s _ {2} ^ {\prime}} p ^ {\pi} \left(s _ {2}, a _ {2}\right) \sum_ {s _ {1} ^ {*}, a _ {1} ^ {*}} p \left(s _ {2} ^ {\prime} \mid s _ {2}, a _ {2}, s _ {1} ^ {*}, a _ {1} ^ {*}\right) p \left(s _ {1} ^ {*} \mid s _ {2}, a _ {2}\right) \nabla_ {\theta_ {1}} \pi_ {\theta_ {1}} \left(a _ {1} ^ {*} \mid s _ {1} ^ {*}\right) (27) \\ = - \sum_ {s _ {2}, a _ {2}} p ^ {\pi} \left(s _ {2}, a _ {2}\right) \sum_ {s _ {1} ^ {*}, a _ {1} ^ {*}} p \left(s _ {1} ^ {*} \mid s _ {2}, a _ {2}\right) \nabla_ {\theta_ {1}} \pi_ {\theta_ {1}} \left(a _ {1} ^ {*} \mid s _ {1} ^ {*}\right) \sum_ {s _ {2} ^ {\prime}} p \left(s _ {2} ^ {\prime} \mid s _ {2}, a _ {2}, s _ {1} ^ {*}, a _ {1} ^ {*}\right) (28) \\ = - \sum_ {s _ {2}, a _ {2}} p ^ {\pi} \left(s _ {2}, a _ {2}\right) \sum_ {s _ {1} ^ {*}, a _ {1} ^ {*}} p \left(s _ {1} ^ {*} \mid s _ {2}, a _ {2}\right) \nabla_ {\theta_ {1}} \pi_ {\theta_ {1}} \left(a _ {1} ^ {*} \mid s _ {1} ^ {*}\right) \underbrace {\sum_ {s _ {2} ^ {\prime}} p \left(s _ {2} ^ {\prime} \mid s _ {2} , a _ {2} , s _ {1} ^ {*} , a _ {1} ^ {*}\right)} _ {= 1} (29) \\ \end{array}
+$$
+
+![](images/65ba4f02f7a1e055aa6c9629a344be297793b4155432b8cde4cc4c33603232bd.jpg)
+
+![](images/db308dbe96b6f4a42583547dc0f8424be002dae6241bae28da22faee3bd85ab8.jpg)
+
+![](images/77f1351cbadea6a43b2115804cfc3581958dcca20d93209af39ad0b4a5bdf186.jpg)  
+Figure 5: Performance of intrinsic EDTI in comparison with EITI and EDTI on Pass, Secret-Room, Push-Box, and Island.
+
+![](images/b640c4466748b943622093b2dee5a76eef53caef7891469b8beb1254a98f0069.jpg)
+
+$$
+\begin{array}{l} = - \sum_ {s _ {2}, a _ {2}} p ^ {\pi} \left(s _ {2}, a _ {2}\right) \sum_ {s _ {1} ^ {*}} p \left(s _ {1} ^ {*} \mid s _ {2}, a _ {2}\right) \nabla_ {\theta_ {1}} \sum_ {a _ {1} ^ {*}} \pi_ {\theta_ {1}} \left(a _ {1} ^ {*} \mid s _ {1} ^ {*}\right) (30) \\ = - \sum_ {s _ {2}, a _ {2}} p ^ {\pi} \left(s _ {2}, a _ {2}\right) \sum_ {s _ {1} ^ {*}} p \left(s _ {1} ^ {*} \mid s _ {2}, a _ {2}\right) \nabla_ {\theta_ {1}} 1 (31) \\ = 0 (32) \\ \end{array}
+$$
+
+# B.2 DEFINITION OF Value of Interaction
+
+To capture both transition and reward interactions between agents and thereby achieve intrinsic reward distribution, we propose a decision-theoretic measure called Value of Interaction. We start from 2-agent cases and the following theorem gives the definition of  $V o I_{2|1}$  in the form of an expectation over trajectories, which is especially helpful in the derivation of the EDTI policy gradient update shown Eq. 15.
+
+Theorem 1. Value of Interaction of agent 1 on agent 2 is:
+
+$$
+\left. V o I _ {2 | 1} ^ {\pi} \left(S _ {2} ^ {\prime}; S _ {1}, A _ {1} \mid S _ {2}, A _ {2}\right) = \mathbb {E} _ {\tau} \left[ \tilde {r} _ {2} (\boldsymbol {s}, \boldsymbol {a}) - \tilde {r} _ {2} ^ {\pi} \left(s _ {2}, a _ {2}\right) + \gamma \left(1 - \frac {p ^ {\pi} \left(s _ {2} ^ {\prime} \mid s _ {2} , a _ {2}\right)}{p \left(s _ {2} ^ {\prime} \mid \boldsymbol {s} , \boldsymbol {a}\right)}\right) V _ {2} ^ {\pi} \left(\boldsymbol {s} ^ {\prime}\right) \right], \right. \tag {33}
+$$
+
+where  $\tilde{r}_2^\pi (s_2,a_2)$  is the counterfactual immediate reward.
+
+$V o I_{2|1}$  can be defined similarly. To lighten notation in the proof, we define
+
+$$
+V _ {2} ^ {\pi} \left(s _ {2} ^ {\prime} \mid s _ {1}, s _ {2}, a _ {1}, a _ {2}\right) = \sum_ {s _ {1} ^ {\prime}} p \left(s _ {1} ^ {\prime} \mid s _ {1}, s _ {2}, a _ {1}, a _ {2}, s _ {2} ^ {\prime}\right) V _ {2} ^ {\pi} \left(s _ {1} ^ {\prime}, s _ {2} ^ {\prime}\right) \tag {34}
+$$
+
+$$
+\tilde {r} _ {2} ^ {\pi} \left(s _ {2}, a _ {2}\right) = \sum_ {s _ {1} ^ {*}, a _ {1} ^ {*}} p ^ {\pi} \left(s _ {1} ^ {*}, a _ {1} ^ {*} \mid s _ {2}, a _ {2}\right) \tilde {r} _ {2} \left(s _ {1} ^ {*}, s _ {2}, a _ {1} ^ {*}, a _ {2}\right), \tag {35}
+$$
+
+$$
+V _ {2} ^ {\pi , *} \left(s _ {2} ^ {\prime} \mid s _ {2}, a _ {2}\right) = \sum_ {s _ {1} ^ {*}, a _ {1} ^ {*}} p ^ {\pi} \left(s _ {1} ^ {*}, a _ {1} ^ {*} \mid s _ {2}, a _ {2}\right) \sum_ {s _ {1} ^ {\prime}} p \left(s _ {1} ^ {\prime} \mid s _ {1} ^ {*}, s _ {2}, a _ {1} ^ {*}, a _ {2}, s _ {2} ^ {\prime}\right) V _ {2} ^ {\pi} \left(s _ {1} ^ {\prime}, s _ {2} ^ {\prime}\right). \tag {36}
+$$
+
+We first prove Lemma 1, which is used in the proof of Theorem 1.
+
+Lemma 1.
+
+$$
+\sum_ {s _ {1}, s _ {2}, a _ {1}, a _ {2}} p ^ {\pi} \left(s _ {1}, s _ {2}, a _ {1}, a _ {2}\right) \gamma \sum_ {s _ {2} ^ {\prime}} p \left(s _ {2} ^ {\prime} \mid s _ {1}, s _ {2}, a _ {1}, a _ {2}\right) V _ {2} ^ {\pi} \left(s _ {2} ^ {\prime} \mid s _ {2}, a _ {2}\right) \tag {37}
+$$
+
+$$
+= \sum_ {s _ {1}, s _ {2}, a _ {1}, a _ {2}} p ^ {\pi} (s _ {1}, s _ {2}, a _ {1}, a _ {2}) \gamma \sum_ {s _ {1} ^ {\prime}, s _ {2} ^ {\prime}} T (s _ {1} ^ {\prime}, s _ {2} ^ {\prime} | s _ {1}, s _ {2}, a _ {1}, a _ {2}) \cdot \frac {p ^ {\pi} (s _ {2} ^ {\prime} | s _ {2} , a _ {2})}{p (s _ {2} ^ {\prime} | s _ {1} , s _ {2} , a _ {1} , a _ {2})} V _ {2} ^ {\pi} (s _ {1} ^ {\prime}, s _ {2} ^ {\prime}).
+$$
+
+Proof.
+
+$$
+\sum_ {s _ {1}, s _ {2}, a _ {1}, a _ {2}} p ^ {\pi} \left(s _ {1}, s _ {2}, a _ {1}, a _ {2}\right) \gamma \sum_ {s _ {2} ^ {\prime}} p \left(s _ {2} ^ {\prime} \mid s _ {1}, s _ {2}, a _ {1}, a _ {2}\right) V _ {2} ^ {\pi} \left(s _ {2} ^ {\prime} \mid s _ {2}, a _ {2}\right) \tag {38}
+$$
+
+$$
+= \sum_ {s _ {1}, s _ {2}, a _ {1}, a _ {2}} p ^ {\pi} \left(s _ {1}, s _ {2}, a _ {1}, a _ {2}\right) \gamma \sum_ {s _ {2} ^ {\prime}} p \left(s _ {2} ^ {\prime} \mid s _ {1}, s _ {2}, a _ {1}, a _ {2}\right) \cdot \tag {39}
+$$
+
+$$
+\sum_ {s _ {1} ^ {*}, a _ {1} ^ {*}} p ^ {\pi} \left(s _ {1} ^ {*}, a _ {1} ^ {*} \mid s _ {2}, a _ {2}\right) \sum_ {s _ {1} ^ {\prime}} p \left(s _ {1} ^ {\prime} \mid s _ {1} ^ {*}, s _ {2}, a _ {1} ^ {*}, a _ {2}, s _ {2} ^ {\prime}\right) V _ {2} ^ {\pi} \left(s _ {1} ^ {\prime}, s _ {2} ^ {\prime}\right) \tag {40}
+$$
+
+$$
+= \sum_ {s _ {1}, s _ {2}, a _ {1}, a _ {2}} p ^ {\pi} \left(s _ {1}, s _ {2}, a _ {1}, a _ {2}\right) \gamma \sum_ {s _ {2} ^ {\prime}} p \left(s _ {2} ^ {\prime} \mid s _ {1}, s _ {2}, a _ {1}, a _ {2}\right) \cdot \tag {41}
+$$
+
+$$
+\sum_ {s _ {1} ^ {*}, a _ {1} ^ {*}} p ^ {\pi} \left(s _ {1} ^ {*}, a _ {1} ^ {*} \mid s _ {2}, a _ {2}\right) \sum_ {s _ {1} ^ {\prime}} \frac {T \left(s _ {1} ^ {\prime} , s _ {2} ^ {\prime} \mid s _ {1} ^ {*} , s _ {2} , a _ {1} ^ {*} , a _ {2}\right)}{p \left(s _ {2} ^ {\prime} \mid s _ {1} ^ {*} , s _ {2} , a _ {1} ^ {*} , a _ {2}\right)} V _ {2} ^ {\pi} \left(s _ {1} ^ {\prime}, s _ {2} ^ {\prime}\right) \tag {42}
+$$
+
+$$
+= \sum_ {s _ {1}, s _ {2}, a _ {1}, a _ {2}} p ^ {\pi} \left(s _ {1}, s _ {2}, a _ {1}, a _ {2}\right) \gamma \sum_ {s _ {1} ^ {\prime}, s _ {2} ^ {\prime}} \frac {T \left(s _ {1} ^ {\prime} , s _ {2} ^ {\prime} \mid s _ {1} , s _ {2} , a _ {1} , a _ {2}\right)}{p \left(s _ {2} ^ {\prime} \mid s _ {1} , s _ {2} , a _ {1} , a _ {2}\right)}. \tag {43}
+$$
+
+$$
+V _ {2} ^ {\pi} \left(s _ {1} ^ {\prime}, s _ {2} ^ {\prime}\right) \sum_ {s _ {1} ^ {*}, a _ {1} ^ {*}} p ^ {\pi} \left(s _ {1} ^ {*}, a _ {1} ^ {*} \mid s _ {2}, a _ {2}\right) p \left(s _ {2} ^ {\prime} \mid s _ {1} ^ {*}, s _ {2}, a _ {1} ^ {*}, a _ {2}\right) \tag {44}
+$$
+
+$$
+= \sum_ {s _ {1}, s _ {2}, a _ {1}, a _ {2}} p ^ {\pi} \left(s _ {1}, s _ {2}, a _ {1}, a _ {2}\right) \gamma \sum_ {s _ {1} ^ {\prime}, s _ {2} ^ {\prime}} T \left(s _ {1} ^ {\prime}, s _ {2} ^ {\prime} \mid s _ {1}, s _ {2}, a _ {1}, a _ {2}\right) \cdot \tag {45}
+$$
+
+$$
+\frac {p ^ {\pi} \left(s _ {2} ^ {\prime} \mid s _ {2} , a _ {2}\right)}{p \left(s _ {2} ^ {\prime} \mid s _ {1} , s _ {2} , a _ {1} , a _ {2}\right)} V _ {2} ^ {\pi} \left(s _ {1} ^ {\prime}, s _ {2} ^ {\prime}\right). \tag {46}
+$$
+
+We now give the proof of Theorem 1:
+
+Proof.
+
+$$
+V o I _ {2 | 1} ^ {\pi} \left(S _ {2} ^ {\prime}; S _ {1}, A _ {1} \mid S _ {2}, A _ {2}\right) \tag {47}
+$$
+
+$$
+= \sum_ {\boldsymbol {s}, \boldsymbol {a}, s _ {2} ^ {\prime} \in (S, A, S _ {2})} p ^ {\pi} (\boldsymbol {s}, \boldsymbol {a}, s _ {2} ^ {\prime}) \left[ Q _ {2} ^ {\pi} (\boldsymbol {s}, \boldsymbol {a}, s _ {2} ^ {\prime}) - Q _ {2 | 1} ^ {\pi , *} (s _ {2}, a _ {2}, s _ {2} ^ {\prime}) \right] \tag {48}
+$$
+
+$$
+= \sum_ {s _ {1}, s _ {2}, a _ {1}, a _ {2}} p ^ {\pi} \left(s _ {1}, s _ {2}, a _ {1}, a _ {2}\right) \left(\tilde {r} _ {2} \left(s _ {1}, s _ {2}, a _ {1}, a _ {2}\right) - \tilde {r} _ {2} ^ {\pi} \left(s _ {2}, a _ {2}\right) + \right. \tag {49}
+$$
+
+$$
+\gamma \sum_ {s _ {2} ^ {\prime}} p \left(s _ {2} ^ {\prime} \mid s _ {1}, s _ {2}, a _ {1}, a _ {2}\right) \left(V _ {2} ^ {\pi} \left(s _ {2} ^ {\prime} \mid s _ {1}, s _ {2}, a _ {1}, a _ {2}\right) - V _ {2} ^ {\pi , *} \left(s _ {2} ^ {\prime} \mid s _ {2}, a _ {2}\right)\right) \tag {50}
+$$
+
+$$
+= \sum_ {s _ {1}, s _ {2}, a _ {1}, a _ {2}} p ^ {\pi} \left(s _ {1}, s _ {2}, a _ {1}, a _ {2}\right) \left(\tilde {r} _ {2} \left(s _ {1}, s _ {2}, a _ {1}, a _ {2}\right) - \tilde {r} _ {2} ^ {\pi} \left(s _ {2}, a _ {2}\right) + \right. \tag {51}
+$$
+
+$$
+\begin{array}{l} \gamma \sum_ {s _ {1} ^ {\prime}, s _ {2} ^ {\prime}} T \left(s _ {1} ^ {\prime}, s _ {2} ^ {\prime} \mid s _ {1}, s _ {2}, a _ {1}, a _ {2}\right) \left(1 - \frac {p ^ {\pi} \left(s _ {2} ^ {\prime} \mid s _ {2} , a _ {2}\right)}{p \left(s _ {2} ^ {\prime} \mid s _ {1} , s _ {2} , a _ {1} , a _ {2}\right)} V _ {2} ^ {\pi} \left(s _ {1} ^ {\prime}, s _ {2} ^ {\prime}\right)\right) (\textbf {L e m m a 1}) (52) \\ = \mathbb {E} _ {\tau} \left[ \tilde {r} _ {2} (\boldsymbol {s}, \boldsymbol {a}) - \tilde {r} _ {2} ^ {\pi} \left(s _ {2}, a _ {2}\right) + \gamma \left(1 - \frac {p ^ {\pi} \left(s _ {2} ^ {\prime} \mid s _ {2} , a _ {2}\right)}{p \left(s _ {2} ^ {\prime} \mid \boldsymbol {s} , \boldsymbol {a}\right)}\right) V _ {2} ^ {\pi} \left(\boldsymbol {s} ^ {\prime}\right) \right]. (53) \\ \end{array}
+$$
+
+![](images/2e9f00a0808078c2c98c681b63f975b37d9cdb84bce5a0205b2b539c681242d4.jpg)
+
+# B.3 CALCULATING GRADIENT OF VOI
+
+In order to optimize  $VoI$  with respect to the parameters of agent policy, in Sec. 3.2.1, we propose to use target function and get:
+
+$$
+\begin{array}{l} \nabla_ {\theta_ {1}} V o I _ {2 | 1} ^ {\pi} (S _ {2} ^ {\prime}; S _ {1}, A _ {1} | S _ {2}, A _ {2}) \approx \sum_ {\pmb {s}, \pmb {a} \in (S, A)} \left(\nabla_ {\theta_ {1}} p ^ {\pi} (\pmb {s}, \pmb {a})\right) \left[ \tilde {r} _ {2} (\pmb {s}, \pmb {a}) - \tilde {r} _ {2} ^ {-} (s _ {2}, a _ {2}) + \right. \\ \gamma \sum_ {\boldsymbol {s} ^ {\prime}} T (\boldsymbol {s} ^ {\prime} | \boldsymbol {s}, \boldsymbol {a}) \left(1 - \frac {p ^ {-} \left(s _ {2} ^ {\prime} \mid s _ {2} , a _ {2}\right)}{p \left(s _ {2} ^ {\prime} \mid \boldsymbol {s} , \boldsymbol {a}\right)}\right) V _ {2} ^ {-} \left(s _ {1} ^ {\prime}, s _ {2} ^ {\prime}\right) ]. \tag {54} \\ \end{array}
+$$
+
+We prove that  $\sum_{\pmb{s},\pmb{a}}\left(\nabla_{\theta_1}p^\pi (\pmb {s},\pmb {a})\right)\tilde{r}_2^-(s_2,a_2)$  is 0 in the following lemma.
+
+Lemma 2.
+
+$$
+\sum_ {s _ {1}, s _ {2}, a _ {1}, a _ {2}} \left(\nabla_ {\theta_ {1}} p ^ {\pi} \left(s _ {1}, s _ {2}, a _ {1}, a _ {2}\right)\right) \tilde {r} _ {2} ^ {-} \left(s _ {2}, a _ {2}\right) = 0. \tag {55}
+$$
+
+Proof. Similar to the way that policy gradient theorem was proved by Sutton et al. (2000),
+
+$$
+\begin{array}{l} \sum_ {s _ {1}, s _ {2}, a _ {1}, a _ {2}} \left(\nabla_ {\theta_ {1}} p ^ {\pi} \left(s _ {1}, s _ {2}, a _ {1}, a _ {2}\right)\right) \tilde {r} _ {2} ^ {-} \left(s _ {2}, a _ {2}\right) (56) \\ = \nabla_ {\theta_ {1}} \sum_ {s _ {1}, s _ {2}, a _ {1}, a _ {2}} p ^ {\pi} \left(s _ {1}, s _ {2}, a _ {1}, a _ {2}\right) \tilde {r} _ {2} ^ {-} \left(s _ {2}, a _ {2}\right) (57) \\ = \sum_ {s _ {1} ^ {0}, s _ {2} ^ {0}} d _ {0} \left(s _ {1} ^ {0}, s _ {2} ^ {0}\right) \prod_ {t = 0} ^ {\infty} \left(\nabla_ {\theta_ {1}} \pi \left(a _ {1} ^ {t}, a _ {2} ^ {t} \mid s _ {1} ^ {t}, s _ {2} ^ {t}\right)\right) T \left(s _ {1} ^ {t + 1}, s _ {2} ^ {t + 1} \mid s _ {1} ^ {t}, a _ {1} ^ {t}, s _ {2} ^ {t}, a _ {2} ^ {t}\right) \tilde {r} _ {2} ^ {-} \left(s _ {2} ^ {t}, a _ {2} ^ {t}\right) (58) \\ = \sum_ {s _ {1} ^ {0}, s _ {2} ^ {0}} d _ {0} \left(s _ {1} ^ {0}, s _ {2} ^ {0}\right) \prod_ {t = 0} ^ {\infty} \left[ \pi \left(a _ {1} ^ {t}, a _ {2} ^ {t} \mid s _ {1} ^ {t}, s _ {2} ^ {t}\right) \left(\nabla_ {\theta_ {1}} \log \pi \left(a _ {1} ^ {t}, a _ {2} ^ {t} \mid s _ {1} ^ {t}, s _ {2} ^ {t}\right)\right) \right. (59) \\ \left. T \left(s _ {1} ^ {t + 1}, s _ {2} ^ {t + 1} \mid s _ {1} ^ {t}, a _ {1} ^ {t}, s _ {2} ^ {t}, a _ {2} ^ {t}\right) \tilde {r} _ {2} ^ {-} \left(s _ {2} ^ {t}, a _ {2} ^ {t}\right) \right] (60) \\ = \sum_ {s _ {1}, s _ {2}, a _ {1}, a _ {2}} p ^ {\pi} \left(s _ {1}, s _ {2}, a _ {1}, a _ {2}\right) \left(\nabla_ {\theta_ {1}} \log \pi \left(a _ {1}, a _ {2} \mid s _ {1}, s _ {2}\right)\right) \tilde {r} _ {2} ^ {-} \left(s _ {2}, a _ {2}\right) (61) \\ = \sum_ {s _ {1}, s _ {2}, a _ {1}, a _ {2}} p ^ {\pi} \left(s _ {1}, s _ {2}, a _ {1}, a _ {2}\right) \left(\nabla_ {\theta_ {1}} \log \pi \left(a _ {1} \mid s _ {1}, s _ {2}\right)\right) \tilde {r} _ {2} ^ {-} \left(s _ {2}, a _ {2}\right) (62) \\ = \sum_ {s _ {2}, a _ {2}} p ^ {\pi} \left(s _ {2}, a _ {2}\right) \sum_ {s _ {1}, a _ {1}} p ^ {\pi} \left(s _ {1}, a _ {1} \mid s _ {2}, a _ {2}\right) \left(\nabla_ {\theta_ {1}} \log \pi \left(a _ {1} \mid s _ {1}, s _ {2}\right)\right) \tilde {r} _ {2} ^ {-} \left(s _ {2}, a _ {2}\right) (63) \\ = \sum_ {s _ {2}, a _ {2}} p ^ {\pi} \left(s _ {2}, a _ {2}\right) \tilde {r} _ {2} ^ {-} \left(s _ {2}, a _ {2}\right) \sum_ {s _ {1}, a _ {1}} p ^ {\pi} \left(s _ {1}, a _ {1} \mid s _ {2}, a _ {2}\right) \left(\nabla_ {\theta_ {1}} \log \pi \left(a _ {1} \mid s _ {1}, s _ {2}\right)\right) (64) \\ = \sum_ {s _ {2}, a _ {2}} p ^ {\pi} \left(s _ {2}, a _ {2}\right) \tilde {r} _ {2} ^ {-} \left(s _ {2}, a _ {2}\right) \sum_ {s _ {1}, a _ {1}} \frac {p ^ {\pi} \left(s _ {1} , a _ {1} \mid s _ {2} , a _ {2}\right)}{\pi \left(a _ {1} \mid s _ {1} , s _ {2}\right)} \left(\nabla_ {\theta_ {1}} \pi \left(a _ {1} \mid s _ {1}, s _ {2}\right)\right) (65) \\ = \sum_ {s _ {2}, a _ {2}} p ^ {\pi} \left(s _ {2}, a _ {2}\right) \tilde {r} _ {2} ^ {-} \left(s _ {2}, a _ {2}\right) \sum_ {s _ {1}, a _ {1}} \frac {p ^ {\pi} \left(s _ {1} \mid s _ {2} , a _ {2}\right) p ^ {\pi} \left(a _ {1} \mid s _ {1} , s _ {2} , a _ {2}\right)}{\pi \left(a _ {1} \mid s _ {1} , s _ {2}\right)} \left(\nabla_ {\theta_ {1}} \pi \left(a _ {1} \mid s _ {1}, s _ {2}\right)\right) (66) \\ = \sum_ {s _ {2}, a _ {2}} p ^ {\pi} \left(s _ {2}, a _ {2}\right) \tilde {r} _ {2} ^ {-} \left(s _ {2}, a _ {2}\right) \sum_ {s _ {1}, a _ {1}} \frac {p ^ {\pi} \left(s _ {1} \mid s _ {2} , a _ {2}\right) \pi \left(a _ {1} \mid s _ {1} , s _ {2}\right)}{\pi \left(a _ {1} \mid s _ {1} , s _ {2}\right)} \left(\nabla_ {\theta_ {1}} \pi \left(a _ {1} \mid s _ {1}, s _ {2}\right)\right) (67) \\ \end{array}
+$$
+
+$$
+\begin{array}{l} = \sum_ {s _ {2}, a _ {2}} p ^ {\pi} \left(s _ {2}, a _ {2}\right) \tilde {r} _ {2} ^ {-} \left(s _ {2}, a _ {2}\right) \sum_ {s _ {1}, a _ {1}} p ^ {\pi} \left(s _ {1} \mid s _ {2}, a _ {2}\right) \left(\nabla_ {\theta_ {1}} \pi \left(a _ {1} \mid s _ {1}, s _ {2}\right)\right) (68) \\ = \sum_ {s _ {2}, a _ {2}} p ^ {\pi} \left(s _ {2}, a _ {2}\right) \tilde {r} _ {2} ^ {-} \left(s _ {2}, a _ {2}\right) \sum_ {s _ {1}} p ^ {\pi} \left(s _ {1} \mid s _ {2}, a _ {2}\right) \sum_ {a _ {1}} \left(\nabla_ {\theta_ {1}} \pi \left(a _ {1} \mid s _ {1}, s _ {2}\right)\right) (69) \\ = \sum_ {s _ {2}, a _ {2}} p ^ {\pi} \left(s _ {2}, a _ {2}\right) \tilde {r} _ {2} ^ {-} \left(s _ {2}, a _ {2}\right) \sum_ {s _ {1}} p ^ {\pi} \left(s _ {1} \mid s _ {2}, a _ {2}\right) \left(\nabla_ {\theta_ {1}} \sum_ {a _ {1}} \pi \left(a _ {1} \mid s _ {1}, s _ {2}\right)\right) (70) \\ = \sum_ {s _ {2}, a _ {2}} p ^ {\pi} \left(s _ {2}, a _ {2}\right) \tilde {r} _ {2} ^ {-} \left(s _ {2}, a _ {2}\right) \sum_ {s _ {1}} p ^ {\pi} \left(s _ {1} \mid s _ {2}, a _ {2}\right) \left(\nabla_ {\theta_ {1}} 1\right) (71) \\ = 0 (72) \\ \end{array}
+$$
+
+![](images/33953d36ebe6952c326c99522f0bbe6ac7b747aaeffeb27b86434ec4bce63e30.jpg)
+
+# B.4 IMMEDIATE REWARD INFLUENCE
+
+Similar to MI and  $VoI$ , we can define influence of agent 1 on the immediate rewards of agent 2 as:
+
+$$
+R I _ {2 | 1} ^ {\pi} \left(S _ {2} ^ {\prime}; S _ {1}, A _ {1} \mid S _ {2}, A _ {2}\right) = \sum_ {\boldsymbol {s}, \boldsymbol {a} \in (S, A)} p ^ {\pi} (\boldsymbol {s}, \boldsymbol {a}) \left[ \tilde {r} _ {2} (\boldsymbol {s}, \boldsymbol {a}) - \tilde {r} _ {2} \left(s _ {2}, a _ {2}\right) \right]. \tag {73}
+$$
+
+Use Lemma 2, we can get:
+
+$$
+\nabla_ {\theta_ {1}} R I _ {2 | 1} ^ {\pi} \left(S _ {2} ^ {\prime}; S _ {1}, A _ {1} \mid S _ {2}, A _ {2}\right) = \sum_ {\boldsymbol {s}, \boldsymbol {a} \in (S, A)} \nabla_ {\theta_ {1}} \left(p ^ {\pi} (\boldsymbol {s}, \boldsymbol {a})\right) \tilde {r} _ {2} (\boldsymbol {s}, \boldsymbol {a}). \tag {74}
+$$
+
+Now we have
+
+$$
+\nabla_ {\theta_ {1}} J _ {\theta_ {1}} (t) \approx \left(\hat {R} _ {1} ^ {t} - \hat {V} _ {1} ^ {\pi} \left(s _ {t}\right)\right) \nabla_ {\theta_ {1}} \log \pi_ {\theta_ {1}} \left(a _ {1} ^ {t} \mid s _ {1} ^ {t}\right), \tag {75}
+$$
+
+where  $\hat{V}_1^\pi (s_t)$  is an augmented value function of  $\hat{R}_1^t = \sum_{t' = t}^{h}\hat{r}_1^{t'}$  and
+
+$$
+\hat {r} _ {1} ^ {t} = r ^ {t} + u _ {1} ^ {t} + \beta u _ {2} ^ {t}. \tag {76}
+$$
+
+# C ESTIMATION OF CONDITIONAL PROBABILITIES
+
+To quantify interdependence among exploration processes of agents, we use mutual information and value of interaction. Calculations of MI and VoI need estimation of  $p(s_2'|s_2,a_2)$  and  $p(s_2'|s,\pmb{a})$ . In practice, we track the empirical frequencies  $p_{emp}(s_2'|s_2,a_2)$  and  $p_{emp}(s_2'|s,\pmb{a})$  and substitute them for the corresponding terms in Eq. 8 and 16.
+
+Estimating  $p_{emp}(s_2'|s_2, a_2)$  and  $p_{emp}(s_2'|s, a)$  is one obstacle to the scalability of our method, we now discuss how to solve this problem. When the state and action space is small, we can use hash table to implement Monte Carlo method (MC) for estimating the distributions accurately. In the MC sampling, we count from the samples the state frequencies  $p(s_2'|s_2, a_2) \equiv \frac{N(s_2', s_2, a_2)}{N(s_2, a_2)}$  and  $p(s_2'|s, a) \equiv \frac{N(s_2', s_1, s_2, a_1, a_2)}{N(s_1, s_2, a_1, a_2)}$ , where  $N(\cdot)$  is the number of times each state-action pair was visited during the learning process. When the problem space becomes large, MC consumes large memory in practice. As an alternative, we adopt variational inference (Fox & Roberts, 2012) to learn variational distributions  $q_{\xi_1}(s_2'|s_2, a_2)$  and  $q_{\xi_2}(s_2'|s, a)$ , parameterized via neural networks with parameters  $\xi_1$  and  $\xi_2$ , by optimizing the evidence lower bound. In Fig. 6, we show the performance of EDTI estimated using variational inference and the changing of associated EDTI rewards on Pass during 9000 PPO updates. Variational inference introduces some noise in EDTI rewards estimation and thus requires slightly more steps to learn the true probability and the strategy. However, estimating using MC sampling consumes 1.6G memory to save the hash table with 100M items each agent while variational inference needs a three-layer fully connected network with 74800 parameters occupying about 0.60M memory. This results highlights the feasibility of estimating EITI and EDTI rewards using variational inference in problem with large state-action space.
+
+![](images/f32d6605c3a457054ba0605ff06f5ef57f4e2fc067f28cc96929f01e6def5d60.jpg)  
+Figure 6: Left: Performance of EDTI (vi) (EDIT estimated using variational inference) compared with EITI and EDTI estimated using MC sampling. Others: Development of EDTI (vi) rewards during exploration process. Top row: EDTI (vi) rewards of agent 1; bottom row: EDTI (vi) rewards of agent 2.
+
+![](images/515792b2ab497fd217eb828a01b7461ec1a51fb123c5e92a0987824183d6940a.jpg)
+
+![](images/33c22ffe6e84c14a5ac717ba3eba0af17b999bbe25f6a48cddcbfcaccf24b30e.jpg)
+
+![](images/b3e3ca5a476bc421bcccbc8747a250c27741724a4bad3cfe54b99346716f7ae9.jpg)
+
+![](images/627e4596f62c806fa247f003c92c2cc3bc68398a3eaba879eac003293d9ddab0.jpg)
+
+Table 2: The scaling weights for different intrinsic reward terms in various tasks.  $\beta_{\mathrm{T}}$  is the weight of term  $T_{1}$  (see Table 1).  $\beta_{\mathrm{int}}$  and  $\beta_{\mathrm{ext}}$  are scaling factors to combine  $r$  and  $u_{i}$  in  $\tilde{r}$ .  $u_{-i}$  in r_influence is scaled by  $\beta_{\mathrm{r}}$  while  $V_{-i}^{int}$  and  $V_{-i}^{ext}$  in plusV are respectively scaled by  $\beta_{\mathrm{int}}^{\mathrm{plusV}}$  and  $\beta_{\mathrm{ext}}^{\mathrm{plusV}}$ .  
+
+<table><tr><td>Task</td><td>η</td><td>βT</td><td>βint</td><td>βext</td><td>βr</td><td>βplusV</td><td>βext</td></tr><tr><td>Pass</td><td>10.</td><td>10</td><td>1.</td><td>0.1</td><td>1.</td><td>0.1</td><td>0.01</td></tr><tr><td>Secret-Room</td><td>10.</td><td>10</td><td>1.</td><td>0.1</td><td>—</td><td>—</td><td>—</td></tr><tr><td>Push-Box</td><td>1.</td><td>100.</td><td>100.</td><td>0.1</td><td>0.1</td><td>0.1</td><td>0.01</td></tr><tr><td>Island</td><td>1.</td><td>10</td><td>10.</td><td>0.5</td><td>0.1</td><td>0.1</td><td>0.01</td></tr><tr><td>Large-Island</td><td>1.</td><td>10</td><td>1.</td><td>0.1</td><td>0.1</td><td>0.1</td><td>0.01</td></tr></table>
+
+# D IMPLEMENTATION DETAILS
+
+# D.1 NETWORK ARCHITECTURE, HYPERPARAMETERS, AND INFRASTRUCTURE
+
+We base our framework on OpenAI implementation of PPO2 (Dhariwal et al., 2017) and use its default parameters to carry out all the experiments. We train our models on an NVIDIA RTX 2080TI GPU using experience sampled from 32 parallel environments. We use visitation count to calculate the intrinsic reward, for its provable effectiveness (Azar et al., 2017; Jin et al., 2018). For all our methods and baselines, we use  $\eta / \sqrt{N(s)}$  as the exploration bonus for  $N(s)$ -th visit to state  $s$ . Specific values of  $\eta$  and scaling weights can be found in Table 2.
+
+As for variational inference, the inference network is a 3-layer fully-connected network coupled with a 64-dimensional reparameterization estimator. ReLU is used as the activation function for the first two layers and the sum of negative log-likelihood and negative Evidence Lower Bound is used as loss. We use Adam optimizer (Kingma & Ba, 2014) with learning rate  $1 \times 10^{-3}$  and batchsize 2048. To speed up the learning of variational distributions estimation, we equip the learning with proportional prioritized experience replay (Schaul et al., 2015).
+
+# D.2 TASK STRUCTURE
+
+In this section, we describe the detailed settings of the experimental tasks.
+
+Pass: There are two agents and two switches to open the door in a  $30 \times 30$  grid. Only when at least one of the switches are occupied will the door open. The agents need navigate from left to right and the team reward, which is 1000, is only provided when all agents reach the target zone. Agents can observe the position of another agents.
+
+Secret-Room: This is an extension of the Pass task with 4 rooms and 4 switches locating in different rooms. The size of the grid is  $25 \times 25$ . When the left switch is occupied, all the three doors are open. And the three switches in each room on the right only control the door of its room. The agents need
+
+![](images/909e4f645fdeaf67a30737c1eb895919bdf2ea789105a63f7a1bf6b2e118b6fa.jpg)  
+Figure 7: Task Push-Box, Island, and Large-Island
+
+![](images/c60d5e15918c4c27caa35513f976f3fecdda424c2d93c84eb4ef3a55522100d7.jpg)
+
+![](images/27e206541dbdb8633bab3c0807a25844ce1b8615d09f631072a22921a9e47129.jpg)
+
+![](images/94533685f35e98829f0dc00118a53e646cf6f47e22b75a770445fa815a60f730.jpg)  
+Figure 8: Comparison of our methods against baselines on Push-Box (left), Island (right).
+
+![](images/9e68ad81166219942c1ee4f47e70978efe27f0f496885f8e8c920e54bf74436e.jpg)
+
+to navigate towards the desired room (in light red of Fig. 1 middle) to achieve the extrinsic team reward 1000. Agents can observe the position of the other agents.
+
+Push-Box: There are two agents and one box in a  $15 \times 15$  grid. Agents need to push the box to the wall. However, the box is so heavy that only when two agents push it in the same direction at the same time can it be moved a grid. The only team reward, 1000, is given when the box is placed right against the wall. Agents can observe the coordinates of their teammate and the location of the box.
+
+Island: A group of two agents are hunting for treasure on an island. However, a random walking beast may attack the agents when they are too close. The agents can also attack the beast within their attack range. This hurt doubles when more than one agent attacks at the same time. Each agent has a maximum health of 5 and will lose  $1/n$  health per step when there are  $n$  agents within the attack range of the beast. Island is a modified version of the classic coordination scenario Stag-Hunt with local optimal, because finding each treasure (9 in total) will trigger a team reward of 10 but catching the beast gives a higher team reward of 300. Agents can observe the position and health of each other, and the coordinates of the beast. Fig. 9 shows the development of the probability of catching the beast and the averaged number of treasures found in an episode during 9000 PPO updates.
+
+Large-Island: Settings are similar to that of Island but with more agents (4), more treasures (16), and a beast with more energy (16) and a higher reward (600) for being caught.
+
+The horizon of one episode is set to 300 timesteps in all these tasks.
+
+# E COMPARISON WITH SINGLE-AGENT EXPLORATION METHODS
+
+In this paper, we study the exploration problem in multi-agent settings from a decentralized perspective. Alternatively, exploration can be carried out in a centralized manner – treating agents as a joint one and using single-agent exploration algorithms. In this section, we compare our methods with centralized exploration strategies using RND (Burda et al., 2018) and EMI (Hyoungseok Kim, 2019), which are among the most cutting-edge exploration algorithms driven by curiosity and based on mutual information, respectively. We use codes published by their authors and carry out a modest
+
+![](images/603bd8647934894b1f99ecb6a8f4191a1d89c5fd20153926f4e521b1644a8114.jpg)
+
+![](images/5381c54beaae0cb896fdf56f265d5e35326e18eb2a387c1361c4bad02ee8c45c.jpg)
+
+![](images/b19283dcb315384ed4164e4ba27746a2e985fa239ed68b30d50e33ba1aa9f035.jpg)  
+Figure 9: Comparison of our methods against baselines and ablations on Island in terms of the probability of catching the beast and the averaged treasures collected in an episode.
+
+![](images/1d955cfbebd6d9429cf8a059cf0948bed8ed87d540817fb00503d2e32db64b25.jpg)
+
+grid search over hyperparameters. For RND, we search intrinsic reward coefficient in the range of [0.005, 1.0] and extrinsic reward coefficient in range [0.05, 2.0]. For EMI, we test difference combinations of loss weights. Results averaged over four random seeds with the best found parameters are shown below.
+
+![](images/5b599dfbdba81280e6191c40d2d873a3d20b00bf4dc173bee3d82c684de0cc4f.jpg)  
+Figure 10: Comparison of our methods against centralized single-agent exploration algorithms on Pass (left), Secret-Room (middle), and Push-Box (right).
+
+![](images/715e96075629849ed8f1ef95fc56911402261cd218cfff42074fd9490e233496.jpg)
+
+![](images/f4bcf22dc85cdfc29a872a8c32ea4b12ea3c752a9022eccb98d9edd6aede3735.jpg)
+
+Performance comparisons on problems of Pass, Secret-Room, and Push-Box are illustrated in Fig. 10. We can observe that our methods significantly outperform centralized exploration strategies using RND or EMI. To better understand this observation, we plot visitation heatmaps over time for RND and EMI, respectively, in Fig. 11 and 12.
+
+Fig. 11 shows visitation heatmaps of RND on the Pass problem. From Fig. 11 (b), we can see that RND seems finding good policies for agents to pass the door in the first 4671 updates. However, agents' policies seem to collapse quickly after that and their visits scatter around rooms again, which explains its learning curve in Fig. 10. From the evolution of its visitation heatmaps, we hypothesize that after visiting the center of the room for many times, agents' curiosity models overfit on a particular set of states and they start to be curious about the relatively unfamiliar transition dynamics around the wall. As the result, the RND intrinsic reward drags the agents to the walls, as shown in Fig. 11(c) and (d), and their performance quickly drops within several updates (i.e., update 4671-4677 shown by Fig. 11(b-d)). After a while, agents then leave from the walls and visit around in the
+
+![](images/90918311c9d00da5713f73bd814b375108af7646bc2431a376cb60339bee9cdc.jpg)
+
+![](images/b5c3a52cf74d73ac728f34162e7d7382fb843c49632485bb83ff641a6d00f88e.jpg)
+
+![](images/e1fea985ee23fe84eb87a04e34a28bd15dabde39dcad5dc8e2ea312085ff877f.jpg)
+
+![](images/3debba41b2383f8cfb244c093ca19531d4a1cad92c6bbcccd163fd6c55014a3b.jpg)
+
+![](images/9bebb6dd6f9ab2f71c74f04d6e56d03a7099ac96d4c97c8fcfd5ae5dd89e5579.jpg)
+
+![](images/076858a1915af89c2952614101439e0bf6948f6aea9935ef1eb3ef936131f22b.jpg)  
+(a) 2500 Updates
+
+![](images/cee818d6917614dee1eb580defcc9771f9278103ef2f086a98266914015d7ec1.jpg)  
+(b) 4671 Updates
+
+![](images/c3cab90132abd775a743badd48ae26cd947da66ac466d83f4ea084e8f1ccb4ad.jpg)  
+(c) 4673 Updates
+
+![](images/b3b9c0c7bf540e44137b421ac82884366ea8ecaa7165d8d1b2c7217c8fbd743f.jpg)  
+(d) 4677 Updates
+
+![](images/59f56a4c8b9174c72d333ac71a85a44971dea75a73ae2e1a043b609a078b6879.jpg)  
+(e) 5500 Updates
+
+![](images/4014d4035f9c29ed722c702e6ed8c626cef19ff5da85bfd6552d71bc76441b0e.jpg)  
+Figure 11: Visitation heatmap of RND agents on Pass of most recent  $1k$  episodes. The brighter the yellow color, the higher the visitation frequency. Top: agent 1, bottom: agent 2.
+
+![](images/ae8ba754c532f055c9700a50e59d5662135e41c9f136aafd8a9995ba12271535.jpg)  
+(a) 10 Updates
+
+![](images/3521290fb0cf7d45ad2a1e4a26d4562ef518ee8a87ca740fb7c93767471930c1.jpg)
+
+![](images/97ec9ce6080cbc90f3d3ee8585a6c1f07174d04ac3cfca8eb214278bdb250c45.jpg)  
+(b) 200 Updates
+
+![](images/d9f221b0b967558c37ce5b3c687f9857bd26d0882a784fd588d7d1c772cb1a54.jpg)
+
+![](images/0a33167a4d27ceac6d5fe549d453f8bc4eafbcd8355e1942114d600173159e12.jpg)  
+(c) 300 Updates
+
+![](images/3c4677c86ec2d5899a6e614acdffa90d45c38aaa6bce9c7299fb202eaf41ac60.jpg)
+
+![](images/0b219ed447a70be5dcf7fa1fb5c8d10e8dcc57d308360ac88dedc77dc4fece20.jpg)  
+(d) 500 Updates  
+Figure 12: Visitation heatmap in most recent  $1k$  episodes of EMI agents on Pass. The brighter the yellow color, the higher the visitation frequency. Top: agent 1, bottom: agent 2.
+
+![](images/e9338d75c47c336d8232bcf6c6647c77ef33b1036693a210a56c64527247eef1.jpg)
+
+![](images/bc8c00b4f9342c9563d5eeaef94330040c66440f0e61f0d1b573f65764597214.jpg)  
+(e) 4000 Updates
+
+room again, as shown in Fig. 11(e). The whole exploration process repeated. Similar behaviors are also observed on the Secret-Room problem.
+
+We also analyze the exploration behaviors of EMI agents on Pass, as illustrated by visitation heatmaps in Fig. 12. EMI tends to explore the state-action pairs where the transition dynamics is relatively complex, such as the edges and corners of the room (Fig. 12(a-c)). For problems where these state-action pairs do not lead to goals, EMI is not very effective. As the (centralized) transition dynamics of the Pass problem is relatively simple, EMI intrinsic reward quickly diminishes, which results in the behaviors of agents keeping unchanged after 500 updates (Fig. 12(d-e)).
+
+In summary, centralized single-agent exploration methods encode some heuristics to facilitate exploration, but they typically do not place a great emphasis on interactions among agents and are thus not very efficient for multi-agent exploration with sparse interactions.
\ No newline at end of file
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+# VIDEOFLOW: A CONDITIONAL FLOW-BASED MODEL FOR STOCHASTIC VIDEO GENERATION
+
+Manoj Kumar*, Mohammad Babaeizadeh, Dumitru Erhan,
+
+Chelsea Finn, Sergey Levine, Laurent Dinh, Durk Kingma
+
+Google Research, Brain Team
+
+{mechcoder,mbz,dumitru,chelseaf,slevine,laurentdinh,durk}@google.com
+
+# ABSTRACT
+
+Generative models that can model and predict sequences of future events can, in principle, learn to capture complex real-world phenomena, such as physical interactions. However, a central challenge in video prediction is that the future is highly uncertain: a sequence of past observations of events can imply many possible futures. Although a number of recent works have studied probabilistic models that can represent uncertain futures, such models are either extremely expensive computationally as in the case of pixel-level autoregressive models, or do not directly optimize the likelihood of the data. To our knowledge, our work is the first to propose multi-frame video prediction with normalizing flows, which allows for direct optimization of the data likelihood, and produces high-quality stochastic predictions. We describe an approach for modeling the latent space dynamics, and demonstrate that flow-based generative models offer a viable and competitive approach to generative modeling of video.
+
+# 1 INTRODUCTION
+
+Exponential progress in the capabilities of computational hardware, paired with a relentless effort towards greater insights and better methods, has pushed the field of machine learning from relative obscurity into the mainstream. Progress in the field has translated to improvements in various capabilities, such as classification of images (Krizhevsky et al., 2012), machine translation (Vaswani et al., 2017) and super-human game-playing agents (Mnih et al., 2013; Silver et al., 2017), among others. However, the application of machine learning technology has been largely constrained to situations where large amounts of supervision is available, such as in image classification or machine translation, or where highly accurate simulations of the environment are available to the learning agent, such as in game-playing agents. An appealing alternative to supervised learning is to utilize large unlabeled datasets, combined with predictive generative models. In order for a complex generative model to be able to effectively predict future events, it must build up an internal representation of the world. For example, a predictive generative model that can predict future frames in a video would need to model complex real-world phenomena, such as physical interactions. This provides an appealing mechanism for building models that have a rich understanding of the physical world, without any labeled examples. Videos of real-world interactions are plentiful and readily available, and a large generative model can be trained on large unlabeled datasets containing many video sequences, thereby learning about a wide range of real-world phenomena. Such a model could be useful for learning representations for further downstream tasks (Mathieu et al., 2016), or could even be used directly in applications where predicting the future enables effective decision making and control, such as robotics (Finn et al., 2016). A central challenge in video prediction is that the future is highly uncertain: a short sequence of observations of the present can imply many possible futures. Although a number of recent works have studied probabilistic models that can represent uncertain futures, such models are either extremely expensive computationally (as in the case of pixel-level autoregressive models), or do not directly optimize the likelihood of the data.
+
+In this paper, we study the problem of stochastic prediction, focusing specifically on the case of conditional video prediction: synthesizing raw RGB video frames conditioned on a short context
+
+of past observations (Ranzato et al., 2014; Srivastava et al., 2015; Vondrick et al., 2015; Xingjian et al., 2015; Boots et al., 2014). Specifically, we propose a new class of video prediction models that can provide exact likelihoods, generate diverse stochastic futures, and accurately synthesize realistic and high-quality video frames. The main idea behind our approach is to extend flow-based generative models (Dinh et al., 2014; 2016) into the setting of conditional video prediction. To our knowledge, flow-based models have been applied only to generation of non-temporal data, such as images (Kingma & Dhariwal, 2018), and to audio sequences (Prenger et al., 2018). Conditional generation of videos presents its own unique challenges: the high dimensionality of video sequences makes them difficult to model as individual datapoints. Instead, we learn a latent dynamical system model that predicts future values of the flow model's latent state. This induces Markovian dynamics on the latent state of the system, replacing the standard unconditional prior distribution. We further describe a practically applicable architecture for flow-based video prediction models, inspired by the Glow model for image generation (Kingma & Dhariwal, 2018), which we call VideoFlow.
+
+Our empirical results show that VideoFlow achieves results that are competitive with the state-of-the-art in stochastic video prediction on the action-free BAIR dataset, with quantitative results that rival the best VAE-based models. VideoFlow also produces excellent qualitative results, and avoids many of the common artifacts of models that use pixel-level mean-squared-error for training (e.g., blurry predictions), without the challenges associated with training adversarial models. Compared to models based on pixel-level autoregressive prediction, VideoFlow achieves substantially faster test-time image synthesis $^{1}$ , making it much more practical for applications that require real-time prediction, such as robotic control (Finn & Levine, 2017). Finally, since VideoFlow directly optimizes the likelihood of training videos, without relying on a variational lower bound, we can evaluate its performance directly in terms of likelihood values.
+
+# 2 RELATED WORK
+
+Early work on prediction of future video frames focused on deterministic predictive models (Ranzato et al., 2014; Srivastava et al., 2015; Vondrick et al., 2015; Xingjian et al., 2015; Boots et al., 2014). Much of this research on deterministic models focused on architectural changes, such as predicting high-level structure (Villegas et al., 2017b), energy-based models (Xie et al., 2017), generative cooperative nets (Xie et al., 2020), ABPTT (Xie et al., 2019), incorporating pixel transformations (Finn et al., 2016; De Brabandere et al., 2016; Liu et al., 2017) and predictive coding architectures (Lotter et al., 2017), as well as different generation objectives (Mathieu et al., 2016; Vondrick & Torralba, 2017; Walker et al., 2015) and disentangling representations (Villegas et al., 2017a; Denton & Birodkar, 2017). With models that can successfully model many deterministic environments, the next key challenge is to address stochastic environments by building models that can effectively reason over uncertain futures. Real-world videos are always somewhat stochastic, either due to events that are inherently random, or events that are caused by unobserved or partially observable factors, such as off-screen events, humans and animals with unknown intentions, and objects with unknown physical properties. In such cases, since deterministic models can only generate one future, these models either disregard potential futures or produce blurry predictions that are the superposition or averages of possible futures.
+
+A variety of methods have sought to overcome this challenge by incorporating stochasticity, via three types of approaches: models based on variational auto-encoders (VAEs) (Kingma & Welling, 2013; Rezende et al., 2014), generative adversarial networks (Goodfellow et al., 2014), and autoregressive models (Hochreiter & Schmidhuber, 1997; Graves, 2013; van den Oord et al., 2016b;c; Van Den Oord et al., 2016).
+
+Among these models, techniques based on variational autoencoders which optimize an evidence lower bound on the log-likelihood have been explored most widely (Babaeizadeh et al., 2017; Denton & Fergus, 2018; Lee et al., 2018; Xue et al., 2016; Li et al., 2018). To our knowledge, the only prior class of video prediction models that directly maximize the log-likelihood of the data are autoregressive models (Hochreiter & Schmidhuber, 1997; Graves, 2013; van den Oord et al., 2016b;c; Van Den Oord et al., 2016), that generate the video one pixel at a time (Kalchbrenner et al., 2017). However, synthesis with such models is typically inherently sequential, making synthesis substantially
+
+![](images/6af4bb4ec1dd263a97238f8736da551dc5b963d841d27ff64a7379cba1a9e52f.jpg)  
+Figure 1: Left: Multi-scale prior The flow model uses a multi-scale architecture using several levels of stochastic variables. Right: Autoregressive latent-dynamic prior The input at each timestep  $\mathbf{x}_t$  is encoded into multiple levels of stochastic variables  $(\mathbf{z}_t^{(1)},\dots,\mathbf{z}_t^{(L)})$ . We model those levels through a sequential process  $\prod_t\prod_lp(\mathbf{z}_t^{(l)}\mid \mathbf{z}_{< t}^{(l)},\mathbf{z}_t^{(>l)})$ .
+
+inefficient on modern parallel hardware. Prior work has aimed to speed up training and synthesis with such auto-regressive models (Reed et al., 2017; Ramachandran et al., 2017). However, (Babaeizadeh et al., 2017) show that the predictions from these models are sharp but noisy and that the proposed VAE model produces substantially better predictions, especially for longer horizons. In contrast to autoregressive models, we find that our proposed method exhibits faster sampling, while still directly optimizing the log-likelihood and producing high-quality long-term predictions.
+
+# 3 PRELIMINARIES: FLOW-BASED GENERATIVE MODELS
+
+Flow-based generative models (Dinh et al., 2014; 2016) have a unique set of advantages: exact latent-variable inference, exact log-likelihood evaluation, and parallel sampling. In flow-based generative models (Dinh et al., 2014; 2016), we infer the latent variable  $\mathbf{z}$  corresponding to a datapoint  $\mathbf{x}$ , by transforming  $\mathbf{x}$  through a composition of invertible functions  $\mathbf{f} = \mathbf{f}_1 \circ \mathbf{f}_2 \circ \dots \circ \mathbf{f}_K$ . We assume a tractable prior  $p_{\theta}(\mathbf{z})$  over latent variable  $\mathbf{z}$ , for eg. a Logistic or a Gaussian distribution. By constraining the transformations to be invertible, we can compute the log-likelihood of  $\mathbf{x}$  exactly using the change of variables rule. Formally,
+
+$$
+\log p _ {\boldsymbol {\theta}} (\mathbf {x}) = \log p _ {\boldsymbol {\theta}} (\mathbf {z}) + \sum_ {i = 1} ^ {K} \log | \det  (d \mathbf {h} _ {i} / d \mathbf {h} _ {i - 1}) | \tag {1}
+$$
+
+where  $\mathbf{h}_0 = \mathbf{x}$ ,  $\mathbf{h}_i = \mathbf{f}_i(\mathbf{h}_{i-1})$ ,  $\mathbf{h}_K = \mathbf{z}$  and  $|\operatorname*{det}(dh_i / dh_{i-1}|)$  is the Jacobian determinant when  $\mathbf{h}_{i-1}$  is transformed to  $\mathbf{h}_i$  by  $\mathbf{f}_i$ . We learn the parameters of  $\mathbf{f}_1 \ldots \mathbf{f}_K$  by maximizing the log-likelihood, i.e. Equation (1), over a training set. Given  $\mathbf{g} = \mathbf{f}^{-1}$ , we can now generate a sample  $\hat{\mathbf{x}}$  from the data distribution, by sampling  $\mathbf{z} \sim p_\theta(\mathbf{z})$  and computing  $\hat{\mathbf{x}} = \mathbf{g}(\mathbf{z})$ .
+
+# 4 PROPOSED ARCHITECTURE
+
+We propose a generative flow for video, using the standard multi-scale flow architecture in (Dinh et al., 2016; Kingma & Dhariwal, 2018) as a building block. In our model, we break up the latent space  $\mathbf{z}$  into separate latent variables per timestep:  $\mathbf{z} = \{\mathbf{z}_t\}_{t=1}^T$ . The latent variable  $\mathbf{z}_t$  at timestep  $t$  is an invertible transformation of a corresponding frame of video:  $\mathbf{x}_t = \mathbf{g}_\theta(\mathbf{z}_t)$ . Furthermore, like in (Dinh et al., 2016; Kingma & Dhariwal, 2018), we use a multi-scale architecture for  $\mathbf{g}_\theta(\mathbf{z}_t)$  (Fig. 1): the latent variable  $\mathbf{z}_t$  is composed of a stack of multiple levels: where each level  $l$  encodes information about frame  $\mathbf{x}_t$  at a particular scale:  $\mathbf{z}_t = \{\mathbf{z}_t^{(l)}\}_{l=1}^L$ , one component  $\mathbf{z}_t^{(l)}$  per level.
+
+# 4.1 INVERTIBLE MULTI-SCALE ARCHITECTURE
+
+We first briefly describe the invertible transformations used in the multi-scale architecture to infer  $\{\mathbf{z}_t^{(l)}\}_{l = 1}^L = \mathbf{f}_\theta (\mathbf{x}_t)$  and refer to (Dinh et al., 2016; Kingma & Dhariwal, 2018) for more details. For convenience, we omit the subscript  $t$  in this subsection. We choose invertible transformations whose Jacobian determinant in Equation 1 is simple to compute, that is a triangular matrix, diagonal matrix or a permutation matrix as explored in prior work (Rezende & Mohamed, 2015; Deco & Brauer, 1995). For permutation matrices, the Jacobian determinant is one and for triangular and diagonal Jacobian matrices, the determinant is simply the product of diagonal terms.
+
+- Actnorm: We apply a learnable per-channel scale and shift with data-dependent initialization.  
+- Coupling: We split the input  $y$  equally across channels to obtain  $y_{1}$  and  $y_{2}$ . We compute  $z_{2} = f(y_{1}) * y_{2} + g(y_{1})$  where  $f$  and  $g$  are deep networks. We concat  $y_{1}$  and  $z_{2}$  across channels.  
+- SoftPermute: We apply a  $1 \times 1$  convolution that preserves the number of channels.  
+- Squeeze: We reshape the input from  $H \times W \times C$  to  $H/2 \times W/2 \times 4C$  which allows the flow to operate on a larger receptive field.
+
+We infer the latent variable  $z^{(l)}$  at level  $l$  using:
+
+$$
+\operatorname {F l o w} (y) = \text {C o u p l i n g} (\text {S o f t P e r m u t e} (\text {A c t n o r m} (y))) \times N \tag {2}
+$$
+
+$$
+\operatorname {F l o w} _ {1} (y) = \operatorname {S p l i t} (\operatorname {F l o w} (\operatorname {S q u e e z e} (y))) \tag {3}
+$$
+
+$$
+\left(\mathbf {h} ^ {(> l)}, \mathbf {z} ^ {l}\right) \leftarrow \operatorname {F l o w} _ {1} \left(\mathbf {h} ^ {(> l - 1)}\right) \tag {4}
+$$
+
+where  $N$  is the number of steps of flow. In Equation (3), via Split, we split the output of Flow equally across channels into  $\mathbf{h}^{(l)}$ , the input to  $\mathrm{Flow}_{(l + 1)}(.)$  and  $z^{(l)}$ , the latent variable at level  $l$ . We, thus enable the flows at higher levels to operate on a lower number of dimensions and larger scales. When  $l = 1$ ,  $\mathbf{h}^{(l - 1)}$  is just the input frame  $x$  and for  $l = L$  we omit the Split operation. Finally, our multi-scale architecture  $\mathbf{f}_{\theta}(\mathbf{x}_t)$  is a composition of the flows at multiple levels from  $l = 1 \dots L$  from which we obtain our latent variables i.e  $\{\mathbf{z}_t^{(l)}\}_{l = 1}^L$ .
+
+# 4.2 AUTOREGRESSIVE LATENT DYNAMICS MODEL
+
+We use the multi-scale architecture described above to infer the set of corresponding latent variables for each individual frame of the video:  $\{\mathbf{z}_t^{(l)}\}_{l = 1}^L = \mathbf{f}_\theta (\mathbf{x}_t)$ ; see Figure 1 for an illustration. As in Equation (1), we need to choose a form of latent prior  $p_{\theta}(\mathbf{z})$ . We use the following autoregressive factorization for the latent prior:
+
+$$
+p _ {\boldsymbol {\theta}} (\mathbf {z}) = \prod_ {t = 1} ^ {T} p _ {\boldsymbol {\theta}} \left(\mathbf {z} _ {t} \mid \mathbf {z} _ {<   t}\right) \tag {5}
+$$
+
+where  $\mathbf{z}_{< t}$  denotes the latent variables of frames prior to the  $t$ -th timestep:  $\{\mathbf{z}_1, \dots, \mathbf{z}_{t-1}\}$ . We specify the conditional prior  $p_{\theta}(\mathbf{z}_t | \mathbf{z}_{< t})$  as having the following factorization:
+
+$$
+p _ {\boldsymbol {\theta}} \left(\mathbf {z} _ {t} \mid \mathbf {z} _ {<   t}\right) = \prod_ {l = 1} ^ {L} p _ {\boldsymbol {\theta}} \left(\mathbf {z} _ {t} ^ {(l)} \mid \mathbf {z} _ {<   t} ^ {(l)}, \mathbf {z} _ {t} ^ {(> l)}\right) \tag {6}
+$$
+
+where  $\mathbf{z}_{<t}^{(l)}$  is the set of latent variables at previous timesteps and at the same level  $l$ , while  $\mathbf{z}_t^{(>l)}$  is the set of latent variables at the same timestep and at higher levels. See Figure 1 for a graphical illustration of the dependencies.
+
+We let each  $p_{\theta}(\mathbf{z}_t^{(l)}|\mathbf{z}_{< t}^{(l)},\mathbf{z}_t^{(>l)})$  be a conditionally factorized Gaussian density:
+
+$$
+p _ {\boldsymbol {\theta}} \left(\mathbf {z} _ {t} ^ {(l)} \mid \mathbf {z} _ {<   t} ^ {(l)}, \mathbf {z} _ {t} ^ {(> l)}\right) = \mathcal {N} \left(\mathbf {z} _ {t} ^ {(l)}; \boldsymbol {\mu}, \sigma\right) \tag {7}
+$$
+
+$$
+\text {w h e r e} (\boldsymbol {\mu}, \log \sigma) = N N _ {\boldsymbol {\theta}} \left(\mathbf {z} _ {<   t} ^ {(l)}, \mathbf {z} _ {t} ^ {(> l)}\right) \tag {8}
+$$
+
+<table><tr><td>Model</td><td>Fooling rate</td></tr><tr><td>SAVP-VAE</td><td>16.4 %</td></tr><tr><td>VideoFlow</td><td>31.8 %</td></tr><tr><td>SV2P</td><td>17.5 %</td></tr></table>
+
+Table 1: We compare the realism of the generated trajectories using a real-vs-fake 2AFC Amazon Mechanical Turk with SAVP-VAE and SV2P.
+
+![](images/132c157101785187fa10c2288e94e2a16b74d90395ccadf112caf5b850630259.jpg)  
+Figure 2: We condition the VideoFlow model with the frame at  $t = 1$  and display generated trajectories at  $t = 2$  and  $t = 3$  for three different shapes.
+
+![](images/de95f8e9532ed5ab0b48b273033d924e91912e1f45914cbe03f6ad0f5d312a38.jpg)
+
+![](images/608b62429553b2e4d89de47431c593bf5fba364fcea2359d98e001942da8a3d6.jpg)
+
+where  $NN_{\theta}(.)$  is a deep 3-D residual network (He et al., 2015) augmented with dilations and gated activation units and modified to predict the mean and log-scale. We describe the architecture and our ablations of the architecture in Section D and E of the appendix.
+
+In summary, the log-likelihood objective of Equation (1) has two parts. The invertible multi-scale architecture contributes  $\sum_{i=1}^{K} \log |\det(d\mathbf{h}_i / d\mathbf{h}_{i-1})|$  via the sum of the log Jacobian determinants of the invertible transformations mapping the video  $\{\mathbf{x}_t\}_{t=1}^T$  to  $\{\mathbf{z}_t\}_{t=1}^T$ ; the latent dynamics model contributes  $\log p_\theta(\mathbf{z})$ , i.e. Equation (5). We jointly learn the parameters of the multi-scale architecture and latent dynamics model by maximizing this objective.
+
+Note that in our architecture we have chosen to let the prior  $p_{\theta}(\mathbf{z})$ , as described in eq. (5), model temporal dependencies in the data, while constraining the flow  $\mathbf{g}_{\theta}$  to act on separate frames of video. We have experimented with using 3-D convolutional flows, but found this to be computationally overly expensive compared to an autoregressive prior; in terms of both number of operations and number of parameters. Further, due to memory limits, we found it only feasible to perform SGD with a small number of sequential frames per gradient step. In case of 3-D convolutions, this would make the temporal dimension considerably smaller during training than during synthesis; this would change the model's input distribution between training and synthesis, which often leads to various temporal artifacts. Using 2-D convolutions in our flow  $\mathbf{f}_{\theta}$  with autoregressive priors, allows us to synthesize arbitrarily long sequences without introducing such artifacts.
+
+# 5 EXPERIMENTS
+
+All our generated videos and qualitative results can be viewed at this website. In the generated videos, a border of blue represents the conditioning frame, while a border of red represents the generated frames.
+
+# 5.1VIDEO MODELLING WITH THE STOCHASTIC MOVEMENT DATASET
+
+We use VideoFlow to model the Stochastic Movement Dataset used in (Babaeizadeh et al., 2017). The first frame of every video consists of a shape placed near the center of a  $64 \times 64 \times 3$  resolution gray background with its type, size and color randomly sampled. The shape then randomly moves in one of eight directions with constant speed. (Babaeizadeh et al., 2017) show that conditioned on the first frame, a deterministic model averages out all eight possible directions in pixel space. Since the shape moves with a uniform speed, we should be able to model the position of the shape at the  $(t + 1)^{th}$  step using only the position of the shape at the  $t^{th}$  step. Using this insight, we extract random temporal patches of 2 frames from each video of 3 frames. We then use VideoFlow to maximize the log-likelihood of the second frame given the first, i.e. the model looks back at just one frame. We observe that the bits-per-pixel on the holdout set reduces to a very low 0.04 bits-per-pixel for this model. On generating videos conditioned on the first frame, we observe that the model consistently predicts the future trajectory of the shape to be one of the eight random directions. We compare our model with two state-of-the-art stochastic video generation models SV2P and SAVP-VAE (Babaeizadeh et al., 2017; Lee et al., 2018) using their Tensor2Tensor implementation (Vaswani et al., 2018). We assess the quality of the generated videos using a real vs fake Amazon Mechanical Turk test. In the test, we inform the rater that a "real" trajectory is one in which the shape is consistent in color and congruent
+
+<table><tr><td>Model</td><td>Bits-per-pixel</td></tr><tr><td>VideoFlow</td><td>1.87</td></tr><tr><td>SAVP-VAE</td><td>≤ 6.73</td></tr><tr><td>SV2P</td><td>≤ 6.78</td></tr></table>
+
+Table 2: Left: We report the average bits-per-pixel across 10 target frames with 3 conditioning frames for the BAIR action-free dataset.
+
+![](images/26a891f58c09594d8d7b42bc73e854e8192dcf39bb8f03102487cfbd2efe8179.jpg)  
+Figure 3: We measure realism using a 2AFC test and diversity using mean pairwise cosine distance between generated samples in VGG perceptual space.
+
+throughout the video. We show that VideoFlow outperforms the baselines in terms of fooling rate in Table 1 consistently generating plausible "real" trajectories at a greater rate.
+
+# 5.2VIDEO MODELING WITH THE BAIR DATASET
+
+We use the action-free version of the BAIR robot pushing dataset (Ebert et al., 2017) that contains videos of a Sawyer robotic arm with resolution  $64 \times 64$ . In the absence of actions, the task of video generation is completely unsupervised with multiple plausible trajectories due to the partial observability of the environment and stochasticity of the robot actions. We train the baseline models, SAVP-VAE, SV2P and SVG-LP to generate 10 target frames, conditioned on 3 input frames. We extract random temporal patches of 4 frames, and train VideoFlow to maximize the log-likelihood of the 4th frame given a context of 3 past frames. We, thus ensure that all models have seen a total of 13 frames during training.
+
+Bits-per-pixel: We estimated the variational bound of the bits-per-pixel on the test set, via importance sampling, from the posteriors for the SAVP-VAE and SV2P models. We find that VideoFlow outperforms these models on bits-per-pixel and report these values in Table 2. We attribute the high values of bits-per-pixel of the baselines to their optimization objective. They do not optimize the variational bound on the log-likelihood directly due to the presence of a  $\beta \neq 1$  term in their objective and scheduled sampling (Bengio et al., 2015).
+
+![](images/ae213bf5e8698051b47e943b76e7a57f0a45ed7a877f7ddb2367b05a1a47fe5f.jpg)  
+Figure 4: For a given set of conditioning frames on the BAIR action-free we sample 100 videos from each of the stochastic video generation models. We choose the video closest to the ground-truth on the basis of PSNR, SSIM and VGG perceptual metrics and report the best possible value for each of these metrics. All the models were trained using ten target frames but are tested to generate 27 frames. For all the reported metrics, higher is better.
+
+![](images/47e1ebffd72a5d3db37c0ee7da429c96d27f52a78a4a6f2afb4302d4142f43de.jpg)
+
+![](images/b813a9408b57380190b386e9f380e3681f6ba17f93ee69db3322a30a6ad3daf3.jpg)
+
+Accuracy of the best sample: The BAIR robot-pushing dataset is highly stochastic and the number of plausible futures are high. Each generated video can be super realistic, can represent a plausible future in theory but can be far from the single ground truth video perceptually. To partially overcome this, we follow the metrics proposed in prior work (Babaeizadeh et al., 2017; Lee et al., 2018; Denton & Fergus, 2018) to evaluate our model. For a given set of conditioning frames in the BAIR action-free test-set, we generate 100 videos from each of the stochastic models. We then compute the closest of these generated videos to the ground truth according to three different metrics, PSNR (Peak Signal to
+
+Noise Ratio), SSIM (Structural Similarity) (Wang et al., 2004) and cosine similarity using features obtained from a pretrained VGG network (Dosovitskiy & Brox, 2016; Johnson et al., 2016) and report our findings in Figure 4. This metric helps us understand if the true future lies in the set of all plausible futures according to the video model.
+
+In prior work, (Lee et al., 2018; Babaeizadeh et al., 2017; Denton & Fergus, 2018) effectively tune the pixel-level variance as a hyperparameter and sample from a deterministic decoder. They obtain training stabilitiy and improve sample quality by removing pixel-level noise using this procedure. We can remove pixel-level noise in our VideoFlow model resulting in higher quality videos at the cost of diversity by sampling videos at a lower temperature, analogous to the procedure in (Kingma & Dhariwal, 2018). For a network trained with additive coupling layers, we can sample the  $t^{th}$  frame  $x_{t}$  from  $P(x_{t}|x_{< t})$  with a temperature  $T$  simply by scaling the standard deviation of the latent gaussian distribution  $P(z_{t}|z_{< t})$  by a factor of  $T$ . We report results with both a temperature of 1.0 and the optimal temperature tuned on the validation set using VGG similarity metrics in Figure 4. Additionally, we also applied low-temperature sampling to the latent gaussian priors of SV2P and SAVP-VAE and empirically found it to hurt performance. We report these results in Figure 12
+
+For SAVP-VAE, we notice that the hyperparameters that perform the best on these metrics are the ones that have disappearing arms. For completeness, we report these numbers as well as the numbers for the best performing SAVP models that do not have disappearing arms. Our model with optimal temperature performs better or as well as the SAVP-VAE and SVG-LP models on the VGG-based similarity metrics, which correlate well with human perception (Zhang et al., 2018) and SSIM. Our model with temperature  $T = 1.0$  is also competent with state-of-the-art video generation models on these metrics. PSNR is explicitly a pixel-level metric, which the VAE models incorporate as part of its optimization objective. VideoFlow on the other-hand models the conditional probability of the joint distribution of frames, hence as expected it underperforms on PSNR.
+
+![](images/352f5f6a979ef6707c99fa19301a68dad8bcafb9dfdbf21593876e162def30b6.jpg)  
+Figure 5: We display three different futures for two sets of conditioning frames (left and right) at  $T = 0.6$  showcasing diversity in outcomes
+
+![](images/044a0837d575f4f01b353cc1d87ae74f99f019cf277479e3bb12235b2fb3833e.jpg)
+
+Diversity and quality in generated samples: For each set of conditioning frames in the test set, we generate 10 videos and compute the mean distance in VGG perceptual space across these 45 different pairs. We average this across the test-set for  $T = 1.0$  and  $T = 0.6$  and report these numbers in Figure 3. We also assess the quality of the generated videos at  $T = 1.0$  and  $T = 0.6$ , using a real vs fake Amazon Mechanical Turk test and report fooling rates. We observe that VideoFlow outperforms diversity values reported in prior work (Lee et al., 2018) while being competitive in the realism axis. We also find that VideoFlow at  $T = 0.6$  has the highest fooling rate while being competent with state-of-the-art VAE models in diversity.
+
+On inspection of the generated videos, we find that at lower temperatures, the arm exhibits less random behaviour with the background objects remaining static and clear achieving higher realism scores. At higher temperatures, the motion of arm is much more stochastic, achieving high diversity scores with the background objects becoming much noisier leading to a drop in realism.
+
+Fréchet Video Distance (FVD): We evaluate VideoFlow using the recently proposed Fréchet Video Distance (FVD) metric (Unterthiner et al., 2018), an adaptation of the Fréchet Inception Distance (FID) metric (Heusel et al., 2017) for video generation. (Unterthiner et al., 2018) report results with models trained on a total of 16 frames with 2 conditioning frames; while we train our VideoFlow
+
+<table><tr><td colspan="5"># Frames Seen: Training</td></tr><tr><td>Conditioning</td><td>3</td><td>3</td><td>3</td><td>2</td></tr><tr><td>Total</td><td>13</td><td>13</td><td>13</td><td>16</td></tr><tr><td colspan="5"># Frames: Evaluation</td></tr><tr><td>Ground truth</td><td>3</td><td>3</td><td>2</td><td>2</td></tr><tr><td>Total</td><td>13</td><td>16</td><td>16</td><td>16</td></tr><tr><td>Model</td><td colspan="4">FVD</td></tr><tr><td>VideoFlow (T=0.8)</td><td>95±4</td><td>127±3</td><td>131±5</td><td>-</td></tr><tr><td>VideoFlow (T=1.0)</td><td>149±6</td><td>221±8</td><td>251±7</td><td>-</td></tr><tr><td>SAVP</td><td>-</td><td>-</td><td>-</td><td>116</td></tr><tr><td>SV2P</td><td>-</td><td>-</td><td>-</td><td>263</td></tr></table>
+
+Table 3: Fréchet Video Distance:. We report the mean and standard deviation across 5 runs for 3 different frame settings. Results are not directly comparable across models due to the differences between the total number of frames seen during training and the number of conditioning frames.
+
+model on a total of 13 frames with 3 conditioning frames, making our results not directly comparable to theirs. We evaluate FVD for both shorter and longer rollouts in Table 3. We show that, even in the settings that are disadvantageous to VideoFlow, where we compute the FVD on a total of 16 frames, when trained on just 13 frames, VideoFlow performs comparable to SAVP.
+
+# 5.3 LATENT SPACE INTERPOLATION
+
+BAIR robot pushing dataset: We encode the first input frame and the last target frame into the latent space using our trained VideoFlow encoder and perform interpolations. We find that the motion of the arm is interpolated in a temporally cohesive fashion between the initial and final position. Further, we use the multi-level latent representation to interpolate representations at a particular level while keeping the representations at other levels fixed. We find that the bottom level interpolates the motion of background objects which are at a smaller scale while the top level interpolates the arm motion.
+
+![](images/2a3fd945397de2fa526e8e55387ffdc22d7c83b8fa5df6fc38cad5150b2b2d73.jpg)
+
+![](images/a2edc183bde1b34d4a7300eda47f0ad69fbc230740360967c8ad5a173e48842d.jpg)  
+Figure 6: Left: We display interpolations between a) a small blue rectangle and a large yellow rectangle b) a small blue circle and a large yellow circle. Right: We display interpolations between the first input frame and the last target frame of two test videos in the BAIR robot pushing dataset.
+
+![](images/eb8d723dd085b23a51f6e0cbf56466fb55322ec0f91a8ce4e36ed19609f002bb.jpg)
+
+![](images/0c990fae5c4c589d2d3dbc8123db9f65a618fc13e87b2c0b7654da6f69bae4c4.jpg)
+
+![](images/f46585e8e27b7b736398aef8dfc3d22cd3a979b6c5a9085c1e70a7c02aa1d20d.jpg)
+
+![](images/7dd5617da7977c8991011a0b548ba441af8e4661b24814e1b68b5b6de0f8fd9a.jpg)
+
+![](images/fa1ca5d074ad10f2147635268a5b572b5fd991ddbecd3e2be2da69403d0e55be.jpg)
+
+![](images/55303b936ab23c50baf3f4f6c5e70edd36fb184c19c729f3365a6a12d023cc70.jpg)
+
+![](images/f83706521ffb72ba0578ff009c9b61f852346e44ed2aed7f952c28185ca11efe.jpg)
+
+![](images/964f5aa5444391dfa458d193092d13ab8edf3a500ea03f8a014590780ebb894f.jpg)
+
+![](images/b8ee5ab56b5851a9e73d9dcb8b9028d7755537cd2881c368df9bdbed4a16aa7a.jpg)
+
+![](images/90b196ed28d0c56aff9ed9494becd759d6644d8f4a387a09d3eaa0ac97acb3b2.jpg)
+
+![](images/9c6fb89006aeea09042f3e4b9cbc70c9c5a35c852e20c3f820ebf26a4eb518ac.jpg)
+
+![](images/4315da70d68e9d67ab06a5f515409cdbf3e5a3099734b5ba013b3317b28a3b09.jpg)
+
+![](images/c76b09ed4f6a2d74d7aee99dc31e1fed8a64aac769575d75aa332c77ef639351.jpg)
+
+![](images/815cf3163a8f4c88fe78f10a1345900fd231b3f4b0b7b415f43b1f07df352690.jpg)
+
+Stochastic Movement Dataset: We encode two different shapes with their type fixed but a different size and color into the latent space. We observe that the size of the shape gets smoothly interpolated. During training, we sample the colors of the shapes from a uniform discrete distribution which is reflected in our experiments. We observe that all the colors in the interpolated space lie in the set of colors in the training set.
+
+# 5.4 LONGER PREDICTIONS
+
+We generate 100 frames into the future using our model trained on 13 frames with a temperature of 0.5 and display our results in Figure 7. On the top, even 100 frames into the future, the generated frames remain in the image manifold maintaining temporal consistency. In the presence of occlusions,
+
+![](images/c695f912d4da06a34e3ff7a09da4855e9ed4292dd0abfb485e1e6f925a3c11ce.jpg)
+
+![](images/e896f73db14516a5b26b116c5b09e4dcef6eb13724a549068640dca4c8ba0a53.jpg)  
+Figure 7: Left: We generate 100 frames into the future with a temperature of 0.5. The top and bottom row correspond to generated videos in the absence and presence of occlusions respectively. Right: We use VideoFlow to detect the plausibility of a temporally inconsistent frame to occur in the immediate future.
+
+![](images/c09453bc2810bcfbc01ceaf90d471567f9e87560d67656ac26af45e2a20a8870.jpg)
+
+![](images/950f7a22a58fe66ecf6eb1e1fc0f34cd295662ec54ff70a02ccd23c33042b8f4.jpg)
+
+![](images/13b8e22fa34bab7307bd04cd7cb40f0abe92fb7745f08a80ed5ad93302392da1.jpg)
+
+![](images/fc3b6f4e409fae67f732b5ae7a625a895cff9f7fb6f631fb341877099f040761.jpg)
+
+![](images/c4ee05a996158b18647c909ab9a2bfbe0a734126a4d294ed4f9f4e8a44cdb54a.jpg)
+
+![](images/a93b5009f7a65bc151cbecf9dc22be36a857f051da4854e06d37b58b2e72111a.jpg)
+
+![](images/47c25d750a70439f9275dc83d26c5a4f077586aff473f18a976d402e2dfb956c.jpg)
+
+the arm remains super-sharp but the background objects become noisier and blurrier. Our VideoFlow model has a bijection between the  $\mathbf{z}_t$  and  $\mathbf{x}_t$  meaning that the latent state  $\mathbf{z}_t$  cannot store information other than that present in the frame  $\mathbf{x}_t$ . This, in combination with the Markovian assumption in our latent dynamics means that the model can forget objects if they have been occluded for a few frames. In future work, we would address this by incorporating longer memory in our VideoFlow model; for example by parameterizing  $NN_{\theta}()$  as a recurrent neural network in our autoregressive prior (eq. 8) or using more memory-efficient backpropagation algorithms for invertible neural networks (Gomez et al., 2017).
+
+# 5.5 OUT-OF-SEQUENCE DETECTION
+
+We use our trained VideoFlow model, conditioned on 3 frames as explained in Section 5.2, to detect the plausibility of a temporally inconsistent frame to occur in the immediate future. We condition the model on the first three frames of a test-set video  $X_{<4}$  to obtain a distribution  $P(X_4|X_{<4})$  over its 4th frame  $X_4$ . We then compute the likelihood of the  $t^{\text{th}}$  frame  $X_t$  of the same video to occur as the 4th time-step using this distribution. i.e.,  $\mathcal{P}(X_4 = X_t|X_{<4})$  for  $t = 4 \ldots 13$ . We average the corresponding bits-per-pixel values across the test set and report our findings in Figure 7. We find that our model assigns a monotonically decreasing log-likelihood to frames that are more far out in the future and hence less likely to occur in the 4th time-step.
+
+# 6 OPEN SOURCE CODE AND CHECKPOINTS
+
+We open-source the implementation of our code in the Tensor2Tensor codebase. We additionally open-source various components of our trained VideoFlow model, to evaluate log-likelihood, to generate frames and compute latent codes as reusable TFHub modules
+
+# 7 CONCLUSION AND DISCUSSION
+
+We describe a practically applicable architecture for flow-based video prediction models, inspired by the Glow model for image generation Kingma & Dhariwal (2018), which we call VideoFlow. We introduce a latent dynamical system model that predicts future values of the flow model's latent state replacing the standard unconditional prior distribution. Our empirical results show that VideoFlow achieves results that are competitive with the state-of-the-art VAE models in stochastic video prediction. Finally, our model optimizes log-likelihood directly making it easy to evaluate while achieving faster synthesis compared to pixel-level autoregressive video models, making our model suitable for practical purposes. In future work, we plan to incorporate memory in VideoFlow to model arbitrary long-range dependencies and apply the model to challenging downstream tasks.
+
+# ACKNOWLEDGEMENTS
+
+We would like to thank Ryan Sepassi and Lukasz Kaiser for their extensive help in using Tensor2Tensor, Oscar Tackström for finding a bug in our evaluation pipeline that improved results across all models, Ruben Villegas for providing code for the SVG-LP baseline and Mostafa Dehghani for providing feedback on a draft of the rebuttal.
+
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+
+Richard Zhang, Phillip Isola, Alexei A Efros, Eli Shechtman, and Oliver Wang. The unreasonable effectiveness of deep features as a perceptual metric. arXiv preprint, 2018.
+
+# A MOVING MNIST - QUALITATIVE EXPERIMENTS
+
+![](images/c6a017957c92c566e42d54eddbc69595fd810255fcefbfc9d36a860305c2b973.jpg)  
+Figure 8: We display ten frame rollouts conditioned on a single frame on the Moving MNIST dataset.
+
+Similar to the Stochastic Movement Dataset as described in Section 5.1, we extract random temporal patches of 2 frames on the Moving MNIST dataset (Srivastava et al., 2015). We train our VideoFlow model to maximize the log-likelihood of the second frame, given the first. Our rollouts over 10 frames capture realistic digit movement.
+
+# B HUMAN3.6M - QUALITATIVE EXPERIMENTS
+
+We model the Human3.6M dataset (Ionescu et al., 2014), by maximizing the log-likelihood of the 4th frame given the first three frames, in a random temporal patch of 4 frames. We observe that on this dataset, our model fails to capture reasonable human motion. We hope that by increasing model capacity and using more expressive priors, we can achieve better performance on this dataset in the future.
+
+# C DISCRETIZATION AND UNIFORM QUANTIZATION
+
+Let  $\mathcal{D} = \{\mathbf{x}^{(i)}\}_{i=1}^{N}$  be our dataset of i.i.d. observations of a random variable  $\mathbf{x}$  with an unknown true distribution  $p^*(\mathbf{x})$ . Our data consist of 8-bit videos, with each dimension rescaled to the domain  $[0,255/256]$ . We add a small amount of uniform noise to the data,  $\mathbf{u} \sim \mathcal{U}(0,1/256)$ , matching its discretization level (Dinh et al., 2016; Kingma & Dhariwal, 2018). Let  $q(\mathbf{x})$  be the resulting empirical distribution corresponding to this scaling and addition of noise. Note that additive noise is required to prevent  $q(\mathbf{x})$  from having infinite densities at the datapoints, which can result in ill-behaved optimization of the log-likelihood; it also allows us to recast maximization of the log-likelihood as minimization of a KL divergence.
+
+![](images/a1da44922dc69d8050465bd392c21033a554cacafb427cbfb3c6ce18321e8143.jpg)  
+Figure 9: We display ten frame rollouts conditioned on 3 frames on the Human3.6M dataset.
+
+# D RESIDUAL NETWORK ARCHITECTURE
+
+Here we'll describe the architecture for the residual network  $NN_{\theta}(.)$  that maps  $\mathbf{z}_{<t}^{(l)}, \mathbf{z}_t^{(>l)}$  to  $(\pmb{\mu}_t^{(l)}, \log \sigma_t^{(l)})$  (Left: Figure 10). As shown in the left of Figure 10, let  $\mathbf{h}_t^{(>l)}$  be the tensor representing  $\mathbf{z}_t^{(>l)}$  after the split operation between levels in the multi-scale architecture. We apply a  $1 \times 1$  convolution over  $\mathbf{h}_t^{(>l)}$  and concatenate this across channels to each latent from the previous time-step and the same-level independently. In this way, we obtain  $((W\mathbf{h}_t^{(>l)}; \mathbf{z}_{t-1}^{(l)}), (W\mathbf{h}_t^{(>l)}; \mathbf{z}_{t-2}^{(l)}) \dots (W\mathbf{h}_t^{(>l)}; \mathbf{z}_{t-n}^{(l)})$ . We transform these values into  $(\pmb{\mu}_t^{(l)}, \log \sigma_t^{(l)})$  via a stack of residual blocks. We obtain a reduction in parameter count by sharing parameters across every 2 time-steps via 3-D convolutions in our residual blocks.
+
+As shown in the right of Figure 10, each 3-D residual block consists of three layers. The first layer has a filter size of  $2 \times 3 \times 3$  with 512 output channels followed by a ReLU activation. The second layer has two  $1 \times 1 \times 1$  convolutions via the Gated Activation Unit Van Den Oord et al. (2016); van den Oord et al. (2016a). The third layer has a filter size of  $2 \times 3 \times 3$  with the number of output channels determined by the level. This block is replicated three times in parallel, with dilation rates 1, 2 and 4, after which the results of each block, in addition to the input of the residual block, are summed.
+
+The first two layers are initialized using a Gaussian distribution and the last layer is initialized to zeroes. In that way, the residual network behaves as an identity network during initialization allowing stable optimization. After applying a sequence of residual blocks, we use the last temporal activation that should capture all context. We apply a final  $1 \times 1$  convolution to this activation to obtain  $(\Delta \mathbf{z}_t^{(l)}, \log \sigma_t^{(l)})$ . We then add  $\Delta \mathbf{z}_t^{(l)}$  to  $\mathbf{z}_{t-1}^{(l)}$  to a temporal skip connection to output  $\boldsymbol{\mu}_t^{(l)}$ . This way, the network learns to predict the change in latent variables for a given level. We have provided visualizations of the network architecture in this website
+
+# E ABLATION STUDIES
+
+Through an ablation study, we experimentally evaluate the importance of the following components of our VideoFlow model: (1) the use of temporal skip connections, (2) the use Gated Activation Unit (GATU) instead of ReLUs in the residual network and (3) the use of dilations in  $NN_{\theta}()$  in Section D
+
+We start with a VideoFlow model with 256 channels in the coupling layer, 16 steps of flow and remove the components mentioned above to create our baseline. We use four different combinations of our components (described in Fig. 11) and keep the rest of the hyperparameters fixed across those combinations. For each combination we plot the mean bits-per-pixel on the holdout BAIR-action
+
+![](images/05fcbc64cd2c6e75d3a9425adef8719a25195ad248397f1f5c7fa230d3ff7a64.jpg)  
+Figure 10: Left: We predict a gaussian distribution over  $\mathbf{z}_t^{(l)}$  via a 3-D Residual network conditioned on  $\mathbf{z}_{< t}^{(l)}$  and  $\mathbf{z}_t^{(>l)}$ . Right: Our 3-D residual network architecture is augmented with dilations and gated activation units improving performance.
+
+![](images/252f813ec06b4841287f1efb17aefdaedc336534544e6c4eb5552e301d93d742.jpg)
+
+free dataset over 300K training steps for both affine and additive coupling in Figure 11. For both the coupling layers, we observe that the VideoFlow model with all the components provide a significant boost in bits-per-pixel over our baseline.
+
+![](images/bcfa6e2bdc0b90f15b70a3d6712b8d20b496b28579ef637b62f32efbb06621c8.jpg)  
+Figure 11: B: baseline, A: Temporal Skip Connection, C: Dilated Convolutions + GATU, D: Dilation Convolutions + Temporal Skip Connection, E: Dilation Convolutions + Temporal Skip Connection + GATU. We plot the holdout bits-per-pixel on the BAIR action-free dataset for different ablations of our VideoFlow model.
+
+![](images/8cbe522c16dc8fa80dca4be6e568a6cde35a0fae4a83b6e6e185a436f3db98c7.jpg)
+
+We also note that other combinations—dilated convolutions + GATU (C) and dilated convolutions + the temporal skip connection—improve over the baseline. Finally, we experienced that increasing the receptive field in  $NN_{\theta}()$  using dilated convolutions alone in the absence of the temporal skip connection or the GATU makes training highly unstable.
+
+# F EFFECT OF TEMPERATURE ON SAVP-VAE AND SV2P
+
+We repeat our evaluations described in Figure 4 applying low temperature to the latent gaussian priors of SV2P and SAVP-VAE. We empirically find that decreasing temperature from 1.0 to 0.0 monotonically decreases the performance of the VAE models. Our insight is that the VideoFlow
+
+![](images/6f134109f65e147bab17a4372b6f1d3796afd5ac04d451b02043f43b40c3d190.jpg)
+
+![](images/e49e8d8ed29ee91ace73e26de9704d219fbe5b9ba9b7a90251cd95a1328512d5.jpg)
+
+![](images/a1d15484b2184699c8b54288c8b1dc32e2db86d073f38e58417a7e2c10ff7251.jpg)
+
+![](images/f81ded8ec4440ee1d39e184e6134d8f3f636334fa8f66500e5123f81941eae11.jpg)  
+Figure 12: We repeat our evaluations described on the SV2P and SAVP-VAE model in Figure 4 using temperatures from 0.0 to 1.0 while sampling from the latent gaussian prior.
+
+![](images/70962df7e5dc9040a6ad2f3a83056a57bd3c35a87c169114724ade91b6195158.jpg)
+
+![](images/4cc6f0d1a45e5c1586ddecc059f842bd679e92c74dc968bdb49517d06854d99a.jpg)
+
+model gains by low-temperature sampling due to the following reason. At lower T, we obtain a tradeoff between a performance gain by noise removal from the background and a performance hit due to reduced stochasticity of the robot arm. On the other hand, the VAE models have a clear but slightly blurry background throughout from  $T = 1.0$  to  $T = 0.0$ . Reducing T in this case, solely reduces the stochasticity of the arm motion thus hurting performance.
+
+# G LIKELIHOOD VS QUALITY
+
+![](images/7c5f0d125078880648709f5e0306e9bd687184de3c00d13f109a1e878de3cdbf.jpg)  
+Figure 13: We provide a comparison between training progression (measured in the mean bits-per-pixel objective on the test-set) and the quality of generated videos.
+
+We show correlation between training progression (measured in bits per pixel) and quality of the generated videos in Figure 13. We display the videos generated by conditioning on frames from the test set for three different values of bits-per-pixel on the test-set. As we approach lower bits-per-pixel, our VideoFlow model learns to model the structure of the arm with high quality as well as its motion resulting in high quality video.
+
+# H VIDEOFLOW - BAIR HYPERPARAMETERS
+
+# H.1 QUANTITATIVE-BITS-PER-PIXEL
+
+To report bits-per-pixel we use the following set of hyperparameters. We use a learning rate schedule of linear warmup for the first 10000 steps and apply a linear-decay schedule for the last 150000 steps.
+
+<table><tr><td>Hyperparameter</td><td>Value</td></tr><tr><td>Flow levels</td><td>3</td></tr><tr><td>Flow steps per level</td><td>24</td></tr><tr><td>Coupling</td><td>Affine</td></tr><tr><td>Number of coupling layer channels</td><td>512</td></tr><tr><td>Optimier</td><td>Adam</td></tr><tr><td>Batch size</td><td>40</td></tr><tr><td>Learning rate</td><td>3e-4</td></tr><tr><td>Number of 3-D residual blocks</td><td>5</td></tr><tr><td>Number of 3-D residual channels</td><td>256</td></tr><tr><td>Training steps</td><td>600K</td></tr></table>
+
+# H.2 QUALITATIVE EXPERIMENTS
+
+For all qualitative experiments and quantitative comparisons with the baselines, we used the following sets of hyperparameters.
+
+<table><tr><td>Hyperparameter</td><td>Value</td></tr><tr><td>Flow levels</td><td>3</td></tr><tr><td>Flow steps per level</td><td>24</td></tr><tr><td>Coupling</td><td>Additive</td></tr><tr><td>Number of coupling layer channels</td><td>392</td></tr><tr><td>Optimier</td><td>Adam</td></tr><tr><td>Batch size</td><td>40</td></tr><tr><td>Learning rate</td><td>3e-4</td></tr><tr><td>Number of 3-D residual blocks</td><td>5</td></tr><tr><td>Number of 3-D residual channels</td><td>256</td></tr><tr><td>Training steps</td><td>500K</td></tr></table>
+
+# I HYPERPARAMETER GRID FOR THE BASELINE VIDEO MODELS.
+
+We train all our baseline models for 300K steps using the Adam optimizer. Our models were tuned using the maximum VGG cosine similarity metric with the ground-truth across 100 decodes.
+
+SAVP-VAE and SV2P: We use three values of latent loss multiplier 1e-3, 1e-4 and 1e-5. For the SAVP-VAE model, we additionally apply linear decay on the learning rate for the last 100K steps.  
+SAVP-GAN: We tune the gan loss multiplier and the learning rate on a logscale from 1e-2 to 1e-4 and 1e-3 to 1e-5 respectively.
+
+# J CORRELATION BETWEEN VGG PERCEPTUAL SIMILARITY AND BITS-PER-PICKEL
+
+We plot correlation between cosine similarity using a pretrained VGG network and bits-per-pixel using our trained VideoFlow model. We compare  $\mathcal{P}(X_4 = X_t|X_{< 4})$  as done in Section 5.5 and the VGG cosine similarity between  $X_{4}$  and  $X_{t}$  for  $t = 4\dots 13$ . We report our results for every video in the test set in Figure 15. We notice a weak correlation between VGG perceptual metrics and bits-per-pixel with a correlation factor of  $-0.51$ .
+
+![](images/258972ff14f7685c6cfe90af9a9373e2b0bba91c1b8df631c3570e81a5a83f00.jpg)  
+Figure 14: We compare  $\mathcal{P}(X_4 = X_t|X_{< 4})$  and VGG cosine similarity between  $X_{4}$  and  $X_{t}$  for  $t = 4\dots 13$
+
+# K VIDEOFLOW: LOW PARAMETER REGIME
+
+We repeated our evaluations described in Figure 4, with a smaller version of our VideoFlow model with  $4\mathrm{x}$  parameter reduction. Our model remains competitive with SVG-LP on the VGG perceptual metrics.
+
+![](images/28fc9a84ebc8bface8572e1557b24229b5aa1cdc98e97e36542c03d44a8ee775.jpg)  
+Figure 15: We repeat our evaluations described in Figure 4 with a smaller version of our VideoFlow model.
\ No newline at end of file
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+# VQ-WAV2VEC: SELF-SUPERVISED LEARNING OF DISCRETE SPEECH REPRESENTATIONS
+
+Alexei Baevski\*
+
+Steffen Schneider*
+
+Michael Auli
+
+$\triangle$  Facebook AI Research, Menlo Park, CA, USA
+
+$\nabla$  University of Tübingen, Germany
+
+# ABSTRACT
+
+We propose vq-wav2vec to learn discrete representations of audio segments through a wav2vec-style self-supervised context prediction task. The algorithm uses either a Gumbel-Softmax or online k-means clustering to quantize the dense representations. Discretization enables the direct application of algorithms from the NLP community which require discrete inputs. Experiments show that BERT pre-training achieves a new state of the art on TIMIT phoneme classification and WSJ speech recognition. $^1$
+
+# 1 INTRODUCTION
+
+Learning discrete representations of speech has gathered much recent interest (Versteegh et al., 2016; Dunbar et al., 2019). A popular approach to discover discrete units is via autoencoding (Tjandra et al., 2019; Eloff et al., 2019; Chorowski et al., 2019) sometimes coupled with an autoregressive model (Chung et al., 2019). Another line of research is to learn continuous speech representations in a self-supervised way via predicting context information (Chung & Glass, 2018; van den Oord et al., 2018; Schneider et al., 2019).
+
+In this paper, we combine these two lines of research by learning discrete representations of speech via a context prediction task instead of reconstructing the input. This enables us to directly apply well performing NLP algorithms to speech data (Figure 1a).
+
+![](images/d77644d8e44601327dc51c73b04c90ccde08199d5d9d443aa6623c1d7f18877e.jpg)  
+(a) vq-wav2vec
+
+![](images/015c22d844c16291c961b47b7b77e42a131785f7b522aa701ef46ef94ab72c14.jpg)  
+(b) Discretized speech training pipeline  
+Figure 1: (a) The vq-wav2vec encoder maps raw audio  $(\mathcal{X})$  to a dense representation  $(\mathcal{Z})$  which is quantized (q) to  $\hat{\mathcal{Z}}$  and aggregated into context representations  $(\mathcal{C})$ ; training requires future time step prediction. (b) Acoustic models are trained by quantizing the raw audio with vq-wav2vec, then applying BERT to the discretized sequence and feeding the resulting representations into the acoustic model to output transcriptions.
+
+Our new discretization algorithm, vq-wav2vec, learns discrete representations of fixed length segments of audio signal by utilizing the wav2vec loss and architecture (Schneider et al, 2019; §2). To
+
+choose the discrete variables, we consider a Gumbel-Softmax approach (Jang et al., 2016) as well as online k-means clustering, similar to VQ-VAE (Oord et al., 2017; Eloff et al., 2019; §3).
+
+We then train a Deep Bidirectional Transformer (BERT; Devlin et al., 2018; Liu et al., 2019) on the discretized unlabeled speech data and input these representations to a standard acoustic model (Figure 1b; §4). Our experiments show that BERT representations perform better than log-mel filterbank inputs as well as dense wav2vec representations on both TIMIT and WSJ benchmarks. Discretization of audio enables the direct application of a whole host of algorithms from the NLP literature to speech data. For example, we show that a standard sequence to sequence model from the NLP literature can be used to perform speech recognition over discrete audio tokens (\$5, §6).
+
+# 2 BACKGROUND
+
+# 2.1 WAV2VEC
+
+wav2vec (Schneider et al., 2019) learns representations of audio data by solving a self-supervised context-prediction task with the same loss function as word2vec (Mikolov et al., 2013; van den Oord et al., 2018). The model is based on two convolutional neural networks where the encoder produces a representation  $\mathbf{z}_i$  for each time step  $i$  at a rate of  $100\mathrm{Hz}$  and the aggregator combines multiple encoder time steps into a new representation  $\mathbf{c}_i$  for each time step  $i$ . Given an aggregated representation  $\mathbf{c}_i$ , the model is trained to distinguish a sample  $\mathbf{z}_{i + k}$  that is  $k$  steps in the future from distractor samples  $\tilde{\mathbf{z}}$  drawn from a distribution  $p_n$ , by minimizing the contrastive loss for steps  $k = 1,\dots ,K$ :
+
+$$
+\mathcal {L} _ {k} ^ {\text {w a v 2 v e c}} = - \sum_ {i = 1} ^ {T - k} \left(\log \sigma \left(\mathbf {z} _ {i + k} ^ {\top} h _ {k} \left(\mathbf {c} _ {i}\right)\right) + \frac {\lambda}{\tilde {\mathbf {z}} \sim p _ {n}} \mathbb {E} \left[ \log \sigma \left(- \tilde {\mathbf {z}} ^ {\top} h _ {k} \left(\mathbf {c} _ {i}\right)\right) \right]\right) \tag {1}
+$$
+
+where  $T$  is the sequence length,  $\sigma(x) = 1 / (1 + \exp(-x))$ , and where  $\sigma(\mathbf{z}_{i+k}^{\top} h_k(\mathbf{c}_i))$  is the probability of  $\mathbf{z}_{i+k}$  being the true sample. We consider a step-specific affine transformation  $h_k(\mathbf{c}_i) = W_k \mathbf{c}_i + \mathbf{b}_k$  that is applied to  $\mathbf{c}_i$  (van den Oord et al., 2018). We optimize the loss  $\mathcal{L} = \sum_{k=1}^{K} \mathcal{L}_k$ , summing (1) over different step sizes. After training, the representations produced by the context network  $\mathbf{c}_i$  are input to the acoustic model instead of log-mel filterbank features.
+
+# 2.2 BERT
+
+BERT (Devlin et al., 2018) is a pre-training approach for NLP tasks, which uses a transformer encoder model to build a representation of text. Transformers use self-attention to encode the input sequence as well as an optional source sequence (Vaswani et al., 2017). The original BERT model combined two tasks for training: first, masked language modeling randomly removes some of the input tokens and the model has to predict those missing tokens. Second, next sentence prediction splices two different text passages together into a single example and the model needs to predict whether the passages are from the same document.
+
+# 3 VQ-WAV2VEC
+
+Our approach, vq-wav2vec, learns vector quantized (VQ) representations of audio data using a future time-step prediction task. We follow the same architectural choices as wav2vec (§2.1) with two convolutional networks  $f: \mathcal{X} \mapsto \mathcal{Z}$  and  $g: \hat{\mathcal{Z}} \mapsto \mathcal{C}$  for feature extraction and aggregation, as well as a new quantization module  $q: \mathcal{Z} \mapsto \hat{\mathcal{Z}}$  to build discrete representations (Figure 1a).
+
+We first map 30ms segments of raw speech to a dense feature representation  $\mathbf{z}$  at a stride of 10ms using the encoder network  $f$ . Next, the quantizer  $(q)$  turns these dense representations into discrete indices which are mapped to a reconstruction  $\hat{\mathbf{z}}$  of the original representation  $\mathbf{z}$ . We feed  $\hat{\mathbf{z}}$  into the aggregator  $g$  and optimize the same context prediction task as wav2vec outlined in §2.1.
+
+The quantization module replaces the original representation  $\mathbf{z}$  by  $\hat{\mathbf{z}} = \mathbf{e}_i$  from a fixed size codebook  $\mathbf{e} \in \mathbb{R}^{V \times d}$  which contains  $V$  representations of size  $d$ . We consider the Gumbel-Softmax which is a differentiable approximation of the argmax for computing one-hot representations (§3.1; Figure 2a)
+
+![](images/c18f879248ae9f1e5fe815031b2d8ed9c93e2155d0f3b82963bb10255e8b1b67.jpg)  
+(a) Gumbel-Softmax
+
+![](images/1a50cf01bb0892a470aea1f4c0adeea568c278a0f89db6d9ad4efd5770941680.jpg)  
+(b) K-means clustering.  
+Figure 2: (a) The Gumbel-Softmax quantization computes logits representing the codebook vectors (e). In the forward pass the argmax codeword  $(\mathbf{e}_2)$  is chosen and for backward (not shown) the exact probabilities are used. (b) K-means vector quantization computes the distance to all codeword vector and chooses the closest (argmin).
+
+as well as online k-means clustering, similar to the vector quantized variational autoencoder (VQ-VAE; Oord et al., 2017; §3.2; Figure 2b). Finally, we perform multiple vector quantizations over different parts of  $\mathbf{z}$  to mitigate mode collapse (§3.3).
+
+# 3.1 GUMBEL- SOFTMAX
+
+The Gumbel-Softmax (Gumbel, 1954; Jang et al., 2016; Maddison et al., 2014) enables selecting discrete codebook variables in a fully differentiable way and we use the straight-through estimator of Jang et al. (2016). Given the dense representation  $\mathbf{z}$ , we apply a linear layer, followed by a ReLU and another linear which outputs  $\mathbf{l} \in \mathbb{R}^V$  logits for the Gumbel-Softmax. At inference, we simply pick the largest index in  $l$ . At training, the output probabilities for choosing the  $j$ -th variable are
+
+$$
+p _ {j} = \frac {\exp \left(l _ {j} + v _ {j}\right) / \tau}{\sum_ {k = 1} ^ {V} \exp \left(l _ {k} + v _ {k}\right) / \tau}, \tag {2}
+$$
+
+where  $v = -\log (-\log (u))$  and  $u$  are uniform samples from  $\mathcal{U}(0,1)$ . During the forward pass,  $i = \operatorname{argmax}_j p_j$  and in the backward pass, the true gradient of the Gumbel-Softmax outputs is used.
+
+# 3.2 K-MEANS
+
+The vector quantization approach of van den Oord et al. (2017) is an alternative to making the index selection procedure fully differentiable. Different to their setup, we optimize a future time step prediction loss instead of the reconstruction loss of an autoencoder.
+
+We choose the codebook variable representation by finding the closest variable to the input features  $\mathbf{z}$  in terms of the Euclidean distance, yielding  $i = \mathrm{argmin}_j\| \mathbf{z} - \mathbf{e}_j\| _2^2$ . During the forward pass, we select  $\hat{\mathbf{z}} = \mathbf{e}_i$  by choosing the corresponding variable from the codebook. We obtain gradients for the encoder network by back-propagating  $\mathrm{d}\mathcal{L}^{\mathrm{wav2vec}} / \mathrm{d}\hat{\mathbf{z}}$  (van den Oord et al., 2017). The final loss has two additional terms:
+
+$$
+\mathcal {L} = \sum_ {k = 1} ^ {K} \mathcal {L} _ {k} ^ {\text {w a v 2 v e c}} + \left(\| \operatorname {s g} (\mathbf {z}) - \hat {\mathbf {z}} \| ^ {2} + \gamma \| \mathbf {z} - \operatorname {s g} (\hat {\mathbf {z}}) \| ^ {2}\right), \tag {3}
+$$
+
+where  $\mathrm{sg}(x) \equiv x$ ,  $\frac{\mathrm{d}}{\mathrm{d}x}\mathrm{sg}(x) \equiv 0$  is the stop gradient operator and  $\gamma$  is a hyperparameter. The first term is the future prediction task and gradients do not change the codebook because of the straight-through gradient estimation of mapping  $\mathbf{z}$  to  $\hat{\mathbf{z}}$ . The second term  $\| \mathrm{sg}(\mathbf{z}) - \hat{\mathbf{z}} \|^{2}$  moves the codebook vectors closer to the encoder output, and the third term  $\| \mathbf{z} - \mathrm{sg}(\hat{\mathbf{z}})\|^{2}$  makes sure that the encoder outputs are close to a centroid (codeword).
+
+# 3.3 VECTOR QUANTIZATION WITH MULTIPLE VARIABLE GROUPS
+
+So far, we considered replacing the encoder feature vector  $\mathbf{z}$  by a single entry  $\mathbf{e}_i$  in the codebook. This is prone to mode collapse where only some of the codewords are actually used. Previously, this problem has been mitigated by workarounds such as re-initializing codewords or applying additional regularizers to the loss function (Caron et al., 2019). In the following, we describe another strategy where we independently quantize partitions of  $\mathbf{z}$ , similar to product quantization (Jegou et al., 2011). This results in larger dictionaries and increased downstream performance (Appendix A).
+
+The dense feature vector  $\mathbf{z} \in \mathbb{R}^d$  is first organized into multiple groups  $G$  into the matrix form  $\mathbf{z}' \in \mathbb{R}^{G \times (d / G)}$ . We then represent each row by an integer index, and hence can represent the full feature vector by the indices  $\mathbf{i} \in [V]^G$ , where  $V$  again denotes the possible number of variables for this particular group and each element  $\mathbf{i}_j$  corresponds to a fixed codebook vector. For each of the  $G$  groups, we apply either one of the two VQ approaches ( $\S 3.1$  and  $\S 3.2$ ).
+
+The codebook itself can be initialized in two possible ways: Codebook variables can be shared across groups, i.e., a particular index in group  $j$  would reference the same vector as the same index in group  $j'$ . This yields a codebook  $\mathbf{e} \in \mathbb{R}^{V \times (d / G)}$ . In contrast, not sharing the codebook variables yields a codebook of size  $\mathbf{e} \in \mathbb{R}^{V \times G \times (d / G)}$ . In practise, we observe that sharing the codebook variables generally yields competitive results to a non-shared representation.
+
+# 4 BERT PRE-TRAINING ON QUANTIZED SPEECH
+
+Once we trained a vq-wav2vec model we can discretize audio data and make it applicable to algorithms that require discrete inputs. One possibility is to use the discretized training data and apply BERT pre-training where the task is to predict masked input tokens based on an encoding of the surrounding context (Devlin et al., 2018). Once the BERT model is trained, we can use it to build representations and feed them into an acoustic model to improve speech recognition. We follow recent advances in BERT training which only use the masked input token prediction (Liu et al., 2019).
+
+Since each of the discretized tokens represents around  $10\mathrm{ms}$  of audio it is likely too easy to predict a single masked input token. We therefore change BERT training by masking spans of consecutive discretized speech tokens, similar to Joshi et al. (2019). To mask the input sequence, we randomly sample  $p = 0.05$  of all tokens to be a starting index, without replacement, and mask  $M = 10$  consecutive tokens from every sampled index; spans may overlap. This makes the masked token prediction harder and we show later that it improves accuracy over masking individual tokens (§6.5).
+
+# 5 EXPERIMENTAL SETUP
+
+# 5.1 DATASETS
+
+We generally pre-train vq-wav2vec and BERT on the full 960h of Librispeech (Panayotov et al., 2015) and after vq-wav2vec training it is discretized to 345M tokens. Where indicated we perform ablations on a clean 100h subset which is discretized to 39.9M tokens. We evaluate models on two benchmarks: TIMIT (Garofolo et al., 1993b) is a 5h dataset with phoneme labels and Wall Street Journal (WSJ; Garofolo et al. 1993a) is a 81h dataset for speech recognition. For TIMIT, we apply the standard evaluation protocol and consider 39 different phonemes. For WSJ, we train acoustic models directly on 31 graphemes, including the English alphabet, the apostrophe, the silence token and tokens for repeating characters.
+
+# 5.2 VQ-WAV2VEC
+
+We adapt the faireq implementation of wav2vec (Schneider et al., 2019; Ott et al., 2019) and use vqvaw2vec/wav2vec models with  $34 \times 10^{6}$  parameters. The encoder has 8 layers with 512 channels each, kernel sizes (10,8,4,4,4,1,1,1) and strides (5,4,2,2,2,1,1,1), yielding a total stride of 160. Each layer contains a convolution, followed by dropout, group normalization with a single group (Wu & He, 2018) and a ReLU non-linearity. The aggregator is composed of 12 layers, with 512 channels, stride 1, and kernel sizes starting at 2 and increasing by 1 for every subsequent layer. The block
+
+structure is the same as for the encoder network, except we introduce skip connections between each subsequent block.
+
+We train with the wav2vec context prediction loss (Equation 1) for  $400\mathrm{k}$  updates, predicting  $K = 8$  steps into the future and sample 10 negatives from the same audio example. Training is warmed up for 500 steps where the learning rate is increased from  $1\times 10^{-7}$  to  $5\times 10^{-3}$ , and then annealed to  $1\times 10^{-6}$  using a cosine schedule (Loshchilov & Hutter, 2016). The batch size is 10, and we crop a random section of  $150\mathrm{k}$  frames for each example (approximately 9.3 seconds for  $16\mathrm{kHz}$  sampling rate). All models are trained on 8 GPUs.
+
+For ablations and experiments on the 100h Librispeech subset, we use a smaller model with kernels (10,8,4,4,4) and strides (5,4,2,2,2) in the encoder and seven convolutional layers with stride one and kernel size three in the aggregator. This model is trained for 40k updates.
+
+Gumbel-Softmax Models. We use  $G = 2$  groups and  $V = 320$  latents per group and the linear layer projects the features produced by the encoder into  $G \cdot V = 640$  logits. The Gumbel-Softmax produces a one-hot vector for each group  $G$ . The temperature  $\tau$  is linearly annealed from 2 to 0.5 over the first  $70\%$  of updates and then kept constant at 0.5. This enables the model to learn which latents work best for each input before committing to a single latent. After training this model on 960h of Librispeech and quantizing the training dataset, we are left with 13.5k unique codewords combinations (out of  $V^{G} = 102\mathrm{k}$  possible codewords).
+
+k-means Models. We use  $G = 2$  groups and  $V = 320$  variables per group. vq-wav2vec on full Librispeech yields 23k unique codewords. Following van den Oord et al. (2017), we found  $\gamma = 0.25$  to be a robust choice for balancing the VQ auxiliary loss.
+
+# 5.3 BERT
+
+BERT base models have 12 layers, model dimension 768, inner dimension (FFN) 3072 and 12 attention heads (Devlin et al., 2018). The learning rate is warmed up over the first 10,000 updates to a peak value of  $1 \times 10^{-5}$ , and then linearly decayed over a total of 250k updates. We train on 128 GPUs with a batch size of 3072 tokens per GPU giving a total batch size of 393k tokens (Ott et al., 2018). Each token represents 10ms of audio data.
+
+BERT small. For ablations we use a smaller setup with model dimension 512, FFN size 2048, 8 attention heads and dropout 0.05. Models are trained for  $250\mathrm{k}$  updates with a batch size of 2 examples per GPU.
+
+# 5.4 ACOUSTIC MODEL
+
+We use wav2letter as acoustic model (Collobert et al., 2016; 2019) and train for 1,000 epochs on 8 GPUs for both TIMIT and WSJ using the auto segmentation criterion. For decoding the emissions from the acoustic model on WSJ we use a lexicon as well as a separate language model trained on the WSJ language modeling data only. We consider a 4-gram KenLM language model (Heafield et al., 2013) and a character based convolutional language model (Likhomanenko et al., 2019) and tune the models with the same protocol as Schneider et al. (2019).
+
+# 6 RESULTS
+
+# 6.1 WSJ SPEECH RECOGNITION
+
+We first evaluate on the WSJ speech recognition benchmark. We train a vq-wav2vec model on the unlabeled version of Librispeech, then discretize the same data with the resulting model to estimate a BERT model. Finally, we train a wav2letter acoustic model on WSJ by inputting either the BERT or vq-wav2vec representations instead of log-mel filterbanks. $^{2}$
+
+We compare to various results from the literature, including wav2vec (Schneider et al., 2019) and we consider three setups: performance without any language model (No LM), with an n-gram LM
+
+<table><tr><td></td><td colspan="2">nov93dev</td><td colspan="2">nov92</td></tr><tr><td></td><td>LER</td><td>WER</td><td>LER</td><td>WER</td></tr><tr><td>Deep Speech 2 (12K h labeled speech; Amodei et al., 2016)</td><td>-</td><td>4.42</td><td>-</td><td>3.1</td></tr><tr><td>Trainable frontend (Zeghidour et al., 2018)</td><td>-</td><td>6.8</td><td>-</td><td>3.5</td></tr><tr><td>Lattice-free MMI (Hadian et al., 2018)</td><td>-</td><td>5.66†</td><td>-</td><td>2.8†</td></tr><tr><td>Supervised transfer-learning (Ghahremani et al., 2017)</td><td>-</td><td>4.99†</td><td>-</td><td>2.53†</td></tr><tr><td>No LM</td><td></td><td></td><td></td><td></td></tr><tr><td>Baseline (log-mel)</td><td>6.28</td><td>19.46</td><td>4.14</td><td>13.93</td></tr><tr><td>wav2vec (Schneider et al., 2019)</td><td>5.07</td><td>16.24</td><td>3.26</td><td>11.20</td></tr><tr><td>vq-wav2vec Gumbel</td><td>7.04</td><td>20.44</td><td>4.51</td><td>14.67</td></tr><tr><td>+ BERT base</td><td>4.13</td><td>13.40</td><td>2.62</td><td>9.39</td></tr><tr><td>4-GRAM LM (Heafield et al., 2013)</td><td></td><td></td><td></td><td></td></tr><tr><td>Baseline (log-mel)</td><td>3.32</td><td>8.57</td><td>2.19</td><td>5.64</td></tr><tr><td>wav2vec (Schneider et al., 2019)</td><td>2.73</td><td>6.96</td><td>1.57</td><td>4.32</td></tr><tr><td>vq-wav2vec Gumbel</td><td>3.93</td><td>9.55</td><td>2.40</td><td>6.10</td></tr><tr><td>+ BERT base</td><td>2.41</td><td>6.28</td><td>1.26</td><td>3.62</td></tr><tr><td>CHAR CONVLM (Likhomanenko et al., 2019)</td><td></td><td></td><td></td><td></td></tr><tr><td>Baseline (log-mel)</td><td>2.77</td><td>6.67</td><td>1.53</td><td>3.46</td></tr><tr><td>wav2vec (Schneider et al., 2019)</td><td>2.11</td><td>5.10</td><td>0.99</td><td>2.43</td></tr><tr><td>vq-wav2vec Gumbel + BERT base</td><td>1.79</td><td>4.46</td><td>0.93</td><td>2.34</td></tr></table>
+
+Table 1: WSJ accuracy of vq-wav2vec on the development (nov93dev) and test set (nov92) in terms of letter error rate (LER) and word error rate (WER) without language modeling (No LM), a 4-gram LM and a character convolutional LM. vq-wav2vec with BERT pre-training improves over the best wav2vec model (Schneider et al., 2019).  
+
+<table><tr><td></td><td colspan="2">nov93dev</td><td colspan="2">nov92</td></tr><tr><td></td><td>LER</td><td>WER</td><td>LER</td><td>WER</td></tr><tr><td>No LM</td><td></td><td></td><td></td><td></td></tr><tr><td>wav2vec (Schneider et al., 2019)</td><td>5.07</td><td>16.24</td><td>3.26</td><td>11.20</td></tr><tr><td>vq-wav2vec Gumbel</td><td>7.04</td><td>20.44</td><td>4.51</td><td>14.67</td></tr><tr><td>+ BERT small</td><td>4.52</td><td>14.14</td><td>2.81</td><td>9.69</td></tr><tr><td>vq-wav2vec k-means (39M codewords)</td><td>5.41</td><td>17.11</td><td>3.63</td><td>12.17</td></tr><tr><td>vq-wav2vec k-means</td><td>7.33</td><td>21.64</td><td>4.72</td><td>15.17</td></tr><tr><td>+ BERT small</td><td>4.31</td><td>13.87</td><td>2.70</td><td>9.62</td></tr><tr><td>4-GRAM LM (Heafield et al., 2013)</td><td></td><td></td><td></td><td></td></tr><tr><td>wav2vec (Schneider et al., 2019)</td><td>2.73</td><td>6.96</td><td>1.57</td><td>4.32</td></tr><tr><td>vq-wav2vec Gumbel</td><td>3.93</td><td>9.55</td><td>2.40</td><td>6.10</td></tr><tr><td>+ BERT small</td><td>2.67</td><td>6.67</td><td>1.46</td><td>4.09</td></tr><tr><td>vq-wav2vec k-means (39M codewords)</td><td>3.05</td><td>7.74</td><td>1.71</td><td>4.82</td></tr><tr><td>vq-wav2vec k-means</td><td>4.37</td><td>10.26</td><td>2.28</td><td>5.71</td></tr><tr><td>+ BERT small</td><td>2.60</td><td>6.62</td><td>1.45</td><td>4.08</td></tr></table>
+
+Table 2: Comparison of Gumbel-Softmax and k-means vector quantization on WSJ (cf. Table 1).
+
+(4-gram LM) and with a character convolutional LM (Char ConvLM). We report the accuracy of wav2letter with log-mel filterbanks as input (Baseline) and wav2vec. For vq-wav2vec we first experiment with the Gumbel-Softmax, with and without a BERT base model (\$5.3).
+
+Table 1 shows that vq-wav2vec together with BERT training can achieve a new state of the art of 2.34WER on nov92. Gains are largest when no language model is used which is the fastest setting. vq-wav2vec with Gumbel-Softmax uses only  $13.5\mathrm{k}$  distinct codewords to represent the audio signal and this limited set of codewords is not sufficient to outperform the baseline. However, it does enable training BERT models which require a relatively small vocabulary.
+
+<table><tr><td></td><td>dev PER</td><td>test PER</td></tr><tr><td>CNN + TD-filterbanks (Zeghidour et al., 2018)</td><td>15.6</td><td>18.0</td></tr><tr><td>Li-GRU + fMLLR (Ravanelli et al., 2018)</td><td>-</td><td>14.9</td></tr><tr><td>wav2vec (Schneider et al., 2019)</td><td>12.9</td><td>14.7</td></tr><tr><td>Baseline (log-mel)</td><td>16.9</td><td>17.6</td></tr><tr><td>vq-wav2vec, Gumbel</td><td>15.34</td><td>17.78</td></tr><tr><td>+ BERT small</td><td>9.64</td><td>11.64</td></tr><tr><td>vq-wav2vec, k-means</td><td>15.65</td><td>18.73</td></tr><tr><td>+ BERT small</td><td>9.80</td><td>11.40</td></tr></table>
+
+Table 3: TIMIT phoneme recognition in terms of phoneme error rate (PER). All our models use the CNN-8L-PReLU-do0.7 architecture (Zeghidour et al., 2018).  
+
+<table><tr><td></td><td>dev clean</td><td>dev other</td><td>test clean</td><td>test other</td></tr><tr><td>Mohamed et al. (2019)</td><td>4.8</td><td>12.7</td><td>4.7</td><td>12.9</td></tr><tr><td>Irie et al. (2019)</td><td>4.4</td><td>13.2</td><td>4.7</td><td>13.4</td></tr><tr><td>Park et al. (2019)</td><td>2.8</td><td>6.8</td><td>2.5</td><td>5.8</td></tr><tr><td>vq-wav2vec Gumbel + Transformer Big</td><td>5.6</td><td>15.5</td><td>6.2</td><td>18.2</td></tr></table>
+
+Table 4: Librispeech results for a standard sequence to sequence model trained on discretized audio without BERT pre-training and results from the literature. All results are without a language model.
+
+Next, we compare Gumbel-Softmax to k-means for vector quantization. For this experiment we use the faster to train BERT small configuration (§5.3). We also train a vq-wav2vec k-means model with a very large number of codewords (39.9M) to test whether a more expressive model can close the gap to wav2vec. Table 2 shows that Gumbel-Softmax and k-means clustering perform relatively comparably: in the no language model setup without BERT, Gumbel-Softmax is more accurate than k-means but these differences disappear with BERT. For 4-gram LM setup, k-means is better but those differences disappear again after BERT training. Finally, the large codeword model can substantially reduce the gap to the original wav2vec model.
+
+# 6.2 TIMIT PHONEME RECOGNITION
+
+Next, we experiment on the much smaller TIMIT phoneme recognition task where we also pre-train vq-wav2vec on the full Librispeech corpus. Table 3 shows that vq-wav2vec and BERT achieve a new state of the art of 11.64 PER which corresponds to a  $21\%$  reduction in error over the previous best result of wav2vec.
+
+# 6.3 SEQUENCE TO SEQUENCE MODELING
+
+So far we used vq-wav2vec to train BERT on discretized speech. However, once the audio is discretized we can also train a standard sequence to sequence model to perform speech recognition. In preliminary experiments, we trained an off-the-shelf Big Transformer (Vaswani et al., 2017; Ott et al., 2019) on the vq-wav2vec Gumbel-Softmax discretized Librispeech corpus and evaluated on the Librispeech dev/test sets; we use a 4k BPE output vocabulary (Sennrich et al., 2016). Table 4 shows that results are promising, even though they are not as good as the state of the art (Park et al., 2019) which depends on data augmentation that we do not use.
+
+# 6.4 ACCURACY VS. BITRATE
+
+Next, we investigate how well vq-wav2vec can compress the audio data. Specifically, we train models with different numbers of groups  $G$  and variables  $V$  to vary the size of the possible codebook size  $V^{G}$  and measure accuracy on TIMIT phoneme recognition without BERT training.
+
+![](images/a0f4acd966bbf51bab82227b844d11972ee187a827ee9d8284a0b592ae5305db.jpg)  
+Figure 3: Comparison of PER on the TIMIT dev set for various audio codecs and vq-wav2vec k-means trained on Librispeech 100h.
+
+We measure compression with the bitrate  $r \cdot G\log_2V$  at sampling rate  $r = 100\mathrm{Hz}$  and report the trade-off between bitrate and accuracy on our phoneme recognition task. We experiment with vq-wav2vec k-means and train models with 1,2,4,8,16 and 32 groups, using 40,80,160,...,1280 variables, spanning a bitrate range from 0.53 kbit/s ( $G = 1$ ,  $V = 40$ ) to 33.03 kbit/s ( $G = 32$ ,  $V = 1280$ ). We place the quantization module after the aggregator module and train all models in the small vq-wav2vec setup (§5.2) on the 100h clean Librispeech subset.
+
+As baselines, we consider various lossy compression algorithms applied to the TIMIT audio data and train wav2letter models on the resulting audio:  $\text{Codec2}^3$  as a low bitrate codec, Opus (Terriberry & Vos, 2012) as a medium bitrate codec and MP3 and Ogg Vorbis (Montgomery, 2004) as high bitrate codec. We use the whole spectrum of both variable and constant bitrate settings of the codec; we encode and decode with ffmpeg (ffmpeg developers, 2016). Figure 3 shows the trade-off between the bitrate and TIMIT accuracy. Acoustic models on vq-wav2vec achieve the best results across most bitrate settings.
+
+# 6.5 ABLATIONS
+
+Table 5a shows that masking entire spans of tokens performs significantly better than individual tokens  $(M = 1)$ . Furthermore, BERT training on discretized audio data is fairly robust to masking large parts of the input (Table 5b).
+
+<table><tr><td>M</td><td>dev</td><td>test</td></tr><tr><td>1</td><td>14.94</td><td>17.38</td></tr><tr><td>5</td><td>13.62</td><td>15.78</td></tr><tr><td>10</td><td>12.65</td><td>15.28</td></tr><tr><td>20</td><td>13.04</td><td>15.56</td></tr><tr><td>30</td><td>13.18</td><td>15.64</td></tr></table>
+
+(a) Mask length.  
+
+<table><tr><td>p</td><td>dev</td><td>test</td></tr><tr><td>0.015</td><td>12.65</td><td>15.28</td></tr><tr><td>0.020</td><td>12.51</td><td>14.43</td></tr><tr><td>0.025</td><td>12.16</td><td>13.96</td></tr><tr><td>0.030</td><td>11.68</td><td>14.48</td></tr><tr><td>0.050</td><td>11.45</td><td>13.62</td></tr></table>
+
+(b) Mask probabilities.
+
+Table 5: TIMIT PER for (a) different mask sizes  $M$  with  $pM = 0.15$  in BERT training and (b) mask probabilities  $p$  for a fixed mask length  $M = 10$ .
+
+# 7 CONCLUSION
+
+vq-wav2vec is a self-supervised algorithm that quantizes unlabeled audio data which makes it amenable to algorithms requiring discrete data. This approach improves the state of the art on the WSJ and TIMIT benchmarks by leveraging BERT pre-training. In future work, we plan to apply
+
+other algorithms requiring discrete inputs to audio data and to explore self-supervised pre-training algorithms which mask part of the continuous audio input. Another future work avenue is to fine-tune the pre-trained model to output transcriptions instead of feeding the pre-trained features to a custom ASR model.
+
+# REFERENCES
+
+Dario Amodei, Sundaram Ananthanarayanan, Rishita Anubhai, Jingliang Bai, Eric Battenberg, Carl Case, Jared Casper, Bryan Catanzaro, Qiang Cheng, Guoliang Chen, et al. Deep speech 2: End-to-end speech recognition in english and mandarin. In Proc. of ICML, 2016.  
+Mathilde Caron, Piotr Bojanowski, Julien Mairal, and Armand Joulin. Unsupervised pre-training of image features on non-curated data. In Proceedings of the International Conference on Computer Vision (ICCV), 2019.  
+Jan Chorowski, Ron J. Weiss, Samy Bengio, and Aïron van den Oord. Unsupervised speech representation learning using wavenet autoencoders. arXiv, abs/1901.08810, 2019.  
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+Yu-An Chung, Wei-Ning Hsu, Hao Tang, and James Glass. An unsupervised autoregressive model for speech representation learning. arXiv, abs/1904.03240, 2019.  
+Ronan Collobert, Christian Puhrsch, and Gabriel Synnaeve. Wav2letter: an end-to-end convnet-based speech recognition system. arXiv, abs/1609.03193, 2016.  
+Ronan Collobert, Awni Hannun, and Gabriel Synnaeve. A fully differentiable beam search decoder. arXiv, abs/1902.06022, 2019.  
+FFmpeg Developers. ffmpeg tool software, 2016. URL http://ffmpeg.org/.  
+Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. Bert: Pre-training of deep bidirectional transformers for language understanding. arXiv, abs/1810.04805, 2018.  
+Ewan Dunbar, Robin Algayres, Julien Karadayi, Mathieu Bernard, Juan Benjumea, Xuan-Nga Cao, Lucie Miskic, Charlotte Dugrain, Lucas Ondel, Alan W Black, et al. The zero resource speech challenge 2019: Tts without t. arXiv, 1904.11469, 2019.  
+Ryan Eloff, André Nortje, Benjamin van Niekerk, Avashna Govender, Leanne Nortje, Arnu Pretorius, Elan Van Biljon, Ewald van der Westhuizen, Lisa van Staden, and Herman Kamper. Unsupervised acoustic unit discovery for speech synthesis using discrete latent-variable neural networks. arXiv, abs/1904.07556, 2019.  
+John S. Garofolo, David Graff, Doug Paul, and David S. Pallett. CSR-I (WSJ0) Complete LDC93S6A. Web Download. Linguistic Data Consortium, 1993a.  
+John S. Garofolo, Lori F. Lamel, William M. Fisher, Jonathon G. Fiscus, David S. Pallett, and Nancy L. Dahlgren. The DARPA TIMIT Acoustic-Phonetic Continuous Speech Corpus CDROM. Linguistic Data Consortium, 1993b.  
+Pegah Ghahremani, Vimal Manohar, Hossein Hadian, Daniel Povey, and Sanjeev Khudanpur. Investigation of transfer learning for asr using lf-mmi trained neural networks. In Proc. of ASRU, 2017.  
+Emil Julius Gumbel. Statistical theory of extreme values and some practical applications: a series of lectures, volume 33. US Government Printing Office, 1954.  
+Hossein Hadian, Hossein Sameti1, Daniel Povey, and Sanjeev Khudanpur. End-to-end speech recognition using lattice-free mmi. In Proc. of Interspeech, 2018.  
+Kenneth Heafield, Ivan Pouzyrevsky, Jonathan H. Clark, and Philipp Koehn. Scalable modified Kneser-Ney language model estimation. In Proc. of ACL, 2013.
+
+Kazuki Irie, Rohit Prabhavalkar, Anjuli Kannan, Antoine Bruguier, David Rybach, and Patrick Nguyen. On the choice of modeling unit for sequence-to-sequence speech recognition. *Interspeech* 2019, Sep 2019. doi: 10.21437/interspeech.2019-2277. URL http://dx.doi.org/10.21437/Interspeech.2019-2277.  
+Eric Jang, Shixiang Gu, and Ben Poole. Categorical reparameterization with gumbel-softmax. arXiv, abs/1611.01144, 2016.  
+Herve Jegou, Matthijs Douze, and Cordelia Schmid. Product quantization for nearest neighbor search. IEEE Trans. Pattern Anal. Mach. Intell., 33(1):117-128, January 2011.  
+Mandar Joshi, Danqi Chen, Yinhan Liu, Daniel S. Weld, Luke Zettlemoyer, and Omer Levy. Spanbert: Improving pre-training by representing and predicting spans. arXiv, abs/1907.10529, 2019.  
+Tatiana Likhomanenko, Gabriel Synnaeve, and Ronan Collobert. Who needs words? lexicon-free speech recognition. In Proc. of Interspeech, 2019.  
+Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Mandar Joshi, Danqi Chen, Omer Levy, Mike Lewis, Luke Zettlemoyer, and Veselin Stoyanov. Roberta: A robustly optimized bert pretraining approach. arXiv preprint arXiv:1907.11692, 2019.  
+Ilya Loshchilov and Frank Hutter. SGDR: stochastic gradient descent with restarts. arXiv, abs/1608.03983, 2016.  
+Chris J Maddison, Daniel Tarlow, and Tom Minka. A* sampling. In Advances in Neural Information Processing Systems, pp. 3086-3094, 2014.  
+Tomas Mikolov, Ilya Sutskever, Kai Chen, Greg S Corrado, and Jeff Dean. Distributed representations of words and phrases and their compositionality. In Proc. of NIPS, 2013.  
+Abdelrahman Mohamed, Dmytro Okhonko, and Luke Zettlemoyer. Transformers with convolutional context for ASR. CoRR, abs/1904.11660, 2019.  
+C Montgomery. Vorbis i specification, 2004.  
+Myle Ott, Sergey Edunov, David Grangier, and Michael Auli. Scaling neural machine translation. In Proc. of WMT, 2018.  
+Myle Ott, Sergey Edunov, Alexei Baevski, Angela Fan, Sam Gross, Nathan Ng, David Grangier, and Michael Auli. *fairseq: A fast, extensible toolkit for sequence modeling.* In *Proc. of NAACL System Demonstrations*, 2019.  
+Vassil Panayotov, Guoguo Chen, Daniel Povey, and Sanjeev Khudanpur. Librispeech: an asr corpus based on public domain audio books. In Proc. of ICASSP, pp. 5206-5210. IEEE, 2015.  
+Daniel S. Park, William Chan, Yu Zhang, Chung-Cheng Chiu, Barret Zoph, Ekin D. Cubuk, and Quoc V. Le. Specaugment: A simple data augmentation method for automatic speech recognition, 2019.  
+Mirco Ravanelli, Philemon Brakel, Maurizio Omologo, and Yoshua Bengio. Light gated recurrent units for speech recognition. IEEE Transactions on Emerging Topics in Computational Intelligence, 2(2):92-102, 2018.  
+Steffen Schneider, Alexei Baevski, Ronan Collobert, and Michael Auli. wav2vec: Unsupervised pre-training for speech recognition. CoRR, abs/1904.05862, 2019. URL http://arxiv.org/abs/1904.05862.  
+Rico Senrrich, Barry Haddow, and Alexandra Birch. Neural machine translation of rare words with subword units. In Proc. of ACL, 2016.  
+Tim Terriberry and Koen Vos. Definition of the opus audio codec, 2012.  
+Andros Tjandra, Berrak Sisman, Mingyang Zhang, Sakriani Sakti, Haizhou Li, and Satoshi Nakamura. Vqae unsupervised unit discovery and multi-scale code2spec inverter for zerospeech challenge 2019. arXiv, 1905.11449, 2019.
+
+Aaron van den Oord, Oriol Vinyals, et al. Neural discrete representation learning. In Advances in Neural Information Processing Systems, pp. 6306-6315, 2017.  
+Aäron van den Oord, Yazhe Li, and Oriol Vinyals. Representation learning with contrastive predictive coding. arXiv, abs/1807.03748, 2018.  
+Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Proc. of NIPS, 2017.  
+Maarten Versteegh, Xavier Anguera, Aren Jansen, and Emmanuel Dupoux. The zero resource speech challenge 2015: Proposed approaches and results. Procedia Computer Science, 81:67-72, 2016.  
+Yuxin Wu and Kaiming He. Group normalization. arXiv, abs/1803.08494, 2018.  
+Neil Zeghidour, Nicolas Usunier, Iasonas Kokkinos, Thomas Schaiz, Gabriel Synnaeve, and Emmanuel Dupoux. Learning filterbanks from raw speech for phone recognition. In Proc. of (ICASSP), 2018.
+
+# APPENDIX A NUMBER OF VARIABLES VS. GROUPS
+
+We investigate the relationship between number of variables  $V$  and groups  $G$ . Table 6 shows that multiple groups are beneficial compared to a single group with a large number of variables. Table 7 shows that with a single group and many variables, only a small number of codewords survive.
+
+<table><tr><td>V</td><td>1 group</td><td>2 groups</td><td>4 groups</td><td>8 groups</td><td>16 groups</td><td>32 groups</td></tr><tr><td>40</td><td>33.44 ± 0.24</td><td>23.52 ± 0.53</td><td>18.76 ± 0.20</td><td>17.43 ± 0.14</td><td>15.97 ± 0.21</td><td>15.44 ± 0.32</td></tr><tr><td>80</td><td>29.14 ± 0.70</td><td>25.36 ± 4.62</td><td>17.32 ± 0.28</td><td>16.36 ± 0.27</td><td>17.55 ± 0.27</td><td>15.49 ± 0.14</td></tr><tr><td>160</td><td></td><td>24.27 ± 0.35</td><td>17.55 ± 0.03</td><td>16.36 ± 0.13</td><td>15.64 ± 0.03</td><td>15.11 ± 0.10</td></tr><tr><td>320</td><td>27.22 ± 0.25</td><td>20.86 ± 0.09</td><td>16.49 ± 0.07</td><td>15.88 ± 0.10</td><td>15.74 ± 0.18</td><td>15.18 ± 0.02</td></tr><tr><td>640</td><td>26.53 ± 2.02</td><td>18.64 ± 0.12</td><td>16.60 ± 0.22</td><td>15.62 ± 0.16</td><td>15.45 ± 0.13</td><td>15.54 ± 0.31</td></tr><tr><td>1280</td><td>32.63 ± 5.73</td><td>18.04 ± 0.26</td><td>16.37 ± 0.07</td><td>15.85 ± 0.05</td><td>15.13 ± 0.29</td><td>15.18 ± 0.05</td></tr></table>
+
+Table 6: PER on TIMIT dev set for vq-wav2vec models trained on Libri100. Results are based on three random seeds.  
+
+<table><tr><td>V</td><td>1 group</td><td>2 groups</td><td>4 groups</td><td>8 groups</td><td>16 groups</td><td>32 groups</td></tr><tr><td>40</td><td>100 % (40)</td><td>95.3 % (1.6k)</td><td>27.4 % (2.56M)</td><td>74.8 % (39.9M)</td><td>99.6 % (39.9M)</td><td>99.9 % (39.9M)</td></tr><tr><td>80</td><td>92.5 % (80)</td><td>78.5 % (6.4k)</td><td>11.8 % (39.9M)</td><td>91.5 % (39.9M)</td><td>99.3 % (39.9M)</td><td>100 % (39.9M)</td></tr><tr><td>160</td><td>95 % (160)</td><td>57.2 % (25.6k)</td><td>35.2 % (39.9M)</td><td>97.6 % (39.9M)</td><td>99.8 % (39.9M)</td><td>100 % (39.9M)</td></tr><tr><td>320</td><td>33.8 % (320)</td><td>24.6 % (102.4k)</td><td>57.3 % (39.9M)</td><td>98.7 % (39.9M)</td><td>99.9 % (39.9M)</td><td>100 % (39.9M)</td></tr><tr><td>640</td><td>24.6 % (640)</td><td>10 % (409.6k)</td><td>60.2 % (39.9M)</td><td>99.3 % (39.9M)</td><td>99.9 % (39.9M)</td><td>100 % (39.9M)</td></tr><tr><td>1280</td><td>7.2 % (1.28k)</td><td>4.9 % (1.63M)</td><td>67.9 % (39.9M)</td><td>99.5 % (39.9M)</td><td>99.9 % (39.9M)</td><td>100 % (39.9M)</td></tr></table>
+
+Table 7: Fraction of used codewords vs. number of theoretically possible codewords  $V^{G}$  in brackets; 39.9M is the number of tokens in Librispeech 100h.
\ No newline at end of file
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+# WATCH THE UNOBSERVED: A SIMPLE APPROACH TO PARALLELIZING MONTE CARLO TREE SEARCH
+
+Anji Liu†, Jianshu Chen*, Mingze Yu†, Yu Zhai†, Xuewen Zhou† & Ji Liu†
+
+†Seattle AI Lab, Kwai Inc., Bellevue, WA 98004, USA
+
+{liuanji03,yumingze,zhaiyu,zhouxuewen,jiliu}@kuaishou.com
+
+*Tencent AI Lab, Bellevue, WA 98004, USA
+
+jianshuchen@tencent.com
+
+# ABSTRACT
+
+Monte Carlo Tree Search (MCTS) algorithms have achieved great success on many challenging benchmarks (e.g., Computer Go). However, they generally require a large number of rollouts, making their applications costly. Furthermore, it is also extremely challenging to parallelize MCTS due to its inherent sequential nature: each rollout heavily relies on the statistics (e.g., node visitation counts) estimated from previous simulations to achieve an effective exploration-exploitation tradeoff. In spite of these difficulties, we develop an algorithm,  $WU-UCT^1$ , to effectively parallelize MCTS, which achieves linear speedup and exhibits only limited performance loss with an increasing number of workers. The key idea in WU-UCT is a set of statistics that we introduce to track the number of on-going yet incomplete simulation queries (named as unobserved samples). These statistics are used to modify the UCT tree policy in the selection steps in a principled manner to retain effective exploration-exploitation tradeoff when we parallelize the most time-consuming expansion and simulation steps. Experiments on a proprietary benchmark and the Atari Game benchmark demonstrate the linear speedup and the superior performance of WU-UCT comparing to existing techniques.
+
+# 1 INTRODUCTION
+
+Recently, Monte Carlo Tree Search (MCTS) algorithms such as UCT (Kocsis et al., 2006) have achieved great success in solving many challenging artificial intelligence (AI) benchmarks, including video games (Guo et al., 2016) and Go (Silver et al., 2016). However, they rely on a large number (e.g. millions) of interactions with the environment emulator to construct search trees for decision-making, which leads to high time complexity (Browne et al., 2012). For this reason, there has been an increasing demand for parallelizing MCTS over multiple workers. However, parallelizing MCTS without degrading its performance is difficult (Segal, 2010; Mirsoleimani et al., 2018a; Chaslot et al., 2008), mainly due to the fact that each MCTS iteration requires information from all previous iterations to provide effective exploration-exploitation tradeoff. Specifically, parallelizing MCTS would inevitably obscure these crucial information, and we will show in Section 2.2 that this loss of information potentially results in a significant performance drop. The key question is therefore how to acquire and utilize more available information to mitigate the information loss caused by parallelization and help the algorithm to achieve better exploration-exploitation tradeoff.
+
+To this end, we propose WU-UCT (Watch the Unovserved in UCT), a novel parallel MCTS algorithm that attains linear speedup with only limited performance loss. This is achieved by a conceptual innovation (Section 3.1) as well as an efficient real system implementation (Section 3.2). Specifically, the key idea in WU-UCT to overcome the aforementioned challenge is a set of statistics that tracks the number of on-going yet incomplete simulation queries (named as unobserved samples). We combine these newly devised statistics with the original statistics of observed samples to modify UCT's policy in the selection steps in a principled manner, which, as we shall show in Section 4, effectively retains exploration-exploitation tradeoff during parallelization. Our proposed
+
+approach has been successfully deployed in a production system for efficiently and accurately estimating the rate at which users pass levels (termed user pass-rate) in a mobile game "Joy City", with the purpose of reducing their design cycles. On this benchmark, we show that WU-UCT achieves near-optimal linear speedup and superior performance in predicting user pass-rate (Section 5.1). We further evaluate WU-UCT on the Atari Game benchmark and compare it to state-of-the-art parallel MCTS algorithms (Section 5.2), which also demonstrate our superior speedup and performance.
+
+# 2 ON THE DIFFICULTIES OF PARALLELIZING MCTS
+
+We first introduce the MCTS and the UCT algorithms, along with their difficulties in parallelization.
+
+# 2.1 MONTE CARLO TREE SEARCH AND UPPER CONFIDENCE BOUND FOR TREES (UCT)
+
+We consider the Markov Decision Process (MDP)  $\langle S, \mathcal{A}, R, P, \gamma \rangle$ , where an agent interacts with the environment in order to maximize a long-term cumulative reward. Specifically, an agent at state  $s_t \in S$  takes an action  $a_t \in \mathcal{A}$  according to a policy  $\pi$ , so that the MDP transits to the next state  $s_{t+1} \sim P(s_{t+1}|s_t, a_t)$  and emits a reward  $R(s_t, a_t)$ . The objective of the agent is to learn an optimal policy  $\pi^*$  such that the long-term cumulative reward is maximized:
+
+$$
+\max  _ {\pi} \mathbb {E} _ {a _ {t} \sim \pi , s _ {t + 1} \sim P} \left[ \sum_ {t = 0} ^ {\infty} \gamma^ {t} R (s _ {t}, a _ {t}) \mid s _ {0} = s \right], \tag {1}
+$$
+
+where  $s \in S$  denotes the initial state and  $\gamma \in (0,1]$  is the discount factor. Many reinforcement learning (RL) algorithms have been developed to solve the above problem (Sutton & Barto, 2018), including model-free algorithms (Mnih et al., 2013; 2016; Williams, 1992; Konda & Tsitsiklis, 2000; Schulman et al., 2015; 2017) and model-based algorithms (Nagabandi et al., 2018; Weber et al., 2017; Bertsekas, 2005; Deisenroth & Rasmussen, 2011). Monte Carlo Tree Search (MCTS) is a model-based RL algorithm that plans the best action at each time step (Browne et al., 2012). Specifically, it uses the MDP model (or its sampler) to identify the best action at each time step by constructing a search tree (Figure 1(a)), where each node  $s$  represents a visited state, each edge from  $s$  denotes an action  $a_{s}$  that can be taken at that state, and the landing node  $s'$  denotes the state it transits to after taking  $a_{s}$ . As shown in Figure 1(a), MCTS repeatedly performs four sequential steps: selection, expansion, simulation and backpropagation. The selection step traverses over the existing search tree until the leaf node (or other termination conditions are satisfied) by choosing actions (edges)  $a_{s}$  at each node  $s$  according to a tree policy. One widely used node-selection policy is the one used in the Upper Confidence bound for Trees (UCT) (Kocsis et al., 2006):
+
+$$
+a _ {s} = \underset {s ^ {\prime} \in \mathcal {C} (s)} {\arg \max } \left\{V _ {s ^ {\prime}} + \beta \sqrt {\frac {2 \log N _ {s}}{N _ {s ^ {\prime}}}} \right\}, \tag {2}
+$$
+
+where  $\mathcal{C}(s)$  denotes the set of all child nodes for  $s$ ; the first term  $V_{s'}$  is an estimate for the long-term cumulative reward that can be received when starting from the state represented by node  $s'$ , and the second term represents the uncertainty (size of the confidence interval) of that estimate. The confidence interval is calculated based on the Upper Confidence Bound (UCB) (Auer et al., 2002; Auer, 2002) using  $N_s$  and  $N_{s'}$ , which denote the number of times that the nodes  $s$  and  $s'$  have been visited, respectively. Therefore, the key idea of the UCT policy (2) is to select the best action according to an optimistic estimation (i.e., the upper confidence bound) of the expected return, which strikes a balance between the exploitation (first term) and the exploration (second term) with  $\beta$  controlling their tradeoff. Once the selection process reaches a leaf node of the search tree (or other termination conditions are met), we will expand the node according to a prior policy by adding a new child node. Then, in the simulation step, we estimate its value function (cumulative reward)  $\hat{V}_s$  by running the environment simulator with a default (simulation) policy. Finally, during backpropagation, we update the statistics  $V_s$  and  $N_s$  from the leaf node  $s_T$  to the root node  $s_0$  of the selected path by recursively performing the following update (i.e., from  $t = T - 1$  to  $t = 0$ ):
+
+$$
+N _ {s _ {t}} \leftarrow N _ {s _ {t}} + 1, \quad \hat {V} _ {s _ {t}} \leftarrow R (s _ {t}, a _ {t}) + \gamma \hat {V} _ {s _ {t + 1}}, \quad V _ {s _ {t}} \leftarrow \big ((N _ {s _ {t}} - 1) V _ {s _ {t}} + \hat {V} _ {s _ {t}} \big) / N _ {s _ {t}}, \qquad (3)
+$$
+
+![](images/887da2720e68243c5851e7fdf647d15f8f5e71aa7a7c399fc8da4cb583d8e9a6.jpg)  
+(a) Each (non-parallel) MCTS rollout consists of four sequential steps: selection, expansion, simulation and backpropagation, where the expansion and the simulation steps are generally most time-consuming.
+
+![](images/75a40adcfdeedeb234bd7e09880777fb855c246dade9442c0a26c47e9cd6d6fe.jpg)  
+(b) Ideal parallelization
+
+![](images/bd7b907397adf5fc1905016ecc57392caed7af5cb80c75078d37e8a4ea1758e8.jpg)  
+(c) Naive parallelization  
+Figure 1: MCTS and its parallelization. (a) An overview of MCTS. (b) The ideal parallelization: the most up-to-date statistics  $\{V_s, N_s\}$  (in chromatic color) are assumed to be available to all workers as soon as a simulation begins (unrealistic in practice). (c) The key challenge in parallelizing MCTS: the workers can only access outdated  $\{V_s, N_s\}$  (in gray-color), leading problems like collapse of exploration. (d) WU-UCT tracks the number of incomplete simulation queries, which is denoted as  $O_s$ , and modifies the UCT policy in a principled manner to retain effective exploration-exploitation tradeoff. It achieves comparable speedup and performance as the ideal parallelization.
+
+![](images/0974d28ff9a98cb300b3f9b32baf8a09b6b94e27e75a4e2a090b0fca810a3654.jpg)  
+(d) WU-UCT
+
+Worker A
+
+$\S$  Simulation in
+
+- Worker B
+
+↑ After Backpropagate
+
+- Worker C
+
+Selection progress
+
+Colored statistics  $(V,N)$  has been updated by the corresponding worker, while gray ones  $(V,N)$  are not updated.
+
+where  $\hat{V}_{s_T}$  is the simulation return of  $s_T$ ;  $a_t$  denotes the action selected following (2) at state  $s_t$ .
+
+# 2.2 THE INTRINSIC DIFFICULTIES OF PARALLELIZING MCTS
+
+The above discussion implies that the MCTS algorithm is intrinsically sequential: each selection step in a new rollout requires the previous rollouts to complete in order to deliver the updated statistics,  $V_{s}$  and  $N_{s}$ , for the UCT tree policy (2). Although this requirement of up-to-date statistics is not mandatory for implementation, it is in practice intensively required to achieve effective exploration-exploitation tradeoff (Auer et al., 2002). Specifically, up-to-date statistics best help the UCT tree policy to identify and prune non-rewarding branches as well as extensively visiting rewarding paths for additional planning depth. Likewise, to achieve the best possible performance, when multiple workers are used, it is also important to ensure that each worker uses the most recent statistics (the colored  $V_{s}$  and  $N_{s}$  in Figure 1(b)) in its own selection step. However, this is impossible in parallelizing MCTS based on the following observations. First, the expansion step and the simulation step are generally more time-consuming compared to the other two steps, because they involve a large number of interactions with the environment (or its simulator). Therefore, as exemplified by Figure 1(c), when a worker C initiates a new selection step, the other workers A and B are most likely still in their simulation or expansion steps. This prevents them from updating the (global) statistics for other workers like C, which happens at their respective backpropagation steps. Using outdated statistics (the gray-colored  $V_{s}$  and  $N_{s}$ ) at different workers could lead to a significant performance loss given a fixed target speedup, due to behaviors like collapses of exploration or exploitation failure, which we shall discuss thoroughly in Section 4. To give an example, Figure 1(c) illustrates the collapse of exploration, where worker C traverses over the same path as the worker A in its selection step due to the determinism of (2). Specifically, if the statistics are unchanged between the moments that worker A and C begin their own selection steps, they will choose the same node according to (2), which greatly reduces the diversity of exploration. Therefore, the key question that we want to address in parallelizing MCTS is how to track the correct statistics and modify the UCT policy in a principled manner, with the hope of retaining effective exploration-exploitation tradeoff at different workers.
+
+# 3 WU-UCT
+
+In this section, we first develop the conceptual idea of our WU-UCT algorithm (Section 3.1), and then we present a real system implementation using a master-worker architecture (Section 3.2).
+
+# 3.1 WATCH THE UNOBSERVED SAMPLES IN UCT TREE POLICY
+
+As we pointed out earlier, the key question we want to address in parallelizing MCTS is how to deliver the most up-to-date statistics  $\{V_s, N_s\}$  to each worker so that they can achieve effective exploration-exploitation tradeoff in its selection step. This is assumed to be the case in the ideal parallelization in Figure 1(b). Algorithmically, it is equivalent to the sequential MCTS except that the rollouts are performed in parallel by different workers. Unfortunately, in practice, the statistics  $\{V_s, N_s\}$  available to each worker are generally outdated because of the slow and incomplete simulation and expansion steps at the other workers. Specifically, since the estimated value  $\hat{V}_s$  is unobservable before simulations complete and workers should not wait for the updated statistics to proceed, the (partial) loss of statistics  $\{V_s, N_s\}$  is unavoidable. Now the question becomes: is there an alternative way to addressing the issue? The answer is in the affirmative and is explained below.
+
+Aiming at bridging the gap between naive parallelization and the ideal case, we closely examine their difference in terms of the availability of statistics. As illustrated by the colors of the statistics, their only difference in  $\{V_s, N_s\}$  is caused by the on-going simulation process. As suggested by (3), although  $V_s$  can only be updated after a simulation step is completed, the newest  $N_s$  information can actually be available as early as a worker initiates a new rollout. This is the key insight that we leverage to enable effective parallelization in our WU-UCT algorithm. Motivated by this, we introduce another quantity,  $O_s$ , to count the number of rollouts that have been initiated but not yet completed, which we name as unobserved samples. That is, our new statistics,  $O_s$ , watch the number of unobserved samples, and are then used to correct the UCT tree policy (2) into the following form:
+
+$$
+a _ {s} = \underset {s ^ {\prime} \in \mathcal {C} (s)} {\arg \max } \left\{V _ {s ^ {\prime}} + \beta \sqrt {\frac {2 \log \left(N _ {s} + O _ {s}\right)}{N _ {s ^ {\prime}} + O _ {s ^ {\prime}}}} \right\}. \tag {4}
+$$
+
+The intuition of the above modified node-selection policy is that when there are  $O_{s}$  workers simulating (querying) node  $s$ , the confidence interval at node  $s$  will eventually be shrunk after they complete. Therefore, adding  $O_{s}$  and  $O_{s'}$  to the exploration term considers such a fact beforehand and let other workers be aware of it. Despite its simple form, (4) provides a principled way to retain effective exploration-exploitation tradeoff under parallel settings; it corrects the confidence bound towards better exploration-exploitation tradeoff. As the confidence level is instantly updated (i.e., at the beginning of simulation), more recent workers are guaranteed to observe additional statistics, which prevent them from extensively querying the same node as well as find better nodes for them to query. For example, when multiple children are in demand for exploration, (4) allows them to be explored evenly. In contrast, when a node has been sufficiently visited (i.e., large  $N_{s}$  and  $N_{s'}$ ), adding  $O_{s}$  and  $O_{s'}$  from the unobserved samples have little effect on (4) because the confidence interval is sufficiently shrunk around  $V_{s'}$ , allowing extensively exploitation of the best-valued child.
+
+# 3.2 SYSTEM IMPLEMENTATION USING MASTER-WORKER ARCHITECTURES
+
+We now proceed to explain the system implementation of WU-UCT, where the overall architecture is shown in Figure 2(a) (see Appendix A for the details). Specifically, we use a master-worker architecture to implement the WU-UCT algorithm with the following considerations. First, since the expansion and the simulation steps are much more time-consuming compared to the selection and the backpropagation steps, they should be intensively parallelized. In fact, they are relatively easy to parallelize (e.g., different simulations could be performed independently). Second, as we discussed earlier, different workers need to access the most up-to-date statistics  $\{V_s, N_s, O_s\}$  in order to achieve successful exploration-exploitation tradeoff. To this end, a centralized architecture for the selection and backpropagation step is more preferable as it allows adding strict restrictions to the retrieval and update of the statistics, making them up-to-date. Specifically, we use a centralized master process to maintain a global set of statistics (in addition to other data such as game states), and let it be in charge of the backpropagation step (i.e., updating the global statistics) and the selection step (i.e., exploiting the global statistics). As shown in Figure 2(a), the master process
+
+![](images/37b0a71d7ac1107c88115c48b5c6bcc8a620127bf0966b32a5ad226a4764b925.jpg)  
+(a) The system architecture that implements WU-UCT.
+
+![](images/3ed3f11c909685a0198ec6bc59237ccec247f27061ce02b432b60c97f5b3f749.jpg)  
+(b) Time consumption in the Tap game.
+
+![](images/5f5a894b6b6743b2b8f07116ff4b2f46274a54c8b0aa88a2cad5505ed0acbc17.jpg)  
+(c) Time consumption in the Atari games.  
+Figure 2: Diagram of WU-UCT's system architecture and its time consumption. (a) the Green blocks and the task buffers are operated at the master, while the blue blocks are executed by the workers. (b-c) the breakdown of the time consumption on two game benchmarks (Section 5).
+
+repeatedly performs rollouts until a predefined number of simulations is reached. During each rollout, it selects nodes to query, assign expansion and simulation tasks to different workers, and collect the returned results to update the global statistics. In particular, we use the following incomplete update and complete update (shown in Figure 2(a)) to track  $N_{s}$  and  $O_{s}$  along the traversed path (see Figure 1(d)):
+
+$$
+[ \text {i n c o m p l e t e u p d a t e} ] \quad O _ {s} \leftarrow O _ {s} + 1, \tag {5}
+$$
+
+$$
+[ \text {c o m p l e t e u p d a t e} ] \quad O _ {s} \leftarrow O _ {s} - 1; N _ {s} \leftarrow N _ {s} + 1, \tag {6}
+$$
+
+where incomplete update is performed before the simulation task starts, allowing the updated statistics to be instantly available globally; complete update is done after the simulation return is available, resembling the backpropagation step in the sequential algorithm. In addition,  $V_{s}$  is also updated in the complete update step using (3). Such a clear division of labor between the master and the workers provides sequential selection and backpropagation steps when we parallelize the costly expansion and simulation steps. It ensures up-to-date statistics for all workers by the centralized master process and achieves linear speedup without much performance degradation (see Section 5 for the experimental results).
+
+To justify the above rationale of our system design, we perform a set of running time analysis for our developed WU-UCT system and report the results in Figure 2(b)-(c). We show the time consumption for different parts at the master and at the workers. First, we focus exclusively on the workers. With a close-to-100% occupancy rate for the simulation workers, the simulation step is fully parallelized. Although the expansion workers are not fully utilized, the expansion step is maximumally parallelized since the number of required simulation and expansion tasks is identical. This suggests the existence of an optimal (task-dependent) ratio between the number of expansion workers and the number of simulation workers that fully parallelize both steps with the least resources (e.g. memory). Returning to the master process, on both benchmarks, we see a clear dominance of the time spent on the simulation and the expansion steps even they are both parallelized by 16 workers. This supports our design to parallelize only the simulation and expansion steps. We finally focus on the communication overhead caused by parallelization. Although more time-consuming compared to simulation and backpropagation, the communication overhead is negligible compared to the time used by the expansion and the simulation steps. Other details in our system, such as the centralized game-state storage, are further discussed in Appendix A.
+
+# 4 THE BENEFITS OF WATCHING UNOBSERVED SAMPLES
+
+In this section, we discuss the benefits of watching unobserved samples in WU-UCT, and compare it with several popular parallel MCTS algorithms (Figure 3), including Leaf Parallelization (LeafP),
+
+(a) Leaf parallelization (LeafP)  
+![](images/abb4c0eef5f94516107100bc7cd04cb78032f2f4c8f256acdfb41bc37206e138.jpg)  
+During simulation, multiple workers (e.g. A, B, C) simultaneously query the same node. Selection, expansion, and backpropagation are sequentially executed (similar to UCT).
+
+Figure 3: Three popular parallel MCTS algorithms. LeafP parallelizes the simulation steps, TreeP uses virtual loss to encourage exploration, and RootP parallelizes the subtrees of the root node.  
+![](images/2ad2fdc4f6bdbb87ad2b47dff2078d9d3e4b6453a42e6afd2675d6f522613a52.jpg)  
+An virtual loss  $r_{VL}$  is subtracted from  $V_{o}$  of nodes that have been traversed by some worker.  $r_{VL}$  will be added back to nodes during backpropagation.
+
+(b) Tree parallelization (TreeP)  
+(c) Root parallelization (RootP)  
+![](images/18fe8d78a49f06da0d6b48058b85888e83a8d32225bc2f8119a89f11c6a34eab.jpg)  
+Different workers perform tree search in a local memory, each starts from different child nodes.
+
+Tree Parallelization (TreeP) with virtual loss, and Root Parallelization (RootP). LeafP parallelizes the leaf simulation, which leads to an effective hex game solver (Wang et al., 2018). TreeP with virtual loss has recently achieved great success in challenging real-world tasks such as Go (Silver et al., 2016). And RootP parallelizes the subtrees of the root node at different workers and aggregates the statistics of the subtrees after all the workers complete their simulations (Soejima et al., 2010).
+
+We argue that, by introducing the additional statistics  $O_{s}$ , WU-UCT achieves a better exploration-exploitation tradeoff than the above methods. First, LeafP and TreeP represent two extremes in such a tradeoff. LeafP lacks diversity in exploration as all its workers are assigned to simulating the same node, leading to performance drop caused by collapse of exploration in much the same way as the naive parallelization (see Figure 1(c)). In contrast, although the virtual loss used in TreeP could encourage exploration diversity, this hard additive penalty could cause exploitation failure: workers will be less likely to co-simulating the same node even when they are certain that it is optimal (Mirsoleimani et al., 2017). RootP tries to avoid these issues by letting workers perform an independent tree search. However, this reduces the equivalent number of rollouts at each worker, decreasing the accuracy of the UCT policy (2). Different from the above three approaches, WU-UCT achieves a much better exploration-exploitation tradeoff in the following manner. It encourages exploration by using  $O_{s}$  to "penalize" the nodes that have many in-progress simulations. Meanwhile, it allows multiple workers to exploit the most rewarding node since this "penalty" vanishes when  $N_{s}$  becomes large (see (4)).
+
+# 5 EXPERIMENTS
+
+This section evaluates the proposed WU-UCT algorithm on a production system to predict the user pass-rate of a mobile game (Section 5.1) as well as on the public Atari Game benchmark (Section 5.2), aiming at demonstrating the superior performance and near-linear speedup of WU-UCT.
+
+# 5.1 EXPERIMENTS ON THE "JOY CITY" GAME
+
+Joy City is a level-oriented game with diverse and challenging gameplay. Players tap to eliminate connected items on the game board. To pass a level, players have to complete certain goals within a given number of steps. The number of steps used to pass a level (termed game step) is the main performance metric, which differentiates masters from beginners. It is a challenging reinforcement learning task due to its large number of game-state (over  $12^{9 \times 9}$ ) and high randomness in the transition. The goal of the production system is to accurately predict the user pass-rate of different game-levels, providing useful and fast feedback for game design. Powered by WU-UCT, the system runs  $16 \times$  faster while accurately predicting user pass-rate (8.6% MAE). In this subsection, we concentrate our analysis on the speedup and performance of WU-UCT using two typical game-levels (Level-35 and Level-58), and refer the readers interested in the user pass-rate prediction system to Appendix C.
+
+We evaluate WU-UCT with different numbers of expansion and simulation workers (from 1 to 16) and report the speedup results in Figures 4(a)-(b). For all experiments, we fix the total number of
+
+![](images/5b9dd321fd239f07331cf3dd97328ca3fa57b0cf1e1ff777ac14ae19c35f84d2.jpg)  
+(a) Speedup (level 35)
+
+![](images/99c374eec4cd8900a92127c36a109cfc0abd918b2c9e30bbef8a6cc3d0967baa.jpg)  
+(b) Speedup (level 58)  
+Figure 4: WU-UCT speedup and performance. Results are averaged over 10 runs. WU-UCT achieves linear speedup with negligible performance loss (measured in game steps).
+
+![](images/3a03a87fcdb64a6de920c2ef427bf09af194a34c3b3da0898dccd6ea5d1259f8.jpg)  
+(c) Performance (level 35)
+
+![](images/5076b8c5e81760484b5f3d3c50d589ab3e8d3c281762d5c4ecb10e34e95fdead.jpg)  
+(d) Performance (level 58)
+
+Table 1: The performance on 15 Atari games. Average episode return (± standard deviation) over 10 trials are reported. The best average scores among parallel algorithms are highlighted in boldface. The mark * indicates that WU-UCT achieves statistically better performance ( $p$ -value  $< 0.0011$  in paired t-test) than TreeP (no mark if both methods perform statistically similar). Similarly, the marks † and ‡ mean that WU-UCT performs statistically better than LeafP and RootP, respectively.  
+
+<table><tr><td>Environment</td><td>WU-UCT</td><td>TreeP</td><td>LeafP</td><td>RootP</td><td>PPO</td><td>UCT</td></tr><tr><td>Alien</td><td>5938±1839</td><td>4200±1086</td><td>4280±1016</td><td>5206±282</td><td>1850</td><td>6820</td></tr><tr><td>Boxing</td><td>100±0*†‡</td><td>99±0</td><td>95±4</td><td>98±1</td><td>94</td><td>100</td></tr><tr><td>Breakout</td><td>408±21†‡</td><td>390±33</td><td>331±45</td><td>281±27</td><td>274</td><td>462</td></tr><tr><td>Centipede</td><td>1163034±403910*†‡</td><td>439433±207601</td><td>162333±69575</td><td>184265±104405</td><td>4386</td><td>652810</td></tr><tr><td>Freeway</td><td>32±0</td><td>32±0</td><td>31±1</td><td>32±0</td><td>32</td><td>32</td></tr><tr><td>Gravitar</td><td>5060±568†</td><td>4880±1162</td><td>3385±155</td><td>4160±1811</td><td>737</td><td>4900</td></tr><tr><td>MsPacman</td><td>19804±2232*†‡</td><td>14000±2807</td><td>5378±685</td><td>7156±583</td><td>2096</td><td>23021</td></tr><tr><td>NameThisGame</td><td>29991±1608*</td><td>23326±2585</td><td>25390±3659</td><td>27440±9533</td><td>6254</td><td>38455</td></tr><tr><td>RoadRunner</td><td>46720±1359*†‡</td><td>24680±3316</td><td>25452±2977</td><td>38300±1191</td><td>25076</td><td>52300</td></tr><tr><td>Robotank</td><td>101±19</td><td>86±13</td><td>80±11</td><td>78±13</td><td>5</td><td>82</td></tr><tr><td>Qbert</td><td>13992±5596</td><td>14620±5738</td><td>11655±5373</td><td>9465±3196</td><td>14293</td><td>17250</td></tr><tr><td>SpaceInvaders</td><td>3393±292</td><td>2651±828</td><td>2435±1159</td><td>2543±809</td><td>942</td><td>3535</td></tr><tr><td>Tennis</td><td>4±1*†‡</td><td>-1±0</td><td>-1±0</td><td>0±1</td><td>-14</td><td>5</td></tr><tr><td>TimePilot</td><td>55130±12474*†</td><td>32600±2165</td><td>38075±2307</td><td>45100±7421</td><td>4342</td><td>52600</td></tr><tr><td>Zaxxon</td><td>39085±6838†‡</td><td>39579±3942†‡</td><td>12300±821</td><td>13380±769</td><td>5008</td><td>46800</td></tr></table>
+
+simulations to 500. First, note that when we have the same number of expansion workers and simulation workers, WU-UCT achieves linear speedup. Furthermore, Figures 4 also suggest that both the expansion workers and the simulation workers are crucial, since lowering the number of workers from either sets decreases the speedup. Besides the near-linear speedup property, WU-UCT suffers negligible performance loss with the increasing number of workers, as shown in Figures 4(c)-(d). The standard deviations of the performance (measured in the average game steps) over different numbers of expansion and simulation workers are only 0.67 and 1.22 for Level-35 and Level-58, respectively, which are much smaller than their average game steps (12 and 30).
+
+# 5.2 EXPERIMENTS ON THE ATARI GAME BENCHMARK
+
+We further evaluate WU-UCT on Atari Games (Bellemare et al., 2013), a classical benchmark for reinforcement learning (RL) and planning algorithms (Guo et al., 2014). The Atari Games are an ideal testbed for MCTS algorithms for its long planning horizon (several thousand), sparse reward, and complex game strategy. We compare WU-UCT to three parallel MCTS algorithms discussed in Section 4: TreeP, LeafP, and RootP (additional experiment results comparing WU-UCT with a variant of TreeP is provided in Appendix E). We also report the results of sequential UCT ( $\approx 16 \times$  slower than WU-UCT) and PPO (Schulman et al., 2017) as reference. Generally, the performance of sequential UCT sets an upper bound for parallel UCT algorithms. PPO is included since we used a distilled PPO policy network (Hinton et al., 2015; Rusu et al., 2015) as the roll-out policy for all other algorithms. It is considered as a performance lower bound for both parallel and sequential UCT algorithms. All experiments are performed with a total of 128 simulation steps, and all parallel algorithms use 16 workers (see Appendix D for the details).
+
+We first compare the performance, measured by average episode reward, between WU-UCT and the baselines on 15 Atari games, which is done with 16 simulation workers and 1 expansion worker (for a fair comparison, since baselines do not parallel the expansion step). Each task is repeated 10 times with the mean and standard deviation reported in Table 1. Due to the better exploration-exploitation tradeoff during selection, WU-UCT out-performs all other parallel algorithms in 12 out of 15 tasks. Pairwise student t-test further show that WU-UCT performs significantly better (adjusted by the Bonferroni method,  $p$ -value  $< 0.0011$ ) than TreeP, LeafP, and RootP in 7, 9, and 7 tasks, respectively. Next, we examine the influence of the number of simulation workers on the speed and the
+
+![](images/a906a46a3f50909faaff5414709d2756c21ae2fb34aa0c6cf084615eb29c56df.jpg)  
+Figure 5: Speed and performance test of our WU-UCT along with three baselines on four Atari games. All experiments are repeated three times and the mean and standard deviation (for episode reward only) are reported. For WU-UCT, the number of expansion workers is fixed to be one.
+
+performance. In Figure 5, we compare the average episode return as well as time consumption (per step) for 4, 8, and 16 simulation workers. The bar plots indicate that WU-UCT experiences little performance loss with an increasing number of workers, while the baselines exhibit significant performance degradation when heavily parallelized. WU-UCT also achieves the fastest speed compared to the baselines, thanks to the efficient master-worker architecture (Section 3.2). In conclusion, our proposed WU-UCT not only out-performs baseline approaches significantly under the same number of workers but also achieves negligible performance loss with the increasing level of parallelization.
+
+# 6 RELATED WORK
+
+MCTS Monte Carlo Tree Search is a planning method for optimal decision making in problems with either deterministic (Silver et al., 2016) or stochastic (Schafer et al., 2008) environments. It has made a profound influence on Artificial Intelligence applications (Browne et al., 2012), and has even been applied to predict and mimic human behavior (van Opheusden et al., 2016). Recently, there has been a wide range of work combining MCTS and other learning methods, providing mutual improvements to both methods. For example, Guo et al. (2014) harnesses the power of MCTS to boost the performance of model-free RL approaches; Shen et al. (2018) bridges the gap between MCTS and graph-based search, outperforming RL and knowledge base completion baselines.
+
+Parallel MCTS Many approaches have been developed to parallelize MCTS methods, with the objective being two-fold: achieve near-linear speedup under a large number of workers while maintaining the algorithm performance. Popular parallelization approaches of MCTS include leaf parallelization, root parallelization, and tree parallelization (Chaslot et al., 2008). Leaf parallelization aims at collecting better statistics by assigning multiple workers to query the same node (Cazenave & Jouandeau, 2007). However, this comes at the cost of wasting diversity of the tree search. Therefore, its performance degrades significantly despite the near-ideal speedup with the help of a client-server network architecture (Kato & Takeuchi, 2010). In root parallelization, multiple search trees are built and assigned to different workers. Additional work incorporates periodical synchronization of statistics from different trees, which results in better performance in real-world tasks (Bourki et al., 2010). However, a case study on Go reveals its inferiority with even a small number of workers (Soejima et al., 2010). On the other hand, tree parallelization uses multiple workers to traverse, perform queries, and update on a shared search tree. It benefits significantly from two techniques. First, a virtual loss is added to avoid querying the same node by different workers (Chaslot et al., 2008). This has been adopted in various successful applications of MCTS such as Go (Silver et al., 2016) and Dou-di-zhu (Whitehouse et al., 2011). Additionally, architecture side improvements such as using pipeline (Mirsoleimani et al., 2018b) or lock-free structure (Mirsoleimani et al., 2018a) speedup the algorithm significantly. However, though being able to increase diversity, virtual loss degrades the performance under even four workers (Mirsoleimani et al., 2017; Bourki et al., 2010). Finally, the idea of counting the unobserved samples to adjust the confidence interval in arm selection also appeared in Zhong et al. (2017). However, it mainly focuses on parallelizing the multi-armed thresholding bandit problem (Chen et al., 2014) instead of the tree search problem as we do.
+
+# 7 CONCLUSION
+
+This paper proposes WU-UCT, a novel parallel MCTS algorithm that addresses the problem of outdated statistics during parallelization by watching the number of unobserved samples. Based on the newly devised statistics, it modifies the UCT node-selection policy in a principled manner, which achieves effective exploration-exploitation tradeoff. Together with our efficiency-oriented system implementation, WU-UCT achieves near-optimal linear speedup with only limited performance loss across a wide range of tasks, including a deployed production system and Atari games.
+
+# 8 ACKNOWLEDGEMENTS
+
+This work is supported by Tencent AI Lab and Seattle AI Lab, Kwai Inc. We thank Xiangru Lian for his help on the system implementation.
+
+# REFERENCES
+
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+Anusha Nagabandi, Gregory Kahn, Ronald S Fearing, and Sergey Levine. Neural network dynamics for model-based deep reinforcement learning with model-free fine-tuning. In 2018 IEEE International Conference on Robotics and Automation (ICRA), pp. 7559-7566. IEEE, 2018.  
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+Jan Schäfer, Michael Buro, and Knut Hartmann. TheUCT algorithm applied to games with imperfect information. *Diploma, Otto-Von-Guericke Univ. Magdeburg, Magdeburg, Germany*, 2008.  
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+Richard B Segal. On the scalability of parallel uct. In International Conference on Computers and Games, pp. 36-47. Springer, 2010.  
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+David Silver, Julian Schrittwieser, Karen Simonyan, Ioannis Antonoglou, Aja Huang, Arthur Guez, Thomas Hubert, Lucas Baker, Matthew Lai, Adrian Bolton, et al. Mastering the game of go without human knowledge. Nature, 550(7676):354, 2017.  
+Yusuke Soejima, Akihiro Kishimoto, and Osamu Watanabe. Evaluating root parallelization in go. IEEE Transactions on Computational Intelligence and AI in Games, 2(4):278-287, 2010.  
+Richard S Sutton and Andrew G Barto. Reinforcement learning: An introduction. MIT press, 2018.  
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+
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+Ronald J Williams. Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine learning, 8(3-4):229-256, 1992.  
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+
+# SUPPLEMENTARY MATERIAL
+
+A ALGORITHM DETAILS FOR WU-UCT
+
+The pseudo-code of WU-UCT is provided in Algorithm 1. Specifically, it provides the workflow of the master process. When the number of completed updates ( $t_{complete}$ ) has not exceeded the maximum simulation step  $T_{max}$  (a pre-defined hyperparameter), the main process repeatedly performs a modified rollout that consists of the following steps: selection, expansion, simulation, and backpropagation. The selection and backpropagation steps are performed in the main process, while the two others are assigned to the workers. The backpropagation step is divided into two sub-routines incomplete update (Algorithm 2) and complete update (Algorithm 3). The former is executed before simulation starts, while the latter is called after receiving simulation results. Task index  $\tau$  is added to help the main process to track different tasks returned from the workers. To maximize efficiency, the master process keeps assigning expansion and simulation tasks until all workers are fully occupied.
+
+Communication overhead of WU-UCT The choice for centralized game-state storage stems from the following observations: (i) size of the game-state is usually small, which allows efficient inter-process transformation, and (ii) each game-state is used at most  $|\mathcal{A}| + 1$  times, thus is inefficient to store in multiple processes. Although this design may not be ideal for all tasks, it is at least a reasonable choice. During rollouts, game-states generated by any expansion worker may be later used by any other expansion and simulation workers. Therefore, either a per-task transformation or decentralized storage is needed. For the latter case, however, since a game-state will be used at most  $|\mathcal{A}| + 1$  times, most workers will not need it, which results in inefficiency of the decentralized storage.
+
+Another possible solution is to store the game-states in shared memory. However, to receive benefit from it, the following conditions should be satisfied: (i) each process can access (read/write) the memory relatively fast even if some collisions may happen, and (ii) the shared memory is big enough to hold all game-states that may be accessed. If the two conditions hold, we may be able to reduce the communication overhead. Since the communication overhead is negligible even with 16 simulation and expansion workers (as shown in Figures 2(b) and 2(c)), we should consider using more workers to speedup the algorithm.
+
+# B ALGORITHM OVERVIEW OF BASELINE APPROACHES
+
+We give an overview of three baseline parallel UCT algorithms: Leaf Parallelization (LeafP), Tree Parallelization (TreeP) with virtual loss, and Root Parallelization (RootP), with the objective of providing a comprehensive view to the readers. We refer readers interested in the details of these algorithms to Chaslot et al. (2008). As suggested by their names, LeafP, TreeP, and RootP parallelized different parts of the search tree. Specifically, LeafP (Algorithm 4) parallelizes only the simulation process: whenever a node (state) is selected to query, all workers perform simulations individually to evaluate it. The main process (master) then waits for all workers to complete simulation and return their respective cumulative rewards, which are used to update the traversed nodes' statistics.
+
+TreeP (Algorithm 5) parallelizes the whole tree search algorithm by allowing different workers to access a shared search tree simultaneously. Each worker individually performs the selection, expansion, simulation, and back-propagation steps and update the nodes' statistics. To discourage querying the same node, individual workers subtract a virtual loss  $r_{VL}$  ( $r_{VL}$  is a hyper-parameter of the algorithm) to each of its traversed node during the selection process, and add it back ( $+r_{VL}$ ) during back-propagation. This allows nodes currently being evaluated by some workers to have lower utility scores (4) and will be less likely to be chosen by other workers, which improves the diversity of the node visited by different workers simultaneously.
+
+Silver et al. (2017) and Segal (2010) introduced an additional way to add pseudo reward into the traversed nodes. See Appendix E for details of this variant of TreeP and more experiments of it on Atari games.
+
+Algorithm 1 WU-UCT  
+Input: environment emulator  $\mathcal{E}$  , root tree node  $s_\mathrm{root}$  , maximum simulation step  $T_{max}$  , maximum simulation depth  $d_{max}$  , number of expansion workers  $N_{exp}$  , and number of simulation workers  $N_{sim}$    
+Initialize: expansion worker pool Wexp, simulation worker pool Wsim, game-state buffer B,  $t\gets 0$  , and  $t_{complete}\gets 0$    
+while  $t_{complete} <   T_{max}$  do   
+    Traverse the tree top down from root node  $s_\mathrm{root}$  following (4) until (i) its depth greater than  $d_{max}$  , (ii) it is a leaf node, or (iii) it is a node that has not been fully expanded and random()  $<  0.5$  if expansion is required then  $\bar{s}\gets$  shallow copy of the current node Assign expansion task  $(t,\bar{s})$  to pool Wexp // t is the task index   
+else assign simulation task  $(t,s)$  to pool Wsim if episode not terminated Call incomplete_update(s); if episode terminated, call complete_update(t,s,0.0)   
+end if   
+if Wexp fully occupied then Wait for a expansion task with return: (task index  $\tau$  , game state s, reward r, terminal signal d, task index  $\tau$  ) ; expand the tree according to s,  $\tau$  r, and d; assign simulation task  $(\tau ,s)$  to pool Wsim Call incomplete_update(t,s)   
+else continue   
+if Wsim fully occupied then Wait for a simulation task with return: (task index  $\tau$  , node s, cumulative reward  $\bar{r}$  ) Call complete_update(  $\tau ,s,\bar{r})$  .  $t_{complete}\gets t_{complete} + 1$    
+else continue   
+ $t\gets t + 1$    
+end while
+
+Algorithm 2 incomplete_update  
+input: node  $s$   
+while  $n \neq$  null do  
+ $O_s \gets O_s + 1$ $s \gets \mathcal{PR}(s) // \mathcal{PR}(s)$  denotes the parent node of  $s$   
+end while
+
+Algorithm 3 complete_update  
+input: task index  $t$  node  $s$  reward  $\bar{r}$    
+while  $n\neq$  null do  $N_{s}\gets N_{s} + 1;O_{s}\gets O_{s} - 1$  Retrieve reward  $r$  according to task index  $t$ $\bar{r}\gets r + \gamma \bar{r};V_s\gets \frac{N_s - 1}{N_s} V_s + \frac{1}{N_s}\bar{r}$ $s\gets \mathcal{PR}(s)$  //  $\mathcal{PR}(s)$  denotes the parent node of  $s$    
+end while
+
+As hinted by its name, RootP (Algorithm 6) parallelizes the root node. Specifically, in an initialization step, all children of the root node is expanded, and different workers are assigned to perform rollouts using the expanded child nodes as the root node of the search tree. The algorithm evenly distribute the workload such that the number of rollouts starting from all child nodes is  $T_{max} / M$ , where  $M$  is the number of workers. After the job assignment, all workers construct search trees in their own local memories and perform sequential tree search until their assigned tasks are finished. Finally, the main process collects statistics from all workers and return the predicted best action of the state represented by the root node of the search tree.
+
+# C EXPERIMENT DETAILS AND SYSTEM DESCRIPTION OF THE JOY CITY TASK
+
+This section describes the basic rules of the Joy City game (Appendix C.1) as well as details about the deployed user pass-rate prediction system (Appendix C.2).
+
+# C.1 DESCRIPTION OF THE JOY CITY GAME
+
+This section serves as an introduction to the basic rules of the tap game. Figure 7 depicts several screenshots of the game. In the main frame, there is a  $9 \times 9$  grid, where each cell contains an item. We can click cells with connected color regions to eliminate them (i.e., if the cell represented by the purple dot in the first screenshot of Figure 6(a) is tapped, the region contains blue boxes will be "eliminated"). The remaining cells then collapse to fill in the gaps of exploded ones. To goal is to fulfill all level requirements (goals) within a fixed number of clicks. Figure 6(a) provides consecutive snapshots for playing level 10 of the game. The goal of this level is depicted on the
+
+![](images/5fd90d19f48e7242ebe55ae2349b6c03006c305c6212835f553071e8f9e4348e.jpg)  
+#
+
+![](images/066be2bc21dd3a94492f82dd2dda8d81a095c763469810b6316b54b2ff3d4a7a.jpg)  
+#
+
+![](images/892b782c2768d49912db765302b5ccc12736dd411a180a6fc3ed946cdef7d29e.jpg)  
+(a) A demonstrated game play in level 10 of the tap game. Purple dots refers to the tapped cell, and red regions indicate directly eliminated cells.
+
+![](images/86073229b13a1dbfcb5975539828bcfdc6f8a3dc9a4d7c32ce9c9c53b30b896d.jpg)  
+#
+
+![](images/3e4778c16da145d0d1e7705c3b46590bc332fe87c8d8aa41536cfa0839a92995.jpg)  
+A "boss level"
+
+![](images/ca7e3d72c6558b3b06de40aa8a3a2f176d0e1c1cf77765a81b0f18a433ea891a.jpg)  
+Small initial connectivity
+
+![](images/468cd098d6cbc4d3faf3898eef335222bb5fea420c45634d859dc323a934aa61.jpg)  
+Various kinds of hard-to-remove obstacles and spiral layout
+
+![](images/6e1cbdb4d171ca7504252aa83ab33a61f56623b8219c50a755226e64e9e26a05.jpg)  
+(b) Examples of levels with different rule, difficulty, and layout.
+
+top, which is 3 "cats" and 24 "balloons". The top-left corner represents the number of remaining steps. Players have to accomplish all given goals before the step runs out. Figure 6(a) demonstrates successful gameplay, where only 6 steps are used to complete the level. In each of the three left frames, the cell noted by the purple circle is clicked. Immediately, the same-color region marked with a red frame is eliminated. Different goal objects/obstacle objects react differently. For instance, when some cell is exploded beside a balloon, it will also explode. Frame two demonstrates the use of props. Tapping regions with connectivity above a certain threshold will provide prop as a bonus. They have special effects that can help players pass the level faster. Finally, in the last screenshot, all goals are completed and we pass the level.
+
+Figure 6(b) further demonstrates the variety of levels. Specifically, the left-most frame depicts a special "boss level", where the goal is the "defeat" the evil cat. The cat will randomly throw objects to the cells, adding additional randomness. Three other frames illustrate relatively hard levels, which is revealed from their low-connectivity, abundance and complexity of the obstacles, and special layout.
+
+![](images/956b762cc2d37da9b64a175df9da878e149065d30cefab8df75289699a6da374.jpg)  
+Figure 6: Snapshots of the Tap-elimination game.
+
+![](images/1213ace9aee1f37542fe9494d14877fe53d503073ed42804b7ce2b03ec6e94e6.jpg)
+
+# C.2 DETAILS OF THE LEVEL PASS-RATE PREDICTION SYSTEM
+
+During a game design cycle, to achieve the desired game pass-rates, a game designer needs to hire many human testers to extensively test
+
+![](images/01ed1d1cddc7b872025365e55e6047d60c28645af2f1de7cbea939a09a5e3b99.jpg)  
+Figure 7: Our deployed user pass-rate prediction system.
+
+all the levels before its release, which generally takes a long time and is inaccurate. Therefore, it would greatly reduce the game design cycle if we can develop a testing system that is able to provide quick and accurate feedback about the user pass-rates. Figure 7 gives an overview of our deployed user pass-rate prediction system, where WU-UCT is used to mimic average user performance and provide features for predicting the human pass-rate. As we have shown in the main paper, it can achieve significant speedup without significant performance loss,<sup>8</sup> allowing the game designer to get the feedback in 20 minutes instead of 12 hours. Specifically, we use WU-UCT with different numbers of rollouts to mimic players with different skill levels, where WU-UCT with 10 rollouts is used to represent average players while the agent with 100 rollouts mimics skillful players. This is verified by the pair-wise sample t-test result provided in Table 2. With 10 simulations, the WU-UCT agent performs statistically similar ( $p$ -value  $>5\%$ ) to human players, while with 100-simulation, the agent performs statistically better ( $p$ -value  $<5\%$ ). Besides, Figure 8 shows that our pass-rate prediction system achieves  $8.6\%$  mean absolute error (MAE) on 130 released game-levels, with  $93\%$  of them having MAE less than  $20\%$ .
+
+The system consists of two working phases, i.e., training and inference. Specifically, training and validation are done on 300 levels that have been released in a test version of the game. In the training phase, the system has access to both the level and players' pass-rate, while only levels are available in the inference phase, and the system needs to give quick and accurate feedback about the (predicted) user pass-rate. In both phases, the levels are first fed into an asynchronous advantage actor-critic (A3C) (Mnih et al., 2016) learner for a base policy  $\pi$ . It is then used by the WU-UCT agent as a prior to select expand action as well as the default policy for simulation. We then use WU-UCT to perform multiple games. The maximum depth and width (maximum number of child nodes for each node) of the search tree is 10 and 5, respectively. The number of simulations is set to 10 and 100 to get AI bots with different skill levels. Six features (three for both the 10-simulation and 100-simulation agent) are extracted from the gameplay results. Specifically, the features are AI's pass-rate, average used step divided by the provided step (the number at the top-left corner in the screenshots in Figure 6), and median used step divided by the provided step. During training, the features, as well as the players' pass-rate, is used to learn a linear regressor, while in the inference phase, the regression model is used to predict user pass-rate.
+
+# C.3 ADDITIONAL EXPERIMENTAL RESULTS
+
+In this section, we list the additional experimental results. In Table 3, we report the specific speedup number for different numbers of expansion worker and simulation workers.
+
+Table 2: Pair-wise sample t-test of pass-rate across 130 levels between two AI bots (different number of MCTS rollouts) and the players. "Avg. diff" means the average difference between the pass-rate of the bot and that of the human players.  $p$ -value measures the likelihood that two sets of paired samples are statistically similar (i.e. larger means similar). Effect size measures the strength of the difference (larger means greater difference).  
+
+<table><tr><td>AI bot</td><td># rollouts</td><td>Avg. diff.</td><td>Effect size</td><td>p-value</td></tr><tr><td>WU-UCT</td><td>10</td><td>-1.54</td><td>0.07</td><td>0.4120</td></tr><tr><td>WU-UCT</td><td>100</td><td>22.18</td><td>0.88</td><td>0.0000</td></tr></table>
+
+![](images/a5569ca174a49ec1aeec54a2081d3d2a0d1f0dfa7881d8b47b19bcac325684cd.jpg)  
+Figure 8: Distribution of the pass-rate prediction error on 130 game-levels.
+
+Table 3: Speedup on two levels of the tap game.  $M_e$  is the number of expansion workers and  $M_s$  is the number of simulation workers.  
+
+<table><tr><td rowspan="2">Lv. \( \overline{M_s} \) \( M_e \)</td><td colspan="5">Level 35</td><td colspan="5">Level 58</td></tr><tr><td>1</td><td>2</td><td>4</td><td>8</td><td>16</td><td>1</td><td>2</td><td>4</td><td>8</td><td>16</td></tr><tr><td>1</td><td>1.0</td><td>2.0</td><td>2.8</td><td>3.6</td><td>4.5</td><td>1.0</td><td>1.8</td><td>4.1</td><td>4.8</td><td>5.1</td></tr><tr><td>2</td><td>1.4</td><td>2.2</td><td>4.1</td><td>5.7</td><td>6.3</td><td>1.1</td><td>3.1</td><td>5.3</td><td>6.7</td><td>8.4</td></tr><tr><td>4</td><td>1.7</td><td>2.5</td><td>4.5</td><td>8.4</td><td>8.8</td><td>1.1</td><td>3.4</td><td>6.1</td><td>10.1</td><td>12.8</td></tr><tr><td>8</td><td>2.3</td><td>3.0</td><td>5.1</td><td>10.1</td><td>12.8</td><td>1.2</td><td>3.6</td><td>6.7</td><td>13.2</td><td>16.1</td></tr><tr><td>16</td><td>2.9</td><td>3.7</td><td>5.7</td><td>11.2</td><td>15.5</td><td>1.2</td><td>3.8</td><td>7.6</td><td>16.1</td><td>20.9</td></tr></table>
+
+# D EXPERIMENT DETAILS OF THE ATARI GAMES
+
+This section provides the implementation details of the experiments on Atari games. Specifically, we first describe the training pipeline of the default policy. We then illustrate how the default policy is connected with MCTS algorithm to perform simulation.
+
+Training default policy for MCTS To allow better overall performance, we used the Proximal Policy Gradient (PPO) (Schulman et al., 2017), one of the state-of-the-art on-policy model-free reinforcement learning (RL) algorithms. We adopted the highest-starred third-party code of PPO on GitHub. The implementation uses the same hyper-parameters with the original paper. The architecture of the policy network is shown in Figure 9. The original PPO network is trained on 10 million frames for each task. To reduce computation count, we reduce the network size using network
+
+Table 4: Performance of the original PPO policy and our distilled policy on 15 Atari games.  
+
+<table><tr><td>Environment</td><td>Origin PPO policy</td><td>Distilled policy</td></tr><tr><td>Alien</td><td>1850</td><td>850</td></tr><tr><td>Boxing</td><td>94</td><td>7</td></tr><tr><td>Breakout</td><td>274</td><td>191</td></tr><tr><td>Centipede</td><td>4386</td><td>1701</td></tr><tr><td>Freeway</td><td>32</td><td>32</td></tr><tr><td>Gravitar</td><td>737</td><td>600</td></tr><tr><td>MsPacman</td><td>2096</td><td>1860</td></tr><tr><td>NameThisGame</td><td>6254</td><td>6354</td></tr><tr><td>RoadRunner</td><td>25076</td><td>26600</td></tr><tr><td>Robotank</td><td>5</td><td>13</td></tr><tr><td>Qbert</td><td>14293</td><td>12725</td></tr><tr><td>SpaceInvaders</td><td>942</td><td>1015</td></tr><tr><td>Tennis</td><td>-14</td><td>-10</td></tr><tr><td>TimePilot</td><td>4342</td><td>4400</td></tr><tr><td>Zaxxon</td><td>5008</td><td>3504</td></tr></table>
+
+![](images/d0166d1a3129e0d961394edc3c0bf28a755152db2543bcf70fdf815838d431f6.jpg)  
+(a) Full-size PPO network.  
+Figure 9: Architecture of the original PPO network (left) and the distilled network (right).
+
+![](images/195fcea6c17580f0e6a741b9c08ea1df2e980f56b3d5460c705e6d9e74fe5287.jpg)  
+(b) Distilled network.
+
+distillation (Hinton et al., 2015). Specifically, it is a teacher-student training framework where the student (distilled) network mimics the output of the teacher network. Samples are collected by the PPO network with the  $\epsilon$ -greedy strategy ( $\epsilon = 0.1$ ). The student network optimizes its parameters to minimize the mean square error of the policy's logits as well as the value. Performance of the original PPO policy network as well as the distilled network is provided in Table 4.
+
+MCTS simulation Both the policy output and the value output of the distilled network is used in the simulation phase. Particularly, if a simulation is started from state  $s$ , rollout is performed using the policy network with an upper bound of 100 steps and reaches the leaf state  $s'$ . If the environment does not terminate, the full return is computed by the intermediate rewards plus the value function at state  $s'$ . Formally, the cumulative reward provided by the simulation is  $R_{simu} = \sum_{i=0}^{99} \gamma^i r_i + \gamma^{100} V(s')$ , where  $V(s)$  denotes the value of state  $s$ . To reduce the variance of Monte Carlo sampling, we average it with the value function  $V(s)$  at state  $s$ . The final simulation return is then  $R = 0.5 R_{simu} + 0.5 V(s)$ .
+
+Hyperparameters and experiment details for WU-UCT For all tree search based algorithms (i.e., WU-UCT, TreeP, LeafP, and RootP), the maximum depth of the search tree is set to 100. The search width is limited by 20 and the maximum number of simulations is 128. The discount factor  $\gamma$  is set to 0.99 (note that the reported score is not discounted). When performing gameplays, a tree search subroutine is called to plan for the best action in each time step. The sub-routine iteratively constructs a search tree from its initialization with a root node only. Experiments are deployed on 4 Intel® Xeon® E5-2650 v4 CPUs and 8 NVIDIA® GeForce® RTX 2080 Ti GPUs. To minimize the speed fluctuation caused by different workloads on the machine, we ensure that the total number of simulation workers is smaller than the total number of CPU cores, allowing each process to fully occupy each single core. The WU-UCT is implemented with multiple processes, with an interprocess pipe between the master process and each worker process.
+
+Hyperparameters and experiments for baseline algorithms Being unable to find appropriate third-party packages for baseline algorithms (i.e., tree parallelization, leaf parallelization, and root parallelization), we built our implementation of them based on the corresponding papers. Building all algorithms in the same package additionally allows us to accurately conduct speed-tests as it elim-
+
+![](images/b96a347ad1ec180fcb6be2ca9b5bdae9d44299883102fa64afa092cb15c7d247.jpg)  
+Figure 10: Relative performance between WU-UCT and three baseline approaches on 15 Atari benchmarks. Relative performance is calculated according to the mean episode reward in 3 trials. The average percentile improvement of WU-UCT on TreeP, LeafP, and RootP is  $49\%$ ,  $104\%$ , and  $82\%$ , respectively.[10]
+
+Table 5: Comparison between WU-UCT and three TreeP variants on 12 Atari games. Average episode return (± standard deviation) over 10 runs are reported. The best average scores among listed algorithms are highlighted in boldface. Hyper-parameter  $r_{VL}$  refers to the virtual loss added before each simulation starts, and  $n_{VL}$  similarly denotes the number of virtual visit counts added before each simulation starts.  
+
+<table><tr><td>Environment</td><td>WU-UCT</td><td>TreeP (rVL = nVL = 1)</td><td>TreeP (rVL = nVL = 2)</td><td>TreeP (rVL = nVL = 3)</td></tr><tr><td>Alien</td><td>5938±1839</td><td>4850±357</td><td>4935±60</td><td>5000±0</td></tr><tr><td>Boxing</td><td>100±0</td><td>99±1</td><td>99±0</td><td>99±1</td></tr><tr><td>Breakout</td><td>408±21</td><td>379±43</td><td>265±50</td><td>463±60</td></tr><tr><td>Freeway</td><td>32±0</td><td>32±0</td><td>32±0</td><td>32±0</td></tr><tr><td>Gravitar</td><td>5060±568</td><td>3500±707</td><td>4105±463</td><td>4950±141</td></tr><tr><td>MsPacman</td><td>19804±2232</td><td>13160±462</td><td>12991±851</td><td>8640±438</td></tr><tr><td>RoadRunner</td><td>46720±1359</td><td>29800±282</td><td>28550±459</td><td>29400±494</td></tr><tr><td>Qbert</td><td>13992±5596</td><td>17055±353</td><td>13425±194</td><td>9075±53</td></tr><tr><td>SpaceInvaders</td><td>3393±292</td><td>2305±176</td><td>3210±127</td><td>3020±42</td></tr><tr><td>Tennis</td><td>4±1</td><td>1±0</td><td>1±0</td><td>1±0</td></tr><tr><td>TimePilot</td><td>55130±12474</td><td>52500±707</td><td>49800±212</td><td>32400±1697</td></tr><tr><td>Zaxxon</td><td>39085±6838</td><td>24300±2828</td><td>24600±424</td><td>37550±1096</td></tr></table>
+
+inates other factors (e.g. different language) that may bias the result. Specifically, leaf parallelization is implemented with a master-worker structure: when the main process enters the simulation step, it assigns the task to all workers. When return from all workers is available, the master process performs backpropagation according to these statistics and begin a new rollout.
+
+As suggested by Browne et al. (2012), tree parallelization is implemented using a decentralized structure, i.e., each worker performs rollouts on a shared search tree. At the selection step, each traversed node is added a fixed virtual loss  $-r_{VL}$  to guarantee diversity of the tree search. When performing backpropagation,  $r_{VL}$  is added back to the traversed nodes.  $r_{VL}$  is chosen from 1.0 and 5.0 for each particular task. In other words, we ran TreeP with  $r_{VL} = 1.0$  and  $r_{VL} = 5.0$  for each task, and report the better result.
+
+Root parallelization is implemented according to Chaslot et al. (2008). Similar to leaf parallelization, root parallelization consists of sub-processes that do not share information with each other. At the beginning of the tree search process, each sub-process is assigned several actions of the root node to query. They then perform sequential UCT rollouts until reaches a pre-defined maximum number of rollouts. When all sub-processes complete the jobs, statistics from them are gathered by the main process, and are used to choose the best action.
+
+# E ADDITIONAL EXPERIMENTS ON THE ATARI GAMES
+
+This section provides additional experiment results to compare WU-UCT with another variant of the Tree Parallelization (TreeP) algorithm. As suggested by Silver et al. (2016), besides pre-adjusting the value  $V$  with virtual loss  $r_{VL}$ , pre-adjusted visit count can also be used to penalize  $V$ . In this variant of TreeP, both the virtual loss  $r_{VL}$  and a hand-crafted count correction  $n_{VL}$  (termed the virtual pseudo-count) is added to adjust  $V$ . Specifically, the value of node  $s$  is adjusted as
+
+$$
+V _ {s} ^ {\prime} \stackrel {\text {d e f}} {=} \frac {N _ {s} V _ {s} - r _ {V L}}{N _ {s} + n _ {V L}}, \tag {7}
+$$
+
+which is used in the UCT selection phase. Table 5 compares WU-UCT with this TreeP variant using both virtual loss and virtual pseudo-count (i.e., Eq. 7). Three sets of hyper-parameters are used in TreeP, which are described in the caption of the table (i.e.,  $r_{VL} = n_{VL} = 1$ ,  $r_{VL} = n_{VL} = 2$ , and  $r_{VL} = n_{VL} = 3$ ). All other experiment setups are the same as Section 5.2 and Appendix D. Table 5 indicates that on 9 out of 12 tasks, WU-UCT out-performs this new baseline (with its best hyper-parameters). Furthermore, we also observe that TreeP does not have an optimal set of hyperparameters that performs uniformly well on all tasks. In other words, to perform well, TreeP needs to conduct per-task hyper-parameter tuning. On the other hand, WU-UCT performs consistently well across different tasks.
+
+Conceptually, WU-UCT is designed based on the fact that on-going simulations (unobserved samples) will eventually return the results, so their number should be tracked and used to adaptively adjust the UCT selection process. On the other hand, TreeP uses artificially designed virtual loss  $r_{VL}$  and virtual pseudo-count  $n_{VL}$  to discourage other threads from simultaneously exploring the same node. Therefore, WU-UCT achieves a better exploration-exploitation tradeoff in parallelization, which leads to better performance as confirmed by the experimental results given in Table 5.
+
+Algorithm 4 Leaf Parallelization (LeafP)  
+Input: environment emulator  $\mathcal{E}$  , prior policy  $\pi$  , root tree node  $s_{root}$  , maximum simulation step  $T_{max}$  maximum simulation depth  $d_{max}$  , and number of workers  $N_{sim}$    
+Initialize:  $t_{complete}\gets 0$    
+while  $t_{complete} <   T_{max}$  do Traverse the tree top down from root node  $s_{root}$  following (2) until (i) its depth greater than  $d_{max}$  , (ii) it is a leaf node, or (iii) it is a node that has not been fully expanded and random()  $<  0.5$ $s^{\prime}\gets \mathrm{expand}(s,\mathcal{E},\pi)$  Each of the simulation workers perform roll-out beginning from  $s^\prime$  Wait until all workers completed simulation and returned cumulative reward  $\{\bar{r}_i\}_{i = 1}^{N_{sim}}$  ( $\bar{r}_i$  is returned by worker i) for  $i = 1:N_{sim}$  do Call back_propagation(s,  $\bar{r}_i$  end for  $t_{complete}\gets t_{complete} + N_{sim}$    
+end while
+
+Algorithm 5 Tree Parallelization (TreeP)  
+Input: environment emulator  $\mathcal{E}$  , prior policy  $\pi$  , root tree node  $s_{root}$  , virtual loss  $r_{VL}$  , maximum simulation step  $T_{max}$  , maximum simulation depth  $d_{max}$  , and number of workers  $N_{sim}$    
+Initialize:  $t_{complete}\gets 0$    
+Initialize:  $N_{sim}$  processes, each with access to the environment emulator, the prior policy, and the search tree   
+Perform asynchronously in each of the  $N_{sim}$  workers Traverse the tree top down from root node  $s_{root}$  following (2) until (i) its depth greater than  $d_{max}$  ,(ii) it is a leaf node, or (iii) it is a node that has not been fully expanded and random()  $<  0.5$  Add virtual loss to each of the traversed node:  $V_{s}\leftarrow V_{s} - r_{VL}$  for each traversed s  $s^{\prime}\gets \mathrm{expand}(s,\mathcal{E},\pi)$  Perform roll-out beginning from  $s^\prime$ $\bar{r}\gets$  the returned cumulative reward of the roll-out Call back_propagation(s,  $\bar{r}_i$  Remove virtual loss from each of the traversed node:  $V_{s}\gets V_{s} + r_{VL}$  for each traversed s  $t_\mathrm{complete}\gets t_\mathrm{complete} + 1$  if  $t_{complete}\geq T_{max}$  Terminate current process end if   
+end
+
+Algorithm 6 Root Parallelization (RootP)  
+Input: environment emulator  $\mathcal{E}$  prior policy  $\pi$  , root tree node  $s_{root}$  , maximum simulation step  $T_{max}$  maximum simulation depth  $d_{max}$  , and number of workers  $N_{sim}$    
+Initialize:  $t_{complete,i}\gets 0$  for  $i = 1:N_{sim}$    
+Initialize:  $N_{sim}$  processes, each with access to the environment emulator, the prior policy, and the search tree   
+Expand all child nodes of  $s_{root}$ $T_{avg}\leftarrow \mathrm{ceil}(T_{max} / |\mathcal{A}|)$  (  $|\mathcal{A}|$  is the number of actions)   
+Averagely distribute the workload (perform tree search  $T_{avg}$  times on each child of  $s_{root}$  ) to the  $N_{sim}$  workers, and copy the corresponding child nodes to the worker's local memory.   
+Perform asynchronously in each of the  $N_{sim}$  workers (i denotes the thread ID)   
+ $s_{root,i}\gets$  Select a child of  $s_{root}$  according to its allocated budget   
+Traverse the tree top down from root node  $s_{root,i}$  following (2) until (i) its depth greater than  $d_{max}$  (ii) it is a leaf node, or (iii) it is a node that has not been fully expanded and random()  $<  0.5$  Add virtual loss to each of the traversed node:  $V_{s}\gets V_{s} - r_{VL}$  for each traversed n   
+ $s^{\prime}\gets \mathrm{expand}(s,\mathcal{E},\pi)$    
+Perform roll-out beginning from  $s^\prime$ $\bar{r}\gets$  the returned cumulative reward of the roll-out   
+Call back_propagation(s,  $\bar{r}_i$  1 Remove virtual loss from each of the traversed node:  $V_{s}\gets V_{s} + r_{VL}$  for each traversed n   
+ $t_{complete,i}\gets t_{complete,i} + 1$    
+if  $t_{complete,i}\geq T_{avg}$  Terminate current process   
+end if   
+end   
+Gather child nodes' statistics from all workers
+
+# Algorithm 7 expansion
+
+input: node  $s$ , environment emulator  $\mathcal{E}$ , prior policy  $\pi$ $a \gets$  random action drawn from  $\pi(\cdot \mid s)$
+
+while  $s$  has expanded  $a$  do
+
+$a\gets$  random action drawn from  $\pi (\cdot \mid s)$
+
+end while
+
+$s^{\prime},r,d\gets$  performing  $a$  in  $s$  according to  $\mathcal{E}$  (d: terminal signal)
+
+$s^{\prime}\gets$  a new node constructed according to  $s$
+
+Store reward signal  $r$  and termination indicator  $d$  in  $s'$
+
+Link  $s'$  as the child of  $s$  by the node corresponding to  $a$ .
+
+# Algorithm 8 back_propagation
+
+input: node  $s$ , cumulative reward  $\bar{r}$
+
+while  $s \neq$  null do
+
+$$
+N _ {s} \leftarrow N _ {s} + 1
+$$
+
+Retrieve the reward  $r$  in the current node  $s$  (which is collected during its expansion)
+
+$$
+\bar {r} \leftarrow r + \gamma \bar {r}
+$$
+
+$$
+V _ {s} \leftarrow \frac {N _ {s} - 1}{N _ {s}} V _ {s} + \frac {1}{N _ {s}} \bar {r}
+$$
+
+$s\gets \mathcal{P}\mathcal{R}(s) / /\mathcal{P}\mathcal{R}(s)$  denotes the parent
+
+node of  $s$
+
+end while
\ No newline at end of file
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+# WATCH, TRY, LEARN: META-LEARNING FROM DEMONSTRATIONS AND REWARDS
+
+Allan Zhou* & Eric Jang
+
+Google Brain
+
+{allanz,ejang}@google.com
+
+Daniel Kappler & Alex Herzog
+
+X
+
+{kappler,alexherzog}@x.team
+
+Mohi Khansari, Paul Wohlart, Yunfei Bai & Mrinal Kalakrishnan
+
+X
+
+{khansari,wohlhart,yunfeibai,kalakris}@x.team
+
+Sergey Levine
+
+Google Brain, UC Berkeley
+
+slevine@google.com
+
+Chelsea Finn
+
+Google Brain, Stanford
+
+chelseaf@google.com
+
+# ABSTRACT
+
+Imitation learning allows agents to learn complex behaviors from demonstrations. However, learning a complex vision-based task may require an impractical number of demonstrations. Meta-imitation learning is a promising approach towards enabling agents to learn a new task from one or a few demonstrations by leveraging experience from learning similar tasks. In the presence of task ambiguity or unobserved dynamics, demonstrations alone may not provide enough information; an agent must also try the task to successfully infer a policy. In this work, we propose a method that can learn to learn from both demonstrations and trial-and-error experience with sparse reward feedback. In comparison to meta-imitation, this approach enables the agent to effectively and efficiently improve itself autonomously beyond the demonstration data. In comparison to meta-reinforcement learning, we can scale to substantially broader distributions of tasks, as the demonstration reduces the burden of exploration. Our experiments show that our method significantly outperforms prior approaches on a set of challenging, vision-based control tasks.
+
+# 1 INTRODUCTION
+
+Imitation learning enables autonomous agents to learn complex behaviors from demonstrations, which are often easy and intuitive for users to provide. However, learning expressive neural network policies from imitation requires a large number of demonstrations, particularly when learning from high-dimensional inputs such as images. Meta-imitation learning has emerged as a promising approach for allowing an agent to leverage data from previous tasks in order to learn a new task from only a handful of demonstrations (Duan et al., 2017; Finn et al., 2017b; James et al., 2018). However, in many practical few-shot imitation settings, there is an identifiability problem: it may not be possible to precisely determine a policy from one or a few demonstrations, especially in a new situation. And even if a demonstration precisely communicates what the task entails, it might not precisely communicate how to accomplish it in new situations. For example, it may be difficult to discern from a single demonstration where to grasp an object when it is in a new position or how much force to apply in order to slide an object without knocking it over. It may be expensive to collect more demonstrations to resolve such ambiguities, and even when we can, it may not be obvious to a human demonstrator where the agent's difficulty is arising from. Alternatively, it is easy for the user to provide success-or-failure feedback, while exploratory interaction is useful for learning how to perform the task. To this end, our goal is to build an agent that can first infer a policy from
+
+![](images/f9277d4967481ea092ab09cea93653bb9a029e95262c265afb4b03fbe94948d5.jpg)  
+Figure 1: Each column displays the first and last frame of an episode top-to-bottom. After watching one demonstration (left), the scene is re-arranged. With one trial episode (middle), our method can learn to solve the task (right) by leveraging both the demo and trial-and-error experience.
+
+![](images/1197dc6118f955e0631f778a7121f92d5fd1d9d0c7331885c8879350c10bb2e5.jpg)
+
+![](images/8c93a99d1659105bf19d786204af84c064ec77f6eac387e02cf54c6f11b4fb22.jpg)
+
+one demonstration, then attempt the task using that policy while receiving binary user feedback, and finally use the feedback to improve its policy such that it can consistently solve the task.
+
+This vision of learning new tasks from a few demonstrations and trials inherently requires some amount of prior knowledge or experience, which we can acquire through meta-learning across a range of previous tasks. To this end, we develop a new meta-learning algorithm that incorporates elements of imitation learning with trial-and-error reinforcement learning. In contrast to previous meta-imitation learning approaches that learn one-shot imitation learning procedures through imitation (Duan et al., 2017; Finn et al., 2017b), our approach enables the agent to improve at the test task through trial-and-error. Further, from the perspective of meta-RL algorithms that aim to learn efficient RL procedures (Duan et al., 2016; Wang et al., 2016; Finn et al., 2017a), our approach also has significant appeal: as we aim to scale meta-RL towards broader task distributions and learn increasingly general RL procedures, exploration and efficiency becomes exceedingly difficult. However, a demonstration can significantly narrow down the search space while also providing a practical means for a user to communicate the goal, enabling the agent to achieve few-shot learning of behavior. While the combination of demonstrations and reinforcement has been studied extensively in single task problems (Kober et al., 2013; Sun et al., 2018; Rajeswaran et al., 2018; Le et al., 2018), this combination is particularly important in meta-learning contexts where few-shot learning of new tasks is simply not possible without demonstrations. Further, we can even significantly improve upon prior methods that study this combination using meta-learning to more effectively integrate the information coming from both sources.
+
+The primary contribution of this paper is a meta-learning algorithm that enables learning of new behaviors with a single demonstration and trial experience. After receiving a demonstration illustrating a new goal, the meta-trained agent can learn to accomplish that goal through a small amount of trial-and-error with only binary success-or-failure labels. We evaluate our algorithm and several prior methods on a challenging, vision-based control problem involving manipulation tasks from four distinct families of tasks: button-pressing, grasping, pushing, and pick and place. We find that our approach can effectively learn tasks with new, held-out objects using one demonstration and a single trial, while significantly outperforming meta-imitation learning, meta-reinforcement learning, and prior methods that combine demonstrations and reward feedback. To our knowledge, our experiments are the first to show that meta-learning can enable an agent to adapt to new tasks with binary reinforcement signals from raw pixel observations, which we show with a single meta-model for a variety of distinct manipulation tasks. We have published videos of our experimental results<sup>1</sup> and the experiment model code<sup>2</sup>.
+
+# 2 RELATED WORK
+
+Learning to learn, or meta-learning, has a long-standing history in the machine learning literature (Thrun & Pratt, 1998; Schmidhuber, 1987; Bengio et al., 1992; Hochreiter et al., 2001). We particularly focus on meta-learning in the context of control. Our approach builds on and signifi
+
+cantly improves upon meta-imitation learning (Duan et al., 2017; Finn et al., 2017b; James et al., 2018; Paine et al., 2018) and meta-reinforcement learning (Duan et al., 2016; Wang et al., 2016; Mishra et al., 2018; Rakelly et al., 2019), extending contextual meta-learning approaches. Unlike prior work in few-shot imitation learning (Duan et al., 2017; Finn et al., 2017b; Yu et al., 2018; James et al., 2018; Paine et al., 2018), our method enables the agent to additionally improve upon trial-and-error experience. In contrast to work in multi-task and meta-reinforcement learning (Duan et al., 2016; Wang et al., 2016; Finn et al., 2017a; Mishra et al., 2018; Houthooft et al., 2018; Sung et al., 2017; Nagabandi et al., 2019; Sæmundsson et al., 2018; Hausman et al., 2017), our approach learns to use one demonstration to address the meta-exploration problem (Gupta et al., 2018; Stadie et al., 2018). Our work also requires only one round of on-policy data collection, collecting only 1,500 trials for the vision-based manipulation tasks, while nearly all prior meta-learning works require thousands of iterations of on-policy data collection, amounting to hundreds of thousands of trials (Duan et al., 2016; Wang et al., 2016; Finn et al., 2017a; Mishra et al., 2018).
+
+Combining demonstrations and trial-and-error experience has long been explored in the machine learning and robotics literature (Kober et al., 2013). This ranges from simple techniques such as demonstration-based pre-training and initialization (Peters & Schaal, 2006; Kober & Peters, 2009; Kormushev et al., 2010; Kober et al., 2013; Silver et al., 2016) to more complex methods that incorporate both demonstration data and reward information in the loop of training (Taylor et al., 2011; Brys et al., 2015; Subramanian et al., 2016; Hester et al., 2018; Sun et al., 2018; Rajeswaran et al., 2018; Nair et al., 2018; Le et al., 2018). The key contribution of this paper is an algorithm that can learn how to learn from both demonstrations and rewards. This is quite different from RL algorithms that incorporate demonstrations: learning from scratch with demonstrations and RL involves a slow, iterative learning process, while fast adaptation with a meta-trained policy involves extracting inherently distinct pieces of information from the demonstration and the trials. The demonstration provides information about what to do, while the small number of RL trials can disambiguate the task and how it's refinement. As a result, we get a procedure that significantly exceeds the efficiency of prior approaches, requiring only one demonstration and one trial to adapt to a new test task, even from pixel observations, by leveraging previous data. In comparison, single-task methods for learning from demonstrations and rewards typically require hundreds or thousands of trials to learn tasks of comparable difficulty (Rajeswaran et al., 2018; Nair et al., 2018).
+
+# 3 META-LEARNING FROM DEMONSTRATIONS AND REWARDS
+
+We first introduce some meta-learning preliminaries, then formalize our particular problem statement before finally describing our approach and its implementation.
+
+# 3.1 PRELIMINARIES
+
+Meta-learning, or learning to learn, aims to learn new tasks from very little data. To achieve this a meta-learning algorithm can first meta-train on a set of tasks  $\{\mathcal{T}_i\}$ , called the meta-train tasks. We then evaluate how quickly the meta-learner can learn an unseen meta-test task  $\mathcal{T}_j$ . We typically assume that the meta-train and meta-test tasks are drawn from some unknown task distribution  $p(\mathcal{T})$  (Finn, 2018). Through meta-training, a meta-learner can learn some common structure between the tasks in  $p(\mathcal{T})$  which it can use to more quickly learn a new meta-test task.
+
+We define a task  $\mathcal{T}_i$  as a finite-horizon Markov decision process (MDP),  $\{S, \mathcal{A}, r_i, P_i, H\}$ , with continuous state space  $S$ , continuous action space  $\mathcal{A}$ , reward function  $r_i: S \times \mathcal{A} \to \mathbb{R}$ , unknown dynamics  $P_i(s_{t+1}|s_t, a_t)$ , and horizon  $H$ . In our manipulation experiments we will restrict the tasks to sparse binary rewards  $r_i: S \to \{0, 1\}$ . The state space  $S$ , reward  $r_i$ , and dynamics  $P_i$  may vary across tasks.
+
+# 3.2 PROBLEM STATEMENT
+
+Our goal is to meta-train an agent such that can quickly learn a new test task  $\mathcal{T}_j$  in two phases:
+
+Phase I: The agent observes and learns from  $k = 1, \dots, K$  task demonstrations  $\mathcal{D}_j^* := \{\mathbf{d}_{j,k}\}$ . It can then attempt the task in  $\ell = 1, \dots, L$  trial episodes  $\{\tau_{j,\ell}\}$ , for which it receives reward labels.
+
+Phase II: The agent learns from those trial episodes and the original demos to succeed at  $\mathcal{T}_j$ .
+
+A demonstration is a trajectory  $\mathbf{d} = \{(s_t, a_t)\}$  of  $T$  states-action tuples that succeeds at the task, while a trial episode is a trajectory  $\pmb{\tau} = \{(s_t, a_t, r_i(s_t, a_t))\}$  that also contains reward information.
+
+# 3.3 LEARNING TO IMITATE AND TRY AGAIN
+
+With the aforementioned problem statement in mind, we aim to develop a method that can learn to learn from both demonstration and trial-and-error experience. We wish to (1) meta-learn a Phase I policy that is suitable for gathering information about a task given demonstrations, and (2) meta-learn a Phase II policy which learns from both demonstrations and trials produced by the Phase I policy. Like prior few shot meta-learning works (Finn et al., 2017b; Duan et al., 2017), we hope to achieve few shot success on a task by explicitly meta-training our Phase I and Phase II policies to learn from very little demonstration and trial data.
+
+We write the Phase I policy as  $\pi_{\theta}^{\mathrm{I}}(a|s,\{\mathbf{d}_{i,k}\})$ , where  $\theta$  represents all the learnable parameters.  $\pi^{\mathrm{I}}$  conditions on the task demonstrations  $\{\mathbf{d}_{i,k}\}$  which helps it infer what the unknown task is before attempting the trials. We condition the Phase II policy on both demonstration and trial data and write it as  $\pi_{\phi}^{\mathrm{II}}(a|s,\{\mathbf{d}_{i,k}\},\{\pmb{\tau}_{i,\ell}\})$ , with parameters  $\phi$ . One simple yet naive approach would be to use a single model across both phases: e.g., using a single MAML (Finn et al., 2017a) policy that sees the demonstration, executes trial(s) in the environment and then adapts its own weights from them. However, a key challenge in meta-training such a single model is that updates based on Phase II behavior (after the trials) will also change Phase I behavior (during the trials). Thus each meta-training update changes the distribution of trial trajectories  $\pmb{\tau}_{i,\ell} \sim \pi_{\theta}^{\mathrm{I}}(a|s,\{\mathbf{d}_{i,k}\})$  that the Phase II policy learns from. Prior meta-reinforcement learning work (Duan et al., 2016; Finn et al., 2017a) have addressed the issue of a changing trial distribution by re-collecting on-policy trial trajectories from the environment after every gradient step during meta-training, but this can be difficult in real-world problem settings with broad task distributions, where it is impractical to collect large amounts of on-policy experience.
+
+Instead, we represent and train  $\pi_{\theta}^{\mathrm{I}}, \pi_{\phi}^{\mathrm{II}}$  separately, decoupling their optimization. In particular, we train  $\pi_{\theta}^{\mathrm{I}}$  first, freeze its weights, and collect trial data  $\{\tau_{i,\ell}\}$  from the environment for each meta-training task  $\mathcal{T}_i$ . We then train  $\pi_{\phi}^{\mathrm{II}}$  using our collected trial data. Crucially,  $\theta$  and  $\phi$  are separate so training  $\pi_{\phi}^{\mathrm{II}}$  will not change  $\pi_{\theta}^{\mathrm{I}}$  behavior, and the distribution of trial data  $\{\tau_{i,l}\}$  for each task remains stationary. How do we train each of these policies with off-policy demonstration and trial data?  $\pi^{\mathrm{I}}$  must be trained in a way that will provide useful exploration for inferring the task. One simple and effective strategy for exploration is posterior or Thompson sampling (Russo et al., 2018; Rakelly et al., 2019), i.e. greedily act according to the policy's current belief of the task. To this end, we train  $\pi^{\mathrm{I}}$  using a meta-imitation learning setup, where for each task  $\mathcal{T}_i$  we assume access to a set of demonstrations  $\mathcal{D}_i^*$ .  $\pi^{\mathrm{I}}$  conditions on  $K$  demonstrations  $\{\mathbf{d}_{i,k}\} \subset \mathcal{D}_i^*$  and aims to
+
+![](images/4b3111f4a0bc5470bf5a4bd1322f4e9b2dfe8fec4360b768c74516f052bec96f.jpg)  
+Figure 2: Meta-training Overview: First, we meta-train  $\pi_{\theta}^{\mathrm{I}}$  according to Eq. 2. Next, we collect  $L$  trial trajectories per meta-training task  $\mathcal{T}_i$  in the environment using our trained  $\pi_{\theta}^{\mathrm{I}}$ . We denote the trial  $\{\pmb{\tau}_{i,\ell}\} \sim \pi_{\theta}^{\mathrm{I}}(a|s,\{\mathbf{d}_{i,k}\})$ , and store the resulting demo-trial pairs  $\{(\{\mathbf{d}_{i,k}\},\{\pmb{\tau}_{i,j}\})\}$  in a new dataset (blue). Finally, we meta-train  $\pi_{\phi}^{\mathrm{II}}$  according to Eq. 4.
+
+maximize the likelihood of the actions under another demonstration of the same task  $\mathbf{d}_i^{\mathrm{test}}\in \mathcal{D}_i^*$  (the test demo is distinct from the conditioning demos). This gives the following Phase I loss for  $\mathcal{T}_i$
+
+$$
+\mathcal {L} ^ {\mathrm {I}} (\theta , \mathcal {D} _ {i} ^ {*}) = \mathbb {E} _ {\left\{\mathbf {d} _ {i, k} \right\} \sim \mathcal {D} _ {i} ^ {*} \mathbb {E} _ {\mathbf {d} _ {i} ^ {\text {t e s t}}} \sim \mathcal {D} _ {i} ^ {*} \backslash \left\{\mathbf {d} _ {i, k} \right\} \mathbb {E} _ {(s _ {t}, a _ {t}) \sim \mathbf {d} _ {i} ^ {\text {t e s t}}} \left[ - \log \pi_ {\theta} ^ {\mathrm {I}} \left(a _ {t} \mid s _ {t}, \left\{\mathbf {d} _ {i, k} \right\}\right) \right] \tag {1}
+$$
+
+Then we can meta-train  $\pi_{\theta}^{\mathrm{I}}$  by minimizing Eq. 1 across the set of meta-train tasks  $\{\mathcal{T}_i\}$ :
+
+$$
+\min  _ {\theta} \frac {1}{| \{\mathcal {T} _ {i} \} |} \sum_ {\mathcal {T} _ {i} \in \left\{\mathcal {T} _ {i} \right\}} \mathcal {L} ^ {\mathrm {I}} (\theta , \mathcal {D} _ {i} ^ {*}) \tag {2}
+$$
+
+We train  $\pi^{\mathrm{II}}$  in a similar fashion, but additionally condition on  $L$  trial trajectories  $\{\tau_{i,\ell}\}$ , which are the result of executing  $\pi^{\mathrm{I}}$  in the environment. Suppose for any task  $\mathcal{T}_i$ , we have a set of demo-trial
+
+Algorithm 1 Watch-Try-Learn: Meta-training  
+1: Input: Training tasks  $\{\mathcal{T}_i\}$   
+2: Input: Demo data  $\mathcal{D}_i^* = \{\mathbf{d}_i\}$  per task  $\mathcal{T}_i$   
+3: Input: Number of training steps  $N$   
+4: Randomly initialize  $\theta, \phi$   
+5: for step  $= 1, \dots, N$  do  
+6: Sample meta-training task  $\mathcal{T}_i$   
+7: Update  $\theta$  with  $\nabla_{\theta} \mathcal{L}^{1}(\theta, \mathcal{D}_{i}^{*})$  (see Eq. 1)  
+8: end for  
+9: for  $\mathcal{T}_i \in \{\mathcal{T}_i\}$  do  
+10: Initialize empty  $\mathcal{D}_i$  for demo-trial pairs.  
+11: while not done do  
+12: Sample  $K$  demonstrations  $\{\mathbf{d}_{i,k}\} \sim \mathcal{D}_i^*$   
+13: Collect  $L$  trials  $\{\pmb{\tau}_{i,l}\} \sim \pi_{\theta}^{T}(a|s, \{\mathbf{d}_{i,k}\})$   
+14: Update  $\mathcal{D}_i \gets \mathcal{D}_i \cup \{(\{\mathbf{d}_{i,k}\}, \{\pmb{\tau}_{i,l}\})\}$   
+15: end while  
+16: end for  
+17: for step  $= 1, \dots, N$  do  
+18: Sample meta-training task  $\mathcal{T}_i$   
+19: Update  $\phi$  with  $\nabla_{\phi} \mathcal{L}^{\mathrm{II}}(\phi, \mathcal{D}_i, \mathcal{D}_i^*)$  (see Eq. 3)  
+20: end for  
+21: return  $\theta, \phi$
+
+Algorithm 2 Watch-Try-Learn: Meta- testing   
+1: Input: Test tasks  $\{\mathcal{T}_j\}$    
+2: Input: Demo data  $D_j^*$  for task  $\mathcal{T}_j$    
+3: for  $\mathcal{T}_j\in \{\mathcal{T}_j\}$  do   
+4: Sample  $K$  demonstrations  $\{\mathbf{d}_{j,k}\} \sim \mathcal{D}_j^*$    
+5: Collect  $L$  trials  $\{\tau_{j,l}\}$  with policy  $\pi_{\theta}^{\mathrm{I}}(a|s,\{\mathbf{d}_{j,k}\})$    
+6: Perform task with re-trial policy  $\pi_{\phi}^{\mathrm{II}}(a|s,\{\mathbf{d}_{j,k}\},\{\tau_{j,l}\})$    
+7: end for
+
+pairs  $\mathcal{D}_i = \{(\{\mathbf{d}_{i,k}\},\{\pmb {\tau}_{i,j}\})\}$ . Then the Phase II objective for  $\mathcal{T}_i$  is:
+
+$$
+\mathcal {L} ^ {\mathrm {I I}} (\phi , \mathcal {D} _ {i}, \mathcal {D} _ {i} ^ {*}) = \mathbb {E} _ {(\{\mathbf {d} _ {i, k} \}, \{\tau_ {i, \ell} \}) \sim \mathcal {D} _ {i}} \mathbb {E} _ {\mathbf {d} _ {i} ^ {\text {t e s t}}} \sim \mathcal {D} _ {i} ^ {*} \backslash \left\{\mathbf {d} _ {i, k} \right\} \mathbb {E} _ {(s _ {t}, a _ {t}) \sim \mathbf {d} _ {i} ^ {\text {t e s t}}} \left[ - \log \pi_ {\phi} ^ {\mathrm {I I}} (a _ {t} | s _ {t}, \left\{\mathbf {d} _ {i, k} \right\}, \left\{\tau_ {i, \ell} \right\}) \right] \tag {3}
+$$
+
+Eq. 3 encourages  $\pi^{\mathrm{II}}$  to use and improve upon the trial experience  $\{\pmb{\tau}_{i,\ell}\}$ , which includes reward information. If a trial has high reward, then  $\pi^{\mathrm{I}}$  likely inferred the task correctly from demonstration alone, and  $\pi^{\mathrm{II}}$  should reinforce that high reward trial behavior. If a trial has low reward, then  $\pi^{\mathrm{I}}$  probably inferred the task incorrectly and  $\pi^{\mathrm{II}}$  should avoid that low reward trial behavior. Hence even failed trials can help  $\pi^{\mathrm{II}}$  correctly infer the task by showing it what not to do. Note that since Eq. 3 evaluates log likelihood at states and actions from the held out demonstration  $\mathbf{d}_i^{\mathrm{est}}$ ,  $\pi^{\mathrm{II}}$  cannot simply copy behavior from high reward trials in  $\{\pmb{\tau}_{i,\ell}\}$  but instead must generalize from the trials to a new instance of the same task. We meta-train  $\pi_{\phi}^{\mathrm{II}}$  by minimizing Eq. 3 across the meta-train tasks:
+
+$$
+\min  _ {\phi} \frac {1}{| \{\mathcal {T} _ {i} \} |} \sum_ {\mathcal {T} _ {i} \in \{\mathcal {T} _ {i} \}} \mathcal {L} ^ {\mathrm {I I}} (\phi , \mathcal {D} _ {i}, \mathcal {D} _ {i} ^ {*}) \tag {4}
+$$
+
+We refer to our approach as Watch-Try-Learn (WTL), and describe our meta-training and meta-test procedures in detail in Alg. 1 and Alg. 2, respectively. We also illustrate the meta-training flow in Fig. 2. In practice we meta-train  $\pi_{\theta}^{\mathrm{I}}$  and  $\pi_{\theta}^{\mathrm{II}}$  by solving Eqs. 2 and 4 using stochastic gradient descent or some variant. We iteratively sample (minibatches of) tasks  $\mathcal{T}_i$  to compute gradient updates on  $\theta$  or  $\phi$ , respectively. At meta-test time, for any test task  $\mathcal{T}_j$  we receive demonstrations  $\{\mathbf{d}_{j,k}\}$  and obtain the demo-conditioned Phase I policy  $\pi_{\theta}^{\mathrm{I}}(a|s,\{\mathbf{d}_{j,k}\})$ . We collect trial(s)  $\{\tau_{j,l}\}$  using  $\pi_{\theta}^{\mathrm{I}}$  in the environment. Then we obtain the Phase II policy  $\pi_{\phi}^{\mathrm{II}}(a|s,\{\mathbf{d}_{j,k}\},\{\tau_{j,l}\})$ . Finally, we execute the  $\pi_{\phi}^{\mathrm{II}}$  in the environment to solve the task.
+
+# 3.4 WATCH-TRY-LEARN IMPLEMENTATION
+
+WTL and Alg. 1 allow for general representations of the Phase I policy  $\pi_{\theta}^{\mathrm{I}}(a|s, \{\mathbf{d}_{i,k}\})$  so long as it conditions on the task demonstrations  $\{\mathbf{d}_{i,k}\}$ . Similarly, the Phase II policy  $\pi^{\mathrm{II}}(a|s, \{\mathbf{d}_{i,k}\}, \{\pmb{\tau}_{i,\ell}\})$  must condition on both  $\{\mathbf{d}_{i,k}\}$  and the trials  $\{\pmb{\tau}_{i,\ell}\}$ . Hence a variety of adaptation mechanisms could be used.
+
+We choose to implement this conditioning in each policy by embedding the demonstration data and (for Phase II) the trial data into context vectors using neural networks. Figure 3 illustrates the  $\pi_{\phi}^{\mathrm{II}}$  architecture assuming a single demonstration and trial. The embedding network first applies a
+
+![](images/be8701cc84a71ac999a040ff402ef9138de1b8a54712b282837b90fe5a808327.jpg)  
+Figure 3: Our vision-based Phase II architecture: Upper left: we pass the RGB observation for each timestep through a 4-layer CNN with ReLU activations and layer normalization, followed by a spatial softmax layer that extracts 2D keypoints (Levine et al., 2016). We flatten the output keypoints and concatenate them with the current gripper pose, gripper velocity, and context embedding. Upper Right: We pass the resulting vector through the actor network, which predicts the parameters of a Gaussian mixture over the commanded end-effector position, axis-angle orientation, and finger angle. Lower left: To produce the context embedding, the embedding network applies a vision network to 40 ordered observations sampled randomly from the demo and trial trajectories. We concatenate the demo and trial outputs with the trial episode rewards along the embedding feature dimension, then apply a  $10 \times 1$  convolution across the time dimension, flatten, and apply a MLP to produce the final context embedding. The Phase I policy architecture (Appendix Fig. 8) is the same as shown here, but omits the concatenation of trial embeddings and trial rewards.
+
+vision network to each demonstration and trial trajectory image  $s_t$ , producing demo and trial feature matrices (respectively) where each row corresponds to one timestep's features. We concatenate the demo and trial feature matrices together on the feature (horizontal) dimension, along with the trial episode rewards to produce a single matrix combining demo and trial observation information and trial reward information. We then apply a 1-D convolution along the time (vertical) dimension of this matrix and flatten to obtain a single vector that integrates and aggregates information across time. Finally we apply a small MLP to produce the context embedding vector which contains information about both the demos  $\{\mathbf{d}_{i,k}\}$  and the trials  $\{\tau_{i,\ell}\}$ . The Phase I policy  $\pi_{\theta}^{\mathrm{I}}$ , illustrated in Appendix Figure 8, produces a similar context embedding but only uses demonstration data. This architectural design resembles prior contextual meta-learning works (Duan et al., 2016; Mishra et al., 2018; Duan et al., 2017; Rakelly et al., 2019), which have previously considered how to meta-learn efficiently from one modality of data (trials or demonstrations), but not how to integrate multiple sources, including off-policy trial data.
+
+Since each policy is additionally conditioned on the current state  $s$ , we concatenate the context embedding with the current state features before feeding both as input to an actor network, which produces a Gaussian mixture distribution (Bishop, 1994) over actions. The current state features include visual features produced by another vision network, distinct from the one used to produce the context embedding. Both vision networks use a fairly standard convolutional neural network architecture with a final spatial softmax layer that extracts keypoints (Levine et al., 2016). The entire  $\pi_{\phi}^{\mathrm{II}}$  architecture, illustrated in Figure 3, is trainable end-to-end using backpropagation on Eq. 4, where the parameters  $\phi$  represent the collective weights of all layers. Similarly, the entire  $\pi_{\theta}^{\mathrm{I}}$  architecture depicted in Appendix Figure 8 is trained by backpropagation on Eq. 2 where  $\theta$  represents the set of weights across all layers. Note that since each neural network architecture is fixed and shared across tasks, we expect the input state dimensions to be the same across tasks, though the content (for example, the objects in the scene) may vary.
+
+# 4 EXPERIMENTS
+
+In our experiments, we aim to evaluate our method on challenging few-shot learning domains that span multiple task families, where the agent must use both demonstrations and trial-and-error to effectively infer a policy for the task. Prior meta-imitation benchmarks (Duan et al., 2017; Finn
+
+![](images/f5b857cf4e4f5b354261e273ba07d6e8ec3697a4c077abf8224566b98c4ac88a.jpg)  
+Figure 5: Illustration of example episodes in four distinct task families: button pressing, grasping, sliding, and pick-and-place. Each column shows the first and last frames of an episode top-to-bottom. We meta-train each model on hundreds of tasks from each of these task families. For each task within a task family, we use a unique pair of kitchenware objects sampled from a set of nearly one hundred different objects.
+
+et al., 2017b) generally contain only a few tasks, and these tasks can be easily disambiguated given a single demonstration. Meanwhile, prior meta-reinforcement learning benchmarks (Duan et al., 2016; Finn et al., 2017a) tend to contain fairly similar tasks that a meta-learner can solve with little exploration and no demonstration at all. Motivated by these shortcomings, we design two new problems where a meta-learner can leverage a combination of demonstration and trial experience: a toy reaching problem and a challenging multitask gripper control problem, described below. We evaluate how the following methods perform in those environments:
+
+BC: A behavior cloning method that does not condition on either demonstration or trial-and-error experience, trained across all meta-training data. We train BC policies using maximum log-likelihood with expert demonstration actions.
+
+MIL (Finn et al., 2017b; James et al., 2018): A meta-imitation learning method that conditions on demonstration data, but does not leverage trial-and-error experience. We train MIL policies to minimize Eq. 1 similar to the WTL Phase I policy, but MIL methods lack a Phase II step. To perform a controlled comparison, we use the same architecture for both MIL and WTL.
+
+WTL: Our Watch-Try-Learn method, which conditions on demonstration and trial experience. In all experiments, the agent receives  $K = 1$  demonstration and can take  $L = 1$  trial.
+
+$\mathbf{BC} + \mathbf{SAC}$ : In the gripper environment we study how much trial-and-error experience soft actor critic (SAC) (Haarnoja et al., 2018), a state of the art reinforcement learning algorithm, would require to solve a single task. While WTL meta-learns a single model that needs just one trial episode per meta-test task, in "BC + SAC" we fine-tune a separate RL agent for each meta-test task and analyze how much trial experience it needs to match WTL's single trial performance. We pre-train a policy similar to BC, then fine-tune for each meta-test task using SAC.
+
+# 4.1 REACHING
+
+# ENVIRONMENT EXPERIMENTS
+
+To first verify that our method can actually leverage demonstration and trial experience in a simplified problem domain, we begin with toy
+
+planar reaching tasks inspired by Finn et al. (2017b) and illustrated in Appendix Fig 7. A demonstrator shows which of two objects to reach towards, but the agent's dynamics are randomized per task and may not match the demonstrator's. This simulates a domain adaptive setting such as a robot imitating a video of a human. Since the demonstrations  $\{\mathbf{d}_{i,k}\}$  do not help identify the unknown
+
+![](images/e798a12b707605082ac2d04e057d6915999068944e7846a9e5ddc8c6625c0665.jpg)  
+Figure 4: Average return of each method on held out meta-test tasks in the reaching environment, after one demonstration and one trial. Our Watch-Try-Learn (WTL) method is quickly able to learn to imitate the demonstrator. Each line shows the average over 5 separate training runs with identical hyperparameters, evaluated on 50 randomly sampled meta-test tasks. Shaded regions indicate  $95\%$  confidence intervals.
+
+![](images/b17ae4c494e1443f3ed61f8aedd8d8d1d6788e45cc2e9d47c7f6f0cd75f2705b.jpg)  
+Figure 6: The average success rate of different methods in the gripper control environment, for both state space (non-vision) and vision based policies. The leftmost column displays aggregate results across all task families. Our Watch-Try-Learn (WTL) method significantly outperforms the meta-imitation (MIL) baseline, which in turn outperforms the behavior cloning (BC) baseline. We conducted 5 training runs of each method with identical hyperparameters and evaluated each run on 40 held out meta-test tasks. Error bars indicate  $95\%$  confidence intervals.
+
+task dynamics, the agent must use trial episodes to successfully reach the target object. During meta-training, WTL meta-learns from demonstrations with the correct dynamics  $\left(\mathbf{d}_i^{\mathrm{est}}\right.$  in Eq. 3) how to adapt to unknown dynamics in new tasks. To obtain expert demonstration data, we first train a reinforcement learning agent using normalized advantage functions (Gu et al., 2016), where the agent receives oracle observations that show only the true target. With our trained expert demonstration agent, we collect 2 demonstrations per task for 10000 meta-training tasks and 1000 meta-test tasks. For these toy experiments we use simplified versions of the architecture in Fig. 3 as described in Appendix C.
+
+The results in Fig 4 show WTL is able to quickly learn to imitate the expert, while methods that do not leverage trial information struggle due to the uncertainty in the task dynamics.
+
+# 4.2 GRIpper ENVIRONMENT EXPERIMENTS
+
+The gripper environment is a realistic 3-D simulation as shown in Figure 5. Gripper tasks fall into four broad task families: button pressing, grasping, pushing, and pick and place. Within a task family, each task involves a different pair of kitchenware objects sampled from a set of nearly one hundred. Unlike prior meta-imitation learning benchmarks, tasks of the same family that are qualitatively similar still have subtle but consequential differences. Some pushing tasks might require the agent to always push the left object towards the right object, while others might require the agent to always push, for example, the cup towards the teapot. The agent controls a free floating gripper with 7-D action space. The vision based policies receive image observations and gripper state, while state-space policies receive a vector of the gripper state and poses for all non-fixed objects in the scene. Gripper environment episodes have a maximum length of 5 seconds and contain sparse binary rewards. In an HTC Vive virtual reality setup, a human demonstrator recorded 1536 demonstrations for 768 distinct tasks involving 96 distinct sets of kitchenware objects. We held out 40 tasks corresponding to 5 sets of kitchenware objects for our meta-validation dataset, which we used for hyperparameter selection. Similarly, we selected and held out 5 object sets of 40 tasks for our meta-test dataset, which we used for final evaluations. Refer to Appendix A for a more detailed description of the scene setup, task families, and reward functions.
+
+We trained and evaluated MIL, BC, and WTL policies with both state-space observations and vision observations. Appendix C describes hyperparameter selection using the meta-validation tasks and Appendix C.3 analyzes the sample and time complexity of WTL. The MIL policy uses an identical
+
+architecture and objective to the WTL trial policy, while the BC policy architecture is the same as the WTL trial policy without the any embedding components. For vision based models, we crop and resize image observations from  $300 \times 220$  to  $100 \times 100$  before providing them as input.
+
+We show the meta-test task success rates in Fig. 6. Overall, in both state space and vision domains, we find that WTL outperforms MIL and BC by a substantial margin, indicating that it can effectively leverage information from the trial and integrate it with that of the demonstration in order to achieve greater performance.
+
+Finally, for the BC + SAC comparison we pre-trained an actor with behavior cloning and fine-tuned 4 RL agents per task with identical hyperparameters using the TFAgents (Guadarrama et al., 2018) SAC implementation. Table 1 shows that BC + SAC fine-tuning typically requires thousands of trial episodes per task to reach the same performance our meta-trained WTL method achieves after one demonstration and a single trial episode. Appendix C.4 shows the BC + SAC training curves averaged across the different meta-test tasks.
+
+# 5 DISCUSSION AND FUTURE WORK
+
+We proposed a meta-learning algorithm that allows an agent to quickly learn new behavior from a single demonstration followed by trial experience and associated (possibly sparse) rewards. The demonstration allows the agent to infer the type of task to be performed, and the trials enable it to improve its performance by resolving ambiguities in new test time situations. We presented experimental results where the agent is meta-trained on a broad distribution of tasks, after which it is able to quickly learn tasks with new held-out objects from just one demonstration and a trial. We showed that our approach outperforms prior meta-imitation approaches in challenging experimental domains.
+
+As illustrated in our qualitative failure analysis (Appendix D), one area of future improvement involves improving the informativeness of even
+
+failed trial trajectories generated by  $\pi^{\mathrm{I}}$ . In future work, we hope to explore alternatives to posterior sampling that produce more informative trials while maintaining sample and computational efficiency. We also plan to explore ways of extending our approach to be meta-trained on a much broader range of tasks, testing the performance of the agent on completely new held-out tasks rather than on held-out objects.
+
+The Watch-Try-Learn (WTL) approach enables a natural way for non-expert users to train agents to perform new tasks: by demonstrating the task and then observing and critiquing the performance of the agent on the task if it initially fails. WTL achieves this through a unique combination of demonstrations and trials in the inner loop of a meta-learning system, where the demonstration guides the exploration process for subsequent trials, and the use of trials allows the agent to learn new task objectives which may not have been seen during meta-training. We hope that this work paves the way towards more practical and general algorithms for meta-learning behavior.
+
+# 6 ACKNOWLEDGEMENTS
+
+We would like to thank Luke Metz and Archit Sharma for reviewing an earlier draft of this paper, and Alex Irpan for valuable discussions. We would also like to thank Murtaza Dalal for finding and correcting an error in the "BC+SAC" experimental results from an earlier version of this paper.
+
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+Ashvin Nair, Bob McGrew, Marcin Andrychowicz, Wojciech Zaremba, and Pieter Abbeel. Overcoming exploration in reinforcement learning with demonstrations. International Conference on Robotics and Automation (ICRA), 2018.  
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+Daniel J Russo, Benjamin Van Roy, Abbas Kazerouni, Ian Osband, Zheng Wen, et al. A tutorial on thompson sampling. Foundations and Trends® in Machine Learning, 11(1):1-96, 2018.  
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+Jürgen Schmidhuber. Evolutionary principles in self-referential learning. PhD thesis, Institut für Informatik, Technische Universität München, 1987.  
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+
+# A GRIpper ENVIRONMENT DETAILS
+
+# A.1 SCENE SETUP
+
+We created the gripper environment using the Bullet physics engine (Coumans & Bai, 2016). The agent controls a floating gripper that can move and rotate freely in space (6 DOFs), and 2 symmetric fingers that open and close together (1 DOF). For vision-based policies, the environment provides  $300 \times 220$  RGB image observations and the 7-D gripper position vector at each timestep. For state-space policies it provides the 7-D gripper state and the 6-D poses (translation + rotation) of all non-fixed objects in the scene. The virtual scene is set up with the same viewpoint and table as in Figure 5 with random initial positions of the gripper, as well as two random kitchenware objects and a button panel that are placed on the table. The button panel contains two press-able buttons of different colors. One of the two kitchenware objects is placed onto a random location on the left half of the table, and the other kitchenware object is placed onto a random location on the right half of the table. After the demonstration episode of each task, the kitchenware objects and button panel are repositioned: a kitchenware object on the left half of the table moves to a random location on the right half of the table, and vice versa. Similarly, the two colored buttons in the button panel swap positions. Swapping the lateral positions of objects and buttons after the demonstration is crucial because otherwise, for example, the difference between the two types of pushing tasks would be meaningless.
+
+# A.2 TASK FAMILIES
+
+Our gripper environment has four broad task families: button pressing, grasping, sliding, and pick and place. But tasks in each family may come in one of two types:
+
+- Button pressing 1: the demonstrator presses the left (respectively, right) button, and the agent must press the left (respectively, right) button.  
+- Button pressing 2: the demonstrator presses one of the two buttons, and the agent must press the button of the same color.  
+- Grasping 1: the demonstrator grasps and lifts the left (respectively, right) object. The agent must grasp and lift the left (respectively, right) object.  
+- Grasping 2: the demonstrator grasps and lifts one of the two objects. The agent must grasp and lift that same object.  
+- Sliding 1: the demonstrator pushes the left object into the right object (respectively, the right object into the left object). The agent must push the left object into the right object (respectively, the right object into the left object).  
+- Sliding 2: the demonstrator pushes one object (A) into the other (B). The agent must push object A into object B.  
+- Pick and Place 1: the demonstrator picks up one of the objects and places it on the near (respectively, far) edge of the table. The agent must pick up the same object and place it on the near (respectively, far) edge of the table.  
+- Pick and Place 2: the demonstrator picks up one of the objects and places it on the left (respectively, right) edge of the table. The agent must pick up the same object and place it on the left (respectively, right) edge of the table.
+
+# A.3 EPISODES AND REWARDS
+
+A gripper environment episode has a maximum length of 50 timesteps, or 5 seconds real time. At each timestep the environment returns a reward of  $+1$  if the agent successfully completes the task, or 0 otherwise. The episode ends immediately upon success, so only the terminal timestep can have nonzero reward. Computing success (and hence the reward) depends on the current task's family. For each task family, the environment determines success at the current timestep if:
+
+- Button pressing: the gripper is in contact with the correct button.  
+- Grasping: the correct object's vertical position is a certain threshold above the table.  
+- Sliding: the two objects are in contact, and object A has moved more than object B.  
+- Pick and Place: the correct object touches the correct edge of the table, and has cleared some vertical threshold at some prior point in the episode.
+
+![](images/78744021145330841601809b32e4b5ed81906f81fff185a34c42566ee9c75f07.jpg)  
+Figure 7: Our reaching toy environment with 2 target objects and 2 degrees of freedom (DOFs), with per-task randomized dynamics.
+
+# B REACHER ENVIRONMENT DETAILS
+
+Figure 7 depicts our reaching toy environment. The agent controls a 2 degree of freedom (DOF) arm and should reach to one of the two randomly positioned objects. Crucially, the agent's dynamics are unknown and randomized at the start of each task: each of the two joints may have reversed orientation with  $50\%$  probability, and the agent's joint orientations may not match that of the task demonstration  $\mathbf{d}_{i,k}$ . This simulates a domain adaptive setting, for example the discrepancy in dynamics when a robot imitates from a human video demonstration. Since simply observing the demonstration is not sufficient to identify the task dynamics, the agent must use its own trials to identify the task dynamics and successfully reach the target object. Following (Brockman et al., 2016), the environment returns a reward at each timestep which penalizes the researcher's distance to the target object and the magnitude of its control inputs at that timestep. Precisely, let  $\phi : S \to \mathbb{R}^3$  map the agent's state to the researcher tip position, and  $x_i \in \mathbb{R}^3$  be the true target object's position. Then:
+
+$$
+r _ {i} (s, a) = - \left\| \phi (s) - x _ {i} \right\| - \left\| a \right\| ^ {2} \tag {5}
+$$
+
+# C EXPERIMENTAL DETAILS
+
+We trained all policies using the ADAM optimizer (Kingma & Ba, 2015), on varying numbers of Nvidia Tesla P100 GPUs. Whenever there is more than 1 GPU, we use synchronized gradient updates and the batch size refers to the batch size of each individual GPU worker.
+
+# C.1 REACHER ENVIRONMENT MODELS
+
+For the reacher toy problem, every neural network in both the WTL policies and the baselines policies uses the same architecture: two hidden layers of 100 neurons each, with ReLU activations on each layer except the output layer, which has no activation.
+
+Rather than mixture density networks, our BC and MIL policies are deterministic policies trained by standard mean squared error (MSE). The WTL Phase II policy  $\pi^{\mathrm{II}}$  is also deterministic and trained by MSE, while the Phase I policy  $\pi^{\mathrm{I}}$  policy is stochastic and samples actions from a Gaussian distribution with learned mean and diagonal covariance (equivalently, it is a simplification of the MDN to a single component). We found that a deterministic policy works best for maximizing the MIL baseline's average return, while a stochastic WTL trial policy achieves lower returns itself but facilitates easier learning in Phase II. Embedding architectures are also simplified: the demonstration "embedding" is simply the state of the last demonstration trajectory timestep, while the trial embedding architecture replaces the temporal convolution and flatten operations with a simple average over per-timestep trial embeddings.
+
+We trained all policies for 50000 steps using a batch size of 100 tasks and a .001 learning rate, using a single GPU operating at 25 gradient steps per second.
+
+![](images/fd37fffbbe79c634977d85e9966b8bef5b5a6289aa3124a06885d22f07e29fc4.jpg)  
+Figure 8: Our vision-based Phase I architecture. This is identical to the architecture in Figure 3, except that there are no trial embeddings or trial rewards to concatenate in the conditioning network.
+
+Table 2: Best Model Hyperparameters for Gripper Environment Experiments  
+
+<table><tr><td>METHOD</td><td>LEARNING RATE</td><td>MDN COMPONENTS</td></tr><tr><td colspan="3">VISION MODELS</td></tr><tr><td>WTL</td><td>9.693 × 10-4</td><td>20</td></tr><tr><td>MIL</td><td>9.397 × 10-4</td><td>20</td></tr><tr><td>BC</td><td>1.093 × 10-3</td><td>20</td></tr><tr><td colspan="3">STATE-SPACE MODELS</td></tr><tr><td>WTL</td><td>2.138 × 10-3</td><td>20</td></tr><tr><td>MIL</td><td>2.659 × 10-3</td><td>20</td></tr><tr><td>BC</td><td>2.138 × 10-3</td><td>20</td></tr></table>
+
+# C.2 GRIPPER ENVIRONMENT MODELS
+
+Algorithm 3 Demonstration and Trial Subsampling  
+```txt
+1: Input: Demonstration  $\mathbf{d} = \{(s_t, a_t)\}_{t=1}^T$  (or WLOG, trial  $\tau$ ), for  $T > 2$   
+2: Input: Output length  $U = 40$   
+3: Initialize  $\hat{\mathbf{d}} = \{(s_1, a_1), (s_T, a_T)\}$   
+4: for step  $= 1, \dots, U - 2$  do  
+5: Sample  $i$  uniformly at random from  $\{2, \dots, T-1\}$ , with replacement.  
+6:  $\hat{\mathbf{d}} \gets \hat{\mathbf{d}} \cup \{(s_i, a_i)\}$   
+7: end for  
+8: return sort( $\hat{\mathbf{d}}$ )
+```
+
+The gripper environment policies illustrated in Figure 3 and Figure 8 produce context embeddings from demonstrations and (for Phase II) trial trajectories. The demonstrations and trials are variable length sequential data, since each gripper environment episode will terminate early if successful. To avoid dealing with variable sized inputs, we subsample 40 timesteps with replacement from any demonstration or trial trajectory to provide as input to our policies. We describe the subsampling process in Alg. 3. Since our gripper environment rewards are sparse and only nonzero on the final timestep, the subsampling process always selects the first and last timesteps. It selects the remaining 38 timesteps uniformly at random from the noninitial and nonterminal timesteps of the original demonstration or trial. It returns the 40 selected timesteps in sorted order for temporal consistency.
+
+For each state-space and vision method we ran a simple hyperparameter sweep. We tried  $K = 10$  and  $K = 20$  MDN mixture components, and for each  $K$  we sampled 3 learning rates log-uniformly from the range [.001,.01] (state-space) or [.0001,.01] (vision), leading to 6 hyperparameter experiments per method. We selected the best hyperparameters by average success rate on the meta
+
+validation tasks, see Table 2. We trained all gripper models with a 11 GPU workers and a batch size of 8 tasks for 500000 (state-space) and 60000 (vision) steps. Training the WTL trial and re-trial policies on the vision-based gripper environment takes 14 hours each.
+
+# C.3 ALGORITHMIC COMPLEXITY OF WTL
+
+WTL's trial and re-trial policies are trained off-policy via imitation learning, so only two phases of data collection are required: the set of demonstrations for training the trial policy, and the set of trial policy rollouts for training the re-trial policy. The time and sample complexity of data collection is linear in the number of tasks, for which only one demo and one trial is required per task. Furthermore, because the optimization of trial and re-trial policies are de-coupled into separate learning stages, the sample complexity is fixed with respect to hyperparameter sweeps. A fixed dataset of demos is used to obtain a good trial policy, and a fixed dataset of trials (from the trial policy) is used to obtain a good re-trial policy.
+
+The forward-backward pass of the vision network is the dominant factor in the computational cost of training the vision-based Phase II architecture. Thus, the time complexity for a single SGD update to WTL is  $O(T)$ , where  $T$  is the number of sub-sampled frames used to compute the demo and trial visual features for forming the contextual embedding. However, in practice the model size and number of sub-sampled frames  $T = 40$  are small enough that computing embeddings for all frames in demos and trials can be vectorized efficiently on GPUs.
+
+# C.4 BC + SAC BASELINE TRAINING CURVE
+
+![](images/b716f3c8fdfc91345cf38f45af475d7b250d64f757596527a955b34faa1d0eab.jpg)  
+Figure 9: For "BC+SAC", we pre-train agents with behavior cloning and use RL to fine-tune on each meta-test task. By comparison, WTL uses a demo and a single trial episode per task (< 50 environment steps). X-axis (log-scale) shows environment experience steps collected per task. Error bars indicate  $95\%$  confidence intervals.
+
+# D FAILURE ANALYSIS
+
+We performed a qualitative failure analysis of our vision-based WTL method on meta-test tasks by manually inspecting videos of Phase I and Phase II policy behavior. Table 3 shows a matrix of different outcomes, where each episode is classified into one of three categories: 1) successfully completing the task, 2) performing the task but on the wrong object or location, 3) moves toward the correct object but misses (often comes close but gets stuck). One common failure mode is when the agent reaches towards but misses the correct object in Phase I, which does not provide the Phase II policy with enough information to disambiguate the correct task.
+
+Table 3: WTL outcome matrix in the vision-based gripper environment. This table shows frequencies of outcomes after Phase I (rows) and after Phase II (columns) over 100 meta-test tasks.  
+
+<table><tr><td></td><td colspan="4">Phase II</td></tr><tr><td rowspan="4">Phase I</td><td></td><td>Missed Object</td><td>Wrong Object</td><td>Success</td></tr><tr><td>Missed Object</td><td>23</td><td>12</td><td>15</td></tr><tr><td>Wrong Object</td><td>5</td><td>11</td><td>8</td></tr><tr><td>Success</td><td>6</td><td>3</td><td>20</td></tr></table>
\ No newline at end of file
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+# WEAKLY SUPERVISED CLUSTERING BY EXPLOITING UNIQUE CLASS COUNT
+
+Mustafa Umit Oner<sup>1,2</sup>, Hwee Kuan Lee<sup>1,2,3,4</sup> & Wing-Kin Sung<sup>1,5</sup>
+
+$^{1}$ School of Computing, National University of Singapore, Singapore 117417,  $^{2}$ A*STAR Bioinformatics Institute, Singapore 138671,  $^{3}$ Image and Pervasive Access Lab (IPAL), CNRS UMI 2955, Singapore 138632,  $^{4}$ Singapore Eye Research Institute, Singapore 169856,  $^{5}$ A*STAR Genome Institute of Singapore, Singapore 138672
+
+{umitoner,ksung}@comp.nus.edu.sg,{leehk}@bii.a-star.edu.sg
+
+# ABSTRACT
+
+A weakly supervised learning based clustering framework is proposed in this paper. As the core of this framework, we introduce a novel multiple instance learning task based on a bag level label called unique class count (ucc), which is the number of unique classes among all instances inside the bag. In this task, no annotations on individual instances inside the bag are needed during training of the models. We mathematically prove that with a perfect ucc classifier, perfect clustering of individual instances inside the bags is possible even when no annotations on individual instances are given during training. We have constructed a neural network based ucc classifier and experimentally shown that the clustering performance of our framework with our weakly supervised ucc classifier is comparable to that of fully supervised learning models where labels for all instances are known. Furthermore, we have tested the applicability of our framework to a real world task of semantic segmentation of breast cancer metastases in histological lymph node sections and shown that the performance of our weakly supervised framework is comparable to the performance of a fully supervised Unet model.
+
+# 1 INTRODUCTION
+
+In machine learning, there are two main learning tasks on two ends of scale bar: unsupervised learning and supervised learning. Generally, performance of supervised models is better than that of unsupervised models since the mapping between data and associated labels is provided explicitly in supervised learning. This performance advantage of supervised learning requires a lot of labelled data, which is expensive. Any other learning tasks reside in between these two tasks, so are their performances. Weakly supervised learning is an example of such tasks. There are three types of supervision in weakly supervised learning: incomplete, inexact and inaccurate supervision. Multiple instance learning (MIL) is a special type of weakly supervised learning and a typical example of inexact supervision (Zhou, 2017). In MIL, data consists of bags of instances and their corresponding bag level labels. Although the labels are somehow related to instances inside the bags, the instances are not explicitly labeled. In traditional MIL, given the bags and corresponding bag level labels, task is to learn the mapping between bags and labels while the goal is to predict labels of unseen bags (Dietterich et al., 1997; Foulds & Frank, 2010).
+
+In this paper, we explore the feasibility of finding out labels of individual instances inside the bags only given the bag level labels, i.e. there is no individual instance level labels. One important application of this task is semantic segmentation of breast cancer metastases in histological lymph node sections, which is a crucial step in staging of breast cancer (Brierley et al., 2016). In this task, each pathology image of a lymph node section is a bag and each pixel inside that image is an instance. Then, given the bag level label that whether the image contains metastases or not, the task is to label each pixel as either metastases or normal. This task can be achieved by asking experts to exhaustively annotate each metastases region in each image. However, this exhaustive annotation process is tedious, time consuming and more importantly not a part of clinical workflow.
+
+![](images/d302dc6c7b90c1f04b89c4d2684d4c0f3a3802a8e52387b30ee4fb6972b66594.jpg)  
+Figure 1: Weakly supervised clustering framework. Our framework (green dashed line) consists of the UCC model (magenta dashed line) and the unsupervised instance clustering branch.
+
+In many complex systems, such as in many types of cancers, measurements can only be obtained at coarse level (bag level), but information at fine level (individual instance level) is of paramount importance. To achieve this, we propose a weakly supervised learning based clustering framework. Given a dataset consisting of instances with unknown labels, our ultimate objective is to cluster the instances in this dataset. To achieve this objective, we introduce a novel MIL task based on a new kind of bag level label called unique class count  $(ucc)$ , which is the number of unique classes or the number of clusters among all the instances inside the bag. We organize the dataset into non-empty bags, where each bag is a subset of individual instances from this dataset. Each bag is associated with a bag level  $ucc$  label. Then, our MIL task is to learn mapping between the bags and their associated bag level  $ucc$  labels and then to predict the  $ucc$  labels of unseen bags. We mathematically show that a  $ucc$  classifier trained on this task can be used to perform unsupervised clustering on individual instances in the dataset. Intuitively, for a  $ucc$  classifier to count the number of unique classes in a bag, it has to first learn discriminant features for underlying classes. Then, it can group the features obtained from the bag and count the number of groups, so the number of unique classes.
+
+Our weakly supervised clustering framework is illustrated in Figure 1. It consists of a neural network based  $ucc$  classifier, which is called as Unique Class Count (UCC) model, and an unsupervised clustering branch. The UCC model accepts any bag of instances as input and uses  $ucc$  labels for supervised training. Then, the trained UCC model is used as a feature extractor and unsupervised clustering is performed on the extracted features of individual instances inside the bags in the clustering branch. One application of our framework is the semantic segmentation of breast cancer metastases in lymph node sections (see Figure 4). The problem can be formulated as follows. The input is a set of images. Each image (bag) has a label of  $ucc1$  (image is fully normal or fully metastases) or  $ucc2$  (image is a mixture of normal and metastases). Our aim is to segment the pixels (instances) in the image into normal and metastases. A UCC model can be trained to predict  $ucc$  labels of individual images in a fully supervised manner; and the trained model can be used to extract features of pixels (instances) inside the images (bags). Then, semantic segmentation masks can be obtained by unsupervised clustering of the pixels (each is represented by the extracted features) into two clusters (metastases or normal). Note that  $ucc$  does not directly provide an exact label for each individual instance. Therefore, our framework is a weakly supervised clustering framework.
+
+Finally, we have constructed  $ucc$  classifiers and experimentally shown that clustering performance of our framework with our  $ucc$  classifiers is better than the performance of unsupervised models and comparable to performance of fully supervised learning models. We have also tested the performance of our model on the real world task of semantic segmentation of breast cancer metastases in lymph node sections. We have compared the performance of our model with the performance of popular medical image segmentation architecture of  $Unet$  (Ronneberger et al., 2015) and shown that our weakly supervised model approximates the performance of fully supervised  $Unet$  model<sup>1</sup>.
+
+Hence, there are three main contributions of this paper:
+
+1. We have defined unique class count as a bag level label in MIL setup and mathematically proved that a perfect  $ucc$  classifier, in principle, can be used to perfectly cluster the individual instances inside the bags.
+
+2. We have constructed a neural network based  $ucc$  classifier by incorporating kernel density estimation (KDE) (Parzen, 1962) as a layer into our model architecture, which provided us with end-to-end training capability.  
+3. We have experimentally shown that clustering performance of our framework is better than the performance of unsupervised models and comparable to performance of fully supervised learning models.
+
+The rest of the paper is organized such that related work is in Section 2, details of our weakly supervised clustering framework are in Section 3, results of the experiments on MNIST, CIFAR10 and CIFAR100 datasets are in Section 4, results of the experiments in semantic segmentation of breast cancer metastases are in Section 5, and Section 6 concludes the paper.
+
+# 2 RELATED WORK
+
+This work is partly related to MIL which was first introduced in (Dietterich et al., 1997) for drug activity prediction. Different types of MIL were derived with different assumptions (Gartner et al., 2002; Zhang & Goldman, 2002; Chen et al., 2006; Foulds, 2008; Zhang & Zhou, 2009; Zhou et al., 2009), which are reviewed in detail in (Foulds & Frank, 2010), and they were used for many different applications such as, image annotation/categorization/retrieval (Chen & Wang, 2004; Zhang et al., 2002; Tang et al., 2010), text categorization (Andrews et al., 2003; Settles et al., 2008), spam detection (Jorgensen et al., 2008), medical diagnosis (Dundar et al., 2007), face/object detection (Zhang et al., 2006; Felzenszwalb et al., 2010) and object tracking (Babenko et al., 2011).
+
+In MIL, different types of pooling layers are used to combine extracted features of instances inside the bags, such as max-pooling and log-sum-exp pooling (Ramon & De Raedt, 2000; Zhou & Zhang, 2002; Wu et al., 2015; Wang et al., 2018). On the other hand, our UCC model uses KDE layer in order to estimate the distribution of extracted features. The advantage of KDE over pooling layers is that it embeds the instance level features into distribution space rather than summarizing them.
+
+There are also methods modeling cardinality and set distributions (Liu et al., 2015; Brukhim & Globerson, 2018; Kipf et al., 2018). However, cardinality of a set and  $ucc$  are completely different from each other. It is also important to state that  $ucc$  is obviously different from object/crowd counting (Idrees et al., 2013; Arteta et al., 2014; Zhang et al., 2015; 2016) since the task in object/crowd counting is to count the instances of the same type of object or people.
+
+Lastly, we compare clustering accuracies of our models with clustering accuracies of unsupervised baseline models: K-means (Wang et al., 2015) and Spectral Clustering (Zelnik-Manor & Perona, 2005); state of the art unsupervised models: JULE (Yang et al., 2016), GMVAE (Dilokthanakul et al., 2016), DAC (Chang et al., 2017), DEPICT (Ghasedi Dizaji et al., 2017) and DEC (Xie et al., 2016); and state of the art semi-supervised models: AAE (Makhzani et al., 2015), CatGAN (Springenberg, 2015), LN (Rasmus et al., 2015) and ADGM (Maaloe et al., 2016).
+
+# 3 WEAKLY SUPERVISED CLUSTERING FRAMEWORK
+
+In this section, we state our machine learning objective and formally define our novel MIL task, which is the core of our weakly supervised clustering framework. Finally, we explain details of the two main components of our framework, namely UCC model and unsupervised clustering branch.
+
+Objective: Let  $\mathcal{X} = \{x_1, x_2, \dots, x_n\}$  be a dataset such that each instance  $x_i \in \mathcal{X}$  belongs to a class, but its label is unknown. In this paper, we assume that total number of classes  $K$  is known. Hence, each instance  $x_i$  is endowed with an underlying, but unknown, label  $\mathcal{L}(x_i) = l_i \in \{1, 2, \dots, K\}$ . Further assume that for each class  $k \in \{1, 2, \dots, K\}$ , there exist at least one element  $x_i \in \mathcal{X}$  such that  $\mathcal{L}(x_i) = l_i = k$ . Our eventual objective is to derive a predicted class label  $\hat{l}_i$  for each instance  $x_i$  that tends towards underlying truth class  $l_i$ , i.e.  $\hat{l}_i \to \mathcal{L}(x_i) = l_i$ .
+
+# 3.1 A NOVEL MIL TASK
+
+In this novel MIL task, unique class count is used as an inexact, weak, bag level label and is defined in Definition 1. Assume that we are given subsets  $\sigma_{\zeta} \subset \mathcal{X}, \zeta = 1,2,\dots,N$  and unique class counts
+
+$\eta_{\sigma_{\zeta}} \forall \sigma_{\zeta}$ . Hence, MIL dataset is  $\mathcal{D} = \{(\sigma_1, \eta_{\sigma_1}), \dots, (\sigma_N, \eta_{\sigma_N})\}$ . Then, our MIL task is to learn the mapping between the bags and their associated bag level  $ucc$  labels while the goal is to predict the  $ucc$  labels of unseen bags.
+
+Definition 1 Given a subset  $\sigma_{\zeta} \subset \mathcal{X}$ , unique class count,  $\eta_{\sigma_{\zeta}}$ , is defined as the number of unique classes that all instances in the subset  $\sigma_{\zeta}$  belong to, i.e.  $\eta_{\sigma_{\zeta}} = |\{\mathcal{L}(x_i)| x_i \in \sigma_{\zeta}\}|$ . Recall that each instance belongs to an underlying unknown class.
+
+Given a dataset  $\mathcal{D}$ , our eventual objective is to assign a label to each instance  $x_{i} \in \mathcal{X}$  such that assigned labels and underlying unknown classes are consistent. To achieve this eventual objective, a deep learning model is designed such that the following intermediate objectives can be achieved while it is being trained on our MIL task:
+
+1. Unique class count: Given an unseen set  $\sigma_{\zeta}$ , the deep learning model, which is trained on  $\mathcal{D}$ , can predict its unique class count  $\eta_{\sigma_{\zeta}}$  correctly.  
+2. Labels on sets: Let  $\sigma_{\zeta}^{pure}$  and  $\sigma_{\xi}^{pure}$  be two disjoint pure sets (Definition 2) such that while all instances in  $\sigma_{\zeta}^{pure}$  belong to one underlying class, all instances in  $\sigma_{\xi}^{pure}$  belong to another class. Given  $\sigma_{\zeta}^{pure}$  and  $\sigma_{\xi}^{pure}$ , the deep learning model should enable us to develop an unsupervised learning model to label instances in  $\sigma_{\zeta}^{pure}$  and  $\sigma_{\xi}^{pure}$  as belonging to different classes. Note that the underlying classes for instances in the sets are unknown.  
+3. Labels on instances: Given individual instances  $x_{i} \in \mathcal{X}$ , the deep learning model should enable us to assign a label to each individual instance  $x_{i}$  such that all instances with different/same underlying unknown classes are assigned different/same labels. This is the eventual unsupervised learning objective.
+
+Definition 2 A set  $\sigma$  is called a pure set if its unique class count equals one. All pure sets is denoted by the symbol  $\sigma^{\mathrm{pure}}$  in this paper.
+
+# 3.2 UNIQUECLASSCOUNTMODEL
+
+In order to achieve the stated objectives, we have designed a deep learning based Unique Class Count (UCC) model. Our UCC model consists of three neural network modules ( $\theta_{\mathrm{feature}}$ ,  $\theta_{\mathrm{dm}}$ ,  $\theta_{\mathrm{decoder}}$ ) and can be trained end-to-end. The first module  $\theta_{\mathrm{feature}}$  extracts features from individual instances; then distributions of features are constructed from extracted features. The second module  $\theta_{\mathrm{dm}}$  is used to predict ucc label from these distributions. The last module  $\theta_{\mathrm{decoder}}$  is used to construct an autoencoder together with  $\theta_{\mathrm{feature}}$  so as to improve the extracted features by ensuring that extracted features contain semantic information for reconstruction.
+
+Formally, for  $x_{i} \in \sigma_{\zeta}, i = \{1,2,\dots,|\sigma_{\zeta}|\}$ , feature extractor module  $\theta_{\mathrm{feature}}$  extracts  $J$  features  $\{f_{\sigma_{\zeta}}^{1,i}, f_{\sigma_{\zeta}}^{2,i}, \dots, f_{\sigma_{\zeta}}^{J,i}\} = \theta_{\mathrm{feature}}(x_{i})$  for each instance  $x_{i} \in \sigma_{\zeta}$ . As a short hand, we write the operator  $\theta_{\mathrm{feature}}$  as operating element wise on the set to generate a feature matrix  $\theta_{\mathrm{feature}}(\sigma_{\zeta}) = f_{\sigma_{\zeta}}$  with matrix elements  $f_{\sigma_{\zeta}}^{j,i} \in \mathbb{R}$ , representing the  $j^{th}$  feature of the  $i^{th}$  instance. After obtaining features for all instances in  $\sigma_{\zeta}$ , a kernel density estimation (KDE) module is used to accumulate feature distributions  $h_{\sigma_{\zeta}} = (h_{\sigma_{\zeta}}^{1}(v), h_{\sigma_{\zeta}}^{2}(v), \dots, h_{\sigma_{\zeta}}^{J}(v))$ . Then,  $h_{\sigma_{\zeta}}$  is used as input to distribution regression module  $\theta_{\mathrm{dn}}$  to predict the ucc label,  $\tilde{\eta}_{\sigma_{\zeta}} = \theta_{\mathrm{dn}}(h_{\sigma_{\zeta}})$  as a softmax vector  $(\tilde{\eta}_{\sigma_{\zeta}}^{1}, \tilde{\eta}_{\sigma_{\zeta}}^{2}, \dots, \tilde{\eta}_{\sigma_{\zeta}}^{K})$ . Concurrently, decoder module  $\theta_{\mathrm{decoder}}$  in autoencoder branch is used to reconstruct the input images from the extracted features in an unsupervised fashion,  $\tilde{x}_{i} = \theta_{\mathrm{decoder}}(\theta_{\mathrm{feature}}(x_{i}))$ . Hence, UCC model, main modules of which are illustrated in Figure 2(a), optimizes two losses concurrently: 'ucc loss' and 'autoencoder loss'. While 'ucc loss' is cross-entropy loss, 'autoencoder loss' is mean square error loss. Loss for one bag is given in Equation 1.
+
+$$
+\alpha \underbrace {\left[ \sum_ {k = 1} ^ {K} \eta_ {\sigma_ {\zeta}} ^ {k} \log \tilde {\eta} _ {\sigma_ {\zeta}} ^ {k} \right]} _ {u c c l o s s} + (1 - \alpha) \underbrace {\left[ \frac {1}{| \sigma_ {\zeta} |} \sum_ {i = 1} ^ {| \sigma_ {\zeta} |} \left(x _ {i} - \tilde {x} _ {i}\right) ^ {2} \right]} _ {\text {a u t o e n c o d e r l o s s}} \text {w h e r e} \alpha \in [ 0, 1 ] \tag {1}
+$$
+
+![](images/67e55343264e2fd245c1a78b8a71b7cb6672a7c48ffa6b1eacc2825be28dbac8.jpg)  
+Figure 2: Weakly supervised clustering framework. (a) UCC model:  $\theta_{\mathrm{feature}}$  extracts  $J$  features, shown in colored nodes. KDE module obtains feature distribution for each feature. Then,  $\theta_{\mathrm{dm}}$  predicts the ucc label  $\tilde{\eta}_{\sigma_{\zeta}}$ . Concurrently, decoder module  $\theta_{\mathrm{decoder}}$  in autoencoder branch reconstructs the input images from the extracted features. (b) Unsupervised clustering: Trained feature extractor,  $\bar{\theta}_{\mathrm{feature}}$ , is used to extract the features of all instances in  $\mathcal{X}$  and unsupervised clustering is performed on extracted features. Note that  $\hat{l}_i$  is clustering label of  $x_i \in \mathcal{X}$ .
+
+# 3.2.1 KERNEL DENSITY ESTIMATION MODULE
+
+In UCC model, input is a set  $\sigma_{\zeta}$  and output is corresponding  $ucc$  label  $\tilde{\eta}_{\sigma_{\zeta}}$ , which does not depend on permutation of the instances in  $\sigma_{\zeta}$ . KDE module provides UCC model with permutation-invariant property. Moreover, KDE module uses the Gaussian kernel and it is differentiable, so our model can be trained end-to-end (Appendix A). KDE module also enables our theoretical analysis thanks to its decomposability property (Appendix B). Lastly, KDE module estimates the probability distribution of extracted features and enables  $\theta_{\mathrm{dm}}$  to fully utilize the information in the shape of the distribution rather than looking at point estimates of distribution obtained by other types of pooling layers (Ramon & De Raedt, 2000; Zhou & Zhang, 2002; Wang et al., 2018) (Appendix C.6).
+
+# 3.2.2 PROPERTIES OF UNIQUE CLASS COUNT MODEL
+
+This section mathematically proves that the UCC model guarantees, in principle, to achieve the stated intermediate objectives in Section 3.1. Proof of propositions are given in Appendix B.
+
+Proposition 1 Let  $\sigma_{\zeta}$ ,  $\sigma_{\xi}$  be disjoint subsets of  $\mathcal{X}$  with predicted unique class counts  $\tilde{\eta}_{\sigma_{\zeta}} = \tilde{\eta}_{\sigma_{\xi}} = 1$ . If the predicted unique class count of  $\sigma_{\nu} = \sigma_{\zeta} \cup \sigma_{\xi}$  is  $\tilde{\eta}_{\sigma_{\nu}} = 2$ , then  $h_{\sigma_{\zeta}} \neq h_{\sigma_{\xi}}$ .
+
+Definition 3 A perfect unique class count classifier takes in any set  $\sigma$  and output the correct predicted unique class count  $\tilde{\eta}_{\sigma} = \eta_{\sigma}$ .
+
+Proposition 2 Given a perfect unique class count classifier. The dataset  $\mathcal{X}$  can be perfectly clustered into  $K$  subsets  $\sigma_{\xi}^{pure}, \xi = 1, 2, \dots, K$ , such that  $\mathcal{X} = \bigcup_{\xi=1}^{K} \sigma_{\xi}^{pure}$  and  $\sigma_{\xi}^{pure} = \{x_i | x_i \in \mathcal{X}, \mathcal{L}(x_i) = \xi\}$ .
+
+Proposition 3 Given a perfect unique class count classifier. Decompose the dataset  $\mathcal{X}$  into  $K$  subsets  $\sigma_{\xi}^{pure}, \xi = 1, \dots, K$ , such that  $\sigma_{\xi}^{pure} = \{x_i | x_i \in \mathcal{X}, \mathcal{L}(x_i) = \xi\}$ . Then,  $h_{\sigma_{\xi}^{pure}} \neq h_{\sigma_{\zeta}^{pure}}$  for  $\xi \neq \zeta$ .
+
+Suppose we have a perfect  $ucc$  classifier. For any two pure sets  $\sigma_{\zeta}^{pure}$  and  $\sigma_{\xi}^{pure}$ , which consist of instances of two different underlying classes,  $ucc$  labels must be predicted correctly by the perfect  $ucc$  classifier. Hence, the conditions of Proposition 1 are satisfied, so we have  $h_{\sigma_{\zeta}^{pure}} \neq h_{\sigma_{\xi}^{pure}}$ . Therefore, we can, in principle, perform an unsupervised clustering on the distributions of the sets without knowing the underlying truth classes of the instances. Hence, the perfect  $ucc$  classifier enables us to achieve our intermediate objective of "Labels on sets". Furthermore, given a perfect  $ucc$  classifier, Proposition 2 states that by performing predictions of  $ucc$  labels alone, without any
+
+knowledge of underlying truth classes for instances, one can in principle perform perfect clustering for individual instances. Hence, a perfect ucc classifier enables us to achieve our intermediate objective of "Labels on instances".
+
+# 3.3 UNSUPERVISEDINSTANCE CLUSTERING
+
+In order to achieve our ultimate objective of developing an unsupervised learning model for clustering all the instances in dataset  $\mathcal{X}$ , we add this unsupervised clustering branch into our framework. Theoretically, we have shown in Proposition 3 that given a perfect  $ucc$  classifier, distributions of pure subsets of instances coming from different underlying classes are different.
+
+In practice, it may not be always possible (probably most of the times) to train a perfect  $ucc$  classifier, so we try to approximate it. First of all, we train our  $ucc$  classifier on our novel MIL task and save our trained model  $(\bar{\theta}_{\mathrm{feature}}, \bar{\theta}_{\mathrm{dm}}, \bar{\theta}_{\mathrm{decoder}})$ . Then, we use trained feature extractor  $\bar{\theta}_{\mathrm{feature}}$  to obtain feature matrix  $f_{\mathcal{X}} = \bar{\theta}_{\mathrm{feature}}(\mathcal{X})$ . Finally, extracted features are clustered in an unsupervised fashion, by using simple k-means and spectral clustering methods. Figure 2(b) illustrates the unsupervised clustering process in our framework. A good feature extractor  $\bar{\theta}_{\mathrm{feature}}$  is of paramount importance in this task. Relatively poor  $\bar{\theta}_{\mathrm{feature}}$  may result in a poor unsupervised clustering performance in practice even if we have a strong  $\bar{\theta}_{\mathrm{dm}}$ . To obtain a strong  $\bar{\theta}_{\mathrm{feature}}$ , we employ an autoencoder branch, so as to achieve high clustering performance in our unsupervised instance clustering task. The autoencoder branch ensures that features extracted by  $\bar{\theta}_{\mathrm{feature}}$  contain semantic information for reconstruction.
+
+# 4 EXPERIMENTS ON MNIST AND CIFAR DATASETS
+
+This section analyzes the performances of our UCC models and fully supervised models in terms of our eventual objective of unsupervised instance clustering on MNIST (10 clusters) (LeCun et al., 1998), CIFAR10 (10 clusters) and CIFAR100 (20 clusters) datasets (Krizhevsky & Hinton, 2009).
+
+# 4.1 MODEL ARCHITECTURES AND DATASETS
+
+To analyze different characteristics of our framework, different kinds of unique class count models were trained during our experiments:  $UCC$ ,  $UCC^{2+}$ ,  $UCC_{\alpha=1}$  and  $UCC_{\alpha=1}^{2+}$ . These unique class count models took sets of instances as inputs and were trained on  $ucc$  labels. While  $UCC$  and  $UCC^{2+}$  models had autoencoder branch in their architecture and they were optimized jointly over both autoencoder loss and  $ucc$  loss,  $UCC_{\alpha=1}$  and  $UCC_{\alpha=1}^{2+}$  models did not have autoencoder branch in their architecture and they were optimized over  $ucc$  loss only (i.e.  $\alpha = 1$  in Equation 1). The aim of training unique class count models with and without autoencoder branch was to show the effect of autoencoder branch in the robustness of clustering performance with respect to  $ucc$  classification performance.  $UCC$  and  $UCC_{\alpha=1}$  models were trained on bags with labels of  $ucc1$  to  $ucc4$ . On the other hand,  $UCC^{2+}$  and  $UCC_{\alpha=1}^{2+}$  models were trained on bags with labels  $ucc2$  to  $ucc4$ . Our models were trained on  $ucc$  labels up to  $ucc4$  instead of  $ucc10$  ( $ucc20$  in CIFAR100) since the performance was almost the same for both cases and training with  $ucc1$  to  $ucc4$  was much faster (Appendix C.2). Please note that for perfect clustering of instances inside the bags, it is enough to have a perfect  $ucc$  classifier that can perfectly discriminate  $ucc1$  and  $ucc2$  bags from Proposition 2. The aim of training  $UCC^{2+}$  and  $UCC_{\alpha=1}^{2+}$  models was to experimentally check whether these models can perform as good as  $UCC$  and  $UCC_{\alpha=1}$  models even if there is no pure subsets during training. In addition to our unique class count models, for benchmarking purposes, we also trained fully supervised models, FullySupervised, and unsupervised autoencoder models, Autoencoder. FullySupervised models took individual instances as inputs and used instance level ground truths as labels during training. On the other hand, Autoencoder models were trained in an unsupervised manner by optimizing autoencoder loss (i.e.  $\alpha = 0$  in Equation 1). It is important to note that all models for a dataset shared the same architecture for feature extractor module and all the modules in our models are fine tuned for optimum performance and training time as explained in Appendix C.1.
+
+We trained and tested our models on MNIST, CIFAR10 and CIFAR100 datasets. We have  $\mathcal{X}_{mnist,tr}$ ,  $\mathcal{X}_{mnist,val}$  and  $\mathcal{X}_{mnist,test}$  for MNIST;  $\mathcal{X}_{cifar10,tr}$ ,  $\mathcal{X}_{cifar10,val}$  and  $\mathcal{X}_{cifar10,test}$  for CIFAR10; and  $\mathcal{X}_{cifar100,tr}$ ,  $\mathcal{X}_{cifar100,val}$  and  $\mathcal{X}_{cifar100,test}$  for CIFAR100. Note that  $tr$ ,  $val$  and  $test$  subscripts stand for 'training', 'validation' and 'test' sets, respectively. All the results presented in this paper were obtained on hold-out test sets  $\mathcal{X}_{mnist,test}$ ,  $\mathcal{X}_{cifar10,test}$  and  $\mathcal{X}_{cifar100,test}$ . Fully Supervised
+
+Table 1: Minimum inter-class JS divergence values,  $ucc$  classification accuracy values and clustering accuracy values of our models (first part), baseline and state of the art unsupervised models (second part) and state of the art semi-supervised models (third part) on different test datasets. The best clustering accuracy values for each kind of models (weakly supervised (our models), unsupervised, semi-supervised) are highlighted in bold. ('x': not applicable, '-': missing')  
+
+<table><tr><td rowspan="2"></td><td colspan="3">min. JS divergence</td><td colspan="3">ucc acc.</td><td colspan="3">clustering acc.</td></tr><tr><td>mnist</td><td>cifar10</td><td>cifar100</td><td>mnist</td><td>cifar10</td><td>cifar100</td><td>mnist</td><td>cifar10</td><td>cifar100</td></tr><tr><td>UCC</td><td>0.222</td><td>0.097</td><td>0.004</td><td>1.000</td><td>0.972</td><td>0.824</td><td>0.984</td><td>0.781</td><td>0.338</td></tr><tr><td>UCC2+</td><td>0.251</td><td>0.005</td><td>0.002</td><td>1.000</td><td>0.936</td><td>0.814</td><td>0.984</td><td>0.545</td><td>0.278</td></tr><tr><td>UCCα=1</td><td>0.221</td><td>0.127</td><td>0.003</td><td>1.000</td><td>0.982</td><td>0.855</td><td>0.981</td><td>0.774</td><td>0.317</td></tr><tr><td>UCC2+α=1</td><td>0.023</td><td>0.002</td><td>0.003</td><td>0.996</td><td>0.920</td><td>0.837</td><td>0.881</td><td>0.521</td><td>0.284</td></tr><tr><td>Autoencoder</td><td>0.101</td><td>0.004</td><td>0.002</td><td>x</td><td>x</td><td>x</td><td>0.930</td><td>0.241</td><td>0.167</td></tr><tr><td>FullySupervised</td><td>0.283</td><td>0.065</td><td>0.019</td><td>x</td><td>x</td><td>x</td><td>0.988</td><td>0.833</td><td>0.563</td></tr><tr><td colspan="7">JULE (Yang et al., 2016)</td><td>0.964</td><td>0.272</td><td>0.137</td></tr><tr><td colspan="7">GMVAE (Dilokthanakul et al., 2016)</td><td>0.885</td><td>-</td><td>-</td></tr><tr><td colspan="7">DAC (Chang et al., 2017)*</td><td>0.978</td><td>0.522</td><td>0.238</td></tr><tr><td colspan="7">DEC (Xie et al., 2016)*</td><td>0.843</td><td>0.301</td><td>0.185</td></tr><tr><td colspan="7">DEPICT (Ghasedi Dizaji et al., 2017)*</td><td>0.965</td><td>-</td><td>-</td></tr><tr><td colspan="7">Spectral (Zelnik-Manor &amp; Perona, 2005)</td><td>0.696</td><td>0.247</td><td>0.136</td></tr><tr><td colspan="7">K-means (Wang et al., 2015)</td><td>0.572</td><td>0.229</td><td>0.130</td></tr><tr><td colspan="7">ADGM (Maaløe et al., 2016)</td><td>0.990</td><td>-</td><td>-</td></tr><tr><td colspan="7">Ladder Networks (Rasmus et al., 2015)</td><td>0.989</td><td>0.796</td><td>-</td></tr><tr><td colspan="7">AAE (Makhzani et al., 2015)</td><td>0.981</td><td>-</td><td>-</td></tr><tr><td colspan="7">CatGAN (Springenberg, 2015)</td><td>0.981</td><td>0.804</td><td>-</td></tr></table>
+
+* Models do not separate training and testing data, i.e. their results are not on hold-out test sets.
+
+models took individual instances as inputs and were trained on instance level ground truths. Unique class count models took sets of instances as inputs, which were sampled from the power sets  $2^{\mathcal{X}_{mnist,tr}}$ ,  $2^{\mathcal{X}_{cifar10,tr}}$  and  $2^{\mathcal{X}_{cifar100,tr}}$ , and were trained on ucc labels (Appendix C.2). While all the models were trained in a supervised setup, either on ucc labels or instance level ground truths, all of them were used to extract features for unsupervised clustering of individual instances.
+
+# 4.2 UNIQUECLASSCOUNTPREDICTION
+
+Preceding sections showed, in theory, that a perfect ucc classifier can perform 'weakly' supervised clustering perfectly. We evaluate ucc prediction accuracy of our unique class count models in accordance with our first intermediate objective that unique class count models should predict ucc labels of unseen subsets correctly. We randomly sampled subsets for each ucc label from the power sets of test sets and predicted the ucc labels by using trained models. Then, we calculated the ucc prediction accuracies by using predicted and truth ucc labels, which are summarized in Table 1 (Appendix C.3). We observed that as the task becomes harder (from MNIST to CIFAR100), it also becomes harder to approximate the perfect ucc classifier. Moreover, UCC and  $UCC_{\alpha = 1}$  models, in general, have higher scores than their counterpart models of  $UCC^{2+}$  and  $UCC_{\alpha = 1}^{2+}$ , which is expected since the ucc prediction task becomes easier at the absence of pure sets and models reach to early stopping condition (Appendix C.1) more easily. This is also supported by another interesting, yet reasonable, observation that  $UCC^{2+}$  models have higher ucc accuracies than  $UCC_{\alpha = 1}^{2+}$  models thanks to the autoencoder branch which makes  $UCC^{2+}$  harder to reach to early stopping condition.
+
+# 4.3 LABELS ON SETS
+
+Jensen-Shannon (JS) divergence (Lin, 1991) value between feature distributions of two pure sets consisting of instances of two different underlying classes is defined as inter-class JS divergence in this paper and used for comparison on 'Labels on sets' objective of assigning labels to pure sets. Higher values of inter-class JS divergence are desired since it means that feature distributions of
+
+![](images/53a48ee944c0cbc22c2b276dc71de98d216843ecc68ab90166832089e1f02135.jpg)  
+Figure 3: Clustering accuracy vs  $ucc$  accuracy plots of  $UCC$  and  $UCC_{\alpha = 1}$  models together with k-means and spectral clustering accuracy baselines on MNIST, CIFAR10 and CIFAR100 datasets.
+
+![](images/6e1c0086e7a443af9591379ef7123a88d5b794f2ca31c551139eb77621b94eff.jpg)
+
+![](images/105c4b9299503cb2b78acca23aa0079ff82d47b8df02eb1d869ffaa9d90b3025.jpg)
+
+pure sets of underlying classes are far apart from each other. The features of all the instances in a particular class are extracted by using a trained model and feature distributions associated to that class obtained by performing kernel density estimation on these extracted features. Then, for each pair of classes, inter-class JS divergence values are calculated (Appendix C.4). For a particular model, which is used in feature extraction, the minimum of these pairwise inter-class JS divergence values is used as a metric in the comparison of models. We have observed that as the task gets more challenging and the number of clusters increases, there is a drop in minimum inter-class JS divergence values, which is summarized in Table 1.
+
+# 4.4 LABELS ON INSTANCES
+
+For our eventual objective of 'Labels on instances', we have used 'clustering accuracy' as a comparison metric, which is calculated similar to Ghasedi Dizaji et al. (2017). By using our trained models, we extracted features of individual instances of all classes in test sets. Then, we performed unsupervised clustering over these features by using k-means and spectral clustering. We used number of classes in ground truth as number of clusters (MNIST: 10, CIFAR10: 10, CIFAR100: 20 clusters) during clustering and gave the best clustering accuracy for each model in Table 1 (Appendix C.5).
+
+In Table 1, we compare clustering accuracies of our models together with baseline and state of the art models in the literature: baseline unsupervised (K-means (Wang et al., 2015), Spectral Clustering (Zelnik-Manor & Perona, 2005)); state of the art unsupervised (JULE (Yang et al., 2016), GMVAE (Dilokthanakul et al., 2016), DAC (Chang et al., 2017), DEPICT (Ghasedi Dizaji et al., 2017), DEC (Xie et al., 2016)) and state of the art semi-supervised (AAE (Makhzani et al., 2015), CatGAN (Springenberg, 2015), LN (Rasmus et al., 2015), ADGM (Maaloe et al., 2016)). Clustering performance of our unique class count models is better than the performance of unsupervised models in all datasets and comparable to performance of fully supervised learning models in MNIST and CIFAR10 datasets. The performance gap gets larger in CIFAR100 dataset as the task becomes harder. Although semi-supervised methods use some part of the dataset with 'exact' labels during training, our models perform on par with AAE and CatGAN models and comparable to LN and ADGM models on MNIST dataset. ADGM and LN even reach to the performance of the Fully Supervised model since they exploit training with 'exact' labeled data. On CIFAR10 dataset, LN and CatGAN models are slightly better than our unique class count models; however, they use  $10\%$  of instances with 'exact' labels, which is not a small portion.
+
+In general, our  $UCC$  and  $UCC_{\alpha = 1}$  models have similar performance, and they are better than their counterpart models of  $UCC^{2+}$  and  $UCC_{\alpha = 1}^{2+}$  due to the absence of pure sets during training. However, in the real world tasks, the absence of pure sets heavily depends on the nature of the problem. In our task of semantic segmentation of breast cancer metastases in histological lymph node sections, for example, there are many pure sets. Furthermore, we observed that there is a performance gap between  $UCC^{2+}$  and  $UCC_{\alpha = 1}^{2+}$  models:  $UCC^{2+}$  models perform better than  $UCC_{\alpha = 1}^{2+}$  models thanks to the autoencoder branch. The effect of autoencoder branch is also apparent in Figure 3, which shows clustering accuracy vs ucc accuracy curves for different datasets. For MNIST dataset, while  $UCC$  model gives clustering accuracy values proportional to ucc accuracy,  $UCC_{\alpha = 1}$  model cannot reach to high clustering accuracy values until it reaches to high ucc accuracies. The reason is that autoencoder branch in  $UCC$  helps  $\theta_{\mathrm{feature}}$  module to extract better features during the initial phases of the training process, where the ucc classification accuracy is low. Compared to other
+
+![](images/793d2614d0d03b340f66fcbd4b34432c6c69a84b78e61814318776a83bfaed09.jpg)  
+Figure 4: Example images from hold-out test dataset with corresponding  $ucc$  labels, ground truth masks and predicted masks by  $UCC_{segment}$ ,  $Unet$  and K-means clustering models.
+
+datasets, this effect is more significant in MNIST dataset since itself is clusterable. Although autoencoder branch helps in CIFAR10 and CIFAR100 datasets as well, improvements in clustering accuracy coming from autoencoder branch seems to be limited, so two models  $UCC$  and  $UCC_{\alpha = 1}$  follow nearly the same trend in the plots. The reason is that CIFAR10 and CIFAR100 datasets are more complex than MNIST dataset, so autoencoder is not powerful enough to contribute to extract discriminant features, which is also confirmed by the limited improvements of Autoencoder models over baseline performance in these datasets.
+
+# 5 SEMANTIC SEGMENTATION OF BREAST CANCER METASTASES
+
+Semantic segmentation of breast cancer metastases in histological lymph node sections is a crucial step in staging of breast cancer, which is the major determinant of the treatment and prognosis (Brierley et al., 2016). Given the images of lymph node sections, the task is to detect and locate, i.e. semantically segment out, metastases regions in the images. We have formulated this task in our novel MIL framework such that each image is treated as a bag and corresponding  $ucc$  label is obtained based on whether the image is from fully normal or metastases region, which is labeled by  $ucc1$ , or from boundary region (i.e. image with both normal and metastases regions), which is labeled by  $ucc2$ . We have shown that this segmentation task can be achieved by using our weakly supervised clustering framework without knowing the ground truth metastases region masks of images, which require experts to exhaustively annotate each metastases region in each image. This annotation process is tedious, time consuming and more importantly not a part of clinical workflow.
+
+We have used  $512 \times 512$  image crops from publicly available CAMELYON dataset (Litjens et al., 2018) and constructed our bags by using  $32 \times 32$  patches over these images. We trained our unique class count model  $UCC_{segment}$  on ucc labels. Then, we used the trained model as a feature extractor and conducted unsupervised clustering over the patches of the images in the hold-out test dataset to obtain semantic segmentation masks. For benchmarking purposes, we have also trained a fully supervised Unet model (Ronneberger et al., 2015), which is a well-known biomedical image segmentation architecture, by using the ground truth masks and predicted the segmentation maps in
+
+Table 2: Semantic segmentation performance statistics of  $UCC_{segment}$ ,  $Unet$  and K-means clustering methods on hold-out test dataset.  
+
+<table><tr><td></td><td>TPR</td><td>FPR</td><td>TNR</td><td>FNR</td><td>PA</td></tr><tr><td>UCCsegment (weakly supervised)</td><td>0.818</td><td>0.149</td><td>0.851</td><td>0.182</td><td>0.863</td></tr><tr><td>Unet (fully supervised)</td><td>0.860</td><td>0.126</td><td>0.874</td><td>0.140</td><td>0.889</td></tr><tr><td>K-means (unsupervised baseline)</td><td>0.370</td><td>0.271</td><td>0.729</td><td>0.630</td><td>0.512</td></tr></table>
+
+the test set. The aim of this comparison was to show that at the absence of ground truth masks, our model can approximate the performance of a fully supervised model. Moreover, we have obtained semantic segmentation maps in the test dataset by using k-means clustering as a baseline study. Example images from test dataset with corresponding ground truth masks,  $ucc$  labels and predicted masks by different models are shown in Figure 4. (Please see Appendix D.1 for more details.)
+
+Furthermore, we have calculated pixel level gross statistics of TPR (True Positive Rate), FPR (False Positive Rate), TNR (True Negative Rate), FNR (False Negative Rate) and PA (Pixel Accuracy) over the images of hold-out test dataset and declared the mean values in Table 2 (Appendix D.2). When we look at the performance of unsupervised baseline method of K-means clustering, it is obvious that semantic segmentation of metastases regions in lymph node sections is not an easy task. Baseline method achieves a very low TPR value of 0.370 and almost random score of 0.512 in PA. On the other hand, both our weakly supervised model  $UCC_{segment}$  and fully supervised model  $Unet$  outperform the baseline method. When we compare our model  $UCC_{segment}$  with  $Unet$  model, we see that both models behave similarly. They have reasonably high TPR and TNR scores, and low FPR and FNR scores. Moreover, they have lower FPR values than FNR values, which is more favorable than vice-versa since pathologists opt to use immunohistochemistry (IHC) to confirm negative cases (Bejnordi et al., 2017). However, there is a performance gap between two models, which is mainly due to the fact that  $Unet$  model is a fully supervised model and it is trained on ground truth masks, which requires exhaustive annotations by experts. On the contrary,  $UCC_{segment}$  model is trained on  $ucc$  labels and approximates to the performance of the  $Unet$  model.  $ucc$  label is obtained based on whether the image is metastatic, non-metastatic or mixture, which is much cheaper and easier to obtain compared to exhaustive mask annotations. Another factor affecting the performance of  $UCC_{segment}$  model is that  $ucc1$  labels can sometimes be noisy. It is possible to have some small portion of normal cells in cancer regions and vice-versa due to the nature of the cancer. However, our  $UCC_{segment}$  is robust to this noise and gives reasonably good results, which approximates the performance of  $Unet$  model.
+
+# 6 CONCLUSION
+
+In this paper, we proposed a weakly supervised learning based clustering framework and introduce a novel MIL task as the core of this framework. We defined  $ucc$  as a bag level label in MIL setup and mathematically proved that a perfect  $ucc$  classifier can be used to perfectly cluster individual instances inside the bags. We designed a neural network based  $ucc$  classifier and experimentally showed that clustering performance of our framework with our  $ucc$  classifiers are better than the performance of unsupervised models and comparable to performance of fully supervised learning models. Finally, we showed that our weakly supervised unique class count model,  $UCC_{segment}$ , can be used for semantic segmentation of breast cancer metastases in histological lymph node sections. We compared the performance of our model  $UCC_{segment}$  with the performance of a Unet model and showed that our weakly supervised model approximates the performance of fully supervised Unet model. In the future, we want to check the performance of our  $UCC_{segment}$  model with other medical image datasets and use it to discover new morphological patterns in cancer that had been overlooked in traditional pathology workflow.
+
+# ACKNOWLEDGEMENTS
+
+This work is supported by the Biomedical Research Council of the Agency for Science, Technology, and Research, Singapore and the National University of Singapore, Singapore.
+
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+
+# A KERNEL DENSITY ESTIMATION
+
+Kernel density estimation is a statistical method to estimate underlying unknown probability distribution in data (Parzen, 1962). It works based on fitting kernels at sample points of an unknown distribution and adding them up to construct the estimated probability distribution. Kernel density estimation process is illustrated in Figure 5.
+
+![](images/c47940a1dce3275f260ac2145e8ae33e91001b2f87c6a4a0c088d7773e1cdcbf.jpg)
+
+![](images/1c1588198111198459dd2a6d8de9246d0834d0972a69d24f1d85098f2cc36948.jpg)  
+Figure 5: KDE module - the Gaussian kernel  $(\kappa (v - f_{\sigma \zeta}^{j,i}))$  for each extracted feature for a sample is illustrated with colored curves and previously accumulated kernels are shown in gray. Estimated feature distributions, which are obtained by employing Equation 2, are sampled at some pre-determined intervals and passed to  $\theta_{\mathrm{dn}}$ .
+
+# A.1 KDE MODULE IS DIFFERENTIABLE
+
+The distribution of the feature  $h_{\sigma_{\zeta}}^{j}(v)$  is obtained by applying kernel density estimation on the extracted features  $f_{\sigma_{\zeta}}^{j,i}$  as in Equation 2. In order to be able to train our unique class count model end-to-end, we need to show that KDE module is differentiable, so that we can pass the gradients from  $\theta_{\mathrm{dn}}$  to  $\theta_{\mathrm{feature}}$  during back-propagation. Derivative of  $h_{\sigma_{\zeta}}^{j}(v)$  with respect to input of KDE
+
+module,  $f_{\sigma_{\zeta}}^{j,i}$ , can be obtained as in Equation 3.
+
+$$
+h _ {\sigma_ {\zeta}} ^ {j} (v) = \frac {1}{| \sigma_ {\zeta} |} \sum_ {i = 1} ^ {| \sigma_ {\zeta} |} \frac {1}{\sqrt {2 \pi \sigma^ {2}}} e ^ {- \frac {1}{2 \sigma^ {2}} \left(v - f _ {\sigma_ {\zeta}} ^ {j, i}\right) ^ {2}} \tag {2}
+$$
+
+$$
+\frac {\partial h _ {\sigma_ {\zeta}} ^ {j} (v)}{\partial f _ {\sigma_ {\zeta}} ^ {j , i}} = \frac {1}{| \sigma_ {\zeta} |} \frac {(v - f _ {\sigma_ {\zeta}} ^ {j , i})}{\sigma^ {2} \sqrt {2 \pi \sigma^ {2}}} e ^ {- \frac {1}{2 \sigma^ {2}} (v - f _ {\sigma_ {\zeta}} ^ {j, i}) ^ {2}} \tag {3}
+$$
+
+After showing that KDE module is differentiable, we can show the weight update process for  $\theta_{\mathrm{feature}}$  module in our model. Feature extractor module  $\theta_{\mathrm{feature}}$  is shared by both autoencoder branch and  $ucc$  branch in our model. During back-propagation phase of the end-to-end training process, the weight updates of  $\theta_{\mathrm{feature}}$  comprise the gradients coming from both branches (Equation 5). Gradients coming from autoencoder branch follow the traditional neural network back-propagation flow through the convolutional and fully connected layers. Different than that, gradients coming from  $ucc$  branch (Equation 6) also back-propagate through the custom KDE layer according to Equation 3.
+
+$$
+L o s s = \alpha \underbrace {L o s s _ {u c c}} _ {\substack {\text {u c c} \\ \text {loss}}} + (1 - \alpha) \underbrace {L o s s _ {a e}} _ {\substack {\text {autoencoder} \\ \text {loss}}} \text {where} \alpha \in [ 0, 1 ] \tag{4}
+$$
+
+$$
+\underbrace {\frac {\partial L o s s}{\partial \theta_ {\text {f e a t u r e}}}} _ {\substack {\text {gradients} \\ \text {for} \\ \theta_ {\text {f e a t u r e}}}} = \alpha \underbrace {\frac {\partial L o s s _ {u c c}}{\partial \theta_ {\text {f e a t u r e}}}} _ {\substack {\text {gradients} \\ \text {from} \\ \text {ucc b r n c h}}} + (1 - \alpha) \underbrace {\frac {\partial L o s s _ {a e}}{\partial \theta_ {\text {f e a t u r e}}}} _ {\substack {\text {gradients} \\ \text {from} \\ \text {autoencoder b r n c h}}} \tag{5}
+$$
+
+$$
+\frac {\partial \text {L o s s} _ {u c c}}{\partial \theta_ {\text {f e a t u r e}}} = \frac {\partial \text {L o s s} _ {u c c}}{\partial h _ {\sigma_ {\zeta}}} \times \underbrace {\frac {\partial h _ {\sigma_ {\zeta}}}{\partial f _ {\sigma_ {\zeta}}}} _ {\substack {\text {back - p r o p a g a t i o n} \\ \text {t h r o u g h} \\ \text {K D E l a y e r}}} \times \underbrace {\frac {\partial f _ {\sigma_ {\zeta}}}{\partial \theta_ {\text {f e a t u r e}}}} _ {\substack {\text {k d - p r o p a g a t i o n} \\ \text {t h r o u g h} \\ \text {K D E l a y e r}}} \tag{6}
+$$
+
+# B PROOFS OF PROPOSITIONS
+
+Before proceeding to the formal proofs, it is helpful to emphasize the decomposability property of kernel density estimation here.
+
+For any set,  $\sigma_{\zeta}$ , one could partition it into a set of  $M$  disjoint subsets  $\sigma_{\zeta} = \sigma_1' \cup \sigma_2' \cup \dots \cup \sigma_M'$  where  $\sigma_{\lambda}' \cap \sigma_{\psi}' = \emptyset$  for  $\lambda \neq \psi$ . It is trivial to show that distribution  $h_{\sigma_{\zeta}}^{j}(v)$  is simply a linear combination of distributions  $h_{\sigma_{\lambda}'}^{j}(v), \lambda = 1, 2, \dots, M$  (Equation 7). As a direct consequence, one could decompose any set into its pure subsets. This is an important decomposition which will be used in the proofs of propositions later.
+
+$$
+h _ {\sigma_ {\zeta}} ^ {j} (v) = \sum_ {\lambda = 1} ^ {M} w _ {\sigma_ {\lambda} ^ {\prime}} h _ {\sigma_ {\lambda} ^ {\prime}} ^ {j} (v), \forall j \text {w h e r e} w _ {\sigma_ {\lambda} ^ {\prime}} = \frac {\left| \sigma_ {\lambda} ^ {\prime} \right|}{\left| \sigma_ {\zeta} \right|} \tag {7}
+$$
+
+Now, we can proceed to formally state our propositions.
+
+Definition 1 Given a subset  $\sigma_{\zeta} \subset \mathcal{X}$ , unique class count,  $\eta_{\sigma_{\zeta}}$ , is defined as the number of unique classes that all instances in the subset  $\sigma_{\zeta}$  belong to, i.e.  $\eta_{\sigma_{\zeta}} = |\{\mathcal{L}(x_i)| x_i \in \sigma_{\zeta}\}|$ . Recall that each instance belongs to an underlying unknown class.
+
+Definition 2 A set  $\sigma$  is called a pure set if its unique class count equals one. All pure sets are denoted by the symbol  $\sigma^{\mathrm{pure}}$  in this paper.
+
+Proposition B.1 For any set  $\sigma_{\zeta} \subset \mathcal{X}$ , the unique class count  $\eta_{\sigma_{\zeta}}$  of  $\sigma_{\zeta}$  does not depend on the number of instances in  $\sigma_{\zeta}$  belonging to a certain class.
+
+Proof: This conclusion is obvious from the definition of unique class count in Definition 1.
+
+Proposition B.2  $\theta_{drn}$  is non-linear.
+
+Proof: We give a proof by contradiction using Proposition B.1. Suppose  $\theta_{\mathrm{dm}}$  is linear, then
+
+$$
+\begin{array}{l} \theta_ {\mathrm {d r n}} \left(h _ {\sigma_ {\nu}}\right) = \theta_ {\mathrm {d r n}} \left(w _ {\zeta} h _ {\sigma_ {\zeta}} + w _ {\xi} h _ {\sigma_ {\xi}}\right) \tag {8} \\ = w _ {\zeta} \theta_ {\mathrm {d m}} \left(h _ {\sigma_ {\zeta}}\right) + w _ {\xi} \theta_ {\mathrm {d m}} \left(h _ {\sigma_ {\xi}}\right) \\ = w _ {\zeta} \tilde {\eta} _ {\sigma_ {\zeta}} + w _ {\xi} \tilde {\eta} _ {\sigma_ {\xi}} = \tilde {\eta} _ {\sigma_ {\nu}} \\ \end{array}
+$$
+
+Hence,  $\theta_{\mathrm{drn}}$  is linear only when Equation 8 holds. However, by Proposition B.1,  $(\theta_{\mathrm{feature}}, \theta_{\mathrm{drn}})$  should count correctly regardless of the proportion of the size of the sets  $|\sigma_{\zeta}|$  and  $|\sigma_{\xi}|$ . Hence, Equation 8 cannot hold true and  $\theta_{\mathrm{drn}}$  by contradiction cannot be linear.
+
+Proposition B.3 Let  $\sigma_{\zeta}$ ,  $\sigma_{\xi}$  be disjoint subsets of  $\mathcal{X}$  with predicted unique class counts  $\tilde{\eta}_{\sigma_{\zeta}}$  and  $\tilde{\eta}_{\sigma_{\xi}}$ , respectively. Let  $\tilde{\eta}_{\sigma_{\nu}}$  be the predicted unique class count of  $\sigma_{\nu} = \sigma_{\zeta} \cup \sigma_{\xi}$ . If  $h_{\sigma_{\zeta}} = h_{\sigma_{\xi}}$ , then  $\tilde{\eta}_{\sigma_{\nu}} = \tilde{\eta}_{\sigma_{\zeta}} = \tilde{\eta}_{\sigma_{\xi}}$ .
+
+Proof: The distribution of set  $\sigma_{\nu}$  can be decomposed into distribution of subsets,
+
+$$
+h _ {\sigma_ {\nu}} = w _ {\zeta} h _ {\sigma_ {\zeta}} + w _ {\xi} h _ {\sigma_ {\xi}} \text {w h e r e} w _ {\zeta} + w _ {\xi} = 1 \tag {9}
+$$
+
+$$
+h _ {\sigma_ {\zeta}} = h _ {\sigma_ {\xi}} \Longrightarrow h _ {\sigma_ {\nu}} = h _ {\sigma_ {\zeta}} \tag {10}
+$$
+
+Hence,  $\tilde{\eta}_{\sigma_{\nu}} = \tilde{\eta}_{\sigma_{\zeta}} = \tilde{\eta}_{\sigma_{\xi}}$
+
+Proposition 1 Let  $\sigma_{\zeta}$ ,  $\sigma_{\xi}$  be disjoint subsets of  $\mathcal{X}$  with predicted unique class counts  $\tilde{\eta}_{\sigma_{\zeta}} = \tilde{\eta}_{\sigma_{\xi}} = 1$ . If the predicted unique class count of  $\sigma_{\nu} = \sigma_{\zeta} \cup \sigma_{\xi}$  is  $\tilde{\eta}_{\sigma_{\nu}} = 2$ , then  $h_{\sigma_{\zeta}} \neq h_{\sigma_{\xi}}$ .
+
+Proof: Proof of this proposition follows immediately from the contra-positive of Proposition B.3.  $\blacksquare$
+
+Definition 3 A perfect unique class count classifier takes in any set  $\sigma$  and output the correct predicted unique class count  $\tilde{\eta}_{\sigma} = \eta_{\sigma}$ .
+
+Proposition 2 Given a perfect unique class count classifier. The dataset  $\mathcal{X}$  can be perfectly clustered into  $K$  subsets  $\sigma_{\xi}^{pure}, \xi = 1, 2, \dots, K$ , such that  $\mathcal{X} = \bigcup_{\xi=1}^{K} \sigma_{\xi}^{pure}$  and  $\sigma_{\xi}^{pure} = \{x_i | x_i \in \mathcal{X}, \mathcal{L}(x_i) = \xi\}$ .
+
+Proof: First note that this proposition holds because the "perfect unique class count classifier" is a very strong condition. Decompose  $\mathcal{X}$  into subsets with single instance and then apply the unique class count on each subset, by definition, unique class counts of all subsets are one. Randomly pair up the subsets and merge them if their union still yield unique class count of one. Recursively apply merging on this condition until no subsets can be merged.
+
+Proposition 3 Given a perfect unique class count classifier. Decompose the dataset  $\mathcal{X}$  into  $K$  subsets  $\sigma_{\xi}^{pure}, \xi = 1, \dots, K$ , such that  $\sigma_{\xi}^{pure} = \{x_i | x_i \in \mathcal{X}, \mathcal{L}(x_i) = \xi\}$ . Then,  $h_{\sigma_{\xi}^{pure}} \neq h_{\sigma_{\zeta}^{pure}}$  for  $\xi \neq \zeta$ .
+
+Proof: Since in Proposition 1, the subsets are arbitrary, it holds for any two subsets with unique class count of one. By pairing up all combinations, one arrives at this proposition. Note that for a perfect unique class count classifier,  $\eta = \tilde{\eta}$ .
+
+# C DETAILS ON EXPERIMENTS WITH MNIST AND CIFAR DATASETS
+
+# C.1 DETAILS OF MODEL ARCHITECTURES
+
+Feature extractor module  $\theta_{\mathrm{feature}}$  has convolutional blocks similar to the wide residual blocks in Zagoruyko & Komodakis (2016). However, the parameters of architectures, number of convolutional and fully connected layers, number of filters in convolutional layers, number of nodes in
+
+fully-connected layers, number of bins and  $\sigma$  value in KDE module, were decided based on models' performance and training times. While increasing number of convolutional layers or filters were not improving performance of the models substantially, they were putting a heavy computation burden. For determining the architecture of  $\theta_{\mathrm{drn}}$ , we checked the performances of different number of fully connected layers. As the number of layers increased, the  $ucc$  classification performance of the models increased. However, we want  $\theta_{\mathrm{feature}}$  to be powerful, so we stopped to increase number of layers as soon as we got good results. For KDE module, we have tried parameters of 11 bins, 21 bins,  $\sigma = 0.1$  and  $\sigma = 0.01$ . Best results were obtained with 11 bins and  $\sigma = 0.1$ . Similarly, we have tested different number of features at the output of  $\theta_{\mathrm{feature}}$  module and we decided to use 10 features for MNIST and CIFAR10 datasets and 16 features for CIFAR100 dataset based on the clustering performance and computation burden.
+
+During training, loss value of validation sets was observed as early stopping criteria. Training of the models was stopped if the validation loss didn't drop for some certain amount of training iterations.
+
+For the final set of hyperparameters and details of architectures, please see the code for our experiments: http://bit.ly/uniqueclasscount
+
+# C.2 DETAILS OF DATASETS
+
+We trained and tested our models on MNIST, CIFAR10 and CIFAR100 datasets. While MNIST and CIFAR10 datasets have 10 classes, CIFAR100 dataset has 20 classes. For MNIST, we randomly splitted 10,000 images from training set as validation set, so we had 50,000, 10,000 and 10,000 images in our training  $\mathcal{X}_{mnist,tr}$ , validation  $\mathcal{X}_{mnist,val}$  and test sets  $\mathcal{X}_{mnist,test}$ , respectively. In CIFAR10 dataset, there are 50,000 and 10,000 images with equal number of instances from each class in training and testing sets, respectively. Similar to MNIST dataset, we randomly splitted 10,000 images from the training set as validation set. Hence, we had 40,000, 10,000 and 10,000 images in our training  $\mathcal{X}_{cifar10,tr}$ , validation  $\mathcal{X}_{cifar10,val}$  and testing  $\mathcal{X}_{cifar10,test}$  sets for CIFAR10, respectively. In CIFAR100 dataset, there are 50,000 and 10,000 images with equal number of instances from each class in training and testing sets, respectively. Similar to other datasets, we randomly splitted 10,000 images from the training set as validation set. Hence, we had 40,000, 10,000 and 10,000 images in our training  $\mathcal{X}_{cifar100,tr}$ , validation  $\mathcal{X}_{cifar100,val}$  and testing  $\mathcal{X}_{cifar100,test}$  sets for CIFAR10, respectively.
+
+FullySupervised models took individual instances as inputs and were trained on instance level ground truths.  $\mathcal{X}_{mnist,tr}$ ,  $\mathcal{X}_{cifar10,tr}$  and  $\mathcal{X}_{cifar100,tr}$  were used for training of FullySupervised models. Unique class count models took sets of instances as inputs and were trained on ucc labels. Inputs to unique class count models were sampled from the power sets of MNIST, CIFAR10 and CIFAR100 datasets, i.e.  $2^{\mathcal{X}_{mnist,tr}}$ ,  $2^{\mathcal{X}_{cifar10,tr}}$  and  $2^{\mathcal{X}_{cifar100,tr}}$ . For MNIST and CIFAR10 datasets, the subsets (bags) with 32 instances and for CIFAR100 dataset, the subsets (bags) with 128 instances are used in our experiments. While UCC and  $UCC_{\alpha = 1}$  models are trained on ucc1 to ucc4 labels,  $UCC^{2+}$  and  $UCC_{\alpha = 1}^{2+}$  models are trained on ucc2 to ucc4 labels.
+
+Our models were trained on ucc labels up to ucc4 instead of ucc10 (ucc20 in CIFAR100) since the performance was almost the same for both cases in our experiment with MNIST dataset, results of which are shown in Table 3. On the other hand, training with ucc1 to ucc4 was much faster than ucc1 to ucc10 because as the ucc label gets larger, the number of instances in a bag is required to be larger in order to represent each class and number of elements in powerset also grows exponentially. Please note that for perfect clustering of instances, it is enough to have a perfect ucc classifier that can discriminate ucc1 and ucc2 from Proposition 2.
+
+All the results presented in this paper were obtained on hold-out test sets  $\mathcal{X}_{mnist,test}$ ,  $\mathcal{X}_{cifar10,test}$  and  $\mathcal{X}_{cifar100,test}$ .
+
+Table 3: Clustering accuracy comparison of training unique class count models with ucc labels of ucc1 to ucc4 and ucc1 to ucc10 on MNIST dataset.  
+
+<table><tr><td></td><td colspan="2">clustering accuracy</td></tr><tr><td></td><td>ucc1 to ucc4</td><td>ucc1 to ucc10</td></tr><tr><td>UCC</td><td>0.984</td><td>0.983</td></tr><tr><td>UCC2+</td><td>0.984</td><td>0.982</td></tr></table>
+
+# C.3 CONFUSION MATRICES FOR ucc PREDICTIONS
+
+We randomly sampled subsets for each  $ucc$  label from the power sets of test sets and predicted the  $ucc$  labels by using trained models. Then, we calculated the  $ucc$  prediction accuracies by using predicted and truth  $ucc$  labels, which are summarized in Table 1. Here, we show confusion matrices of our  $UCC$  and  $UCC^{2+}$  models on MNIST, CIFAR10 and CIFAR100 datasets as examples in Figure 6, 7 and 8, respectively.
+
+![](images/644bf1ea88a43a51c872530d84ee69d27831074c930590852b3a5427fe03647c.jpg)  
+(a) UCC
+
+![](images/22f9981847bc91044aa6143da7a7cb6fd41ae6cc4267828c14aaed0c571c420f.jpg)  
+(b)  $UCC^{2+}$
+
+![](images/b3c57dd1cd44b3d4fadd40cf41d61dcdf47e2e15dafc859d1336eed1bc84885e.jpg)  
+Figure 6: Confusion matrices of our UCC and  $UCC^{2+}$  models for ucc prediction on MNIST.  
+(a) UCC
+
+![](images/2f2923b546cbbf4a4fe033beecca0967965b89dfc462769a90c6cba882f4e9e4.jpg)  
+(b)  $UCC^{2+}$  
+Figure 7: Confusion matrices of our UCC and  $UCC^{2+}$  models for ucc prediction on CIFAR10.
+
+![](images/66dc216373abe8057de7b1c599e31e24df86c80f673d17a464894dce03341c4d.jpg)  
+(a) UCC
+
+![](images/910312f7cae4c84c31eb1bf1a023e482393090e5ed9c3061009b1a822b1bcdeb.jpg)  
+(b)  $UCC^{2+}$  
+Figure 8: Confusion matrices of our  $UCC$  and  $UCC^{2+}$  models for  $ucc$  prediction on CIFAR100.
+
+# C.4 FEATURE DISTRIBUTIONS AND INTER-CLASS JS DIVERGENCE MATRICES
+
+The features of all the instances in a particular class are extracted by using a trained model and feature distributions associated to that class obtained by performing kernel density estimation on these extracted features. Then, for each pair of classes, inter-class JS divergence values are calculated. We show inter-class JS divergence matrices for our FullySupervised and UCC models on MNIST test dataset in Figure 9. We also show the underlying distributions for FullySupervised and UCC models in Figure 10 and 11, respectively.
+
+![](images/5c6820ec8b0dfd6bef75bf6a1ff41dc6fc289e406ff6eddd15bbd49639fe655b.jpg)  
+(a) Fully Supervised  
+Figure 9: Inter-class JS divergence matrix calculated over the distributions of features extracted by our FullySupervised and UCC models on MNIST test dataset.
+
+![](images/f25a006c919ee2ba62664d49d4e3668a018609315ec3080a2167d0f274bf0bab.jpg)  
+(b) UCC
+
+![](images/2847371f6b11daea4eed61d76013df2299ab07e137710bdd7d32740916126286.jpg)  
+Figure 10: Distributions of extracted features by our Fully Supervised model on MNIST test dataset. Each column corresponds to a feature learned by model and each row corresponds to an underlying class in the test dataset.
+
+![](images/c1b833d4bd445ea6668dd2dbda731beeeaa5f54d086b0fa2810ab75799da5935.jpg)  
+Figure 11: Distributions of extracted features by our UCC model on MNIST test dataset. Each column corresponds to a feature learned by model and each row corresponds to an underlying class in the test dataset.
+
+# C.5 K-MEANS AND SPECTRAL CLUSTERING ACCURACIES OF OUR MODELS
+
+We performed unsupervised clustering by using k-means and spectral clustering and gave the best clustering accuracy for each model on each dataset in Table 1 in the main text. Here, we present all the clustering accuracies for our models in Table 4.
+
+Table 4: Clustering accuracy values of our models with K-means and Spectral clustering methods on different test datasets. Best value for each model in each dataset is highlighted in bold.  
+
+<table><tr><td></td><td colspan="2">MNIST</td><td colspan="2">CIFAR10</td><td colspan="2">CIFAR100</td></tr><tr><td></td><td>K-means</td><td>Spectral</td><td>K-means</td><td>Spectral</td><td>K-means</td><td>Spectral</td></tr><tr><td>UCC</td><td>0.979</td><td>0.984</td><td>0.781</td><td>0.680</td><td>0.338</td><td>0.261</td></tr><tr><td>UCC2+</td><td>0.977</td><td>0.984</td><td>0.545</td><td>0.502</td><td>0.278</td><td>0.225</td></tr><tr><td>UCCα=1</td><td>0.981</td><td>0.984</td><td>0.774</td><td>0.635</td><td>0.317</td><td>0.249</td></tr><tr><td>UCC2+α=1</td><td>0.881</td><td>0.832</td><td>0.521</td><td>0.463</td><td>0.284</td><td>0.237</td></tr><tr><td>Autoencoder</td><td>0.930</td><td>0.832</td><td>0.241</td><td>0.230</td><td>0.167</td><td>0.140</td></tr><tr><td>FullySupervised</td><td>0.988</td><td>0.106</td><td>0.833</td><td>0.464</td><td>0.563</td><td>0.328</td></tr></table>
+
+# C.6 UCC MODELS WITH AVERAGING LAYER AND KDE LAYER
+
+KDE layer is chosen as MIL pooling layer in UCC model because of its four main properties, first three of which are essential for the proper operation of proposed framework and validity of the propositions in the paper:
+
+1. KDE layer is permutation-invariant, i.e. the output of KDE layer does not depend on the permutation of its inputs, which is important for the stability of  $\theta_{\mathrm{drn}}$  module.  
+2. KDE layer is differentiable, so  $UCC$  model can be trained end-to-end.  
+3. KDE layer has decomposability property which enables our theoretical analysis (Appendix B).  
+4. KDE layer enables  $\theta_{\mathrm{dn}}$  to fully utilize the information in the shape of the distribution rather than looking at point estimates of distribution.
+
+Averaging layer (Wang et al., 2018) as an MIL pooling layer, which also has the first three properties, can be an alternative to KDE layer in UCC model. We have conducted additional experiments by replacing KDE layer with 'averaging layer' and compare the clustering accuracy values of the models with averaging layer and the models with KDE layer in Table 5.
+
+Table 5: Clustering accuracy values of the models with averaging layer and the models with KDE layer.  
+
+<table><tr><td></td><td colspan="2">clustering acc.</td></tr><tr><td></td><td>mnist</td><td>cifar10</td></tr><tr><td>UCC (KDE layer)</td><td>0.984</td><td>0.781</td></tr><tr><td>UCC (Averaging layer)</td><td>0.987</td><td>0.638</td></tr><tr><td>UCCα=1 (KDE layer)</td><td>0.981</td><td>0.774</td></tr><tr><td>UCCα=1 (Averaging layer)</td><td>0.943</td><td>0.508</td></tr></table>
+
+# D DETAILS ON SEMANTIC SEGMENTATION TASK
+
+# D.1 DETAILS OF MODEL AND DATASET
+
+Our model  $UCC_{segment}$  has the same architecture with the UCC model in CIFAR10 dataset, but this time we have used 16 features. We have also constructed the Unet model with the same blocks used in  $UCC_{segment}$  model in order to ensure a fair comparison. The details of the models can be seen in our code: http://bit.ly/uniqueclasscount
+
+We have used  $512 \times 512$  image crops from publicly available CAMELYON dataset (Litjens et al., 2018). CAMELYON dataset is a public Whole Slide Image (WSI) dataset of histological lymph node sections. It also provides the exhaustive annotations for metastases regions inside the slides which enables us to train fully supervised models for benchmarking of our weakly supervised unique class count model.
+
+We randomly crop  $512 \times 512$  images over the WSIs of CAMELYON dataset and associate a  $ucc$  label to each image based on whether it is fully metastases/normal ( $ucc1$ ) or mixture ( $ucc2$ ). We assigned  $ucc$  labels based on provided ground truths since they are readily available. However, please note that in case no annotations provided, obtaining  $ucc$  labels is much cheaper and easier compared to tedious and time consuming exhaustive metastases region annotations. We assigned  $ucc1$  label to an image if the metastases region in the corresponding ground truth mask is either less than  $20\%$  (i.e. normal) or more than  $80\%$  (i.e. metastases). On the other hand, we assigned  $ucc2$  label to an image if the metastases region in the corresponding ground truth mask is more than  $30\%$  and less than  $70\%$  (i.e. mixture). Actually, this labeling scheme imitates the noise that would have been introduced if  $ucc$  labeling had been done directly by the user instead of using ground truth masks. Beyond that,  $ucc1$  labels in this task can naturally be noisy since it is possible to have some small portion of normal cells in cancer regions and vice-versa due to the nature of the cancer. In this way, we have constructed our segmentation dataset consisting of training, validation and testing sets. The images in training and validation sets are cropped randomly over the WSIs in training set of CAMELYON dataset and the images in testing set are cropped randomly over the test set of CAMELYON dataset. Then, the bags in our MIL dataset to train  $UCC_{segment}$  model are constructed by using  $32 \times 32$  patches over these images. Each bag contains 32 instances, where each instance is a  $32 \times 32$  patch. The details of our segmentation dataset are shown in Table 6.
+
+We have provided the segmentation dataset under“.data/camelyon/” folder inside our code folder. If you want to use this dataset for benchmarking purposes please cite our paper (referenced later) together with the original CAMELYON dataset paper of Litjens et al. (2018).
+
+Table 6: Details of our segmentation dataset: number of WSIs used to crop the images in each set, number of images in each set and corresponding label distributions in each set  
+
+<table><tr><td></td><td colspan="3">ucc1</td><td colspan="3">ucc2</td></tr><tr><td></td><td>normal</td><td>metastases</td><td>total</td><td>mixture</td><td># of images</td><td># of WSIs</td></tr><tr><td>Training</td><td>461</td><td>322</td><td>783</td><td>310</td><td>1093</td><td>159</td></tr><tr><td>Validation</td><td>278</td><td>245</td><td>523</td><td>211</td><td>734</td><td>106</td></tr><tr><td>Testing</td><td>282</td><td>668</td><td>950</td><td>228</td><td>1178</td><td>126</td></tr></table>
+
+We have given confusion matrix for  $ucc$  predictions of our  $UCC_{segment}$  model in Figure 12. For  $Unet$  model, we have shown loss curves of training and validation sets during training in Figure 13.
+
+![](images/c8ca38ae32d24c525cd675aad7dd9a6249b9997c3aeb60dc426180bbc8d126c0.jpg)  
+Figure 12: Confusion matrix of our  $UCC_{segment}$  model for  $ucc$  predictions on our segmentation dataset.
+
+![](images/5969f3a55584986ce193cd9589556d7c1fd60482d5e12feb689dc69c5cd1007e.jpg)  
+Figure 13: Training and validation loss curves during training of our Unet model. We have used the best model weights, which were saved at iteration 58000, during training. Models starts to overfit after iteration 60000 and early stopping terminates the training.
+
+# D.2 DEFINITIONS OF EVEALUTION METRICS
+
+In this section, we have defined our pixel level evaluation metrics used for performance comparison of our weakly supervised  $UCC_{segment}$  model, fully supervised  $Unet$  model and unsupervised baseline  $K - means$  model. Table 7 shows the structure of pixel level confusion matrix together with basic statistical terms. Then, our pixel level evaluation metrics TPR (True Positive Rate), FPR (False Positive Rate), TNR (True Negative Rate), FNR (False Negative Rate) and PA (Pixel Accuracy) are defined in Equation 11, 12, 13, 14 and 15, respectively.
+
+Table 7: Structure of pixel level confusion matrix together with basic statistical terms  
+
+<table><tr><td rowspan="2" colspan="2"></td><td colspan="2">Ground Truth</td></tr><tr><td>Positive (P)</td><td>Negative (N)</td></tr><tr><td rowspan="2">Predicted</td><td>Positive (P)</td><td>True Positive (TP)</td><td>False Positive (FP)</td></tr><tr><td>Negative (N)</td><td>False Negative (FN)</td><td>True Negative (TN)</td></tr></table>
+
+$$
+T P R = \frac {T P}{T P + F N} \tag {11}
+$$
+
+$$
+F P R = \frac {F P}{F P + T N} \tag {12}
+$$
+
+$$
+T N R = \frac {T N}{T N + F P} \tag {13}
+$$
+
+$$
+F N R = \frac {F N}{F N + T P} \tag {14}
+$$
+
+$$
+P A = \frac {T P + T N}{T P + F P + T N + F N} \tag {15}
+$$
\ No newline at end of file
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+# WEAKLY SUPERVISED DISENTANGLEMENT WITH GUARANTEES
+
+Rui Shu†*, Yining Chen†, Abhishek Kumar‡, Stefano Ermon‡ & Ben Poole‡
+
+†Stanford University, ‡Google Brain
+
+$\dagger$  {ruishu, cynnjjs, ermon}@stanford.edu
+
+$\ddagger$  {abhishk,pooleb}@google.com
+
+# ABSTRACT
+
+Learning disentangled representations that correspond to factors of variation in real-world data is critical to interpretable and human-controllable machine learning. Recently, concerns about the viability of learning disentangled representations in a purely unsupervised manner has spurred a shift toward the incorporation of weak supervision. However, there is currently no formalism that identifies when and how weak supervision will guarantee disentanglement. To address this issue, we provide a theoretical framework to assist in analyzing the disentanglement guarantees (or lack thereof) conferred by weak supervision when coupled with learning algorithms based on distribution matching. We empirically verify the guarantees and limitations of several weak supervision methods (restricted labeling, match-pairing, and rank-pairing), demonstrating the predictive power and usefulness of our theoretical framework.
+
+# 1 INTRODUCTION
+
+Many real-world datasets can be intuitively described via a data-generating process that first samples an underlying set of interpretable factors, and then—conditional on those factors—generates an observed data point. For example, in image generation, one might first sample the object identity and pose, and then render an image with the object in the correct pose. The goal of disentangled representation learning is to learn a representation where each dimension of the representation corresponds to a distinct factor of variation in the dataset (Bengio et al., 2013). Learning such representations that align with the underlying factors of variation may be critical to the development of machine learning models that are explainable or human-controllable (Gilpin et al., 2018; Lee et al., 2019; Klys et al., 2018).
+
+In recent years, disentanglement research has focused on the learning of such representations in an unsupervised fashion, using only independent samples from the data distribution without access to the true factors of variation (Higgins et al., 2017; Chen et al., 2018a; Kim & Mnih, 2018; Esmaeili et al., 2018). However, Locatello et al. (2019) demonstrated that many existing methods for the unsupervised learning of disentangled representations are brittle, requiring careful supervision-based hyperparameter tuning. To build robust disentangled representation learning methods that do not require large amounts of supervised data, recent work has turned to forms of weak supervision (Chen & Batmanghelich, 2019; Gabbay & Hoshen, 2019). Weak supervision can allow one to build models that have interpretable representations even when human labeling is challenging (e.g., hair style in face generation, or style in music generation). While existing methods based on weakly-supervised learning demonstrate empirical gains, there is no existing formalism for describing the theoretical guarantees conferred by different forms of weak supervision (Kulkarni et al., 2015; Reed et al., 2015; Bouchacourt et al., 2018).
+
+In this paper, we present a comprehensive theoretical framework for weakly supervised disentanglement, and evaluate our framework on several datasets. Our contributions are several-fold.
+
+1. We formalize weakly-supervised learning as distribution matching in an extended space.
+
+2. We propose a set of definitions for disentanglement that can handle correlated factors and are inspired by many existing definitions in the literature (Higgins et al., 2018; Suter et al., 2018; Ridgeway & Mozer, 2018).  
+3. Using these definitions, we provide a conceptually useful and theoretically rigorous calculus of disentanglement.  
+4. We apply our theoretical framework of disentanglement to analyze three notable classes of weak supervision methods (restricted labeling, match pairing, and rank pairing). We show that although certain weak supervision methods (e.g., style-labeling in style-content disentanglement) do not guarantee disentanglement, our calculus can determine whether disentanglement is guaranteed when multiple sources of weak supervision are combined.  
+5. Finally, we perform extensive experiments to systematically and empirically verify our predicted guarantees.<sup>1</sup>
+
+# 2 FROM UNSUPERVISED TO WEAKLY SUPERVISED DISTRIBUTION MATCHING
+
+Our goal in disentangled representation learning is to identify a latent-variable generative model whose latent variables correspond to ground truth factors of variation in the data. To identify the role that weak supervision plays in providing guarantees on disentanglement, we first formalize the model families we are considering, the forms of weak supervision, and finally the metrics we will use to evaluate and prove components of disentanglement.
+
+We consider data-generating processes where  $S \in \mathbb{R}^n$  are the factors of variation, with distribution  $p^{*}(s)$ , and  $X \in \mathbb{R}^{m}$  is the observed data point which is a deterministic function of  $S$ , i.e.,  $X = g^{*}(S)$ . Many existing algorithms in unsupervised learning of disentangled representations aim to learn a latent-variable model with prior  $p(z)$  and generator  $g$ , where  $g(Z) \stackrel{d}{=} g^{*}(S)$ . However, simply matching the marginal distribution over data is not enough: the learned latent variables  $Z$  and the true generating factors  $S$  could still be entangled with each other (Locatello et al., 2019).
+
+To address the failures of unsupervised learning of disentangled representations, we leverage weak supervision, where information about the data-generating process is conveyed through additional observations. By performing distribution matching on an augmented space (instead of just on the observation  $X$ ), we can provide guarantees on learned representations.
+
+![](images/a3eceaf200b0ed038b66ca011ab18bfb7acce144476d4b7e45a97f828ec0f897.jpg)  
+Unsupervised  $p(x)$
+
+![](images/1b4cef873c7cc2a93ef6a485e48cdf8de4126c3214c198256a184d6d8a9f81bb.jpg)  
+Figure 1: Augmented data distributions derived from weak supervision. Shaded nodes denote observed quantities, and unshaded nodes represent unobserved (latent) variables.
+
+![](images/50a04a5a1ff8523ab24a4b6e7e392ae1118e9f21b4ac1249a427eddec5a36453.jpg)  
+Restricted Labeling  $p(x,s_I)$  
+Match Pairing  $p(x,x^{\prime})$
+
+![](images/c1e4d58208bf36fefd32ed99fb4feb8d6484e67228670a0f31456876666506ed.jpg)  
+Rank Pairing  $p(x,x^{\prime},y)$
+
+We consider three practical forms of weak supervision: restricted labeling, match pairing, and rank pairing. All of these forms of supervision can be thought of as augmented forms of the original joint distribution, where we partition the latent variables in two  $S = (S_I, S_{\backslash I})$ , and either observe a subset of the latent variables or share latents between multiple samples. A visualization of these augmented distributions is presented in Figure 1, and below we detail each form of weak supervision.
+
+In restricted labeling, we observe a subset of the ground truth factors,  $S_I$  in addition to  $X$ . This allows us to perform distribution matching on  $p^*(s_I, x)$ , the joint distribution over data and observed factors, instead of just the data,  $p^*(x)$ , as in unsupervised learning. This form of supervision is often leveraged in style-content disentanglement, where labels are available for content but not style (Kingma et al., 2014; Narayanaswamy et al., 2017; Chen et al., 2018b; Gabbay & Hoshen, 2019).
+
+Match Pairing uses paired data,  $(x,x^{\prime})$  that share values for a known subset of factors,  $I$ . For many data modalities, factors of variation may be difficult to explicitly label. Instead, it may be easier to collect pairs of samples that share the same underlying factor (e.g., collecting pairs of images of different people wearing the same glasses is easier than defining labels for style of glasses). Match pairing is a weaker form of supervision than restricted labeling, as the learning algorithm no longer depends on the underlying value  $s_I$ , and only on the indices of shared factors  $I$ . Several variants of match pairing have appeared in the literature (Kulkarni et al., 2015; Bouchacourt et al., 2018; Ridgeway & Mozer, 2018), but typically focus on groups of observations in contrast to the paired setting we consider in this paper.
+
+Rank Pairing is another form of paired data generation where the pairs  $(x, x')$  are generated in an i.i.d. fashion, and an additional indicator variable  $y$  is observed that determines whether the corresponding latent  $s_i$  is greater than  $s_i'$ :  $y = \mathbf{1}\{s_i \geq s_i'\}$ . Such a form of supervision is effective when it is easier to compare two samples with respect to an underlying factor than to directly collect labels (e.g., comparing two object sizes versus providing a ruler measurement of an object). Although supervision via ranking features prominently in the metric learning literature (McFee & Lanckriet, 2010; Wang et al., 2014), our focus in this paper will be on rank pairing in the context of disentanglement guarantees.
+
+For each form of weak supervision, we can train generative models with the same structure as in Figure 1, using data sampled from the ground truth model and a distribution matching objective. For example, for match pairing, we train a generative model  $(p(z), g)$  such that the paired random variable  $(g(Z_I, Z_{\backslash I}), g(Z_I, Z_{\backslash I}'))$  from the generator matches the distribution of the corresponding paired random variable  $(g^*(S_I, S_{\backslash I}), g^*(S_I, S_{\backslash I}'))$  from the augmented data distribution.
+
+# 3 DEFINING DISENTANGLEMENT
+
+To identify the role that weak supervision plays in providing guarantees on disentanglement, we introduce a set of definitions that are consistent with our intuitions about what constitutes "disentanglement" and amenable to theoretical analysis. Our new definitions decompose disentanglement into two distinct concepts: consistency and restrictiveness. Different forms of weak supervision can enable consistency or restrictiveness on subsets of factors, and in Section 4 we build up a calculus of disentanglement from these primitives. We discuss the relationship to prior definitions of disentanglement in Appendix A.
+
+# 3.1 DECOMPOSING DISENTANGLEMENT INTO CONSISTENCY AND RESTRICTIVENESS
+
+![](images/a23c990f60d8ada6dbbc18ee1367ca56ea78ce773af20cdf1285686f99ed9ff9.jpg)  
+(a) Disentanglement
+
+![](images/bd5cd7e4ddb19c492571df1a56a561b9bcd73d8772c4731586851bbbfd04c742.jpg)  
+(b) Consistency  
+Figure 2: Illustration of disentanglement, consistency, and restrictiveness of  $z_{1}$  with respect to the factor of variation size. Each image of a shape represents the decoding  $g(z_{1:3})$  by the generative model. Each column denotes a fixed choice of  $z_{1}$ . Each row denotes a fixed choice of  $(z_{2}, z_{3})$ . A demonstration of consistency versus restrictiveness on models from disentanglement_lib is available in Appendix B.
+
+![](images/4504af1d7bcc0f4e612692813b0a3b4c92927bd91644e6016b63574f2752a723.jpg)  
+(c) Restrictiveness
+
+To ground our discussion of disentanglement, we consider an oracle that generates shapes with factors of variation for size  $(S_1)$ , shape  $(S_2)$ , and color  $(S_3)$ . How can we determine whether  $Z_1$  of our generative model "disentangles" the concept of size? Intuitively, one way to check whether  $Z_1$  of the generative model disentangles size  $(S_1)$  is to visually inspect what happens as we vary  $Z_1$ ,  $Z_2$ , and  $Z_3$ , and see whether the resulting visualizations are consistent with Figure 2a. In doing so, our visual inspection checks for two properties:
+
+1. When  $Z_{1}$  is fixed, the size  $(S_{1})$  of the generated object never changes.  
+2. When only  $Z_{1}$  is changed, the change is restricted to the size  $(S_{1})$  of the generated object, meaning that there is no change in  $S_{j}$  for  $j \neq 1$ .
+
+We argue that disentanglement decomposes into these two properties, which we refer to as generator consistency and generator restrictiveness. Next, we formalize these two properties.
+
+Let  $\mathcal{H}$  be a hypothesis class of generative models from which we assume the true data-generating function is drawn. Each element of the hypothesis class  $\mathcal{H}$  is a tuple  $(p(s), g, e)$ , where  $p(s)$  describes the distribution over factors of variation, the generator  $g$  is a function that maps from the factor space  $S \in \mathbb{R}^n$  to the observation space  $\mathcal{X} \in \mathbb{R}^m$ , and the encoder  $e$  is a function that maps from  $\mathcal{X} \to S$ .  $S$  and  $X$  can consist of both discrete and continuous random variables. We impose a few mild assumptions on  $\mathcal{H}$  (see Appendix I.1). Notably, we assume every factor of variation is exactly recoverable from the observation  $X$ , i.e.  $e(g(S)) = S$ .
+
+Given an oracle model  $h^* = (p^*, g^*, e^*) \in \mathcal{H}$ , we would like to learn a model  $h = (p, g, e) \in \mathcal{H}$  whose latent variables disentangle the latent variables in  $h^*$ . We refer to the latent-variables in the oracle  $h^*$  as  $S$  and the alternative model  $h$ 's latent variables as  $Z$ . If we further restrict  $h$  to only those models where  $g(Z) \stackrel{d}{=} g^*(S)$  are equal in distribution, it is natural to align  $Z$  and  $S$  via  $S = e^* \circ g(Z)$ . Under this relation between  $Z$  and  $S$ , our goal is to construct definitions that describe whether the latent code  $Z_i$  disentangles the corresponding factor  $S_i$ .
+
+Generator Consistency. Let  $I$  denote a set of indices and  $p_I$  denote the generating process
+
+$$
+z _ {I} \sim p \left(z _ {I}\right) \tag {1}
+$$
+
+$$
+z _ {\backslash I}, z _ {\backslash I} ^ {\prime} \stackrel {{\mathrm {i i d}}} {{\sim}} p \left(z _ {\backslash I} \mid z _ {I}\right). \tag {2}
+$$
+
+This generating process samples  $Z_{I}$  once and then conditionally samples  $Z_{I}$  twice in an i.i.d. fashion. We say that  $Z_{I}$  is consistent with  $S_{I}$  if
+
+$$
+\mathbb {E} _ {p _ {I}} \left\| e _ {I} ^ {*} \circ g \left(z _ {I}, z _ {\backslash I}\right) - e _ {I} ^ {*} \circ g \left(z _ {I}, z _ {\backslash I} ^ {\prime}\right) \right\| ^ {2} = 0, \tag {3}
+$$
+
+where  $e_I^*$  is the oracle encoder restricted to the indices  $I$ .
+
+Intuitively, Equation (3) states that, for any fixed choice of  $Z_{I}$ , resampling of  $Z_{\backslash I}$  will not influence the oracle's measurement of the factors  $S_{I}$ . In other words,  $S_{I}$  is invariant to changes in  $Z_{\backslash I}$ . An illustration of a generative model where  $Z_{1}$  is consistent with size  $(S_{1})$  is provided in Figure 2b. A notable property of our definition is that the prescribed sampling process  $p_{I}$  does not require the underlying factors of variation to be statistically independent. We characterize this property in contrast to previous definitions of disentanglement in Appendix A.
+
+Generator Restrictiveness. Let  $p_{\backslash I}$  denote the generating process
+
+$$
+z _ {\backslash I} \sim p (z _ {\backslash I}) \tag {4}
+$$
+
+$$
+z _ {I}, z _ {I} ^ {\prime} \stackrel {\text {i i d}} {\sim} p \left(z _ {I} \mid z _ {\backslash I}\right). \tag {5}
+$$
+
+We say that  $Z_{I}$  is restricted to  $S_{I}$  if
+
+$$
+\mathbb {E} _ {p \backslash I} \| e _ {\backslash I} ^ {*} \circ g (z _ {I}, z _ {\backslash I}) - e _ {\backslash I} ^ {*} \circ g (z _ {I} ^ {\prime}, z _ {\backslash I}) \| ^ {2} = 0. \tag {6}
+$$
+
+Equation (6) states that, for any fixed choice of  $Z_{\backslash I}$ , resampling of  $Z_I$  will not influence the oracle's measurement of the factors  $S_{\backslash I}$ . In other words,  $S_{\backslash I}$  is invariant to changes in  $Z_I$ . Thus, changing  $Z_I$  is restricted to modifying only  $S_I$ . An illustration of a generative model where  $Z_1$  is restricted to size  $(S_1)$  is provided in Figure 2c.
+
+Generator Disentanglement. We now say that  $Z_{I}$  disentangles  $S_{I}$  if  $Z_{I}$  is consistent with and restricted to  $S_{I}$ . If we denote consistency and restrictiveness via Boolean functions  $C(I)$  and  $R(I)$ , we can now concisely state that
+
+$$
+D (I) := C (I) \wedge R (I), \tag {7}
+$$
+
+where  $D(I)$  denotes whether  $Z_{I}$  disentangles  $S_{I}$ . An illustration of a generative model where  $Z_{1}$  disentangles size  $(S_{1})$  is provided in Figure 2a. Note that while size increases monotonically with  $Z_{1}$  in the schematic figure, we wish to clarify that monotonicity is unrelated to the concepts of consistency and restrictiveness.
+
+# 3.2 RELATIONTO BIJECTIVITY-BASED DEFINITION OF DISENTANGLEMENT
+
+Under our mild assumptions on  $\mathcal{H}$ , distribution matching on  $g(Z) \stackrel{d}{=} g(S)$  combined with generator disentanglement on factor  $I$  implies the existence of two invertible functions  $f_{I}$  and  $f_{\backslash I}$  such that the alignment via  $S = e^{*} \circ g(Z)$  decomposes into
+
+$$
+\left[ \begin{array}{l} S _ {I} \\ S _ {\backslash I} \end{array} \right] = \left[ \begin{array}{l} f _ {I} \left(Z _ {I}\right) \\ f _ {\backslash I} \left(Z _ {\backslash I}\right) \end{array} \right]. \tag {8}
+$$
+
+This expression highlights the connection between disentanglement and invariance, whereby  $S_I$  is only influenced by  $Z_I$ , and  $S_{\backslash I}$  is only influenced by  $Z_{\backslash I}$ . However, such a bijectivity-based definition of disentanglement does not naturally expose the underlying primitives of consistency and restrictiveness, which we shall demonstrate in our theory and experiments to be valuable concepts for describing disentanglement guarantees under weak supervision.
+
+# 3.3 ENCODER-BASED DEFINITIONS FOR DISENTANGLEMENT
+
+Our proposed definitions are asymmetric—measuring the behavior of a generative model against an oracle encoder. So far, we have chosen to present the definitions from the perspective of a learned generator  $(p,g)$  measured against an oracle encoder  $e^{*}$ . In this sense, they are generator-based definitions. We can also develop a parallel set of definitions for encoder-based consistency, restrictiveness, and disentanglement within our framework simply by using an oracle generator  $(p^{*},g^{*})$  measured against a learned encoder  $e$ . Below, we present the encoder-based perspective on consistency.
+
+Encoder Consistency. Let  $p_I^*$  denote the generating process
+
+$$
+s _ {I} \sim p ^ {*} (s _ {I}) \tag {9}
+$$
+
+$$
+s _ {\backslash I}, s _ {\backslash I} ^ {\prime} \stackrel {{\mathrm {i i d}}} {{\sim}} p ^ {*} (s _ {\backslash I}, | s _ {I}). \tag {10}
+$$
+
+This generating process samples  $S_I$  once and then conditionally samples  $S_I$  twice in an i.i.d. fashion. We say that  $S_I$  is consistent with  $Z_I$  if
+
+$$
+\mathbb {E} _ {p _ {I} ^ {*}} \left\| e _ {I} \circ g ^ {*} \left(s _ {I}, s _ {\backslash I}\right) - e _ {I} \circ g ^ {*} \left(s _ {I}, s _ {\backslash I} ^ {\prime}\right) \right\| ^ {2} = 0. \tag {11}
+$$
+
+We now make two important observations. First, a valuable trait of our encoder-based definitions is that one can check for encoder consistency / restrictiveness / disentanglement as long as one has access to match pairing data from the oracle generator. This is in contrast to the existing disentanglement definitions and metrics, which require access to the ground truth factors (Higgins et al., 2017; Kumar et al., 2018; Kim & Mnih, 2018; Chen et al., 2018a; Suter et al., 2018; Ridgeway & Mozer, 2018; Eastwood & Williams, 2018). The ability to check for our definitions in a weakly supervised fashion is the key to why we can develop a theoretical framework using the language of consistency and restrictiveness. Second, encoder-based definitions are tractable to measure when testing on synthetic data, since the synthetic data directly serves the role of the oracle generator. As such, while we develop our theory to guarantee both generator-based and the encoder-based disentanglement, all of our measurements in the experiments will be conducted with respect to a learned encoder.
+
+We make three remarks on notations. First,  $D(i) \coloneqq D(\{i\})$ . Second,  $D(\varnothing)$  evaluates to true. Finally,  $D(I)$  is implicitly dependent on either  $(p,g,e^{*})$  (generator-based) or  $(p^{*},g^{*},e)$  (encoder-based). Where important, we shall make this dependency explicit (e.g., let  $D(I;p,g,e^{*})$  denote generator-based disentanglement). We apply these conventions to  $C$  and  $R$  analogously.
+
+# 4 A CALCULUS OF DISENTANGLEMENT
+
+There are several interesting relationships between restrictiveness and consistency. First, by definition,  $C(I)$  is equivalent to  $R(\backslash I)$ . Second, we can see from Figures 2b and 2c that  $C(I)$  and  $R(I)$  do not imply each other. Based on these observations and given that consistency and restrictiveness operate over subsets of the random variables, a natural question that arises is whether consistency or restrictiveness over certain sets of variables imply additional properties over other sets of variables.
+
+We develop a calculus for discovering implied relationships between learned latent variables  $Z$  and ground truth factors of variation  $S$  given known relationships as follows.
+
+# Calculus of Disentanglement
+
+# Consistency and Restrictiveness
+
+$$
+C (I) \not \Rightarrow R (I)
+$$
+
+$$
+R (I) \Rightarrow C (I)
+$$
+
+$$
+C (I) \Longleftrightarrow R (\backslash I)
+$$
+
+# Union Rules
+
+$$
+C (I) \wedge C (J) \Rightarrow C (I \cup J)
+$$
+
+$$
+R (I) \wedge R (J) \Rightarrow R (I \cup J)
+$$
+
+# Intersection Rules
+
+$$
+C (I) \wedge C (J) \Rightarrow C (I \cap J)
+$$
+
+$$
+R (I) \wedge R (J) \Rightarrow R (I \cap J)
+$$
+
+# Full Disentanglement
+
+$$
+\bigwedge_ {i = 1} ^ {n} \bar {C} (i) \Longleftrightarrow \bigwedge_ {i = 1} ^ {n} D (i)
+$$
+
+$$
+\bigwedge_ {i = 1} ^ {n} R (i) \Longleftrightarrow \bigwedge_ {i = 1} ^ {n} D (i)
+$$
+
+Our calculus provides a theoretically rigorous procedure for reasoning about disentanglement. In particular, it is no longer necessary to prove whether the supervision method of interest satisfies consistency and restrictiveness for each and every factor. Instead, it suffices to show that a supervision method guarantees consistency or restrictiveness for a subset of factors, and then combine multiple supervision methods via the calculus to guarantee full disentanglement. We can additionally use the calculus to uncover consistency or restrictiveness on individual factors when weak supervision is available only for a subset of variables. For example, achieving consistency on  $S_{1,2}$  and  $S_{2,3}$  implies consistency on the intersection  $S_{2}$ . Furthermore, we note that these rules are agnostic to using generator or encoder-based definitions. We defer the complete proofs to Appendix I.2.
+
+# 5 FORMALIZING WEAK SUPERVISION WITH GUARANTEES
+
+In this section, we address the question of whether disentanglement arises from the supervision method or model inductive bias. This challenge was first put forth by Locatello et al. (2019), who noted that unsupervised disentanglement is heavily reliant on model inductive bias. As we transition toward supervised approaches, it is crucial that we formalize what it means for disentanglement to be guaranteed by weak supervision.
+
+Sufficiency for Disentanglement. Let  $\mathcal{P}$  denote a family of augmented distributions. We say that a weak supervision method  $\mathbf{S}:\mathcal{H}\to \mathcal{P}$  is sufficient for learning a generator whose latent codes  $Z_{I}$  disentangle the factors  $S_{I}$  if there exists a learning algorithm  $\mathcal{A}:\mathcal{P}\rightarrow \mathcal{H}$  such that for any choice of  $(p^{*}(s),g^{*},e^{*})\in \mathcal{H}$ , the procedure  $\mathcal{A}\circ \mathbf{S}(p^{*}(s),g^{*},e^{*})$  returns a model  $(p(z),g,e)$  for which both  $D(I;p,g,e^{*})$  and  $D(I;p^{*},g^{*},e)$  hold, and  $g(Z)\stackrel {d}{=}g^{*}(S)$ .
+
+The key insight of this definition is that we force the strategy and learning algorithm pair  $(\mathbf{S},\mathcal{A})$  to handle all possible oracles drawn from the hypothesis class  $\mathcal{H}$ . This prevents the exploitation of model inductive bias, since any bias from the learning algorithm  $\mathcal{A}$  toward a reduced hypothesis class  $\hat{\mathcal{H}}\subset \mathcal{H}$  will result in failure to handle oracles in the complementary hypothesis class  $\mathcal{H}\setminus \hat{\mathcal{H}}$ .
+
+The distribution matching requirement  $g(Z) \stackrel{d}{=} g^{*}(S)$  ensures latent code informativeness, i.e., preventing trivial solutions where the latent code is uninformative (see Proposition 6 for formal statement). Intuitively, distribution matching paired with a deterministic generator guarantees invertibility of the learned generator and encoder, enforcing that  $Z_{I}$  cannot encode less information than  $S_{I}$  (e.g., only encoding age group instead of numerical age) and vice versa.
+
+# 6 ANALYSIS OF WEAK SUPERVISION METHODS
+
+We now apply our theoretical framework to three practical weak supervision methods: restricted labeling, match pairing, and rank pairing. Our main theoretical findings are that: (1) these methods can be applied in a targeted manner to provide single factor consistency or restrictiveness guarantees; (2) by enforcing consistency (or restrictiveness) on all factors, we can learn models with strong
+
+disentanglement performance. Correspondingly, Figure 3 and Figure 5 are our main experimental results, demonstrating that these theoretical guarantees have predictive power in practice.
+
+# 6.1 THEORETICAL GUARANTEES FROM WEAK SUPERVISION
+
+We prove that if a training algorithm successfully matches the generated distribution to data distribution generated via restricted labeling, match pairing, or rank pairing of factors  $S_{I}$ , then  $Z_{I}$  is guaranteed to be consistent with  $S_{I}$ :
+
+Theorem 1. Given any oracle  $(p^{*}(s),g^{*},e^{*})\in \mathcal{H}$ , consider the distribution-matching algorithm  $\mathcal{A}$  that selects a model  $(p(z),g,e)\in \mathcal{H}$  such that:
+
+1.  $(g^{*}(S), S_{I}) \stackrel{d}{=} (g(Z), Z_{I})$  (Restricted Labeling); or  
+2.  $\left(g^{*}(S_{I},S_{\backslash I}),g^{*}(S_{I},S_{\backslash I}^{\prime})\right)\stackrel {d}{=}\left(g(Z_{I},Z_{\backslash I}),g(Z_{I},Z_{\backslash I}^{\prime})\right)$  (Match Pairing); or  
+3.  $(g^{*}(S), g^{*}(S'), \mathbf{1}\{S_{I} \leq S_{I}'\}) \stackrel{d}{=} (g(Z), g(Z'), \mathbf{1}\{Z_{I} \leq Z_{I}'\})$  (Rank Pairing).
+
+Then  $(p,g)$  satisfies  $C(I;p,g,e^{*})$  and  $e$  satisfies  $C(I;p^{*},g^{*},e)$ .
+
+Theorem 1 states that distribution-matching under restricted labeling, match pairing, or rank pairing of  $S_I$  guarantees both generator and encoder consistency for the learned generator and encoder respectively. We note that while the complement rule  $C(I) \Rightarrow R(\backslash I)$  further guarantees that  $Z_{\backslash I}$  is restricted to  $S_{\backslash I}$ , we can prove that the same supervision does not guarantee that  $Z_I$  is restricted to  $S_I$  (Theorem 2). However, if we additionally have restricted labeling for  $S_{\backslash I}$ , or match pairing for  $S_{\backslash I}$ , then we can see from the calculus that we will have guaranteed  $R(I) \wedge C(I)$ , thus implying disentanglement of factor  $I$ . We also note that while restricted labeling and match pairing can be applied on a set of factors at once (i.e.  $|I| \geq 1$ ), rank pairing is restricted to one-dimensional factors for which an ordering exists. In the experiments below, we empirically verify the theoretical guarantees provided in Theorem 1.
+
+# 6.2 EXPERIMENTS
+
+We conducted experiments on five prominent datasets in the disentanglement literature: *Shapes3D* (Kim & Mnih, 2018), *dSprites* (Higgins et al., 2017), *Scream-dSprites* (Locatello et al., 2019), *Small-NORB* (LeCun et al., 2004), and *Cars3D* (Reed et al., 2015). Since some of the underlying factors are treated as nuisance variables in *SmallNORB* and *Scream-dSprites*, we show in Appendix C that our theoretical framework can be easily adapted accordingly to handle such situations. We use generative adversarial networks (GANs, Goodfellow et al. (2014)) for learning  $(p, g)$  but any distribution matching algorithm (e.g., maximum likelihood training in tractable models, or VI in latent-variable models) could be applied. Our results are collected over a broad range of hyperparameter configurations (see Appendix H for details).
+
+Since existing quantitative metrics of disentanglement all measure the performance of an encoder with respect to the true data generator, we trained an encoder post-hoc to approximately invert the learned generator, and measured all quantitative metrics (e.g., mutual information gap) on the encoder. Our theory assumes that the learned generator must be invertible. While this is not true for conventional GANs, our empirical results show that this is not an issue in practice (see Appendix G).
+
+We present three sets of experimental results: (1) Single-factor experiments, where we show that our theory can be applied in a targeted fashion to guarantee consistency or restrictiveness of a single factor. (2) Consistency versus restrictiveness experiments, where we show the extent to which single-factor consistency and restrictiveness are correlated even when the models are only trained to maximize one or the other. (3) Full disentanglement experiments, where we apply our theory to fully disentangle all factors. A more extensive set of experiments can be found in the Appendix.
+
+# 6.2.1 SINGLE-FACTOR CONSISTENCY AND RESTRICTIVENESS
+
+We empirically verify that single-factor consistency or restrictiveness can be achieved with the supervision methods of interest. Note there are two special cases of match pairing: one where  $S_{i}$  is
+
+![](images/7ff77e8e03e95c22106b45ea47148fb9c710cb63a8352e2dbf93fd0ddee177c7.jpg)  
+Figure 3: Heatmap visualization of ablation studies that measure either single-factor consistency or single-factor restrictiveness as a function of various supervision methods, conducted on Shapes3D. Our theory predicts the diagonal components to achieve the highest scores. Note that share pairing, change pairing, and change pair intersection are special cases of match pairing.
+
+![](images/8be564d75319bd8827ce9e0e5a2da86a86c89844a24105c0bf8e3203ccde7525.jpg)
+
+![](images/b57fd8770274be9d8a1b2b4cb3c266fc31e7f3a9ccf7ccf45e1773c033e954f1.jpg)
+
+![](images/e387efc84006a74bd36a7a3f5c0357e123e4537bd6710ee56625f167df108330.jpg)
+
+![](images/aef4725282092d6a4973d9f84622a37b1a11a0c07ffc459d46069b7eecce2521.jpg)
+
+the only factor that is shared between  $x$  and  $x'$  and one where  $S_i$  is the only factor that is changed. We distinguish these two conditions as share pairing and change pairing, respectively. Theorem 1 shows that restricted labeling, share pairing, and rank pairing of the  $i^{\text{th}}$  factor are each sufficient supervision strategies for guaranteeing consistency on  $S_i$ . Change pairing at  $S_i$  is equivalent to share pairing at  $S_{\backslash i}$ ; the complement rule  $C(I) \Longleftrightarrow R(\backslash I)$  allows us to conclude that change pairing guarantees restrictiveness. The first four heatmaps in Figure 3 show the results for restricted labeling, share pairing, change pairing, and rank pairing. The numbers shown in the heatmap are the normalized consistency and restrictiveness scores. We define the normalized consistency score as
+
+$$
+\tilde {c} (I; p ^ {*}, g ^ {*}, e) = 1 - \frac {\mathbb {E} _ {p _ {I} ^ {*}} \| e _ {I} \circ g ^ {*} \left(s _ {I} , s _ {\backslash I}\right) - e _ {I} \circ g ^ {*} \left(s _ {I} , s _ {\backslash I} ^ {\prime}\right) \| ^ {2}}{\mathbb {E} _ {s , s ^ {\prime} \sim p ^ {*}} \| e _ {I} \circ g ^ {*} (s) - e _ {I} \circ g ^ {*} \left(s ^ {\prime}\right) \| ^ {2}}. \tag {12}
+$$
+
+This score is bounded on the interval  $[0,1]$  (a consequence of Lemma 1) and is maximal when  $C(I;p^{*},g^{*},e)$  is satisfied. This normalization procedure is similar in spirit to the Interventional Robustness score in Suter et al. (2018). The normalized restrictiveness score  $\tilde{r}$  can be analogously defined. In practice, we estimate this score via Monte Carlo estimation.
+
+The final heatmap in Figure 3 demonstrates the calculus of intersection. In practice, it may be easier to acquire paired data where multiple factors change simultaneously. If we have access to two kinds of datasets, one where  $S_{I}$  are changed and one where  $S_{J}$  are changed, our calculus predicts that training on both datasets will guarantee restrictiveness on  $S_{I \cap J}$ . The final heatmap shows six such intersection settings and measures the normalized restrictiveness score; in all but one setting, the results are consistent with our theory. We show in Figure 7 that this inconsistency is attributable to the failure of the GAN to distribution-match due to sensitivity to a specific hyperparameter.
+
+# 6.2.2 CONSISTENCY VERSUS RESTRICTIVENESS
+
+![](images/81ed25c631d6aae20f823e8aaf327c0aa869b70911adcba138ed1f8efbbdf3ab.jpg)  
+Figure 4: Correlation plot and scatterplots demonstrating the empirical relationship between  $\tilde{c}(i)$  and  $\tilde{r}(i)$  across all 864 models trained on Shapes3D.
+
+![](images/3f5727642c45728622f80d49ff03f56ddc2f09a3e3e231aa1b685cac8c8959bf.jpg)
+
+![](images/b24d2ee4a756a0a49a0506959dec7fa735996feb75b772166215103afff0d152.jpg)
+
+![](images/ce81f8af53f25a2630cf63897f0b08629fffee7733279098b14fd3d1cc095a34.jpg)
+
+![](images/bee0f2ec2e77e070112bd7119adbee0082bece65c3d39b609557fd7794c7bce0.jpg)
+
+![](images/fa14f0376c11eb54e6ac40038b89bd2ba66a3ddc60f1509e9e1e8b92b628f4eb.jpg)
+
+![](images/20dbf2c8da06979ed7a0f9bb5f785a2dd38c5efb16dc22bec04cc3f0407f3ef9.jpg)
+
+We now determine the extent to which consistency and restrictiveness are correlated in practice. In Figure 4, we collected all 864 Shapes3D models that we trained in Section 6.2.1 and measured the consistency and restrictiveness of each model on each factor, providing both the correlation plot and scatterplots of  $\tilde{c}(i)$  versus  $\tilde{r}(i)$ . Since the models trained in Section 6.2.1 only ever targeted the consistency or restrictiveness of a single factor, and since our calculus demonstrates that consistency and restrictiveness do not imply each other, one might a priori expect to find no correlation in Figure 4. Our results show that the correlation is actually quite strong. Since this correlation is not guaranteed by our choice of weak supervision, it is necessarily a consequence of model inductive
+
+bias. We believe this correlation between consistency and restrictiveness to have been a general source of confusion in the disentanglement literature, causing many to either observe or believe that restricted labeling or share pairing on  $S_{i}$  (which only guarantees consistency) is sufficient for disentangling  $S_{i}$  (Kingma et al., 2014; Chen & Batmanghelich, 2019; Gabbay & Hoshen, 2019; Narayanaswamy et al., 2017). It remains an open question why consistency and restrictiveness are so strongly correlated when training existing models on real-world data.
+
+# 6.2.3 FULL DISENTANGLEMENT
+
+![](images/b329277c57cf96844c4b24ceef741e8a965192db1d9483a7a71be38a8bc5a50f.jpg)  
+Figure 5: Disentanglement performance of a vanilla GAN, share pairing GAN, change pairing GAN, rank pairing GAN, and fully-labeled GAN, as measured by the mutual information gap across several datasets. A comprehensive set of performance evaluations on existing disentanglement metrics is available in Figure 13.
+
+If we have access to share / change / rank-pairing data for each factor, our calculus states that it is possible to guarantee full disentanglement. We trained our generative model on either complete share pairing, complete change pairing, or complete rank pairing, and measured disentanglement performance via the discretized mutual information gap (Chen et al., 2018a; Locatello et al., 2019). As negative and positive controls, we also show the performance of an unsupervised GAN and a fully-supervised GAN where the latents are fixed to the ground truth factors of variation. Our results in Figure 5 empirically verify that combining single-factor weak supervision datasets leads to consistently high disentanglement scores.
+
+# 7 CONCLUSION
+
+In this work, we construct a theoretical framework to rigorously analyze the disentanglement guarantees of weak supervision algorithms. Our paper clarifies several important concepts, such as consistency and restrictiveness, that have been hitherto confused or overlooked in the existing literature, and provides a formalism that precisely distinguishes when disentanglement arises from supervision versus model inductive bias. Through our theory and a comprehensive set of experiments, we demonstrated the conditions under which various supervision strategies guarantee disentanglement. Our work establishes several promising directions for future research. First, we hope that our formalism and experiments inspire greater theoretical and scientific scrutiny of the inductive biases present in existing models. Second, we encourage the search for other learning algorithms (besides distribution-matching) that may have theoretical guarantees when paired with the right form of supervision. Finally, we hope that our framework enables the theoretical analysis of other promising weak supervision methods.
+
+# ACKNOWLEDGMENTS
+
+We would like to thank James Brofos and Honglin Yuan for their insightful discussions on the theoretical analysis in this paper, and Aditya Grover and Hung H. Bui for their helpful feedback during the course of this project.
+
+# REFERENCES
+
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+Diane Bouchacourt, Ryota Tomioka, and Sebastian Nowozin. Multi-level variational autoencoder: Learning disentangled representations from grouped observations. In Thirty-Second AAAI Conference on Artificial Intelligence, 2018.  
+Junxiang Chen and Kayhan Batmanghelich. Weakly supervised disentanglement by pairwise similarities. arXiv preprint arXiv:1906.01044, 2019.  
+Tian Qi Chen, Xuechen Li, Roger B Grosse, and David K Duvenaud. Isolating sources of disentanglement in variational autoencoders. Advances in Neural Information Processing Systems, pp. 2610-2620, 2018a.  
+Yutian Chen, Yannis Assael, Brendan Shillingford, David Budden, Scott Reed, Heiga Zen, Quan Wang, Luis C Cobo, Andrew Trask, Ben Laurie, et al. Sample efficient adaptive text-to-speech. arXiv preprint arXiv:1809.10460, 2018b.  
+Cian Eastwood and Christopher KI Williams. A framework for the quantitative evaluation of disentangled representations. *ICLR*, 2018.  
+Babak Esmaeili, Hao Wu, Sarthak Jain, Alican Bozkurt, Narayanaswamy Siddharth, Brooks Paige, Dana H Brooks, Jennifer Dy, and Jan-Willem van de Meent. Structured disentangled representations. arXiv preprint arXiv:1804.02086, 2018.  
+Aviv Gabbay and Yedid Hoshen. Latent optimization for non-adversarial representation disentanglement. arXiv preprint arXiv:1906.11796, 2019.  
+Leilani H Gilpin, David Bau, Ben Z Yuan, Ayesha Bajwa, Michael Specter, and Lalana Kagal. Explaining explanations: An overview of interpretability of machine learning. In 2018 IEEE 5th International Conference on data science and advanced analytics (DSAA), pp. 80-89. IEEE, 2018.  
+Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Advances in neural information processing systems, pp. 2672-2680, 2014.  
+Luigi Gresele, Paul K Rubenstein, Arash Mehrjou, Francesco Locatello, and Bernhard Scholkopf. The incomplete rosetta stone problem: Identifiability results for multi-view nonlinear ica. arXiv preprint arXiv:1905.06642, 2019.  
+Irina Higgins, Loic Matthew, Arka Pal, Christopher Burgess, Xavier Glorot, Matthew Botvinick, Shakir Mohamed, and Alexander Lerchner. beta-vae: Learning basic visual concepts with a constrained variational framework. *ICLR*, 2(5):6, 2017.  
+Irina Higgins, David Amos, David Pfau, Sebastien Racaniere, Loic Matthew, Danilo Rezende, and Alexander Lerchner. Towards a definition of disentangled representations. arXiv preprint arXiv:1812.02230, 2018.  
+Hyunjik Kim and Andriy Mnih. Disentangling by factorising. ICML, 2018.  
+Durk P Kingma, Shakir Mohamed, Danilo Jimenez Rezende, and Max Welling. Semi-supervised learning with deep generative models. Advances in neural information processing systems, pp. 3581-3589, 2014.  
+Jack Klys, Jake Snell, and Richard Zemel. Learning latent subspaces in variational autoencoders. In Advances in Neural Information Processing Systems, pp. 6444-6454, 2018.  
+Tejas D Kulkarni, William F Whitney, Pushmeet Kohli, and Josh Tenenbaum. Deep convolutional inverse graphics network. In Advances in neural information processing systems, pp. 2539-2547, 2015.
+
+Abhishek Kumar, Prasanna Sattigeri, and Avinash Balakrishnan. Variational inference of disentangled latent concepts from unlabeled observations. In ICLR, 2018.  
+Yann LeCun, Fu Jie Huang, Leon Bottou, et al. Learning methods for generic object recognition with invariance to pose and lighting. In CVPR (2), pp. 97-104. CiteSeer, 2004.  
+Hsin-Ying Lee, Hung-Yu Tseng, Qi Mao, Jia-Bin Huang, Yu-Ding Lu, Maneesh Singh, and Ming-Hsuan Yang. Drit++: Diverse image-to-image translation via disentangled representations. arXiv preprint arXiv:1905.01270, 2019.  
+Francesco Locatello, Stefan Bauer, Mario Lucic, Sylvain Gelly, Bernhard Schölkopf, and Olivier Bachem. Challenging common assumptions in the unsupervised learning of disentangled representations. ICML, 2019.  
+Brian McFee and Gert R Lanckriet. Metric learning to rank. In Proceedings of the 27th International Conference on Machine Learning (ICML-10), pp. 775-782, 2010.  
+Takeru Miyato and Masanori Koyama. cgans with projection discriminator. arXiv preprint arXiv:1802.05637, 2018.  
+Siddharth Narayanaswamy, T Brooks Paige, Jan-Willem Van de Meent, Alban Desmaison, Noah Goodman, Pushmeet Kohli, Frank Wood, and Philip Torr. Learning disentangled representations with semi-supervised deep generative models. In Advances in Neural Information Processing Systems, pp. 5925-5935, 2017.  
+Scott E Reed, Yi Zhang, Yuting Zhang, and Honglak Lee. Deep visual analogy-making. In Advances in neural information processing systems, pp. 1252-1260, 2015.  
+Karl Ridgeway and Michael C Mozer. Learning deep disentangled embeddings with the f-statistic loss. Advances in Neural Information Processing Systems, pp. 185-194, 2018.  
+Raphael Suter, Dorde Miladinovic, Stefan Bauer, and Bernhard Schölkopf. Interventional robustness of deep latent variable models. ICML, 2018.  
+Jiang Wang, Yang Song, Thomas Leung, Chuck Rosenberg, Jingbin Wang, James Philbin, Bo Chen, and Ying Wu. Learning fine-grained image similarity with deep ranking. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 1386-1393, 2014.
+
+# APPENDIX
+
+Our appendix consists of nine sections. We provide a brief summary of each section below.
+
+Appendix A: We elaborate on the connections between existing definitions of disentanglement and our definitions of consistency / restrictiveness / disentanglement. In particular, we highlight three notable properties of our definitions not present in many existing definitions.
+
+Appendix B: We evaluate our consistency and restrictiveness metrics on the 10800 models in the disentanglement_lib, and identify models where consistency and restrictiveness are not correlated.
+
+Appendix C: We adapt our definitions to be able to handle nuisance variables. We do so through a simple modification of the definition of restrictiveness.
+
+Appendix D: We show several additional single-factor experiments. We first address one of the results in the main text that is not consistent with our theory, and explain why it can be attributed to hyperparameter sensitivity. We next unwrap the heatmaps into more informative boxplots.
+
+Appendix E: We provide an additional suite of consistency versus restrictiveness experiments by comparing the effects of training with share pairing (which guarantees consistency), change pairing (which guarantees restrictiveness), and both.
+
+Appendix F: We provide full disentanglement results on all five datasets as measured according to six different metrics of disentanglement found in the literature.
+
+Appendix G: We show visualizations of a weakly supervised generative model trained to achieve full disentanglement.
+
+Appendix H: We describe the set of hyperparameter configurations used in all our experiments.
+
+Appendix I: We provide the complete set of assumptions and proofs for our theoretical framework.
+
+# A CONNECTIONS TO EXISTING DEFINITIONS
+
+Numerous definitions of disentanglement are present in the literature (Higgins et al., 2017; 2018; Kim & Mnih, 2018; Suter et al., 2018; Ridgeway & Mozer, 2018; Eastwood & Williams, 2018; Chen et al., 2018a). We mostly defer to the terminology suggested by Ridgeway & Mozer (2018), which decomposes disentanglement into modularity, compactness, and explicitness. Modularity means a latent code  $Z_{i}$  is predictive of at most one factor of variation  $S_{j}$ . Compactness means a factor of variation  $S_{i}$  is predicted by at most one latent code  $Z_{j}$ . And explicitness means a factor of variation  $S_{j}$  is predicted by the latent codes via a simple transformation (e.g. linear). Similar to Eastwood & Williams (2018); Higgins et al. (2018), we suggest a further decomposition of Ridgeway & Mozer (2018)'s explicitness into latent code informativeness and latent code simplicity. In this paper, we omit latent code simplicity from consideration. Since informativeness of the latent code is already enforced by our requirement that  $g(Z)$  is equal in distribution to  $g^{*}(S)$  (see Proposition 6), we focus on comparing our proposed concepts of consistency and restrictiveness to modularity and compactness. We make note of three important distinctions.
+
+Restrictiveness is not synonymous with either modularity or compactness. In Figure 2c, it is evident the factor of variation size is not predictable any individual  $Z_{i}$  (conversely,  $Z_{1}$  is not predictable from any individual factor  $S_{i}$ ). As such,  $Z_{1}$  is neither a modular nor compact representation of size, despite being restricted to size. To our knowledge, no existing quantitative definition of disentanglement (or its decomposition) specifically measures restrictiveness.
+
+Consistency and restrictiveness are invariant to statistically dependent factors of variation. Many existing definitions of disentanglement are instantiated by measuring the mutual information between  $Z$  and  $S$ . For example, Ridgeway & Mozer (2018) defines that a latent code  $Z_{i}$  to be "ideally modular" if it has high mutual information with a single factor  $S_{j}$  and zero mutual information with all other factors  $S_{\backslash j}$ . This presents a issue when the true factors of variation themselves are statistically dependent; even if  $Z_{1} = S_{1}$ , the latent code  $Z_{1}$  would violate modularity if  $S_{1}$  itself has positive mutual information with  $S_{2}$ . Consistency and restrictiveness circumvent this issue by relying on conditional resampling. Consistency, for example, only measures the extent to which  $S_{I}$
+
+is invariant to resampling of  $Z_{\backslash I}$  when conditioned on  $Z_{I}$  and is thus achieved as long as  $s_I$  is a function of only  $z_{I}$  —irrespective of whether  $s_I$  and  $s_{\backslash I}$  are statistically dependent. In this regard, our definitions draw inspiration from Suter et al. (2018)'s intervention-based definition but replaces the need for counterfactual reasoning with the simpler conditional sampling. Because we do not assume the factors of variation are statistically independent, our theoretical analysis is also distinct from the closely-related match pairing analysis in Gresele et al. (2019).
+
+Consistency and restrictiveness arise in weak supervision guarantees. One of our goals is to propose definitions that are amenable to theoretical analysis. As we can see in Section 4, consistency and restrictiveness serve as the core primitive concepts that we use to describe disentanglement guarantees conferred by various forms of weak supervision.
+
+# B EVALUATING CONSISTENCY AND RESTRICTIVENESS ON DISENTANGLEMENT-LIB MODELS
+
+To better understand the empirical relationship between consistency and restrictiveness, we calculated the normalized consistency and restrictiveness scores on the suite of 12800 models from disentanglement.lib for each ground-truth factor. By using the normalized consistency and restrictiveness scores as probes, we were able to identify models that achieve high consistency but low restrictiveness (and vice versa). In Fig. 6, we highlight two models that are either consistent or restrictive for object color on the Shapes3D dataset.
+
+![](images/3603c29ff5c42cc3cbd44d76f647ba191143b699f1cbfc224b9ddce8e12ed354.jpg)  
+(a) Consistent but not Restrictive  
+Figure 6: Visualization of two models from disentanglement.lib (model ids 11964 and 12307), matching the schematic in Fig. 2. For each panel, we visualize an interpolation along a single latent across rows, with each row corresponding to a fixed set of values for all other factors. In Fig. 6a, we can see that this factor consistently represents object color, i.e. each column of images has the same object color, but as we move along rows we see that other factors change as well, e.g. object type, thus this factor is not restricted to object color. In Fig. 6b, we see that varying the factor along each row results in changes to object color but to no other attributes. However if we look across columns, we see that the representation of color changes depending on the setting of other factors, thus this factor is not consistent for object color.
+
+![](images/83c2f184e618ad86e9e74c3d8cb41f91de0c6bdce37d3d4e63193155b8037419.jpg)  
+(b) Restrictive but not Consistent
+
+# C HANDLING NUISANCE VARIABLES
+
+Our theoretical framework can handle nuisance variables, i.e., variables we cannot measure or perform weak supervision on. It may be impossible to label, or provide match-pairing on that factor of variation. For example, while many features of an image are measurable (such as brightness and
+
+coloration), we may not be able to measure certain factors of variation or generate data pairs where these factors are kept constant. In this case, we can let one additional variable  $\eta$  act as nuisance variable that captures all additional sources of variation / stochasticity.
+
+Formally, suppose the full set of true factors is  $S \cup \{\eta\} \in \mathbb{R}^{n+1}$ . We define  $\eta$ -consistency  $C_{\eta}(I) = C(I)$  and  $\eta$ -restrictiveness  $R_{\eta}(I) = R(I \cup \{\eta\})$ . This captures our intuition that, with nuisance variable, for consistency, we still want changes to  $Z_{\backslash I} \cup \{\eta\}$  to not modify  $S_I$ ; for restrictiveness, we want changes to  $Z_I \cup \{\eta\}$  to only modify  $S_I \cup \{\eta\}$ . We define  $\eta$ -disentanglement as  $D_{\eta}(I) = C_{\eta}(I) \wedge R_{\eta}(I)$ .
+
+All of our calculus still holds where we substitute  $C_{\eta}(I), R_{\eta}(I), D_{\eta}(I)$  for  $C(I), R(I), D(I)$ ; we prove one of the new full disentanglement rule as an illustration:
+
+Proposition 1.  $\bigwedge_{i = 1}^{n}C_{\eta}(i)\Longleftrightarrow \bigwedge_{i = 1}^{n}D_{\eta}(i).$
+
+Proof. On the one hand,  $\bigwedge_{i=1}^{n} C_{\eta}(i) \Longleftrightarrow \bigwedge_{i=1}^{n} C(i) \Longrightarrow C(1:n) \Longrightarrow R(\eta)$ . On the other hand,  $\bigwedge_{i=1}^{n} C(i) \Longrightarrow \bigwedge_{i=1}^{n} D(i) \Longrightarrow \bigwedge_{i=1}^{n} R(i)$ . Therefore  $LHS \Longrightarrow \forall i \in [n], R(i) \land R(\eta) \Longrightarrow R_{\eta}(i)$ . The reverse direction is trivial.
+
+In (Locatello et al., 2019), the "instance" factor in SmallNORB and the background image factor in Scream-dSprites are treated as nuisance variables. By Proposition 1, as long as we perform weak supervision on all of the non-nuisance variables (via sharing-pairing, say) to guarantee their consistency with respect to the corresponding true factor of variation, we still have guaranteed full disentanglement despite the existence of nuisance variable and the fact that we cannot measure or perform weak supervision on nuisance variable.
+
+# D SINGLE-FACTOR EXPERIMENTS
+
+![](images/153ad311eeae53a4be4122f772b90ec7bb076a12a6266a3eaafe66f633d97a0d.jpg)  
+Figure 7: This is the same plot as Figure 7, but where we restrict our hyperparameter sweep to always set extra dense  $=$  False. See Appendix H for details about hyperparameter sweep.
+
+![](images/23507aeeada2ecd862c6d525096c9d41ff1c20e20ac447151ec25044589d0e8e.jpg)
+
+![](images/277235d26ecca270d0876225baa1e6563cefb1906676eb1cb81bebee4da520a0.jpg)
+
+![](images/c61d4b68475c7d2b66d7693271ac4bcfe936da5e522c51bed220f1cebaa11e6a.jpg)
+
+![](images/208f7d62b9b748bab9c327b0169b17943d1c8dd680913bec0353a554c4b3ccf2.jpg)
+
+![](images/1ee2a9b1c3d2e204a9d958e316a842ab810c7b353c1e9336119484ee714f14e8.jpg)  
+Figure 8: Restricted pairing guarantees consistency. Each plot shows the normalized consistency score of each model for each factor of variation. Our theory predicts each boxplot highlighted in red to achieve the highest consistency. Due to the prevalence of restricted pairing in the existing literature, we chose to only conduct the single-factor restricted labeling experiment on Shapes3D.
+
+![](images/60370f08a87a94f080bb14621d792c020a9e42de171e51a216904040496546c2.jpg)  
+Figure 9: Change pairing guarantees restrictiveness. Each plot shows normalized restrictiveness score of each model for each factor of variation (row) across different datasets (columns). Different colors indicate models trained with change pairing on different factors. The appropriately-supervised model for each factor is marked in red.
+
+![](images/c607c197179ff1c9d02303e2d6b323488f0d1d4f68ce04cc26921721da496e13.jpg)  
+Figure 10: Share pairing guarantees consistency. Each plot shows normalized consistency score of each model for each factor of variation (row) across different datasets (columns). Different colors indicate models trained with share pairing on different factors. The appropriately-supervised model for each factor is marked in red.
+
+![](images/36860076b97d8d916b76fe8a97724bc7011796cb052fd566586881f1dbb9844b.jpg)  
+Figure 11: Rank pairing guarantees consistency. Each plot shows normalized consistency score of each model for each factor of variation (row) across different datasets (columns). Different colors indicate models trained with rank pairing on different factors. The appropriately-supervised model for each factor is marked in red.
+
+# E CONSISTENCY VERSUS RESTRICTIVENESS
+
+![](images/b77f87bb9b54eac2e047936096298d6ab80d070fbc8c5455a244941931b0c794.jpg)  
+Figure 12: Normalized consistency vs. restrictiveness score of different models on each factor (row) across different datasets (columns). In many of the plots, we see that models trained via change-sharing (blue) achieve higher restrictiveness; models trained via share-sharing (orange) achieve higher consistency; models trained via both techniques (green) simultaneously achieve restrictiveness and consistency in most cases.
+
+# F FULL DISENTANGLEMENT EXPERIMENTS
+
+![](images/bf2d39fbdcd0766b3b455664a4d852b70ba186d23564048e6c83367446829b6f.jpg)  
+Figure 13: Disentanglement performance of a vanilla GAN, share pairing GAN, change pairing GAN, rank pairing GAN, and fully-labeled GAN, as measured by multiple disentanglement metrics in existing literature (rows) across multiple datasets (columns). According to almost all metrics, our weakly supervised models surpass the baseline, and in some cases, even outperform the fully-labeled model.
+
+![](images/bdf48c70582925f34f8149c3ba53d59e3a635bffaaea107ce42897f555bd866b.jpg)  
+Figure 14: Performance of a vanilla GAN (blue), share pairing GAN (orange), change pairing GAN (green), rank pairing GAN (red), and fully-labeled GAN (purple), as measured by normalized consistency score of each factor (rows) across multiple datasets (columns). Factors  $\{3,4,5\}$  in the first column shows that distribution matching to all six change / share pairing datasets is particularly challenging for the models when trained on certain hyperparameter choices. However, since consistency and restrictiveness can be measured in weakly supervised settings, it suffices to use these metrics for hyperparameter selection. We see in Figure 16 and Appendix G that using consistency and restrictiveness for hyperparameter selection serves as a viable weakly-supervised surrogate for existing fully-supervised disentanglement metrics.
+
+![](images/e760f9d3f75c109b60b759fd3173233b48a3d23f84aed65e4a950ae5f5cb605b.jpg)
+
+![](images/0a29ea3c9c0916beeccaebbf44a760458906d7933bd01ec2c472974639215148.jpg)
+
+![](images/652de5a2630fdc753cd0d638e305c8805fdbfe7cc44617e987062ea2a137d86f.jpg)
+
+![](images/b1bf48dad0b75be7defa68d1bc4ad7e611c28336bf77ce7a68a1fce13668a370.jpg)
+
+![](images/226a2ba3649a661bb94a0ae4ada88a5e68e2b542eb49454c9d96e2e2715a41de.jpg)
+
+![](images/87ff0857a84355887e1397e4e4fa409268b433693e909e0cbc431c7ae5366b30.jpg)
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+![](images/a8047de9a9cb38752e99fd39cb93b738e17ec3a99d39b82575318baf5c79341f.jpg)
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+![](images/9c56a066f620dd8f910b1dfe466b329ee4a09a1877122f8dff614c3c24dcfe23.jpg)
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+![](images/8de2e329979371284e6fbd3e65a913136af95c341b87fe61e6d969cb23842089.jpg)
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+![](images/681bc9efdf6480d6be913db72ffbf6cddceb43a0b6eac543f868d3f5204bc70b.jpg)
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+![](images/59e3b54f070edb637801efd03c2ed3e51f6510cc7a6119d738c0de5b0bef0a10.jpg)
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+![](images/ead8cbbcdc164481b45841cb6add90f63e717ad68a86eaedad43173f82eecc01.jpg)
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+![](images/97a00e4634c99e5d5247b31f53b2c972dae9120169445655ebeb3840fa092cbb.jpg)
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+![](images/42b6dfc4c1090803848801cc0f2ded782d0ad1ba064387e7c06baf42f04496b7.jpg)
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+![](images/abdb96f608da9b1bcc78f752da46d8993db45c434f962f0169ed49d331d85816.jpg)
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+![](images/0a251b0ed277c0f231cb6acddf1e058555aae1c2878030a7d631df0cb0ad1366.jpg)
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+![](images/5bfa64056f5d407003109324b58b1814f2f8fdfed8adddb903f04807194ea150.jpg)
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+![](images/f324498d842f76db03acc95637a3c61dc1585a369e20aa10f47ac927d3272e03.jpg)
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+![](images/8141826dbaeb7f0c6873c71f46e1468c3adf8862f397f9f93782d911fca37b33.jpg)
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+![](images/0f427013e988138c6d8c934f80627d3c084d68b942f0ffb4d097a7ff2f040f67.jpg)
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+![](images/f374311f597a58e53e4d5180438e29c4bacf8b22aea99ceb442ef064c80dc05b.jpg)
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+![](images/0ed126f15db01324dc4af9fcd9c693c2148b6f7b28253f41e2e291d2298e6f19.jpg)
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+![](images/6726cd1e08ccdc971610b1a89d6c1ab67e02e912cb21bbd8eb33d4b4f7eac978.jpg)  
+Figure 15: Performance of a vanilla GAN (blue), share pairing GAN (orange), change pairing GAN (green), rank pairing GAN (red), and fully-labeled GAN (purple), as measured by normalized restrictiveness score of each factor (rows) across multiple datasets (columns). Since restrictiveness and consistency are complementary, we see that the anomalies in Figure 14 are reflected in the complementary factors in this figure.
+
+![](images/09ac7b6cae975545fa8a92bc3733305b0bf1ee6c1e54e01b93213a1e239edfa8.jpg)
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+![](images/b2a4f32639035c7efa438f72607cbd05250b04e6e839cba94a45324e7b11a5aa.jpg)
+
+![](images/b2b662a3bd344474e60b33c3452b11bc7f9b1abbcf459f93b6555a6df6f07bb0.jpg)  
+(a) Shapes3D  
+(c) Scream-dSprites  
+Figure 16: Scatterplot of existing disentanglement metrics versus average normalized consistency and restrictiveness. Whereas existing disentanglement metrics are fully-supervised, it is possible to measure average normalized consistency and restrictiveness with weakly supervised data (share-pairing and match-pairing respectively), making it viable to perform hyperparameter tuning under weakly supervised conditions.
+
+![](images/e25a7ca3288599bd087909ddc563f4d4163a35ede8401821112f2244d39cb60d.jpg)  
+(b) dSprites  
+(d) SmallNORB
+
+![](images/fce60cf848665118cf708a36487f10e27a231ceff77cd0f9aa2aa0a4500da5e8.jpg)  
+(e) Cars3D
+
+# G FULL DISENTANGLEMENT VISUALIZATIONS
+
+As a demonstration of the weakly-supervised generative models, we visualize our best-performing match-pairing generative models (as selected according to the normalized consistency score averaged across all the factors). Recall from Figures 2a to 2c that, to visually check for consistency and restrictiveness, it is important that we not only ablate a single factor (across the column), but also show that the factor stays consistent (down the row). Each block of  $3 \times 12$  images in Figures 17 to 21 checks for disentanglement of the corresponding factor. Each row is constructed by random sampling of  $Z_{\backslash i}$  and then ablating  $Z_{i}$ .
+
+![](images/03ee9e31be8893bab96c9c46983871f379bd234ceebe4c798737b7df45551b8f.jpg)  
+Figure 17: Cars3D. Ground truth factors: elevation, azimuth, object type.
+
+![](images/07dd5ac8ce0fab0b612b927741bcc467b110c24a400de8a9f8f252a5269df86d.jpg)  
+Figure 18: Shapes3D. Ground truth factors: floor color, wall color, object color, object size, object type, and azimuth.
+
+![](images/d1a90e263373f359ef0d562aa79ba20d7370ba41023f6b117d49d42287009181.jpg)  
+Figure 19: dSprites. Ground truth factors: shape, scale, orientation, X-position, Y-position.
+
+![](images/801cc4a99a11494324f7f6b436ea6096e5910081586f670f4520fd6d35ca223f.jpg)  
+Figure 20: Scream-dSprites. Ground truth factors: shape, scale, orientation, X-position, Y-position.
+
+![](images/27ad8e505bf9bf2acb3020b3e38eedf860ebdc85e39df0b5d2e632302d25ec5f.jpg)  
+Figure 21: SmallNORB. Ground truth factors: category, elevation, azimuth, lighting condition.
+
+# H HYPERPARAMETERS
+
+Table 1: We trained a probabilistic Gaussian encoder to approximately invert the generative model. The encoder is not trained jointly with the generator, but instead trained separately from the generative model (i.e. encoder gradient does not backpropagate to generative model). During training, the encoder is only exposed to data generated by the learned generative model.
+
+<table><tr><td>Encoder</td></tr><tr><td>4 × 4 spectral norm conv. 32. lReLU</td></tr><tr><td>4 × 4 spectral norm conv. 32. lReLU</td></tr><tr><td>4 × 4 spectral norm conv. 64. lReLU</td></tr><tr><td>4 × 4 spectral norm conv. 64. lReLU</td></tr><tr><td>flatten</td></tr><tr><td>128 spectral norm dense. lReLU</td></tr><tr><td>2 × z-dim spectral norm dense</td></tr></table>
+
+Table 2: Generative model architecture.  
+
+<table><tr><td>Generator</td></tr><tr><td>128 dense. ReLU. batchnorm.</td></tr><tr><td>1024 dense. ReLU. batchnorm.</td></tr><tr><td>4 × 4 × 64 reshape.</td></tr><tr><td>4 × 4 conv. 64. lReLU. batchnorm.</td></tr><tr><td>4 × 4 conv. 32. lReLU. batchnorm.</td></tr><tr><td>4 × 4 conv. 32. lReLU. batchnorm.</td></tr><tr><td>4 × 4 conv. 3. sigmoid</td></tr></table>
+
+Table 3: Discriminator used for restricted labeling. Parts in red are part of hyperparameter search.  
+
+<table><tr><td>Discriminator Body</td></tr><tr><td>4 × 4 spectral norm conv. 32 × width. lReLU</td></tr><tr><td>4 × 4 spectral norm conv. 32 × width. lReLU</td></tr><tr><td>4 × 4 spectral norm conv. 64 × width. lReLU</td></tr><tr><td>4 × 4 spectral norm conv. 64 × width. lReLU</td></tr><tr><td>flatten</td></tr><tr><td>if extra dense: 128 × width spectral norm dense. lReLU</td></tr><tr><td>Discriminator Auxiliary Channel for Label</td></tr><tr><td>128 × width spectral norm dense. lReLU</td></tr><tr><td>If extra dense: 128 × width spectral norm dense. lReLU</td></tr><tr><td>Discriminator head</td></tr><tr><td>concatenate body and auxiliary.</td></tr><tr><td>128 × width spectral norm dense. lReLU</td></tr><tr><td>128 × width spectral norm dense. lReLU</td></tr><tr><td>1 spectral norm dense with bias.</td></tr></table>
+
+Table 4: Discriminator used for match pairing. We use a projection discriminator (Miyato & Koyama, 2018) and thus have an unconditional and conditional head. Parts in red are part of hyperparameter search.  
+
+<table><tr><td>Discriminator Body Applied Separately to x and x&#x27;</td></tr><tr><td>4 × 4 spectral norm conv. 32 × width. lReLU</td></tr><tr><td>4 × 4 spectral norm conv. 32 × width. lReLU</td></tr><tr><td>4 × 4 spectral norm conv. 64 × width. lReLU</td></tr><tr><td>4 × 4 spectral norm conv. 64 × width. lReLU</td></tr><tr><td>flatten</td></tr><tr><td>If extra dense: 128 × width spectral norm dense. lReLU</td></tr><tr><td>concatenate the pair.</td></tr><tr><td>128 × width spectral norm dense. lReLU</td></tr><tr><td>128 × width spectral norm dense. lReLU</td></tr><tr><td>Unconditional Head</td></tr><tr><td>1 spectral norm dense with bias</td></tr><tr><td>Conditional Head</td></tr><tr><td>128 × width spectral norm dense</td></tr></table>
+
+Table 5: Discriminator used for rank pairing. For rank-pairing, we use a special variant of the projection discriminator, where the conditional logit is computed via taking the difference between the two pairs and multiplying by  $y \in \{-1, +1\}$ . The discriminator is thus implicitly taking on the role of an adversarially trained encoder that checks for violations of the ranking rule in the embedding space. Parts in red are part of hyperparameter search.  
+
+<table><tr><td>Discriminator Body Applied Separately to x and x&#x27;</td></tr><tr><td>4 × 4 spectral norm conv. 32 × width. lReLU</td></tr><tr><td>4 × 4 spectral norm conv. 32 × width. lReLU</td></tr><tr><td>4 × 4 spectral norm conv. 64 × width. lReLU</td></tr><tr><td>4 × 4 spectral norm conv. 64 × width. lReLU</td></tr><tr><td>flatten</td></tr><tr><td>If extra dense: 128 × width spectral norm dense. lReLU concatenate the pair.</td></tr><tr><td>Unconditional Head Applied Separately to x and x&#x27;</td></tr><tr><td>1 spectral norm dense with bias.</td></tr><tr><td>Conditional Head Applied Separately to x and x&#x27;</td></tr><tr><td>y-dim spectral norm dense.</td></tr></table>
+
+For all models, we use the Adam optimizer with  $\beta_{1} = 0.5$ ,  $\beta_{2} = 0.999$  and set the generator learning rate to  $1\times 10^{-3}$ . We use a batch size of 64 and set the leaky ReLU negative slope to 0.2.
+
+To demonstrate some degree of robustness to hyperparameter choices, we considered five different ablations:
+
+1. Width multiplier on the discriminator network  $(\{1,2\})$  
+2. Whether to add an extra fully-connected layer to the discriminator ( $\{\mathrm{True}, \mathrm{False}\}$ ).  
+3. Whether to add a bias term to the head ( $\{\text{True}, \text{False}\}$ ).  
+4. Whether to use two-time scale learning rate by setting encoder+discriminator learning rate multiplier to  $(\{1,2\})$  
+5. Whether to use the default PyTorch or Keras initialization scheme in all models.
+
+As such, each of our experimental setting trains a total of 32 distinct models. The only exception is the intersection experiments where we fixed the width multiplier to 1.
+
+To give a sense of the scale of our experimental setup, note that the 864 models in Figure 4 originate as follows:
+
+1. 32 hyperparameter conditions  $\times 6$  restricted labeling conditions.  
+2. 32 hyperparameter conditions  $\times 6$  match pairing conditions.  
+3. 32 hyperparameter conditions  $\times 6$  share pairing conditions.  
+4. 32 hyperparameter conditions  $\times 6$  rank pairing conditions.  
+5. 16 hyperparameter conditions  $\times 6$  intersection conditions.
+
+# I PROOFS
+
+# I.1 ASSUMPTIONS ON  $\mathcal{H}$
+
+Assumption 1. Let  $D \subseteq [n]$  indexes discrete random variables  $S_{D}$ . Assume that the remaining random variables  $S_{C} = S_{\backslash D}$  have probability density function  $p(s_{C}|s_{D})$  for any set of values  $s_{D}$  where  $p(S_{D} = s_{D}) > 0$ .
+
+Assumption 2. Without loss of generality, suppose  $S_{1:n} = [S_C, S_D]$  is ordered by concatenating the continuous variables with the discrete variables. Let  $\mathcal{B}(s_D) = [\mathrm{int}(\mathrm{supp}(p(s_C \mid s_D))), s_D]$  denote the interior of the support of the continuous conditional distribution of  $S_C$  concatenated with its conditioning variable  $s_D$  drawn from  $S_D$ . With a slight abuse of notation, let  $\mathcal{B}(S) = \bigcup_{s_D: p(s_D > 0)} \mathcal{B}(s_D)$ . We assume  $\mathcal{B}(S)$  is zig-zag connected, i.e., for any  $I, J \subseteq [n]$ , for any two points  $s_{1:n}, s_{1:n}' \in \mathcal{B}(S)$  that only differ in coordinates in  $I \cup J$ , there exists a path  $\{s_{1:n}^t\}_{t=0:T}$  contained in  $\mathcal{B}(S)$  such that
+
+$$
+s _ {1: n} ^ {0} = s _ {1: n} \tag {13}
+$$
+
+$$
+s _ {1: n} ^ {T} = s _ {1: n} ^ {\prime} \tag {14}
+$$
+
+$$
+\forall 0 \leq t <   T, \text {e i t h e r} s _ {\backslash I} ^ {t} = s _ {\backslash I} ^ {t + 1} \text {o r} s _ {\backslash J} ^ {t} = s _ {\backslash J} ^ {t + 1}, \tag {15}
+$$
+
+Intuitively, this assumption allows transition from  $s_{1:n}$  to  $s_{1:n}'$  via a series of modifications that are only in  $I$  or only in  $J$ . Note that zig-zag connectedness is necessary for restrictiveness union (Proposition 3) and consistency intersection (Proposition 4). Fig. 22 gives examples where restrictiveness union is not satisfied when zig-zag connectedness is violated.
+
+Assumption 3. For arbitrary coordinate  $j \in [m]$  of  $g$  that maps to a continuous variable  $X_{j}$ , we assume that  $g_{j}(s)$  is continuous at  $s, \forall s \in \mathcal{B}(S)$ ; For arbitrary coordinate  $j \in [m]$  of  $g$  that maps to a discrete variable  $X_{j}, \forall s_{D}$  where  $p(s_{D}) > 0$ , we assume that  $g_{j}(s)$  is constant over each connected component of  $\operatorname{int}(\operatorname{supp}(p(s_{C} \mid s_{D}))$ .
+
+Define  $\mathcal{B}(X)$  analogously to  $\mathcal{B}(S)$ . Symmetrically, for arbitrary coordinate  $i\in [n]$  of  $e$  that maps to a continuous variable  $S_{i}$ , we assume that  $e_i(x)$  is continuous at  $x$ ,  $\forall x\in \mathcal{B}(X)$ ; For arbitrary coordinate  $i\in [n]$  of  $e$  that maps to a discrete  $S_{i}$ ,  $\forall x_{D}$  where  $p(x_D) > 0$ , we assume that  $e_i(x)$  is constant over each connected component of  $\operatorname {int}(\operatorname {supp}(p(x_C\mid x_D)))$
+
+Assumption 4. Assume that every factor of variation is recoverable from the observation  $\mathcal{X}$ . Formally,  $(p,g,e)$  satisfies the following property
+
+$$
+\mathbb {E} _ {p \left(s _ {1: n}\right)} \| e \circ g \left(s _ {1: n}\right) - s _ {1: n} \| ^ {2} = 0. \tag {16}
+$$
+
+# I.2 CALCULUS OF DISENTANGLEMENT
+
+# I.2.1 EXPECTED-NORM REDUCTION LEMMA
+
+Lemma 1. Let  $x, y$  be two random variables with distribution  $p$ ,  $f(x, y)$  be arbitrary function. Then
+
+$$
+\mathbb {E} _ {x \sim p (x)} \mathbb {E} _ {y, y ^ {\prime} \sim p (y | x)} \| f (x, y) - f (x, y ^ {\prime}) \| ^ {2} \leq \mathbb {E} _ {(x, y), (x ^ {\prime}, y ^ {\prime}) \sim p (x, y)} \| f (x, y) - f (x ^ {\prime}, y ^ {\prime}) \| ^ {2}.
+$$
+
+Proof. Assume w.l.o.g that  $\mathbb{E}_{(x,y)\sim p(x,y)}f(x,y) = 0$
+
+$$
+\begin{array}{l} L H S = 2 \mathbb {E} _ {(x, y) \sim p (x, y)} \| f (x, y) \| ^ {2} - 2 \mathbb {E} _ {x \sim p (x)} \mathbb {E} _ {y, y ^ {\prime} \sim p (y | x)} f (x, y) ^ {T} f (x, y ^ {\prime}) (17) \\ = 2 \mathbb {E} _ {(x, y) \sim p (x, y)} \| f (x, y) \| ^ {2} - 2 \mathbb {E} _ {x \sim p (x)} \mathbb {E} _ {y \sim p (y | x)} f (x, y) ^ {T} \mathbb {E} _ {y ^ {\prime} \sim p (y | x)} f (x, y ^ {\prime}) (18) \\ = 2 \mathbb {E} _ {(x, y) \sim p (x, y)} \| f (x, y) \| ^ {2} - 2 \mathbb {E} _ {x \sim p (x)} \| \mathbb {E} _ {y \sim p (y | x)} f (x, y) \| ^ {2} (19) \\ \leq 2 \mathbb {E} _ {(x, y) \sim p (x, y)} \| f (x, y) \| ^ {2} (20) \\ = 2 \mathbb {E} _ {(x, y) \sim p (x, y)} \| f (x, y) \| ^ {2} - 2 \| \mathbb {E} _ {(x, y) \sim p (x, y)} f (x, y) \| ^ {2} (21) \\ = 2 \mathbb {E} _ {(x, y) \sim p (x, y)} \| f (x, y) \| ^ {2} - 2 \mathbb {E} _ {(x, y), \left(x ^ {\prime}, y ^ {\prime}\right) \sim p (x, y)} f (x, y) ^ {T} f \left(x ^ {\prime}, y ^ {\prime}\right) (22) \\ = R H S. (23) \\ \end{array}
+$$
+
+□
+
+# I.2.2 CONSISTENCY UNION
+
+Let  $L = I \cap J, K = \backslash (I \cup J)$ ,  $M = I - L, N = J - L$ .
+
+Proposition 2.  $C(I)\wedge C(J)\Rightarrow C(I\cup J).$
+
+Proof.
+
+$$
+C (I) \Rightarrow \mathbb {E} _ {z _ {M}, z _ {L}} \mathbb {E} _ {z _ {N}, z _ {N} ^ {\prime}, z _ {K}, z _ {K} ^ {\prime}} \| r _ {I} \circ G \left(z _ {M}, z _ {L}, z _ {N}, z _ {K}\right) - r _ {I} \circ G \left(z _ {M}, z _ {L}, z _ {N} ^ {\prime}, z _ {K} ^ {\prime}\right) \| ^ {2} = 0. \tag {24}
+$$
+
+For any fixed value of  $z_{M},z_{L}$
+
+$$
+\mathbb {E} _ {z _ {N}, z _ {N} ^ {\prime}, z _ {K}, z _ {K} ^ {\prime}} \left\| r _ {I} \circ G \left(z _ {M}, z _ {L}, z _ {N}, z _ {K}\right) - r _ {I} \circ G \left(z _ {M}, z _ {L}, z _ {N} ^ {\prime}, z _ {K} ^ {\prime}\right) \right\| ^ {2} \tag {25}
+$$
+
+$$
+\geq \mathbb {E} _ {z _ {N}} \mathbb {E} _ {z _ {K}, z _ {K} ^ {\prime}} \| r _ {I} \circ G (z _ {M}, z _ {L}, z _ {N}, z _ {K}) - r _ {I} \circ G (z _ {M}, z _ {L}, z _ {N}, z _ {K} ^ {\prime}) \| ^ {2}. \tag {26}
+$$
+
+by plugging in  $x = z_{N},y = z_{K}$  into Lemma 1. Therefore
+
+$$
+C (I) \Longrightarrow \mathbb {E} _ {z _ {M}, z _ {L}, z _ {N}} \mathbb {E} _ {z _ {K}, z _ {K} ^ {\prime}} \| r _ {I} \circ G (z _ {M}, z _ {L}, z _ {N}, z _ {K}) - r _ {I} \circ G (z _ {M}, z _ {L}, z _ {N}, z _ {K} ^ {\prime}) \| ^ {2} = 0. \tag {27}
+$$
+
+Similarly we have
+
+$$
+\begin{array}{l} C (J) \Longrightarrow \mathbb {E} _ {z _ {M}, z _ {L}, z _ {N}} \mathbb {E} _ {z _ {K}, z _ {K} ^ {\prime}} \| r _ {J} \circ G \left(z _ {M}, z _ {L}, z _ {N}, z _ {K}\right) - r _ {J} \circ G \left(z _ {M}, z _ {L}, z _ {N}, z _ {K} ^ {\prime}\right) \| ^ {2} = 0 (28) \\ \Rightarrow \mathbb {E} _ {z _ {M}, z _ {L}, z _ {N}} \mathbb {E} _ {z _ {K}, z _ {K} ^ {\prime}} \| r _ {N} \circ G (z _ {M}, z _ {L}, z _ {N}, z _ {K}) - r _ {N} \circ G (z _ {M}, z _ {L}, z _ {N}, z _ {K} ^ {\prime}) \| ^ {2} = 0. (29) \\ \end{array}
+$$
+
+As  $I \cap N = \emptyset$ ,  $I \cup N = I \cup J$ , adding the above two equations gives us  $C(I \cup J)$ .
+
+![](images/301914e4e3eaec57c8b0a0549123f46fa410999a24ad9502e75f9e0a45946a8d.jpg)
+
+# I.2.3 RESTRICTIVENESS UNION
+
+![](images/a6a8dc32481163ba16e55e037c281c81572e9ec3317d86586799c956208e01fe.jpg)  
+Figure 22: Zig-zag connectedness is necessary for restriveness union. Here  $n = m = 3$ . Colored areas indicate the support of  $p(z_1, z_2)$ ; the marked numbers indicate the measurement of  $s_3$  given  $(z_1, z_2)$ . Left two panels satisfy zig-zag connectedness (the paths are marked in gray) while the right two do not (indeed  $R(1) \wedge R(2) \neq R(\{1, 2\})$ ). In the right-most panel, any zig-zag path connecting two points from blue and orange areas has to pass through boundary of the support (disallowed).
+
+![](images/6cc9afb7a6287ab475d5354efa35db64e813f55ed124c8b3af9052beb14f0bda.jpg)
+
+![](images/443d187b07d9e8dcfba22bc89e9c1aba17d664e88430f322db4fd6ed45f5deb0.jpg)
+
+![](images/46f0a206eff8c94497ebc6dd2f9564dfb013fb5756b159cd93906d405893cfe8.jpg)
+
+Similarly define index sets  $L, K, M, N$ .
+
+Proposition 3. Under assumptions specified in Appendix I.1,  $R(I) \wedge R(J) \Rightarrow R(I \cup J)$ .
+
+Proof. Denote  $f = e_{K}^{*}\circ g$ . We claim that
+
+$$
+\begin{array}{l} R (I) \Longleftrightarrow \mathbb {E} _ {z _ {\backslash I}} \mathbb {E} _ {z _ {I}, z _ {I} ^ {\prime}} \| f \left(z _ {I}, z _ {\backslash I}\right) - f \left(z _ {I} ^ {\prime}, z _ {\backslash I}\right) \| ^ {2} = 0. (30) \\ \Longleftrightarrow \forall \left(z _ {I}, z _ {\backslash I}\right), \left(z _ {I} ^ {\prime}, z _ {\backslash I}\right) \in \mathcal {B} (Z), f \left(z _ {I}, z _ {\backslash I}\right) = f \left(z _ {I} ^ {\prime}, z _ {\backslash I}\right). (31) \\ \end{array}
+$$
+
+We first prove the backward direction: When we draw  $z_{\backslash I} \sim p(z_{\backslash I}), z_I, z_I' \sim p(z_I | z_{\backslash I})$ , let  $E_1$  denote the event that  $(z_I, z_{\backslash I}) \notin \mathcal{B}(Z)$ , and  $E_2$  denote the event that  $(z_I', z_{\backslash I}) \notin \mathcal{B}(Z)$ . Reorder the indices of  $(z_I, z_{\backslash I})$  as  $(z_C, z_D)$ . The probability that  $(z_I, z_{\backslash I}) \notin \mathcal{B}(Z)$  (i.e.,  $z_C$  is on the boundary of  $\mathcal{B}(z_D)$ ) is 0. Therefore  $\operatorname*{Pr}[E_1] = \operatorname*{Pr}[E_2] = 0$ . Therefore  $\operatorname*{Pr}[E_1 \cup E_2] \leq \operatorname*{Pr}[E_1] + \operatorname*{Pr}[E_2] = 0$ , i.e., with probability 1,  $\| f(z_I, z_{\backslash I}) - f(z_I', z_{\backslash I}) \|^2 = 0$ .
+
+Now we prove the forward direction: Assume for the sake of contradiction that  $\exists (z_I,z_{\backslash I}),(z_I',z_{\backslash I})\in \mathcal{B}(Z)$  such that  $f(z_I,z_{\backslash I}) < f(z_I',z_{\backslash I})$ . Denote  $U = I\cap D$ ,  $V = I\cap C$ ,  $W = \backslash I\cap D$ ,  $Q = \backslash I\cap C$ . We have  $f(z_U,z_V,z_W,z_Q) < f(z_U',z_V',z_W,z_Q)$ . Since  $f$  is continuous (or constant) at  $(z_U,z_V,z_W,z_Q)$  in the interior of  $\mathcal{B}([z_U,z_W])$ , and  $f$  is also continuous (or constant) at  $(z_U',z_V',z_W,z_Q)$  in the interior of  $\mathcal{B}([z_U',z_W])$ , we can draw open balls of radius  $r > 0$  around each point, i.e.,  $B_r(z_V,z_Q)\subset \mathcal{B}([z_U,z_W])$  and  $B_r(z_V',z_Q)\subset \mathcal{B}([z_U',z_W])$ , where
+
+$$
+\forall \left(z _ {V} ^ {*}, z _ {Q} ^ {*}\right) \in B _ {r} \left(z _ {V}, z _ {Q}\right), \forall \left(z _ {V} ^ {\Delta}, z _ {Q} ^ {\Delta}\right) \in B _ {r} \left(z _ {V} ^ {\prime}, z _ {Q}\right), f \left(z _ {U}, z _ {V} ^ {*}, z _ {W}, z _ {Q} ^ {*}\right) <   f \left(z _ {U} ^ {\prime}, z _ {V} ^ {\Delta}, z _ {W}, z _ {Q} ^ {\Delta}\right). \tag {32}
+$$
+
+When we draw  $z_{\backslash I} \sim p(z_{\backslash I}), z_I, z_I' \sim p(z_I | z_{\backslash I})$ , let  $C$  denote the event that  $(z_I, z_{\backslash I}) = (z_V^*, z_U, z_Q^\#, z_W)$ ,  $(z_I', z_{\backslash I}) = (z_V^\Delta, z_U', z_Q^\#, z_W)$  where  $(z_V^*, z_Q^\#) \in B_r(z_V, z_Q)$  and  $(z_V^\Delta, z_Q^\#) \in B_r(z_V', z_Q)$ . Since both balls have positive volume,  $\operatorname{Pr}[C] > 0$ . However,  $\| f(z_I, z_{\backslash I}) - f(z_I', z_{\backslash I}) \|^2 > 0$  whenever event  $C$  happens, which contradicts  $R(I)$ . Therefore  $\forall (z_I, z_{\backslash I}), (z_I', z_{\backslash I}) \in \mathcal{B}(Z), f(z_I, z_{\backslash I}) = f(z_I', z_{\backslash I})$ .
+
+We have shown that
+
+$$
+R (I) \Longleftrightarrow \forall \left(z _ {M}, z _ {L}, z _ {N}, z _ {K}\right), \left(z _ {M} ^ {\prime}, z _ {L} ^ {\prime}, z _ {N}, z _ {K}\right) \in \mathcal {B} (Z), f \left(z _ {M}, z _ {L}, z _ {N}, z _ {K}\right) = f \left(z _ {M} ^ {\prime}, z _ {L} ^ {\prime}, z _ {N}, z _ {K}\right). \tag {33}
+$$
+
+Similarly
+
+$$
+R (J) \Longleftrightarrow \forall \left(z _ {M}, z _ {L}, z _ {N}, z _ {K}\right), \left(z _ {M}, z _ {L} ^ {\prime}, z _ {N} ^ {\prime}, z _ {K}\right) \in \mathcal {B} (Z), f \left(z _ {M}, z _ {L}, z _ {N}, z _ {K}\right) = f \left(z _ {M}, z _ {L} ^ {\prime}, z _ {N} ^ {\prime}, z _ {K}\right). \tag {34}
+$$
+
+$$
+R (I \cup J) \Longleftrightarrow \forall \left(z _ {M}, z _ {L}, z _ {N}, z _ {K}\right), \left(z _ {M} ^ {\prime}, z _ {L} ^ {\prime}, z _ {N} ^ {\prime}, z _ {K}\right) \in \mathcal {B} (Z), f \left(z _ {M}, z _ {L}, z _ {N}, z _ {K}\right) = f \left(z _ {M} ^ {\prime}, z _ {L} ^ {\prime}, z _ {N} ^ {\prime}, z _ {K}\right) \tag {35}
+$$
+
+Let the zig-zag path between  $(z_{M}, z_{L}, z_{N}, z_{K})$  and  $(z_{M}', z_{L}', z_{N}', z_{K}') \in \mathcal{B}(Z)$  be  $\{(z_{M}^{t}, z_{L}^{t}, z_{N}^{t}, z_{K})\}_{t=0}^{T}$ . Repeatedly applying the equivalent conditions of  $R(I)$  and  $R(J)$  gives us
+
+$$
+f \left(z _ {M}, z _ {L}, z _ {N}, z _ {K}\right) = f \left(z _ {M} ^ {1}, z _ {L} ^ {1}, z _ {N} ^ {1}, z _ {K}\right) = \dots = f \left(z _ {M} ^ {T - 1}, z _ {L} ^ {T - 1}, z _ {N} ^ {T - 1}, z _ {K}\right) = f \left(z _ {M} ^ {\prime}, z _ {L} ^ {\prime}, z _ {N} ^ {\prime}, z _ {K}\right). \tag {36}
+$$
+
+![](images/5cca0510e8c9ce1f5369438b0c6e113b94e5234947b4694083384afb996d1592.jpg)
+
+# I.3 CONSISTENCY AND RESTIVENESS INTERSECTION
+
+Proposition 4. Under the same assumptions as restrictiveness union,  $C(I) \wedge C(J) \Rightarrow C(I \cap J)$ .
+
+Proof.
+
+$$
+\begin{array}{l} C (I) \wedge C (J) \Longrightarrow R (\backslash I) \wedge R (\backslash J) (37) \\ \Longrightarrow R (\backslash I \cup \backslash J) (38) \\ \Longrightarrow C (\backslash (\backslash I \cup \backslash J)) (39) \\ \Longrightarrow C (I \cap J). (40) \\ \end{array}
+$$
+
+![](images/05a4eb63f1834a25a79e3d9d5901a4cd561943dd8d84c689756f878c02ca9346.jpg)
+
+Proposition 5.  $R(I)\wedge R(J)\Rightarrow R(I\cap J).$
+
+Proof is analogous to Proposition 4.
+
+# I.4 DISTRIBUTION MATCHING GUARANTEES LATENT CODE INFORMATIVENESS
+
+Proposition 6. If  $(p^{*},g^{*},e^{*})\in \mathcal{H}$ , and  $(p,g,e)\in \mathcal{H}$ , and  $g^{*}(S)\stackrel {d}{=}g(Z)$ , then there exists a continuous function  $r$  such that
+
+$$
+\mathbb {E} _ {p \left(s _ {1: n}\right)} \| r \circ e \circ g ^ {*} (s) - s \| = 0. \tag {41}
+$$
+
+Proof. We show that  $r = e^{*}\circ g$  satisfies Proposition 6. By Assumption 4,
+
+$$
+\mathbb {E} _ {s} \| e ^ {*} \circ g ^ {*} (s) - s \| ^ {2} = 0. \tag {42}
+$$
+
+$$
+\mathbb {E} _ {z} \| e \circ g (z) - z \| ^ {2} = 0. \tag {43}
+$$
+
+By the same reasoning as in the proof of Proposition 3,
+
+$$
+\mathbb {E} _ {s} \| e ^ {*} \circ g ^ {*} (s) - s \| ^ {2} = 0 \Longrightarrow \forall s \in \mathcal {B} (S), e ^ {*} \circ g ^ {*} (s) = s. \tag {44}
+$$
+
+$$
+\mathbb {E} _ {z} \| e \circ g (z) - z \| ^ {2} = 0 \Longrightarrow \forall z \in \mathcal {B} (Z), e \circ g (z) = z. \tag {45}
+$$
+
+Let  $s \sim p(s)$ . We claim that  $\operatorname{Pr}[E_1] = 1$ , where  $E_1$  denote the event that  $\exists z \in \mathcal{B}(Z)$  such that  $g^{*}(s) = g(z)$ . Suppose to the contrary that there is a measure-non-zero set  $S \subseteq \operatorname{supp}(p(s))$  such that  $\forall s \in S$ , no  $z \in \mathcal{B}(Z)$  satisfies  $g^{*}(s) = g(z)$ . Let  $\mathcal{X} = \{g(s) : s \in S\}$ . As  $g^{*}(S) \stackrel{d}{=} g(Z)$ ,  $\operatorname{Pr}_s[g^*(s) \in \mathcal{X}] = \operatorname{Pr}_z[g(z) \in \mathcal{X}] > 0$ . Therefore  $\exists \mathcal{Z} \subseteq \operatorname{supp}(p(z)) - \mathcal{B}(Z)$  such that  $\mathcal{X} \subseteq \{g(z) : z \in \mathcal{Z}\}$ . But  $\operatorname{supp}(p(z)) - \mathcal{B}(Z)$  has measure 0. Contradiction.
+
+When we draw  $s$ , let  $E_2$  denote the event that  $s \in \mathcal{B}(S)$ .  $\operatorname*{Pr}[E_2] = 1$ , so  $\operatorname*{Pr}[E_1 \wedge E_2] = 1$ . When  $E_1 \wedge E_2$  happens,  $e^* \circ g \circ e \circ g^*(s) = e^* \circ g \circ e \circ g(z) = e^* \circ g(z) = e^* \circ g^*(s) = s$ . Therefore
+
+$$
+\mathbb {E} _ {s} \left\| e ^ {*} \circ g \circ e \circ g ^ {*} (s) - s \right\| = 0. \tag {46}
+$$
+
+![](images/63eef6b26fafc7ba1ac43c3a26ad549e35ce50575490ff75c4b15bed34f8358d.jpg)
+
+# I.5 WEAK SUPERVISION GUARANTEE
+
+Theorem 1. Given any oracle  $(p^{*}(s),g^{*},e^{*})\in \mathcal{H}$ , consider the distribution-matching algorithm  $\mathcal{A}$  that selects a model  $(p(z),g,e)\in \mathcal{H}$  such that:
+
+1.  $(g^{*}(S), S_{I}) \stackrel{d}{=} (g(Z), Z_{I})$  (Restricted Labeling); or  
+2.  $\left(g^{*}(S_{I},S_{\backslash I}),g^{*}(S_{I},S_{\backslash I}^{\prime})\right)\stackrel {d}{=}\left(g(Z_{I},Z_{\backslash I}),g(Z_{I},Z_{\backslash I}^{\prime})\right)$  (Match Pairing); or  
+3.  $(g^{*}(S), g^{*}(S^{\prime}), \mathbf{1}\{S_{I} \leq S_{I}^{\prime}\}) \stackrel{d}{=} (g(Z), g(Z^{\prime}), \mathbf{1}\{Z_{I} \leq Z_{I}^{\prime}\})$  (Rank Pairing).
+
+Then  $(p,g)$  satisfies  $C(I;p,g,e^{*})$  and  $e$  satisfies  $C(I;p^{*},g^{*},e)$ .
+
+Proof. We prove the three cases separately:
+
+1. Since  $(x_d, s_I) \stackrel{d}{=} (x_g, z_I)$ , consider the measurable function
+
+$$
+f (a, b) = \left\| e _ {I} ^ {*} (a) - b \right\| ^ {2}. \tag {47}
+$$
+
+We have
+
+$$
+\mathbb {E} \| e _ {I} ^ {*} (x _ {d}) - s _ {I} \| ^ {2} = \mathbb {E} \| e _ {I} ^ {*} (x _ {g}) - z _ {I} \| ^ {2} = 0. \tag {48}
+$$
+
+By the same reasoning as in the proof of Proposition 3,
+
+$$
+\mathbb {E} _ {z} \left\| e _ {I} ^ {*} \circ g (z) - z _ {I} \right\| ^ {2} = 0 \Longrightarrow \forall z \in \mathcal {B} (Z), e _ {I} ^ {*} \circ g (z) = z _ {I}. \tag {49}
+$$
+
+Therefore
+
+$$
+\mathbb {E} _ {z _ {I}} \mathbb {E} _ {z _ {\backslash I}, z _ {\backslash I} ^ {\prime}} \| e _ {I} ^ {*} \circ g (z _ {I}, z _ {\backslash I}) - e _ {I} ^ {*} \circ g (z _ {I}, z _ {\backslash I} ^ {\prime}) \| ^ {2} = 0. \tag {50}
+$$
+
+i.e.,  $g$  satisfies  $C(I;p,g,e^{*})$ . By symmetry,  $e$  satisfies  $C(I;p^{*},g^{*},e)$
+
+2.
+
+$$
+\left(g ^ {*} \left(S _ {I}, S _ {\backslash I}\right), g ^ {*} \left(S _ {I}, S _ {\backslash I} ^ {\prime}\right)\right) \stackrel {{d}} {{=}} \left(g \left(Z _ {I}, Z _ {\backslash I}\right), g \left(Z _ {I}, Z _ {\backslash I} ^ {\prime}\right)\right) \tag {51}
+$$
+
+$$
+\Rightarrow \left\| e _ {I} ^ {*} \circ g ^ {*} \left(S _ {I}, S _ {\backslash I}\right) - e _ {I} ^ {*} \circ g ^ {*} \left(S _ {I}, S _ {\backslash I} ^ {\prime}\right) \right\| ^ {2} \stackrel {{d}} {{=}} \left\| e _ {I} ^ {*} \circ g \left(Z _ {I}, Z _ {\backslash I}\right) - e _ {I} ^ {*} \circ g \left(Z _ {I}, Z _ {\backslash I} ^ {\prime}\right) \right\| ^ {2} \tag {52}
+$$
+
+$$
+\Longrightarrow \mathbb {E} _ {z _ {I}} \mathbb {E} _ {z _ {\backslash I}, z _ {\backslash I} ^ {\prime}} \| e _ {I} ^ {*} \circ g (z _ {I}, z _ {\backslash I}) - e _ {I} ^ {*} \circ g (z _ {I}, z _ {\backslash I} ^ {\prime}) \| ^ {2} = 0. \tag {53}
+$$
+
+So  $g$  satisfies  $C(I; p, g, e^{*})$ . By symmetry,  $e$  satisfies  $C(I; p^{*}, g^{*}, e)$ .
+
+3. Let  $I = \{i\}$ ,  $f = e_I^* \circ g$ . Distribution matching implies that, with probability 1 over random draws of  $Z$ ,  $Z'$ , the following event  $P$  happens:
+
+$$
+Z _ {I} <   = Z _ {I} ^ {\prime} \Longrightarrow f (Z) <   = f \left(Z ^ {\prime}\right). \tag {54}
+$$
+
+i.e.,
+
+$$
+\mathbb {E} _ {z, z ^ {\prime}} \mathbf {1} [ \neg P ] = 0. \tag {55}
+$$
+
+Let  $W = \backslash I \cap D$ ,  $Q = \backslash I \cap \backslash D$ . We showed in the proof of Proposition 3 that
+
+$$
+C (I) \Longleftrightarrow \forall \left(z _ {I}, z _ {W}, z _ {Q}\right), \left(z _ {I}, z _ {W} ^ {\prime}, z _ {Q} ^ {\prime}\right) \in \mathcal {B} (Z), f \left(z _ {I}, z _ {W}, z _ {Q}\right) = f \left(z _ {I}, z _ {W} ^ {\prime}, z _ {Q} ^ {\prime}\right). \tag {56}
+$$
+
+We prove by contradiction. Suppose  $\exists (z_I, z_W, z_Q), (z_I, z_W', z_Q') \in \mathcal{B}(Z)$  such that  $f(z_I, z_W, z_Q) < f(z_I, z_W', z_Q')$ .
+
+(a) Case 1:  $Z_I$  is discrete.
+
+Since  $f$  is constant both at  $(z_I, z_W, z_Q)$  in the interior of  $\mathcal{B}([z_I, z_W])$ , and at  $(z_I, z_W', z_Q')$  in the interior of  $\mathcal{B}([z_I, z_W'])$ , we can draw open balls of radius  $r > 0$  around each point, i.e.,  $B_r(z_Q) \subset \mathcal{B}([z_I, z_W])$  and  $B_r(z_Q') \subset \mathcal{B}([z_I, z_W'])$ , where
+
+$$
+\forall z _ {Q} ^ {*} \in B _ {r} \left(z _ {Q}\right), \forall z _ {Q} ^ {\Delta} \in B _ {r} \left(z _ {Q} ^ {\prime}\right), f \left(z _ {I}, z _ {W}, z _ {Q} ^ {*}\right) <   f \left(z _ {I}, z _ {W} ^ {\prime}, z _ {Q} ^ {\Delta}\right). \tag {57}
+$$
+
+When we draw  $z, z' \sim p(z)$ , let  $C$  denote the event that this specific value of  $z_I$  is picked for both  $z, z'$ , and we picked  $z_{\backslash I} \in B_r(z_Q')$ ,  $z_{\backslash I}' \in B_r(z_Q)$ . Since both balls have positive volume,  $\operatorname*{Pr}[C] > 0$ . However,  $P$  does not happen whenever event  $C$  happens, since  $z_I = z_I'$  but  $f(z) > f(z')$ , which contradicts  $\operatorname*{Pr}[P] = 1$ .
+
+(b) Case 2:  $z_{I}$  is continuous.
+
+Similar to case 1, we can draw open balls of radius  $r > 0$  around each point, i.e.,  $B_{r}(z_{I},z_{Q})\subset \mathcal{B}(z_{W})$  and  $B_{r}(z_{I},z_{Q}^{\prime})\subset \mathcal{B}(z_{W}^{\prime})$ , where
+
+$$
+\forall \left(z _ {I} ^ {*}, z _ {Q} ^ {*}\right) \in B _ {r} \left(z _ {I}, z _ {Q}\right), \forall \left(z _ {I} ^ {\Delta}, z _ {Q} ^ {\Delta}\right) \in B _ {r} \left(z _ {I}, z _ {Q} ^ {\prime}\right), f \left(z _ {I} ^ {*}, z _ {W}, z _ {Q} ^ {*}\right) <   f \left(z _ {I} ^ {\Delta}, z _ {W} ^ {\prime}, z _ {Q} ^ {\Delta}\right). \tag {58}
+$$
+
+Let  $H^{1} = \{(z_{I}^{*}, z_{Q}^{*}) \in B_{r}(z_{I}, z_{Q}) : z_{I}^{*} > = z_{I}\}$ ,  $H^{2} = \{(z_{I}^{\Delta}, z_{Q}^{\Delta}) \in B_{r}(z_{I}, z_{Q}') : z_{I}^{\Delta} < = z_{I}\}$ . When we draw  $z, z' \sim p(z)$ , let  $C$  denote the event that we picked  $z' \in H^{1} \times \{z_{W}\}$ ,  $z \in H^{2} \times \{z_{W}'\}$ . Since  $H^{1}, H^{2}$  have positive volume,  $\operatorname{Pr}[C] > 0$ . However,  $P$  does not happen whenever event  $C$  happens, since  $z_{I} < = z_{I}'$  but  $f(z) > f(z')$ , which contradicts  $\operatorname{Pr}[P] = 1$ .
+
+Therefore we showed
+
+$$
+\forall \left(z _ {I}, z _ {W}, z _ {Q}\right), \left(z _ {I}, z _ {W} ^ {\prime}, z _ {Q} ^ {\prime}\right) \in \mathcal {B} (Z), f \left(z _ {I}, z _ {W}, z _ {Q}\right) = f \left(z _ {I}, z _ {W} ^ {\prime}, z _ {Q} ^ {\prime}\right), \tag {59}
+$$
+
+i.e.,  $g$  satisfies  $C(I; p, g, e^{*})$ . By symmetry,  $e$  satisfies  $C(I; p^{*}, g^{*}, e)$ .
+
+![](images/8fa923d0fbc14fd28673ed23c66e5e9790e41d8c840040d217d9d20498cc1387.jpg)
+
+# I.6 WEAK SUPERVISION IMPOSSIBILITY RESULT
+
+Theorem 2. Weak supervision via restricted labeling, match pairing, or ranking on  $s_I$  is not sufficient for learning a generative model whose latent code  $Z_I$  is restricted to  $S_I$ .
+
+Proof. We construct the following counterexample. Let  $n = m = 3$  and  $I = \{1\}$ . The data generation process is  $s_1 \sim \mathrm{unif}([0,2\pi))$ ,  $(s_2, s_3) \sim \mathrm{unif}(\{(x,y): x^2 + y^2 \leq 1\})$ ,  $g^*(s) = [s_1, s_2, s_3]$ . Consider a generator  $z_1 \sim \mathrm{unif}([0,2\pi))$ ,  $(s_2, s_3) \sim \mathrm{unif}(\{(x,y): x^2 + y^2 \leq 1\})$ ,  $g(z) = [z_1, \cos(z_1)z_2 - \sin(z_1)z_3, \sin(z_1)z_2 + \cos(z_1)z_3]$ . Then  $(x_d, s_I) \stackrel{d}{=} (x_g, z_I)$  but  $R(I; p, g, e^*)$  and  $R(I; p^*, g^*, e)$  does not hold. The same counterexample is applicable for match pairing and rank pairing.
\ No newline at end of file
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+# WHAT GRAPH NEURAL NETWORKS CANNOT LEARN: DEPTH VS WIDTH
+
+Andreas Loukas
+
+École Polytechnique Fédérale Lausanne andreas.loukas@epfl.ch
+
+# ABSTRACT
+
+This paper studies the expressive power of graph neural networks falling within the message-passing framework  $(\mathtt{GNN}_{\mathtt{mp}})$ . Two results are presented. First,  $\mathtt{GNN}_{\mathtt{mp}}$  are shown to be Turing universal under sufficient conditions on their depth, width, node attributes, and layer expressiveness. Second, it is discovered that  $\mathtt{GNN}_{\mathtt{mp}}$  can lose a significant portion of their power when their depth and width is restricted. The proposed impossibility statements stem from a new technique that enables the repurposing of seminal results from distributed computing and leads to lower bounds for an array of decision, optimization, and estimation problems involving graphs. Strikingly, several of these problems are deemed impossible unless the product of a  $\mathtt{GNN}_{\mathtt{mp}}$  's depth and width exceeds a polynomial of the graph size; this dependence remains significant even for tasks that appear simple or when considering approximation.
+
+# 1 INTRODUCTION
+
+A fundamental question in machine learning is to determine what a model can and cannot learn. In deep learning, there has been significant research effort in establishing expressivity results for feedforward (Cybenko, 1989; Hornik et al., 1989; Lu et al., 2017) and recurrent neural networks (Neto et al., 1997), as well as more recently for Transformers and Neural GPUs (Pérez et al., 2019). We have also seen the first results studying the universality of graph neural networks, i.e., neural networks that take graphs as input. Maron et al. (2019b) derived a universal approximation theorem over invariant functions targeted towards deep networks whose layers are linear and equivariant to permutation of their input. Universality was also shown for equivariant functions and a particular shallow architecture (Keriven & Peyré, 2019).
+
+Universality statements allow us to grasp the expressive power of models in the limit. In theory, given enough data and the right training procedure, a universal network will be able to solve any task that it is presented with. Nevertheless, the insight brought by such results can also be limited. Knowing that a sufficiently large network can be used to solve any problem does not reveal much about how neural networks should be designed in practice. It also cannot guarantee that said network will be able to solve a given task given a particular training procedure, such as stochastic gradient descent.
+
+On the other hand, it might be easier to obtain insights about models by studying their limitations. After all, the knowledge of what cannot be computed (and thus learned) by a network of specific characteristics applies independently of the training procedure. Further, by helping us comprehend the difficulty of a task in relation to a model, impossibility results can yield practical advice on how to select model hyperparameters. Take, for instance, the problem of graph classification. Training a graph classifier entails identifying what constitutes a class, i.e., finding properties shared by graphs in one class but not the other, and then deciding whether new graphs abide to said learned properties. However, if the aforementioned decision problem is shown to be impossible by a graph neural network of certain depth then we can be certain that the same network will not learn how to classify a sufficiently diverse test set correctly, independently of which learning algorithm is employed. We should, therefore, focus on networks deeper that the lower bound when performing experiments.
+
+<table><tr><td>problem</td><td>bound</td><td>problem</td><td>bound</td></tr><tr><td>cycle detection (odd)</td><td>dw = Ω(n/log n)</td><td>shortest path</td><td>d√w = Ω(√n/ log n)</td></tr><tr><td>cycle detection (even)</td><td>dw = Ω(√n/ log n)</td><td>max. indep. set</td><td>dw = Ω(n2/ log2n) for w = O(1)</td></tr><tr><td>subgraph verification*</td><td>d√w = Ω(√n/ log n)</td><td>min. vertex cover</td><td>dw = Ω(n2/ log2n) for w = O(1)</td></tr><tr><td>min. spanning tree</td><td>d√w = Ω(√n/ log n)</td><td>perfect coloring</td><td>dw = Ω(n2/ log2n) for w = O(1)</td></tr><tr><td>min. cut</td><td>d√w = Ω(√n/ log n)</td><td>girth 2-approx.</td><td>dw = Ω(√n/ log n)</td></tr><tr><td>diam. computation</td><td>dw = Ω(n/log n)</td><td>diam. 3/2-approx.</td><td>dw = Ω(√n/ log n)</td></tr></table>
+
+Table 1: Summary of main results. Subgraph verification* entails verifying one of the following predicates for a given subgraph: is connected, contains a cycle, forms a spanning tree, is bipartite, is a cut, is an  $s-t$  cut. All problems are defined in Appendix A.
+
+# 1.1 MAIN RESULTS
+
+This paper studies the expressive power of message-passing graph neural networks  $(\mathsf{GNN}_{\mathsf{mp}})$  (Gilmer et al., 2017). This model encompasses several state-of-the-art networks, including GCN (Kipf & Welling, 2016), gated graph neural networks (Li et al., 2015), molecular fingerprints (Duvenaud et al., 2015), interaction networks (Battaglia et al., 2016), molecular convolutions (Kearnes et al., 2016), among many others. Networks using a global state (Battaglia et al., 2018) or looking at multiple hops per layer (Morris et al., 2019; Liao et al., 2019; Isufi et al., 2020) are not directly  $\mathsf{GNN}_{\mathsf{mp}}$ , but they can often be re-expressed as such. The provided contributions are two-fold:
+
+I. What  $\mathsf{GNN}_{\mathsf{mp}}$  can compute. Section 3 derives sufficient conditions such that a  $\mathsf{GNN}_{\mathsf{mp}}$  can compute any function on its input that is computable by a Turing machine. This result compliments recent universality results (Maron et al., 2019b; Keriven & Peyré, 2019) that considered approximation (rather than computability) over specific classes of functions (permutation invariant and equivariant) and particular architectures. The claim follows in a straightforward manner by establishing the equivalence of  $\mathsf{GNN}_{\mathsf{mp}}$  with LOCAL (Angluin, 1980; Linial, 1992; Naor & Stockmeyer, 1993), a classical model in distributed computing that is itself Turing universal. In a nutshell,  $\mathsf{GNN}_{\mathsf{mp}}$  are shown to be universal if four strong conditions are met: there are enough layers of sufficient expressiveness and width, and nodes can uniquely distinguish each other. Since Turing universality is a strictly stronger property than universal approximation, Chen et al. (2019)'s argument further implies that a Turing universal  $\mathsf{GNN}_{\mathsf{mp}}$  can solve the graph isomorphism problem (a sufficiently deep and wide network can compute the isomorphism class of its input).
+
+II. What  $\mathsf{GNN}_{\mathsf{mp}}$  cannot compute (and thus learn). Section 4 analyses the implications of restricting the depth  $d$  and width  $w$  of  $\mathsf{GNN}_{\mathsf{mp}}$  that do not use a readout function. Specifically, it is proven that  $\mathsf{GNN}_{\mathsf{mp}}$  lose a significant portion of their power when the product  $dw$ , which I call capacity, is restricted. The analysis relies on a new technique that enables repurposing impossibility results from the context of distributed computing to the graph neural network setting. Specifically, lower bounds for the following problems are presented: (i) detecting whether a graph contains a cycle of specific length; (ii) verifying whether a given subgraph is connected, contains a cycle, is a spanning tree, is bipartite, is a simple path, corresponds to a cut or Hamiltonian cycle; (iii) approximating the shortest path between two nodes, the minimum cut, and the minimum spanning tree; (iv) finding a maximum independent set, a minimum vertex cover, or a perfect coloring; (v) computing or approximating the diameter and girth. The bounds are summarized in Table 1 and the problem definitions can be found in Appendix A. Section 5 presents some empirical evidence of the theory.
+
+Though formulated in a graph-theoretic sense, the above problems are intimately linked to machine learning on graphs. Detection, verification, and computation problems are relevant to classification: knowing what properties of a graph a  $\mathsf{GNN}_{\mathsf{mp}}$  cannot see informs us also about which features of a graph can it extract. Further, there have been attempts to use  $\mathsf{GNN}_{\mathsf{mp}}$  to devise heuristics for graph-based optimization problems (Khalil et al., 2017; Battaglia et al., 2018; Li et al., 2018; Joshi et al., 2019; Bianchi et al., 2019), such as the ones discussed above. The presented results can then be taken as a worst-case analysis for the efficiency of  $\mathsf{GNN}_{\mathsf{mp}}$  in such endeavors.
+
+# 1.2 DISCUSSION
+
+The results of this paper carry several intriguing implications. To start with, it is shown that the capacity  $dw$  of a  $GNN_{mp}$  plays a significant role in determining its power. Solving many problems
+
+is shown to be impossible unless  $dw = \tilde{\Omega}(n^{\delta})$ , where  $\delta \in [1/2, 2]$ ,  $n$  is the number of nodes of the graph, and  $f(n) = \tilde{\Omega}(g(n))$  is interpreted as  $f(n)$  being, up to logarithmic factors, larger than  $g(n)$  as  $n$  grows. This reveals a direct trade-off between the depth and width of a graph neural network. Counter-intuitively, the dependence on  $n$  can be significant even if the problem appears local in nature or one only looks for approximate solutions. For example, detecting whether  $G$  contains a short cycle of odd length cannot be done unless  $dw = \tilde{\Omega}(n)$ . Approximation helps, but only to a limited extent; computing the graph diameter requires  $dw = \tilde{\Omega}(n)$  and this reduces to  $dw = \tilde{\Omega}(\sqrt{n})$  for any  $3/2$ -factor approximation. Further, it is impossible to approximate within any constant factor the shortest path, the minimum cut, and the minimum spanning tree, all three of which have polynomial-time solutions, unless  $d\sqrt{w} = \tilde{\Omega}(\sqrt{n})$ . Finally, for truly hard problems, the capacity may even need to be super-linear on  $n$ . Specifically, it is shown that, even if the layers of the  $\mathsf{GNN}_{\mathsf{mp}}$  are allowed to take exponential time, solving certain NP-hard problems necessitates  $d = \tilde{\Omega}(n^{2})$  depth for any constant-width network.
+
+Relation to previous impossibility results. In contrast to universality (Maron et al., 2019b; Keriven & Peyre, 2019), the limitations of  $\mathsf{GNN}_{\mathsf{mp}}$  have been much less studied. In particular, the bounds presented here are the first impossibility results that (i) explicitly connect  $\mathsf{GNN}_{\mathsf{mp}}$  properties (depth and width) with graph properties and that (ii) go beyond isomorphism by addressing decision, optimization, and estimation graph problems. Three main directions of related work can be distinguished. First, Dehmamy et al. (2019) bounded the ability of graph convolutional networks (i.e.,  $\mathsf{GNN}_{\mathsf{mp}}$  w/o messaging functions) to compute specific polynomial functions of the adjacency matrix, referred to as graph moments by the authors. Second, Xu et al. (2018) and Morris et al. (2019) established the equivalence of anonymous  $\mathsf{GNN}_{\mathsf{mp}}$  (those that do not rely on node identification) to the Weisfeiler-Lehman (WL) graph isomorphism test. The equivalence implies that anonymous networks are blind to the many graph properties that WL cannot see: e.g., any two regular graphs with the same number of nodes are identical from the perspective of the WL test (Arvind et al., 2015; Kiefer et al., 2015). Third, in parallel to this work, Sato et al. (2019) utilized a connection to LOCAL to derive impossibility results for the ability of a class of novel partially-labeled  $\mathsf{GNN}_{\mathsf{mp}}$  to find good approximations for three NP-hard optimization problems. Almost all of the above negative results occur due to nodes being unable to distinguish between neighbors at multiple hops (see Appendix D). With discriminative attributes  $\mathsf{GNN}_{\mathsf{mp}}$  become significantly more powerful (without necessarily sacrificing permutation in/equivalence). Still, as this work shows, even in this setting certain problems remain impossible when the depth and width of the  $\mathsf{GNN}_{\mathsf{mp}}$  is restricted. For instance, though cycles can be detected (something impossible in anonymous networks (Xu et al., 2018; Morris et al., 2019)), even for short cycles one now needs  $dw = \tilde{\Omega}(n)$ . Further, in contrast to Sato et al. (2019), an approximation ratio below 2 for the minimum vertex cover is not impossible, but necessitates  $dw = \tilde{\Omega}(n^2)$ .
+
+Limitations. First, all lower bounds are of a worst-case nature: a problem is deemed impossible if there exists a graph for which it cannot be solved. The discovery of non worst-case capacity bounds remains an open problem. Second, rather than taking into account specific parametric functions, each layer is assumed to be sufficiently powerful to compute any function of its input. This strong assumption does not significantly limit the applicability of the results, simply because all lower bounds that hold with universal layers also apply to those that are limited computationally. Lastly, it will be assumed that nodes can uniquely identify each other. Node identification is compatible with permutation invariance/equivalence as long as the network output is asked to be invariant to the particular way the ids have been assigned. In the literature, one-hot encoded node ids are occasionally useful (Kipf & Welling, 2016; Berg et al., 2017). When attempting to learn functions across multiple graphs, ids should be ideally substituted by sufficiently discriminative node attributes (attributes that uniquely identify each node within each receptive field it belongs to can serve as ids). Nevertheless, similar to the unbounded computation assumption, if a problem cannot be solved by a graph neural network in the studied setting, it also cannot be solved without identifiers and discriminative attributes. Thus, the presented lower bounds also apply to partially and fully anonymous networks.
+
+Notation. I consider connected graphs  $G = (\mathcal{V}, \mathcal{E})$  consisting of  $n = |\mathcal{V}|$  nodes. The edge going from  $v_{j}$  to  $v_{i}$  is written as  $e_{i \leftarrow j}$  and it is asserted that if  $e_{i \leftarrow j} \in \mathcal{E}$  then also  $e_{j \leftarrow i} \in \mathcal{E}$ . The neighborhood  $\mathcal{N}_{i}$  of a node  $v_{i} \in \mathcal{V}$  consists of all nodes  $v_{j}$  for which  $e_{i \leftarrow j} \in \mathcal{E}$ . The degree of  $v_{i}$  is denoted by  $\deg_{i}$ ,  $\Delta$  is the maximum degree of all nodes and the graph diameter  $\delta_{G}$  is the length of the longest
+
+shortest path between any two nodes. In the self-loop graph  $G^{*} = (\mathcal{V},\mathcal{E}^{*})$ , the neighborhood set of  $v_{i}$  is given by  $\mathcal{N}_i^* = \mathcal{N}_i\cup v_i$ .
+
+# 2 THE GRAPH NEURAL NETWORK COMPUTATIONAL MODEL
+
+Graph neural networks are parametric and differentiable learning machines. Their input is usually an attributed graph  $G_{a} = (G, (a_{i}: v_{i} \in \mathcal{V}), (a_{i \leftarrow j}: e_{i \leftarrow j} \in \mathcal{E}))$ , where vectors  $a_{1}, \ldots, a_{n}$  encode relevant node attributes and  $a_{i \leftarrow j}$  are edge attributes, e.g., encoding edge direction.
+
+Model 1 formalizes the graph neural network operation by placing it in the message passing model (Gilmer et al., 2017). The computation proceeds in layers, within which a message  $m_{i \leftarrow j}$  is passed along each directed edge  $e_{i \leftarrow j} \in \mathcal{E}$  going from  $v_j$  to  $v_i$  and each node updates its internal representation by aggregating its state with the messages sent by its incoming neighbors  $v_j \in \mathcal{N}_i$ . The network output can be either of two things: a vector  $x_i$  for each node  $v_i$  or a single vector  $x_G$  obtained by combining the representations of all nodes using a readout function. Vectors  $x_i / x_G$  could be scalars (node/graph regression), binary variables (node/graph classification) or multi-dimensional (node/graph embedding). I use the symbols  $\mathtt{GNN}_{\mathtt{mp}}^{\mathtt{n}}$  and  $\mathtt{GNN}_{\mathtt{mp}}^{\mathtt{g}}$  to distinguish between models that return a vector per node and one per graph, respectively.
+
+Computational model 1 Message passing graph neural network (GNN<sub>mp</sub>)
+
+Initialization: Set  $x_{i}^{(0)} = a_{i}$  for all  $v_{i} \in \mathcal{V}$ .
+
+for layer  $\ell = 1,\dots ,d$  do
+
+for every edge  $e_{i\leftarrow j}\in \mathcal{E}^*$  (in parallel) do
+
+$$
+m _ {i \leftarrow j} ^ {(\ell)} = \operatorname {M S G} _ {\ell} \left(x _ {i} ^ {(\ell - 1)}, x _ {j} ^ {(\ell - 1)}, v _ {i}, v _ {j}, a _ {i \leftarrow j}\right)
+$$
+
+for every node  $v_{i} \in \mathcal{V}$  (in parallel) do
+
+$$
+x _ {i} ^ {(\ell)} = \mathrm {U P} _ {\ell} \Big (\sum_ {v _ {j} \in \mathcal {N} _ {i} ^ {*}} m _ {i \leftarrow j} ^ {(\ell)} \Big)
+$$
+
+Set  $x_{i} = x_{i}^{(d)}$
+
+return Either  $x_{i}$  for every  $v_{i} \in \mathcal{V}$  ( $\mathsf{GNN}_{\mathsf{mp}}^{\mathsf{n}}$ ) or  $x_{G} = \mathrm{READ}\left(\{x_{i}:v_{i} \in \mathcal{V}\}\right)$  ( $\mathsf{GNN}_{\mathsf{mp}}^{\mathsf{g}}$ ).
+
+The operation of a  $\mathrm{GNN}_{\mathrm{mp}}$  is primarily determined by the messaging, update, and readout functions. I assume that  $\mathrm{MSG}_{\ell}$  and  $\mathrm{UP}_{\ell}$  are general functions that act on intermediate node representations and node ids (the notation is overloaded such that  $v_{i}$  refers to both the  $i$ -th node as well as its unique id). As is common in the literature (Lu et al., 2017; Battaglia et al., 2018), these functions are instantiated by feed-forward neural networks. Thus, by the universal approximation theorem and its variants (Cybenko, 1989; Hornik et al., 1989), they can approximate any general function that maps vectors onto vectors, given sufficient depth and/or width. Function READ is useful when one needs to retrieve a representation that is invariant of the number of nodes. The function takes as an input a multiset, i.e., a set with possibly repeating elements, and returns a vector. Commonly, READ is chosen to be a dimension squashing operator, such as a sum or a histogram, followed by a feed-forward neural network (Xu et al., 2018; Seo et al., 2019).
+
+Depth and width. The depth  $d$  is equal to the number of layers of the network. Larger depth means that each node has the opportunity to learn more about the rest of the graph (i.e., it has a larger receptive field). The width  $w$  of a  $\mathsf{GNN}_{\mathsf{mp}}$  is equal to the largest dimension of state  $x_{i}^{(l)}$  over all layers  $l$  and nodes  $v_{i}\in \mathcal{V}$ . Since nodes need to be able to store their own unique ids, in the following it is assumed that each variable manipulated by the network is represented in finite-precision using  $p = \Theta (\log n)$  bits (though this is not strictly necessary for the analysis).
+
+# 3 SUFFICIENT CONDITIONS FOR TURING UNIVERSALITY
+
+This section studies what graph neural networks can compute. It is demonstrated that, even without readout function, a network is computationally universal if it has enough layers of sufficient
+
+width, nodes can uniquely distinguish each other, and the functions computed within each layer are sufficiently expressive. The derivation entails establishing that  $\mathsf{GNN}_{\mathsf{mp}}^{\mathsf{n}}$  is equivalent to LOCAL, a classical model used in the study the distributed algorithms that is itself Turing universal.
+
+# 3.1 THE LOCAL COMPUTATIONAL MODEL
+
+A fundamental question in theoretical computer science is determining what can and cannot be computed efficiently by a distributed algorithm. The LOCAL model, initially studied by Angluin (1980), Linial (1992), and Naor & Stockmeyer (1993), provides a common framework for analyzing the effect of local decision. Akin to  $\mathsf{GNN}_{\mathsf{mp}}$ , in LOCAL a graph plays a double role: it is both the input of the system and captures the network topology of the distributed system that solves the problem. In this spirit, the nodes of the graph are here both the machines where computation takes place as well as the variables of the graph-theoretic problem we wish to solve—similarly, edges model communication links between machines as well as relations between nodes. Each node  $v_{i} \in \mathcal{V}$  is given a problem-specific local input and has to produce a local output. The input contains necessary the information that specifies the problem instance. All nodes execute the same algorithm, they are fault-free, and they are provided with unique identifiers.
+
+A pseudo-code description is given in Model 2. Variables  $s_i^{(l)}$  and  $s_{i\leftarrow j}^{(l)}$  refer respectively to the state of  $v_{i}$  in round  $l$  and to the message sent by  $v_{j}$  to  $v_{i}$  in the same round. Both are represented as strings. The computation starts simultaneously and unfolds in synchronous rounds  $l = 1,\dots ,d$ . Three things can occur within each round: each node receives a string of unbounded size from its incoming neighbors; each node updates its internal state by performing some local computation; and each node sends a string to every one of its outgoing neighbors. Functions  $\mathrm{ALG}_l^1$  and  $\mathrm{ALG}_l^2$  are algorithms computed locally by a Turing machine running on node  $v_{i}$ . Before any computation is done, each node  $v_{i}$  is aware of its own attribute  $a_{i}$  as well as of all edge attributes  $\{a_{i\leftarrow j}:v_j\in \mathcal{N}_i^*\}$ .
+
+Computational model 2 LOCAL (computed distributedly by each node  $v_{i} \in \mathcal{V}$ ).  
+Initialization: Set  $s_{i\leftarrow i}^{(0)} = (a_i,v_i)$  and  $s_{i\leftarrow j}^{(0)} = (a_j,v_j)$  for all  $e_{i\leftarrow j}\in \mathcal{E}$    
+for round  $\ell = 1,\ldots ,d$  do Receive  $s_{i\leftarrow j}^{(\ell -1)}$  from  $v_{j}\in \mathcal{N}_{i}^{*}$  ,compute  $s_i^{(\ell)} = \mathrm{ALG}_\ell^1\left(\left\{\left(s_{i\leftarrow j}^{(\ell -1)},a_{i\leftarrow j}\right):v_j\in \mathcal{N}_i^*\right\} ,v_i\right)$  and send  $s_{j\leftarrow i}^{(\ell)} = \mathrm{ALG}_{\ell}^{2}\left(s_{i}^{(\ell)},v_{i}\right)$  to  $v_{j}\in \mathcal{N}_{i}^{*}$    
+return  $s_i^{(d)}$
+
+In LOCAL, there are no restrictions on how much information a node can send at every round. Asserting that each message  $s_{i \leftarrow j}^{(\ell)}$  is at most  $b$  bits yields the CONGEST model (Peleg, 2000).
+
+# 3.2 TURING UNIVERSALITY
+
+The reader might have observed that LOCAL resembles closely  $\mathtt{GNN}_{\mathtt{mp}}^{\mathtt{n}}$  in its structure, with only a few minor differences: firstly, whereas a LOCAL algorithm A may utilize messages in any way it chooses, a  $\mathtt{GNN}_{\mathtt{mp}}^{\mathtt{n}}$  network N always sums received messages before any local computation. The two models also differ in the arguments of the messaging function and the choice of information representation (string versus vector). Yet, as the following theorem shows, the differences between  $\mathtt{GNN}_{\mathtt{mp}}^{\mathtt{n}}$  and LOCAL are inconsequential when seen from the perspective of their expressive power:
+
+Theorem 3.1 (Equivalence). Let  $\mathcal{N}_{\ell}(G_{a})$  be the binary representation of the state  $(x_{1}^{(\ell)}, \ldots, x_{n}^{(\ell)})$  of a  $GNN_{mp}^{n}$  network  $N$  and  $A_{\ell}(G_{a}) = (s_{1}^{(\ell)}, \ldots, s_{n}^{(\ell)})$  that of a LOCAL algorithm  $A$ . If  $\mathrm{MSG}_{\ell}$  and  $\mathrm{UP}_{\ell}$  are Turing complete functions, then, for any algorithm  $A$  there exists  $N$  (resp. for any  $N$  there exists  $A$ ) such that
+
+$$
+\mathcal {A} _ {\ell} (G _ {a}) = \mathcal {N} _ {\ell} (G _ {a}) \quad f o r e v e r y l a y e r \ell a n d G _ {a} \in \mathcal {G} _ {a},
+$$
+
+where  $\mathcal{G}_a$  is the set of all attributed graphs.
+
+This equivalence enables us to reason about the power of  $\mathsf{GNN}_{\mathsf{mp}}^{\mathsf{n}}$  by building on the well-studied properties of LOCAL. In particular, it is well known in distributed computing that, as long as the number of rounds  $d$  of a distributed algorithm is larger than the graph diameter  $\delta_G$ , every node in a LOCAL can effectively make decisions based on the entire graph (Linial, 1992). Together with Theorem 3.1, the above imply that, if computation and memory are not an issue, one may construct a  $\mathsf{GNN}_{\mathsf{mp}}^{\mathsf{n}}$  that effectively computes any computable function w.r.t. its input.
+
+Corollary 3.1.  $GNN_{mp}^{n}$  can compute any Turing computable function over connected attributed graphs if the following conditions are jointly met: each node is uniquely identified;  $\mathrm{MSG}_l$  and  $\mathrm{UP}_l$  are Turing-complete for every layer  $\ell$ ; the depth is at least  $d \geq \delta_G$  layers; and the width is unbounded.
+
+Why is this result relevant? From a cursory review, it might seem that universality is an abstract result with little implication to machine learning architects. After all, the utility of a learning machine is usually determined not with regards to its expressive power but with its ability to generalize to unseen examples. Nevertheless, it can be argued that universality is an essential property of a good learning model. This is for two main reasons: First, universality guarantees that the learner does not have blind-spots in its hypothesis space. No matter how good the optimization algorithm is, how rich the dataset, and how overparameterized the network is, there will always be functions which are non universal learners cannot learn. Second, a universality result provides a glimpse on how the size of the learner's hypothesis space is affected by different design choices. For instance, Corollary 3.1 puts forth four necessary conditions for universality: the  $\mathsf{GNN}_{\mathfrak{mp}}^{\mathsf{n}}$  should be sufficiently deep and wide, nodes should be able to uniquely and consistently identify each other, and finally, the functions utilized in each layer should be sufficiently complex. The following section delves further into the importance of two of these universality conditions. It will be shown that  $\mathsf{GNN}_{\mathfrak{mp}}^{\mathsf{n}}$  lose a significant portion of their power when the depth and width conditions are relaxed.
+
+The universality of  $\mathsf{GNN}_{\mathsf{mp}}^{\mathsf{g}}$ . Though a universality result could also be easily derived for networks with a readout function, the latter is not included as it deviates from how graph neural networks are meant to function: given a sufficiently powerful readout function, a  $\mathsf{GNN}_{\mathsf{mp}}^{\mathsf{g}}$  of  $d = 1$  depth and  $O(\Delta)$  width can be used to compute any Turing computable function. The nodes should simply gather one hop information about their neighbors; the readout function can then reconstruct the problem input based on the collective knowledge and apply any computation needed.
+
+# 4 IMPOSSIBILITY RESULTS AS A FUNCTION OF DEPTH AND WIDTH
+
+This section analyzes the effect of depth and width in the expressive power of  $\mathsf{GNN}_{\mathsf{mp}}^{\mathsf{n}}$ . Specifically, I will consider problems that cannot be solved by a network of a given depth and width.
+
+To be able to reason in terms of width, it will be useful to also enforce that the message size in LOCAL at each round is at most  $b$  bits. This model goes by the name CONGEST in the distributed computing literature (Peleg, 2000). In addition, it will be assumed that nodes do not have access to a random generator. With this in place, the following theorem shows us how to translate impossibility results from CONGEST to  $\mathrm{GNN}_{\mathrm{mp}}^{\mathrm{n}}$ :
+
+Theorem 4.1. If a problem  $P$  cannot be solved in less than  $d$  rounds in CONGEST using messages of at most  $b$  bits, then  $P$  cannot be solved by a  $GNN_{mp}^{n}$  of width  $w \leq (b - \log_2 n) / p = O(b / \log n)$  and depth  $d$ .
+
+The  $p = \Theta (\log n)$  factor corresponds to the length of the binary representation of every variable—the precision needs to depend logarithmically on  $n$  for the node ids to be unique. With this result in place, the following sections re-state several known lower bounds in terms of a  $\mathsf{GNN}_{\mathsf{mp}}^{\mathsf{n}}$ 's depth and width.
+
+# 4.1 IMPOSSIBILITY RESULTS FOR DECISION PROBLEMS
+
+I first consider problems where one needs to decide whether a given graph satisfies a certain property (Feuilleoy & Fraigniaud, 2016). Concretely, given a decision problem  $P$  and a graph  $G$ , the  $\mathsf{GNN}_{\mathsf{mp}}^{\mathsf{n}}$  should output  $x_{i} \in \{\text{true,false}\}$  for all  $v_{i} \in \mathcal{V}$ . The network then accepts the premise if the logical conjunction of  $\{x_{1}, \ldots, x_{n}\}$  is true and rejects it otherwise. Such problems are intimately connected to graph classification: classifying a graph entails identifying what constitutes a class from some training set and using said learned definition to decide the label of graphs sampled from
+
+the test set. Instead, I will suppose that the class definition is available to the classifier and I will focus on the corresponding decision problem. As a consequence, every lower bound presented below for a decision problem must also be respected by a  $\mathsf{GNN}_{\mathsf{mp}}^{\mathsf{n}}$  classifier that attains zero error on the corresponding graph classification problem.
+
+Subgraph detection. In this type of problems, the objective is to decide whether  $G$  contains a subgraph belonging to a given family. I focus specifically on detecting whether  $G$  contains a cycle  $C_k$ , i.e., a simple undirected graph of  $k$  nodes each having exactly two neighbors. As the following result shows, even with ids, cycle detection remains relatively hard:
+
+Corollary 4.1 (Repurposed from (Drucker et al., 2014; Korhonen & Rybicki, 2018)). There exists graph  $G$  on which every  $\mathsf{GNN}_{mp}^{n}$  of width  $w$  requires depth at least  $d = \Omega(\sqrt{n} / (w \log n))$  and  $d = \Omega(n / (w \log n))$  to detect if  $G$  contains a cycle  $C_k$  for even  $k \geq 4$  and odd  $k \geq 5$ , respectively.
+
+Whereas an anonymous  $\mathsf{GNN}_{\mathsf{mp}}$  cannot detect cycles (e.g., distinguish between two  $C_3$  vs one  $C_6$  (Maron et al., 2019a)), it seems that with ids the product of depth and width should exhibit an (at least) linear dependence on  $n$ . The intuition behind this bound can be found in Appendix C and empirical evidence in support of the theory are presented in Section 5.
+
+Subgraph verification. Suppose that the network is given a subgraph  $H = (\mathcal{V}_H, \mathcal{E}_H)$  of  $G$  in its input. This could, for instance, be achieved by selecting the attributes of each node and edge to be a one-hot encoding of their membership on  $\mathcal{V}_H$  and  $\mathcal{E}_H$ , respectively. The question considered is whether the neural network can verify a certain property of  $H$ . More concretely, does a graph neural network exist that can successfully verify  $H$  as belonging to a specific family of graphs w.r.t.  $G$ ? In contrast to the standard decision paradigm, here every node should reach the same decision—either accepting or rejecting the hypothesis. The following result is a direct consequence of the seminal work by Sarma et al. (2012):
+
+Corollary 4.2 (Repurposed from (Sarma et al., 2012)). There exists a graph  $G$  on which every  $GNN_{mp}^{n}$  of width  $w$  requires depth at least  $d = \Omega \left( \sqrt{\frac{n}{w\log^2 n}} + \delta_G \right)$  to verify if some subgraph  $H$  of  $G$  is connected, contains a cycle, forms a spanning tree of  $G$ , is bipartite, is a cut of  $G$ , or is an  $s-t$  cut of  $G$ . Furthermore, the depth should be at least  $d = \Omega \left( \left( \frac{n}{w\log n} \right)^{\gamma} + \delta_G \right)$  with  $\gamma = \frac{1}{2} - \frac{1}{2(\delta_{G'} - 1)}$  to verify if  $H$  is a Hamiltonian cycle or a simple path.
+
+Therefore, even if one knows where to look in  $G$ , verifying whether a given subgraph meets a given property can be non-trivial, and this holds for several standard graph-theoretic properties. For instance, if we constrain ourselves to networks of constant width, detecting whether a subgraph is connected can, up to logarithmic factors, require  $\Omega(\sqrt{n})$  depth in the worst case.
+
+# 4.2 IMPOSSIBILITY RESULTS FOR OPTIMIZATION PROBLEMS
+
+I turn my attention to the problems involving the exact or approximate optimization of some graph-theoretic objective function. From a machine learning perspective, the considered problems can be interpreted as node/edge classification problems: each node/edge is tasked with deciding whether it belongs to the optimal set or not. Take, for instance, the maximum independent set, where one needs to find the largest cardinality node set, such that no two of them are adjacent. Given only information identifying nodes,  $\mathsf{GNN}_{\mathsf{mp}}^{\mathsf{n}}$  will be asked to classify each node as being part of the maximum independent set or not.
+
+Polynomial-time problems. Let me first consider three problems that possess known polynomial-time solutions. To make things easier for the  $\mathsf{GNN}_{\mathsf{mp}}^{\mathsf{n}}$ , I relax the objective and ask for an approximate solution rather than optimal. An algorithm (or neural network) is said to attain an  $\alpha$ -approximation if it produces a feasible output whose utility is within a factor  $\alpha$  of the optimal. Let OPT be the utility of the optimal solution and ALG that of the  $\alpha$ -approximation algorithm. Depending on whether the problem entails minimization or maximization, the ratio ALG/OPT is at most  $\alpha$  and at least  $1 / \alpha$ , respectively.
+
+According to the following corollary, it is non-trivial to find good approximate solutions:
+
+Corollary 4.3 (Repurposed from (Sarma et al., 2012; Ghaffari & Kuhn, 2013)). There exists graphs  $G$  and  $G'$  of diameter  $\delta_G = \Theta(\log n)$  and  $\delta_{G'} = O(1)$  on which every  $GNN_{mp}^n$  of width  $w$  requires
+
+depth at least  $d = \Omega(\sqrt{\frac{n}{w\log^2n}})$  and  $d' = \Omega((\frac{n}{w\log n})^\gamma)$  with  $\gamma = \frac{1}{2} - \frac{1}{2(\delta_{G'} - 1)}$ , respectively, to approximate within any constant factor: the minimum cut problem, the shortest s-t path problem, or the minimum spanning tree problem.
+
+Thus, even for simple problems (complexity-wise), in the worst case a constant width  $\mathsf{GNN}_{\mathsf{mp}}^{\mathsf{n}}$  should be almost  $\Omega (\sqrt{n})$  deep even if the graph diameter is exponentially smaller than  $n$ .
+
+NP-hard problems. So what about truly hard problems? Clearly, one cannot expect a  $\mathsf{GNN}_{\mathsf{mp}}$  to solve an NP-hard time in polynomial time². However, it might be interesting as a thought experiment to consider a network whose layers take exponential time on the input size—e.g., by selecting the  $\mathsf{MSG}_l$  and  $\mathsf{UP}_l$  functions to be feed-forward networks of exponential depth and width. Could one ever expect such a  $\mathsf{GNN}_{\mathsf{mp}}^{\mathsf{n}}$  to arrive at the optimal solution?
+
+The following corollary provides necessary conditions for three well-known NP-hard problems:
+
+Corollary 4.4 (Repurposed from (Censor-Hillel et al., 2017)). There exists a graph  $G$  on which every  $GNN_{mp}^{n}$  of width  $w = O(1)$  requires depth at least  $d = \Omega(n^2 / \log^2 n)$  to solve: the minimum vertex cover problem; the maximum independent set problem; the perfect coloring problem.
+
+Thus, even if each layer is allowed to take exponential time, the depth should be quadratically larger than the graph diameter  $\delta_G = O(n)$  to have a chance of finding the optimal solution. Perhaps disappointingly, the above result suggests that it may not be always possible to exploit the distributed decision making performed by  $\mathsf{GNN}_{\mathsf{mp}}$  architectures to find solutions faster than classical (centralized) computational paradigms.
+
+# 4.3 IMPOSSIBILITY RESULTS FOR ESTIMATION PROBLEMS
+
+Finally, I will consider problems that involve the computation or estimation of some real function that takes as an input the graph and attributes. The following corollary concerns the computation of two well-known graph invariants: the diameter  $\delta_G$  and the girth. The latter is defined as the length of the shortest cycle and is infinity if the graph has no cycles.
+
+Corollary 4.5 (Repurposed from (Frischknecht et al., 2012)). There exists a graph  $G$  on which every  $\mathsf{GNN}_{mp}^{n}$  of width  $w$  requires depth at least  $d = \Omega(n / (w\log n) + \delta_G)$  to compute the graph diameter  $\delta_G$  and  $d = \Omega(\sqrt{n} / (w\log n) + \delta_G)$  to approximate the graph diameter and girth within a factor of  $\frac{3}{2}$  and 2, respectively.
+
+Term  $\delta_G$  appears in the lower bounds because both estimation problems require global information. Further, approximating the diameter within a  $3/2$  factor seems to be simpler than computing it. Yet, in both cases, one cannot achieve this using a  $\mathsf{GNN}_{\mathsf{mp}}^{\mathsf{n}}$  whose capacity is constant. As a final remark, the graphs giving rise to the lower bounds of Corollary 4.5 have constant diameter and  $\Theta(n^2)$  edges. However, similar bounds can be derived also for graphs with  $O(n \log n)$  edges (Abboud et al., 2016). For the case of exact computation, the lower bound is explained in Appendix C.
+
+# 5 EMPIRICAL EVIDENCE
+
+This section aims to empirically test the connection between the capacity  $dw$  of a  $\mathrm{GNN}_{\mathrm{mp}}$ , the number of nodes  $n$  of its input, and its ability to solve a given task. In particular, I considered the problem of 4-cycle classification and tasked the neural network with classifying graphs based on whether they contained a cycle of length four. Following the lower bound construction described in Appendix A, I generated five distributions over graphs with  $n \in (8,16,24,32,44)$  nodes and an average diameter of  $(4,6,8,9,11)$ , respectively (this was achieved by setting  $p \in (6,8,10,12,14)$ , see Figure 3a). For each such distribution, I generated a training and test set consisting respectively of 1000 and 200 examples. Both sets were exactly balanced, i.e., any example graph from the training and test set had exactly  $50\%$  chance of containing a 4-cycle.
+
+The experiment aimed to evaluate how able were  $\mathsf{GNN}_{\mathsf{mp}}$  of different capacities to attain high accuracy on the test set. To this end, I performed grid search over the hyperparameters  $w \in (2, 10, 20)$  and
+
+![](images/5dfb93e9b230c0bb5514a9c1f2034ffb1530c77b1d3818598fcce7041174fb98.jpg)
+
+![](images/8e8fdb77cace8994065c8a822d615684da48c8762e81548c8b4df56e80f762bd.jpg)  
+(a) training accuracy of all trained networks
+
+![](images/c3f3094df79b034170397acf86a71abe53a5089dce8932a6a1a8c2c5c2c502e8.jpg)
+
+![](images/c4b864dfb8588a3ee98188a44d9a59c6cbe777aacf17a85b6ee029c6274b85bc.jpg)
+
+![](images/60e79d2df684689610d9ffb85d5e1a8a39adb469b1e2736cfd3fccbf40e84357.jpg)
+
+![](images/8c396614e4308a6dbb886e98d985aedc85b9999da8d178748df765ec68c4907d.jpg)
+
+![](images/0896fd8ca2030d30fda0d14f91bda84d2382e5dbe8eaa4d25c383de61ff1cf08.jpg)  
+(b) test accuracy of all trained networks
+
+![](images/7cfffa156cf08f45fe43ac399ce3da9e67f2bfb0a990872a9ad4623e7f66b976.jpg)
+
+![](images/71dada4e8ad63c5430489127a2c26c8d65a6f0721d87ca32159d52060b53e3cb.jpg)
+
+![](images/aa3eb0ca0e28e7a083c4894bec0632729aef43e05eb711e21228c77af35bb198.jpg)
+
+![](images/da365cd3a32a7462aaebd2e6c9fd4412419587b32f668a1e2e3b1ca48a990383.jpg)  
+(c) best training accuracy  
+Figure 1: Accuracy as a function of  $\mathsf{GNN}_{\mathsf{mp}}$  capacity  $dw$  and  $n$ . (Best seen in color.)
+
+![](images/e73fa9d7a8001ea1c48bc72faa3e8028bed5be3aedd9fcd91d5b78e186dc52a4.jpg)  
+(d) best test accuracy
+
+$d \in (5, 10, 20, 15)$ . To reduce the dependence on the initial conditions and training length, for each hyperparameter combination, I trained 4 networks independently (using Adam and learning rate decay) for 4000 epochs. The  $\mathrm{GNN}_{\mathrm{mp}}$  chosen was that proposed by Xu et al. (2018), with the addition of residual connections—this network outperformed all others that I experimented with.
+
+It is important to stress that empirically verifying lower bounds for neural networks is challenging, because it involves searching over the space of all possible networks in order to find the ones that perform the best. For this reason, an experiment such as the one described above cannot be used to verify<sup>3</sup> the tightness of the bounds: we can never be certain whether the results obtained are the best possible or whether the optimization resulted in a local minimum. In that view, the following results should be interpreted in a qualitative sense. The question that I will ask is: to which extend do the trends uncovered match those predicted by the theory? More specifically, does the ability of a network to detect 4-cycles depend on the relation between  $dw$  and  $n$ ?
+
+To answer this question, Figure 1 depicts the training and test accuracy as a function of the capacity  $dw$  for all the 240 networks trained (5 distributions × 3 widths × 4 depths × 4 iterations). The accuracy of the best performing networks with the smallest capacity is shown in Figures 1c and 1d. It is important to stress that, based on Weisfeiler-Lehman analyses, anonymous  $\mathsf{GNN}_{\mathsf{mp}}$  cannot solve the considered task. However, as it seen in the figures, the impossibility is annulled when using node  $\mathrm{id}_s^4$ . Indeed, even small neural networks could consistently classify all test examples perfectly (i.e., achieving  $100\%$  test accuracy) when  $n \leq 16$ . Moreover, as the theoretical results predicted, there is a strong correlation between the test accuracy,  $dw$  and  $n$  (recall that Corollary 4.1 predicts  $dw = \tilde{\Omega}(\sqrt{n})$ ). Figure 1d shows that networks of the same capacity were consistently less accurate on the test set as  $n$  increased (even though the cycle length remained 4 in all experiments). It is also
+
+![](images/142025d2459f229747bc8bef4a9f906e25458fb19a19de19a07615b2af9ad154.jpg)  
+(a) effect of anonymity
+
+![](images/768aa01acbc793bd3433d48b75d789fa0d15ea1c3cf1180e9a060ee56aa29457.jpg)  
+Figure 2: (a) GNNs are significantly more powerful when given discriminative node attributes. (b) Test accuracy indicated by color as a function of normalized depth and width. Points in highlighted areas correspond to networks with super-critical capacity, whereas the diagonal line separates networks that more deep than wide. (For improved visibility, points are slightly perturbed. Best seen in color.)
+
+![](images/571dc9eda4b03b8ce6f58d9c4d8e5806c469226b287048dfcb7897be5e4e290b.jpg)  
+(b) depth vs width
+
+striking to observe that even the most powerful networks considered could not achieve a test accuracy above  $95\%$  for  $n > 16$ ; for  $n = 40$  their best accuracy was below  $80\%$ .
+
+Effect of anonymity. Figure 2a plots example training and test curves for  $\mathsf{GNN}_{\mathsf{mp}}$  trained with four different node attributes: no attributes (anonymous), a one-hot encoding of the node degrees (degree), a one-hot encoding of node ids (unique id), and a one-hot encoding of node ids that changed across graphs (random unique id). It can be clearly observed that there is a direct correlation between accuracy and the type of attributes used. With non- or partially-discriminative attributes, the network could not detect cycles even in the training set. The cycle detection problem was solved exactly with unique ids, but when the latter were inconsistently assigned, the network could not learn to generalize.
+
+Exchangeability of depth and width. Figure 2b examines further the relationship between depth, width, and test accuracy. This time, networks were separated depending on their depth and width normalized by the square root of the "critical capacity". For each  $n$ , the critical capacity is the minimum  $dw$  of a network that was able to solve the task on a graph of  $n$  nodes—here, solving amounts to a test accuracy above  $95\%$ . In this way, a network of depth  $d$  and width  $w$  tested on  $n$  nodes corresponds to a point positioned at  $x = d / \sqrt{\text{critical}}$ ,  $y = w / \sqrt{\text{critical}}$  and no network positioned at  $xy < 1$  can solve the task (non-highlighted region in the bottom left corner). As seen, there is a crisp phase transition between the regime of under- and super-critical capacity: almost every network meeting the condition  $dw \geq$  critical was able to solve the task, irrespective of whether the depth or width was larger. Note that, the exchangeability of depth and width cannot be guaranteed by the proposed theory which asserts that the condition  $dw = \tilde{\Omega}(\sqrt{n})$  is necessary—but not sufficient. The empirical results however do agree with the hypothesis that, for 4-cycle classification, depth and width are indeed exchangeable.
+
+# 6 CONCLUSION
+
+This work studied the expressive power of graph neural networks falling within the message-passing framework. Two results were derived. First, sufficient conditions were provided such that  $\mathsf{GNN}_{\mathsf{mp}}$  can compute any function computable by a Turing machine with the same connected graph as input. Second, it was discovered that the product of a  $\mathsf{GNN}_{\mathsf{mp}}$  's depth and width plays a prominent role in determining whether the network can solve various graph-theoretic problems. Specifically, it was shown that  $\mathsf{GNN}_{\mathsf{mp}}^{\mathsf{n}}$  with  $dw = \tilde{\Omega}(n^{\delta})$  and  $\delta \in [0.5, 2]$  cannot solve a range of decision, optimization, and estimation problems involving graphs. Overall, the proposed results demonstrate that the power of graph neural networks depends critically on their capacity and illustrate the importance of using discriminative node attributes.
+
+Acknowledgements. I thank the Swiss National Science Foundation for supporting this work in the context of the project "Deep Learning for Graph-Structured Data" (grant number PZ00P2 179981).
+
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+
+# A GRAPH THEORY DEFINITIONS
+
+The main graph-theoretic terms encountered in this work are:
+
+-  $k$ -cycle detection: a  $k$ -cycle is a subgraph of  $G$  consisting of  $k$  nodes, each with degree two. The  $k$ -cycle detection problem entails determining if  $G$  contains a  $k$ -cycle.  
+- Hamiltonian cycle: a cycle of length  $n$  
+- (minimum) spanning tree: a spanning tree is a tree subgraph of  $G$  consisting of  $n$  nodes. The minimum spanning tree problem entails finding the spanning tree of  $G$  of minimum weight (the weight of a tree is equal to the sum of its edge weights).  
+- (minimum) cut: a cut is a subgraph of  $G$  that when deleted leaves  $G$  disconnected. The minimum cut problem entails finding the cut of minimum weight (the weight of a cut is equal to the sum of its edge weights).  
+-  $s - t$  cut: a subgraph of  $G$  such that removing all subgraph edges from  $G$  will leave the nodes  $s$  and  $t$  of  $G$  disconnected.  
+- (shortest) path: a simple path is subgraph of  $G$  where all nodes have degree 2 except from the two endpoint nodes whose degree is one. The shortest path problem entails finding the simple path of minimum weight that connects two given nodes (the weight of a path is equal to the sum of its edge weights).
+
+- (maximum) independent set: an independent set is a set of nodes in a graph no two of which are adjacent. The maximum independent set problem entails finding the independent set of maximum cardinality.  
+- (minimum) vertex cover: a vertex cover of  $G$  is a set of nodes such that each edge of  $G$  is incident to at least one node in the set. The minimum vertex cover problem entails finding the vertex cover of minimum cardinality.  
+- (perfect) coloring: a coloring of  $G$  is a labeling of the nodes with distinct colors such that no two adjacent nodes are colored using same color. The perfect coloring problem entails finding a coloring with the smallest number of colors.  
+- diameter: the diameter  $\delta_G$  of  $G$  equals the length of the longest shortest path.  
+- girth: the girth of  $G$  equals the length of the shortest cycle. It is infinity if no cycles are present.
+
+# B DEFERRED PROOFS
+
+# B.1 PROOF OF THEOREM 3.1
+
+The claim is proven by expressing the state of node  $v_{i}$  in the two models in the same form. It is not difficult to see that for each layer of the  $\mathsf{GNN}_{\mathsf{mp}}^{\mathsf{n}}$  one has
+
+$$
+\begin{array}{l} x _ {i} ^ {(l)} = \mathrm {U P} _ {\ell} \left(\sum_ {v _ {j} \in \mathcal {N} _ {i} ^ {*}} m _ {i \leftarrow j} ^ {(\ell)}\right) \quad (\text {b y}) \\ = \mathrm {U P} _ {\ell} \left(\sum_ {v _ {j} \in \mathcal {N} _ {i} ^ {*}} \mathrm {M S G} _ {\ell} \left(x _ {i} ^ {(\ell - 1)}, x _ {j} ^ {(\ell - 1)}, v _ {i}, v _ {j}, a _ {i \leftarrow j}\right)\right) \quad (\text {s u b s t i t u t e d} m _ {i \leftarrow j} ^ {(\ell)}) \\ = \operatorname {A G G} _ {\ell} \left(\left\{\left(x _ {i} ^ {(\ell - 1)}, x _ {j} ^ {(\ell - 1)}, v _ {i}, v _ {j}, a _ {i \leftarrow j}\right): v _ {j} \in \mathcal {N} _ {i} ^ {*} \right\}\right), \quad (\text {f r o m (X u e t a l , 2 0 1 8 , L e m m a 5)}) \\ \end{array}
+$$
+
+where  $\mathrm{AGG}_{\ell}$  is an aggregation function, i.e., a map from the set of multisets onto some vector space. In the last step, I used a result of Xu et al. Xu et al. (2018) stating that each aggregation function can be decomposed as an element-wise function over each element of the multiset, followed by summation of all elements, and then a final function.
+
+Similarly, one may write:
+
+$$
+\begin{array}{l} s _ {i} ^ {(\ell)} = \operatorname {A L G} _ {\ell} ^ {1} \left(\left\{\left(s _ {i \leftarrow j} ^ {(\ell - 1)}, a _ {i \leftarrow j}\right): v _ {j} \in \mathcal {N} _ {i} ^ {*} \right\}, v _ {i}\right) \quad (\text {b y}) \\ = \operatorname {A L G} _ {\ell} ^ {1} \left(\left\{\left(\operatorname {A L G} _ {\ell - 1} ^ {2} \left(s _ {j} ^ {(\ell - 1)}, v _ {j}\right), a _ {i \leftarrow j}\right): v _ {j} \in \mathcal {N} _ {i} ^ {*} \right\}, v _ {i}\right) \quad (\text {s u b s t i t u t e d} s _ {i \leftarrow j} ^ {(\ell - 1)}) \\ = \operatorname {A L G} _ {\ell} \left(\left\{\left(s _ {j} ^ {(\ell - 1)}, v _ {i}, v _ {j}, a _ {i \leftarrow j}\right): v _ {j} \in \mathcal {N} _ {i} ^ {*} \right\}\right), \\ \end{array}
+$$
+
+with the last step following by restructuring the input and defining  $\mathrm{ALG}_{\ell}$  as the Turning machine that simulates the action of both  $\mathrm{ALG}_{\ell}^{2}$  and  $\mathrm{ALG}_{\ell -1}^{1}$ .
+
+Since one may encode any vector into a string and vice versa, w.l.o.g. one may assume that the state of each node in LOCAL is encoded as a vector  $x_{i}$ . Then, to complete the proof, one still needs to demonstrate that the functions
+
+$$
+\operatorname {A G G} \left(\left\{\left(x _ {i}, x _ {j}, v _ {i}, v _ {j}, a _ {i \leftarrow j}\right): v _ {j} \in \mathcal {N} _ {i} ^ {*} \right\}\right) \quad \text {a n d} \quad \operatorname {A L G} \left(\left\{\left(x _ {j}, v _ {i}, v _ {j}, a _ {i \leftarrow j}\right): v _ {j} \in \mathcal {N} _ {i} ^ {*} \right\}\right)
+$$
+
+are equivalent (in the interest of brevity the layer/round indices have been dropped). If this holds then each layer of  $\mathsf{GNN}_{\mathsf{mp}}^{\mathsf{n}}$  is equivalent to a round of LOCAL and the claim follows.
+
+I first note that, since its input is a multiset,  $\mathrm{ALG}_l$  is also an aggregation function. To demonstrate equivalence, one thus needs to show that, despite not having identical inputs, each of the two aggregation functions can be used to replace the other. For the forward direction, it suffices to show that for every aggregation function AGG there exists ALG with the same output. Indeed, one may always construct  $\mathrm{ALG} = \mathrm{AGG} \circ g$ , where  $g$  takes as input  $\{(x_j, v_i, v_j, a_{i \leftarrow j}) : v_j \in \mathcal{N}_i^*\}$ , identifies  $x_i$  (by searching for  $v_i, v_i$ ) and appends it to each element of the multiset yielding  $\{(x_i, x_j, v_i, v_j, a_{i \leftarrow j}) : v_j \in \mathcal{N}_i^*\}$ . The backward direction can also be proven with an elementary construction: given ALG, one sets AGG = ALG  $\circ h$ , where  $h$  deletes  $x_i$  from each element of the multiset.
+
+# B.2 PROOF OF COROLLARY 3.1
+
+In the LOCAL model the reasoning is elementary (Linial, 1992; Fraigniaud et al., 2013; Seidel, 2015): suppose that the graph is represented by a set of edges and further consider that  $\mathrm{ALG}_l$  amounts to a union operation. Then in  $d = \delta_G$  rounds, the state of each node will contain the entire graph. The function  $\mathrm{ALG}_d^1$  can then be used to make the final computation. This argument also trivially holds for node/edge attributes. The universality of  $\mathsf{GNN}_{\mathsf{mp}}^{\mathsf{n}}$  then follows by the equivalence of LOCAL and  $\mathsf{GNN}_{\mathsf{mp}}^{\mathsf{n}}$ .
+
+# B.3 PROOF OF THEOREM 4.1
+
+First note that, since the  $\mathsf{GNN}_{\mathsf{mp}}^{\mathsf{n}}$  and LOCAL models are equivalent, if no further memory/width restrictions are placed, an impossibility for one implies also an impossibility for the other. It can also be seen in Theorem 3.1 that there is a one-to-one mapping between the internal state of each node at each level between the two models (i.e., variables  $x_{i}^{(l)}$  and  $s_i^{(l)}$ ). As such, impossibility results that rely on restrictions w.r.t. state size (in terms of bits) also transfer between the models.
+
+To proceed, I demonstrate that a depth lower bound in the CONGEST model (i.e., in the LOCAL model with bounded message size) also implies the existence of a depth lower bound in the LOCAL model with a bounded state size—with this result in place, the proof of the main claim follows directly. As in the statement of the theorem, one starts by assuming that  $P$  cannot be solved in less than  $d$  rounds when messages are bounded to be at most  $b$  bits. Then, for the sake of contradiction, it is supposed that there exists an algorithm  $A \in \mathrm{LOCAL}$  that can solve  $P$  in less than  $d$  rounds with a state of at most  $c$  bits, but unbounded message size. I argue that the existence of this algorithm also implies the existence of a second algorithm  $A'$  whose messages are bounded by  $c + \log_2 n$ : since each message  $s_{j\leftarrow i}^{(l)}$  is the output of a universal Turing machine  $\mathrm{ALG}_l^2$  that takes as input the tuple  $(s_i^{(l)}, v_i)$ , algorithm  $A'$  directly sends the input and relies on the universality of  $\mathrm{ALG}_{l+1}^1$  to simulate the action of  $\mathrm{ALG}_l^2$ . The message size bound follows by adding the size  $c$  of the state with that of representing the node id ( $\log_2 n$  bits suffice for unique node ids). This line of reasoning leads to a contradiction when  $c \leq b - \log_2 n$ , as it implies that there exists an algorithm (namely  $A'$ ) that can solve  $P$  in less than  $d$  rounds while using messages of at most  $b$  bits. Hence, no algorithm whose state is less than  $b - \log_2 n$  bits can solve  $P$  in LOCAL, and the width of  $\mathsf{GNN}_{\mathsf{mp}}^n$  has to be at least  $(b - \log_2 n)/p$ .
+
+# C AN EXPLANATION OF THE LOWER BOUNDS FOR CYCLE DETECTION AND DIAMETER ESTIMATION
+
+A common technique for obtaining lower bounds in the CONGEST model is by reduction to the set-disjointness problem in two-player communication complexity: Suppose that Alice and Bob are each given some secret string  $(s_a$  and  $s_b)$  of  $q$  bits. The two players use the string to construct a set by selecting the elements from the base set  $\{1,2,\ldots ,q\}$  for which the corresponding bit is one. It is known that Alice and Bob cannot determine whether their sets are disjoint or not without exchanging at least  $\Omega (q)$  bits (Kalyanasundaram & Schintger, 1992; Chor & Goldreich, 1988).
+
+The reduction involves constructing a graph that is partially known by each player. Usually, Alice and Bob start knowing half of the graph (red and green induced subgraphs in Figure 3). The players then use their secret string to control some aspect of their private topology (subgraphs annotated in dark gray). Let the resulting graph be  $G(s_{a}, s_{b})$  and denote by cut the number of edges connecting the subgraphs controlled by Alice and Bob. To derive a lower bound for some problem  $P$ , one needs to prove that a solution for  $P$  in  $G(s_{a}, s_{b})$  would also reveal whether the two sets are disjoint or not. Since each player can exchange at most  $O(b \cdot \mathrm{cut})$  bits per round, at least  $\Omega(q / (b \cdot \mathrm{cut}))$  rounds are needed in total in CONGEST. By Theorem 4.1, one then obtains a  $d = \Omega(q / (w \log n \cdot \mathrm{cut}))$  depth lower bound for  $\mathtt{GNN}_{\mathtt{mp}}^{\mathtt{n}}$ .
+
+The two examples in Figure 3 illustrate the graphs  $G(s_{a}, s_{b})$  giving rise to the lower bounds for even  $k$ -cycle detection and diameter estimation. To reduce occlusion, only a subset of the edges are shown.
+
+![](images/6597c250ad61ab8c558f04f08c9b174a5a520b55630b90790045f49cd19e414b.jpg)  
+(a) 10-cycle lower bound graph  
+Figure 3: Examples of graphs giving rise to lower bounds.
+
+![](images/5733704f94bcb0d96317eb07f27e866dbbbe099b70382def853c734d6f58c3a3.jpg)  
+(b) diameter lower bound graph
+
+(a) In the construction of Korhonen & Rybicki (2018), each player starts from a complete bipartite graph of  $p = \sqrt{q}$  nodes (nodes annotated in dark grey) with nodes numbered from 1 to  $2p$ . The nodes with the same id are connected yielding a cut of size  $2p$ . Each player then uses its secret (there are as many bits as bipartite edges) to decide which of the bipartite edges will be deleted (corresponding to zero bits). Remaining edges are substituted by a path of length  $k / 2 - 1$ . This happens in a way that ensures that  $G(s_{a}, s_{b})$  contains a cycle of length  $k$  (half known by Alice and half by Bob) if and only if the two sets are disjoint: the cycle will pass through nodes  $t$  and  $p + t$  of each player to signify that the  $t$ -th bits of  $s_{a}$  and  $s_{b}$  are both one. It can then be shown that  $n = \Theta(p^{2})$  from which it follows that: CONGEST requires at least  $d = \Omega(q / (b \cdot \mathrm{cut})) = \Omega(n / (b \cdot p)) = \Omega(\sqrt{n} / b)$  bits to decide if there is a cycle of length  $k$ ; and  $\mathsf{GNN}_{\mathsf{mp}}^{\mathsf{n}}$  has to have  $d = \Omega(\sqrt{n} / (w \log n))$  depth to do the same.  
+(b) In the construction of Abboud et al. (2016), each string consists of  $q = \Omega(n)$  bits. The strings are used to encode the connectivity of subgraphs annotated in dark gray: an edge exists between the red nodes  $i$  and  $q$  if and only if the  $i$ -th bit of  $s_a$  is one (and similarly for green). Due to the graph construction, the cut between Alice and Bob has  $O(\log q)$  edges. Moreover, About et al. proved that  $G(s_a, s_b)$  has diameter at least five if and only if the sets defined by  $s_a$  and  $s_b$  are disjoint. This implies that  $d = \Omega(n / (w\log^2 n))$  depth is necessary to compute the graph diameter in  $\mathsf{GNN}_{\mathsf{mp}}^{\mathsf{n}}$ .
+
+# D THE COST OF ANONYMITY
+
+There is a striking difference between the power of anonymous networks and those in which nodes have the ability to uniquely identify each other, e.g., based on ids or discriminative attributes (see the survey by Suomela (2013)).
+
+To illustrate this phenomenon, I consider a thought experiment where a node is tasked with reconstructing the graph topology in the LOCAL model. In the left, Figure 4 depicts the red node's knowledge after two rounds (equivalent to a  $\mathsf{GNN}_{\mathsf{mp}}^{\mathsf{n}}$  having  $d = 2$ ) when each node has a unique identifier (color). At the end of the first round, each node is aware of its neighbors and after two rounds the entire graph has been successfully reconstructed in red's memory.
+
+In the right subfigure, nodes do not possess ids (as in the analysis of (Xu et al., 2018; Morris et al., 2019)) and thus cannot distinguish which of their neighbors are themselves adjacent. As such, the red node cannot tell whether the graph contains cycles: after two rounds there are at least two plausible topologies that could explain its observations.
+
+![](images/883a0adc4ac80d5b1be47744853f856ec9f8257a66408b5ebfff62e62017b562.jpg)  
+(a) nodes can identify each other
+
+![](images/5fbd4f7948114506ee4d56fb08c7befa70fcf4a84622a5710633e4b362f7eb25.jpg)
+
+![](images/f0ba5892861d1c4ed62c0e873e62f68a6e1207f54dada007f906c4b0513e2942.jpg)  
+(b) nodes cannot identify each other (WL test).
+
+![](images/f13a87c96e563e912e5a89b8693ef04231503286edd9f8c3aa283700b080a363.jpg)  
+Figure 4: Toy example of message exchange from the perspective of the red node. The arrows show where each received message comes from and the message content is shown in light gray boxes. Red's knowledge of the graph topology is depicted at the bottom.
\ No newline at end of file
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+# WHY GRADIENT CLIPPING ACCELERATES TRAINING: A THEORETICAL JUSTIFICATION FOR ADAPTIVITY
+
+Jingzhao Zhang, Tianxing He, Suvrit Sra & Ali Jadbabaie
+
+Massachusetts Institute of Technology
+
+Cambridge, MA 02139, USA
+
+{jzhzhang, tianxing, suvrit, jadbabai}@mit.edu
+
+# ABSTRACT
+
+We provide a theoretical explanation for the effectiveness of gradient clipping in training deep neural networks. The key ingredient is a new smoothness condition derived from practical neural network training examples. We observe that gradient smoothness, a concept central to the analysis of first-order optimization algorithms that is often assumed to be a constant, demonstrates significant variability along the training trajectory of deep neural networks. Further, this smoothness positively correlates with the gradient norm, and contrary to standard assumptions in the literature, it can grow with the norm of the gradient. These empirical observations limit the applicability of existing theoretical analyses of algorithms that rely on a fixed bound on smoothness. These observations motivate us to introduce a novel relaxation of gradient smoothness that is weaker than the commonly used Lipschitz smoothness assumption. Under the new condition, we prove that two popular methods, namely, gradient clipping and normalized gradient, converge arbitrarily faster than gradient descent with fixed stepsize. We further explain why such adaptively scaled gradient methods can accelerate empirical convergence and verify our results empirically in popular neural network training settings.
+
+# 1 INTRODUCTION
+
+We study optimization algorithms for neural network training and aim to resolve the mystery of why adaptive methods converge fast. Specifically, we study gradient-based methods for minimizing a differentiable nonconvex function  $f: \mathbb{R}^d \to \mathbb{R}$ , where  $f(x)$  can potentially be stochastic, i.e.,  $f(x) = \mathbb{E}_{\xi}[F(x, \xi)]$ . Such choices of  $f$  cover a wide range of problems in machine learning, and their study motivates a vast body of current optimization literature.
+
+A widely used (and canonical) approach for minimizing  $f$  is the (stochastic) gradient descent (GD) algorithm. Despite its simple form, GD often achieves superior empirical (Wilson et al., 2017) performances and theoretical (Carmon et al., 2017) guarantees. However, in many tasks such as reinforcement learning and natural language processing (NLP), adaptive gradient methods (e.g., Adagrad (Duchi et al., 2011), ADAM (Kingma and Ba, 2014), and RMSProp (Tieleman and Hinton, 2012)) outperform SGD. Despite their superior empirical performance, our understanding of the fast convergence of adaptive methods is limited. Previous analysis has shown that adaptive methods are more robust to variation in hyper-parameters (Ward et al., 2018) and adapt to sparse gradients (Duchi et al., 2011) (a more detailed literature review is in Appendix A). However, in practice, the gradient updates are dense, and even after extensively tuning the SGD hyperparameters, it still converges much slower than adaptive methods in NLP tasks.
+
+We analyze the convergence of clipped gradient descent and provide an explanation for its fast convergence. Even though gradient clipping is a standard practice in tasks such as language models (e.g. Merity et al., 2018; Gehring et al., 2017; Peters et al., 2018), it lacks a firm theoretical grounding. Goodfellow et al. (2016); Pascanu et al. (2013; 2012) discuss the gradient explosion problem in recurrent models and consider clipping as an intuitive work around. We formalize this intuition and prove that clipped GD can converge arbitrarily faster than fixed-step gradient descent. This result is shown to hold under a novel smoothness condition that is strictly weaker than the standard Lipschitz-gradient assumption pervasive in the literature. Hence our analysis captures many functions that are not globally Lipschitz smooth. Importantly, the proposed smoothness condition is derived on the
+
+basis of extensive NLP training experiments, which are precisely the same type of experiments for which adaptive gradient methods empirically perform superior to gradient methods.
+
+By identifying a new smoothness condition through experiments and then using it to analyze the convergence of adaptively-scaled methods, we reduce the following gap between theory and practice. On one hand, powerful techniques such as Nesterov's momentum and variance reduction theoretically accelerate convex and nonconvex optimization. But, at least for now, they seem to have limited applicability in deep learning (Defazio and Bottou, 2018). On the other hand, some widely used techniques (e.g., heavy-ball momentum, adaptivity) lack theoretical acceleration guarantees. We suspect that a major reason here is the misalignment of the theoretical assumptions with practice. Our work demonstrates that the concept of acceleration critically relies on the problem assumptions and that the standard global Lipschitz-gradient condition may not hold in the case of some applications and thus must be relaxed to admit a wider class of objective functions.
+
+# 1.1 CONTRIBUTIONS
+
+In light of the above background, the main contributions of this paper are the following:
+
+Inspired and supported by neural network training experiments, we introduce a new smoothness condition that allows the local smoothness constant to increase with the gradient norm. This condition is strictly weaker than the pervasive Lipschitz-gradient assumption.  
+We provide a convergence rate for clipped GD under our smoothness assumption (Theorem 3).  
+- We prove an upper-bound (Theorem 6) and a lower-bound (Theorem 4) on the convergence rate of GD under our relaxed smoothness assumption. The lower-bound demonstrates that GD with fixed step size can be arbitrarily slower than clipped GD.  
+- We provide upper bounds for stochastic clipped GD (Theorem 7) and SGD (Theorem 8). Again, stochastic clipped GD can be arbitrarily faster than SGD with a fixed step size.
+
+We support our proposed theory with realistic neural network experiments. First, in the state of art LSTM language modeling (LM) setting, we observe the function smoothness has a strong correlation with gradient norm (see Figure 2). This aligns with the known fact that gradient clipping accelerates LM more effectively compared to computer vision (CV) tasks. Second, our experiments in CV and LM demonstrate that clipping accelerates training error convergence and allows the training trajectory to cross non-smooth regions of the loss landscape. Furthermore, gradient clipping can also achieve good generalization performance even in image classification (e.g.,  $95.2\%$  test accuracy in 200 epochs for ResNet20 on Cifar10). Please see Section 5 for more details.
+
+# 2 A NEW RELAXED SMOOTHNESS CONDITION
+
+In this section, we motivate and develop a relaxed smoothness condition that is weaker (and thus, more general) than the usual global Lipschitz smoothness assumption. We start with the traditional definition of smoothness.
+
+# 2.1 FUNCTION SMOOTHNESS (LIPSCHITZ GRADIENTS)
+
+Recall that  $f$  denotes the objective function that we want to minimize. We say that  $f$  is  $L$ -smooth if
+
+$$
+\left\| \nabla f (x) - \nabla f (y) \right\| \leq L \| x - y \|, \quad \text {f o r a l l} x, y \in \mathbb {R} ^ {d}. \tag {1}
+$$
+
+For twice differentiable functions, condition (1) is equivalent to  $\| \nabla^2 f(x)\| \leq L,\forall x\in \mathbb{R}^d$ . This smoothness condition enables many important theoretical results. For example, Carmon et al. (2017) show that GD with  $h = 1 / L$  is up to a constant optimal for optimizing smooth nonconvex functions.
+
+But the usual  $L$ -smoothness assumption (1) also has its limitations. Assuming existence of a global constant  $L$  that upper bounds the variation of the gradient is very restrictive. For example, simple polynomials such as  $f(x) = x^3$  break the assumption. One workaround is to assume that  $L$  exists in a compact region, and either prove that the iterates do not escape the region or run projection-based algorithms. However, such assumptions can make  $L$  very large and slow down the theoretical convergence rate. In Section 4, we will show that a slow rate is unavoidable for gradient descent with fixed step size, whereas clipped gradient descent can greatly improve the dependency on  $L$ .
+
+The above limitations force fixed-step gradient descent (which is tailored for Lipschitz smooth functions) to converge slowly in many tasks. In Figure 1, we plot the estimated function smoothness at different iterations during training neural networks. We find that function smoothness varies greatly at different iterations. From Figure 1, we further find that local smoothness positively correlates with the full gradient norm, especially in the language modeling experiment. A natural question is:
+
+Can we find a fine-grained smoothness condition under which we can design theoretically and empirically fast algorithms at the same time?
+
+To answer this question, we introduce the relaxed smoothness condition in the next section, which is developed on the basis of extensive experiments—Figure 1 provides an illustrative example.
+
+![](images/5adac277dfa0afacd26be5a118cd5c403f082d1f12bc640e9b2fddd243941bac.jpg)  
+Figure 1: Gradient norm vs local gradient Lipschitz constant on a log-scale along the training trajectory for AWD-LSTM (Merit et al., 2018) on PTB dataset. The colorbar indicates the number of iterations during training. More experiments can be found in Section 5. Experiment details are in Appendix H.
+
+# 2.2 A NEW RELAXED SMOOTHNESS CONDITION
+
+We observe strong positive correlation between function smoothness and gradient norm in language modeling experiments (Figure 1(a)). This observation leads us to propose the following smoothness condition that allows local smoothness to grow with function gradients.
+
+Definition 1. A second order differentiable function  $f$  is  $(L_0, L_1)$ -smooth if
+
+$$
+\left\| \nabla^ {2} f (x) \right\| \leq L _ {0} + L _ {1} \| \nabla f (x) \|. \tag {2}
+$$
+
+Definition 1 strictly relaxes the usual (and widely used)  $L$ -smoothness. There are two ways to interpret the relaxation: First, when we focus on a compact region, we can balance the constants  $L_0$  and  $L_1$  such that  $L_0 \ll L$  while  $L_1 \ll L$ . Second, there exist functions that are  $(L_0, L_1)$ -smooth globally, but not  $L$ -smooth. Hence the constant  $L$  for  $L$ -smoothness gets larger as the compact set increases but  $L_0$  and  $L_1$  stay fixed. An example is given in Lemma 2.
+
+Remark 1. It is worth noting that we do not need the Hessian operator norm and gradient norm to necessarily satisfy the linear relation (2). As long as these norms are positively correlated, gradient clipping can be shown to achieve faster rate than fixed step size gradient descent. We use the linear relationship (2) for simplicity of exposition.
+
+Lemma 2. Let  $f$  be the univariate polynomial  $f(x) = \sum_{i=1}^{d} a_i x^i$ . When  $d \geq 3$ , then  $f$  is  $(L_0, L_1)$ -smooth for some  $L_0$  and  $L_1$  but not  $L$ -smooth.
+
+Proof. The first claim follows from  $\lim_{x\to \infty}\left|\frac{f'(x)}{f''(x)}\right| = \lim_{x\to -\infty}\left|\frac{f'(x)}{f''(x)}\right| = \infty$ . The second claim follows by the unboundedness of  $f''(x)$ .
+
+# 2.3 SMOOTHNESS IN NEURAL NETWORKS
+
+We saw that our smoothness condition relaxes the traditional smoothness assumption and is motivated empirically (Figure 1). Below we develop some intuition for this phenomenon. We conjecture that the proposed positive correlation results from the common components in expressions of the gradient and the Hessian. We illustrate the reasoning behind this conjecture by considering an  $\ell$ -layer linear network with quadratic loss—a similar computation also holds for nonlinear networks.
+
+The  $L_{2}$  regression loss of a deep linear network is  $\mathcal{L}(Y, f(X)) \coloneqq \| Y - W_{\ell} \cdots W_{1}X\|^{2}$ , where  $Y$  denotes labels,  $X$  denotes the input data matrix, and  $W_{i}$  denotes the weights in the  $i^{\text{th}}$  layer. By (Lemma 4.3 Kawaguchi, 2016), we know that
+
+$$
+\nabla_ {\operatorname {v e c} (w _ {i})} \mathcal {L} (Y, f (X)) = ((W _ {\ell} \dots W _ {i + 1}) \otimes (W _ {i - 1} \dots W _ {2} W _ {1} X) ^ {T}) ^ {T} \operatorname {v e c} (f (X) - Y),
+$$
+
+where  $\operatorname{vec}(\cdot)$  flattens a matrix in  $\mathbb{R}^{m\times n}$  into a vector in  $\mathbb{R}^{mn}$ ;  $\otimes$  denotes the Kronecker product. For constants  $i,j$  such that  $\ell \geq j > i > 0$ , the second order derivative
+
+$$
+\begin{array}{l} \nabla_ {\operatorname {v e c} \left(w _ {j}\right)} \nabla_ {\operatorname {v e c} \left(w _ {i}\right)} \mathcal {L} (Y, f (X)) = \\ \left(\left(W _ {\ell} \dots W _ {i + 1}\right) \otimes \left(W _ {i - 1} \dots W _ {2} W _ {1} X\right) ^ {T}\right) ^ {T} \left(\left(W _ {\ell} \dots W _ {j + 1}\right) \otimes \left(W _ {j - 1} \dots W _ {2} W _ {1} X\right) ^ {T}\right) + \\ \left(\left(W _ {j - 1} \dots W _ {i + 1}\right) \otimes \left(W _ {i - 1} \dots W _ {2} W _ {1} X\right)\right) (I \otimes \left(\left(f (X) - Y\right) W _ {\ell} \dots W _ {j + 1}\right)). \\ \end{array}
+$$
+
+When  $j = i$ , the second term equals 0. Based on the above expressions, we notice that the gradient norm and Hessian norm may be positively correlated due to the following two observations. First, the gradient and the Hessian share many components such as the matrix product of weights across layers. Second, if one naively upper bounds the norm using Cauchy-Schwarz, then both upper bounds would be monotonically increasing with respect to  $\| W_i\|$  and  $\| f(X) - Y\|$ .
+
+# 3 PROBLEMS SETUP AND ALGORITHMS
+
+In this section, we state the optimization problems and introduce gradient based algorithms for them that work under the new smoothness condition (2). Convergence analysis follows in Section 4.
+
+Recall that we wish to solve the nonconvex optimization problem  $\min_{x\in \mathbb{R}^d}f(x)$ . Since in general this problem is intractable, following common practice we also seek an  $\epsilon$ -stationary point, i.e., a point  $x$  such that  $\| \nabla f(x)\| \leq \epsilon$ . Furthermore, we make the following assumptions to regularize the function class studied and subsequently provide nonasymptotic convergence rate analysis.
+
+Assumption 1. The function  $f$  is lower bounded by  $f^{*} > -\infty$ .
+
+Assumption 2. The function  $f$  is twice differentiable.
+
+Assumption 3  $((L_0,L_1)$ -smoothness). The function  $f$  is  $(L_0,L_1)$ -smooth, i.e., there exist positive constants  $L_{0}$  and  $L_{1}$  such that  $\| \nabla^2 f(x)\| \leq L_0 + L_1\| \nabla f(x)\|$  see condition (2).
+
+The first assumption is standard. Twice differentiability in Assumption 2 can be relaxed to first-order differentiability by modifying the definition of  $(L_0, L_1)$ -smoothness as
+
+$$
+\operatorname * {l i m s u p} _ {\delta \to \bar {0}} \frac {\| \nabla f (x) - \nabla f (x + \delta) \|}{\| \delta \|} \leq L _ {1} \| \nabla f (x) \| + L _ {0}.
+$$
+
+The above inequality implies  $\nabla f(x)$  is locally Lipschitz, and hence almost everywhere differentiable. Therefore, all our results can go through by handling the integrations more carefully. But to avoid complications and simplify exposition, we assume that the function is twice differentiable.
+
+To further relax the global assumptions, by showing that GD and clipped GD are monotonically decreasing in function value, we require the above assumptions to hold just in a neighborhood determined by the sublevel set  $S^1$  for a given initialization  $x_0$ , where
+
+$$
+\mathcal {S} := \{x \mid \exists y \text {s u c h t h a t} f (y) \leq f \left(x _ {0}\right), \text {a n d} \| x - y \| \leq 1 \}. \tag {3}
+$$
+
+# 3.1 GRADIENT DESCENT ALGORITHMS
+
+In this section, we review a few well-known variants of gradient based algorithms that we analyze. We start with the ordinary gradient descent with a fixed step size  $\eta$ ,
+
+$$
+x _ {k + 1} = x _ {k} - \eta \nabla f (x _ {k}). \tag {4}
+$$
+
+This algorithm (pedantically, its stochastic version) is widely used in neural network training. Many modifications of it have been proposed to stabilize or accelerate training. One such technique of particular importance is clipped gradient descent, which performs the following updates:
+
+$$
+x _ {k + 1} = x _ {k} - h _ {c} \nabla f (x _ {k}), \quad \text {w h e r e} h _ {c} := \min  \left\{\eta_ {c}, \frac {\gamma \eta_ {c}}{\| \nabla f (x) \|} \right\}. \tag {5}
+$$
+
+Another algorithm that is less common in practice but has attracted theoretical interest is normalized gradient descent. The updates for normalized GD method can be written as
+
+$$
+x _ {k + 1} = x _ {k} - h _ {n} \nabla f (x _ {k}), \quad \text {w h e r e} h _ {n} := \frac {\eta_ {n}}{\| \nabla f (x) \| + \beta}. \tag {6}
+$$
+
+The stochastic version of the above algorithms replace the gradient with a stochastic estimator.
+
+We note that Clipped GD and NGD are almost equivalent. Indeed, for any given  $\eta_{n}$  and  $\beta$ , if we set  $\gamma \eta_{c} = \eta_{n}$  and  $\eta_{c} = \eta_{n} / \beta$ , then we have
+
+$$
+\frac {1}{2} h _ {c} \leq h _ {n} \leq 2 h _ {c}.
+$$
+
+Therefore, clipped GD is equivalent to NGD up to a constant factor in the step size choice. Consequently, the nonconvex convergence rates in Section 4 and Section 4.2 for clipped GD also apply to NGD. We omit repeating the theorem statements and the analysis for conciseness.
+
+# 4 THEORETICAL ANALYSIS
+
+In this section, we analyze the oracle complexities of GD and clipped GD under our relaxed smoothness condition. All the proofs are in the appendix. We highlight the key theoretical challenges that needed to overcome in Appendix B (e.g., due to absence of Lipschitz-smoothness, already the first-step of analysis, the so-called "descent lemma" fails).
+
+Since we are analyzing the global iteration complexity, let us recall the formal definition being used. We follow the notation from Carmon et al. (2017). For a deterministic sequence  $\{x_{k}\}_{k\in \mathbb{N}}$ , define the complexity of  $\{x_{k}\}_{k\in \mathbb{N}}$  for a function  $f$  as
+
+$$
+T _ {\epsilon} \left(\left\{x _ {t} \right\} _ {t \in \mathbb {N}}, f\right) := \inf  \left\{t \in \mathbb {N} | \| \nabla f \left(x _ {t}\right) \| \leq \epsilon \right\}. \tag {7}
+$$
+
+For a random process  $\{x_{k}\}_{k\in \mathbb{N}}$ , we define the complexity of  $\{x_{k}\}_{k\in \mathbb{N}}$  for function  $f$  as
+
+$$
+T _ {\epsilon} \left(\left\{x _ {t} \right\} _ {t \in \mathbb {N}}, f\right) := \inf  \left\{t \in \mathbb {N} | \operatorname {P r o b} \left(\| \nabla f \left(x _ {k}\right) \| \geq \epsilon \text {f o r a l l} k \leq t\right) \leq \frac {1}{2} \right\}. \tag {8}
+$$
+
+In particular, if the condition is never satisfied, then the complexity is  $\infty$ . Given an algorithm  $A_{\theta}$  where  $\theta$  denotes hyperparameters such as step size and momentum coefficient, we denote  $A_{\theta}[f,x_0]$  as the sequence of (potentially stochastic) iterates generated by  $A$  when operating on  $f$  with initialization  $x_0$ . Finally, we define the iteration complexity of an algorithm class parameterized by  $p$  hyperparameters,  $\mathcal{A} = \{A_{\theta}\}_{\theta \in \mathbb{R}^p}$  on a function class  $\mathcal{F}$  as
+
+$$
+\mathcal {N} (\mathcal {A}, \mathcal {F}, \epsilon) := \inf  _ {A _ {\theta} \in \mathcal {A}} \sup  _ {x _ {0} \in \mathbb {R} ^ {d}, f \in \mathcal {F}} T _ {\epsilon} \left(A _ {\theta} [ f, x _ {0} ], f\right). \tag {9}
+$$
+
+The definition in the stochastic setting simply replaces the expression (7) with the expression (8). In the rest of the paper, "iteration complexity" refers to the quantity defined above.
+
+# 4.1 CONVERGENCE IN THE DETERMINISTIC SETTING
+
+In this section, we present the convergence rates for GD and clipped GD under deterministic setting. We start by analyzing the clipped GD algorithm with update defined in equation (5).
+
+Theorem 3. Let  $\mathcal{F}$  denote the class of functions that satisfy Assumptions 1, 2, and 3 in set  $S$  defined in (3). Recall  $f^{*}$  is a global lower bound for function value. With  $\eta_{c} = \frac{1}{10L_{0}}$ ,  $\gamma = \min \left\{\frac{1}{\eta_{c}}, \frac{1}{10L_{1}\eta_{c}}\right\}$ , we can prove that the iteration complexity of clipped GD (Algorithm 5) is upper bounded by
+
+$$
+\frac {2 0 L _ {0} (f (x _ {0}) - f ^ {*})}{\epsilon^ {2}} + \frac {2 0 \max \{1 , L _ {1} ^ {2} \} (f (x _ {0}) - f ^ {*})}{L _ {0}}.
+$$
+
+The proof of Theorem 3 is included in Appendix C.
+
+Now, we discuss the convergence of vanilla GD. The standard GD is known to converge to first order  $\epsilon$ -stationary points in  $\mathcal{O}((L(f(x_0) - f^*))\epsilon^{-2})$  iterations for  $(L,0)$ -smooth nonconvex functions. By Theorem 1 of Carmon et al. (2017), this rate is up to a constant optimal.
+
+However, we will show below that gradient descent is suboptimal under our relaxed  $(L_0,L_1)$ -smoothness condition. In particular, to prove the convergence rate for gradient descent with fixed step size, we need to permit it benefit from an additional assumption on gradient norms.
+
+Assumption 4. Given an initialization  $x_0$ , we assume that
+
+$$
+M := \sup  \left\{\| \nabla f (x) \| \mid x \text {s u c h} f (x) \leq f \left(x _ {0}\right) \right\} <   \infty .
+$$
+
+This assumption is in fact necessary, as our next theorem reveals.
+
+Theorem 4. Let  $\mathcal{F}$  be the class of objectives satisfying Assumptions 1, 2, 3, and 4 with fixed constants  $L_0 \geq 1$ ,  $L_1 \geq 1$ ,  $M > 1$ . The iteration complexity for the fixed-step gradient descent algorithms parameterized by step size  $h$  is at least
+
+$$
+\frac {L _ {1} M (f (x _ {0}) - f ^ {*} - 5 \epsilon / 8)}{8 \epsilon^ {2} (\log M + 1)}.
+$$
+
+The proof can be found in Appendix D.
+
+Remark 5. Theorem 1 of Carmon et al. (2017) and Theorem 4 together show that gradient descent with a fixed step size cannot converge to an  $\epsilon$ -stationary point faster than  $\Omega\left((L_1M / \log (M) + L_0)(f(x_0) - f^*)\epsilon^{-2}\right)$ . Recall that clipped GD algorithm converges as  $\mathcal{O}\left(L_0(f(x_0) - f^*)\epsilon^{-2} + L_1^2 (f(x_0) - f^*)L_0^{-1}\right)$ . Therefore, clipped GD can be arbitrarily faster than GD when  $L_{1}M$  is large, or in other words, when the problem has a poor initialization.
+
+Below, we provide an iteration upper bound for the fixed-step gradient descent update (4).
+
+Theorem 6. Suppose assumptions 1, 2, 3 and 4 hold in set  $S$  defined in (3). If we pick parameters such that  $h = \frac{1}{(2(ML_1 + L_0))}$ , then we can prove that the iteration complexity of  $GD$  with a fixed step size defined in Algorithm 4 is upper bounded by
+
+$$
+4 (M L _ {1} + L _ {0}) (f (x _ {0}) - f ^ {*}) \epsilon^ {- 2}.
+$$
+
+Please refer to Appendix E for the proof. Theorem 6 shows that gradient descent with a fixed step size converges in  $\mathcal{O}((ML_1 + L_0)(f(x_0) - f^*) / \epsilon^2)$  iterations. This suggests that the lower bound in Remark 5 is tight up to a log factor in  $M$ .
+
+# 4.2 CONVERGENCE IN THE STOCHASTIC SETTING
+
+In the stochastic setting, we assume GD and clipped GD have access to an unbiased stochastic gradient  $\nabla \hat{f}(x)$  instead of the exact gradient  $\nabla f(x)$ . For simplicity, we denote  $g_k = \nabla \hat{f}(x_k)$  below. To prove convergence, we need the following assumption.
+
+Assumption 5. There exists  $\tau >0$ , such that  $\| \nabla \hat{f} (x) - \nabla f(x)\| \leq \tau$  almost surely.
+
+Bounded noise can be relaxed to sub-gaussian noise if the noise is symmetric. Furthermore, up to our knowledge, this is the first stochastic nonconvex analysis of adaptive methods that does not require the gradient norm  $\| \nabla f(x)\|$  to be bounded globally.
+
+The main result of this section is the following convergence guarantee for stochastic clipped GD (based on the stochastic version of the update (5)).
+
+Theorem 7. Let Assumptions 1-3 and 5 hold globally with  $L_{1} > 0$ . Let  $h = \min \left\{\frac{1}{16\eta L_{1}\left(\|g_{k}\| + \tau\right)}, \eta \right\}$  where  $\eta = \min \left\{\frac{1}{20L_{0}}, \frac{1}{128L_{1}\tau}, \frac{1}{\sqrt{T}}\right\}$ . Then we can show that iteration complexity for stochastic clipped GD after of update (5) is upper bounded by
+
+$$
+\Delta \max \{\frac {1 2 8 L _ {1}}{\epsilon}, \frac {4 \Delta}{\epsilon^ {4}}, \frac {8 0 L _ {0} + 5 1 2 L _ {1} \tau}{\epsilon^ {2}} \},
+$$
+
+where  $\Delta = (f(x_0) - f^* + (5L_0 + 2L_1\tau)\tau^2 + 9\tau L_0^2 / L_1)$ .
+
+In comparison, we have the following upper bound for ordinary SGD.
+
+Theorem 8. Let Assumptions 1-3, and 5 hold globally with  $L_{1} > 0$ . Let  $h = \min \left\{\frac{1}{\sqrt{T}}, \frac{1}{L_{1}(M + \tau)}\right\}$ . Then the iteration complexity for the stochastic version of  $GD$  (4) is upper bounded by
+
+$$
+\left(f (x _ {0}) - f ^ {*} + (5 L _ {0} + 4 L _ {1} M) (M + \tau) ^ {2}\right) ^ {2} \epsilon^ {- 4}.
+$$
+
+We cannot provide a lower bound for this algorithm. In fact, lower bound is not known for SGD even in the global smoothness setting. However, the deterministic lower bound in Theorem 4 is still valid, though probably loose. Therefore, the convergence of SGD still requires additional assumption and can again be arbitrarily slower compared to clipped SGD when  $M$  is large.
+
+![](images/74c6d049aa8b0eb5299bea8f64ff259953a3b50bc152c7af53373eb3d47f2325.jpg)  
+(a) Learning rate 30, with clipping.
+
+![](images/2da26e2c1168c1015050646dc92f1b5ec579144d3a28043f38d0c20e2bfc8f84.jpg)  
+(b) Learning rate 2, without clipping.
+
+![](images/f1d94e85189c932d2654dc77348cdc2103667b452e151599b1d95c40489d36de.jpg)  
+(c) Learning rate 2, with clipping.
+
+![](images/79b98ca380a880201fee72a24cb3e7248897a04b8fdf71d56817dff384d847f5.jpg)  
+Figure 2: Gradient norm vs smoothness on log scale for LM training. The dot color indicates the iteration number. Darker ones correspond to earlier iterations. Note that the spans of  $x$  and  $y$  axis are not fixed.  
+(a) SGD with momentum.  
+Figure 3: Gradient norm vs smoothness on log scale for ResNet20 training. The dot color indicates the iteration number.
+
+![](images/3a08710910df9d2901d578c240e1c789f6a9f29b894f84d506d8d7e6be25ea04.jpg)  
+(b) Learning rate 1, without clipping.
+
+![](images/1f5e8968e64ee8a839a3fb4bf4b2b3af1dd2c99b728150e1d2d58dca2a3900be.jpg)  
+(c) Learning rate 5, with clipping.
+
+# 5 EXPERIMENTS
+
+In this section, we summarize our empirical findings on the positive correlation between gradient norm and local smoothness. We then show that clipping accelerates convergence during neural network training. Our experiments are based on two tasks: language modeling and image classification. We run language modeling on the Penn Treebank (PTB) (Mikolov et al., 2010) dataset with AWD-LSTM models (Merit et al., 2018). We train ResNet20 (He et al., 2016) on the Cifar10 dataset (Krizhevsky and Hinton, 2009). Details about the smoothness estimation and experimental setups are in Appendix H. An additional synthetic experiment is discussed in Appendix I.
+
+First, our experiments test whether the local smoothness constant increases with the gradient norm, as suggested by the relaxed smoothness conditions defined in (2) (Section 2). To do so, we evaluate both quantities at points generated by the optimization procedure. We then scatter the local smoothness constants against the gradient norms in Figure 2 and Figure 3. Note that the plots are on a log-scale. A linear scale plot is shown in Appendix Figure 5.
+
+We notice that the correlation exists in the default training procedure for language modeling (see Figure 2a) but not in the default training for image classification (see Figure 3a). This difference aligns with the fact that gradient clipping is widely used in language modeling but is less popular in ResNet training, offering empirical support to our theoretical findings.
+
+We further investigate the cause of correlation. The plots in Figures 2 and 3 show that correlation appears when the models are trained with clipped GD and large learning rates. We propose the following explanation. Clipping enables the training trajectory to stably traverse non-smooth regions. Hence, we can observe that gradient norms and smoothness are positively correlated in Figures 2a and 3c. Without clipping, the optimizer has to adopt a small learning rate and stays in a region where local smoothness does not vary much, otherwise the sequence diverges, and a different learning rate is used. Therefore, in other plots of Figures 2 and 3, the correlation is much weaker.
+
+As positive correlations are present in both language modeling and image classification experiments with large step sizes, our next set of experiments checks whether clipping helps accelerate convergence as predicted by our theory. From Figure 4, we find that clipping indeed accelerates convergence. Because gradient clipping is a standard practice in language modeling, the LSTM models trained with clipping achieve the best validation performance and the fastest training loss conver
+
+![](images/5b6f332a966fca8182b7e30a1d55b4a5afeec29f85f9055e1a125f1508e7c322.jpg)  
+(a) Training loss of LSTM with different optimization parameters.  
+(b) Validation loss of LSTM with different optimization parameters.
+
+![](images/f1f12fc125b518049cabe1c668f87e6bbb9f964f1719c21ceb3c6a2430aa0e6e.jpg)  
+(c) Training loss of ResNet20 with different optimization parameters.  
+(d) Test accuracy of ResNet20 with different optimization parameters.  
+Figure 4: Training and validation loss obtained with different training methods for LSTM and ResNet training. The validation loss plots the cross entropy. The training loss additionally includes the weight regularization term. In the legend, 'lr30clip0.25' denotes that clipped SGD uses step size 30 and that the  $L_{2}$  norm of the stochastic gradient is clipped by 0.25. In ResNet training, we threshold the stochastic gradient norm at 0.25 when clipping is applied.
+
+gence as expected. For image classification, surprisingly, clipped GD also achieves the fastest convergence and matches the test performance of  $\mathrm{SGD + }$  momentum. These plots show that clipping can accelerate convergence and achieve good test performance at the same time.
+
+# 6 DISCUSSION
+
+Much progress has been made to close the gap between upper and lower oracle complexities for first order smooth optimization. The works dedicated to this goal provide important insights and tools for us to understand the optimization procedures. However, there is another gap that separates theoretically accelerated algorithms from empirically fast algorithms.
+
+Our work aims to close this gap. Specifically, we propose a relaxed smoothness assumption that is supported by empirical evidence. We analyze a simple but widely used optimization technique known as gradient clipping and provide theoretical guarantees that clipping can accelerate gradient descent. This phenomenon aligns remarkably well with empirical observations.
+
+There is still much to be explored in this direction. First, though our smoothness condition relaxes the usual Lipschitz assumption, it is unclear if there is an even better condition that also matches the experimental observations while also enabling a clean theoretical analysis. Second, we only study convergence of clipped gradient descent. Studying the convergence properties of other techniques such as momentum, coordinate-wise learning rates (more generally, preconditioning), and variance reduction is also interesting. Finally, the most important question is: "can we design fast algorithms based on relaxed conditions that achieve faster convergence in neural network training?"
+
+Our experiments also have noteworthy implications. First, though advocating clipped gradient descent in ResNet training is not a main point of this work, it is interesting to note that gradient
+
+descent and clipped gradient descent with large step sizes can achieve a similar test performance as momentum-SGD. Second, we learned that the performance of the baseline algorithm can actually beat some recently proposed algorithms. Therefore, when we design or learn about new algorithms, we need to pay extra attention to check whether the baseline algorithms are properly tuned.
+
+# 7 ACKNOWLEDGEMENT
+
+SS acknowledges support from an NSF-CAREER Award (Number 1846088) and an Amazon Research Award. AJ acknowledges support from an MIT-IBM-Exploratory project on adaptive, robust, and collaborative optimization.
+
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+
+# A MORE RELATED WORK ON ACCELERATING GRADIENT METHODS
+
+Variance reduction. Many efforts have been made to accelerate gradient-based methods. One elegant approach is variance reduction (e.g. Schmidt et al., 2017; Johnson and Zhang, 2013; Defazio et al., 2014; Bach and Moulines, 2013; Konečný and Richtárik, 2013; Xiao and Zhang, 2014; Gong and Ye, 2014; Fang et al., 2018; Zhou et al., 2018b). This technique aims to solve stochastic and finite sum problems by averaging the noise in the stochastic oracle via utilizing the smoothness of the objectives.
+
+Momentum methods. Another line of work focuses on achieving acceleration with momentum. Polyak (1964) showed that momentum can accelerate optimization for quadratic problems; later, Nesterov (1983) designed a variation that provably accelerates any smooth convex problems. Based on Nesterov's work, much theoretical progress was made to accelerate different variations of the original smooth convex problems (e.g. Ghadimi and Lan, 2016; 2012; Beck and Teboulle, 2009; Shalev-Shwartz and Zhang, 2014; Jin et al., 2018; Carmon et al., 2018; Allen-Zhu, 2017; Lin et al., 2015; Nesterov, 2012).
+
+Adaptive step sizes. The idea of varying step size in each iteration has long been studied. Armijo (1966) proposed the famous backtracking line search algorithm to choose step size dynamically. Polyak (1987) proposed a strategy to choose step size based on function suboptimality and gradient norm. More recently, Duchi et al. (2011) designed the Adagrad algorithm that can utilize the sparsity in stochastic gradients.
+
+Since 2018, there has been a surge in studying the theoretical properties of adaptive gradient methods. One starting point is (Reddi et al., 2019), which pointed out that ADAM is not convergent and proposed the AMSGrad algorithm to fix the problem. Ward et al. (2018); Li and Orabona (2018) prove that Adagrad converges to stationary point for nonconvex stochastic problems. Zhou et al. (2018a) generalized the result to a class of algorithms named Padam. Zou et al. (2018); Staib et al. (2019); Chen et al. (2018); Zhou et al. (2018c); Agarwal et al. (2018); Zhou et al. (2018b); Zou and Shen (2018) also studied different interesting aspects of convergence of adaptive methods. In addition, Levy (2016) showed that normalized gradient descent may have better convergence rate in presence of injected noise. However, the rate comparison is under dimension dependent setting. Hazan et al. (2015) studied the convergence of normalized gradient descent for quasi-convex functions.
+
+# B CHALLENGES IN THE PROOFS
+
+In this section, we highlight a few key challenges in our proofs. First, the analysis convergence under the relaxed smoothness condition is more difficult than the traditional setup. In particular, classical analyses based on Lipschitz-smooth gradients frequently exploit the descent condition:
+
+$$
+f (y) \leq f (x) + \left\langle \nabla f (x), y - x \right\rangle + \frac {L}{2} \| y - x \| ^ {2}. \tag {10}
+$$
+
+However, under our relaxed smoothness condition, the last term will increase exponentially in  $\| y - x \|^2$ . To solve this challenge, we bound the distance moved by clipping and apply Gronwall's inequality.
+
+Second, our algorithm specific lower bound proved in Theorem 4 is novel and tight up to a log factor. To our knowledge, the worst case examples used have not been studied before.
+
+Last, proving the convergence of adaptive methods in the nonconvex stochastic setting suffers from a fundamental challenge: the stochastic gradient is dependent on the update step size. This problem is usually circumvented by either assuming gradients have bounded norms or by using a lagging-by-one step-size to decouple the correlation. The situation is even worse under the relaxed smoothness assumption. In our case, we overcome this challenge by a novel analysis that divides the proof into the large gradient scenario and the small gradient scenario.
+
+# C PROOF OF THEOREM 3
+
+We start by proving a lemma that is repeatedly used in later proofs. The lemma bounds the gradient in a neighborhood of the current point by Gronwall's inequality (integral form).
+
+Lemma 9. Given  $x$  such that  $f(x) \leq f(x_0)$ , for any  $x^+$  such that  $\| x^+ - x \| \leq \min \{1 / L_1, 1\}$ , we have  $\| \nabla f(x^+) \| \leq 4(L_0 / L_1 + \| \nabla f(x) \|)$ .
+
+Remark 10. Note that the constant "1" comes from the definition of  $S$  in (3). If Assumption 3 holds globally, then we do not need to constrain  $\| x^{+} - x\| \leq 1$ . This version will be used in Theorem 7.
+
+Proof. Let  $\gamma(t)$  be a curve defined below,
+
+$$
+\gamma (t) = t \left(x ^ {+} - x\right) + x, t \in [ 0, 1 ].
+$$
+
+Then we have
+
+$$
+\nabla f (\gamma (t)) = \int_ {0} ^ {t} \nabla^ {(2)} f (\gamma (\tau)) (x ^ {+} - x) d \tau + \nabla f (\gamma (0)).
+$$
+
+By Cauchy-Schwarz's inequality, we get
+
+$$
+\begin{array}{l} \| \nabla f (\gamma (t)) \| \leq \| x ^ {+} - x \| \int_ {0} ^ {t} \| \nabla^ {(2)} f (\gamma (\tau)) \| d \tau + \| \nabla f (x) \| \\ \leq \frac {1}{L _ {1}} \int_ {0} ^ {t} \left(L _ {0} + L _ {1} \| \nabla f (\gamma (\tau)) \|\right) d \tau + \| \nabla f (x) \|. \\ \end{array}
+$$
+
+The second inequality follows by Assumption 3. Then we can apply the integral form of Gronwall's inequality and get
+
+$$
+\| \nabla f (\gamma (t)) \| \leq \frac {L _ {0}}{L _ {1}} + \| \nabla f (x) \| + \int_ {0} ^ {t} \left(\frac {L _ {0}}{L _ {1}} + \| \nabla f (x) \|\right) \exp (t - \tau) d \tau .
+$$
+
+The Lemma follows by setting  $t = 1$ .
+
+![](images/cfd75625718805ce14b871fe9b412223bc42e0b4b017cda80210d82a57463394.jpg)
+
+# C.1 PROOF OF THE THEOREM
+
+We parameterize the path between  $x_{k}$  and its updated iterate  $x_{k + 1}$  as follows:
+
+$$
+\gamma (t) = t \left(x _ {k + 1} - x _ {k}\right) + x _ {k}, \forall t \in [ 0, 1 ].
+$$
+
+Since  $x_{k + 1} = x_k - h_k\nabla f(x_k)$ , using Taylor's theorem, the triangle inequality, and Cauchy-Schwarz, we obtain
+
+$$
+f (x _ {k + 1}) \leq f (x _ {k}) - h _ {k} \| \nabla f (x _ {k}) \| ^ {2} + \frac {\| x _ {k + 1} - x _ {k} \| ^ {2}}{2} \int_ {0} ^ {1} \| \nabla^ {2} f (\gamma (t)) \| d t.
+$$
+
+Since
+
+$$
+h _ {k} \leq \frac {\gamma \eta}{\| \nabla f (x) \|} \leq \min  \left\{\frac {1}{\| \nabla f (x) \|}, \frac {1}{L _ {1} \| \nabla f (x _ {k}) \|} \right\},
+$$
+
+we know by Lemma 9
+
+$$
+\| \nabla f (\gamma (t) \| \leq 4 \left(\frac {L _ {0}}{L _ {1}} + \| \nabla f (x) \|\right).
+$$
+
+Then by Assumption 3, we obtain the "descent inequality":
+
+$$
+f (x _ {k + 1}) \leq f (x _ {k}) - h _ {k} \| \nabla f (x _ {k}) \| ^ {2} + \frac {5 L _ {0} + 4 L _ {1} \| \nabla f (x _ {k}) \|}{2} \| \nabla f (x _ {k}) \| ^ {2} h _ {k} ^ {2}.
+$$
+
+Therefore, as long as  $h_k \leq 1 / (5L_0 + 4L_1\| \nabla f(x_k)\|)$  (which follows by our choice of  $\eta, \gamma$ ), we can quantify the descent to be
+
+$$
+f (x _ {k + 1}) \leq f (x _ {k}) - \frac {h _ {k} \| \nabla f (x _ {k}) \| ^ {2}}{2}.
+$$
+
+When  $\| \nabla f(x_k)\| \geq L_0 / L_1$ , we have
+
+$$
+\frac {h _ {k} \| \nabla f (x _ {k}) \| ^ {2}}{2} \geq \frac {L _ {0}}{2 0 \max  \{1 , L _ {1} ^ {2} \}}.
+$$
+
+When  $\epsilon \leq \| \nabla f(x_k)\| \leq L_0 / L_1$ , we have
+
+$$
+\frac {h _ {k} \| \nabla f (x _ {k}) \| ^ {2}}{2} \geq \frac {\| \nabla f (x _ {k}) \| ^ {2}}{2 0 L _ {0}} \geq \frac {\epsilon^ {2}}{2 0 L _ {0}}.
+$$
+
+Therefore,
+
+$$
+f (x _ {k + 1}) \leq f (x _ {k}) - \min \left\{\frac {L _ {0}}{2 0 \max \{1 , L _ {1} ^ {2} \}}, \frac {\epsilon^ {2}}{2 0 L _ {0}} \right\}.
+$$
+
+Assume that  $\epsilon \leq \| \nabla f(x_k)\|$  for  $k\leq T$  iterations. By doing a telescopic sum, we get
+
+$$
+\sum_ {k = 0} ^ {T - 1} f (x _ {k + 1}) - f (x _ {k}) \leq - T \min \left\{\frac {L _ {0}}{2 0 \max \{1 , L _ {1} ^ {2} \}}, \frac {\epsilon^ {2}}{2 0 L _ {0}} \right\}.
+$$
+
+Rearranging we get
+
+$$
+T \leq \frac {2 0 L _ {0} (f (x _ {0}) - f ^ {*})}{\epsilon^ {2}} + \frac {2 0 \max \{1 , L _ {1} ^ {2} \} (f (x _ {0}) - f ^ {*})}{L _ {0}}.
+$$
+
+# D PROOF OF THEOREM 4
+
+We will prove a lower bound for the iteration complexity of GD with fixed step size. The high level idea is that if GD converges for all functions satisfying the assumptions, then the step size needs to be small. However, this small step size will lead to very slow convergence for another function.
+
+Recall that the fixed step size GD algorithm is parameterized by the scalar: step size  $h$ . First, we show that when  $h > \frac{2\log(M) + 2}{ML_1}$ ,
+
+$$
+\sup  T _ {\epsilon} \left(A _ {h} [ f, x _ {0} ], f\right) = \infty
+$$
+
+$$
+\begin{array}{c} x _ {0} \in \mathbb {R} ^ {d}, \\ f \in \mathcal {F} \end{array}
+$$
+
+We start with a function that grows exponentially. Let  $L_{1} > 1$ ,  $M > 1$  be fixed constants. Pick the initial point  $x_0 = (\log (M) + 1) / L_1$ . Let the objective be defined as follows,
+
+$$
+f (x) = \left\{ \begin{array}{l l} \frac {e ^ {- L _ {1} x}}{L _ {1} e}, & \text {f o r} x <   - \frac {1}{L _ {1}}, \\ \frac {L _ {1} x ^ {2}}{2} + \frac {1}{2 L _ {1}}, & \text {f o r} x \in [ - \frac {1}{L _ {1}}, \frac {1}{L _ {1}} ], \\ \frac {e ^ {L _ {1} x}}{L _ {1} e}, & \text {f o r} x > \frac {1}{L _ {1}}. \end{array} \right.
+$$
+
+We notice that the function satisfies the assumptions with constants
+
+$$
+L _ {0} = 1, \quad L _ {1} > 1, \quad M > 1. \tag {11}
+$$
+
+When  $h > 2x_0 / M$ , we would have  $|x_{1}| > |x_{0}|$ . By symmetry of the function and the super-linear growth of the gradient norm, we know that the iterates will diverge. Hence, in order for gradient descent with a fixed step size  $h$  to converge,  $h$  must be small enough. Formally,
+
+$$
+h \leq \frac {2 x _ {0}}{M} = \frac {2 \log (M) + 2}{M L _ {1}}.
+$$
+
+Second, we show that when  $h \leq \frac{2\log(M) + 2}{ML_1}$ ,
+
+$$
+\sup  T _ {\epsilon} \left(A _ {h} [ f, x _ {0} ], f\right) \geq \Delta L _ {1} M / \left(4 \epsilon^ {2} (\log M + 1)\right)
+$$
+
+$$
+\begin{array}{c} x _ {0} \in \mathbb {R} ^ {d}, \\ f \in \mathcal {F} \end{array}
+$$
+
+Now, let's look at a different objective that grows slowly.
+
+$$
+f (x) = \left\{ \begin{array}{l l} - 2 \epsilon (x + 1) + \frac {5 \epsilon}{4}, & \text {f o r} x <   - 1, \\ \frac {\epsilon}{4} (6 x ^ {2} - x ^ {4}), & \text {f o r} x \in [ - 1, 1 ], \\ 2 \epsilon (x - 1) + \frac {5 \epsilon}{4}, & \text {f o r} x > 1. \end{array} \right.
+$$
+
+This function is also second order differentiable and satisfies the assumptions with constants in (11). If we set  $x_0 = 1 + \Delta/\epsilon$  for some constant  $\Delta > 0$ , we know that  $f(x_0) - f^* = 2\Delta + 5\epsilon/4$ . With the step size choice  $h \leq (2\log M + 2)/(ML_1)$ , we know that in each step,  $x_{k+1} \geq x_k - (4\epsilon(\log M + 1))/(L_1M)$ . Therefore, for  $k \leq \Delta L_1M/(4\epsilon^2(\log M + 1))$ ,
+
+$$
+\| \nabla f (x _ {k}) \| = 2 \epsilon .
+$$
+
+After combining these two points, we proved the theorem by definition (9).
+
+# E PROOF OF THEOREM 6
+
+We start by parametrizing the function value along the update,
+
+$$
+f (\gamma (t)) := f \left(x _ {k} - t h \nabla f \left(x _ {k}\right)\right), t \in [ 0, 1 ].
+$$
+
+Note that with this parametrization, we have  $\gamma(0) = x_k$ ,  $\gamma(1) = x_{k+1}$ . Now we would like to argue that if  $f(x_k) \leq f(x_0)$ , then  $\|\nabla f(x(t))\| \leq M, \forall t \leq 1$ . Assume by contradiction that this is not true. Then there exists  $\epsilon > 0, t \in [0,1]$  such that  $\|\nabla f(x(t))\| \geq M + \epsilon$ . Since  $\epsilon$  can be made arbitrarily small below a threshold, we assume  $\epsilon < M$ . Denote
+
+$$
+t ^ {*} = \inf  \{t \mid \| \nabla f (x (t)) \| \geq M + \epsilon \}.
+$$
+
+The value  $t^*$  exists by continuity of  $\| \nabla f(x(t)) \|$  as a function of  $t$ . Then we know by Assumption 4 that  $f(x(t^*)) > f(x_k)$ . However, by Taylor expansion, we know that
+
+$$
+\begin{array}{l} f (x (t ^ {*})) \leq f (x _ {k}) - t h \| \nabla f (x _ {k}) \| ^ {2} + (t h) ^ {2} \| \nabla f (x _ {k}) \| ^ {2} \int_ {0} ^ {t} \| \nabla^ {(2)} f (x (\tau)) \| d \tau \\ \leq f \left(x _ {k}\right) - t h \| \nabla f \left(x _ {k}\right) \| ^ {2} + (t h) ^ {2} \| \nabla f \left(x _ {k}\right) \| ^ {2} \left(L _ {1} (M + \epsilon) + L _ {0}\right) \\ \leq f (x _ {k}). \\ \end{array}
+$$
+
+The last inequality follows by  $h = 1 / (2(ML_1 + L_0))$ . Hence we get a contradiction and conclude that for all  $t \leq 1$ ,  $\| \nabla f(x(t)) \| \leq M$ . Therefore, following the above inequality and Assumption 3, we get
+
+$$
+\begin{array}{l} f (x _ {k + 1}) \leq f (x _ {k}) - h \| \nabla f (x _ {k}) \| ^ {2} + h ^ {2} \frac {L _ {1} M + L _ {0}}{2} \| \nabla f (x _ {k}) \| ^ {2} \\ \leq f (x _ {k}) - \frac {\epsilon^ {2}}{4 (M L _ {1} + L _ {0})}. \\ \end{array}
+$$
+
+The conclusion follows by the same argument as in Theorem 3 via a telescopic sum over  $k$ .
+
+# F PROOF OF THEOREM 7
+
+Recall that we set the following parameters
+
+$$
+h _ {k} = \min  \left\{\frac {1}{1 6 \eta L _ {1} \left(\left\| g _ {k} \right\| + \tau\right)}, \eta \right\}
+$$
+
+$$
+\eta = \min  \left\{\frac {1}{2 0 L _ {0}}, \frac {1}{1 2 8 L _ {1} \tau}, \frac {1}{\sqrt {T}} \right\} \tag {12}
+$$
+
+Similar to proof of Theorem 3, we have
+
+$$
+\begin{array}{l} \mathbb {E} [ f (x _ {k + 1}) | ] \leq f (x _ {k}) - \mathbb {E} [ h _ {k} \langle g _ {k}, \nabla f (x _ {k}) \rangle ] + \frac {5 L _ {0} + 4 L _ {1} \| \nabla f (x _ {k}) \|}{2} \mathbb {E} [ h _ {k} ^ {2} \| g _ {k} \| ^ {2} ] \tag {13} \\ \leq f (x _ {k}) - \mathbb {E} [ h _ {k} \langle g _ {k}, \nabla f (x _ {k}) \rangle ] + \frac {5 L _ {0} + 4 L _ {1} \| \nabla f (x _ {k}) \|}{2} \mathbb {E} [ h _ {k} ^ {2} (\| \nabla f (x _ {k}) \| ^ {2} \\ \left. + \left\| g _ {k} - \nabla f (x _ {k}) \right\| ^ {2} + 2 \langle \nabla f (x _ {k}), g _ {k} - \nabla f (x _ {k}) \rangle\right) ] \\ \leq f (x _ {k}) + \mathbb {E} \left[ - h _ {k} + \frac {5 L _ {0} + 4 L _ {1} \| \nabla f (x _ {k}) \|}{2} h _ {k} ^ {2} \right] \| \nabla f (x _ {k}) \| ^ {2} \\ + \mathbb {E} [ h _ {k} (- 1 + (5 L _ {0} + 4 L _ {1} \| \nabla f (x _ {k}) \|) h _ {k}) \langle \nabla f (x _ {k}), g _ {k} - \nabla f (x _ {k}) \rangle ] \\ + \frac {5 L _ {0} + 4 L _ {1} \| \nabla f (x _ {k}) \|}{2} \mathbb {E} \left[ h _ {k} ^ {2} \left(\| g _ {k} - \nabla f (x _ {k}) \| ^ {2}\right) \right] \\ \end{array}
+$$
+
+First we show  $(5L_0 + 4L_1\| \nabla f(x_k)\|)h_k\leq \frac{1}{2}$ . This follows by  $5L_{0}h_{k}\leq \frac{1}{4},h_{k}4L_{1}\| \nabla f(x_{k})\| \leq h_{k}4L_{1}(\| g_{k}\| +\tau)\leq \frac{1}{4}$ . Substitute in (15) and we get
+
+$$
+\begin{array}{l} \mathbb {E} [ f (x _ {k + 1}) | ] \leq f (x _ {k}) + \mathbb {E} \left[ - \frac {3 h _ {k}}{4} \right] \| \nabla f (x _ {k}) \| ^ {2} \tag {14} \\ + \underbrace {\mathbb {E} [ - h _ {k} \langle \nabla f (x _ {k}) , g _ {k} - \nabla f (x _ {k}) \rangle ]} _ {T _ {1}} \\ + \underbrace {\mathbb {E} \left[ \left(5 L _ {0} + 4 L _ {1} \| \nabla f (x _ {k}) \|\right) h _ {k} ^ {2} \langle \nabla f (x _ {k}) , g _ {k} - \nabla f (x _ {k}) \rangle \right]} _ {T _ {2}} \\ + \underbrace {\frac {5 L _ {0} + 4 L _ {1} \| \nabla f (x _ {k}) \|}{2} \mathbb {E} [ h _ {k} ^ {2} (\| g _ {k} - \nabla f (x _ {k}) \| ^ {2}) ]} _ {T _ {3}} \\ \end{array}
+$$
+
+Then we bound  $T_{1}, T_{2}, T_{3}$  in Lemma 11,12,13 and get
+
+$$
+\begin{array}{l} \mathbb {E} [ f (x _ {k + 1}) | ] \leq f (x _ {k}) + \mathbb {E} \left[ - \frac {h _ {k}}{4} \right] \| \nabla f (x _ {k}) \| ^ {2} \tag {15} \\ + \left(5 L _ {0} + 2 L _ {1} \tau\right) \eta^ {2} \tau^ {2} + 9 \eta^ {2} \tau L _ {0} ^ {2} / L _ {1} \\ \end{array}
+$$
+
+Rearrange and do a telescopic sum, we get
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \sum_ {k \leq T} \frac {h _ {k}}{4} \| \nabla f (x _ {k}) \| ^ {2} \right] \leq f (x _ {0}) - f ^ {*} + \eta^ {2} T ((5 L _ {0} + 2 L _ {1} \tau) \tau^ {2} + 9 \tau L _ {0} ^ {2} / L _ {1}) \\ \leq f \left(x _ {0}\right) - f ^ {*} + \left(\left(5 L _ {0} + 2 L _ {1} \tau\right) \tau^ {2} + 9 \tau L _ {0} ^ {2} / L _ {1}\right) \\ \end{array}
+$$
+
+Furthermore, we know
+
+$$
+\begin{array}{l} h _ {k} \| \nabla f _ {k} \| ^ {2} = \min  \{\eta , \frac {1}{1 6 L _ {1} (\| \nabla f _ {k} \| + \tau)} \} \| \nabla f _ {k} \| ^ {2} \\ \geq \min  \left\{\eta , \frac {1}{3 2 L _ {1} \| \nabla f _ {k} \|}, \frac {1}{3 2 L _ {1} \tau} \right\} \| \nabla f _ {k} \| ^ {2} \\ \geq \min \{\eta , \frac {1}{3 2 L _ {1} \| \nabla f _ {k} \|} \} \| \nabla f _ {k} \| ^ {2} \\ \end{array}
+$$
+
+Hence along with  $\eta \leq T^{-1 / 2}$ , we get
+
+$$
+\mathbb {E} \left[ \sum_ {k \leq T} \min  \left\{\eta \| \nabla f _ {k} \| ^ {2}, \frac {\| \nabla f _ {k} \|}{3 2 L _ {1}} \right\} \right] \leq f (x _ {0}) - f ^ {*} + ((5 L _ {0} + 2 L _ {1} \tau) \tau^ {2} + 9 \tau L _ {0} ^ {2} / L _ {1})
+$$
+
+Let  $\mathcal{U} = \{k|\eta \| \nabla f_k\| ^2\leq \frac{\|\nabla f_k\|}{16L_1}\}$ , we know that
+
+$$
+\mathbb {E} \left[ \sum_ {k \in \mathcal {U}} \eta \| \nabla f _ {k} \| ^ {2} \right] \leq f (x _ {0}) - f ^ {*} + ((5 L _ {0} + 2 L _ {1} \tau) \tau^ {2} + 9 \tau L _ {0} ^ {2} / L _ {1}),
+$$
+
+and
+
+$$
+\mathbb {E} \left[ \sum_ {k \in \mathcal {U} ^ {c}} \frac {\| \nabla f _ {k} \|}{3 2 L _ {1}} \right] \leq f (x _ {0}) - f ^ {*} + \left(\left(5 L _ {0} + 2 L _ {1} \tau\right) \tau^ {2} + 9 \tau L _ {0} ^ {2} / L _ {1}\right).
+$$
+
+Therefore,
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \min  _ {k} \| \nabla f (x _ {k}) \| \right] \leq \mathbb {E} \left[ \min  \left\{\frac {1}{| \mathcal {U} |} \sum_ {k \in \mathcal {U}} \| \nabla f _ {k} \| \right], \frac {1}{| \mathcal {U} ^ {c} |} \sum_ {k \in \mathcal {U} ^ {c}} \| \nabla f _ {k} \| \right\} ] \\ \leq \mathbb {E} [ \min  \{\sqrt {\frac {1}{| \mathcal {U} |} \sum_ {k \in \mathcal {U}} \| \nabla f _ {k} \| ^ {2}}, \frac {1}{| \mathcal {U} ^ {c} |} \sum_ {k \in \mathcal {U} ^ {c}} \| \nabla f _ {k} \| \} ] \\ \leq \max  \{\sqrt {f (x _ {0}) - f ^ {*} + ((5 L _ {0} + 2 L _ {1} \tau) \tau^ {2} + 9 \tau L _ {0} ^ {2} / L _ {1}) \frac {\sqrt {T} + 2 0 L _ {0} + 1 2 8 L _ {1} \tau}{T}}, \\ \left(f \left(x _ {0}\right) - f ^ {*} + \left(5 L _ {0} + 2 L _ {1} \tau\right) \tau^ {2} + 9 \tau L _ {0} ^ {2} / L _ {1}\right) \frac {6 4 L _ {1}}{T} \}. \\ \end{array}
+$$
+
+The last inequality follows by the fact that either  $|\mathcal{U}| \geq T/2$  or  $|\mathcal{U}^c| \geq T/2$ . This implies that  $\mathbb{E}[\min_{k \leq T} \|\nabla f(x_k)\|] \leq 2\epsilon$  when
+
+$$
+T \geq \Delta \max \{\frac {1 2 8 L _ {1}}{\epsilon}, \frac {4 \Delta}{\epsilon^ {4}}, \frac {8 0 L _ {0} + 5 1 2 L _ {1} \tau}{\epsilon^ {2}} \},
+$$
+
+where  $\Delta = (f(x_0) - f^* + (5L_0 + 2L_1\tau)\tau^2 + 9\tau L_0^2 / L_1)$ . By Markov inequality,
+
+$$
+\mathbb {P} \left\{\min  _ {k \leq T} \| \nabla f (x _ {k}) \| \leq \epsilon \right\} \geq \frac {1}{2}.
+$$
+
+The theorem follows by the definition in (9).
+
+# F.1 TECHNICAL LEMMAS
+
+Lemma 11.
+
+$$
+\mathbb {E} \left[ - h _ {k} \langle \nabla f (x _ {k}), g _ {k} - \nabla f (x _ {k}) \rangle \right] \leq \frac {1}{4} \mathbb {E} \left[ h _ {k} \right] \| \nabla f (x _ {k}) \| ^ {2}.
+$$
+
+Proof. By unbiasedness of  $g_{k}$  and the fact that  $\eta$  is a constant, we have
+
+$$
+\begin{array}{l} \mathbb {E} \left[ - h _ {k} \langle \nabla f (x _ {k}), g _ {k} - \nabla f (x _ {k}) \rangle \right] = \mathbb {E} \left[ (\eta - h _ {k}) \langle \nabla f (x _ {k}), g _ {k} - \nabla f (x _ {k}) \rangle \right] \\ = \mathbb {E} \left[ \left(\eta - h _ {k}\right) \langle \nabla f (x _ {k}), g _ {k} - \nabla f (x _ {k}) \rangle \mathbb {1} _ {\left\{\| g _ {k} \| \geq \frac {1}{1 6 L _ {1} \eta} - \tau \right\}} \right] \\ \leq \eta \| \nabla f (x _ {k}) \| \mathbb {E} [ \| g _ {k} - \nabla f (x _ {k}) \| \mathbb {1} _ {\{\| g _ {k} \| \geq \frac {1}{1 6 L _ {1} \eta} - \tau \}} ] \\ \leq \eta \| \nabla f (x _ {k}) \| ^ {2} 3 2 L _ {1} \mathbb {E} [ h _ {k} ] \tau \\ \end{array}
+$$
+
+The second last inequality follows by  $h_k \leq \eta$  and Cauchy-Schwartz inequality. The last inequality follows by
+
+$$
+\left\| \nabla f \left(x _ {k}\right) \right\| \geq \left\| g _ {k} \right\| - \tau = \frac {1}{1 6 L _ {1} h _ {k}} - 2 \tau \geq \frac {1}{1 6 L _ {1} h _ {k}} - \frac {1}{3 2 L _ {1} \eta} \geq \frac {1}{3 2 L _ {1} h _ {k}}. \tag {16}
+$$
+
+The equality above holds because  $h_k = \frac{1}{16\eta L_1(\|g_k\| + \tau)}$ . The lemma follows by  $32\eta L_1\tau \leq 1/4$ .
+
+Lemma 12.
+
+$$
+\mathbb {E} \left[ \left(5 L _ {0} + 4 L _ {1} \| \nabla f (x _ {k}) \|\right) h _ {k} ^ {2} \langle \nabla f (x _ {k}), g _ {k} - \nabla f (x _ {k}) \rangle \right] \leq 9 \eta^ {2} \tau L _ {0} ^ {2} / L _ {1} + \frac {1}{8} \mathbb {E} \left[ h _ {k} \right] \| \nabla f (x _ {k}) \| ^ {2}.
+$$
+
+Proof. When  $\| \nabla f(x_k)\| \geq L_0 / L_1$
+
+$$
+\begin{array}{l} \mathbb {E} \left[ \left(5 L _ {0} + 4 L _ {1} \| \nabla f (x _ {k}) \|\right) h _ {k} ^ {2} \langle \nabla f (x _ {k}), g _ {k} - \nabla f (x _ {k}) \rangle \right] \leq 9 L _ {1} \| \nabla f (x _ {k}) \| \mathbb {E} \left[ h _ {k} ^ {2} \right] \| \nabla f (x _ {k}) \| \tau \\ \leq \frac {1}{8} \mathbb {E} [ h _ {k} ] \| \nabla f (x _ {k}) \| ^ {2} \\ \end{array}
+$$
+
+The last inequality follows by (12).
+
+When  $\| \nabla f(x_k)\| \leq L_0 / L_1$
+
+$$
+\mathbb {E} \left[ \left(5 L _ {0} + 4 L _ {1} \| \nabla f (x _ {k}) \|\right) h _ {k} ^ {2} \langle \nabla f (x _ {k}), g _ {k} - \nabla f (x _ {k}) \rangle \right] \leq 9 \eta^ {2} \tau L _ {0} ^ {2} / L _ {1}
+$$
+
+Lemma 13.
+
+$$
+\frac {5 L _ {0} + 4 L _ {1} \| \nabla f (x _ {k}) \|}{2} \mathbb {E} [ h _ {k} ^ {2} (\| g _ {k} - \nabla f (x _ {k}) \| ^ {2}) ] \leq (5 L _ {0} + 2 L _ {1} \tau) \eta^ {2} \tau^ {2} + \frac {1}{8} \| \nabla f (x _ {k}) \| ^ {2} \mathbb {E} [ h _ {k} ].
+$$
+
+Proof. When  $\| \nabla f(x_k) \| \geq L_0 / L_1 + \tau$ , we get
+
+$$
+\frac {5 L _ {0} + 4 L _ {1} \| \nabla f (x _ {k}) \|}{2} \mathbb {E} [ h _ {k} ^ {2} (\| g _ {k} - \nabla f (x _ {k}) \| ^ {2}) ] \leq 5 L _ {1} \| \nabla f (x _ {k}) \| ^ {2} \mathbb {E} [ h _ {k} ] \eta \tau \leq \frac {1}{8} \| \nabla f (x _ {k}) \| ^ {2} \mathbb {E} [ h _ {k} ].
+$$
+
+The first inequality follows by  $h_k \leq \eta$  and  $\| g_k - \nabla f(x_k) \| \leq \tau \leq \| \nabla f(x_k) \|$ . The last inequality follows by (12).
+
+When  $\| \nabla f(x_k)\| \leq L_0 / L_1 + \tau$ , we get
+
+$$
+\frac {5 L _ {0} + 4 L _ {1} \| \nabla f (x _ {k}) \|}{2} \mathbb {E} [ h _ {k} ^ {2} (\| g _ {k} - \nabla f (x _ {k}) \| ^ {2}) ] \leq (5 L _ {0} + 2 L _ {1} \tau) \eta^ {2} \tau^ {2}.
+$$
+
+# G PROOF OF THEOREM 8
+
+Similar to proof of Theorem 3, we have
+
+$$
+\begin{array}{l} \mathbb {E} \big [ f (x _ {k + 1}) | \big ] \leq f (x _ {k}) - \mathbb {E} \big [ h _ {k} \langle g _ {k}, \nabla f (x _ {k}) \rangle \big ] + \frac {5 L _ {0} + 4 L _ {1} \| \nabla f (x _ {k}) \|}{2} \mathbb {E} \big [ h _ {k} ^ {2} \| g _ {k} \| ^ {2} \big ] \\ \leq f (x _ {k}) - \frac {1}{\sqrt {T}} \| \nabla f (x _ {k}) \| ^ {2} + \frac {5 L _ {0} + 4 L _ {1} M (M + \tau) ^ {2}}{2 T} \\ \end{array}
+$$
+
+Sum across  $k \in \{0, \dots, T - 1\}$  and take expectations, then we can get
+
+$$
+0 \leq f (x _ {0}) - \mathbb {E} [ f (x _ {T}) ] - \frac {1}{\sqrt {T}} \sum_ {k = 1} ^ {T} \mathbb {E} \left[ \| \nabla f (x _ {k}) \| ^ {2} \right] + \frac {5 L _ {0} + 4 L _ {1} M (M + \tau) ^ {2}}{2}
+$$
+
+Rearrange and we get
+
+$$
+\frac {1}{T} \sum_ {k = 1} ^ {T} \mathbb {E} \left[ \| \nabla f (x _ {k}) \| ^ {2} \right] \leq \frac {1}{\sqrt {T}} \left(f (x _ {0}) - f ^ {*} + \frac {5 L _ {0} + 4 L _ {1} M (M + \tau) ^ {2}}{2}\right)
+$$
+
+By Jensen's inequality,
+
+$$
+\frac {1}{T} \sum_ {k = 1} ^ {T} \mathbb {E} [ \| \nabla f (x _ {k}) \| ] \leq \sqrt {\frac {1}{\sqrt {T}} \left(f (x _ {0}) - f ^ {*} + \frac {5 L _ {0} + 4 L _ {1} M (M + \tau) ^ {2}}{2}\right)}
+$$
+
+By Markov inequality,
+
+$$
+\mathbb {P} \left\{\frac {1}{T} \sum_ {k = 1} ^ {T} \left[ \| \nabla f (x _ {k}) \| ^ {2} \right] > \frac {2}{\sqrt {T}} \left(f (x _ {0}) - f ^ {*} + \frac {5 L _ {0} + 4 L _ {1} M (M + \tau) ^ {2}}{2}\right) \right\} \leq 0. 5
+$$
+
+The theorem follows by the definition in (9) and Jensen's inequality.
+
+# H EXPERIMENT DETAILS
+
+In this section, we first briefly overview the tasks and models used in our experiment. Then we explain how we estimate smoothness of the function. Lastly, we describe some details for generating the plots in Figure 2 and Figure 3.
+
+# H.1 LANGUAGE MODELLING
+
+Clipped gradient descent was introduced in (vanilla) recurrent neural network (RNN) language model (LM) (Mikolov et al., 2010) training to alleviate the exploding gradient problem, and has been used in more sophisticated RNN models (Hochreiter and Schmidhuber, 1997) or seq2seq models for language modelling or other NLP applications (Sutskever et al., 2014; Cho et al., 2014). In this work we experiment with LSTM LM (Sundermeyer et al., 2012), which has been an important building block for many popular NLP models (Young et al., 2017).
+
+The task of language modelling is to model the probability of the next word  $w_{t+1}$  based on word history (or context). Given a document of length  $T$  (words) as training data, the training objective is to minimize negative log-likelihood of the data  $\frac{-1}{T}\Sigma_{t=1}^{T}\log P(w_t|w_1\ldots w_{t-1})$ .
+
+We run LM experiments on the Penn Treebank (PTB) (Mikolov et al., 2010) dataset, which has been a popular benchmark for language modelling. It has a vocabulary of size  $10\mathrm{k}$ , and  $887\mathrm{k} / 70\mathrm{k} / 78\mathrm{k}$  words for training/validation/testing.
+
+To train the LSTM LM, we follow the training recipe from  $^{3}$  (Merit et al., 2018). The model is a 3-layer LSTM LM with hidden size of 1150 and embedding size of 400. Dropout (Srivastava et al., 2014) of rate 0.4 and DropConnect (Wan et al., 2013) of rate 0.5 is applied. For optimization, clipped SGD with clip value of 0.25 and a learning rate of 30 is used, and the model is trained for 500 epochs. After training, the model reaches a text-set perplexity of 56.5, which is very close to the current state-of-art result (Dai et al., 2019) on the PTB dataset.
+
+# H.2 IMAGE CLASSIFICATION
+
+As a comparison, we run the same set of experiments on image classification tasks. We train the ResNet20 (He et al., 2016) model on Cifar10 (Krizhevsky and Hinton, 2009) classification dataset. The dataset contains 50k training images and 10k testing images in 10 classes.
+
+Unless explicitly state, we use the standard hyper-parameters based on the Github repository<sup>4</sup>. Our baseline algorithm runs SGD momentum with learning rate 0.1, momentum 0.9 for 200 epochs. We choose weight decay to be  $5e - 4$ . The learning rate is reduced by 10 at epoch 100 and 150. Up to our knowledge, this baseline achieves the best known test accuracy  $(95.0\%)$  for Resnet20 on Cifar10. The baseline already beats some recently proposed algorithms which claim to improve upon SGD momentum.
+
+![](images/ac1a4611f0f8b2dbf753a620b4528f6308a42797b819e252fa30ee75c2047b67.jpg)  
+(a) Figure 2a plotted on a linear scale.
+
+![](images/a04f4fdb96195e488e71af7afc7a05c5c12c0dc82da2b7f04e250192ba3931c5.jpg)  
+(b) Figure 2a with 200 epochs.  
+Figure 5: Auxiliary plots for Figure 2a. The left subfigure shows the values scattered on linear scale. The right subfigure shows more data points from 200 epochs.
+
+![](images/42d2e5578560ed00f468ee36445019e0ea78f2bbd6e17da55e0e8a7e9e2d7a68.jpg)  
+(a) Estimation for gradient norm.
+
+![](images/1cc8f87f1566241f2fd93a2751a1a8d10664748e90182ab21b48954d19933b76.jpg)  
+(b) Estimation for local smoothness.  
+Figure 6: Estimated gradient norm and smoothness using  $10\%$  data versus all data. The values are computed from checkpoints of the LSTM LM model in the first epoch. This shows that statistics evaluated from  $10\%$  of the entire dataset provides accurate estimation.
+
+# H.3 ESTIMATING SMOOTHNESS
+
+Our smoothness estimator follows a similar implementation as in (Santurkar et al., 2018). More precisely, given a sequence of iterates generated by training procedure  $\{x_{k}\}_{k}$ , we estimate the smoothness  $\hat{L}(x_{k})$  as follows. For some small value  $\delta \in (0,1)$ ,  $d = x_{k+1} - x_{k}$
+
+$$
+\hat {L} \left(x _ {k}\right) = \max  _ {\gamma \in \{\delta , 2 \delta , \dots , 1 \}} \frac {\left\| \nabla f (x + \gamma d) - \nabla f (x) \right\|}{\| \gamma d \|}. \tag {17}
+$$
+
+This suggests that we only care about the variation of gradient along  $x_{k + 1} - x_k$ . The motivation is based on the function upper bound (10), which shows that the deviation of the objective from its linear approximation is determined by the variation of gradient between  $x_{k + 1}$  and  $x_k$ .
+
+# H.4 ADDITIONAL PLOTS
+
+The plots in Figure 2a show log-scale scattered data for iterates in the first epoch. To supplement this result, we show in Figure 5a the linear scale plot of the same data as in Figure 2a. In Figure 5b, we run the same experiment as in Figure 2a for 200 epochs instead of 1 epoch and plot the gradient norm and estimated smoothness along the trajectory.
+
+In Figure 2, we plot the correlation between gradient norm and smoothness in LSTM LM training. We take snapshots of the model every 5 iterations in the first epoch, and use  $10\%$  of training data to estimate gradient norm and smoothness. As shown in Figure 6, using  $10\%$  of the data provides a very accurate estimate of the smoothness computed from the entire data.
+
+# I A SYNTHETIC EXPERIMENT
+
+In this section, we demonstrate the different behaviors of gradient descent versus clipped gradient descent by optimizing a simple polynomial  $f(x) = x^4$ . We initialize the point at  $x_0 = 30$  and run both algorithms. Within the sublevel set  $[-30, 30]$ , the function satisfies
+
+$$
+f ^ {\prime \prime} (x) \leq 1 2 \times 3 0 ^ {2} = 1. 0 8 \times 1 0 ^ {4}
+$$
+
+$$
+f ^ {\prime \prime} (x) \leq 1 0 f ^ {\prime} (x) + 0. 1.
+$$
+
+Therefore, we can either pick  $L_{1} = 0$ ,  $L_{0} = 1.08^{4}$  for gradient descent or  $L_{1} = 10 \cdot L_{0} = 0.1$  for clipped GD. Since the theoretical analysis is not tight with respect to constants, we scan the step sizes to pick the best parameter for both algorithms. For gradient descent, we scan step size by halving the current steps. For clipped gradient descent, we fix threshold to be 0.01 and pick the step size in the same way. The convergence results are shown in Figure 7. We can conclude that clipped gradient descent converges much faster than vanilla gradient descent, as the theory suggested.
+
+![](images/40f2188b5b5d83fb43412334e635d2df5e787b6a2d34ade76658e1754d29a35e.jpg)
+
+![](images/56d2828e407fdb76d4bf96384a95b0ce4b4a265807c00ea1a5bbc32dccb3b189.jpg)
+
+![](images/7a53c0751672e582fa04dfc54ebac20ddbadda78073dbbf9d39eca9eb97cdc18.jpg)
+
+![](images/d8604556a6534e1f8a61314af27ad01d371e77b926c5fe236e30bdaadca11f6b.jpg)  
+(a) Gradient descent  $|x|$ .  
+(d) Clipped GD  $|x|$ .  
+Figure 7: An synthetic experiment to optimize  $f(x) = x^4$ . (a) Gradient descent with different step size. (b) Clipped gradient descent with different step size and threshold  $= 0.01$ .
+
+![](images/574ceaefaf16710c8ecf1c9ef2351cab471dce38531eebfb1f89ed56b68fd8ed.jpg)  
+(b) Gradient descent  $f(x) = x^4$ .  
+(e) Clipped GD  $f(x) = x^4$ .
+
+![](images/7dc1f0245db92217f5463be1f20d51c951cfde19ba50b9604cc9e7ed773ef52f.jpg)  
+(c) Gradient descent  $f^{\prime}(x) = 4x^{3}$ .  
+(f) Clipped GD  $f'(x) = 4x^3$ .
+
+# J A QUANTITATIVE COMPARISON OF THEOREMS AND EXPERIMENTS
+
+To quantify how much the result align with the theorem, we assume that the leading term in the iteration complexity is the  $\epsilon$  dependent term. For GD, the term scales as  $\mathcal{O}\left(\frac{ML_1 + L_0}{\sqrt{T}}\right)$ , while for Clipped GD, the term scales as  $\mathcal{O}\left(\frac{L_0}{\sqrt{T}}\right)$ .
+
+First, we start with the synthetic experiment presented in I. From theory, we infer that the improvement of  $\frac{f'(x_{GD})}{f'(x_{CGD})} \approx \frac{L_0}{ML_1 + L_0} \approx 1e5$ . In experiment, the best performing GD reaches  $f'(x_T) = 0.36$ , while for clipped GD, the value is  $1.3e - 8$ , and the ratio is  $\frac{f'(x_{GD})}{f'(x_{CGD})} \approx 1e7$ . This suggests that in this very adversarial (against vanilla GD) synthetic experiment, the theorem is correct but conservative.
+
+Next, we test how well the theory can align with practice in neural network training. To do so, we rerun the PTB experiment in 5 with a smaller architecture (2-Layer LSTM with 200 embedding dimension and 512 inner dimension). We choose hyperparameters based on Figure 4a. For clipped GD, we choose clipping threshold to be 0.25 and learning rate to be 10. For GD, we use a learning rate 2. One interesting observation is that, though GD makes steady progress in minimizing function value, its gradient norm is not decreasing.
+
+To quantify the difference between theory and practice, we follow the procedure as in the synthetic experiment. First, we estimate  $ML_{1} + L_{0} = 25$  for GD from Figure 8(c). Second, we estimate  $L_{1} = 10$ ,  $L_{0} = 5$  for clipped GD's trajectory from subplot (d). Then the theory predicts that the ratio between gradients should be roughly  $25 / 5 = 5$ . Empirically, we found the ratio to be  $\approx 3$  by taking the average (as in Theorem 3 and Theorem 6). This doesn't exactly align but is of the same scale. From our view, the main reason for the difference could be the presence of noise in this experiment. As theorems suggested, noise impacts convergence rates but is absent in our rough estimates  $\frac{ML_{1} + L_{0}}{L_{0}}$ .
+
+![](images/3d1e88acb6567172a0bdaae6d98fbcaabdaff519c58ab750eb4602ef6b011728.jpg)  
+(a) Gradient descent.
+
+![](images/e97cdc7143328c56ef84d4f88591ad55f0d4de52d0845fe64eaf00b4fea4abbd.jpg)  
+(b) Clipped gradient descent.
+
+![](images/39343e61d774fe5b9c370de8394160183df1bb778aa09646efc459e068857da2.jpg)  
+(c)
+
+![](images/a627aaac77c73a16534d25053e2fe621c85507f81c09296b3081b46979cae801.jpg)  
+(d)  
+Figure 8: (a) Gradient norm. (b) Loss curves. (c) The scatter points of smoothness vs gradient norm for the model trained with gradient descent. (d) The scatter points of smoothness vs gradient norm for the model trained with clipped GD.
\ No newline at end of file
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+# WHY NOT TO USE ZERO IMPUTATION? CORRECTING SPARSITY BIAS IN TRAINING NEURAL NETWORKS
+
+Joonyoung Yi $^{1}$ , Juhyuk Lee $^{1}$ , Kwang Joon Kim $^{3}$ , Sung Ju Hwang $^{1,2}$ , Eunho Yang $^{1,2}$ , KAIST $^{1}$ , AITRICS $^{2}$ , Yonsei University College of Medicine $^{3}$ , South Korea {joonyoung.yi, sehkmg, sjhwang82, eunhoy} @kaist.ac.kr, preppie@yuhs.ac
+
+# ABSTRACT
+
+Handling missing data is one of the most fundamental problems in machine learning. Among many approaches, the simplest and most intuitive way is zero imputation, which treats the value of a missing entry simply as zero. However, many studies have experimentally confirmed that zero imputation results in suboptimal performances in training neural networks. Yet, none of the existing work has explained what brings such performance degradations. In this paper, we introduce the variable sparsity problem (VSP), which describes a phenomenon where the output of a predictive model largely varies with respect to the rate of missingness in the given input, and show that it adversarially affects the model performance. We first theoretically analyze this phenomenon and propose a simple yet effective technique to handle missingness, which we refer to as Sparsity Normalization (SN), that directly targets and resolves the VSP. We further experimentally validate SN on diverse benchmark datasets, to show that debiasing the effect of input-level sparsity improves the performance and stabilizes the training of neural networks.
+
+# 1 INTRODUCTION
+
+Many real-world datasets often contain data instances whose subset of input features is missing. While various imputing techniques, from imputing using global statistics such as mean, to individually imputing by learning auxiliary models such as GAN, can be applied with their own pros and cons, the most simple and natural way to do this is zero imputation, where we simply treat a missing feature as zero. In neural networks, at first glance, zero imputation can be thought of as a reasonable solution since it simply drops missing input nodes by preventing the weights associated with them from being updated. Some what surprisingly, however, many previous studies have reported that this intuitive approach has an adverse effect on model performances (Hazan et al., 2015; Luo et al., 2018; Smieja et al., 2018), and none of them has investigated the reasons of such performance degradations.
+
+In this work, we find that zero imputation causes the output of a neural network to largely vary with respect to the number of missing entries in the input. We name this phenomenon Variable Sparsity Problem (VSP), which should be avoided in many real-world tasks. Consider a movie recommender system, for instance. It is not desirable that users get different average of predicted ratings just because they have rated different number of movies (regardless of their actual rating values). One might argue that people with less ratings do not like movies in general and it is natural to give higher predicted values to people with more ratings. This might be partially true for users of some sparsity levels, but it is not a common case uniformly applicable for a wider range of sparsity levels. This can be verified in real collaborative filtering datasets as shown in Figure 1 (upper left corner) where users have a similar average rating for test data regardless of the number of known ratings (see also other two examples in Figure 1). However, in standard neural networks with zero imputation, we observe that the model's inference correlates with the number of known entries of the data instance as shown in
+
+![](images/a7b24842b01afb51278cf7647cd1b2172a87ea9e6f23deaab26dd4455584ccaa.jpg)
+
+![](images/2412249793c18ce3fe2c2e2c1b02a1e0e1e3605ed5f85e489e02aabb68b73478.jpg)
+
+![](images/b67d3af231840646311fab552367f01123cbdcce8d85e87a2d1a6bad7ba8dc35.jpg)
+
+![](images/db3734efe1fcd24cc0b77c35232c7250df1a3ac57ffa63e2cfec0784d257a978.jpg)
+
+![](images/629c07609485914092c654db26f4a3616f7b77d5adbce567ad3d48567b6dcd76.jpg)
+
+![](images/8c6d72a404e9cbba860aa09a66fdd68e2aaac9fbcf67aa61deae27282d52144f.jpg)
+
+![](images/29352d1024d1501977c82c86e65a967df35d4cf2f78e1c00ed3b279ba1f0e30a.jpg)  
+Figure 1: First column: Sedhain et al. (2015) on Movielens 100K (collaborative filtering) dataset. Second column: LSTM on Physionet 2012 (electronic medical records) dataset. Third column: Talwar et al. (2018) on Blakeley (single cell RNA sequence) dataset. First row: Mean of target values according to the number of known entries in training set. Second row: Predicted values of models with zero imputation according to the number of known entries for a randomly selected test point. Input masks are randomly sampled (to artificially control its sparsity level). For each target sparsity level through x-axis, 50 samples are drawn, scattering the predicted values and plotting the average in solid line. Third row: Figures on how plots in second row are corrected by Sparsity Normalization.
+
+![](images/76d9e6aeab9a446c6abcecef0ffdb87b83ea5fcbd50ab4cdb59b99abbe1e6a6d.jpg)
+
+![](images/9660c252b21fc4577d02b6db828a53bbdb5a067ced93cd6e1dab2c413ea98f68.jpg)
+
+the second row of Figure  $1^{1}$ . It would be fatal in some safety-critical applications such as a medical domain: a patient's probability of developing disease for example should not be evaluated differently depending on the number of medical tests they received (we do not want our model to predict the probability of death is high just because some patient has been screened a lot!).
+
+In addition, we theoretically analyze the existence of VSP under several circumstances and propose a simple yet effective means to suppress VSP while retaining the intuitive advantages of zero imputation: normalizing with the number of non-zero entries for each data instance. We refer to this regularization as Sparsity Normalization, and show that it effectively deals away with the VSP, resulting in significant improvements in both the performance and the stability of training neural networks.
+
+Our contribution in this paper is threefold:
+
+- To best of our knowledge, we are the first in exploring the adverse effect of zero imputation, both theoretically and empirically.
+
+![](images/696eaa0313566bbc53c5e1e2129c61feb01346eab2ae867ae780f312eb8adda0.jpg)  
+(a) Neural network with variable sparsity levels
+
+![](images/305ab68366b230611301ffe92f23d51a59f064e90d7c1b46becdd9037a6f9dd9.jpg)  
+Figure 2: The change of output values according to the sparsity level of inputs (darker color indicates greater absolute value and dotted circles indicate missing nodes with zero imputation). SN makes the possible output range of a network be more stable with respect to the sparsity level.
+
+![](images/e39c07a9288e6386fc12f53de5114cc2a30dbbf29c98e42d597133e6e9b99cc5.jpg)
+
+![](images/c6151c7dd4da801659e17b9ff6b4d8a4d325cc727961a9c1d7820eea75a5ac8e.jpg)  
+(b) with Sparsity Normalization
+
+![](images/3a29ce148a87f9a188bc3c5c50cf602f606391fca636a1f99823fca3270807cb.jpg)
+
+![](images/52cebfab6472ab110ac12c526c6949e1dc44173013923a4ba3a6db6ad496438d.jpg)
+
+- We identify the cause of adverse effect of zero imputation, which we refer to as variable sparsity problem, and formally describe how this problem actually affects training and inference of neural networks (Section 2). We further provide new perspectives using VSP to understand phenomena that have not been clearly explained or that we have misunderstood (Section 4 and 5).  
+- We present Sparsity Normalization (SN) and theoretically show that SN can solve the VSP under certain conditions (Section 3). We also experimentally reaffirm that simply applying SN can effectively alleviate or solve the VSP yielding significant performance gains (Section 4).
+
+# 2 VARIABLE SPARSITY PROBLEM
+
+We formally define the Variable Sparsity Problem (VSP) as follows: a phenomenon in which the expected value of the output layer of a neural network (over the weight and input distributions) depends on the sparsity (the number of zero values) of the input data (Figure 2a). With VSP, the activation values of neural networks could become largely different for exactly the same input instance, depending on the number of zero entries; this makes training more difficult and may mislead the model into incorrect predictions.
+
+While zero imputation is intuitive in the sense that it drops the missing input features, we will show that it causes variable sparsity problem for several example cases. Specifically, we show the VSP under assumptions with increasing generality: (Case 1) where activation function is an identity mapping with no bias, (Case 2) where activation function is an affine function, and (Case 3) where activation function is a non-decreasing convex function such as ReLU (Glorot et al., 2011), leaky ReLU (Maas et al., 2013), ELU (Clevert et al., 2016), or Softplus (Dugas et al., 2001).
+
+Here, we summarize the notation for clarity. For a  $L$ -layer deep network with non-linearity  $\sigma$ , we use  $W^{i} \in \mathbb{R}^{n_{i} \times n_{i-1}}$  to denote the weight matrix of  $i$ -th layer,  $\mathbf{b}^{i} \in \mathbb{R}^{n_{i}}$  to denote the bias,  $\mathbf{h}^{i} \in \mathbb{R}^{n_{i}}$  to denote the activation vector. For simplicity, we use  $\mathbf{h}^{0} \in \mathbb{R}^{n_{0}}$  and  $\mathbf{h}^{L} \in \mathbb{R}^{n_{L}}$  to denote input and output layer, respectively. Then, we have
+
+$$
+\mathbf {h} ^ {i} = \sigma \left(W ^ {i} \mathbf {h} ^ {i - 1} + \mathbf {b} ^ {i}\right), \quad \text {f o r} i = 1, \dots , L.
+$$
+
+Our goal in this section is to observe the change in  $\mathbf{h}^L$  as the sparsity of  $\mathbf{h}^0$  (input  $\mathbf{x}$ ) changes. To simplify the discussion, we consider the following assumption:
+
+Assumption 1. (i) Every coordinate of input vector,  $h_l^0$ , is generated by the element-wise multiplication of two random variables  $\tilde{h}_l^0$  and  $m_l$  where  $m_l$  is binary mask indicating missing value and  $\tilde{h}_l^0$  is a (possibly unobserved) feature value. Here, missing mask  $m_l$  is MCAR (missing completely at random), with no dependency with other mask variables or their values  $\tilde{\mathbf{h}}^0$ . All  $m_l$  follow some identical distribution with mean  $\mu_m$ . (ii) The elements of matrix  $W^i$  are mutually independent and follow the identical distribution with mean  $\mu_w^i$ . Similarly,  $\mathbf{b}^i$  and  $\tilde{\mathbf{h}}^0$  consist of i.i.d. coordinates with mean  $\mu_b^i$  and  $\mu_x$ , respectively. (iii)  $\mu_w^i$  is not zero uniformly over all  $i$ .
+
+(i) assumes the simplest missing mechanism. (ii) is similarly defined in Glorot & Bengio (2010) and He et al. (2015) in studying weight initialization techniques. (iii) may not hold under some initialization strategies, but as the learning progresses, it is very likely to hold.
+
+(Case 1) For simplicity, let us first consider networks without the non-linearity nor the bias term. Theorem 1 shows that the average value of the output layer  $E[h_l^L]$  is directly proportional to the expectation of the mask vector  $\mu_m$ :  
+Theorem 1. Suppose that activation  $\sigma$  is an identity function and that  $b_{l}^{i}$  is uniformly fixed as zero under Assumption 1. Then, we have  $E[h_l^L ] = \prod_{i = 1}^{L}n_{i - 1}\mu_w^i\mu_x\mu_m$ .  
+(Case 2) When the activation function is affine but now with a possibly nonzero bias,  $E[h_l^L]$  is influenced by  $\mu_{m}$  in the following way:  
+Theorem 2. Suppose that activation  $\sigma$  is an affine function under Assumption 1. Suppose further that  $f_{i}(x)$  is defined as  $\sigma (n_{i - 1}\mu_w^i x + \mu_b^i)$ . Then,  $E[h_l^L ] = f_L\circ \dots \circ f_1(\mu_x\mu_m)$ .  
+(Case 3) Finally, when the activation function is non-linear but non-decreasing and convex, we can show that  $E[h_l^L]$  is lower-bounded by some quantity involving  $\mu_m$ :  
+Theorem 3. Suppose that  $\sigma$  is a non-decreasing convex function under Assumption 1. Suppose further that  $f_{i}(x)$  is defined as  $\sigma (n_{i - 1}\mu_w^i x + \mu_b^i)$  and  $\mu_w^i >0$ . Then,  $E[h_l^L ]\geq f_L\circ \dots \circ f_1(\mu_x\mu_m)$ .
+
+If the expected value of the output layer (or the lower bound of it) depends on the level of sparsity/missingness as in Theorem 1-3, even similar data instances may have different output values depending on their sparsity levels, which would hinder fair and correct inference of the model. As shown in Figure 1 (second row), the VSP can easily occur even in practical settings of training neural networks where the above conditions do not hold.
+
+# 3 SPARSITY NORMALIZATION
+
+In this section, we propose a simple yet surprisingly effective method to resolve the VSP. We first revisit (Case 2) to find a way of making expected output independent of input sparsity level since the linearity in activation simplifies the correction. Recalling the notation of  $\mathbf{h}^0 = \tilde{\mathbf{h}}^0\odot \mathbf{m}$  ( $\odot$  represents the element-wise product), we find that simply normalizing via  $\mathbf{h}_{\mathrm{SN}}^0 = (\tilde{\mathbf{h}}^0\odot \mathbf{m})\cdot K_1 / \mu_m$  for any fixed constant  $K_{1}$ , can debias the dependency on the input sparsity level. We name this simple normalizing technique Sparsity Normalization (SN) and describe it in Algorithm 1. Conceptually, this method scales the size of each input value according to its sparsity level so that the change in output
+
+# Algorithm 1 Sparsity Normalization (SN)
+
+Input: Dataset  $\mathcal{D}$ , constant  $K$ .
+
+Output: Sparsity Normalized Dataset  $\mathcal{D}_{\mathrm{SN}}$
+
+Empty set  $S = \phi$
+
+for each  $(\mathbf{h}^0,\mathbf{m})\in \mathcal{D}$  do
+
+$$
+\mathbf {h} _ {\mathrm {S N}} ^ {0} \leftarrow K \cdot \mathbf {h} ^ {0} / \| \mathbf {m} \| _ {1}
+$$
+
+$$
+\mathcal {S} \leftarrow \mathcal {S} \cup \left\{\mathbf {h} _ {\mathrm {S N}} ^ {0} \right\}
+$$
+
+end for
+
+$$
+\mathcal {D} _ {\mathrm {S N}} \leftarrow \mathcal {S}
+$$
+
+size is less sensitive to the sparsity level (Figure 2b). The formal description on correcting sparsity bias by SN is as follows in this particular case:
+
+Theorem 4. (With Sparsity Normalization) Suppose that activation  $\sigma$  is an affine function under Assumption 1. Suppose further that  $f_{i}(x) = \sigma (n_{i - 1}\mu_{w}^{i}x + \mu_{b}^{i})$  and replace the input layer using SN, i.e.  $\mathbf{h}_{SN}^0 = (\tilde{\mathbf{h}}^0\odot \mathbf{m})\cdot K_1 / \mu_m$  for any fixed constant  $K_{1}$ . Then, we have  $E[h_l^L ] = f_L\circ \dots \circ f_1(\mu_x\cdot K_1)$ .
+
+Unlike in Theorem 2, SN in Theorem 4 makes average activation to be independent of  $\mu_{m}$ , which determines the sparsity levels of input. It is not trivial to show the counterpart of (Case 3) using SN since  $E[\sigma(x)] = \sigma(E[x])$  does not hold in general. However, we show through extensive experiments in Section 4 that SN is practically effective even in more general cases.
+
+Table 1: Debiasing variable sparsity using SN on Movielens datasets. Test RMSE with  $95\%$  confidence interval of 5-runs is provided.  
+
+<table><tr><td colspan="3">Datasets</td><td>Movielens 100K</td><td>Movielens 1M</td><td>Movielens 10M</td></tr><tr><td rowspan="4">AutoRec(Sedhain et al., 2015)</td><td rowspan="2">user vector</td><td rowspan="2">w/o SN w/ SN</td><td>0.9346 ± 0.0007</td><td>0.8831 ± 0.0002</td><td>0.8859 ± 0.0014</td></tr><tr><td>0.9208 ± 0.0023</td><td>0.8742 ± 0.0003</td><td>0.8462 ± 0.0005</td></tr><tr><td rowspan="2">item vector</td><td rowspan="2">w/o SN w/ SN</td><td>0.8835 ± 0.0003</td><td>0.8320 ± 0.0003</td><td>0.7807 ± 0.0017</td></tr><tr><td>0.8809 ± 0.0011</td><td>0.8294 ± 0.0004</td><td>0.7706 ± 0.0023</td></tr><tr><td rowspan="4">CF-NADE(Zheng et al., 2016)</td><td rowspan="2">user vector</td><td rowspan="2">w/o SN w/ SN</td><td>0.9253 ± 0.0010</td><td>0.8530 ± 0.0006</td><td>0.8113 ± 0.0058</td></tr><tr><td>0.9231 ± 0.0012</td><td>0.8525 ± 0.0006</td><td>0.7854 ± 0.0006</td></tr><tr><td rowspan="2">item vector</td><td rowspan="2">w/o SN w/ SN</td><td>0.8982 ± 0.0005</td><td>0.8405 ± 0.0007</td><td>N/A</td></tr><tr><td>0.8900 ± 0.0018</td><td>0.8366 ± 0.0008</td><td>N/A</td></tr><tr><td rowspan="2" colspan="2">CF-UIcA (Du et al., 2018)</td><td rowspan="2">w/o SN w/SN</td><td>0.8945 ± 0.0024</td><td>0.8223 ± 0.0016</td><td>N/A</td></tr><tr><td>0.8793 ± 0.0017</td><td>0.8178 ± 0.0007</td><td>N/A</td></tr></table>
+
+Table 2: Comparison of AutoRec with SN and CF-UIcA with SN against state-of-the-art collaborative filtering methods for each datasets. Bold font indicates neural network based models. The results marked  $\dagger$  are taken from Zhang et al. (2017) and all other baselines are taken from original papers. The result of CF-UIcA with SN on Movielens 10M is not provided because the authors of CF-UIcA did not provide the results for it due to the complexity of the model.  
+
+<table><tr><td>Models</td><td>Movielens 100K</td><td>Models</td><td>Movielens 1M</td><td>Models</td><td>Movielens 10M</td></tr><tr><td>Koren (2008)</td><td>0.913†</td><td>Fu et al. (2018)</td><td>0.836</td><td>Sedhain et al. (2015)</td><td>0.782</td></tr><tr><td>Zhuang et al. (2017)</td><td>0.9114 ± 0.0093</td><td>Lee et al. (2016)</td><td>0.8333</td><td>Berg et al. (2018)</td><td>0.777</td></tr><tr><td>Koren et al. (2009)</td><td>0.911†</td><td>Yi et al. (2019)</td><td>0.8321</td><td>Chen et al. (2016)</td><td>0.7712 ± 0.0002</td></tr><tr><td>Dziugaite &amp; Roy (2015)</td><td>0.903</td><td>Berg et al. (2018)</td><td>0.832</td><td>Zheng et al. (2016)</td><td>0.771</td></tr><tr><td>Zhang et al. (2017)</td><td>0.901</td><td>Sedhain et al. (2015)</td><td>0.831</td><td>AutoRec w/ SN (ours.)</td><td>0.7690 ± 0.0023</td></tr><tr><td>Yi et al. (2019)</td><td>0.8889</td><td>Zheng et al. (2016)</td><td>0.829</td><td>Li et al. (2016)</td><td>0.7682 ± 0.0003</td></tr><tr><td>Lee et al. (2016)</td><td>0.8881 ± 0.0017</td><td>AutoRec w/ SN (ours.)</td><td>0.8260 ± 0.0023</td><td>Chen et al. (2017)</td><td>0.7672 ± 0.0001</td></tr><tr><td>AutoRec w/ SN (ours.)</td><td>0.8816 ± 0.0087</td><td>Du et al. (2018)</td><td>0.823</td><td>Fu et al. (2018)</td><td>0.766</td></tr><tr><td>CF-UIcA w/ SN (ours.)</td><td>0.8779 ± 0.0159</td><td>CF-UIcA w/ SN (ours.)</td><td>0.8215 ± 0.0037</td><td>Li et al. (2017)</td><td>0.7634 ± 0.0002</td></tr></table>
+
+While Theorem 4 assumes that  $\mu_{m}$  is known and fixed across all data instances, we relax this assumption in practice and consider varying  $\mu_{m}$  across data instances. Specifically, by a maximum likelihood principle, we can estimate  $\mu_{m}$  for each instance by  $\left\| \mathbf{h}^{0}\right\|_{0} / n_{0} = \left\| \mathbf{m}\right\|_{1} / n_{0}$ . Thus, we have  $\mathbf{h}_{\mathrm{SN}}^{0} = K\cdot \mathbf{h}^{0} / \left\| \mathbf{m}\right\|_{1}$  where  $K = n_0\cdot K_1$  (see Algorithm 1) $^2$ . In practice, we recommend using  $K$  as the average of  $\| \mathbf{m}\| _1$  over all instances in the training set. We could encounter the dying ReLU phenomenon (He et al., 2015) if  $K$  is too small (e.g.,  $K = 1$ ). Since the hyper-parameter  $K$  can bring in a regularization effect via controlling the magnitude of gradient (Salimans & Kingma, 2016), we define  $K = E_{(\mathbf{h}^0,\mathbf{m})\in \mathcal{D}}[\| \mathbf{m}\| _1]$  so that the average scales remain constant before and after the normalization, minimizing such side effects caused by SN.
+
+# 4 EXPERIMENTS
+
+In this section, we empirically show that VSP occurs in various machine learning tasks and it can be alleviated by SN. In addition, we also show that resolving VSP leads to improved model performance on diverse scenarios.
+
+# 4.1 COLLABORATIVE FILTERING (RECOMMENDATION) DATASETS
+
+We identify VSP and the effect of SN on several popular benchmark datasets for collaborative filtering with extremely high missing rates. We train an AutoRec (Sedhain et al., 2015) using user vector on Movielens (Harper & Konstan, 2016) 100K dataset for validating VSP and SN. Going back to the first column of Figure 1, the prediction with SN is almost constant regardless of  $\| \mathbf{m}\| _1$ . Another perceived phenomenon in Figure 1 is that the higher the  $\| \mathbf{m}\| _1$ , the smaller the variation in the prediction of
+
+Table 3: Debiasing variable sparsity using SN and comparisons against other missing handling techniques on five disease identification tasks of NHIS dataset. Test AUROC with  $95\%$  confidence interval of 5-runs is provided.  
+
+<table><tr><td>Dataset</td><td>Cardiovascular</td><td>Fatty Liver</td><td>Hypertension</td><td>Heart Failure</td><td>Diabetes</td></tr><tr><td>Zero Imputation w/o SN</td><td>0.7057 ± 0.0027</td><td>0.6750 ± 0.0050</td><td>0.7977 ± 0.0027</td><td>0.7834 ± 0.0036</td><td>0.9121 ± 0.0097</td></tr><tr><td>Zero Imputation w/ SN (ours.)</td><td>0.7106 ± 0.0005</td><td>0.6911 ± 0.0022</td><td>0.8096 ± 0.0010</td><td>0.7914 ± 0.0012</td><td>0.9283 ± 0.0011</td></tr><tr><td>Zero Imputation w/ BN</td><td>0.7033 ± 0.0035</td><td>0.6691 ± 0.0081</td><td>0.7828 ± 0.0136</td><td>0.7749 ± 0.0083</td><td>0.9026 ± 0.0105</td></tr><tr><td>Zero Imputation w/ LN</td><td>0.7049 ± 0.0035</td><td>0.6811 ± 0.0062</td><td>0.7944 ± 0.0015</td><td>0.7820 ± 0.0007</td><td>0.9127 ± 0.0056</td></tr><tr><td>Dropout</td><td>0.7047 ± 0.0021</td><td>0.6747 ± 0.0069</td><td>0.7926 ± 0.0038</td><td>0.7802 ± 0.0049</td><td>0.9101 ± 0.0054</td></tr><tr><td>Mean Imputation</td><td>0.7054 ± 0.0025</td><td>0.6766 ± 0.0067</td><td>0.7913 ± 0.0019</td><td>0.7839 ± 0.0021</td><td>0.9117 ± 0.0075</td></tr><tr><td>Median Imputation</td><td>0.7056 ± 0.0012</td><td>0.6713 ± 0.0058</td><td>0.7865 ± 0.0034</td><td>0.7818 ± 0.0014</td><td>0.8975 ± 0.0060</td></tr><tr><td>k-NN</td><td>0.7052 ± 0.0010</td><td>0.6874 ± 0.0045</td><td>0.8000 ± 0.0050</td><td>0.7843 ± 0.0024</td><td>0.9107 ± 0.0075</td></tr><tr><td>MICE</td><td>0.7075 ± 0.0022</td><td>0.6902 ± 0.0058</td><td>0.7957 ± 0.0037</td><td>0.7875 ± 0.0032</td><td>0.9224 ± 0.0021</td></tr><tr><td>SoftImpute</td><td>0.7014 ± 0.0030</td><td>0.6868 ± 0.0040</td><td>0.7990 ± 0.0050</td><td>0.7828 ± 0.0042</td><td>0.9224 ± 0.0019</td></tr><tr><td>GMMC</td><td>0.7072 ± 0.0016</td><td>0.6816 ± 0.0067</td><td>0.7984 ± 0.0062</td><td>0.7884 ± 0.0029</td><td>0.9109 ± 0.0045</td></tr><tr><td>GAIN</td><td>0.7065 ± 0.0015</td><td>0.6765 ± 0.0060</td><td>0.7956 ± 0.0020</td><td>0.7847 ± 0.0031</td><td>0.9091 ± 0.0067</td></tr></table>
+
+the model with SN. Note that the same tendency has been observed regardless of test instances or datasets. This implies that models with SN yield more calibrated predictions; as more features are known for a particular instance, the variance of prediction for that instance should decrease (since we generated independent masks in Figure 1). It is also worthwhile to note that an AutoRec is a sigmoid-based network and Movielens datasets are known to have no MCAR hypothesis (Wang et al., 2018), in which Assumption 1 does not hold at all.
+
+It is also validated that performance gains can be obtained by putting SN into AutoRec (Sedhain et al., 2015), CF-NADE (Zheng et al., 2016), and CF-Ulca (Du et al., 2018), which are the state-of-the-art among neural network based collaborative filtering models on several Movielens datasets (see Appendix B for detailed settings). In Table  $1^{3}$ , we consider three different sized Movielens datasets. Note that AutoRec and CF-NADE allow two types of models according to data encoding (user- or item-rating vector based) and we consider both types. While we obtain performance improvements with SN in most cases, it is more prominent in user-rating based model in case of AutoRec and CF-NADE.
+
+Furthermore, Table 2 compares our simply modification using SN on AutoRec and CF-UiCA with other state-of-the-art collaborative filtering models beyond neural networks. Unlike experiments of AutoRec in Table 1, which use the same network architectures proposed in original papers, here we could successfully learn more expressive network due to the stability obtained by using  $\mathrm{SN}^4$ . For Movielens 100K and 1M datasets, applying SN yields better or similar performance compared to other state-of-the-art collaborative filtering methods beyond neural network based models. It is important to note that all models outperforming AutoRec with SN on Movielens 10M are ensemble models while AutoRec with SN is a single model and it shows consistently competitive results across all datasets.
+
+# 4.2 ELECTRONIC MEDICAL RECORDS (EMR) DATASETS
+
+We further test VSP and SN for clinical time-series prediction with two Electronic Medical Records (EMR), namely PhysioNet Challenge 2012 (Silva et al., 2012) and the National Health Insurance Service (NHIS) datasets, which have intrinsic missingness since patients will only receive medical examinations that are considered to be necessary. We identify whether the VSP exists with the PhysioNet Challenge 2012 dataset (Silva et al., 2012). We randomly select one test point and plot in-hospital death probabilities as the number of examinations varies (Second column of Figure 1). Without SN, the in-hospital death probability increases as the number of examinations increases,
+
+![](images/9d750564c1cdf0ce7d21980f642dcbb7152944a9b184f890fc97aa0bc1fa5d69.jpg)
+
+![](images/dca1a3768697a3d37461ac3b16e6f50554064bbad11ffeecfed2098bf7eadee5.jpg)
+
+![](images/55bc7531973b3927a1ddc0b3a3ef8375df1fc3cd30e685c1e9bcd40071ee21b1.jpg)
+
+![](images/31b046fa9c314a1e33dfc489024bb19f3da93612409860a61095aa0c63ac4f86.jpg)  
+Figure 3: Debiasing variable sparsity using SN according to test set ratio on six imputation tasks of single cell RNA sequence dataset. Test RMSE with  $95\%$  confidence interval of 10-runs is provided.
+
+![](images/89e418bd3adbb6087cece8652101e8e0a3e9abde9838e2547f9ba40db733e129.jpg)
+
+![](images/69b96782ede8814ee7bb998979481bc8898f27dbd75aaa00c3211ac3707417a8.jpg)
+
+![](images/fb34dad30b92cdcdf070bbffa3e9367fbafe6ef9011f7b84f415143c7cd3538d.jpg)  
+Figure 4: Debiasing variable sparsity using SN with respect to drop rates on three popular UCI regression datasets. Test RMSE with  $95\%$  confidence interval of 5-runs is provided.
+
+![](images/a0a2a3a2b99be1afda9ada4e9599ace9cd39fb4b77cd2b655113530f2f20d8e7.jpg)  
+w/o SN w/ SN
+
+![](images/9bd065293d370ca1a1ba0ef5b091290aba392ddc9b38a4d7f2194da3c5551f7d.jpg)
+
+even though there is no such tendency in the dataset statistics. However, SN corrects this bias so that in-hospital death probability is consistent regardless of the number of examinations. We observe a similar tendency for examples from the NHIS dataset as well.
+
+Although SN corrects the VSP in both datasets, we perceive different behaviors in both cases in terms of actual performance changes. While SN significantly outperforms its counterpart without SN on NHIS dataset as shown in Table 3, it just performs similarly on PhysioNet dataset (results and detailed settings are deferred to Appendix C). However, SN is still valuable for its ability to prevent biased predictions in this mission-critical area.
+
+In addition, we compare SN with other missing handling techniques to show the efficacy of SN although our main purpose is to provide a deeper understanding and the corresponding solution about the issue that the zero imputation, which is the simplest and most intuitive way of handling missing data, degrades the performance in training neural networks. Even with its simplicity, SN exhibits better or similar performances compared to other more complex techniques. Detailed descriptions of other missing handling techniques are deferred to Appendix H.
+
+![](images/72deb5c1338e26d174bdf269ea9ebcdefc6374e8c7b6137b66de558f1de7c784.jpg)  
+Figure 5: Negative log likelihood of MADE on binarized MNIST with and without SN.
+
+# 4.3 SINGLE-CELL RNA SEQUENCE DATASETS
+
+Single-cell RNA sequence datasets contain expression levels of specific RNA for each single cells. AutoImpute (Talwar et al., 2018) is one of the state-of-the-art methods that imputes missing data on single-cell RNA sequence datasets. We reproduce their experiments using authors' official implementation, and follow most of their experimental settings (see Appendix D for details).
+
+As before, we first check whether VSP occurs in AutoImpute model. The third column in Figure 1 shows how the prediction of a AutoImpute model changes as the number of known entries changes. Although the number of RNAs found in the specific cell is less related to cell characteristics (upper right corner in Figure 1), the prediction increases as the number of RNAs found in the cell increases. This tendency is dramatically reduced with SN.
+
+Figure 3 shows how imputation performance changes by being worked with SN to several single cell RNA sequence datasets with respect to the portion of train set (see Appendix D for more results). As we can see in Figure 3, SN significantly increases the imputation performance of AutoImpute model. In particular, the smaller the train data, the better the effect of SN in all datasets consistently. AutoImpute model is a sigmoid-based function and single cell RNA datasets (Talwar et al., 2018) do not have the MCAR hypothesis, unlike Assumption 1. Nevertheless, VSP occurs even here and it can be successfully alleviated by SN with huge performance gain. It implies that SN would work for other neural network based imputation techniques.
+
+# 4.4 DROPOUT ON UCI DATASETS
+
+While SN primarily targets to fix the VSP in the input layer, it could be also applied to any layers of deep neural networks to resolve VSP. The typical example of having heterogeneous sparsity in hidden layers is when we use dropout (Srivastava et al., 2014), which can be understood as another form of zero imputation but at the hidden layers; with Bernoulli dropout, the variance of the number of zero units across instances is  $np(1 - p)$  ( $n$ : the dimension of hidden layer,  $p$ : drop rate). While dropout partially handles this VSP issue by scaling  $1 / (1 - p)$  in the training phase $^5$ , SN can exactly correct VSP of hidden layers by considering individual level sparsity (Note that the scaling of dropout can be viewed as applying SN in an average sense:  $E[\| \mathbf{m}\|_1] = n(1 - p)$  and  $K = n$ ).
+
+Figure 4 shows how the RMSE changes as the drop rate changes with and without SN on three popular UCI regression datasets (Boston Housing, Diabetes, and California Housing). As illustrated in Figure 4, the larger the drop rate, the greater the difference of RMSE between with and without SN. To explain this phenomenon, we define the degree of VSP as the inverse of signal-to-noise ratio with respect to the number of active units in hidden layers:  $\sqrt{p / (n(1 - p))}$  (expected number of active units over its standard deviation). As can be seen from the figure, the larger drop rate  $p$  is, the more severe degree of VSP is and thus the greater protection by SN against performance degradation.
+
+# 4.5 DENSITY ESTIMATION
+
+In the current literature of estimating density based on deep models, inputs with missing features are in general not largely considered. However, we still may experience the VSP since the proportion of zeros in the data itself can vary greatly from instance to instance. In this experiment, we apply SN to MADE (Germain et al., 2015), which is one of the modern architectures in neural network-based density estimation. We reproduce binarized MNIST (LeCun, 1998) experiments of Germain et al. (2015) measuring negative log likelihood (the lower the better) of test dataset while increasing the number of masks. Figure 5 illustrates the effect of using SN. Note that MADE uses masks in the hidden layer that are designed to produce proper autoregressive conditional probabilities and variable sparsity arises across hidden nodes. SN can be trivially extended to handle this case as well and the corresponding result is given in the figure denoted as  $w / \text{SN}(\text{all})^7$ . We reaffirm that SN is effective even when MCAR assumption is not established.
+
+# 5 RELATED WORKS
+
+Missing handling techniques Missing imputation can be understood as a technique to increase the generalization performance by injecting plausible noise into data. Noise injection using global statistics like mean or median values is the simplest way to do this (Lipton et al., 2016; Smieja et al., 2018). However, it could lead to highly incorrect estimation since they do not take into consideration the characteristics of each data instance (Tresp et al., 1994; Che et al., 2018). To overcome this limitation, researchers have proposed various ways to model individualized noise using autoencoders (Pathak et al., 2016; Gondara & Wang, 2018), or GANs (Yoon et al., 2018; Li et al., 2019). However, those model based imputation techniques have not properly worked for high dimensional datasets with the large number of features and/or extremely high missing rates (Yoon et al., 2018) because excessive noise can ruin the training of neural networks rather increasing generalization performance.
+
+For this reason, in the case of high dimensional datasets such as collaborative filtering or single cell RNA sequences, different methods of handling missing data have been proposed. A line of work simply used zero imputation by minimizing noise level and achieved state-of-the-art performance on their target datasets (Sedhain et al., 2015; Zheng et al., 2016; Talwar et al., 2018). In addition, methods using low-rank matrix factorization have been proposed to reduce the input dimension, but these methods not only cause lots of information loss but also fail to capture non-linearity of the input data (Hazan et al., 2015; Bachman et al., 2017; He et al., 2017). Vinyals et al. (2016); Monti et al. (2017) proposed recurrent neural network (RNN) based methods but computational costs for these methods are outrageous for high dimensional datasets. Also, it is not natural to use RNN-based models for non-sequential datasets.
+
+Other forms of Sparsity Normalization We discuss other forms of SN to alleviate VSP, already in use unwittingly due to empirical performance improvements. DropBlock (Ghiasi et al., 2018) compensates activation from dropped features by exactly counting mask vector similar to SN (similar approach discussed in Section 4.4.) It is remarkable that we can find models using SN-like normalization even in handling datasets without missing features. For example, in CBOW model (Mikolov et al., 2013) where the number of words used as an input depends on the position in the sentence, it was later revealed that SN like normalization has a practical performance improvement. As an another example, Kipf & Welling (2017) applied Laplacian normalization which is the standard way of representing a graph in graph theory, can naturally handle heterogeneous node degrees and precisely matches the SN operation. In this paper, we explicitly extend SN, which was limited and unintentionally applied to only few settings, to a model agnostic technique.
+
+# 6 CONCLUSION
+
+We identified variable sparsity problem (VSP) caused by zero imputation that has not been explicitly studied before. To best of our knowledge, this paper provided the first theoretical analysis on why zero imputation is harmful to inference of neural networks. We showed that variable sparsity problem actually exists in diverse real-world datasets. We also confirmed that theoretically inspired normalizing method, Sparsity Normalization, not only reduces the VSP but also improves the generalization performance and stability of feed-forwarding of neural networks with missing values, even in areas where existing missing imputation techniques do not cover well (e.g., collaborative filtering, single cell RNA datasets).
+
+# ACKNOWLEDGMENTS
+
+This work was supported by the Institute of Information & Communications Technology Planning & Evaluation (IITP) grants (No.2016-0-00563, No.2017-0-01779, and No.2019-0-01371), the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) grants (No.2018R1A5A1059921 and No.2019R1C1C1009192), the Samsung Research Funding & Incubation Center of Samsung Electronics via SRFC-IT1702-15, the National IT Industry Promotion Agency grant funded by the Ministry of Science and ICT, and the Ministry of Health and Welfare (NO. S0310-19-1001, Development Project of The Precision Medicine Hospital Information System (P-HIS)).
+
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+
+# A PROOFS
+
+# A.1 PROOF OF THEOREM 1
+
+Proof. From the definition of  $h_l, w_l^1, h_l^0, \tilde{h}_l^0, m_l$ , the following equation holds.
+
+$$
+E [ h _ {l} ^ {1} ] = n _ {0} E [ w _ {l} ^ {1} h _ {1} ^ {0} ] = n _ {0} E [ w _ {l} ^ {1} \tilde {h} _ {l} ^ {0} m _ {l} ]
+$$
+
+From the Assumption 1,  $w_{l}^{1},\tilde{h}_{l}^{0}$  , and  $m_l$  are independent of each other. Thus,
+
+$$
+E [ h _ {l} ^ {1} ] = n _ {0} E [ w _ {l} ^ {1} ] E [ \bar {h} _ {l} ^ {0} ] E [ m _ {l} ]
+$$
+
+Similarly, the following holds.
+
+$$
+E \left[ h _ {l} ^ {i} \right] = n _ {i - 1} E \left[ w _ {l} ^ {i} h _ {l} ^ {i - 1} \right] \text {f o r} i = 1, \dots , L
+$$
+
+Since  $h_l^{i-1}$  and  $w_l^i$  are independent of each other by the Assumption 1 and the definition of  $h_l^{i-1}$ ,  $E[h_l^i] = n_{i-1}E[w_l^i]E[h_l^{i-1}]$ . Therefore,
+
+$$
+E [ h _ {l} ^ {L} ] = \prod_ {i = 1} ^ {L} n _ {i - 1} E [ w _ {l} ^ {i} ] E [ \tilde {h} _ {l} ^ {0} ] E [ m _ {l} ] = \prod_ {i = 1} ^ {L} n _ {i - 1} \mu_ {w} ^ {i} \mu_ {x} \mu_ {m}
+$$
+
+![](images/f4fa58fbb51a0f179aae145f73c0aad6c5ff9e813dda5498f393b14fb4c2ff40.jpg)
+
+# A.2 PROOF OF THEOREM 2
+
+Proof. From the definition of  $h_l, w_l^1, h_l^0, \tilde{h}_l^0, m_l$  and the property of an affine function  $\sigma(E[\cdot]) = E[\sigma(\cdot)]$ , the following equation holds.
+
+$$
+E [ h _ {l} ^ {1} ] = \sigma \left(n _ {0} E [ w _ {l} ^ {1} h _ {l} ^ {0} ] + E [ b _ {l} ^ {1} ]\right) = \sigma \left(n _ {0} E [ w _ {l} ^ {1} \tilde {h} _ {l} ^ {0} m _ {l} ] + E [ b _ {l} ^ {1} ]\right)
+$$
+
+From the Assumption 1,  $w_{l}^{1},\tilde{h}_{l}^{0}$  , and  $m_l$  are independent of each other. Thus,
+
+$$
+\begin{array}{l} E \left[ h _ {l} ^ {1} \right] = \sigma \left(n _ {0} E \left[ w _ {l} ^ {1} \right] E \left[ \tilde {h} _ {l} ^ {0} \right] E \left[ m _ {l} \right] + E \left[ b _ {l} ^ {1} \right]\right) \\ = \sigma \left(n _ {0} \mu_ {w} ^ {1} \mu_ {x} \mu_ {m} + \mu_ {b} ^ {1}\right) \\ = f _ {1} \left(\mu_ {x} \mu_ {m}\right) \\ \end{array}
+$$
+
+Similarly, the following holds.
+
+$$
+E \left[ h _ {l} ^ {i} \right] = \sigma \left(n _ {i - 1} E \left[ w _ {l} ^ {i} h _ {l} ^ {i - 1} \right] + E \left[ b _ {l} ^ {i} \right]\right) \text {f o r} i = 1, \dots , L
+$$
+
+Since  $h_l^{i-1}$  and  $w_l^i$  are independent of each other by the Assumption 1 and the definition of  $h_l^{i-1}$ ,
+
+$$
+\begin{array}{l} E \left[ h _ {l} ^ {i} \right] = \sigma \left(n _ {i - 1} E \left[ w _ {l} ^ {i} \right] E \left[ h _ {l} ^ {i - 1} \right] + E \left[ b _ {l} ^ {i} \right]\right) \\ = \sigma \left(n _ {i - 1} \mu_ {w} ^ {i} E \left[ h _ {l} ^ {i - 1} \right] + \mu_ {b} ^ {i}\right) \\ = f _ {i} \left(E \left[ h _ {l} ^ {i} \right]\right) \\ \end{array}
+$$
+
+Therefore,
+
+$$
+E \left[ h _ {l} ^ {L} \right] = f _ {L} \circ \dots \circ f _ {1} \left(\mu_ {x} \mu_ {m}\right)
+$$
+
+![](images/9c268bf846cc75e18e577a29d7036a88d324a5e5216995d436a2f6bbd24b17a4.jpg)
+
+# A.3 PROOF OF THEOREM 3
+
+Proof. From the definition of  $h_l, w_l^1, h_l^0, \tilde{h}_l^0, m_l$  and the property of a convex function  $E[\sigma(\cdot)] \geq \sigma(E[\cdot])$ , the following equation holds.
+
+$$
+E [ h _ {l} ^ {1} ] \geq \sigma \left(n _ {0} E [ w _ {l} ^ {1} h _ {l} ^ {0} ] + E [ b _ {l} ^ {1} ]\right) = \sigma \left(n _ {0} E [ w _ {l} ^ {1} \tilde {h} _ {l} ^ {0} m _ {l} ] + E [ b _ {l} ^ {1} ]\right)
+$$
+
+From the Assumption 1,  $w_{l}^{1},\tilde{h}_{l}^{0}$  , and  $m_l$  are independent of each other. Thus,
+
+$$
+\begin{array}{l} E \left[ h _ {l} ^ {1} \right] \geq \sigma \left(n _ {0} E \left[ w _ {l} ^ {1} \right] E \left[ \tilde {h} _ {l} ^ {0} \right] E [ m _ {l} ] + E \left[ b _ {l} ^ {1} \right]\right) \\ = \sigma \left(n _ {0} \mu_ {w} ^ {1} \mu_ {x} \mu_ {m} + \mu_ {b} ^ {1}\right) \\ = f _ {1} \left(\mu_ {x} \mu_ {m}\right) \\ \end{array}
+$$
+
+Similarly, the following holds.
+
+$$
+E \left[ h _ {l} ^ {i} \right] \geq \sigma \left(n _ {i - 1} E \left[ w _ {l} ^ {i} h _ {l} ^ {i - 1} \right] + E \left[ b _ {l} ^ {i} \right]\right) \text {f o r} i = 1, \dots , L
+$$
+
+Since  $h_l^{i-1}$  and  $w_l^i$  are independent of each other by the Assumption 1 and the definition of  $h_l^{i-1}$ ,
+
+$$
+\begin{array}{l} E \left[ h _ {l} ^ {i} \right] \geq \sigma \left(n _ {i - 1} E \left[ w _ {l} ^ {i} \right] E \left[ h _ {l} ^ {i - 1} \right] + E \left[ b _ {l} ^ {i} \right]\right) \\ = \sigma \left(n _ {i - 1} \mu_ {w} ^ {i} E \left[ h _ {l} ^ {i - 1} \right] + \mu_ {b} ^ {i}\right) \\ = f _ {i} (E [ h _ {l} ^ {i - 1} ]) \\ \end{array}
+$$
+
+Since we assume that  $\sigma$  is non-decreasing, we finally get
+
+$$
+E \left[ h _ {l} ^ {L} \right] \geq f _ {L} \circ \dots \circ f _ {1} \left(\mu_ {x} \mu_ {m}\right)
+$$
+
+![](images/85fecbea019c6fdeeab8cdff6e7558f26a00137fc455eb30744f58319827f1d2.jpg)
+
+# A.4 PROOF OF THEOREM 4
+
+Proof. By theorem 2,  $E[h_l^L] = f_L \circ \dots \circ f_1(E[\mathbf{h}_{\mathrm{SN}}^0])$ . Since  $E[\mathbf{h}_{\mathrm{SN}}^0] = E[\mathbf{h}^0] \cdot K / \mu_m$ ,
+
+$$
+E \left[ h _ {l} ^ {L} \right] = f _ {L} \circ \dots \circ f _ {1} \left(\mu_ {x} \mu_ {m} \cdot K / \mu_ {m}\right) = f _ {L} \circ \dots \circ f _ {1} \left(\mu_ {x} \cdot K\right)
+$$
+
+![](images/9f98814f787bb3746478357337ec99f4580344e7e5004d5d0f89e552160e22ff.jpg)
+
+# B COLLABORATIVE FILTERING (RECOMMENDATION) DATASETS
+
+# B.1 DETAILED EXPERIMENTAL SETTINGS OF TABLE 1
+
+This subsection describes the experimental settings of detailed collaborative filtering tasks in Section 4. As already mentioned, we follow the settings of AutoRec, CF-NADE, and CF-UiCAs as much as possible. We perform each experiments by 5 times, and report mean and  $95\%$  confidence intervals. We randomly select  $10\%$  of the ratings of each datasets for the test set (Harper & Konstan, 2016). As the dataset is too small for Movielens 100K and 1M datasets, the confidence interval tends to be large by changing the dataset split. Hence, the same dataset split is used in each 5 experiments in Table 1.
+
+AutoRec (Sedhain et al., 2015) We use two layer AutoRec model with 500 hidden units. For fair comparisons, we tune the hyper-parameters for weight decay in all experiments to have only one significant digit, and use a learning rate of  $10^{-3}$  except on Movielens 10M where we use a learning rate of  $10^{-4}$ . We use full batch on Movielens 100K and 1M, mini-batch (1000) on Movielens 10M. Besides, we use Adam optimizer instead of Resilient Propagation (RProp) unlike the AutoRec paper. The RProp optimizer shows fast convergence but can only be used in full batch scenario. It is not possible for 12GB of GPU memory to use full batch in training a large dataset such as Movielens 10M. Thus, we decide to use Adam optimizer rather than RProp optimizer. Fortunately, although the optimizer is changed to Adam, the prediction performance is not degraded in most cases. The experimental results of comparing both optimizers are summarized in Table 4.
+
+Table 4: Comparison of Test RMSE between Adam and Resilient Propagation optimizer on Movielens 100K, 1M, and 10M datasets.  
+
+<table><tr><td>Datasets</td><td colspan="2">Movielens 100K</td><td colspan="2">Movielens 1M</td><td colspan="2">Movielens 10M</td></tr><tr><td>input vector</td><td>item vector</td><td>user vector</td><td>item vector</td><td>user vector</td><td>item vector</td><td>user vector</td></tr><tr><td>RProp</td><td>0.8861</td><td>0.9437</td><td>0.8358</td><td>0.8804</td><td>0.782†</td><td>0.867†</td></tr><tr><td>Adam</td><td>0.8831</td><td>0.9343</td><td>0.8306</td><td>0.8832</td><td>0.7807</td><td>0.8859</td></tr></table>
+
+†: Taken from Sedhain et al. (2015).
+
+CF-NADE (Zheng et al., 2016) We use two layer CF-NADE model with 500 hidden units. For fair comparisons, we tune the hyper-parameters for weight decay in all experiments to have only one significant digit, and use a learning rate of 0.001. Also, we use mini-batch (512) just following CF-NADE. Although the CF-NADE used weight sharing and averaging possible choices in addition to weight decay, we report the results without weight sharing and averaging possible choices in Table 1 because it is not clear how to apply SN with weight sharing, and there is almost no performance gain with averaging possible choices despite of its high computational costs. Furthermore, we do not experiment on Movielens 10M with item vector encoding because the authors of CF-NADE did not provide the results for it due to the complexity of the model.
+
+CF-UIcA (Du et al., 2018) We use the authors' official code and the train/test dataset splits<sup>9</sup> for these experiments. Since CF-UIcA is a model that accepts both user and item vector as input, it is not necessary to consider two types of encoding as in AutoRec or CF-NADE. On the other hand, it is reasonable to take different  $K$  values for user and item vector with SN. We treat  $K$  as 66 for the user vector and 110 for the item vector. For models without SN, 0.0001 is used as the parameter  $\lambda$  for weight decay as suggested by the CF-UIcA. It is natural to use different  $\lambda$  values for SN because the optimal  $\lambda$  must be changed along with SN. Hence, we use  $\lambda = 0.0005$  for Movielens 100K and  $\lambda = 0.00006$  for Movielens 1M in the case of SN. As in CF-NADE, we do not test Movielens 10M because the authors of CF-UIcA also did not report for it owing to its high computational cost.
+
+# B.2 DETAILED EXPERIMENTAL SETTINGS OF TABLE 2
+
+In comparison with other state-of-the-art models, we used 1000 hidden units in AutoRec (Sedhain et al., 2015) with SN. While Sedhain et al. (2015) claimed that they were able to achieve enough performance only with 500 hidden units, 500 hidden units did not achieve sufficient performance when applying SN. The Figure 6 plots the test RMSE, changing the number of hidden units for Movielens 100K and 1M. We can see that 600 units for Movielens 100K and 900 units for Movielens 1M are necessary for getting better performance. Obviously, as datasets become more complex and larger, we need more network capacity. Therefore, we decide to use two times larger network capacity (1000 hidden units) to get better performance. The number of hidden units can also be viewed and tuned as a hyper-parameter, and we have not tuned much for the number of hidden units. Note that, unlike Table 1, we report the results with five random splits for all datasets in Table 2 to compare fairly with other state-of-the-art methods.
+
+![](images/4771bb2a3857bf78f31452c228a763a7b56f49e29a5c7eb4e0a2457a9ef6a866.jpg)  
+Figure 6: Test RMSE of AutoRec with SN on Movielens 100K (left) and Movielens 1M (right) with respect to the number of hidden units.
+
+# C ELECTRONIC MEDICAL RECORDS (EMR) DATASETS
+
+# C.1 NHIS DATASET
+
+Table 5: Debiasing variable sparsity in the case of applying dropout using SN on five disease identification tasks of NHIS dataset. Test AUROC with  $95\%$  confidence interval of 5-runs is provided.  
+
+<table><tr><td>Dataset</td><td>Cardiovascular</td><td>Fatty Liver</td><td>Hypertension</td><td>Heart Failure</td><td>Diabetes</td></tr><tr><td>Zero Imputation w/o SN</td><td>0.7084 ± 0.0005</td><td>0.6858 ± 0.0065</td><td>0.8023 ± 0.0054</td><td>0.7876 ± 0.0012</td><td>0.9263 ± 0.0026</td></tr><tr><td>Zero Imputation w/ SN (ours.)</td><td>0.7105 ± 0.0009</td><td>0.6941 ± 0.0011</td><td>0.8086 ± 0.0016</td><td>0.7922 ± 0.0015</td><td>0.9303 ± 0.0029</td></tr></table>
+
+The NHIS dataset, which is from National Health Insurance Service (NHIS), consists of medical diagnosis of around 300,000 people. The goal is to predict the occurrence of 5 diseases. Each patient takes 34 examinations over 5 years. We split dataset into two sets (train and test), where the ratio of train and test split is 3:1. We pre-process input data with min-max normalization, which makes min and max values of each features be zero and one following GAIN (Yoon et al., 2018). We train 2 layer neural networks which have 50 and 30 hidden units each, and evaluate the model with AUROC. We use ReLU activation, and dropout rate as 0.8 (if applied). Since these datasets is too imbalance, we apply class weight to the loss function for handling label imbalance. Besides, we use Adam optimizer with learning rate  $10^{-2}$  without weight decay, and full batch. We evaluate the model on 5 times and report mean and  $95\%$  confidence interval. We also observe that Sparsity Normalization makes performance gain even when the dropout is integrated in the networks (See Table 5).
+
+Data source This study used the National Health Insurance System-National Health Screening Cohort (NHIS-HEALS)* data derived from a national health screening program and the national health insurance claim database in the National Health Insurance System (NHIS) of South Korea. Data from the NHIS-HEALS $^{10}$  was fully anonymized for all analyses and informed consent was not specifically obtained from each participant. This study was approved and exempt from informed consent by the Institutional Review Board of Yonsei University, Severance Hospital in Seoul, South Korea (IRB no.4-2016-0383).
+
+Data Availability Data cannot be shared publicly because of the provisions of the National Health Insurance Service (NHIS). Korean legal restrictions prohibit authors from making the data publicly available, and the authority implemented the restrictions is NHIS (National Health Insurance Service), one of the government agency of Republic of Korea. NHIS provides limited portion of anonymized data to the researchers for the purpose of the public interest. However, they exclusively provide data to whom made direct contact of the NHIS and agreed to policies of NHIS. Redistribution of the data is not permitted for the researchers. The contact name and the information to which the data request can be sent: Haeryoung Park Information analysis department Big data operation room NHISS Tel: +82-33-736-2430. E-mail: lumen77@nhis.or.kr.
+
+# C.2 PHYSIONET CHALLENGE 2012 DATASET
+
+Table 6: Debiasing variable sparsity using SN on mortality prediction task of PhysioNet Challenge 2012. Test AUROC with  $95\%$  confidence interval of 5-runs is provided.  
+
+<table><tr><td>Model</td><td>Test AUROC</td></tr><tr><td>Zero Imputation w/o SN</td><td>0.8177 ± 0.0071</td></tr><tr><td>Zero Imputation w/ SN (ours.)</td><td>0.8152 ± 0.0010</td></tr></table>
+
+PhysioNet Challenge 2012 dataset (Silva et al., 2012) consists of 48 hours-long multivariate clinical time series data from intensive care unit (ICU). Most of the experimental settings are followed by BRITS (Cao et al., 2018). We divide 48 hours into 288 timesteps, which contain 35 examinations each. The goal of this task is to predict in-hospital death. We use dataset split given by PhysioNet Challenge 2012 (Silva et al., 2012) where each of them contains 4000 data points. In the preprocessing phase, we standardize (make mean as zero and standard deviation as one for each features) the input features and fill zero for missing values (zero imputation). We train single layer LSTM network which has 108 hidden units, and evaluate the model with AUROC. We apply class weight to handle imbalance problem in the dataset like in setting above. We use Adam optimizer with learning rate  $2 \times 10^{-4}$ , 512 batch size, and early stopping method based on AUROC of validation set. We evaluate the model on 5 times and report mean and  $95\%$  confidence interval. Note that input of LSTM model is the results for 35 medical examinations at a specific timestamp. Hence, we apply SN separately for each timestamp. As the aforementioned state of Section 4.2, SN could not make significant performance gain in PhysioNet Challenge 2012 dataset (Silva et al., 2012) though Sparsity Normalization eases the VSP (See Table 6). Nevertheless, SN is still valuable for its ability to prevent biased predictions in this mission-critical area.
+
+# D SINGLE-CELL RNA SEQUENCE DATASETS
+
+In the AutoImpute experiments, we run the experiments using the author's public code $^{11}$ . Talwar et al. (2018) reported experimental results on eight datasets (Blakeley, Jurkat, Kolodziejczyk, Preimplantation, Quake, Usoskin, Zeisel, and PBMC). Since we can obtain preprocessed datasets only for seven datasets except PBMC, we run experiments on seven single cell RNA sequence datasets $^{12}$ . All experimental settings without SN are exactly followed by the author's code and the hyper-parameter settings published in Table 2 of their paper. For the model integrated with SN, all the experimental settings are followed by the original model except that the smaller threshold for early stopping is taken because the models with SN tends to be underfitted by using the threshold as suggested by the authors $^{13}$ . In addition, Talwar et al. (2018) conducted experiments only on cases with test set ratios of  $\{0.1, 0.2, 0.3, 0.4, 0.5\}$ , but we explored more test set ratios  $\{0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9\}$  to show that the SN works well under more sparsity (extremely high test set ratio). Like the authors' settings, we perform 10 experiments and report mean and  $95\%$  confidence intervals. We change the test set split on every trials. It is remarkable that although hyper-parameters are not favorable for SN, SN performs similarly or better on all seven datasets that we concerned (See Figure 3 and 7).
+
+![](images/d134bea28e52141317b88f223e8095b76a1b5399ad8a6b24759b8f49ccad4422.jpg)  
+Figure 7: Debiasing variable sparsity using SN according to test set ratio on imputation task of Preimplantation dataset. Test RMSE with  $95\%$  confidence interval of 10-runs is provided.
+
+# E DROPOUT ON UCI DATASETS
+
+Most experimental settings are adopted from Klambauer et al. (2017); Littwin & Wolf (2018)'s UCI experiments. We use ReLU (Glorot et al., 2011) networks of 4 hidden layers with 256 units. We use Mean Square Error (MSE) for loss function and Adam optimizer without weight decay,  $\epsilon = 10^{-8}$  and learning rate  $10^{-4}$ . The batch size used for training is 128. We split  $10\%$  from the dataset for test and use  $20\%$  of training set as the validation set. For all inputs, we applied min-max normalization, which makes min and max values of each features be 0 and 1. Note that, we use dense datasets without any missing attributes to focus on the effects of dropout. All the datasets are used as provided in the package in sklearn.datasets.
+
+# F DENSITY ESTIMATION
+
+To reproduce binarized MNIST experiments of MADE (Germain et al., 2015), we adopt and slightly modify a public implementation of  $\mathsf{MADE}^{14}$ . Figure 5 shows our reproduction results of Figure 2 in the original paper. We follow most settings of the MADE. We use a single hidden layer MADE networks with 500 units, learning rate 0.005, and test on authors' binarized MNIST dataset<sup>15</sup>. The only difference from the original authors' implementation is that we used Adam ( $\epsilon = 10^{-8}$ , and no weight decay), rather using Adagrad. Because it is observed that the model is undefitted with Adagrad while the learning rates of each element of weight matrix is dropped so rapidly.
+
+Since there is no missingness in the Binarized MNIST dataset, it cannot be divided by  $\| M\| _1$  as suggested by Algorithm 1. In this experiment, we regard  $\| M\| _1$  as  $\| \mathbf{h}^0\| _0$  so as not to lose the generality. That is, all pixels that are 0 in the binarized MNIST dataset are regarded as missing. We label results of this with w/ SN and plot them in Figure 5. As the aforementioned state in Section 4.5, MADE can also cause variable sparsity by mask matrices of each weight. In MADE, the connection between specific units is forcibly controlled through a mechanism of element-wise product of a specific mask matrix  $M^{i}$  in weight  $W^{i}\in \mathbb{R}^{n_{i}\times n_{i - 1}}$ . These mask matrices also cause variation in sparsity. We plot the results of using the new mask matrix  $M_{\mathrm{SN}}^{i}$  in place of the mask matrix  $M^{i}$  used by MADE with w/ SN (all) in Figure 5. The method of calculating the new mask matrix  $M_{\mathrm{SN}}^{i}$  using the existing mask matrix  $M^{i}$  is as follows:
+
+$$
+M _ {\mathrm {S N}} ^ {i} \leftarrow \left(\mathbf {1} ^ {T} M ^ {i} \mathbf {1} / n _ {i}\right) \cdot M ^ {i} \oslash \left(M ^ {i} \mathbf {1} \mathbf {1} ^ {T}\right)
+$$
+
+where  $\varnothing$  denotes element-wise division and  $\mathbf{1}$  is a column vector where all elements are 1.  $(\mathbf{1}^T M^i\mathbf{1} / n_i)$  corresponds to  $K$  and  $(M^{i}\mathbf{1}\mathbf{1}^{T})$  does to  $\| \mathbf{m}\| _1$  in Algorithm 1.
+
+We demonstrate the effectiveness of SN even in situations where the learning rate is smaller. Smaller learning rate (0.001) shows the effect of SN more clearly despite of their longer training time as show in Figure 8.
+
+![](images/efe13d0636fdc85a8599417aec326f92ebc3eaa9dc7acecff8c382bf1976dcf6.jpg)  
+Figure 8: Negative log-likelihood of MADE on binarized MNIST with and without SN in the case of small learning rate (0.001).
+
+# G MACHINE DESCRIPTION
+
+We perform all the experiments on a Titan X with 12GB of VRAM. 12 GB of VRAM is not always necessary, and most experiments require smaller VRAM.
+
+# H COMPARISON TO OTHER MISSING HANDLING TECHNIQUES
+
+In this section, we compare Sparsity Normalization with other missing handling techniques. Although the main contribution of our paper is to provide a deeper understanding and the corresponding solution about the issue that the zero imputation, the simplest and most intuitive way of handling missing data, degrades the performance in training neural networks. Nonetheless, we show that Sparsity Normalization is effective for high dimensional datasets (with a large number of features and high missing rates) via a collaborative filtering dataset, while showing that Sparsity Normalization has competitive results with other modern missing handling techniques for datasets with non-high dimensional setting (electronic medical records datasets, UCI datasets w/ and w/o MCAR assumption). For fair comparisons, we consider the tasks in Section 4, as well as the tasks used by modern missing handling techniques such as GAIN (Yoon et al., 2018) and Šmieja et al. (2018).
+
+As baselines, we consider modern missing handling techniques such as GAIN, Smieja et al. (2018) and their baselines as well.
+
+- Zero Imputation w/o Sparsity Normalization: Missing values are replaced with zero.  
+- Zero Imputation w/ Sparsity Normalization (ours.): Based on zero imputation, apply Sparsity Normalization (SN).  
+- Zero Imputation with Batch Normalization (Ioffe & Szegedy, 2015): Based on zero imputation, apply Batch Normalization (BN) only on the first layer.  
+- Zero Imputation with Layer Normalization (Lei Ba et al., 2016): Based on zero imputation, apply Layer Normalization (LN) only on the first layer.  
+- Dropout<sup>16</sup>: Missing values are replaced with zero and other values are divided by  $\left(E_{(\mathbf{h}^0,\mathbf{m})\in \mathcal{D}}[\| \mathbf{m}\| _1] / n_0\right)$  like standard dropout (Srivastava et al., 2014). Dropout uses a single missing (drop) probability uniformly across all instances of the dataset while SN normalizes each data instance with its own missing rate.  
+- Mean Imputation: Missing values are replaced with the mean of those features.  
+- Median Imputation: Missing values are replaced with the median of those features.  
+-  $k$ -Nearest Neighbors ( $k$ -NN): Missing values are replaced with the mean of those features from  $k$  nearest neighbor samples. We use  $k = 5$  following Smieja et al. (2018)'s experimental setting.  
+- Multivariate Imputation by Chained Equations (MICE): Proposed by Buuren & Groothuis-Oudshoorn (2010).  
+- SoftImpute (Mazumder et al., 2010) $^{17}$  
+- Gaussian Mixture Model Compensator (GMMC): Proposed by Smieja et al. (2018). In the case of GMMC, any activation functions except ReLU and RBF (Radial Basis Function) is prohibitive on the first hidden layer. Thus, the activation function of the first hidden layer is replaced by ReLU in all base architectures without ReLU.  
+GAIN (Yoon et al., 2018)
+
+We implement Mean Imputation, Median Imputation, MICE,  $k$ -NN and SoftImpute by python package fancyimpute. We use authors' official codes for GMMC<sup>18</sup> and GAIN<sup>19</sup>. Layer Normalization and Batch Normalization are not commonly considered in studies of handling missing data. However, we additionally take these as baselines because they have similarities to Sparsity Normalization in terms of stabilizing the statistics of a hidden layer<sup>20</sup> (See Appendix H.1.2 for deeper analysis).
+
+# H.1 COLLABORATIVE FILTERING (RECOMMENDATION) DATASET
+
+In this section, we compare the SN with other missing handling techniques using the collaborative filtering dataset. Appendix H.1.1 compares the prediction performance of the baseline methods and the SN, while Appendix H.1.2 deeply analyzes the characteristics of the SN in comparison with Layer Normalization (Lei Ba et al., 2016) and Batch Normalization (Ioffe & Szegedy, 2015).
+
+# H.1.1 COMPARISON OF PREDICTION PERFORMANCE
+
+It is considered training an AutoRec (Sedhain et al., 2015) model on the Movielens 100K dataset. Most experimental settings are adopted from Section  $4.1^{21}$ . We evaluate each missing handling techniques on both data encoding (user- or item-rating vector) as shown in Table 7. In both encoding, Sparsity Normalization performs better or similar to other missing handling techniques. While some missing handling techniques perform poorly rather than zero imputation depending on the encoding, Sparsity Normalization improves performance consistently for both data encodings. It is worth mentioning that Sparsity Normalization performs statistically significantly better than all other baselines with item vector encoding, which is considered a better encoding scheme in most collaborative filtering models (Salakhutdinov et al., 2007; Sedhain et al., 2015; Zheng et al., 2016).
+
+Table 7: Comparison between Sparsity Normalization and other missing handling techniques using AutoRec on Movielens 100K dataset. Test RMSE with  $95\%$  confidence interval of 5-runs is provided.  
+
+<table><tr><td>Models</td><td>User vector</td><td>Item vector</td></tr><tr><td>Zero Imputation w/o Sparsity Normalization</td><td>0.9346 ± 0.0007</td><td>0.8835 ± 0.0003</td></tr><tr><td>Zero Imputation w/ Sparsity Normalization (ours.)</td><td>0.9280 ± 0.0023</td><td>0.8809 ± 0.0011</td></tr><tr><td>Zero Imputation w/ Batch Normalization</td><td>0.9929 ± 0.0088</td><td>0.9205 ± 0.0081</td></tr><tr><td>Zero Imputation w/ Layer Normalization</td><td>0.9996 ± 0.0131</td><td>0.9396 ± 0.0141</td></tr><tr><td>Dropout</td><td>0.9252 ± 0.0019</td><td>0.9268 ± 0.0261</td></tr><tr><td>Mean Imputation</td><td>0.9310 ± 0.0017</td><td>0.9206 ± 0.0012</td></tr><tr><td>Median Imputation</td><td>0.9333 ± 0.0014</td><td>0.9196 ± 0.0017</td></tr><tr><td>k-NN</td><td>0.9346 ± 0.0007</td><td>0.9133 ± 0.0011</td></tr><tr><td>MICE (Buuren &amp; Groothuis-Oudshoorn, 2010)</td><td>0.9318 ± 0.0015</td><td>0.9209 ± 0.0022</td></tr><tr><td>SoftImpute (Mazumder et al., 2010)</td><td>0.9262 ± 0.0003</td><td>0.8867 ± 0.0007</td></tr><tr><td>GMMC (Śmieja et al., 2018)</td><td>0.9331 ± 0.0067</td><td>0.9109 ± 0.0166</td></tr><tr><td>GAIN (Yoon et al., 2018)</td><td>1.0470 ± 0.0098</td><td>1.0354 ± 0.0101</td></tr></table>
+
+# H.1.2 IS BATCH NORMALIZATION OR LAYER NORMALIZATION ABLE TO SOLVE VSP?
+
+Someone might wonder if Batch Normalization (Ioffe & Szegedy, 2015) or Layer Normalization (Lei Ba et al., 2016) have a similar effect to Sparsity Normalization by alleviating VSP. However, BN or LN can not solve VSP even though these three methods have something in common in terms of stabilizing the statistics of the hidden layer. To validate this, we compare SN with LN and BN by controlling the strength of weight decay regularization. We use the AutoRec model with Movielens 100K for these experiments as in Appendix H.1.1.
+
+Figure 9 shows that VSP occurs in all cases except SN with weak regularization (left column). The model's prediction highly correlates the number of known entries in all methods except SN. On the other hand, strong regularization might seem to solve the VSP while the model shows relatively constant inference regardless of the number of known entries. However, the strong regularization is not an acceptable solution because it gives the less freedom of the model's inference making constant prediction (right column). It must not be natural that the predicted values of the model are almost constant regardless of input sample (we do not want a model that recommends the same movie no
+
+![](images/d07114e13df77a800d434c27d5ea0347487b0a3fc596d5db9a8744e2b22adb72.jpg)  
+Figure 9: Predicted values of AutoRec (Sedhain et al., 2015) (user vector encoding) on Movielens 100K (collaborative filtering) dataset with zero imputation (w/ or w/o a normalization) according to the number of known entries for a randomly selected test point. Input masks are randomly sampled (to artificially control its sparsity level). For each target sparsity level through x-axis, 100 samples are drawn, scattering the predicted values and plotting the average in solid line. Predicted values of the model is also plotted according to the strength of the weight decay controlling  $\lambda$  of the hyper-parameter for weight decay. First row: The vanilla zero imputation. Second row (ours.): Zero imputation with Sparsity Normalization. Third row: Zero imputation with Layer Normalization (Lei Ba et al., 2016). Fourth row: Zero imputation with Batch Normalization (Ioffe & Szegedy, 2015). First column: Weak regularization ( $\lambda = 50$ ). Second column: Moderate regularization ( $\lambda = 500$ ). Third column: Strong regularization ( $\lambda = 5000$ ).
+
+matter which movies a user like/dislike!). This trend can also be seen in the process of tuning the hyper-parameter  $\lambda$  for each model through that the optimal  $\lambda$  value of each model except LN and BN is determined at around 500, whereas that of BN and LN is determined above 500000 (inordinate regularization). In other words, unlike SN, the VSP is not solved with LN or BN. Rather, strong regularization is able to solve the VSP, but this is not a direct solution to VSP forcing the model to choice constant predicted values irrespectively of the input. It is instructive note that the trend of Figure 9 is extremely consistent with the test points as Figure 1.
+
+# H.2 ELECTRONIC MEDICAL RECORDS (EMR) DATASETS
+
+We also compare Sparsity Normalization and baselines for the five disease identification tasks in the NHIS dataset used in Section 4.2. The results are described in Table 3. Sparsity Normalization shows better or similar performance compared to other baseline methods as well.
+
+# H.3 UCI DATASETS
+
+We further compare Sparsity Normalization with other missing handling techniques on UCI datasets which have relatively low missing rates and small feature dimension (non-high dimensional datasets). We consider the UCI datasets used in GAIN (Yoon et al., 2018) and GMMC (Śmieja et al., 2018). The datasets used in the both papers can be divided into two categories: missing features are intentionally injected (w/ MCAR assumption) or missing features exist inherently (w/o MCAR assumption).
+
+We adopt the settings of Klambauer et al. (2017); Littwin & Wolf (2018)'s UCI experiments to use the same Multi Layer Perceptron (MLP) architecture as in Section 4.4: ReLU (Glorot et al., 2011) networks of 4 hidden layers with 256 units. The main purpose of imputation should be to improve prediction performance rather than imputation performance. In this reason, we just focus on the prediction performance of each missing handling techniques for UCI datasets. Because all UCI datasets used in this section are for imbalanced binary classification tasks, prediction performance is reported with AUROC rather than accuracy, and the class weight is considered in loss function. On top of that, we use Adam Optimizer in all experiments for fair comparison with baselines.
+
+Though we adopt datasets used in GAIN (Yoon et al., 2018) and GMMC (Śmieja et al., 2018), we report quite different performance from that of the papers. Several possible reasons are as follows. First, GAIN and GMMC did not publish the train/test dataset split, thus we use our own split which is made under the similar settings of both papers. Second, MLP is used rather than logistic regression or Radial Basis Function Network (RBFN) which are used in GAIN and GMMC respectively. It is because we think that MLP is more reasonable and widely acceptable architecture nowadays[22] than the others. Furthermore, we use AUROC and class weights, unlike the GAIN and GMMC. The final possible reason is that we use Adam Optimizer for all models because SGD with learning rate decay is difficult for fair comparison when hyper-parameters are set in favor to a particular model. In these ways, we do our best to compare Sparsity Normalization and other missing handling techniques including GMMC and GAIN in the most fair and reasonable setting.
+
+Table 8: Summary of UCI datasets with and without MCAR assumption.  
+
+<table><tr><td rowspan="2">Dataset</td><td colspan="5">w/ MCAR assumption</td><td colspan="5">w/o MCAR assumption</td></tr><tr><td>Breast</td><td>Spam</td><td>Credit</td><td>Crashes</td><td>Heart</td><td>Bands</td><td>Hepartitis</td><td>Horse</td><td>Mammographics</td><td>Pima</td></tr><tr><td># Instances</td><td>569</td><td>4601</td><td>30000</td><td>540</td><td>270</td><td>539</td><td>155</td><td>368</td><td>961</td><td>768</td></tr><tr><td># Attributes</td><td>30</td><td>57</td><td>23</td><td>20</td><td>13</td><td>19</td><td>19</td><td>22</td><td>5</td><td>8</td></tr><tr><td>Missing Rate (%)</td><td>20.0</td><td>20.0</td><td>20.0</td><td>50.0</td><td>50.0</td><td>5.38</td><td>5.67</td><td>23.8</td><td>3.37</td><td>12.2</td></tr></table>
+
+# H.3.1 UCI DATASETS WITH MCAR ASSUMPTION
+
+We deliberately inject missing values into the datasets which don't have any missing attributes internally (w/ MCAR assumption) to perform binary classification tasks. We consider Breast, Spam, and Credit datasets from GAIN (Yoon et al., 2018) and Crashes and Heart datasets from GMMC (Śmieja et al., 2018). We make  $20\%$  of all features be missing for the Breast, Spam, and Credit datasets following GAIN, and  $50\%$  for the Crashes and Heart datasets following GMMC. The summary of the datasets are described in Table 8. For Breast, Spam, and Credit datasets taken by GAIN, we perform min-max normalization which makes min and max values of each features be 0 and 1 following GAIN paper, and for Crashes and Heart datasets taken by GMMC, we perform another kind of min-max normalization which makes min and max values of each features be -1 and 1 following GMMC paper.
+
+Table 9: Comparison between Sparsity Normalization and other missing handling techniques on five imbalanced binary classification tasks in UCI datasets with MCAR assumption. Test AUROC with  $95\%$  confidence interval of 5-runs is provided.  
+
+<table><tr><td>Models</td><td>Breast</td><td>Spam</td><td>Credit</td><td>Crashes</td><td>Heart</td></tr><tr><td>Zero Imputation w/o SN</td><td>0.9958 ± 0.0067</td><td>0.9699 ± 0.0040</td><td>0.7344 ± 0.0135</td><td>0.9070 ± 0.1050</td><td>0.8967 ± 0.0268</td></tr><tr><td>Zero Imputation w/ SN (ours.)</td><td>0.9992 ± 0.0023</td><td>0.9700 ± 0.0032</td><td>0.7292 ± 0.0098</td><td>0.8905 ± 0.0934</td><td>0.9055 ± 0.0327</td></tr><tr><td>Zero Imputation w/ BN</td><td>1.0000 ± 0.0000</td><td>0.9666 ± 0.0051</td><td>0.7178 ± 0.0143</td><td>0.9070 ± 0.0645</td><td>0.8857 ± 0.0268</td></tr><tr><td>Zero Imputation w/ LN</td><td>0.9979 ± 0.0059</td><td>0.9704 ± 0.0031</td><td>0.7353 ± 0.0051</td><td>0.9095 ± 0.0972</td><td>0.8863 ± 0.0177</td></tr><tr><td>Dropout</td><td>0.9897 ± 0.0209</td><td>0.9697 ± 0.0031</td><td>0.7313 ± 0.0066</td><td>0.9220 ± 0.0553</td><td>0.8857 ± 0.0335</td></tr><tr><td>Mean Imputation</td><td>0.9966 ± 0.0089</td><td>0.9690 ± 0.0028</td><td>0.7349 ± 0.0066</td><td>0.8990 ± 0.1081</td><td>0.9011 ± 0.0522</td></tr><tr><td>Median Imputation</td><td>0.9987 ± 0.0045</td><td>0.9695 ± 0.0024</td><td>0.7353 ± 0.0044</td><td>0.9080 ± 0.0892</td><td>0.9253 ± 0.0407</td></tr><tr><td>k-NN</td><td>1.0000 ± 0.0000</td><td>0.9613 ± 0.0035</td><td>0.5270 ± 0.0171</td><td>0.8940 ± 0.0356</td><td>0.8703 ± 0.0435</td></tr><tr><td>MICE</td><td>1.0000 ± 0.0000</td><td>0.9680 ± 0.0041</td><td>0.5315 ± 0.0187</td><td>0.9760 ± 0.0424</td><td>0.9066 ± 0.0410</td></tr><tr><td>SoftImpute</td><td>1.0000 ± 0.0000</td><td>0.9725 ± 0.0021</td><td>0.5375 ± 0.0184</td><td>0.9390 ± 0.0754</td><td>0.8846 ± 0.0437</td></tr><tr><td>GMMC</td><td>0.9987 ± 0.0026</td><td>0.9679 ± 0.0029</td><td>0.7355 ± 0.0053</td><td>0.9400 ± 0.0422</td><td>0.9264 ± 0.0223</td></tr><tr><td>GAIN</td><td>1.0000 ± 0.0000</td><td>0.9688 ± 0.0029</td><td>0.7356 ± 0.0045</td><td>0.9650 ± 0.0360</td><td>0.8879 ± 0.0196</td></tr></table>
+
+The experimental results are summarized in Table 9. It is difficult to find a significant difference in prediction performance among each missing handling techniques for datasets with small feature dimensions. The results of these experiments are also consistent with experiments of GAIN (Yoon et al., 2018). Though the GAIN showed significantly better imputation performance compared to their baseline methods, prediction performances were not statistically significant (See Table 3 of the GAIN paper, and note that they didn't report  $95\%$  confidence interval but standard deviation). From these overall results, we conclude that SN is quite comparable for the datasets of low dimension/missing rate with MCAR assumption.
+
+# H.3.2 UCI DATASETS WITHOUT MCAR ASSUMPTION
+
+We compare SN and each missing handling techniques on the datasets that have internal missingness (w/o MCAR assumption). We consider Bands, Hepartitis, Horse, Mammographics, and Pima datasets experimented in the GMMC (Śmieja et al., 2018) paper (See Table 8 for statistics of the datasets). Following the GMMC paper, the min-max normalization is performed, which makes min and max values of each features be -1 and 1. As shown in Table 10, it is concluded that even without MCAR assumption, SN shows comparable results for the datasets of low dimension/missing rate.
+
+Table 10: Comparison between Sparsity Normalization and other missing handling techniques on five imbalanced binary classification tasks in UCI datasets without MCAR assumption. Test AUROC with  $95\%$  confidence interval of 5-runs is provided.  
+
+<table><tr><td>Models</td><td>Bands</td><td>Hepartitis</td><td>Horse</td><td>Mammographics</td><td>Pima</td></tr><tr><td>Zero Imputation w/o SN</td><td>0.7851 ± 0.0962</td><td>0.8222 ± 0.0826</td><td>0.9090 ± 0.0331</td><td>0.8835 ± 0.0083</td><td>0.8466 ± 0.0180</td></tr><tr><td>Zero Imputation w/ SN (ours.)</td><td>0.7939 ± 0.1011</td><td>0.8556 ± 0.1300</td><td>0.9238 ± 0.0311</td><td>0.8787 ± 0.0063</td><td>0.8355 ± 0.0233</td></tr><tr><td>Zero Imputation w/ BN</td><td>0.7512 ± 0.0716</td><td>0.7889 ± 0.1061</td><td>0.8827 ± 0.0362</td><td>0.8933 ± 0.0055</td><td>0.8614 ± 0.0106</td></tr><tr><td>Zero Imputation w/ LN</td><td>0.8000 ± 0.0789</td><td>0.8222 ± 0.0911</td><td>0.9115 ± 0.0404</td><td>0.8854 ± 0.0097</td><td>0.7972 ± 0.0074</td></tr><tr><td>Dropout</td><td>0.7956 ± 0.1012</td><td>0.8194 ± 0.1089</td><td>0.9034 ± 0.0507</td><td>0.8816 ± 0.0059</td><td>0.8408 ± 0.0292</td></tr><tr><td>Mean Imputation</td><td>0.7765 ± 0.0504</td><td>0.8556 ± 0.0895</td><td>0.9009 ± 0.0271</td><td>0.8834 ± 0.0113</td><td>0.8382 ± 0.0142</td></tr><tr><td>Median Imputation</td><td>0.7997 ± 0.0682</td><td>0.8889 ± 0.0544</td><td>0.9133 ± 0.0379</td><td>0.8783 ± 0.0070</td><td>0.8334 ± 0.0099</td></tr><tr><td>k-NN</td><td>0.7571 ± 0.0430</td><td>0.8833 ± 0.1047</td><td>0.9300 ± 0.0363</td><td>0.8840 ± 0.0141</td><td>0.6237 ± 0.0399</td></tr><tr><td>MICE</td><td>0.7618 ± 0.0297</td><td>0.8333 ± 0.0667</td><td>0.9195 ± 0.0431</td><td>0.8814 ± 0.0087</td><td>0.6477 ± 0.0655</td></tr><tr><td>SoftImpute</td><td>0.7644 ± 0.0475</td><td>0.8139 ± 0.1047</td><td>0.9232 ± 0.0124</td><td>0.8848 ± 0.0089</td><td>0.5912 ± 0.0341</td></tr><tr><td>GMMC</td><td>0.7985 ± 0.0605</td><td>0.8500 ± 0.0621</td><td>0.9000 ± 0.0124</td><td>0.8844 ± 0.0090</td><td>0.8391 ± 0.0201</td></tr><tr><td>GAIN</td><td>0.7993 ± 0.0785</td><td>0.8222 ± 0.0826</td><td>0.8941 ± 0.0322</td><td>0.8859 ± 0.0091</td><td>0.8457 ± 0.0178</td></tr></table>
+
+# H.4 CONCLUSION
+
+In conclusion, Sparsity Normalization is significantly superior to other missing handling techniques for the high dimensional/missing rate datasets. Sparsity Normalization performs well compared to other modern missing handling techniques even on non-high dimensional datasets. Sparsity Normalization is valuable in that it performs better than other models and does not require additional training or parameters. Moreover, Sparsity Normalization is computationally inexpensive compared to Mean or Median Imputation because sparse tensors can be used to save computational costs to calculate first hidden layer not with mean or median imputation but with zero imputation (w/ or w/o SN). The reduced computational cost by using sparse tensors is relatively higher when we deal with high-dimensional datasets.
\ No newline at end of file
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+# YOU CAN TEACH AN OLD DOG NEW TRICKS! ON TRAINING KNOWLEDGE GRAPH EMBEDDINGS
+
+Daniel Ruffinelli*
+
+Data and Web Science Group
+
+University of Mannheim, Germany
+
+Samuel Broscheit*
+
+Data and Web Science Group
+
+University of Mannheim, Germany
+
+Rainer Gemulla†
+
+Data and Web Science Group
+
+University of Mannheim, Germany
+
+# ABSTRACT
+
+Knowledge graph embedding (KGE) models learn algebraic representations of the entities and relations in a knowledge graph. A vast number of KGE techniques for multi-relational link prediction have been proposed in the recent literature, often with state-of-the-art performance. These approaches differ along a number of dimensions, including different model architectures, different training strategies, and different approaches to hyperparameter optimization. In this paper, we take a step back and aim to summarize and quantify empirically the impact of each of these dimensions on model performance. We report on the results of an extensive experimental study with popular model architectures and training strategies across a wide range of hyperparameter settings. We found that when trained appropriately, the relative performance differences between various model architectures often shrinks and sometimes even reverses when compared to prior results. For example, RESCAL (Nickel et al., 2011), one of the first KGE models, showed strong performance when trained with state-of-the-art techniques; it was competitive to or outperformed more recent architectures. We also found that good (and often superior to prior studies) model configurations can be found by exploring relatively few random samples from a large hyperparameter space. Our results suggest that many of the more advanced architectures and techniques proposed in the literature should be revisited to reassess their individual benefits. To foster further reproducible research, we provide all our implementations and experimental results as part of the open source LibKGE framework.
+
+# 1 INTRODUCTION
+
+Knowledge graph embedding (KGE) models learn algebraic representations, termed embeddings, of the entities and relations in a knowledge graph. They have been successfully applied to knowledge graph completion (Nickel et al., 2015) as well as in downstream tasks and applications such as recommender systems (Wang et al., 2017) or visual relationship detection (Baier et al., 2017).
+
+A vast number of different KGE models for multi-relational link prediction have been proposed in the recent literature; e.g., RESCAL (Nickel et al., 2011), TransE (Bordes et al., 2013), DistMult, ComplEx (Trouillon et al., 2016), ConvE (Dettmers et al., 2018), TuckER (Balazevic et al., 2019), RotatE (Sun et al., 2019a), SACN (Shang et al., 2019), and many more. Model architectures generally differ in the way the entity and relation embeddings are combined to model the presence or absence of an edge (more precisely, a subject-predicate-object triple) in the knowledge graph; they include factorization models (e.g., RESCAL, DistMult, ComplEx, TuckER), translational models (TransE, RotatE), and more advanced models such as convolutional models (ConvE). In many cases, the introduction of new models went along with new approaches for training these models—e.g., new training types (such as negative sampling or 1vsAll scoring), new loss functions (such as
+
+<table><tr><td>Publication</td><td>Model</td><td>Loss</td><td>Training</td><td>Regularizer</td><td>Optimizer</td><td>Reciprocal</td></tr><tr><td>Nickel et al. (2011)</td><td>RESCAL</td><td>MSE</td><td>Full</td><td>L2</td><td>ALS</td><td>No</td></tr><tr><td>Bordes et al. (2013)</td><td>TransE</td><td>MR</td><td>NegSamp</td><td>Normalization</td><td>SGD</td><td>No</td></tr><tr><td>Yang et al. (2015)</td><td>DistMult</td><td>MR</td><td>NegSamp</td><td>Weighted L2</td><td>Adagrad</td><td>No</td></tr><tr><td>Trouillon et al. (2016)</td><td>ComplEx</td><td>BCE</td><td>NegSamp</td><td>Weighted L2</td><td>Adagrad</td><td>No</td></tr><tr><td>Kadlec et al. (2017)</td><td>DistMult</td><td>CE</td><td>NegSamp</td><td>Weighted L2</td><td>Adam</td><td>No</td></tr><tr><td>Dettmers et al. (2018)</td><td>ConvE</td><td>BCE</td><td>KvsAll</td><td>Dropout</td><td>Adam</td><td>Yes</td></tr><tr><td>Lacroix et al. (2018)</td><td>ComplEx</td><td>CE</td><td>1vsAll</td><td>Weighted L3</td><td>Adagrad</td><td>Yes</td></tr></table>
+
+MSE = mean squared error, MR = margin ranking, BCE = binary cross entropy, CE = cross entropy
+
+Table 1: Selected KGE models and training strategies from the literature. Entries marked in bold were introduced (or first used) in the context of KGE in the corresponding publication.
+
+pairwise margin ranking or binary cross entropy), new forms of regularization (such as unweighted and weighted L2), or the use of reciprocal relations (Kazemi & Poole, 2018; Lacroix et al., 2018)—and ablation studies were not always performed. Table 1 shows an overview of selected models and training techniques along with the publications that introduced them.
+
+The diversity in model training makes it difficult to compare performance results for various model architectures, especially when results are reproduced from prior studies that used a different experimental setup. Model hyperparameters are commonly tuned using grid search on a small grid involving hand-crafted parameter ranges or settings known to "work well" from prior studies. A grid suitable for one model may be suboptimal for another, however. Indeed, it has been observed that newer training strategies can considerably improve model performance (Kadlec et al., 2017; Lacroix et al., 2018; Salehi et al., 2018).
+
+In this paper, we take a step back and aim to summarize and quantify empirically the impact of different model architectures and different training strategies on model performance. We performed an extensive set of experiments using popular model architectures and training strategies in a common experimental setup. In contrast to most prior work, we considered many training strategies as well as a large hyperparameter space, and we performed model selection using quasi-random search (instead of grid search) followed by Bayesian optimization. We found that this approach was able to find good (and often superior to prior studies) model configurations with relatively low effort. Our study complements and expands on the results of Kotnis & Nastase (2018) (focus on negative sampling) and Mohamed et al. (2019) (focus on loss functions) as well as similar studies in other areas, including language modeling (Melis et al., 2017), generative adversarial networks (Lucic et al., 2018), or sequence tagging (Reimers & Gurevych, 2017).
+
+We found that when trained appropriately, the performance of a particular model architecture can by far exceed the performance observed in older studies. For example, RESCAL (Nickel et al., 2011), which constitutes one of the first KGE models but is rarely considered in newer work, showed very strong performance in our study: it was competitive to or outperformed more recent architectures such as ConvE (Dettmers et al., 2018) and TuckER (Balazevic et al., 2019). More generally, we found that the relative performance differences between various model architectures often shrunk and sometimes even reversed when compared to prior results. This suggests that (at least currently) training strategies have a significant impact on model performance and may account for a substantial fraction of the progress made in recent years. We also found that suitable training strategies and hyperparameter settings vary significantly across models and datasets, indicating that a small search grid may bias results on model performance. Fortunately, as indicated above, large hyperparameter spaces can be (and should be) used with little additional training effort. To facilitate such efforts, we provide implementations of relevant training strategies, models, and evaluation methods as part of the open source LibKGE framework, $^{1}$  which emphasizes reproducibility and extensibility.
+
+Our study focuses solely on pure KGE models, which do not exploit auxiliary information such as textual data or logical rules (Wang et al., 2017). Since many of the studies on these non-pure models did not (and, to be fair, could not) use current training strategies and consequently underestimated the performance of pure KGE models, their results and conclusions need to be revisited.
+
+# 2 KNOWLEDGE GRAPH EMBEDDINGS: MODELS, TRAINING, EVALUATION
+
+The literature on KGE models is expanding rapidly. We review selected architectures, training methods, and evaluation protocols; see Table 1. The table exemplarily indicates that new model architectures are sometimes introduced along with new training strategies (marked bold). Reasonably recent survey articles about KGE models include Nickel et al. (2015) and Wang et al. (2017).
+
+Multi-relational link prediction. KGE models are typically trained and evaluated in the context of multi-relational link prediction for knowledge graphs (KG). Generally, given a set  $\mathcal{E}$  of entities and a set  $\mathcal{R}$  of relations, a knowledge graph  $\mathcal{K} \subseteq \mathcal{E} \times \mathcal{R} \times \mathcal{E}$  is a set of subject-predicate-object (spo) triples. The goal of multi-relational link prediction is to "complete the KG", i.e., to predict true but unobserved triples based on the information in  $\mathcal{K}$ . Common approaches include rule-based methods (Galarraga et al., 2013; Meilicke et al., 2019), KGE methods (Nickel et al., 2011; Bordes et al., 2013; Trouillon et al., 2016; Dettmers et al., 2018), and hybrid methods (Guo et al., 2018).
+
+Knowledge graph embeddings (KGE). A KGE model associates with each entity  $i \in \mathcal{E}$  and relation  $k \in \mathcal{R}$  an embedding  $\boldsymbol{e}_i \in \mathbb{R}^{d_e}$  and  $\boldsymbol{r}_k \in \mathbb{R}^{d_r}$  in a low-dimensional vector space, respectively; here  $d_e, d_r \in \mathbb{N}^+$  are hyperparameters for the embedding size. Each particular model uses a scoring function  $s: \mathcal{E} \times \mathcal{R} \times \mathcal{E} \to \mathbb{R}$  to associate a score  $s(i, k, j)$  with each potential triple  $(i, k, j) \in \mathcal{E} \times \mathcal{R} \times \mathcal{E}$ . The higher the score of a triple, the more likely it is considered to be true by the model. The scoring function takes form  $s(i, k, j) = f(\boldsymbol{e}_i, \boldsymbol{r}_k, \boldsymbol{e}_j)$ , i.e., depends on  $i, k$ , and  $j$  only through their respective embeddings. Here  $f$ , which represents the model architecture, may be either a fixed function or learned (e.g.,  $f$  may be a parameterized function).
+
+Evaluation. The most common evaluation task for KGE methods is entity ranking, which is a form of question answering. The available data is partitioned into a set of training, validation, and test triples. Given a test triple  $(i,k,j)$  (unseen during training), the entity ranking task is to determine the correct answer—i.e., the missing entity  $j$  and  $i$ , resp.—to questions  $(i,k,?)$  and  $(?,k,j)$ . To do so, potential answer triples are first ranked by their score in descending order. All triples but  $(i,k,j)$  that occur in the training, validation, or test data are subsequently filtered out so that other triples known to be true do not affect the ranking. Finally, metrics such as the mean reciprocal rank (MRR) of the true answer or the average HITS@ $k$  are computed; see the appendix.
+
+KGE models. KGE model architectures differ in their scoring function. We can roughly classify models as decomposable or monolithic: the former only allow element-wise interactions between (relation-specific) subject and object embeddings, whereas the latter allow arbitrary interactions. More specifically, decomposable models use scoring functions of form  $s(i,k,j) = f(\sum_{z} g([h_1(e_i,e_r) \circ h_2(e_r,e_j)]_z))$ , where  $\circ$  is any element-wise function (e.g., multiplication),  $h_1$  and  $h_2$  are functions that obtain relation-specific subject and object embeddings, resp., and  $g$  and  $f$  are scalar functions (e.g., identity or sigmoid). The most popular models in this category are perhaps RESCAL (Nickel et al., 2011), TransE (Bordes et al., 2013), DistMult (Yang et al., 2015), ComplEx (Trouillon et al., 2016), and ConvE (Dettmers et al., 2018). RESCAL's scoring function is bilinear in the entity embeddings: it uses  $s(i,k,j) = e_i^T R_k e_j$ , where  $R_k \in \mathbb{R}^{d_e \times d_e}$  is a matrix formed from the entries of  $r_k \in \mathbb{R}^{d_r}$  (where  $d_r = d_e^2$ ). DistMult and ComplEx can be seen as constrained variants of RESCAL with smaller relation embeddings ( $d_r = d_e$ ). TransE is a translation-based model and uses negative distance  $-\| e_i + r_k - e_j\|_p$  between  $e_i + r_k$  and  $e_j$  as score, commonly using the L1 norm ( $p = 1$ ) or the L2 norm ( $p = 2$ ). Finally, ConvE uses a 2D convolutional layer and a large fully connected layer to obtain relation-specific entity embeddings (i.e., in  $h_1$  above). Other recent examples for decomposable models include TuckER (Balazevic et al., 2019), RotatE (Sun et al., 2019a), and SACN (Shang et al., 2019). Decomposable models are generally fast to use: once the relation-specific embeddings are (pre-)computed, score computations are cheap. Monolithic models—such as ConvKB or KBGAT—do not decompose into relation-specific embeddings: they take form  $s(i,k,j) = f(e_i,r_k,e_j)$ . Such models are more flexible, but they are also considerably more costly to train and use. It is currently unclear whether monolithic models can achieve comparable or better performance than decomposable models Sun et al. (2019b).
+
+Training type. There are three commonly used approaches to train KGE models, which differ mainly in the way negative examples are generated. First, training with negative sampling (NegSamp) (Bordes et al., 2013) obtains for each positive triple  $t = (i,k,j)$  from the training data a set of (pseudo-)negative triples obtained by randomly perturbing the subject, relation, or object position in  $t$  (and optionally verifying that the so-obtained triples do not exist in the KG). An
+
+alternative approach (Lacroix et al., 2018), which we term  $I$  vsAll, is to omit sampling and take all triples that can be obtained by perturbing the subject and object positions as negative examples for  $t$  (even if these tuples exist in the KG).  $I$  vsAll is generally more expensive than NegSamp, but it is feasible (and even surprisingly fast in practice) if the number of entities is not excessively large. Finally, Dettmers et al. (2018) proposed a training type that we term  $K$  vsAll²: this approach (i) constructs batches from non-empty rows  $(i, k, *)$  or  $(*, k, j)$  instead of from individual triples, and (ii) labels all such triples as either positive (occurs in training data) or negative (otherwise).
+
+Loss functions. Several loss functions for training KGEs have been introduced so far. RESCAL originally used squared error between the score of each triple and its label (positive or negative). TransE used pairwise margin ranking with hinge loss (MR), where each pair consists of a positive triple and one of its negative triples (only applicable to NegSamp and 1vsAll) and the margin  $\eta$  is a hyperparameter. Trouillon et al. (2016) proposed to use binary cross entropy (BCE) loss: it applies a sigmoid to the score of each (positive or negative) triple and uses the cross entropy between the resulting probability and that triple's label as loss. BCE is suitable for multi-class and multi-label classification. Finally, Kadlec et al. (2017) used cross entropy (CE) between the model distribution (softmax distribution over scores) and the data distribution (labels of corresponding triples, normalized to sum to 1). CE is more suitable for multi-class classification (as in NegSamp and 1vsAll), but it has also been used in the multi-label setting (KvsAll). Mohamed et al. (2019) found that the choice of loss function can have a significant impact on model performance, and that the best choice is data and model dependent. Our experimental study provides additional evidence for this finding.
+
+Reciprocal relations. Kazemi & Poole (2018) and Lacroix et al. (2018) introduced the technique of reciprocal relations into KGE training. Observe that during evaluation and also most training methods discussed above, the model is solely asked to score subjects (for questions of form  $(?, k, j)$ ) or objects (for questions of form  $(i, k, ?)$ ). The idea of reciprocal relations is to use separate scoring functions  $s_{\mathrm{sub}}$  and  $s_{\mathrm{obj}}$  for each of these tasks, resp. Both scoring functions share entity embeddings, but they do not share relation embeddings: each relation thus has two embeddings. The use of reciprocal relations may decrease the computational cost (as in the case of ConvE), and it may also lead to better model performance Lacroix et al. (2018) (e.g., for relations in which one direction is easier to predict). On the downside, the use of reciprocal relations means that a model does not provide a single triple score  $s(i, k, j)$  anymore; generally,  $s_{\mathrm{sub}}(i, k, j) \neq s_{\mathrm{obj}}(i, k, j)$ . Kazemi & Poole (2018) proposed to take the average of the two triple scores and explored the resulting models.
+
+Regularization. The most popular form of regularization in the literature is L2 regularization on the embedding vectors, either unweighted or weighted by the frequency of the corresponding entity or relation (Yang et al., 2015). Lacroix et al. (2018) proposed to use L3 regularization. TransE normalized the embeddings to unit norm after each update. ConvE used dropout (Srivastava et al., 2014) in its hidden layers (and only in those). In our study, we additionally consider L1 regularization and the use of dropout on the entity and/or relation embeddings.
+
+Hyperparameters. Many more hyperparameters have been used in prior work. This includes, for example, different methods to initialize model parameters, different optimizers, different optimizer parameters such as the learning rate or batch size, the number of negative examples for NegSamp, the regularization weights for entities and relations, and so on. To deal with such a large search space, most prior studies favor grid search over a small grid where most of these hyperparameters remain fixed. As discussed before, this approach may lead to bias in the results, however.
+
+# 3 EXPERIMENTAL STUDY
+
+In this section, we report on the design and results of our experimental study. We focus on the most salient points here; more details can be found in the appendix.
+
+# 3.1 EXPERIMENTAL SETUP
+
+Datasets. We used the FB15K-237 (Toutanova & Chen, 2015) (extracted from Freebase) and WNRR (Dettmers et al., 2018) (extracted from WordNet) datasets in our study. We chose these datasets because (i) they are frequently used in prior studies, (ii) they are "hard" datasets that have been designed specifically to evaluate multi-relational link prediction techniques, (iii) they are diverse in that relative model performance often differs, and (iv) they are of reasonable size for a large study. Dataset statistics are given in Table 4 in the appendix.
+
+Models. We selected RESCAL (Nickel et al., 2011), TransE (Bordes et al., 2013), DistMult (Yang et al., 2015), ComplEx (Trouillon et al., 2016) and ConvE (Dettmers et al., 2018) for our study. These models are perhaps the most well-known KGE models and include both early and more recent models. We did not consider monolithic models due to their excessive training cost.
+
+Evaluation. We report filtered MRR (\%) and filtered HITS@10 (\%); see Sec. 2 and the appendix for details. We use test data to compare final model performance (Table 2) and validation data for our more detailed analysis.
+
+Hyperparameters. We used a large hyperparameter space to ensure sure that suitable hyperparameters for each model are not excluded a priori. We included all major training types (NegSamp, 1vsAll, KvsAll), use of reciprocal relations, loss functions (MR, BCE, CE), regularization techniques (none/L1/L2/L3, dropout), optimizers (Adam, Adagrad), and initialization methods (4 in total) used in the KGE community as hyperparameters. We considered three embeddings sizes (128, 256, 512) and used separate weights for dropout/regularization for entity and relation embeddings. Table 5 in the appendix provides additional details. To the best of our knowledge, no prior study used such a large hyperparameter search space.
+
+Training. All models were trained for a maximum of 400 epochs. We validated models using filtered MRR (on validation data) every five epochs and performed early stopping with a patience of 50 epochs. To keep search tractable, we stopped training on models that did not reach  $\geq 5\%$  filtered MRR after 50 epochs; in our experience, such configurations did not produce good models.
+
+Model selection. Model selection was performed using filtered MRR on validation data. We used the Ax framework (https://ax.dev/) to conduct quasi-random hyperparameter search via a Sobol sequence. Quasi-random search methods aim to distribute hyperparameter settings evenly and try avoid "clumping" effects (Bergstra & Bengio, 2012). More specifically, for each dataset and model, we generated 30 different configurations per valid combination of training type and loss function (2 for TransE, which only supports NegSamp+MR and NegSamp+CE; 7 for all other models). After the quasi-random hyperparameter search, we added a short Bayesian optimization phase (best configuration so far + 20 new trials, using expected improvement; also provided by Ax) to tune the numerical hyperparameters further. Finally, we trained five models with the so-obtained hyperparameter configuration and selected the best-performing model according to validation MRR as the final model. The standard deviation of the validation MRR was relatively low; see Table 9.
+
+Reproducibility. We implemented all models, training strategies, evaluation, and hyperparameter search in the LibKGE framework. The framework emphasizes reproducibility and extensibility and is highly configurable. We provide all configurations used in this study as well as the detailed data of our experiments for further analysis.4 We hope that these resources facilitate the evaluation of KGE models in a comparable environment.
+
+# 3.2 COMPARISON OF MODEL PERFORMANCE
+
+Table 2 shows the filtered MRR and filtered Hits@10 on test data of various models both from prior studies and the best models (according filtered validation MRR) found in our study.
+
+First reported performance vs. our observed performance. We compared the first reported performance on FB15K-237 and WNRR ("First" block of Table 2) with the performance obtained in our study ("Ours" block). We found that the performance of a single model can vary wildly across studies. For example, ComplEx was first run on FB15K-237 by Dettmers et al. (2018), where it achieved a filtered MRR of  $24.7\%$ . This is a relatively low number by today's standards. In our
+
+<table><tr><td rowspan="2" colspan="2"></td><td colspan="2">FB15K-237</td><td colspan="2">WNRR</td></tr><tr><td>MRR</td><td>Hits@10</td><td>MRR</td><td>Hits@10</td></tr><tr><td rowspan="5">First</td><td>RESCAL (Wang et al., 2019)</td><td>27.0</td><td>42.7</td><td>42.0</td><td>44.7</td></tr><tr><td>TransE (Nguyen et al., 2018)</td><td>29.4</td><td>46.5</td><td>22.6</td><td>50.1</td></tr><tr><td>DistMult (Dettmers et al., 2018)</td><td>24.1</td><td>41.9</td><td>43.0</td><td>49.0</td></tr><tr><td>ComplEx (Dettmers et al., 2018)</td><td>24.7</td><td>42.8</td><td>44.0</td><td>51.0</td></tr><tr><td>ConvE (Dettmers et al., 2018)</td><td>32.5</td><td>50.1</td><td>43.0</td><td>52.0</td></tr><tr><td rowspan="5">Ours</td><td>RESCAL</td><td>35.7</td><td>54.1</td><td>46.7</td><td>51.7</td></tr><tr><td>TransE</td><td>31.3</td><td>49.7</td><td>22.8</td><td>52.0</td></tr><tr><td>DistMult</td><td>34.3</td><td>53.1</td><td>45.2</td><td>53.1</td></tr><tr><td>ComplEx</td><td>34.8</td><td>53.6</td><td>47.5</td><td>54.7</td></tr><tr><td>ConvE</td><td>33.9</td><td>52.1</td><td>44.2</td><td>50.4</td></tr><tr><td rowspan="3">Recent</td><td>TuckER (Balazevic et al., 2019)</td><td>35.8</td><td>54.4</td><td>47.0</td><td>52.6</td></tr><tr><td>RotatE (Sun et al., 2019a)</td><td>33.8</td><td>53.3</td><td>47.6</td><td>57.1</td></tr><tr><td>SACN (Shang et al., 2019)</td><td>35.0</td><td>54.0</td><td>47.0</td><td>54.4</td></tr><tr><td rowspan="2">Large</td><td>DistMult (Salehi et al., 2018)</td><td>35.7</td><td>54.8</td><td>45.5</td><td>54.4</td></tr><tr><td>ComplEx-N3 (Lacroix et al., 2018)</td><td>37.0</td><td>56.0</td><td>49.0</td><td>58.0</td></tr></table>
+
+Table 2: Model performance in prior studies and our study (as percentages, on test data). First: first reported performance on the respective datasets (oldest models first); Ours: performance in our study; Recent: best performance results obtained in prior studies of selected recent models; Large: best performance achieved in prior studies using very large models (not part of our search space). Bold numbers indicate best performance in group. References indicate where the performance number was reported.
+
+study, ComplEx achieved a competitive MRR of  $34.8\%$ , which is a large improvement over the first reports. Studies that report the lower performance number of ComplEx (i.e.,  $24.7\%$ ) thus do not adequately capture its performance. Similar remarks hold for RESCAL and DistMult as well as (albeit to a smaller extent) ConvE and TransE. To the best of our knowledge, our results for RESCAL and ConvE constitute the best results for these models obtained so far.
+
+Relative model performance (our study). Next, we compared the performance across the models models used in our study ("Ours" block). We found that the relative performance differences between model architectures often shrunk and sometimes reversed when compared to prior results ("First" block). For example, ConvE showed the best overall performance in prior studies, but is consistently outperformed by RESCAL, DistMult, and ComplEx in our study. As another example, RESCAL (Nickel et al., 2011), which constitutes one of the first KGE models but is rarely considered in newer work, showed strong performance and outperformed all models except ComplEx. This suggests that (at least currently) training strategies have a significant impact on model performance and may account for a large fraction of the progress made in recent years.
+
+Relative model performance (overall). Table 2 additionally shows the best performance results obtained in prior studies of selected recent models ("Recent" block) and very large models with very good performance ("Large" block). We found that TuckER (Balazevic et al., 2019), RotatE (Sun et al., 2019a), and SACN (Shang et al., 2019) all achieve state-of-the-art performance, but the performance difference to the best prior models ("Ours" block) in terms of MRR is small or even vanishes. Even for HITS@10, which we did not consider for model selection, the advantage of more recent models is often small, if present. The models in the last block ("Large") show the best performance numbers we have seen in the literature. For DistMult and ComplEx, these numbers have been obtained using very large embedding sizes (up to 4000). Such large models were not part of our search space.
+
+Limitations. We note that all models used in our study are likely to yield better performance when hyperparameter tuning is further improved. For example, we found a ComplEx configuration (of size 100) that achieved  $49.0\%$  MRR and  $56.5\%$  Hits@10 on WNRR, and a RESCAL configuration that achieved  $36.3\%$  MRR and  $54.6\%$  Hits@10 on FB15K-237 in preliminary experiments. Sun et al. (2019a) report on a TransE configuration that achieved  $33.3\%$  MRR ( $52.2\%$  Hits@10) on
+
+![](images/1130fb63fb58fdedae323ccc185330a04aa565b52e6ac6461c8032308955a7b6.jpg)  
+Figure 1: Distribution of filtered MRR (\%) on validation data over the quasi-random hyperparameter configurations explored in our study.
+
+![](images/cbc09b2865c85cfbb677f6e621d6fe2dd6610602ce87fd5c5ffa08e8a7ac77a7.jpg)
+
+FB15K-237. Since the focus of this study is to compare models in a common experimental setup and without manual intervention, we do not further report on these results.
+
+# 3.3 IMPACT OF HYPERPARAMETERS
+
+Anatomy of search space. Figure 1 shows the distribution of filtered MRR for each model and dataset (on validation data, Sobol configurations only). Each distribution consists of roughly 200 different quasi-random hyperparameter configurations (except TransE, for which 60 configurations have been used). Perhaps unsurprisingly, we found that all models showed a wide dispersion on both datasets and only very few configurations achieved state-of-the-art results. There are, however, notable differences across datasets and models. On FB15K-237, for example, the median ConvE configuration is best; on WNRR, DistMult and Complex have a much higher median MRR. Generally, the impact of the hyperparameter choice is more pronounced on WNRR (higher variance) than on FB15K-237.
+
+Best configurations (quasi-random search). The hyperparameters of the best performing models during our quasi-random search are reported in Table 3 (selected hyperparameters) and Tables 6 and 7 (in the appendix, all hyperparameters). First, we found that the optimum choice of hyperparameters is often model and dataset dependent: with the exception of the loss function (discussed below), almost no hyperparameter setting was consistent across datasets and models. For example, the NegSample training type was frequently selected on FB15K-237 but not on WNRR, where KvsAll was more prominent. Our findings provide further evidence that grid search on a small search space is not suitable to compare model performance because the result may be heavily influenced by the specific grid points being used. Moreover, the budget that we used for quasi-random search was comparable to a grid search over a small (or at least not very large) grid: we tested merely 30 different model configurations for every combination of training type and loss function. It is therefore both feasible and beneficial to work with a large search space.
+
+Best configurations (Bayesian optimization). We already obtained good configurations solely from quasi-random search. After Bayesian optimization (which we used to tune the numerical hyperparameters), the performance on test data almost always improved only marginally. The best hyperparameters after the Bayesian optimization phase are reported in Table 8 in the appendix.
+
+Impact of specific hyperparameters. It is difficult to assess the impact of each hyperparameter individually. As a proxy for the importance of each hyperparameter setting, we report in Table 3 and Tables 6 and 7 (appendix) the best performance in terms of filtered MRR on validation data for a different choice of value for each hyperparameter (difference in parenthesis). For example, the best configuration during quasi-random search for RESCAL on FB15K-237 did not use reciprocal relations. The best configuration that did use reciprocal relations had an MRR that was 0.5 points lower (35.6 instead of 36.1). This does not mean that the gap is explained by the use of the reciprocal relations alone, as other hyperparameters may also be different, but it does show the best performance
+
+<table><tr><td></td><td></td><td>RESCAL</td><td>TransE</td><td>DistMult</td><td>ComplEx</td><td>ConvE</td></tr><tr><td rowspan="9">FB15K-237</td><td>Valid. MRR</td><td>36.1</td><td>31.5</td><td>35.0</td><td>35.3</td><td>34.3</td></tr><tr><td>Emb. size</td><td>128 (-0.5)</td><td>512 (-3.7)</td><td>256 (-0.2)</td><td>256 (-0.3)</td><td>256 (-0.4)</td></tr><tr><td>Batch size</td><td>512 (-0.5)</td><td>128 (-7.1)</td><td>1024 (-0.2)</td><td>1024 (-0.3)</td><td>1024 (-0.4)</td></tr><tr><td>Train type</td><td>1vsAll (-0.8)</td><td>NegSamp -</td><td>NegSamp (-0.2)</td><td>NegSamp (-0.3)</td><td>1vsAll (-0.4)</td></tr><tr><td>Loss</td><td>CE (-0.9)</td><td>CE (-7.1)</td><td>CE (-3.1)</td><td>CE (-3.8)</td><td>CE (-0.4)</td></tr><tr><td>Optimizer</td><td>Adam (-0.5)</td><td>Adagrad (-3.7)</td><td>Adagrad (-0.2)</td><td>Adagrad (-0.5)</td><td>Adagrad (-1.5)</td></tr><tr><td>Initializer</td><td>Normal (-0.8)</td><td>XvNorm (-3.7)</td><td>Unif. (-0.2)</td><td>Unif. (-0.5)</td><td>XvNorm (-0.4)</td></tr><tr><td>Regularizer</td><td>None (-0.5)</td><td>L2 (-3.7)</td><td>L3 (-0.2)</td><td>L3 (-0.3)</td><td>L3 (-0.4)</td></tr><tr><td>Reciprocal</td><td>No (-0.5)</td><td>Yes (-9.5)</td><td>Yes (-0.3)</td><td>Yes (-0.3)</td><td>Yes -</td></tr><tr><td rowspan="9">WNRR</td><td>Valid. MRR</td><td>46.8</td><td>22.6</td><td>45.4</td><td>47.6</td><td>44.3</td></tr><tr><td>Emb. size</td><td>128 (-1.0)</td><td>512 (-5.1)</td><td>512 (-1.1)</td><td>128 (-1.0)</td><td>512 (-1.2)</td></tr><tr><td>Batch size</td><td>128 (-1.0)</td><td>128 (-5.1)</td><td>1024 (-1.1)</td><td>512 (-1.0)</td><td>1024 (-1.3)</td></tr><tr><td>Train type</td><td>KvsAll (-1.0)</td><td>NegSamp -</td><td>KvsAll (-1.1)</td><td>1vsAll (-1.0)</td><td>KvsAll (-1.2)</td></tr><tr><td>Loss</td><td>CE (-2.0)</td><td>CE (-5.1)</td><td>CE (-2.4)</td><td>CE (-3.5)</td><td>CE (-1.4)</td></tr><tr><td>Optimizer</td><td>Adam (-1.2)</td><td>Adagrad (-5.8)</td><td>Adagrad (-1.5)</td><td>Adagrad (-1.5)</td><td>Adam (-1.4)</td></tr><tr><td>Initializer</td><td>Unif. (-1.0)</td><td>XvNorm (-5.1)</td><td>Unif. (-1.3)</td><td>Unif. (-1.5)</td><td>XvNorm (-1.4)</td></tr><tr><td>Regularizer</td><td>L3 (-1.2)</td><td>L2 (-5.1)</td><td>L3 (-1.1)</td><td>L2 (-1.0)</td><td>L1 (-1.2)</td></tr><tr><td>Reciprocal</td><td>Yes (-1.0)</td><td>Yes (-5.9)</td><td>Yes (-1.1)</td><td>No (-1.0)</td><td>Yes -</td></tr></table>
+
+Table 3: Hyperparameters of best performing models after quasi-random hyperparameter search and Bayesian optimization w.r.t. filtered MRR (on validation data). For each hyperparameter, we also give the reduction in filtered MRR for the best configuration that does not use this value of the hyperparameter (in parenthesis).
+
+![](images/313884c1810054c7cac667e49d3fcebe4a2a0edf667af9165f663d16e2cd7d10.jpg)  
+Figure 2: Distribution of filtered MRR (\%) on validation data over the quasi-random hyperparameter configurations for different training type and loss functions.
+
+we obtained when enforcing reciprocal relations. We can see that most hyperparameters appear to have moderate impact on model performance: many hyperparameter settings did not produce a significant performance improvement that others did not also produce. A clear exception here is the use of the loss function, which is consistent across models and datasets (always CE), partly with large performance improvements over alternative loss functions.
+
+![](images/51d5bee87db0bd298eab2c28d2750f7f5423404a16c690599e4d0f09230fc97b.jpg)  
+Figure 3: Best filtered MRR  $(\%)$  on validation data achieved during quasi-random search as a function of the number of training epochs. Changes in the corresponding "best-so-far" hyperparameter configuration are marked with a dot.
+
+![](images/aaff63bec3886b54d3c54fba6d5082b45bad153a337d8d8947045a7b0c358546.jpg)
+
+Impact of loss function. In order to further explore the impact of the loss function, we show the distribution of the filtered MRR on validation data for every (valid) combination of training type and loss function in Figure 10. We found that the use of CE (the three rightmost bars) generally lead to better configurations than using other losses. This is surprising (and somewhat unsatisfying) in that we use the CE loss, which is a multi-class loss, for a multi-label task. Similar observations have been made for computer vision tasks (Joulin et al., 2016; Mahajan et al., 2018) though. Note that the combination of KvsAll with CE (on the very right in figure) has not been explored previously.
+
+# 3.4 MODEL SELECTION
+
+We briefly summarize the behavior of various models during model selection. Figure 3 shows the best validation MRR (over all quasi-random hyperparameter configurations) achieved by each model when trained for the specified number of epochs. The corresponding hyperparameter configuration may change as models were trained longer; in the figure, we marked such changes with a dot. We found that a suitable hyperparameter configuration as well as a good model can be found using significantly less than the 400 epochs used in our study. Nevertheless, longer training did help in many settings and some models may benefit when trained for even more than 400 epochs.
+
+# 4 CONCLUSION
+
+We reported the results of an extensive experimental study with popular KGE model architectures and training strategies across a wide range of hyperparameter settings. We found that when trained appropriately, the relative performance differences between various model architectures often shrunk and sometimes even reversed when compared to prior results. This suggests that (at least currently) training strategies have a significant impact on model performance and may account for a substantial fraction of the progress made in recent years. To enable an insightful model comparison, we recommend that new studies compare models in a common experimental setup using a sufficiently large hyperparameter search space. Our study shows that such an approach is possible with reasonable computational cost, and we provide our implementation as part of the LibKGE framework to facilitate such model comparison. Moreover, we feel that many of the more advanced architectures and techniques proposed in the literature need to be revisited to reassess their individual benefits.
+
+# REFERENCES
+
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+
+# APPENDIX
+
+Evaluation metrics. We formally define the evaluation metrics used in our study. Given a triple  $(i,k,j)$ , denote by  $\mathrm{rank}(j|i,k)$  the filtered rank of object  $j$ , i.e., the rank of model score  $s(i,k,j)$  among the collection of all pseudo-negative object scores
+
+$\{s(i,k,j^{\prime}):j^{\prime}\in \mathcal{E}$  and  $(i,k,j^{\prime})$  does not occur in training, validation, or test data  $\}$ .
+
+If there are ties, we take the mean rank of all triples with score  $s(i,k,j)$ . Define  $\mathrm{rank}(i|k,j)$  likewise. Denote by  $\mathcal{K}^{\mathrm{test}}$  the set of test triples. Then
+
+$$
+\mathrm{MRR} = \frac{1}{2|\mathcal{K}^{\text{test}}|}\sum_{(i,k,j)\in \mathcal{K}^{\text{test}}}\left(\frac{1}{\operatorname{rank}(i|k,j)} +\frac{1}{\operatorname{rank}(j|i,k)}\right),
+$$
+
+$$
+\text {H i t s} @ K = \frac {1}{2 | \mathcal {K} ^ {\text {t e s t}} |} \sum_ {(i, k, j) \in \mathcal {K} ^ {\text {t e s t}}} \left(\mathbb {1} (\operatorname {r a n k} (i | k, j) \leq K) + \mathbb {1} (\operatorname {r a n k} (j | i, k) \leq K)\right),
+$$
+
+where indicator  $\mathbb{1}(E)$  is 1 if condition  $E$  is true, else 0.
+
+<table><tr><td>Dataset</td><td>Entities</td><td>Relations</td><td>Train</td><td>Validation</td><td>Test</td></tr><tr><td>FB-237</td><td>14505</td><td>237</td><td>272115</td><td>17535</td><td>20466</td></tr><tr><td>WNRR</td><td>40559</td><td>11</td><td>86835</td><td>3034</td><td>3134</td></tr></table>
+
+Table 4: Dataset statistics  
+
+<table><tr><td>Hyperparameter</td><td>Quasi-random search</td><td>Bayesian optimization</td></tr><tr><td>Embedding size</td><td>{128, 256, 512}</td><td>Value from Tab. 6</td></tr><tr><td>Training type</td><td>{NegSamp, 1vsAll, KvsAll}</td><td>Value from Tab. 6</td></tr><tr><td>Reciprocal</td><td>{True, False}</td><td>Value from Tab. 6</td></tr><tr><td>No. subject samples (NegSamp)</td><td>[1, 1000], log scale</td><td>[1, 1000], log scale</td></tr><tr><td>No. object samples (NegSamp)</td><td>[1, 1000], log scale</td><td>[1, 1000], log scale</td></tr><tr><td>Label smoothing (KvsAll)</td><td>[0.0, 0.3]</td><td>[0.0, 0.3]</td></tr><tr><td>Loss</td><td>{BCE, MR, CE}</td><td>Value from Tab. 6</td></tr><tr><td>Margin (MR)</td><td>[0, 10]</td><td>[0, 10]</td></tr><tr><td>Lp-norm (TransE)</td><td>{1, 2}</td><td>Value from Tab. 6</td></tr><tr><td>Optimizer</td><td>{Adam, Adagrad}</td><td>Value from Tab. 6</td></tr><tr><td>Batch size</td><td>{128, 256, 512, 1024}</td><td>Value from Tab. 6</td></tr><tr><td>Learning rate</td><td>[10-4, 1], log scale</td><td>[10-4, 1], log scale</td></tr><tr><td>LR scheduler patience</td><td>[0, 10]</td><td>[0, 10]</td></tr><tr><td>Lp regularization</td><td>{L1, L2, L3, None}</td><td>Value from Tab. 6</td></tr><tr><td>Entity emb. weight</td><td>[10-20, 10-5]</td><td>[10-20, 10-5]</td></tr><tr><td>Relation emb. weight</td><td>[10-20, 10-5]</td><td>[10-20, 10-5]</td></tr><tr><td>Frequency weighting</td><td>{True, False}</td><td>Value from Tab. 6</td></tr><tr><td>Embedding normalization (TransE)</td><td></td><td></td></tr><tr><td>Entity</td><td>{True, False}</td><td>Value from Tab. 6</td></tr><tr><td>Relation</td><td>{True, False}</td><td>Value from Tab. 6</td></tr><tr><td>Dropout</td><td></td><td></td></tr><tr><td>Entity embedding</td><td>[0.0, 0.5]</td><td>[0.0, 0.5]</td></tr><tr><td>Relation embedding</td><td>[0.0, 0.5]</td><td>[0.0, 0.5]</td></tr><tr><td>Feature map (ConvE)</td><td>[0.0, 0.5]</td><td>[0.0, 0.5]</td></tr><tr><td>Projection (ConvE)</td><td>[0.0, 0.5]</td><td>[0.0, 0.5]</td></tr><tr><td>Embedding initialization</td><td>{Normal, Unif, XvNorm, XvUnif}</td><td>Value from Tab. 6</td></tr><tr><td>Std. deviation (Normal)</td><td>[10-5, 1.0]</td><td>[10-5, 1.0]</td></tr><tr><td>Interval (Unif)</td><td>[-1.0, 1.0]</td><td>[-1.0, 1.0]</td></tr><tr><td>Gain (XvNorm)</td><td>1.0</td><td>1.0</td></tr><tr><td>Gain (XvUnif)</td><td>1.0</td><td>1.0</td></tr></table>
+
+Table 5: Hyperparameter search space used in our study. Settings that apply only to certain configurations are indicated in parenthesis.
+
+<table><tr><td></td><td colspan="2">RESCAL</td><td colspan="2">TransE</td><td colspan="2">DistMult</td><td colspan="2">ComplEx</td><td colspan="2">ConvE</td></tr><tr><td>Mean MRR</td><td colspan="2">36.1</td><td colspan="2">31.5</td><td colspan="2">35.0</td><td colspan="2">35.3</td><td colspan="2">34.3</td></tr><tr><td>Embedding size</td><td>128</td><td>(-0.5)</td><td>512</td><td>(-3.7)</td><td>256</td><td>(-0.2)</td><td>256</td><td>(-0.3)</td><td>256</td><td>(-0.4)</td></tr><tr><td>Training type</td><td>1vsAll</td><td>(-0.8)</td><td>NegSamp</td><td>-</td><td>NegSamp</td><td>(-0.2)</td><td>NegSamp</td><td>(-0.3)</td><td>1vsAll</td><td>(-0.4)</td></tr><tr><td>Reciprocal</td><td>No</td><td>(-0.5)</td><td>Yes</td><td>(-9.5)</td><td>Yes</td><td>(-0.3)</td><td>Yes</td><td>(-0.3)</td><td>Yes</td><td>-</td></tr><tr><td>No. subject samples (NegSamp)</td><td>-</td><td></td><td>2</td><td></td><td>557</td><td></td><td>557</td><td></td><td>-</td><td></td></tr><tr><td>No. object samples (NegSamp)</td><td>-</td><td></td><td>56</td><td></td><td>367</td><td></td><td>367</td><td></td><td>-</td><td></td></tr><tr><td>Label Smoothing (KvsAll)</td><td>-</td><td></td><td>-</td><td></td><td>-</td><td></td><td>-</td><td></td><td>-</td><td></td></tr><tr><td>Loss</td><td>CE</td><td>(-0.9)</td><td>CE</td><td>(-7.1)</td><td>CE</td><td>(-3.1)</td><td>CE</td><td>(-3.8)</td><td>CE</td><td>(-0.4)</td></tr><tr><td>Margin (MR)</td><td>-</td><td></td><td>-</td><td></td><td>-</td><td></td><td>-</td><td></td><td>-</td><td></td></tr><tr><td>\(L_{p}\)-norm (TransE)</td><td>-</td><td></td><td>L2</td><td></td><td>-</td><td></td><td>-</td><td></td><td>-</td><td></td></tr><tr><td>Optimizer</td><td>Adam</td><td>(-0.5)</td><td>Adagrad</td><td>(-3.7)</td><td>Adagrad</td><td>(-0.2)</td><td>Adagrad</td><td>(-0.5)</td><td>Adagrad</td><td>(-1.5)</td></tr><tr><td>Batch size</td><td>512</td><td>(-0.5)</td><td>128</td><td>(-7.1)</td><td>1024</td><td>(-0.2)</td><td>1024</td><td>(-0.3)</td><td>1024</td><td>(-0.4)</td></tr><tr><td>Learning Rate</td><td>0.00063</td><td></td><td>0.04122</td><td></td><td>0.14118</td><td></td><td>0.14118</td><td></td><td>0.00373</td><td></td></tr><tr><td>Scheduler patience</td><td>1</td><td>(-0.5)</td><td>6</td><td>(-3.7)</td><td>9</td><td>(-0.2)</td><td>9</td><td>(-0.3)</td><td>5</td><td>(-0.4)</td></tr><tr><td>\(L_{p}\) regularization</td><td>None</td><td>(-0.5)</td><td>L2</td><td>(-3.7)</td><td>L3</td><td>(-0.2)</td><td>L3</td><td>(-0.3)</td><td>L3</td><td>(-0.4)</td></tr><tr><td>Entity emb. weight</td><td>-</td><td></td><td>\(1.32^{-07}\)</td><td></td><td>\(1.55^{-10}\)</td><td></td><td>\(1.55^{-10}\)</td><td></td><td>\(1.55^{-11}\)</td><td></td></tr><tr><td>Relation emb. weight</td><td>-</td><td></td><td>\(3.72^{-18}\)</td><td></td><td>\(3.93^{-15}\)</td><td></td><td>\(3.93^{-15}\)</td><td></td><td>\(7.91^{-12}\)</td><td></td></tr><tr><td>Frequency weighting</td><td>Yes</td><td>(-0.5)</td><td>No</td><td>(-11.0)</td><td>Yes</td><td>(-0.3)</td><td>Yes</td><td>(-0.3)</td><td>Yes</td><td>(-1.5)</td></tr><tr><td colspan="11">Embedding normalization (TransE)</td></tr><tr><td>Entity</td><td>-</td><td></td><td>No</td><td></td><td>-</td><td></td><td>-</td><td></td><td>-</td><td></td></tr><tr><td>Relation</td><td>-</td><td></td><td>No</td><td></td><td>-</td><td></td><td>-</td><td></td><td>-</td><td></td></tr><tr><td colspan="11">Dropout</td></tr><tr><td>Entity embedding</td><td>0.37</td><td></td><td>0.00</td><td></td><td>0.46</td><td></td><td>0.46</td><td></td><td>0.00</td><td></td></tr><tr><td>Relation embedding</td><td>0.28</td><td></td><td>0.00</td><td></td><td>0.36</td><td></td><td>0.36</td><td></td><td>0.10</td><td></td></tr><tr><td>Projection (ConvE)</td><td>-</td><td></td><td>-</td><td></td><td>-</td><td></td><td>-</td><td></td><td>0.19</td><td></td></tr><tr><td>Feature map (ConvE)</td><td>-</td><td></td><td>-</td><td></td><td>-</td><td></td><td>-</td><td></td><td>0.49</td><td></td></tr><tr><td>Embedding initialization</td><td>Normal</td><td>(-0.8)</td><td>XvNorm</td><td>(-3.7)</td><td>Unif.</td><td>(-0.2)</td><td>Unif.</td><td>(-0.5)</td><td>XvNorm</td><td>(-0.4)</td></tr><tr><td>Std. deviation (Normal)</td><td>0.80620</td><td></td><td>-</td><td></td><td>-</td><td></td><td>-</td><td></td><td>-</td><td></td></tr><tr><td>Interval (Unif)</td><td>-</td><td></td><td>-</td><td></td><td>[-0.85, 0.85]</td><td></td><td>[-0.85, 0.85]</td><td></td><td>-</td><td></td></tr></table>
+
+Table 6: Hyperparameters of best performing models found with quasi-random hyperparameter optimization on FB15K-237. Settings that apply only to certain configurations are indicated in parenthesis. For each hyperparameter, we also give the reduction in filtered MRR for the best configuration that does not use this value of the hyperparameter (in parenthesis).
+
+<table><tr><td></td><td>RESCAL</td><td>TransE</td><td>DistMult</td><td>ComplEx</td><td>ConvE</td></tr><tr><td>Mean MRR</td><td>46.8</td><td>22.6</td><td>45.4</td><td>47.6</td><td>44.3</td></tr><tr><td>Embedding size</td><td>128 (-1.0)</td><td>512 (-5.1)</td><td>512 (-1.1)</td><td>128 (-1.0)</td><td>512 (-1.2)</td></tr><tr><td>Training type</td><td>KvsAll (-1.0)</td><td>NegSamp -</td><td>KvsAll (-1.1)</td><td>1vsAll (-1.0)</td><td>KvsAll (-1.2)</td></tr><tr><td>Reciprocal</td><td>Yes (-1.0)</td><td>Yes (-5.9)</td><td>Yes (-1.1)</td><td>No (-1.0)</td><td>Yes -</td></tr><tr><td>No. subject samples (NegSamp)</td><td>-</td><td>2</td><td>-</td><td>-</td><td>-</td></tr><tr><td>No. object samples (NegSamp)</td><td>-</td><td>56</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Label Smoothing (KvsAll)</td><td>0.30</td><td>-</td><td>0.21</td><td>-</td><td>-0.29</td></tr><tr><td>Loss</td><td>CE (-2.0)</td><td>CE (-5.1)</td><td>CE (-2.4)</td><td>CE (-3.5)</td><td>CE (-1.4)</td></tr><tr><td>Margin (MR)</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>\( L_p \)-norm (TransE)</td><td>-</td><td>L2</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Optimizer</td><td>Adam (-1.2)</td><td>Adagrad (-5.8)</td><td>Adagrad (-1.5)</td><td>Adagrad (-1.5)</td><td>Adam (-1.4)</td></tr><tr><td>Batch size</td><td>128 (-1.0)</td><td>128 (-5.1)</td><td>1024 (-1.1)</td><td>512 (-1.0)</td><td>1024 (-1.3)</td></tr><tr><td>Learning Rate</td><td>0.00160</td><td>0.04122</td><td>0.25575</td><td>0.50338</td><td>0.00160</td></tr><tr><td>Scheduler patience</td><td>8 (-1.0)</td><td>6 (-5.1)</td><td>6 (-1.1)</td><td>7 (-1.0)</td><td>1 (-1.2)</td></tr><tr><td>\( L_p \) regularization</td><td>L3 (-1.2)</td><td>L2 (-5.1)</td><td>L3 (-1.1)</td><td>L2 (-1.0)</td><td>L1 (-1.2)</td></tr><tr><td>Entity emb. weight</td><td>\( 3.82^{-20} \)</td><td>\( 1.32^{-07} \)</td><td>\( 1.34^{-10} \)</td><td>\( 1.48^{-18} \)</td><td>\( 3.70^{-12} \)</td></tr><tr><td>Relation emb. weight</td><td>\( 7.47^{-05} \)</td><td>\( 3.72^{-18} \)</td><td>\( 6.38^{-16} \)</td><td>\( 1.44^{-18} \)</td><td>\( 6.56^{-10} \)</td></tr><tr><td>Frequency weighting</td><td>No (-1.0)</td><td>No (-7.4)</td><td>Yes (-1.1)</td><td>No (-1.0)</td><td>No (-1.4)</td></tr><tr><td colspan="6">Embedding normalization (TransE)</td></tr><tr><td>Entity</td><td>-</td><td>No</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Relation</td><td>-</td><td>No</td><td>-</td><td>-</td><td>-</td></tr><tr><td colspan="6">Dropout</td></tr><tr><td>Entity embedding</td><td>0.00</td><td>0.00</td><td>0.12</td><td>0.05</td><td>0.00</td></tr><tr><td>Relation embedding</td><td>0.00</td><td>0.00</td><td>0.36</td><td>0.44</td><td>0.23</td></tr><tr><td>Projection (ConvE)</td><td>-</td><td>-</td><td>-</td><td>-</td><td>0.09</td></tr><tr><td>Feature map (ConvE)</td><td>-</td><td>-</td><td>-</td><td>-</td><td>0.42</td></tr><tr><td>Embedding initialization</td><td>Unif. (-1.0)</td><td>XvNorm (-5.1)</td><td>Unif. (-1.3)</td><td>Unif. (-1.5)</td><td>XvNorm (-1.4)</td></tr><tr><td>Std. deviation (Normal)</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>Interval (Unif)</td><td>[-0.31, 0.31]</td><td>-</td><td>[-0.81, 0.81]</td><td>[-0.31, 0.31]</td><td>-</td></tr></table>
+
+Table 7: Hyperparameters of best performing models found with quasi-random hyperparameter optimization on WNRR. Settings that apply only to certain configurations are indicated in parenthesis. For each hyperparameter, we also give the reduction in filtered MRR for the best configuration that does not use this value of the hyperparameter (in parenthesis).
+
+<table><tr><td></td><td></td><td>RESCAL</td><td>TransE</td><td>DistMult</td><td>ComplEx</td><td>ConvE</td></tr><tr><td rowspan="8">FB15K-237</td><td>Learning rate</td><td>0.00074</td><td>0.00030</td><td>0.15954</td><td>0.18255</td><td>0.00428</td></tr><tr><td>Scheduler patience</td><td>1</td><td>5</td><td>6</td><td>7</td><td>9</td></tr><tr><td>Entity reg. weight</td><td>-</td><td>-</td><td>\( 1.41^{-09} \)</td><td>\( 6.70^{-09} \)</td><td>\( 4.30^{-15} \)</td></tr><tr><td>Relation reg. weight</td><td>-</td><td>-</td><td>\( 4.10^{-15} \)</td><td>\( 4.13^{-14} \)</td><td>\( 1.12^{-14} \)</td></tr><tr><td>Entity emb. dropout</td><td>0.43</td><td>0.00</td><td>0.42</td><td>0.50</td><td>0.00</td></tr><tr><td>Relation emb. dropout</td><td>0.16</td><td>0.00</td><td>0.41</td><td>0.23</td><td>0.12</td></tr><tr><td>Projection dropout (ConvE)</td><td>-</td><td>-</td><td>-</td><td>-</td><td>0.50</td></tr><tr><td>Feature map dropout (ConvE)</td><td>-</td><td>-</td><td>-</td><td>-</td><td>0.49</td></tr><tr><td rowspan="8">WRR</td><td>Learning rate</td><td>0.00085</td><td>0.25327</td><td>0.33127</td><td>0.52558</td><td>0.00162</td></tr><tr><td>Scheduler patience</td><td>8</td><td>5</td><td>7</td><td>5</td><td>1</td></tr><tr><td>Entity reg. weight</td><td>\( 1.37^{-14} \)</td><td>\( 1.06^{-07} \)</td><td>\( 1.25^{-12} \)</td><td>\( 4.52^{-06} \)</td><td>\( 1.08^{-08} \)</td></tr><tr><td>Relation reg. weight</td><td>\( 2.57^{-15} \)</td><td>\( 4.50^{-13} \)</td><td>\( 1.53^{-14} \)</td><td>\( 4.19^{-10} \)</td><td>\( 9.52^{-11} \)</td></tr><tr><td>Entity emb. dropout</td><td>0.00</td><td>0.25</td><td>0.37</td><td>0.36</td><td>0.00</td></tr><tr><td>Relation emb. dropout</td><td>0.24</td><td>0.00</td><td>0.50</td><td>0.31</td><td>0.23</td></tr><tr><td>Projection dropout (ConvE)</td><td>-</td><td>-</td><td>-</td><td>-</td><td>0.15</td></tr><tr><td>Feature map dropout (ConvE)</td><td>-</td><td>-</td><td>-</td><td>-</td><td>0.33</td></tr></table>
+
+Table 8: Hyperparameters of best performing models found after quasi-random hyperparameter optimization and Bayesian optimization. All other hyperparameters are the same as in Tables 6 (FB15K-237) and 7 (WNRR).  
+
+<table><tr><td></td><td></td><td>RESCAL</td><td>TransE</td><td>DistMult</td><td>ComplEx</td><td>ConvE</td></tr><tr><td rowspan="2">FB15K-237</td><td>MRR</td><td>36.1±0.3</td><td>31.5±0.1</td><td>35.0±0.0</td><td>35.3±0.1</td><td>34.3±0.1</td></tr><tr><td>Hits@10</td><td>54.3±0.5</td><td>49.8±0.2</td><td>53.5±0.1</td><td>53.9±0.2</td><td>52.4±0.2</td></tr><tr><td rowspan="2">WNRR</td><td>MRR</td><td>46.8±0.2</td><td>22.6±0.0</td><td>45.4±0.1</td><td>47.6±0.1</td><td>44.3±0.1</td></tr><tr><td>Hits@10</td><td>51.8±0.2</td><td>51.5±0.1</td><td>52.4±0.3</td><td>54.1±0.4</td><td>50.4±0.2</td></tr></table>
+
+Table 9: Mean and standard deviation of the validation data performance of each model's best hyperparameter configuration when trained from scratch five times.
+
+![](images/45e58046c4ab00e126a64e67e198d4aa3457892c1bb4f894be84d4986dfbbe5a.jpg)  
+Figure 4: Distribution of filtered MRR (on validation data, quasi-random search only) for different embedding sizes (top row: FB15K-237, bottom row: WNRR)
+
+![](images/59822c72bfd33c03ae082637fa59fecaf58bb5ae59e18792316661e349df4cd8.jpg)  
+Figure 5: Distribution of filtered MRR (on validation data, quasi-random search only) for different batch sizes (top row: FB15K-237, bottom row: WNRR)
+
+![](images/7cb2ee686217b4b4a3293332dcebc86737060a278f408d22d6af25fff1e3eeb7.jpg)  
+Figure 6: Distribution of filtered MRR (on validation data, quasi-random search only) for different optimizers (top row: FB15K-237, bottom row: WNRR)
+
+![](images/db7be2eb8f02be0aaa6a840470470644947df28da7cc4fee5cd31d2ce925da8b.jpg)  
+Figure 7: Distribution of filtered MRR (on validation data, quasi-random search only) for different initializers (top row: FB15K-237, bottom row: WNRR)
+
+![](images/d6bbbfd7421691cbd931ab95a77cd425c5e09629757e5b28b96aae2af93c4972.jpg)  
+Figure 8: Distribution of filtered MRR (on validation data, quasi-random search only) with and without reciprocal relations (top row: FB15K-237, bottom row: WNRR)
+
+![](images/9c97a5821542afed3d4a3dad2b81af1ede9188753b42f549d954990477665746.jpg)  
+Figure 9: Distribution of filtered MRR (on validation data, quasi-random search only) for different penalty terms in the loss function (top row: FB15K-237, bottom row: WNRR)
+
+![](images/e8684d349acc50c66ac9d91fb7f1b4706109a0ad6320518700703714f3f69b14.jpg)  
+Figure 10: Distribution of filtered MRR (on validation data, quasi-random search only) with and without dropout (top row: FB15K-237, bottom row: WNRR)
\ No newline at end of file
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+# YOU ONLY TRAIN ONCE:  
+LOSS-CONDITIONAL TRAINING OF DEEP NETWORKS
+
+Alexey Dosovitskiy & Josip Djolonga
+
+Google Research, Brain Team
+
+{adosovitskiy, josipd}@google.com
+
+# ABSTRACT
+
+In many machine learning problems, loss functions are weighted sums of several terms. A typical approach to dealing with these is to train multiple separate models with different selections of weights and then either choose the best one according to some criterion or keep multiple models if it is desirable to maintain a diverse set of solutions. This is inefficient both at training and at inference time. We propose a method that allows replacing multiple models trained on one loss function each by a single model trained on a distribution of losses. At test time a model trained this way can be conditioned to generate outputs corresponding to any loss from the training distribution of losses. We demonstrate this approach on three tasks with parameterized losses:  $\beta$ -VAE, learned image compression, and fast style transfer.
+
+# 1 INTRODUCTION
+
+When designing a model for a machine learning problem at hand, a practitioner often thinks of multiple qualities that the model should have. For example, in addition to requiring a good prediction performance, it may be desirable to have a classifier that is simple, so that it would better generalize to unseen data. Similarly, when training a lossy image compression model, one optimizes both the size of the compressed images and their quality. In such scenarios, multiple loss functions have to be minimized simultaneously, each of them modelling a different facet of the considered problem.
+
+Typically, it is not possible to simultaneously optimize all these losses — either due to limited model capacity, or because some of the losses may be fundamentally in conflict (for instance, image quality and the compression ratio in image compression). One thus has to decide how to balance the losses during optimization. Such multi-objective problems are most commonly scalarized by linearly combining the losses, with weights defining the trade-off between the loss terms. However, the exact values of the chosen weights can strongly affect the performance of the model and their tuning can be cumbersome. It might be also unpredictable how the weights affect the final values of the individual losses; moreover, the actual goal may be to optimize a down-stream metric that is not being explicitly minimized during training (for instance, classification accuracy). Furthermore, in many tasks there may not be a single optimal model, but it is instead important to make different trade-offs at test time. For instance, in image compression, to achieve a fine-grained coverage of rate-distortion trade-offs, one would first have to train at least ten models with different parameter settings, and then at inference time store them all in memory and select the appropriate one depending on the required compression level. This is inefficient both at training and at inference time. However, intuitively, the models trained for different loss variants are related and could share a large fraction of computation. Is it possible to improve the efficiency by making use of this model redundancy?
+
+In this paper we propose a simple and broadly applicable approach that efficiently deals with multi-term loss functions and, more generally, arbitrarily parameterized loss functions. Instead of training one model for each parameter combination of the loss, we train one model that covers the whole space of different loss weightings. We achieve this by (i) training the model on a distribution of losses instead of a single loss, and (ii) conditioning the model outputs on the parameters of the
+
+loss function. This way, at inference time the conditioning vector can be varied, allowing us to traverse the space of models corresponding to different loss functions. We dub the method You Only Train Once (YOTO). The conceptual simplicity of the approach makes it easily applicable to many problem domains, with only minimal changes to existing codebases.
+
+We experimentally showcase the efficacy of the proposed optimization scheme on image synthesis problems, where multi-loss problems are especially widespread. We first show that  $(\beta)$ -variational autoencoders (Kingma & Welling, 2014; Higgins et al., 2017) can be trained for a large range of  $\beta$  parameters at the cost of training a single fixed- $\beta$  model. Then, we train a single deep image compression model (Balle et al., 2018) that can adjust the rate-distortion trade-off on the fly at test time. Finally, we apply YOTO to style transfer (Gatys et al., 2016; Ghiasi et al., 2017), where it is provides a clear benefit given that there is no clear single optimum and that the effect of the various losses on the stylized image is difficult to characterize.
+
+# 2 RELATED WORK
+
+Losses with multiple terms are most commonly associated with multi-task learning (Caruana, 1997; Kokkinos, 2017; Zamir et al., 2018), where the goal is to train a single model that simultaneously solves several learning tasks. We refer the reader to Ruder (2017) for a recent review. Most related to our approach are methods that condition the network architecture on the task at hand (Rebuffi et al., 2017; Mallya et al., 2018), which can be seen as a special case of our method with "one hot" loss weight vectors. While ideas from our work could also be applied to the multi-task learning scenario, here we choose to focus on problems where it is desirable to learn a diverse set of models, rather than a single well-performing model. Recently, Brault et al. (2019) studied simultaneous optimization of multiple loss functions in the context of multi-task kernel learning. Our work is similar in spirit, but differs drastically both technically and in the application domains.
+
+Conditional normalization schemes have recently been used for a range of supervised learning tasks, including Conditional Batch Normalization (CBN) for visual question answering (de Vries et al., 2017), Conditional Instance Normalization and Adaptive Instance Normalization (AdaIN) for fast style transfer (Dumoulin et al., 2017; Huang & Belongie, 2017; Ghiasi et al., 2017), Feature-wise Linear Modulation (FiLM) for visual reasoning (Perez et al., 2018), and a variety of conditioning schemes for generative models (Karras et al., 2019; Brock et al., 2019). A comprehensive overview of conditioning methods is provided by Dumoulin et al. (2018). All these methods condition a convolutional network on some side information by linearly transforming the activations with coefficients computed from the side information. The side information typically informs the model about the prediction target: for instance, it can be the text of a question in the question answering setting or the style image in style transfer. The only exception we are aware of is the recent work of Babaeizadeh & Ghiasi (2018), which conditions a style transfer network on the parameters of the loss function, same as in our method. The approach is technically very similar, but we formulate it as a generally applicable technique and evaluate on several tasks both qualitatively and quantitatively.
+
+Conditional models are also often used in sensorimotor control to train a single model capable of addressing a family of tasks, such as throwing darts at different targets (da Silva et al., 2012), block stacking and ball hitting (Deisenroth et al., 2014), navigating towards different locations (Schaul et al., 2015), moving objects with a robotic arm (Andrychowicz et al., 2017), driving in the desired direction (Codevilla et al., 2018) or trading off the reward components (Dosovitskiy & Koltun, 2017). A single model, conditioned on the task parameters, is trained to simultaneously maximize the rewards for a distribution of tasks. While technically similar to our setting, the conceptual difference is that between the tasks the training targets are changing (either rewards or expert actions used as labels for imitation learning), rather than the loss function itself, as in our case.
+
+Another recent line of related work explores the use hypernetworks — networks that generate the weights of other networks — for hyperparameter optimization. Lorraine & Duvenaud (2018) propose to parametrize the weights of a network by a hypernetwork, which is conditioned on the values of the hyperparameters. The model can then be trained on a distribution of hyperparameters, and the hyperparameters can be optimized via gradient descent on the validation data. MacKay et al. (2019) improve the scalability of the method by proposing efficient approximations to hypernetworks. These methods are technically similar to our approach, but they focus on hyperparameter
+
+![](images/af4eb1d24e5f39ec43bd2d551365420140bbae9a789c1fc81ff8a9fef7b53f06.jpg)  
+Figure 1: Illustration of the proposed method. At training time (left), for each training data point loss parameters  $\lambda$  are sampled from a distribution  $P_{\lambda}$ , and the model is conditioned on these. Then, at test time (right), the model can be conditioned on any desired loss parameters to achieve behavior corresponding to the loss function with these parameters.
+
+![](images/88d3dbb3ec50a4989adc7da09d1bd7e8ab3f5ee893ca8e86823f02399049f5bf.jpg)
+
+optimization, while we are more interested in obtaining the full set of solutions corresponding to the different loss coefficients.
+
+Recently, Shoshan et al. (2019) and Wang et al. (2019) addressed the problem of interpolating between predictions of two or more image generation models. Both methods start by training several separate models and then propose different ways of interpolating between them: either by using specialized adaptor blocks in the networks, or by directly interpolating the network weights. We, in contrast, train only a single model by directly optimizing the performance over a distribution of losses. This is conceptually simpler, can be easily integrated in existing models, and requires training a single model instead of several.
+
+# 3 METHOD
+
+We consider the following learning scenario: given a training distribution of pairs  $\boldsymbol{x}, \boldsymbol{y} \sim P_{\boldsymbol{x}, \boldsymbol{y}}$  with  $\boldsymbol{x} \in \mathbb{X} \subset \mathbb{R}^{d_{\mathbb{X}}}, \boldsymbol{y} \in \mathbb{Y} \subset \mathbb{R}^{d_{\mathbb{Y}}}$ , and a loss function  $\mathcal{L}(\cdot, \cdot): \mathbb{Y} \times \mathbb{Y} \to \mathbb{R}$ , we aim to learn a model  $F: \mathbb{X} \to \mathbb{Y}$ , with parameters  $\theta$ , such that its predictions  $\hat{\boldsymbol{y}} = F(\boldsymbol{x}, \boldsymbol{\theta})$  minimize the expected value of the loss  $\mathcal{L}(\boldsymbol{y}, \hat{\boldsymbol{y}})$  over the dataset. This formulation is very broad and covers a variety of machine learning problems, including (self-)supervised learning learning, and some types of generative models. We focus on this setting for a more streamlined presentation, although the proposed method can be applied to models that go beyond this formulation.
+
+Instead of a single fixed loss function  $\mathcal{L}(\cdot, \cdot)$ , assume that we are interested in a family of losses  $\mathcal{L}(\cdot, \cdot, \lambda)$ , parameterized by a vector  $\lambda \in \Lambda \subset \mathbb{R}^{d_{\lambda}}$ . The most common case of such a family of losses is a weighted sum of several loss terms:  $\mathcal{L}(\cdot, \cdot, \lambda) = \sum_{i} \lambda_{i} \mathcal{L}^{i}(\cdot, \cdot)$ . However, other parameters, such as, for instance, the power  $p$  in  $\ell^p$  losses, might also be included in the vector  $\lambda$ . In practice, one then typically minimizes the loss independently for each choice of the parameter vector  $\lambda$ :
+
+$$
+\boldsymbol {\theta} _ {\boldsymbol {\lambda}} ^ {*} = \underset {\boldsymbol {\theta}} {\arg \min } \mathbb {E} _ {\boldsymbol {x}, \boldsymbol {y} \sim P _ {\boldsymbol {x}, \boldsymbol {y}}} \mathcal {L} (\boldsymbol {y}, F (\boldsymbol {x}, \boldsymbol {\theta}), \boldsymbol {\lambda}). \tag {1}
+$$
+
+Instead of fixing  $\lambda$ , we propose to solve an optimization problem where the parameters  $\lambda$  are sampled from a distribution  $P_{\lambda}$ . Hence, during training time the model observes many losses, and can learn to utilize the relationships between them to optimize all of them simultaneously. The distribution  $P_{\lambda}$  can be seen as a prior over the losses, and we discuss its choice in the next subsection. At inference time, the joint model can be conditioned on an arbitrary desired parameter value  $\lambda$ , yielding the corresponding predictions. We thus solve the following optimization problem:
+
+$$
+\boldsymbol {\theta} ^ {*} = \underset {\boldsymbol {\theta}} {\arg \min } \mathbb {E} _ {\boldsymbol {\lambda} \sim P _ {\boldsymbol {\lambda}}} \mathbb {E} _ {\boldsymbol {x}, \boldsymbol {y} \sim P _ {\boldsymbol {x}, \boldsymbol {y}}} \mathcal {L} (\boldsymbol {y}, F _ {c} (\boldsymbol {x}, \boldsymbol {\theta}, \boldsymbol {\lambda}), \boldsymbol {\lambda}). \tag {2}
+$$
+
+In the limit of infinite model capacity, this new proposed optimization problem is equivalent to the original independent optimization scheme, in the sense that the minimal values for both problems coincide. Further discussion, as well as the proof, are provided in the appendix.
+
+Proposition 1. Consider two optimization problems:
+
+(1)  $\min_{F\in C(\mathbb{X})}\mathbb{E}_{\boldsymbol {x},\boldsymbol {y}\sim P_{\boldsymbol{x},\boldsymbol{y}}}\mathcal{L}(\boldsymbol {y},F(\boldsymbol {x}),\boldsymbol {\lambda})$
+
+(2)  $\min_{G\in C(\mathbb{X}\times \Lambda)}\mathbb{E}_{\pmb {\lambda}\sim P_{\pmb{\lambda}}}\mathbb{E}_{\pmb {x},\pmb {y}\sim P_{\pmb{x},\pmb{y}}}\mathcal{L}(\pmb {y},G(\pmb {x},\lambda),\pmb {\lambda}),$
+
+where  $C(S)$  denotes the space of continuous functions on  $S$ . Assume there exists a continuous function  $F^{*}(\pmb{x},\pmb{\lambda})$  such that for every  $\pmb{\lambda} \in \Lambda$  the function  $F^{*}(\cdot ,\pmb{\lambda})$  solves the problem (1). Then, if  $G^{*}$  is a solution of the problem (2),  $G^{*}(\cdot ,\lambda)$  also minimizes the problem (1) almost surely w.r.t.  $P_{\lambda}$ .
+
+This theoretical result shows that the method is fundamentally as powerful as the standard per-parameter training, while only requiring a single model. However, in practice the infinite capacity assumption does not hold, so implementation details become important, which we describe next.
+
+# 3.1 PRACTICAL DETAILS
+
+In this work we focus on training deep networks with methods based on stochastic gradient descent. We therefore approximate the expectations in equation 2 using Monte Carlo estimates: we estimate the expectation over the data distribution by averaging over a training set  $D = \{(x_{i},y_{i})\}_{i = 1}^{N}$ , and the expectation over the loss parameters by sampling from the loss parameter distribution:
+
+$$
+\boldsymbol {\theta} ^ {*} = \arg \min  _ {\boldsymbol {\theta}} \sum_ {i = 1} ^ {n} L (\boldsymbol {y} _ {i}, F (\boldsymbol {x} _ {i}, \boldsymbol {\theta}, \boldsymbol {\lambda} _ {i}), \boldsymbol {\lambda} _ {i}), \quad \boldsymbol {\lambda} _ {i} \sim P _ {\boldsymbol {\lambda}}. \tag {3}
+$$
+
+In our experiments, we sample a new loss parameter for each training data point every time it is encountered during training.
+
+The distribution of the loss parameters to be used for training is typically not given and has to be chosen. Clearly, the support of the training distribution has to include the range of parameter values we are interested in covering, but this still leaves a lot of freedom in the specific choice of the distribution. In our experiments we use the log-uniform distribution, which is commonly used for sampling the parameters of machine learning models (Bergstra & Bengio, 2012).
+
+In our experiments the model  $F$  is a convolutional network. To condition the network on the loss parameters, we use Feature-wise Linear Modulation (FiLM) (Perez et al., 2018). First, we select the layers of the network to be conditioned (can be all layers or a subset). Next, we condition each of the layers on the given weight parameters  $\lambda$ . Assume the layer outputs a feature map  $\pmb{f}$  of dimensions  $W\times H\times C$ , with  $W$  and  $H$  corresponding to the spatial dimensions and  $C$  to the channels. We feed the parameter vector  $\lambda$  to two multi-layer perceptrs (MLPs)  $M_{\sigma}$  and  $M_{\mu}$  to generate two vectors,  $\sigma$  and  $\mu$ , of dimensionality  $C$  each. We then multiply the feature map channel-wise by  $\sigma$  and add  $\mu$  to get the transformed feature map  $\tilde{\pmb{f}}$ :
+
+$$
+\tilde {f} _ {i j k} = \sigma_ {k} f _ {i j k} + \mu_ {k}, \quad \boldsymbol {\sigma} = M _ {\sigma} (\boldsymbol {\lambda}), \quad \boldsymbol {\mu} = M _ {\mu} (\boldsymbol {\lambda}). \tag {4}
+$$
+
+# 4 EXPERIMENTS
+
+We evaluate the proposed method both quantitatively and qualitatively on three problems with multi-term loss functions:  $\beta$ -VAE, learned image compression, and fast style transfer. All details regarding the architectures and the training, as well as additional results, are provided in the appendix.
+
+# 4.1  $\beta$ -VARIATIONAL AUTOENCODERS
+
+Our first problem is the loss of the  $\beta$ -variational autoencoder (Higgins et al., 2017)
+
+$$
+\mathcal {L} _ {\beta - \mathrm {V A E}} (\boldsymbol {x}, \boldsymbol {z}, \phi , \theta , \beta) = \mathbb {E} _ {q _ {\phi} (\boldsymbol {z} | \boldsymbol {x})} \log p _ {\theta} (\boldsymbol {x} \mid \boldsymbol {z}) - \beta D _ {\mathrm {K L}} \left(q _ {\phi} (\boldsymbol {z} \mid \boldsymbol {x}) \| p (\boldsymbol {z})\right), \tag {5}
+$$
+
+where  $q_{\phi}(\boldsymbol{z} \mid \boldsymbol{x})$  is the amortized approximate posterior implemented by an encoder network with parameters  $\phi$ ,  $p_{\theta}(\boldsymbol{x} \mid \boldsymbol{z})$  is a stochastic decoder network with parameters  $\theta$ , and  $p(\boldsymbol{z})$  is the prior distribution, in our case a standard multivariate Gaussian. The loss has a single positive parameter  $\beta$  trading off between the reconstruction quality and the divergence between the approximate posterior and the prior.
+
+We consider two settings: the CIFAR-10 dataset (Krizhevsky, 2009) with Gaussian outputs, and the Shapes3D dataset (Burgess & Kim, 2018) with Bernoulli outputs. In both cases we use convolutional encoder-decoder networks with Gaussian latents. We condition after each downsampling or upsampling operation. Both for YOTO and fixed-weight models, we train a set of networks with varying capacities, by proportionally changing the width of all layers. This is done with the goal
+
+![](images/ad8b785260767f9afe00d83f0a0e8e931de3bea85d2176651f5daf270635e9bf.jpg)  
+(a) Frontier on CIFAR-10.
+
+![](images/3f7c0309501e482b80e642ad988a53784db2089414ef641bea616605de59438a.jpg)  
+(b) Full losses on CIFAR-10.
+
+![](images/4098d061831f31519d5b6f4eb9e91de4841e369b1655b00f724ef0433cb8ae9f.jpg)  
+(c) Frontier on Shapes3d.
+
+![](images/037f38d7a093bd76a2180e61fa94a54b659a3f7ce271fb919fbe21cff24c3861.jpg)  
+(d) Full losses on Shapes3D.  
+Figure 2: Quantitative  $\beta$ -VAE results on CIFAR-10 (a,b), and Shapes3D (c,d) for models of varying capacity (width). In most cases, the proposed method performs close to models trained independently for each loss weight value. This is especially the case for high capacity models.
+
+of measuring how does the capacity of a YOTO model affect its performance relative to a set of fixed-weight models. We train each network 3 times and report means and standard deviations over these three runs.
+
+Quantitative results are shown in Figure 2. For each dataset, we provide two plots: the frontier of the two loss components plotted against each other ((a), (c)), and the value of the full loss plotted against the value of the parameter  $\beta$  ((b), (d)). The first plot allows to assess how well the proposed model fits the two loss terms, while the second one shows the actual optimization objective — the weighted sum of the losses. To better highlight the differences between the methods, when plotting the full loss, we normalize it by subtracting the loss of one of the models from all other models.
+
+On both datasets, we observe several common tendencies. First, for small-capacity models the proposed method somewhat underperforms relative to the fixed-weight models, especially on CIFAR-10. This is not surprising, since it is difficult for a small model to cover all loss variants. Second, both the YOTO models and the fixed-weight ones improve substantially with increasing model capacity, so a set of fixed-weight models of a certain width is consistently matched or outperformed by a twice wider YOTO model (this is additionally highlighted in Figure 4 that plots the average full loss versus the width of the network). Third, the larger the network capacity, the closer the performance of fixed-models and the YOTO model. In particular, on both datasets for the widest networks the two frontiers become very close. This is quite intuitive and matches the theoretical result that in the limit of infinite model capacity the proposed method should be as powerful as training separate fixed-weight models.
+
+Qualitative results are shown in Figure 3. Both for reconstruction and sampling, for each value of the weight  $\beta$  that we show YOTO produces results closely resembling those of the corresponding
+
+![](images/2ace9cbfd168eb43f8b49d4b3a3bb4d62d9469b57ba2b164349fa5204784978d.jpg)  
+(b) Samples from the trained  $\beta$ -VAE models.
+
+![](images/5fe5d9f7c5bb88cd4477f900679ff0867af4a55cb534f6a5400c8de018cc850f.jpg)  
+(a) CIFAR-10.  
+Figure 4: Loss averaged over all loss weight values as a function of the model capacity. We compare the proposed method to a set of fixed-weight models on two datasets: CIFAR-10 (a) and Shapes3D (b). We vary the capacity of the network by multiplying the width of the network by a factor. The performance of the proposed method overall matches that of fixed-based models with a roughly  $1.5\mathrm{x}$  wider model. Moreover, the performance of the proposed method gets closer to that of the fixed-weight models when the capacity of the model is increased.
+
+![](images/e9127a378b19100b105978bfc63bb7c1914404042d041302ee6878a3e437311b.jpg)  
+Figure 3: Qualitative  $\beta$ -VAE results on the Shapes3D dataset. For each loss weight  $\beta$ , the reconstruction and sampling results of YOTO are very similar to those of the separately trained models.  
+(b) Shapes3D.
+
+independently trained model. An interesting property of the YOTO model is that sampling with the same noise vector for different beta values generates similar images at different quality levels.
+
+Additional results, including a comparison to two simple baselines (a single fixed-weight model and a baseline based on interpolation), an ablation study of the proposed method, and qualitative results on CIFAR-10, are shown in the appendix.
+
+# 4.2 LEARNED IMAGE COMPRESSION
+
+We build on the learned compression model of Balle et al. (2018). The model is similar to a variational autoencoder, but with a few modifications that allow for the quantization of the latent representation and its use as a compressed image encoding. Moreover, it includes a "hyperprior" learned
+
+![](images/58a6180ffc51e972993ab71e902c5e9acc25e57430f62673327f1faf9584235c.jpg)  
+(a) PSNR on Tecnick.
+
+![](images/002734351daca4b4445d7639512e1b6f0162340c4a94acea840536fca9ea97e7.jpg)  
+(b) PSNR on Kodak.  
+Figure 5: Quantitative compression results on the Tecnick and Kodak datasets. A basic YOTO model under-performs compared to a set of fixed-weight models, but a wider network nearly matches the performance of the fixed-weight models, especially in the high-quality regime.
+
+by a second variational autoencoder on top of the latents of the first one. The loss function has a single parameter — the "rate-distortion weight", trading off compression rate versus reconstruction quality. We refer the reader to (Balle et al., 2018) for details. The architecture of the model includes four networks: a convolutional encoder and hyper-encoder, as well as a transposed convolutional decoder and a hyper-decoder. The networks are relatively shallow, with three convolutional layers each. We apply conditioning to all layers of the model. Same as for VAEs, we train three models for each setting to estimate the stability of training. Further details are provided in the appendix.
+
+We evaluate the compression models on two datasets: Kodak (Kodak, 1993) and Tecnick (Asuni & Giachetti, 2014). The quantitative results on Tecnick are shown in Figure 5, while the results on Kodak are deferred to the appendix. We show rate-distortion plots with standard PSNR on pixel values as a measure of compression quality and bits per pixel as a measure of compression rate. We plot three models per method, but since the training is stable, there difference between them is almost not visible. The gap between separate fixed-weight models ("Fixed weight") and a single joint model of the same size ("YOTO") is fairly small, but consistent across all compression rates and is especially pronounced in the higher quality regime. However, making the model larger ("YOTO width x1.33") recovers a large fraction of the performance, in particular the wider model matches the performance of a smaller fixed-weight model in the high quality regime. However, in the high-compression regime there is still a small gap from the fixed-weight models.
+
+We hypothesize that the reason for relatively worse performance of YOTO on the compression task compared to  $\beta$ -VAEs is that the networks used by Ballé et al. (2018) are relatively shallow. They therefore lack the representational power to efficiently make use of the conditioning input. We expect that one could devise combinations of architectures and conditioning methods for the compression task which would lead to a smaller gap between the fixed-weight and YOTO models; however, this additional tuning is beyond the scope of this paper.
+
+# 4.3 STYLE TRANSFER
+
+The final problem we consider is style transfer (Gatys et al., 2016) — given a content image  $\boldsymbol{x}_c$  and a style image  $\boldsymbol{x}_s$ , the goal is to synthesize a stylized image  $\boldsymbol{y}$  that contains the same content as  $\boldsymbol{x}_c$ , but in a style that resembles that of  $\boldsymbol{x}_s$ . This is typically cast as an optimization problem — we want to find an image  $\boldsymbol{y}$ , that when passed through a pre-trained neural network has activations similar to those of  $\boldsymbol{x}_c$ , but whose aggregate feature statistics resemble those of  $\boldsymbol{x}_s$ . There is a lot of freedom in deciding which layers of the network to use and how to weigh them. These parameters are not easy to tune, since we are optimizing for visual appearance, which cannot be easily quantified. Hence, a single model capable of representing all loss variants would be of great benefit. These experiments are similar to those presented by Babaeizadeh & Ghiasi (2018), but with more focus on a quantitative evaluation, same as in other applications reported above.
+
+![](images/719d3e5b650388029518f6839942e13fadef63c46e0057a824f31a09912bde12.jpg)
+
+![](images/c689a31b1cf8fb4e350110437186fb9c8c867c18034f3d079695705bfc7ede63.jpg)  
+(a) Performance of YOTO when we unilaterally vary each weight. We show means and standard deviations over four runs.  
+(b) Sum of all style losses plotted against the content loss.
+
+![](images/4b396fc4b506f1300907c3a5e0bd58287901596c3c7da14fb0168a4ada2042b8.jpg)  
+Figure 6: Quantitative results on style transfer. In both cases YOTO performance approaches or matches that of separate models trained per loss parameter vector.  
+(a) Varying the content coefficient  $\lambda_{c}$  
+Figure 7: Qualitative comparison of image stylization models on an image from the validation set of ImageNet. The first row shows the results of YOTO trained on all parameters, and the second row shows the results of fixed-weight models trained independently for each loss parameters. In both cases the parameter values increase from left to right. YOTO generates results very similar to the separate models, while training just a single model.
+
+![](images/0fd36b54c4963698b79df3d97d171969c3e110bd17880366a973b59d57607cee.jpg)  
+(b) Varying one of the style coefficients  $\lambda_{s,3}$
+
+We build on the work of Ghiasi et al. (2017), who propose training a single deep network that can stylize images in arbitrary styles. To this end, the content image is fed to the stylization network itself, while the style image is fed to a different "style prediction" network that extracts a feature embedding. This embedding is then used to modulate the stylization network via conditional instance normalization. We refer the reader to (Ghiasi et al., 2017) for a detailed explanation.
+
+Following Ghiasi et al. (2017), we extract the activations from the VGG network (Simonyan & Zisserman, 2015), and define a total of six losses: four style losses  $\mathcal{L}_{s,1}(\pmb{y},\pmb{x}_s),\dots,\mathcal{L}_{s,4}(\pmb{y},\pmb{x}_s)$  one content loss  $\mathcal{L}_c(\pmb{y},\pmb{x}_c)$ , and a total variation loss  $\mathcal{L}_{\mathrm{tv}}(\pmb{y})$  encouraging smoothness between the neighbouring pixels of  $\pmb{y}$ . The final loss is then a linear weighting of these, which we will denote as  $\mathcal{L}(\pmb{y}) = \sum_{i=1}^{n}\lambda_{s,i}\mathcal{L}_{s,i}(\pmb{y},\pmb{x}_s) + \lambda_c\mathcal{L}_c(\pmb{x},\pmb{c}) + \lambda_{\mathrm{tv}}\mathcal{L}_{\mathrm{tv}}(\pmb{y})$ . We train a single YOTO model over five of these parameters (while keeping the TV loss fixed to  $10^4$ ), with log-uniform distributions. To this end, in addition to the style embeddings we condition the stylization network on the vector of loss parameters  $\lambda$ . We sample the content images form ImageNet (Deng et al., 2009) and use 14 pointillism paintings as the style images. Further details are provided in the appendix.
+
+We start by reporting quantitative results. As the problem has 5 loss parameters, it is not possible to visualize the frontier with respect to all losses. Instead, we take two visualization approaches. First, we show the full loss achieved by different methods on a set of loss parameter values selected as follows: we start with the default parameter settings of Ghiasi et al. (2017)  $\lambda_{s,1} = \lambda_{s,2} = \lambda_{s,3} = 10^{-3},\lambda_{s,4} = 10^{-2},\lambda_c = 2$  and then either increase or decrease one of these parameters by a factor of 10 while keeping the rest fixed. This adds up to a total of 10 parameter values: 2 variants per each of the 5 parameters. To better understand the variability of the learned models due to the noisy training process we further trained each model four times. We provide the results in Figure 6a. As a baseline, for each parameter configuration we report the full loss value for the model trained with the default parameters ("Single model baseline"). The YOTO model trained on all parameters closely matches the performance of fixed-weight models trained for each of the parameter configurations.
+
+Next, we focus on just one aspect of the loss surface: the trade-off between the content loss and the sum of style losses. In Figure 6b we plot the frontier of these two values corresponding to different
+
+methods. Here we add a YOTO "specialist" model to the comparison, which has been trained with only 1 parameter, the content loss  $\lambda_{c}$ . Moreover, we report a baseline method that simply interpolates the images produced by two models trained with the extreme values of the content weight. Both YOTO models are close to the fixed-wight models, with the frontier of the "specialist" model closely tracing the fixed models. The interpolation baseline performs very poorly.
+
+Finally, we show select qualitative results in Figure 7, where we compare fixed weight models to a YOTO model trained on all five parameters. We vary the content weight  $\lambda_{c}$  and the third style weight  $\lambda_{s,3}$ , which contribute most to the full loss. We can see that YOTO captures the effect of changing each of these weights and obtains results similar to those of the separately trained models.
+
+# 5 CONCLUSION
+
+We have presented an approach to training a single deep model to fit a parametric family of loss functions. This way, instead of training multiple models corresponding to different loss variants, we can train a single model that provides solutions for all loss variants. We demonstrated the successful application of the method on three applications:  $\beta$ -VAE, learned image compression, and fast style transfer. These initial results are promising, but several challenges remain. First, it would be interesting to devise provably effective strategies of selecting the loss parameter distribution, especially for losses with multiple parameters. Second, further experiments with various architectures and conditioning methods can lead to an even smaller discrepancy between the separate models and our YOTO model. Third, the method could be used for many other applications, within or beyond the domain of image generation. Fourth, one might use YOTO-trained models to initialize usual fixed-loss training or, vice-versa, use a pre-trained fixed-loss model to initialize the training of YOTO. Finally, we believe that this work can also open a new family of theoretical questions centered around the analysis of problems that have to optimize a continuum of loss functions. We see all these directions as exciting avenues for future research.
+
+# ACKNOWLEDGEMENTS
+
+We sincerely thank Johannes Ballé for the help with setting up the image compression experiments and insightful discussions. We also thank Sebastian Nowozin, Rodolphe Jenatton, Joan Puigcerver, Sylvain Gelly, and others in Google Brain for useful discussions and the support of the project.
+
+# REFERENCES
+
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+
+# APPENDIX
+
+# A ARCHITECTURE AND TRAINING DETAILS
+
+# A.1 BETA-VAE
+
+We use convolutional encoder-decoder networks on both datasets. We use leaky rectified linear layer (ReLU) non-linearity in all networks, except for units predicting stangard deviation of Gaussians where we use softmax non-linearity. On CIFAR-10 the network inputs have size  $32 \times 32$  pixels, on Shapes3D -  $64 \times 64$  pixels. We now describe the base network architectures, corresponding to "width x1.0".
+
+On CIFAR-10 both the encoder and the decoder consist of 3 blocks of 2 convolutional layers each, with the first layer in each block having stride 2. All kernels have spatial size  $3 \times 3$ , and the number of channels is, respectively, 8, 8, 16, 16, 32, 32 (and increased proportionally for the "wider" models). The convolutional encoder is followed by fully connected layers with 256 and 512 units respectively, where the output of the latter layer is split in two 256-dimensional vectors encoding the mean and the variance of the approximate posterior. The decoder starts two fully connected layers with 256 and 512 units, the output of the latter layer is reshaped to a  $4 \times 4 \times 32$  feature map and further processed by a decoder that is symmetric to the encoder.
+
+On Shapes3D both the encoder and the decoder consist of 4 convolutional layers each, all with stride 2. All kernels have spatial size  $3 \times 3$ , and the number of channels is, respectively, 8, 16, 32, 64 (and increased proportionally for the "wider" models). The convolutional encoder is followed by fully connected layers with 256 and 20 units respectively, where the output of the latter layer is split in two 10-dimensional vectors encoding the mean and the variance of the approximate posterior. The decoder starts two fully connected layers with 256 and 1024 units, the output of the latter layer is reshaped to a  $4 \times 4 \times 64$  feature map and further processed by a decoder that is symmetric to the encoder.
+
+On Shapes3D we train all models for a total of 600,000 mini-batch iterations, and we multiply the learning rate by 0.5 after 300,000, 390,000, 480,000, and 570,000 iterations. On CIFAR-10 we use a proportionally twice longer schedule. We tuned the learning rates by sweeping over the values  $\{5\cdot 10^{-5},1\cdot 10^{-4},2\cdot 10^{-4},4\cdot 10^{-4},8\cdot 10^{-4}\}$  and ended up using the learning rates  $1\cdot 10^{-4}$  on CIFAR-10 and  $2\cdot 10^{-4}$  on Shapes3D. We use mini-batches of 128 samples on CIFAR-10 and 64 samples on Shapes3D. We use weight decay of  $10^{-5}$  in all models.
+
+For YOTO models, we condition the last layer of each convolutional block. The conditioning MLP has one hidden layer with 256 units on Shapes3D and 512 units on CIFAR-10. At training time we sample the  $\beta$  parameter from log-normal distribution on the interval [0.125, 1024.] for Shapes3D and on the interval [0.125, 512.] for CIFAR-10.
+
+# A.2 IMAGE COMPRESSION
+
+We train the model as proposed by Ballé et al. (2018), and we refer the reader to that paper for architecture details. We train the models with the standard MSE reconstruction loss. All models have 192 channels in all layers, except for the "width x1.33" ones that have 256 channels. We use mini-batches of 8 samples. We train for 2 million mini-batch iterations with the learning rate of  $10^{-4}$ . We condition all convolutional layers in all networks using MLPs with 1 hidden layer of 128 units. During training we sample the rate-distortion weight log-uniformly from the interval  $[1.2 \cdot 10^{-3}, 2.6 \cdot 10^{-1}]$ .
+
+# A.3 FAST STYLE TRANSFER
+
+We build on the work of Ghiasi et al. (2017), and use the exact same network architecture and training protocol. We optimize for 2 million steps with a learning rate of  $10^{-5}$ . The ranges that we used were  $[0.1, 100]$  for the content parameters,  $[10^{-4}, 10^{-1}]$  for the first three style parameters and  $[10^{-3}, 1]$  for the fourth one — we always sampled from a log-uniform distribution.
+
+We condition the models by 1) sending the logarithms of the weights to an MLP with a single hidden layer of size 512 with ReLu activations and an output dimension of size, 2) concatenating these inputs to the inception features of the style image, 3) for each layer modulation we use an MLP
+
+![](images/7e35af775dd79eb5ad219f57ec3ce6e39aacad59bec46ca10b295cee7e537d18.jpg)  
+Figure 8: Qualitative VAE results on the CIFAR-10 dataset. Both for reconstruction and sampling, the YOTO results are qualitatively very similar to those of separate models trained for each value of the loss weight.
+
+with a single layer of dimension 256 and ReLU activations, on which two affine maps are applied to compute the scale and shift.
+
+# B ADDITIONAL RESULTS
+
+# B.1 BETA-VAE
+
+Figure 8 shows qualitative results for  $\beta$ -VAE on CIFAR-10. Same as for Shapes3D, both reconstructions and samples from the YOTO model are qualitatively very similar to those generated with models trained separately for each weight.
+
+We compare the proposed method to two baselines. The first one is a naive baseline — training a VAE with a fixed  $\beta$ , and computing the loss value of this fixed model for all other  $\beta$  values ("Single model baseline"). We select the fixed  $\beta$  so that it minimizes the average validation loss over all  $\beta$  values. The second baseline interpolates between two models trained with fixed weights. Namely, we train two beta-VAEs for the extreme values of beta, and interpolate both the latents and the reconstructions produced by these models. We then measure the KL and reconstruction losses for these interpolated values. Note that this interpolation baseline is not a VAE any more: the latents and the reconstructions are averaged independently, so the interpolated reconstruction cannot be generated from the interpolated latent. Therefore, it is not clear how would one sample from such a model, and we do not report these results in the main paper to avoid confusion.
+
+The results are shown in Figure 9. Surprisingly, on Shapes3D the baseline performs very close to the fixed-model and YOTO frontiers, and sometimes even outperforms them. In contrast, on CIFAR-10 the baseline is much worse than the other methods. We believe the reason is that while averaging the logits for Bernoulli outputs on Shapes3D is quite meaningful, averaging the standard deviations of Gaussian outputs on CIFAR-10 strongly affects Gaussian log-likelihood.
+
+Timing. We timed the training of a fixed-weight  $\beta$ -VAE and a YOTO model. To this end, we trained both models on a single CPU core over 200 mini-batch iterations. We found the YOTO model to be only  $8\%$  slower than its fixed-weight counterpart: 4.64 vs 5.04 training steps per second with the "width x2" network architecture.
+
+![](images/6a75ee86cda9b43955b078878cfd366c5d3357da79d09facba3e4ee424ec9e11.jpg)  
+(a) CIFAR-10 frontier
+
+![](images/6093e0a3781dc4811cf65272da5c74d74887cd5dabe5e87b1d9407a8d9f45908.jpg)  
+(b) CIFAR-10 full loss
+
+![](images/ee98a6db31c29f7ed9bbeb9e47098ecc5d8aafb94f60a50d5c9f7aa4441a444e.jpg)  
+(c) 3DShapes frontier
+
+![](images/3d041720a0740c18778f1fd7275f542ff31881257f6d4fc38a98a2144c8cbbeb.jpg)  
+(d) 3DShapes full loss  
+Figure 9: Quantitative comparison of the proposed method with baselines for beta-VAE on CIFAR-10 (a,b), and Shapes3D (c,d). The proposed method outperforms the "single model" baseline by a large model. The "interpolation" baseline is reported for completeness, but it is not a fair baseline, since it does not correspond to a VAE.
+
+![](images/8795fdd03dd3c720b7bca1ff68f2cc548cba2170134b7f95f5b17554b690d8db.jpg)  
+(a) MS-SSIM on Tecnick.  
+Figure 10: Quantitative results of compression with the MS-SSIM metric on Tecnick (a) and Kodak (b).
+
+![](images/af10542e05f5812f62be500b41d7058318dd6ad45a409d58acf8d6602b3a6df9.jpg)  
+(b) MS-SSIM on Kodak.
+
+![](images/7c0336fc8d9c18983a73a34fa18aa6ecccc42a859d8e53a4e3d09922e17f6c67.jpg)  
+(a) Frontier.
+
+![](images/337c2892c4352bf57fda26393d75bebff1f1b8779c0b426db030ad7e765b3c66.jpg)  
+(b) Full losses.  
+Figure 11: Ablation study of the CIFAR-10  $\beta$ -VAE. The distribution of the loss weight used at training time has the largest impact on the performance. Moreover, conditioning the model by providing the loss weights as inputs to the model underperforms compared to more advanced conditioning methods. Other variants all perform close to the full method.
+
+# B.2 ABLATION STUDY
+
+We now ablate different design choices made in our implementation of YOTO to check their importance. We perform this analysis on the CIFAR-10 dataset, with the "width x4" architecture. The design decisions we study are: the weight distribution used during training (log-uniform or uniform), the method of sampling the loss weights (per sample or per mini-batch), the method of conditioning the network on the weights (FiLM, concatenating the conditioning vectors with the feature maps, or feeding the conditioning vectors to the network as an additional input), pre-processing of the conditioning inputs (taking their logarithm or not).
+
+The results are presented in Figure 11. Clearly, the choice of the parameter distribution during training is the most important choice, and log-uniform distribution performs much better than uniform. Other choices are of less importance, but more advanced conditioning schemes have an advantage over simply feeding the loss parameters as one of the network inputs.
+
+# B.3 STYLE TRANSFER
+
+Figure 12 shows additional qualitative results on style transfer for different content and style images.
+
+# C THEORY
+
+We start by proving Proposition 1 from the main paper and then comment on the cases when the conditions of the proposition are satisfied.
+
+Proof of Proposition 1. Denote the values minimized in optimization problems (1) and (2) by  $V_{1}(\cdot)$  and  $V_{2}(\cdot)$  respectively. From the definition of  $F^{*}$  it follows that  $V_{1}(G(\cdot, \lambda)) \geqslant V_{1}(F^{*}(\cdot, \lambda))$  for any  $\lambda \in \Lambda$  and  $G \in C(\mathbb{X} \times \Lambda)$ . Therefore,  $V_{2}(G) \geqslant V_{2}(F^{*})$ , so  $F^{*}$  minimizes (2). Thus, for any  $G^{*}$  that minimizes (2):
+
+$$
+0 = V _ {2} \left(G ^ {*}\right) - V _ {2} \left(F ^ {*}\right) = \mathbb {E} _ {\lambda \sim P _ {\lambda}} \left[ V _ {1} \left(G ^ {*} (\cdot , \lambda)\right) - V _ {1} \left(F ^ {*} (\cdot , \lambda)\right) \right]. \tag {6}
+$$
+
+Moreover,  $V_{1}(G^{*}(\cdot ,\lambda)) - V_{1}(F^{*}(\cdot ,\lambda)) \geqslant 0$  for all  $\lambda \in \Lambda$ . Expectation of a non-negative function is 0 only if it equals 0 almost surely, which means that  $V_{1}(G^{*}(\cdot ,\lambda)) = V_{1}(F^{*}(\cdot ,\lambda))$  a.s., q.e.d.
+
+The most restrictive condition of the proposition is the continuity of the function  $F^{*}(\cdot, \cdot)$  as a function of  $\lambda$ . Indeed, generally  $\arg \max$  is a multi-valued mapping that can be shown to be hemicontinuous using Berge's Maximum theorem (Berge, 1963) under weak technical conditions. However, there does not necessarily exist a continuous selection of this multi-valued mapping. Its existence can be proven under very restrictive conditions, such as the uniqueness of the minimizer,
+
+![](images/af5c164ad105d9a5e6410fb919fb79a581598a07a166996cab406f34ad065731.jpg)  
+Figure 12: Additional qualitative comparison of image stylization models on images from the validation set of ImageNet. The first row shows the results of YOTO trained on all parameters, the second row - of models trained independently per configuration. In both cases parameter value increases from left or right. YOTO results are very similar to those of independently trained models.
+
+![](images/f6e34a139c227ac1641f0c6cf70bfead6cefb856af0a5f0e3a78db958cbb2630.jpg)
+
+which clearly do not hold for optimization problems encountered in deep learning. While empirically the existence of a continuous  $F^{*}$  seems natural, it remains to be seen if theoretical guarantees can be provided for realistically complex scenarios.
+
+Another direction for extending Proposition 1 is explicitly incorporating function approximation. With function approximation, it would likely not be possible to prove that minimizers of the two problems coincide almost surely, but rather that they can be made arbitrarily close with a probability arbitrarily close to 1. Overall, we believe our work raises a range of interesting theoretical questions to be addressed in future research.
\ No newline at end of file
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+# YOUR CLASSIFIER IS SECRETLY AN ENERGY BASED MODEL AND YOU SHOULD TREAT IT LIKE ONE
+
+# Will Grathwohl
+
+University of Toronto & Vector Institute Google Research wgrathwohl@cs.toronto.edu
+
+# Kuan-Chieh Wang* & Jörn-Henrik Jacobsen*
+
+University of Toronto & Vector Institute wangkual@cs.toronto.edu j.jacobsen@vectorinstitute.ai
+
+# David Duvenaud
+
+University of Toronto & Vector Institute duvenaud@cs.toronto.edu
+
+# Kevin Swersky & Mohammad Norouzi
+
+Google Research {kswersky, mnorouzi}  $@$  google.com
+
+# ABSTRACT
+
+We propose to reinterpret a standard discriminative classifier of  $p(y|\mathbf{x})$  as an energy based model for the joint distribution  $p(\mathbf{x},y)$ . In this setting, the standard class probabilities can be easily computed as well as unnormalized values of  $p(\mathbf{x})$  and  $p(\mathbf{x}|y)$ . Within this framework, standard discriminative architectures may be used and the model can also be trained on unlabeled data. We demonstrate that energy based training of the joint distribution improves calibration, robustness, and out-of-distribution detection while also enabling our models to generate samples rivaling the quality of recent GAN approaches. We improve upon recently proposed techniques for scaling up the training of energy based models and present an approach which adds little overhead compared to standard classification training. Our approach is able to achieve performance rivaling the state-of-the-art in both generative and discriminative learning within one hybrid model.
+
+# 1 INTRODUCTION
+
+For decades, research on generative models has been motivated by the promise that generative models can benefit downstream problems such as semi-supervised learning, imputation of missing data, and calibration of uncertainty (e.g., Chapelle et al. (2006); Dempster et al. (1977)). Yet, most recent research on deep generative models ignores these problems, and instead focuses on qualitative sample quality and log-likelihood on heldout validation sets.
+
+Currently, there is a large performance gap between the strongest generative modeling approach to downstream tasks of interest and hand-tailored solutions for each specific problem. One potential explanation is that most downstream tasks are discriminative in nature and state-of-the-art generative models have diverged quite heavily from state-of-the-art discriminative architectures. Thus, even when trained solely as classifiers, the performance of generative models is far below the performance of the best discriminative models. Hence, the potential benefit from the generative component of the model is far outweighed by the decrease in discriminative performance. Recent work (Behrmann et al., 2018; Chen et al., 2019) attempts to improve the discriminative performance of generative models by leveraging invertible architectures, but these methods still underperform their purely discriminative counterparts jointly trained as generative models.
+
+This paper advocates the use of energy based models (EBMs) to help realize the potential of generative models on downstream discriminative problems. While EBMs are currently challenging to work with, they fit more naturally within a discriminative framework than other generative models and facilitate the use of modern classifier architectures. Figure 1 illustrates an overview of the architecture, where the logits of a classifier are re-interpreted to define the joint density of data points and labels and the density of data points alone.
+
+The contributions of this paper can be summarized as: 1) We present a novel and intuitive framework for joint modeling of labels and data. 2) Our models considerably outperform previous state-of-the-art hybrid models at both generative and discriminative modeling. 3) We show that the incorporation of generative modeling gives our models improved calibration, out-of-distribution detection, and adversarial robustness, performing on par with or better than hand-tailored methods for multiple tasks.
+
+# 2 ENERGY BASED MODELS
+
+Energy based models (LeCun et al., 2006) hinge on the observation that any probability density  $p(\mathbf{x})$  for  $\mathbf{x} \in \mathbb{R}^D$  can be expressed as
+
+$$
+p _ {\theta} (\mathbf {x}) = \frac {\exp \left(- E _ {\theta} (\mathbf {x})\right)}{Z (\theta)}, \tag {1}
+$$
+
+where  $E_{\theta}(\mathbf{x}) : \mathbb{R}^{D} \to \mathbb{R}$ , known as the energy function, maps each point to a scalar, and  $Z(\theta) = \int_{\mathbf{x}} \exp(-E_{\theta}(\mathbf{x}))$  is the normalizing constant (with respect to  $\mathbf{x}$ ) also known as the partition function. Thus, one can parameterize an EBM using any function that takes  $\mathbf{x}$  as the input and returns a scalar.
+
+For most choices of  $E_{\theta}$ , one cannot compute or even reli
+
+ably estimate  $Z(\theta)$ , which means estimating normalized densities is intractable and standard maximum likelihood estimation of the parameters,  $\theta$ , is not straightforward. Thus, we must rely on other methods to train EBMs. We note that the derivative of the log-likelihood for a single example  $x$  with respect to  $\theta$  can be expressed as
+
+$$
+\frac {\partial \log p _ {\theta} (\mathbf {x})}{\partial \theta} = \mathbb {E} _ {p _ {\theta} \left(\mathbf {x} ^ {\prime}\right)} \left[ \frac {\partial E _ {\theta} \left(\mathbf {x} ^ {\prime}\right)}{\partial \theta} \right] - \frac {\partial E _ {\theta} (\mathbf {x})}{\partial \theta}, \tag {2}
+$$
+
+where the expectation is over the model distribution. Unfortunately, we cannot easily draw samples from  $p_{\theta}(\mathbf{x})$ , so we must resort to MCMC to use this gradient estimator. This approach was used to train some of the earliest EBMs. For example, Restricted Boltzmann Machines (Hinton, 2002) were trained using a block Gibbs sampler to approximate the expectation in Eq. (2).
+
+Despite a long period of little development, there has been recent work using this method to train large-scale EBMs on high-dimensional data, parameterized by deep neural networks (Nijkamp et al., 2019b;a; Du & Mordatch, 2019; Xie et al., 2016). These recent successes have approximated the expectation in Eq. (2) using a sampler based on Stochastic Gradient Langevin Dynamics (SGLD) (Welling & Teh, 2011) which draws samples following
+
+$$
+\mathbf {x} _ {0} \sim p _ {0} (\mathbf {x}), \quad \mathbf {x} _ {i + 1} = \mathbf {x} _ {i} - \frac {\alpha}{2} \frac {\partial E _ {\theta} \left(\mathbf {x} _ {i}\right)}{\partial \mathbf {x} _ {i}} + \epsilon , \quad \epsilon \sim \mathcal {N} (0, \alpha) \tag {3}
+$$
+
+where  $p_0(\mathbf{x})$  is typically a Uniform distribution over the input domain and the step-size  $\alpha$  should be decayed following a polynomial schedule. In practice the step-size,  $\alpha$ , and the standard deviation of  $\epsilon$  is often chosen separately leading to a biased sampler which allows for faster training. See Appendix H.1 for further discussion of samplers for EBM training.
+
+# 3 WHAT YOUR CLASSIFIER IS HIDING
+
+In modern machine learning, a classification problem with  $K$  classes is typically addressed using a parametric function,  $f_{\theta} : \mathbb{R}^{D} \to \mathbb{R}^{K}$ , which maps each data point  $\mathbf{x} \in \mathbb{R}^{D}$  to  $K$  real-valued numbers known as logits. These logits are used to parameterize a categorical distribution using the so-called Softmax transfer function:
+
+$$
+p _ {\theta} (y \mid \mathbf {x}) = \frac {\exp \left(f _ {\theta} (\mathbf {x}) [ y ]\right)}{\sum_ {y ^ {\prime}} \exp \left(f _ {\theta} (\mathbf {x}) \left[ y ^ {\prime} \right]\right)}, \tag {4}
+$$
+
+![](images/44269a4abe7402fc2f32be3132ae7653be60226b94ec770f75eaa62846618707.jpg)  
+Figure 1: Visualization of our method, JEM, which defines a joint EBM from classifier architectures.
+
+where  $f_{\theta}(\mathbf{x})[y]$  indicates the  $y^{\mathrm{th}}$  index of  $f_{\theta}(\mathbf{x})$ , i.e., the logit corresponding the the  $y^{\mathrm{th}}$  class label.
+
+Our key observation in this work is that one can slightly re-interpret the logits obtained from  $f_{\theta}$  to define  $p(\mathbf{x},y)$  and  $p(\mathbf{x})$  as well. Without changing  $f_{\theta}$ , one can re-use the logits to define an energy based model of the joint distribution of data point  $\mathbf{x}$  and labels  $y$  via:
+
+$$
+p _ {\theta} (\mathbf {x}, y) = \frac {\exp \left(f _ {\theta} (\mathbf {x}) [ y ]\right)}{Z (\theta)}, \tag {5}
+$$
+
+where  $Z(\theta)$  is the unknown normalizing constant and  $E_{\theta}(\mathbf{x},y) = -f_{\theta}(\mathbf{x})[y]$ .
+
+By marginalizing out  $y$ , we obtain an unnormalized density model for  $\mathbf{x}$  as well,
+
+$$
+p _ {\theta} (\mathbf {x}) = \sum_ {y} p _ {\theta} (\mathbf {x}, y) = \frac {\sum_ {y} \exp \left(f _ {\theta} (\mathbf {x}) [ y ]\right)}{Z (\theta)}. \tag {6}
+$$
+
+Notice now that the LogSumExp(\cdot) of the logits of any classifier can be re-used to define the energy function at a data point  $\mathbf{x}$  as
+
+$$
+E _ {\theta} (\mathbf {x}) = - \operatorname {L o g S u m E x p} _ {y} \left(f _ {\theta} (\mathbf {x}) [ y ]\right) = - \log \sum_ {y} \exp \left(f _ {\theta} (\mathbf {x}) [ y ]\right). \tag {7}
+$$
+
+Unlike typical classifiers, where shifting the logits  $f_{\theta}(\mathbf{x})$  by an arbitrary scalar does not affect the model at all, in our framework, shifting the logits for a data point  $\mathbf{x}$  will affect  $\log p_{\theta}(\mathbf{x})$ . Thus, we are making use of the extra degree of freedom hidden within the logits to define the density function over input examples as well as the joint density among examples and labels. Finally, when we compute  $p_{\theta}(y \mid \mathbf{x})$  via  $p_{\theta}(\mathbf{x}, y) / p_{\theta}(\mathbf{x})$  by dividing Eq. (5) to Eq. (6), the normalizing constant cancels out, yielding the standard Softmax parameterization in Eq. (4). Thus, we have found a generative model hidden within every standard discriminative model! Since our approach proposes to reinterpret a classifier as a Joint Energy-based Model we refer to it throughout this work as JEM.
+
+# 4 OPTIMIZATION
+
+We now wish to take advantage of our new interpretation of classifier architectures to gain the benefits of generative models while retaining strong discriminative performance. Since our model's parameterization of  $p(y|\mathbf{x})$  is normalized over  $y$ , it is simple to maximize its likelihood as in standard classifier training. Since our models for  $p(\mathbf{x})$  and  $p(\mathbf{x},y)$  are unnormalized, maximizing their likelihood is not as easy. There are many ways we could train  $f_{\theta}$  to maximize the likelihood of the data under this model. We could apply the gradient estimator of Equation 2 to the likelihood under the joint distribution of Equation 5. Using Equations 6 and 4, we can also factor the likelihood as
+
+$$
+\log p _ {\theta} (\mathbf {x}, y) = \log p _ {\theta} (\mathbf {x}) + \log p _ {\theta} (y | \mathbf {x}). \tag {8}
+$$
+
+The estimator of Equation 2 is biased when using a MCMC sampler with a finite number of steps. Given that the goal of our work is to incorporate EBM training into the standard classification setting, the distribution of interest is  $p(y|\mathbf{x})$ . For this reason we propose to train using the factorization of Equation 8 to ensure this distribution is being optimized with an unbiased objective. We optimize  $p(y|\mathbf{x})$  using standard cross-entropy and optimize  $\log p(\mathbf{x})$  using Equation 2 with SGLD where gradients are taken with respect to  $\mathrm{LogSumExp}_y(f_\theta(x)[y])$ . We find alternative factorings of the likelihood lead to considerably worse performance as can be seen in Section 5.1.
+
+Following Du & Mordatch (2019) we use persistent contrastive divergence (Tieleman, 2008) to estimate the expectation in the right-hand-side of Equation 2 since it gives an order of magnitude savings in computation compared to seeding new chains at each iteration as in Nijkamp et al. (2019b). This comes at the cost of decreased training stability. These trade-offs are discussed in Appendix H.2.
+
+# 5 APPLICATIONS
+
+We completed a thorough empirical investigation to demonstrate the benefits of JEM over standard classifiers. First, we achieved performance rivaling the state of the art in both discriminative and generative modeling. Even more interesting, we observed a number of benefits related to the practical application of discriminative models including improved uncertainty quantification, out-of-distribution detection, and robustness to adversarial examples. Generative models have been long-expected to provide these benefits but have never been demonstrated to do so at this scale.
+
+<table><tr><td>Class</td><td>Model</td><td>Accuracy% ↑</td><td>IS↑</td><td>FID↓</td></tr><tr><td rowspan="5">Hybrid</td><td>Residual Flow</td><td>70.3</td><td>3.6</td><td>46.4</td></tr><tr><td>Glow</td><td>67.6</td><td>3.92</td><td>48.9</td></tr><tr><td>IGEBM</td><td>49.1</td><td>8.3</td><td>37.9</td></tr><tr><td>JEM p(x|y) factored</td><td>30.1</td><td>6.36</td><td>61.8</td></tr><tr><td>JEM (Ours)</td><td>92.9</td><td>8.76</td><td>38.4</td></tr><tr><td>Disc.</td><td>Wide-Resnet</td><td>95.8</td><td>N/A</td><td>N/A</td></tr><tr><td rowspan="2">Gen.</td><td>SNGAN</td><td>N/A</td><td>8.59</td><td>25.5</td></tr><tr><td>NCSN</td><td>N/A</td><td>8.91</td><td>25.32</td></tr></table>
+
+![](images/6454f561a794a3c673f1c299b0994d4fcc8a6195438ba1c57a16ff76125f435a.jpg)  
+Figure 2: CIFAR10 class-conditional samples.
+
+Table 1: CIFAR10 Hybrid modeling Results. Residual Flow (Chen et al., 2019), Glow (Kingma & Dhariwal, 2018), IGEBM (Du & Mordatch, 2019), SNGAN (Miyato et al., 2018), NCSN (Song & Ermon, 2019)
+
+All architectures used are based on Wide Residual Networks (Zagoruyko & Komodakis, 2016) where we have removed batch-normalization<sup>1</sup> to ensure that our models' outputs are deterministic functions of the input. This slightly increases classification error of a WRN-28-10 from  $4.2\%$  to  $6.4\%$  on CIFAR10 and from 2.3 to  $3.4\%$  on SVHN.
+
+All models were trained in the same way with the same hyper-parameters which were tuned on CIFAR10. Intriguingly, the SGLD sampler parameters found here generalized well across datasets and model architectures. All models are trained on a single GPU in approximately 36 hours. Full experimental details can be found in Appendix A. SVHN
+
+# 5.1 HYBRID MODELING
+
+First, we show that a given classifier architecture can be trained as an EBM to achieve competitive performance as both a classifier and a generative model. We train JEM on CIFAR10, SVHN, and CIFAR100 and compare against other hybrid models as well as standalone generative and discriminative models. We find JEM performs near the state of the art in both tasks simultaneously, outperforming other hybrid models (Table 1).
+
+Given that we cannot compute normalized likelihoods, we present inception scores (IS) (Salimans et al., 2016) and Frechet Inception Distance (FID) (Heusel et al., 2017) as a proxy for this quantity. We find that JEM is competitive with SOTA generative models at these metrics. These metrics are not commonly reported on CIFAR100 and SVHN so we present accuracy and qualitative samples on these datasets. Our models achieve  $96.7\%$  and  $72.2\%$  accuracy on SVHN and CIFAR100, respectively. Samples from JEM can be seen in Figures 2, 3 and in Appendix C.
+
+![](images/b8140171feaac3d816b05a9b49aa15ce8f3e75b0bb8a06e56f3e1e4ad0117f34.jpg)  
+CIFAR100
+
+![](images/5990912c104042ea4c36dd99645811da25f947f2826f31cada5a831ab54d5f54.jpg)  
+Figure 3: Class-conditional samples.
+
+JEM is trained to maximize the likelihood factorization shown in Eq. 8. This was to ensure that no bias is added into our estimate of  $\log p(y|\mathbf{x})$
+
+which can be computed exactly in our setup. Prior work (Du & Mordatch, 2019; Xie et al., 2016) proposes to factorize the objective as  $\log p(\mathbf{x}|y) + \log p(y)$ . In these works, each  $p(\mathbf{x}|y)$  is a separate EBM with a distinct, unknown normalizing constant, meaning that their model cannot be used to compute  $p(y|\mathbf{x})$  or  $p(\mathbf{x})$ . This explains why the model of Du & Mordatch (2019) (we will refer to this model as IGEBM) is not a competitive classifier. As an ablation, we trained JEM to maximize this objective and found a considerable decrease in discriminative performance (see Table 1, row 4).
+
+# 5.2 CALIBRATION
+
+A classifier is considered calibrated if its predictive confidence,  $\max_y p(y|\mathbf{x})$ , aligns with its misclassification rate. Thus, when a calibrated classifier predicts label  $y$  with confidence .9 it should have a  $90\%$  chance of being correct. This is an important feature for a model to have when deployed in real-world scenarios where outputting an incorrect decision can have catastrophic consequences. The classifier's confidence can be used to decide when to output a prediction or defer to a human,
+
+for example. Here, a well-calibrated, but less accurate classifier can be considerably more useful than a more accurate, but less-calibrated model.
+
+While classifiers have grown more accurate in recent years, they have also grown considerably less calibrated (Guo et al., 2017). Contrary to this behavior, we find that JEM notably improves classification while retaining high accuracy.
+
+We focus on CIFAR100 since SOTA classifiers achieve approximately  $80\%$  accuracy. We train JEM on this dataset and compare to a baseline of the same architecture without EBM training. Our baseline model achieves
+
+![](images/f6708feec46ca0fca25bd2c03f8d3f8e8b7cb7f0a32c2328187e460135ca9faa.jpg)  
+Figure 4: CIFAR100 calibration results. ECE = Expected Calibration Error (Guo et al., 2017), see Appendix E.1.
+
+![](images/4f694a4fdbd7d0b8e1a561718a2ce3c0e608e9c6ea4c04682df63daf9194ad2e.jpg)
+
+$74.2\%$  accuracy and JEM achieves  $72.2\%$  (for reference, a ResNet-110 achieves  $74.8\%$  accuracy (Zagoruyko & Komodakis, 2016)). We find the baseline model is very poorly calibrated outputting highly over-confident predictions. Conversely, we find JEM produces a nearly perfectly calibrated classifier when measured with Expected Calibration Error (see Appendix E.1). Compared to other calibration methods such as Platt scaling (Guo et al., 2017), JEM requires no additional training data. Results can be seen in Figure 4 and additional results can be found in Appendix E.2.
+
+# 5.3 OUT-OF-DISTRIBUTION DETECTION
+
+In general, out-of-distribution (OOD) detection is a binary classification problem, where the model is required to produce a score
+
+$$
+s _ {\theta} (\mathbf {x}) \in \mathbb {R},
+$$
+
+where  $\mathbf{x}$  is the query, and  $\theta$  is the set of learnable parameters. We desire that the scores for in-distribution examples are higher than that out-of-distribution examples. Typically for evaluation, threshold-free metrics are used, such as the area under the receiver-operating curve (AU-ROC) (Hendrycks & Gimpel, 2016). There exist a number of distinct OOD detection approaches to which JEM can be applied. We expand on them below. Further results and experimental details can be found in Appendix F.2.
+
+# 5.3.1 INPUT DENSITY
+
+A natural approach to OOD detection is to fit a density model on the data and consider examples with low likelihood to be OOD. While intuitive, this approach is currently not competitive on high-dimensional data. Nalisnick et al. (2018) showed that tractable deep generative models such as Kingma & Dhariwal (2018) and Salimans et al. (2017) can assign higher densities to OOD examples than in-distribution examples. Further work (Nalisnick et al., 2019) shows examples where the densities of an OOD dataset are completely indistinguishable from the in-distribution set, e.g., see Table 2, column 1. Conversely, Du & Mordatch (2019) have shown that the likelihoods from EBMs can be reliably used as a predictor for OOD inputs. As can be seen in Table 2 column 2, JEM consistently assigns higher likelihoods to in-distribution data than OOD data. One possible explanation for JEM's further improvement over IGEBM is its ability to incorporate labeled information during training while also being able to derive a principled model of  $p(\mathbf{x})$ . Intriguingly, Glow does not appear to benefit in the same way from this supervision as is demonstrated by the little difference between our unconditional and class-conditional Glow results. Quantitative results can be found in Table 3 (top).
+
+# 5.3.2 PREDICTIVE DISTRIBUTION
+
+Many successful approaches have utilized a classifier's predictive distribution for OOD detection (Gal & Ghahramani, 2016; Wang et al., 2018; Liang et al., 2017). A useful OOD score that can be derived from this distribution is the maximum prediction probability:  $s_{\theta}(\mathbf{x}) = \max_y p_{\theta}(y|\mathbf{x})$
+
+(Hendrycks & Gimpel, 2016). It has been demonstrated that OOD performance using this score is highly correlated with a model's classification accuracy. Since JEM is a competitive classifier, we find it performs on par (or beyond) the performance of a strong baseline classifier and considerably outperforms other generative models. Results can be seen in Table 3 (middle).
+
+# 5.3.3 A NEW SCORE: APPROXIMATE MASS
+
+It has been recently proposed that likelihood may not be enough for OOD detection in high dimensions (Nalisnick et al., 2019). It is possible for a point to have high likelihood under a distribution yet be nearly impossible to be sampled. Real samples from a distribution lie in what is known as the "typical" set. This is the area of high probability mass. A single point may have high density but if the surrounding areas have very low density, then that point is likely not in the typical set and therefore likely not a sample from the data distribution. For a high-likelihood datapoint outside of the typical set, we expect the density to change rapidly around it, thus the norm of the gradient of the log-density will be large compared to examples in the typical set (otherwise it would be in an area of high mass). We propose an alternative OOD score based on this quantity:
+
+$$
+s _ {\theta} (\mathbf {x}) = - \left\| \frac {\partial \log p _ {\theta} (\mathbf {x})}{\partial \mathbf {x}} \right\| _ {2}. \tag {9}
+$$
+
+For EBMs (JEM and IGEBM), we find this predictor greatly outperforms our own and other generative model's likelihoods - see Table 2 column 3. For tractable likelihood methods we find this predictor is anti-correlated with the model's likelihood and neither is reliable for OOD detection. Results can be seen in Table 3 (bottom).
+
+# 5.4 ROBUSTNESS
+
+Recent work (Athalye et al., 2017) has demonstrated that classifiers trained to be adversarially robust can be re-purposed to generate convincing images, do in-painting, and translate examples from one class to another. This is done through an iterative refinement procedure, quite similar to the SGLD used to sample from EBMs. We also note that adversarial training (Goodfellow et al., 2014) bears many similarities to SGLD training of EBMs. In both settings, we use a gradient-based optimization procedure to generate examples which activate a specific high-level network activation, then optimize the weights of the network to minimize the generated example's effect on that activation. Further connections have been drawn between adversarial training and regularizing the gradients of the network's activations around the data (Simon-Gabriel et al., 2018). This is similar to the objective of Score Matching (Hyvärinen, 2005) which can also be used to train EBMs (Kingma & Lecun, 2010; Song & Ermon, 2019).
+
+Given these connections one may wonder if a classifier derived from an EBM would be more robust to adversarial examples than a standard model. This behavior has been demonstrated in prior work on EBMs (Du & Mordatch, 2019) but their work did not produce a competitive discriminative model
+
+![](images/b7fd01153d6102cc85d854f660003264e1455ec9680ddeaab137e8910a45d130.jpg)  
+Table 2: Histograms for OOD detection. All models trained on CIFAR10. Green corresponds to the score on (in-distribution) CIFAR10, and red corresponds to the score on the OOD dataset.
+
+<table><tr><td>sθ(x)</td><td>Model</td><td>SVHN</td><td>CIFAR10Interp</td><td>CIFAR100</td><td>CelebA</td></tr><tr><td rowspan="4">log p(x)</td><td>Unconditional Glow</td><td>.05</td><td>.51</td><td>.55</td><td>.57</td></tr><tr><td>Class-Conditional Glow</td><td>.07</td><td>.45</td><td>.51</td><td>.53</td></tr><tr><td>IGEBM</td><td>.63</td><td>.70</td><td>.50</td><td>.70</td></tr><tr><td>JEM (Ours)</td><td>.67</td><td>.65</td><td>.67</td><td>.75</td></tr><tr><td rowspan="4">maxy p(y|x)</td><td>Wide-ResNet</td><td>.93</td><td>.77</td><td>.85</td><td>.62</td></tr><tr><td>Class-Conditional Glow</td><td>.64</td><td>.61</td><td>.65</td><td>.54</td></tr><tr><td>IGEBM</td><td>.43</td><td>.69</td><td>.54</td><td>.69</td></tr><tr><td>JEM (Ours)</td><td>.89</td><td>.75</td><td>.87</td><td>.79</td></tr><tr><td rowspan="4">||∂ log p(x)/∂x||</td><td>Unconditional Glow</td><td>.95</td><td>.27</td><td>.46</td><td>.29</td></tr><tr><td>Class-Conditional Glow</td><td>.47</td><td>.01</td><td>.52</td><td>.59</td></tr><tr><td>IGEBM</td><td>.84</td><td>.65</td><td>.55</td><td>.66</td></tr><tr><td>JEM (Ours)</td><td>.83</td><td>.78</td><td>.82</td><td>.79</td></tr></table>
+
+Table 3: OOD Detection Results. Models trained on CIFAR10. Values are AUROC.
+
+and is therefore of limited practical application for this purpose. Similarly, we find JEM achieves considerable robustness without sacrificing discriminative performance.
+
+# 5.4.1 IMPROVED ROBUSTNESS THROUGH EBM TRAINING
+
+A common threat model for adversarial robustness is that of perturbation-based adversarial examples with an  $L_{p}$ -norm constraint (Goodfellow et al., 2014). They are defined as perturbed inputs  $\tilde{\mathbf{x}} = \mathbf{x} + \delta$ , which change a model's prediction subject to  $||\tilde{\mathbf{x}} - \mathbf{x}||_p < \epsilon$ . These examples exploit semantically meaningless perturbations to which the model is overly sensitive. However, closeness to real inputs in terms of a given metric does not imply that adversarial examples reside within areas of high density according to the model distribution, hence it is not surprising that the model makes mistakes when asked to classify inputs it has rarely or never encountered during training.
+
+This insight has been used to detect and robustly classify adversarial examples with generative models (Song et al., 2017; Li et al., 2018; Fetaya et al., 2019). The state-of-the-art method for adversarial robustness on MNIST classifies by comparing an input to samples generated from a class-conditional generative model (Schott et al., 2018). This can be thought of as classifying an example similar to the input but from an area of higher density under the model's learned distribution. This refined input resides in areas where the model has already "seen" sufficient data and is thus able to accurately classify. Albeit promising, this family of methods has not been able to scale beyond MNIST due to a lack of sufficiently powerful conditional generative models. We believe JEM can help close this gap. We propose to run a few iterations of our model's sampling procedure seeded at
+
+![](images/f1cde4c98213c385d015e56a9ed03d428f36f0251784a5732b0d2dd7e83a2a39.jpg)  
+(a)  $L_{\infty}$  Robustness
+
+![](images/11be197a31e8f298b79c1bf68ecf3769050b83520ef010bfab58ea12759eca87.jpg)  
+(b)  $L_{2}$  Robustness  
+Figure 5: Adversarial Robustness Results with PGD attacks. JEM adds considerable robustness.
+
+a given input. This should be able to transform low-probability inputs to a nearby point of high probability, "undoing" any adversarial attack and enabling the model to classify robustly.
+
+Perturbation Robustness We run a number of powerful adversarial attacks on our CIFAR10 models. We run a white-box PGD attack, giving the attacker access to the gradients through our sampling procedure<sup>2</sup>. Because our sampling procedure is stochastic, we compute the "expectation over transformations" Athalye et al. (2018), the expected gradient over multiple runs of the sampling
+
+procedure. We also run gradient-free black-box attacks; the boundary attack (Brendel et al., 2017) and the brute-force pointwise attack (Rauber et al., 2017). All attacks are run with respect to the  $L_{2}$  and  $L_{\infty}$  norms and we test JEM with 0, 1, and 10 steps of sampling seeded at the input.
+
+Results from the PGD experiments can be seen in Figure 5. Experimental details and remaining results, including gradient-free attacks, can be found in Appendix G. Our model is considerably more robust than a baseline with standard classifier training. With respect to both norms, JEM delivers considerably improved robustness when compared to the baseline but for many epsilons falls below state-of-the-art adversarial training (Madry et al., 2017; Santurkar et al., 2019) and the state-of-the-art certified robustness method of Salman et al. (2019) ("RandAdvSmooth" in Figure 5). We note that each of these baseline methods is trained to be robust to the norm through which it is being attacked and it has been shown that attacking an  $L_{\infty}$  adversially trained model with an  $L_{2}$  adversary decreases robustness considerably (Madry et al., 2017). However, we attack the same JEM model with both norms and observe competitive robustness in both cases.
+
+JEM with 0 steps refinement is noticeably more robust than the baseline model trained as a standard classifier, thus simply adding EBM training can produce more robust models. We also find that increasing the number of refinement steps further increases robustness to levels at robustness-specific approaches. We expect that increasing the number of refinement steps will lead to more robust models but due to computational constraints we could not run attacks in this setting.
+
+Distal Adversarials Another common failure mode of non-robust models is their tendency to classify nonsensical inputs with high confidence. To analyze this property, we follow Schott et al. (2018). Starting from noise we generate images to maximize  $p(y = \text{"car"}|\mathbf{x})$ . Results are shown in figure 6. The baseline confidently classifies unstructured noise images. The  $L_{2}$  adversari-ally trained ResNet with  $\epsilon = 0.5$  (Santurkar et al., 2019) confidently classifies somewhat structured, but unrealistic images. JEM does not confidently classify nonsensical images, so instead, car attributes and natural image properties visibly emerge.
+
+![](images/e50f629d5de3efa1d6e10b6f7ba70c972b3a7af01ba6a07d14e24bf1bb4629a6.jpg)  
+Figure 6: Distal Adversarials. Confidently classified images generated from noise, such that:  $p(y = \text{"car"}|\mathbf{x}) > .9$ .
+
+# 6 LIMITATIONS
+
+Energy based models can be very challenging to work with. Since normalized likelihoods cannot be computed, it can be hard to verify that learning is taking place at all. When working in domains such as images, samples can be drawn and checked to assess learning, but this is far from a generalizable strategy. Even so, these samples are only samples from an approximation to the model so they can only be so useful. Furthermore, the gradient estimators we use to train JEM are quite unstable and are prone to diverging if the sampling and optimization parameters are not tuned correctly. Regularizers may be added (Du & Mordatch, 2019) to increase stability but it is not clear what effect they have on the final model. The models used to generate the results in this work regularly diverged throughout training, requiring them to be restarted with lower learning rates or with increased regularization. See Appendix H.3 for a detailed description of how these difficulties were handled.
+
+While this may seem prohibitive, we believe the results presented in this work are sufficient to motivate the community to find solutions to these issues as any improvement in the training of energy based models will further improve the results we have presented in this work.
+
+# 7 RELATED WORK
+
+Prior work (Xie et al., 2016) made a similar observation to ours about classifiers and EBMs but define the model differently. They reinterpret the logits to define a class-conditional EBM  $p(\mathbf{x} | y)$ , similar to Du & Mordatch (2019). This setting requires additional parameters to be learned to derive a classifier and an unconditional model. We believe this subtle distinction is responsible for our model's success. The model of (Song & Ou, 2018) is similar as well but is trained using a GAN-like generator and is applied to different applications. Also related are Introspective Networks (Jin et al., 2017; Lee et al., 2018) which have drawn a similar connection between discriminative classifiers and generative models. They derive a generative model from a classifier which learns to distinguish
+
+between data and negative examples generative via an MCMC-like procedure. Training in this way has also been shown to improve adversarial robustness.
+
+Our work builds heavily on Nijkamp et al. (2019b;a); Du & Mordatch (2019) which scales the training of EBMs to high-dimensional data using Contrastive Divergence and SGLD. While these works have pushed the boundaries of the types of data to which we can apply EBMs, many issues still exist. These methods require many steps of SGLD to take place at each training iteration. Each step requires approximately the same amount of computation as one iteration of standard discriminative model training, therefore training EBMs at this scale is orders of magnitude slower than training a classifier - limiting the size of problems we can attack with these methods. There exist orthogonal approaches to training EBMs which we believe have promise to scale more gracefully.
+
+Score matching (Hyvärinen, 2005) attempts to match the derivative of the model's density with the derivative of the data density. This approach saw some development towards high-dimensional data (Kingma & Lecun, 2010) and recently has been successfully applied to large natural images (Song & Ermon, 2019). This approach required a model to output the derivatives of the density function, not the density function itself, so it is unclear what utility this model can provide to the applications we have discussed in this work. Regardless, we believe this is a promising avenue for further research. Noise Contrastive Estimation (Gutmann & Hyvärinen, 2010) rephrases the density estimation problem as a classification problem, attempting to distinguish data from a known noise distribution. If the classifier is properly structured, then once the classification problem is solved, an unnormalized density estimator can be derived from the classifier and noise distribution. While this method has been recently extended (Ceylan & Gutmann, 2018), these methods are challenging to extend to high-dimensional data.
+
+# 8 CONCLUSION AND FURTHER WORK
+
+In this work we have presented JEM, a novel reinterpretation of standard classifier architectures which retains the strong performance of SOTA discriminative models while adding the benefits of generative modeling approaches. Our work is enabled by recent work scaling techniques for training EBMs to high dimensional data. We have demonstrated the utility of incorporating this type of training into discriminative models. While there exist many issues in training EBMs we hope the results presented here will encourage the community to improve upon current approaches.
+
+# 9 ACKNOWLEDGEMENTS
+
+We would like to thank Ying Nian Wu and Mitch Hill for providing some EBM training tips and tricks which were crucial in getting this project off the ground. We would also like to thank Jeremy Cohen for his useful feedback which greatly strengthened our adversarial robustness results. We would like to thank Lukas Schott for feedback on the robustness evaluation, Alexander Meinke and Francesco Croce for spotting some typos and suggesting the transfer attack. We would also like to thank Zhuowen Tu and Kwonjoon Lee for bringing related work to our attention.
+
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+Yang Song, Taesup Kim, Sebastian Nowozin, Stefano Ermon, and Nate Kushman. Pixeldefend: Leveraging generative models to understand and defend against adversarial examples. arXiv preprint arXiv:1710.10766, 2017.  
+Yunfu Song and Zhijian Ou. Learning neural random fields with inclusive auxiliary generators. arXiv preprint arXiv:1806.00271, 2018.
+
+Tijmen Tieleman. Training restricted boltzmann machines using approximations to the likelihood gradient. In Proceedings of the 25th international conference on Machine learning, pp. 1064-1071. ACM, 2008.  
+Kuan-Chieh Wang, Paul Vicol, James Lucas, Li Gu, Roger Grosse, and Richard Zemel. Adversarial distillation of bayesian neural network posteriors. In International Conference on Machine Learning (ICML), 2018.  
+Max Welling and Yee W Teh. Bayesian learning via stochastic gradient Langevin dynamics. In Proceedings of the 28th international conference on machine learning (ICML-11), pp. 681-688, 2011.  
+Jianwen Xie, Yang Lu, Song-Chun Zhu, and Yingnian Wu. A theory of generative convnet. In International Conference on Machine Learning, pp. 2635-2644, 2016.  
+Sergey Zagoruyko and Nikos Komodakis. Wide residual networks. arXiv preprint arXiv:1605.07146, 2016.
+
+# A TRAINING DETAILS
+
+We train all models with the Adam optimizer (Kingma & Ba, 2014) for 150 epochs through the dataset using a staircase decay schedule. All network architectures are based on WideResNet-28-10 with no batch normalization. We generate samples using PCD with hyperparameters in Table 4. We evolve the chains with 20-steps of SGLD per iteration and with probability .05 we reinitilize the chains with uniform random noise. For preprocessing, we scale images to the range  $[-1,1]$  and add Gaussian noise of stddev = .03. Pseudo-code for our training procedure is in Algorithm 1.
+
+When training via contrastive divergence there are a few different ways one could potentially draw samples from  $p_{\theta}(\mathbf{x})$ . We could:
+
+1. Sample  $y \sim p(y)$  then sample  $\mathbf{x} \sim p_{\theta}(\mathbf{x}|y)$  via SGLD with energy  $E(\mathbf{x}|y) = -f_{\theta}(\mathbf{x})[y]$  then throw away  $y$ .  
+2. Sample  $\mathbf{x} \sim p_{\theta}(\mathbf{x})$  via SGLD with energy  $E(x) = -\mathrm{LogSumExp}_y f_\theta(\mathbf{x})[y]$ .
+
+We experimented with both methods during training and found that while method 1 produced more visually appealing samples (from a human's perspective), method 2 produced slightly stronger discriminative performance –  $92.9\%$  vs.  $91.2\%$  accuracy on CIFAR10. For this reason we use method 2 in all results presented.
+
+Algorithm 1 JEM training: Given network  $f_{\theta}$ , SGLD step-size  $\alpha$ , SGLD noise  $\sigma$ , replay buffer  $B$ , SGLD steps  $\eta$ , reinitialization frequency  $\rho$  
+1: while not converged do  
+2: Sample  $\mathbf{x}$  and  $y$  from dataset  
+3:  $L_{\mathrm{clf}}(\theta) = \mathrm{xent}(f_{\theta}(\mathbf{x}), y)$   
+4: Sample  $\widehat{\mathbf{x}}_0 \sim B$  with probability  $1 - \rho$ , else  $\widehat{\mathbf{x}}_0 \sim \mathcal{U}(-1, 1)$   
+5: for  $t \in [1, 2, \ldots, \eta]$  do  
+6:  $\widehat{\mathbf{x}}_t = \widehat{\mathbf{x}}_{t-1} + \alpha \cdot \frac{\partial \text{LogSumExp}_{y'}(f_{\theta}(\widehat{\mathbf{x}}_{t-1})[y'])}{\partial \widehat{\mathbf{x}}_{t-1}} + \sigma \cdot \mathcal{N}(0, I)$   
+7: end for  
+8:  $L_{\mathrm{gen}}(\theta) = \text{LogSumExp}_{y'}(f(\mathbf{x})[y']) - \text{LogSumExp}_{y'}(f(\widehat{\mathbf{x}}_t)[y'])$   
+9:  $L(\theta) = L_{\mathrm{clf}}(\theta) + L_{\mathrm{gen}}(\theta)$   
+10: Obtain gradients  $\frac{\partial L(\theta)}{\partial \theta}$  for training  
+11: Add  $\widehat{\mathbf{x}}_t$  to  $B$   
+12: end while
+
+<table><tr><td>Variable</td><td>Values</td></tr><tr><td>initial learning rate</td><td>.0001</td></tr><tr><td>learning epochs</td><td>150</td></tr><tr><td>learning rate decay</td><td>.3</td></tr><tr><td>learning rate decay epochs</td><td>50, 100</td></tr><tr><td>SGLD steps η</td><td>20</td></tr><tr><td>Buffer-size</td><td>10000</td></tr><tr><td>reinitialization frequency ρ</td><td>.05</td></tr><tr><td>SGLD step-size α</td><td>1</td></tr><tr><td>SGLD noise σ</td><td>.01</td></tr></table>
+
+Table 4: Hyperparameters
+
+# B SAMPLE QUALITY EVALUTION
+
+In this section we describe the details for reproducing the Inception Score (IS) and FID results reported in the paper. First we note that both IS and FID are scores computed based on a pretrained classifier network, and thus can be very dependent on the exact model/code repository used. For a more detailed discussion on the variability of IS, please refer to Barratt & Sharma (2018). To gauge our model against the other papers, we document our attempt to fairly compare the scores across
+
+papers in Table 6. As a direct comparison of IS, we got 8.76 using the code provided by Du & Mordatch (2019), and is better than their best reported score of 8.3. For FID, we used the official implementation from Heusel et al. (2017). Note that FID computed from this repository assigned much worse FID than reported in Chen et al. (2019).
+
+Conditional vs unconditional samples. Since we are interested in training a Hybrid model, our model, by definition, is a conditional generative model as it has access to label information. In Table 5, unconditional samples mean samples directly obtained from running SGLD using  $p(x)$ . Conditional samples are obtained by taking the max of our  $p(y|x)$  model. The reported scores are obtained by keeping the top 10 percentile samples with the highest  $p(y|x)$  values. Scores obtained on a "single" model are computed directly on the training replay buffer of the last checkpoint. "Ensemble" here are obtained by lumping together 5 buffers over the last few epochs of training. As we initialize SGLD with uniform noise, using the training buffer is exactly the same as re-sampling from the model.
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Conditional</td><td colspan="2">Unconditional</td></tr><tr><td>single</td><td>ensemble</td><td>single</td><td>ensemble</td></tr><tr><td>JEM (Ours)</td><td>-</td><td>8.76</td><td>7.82</td><td>7.79</td></tr><tr><td>EBM (D&amp;M)</td><td>8.3</td><td>X</td><td>6.02</td><td>6.78</td></tr></table>
+
+Table 5: Conditional vs. unconditional Inception Scores.  
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">Inception Score</td><td colspan="3">FID</td></tr><tr><td>from paper</td><td>B&amp;S</td><td>D&amp;M</td><td>from paper</td><td>H</td><td>D&amp;M</td></tr><tr><td>Residual Flow</td><td>X</td><td>3.6</td><td>-</td><td>46.4</td><td>-</td><td>-</td></tr><tr><td>Glow</td><td>X</td><td>-</td><td>3.9</td><td>48.9*</td><td>107</td><td>-</td></tr><tr><td>JEM (Ours)</td><td>X</td><td>7.13</td><td>8.76</td><td>X</td><td>38.4</td><td>-</td></tr><tr><td>JEM p(x|y) factored</td><td>X</td><td>-</td><td>6.36</td><td>X</td><td>61.8</td><td>-</td></tr><tr><td>EBM (D&amp;M)</td><td>8.3</td><td>-</td><td>8.3</td><td>37.9</td><td>-</td><td>37.9</td></tr><tr><td>SNGAN</td><td>8.59</td><td>-</td><td>-</td><td>25.5</td><td>-</td><td>-</td></tr><tr><td>NCSN</td><td>8.91</td><td>-</td><td>-</td><td>25.3</td><td>-</td><td>-</td></tr></table>
+
+Table 6: The headings: B&S, D&M, and H denotes scores computed using code provided by Barratt & Sharma (2018), Du & Mordatch (2019), Heusel et al. (2017). *denotes numbers copied from Chen et al. (2019), but not the original papers. As unfortunate as the case is with Inception Score and FID (i.e., taking different code repository yields vastly different results), from this table we can still see that our model performs well. Using D&M Inception Score we beat their own model, and using the official repository for FID we beat the Glow  ${}^{3}$  model by a big margin.
+
+# C FURTHER HYBRID MODEL SAMPLES
+
+Additional samples from CIFAR10 and SVHN can be seen in Figure 7 and samples from CIFAR100 can be seen in Figure 8
+
+![](images/c0677aa484fbdeb5c0066fd398a16a19112c9f5907c7ab88f601c5e3d50a4326.jpg)  
+Figure 7: Class-conditional Samples. Left to right: CIFAR10, SVHN.
+
+![](images/7a9ba2581aa6c524e0905d2e01ccbd4ee441755b3cceabbcc14b6fb49ba46459.jpg)
+
+![](images/64735e17a67e4c3311564f43b798c5f2067366cb4b263cdf0d6e32de1cc73f5d.jpg)  
+Figure 8: CIFAR100 Class-conditional Samples.
+
+![](images/e10658c03d6fa6f6fc90ad8b43798522dea2c3bea8f8070ac9da51488ac35a54.jpg)
+
+# D QUALITATIVE ANALYSIS OF SAMPLES
+
+Visual quality is difficult to quantify. Of the known metrics like IS and FID, using samples that have higher  $p(y|\mathbf{x})$  values results in higher scores, but not necessary if we use samples with higher  $\log p(\mathbf{x})$ . However, this is likely because of the downfalls of the evaluation metrics themselves rather than reflecting true sample quality.
+
+Based on our analysis (below), we find
+
+1. Our  $\log p(\mathbf{x})$  model assigns values that cluster around different means for different classes. The class automobiles has the highest  $\log p(\mathbf{x})$ . Of all generated samples, all top 100 samples are of this class.  
+2. Given the class, the samples that have higher  $\log p(\mathbf{x})$  values all have white background and centered object, and lower  $\log p(\mathbf{x})$  samples have colorful (e.g., forest-like) background.  
+3. Of all samples, higher  $p(y|\mathbf{x})$  values means clearly centered objects, and lower  $p(y|\mathbf{x})$  otherwise.
+
+![](images/9ea42b33160fe2b0bd4721be79dc23af6c5912e70fd13b5f4fd6f4c4d41e5f9c.jpg)  
+Figure 9: Each row corresponds to 1 class, subfigures corresponds to different values of  $\log p(\mathbf{x})$ . left: highest, mid: random, right: lowest.
+
+![](images/4400c380b95f3ec77f4d9761aedf2a214567f3eda55a4e89c4d8c45dbdef5a98.jpg)
+
+![](images/f0bd6876755eef6aa8d8499627f976cdaac59bc21fe2041373f8c115704bcbd3.jpg)
+
+![](images/0328d78d51c09312c3aabaf53390f574960902d30ccc32ad2fa2d2020b01dca7.jpg)  
+Figure 10: Histograms (oriented horizontally for easier visual alignment) of  $\log p(\mathbf{x})$  arranged by class.
+
+![](images/0f3d649efec70dd9ea669a989f3753a845bd61bd21a7a04e99e2e8c0c9b403b0.jpg)
+
+![](images/b9ef40c9166f53ffc15a4ff48e247bd3bc2ca753b335b57c77795d46b9f47225.jpg)
+
+![](images/0711850705de77e367ac9f334494d6575a96623ff9c0991529d0f37cd0cc996b.jpg)
+
+![](images/cf54795d35e8d06b48c0714c77c37e6dc4de628a38cedba4609841e4770f595b.jpg)
+
+![](images/721f3669d282949ff21b4b38f16f8df12536f20a18a74100e9508be8db1cce69.jpg)  
+Figure 11: left: samples with highest  $\log p(\mathbf{x})$ , right: left: samples with lowest  $\log p(\mathbf{x})$
+
+![](images/f429762a758757a517f5ba7f6d29ab9e7439311ec50e22b0d31a4bb97510a1d9.jpg)
+
+![](images/88f45b72c165981a47582c1e595ea02bd011c2911ca59ebcaf98860ca3d7ba3a.jpg)  
+Figure 12: left: samples with highest  $p(y|x)$ , right: left: samples with lowest  $p(y|x)$
+
+![](images/5e99c306c16656f78b0a4ae71295f6a0f2abe59ac307a11e648846ea46c324f6.jpg)
+
+# E CALIBRATION
+
+# E.1 EXPECTED CALIBRATION ERROR
+
+Expected Calibration Error (ECE) is a metric to measure the calibration of a classifier. It works by first computing the confidence,  $\max_y p(y|\mathbf{x}_i)$ , for each  $\mathbf{x}_i$  in some dataset. We then group the items into equally spaced buckets  $\{B_m\}_{m=1}^M$  based on the classifier's output confidence. For example, if  $M = 20$ , then  $B_0$  would represent all examples for which the classifier's confidence was between 0.0 and 0.05.
+
+We then define:
+
+$$
+\mathrm {E C E} = \sum_ {m = 1} ^ {M} \frac {\left| B _ {m} \right|}{n} \left| \operatorname {a c c} \left(B _ {m}\right) - \operatorname {c o n f} \left(B _ {m}\right) \right| \tag {10}
+$$
+
+where  $n$  is the number of examples in the dataset,  $\mathrm{acc}(B_m)$  is the averaged accuracy of the classifier of all examples in  $B_{m}$  and  $\operatorname{conf}(B_m)$  is the averaged confidence over all examples in  $B_{m}$ .
+
+For a perfectly calibrated classifier, this value will be 0 for any choice of  $M$ . In our analysis, we choose  $M = 20$  throughout.
+
+# E.2 FURTHER RESULTS
+
+We find that JEM also improves calibration on CIFAR10 as can be seen in Table 13. There we see an improvement in calibration, but both classifiers are well calibrated because their accuracy is so high. In a more interesting experiment, we limit the size of the training set to 4,000 labeled examples. In this setting the accuracy drops to  $78.0\%$  and  $74.9\%$  in the baseline and JEM, respectively. Given the JEM can be trained on unlabeled data, we treat the remainder of the training set as unlabeled and train in a semi-supervised manner. We find this gives a noticeable boost in the classifier's calibration as seen in Figure 13. Surprisingly this did not improve generalization. We leave exploring this phenomenon for future work.
+
+![](images/00e8be33e612ed53f85f0bfea1649f9fd061a38cb962390fdbb34bfbfc552414.jpg)  
+(a) CIFAR10 Baseline
+
+![](images/2266bd095ab778f97a9ebb0a5c522df80cb3dd8715fae1998bedf0d3a45bc9a9.jpg)  
+(b) CIFAR10 JEM
+
+![](images/f4456a5be2073c9d810cb77dd886935d6f6a370012d12b461cdb0b38fbe351f6.jpg)  
+(c) CIFAR100 Baseline (4k labels)  
+Figure 13: CIFAR10 Calibration results
+
+![](images/3e3f94ebca741417f39b264312e4460150676668756d06445c3ef3ca6b6005b6.jpg)  
+(d) CIFAR100 JEM (4k labels)
+
+# F OUF-OF-DISTRIBUTION DETECTION
+
+# F.1 EXPERIMENTAL DETAILS
+
+To obtain OOD results for unconditional Glow, we used the pre-trained model and implementation of https://github.com/y0ast/Glow-PyTorch. We trained a Class-Conditional model as well using this codebase which was used to generate the class-conditional OOD results.
+
+We obtained the IGEBM of Du & Mordatch (2019) from their open-source implementation at https://github.com/openai/ebm_code_release. For likelihood and likelihood-gradient OOD scores we used their pre-trained CIFar10_large_model_uncond model. We were able to replicate the likelihood based OOD results presented in their work. We implemented our likelihood-gradient approximate-mass score on top of their codebase. For predictive distribution based OOD scores we used their CIFar_cond model which was the model used in their work to generate their robustness results.
+
+# F.2 FURTHER RESULTS
+
+Figure 7 contains results on two datasets, Constant and Uniform, which were omitted for space. Most models perform very well at the Uniform dataset. On the Constant dataset (all examples = 0) generative models mainly fail – with JEM being the only one whose likelihoods can be used to derive a predictive score function for OOD detection. Interestingly, we could not obtain approximate mass scores on this dataset from the Glow models due to numerical stability issues.
+
+<table><tr><td rowspan="2">Score</td><td rowspan="2">Model</td><td colspan="6">CIFAR10</td></tr><tr><td>SVHN</td><td>Uniform</td><td>Constant</td><td>Interp</td><td>CIFAR100</td><td>CelebA</td></tr><tr><td rowspan="4">log p(x)</td><td>Unconditional Glow</td><td>.05</td><td>1.0</td><td>0.0</td><td>.51</td><td>.55</td><td>.57</td></tr><tr><td>Glow Supervised</td><td>.07</td><td>1.0</td><td>0.0</td><td>.45</td><td>.51</td><td>.53</td></tr><tr><td>IGEBM</td><td>.63</td><td>1.0</td><td>.30</td><td>.70</td><td>.50</td><td>.70</td></tr><tr><td>JEM (Ours)</td><td>.67</td><td>1.0</td><td>.51</td><td>.65</td><td>.67</td><td>.75</td></tr><tr><td rowspan="4">maxy p(y|x)</td><td>WRN-baseline</td><td>.93</td><td>.97</td><td>.99</td><td>.77</td><td>.85</td><td>.62</td></tr><tr><td>Class-Conditional Glow</td><td>.64</td><td>0.0</td><td>.82</td><td>.61</td><td>.65</td><td>.54</td></tr><tr><td>IGEBM</td><td>.43</td><td>.05</td><td>.60</td><td>.69</td><td>.54</td><td>.69</td></tr><tr><td>JEM (Ours)</td><td>.89</td><td>.41</td><td>.84</td><td>.75</td><td>.87</td><td>.79</td></tr><tr><td rowspan="4">|| ∂ log p(x)/∂x ||</td><td>Unconditional Glow</td><td>.95</td><td>.99</td><td>NaN</td><td>.27</td><td>.46</td><td>.29</td></tr><tr><td>Class-Conditional Glow</td><td>.47</td><td>.99</td><td>NaN</td><td>.01</td><td>.52</td><td>.59</td></tr><tr><td>IGEBM</td><td>.84</td><td>.99</td><td>0.0</td><td>.65</td><td>.55</td><td>.66</td></tr><tr><td>JEM (Ours)</td><td>.83</td><td>1.0</td><td>.75</td><td>.78</td><td>.82</td><td>.79</td></tr></table>
+
+Table 7: OOD Detection Results. Values are AUROC.
+
+# G ATTACK DETAILS AND FURTHER ROBUSTNESS RESULTS
+
+We use foolbox (Rauber et al., 2017) for our experiments. PGD uses binary search to determine minimal epsilons for every input and we plot the resulting robustness-distortion curves. PGD runs with 20 random restarts and 40 iterations. For the boundary attack, we run default foolbox settings with one important difference. The random initialization often fails for JEM and thus we initialize the attack with a correctly classified input of another class. This other class is chosen based on the top-2 prediction for the image to be attacked. As all our attacks are expensive to run, we only attacked 300 randomly chosen inputs. The same randomly chosen inputs were used to attack each model.
+
+In Figure 14 we see the results of the boundary attack and pointwise attack on JEM and a baseline. The main point to running these attacks was to demonstrate that our model was not able to "cheat" by having vanishing gradients through our gradient-based sampling procedure. Since PGD was more successful than these gradient-free methods, this is clearly not the case and the attacker was able to use the gradients of the sampling procedure to attack our model. Further, we observe the same
+
+behavior across all attacks; the EBM with 0 steps sampling is more robust than the baseline and the robustness increases as we add more steps of sampling.
+
+We also compare JEM to the IGEBM of Du & Mordatch (2019) with 10 steps of sampling refinement, see Figure 15. We run the same gradient-based attacks on their model and find that despite not having competitive clean accuracy, it is quite robust to large  $\epsilon$  attacks - especially with respect to the  $L_{\infty}$  norm. After  $\epsilon = 12$  their model is more robust than ours and after  $\epsilon = 18$  it is more robust than the adversarial training baseline. With respect to the  $L_{2}$  norm their model is more robust than the adversarial training baseline above  $\epsilon = 280$  but remains less robust than JEM until  $\epsilon = 525$ .
+
+We believe these results demonstrate that EBMs are a compelling class of models to explore for further work on building robust models.
+
+![](images/00078d8a62160959c7891080ff1d26929fa238934ce97d819f87336686510823.jpg)  
+(a) Boundary  $L_{\infty}$
+
+![](images/f049915e3ce76d9f8b7d691f986e62882bb98bbe2aee7aa3bcb450436927a803.jpg)  
+(b) Boundary  $L_{2}$
+
+![](images/65646b653abd92b5824bf01a92b7472301ba2c0d26574af37ecbca4fc316df1d.jpg)  
+(c) Pointwise  $L_{\infty}$  
+Figure 14: Gradient-free adversarial attacks.
+
+![](images/97bbcfa6207c14dd0dd2a5a3f9b9bbf11a6cbefacaae8906622e23bb04c6040a.jpg)  
+(d) Pointwise  $L_{2}$
+
+# G.1 EXPECTATION OVER TRANSFORMATIONS
+
+Our SGLD-based refinement procedure is stochastic in nature and it has been shown that stochastic defenses to adversarial attacks can provide a false sense of security (Athalye et al., 2018). To deal with this, when we attack our stochastically refined classifiers, we average the classifier's predictions over multiple samples of this refinement procedure. This makes the defense more deterministic and easier to attack. We redefine the logits of our classifier as:
+
+$$
+\log p _ {n} ^ {k} (y | \mathbf {x}) = \frac {1}{n} \sum_ {i = 1} ^ {n} \log p (y | \mathbf {x} _ {i}), \quad \mathbf {x} _ {i} \sim \operatorname {S G L D} (\mathbf {x}, k) \tag {11}
+$$
+
+where we have defined  $\mathrm{SGLD}(\mathbf{x},k)$  as an SGLD chain run for  $k$  steps seeded at  $\mathbf{x}$ . Intuitively, we draw  $n$  different samples  $\{\mathbf{x}_i\}_{i = 1}^n$  from our model seeded at input  $\mathbf{x}$ , then compute  $\log p(y|\mathbf{x}_i)$  for each of these samples, then average the results. We then attack these averaged logits with PGD to generate the results in Figure 5. We experimented with different numbers of samples and found that
+
+![](images/fe317eec071703cd18fb5692296c1186024e5dc0c9109d40d7d22cd4c3a11060.jpg)  
+(a) PGD  $L_{\infty}$
+
+![](images/a52c1e30dfe5f3f63c466f90e3923d05610f4547b870e550cd1233931f9f710a.jpg)  
+(b) PGD  $L_{2}$  
+Figure 15: PGD attacks comparing JEM to the IG EBM of Du & Mordatch (2019).
+
+10 samples yields very similar results to 5 samples on JEM with one refinement step (see Figure 16). Because 10 samples took very long to run on the JEM model with ten refinement steps, we settled on using 5 samples in the results reported in the main body of the paper.
+
+![](images/7b0392b60de3397cbfaf0b95b8b29f306b0245a16f78447510dcee0b658c4a10.jpg)  
+(a) PGD  $L_{\infty}$  
+Figure 16: Comparing the effect of the number of samples in the EOT attack. We find negligible difference between 5 and 10 for JEM-1 (red and green curves).
+
+![](images/b6af49cf8a6388624ea1440960eb2055248c9fa3353507c78cd2f62dd529fad4.jpg)  
+(b) PGD  $L_{2}$
+
+# G.2 TRANSFER ATTACKS
+
+We would like to see if JEM's refinement procedure can correct adversarial perturbed inputs - inputs which cause the model to fail. To do this, we generate a series of adversarial examples for JEM-0, with respect to the  $l_{\infty}$  norm, and test the accuracy of JEM-\{1,10\} on these examples. Ideally, with further refinement the accuracy will increase. The results of this experiment can be seen in Figure 17. We see here that JEM's refinement procedure can correct for adversarial perturbations.
+
+# H A DISCUSSION ON SAMPLERS
+
+# H.1 IMPROPER SGLD
+
+Recall the transition kernel of SGLD:
+
+$$
+\begin{array}{l} \mathbf {x} _ {0} \sim p _ {0} (\mathbf {x}) \\ \mathbf {x} _ {i + 1} = \mathbf {x} _ {i} - \frac {\alpha}{2} \frac {\partial E _ {\theta} (\mathbf {x} _ {i})}{\partial \theta} + \epsilon , \quad \epsilon \sim \mathcal {N} (0, \alpha) \\ \end{array}
+$$
+
+![](images/02be853003712c053765b907f6260ad0690d1723db374245c333dcef18ce9be7.jpg)  
+(a) PGD  $L_{\infty}$  
+Figure 17: PGD transfer attack  $L_{\infty}$ . We attack JEM-0 and evaluate success of the same adversarial examples under JEM-1 and JEM-10. Whenever an adversarial example is refined back to its correct class, we set the distance to infinity. Note that the adversarial examples do not transfer well from JEM-0 to JEM-1/-10.
+
+In the proper formulation of this sampler (Welling & Teh, 2011), the step-size and the variance of the Gaussian noise are related  $\mathrm{Var}(\epsilon) = \alpha$ . If the stepsize is decayed with a polynomial schedule, then samples from SGLD converge to samples from our unnormalized density as the number of steps goes to  $\infty$ .
+
+In practice, we approximate these samples with a sampler that runs for a finite number of steps. When using the proper step-size to noise ratio, the signal from the gradient is overtaken by the noise when step-sizes are large enough to be informative. In practice the sampler is typically "relaxed" in that different values are used for the step-size and the amount of Gaussian noise added – typically the amount of noise is significantly reduced.
+
+While we are no longer working with a valid MCMC sampler, this approximation has been successfully applied in practice in most recent work scaling EBM training to high dimensional data (Nijkamp et al., 2019b;a; Du & Mordatch, 2019) with the exception of Song & Ermon (2019) (which develops a clever work-around). The model they train is actually an ensemble of models trained on data with different amounts of noise added. They use a proper SGLD sampler decaying the step size as they sample, moving from their high-noise models to their low-noise models. This provides one possible explanation for the compelling results of their model.
+
+In our work we have set the step-size  $\alpha = 2$  and draw  $\epsilon \sim \mathcal{N}(0, .01^2)$ . We have found these parameters to work well across a variety of datasets, domains, architectures, and sampling procedures (persistent vs. short-run). We believe they are a decent "starting place" for energy-functions parameterized by deep neural networks.
+
+# H.2 PERSISTENT OR SHORT-RUN CHAINS?
+
+Both persistent and short-run markov chains have been able to successfully train EBMs. Nijkamp et al. (2019a) presents a careful study of various samplers which can be used and the tradeoffs one makes when choosing one sampler over another. In our work we have found that if computation allows, short-run MCMC chains are preferable in terms of training stability. Given that each step of SGLD requires approximately the computation of 1 training iteration of a standard classifier we are incentivized to find a sampler which can stably train EBMs requiring as few steps as possible per training iteration.
+
+In our experiments we found the smallest number of SGLD steps we could take to stably train an EBM at the scale of this work was 80 steps. Even so, these models eventually would diverge late into training. At 80 steps, we found the cost of training to be prohibitively high compared to a standard classifier.
+
+We found that by using persistent markov chains, we could further reduce the number of steps per iteration to 20 and still allow for relatively stable training. This gave a  $4\mathrm{x}$  speedup over our fastest short-run MCMC sampler. Still, this PCD sampler was noticeably less stable than the fastest short-run sampler we could use but we found the multiple factor increase in speed to be a worth-while trade-off.
+
+If time allows, we recommend using a short-run MCMC sampler with a large enough number of steps to be stable. Given that is not always possible on problems of scale, PCD can be made to work more efficiently, but at the cost of a greater number of stability-related hyper-parameters. These additional parameters include the buffer size and the re-initialization frequency of the Markov chains. We found both to be important for training stability and found no general recipe for which to set them. We ran most of our experiments with re-initialization frequency at  $5\%$ .
+
+A particularly interesting observation we discovered while using PCD is that the model would use the length of the markov chains to encode semantic information. We found that when training models on CIFAR10, when chains were young they almost always could be identified as frogs. When chains were old they could almost always be identified as cars. This behavior is likely some degeneracy of PCD which would not be possible with a short-run MCMC since all chains have the same length.
+
+# H.3 DEALING WITH INSTABILITY
+
+Training a model with the gradient estimator of Eq. (2) can be quite unstable - especially when combined with other objective as was the case with all models presented in this work. There exists a "stable region" of sorts when training these models where the energy values of the true data are in the same range as the energy values of the generated samples. Intuitively, if the generated samples create energies that are not trivially separated from the training data, then real learning has to take place. Nijkamp et al. (2019b;a) provide a careful analysis of this and we refer the reader there for a more in-depth analysis.
+
+We find that when using PCD occasionally throughout training a sample will be drawn from the replay buffer that has a considerably higher-than average energy (higher than the energy of a random initialization). This causes the gradients w.r.t this example to be orders of magnitude larger than gradients w.r.t the rest of the examples and causes the model to diverge. We tried a number of heuristic approaches such as gradient clipping, energy clipping, ignoring examples with atypical energy values, and many others but could not find an approach that stabilized training and did not hurt generative and discriminative performance.
+
+The only two approaches we found to consistently work to increase stability of a model which has diverged is to 1) decrease the learning rate and 2) increase the number of SGLD steps in each PCD iteration. Unfortunately, both of these approaches slow down learning. We also had some success simply restarting models from a saved checkpoint with a different random seed. This was the main approach taken unless the model was late into training. In this case, random restarts were less effective and we increased the number of SGLD steps from 20 to 40 which stabilized training.
+
+While we are very optimistic about the future of large-scale EBMs we believe these are the most important issues that must be addressed in order for these models to be successful.
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