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+# AMORTIZED CONTEXT VECTOR INFERENCE FOR SEQUENCE-TO-SEQUENCE NETWORKS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Neural attention (NA) has become a key component of sequence-to-sequence models that yield state-of-the-art performance in as hard tasks as abstractive document summarization (ADS), machine translation (MT), and video captioning (VC). NA mechanisms perform inference of context vectors; these constitute weighted sums of deterministic input sequence encodings, adaptively sourced over long temporal horizons. Inspired from recent work in the field of amortized variational inference (AVI), in this work we consider treating the context vectors generated by softattention (SA) models as latent variables, with approximate finite mixture model posteriors inferred via AVI. We posit that this formulation may yield stronger generalization capacity, in line with the outcomes of existing applications of AVI to deep networks. To illustrate our method, we implement it and experimentally evaluate it considering challenging ADS, VC, and MT benchmarks. This way, we exhibit its improved effectiveness over state-of-the-art alternatives.
+
+# 1 INTRODUCTION
+
+Sequence-to-sequence $( s e q 2 s e q )$ or encoder-decoder models (Sutskever et al., 2014) constitute a novel solution to inferring relations between sequences of different lengths. They are broadly used for addressing tasks including machine translation (MT) (Bahdanau et al., 2015; Luong et al., 2015), abstractive document summarization (ADS), descriptive caption generation (DCG) (Xu et al., 2016), and question answering (QA) (Sukhbaatar et al., 2015), to name just a few. Seq2seq models comprise two distinct RNN models: an encoder RNN, and a decoder RNN. Their main principle of operation is based on the idea of learning to infer an intermediate context vector representation, c, which is “shared” among the two RNN modules of the model, i.e., the encoder and the decoder. Specifically, the encoder converts the source sequence to a context vector (e.g., the final state of the encoder RNN), while the decoder is presented with the inferred context vector to produce the target sequence.
+
+Despite these merits, though, baseline seq2seq models cannot learn temporal dynamics over long horizons. This is due to the fact that a single context vector $^ c$ is capable of encoding rather limited temporal information. This major limitation has been addressed via the development of neural attention (NA) mechanisms (Bahdanau et al., 2015). NA has been a major breakthrough in Deep Learning for Natural Language Processing, as it enables the decoder modules of seq2seq models to adaptively focus on temporally-varying subsets of the source sequence. This capacity, in turn, enables flexibly capturing long temporal dynamics in a computationally efficient manner.
+
+Among the large collection of recently devised NA variants, the vast majority build upon the concept of Soft Attention (SA) (Xu et al., 2016). Under this rationale, at each sequence generation (decoding) step, NA-obtained context vectors essentially constitute deterministic representations of the dynamics between the source sequence and the decodings obtained thus far. However, recent work in the field of amortized variational inference (AVI) (Jimenez Rezende & Mohamed, 2015; Kingma & Welling, 2013) has shown that it is often useful to treat representations generated by deep networks as latent random variables. Indeed, it is now well-understood that, under such an inferential setup, the trained deep learning models become more effective in inferring representations that offer stronger generalizaton capacity, instead of getting trapped to representations of poor generalizaton quality. Then, model training reduces to inferring posterior distributions over the introduced latent variables. This can be performed by resorting to variational inference (Attias, 2000), where the sought variational posteriors are parameterized via appropriate deep networks.
+
+Motivated from these research advances, in this paper we consider a novel formulation of SA. Specifically, we propose an NA mechanism formulation where the generated context vectors are considered random latent variables with finite mixture model posteriors, over which AVI is performed. We dub our approach amortized context vector inference (ACVI). To exhibit the efficacy of ACVI, we implement it into: (i) Pointer-Generator Networks (See et al., 2017), which constitute a state-of-the-art approach for addressing ADS tasks; (ii) baseline seq2seq models with additive SA, applied to the task of VC; and (iii) baseline seq2seq models with multiplicative SA, applied to MT.
+
+The remainder of this paper is organized as follows: In Section 2, we briefly present the seq2seq model variants in the context of which we implement our method and exhibit its efficacy. In Section 3, we introduce the proposed approach, and elaborate on its training and inference algorithms. In Section 4, we perform an extensive experimental evaluation of our approach using benchmark ADS, MT, and VC datasets. Finally, in the concluding Section, we summarize the contribution of this work.
+
+# 2 METHODOLOGICAL BACKGROUND
+
+# 2.1 ABSTRACTIVE DOCUMENT SUMMARIZATION
+
+ADS consists in not only copying from an original document, but also learning to generate new sentences or novel words during the summarization process. The introduction of seq2seq models has rendered ADS both feasible and effective (Rush et al., 2015; Zeng et al., 2017). Dealing with out-ofvocabulary (OOV) words was one of the main difficulties that early ADS models were confronted with. Word and/or phrase repetition was a second issue. The pointer-generator model presented in (See et al., 2017) constitutes one of the most comprehensive efforts towards ameliorating these issues.
+
+In a nutshell, this model comprises one bidirectional LSTM (Hochreiter & Schmidhuber, 1997) (BiLSTM) encoder, and a unidirectional LSTM decoder, which incorporates an SA mechanism (Bahdanau et al., 2015). The word embedding of each token, $\pmb { x } _ { i } , \ i \in \mathsf { \bar { \{ 1 , \dots , N \} } }$ , in the source sequence (document) is presented to the encoder BiLSTM; this obtains a representation (encoding) $h _ { i } = [ \overrightarrow { { h } } _ { i } ; \overleftarrow { { h } } _ { i } ]$ , where $\vec { \boldsymbol { h } _ { i } }$ is the corresponding forward LSTM state, and $\overleftarrow { \overline { { h } } } _ { i }$ is the corresponding backward LSTM state. Then, at each generation step, $t$ , the decoder LSTM gets as input the (word embedding of the) previous token in the target sequence. During training, this is the previous word in the available reference summary; during inference, this is the previous generated word. On this basis, the decoder updates its internal state, $\mathbf { \Delta } _ { \mathbf { \mathcal { S } } _ { t } }$ , which is then presented to the postulated SA network. Specifically, the attention distribution, $\mathbf { } \mathbf { a } _ { t }$ , is given by:
+
+$$
+e _ { t } ^ { i } = v ^ { T } \operatorname { t a n h } ( W _ { h } h _ { i } + W _ { s } s _ { t } + b _ { a t t n } )
+$$
+
+$$
+{ \pmb a } _ { t } = \mathrm { s o f t m a x } ( { \pmb e } _ { t } ) , { \pmb e } _ { t } = [ { \pmb e } _ { t } ^ { i } ] _ { i }
+$$
+
+where the $W .$ · are trainable weight matrices, $b _ { a t t n }$ is a trainable bias vector, and $\textbf {  { v } }$ is a trainable parameter vector of the same size as $\pmb { b } _ { a t t r }$ . Then, the model updates the maintained context vector, $\mathbf { } _ { c _ { t } }$ , by taking an weighted average of all the source token encodings; in that average, the used weights are the inferred attention probabilities. We obtain:
+
+$$
+c _ { t } = \sum _ { i } a _ { t } ^ { i } h _ { i }
+$$
+
+Eventually, the predictive distribution over the next generated word yields:
+
+$$
+P _ { t } ^ { v o c a b } = \mathrm { s o f t m a x } ( V ^ { \prime } \mathrm { t a n h } ( V [ s _ { t } ; c _ { t } ] + b ) + b ^ { ' } )
+$$
+
+where $V$ and $V ^ { \prime }$ are trainable weight matrices, while $^ { b }$ and $\pmb { b } ^ { \prime }$ are trainable bias vectors.
+
+In parallel, the network also computes an additional probability, $p _ { t } ^ { g e n }$ , which expresses whether the next output should be generated by sampling from the predictive distribution, $\dot { P } _ { t } ^ { v o c a b }$ , or the model should simply copy one of the already available source sequence tokens. This mechanism allows for the model to cope with OOV words; it is defined via a simple sigmoid layer of the form:
+
+$$
+p _ { t } ^ { g e n } = \sigma ( \boldsymbol { w } _ { c } ^ { T } \boldsymbol { c } _ { t } + \boldsymbol { w } _ { s } ^ { T } \boldsymbol { s } _ { t } + \boldsymbol { w } _ { x } ^ { T } \boldsymbol { x } _ { t } + b _ { p t r } )
+$$
+
+where $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { t } }$ is the decoder input, while the $\pmb { w }$ · and $b _ { p t r }$ are trainable parameter vectors. The probability of copying the $i$ th source sequence token is considered equal to the corresponding attention probability,
+
+$a _ { t } ^ { i }$ . Eventually, the obtained probability that the next output word will be $\beta$ (found either in the vocabulary or among the source sequence tokens) yields:
+
+$$
+P _ { t } ( \beta ) = p _ { t } ^ { g e n } P _ { t } ^ { v o c a b } ( \beta ) + ( 1 - p _ { t } ^ { g e n } ) \sum _ { i : \beta _ { i } = \beta } a _ { t } ^ { i }
+$$
+
+Finally, a coverage mechanism may also be employed (Tu et al., 2016), as a means of penalizing words that have already received attention in the past, to prevent repetition. Specifically, the coverage vector, $k _ { t }$ , is defined as:
+
+$$
+k _ { t } = [ k _ { t } ^ { i } ] _ { i = 1 } ^ { N } = \sum _ { \tau = 0 } ^ { t - 1 } { \mathbf a } _ { \tau }
+$$
+
+Using the so-obtained coverage vector, expression (1) is modified as follows:
+
+$$
+e _ { t } ^ { i } = v ^ { T } \operatorname { t a n h } ( W _ { h } h _ { i } + W _ { s } s _ { t } + w _ { k } k _ { t } ^ { i } + b _ { a t t n } )
+$$
+
+where ${ \pmb w } _ { k }$ is a trainable parameter vector of size similar to $\textbf {  { v } }$ . Model training is performed via minimization of the categorical cross-entropy, augmented with a coverage term of the form:
+
+$$
+\lambda \sum _ { i } \sum _ { t } \operatorname* { m i n } ( a _ { t } ^ { i } , c _ { t } ^ { i } )
+$$
+
+Here, $\lambda$ controls the influence of the coverage term; in the remainder of this work, we set $\lambda = 1$ .
+
+# 2.2 VIDEO CAPTIONING
+
+Seq2seq models with attention have been successfully applied to several datasets of multimodal nature. Video captioning constitutes a popular such application. In this work, we consider a simple seq2seq model with additive SA that comprises a BiLSTM encoder, an LSTM decoder, and an output distribution of the form (4). The used encoder is presented with visual features obtained from a pretrained convolutional neural network (CNN). Using a pretrained CNN as our employed visual feature extractor ensures that all the evaluated attention models are presented with identical feature descriptors of the available raw data. Hence, it facilitates fairness in the comparative evaluation of our proposed attention mechanism. We elaborate on the specific model configuration in Section 4.2.
+
+# 2.3 MACHINE TRANSLATION
+
+Machine translation constitutes one of the first sequential data modeling applications where seq2seq models were shown to obtain state-of-the-art performance. In this work, we perform MT by means of a baseline seq2seq model comprising a BiLSTM encoder, an LSTM decoder, a predictive distribution over the next generated word which is given by (4), and a multiplicative SA mechanism. The latter is described by (Luong et al., 2015):
+
+$$
+e _ { t } ^ { i } = h _ { i } W s _ { t }
+$$
+
+in conjunction with Eq. (2); therein, $W$ is a trainable weights matrix. Our consideration here of multiplicative SA both serves the purpose of implementing and evaluating our approach under diverse SA variants, and is congruent with the best reported results in the related literature.
+
+# 3 PROPOSED APPROACH
+
+We begin by introducing the core assumption that the computed context vectors, $\mathbf { } c _ { t }$ , constitute latent random variables. Further, we assume that, at each decoding step, $t$ , the corresponding context vector, $\mathbf { } _ { c _ { t } }$ $\{ h _ { i } \} _ { i = 1 } ^ { N }$ , is drawn from a distribution associated with one of the available source sequence encodings, . The selection of the source sequence encoding to associate with is determined from the output sequence via the decoder state, $\mathbf { \boldsymbol { s } } _ { t }$ , as we explain next.
+
+Let us introduce the set of binary latent indicator variables, $\{ z _ { t } ^ { i } \} _ { i = 1 } ^ { N }$ , $z _ { t } ^ { i } \in \{ 0 , 1 \}$ , with $z _ { t } ^ { i } = 1$ denoting that the context vector $\mathbf { } c _ { t }$ is drawn from the $i$ th density, that is the density associated with the ith source encoding, $\boldsymbol { h } _ { i }$ , and $z _ { t } ^ { i } = 0$ otherwise. Then, we postulate the following hierarchical model:
+
+$$
+\begin{array} { r } { \begin{array} { r } { \boldsymbol { c } _ { t } | \boldsymbol { z } _ { t } ^ { i } = 1 ; \mathcal { D } \sim p ( \boldsymbol { \theta } ( h _ { i } ) ) } \end{array} } \end{array}
+$$
+
+$$
+z _ { t } ^ { i } = 1 | \mathcal { D } \sim \pi _ { t } ^ { i } ( a _ { t } ^ { i } )
+$$
+
+where $\mathcal { D }$ comprises the set of both the source and target training sequences, $\pmb \theta$ denotes the parameters set of the context vector conditional density, and $\hat { \pi _ { t } ^ { i } }$ denotes the probability of drawing from the ith conditional at time $t$ . Notably, we assume that the component assignment probabilities, $\pi _ { t } ^ { i }$ , are functions of the attention probabilities, $a _ { t } ^ { i }$ . Thus, the selection of the mixture component density that we draw the context vector from at decoding time $t$ is directly determined from the value of the current decoder state, $\mathbf { \Delta } _ { \mathbf { \mathcal { S } } _ { t } }$ , via the corresponding attention probabilities. A higher affinity of the current decoder state $\mathbf { \Delta } _ { \mathbf { \mathcal { S } } _ { t } }$ with the $i$ th encoding, $\boldsymbol { h } _ { i }$ , at time $t$ , results in higher probability that the context vector be drawn from the corresponding conditional density.
+
+Having defined the hierarchical model (11)-(12), it is important that we examine the resulting expression of the posterior density $p ( c _ { t } ; \mathcal { D } )$ . By marginalizing over (11) and (12), we obtain:
+
+$$
+p ( \boldsymbol { c } _ { t } ; \mathcal { D } ) = \sum _ { i = 1 } ^ { N } \pi _ { t } ^ { i } ( a _ { t } ^ { i } ) p ( \pmb { \theta } ( \boldsymbol { h } _ { i } ) )
+$$
+
+In other words, we obtain a finite mixture model posterior over the context vectors, with mixture conditional densities associated with the available source sequence encodings, and mixture weights that are functions of the corresponding attention vectors, and are therefore determined by the target sequences.
+
+In addition, it is interesting to compare this expression to the definition of context vectors under the conventional SA scheme. From (3), we observe that conventional SA is merely a special case of our proposed model, obtained by introducing two assumptions: (i) that the postulated mixture component assignment probabilities are identity functions of the associated attention probabilities, i.e.
+
+$$
+\begin{array} { r l } & { p ( z _ { t } ; \mathcal { D } ) = \mathrm { C a t } ( z _ { t } | \pi _ { t } ) , z _ { t } = [ z _ { t } ^ { i } ] _ { i = 1 } ^ { N } , \pi _ { t } = [ \pi _ { t } ^ { i } ( a _ { t } ^ { i } ) ] _ { i = 1 } ^ { N } } \\ & { \mathrm { s . t . } \quad \pi _ { t } ^ { i } ( a _ { t } ^ { i } ) \triangleq p ( z _ { t } ^ { i } = 1 ; \mathcal { D } ) = a _ { t } ^ { i } = \mathrm { s o f t m a x } ( e _ { t } ) ; } \end{array}
+$$
+
+and (ii) that the conditional densities of the context vectors have all their mass concentrated on $\boldsymbol { h } _ { i }$ , that is they collapse onto the single point, $\boldsymbol { h } _ { i }$ :
+
+$$
+p ( \pmb { c } _ { t } | z _ { t } ^ { i } = 1 ; \mathcal { D } ) = \delta ( \pmb { h } _ { i } )
+$$
+
+Indeed, by combining (13) - (15), we yield:
+
+$$
+p ( \boldsymbol { c } _ { t } ; \mathcal { D } ) = \sum _ { i = 1 } ^ { N } a _ { t } ^ { i } \delta ( \boldsymbol { h } _ { i } )
+$$
+
+whence we obtain (3) with probability 1.
+
+Thus, our approach replaces the simplistic conditional density expression (15) with a more appropriate family $p ( \pmb { \theta } ( h _ { i } ) )$ , as in (13). Based on the literature of AVI, e.g. (Jimenez Rezende & Mohamed, 2015; Kingma & Welling, 2013; Sønderby et al., 2016), we posit that such a stochastic latent variable consideration may result in significant advantages for the postulated seq2seq model. Specifically, our trained model becomes more agile in searching for effective context representations, as opposed to getting trapped to poor local solutions.
+
+In the following, we examine conditional densities of Gaussian form. Adopting the inferential rationale of AVI, we consider that these conditional Gaussians are parameterized via the postulated BiLSTM encoder. Specifically, we assume:
+
+$$
+p ( \pmb { c } _ { t } | z _ { t } ^ { i } = 1 ; \mathcal { D } ) = \mathcal { N } \big ( \pmb { c } _ { t } | \pmb { h } _ { i } , \mathrm { d i a g } ( \pmb { \sigma } ^ { 2 } ( \pmb { h } _ { i } ) ) \big )
+$$
+
+where
+
+$$
+\log \sigma ^ { 2 } ( h ) = \mathrm { R e L U } ( h )
+$$
+
+ReLU(·) is a trainable ReLU layer of size $\dim ( h )$ , and the encodings, $\boldsymbol { h } _ { i }$ , are obtained from a BiLSTM encoder, similar to conventional models. Hence:
+
+$$
+p ( { c } _ { t } ; \mathcal { D } ) = \sum _ { i = 1 } ^ { N } a _ { t } ^ { i } \mathcal { N } \big ( c _ { t } | h _ { i } , \mathrm { d i a g } ( \sigma ^ { 2 } ( h _ { i } ) ) \big )
+$$
+
+Thus, we have arrived at an approximate (variational) posterior expression for the context vectors, $\mathbf { } _ { c _ { t } }$ . In our variational treatment, both the component-conditional means, $\boldsymbol { h } _ { i }$ , and their variances, $\sigma ^ { 2 } ( h _ { i } )$ , are obtained from (amortizing) neural networks presented with the source sequences. On the other hand, though, the assignment probabilities, $\pi _ { t } ^ { i }$ , in the variational posterior are taken as the attention probabilities, $a _ { t } ^ { i }$ . Thus, they are determined by the target sequences, which are generated from the decoder of the model. Hence, our treatment represents a valid approximate posterior formulation, overall conditioned on both the source and target sequences.
+
+This concludes the formulation of ACVI.
+
+Relation to Recent Work. From the above exhibition, it becomes apparent that our approach generalizes the concept of neural attention by introducing stochasticity in the computation of context vectors. As we have already discussed, the ultimate goal of this construction is to allow for inferring representations of better generalization capacity, by leveraging Bayesian inference arguments.
+
+We emphasize that this is in stark contrast to recent efforts toward generalizing neural attention by deriving more complex attention distributions. For instance, (Kim et al., 2017) have recently introduced structured attention. In that work, the model infers complex posterior probabilities over the assignment latent variables, as opposed to using a simplistic gating function. Specifically, instead of considering independent assignments, they postulate the first-order Markov dynamics assumption:
+
+$$
+p ( \{ \boldsymbol { z } _ { t } \} _ { t = 1 } ^ { T } ; \mathcal { D } ) = p ( \boldsymbol { z } _ { 1 } ; \mathcal { D } ) \prod _ { t = 1 } ^ { T - 1 } p ( \boldsymbol { z } _ { t + 1 } | \boldsymbol { z } _ { t } ; \mathcal { D } )
+$$
+
+Thus, (Kim et al., 2017) compute posterior distributions over the attention assignments, while ACVI provides a method for obtaining improved representations through the inferred context vectors. Note also that Eq. (20) gives rise to the need of executing much more computationally complex algorithms to perform attention distribution inference, e.g. the forward-backward algorithm (Rabiner, 1989). In contrast, our method imposes computational costs comparable to conventional SA.
+
+Similar is the innovation in the variational attention method, recently presented (Deng et al., 2018). In essence, its key conceptual difference from structured attention is the consideration of full independence between the attention assignments $\{ z _ { t } \} _ { t = 1 } ^ { T }$ . Among the several alternatives considered in (Deng et al., 2018) to obtain stochastic gradient estimators of low variance, it was found that an approach using REINFORCE (Williams, 1992) along with a specialized baseline was effective.
+
+Another noteworthy recent work, closer related to ACVI, is the variational encoder-decoder (VED) method presented in (Bahuleyan et al., 2018). Among the several alternative formulations considered in that paper, the one that clearly outperformed the baselines in terms of the obtained accuracy (BLEU scores) combined seq2seq models with SA with an extra variational autoencoder (VAE) module. This way, apart from the context vector, which is computed under the standard SA scheme, an additional latent vector $\boldsymbol { \xi }$ is essentially inferred. The imposed prior over it is a standard $\mathcal { N } ( \mathbf { 0 } , \pmb { I } )$ , while the inferred posterior is a diagonal Gaussian parameterised by a BiLSTM network presented with the input sequence; the final BiLSTM state vector is presented to dense layers that output the posterior means and variances of the latent vectors $\boldsymbol { \xi }$ . Both the context vector, $^ c$ , as well as the latent vectors, $\boldsymbol { \xi }$ , are fed to the final softmax layer of the model that yields the generated output symbols.
+
+We shall provide comparisons to all these related approaches in the experimental section of our paper.
+
+Training Algorithm. To perform training of a seq2seq model equipped with the ACVI mechanism, we resort to maximization of the resulting evidence lower-bound (ELBO) expression. To this end, we need first to introduce some prior assumption over the context latent variables, $\mathbf { } c _ { t }$ . To serve the purpose of simplicity, and also offer a valid way to effect model regularization, we consider:
+
+$$
+p ( \pmb { c } _ { t } ) = \mathcal { N } \big ( \pmb { c } _ { t } | \mathbf { 0 } , I \big )
+$$
+
+On the grounds of these assumptions, it is easy to show that the resulting ELBO expression becomes:
+
+$$
+\mathcal { L } = \sum _ { t } \left\{ \mathbb { E } _ { p ( \boldsymbol { c } _ { t } ; \mathcal { D } ) } [ - J _ { t } ] - \mathrm { K L } [ p ( \boldsymbol { c } _ { t } ; \mathcal { D } ) | | p ( \boldsymbol { c } _ { t } ) ] \right\}
+$$
+
+In this expression, $\mathbb { E } _ { p ( \pmb { c } _ { t } ; \mathcal { D } ) } [ - J _ { t } ]$ is the posterior expectation of the model log-likelihood, which is an integral part of the ELBO definition. In the following, we approximate all the entailed ELBO terms by drawing MC samples from the context vector posterior. In this work, we are dealing with a one-out-of-many predictive selection; hence, the model likelihood is a simple Categorical. As such, $\mathbb { E } _ { p ( \pmb { c } _ { t } ; \mathcal { D } ) } [ - J _ { t } ]$ essentially reduces to the negative categorical cross-entropy of the model, averaged over multiple MC samples of the context vectors, drawn from (19). Besides, to ensure that the resulting MC estimators will be of low variance, we adopt the reparameterization trick. To this end, we rely on the posterior expressions (17) and (14); we express the drawn MC samples as follows:
+
+Table 1: Abstractive Document Summarization: Scores on the test set.   
+
+<table><tr><td rowspan=2 colspan=1>Method</td><td rowspan=1 colspan=3>ROUGE</td><td rowspan=1 colspan=2>METEOR</td></tr><tr><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>L</td><td rowspan=1 colspan=1>ExactMatch</td><td rowspan=1 colspan=1>+ stem/syn/para</td></tr><tr><td rowspan=2 colspan=1>seq2seq with SApointer-generator + coverage: SA</td><td rowspan=1 colspan=1>31.33</td><td rowspan=1 colspan=1>11.81</td><td rowspan=1 colspan=1>28.83</td><td rowspan=1 colspan=1>12.03</td><td rowspan=1 colspan=1>13.20</td></tr><tr><td rowspan=1 colspan=1>39.53</td><td rowspan=1 colspan=1>17.28</td><td rowspan=1 colspan=1>36.38</td><td rowspan=1 colspan=1>17.32</td><td rowspan=1 colspan=1>18.72</td></tr><tr><td rowspan=1 colspan=1>transformer</td><td rowspan=1 colspan=1>24.40</td><td rowspan=1 colspan=1>5.89</td><td rowspan=1 colspan=1>17.60</td><td rowspan=1 colspan=1>10.38</td><td rowspan=1 colspan=1>10.72</td></tr><tr><td rowspan=1 colspan=1>pointer-generator + coverage:structured attention</td><td rowspan=1 colspan=1>40.12</td><td rowspan=1 colspan=1>17.61</td><td rowspan=1 colspan=1>36.74</td><td rowspan=1 colspan=1>17.38</td><td rowspan=1 colspan=1>18.93</td></tr><tr><td rowspan=1 colspan=1>pointer-generator + coverage:variational attention</td><td rowspan=1 colspan=1>40.04</td><td rowspan=1 colspan=1>17.37</td><td rowspan=1 colspan=1>36.45</td><td rowspan=1 colspan=1>17.14</td><td rowspan=1 colspan=1>18.66</td></tr><tr><td rowspan=1 colspan=1>pointer-generator+coverage:VED</td><td rowspan=1 colspan=1>41.28</td><td rowspan=1 colspan=1>18.05</td><td rowspan=1 colspan=1>38.12</td><td rowspan=1 colspan=1>17.63</td><td rowspan=1 colspan=1>18.87</td></tr><tr><td rowspan=1 colspan=1>pointer-generator+coverage:ACVI</td><td rowspan=1 colspan=1>42.71</td><td rowspan=1 colspan=1>19.24</td><td rowspan=1 colspan=1>39.05</td><td rowspan=1 colspan=1>18.47</td><td rowspan=1 colspan=1>20.09</td></tr></table>
+
+$$
+c _ { t } ^ { ( k ) } = \sum _ { i = 1 } ^ { N } z _ { t i } ^ { ( k ) } c _ { t i } ^ { ( k ) }
+$$
+
+In this expression, the c(k)ti are samples from the conditional Gaussians (17), which employ the standard reparameterization trick rationale, as applied to Gaussian variables:
+
+$$
+{ \boldsymbol { c } } _ { t i } ^ { ( k ) } = h _ { i } + \sigma ( h _ { i } ) \circ \epsilon _ { t i } ^ { ( k ) } , \ \epsilon \sim \mathcal { N } ( \mathbf { 0 } , I )
+$$
+
+On the other hand, the z(k)ti are samples from the Categorical distribution (14). To allow for performing backpropagation through these samples, while ensuring that the obtained gradients will be of low variance, we may draw $z _ { t i } ^ { ( k ) }$ by making use of the Gumbel-Softmax relaxation (Jang et al., 2017). We have empirically found it suffices that we employ the Gumbel-Softmax trick for the last $10 \%$ of the model training iterations1; previously, we merely adopt the following heuristic, without any statistically significant performance deviation: We use a simple weighted average of the samples $\mathbf { \Delta } _ { c _ { t i } } ^ { ( k ) }$ , with the weights being the attention probabilities, $a _ { t } ^ { i }$ :
+
+$$
+\boldsymbol { c } _ { t } ^ { ( k ) } \gets \sum _ { i = 1 } ^ { N } a _ { t } ^ { i } \boldsymbol { c } _ { t i } ^ { ( k ) }
+$$
+
+This way, we alleviate the computational costs of employing the Gumbel-Softmax relaxation, which dominates the costs of sampling from the mixture posterior (19).
+
+Having obtained a reparameterization of the model ELBO that guarantees low variance estimators, we proceed to its maximization by resorting to a modern, off-the-shelf, stochastic gradient optimizer. Specifically, we adopt simple stochastic gradient descent (SGD) for the MT tasks, and Adam with its default settings (Kingma & Ba, 2015) for the rest.
+
+Inference Algorithm. To perform target decoding by means of a seq2seq model that employs the ACVI mechanism, we resort to Beam search (Russel & Norvig). In our experiments, Beam width is set to five for the ADS and VC tasks (Sections 4.1 and 4.2), and to ten for the MT tasks (Section 4.3).
+
+# 4 EXPERIMENTAL EVALUATION2
+
+# 4.1 ABSTRACTIVE DOCUMENT SUMMARIZATION
+
+Our experiments are based on the non-anonymized CNN/Daily Mail dataset, similar to the experiments of (See et al., 2017). To obtain some comparative results, we use pointer-generator networks as our evaluation platform (See et al., 2017); therein, we employ our ACVI mechanism, the standard SA mechanism used in (See et al., 2017), VED (Bahuleyan et al., 2018), variational attention (Deng et al., 2018), as well as structured attention using the first-order Markov assumption (20) (Kim et al., 2017). The observations presented to the encoder modules constitute 128-dimensional word embeddings of the original 50K-dimensional one-hot-vectors of the source tokens. Similarly, the observations presented to the decoder modules are 128-dimensional word embeddings pertaining to the summary tokens (reference tokens during training; generated tokens during inference). Both these embeddings are trained, as part of the overall training procedure of the evaluated models. To allow for faster training convergence, we split training into five phases, as suggested in (See et al., 2017). Following the suggestions in (See et al., 2017), we evaluate all approaches with LSTMs that comprise 256-dimensional states and do not employ Dropout. We have tested VED with various selections of the dimensionality of the autoencoder latent vectors, $\boldsymbol { \xi }$ ; we report results with 128-dimensional latent vectors, which yielded the best performance in our experiments3.
+
+Table 2: Abstractive Document Summarization: Novel words generation rate and OOV words adoption rate obtained by using pointer-generator networks.   
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>SA</td><td rowspan=1 colspan=1>Structured Attention</td><td rowspan=1 colspan=1>Variational Attention</td><td rowspan=1 colspan=1>VED</td><td rowspan=1 colspan=1>ACVI</td></tr><tr><td rowspan=1 colspan=1>RateofNovel Words</td><td rowspan=1 colspan=1>0.05</td><td rowspan=1 colspan=1>0.05</td><td rowspan=1 colspan=1>0.05</td><td rowspan=1 colspan=1>0.12</td><td rowspan=1 colspan=1>0.38</td></tr><tr><td rowspan=1 colspan=1>Rate of OOVWordsAdoption</td><td rowspan=1 colspan=1>1.16</td><td rowspan=1 colspan=1>1.18</td><td rowspan=1 colspan=1>1.18</td><td rowspan=1 colspan=1>1.21</td><td rowspan=1 colspan=1>1.25</td></tr></table>
+
+Table 3: Video Captioning: Performance of the considered alternatives.   
+
+<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>ROUGE:Valid. Set</td><td rowspan=1 colspan=1>ROUGE:Test Set</td><td rowspan=1 colspan=1>CIDEr:Valid.Set</td><td rowspan=1 colspan=1>CIDEr: Test Set</td></tr><tr><td rowspan=1 colspan=1>SA</td><td rowspan=1 colspan=1>0.5628</td><td rowspan=1 colspan=1>0.5701</td><td rowspan=1 colspan=1>0.4575</td><td rowspan=1 colspan=1>0.421</td></tr><tr><td rowspan=1 colspan=1>Structured Attention</td><td rowspan=1 colspan=1>0.5804</td><td rowspan=1 colspan=1>0.5712</td><td rowspan=1 colspan=1>0.5071</td><td rowspan=1 colspan=1>0.4283</td></tr><tr><td rowspan=1 colspan=1>Variational Attention</td><td rowspan=1 colspan=1>0.5809</td><td rowspan=1 colspan=1>0.5716</td><td rowspan=1 colspan=1>0.5103</td><td rowspan=1 colspan=1>0.4289</td></tr><tr><td rowspan=1 colspan=1>VED</td><td rowspan=1 colspan=1>0.5839</td><td rowspan=1 colspan=1>0.5749</td><td rowspan=1 colspan=1>0.5421</td><td rowspan=1 colspan=1>0.4298</td></tr><tr><td rowspan=1 colspan=1>ACVI</td><td rowspan=1 colspan=1>0.5968</td><td rowspan=1 colspan=1>0.5766</td><td rowspan=1 colspan=1>0.6039</td><td rowspan=1 colspan=1>0.4375</td></tr></table>
+
+Finally, for completeness sake, we also evaluate the Transformer network (Vaswani et al., 2017), which is a popular alternative to seq2seq models with SA, based on the notion of self-attention. Following Fevry (2018), Transformer is evaluated with 256-dimensional word embeddings, 4 encoding and decoding layers of 256 units each, 4 heads, and a Dropout rate of 0.2.
+
+We use ROUGE4 (Lin, 2004) and METEOR5 (Denkowski & Lavie, 2014) as our performance metrics. METEOR is evaluated both in exact match mode (rewarding only exact matches between words) and full mode (additionally rewarding matching stems, synonyms and paraphrases). In all our experiments, we restrict the used vocabulary to the 50K most common words in the considered dataset, similar to (See et al., 2017). Note that this is significantly smaller than typical in the literature (Nallapati et al., 2016). Our quantitative evaluation is provided in Table 1. Some indicative examples of generated summaries can be found in Appendix A (Tables 7-10).
+
+As we observe, utilization of ACVI outperforms all the alternatives by a large margin. It is also interesting that the Transformer network yields the lowest performance among the considered alternatives; the obtained results are actually very poor. This is commensurate with the results reported by other researchers, e.g. Fevry (2018).
+
+Finally, it is interesting to examine whether ACVI increases the propensity of a trained model towards generating novel words, that is words that are not found in the source document, as well as the capacity to adopt OOV words. The related results are provided in Table 2. We observe that ACVI increases the number of generated novel words by 3 times compared to the best performing alternative, that is VED (Bahuleyan et al., 2018). In a similar vein, ACVI appears to help the model better cope with OOV words.
+
+Table 4: Translation results on the (En, Vi) and (En, Ro) pairs.   
+
+<table><tr><td rowspan=1 colspan=3></td><td rowspan=1 colspan=8>BLEU</td></tr><tr><td rowspan=1 colspan=3>Source-&gt;Target Language</td><td rowspan=1 colspan=2>En→Vi</td><td rowspan=1 colspan=2>Vi-→En</td><td rowspan=1 colspan=2>En-→Ro</td><td rowspan=1 colspan=2>Ro→En</td></tr><tr><td rowspan=5 colspan=1></td><td rowspan=5 colspan=2>BaselineStructured Attention</td><td rowspan=1 colspan=1>dev</td><td rowspan=1 colspan=1>test</td><td rowspan=1 colspan=1>dev</td><td rowspan=1 colspan=1>test</td><td rowspan=1 colspan=1>dev</td><td rowspan=1 colspan=1>test</td><td rowspan=1 colspan=1>dev</td><td rowspan=1 colspan=1>test</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>23.21</td><td rowspan=1 colspan=1>25.18</td><td rowspan=1 colspan=1>20.89</td><td rowspan=1 colspan=1>23.28</td><td rowspan=1 colspan=1>12.87</td><td rowspan=1 colspan=1>14.40</td><td rowspan=1 colspan=1>15.87</td><td rowspan=1 colspan=1>15.78</td></tr><tr><td rowspan=7 colspan=2>Structured AttentionVariational AttentionVEDACVITransformer</td><td rowspan=3 colspan=1>ntion</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan=1 colspan=1>ion</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td rowspan=1 colspan=1>23.81</td><td rowspan=1 colspan=1>25.00</td><td rowspan=1 colspan=1>21.19</td><td rowspan=1 colspan=1>23.08</td><td rowspan=1 colspan=1>14.04</td><td rowspan=1 colspan=1>15.08</td><td rowspan=1 colspan=1>17.02</td><td rowspan=1 colspan=1>17.68</td></tr><tr><td rowspan=4 colspan=1>Method</td><td rowspan=1 colspan=1>23.48</td><td rowspan=1 colspan=1>25.54</td><td rowspan=1 colspan=1>21.13</td><td rowspan=1 colspan=1>23.61</td><td rowspan=1 colspan=1>14.02</td><td rowspan=1 colspan=1>15.51</td><td rowspan=1 colspan=1>17.49</td><td rowspan=1 colspan=1>17.40</td></tr><tr><td rowspan=1 colspan=1>24.47</td><td rowspan=1 colspan=1>25.31</td><td rowspan=1 colspan=1>21.32</td><td rowspan=1 colspan=1>23.80</td><td rowspan=1 colspan=1>12.84</td><td rowspan=1 colspan=1>12.76</td><td rowspan=1 colspan=1>15.18</td><td rowspan=1 colspan=1>15.56</td></tr><tr><td rowspan=1 colspan=1>24.08</td><td rowspan=1 colspan=1>26.16</td><td rowspan=1 colspan=1>21.26</td><td rowspan=1 colspan=1>24.47</td><td rowspan=1 colspan=1>14.15</td><td rowspan=1 colspan=1>15.78</td><td rowspan=1 colspan=1>18.07</td><td rowspan=1 colspan=1>17.78</td></tr><tr><td rowspan=1 colspan=1>24.34</td><td rowspan=1 colspan=1>25.68</td><td rowspan=1 colspan=1>21.40</td><td rowspan=1 colspan=1>23.92</td><td rowspan=1 colspan=1>13.90</td><td rowspan=1 colspan=1>15.30</td><td rowspan=1 colspan=1>17.66</td><td rowspan=1 colspan=1>17.91</td></tr></table>
+
+# 4.2 VIDEO CAPTIONING
+
+Our evaluation of the proposed approach in the context of a VC application is based on the Youtube2Text video corpus (Yao et al., 2015). We split the available dataset into a training set comprising the first 1,200 video clips, a validation set composed of 100 clips, and a test set comprising the last 600 clips in the dataset. To reduce the entailed memory requirements, we process only the first 240 frames of each video. To obtain some initial video frame descriptors, we employ a pretrained GoogLeNet CNN (Szegedy et al., 2015) (implementation provided in Caffe (Jia et al., 2014)). Specifically, we use the features extracted at the pool5/7x7_s1 layer of this pretrained model. We select 24 equally-spaced frames out of the first 240 from each video, and feed them into the prescribed CNN to obtain a 1024 dimensional frame-wise feature vector. These are the visual inputs presented to the trained models. All employed LSTMs entail 1000-dimensional states. These are mapped to 100-dimensional features via the matrices $W _ { h }$ and $W _ { s }$ in Eq. (1). The autoencoder latent variables, $\boldsymbol { \xi }$ , of VED are also selected to be 100-dimensional vectors. The decoders are presented with 256-dimensional word embeddings, obtained in a fashion similar to our ADS experiments. In all cases, we use Dropout with a rate of 0.5.
+
+We yield some comparative results by evaluating seq2seq models configured as described in Section 2.2; we use ACVI, structured attention in the form (20), VED, variational attention, or the conventional SA mechanism. Our quantitative evaluation is performed on the grounds of the ROUGE-L and CIDEr (Vedantam et al., 2015) scores, on both the validation set and the test set. The obtained results are depicted in Table 3; they show that our method outperforms the alternatives by an important margin. It is also characteristic that Structured Attention yields essentially identical results with Variational Attention. Thus, the first-order Markovian assumption does not offer practical benefits when generating short sequences like the ones involved in VC. Finally, we provide some indicative examples of the generated results in Appendix B (Figs. 1-8).
+
+# 4.3 MACHINE TRANSLATION
+
+Our experiments make use of publicly available corpora, namely WMT’16 English-to-Romanian $( \mathrm { E n \to R o } )$ and Romanian-to-English $( \mathrm { R o } \to \mathrm { E n } ) ,$ ), as well as IWSLT’15 English-to-Vietnamese $( \mathrm { E n \to V i } )$ ) and Vietnamese-to-English $( { \mathrm { V i } } \to { \mathrm { E n } } )$ ). We benchmark the evaluated models against word-based vocabularies, and present our results in terms of the BLEU score (Papineni et al., 2002).
+
+Following the related literature, we utilize byte pair encoding (BPE) (Sennrich et al., 2016) in the case of the (En, Ro) pair. This allows for seamlessly handling rare words, by breaking a given vocabulary into a fixed-size vocabulary of variable-length character sequences (subwords). Subword vocabularies are shared among the languages of a source/destination pair. This way, we promote frequent subword units, thus improving the coverage of the available dictionary words.
+
+We obtain some comparative performance results by evaluating seq2seq models using ACVI, conventional SA, structured attention, as well as both the variational alternatives (Bahuleyan et al., 2018; Deng et al., 2018) discussed in Section 3. The trained architecture is homogeneous across all our comparisons. Specifically, both the encoders and the decoders of the evaluated models are presented with 256-dimensional trainable word embeddings. We utilize 2-layer BiLSTM encoders, and 2-layer LSTM decoders; all comprise 256-dimensional hidden states on each layer, similar to the summarization task, and employ a Dropout rate of 0.2. For VED, we employ 100-dimensional latent variables $\boldsymbol { \xi }$ , following Bahuleyan et al. (2018). Finally, we also provide the performance of the
+
+Table 5: Abstractive Document Summarization: Domain Adaptation Performance on DUC2004.   
+
+<table><tr><td rowspan=1 colspan=1>Model</td><td rowspan=1 colspan=1>ROUGE-1</td><td rowspan=1 colspan=1>ROUGE-2</td><td rowspan=1 colspan=1>ROUGE-L</td></tr><tr><td rowspan=1 colspan=1>SA</td><td rowspan=1 colspan=1>27.02</td><td rowspan=1 colspan=1>7.44</td><td rowspan=1 colspan=1>22.69</td></tr><tr><td rowspan=1 colspan=1>Variational Attention</td><td rowspan=1 colspan=1>27.65</td><td rowspan=1 colspan=1>7.58</td><td rowspan=1 colspan=1>23.50</td></tr><tr><td rowspan=1 colspan=1>VED</td><td rowspan=1 colspan=1>30.68</td><td rowspan=1 colspan=1>9.97</td><td rowspan=1 colspan=1>27.02</td></tr><tr><td rowspan=1 colspan=1>ACVI</td><td rowspan=1 colspan=1>32.09</td><td rowspan=1 colspan=1>10.88</td><td rowspan=1 colspan=1>28.14</td></tr></table>
+
+Transformer network. The trained Transformer network comprises 4 heads, 4 encoder/decoder layers of 256 units, and a Dropout rate of 0.1.
+
+Our results in Table 4 show inferior performance for our variational inference-based competitors. We observe that VED is competitive to ACVI in two of the four development sets, but fails to generalize as well across test sets. Some indicative examples of generated outputs from the considered variational alternatives are provided in Appendix C, Tables 11-16.
+
+# 4.4 FURTHER INVESTIGATION: DOMAIN ADAPTATION
+
+Finally, we wish to examine the capability of ACVI to generalize across domains. We have already elaborated on our expectation that modeling the context vectors as latent random variables should yield improved generalization performance. We attribute to this fact the improved accuracy ACVI obtained in our experimental evaluations. However, if this is the case, one would probably expect the method to also generalize better across different domains.
+
+To investigate this aspect, we use the trained ADS models described in Section 4.1 to generate summaries for the documents of the DUC2004 dataset 6. This is an English dataset comprising 500 documents. Each document contains 4 model summaries written by experts. In Table 5, we show how our method performs in this setting, and how it compares to the alternative variational methods considered in Section 4.1. We observe that ACVI yields a clear improvement over the alternatives, while all variational methods perform significantly better than baseline SA. These findings seem to support our theoretical intuitions.
+
+# 5 CONCLUSIONS
+
+In this work, we cast the problem of context vector computation for seq2seq-type models employing SA into amortized variational inference. We made this possible by considering that the sought context vectors are latent variables following a Gaussian mixture posterior; therein, the mixture component densities depend on the source sequence encodings, while the mixture weights depend on the target sequence attention probabilities. We exhibited the merits of our approach on seq2seq architectures addressing ADS, VC, and MT tasks; we used benchmark datasets in all cases.
+
+We underline that our approach induces only negligible computational overheads compared to conventional SA. Specifically, the only extra trainable parameters that our approach postulates stem from Eq. (17); these are of extremely limited size compared to the overall model size, and correspond to merely few extra feedforward computations at inference time. Besides, our sampling strategy does not induce significant computational costs, since we adopt the reparameterization (25) for the most part of the model training algorithm. In the future, we aim to consider how ACVI can cope with power-law distributions (Chatzis & Demiris, 2012; Chatzis & Kosmopoulos, 2015); such a capacity is of importance to real-world natural language generation.
+
+# REFERENCES
+
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+Abigail See, Peter J. Liu, and Christopher D. Manning. Get to the point: Summarization with pointer-generator networks. In Proc. ACL, 2017.   
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+
+Table 6: Abstractive Document Summarization: Training phases.   
+
+<table><tr><td rowspan=1 colspan=1>Phase</td><td rowspan=1 colspan=1>Iterations</td><td rowspan=1 colspan=1>Max encoding steps</td><td rowspan=1 colspan=1>Max decoding steps</td></tr><tr><td rowspan=1 colspan=1>1</td><td rowspan=1 colspan=1>0-71k</td><td rowspan=1 colspan=1>10</td><td rowspan=1 colspan=1>10</td></tr><tr><td rowspan=1 colspan=1>2</td><td rowspan=1 colspan=1>71k - 116k</td><td rowspan=1 colspan=1>50</td><td rowspan=1 colspan=1>50</td></tr><tr><td rowspan=1 colspan=1>3</td><td rowspan=1 colspan=1>116k - 184k</td><td rowspan=1 colspan=1>100</td><td rowspan=1 colspan=1>50</td></tr><tr><td rowspan=1 colspan=1>4</td><td rowspan=1 colspan=1>184k - 223k</td><td rowspan=1 colspan=1>200</td><td rowspan=1 colspan=1>50</td></tr><tr><td rowspan=1 colspan=1>5</td><td rowspan=1 colspan=1>223k-250k</td><td rowspan=1 colspan=1>400</td><td rowspan=1 colspan=1>100</td></tr></table>
+
+# APPENDIX A
+
+We provide some further details on the experimental setup of Section 4.1. To begin with, the used dataset comprises 287,226 training pairs of documents and reference summaries, 13,368 validation pairs, and 11,490 test pairs. In this dataset, the average article length is 781 tokens; the average summary length is 3.75 sentences, with the average summary being 56 tokens long.
+
+To allow for faster training convergence, we split it into five phases, following See et al. (2017). On each phase, we employ a different number of maximum encoding steps for the evaluated models (i.e., the size of the inferred attention vectors), as well as for the maximum allowed number of decoding steps. We provide the related details in Table 6. During these phases, we train the employed models with the coverage mechanism being disabled; that is, we set $\pmb { w } _ { k } = \mathbf { 0 }$ . We enable this mechanism only after these five training phases conclude. Specifically, we perform a final 3K iterations of model training, during which we train the ${ \pmb w } _ { k }$ weights along with the rest of the model parameters.
+
+In Tables 7-10, we provide some indicative examples of produced summaries. We also show what the initial document has been, as well as the available reference summary used for quantitative performance evaluation. In all cases, we annotate OOV words in italics, we highlight novel words in purple, we show contextual understanding in bold, while article fragments also included in the generated summary are highlighted in green.
+
+Table 7: Example 223.   
+
+<table><tr><td>Article</td></tr><tr><td>lagos , nigeria -lrb- cnn -rrb- a day after winning nigeria &#x27;s presidency , muhammadu buhari told cnn &#x27;s christiane amanpour that he plans to aggressively fight corruption that has long plagued nigeria and go after the root of the nation ’s unrest . buhari said he &#x27;ll&quot; rapidly give attention ” to curbing violence in the northeast part of nigeria ,where the terrorist group boko haram operates . by cooperating with neighboring nations chad ,cameroon and niger, he said his administration is confident it will be able to thwart criminals and others contributing to nigeria’s instability. for the first time in nigeria &#x27;s history , the opposition defeated the ruling party in democratic elections . buhari defeated incumbent goodluck jonathan by about 2 million votes , according to nigeria &#x27;s independent national electoral commission .the win comes after a long history of military rule , coups and botched attempts at democracy in africa &#x27;s most populous nation . in an exclusive live interview from abuja ,buhari told amanpour he was not concerned about reconciling the nation after a divisive campaign . he said now that he has been elected he will turn his focus to boko haram and“ plug holes ”in the“ corruption infrastructure”in the country.“ a new day and</td></tr><tr><td>a new nigeria are upon us ,”buhari said after his win tuesday .“ the victory is yours ,and the glory is that of our nation . earlier, jonathan phoned buhari to concede defeat . the outgoing president also offered a written statement to his nation .“i thank allnigerians once again for the</td></tr><tr><td>great opportunity i was given to lead this country ,and assure you that i will continue to do my best at the helm of national affairs until the end of my tenure ,” jonathan said .“ i promised the country free and fair elections . (...) ReferenceSummary</td></tr><tr><td>muhammadu buhari tells cnn &#x27;s christiane amanpour that he will fight corruption in nigeria . nigeria is the most populous country in africa and is grappling with violent boko haram extremists</td></tr><tr><td>. nigeria is also africa&#x27;s biggest economy,but up to 7O % of nigerians live on less than a dollar a day. Generated Summary muhammadu buhari talks to cnn &#x27;s christiane amanpour about the nation &#x27;s unrest .for the</td></tr></table>
+
+Table 8: Example 89.   
+
+<table><tr><td>Article</td></tr><tr><td>lrb- cnn -rrb- eyewitness video showing white north charleston police officer michael slager shooting to death an unarmed black man has exposed discrepancies in the reports of the first officers on the scene . slager has been fired and charged with murder in the death of 50-year-old walter scott . a bystander &#x27;s cell phone video ， which began after an alleged struggle on the ground between slager and scot ,shows the five-year police veteran shooting at scott eight times as scott runs away . scott was hit five times . if words were exchanged between the men ,they &#x27;re are not audible on the tape . it ’s unclear what happened before scott ran ,or why he ran . the officer initially said that he used a taser on scot ， who ,slager said ,tried to take the weapon . before slager opens fire ,the video shows a dark object falling behind scott and hiting the ground</td></tr><tr><td>. it &#x27;s unclear whether that is the taser .(...)</td></tr><tr><td>ReferenceSummary more questions than answers emerge in controversial s. c. police shooting . oficer michael slager</td></tr><tr><td>,charged with murder,was fired from the north charleston police department . Generated Summary</td></tr><tr><td>video shows white north charleston police offcer michael slager shooting to death . slager has been charged with murder in the death of 50-year-old walter scot . the video shows a dark object falling behind scott and hitting the ground .</td></tr></table>
+
+Table 9: Example 1305.   
+
+<table><tr><td rowspan=1 colspan=1>Article</td></tr><tr><td rowspan=1 colspan=1> andy murray came close to giving himself some extra preparation time for his wedding nextweek before ensuring that he still has unfinished tennis business to attend to .the world no 4 is into the semi-finals of the miami open , but not before geting a scare from 21 year-old austriandominic thiem,who pushed him to 4-4 in the second set before going down 3-6 6-4,6-1 in an hour and three quarters . murray was awaiting the winner from the last eight match betweentomas berdych and argentina &#x27;s juan monaco . prior to this tournament thiem lost in the secondround of a challenger event to soon-to-be new brit aljaz bedene . andy murray pumps his firstafter defeating dominic thiem to reach the miami open semi finals . muray throws his sweatband into the crowd after completing a 3-6 , 6-4 , 6-1 victory in florida . murray shakes hands withthiem who he described as a‘ strong guy ’after the game . (..)</td></tr><tr><td rowspan=1 colspan=1>ReferenceSummary</td></tr><tr><td rowspan=1 colspan=1>british no 1 defeated dominic thiem in miami open quarter finals . andy murray celebrated his500th career win in the previous round . third seed will play the winner of tomas berdych and juan monaco in the semi finals of the atp masters 1OoO event in key biscayne</td></tr><tr><td rowspan=1 colspan=1>Generated Summary</td></tr><tr><td rowspan=1 colspan=1>the world no 4 is into the semi-finals of the miami open . murray is still ahead of his careerthrough the season .andy murray was awaiting the winner from the last eight match .murraythrows his sweatband into the crowd after a 6-4 6-1 victory in florida .</td></tr></table>
+
+Table 10: Example 1710.   
+
+<table><tr><td>Article</td></tr><tr><td>steve clarke afforded himself a few smiles on the touchline and who could blame him ? this has been a strange old season for reading ,who are one win away from an fa cup semi-final against arsenal but have spent too long being too close to a championship relegation batle .at least this win will go some way to easing that load . they made it hard for themselves , but they had an in-form player in jamie mackie who was able to get the job done . he put reading in front in the first half and then scored a briliant winner just moments after chris o&#x27;grady had levelled with a penalty -- one of the only legitimate chances brighton had all night ,even if clarke was angry about the decision . reading frontman jamie mackie fires the royals ahead against brighton in tuesday &#x27;s championship fixture . mackie -lrb- centre -rrb- is congratulated by nathaniel chalobah and garath mccleary after neting reading &#x27;s opener . reading -lrb- 4-1-3-2 -rrb- : federici ; gunter ,hector,cooper,chalobah ; akpan ; mcleary,williams -lrb- keown 92 -rrb-,robson-kanu -lrb- pogrebnyak 76 -rrb- ; blackman ,mackie -lrb- norwood 79 -rrb- . subs not used : cox,yakubu,</td></tr><tr><td>andersen,taylor.scorer : mackie,24,56.booked : mcleary,pogrebnyak .brighton -lrb-4-3-3 -rrb- :stockdale ; halford,greer,dunk,bennet ; ince -lrb-best 75 -rrb-,kayal,forster-caskey; ledesma -lrb- bruno 86 -rrb-,o&#x27;grady,lualua .subs not used :ankergren,calderon,hughes ,</td></tr><tr><td>holla ,teixeira .scorer : o&#x27;grady -lrb- pen -rrb-,53 . booked : ince ,dunk ,bennett, greer . ref : andy haines .attendance : 14,748 .ratings by riath al-samarrai .(...)</td></tr><tr><td>ReferenceSummary reading are now 13 points above the championship drop zone .frontman jamie mackie scored twice to earn royals all three points .chris o&#x27;grady scored for chris hughton &#x27;s brighton from the</td></tr><tr><td>penalty spot . niall keown - son of sportsmail columnist martin - made reading debut . Generated Summary jamie mackie opened the scoring against brighton in tuesday &#x27;s championship fixture</td></tr></table>
+
+# APPENDIX B
+
+The considered Video Captioning task utilizes a dataset that comprises 1,970 video clips, each associated with multiple natural language descriptions. This results in a total of approximately 80,000 video / description pairs; the used vocabulary comprises approximately 16,000 unique words. The constituent topics cover a wide range of domains, including sports, animals and music. We preprocess the available descriptions only using the wordpunct tokenizer from the NLTK toolbox7.
+
+Moving on, we provide some characteristic examples of generated video descriptions. In the captions of the figures that follow, we annotate minor deviations with blue color, and use red color to indicate major mistakes which imply wrong perception of the scene.
+
+![](images/ca71c2a237bc9cd2999b306f0d68ec98b33f3c053122fb39b1ef55da160dcbca.jpg)
+
+![](images/a9f1d8b91691fae1fe18bc43f5f07ff3b9dc3ba0f63a3faee09f032af1e3e3ca.jpg)
+
+Figure 1: ACVI: a man is firing a gun VED: a man is firing a gun   
+Structured Attention: a man is firing a gun   
+Variational Attention: a man is firing a gun SA: a man is firing a gun Reference Description: a man is firing a gun at targets
+
+![](images/1055d060302d9b8444ad5dc2cc7cb1260bbae2e1d6abe5f2f425e8fa9c1f893b.jpg)
+
+![](images/3cc8715431eb916fc54286331531b2fa58eaeca964650c984b96d4b00e62539e.jpg)
+
+Figure 2: ACVI: a woman is cutting a piece of pork VED: a woman is cutting a bed Structured Attention: a woman is cutting pork Variational Attention: a woman is cutting pork SA: a woman is putting butter on a bed Reference Description: someone is cutting a piece of meat
+
+![](images/987541ab9963446b382267030a2999ef9c9f4900ba269e766e84b82e0358f44e.jpg)
+
+![](images/9f2fd2dc98badf8770f19ef87248988f77a19faefb10b25de3983db47d2c2af3.jpg)  
+Figure 3: ACVI: a small animal is eating VED: a small woman is talking Structured Attention: a small woman is eating Variational Attention: a small woman is eating SA: a small woman is talking Reference Description: a hamster is eating
+
+![](images/7c65f67e59362e4daa61672347c86cf0f859a70085227a0c587e68a63a8d19dd.jpg)
+
+Figure 4: ACVI: the lady poured the something into a bowl VED: a woman is cracking an egg   
+Structured Attention: a woman poured an egg into a bowl   
+Variational Attention: a woman poured an egg into a bowl SA: a woman is cracking an egg Reference Description: someone is pouring something into a bowl Figure 5: ACVI: a woman is riding a horse VED: a woman is riding a horse   
+Structured Attention: a woman is riding a horse   
+Variational Attention: a woman is riding a horse SA: a woman is riding a horse Reference Description: a woman is riding a horse Figure 6: ACVI: several people are driving down a street VED: several people trying to jump   
+Structured Attention: several people are driving down the avenue   
+Variational Attention: several people are driving down the avenue SA: a boy trying to jump Reference Description: a car is driving down the road Figure 8: ACVI: the man is riding a bicycle VED: the man is riding a motorcycle   
+Structured Attention: the man is riding a motorcycle   
+Variational Attention: the man is riding a motorcycle SA: a man rides a motorcycle Reference Description: a girl is riding a bicycle
+
+![](images/69bf4026df5c22fbf1770130de94de8528e8bdd2ecdc8f220eb7f81b686d1831.jpg)
+
+![](images/007a64a7f29f3269f4bbc9e5e2d1f557b787c6b479cbb386bb48ff11ed5e85ff.jpg)
+
+![](images/c18c8d4f7205ee75bd0a7708fa8ce01c50106abb3a35df0c0c65bc5b5428c62f.jpg)
+
+![](images/a4ad2efe09c95b388f1c1c18f964b43c0c22f2df868d8c0917d17bfb7e800ccc.jpg)
+
+![](images/25ea6d7c7ff594cfdc2812e44445b677b33f4c5657ed0cd4a411f2711b8064d9.jpg)  
+Figure 7: ACVI: a man is playing the guitar VED: a man is dancing Structured Attention: a high man is playing the guitar Variational Attention: a man is dancing SA: a high man is dancing Reference Description: a boy is playing the guitar
+
+![](images/62fc124d41af297d0ed6a7ea63215db6193ffb0d1aa44a2c20e09ed8e2666839.jpg)
+
+Table 11: V En, tst2012 - Example 84.   
+
+<table><tr><td rowspan=1 colspan=1>Source sentence</td></tr><tr><td rowspan=1 colspan=1>Hau hét y tuong cüa chung toi deu dien khung, nhung vai y tuong vo cung tuyet voi, va chung toitao ra dot pha.</td></tr><tr><td rowspan=1 colspan=1>Reference Translation</td></tr><tr><td rowspan=1 colspan=1>Most of our ideas were crazy, but a few were brilliant, and we broke through.</td></tr><tr><td rowspan=1 colspan=1>Generated Translation - Baseline</td></tr><tr><td rowspan=1 colspan=1>Most of our ideas were crazy, but some incredible ideas were awesome, and we created thebreakthrough.</td></tr><tr><td rowspan=1 colspan=1>Generated Translation - Structured Attention</td></tr><tr><td rowspan=1 colspan=1>Most of our ideas are crazy, but some [missing: verb] really wonderful ideas, and we created asudden.</td></tr><tr><td rowspan=1 colspan=1>Generated Translation-VED</td></tr><tr><td rowspan=1 colspan=1>Most of our ideas were crazy, but some [missing: verb] wonderful ideas, and we created abreakthrough.</td></tr><tr><td rowspan=1 colspan=1>Generated Translation - Variational Attention</td></tr><tr><td rowspan=1 colspan=1>Most of our ideas were insane, but some ideas were wonderful, and we created &lt;unk&gt;.</td></tr><tr><td rowspan=1 colspan=1>Generated Translation- ACVI</td></tr><tr><td rowspan=1 colspan=1>Most of our ideas were crazy, but some [missing: verb] wonderful ideas,and we made abreakthrough.</td></tr></table>
+
+# APPENDIX C
+
+Let us first provide some details on the datasets used in the context of our MT experiments. The WMT’16 task comprises of data from combining the Europarl v7, News Commentary v10 and Common Crawl corpora. For the (En, Ro) pair, this amounts to ${ \sim } 4 0 0 \mathrm { K }$ parallel sentences. The shared vocabulary sizes (obtained from BPE) total ${ \sim } 3 1 . 7 \mathrm { K }$ words. We use newsdev2016 as our development set ${ \mathrm { \Omega } } ^ { \prime } { \sim } 1 . 9 \mathrm { K }$ sentences), and newstest2016 as our test set ${ \mathrm { \Omega } } ^ { \prime } { \sim } 1 . 9 \mathrm { K }$ sentences) for the (En, Ro) pair.
+
+On the other hand, the IWSLT’15 task boasts a dataset with ${ \sim } 1 3 3 \mathrm { K }$ training sentence pairs from translated TED talks, provided by the IWSLT 2015 Evaluation Campaign (Cettolo et al., 2015). Following the same preprocessing steps as in (Luong et al., 2015), we use TED tst2012 ${ \mathrm { \Omega } } ^ { \prime } { \sim } 1 . 5 \mathrm { K }$ sentences) as our validation set for hyperparameter tuning, and TED tst2013 $\mathrm { \sim } 1 . 3 \mathrm { K }$ sentences) as our test set. The Vietnamese and English vocabulary sizes are ${ \sim } 7 . 7 \mathrm { K }$ and ${ \sim } 1 7 . 2 \mathrm { K }$ , respectively.
+
+We prefer default settings for the hyperparameters of the trained seq2seq models, as used in the code8. These hyper-parameters remain unchanged for the VED and Variational Attention implementations as well. We have migrated the code9 of the former, provided from the authors, to ensure identical data processing. For the latter, we use their codebase10 directly.
+
+In conclusion, we provide some characteristic examples of generated translations for all examined models. In the Tables that follow, we annotate minor and major deviations from the reference translation with blue and red respectively. Synonyms are highlighted with green. We also indicate missing tokens by adding the [missing] identifier mid-sentence, i.e. verbs, articles, adjectives, etc.
+
+Table 12: V $\dot { \lfloor  \rfloor }$ En, tst2012 - Example 165.   
+
+<table><tr><td rowspan=1 colspan=1>Source sentence</td></tr><tr><td rowspan=1 colspan=1>diéu dau tien ba muon con hua la con phai luon yeu thuong me con</td></tr><tr><td rowspan=1 colspan=1>Reference Translation</td></tr><tr><td rowspan=1 colspan=1> She said, &quot; The first thing I want you to promise me is that you &#x27;ll always love your mom. &quot;</td></tr><tr><td rowspan=1 colspan=1>Generated Translation-Baseline</td></tr><tr><td rowspan=1 colspan=1>&quot; The first thing she wants to revenge is she always loves her mother. &quot;</td></tr><tr><td rowspan=1 colspan=1>Generated Translation - Structured Attention</td></tr><tr><td rowspan=1 colspan=1>&quot; The first thing she wants to do is always love her. &quot;</td></tr><tr><td rowspan=1 colspan=1>Generated Translation - VED</td></tr><tr><td rowspan=1 colspan=1>&quot; The first thing she wanted me to do is to love my mother. &quot;</td></tr><tr><td rowspan=1 colspan=1>Generated Translation - Variational Attention</td></tr><tr><td rowspan=1 colspan=1>&quot; The first thing she wants to promise is that you have to love her mother. &quot;</td></tr><tr><td rowspan=1 colspan=1>Generated Translation- ACVI</td></tr><tr><td rowspan=1 colspan=1> &quot; The first thing she wanted you to promise you would have to do is to love your mother.</td></tr></table>
+
+Table 13: Vi En, tst2012 - Example 1542.   
+
+<table><tr><td rowspan=1 colspan=1>Source sentence</td></tr><tr><td rowspan=1 colspan=1> Ho tham chi sé su dung nhung cong cu nhu Trojan Scuinst de lay nhiem vao may tinh cua ban ,va tu d6 ho c6 thé có dugc moi thong tin ban trao doi,có dugc moi cuoc hoi thoai qua mang cuaban , va có dugc mat khau cua ban .</td></tr><tr><td rowspan=1 colspan=1>Reference Translation</td></tr><tr><td rowspan=1 colspan=1>They will even use tools like State Trojan to infect your computer with a trojan , which enables them to watch all your communication ,to listen to your online discussions ,to collect yourpasswords .</td></tr><tr><td rowspan=1 colspan=1>Generated Translation - Baseline</td></tr><tr><td rowspan=1 colspan=1>They&#x27;re even going to use tools like &lt;unk&gt; &lt;unk&gt; to infect your computer, and from that they can get all sorts of information that you traded, you get all the conversation through your lives, and there&#x27;s been available to be able to get all of [missing: end of sentence]</td></tr><tr><td rowspan=1 colspan=1>Generated Translation - Structured Attention</td></tr><tr><td rowspan=1 colspan=1>They&#x27;re even going to use your tools like &lt;unk&gt; &lt;unk&gt; to infect your computer, and from that they can get allte information you communicate, there&#x27;s all kinds of conversations through your network, and you get your password.</td></tr><tr><td rowspan=1 colspan=1>Generated Translation-VED</td></tr><tr><td rowspan=1 colspan=1>They&#x27;re even going to use tools like &lt;unk&gt; &lt;unk&gt; to infect your computer, and then they can get all sorts of information that you share, whether you can get all your &lt;unk&gt;.</td></tr><tr><td rowspan=1 colspan=1>Generated Translation- Variational Attention</td></tr><tr><td rowspan=1 colspan=1>They&#x27;re even going to use tools like &lt;unk&gt; &lt;unk&gt; to infect your computer, and then they can be able to get allof the information that you can change, there&#x27;s your conversation through your online, and there&#x27;s your password.</td></tr><tr><td rowspan=1 colspan=1>Generated Translation- ACVI</td></tr><tr><td rowspan=1 colspan=1>They&#x27;re even going to use tools like &lt;unk&gt; &lt;unk&gt; to infect your computer, and then they can get all the information you communicate, get all your conversations through your network, and getyour password.</td></tr></table>
+
+Table 14: Ro En, newsdev2016 - Example 5.   
+
+<table><tr><td rowspan=1 colspan=1>Source sentence</td></tr><tr><td rowspan=1 colspan=1>Dirceu este cel mai vechi membru al Partidului Muncitorilor aflat la guvernare luat in custodie pentru legaturile cu aceasta schema.</td></tr><tr><td rowspan=1 colspan=1>Reference Translation</td></tr><tr><td rowspan=1 colspan=1>Dirceu is the most senior member of the ruling Workers ’ Party to be taken into custody inconnection with the scheme.</td></tr><tr><td rowspan=1 colspan=1>Generated Translation - Baseline</td></tr><tr><td rowspan=1 colspan=1>That is the most old Member of the People &#x27;s Party of Maiers to government in custody for tieswith this scheme.</td></tr><tr><td rowspan=1 colspan=1>Generated Translation - Structured Attention</td></tr><tr><td rowspan=1 colspan=1> It is the oldest member of the Mandi of the Massi in the government in the government.</td></tr><tr><td rowspan=1 colspan=1>Generated Translation - VED</td></tr><tr><td rowspan=1 colspan=1>(RO) Mr President, it is the oldest member of the Dutch Party on the government in custody forthe ties with this scheme .</td></tr><tr><td rowspan=1 colspan=1>Generated Translation - Variational Attention</td></tr><tr><td rowspan=1 colspan=1>It is the oldest Member of the Party of Women&#x27;s Party of Government in custody for the ties withthis scheme.</td></tr><tr><td rowspan=1 colspan=1>Generated Translation- ACVI</td></tr><tr><td rowspan=1 colspan=1>Dirse is the oldest member of the People &#x27;s Party on government in custody for the links withthis scheme.</td></tr></table>
+
+Table 15: Ro En, newsdev2016 - Example 7.   
+
+<table><tr><td rowspan=1 colspan=1>Source sentence</td></tr><tr><td rowspan=1 colspan=1>A fost arestat la inceputul lui august de acasa, unde deja se afla sub arest la domiciliu, cu o pedeapsä de 11 ani pentru implicarea intr-o schemä de cumpärare a voturilor in Congres cu peste10 ani in urma.</td></tr><tr><td rowspan=1 colspan=1>Reference Translation</td></tr><tr><td rowspan=1 colspan=1>He was arrested early August in his home, where he already was under house arrest serving an11-year sentence for his involvement in a cash-for-votes scheme in Congress more than 1O yearsago.</td></tr><tr><td rowspan=1 colspan=1>Generated Translation-Baseline</td></tr><tr><td rowspan=1 colspan=1>He was arrested at the beginning of August at home, where it is already under arrest at home, with a death penalty for the involvement of the votes in Congress on 10 years ago.</td></tr><tr><td rowspan=1 colspan=1>Generated Translation - Structured Attention</td></tr><tr><td rowspan=1 colspan=1>It has been arrested at the beginning of last August, which is already being found in home, with aban on a 11 years for the involvement of a ban in the reception scheme for more than 1O yearsago.</td></tr><tr><td rowspan=1 colspan=1>Generated Translation - VED</td></tr><tr><td rowspan=1 colspan=1>He was arrested at the beginning of August at home, where it is already under arrest at home, with a three-11 sentence for the involvement in a no-fly scheme on 1O years ago.</td></tr><tr><td rowspan=1 colspan=1>Generated Translation - Variational Attention</td></tr><tr><td rowspan=1 colspan=1> It was arrested at the beginning of August August, where already under home, with a 11 years[missing: noun], with a 11 years [missing: noun] for the involvement of a purchasing votes in10 years ago.</td></tr><tr><td rowspan=1 colspan=1>Generated Translation - ACVI</td></tr><tr><td rowspan=1 colspan=1>He was arrested at the beginning of August at home, where he is under house arrest, with apunishment of 1l years for involving a purchasing scheme in Congress over 1O years ago.</td></tr></table>
+
+Table 16: Ro En, newsdev2016 - Example 182.   
+
+<table><tr><td rowspan=1 colspan=1>Source sentence</td></tr><tr><td rowspan=1 colspan=1>Reprezentantii grupurilor de interese au vorbit la unison despre speranta lor in abilitatea luiTurnbullde a satisface interesul public, de a ajunge la un acord politic si de a face lucrurile bine.</td></tr><tr><td rowspan=1 colspan=1>Reference Translation</td></tr><tr><td rowspan=1 colspan=1>With one voice the lobbyists talked about a hoped-for ability in Turnbul to make the public argument, to cut the political deal and get tough things done.</td></tr><tr><td rowspan=1 colspan=1>Generated Translation - Baseline</td></tr><tr><td rowspan=1 colspan=1>The representatives of interest groups have spoken about their hope in the capacity of tourism to meet public interest, to reach a political agreement and to do things well.</td></tr><tr><td rowspan=1 colspan=1>Generated Translation - Structured Attention</td></tr><tr><td rowspan=1 colspan=1>The representatives of the interest groups have spoken in mind about their hope to meet the public interest, to achieve a political and good thing.</td></tr><tr><td rowspan=1 colspan=1>Generated Translation - VED</td></tr><tr><td rowspan=1 colspan=1>The representatives of interest groups have spoken in unity about their hope in Turkey&#x27;s ability to satisfy the public interest, to reach a political agreement and to make things right.</td></tr><tr><td rowspan=1 colspan=1>Generated Translation- Variational Attention</td></tr><tr><td rowspan=1 colspan=1>The representatives of the interest groups have talked about their hope about their hope of their Turk hope to meet the public interest, to reach a political agreement and to do so well.</td></tr><tr><td rowspan=1 colspan=1>Generated Translation- ACVI</td></tr><tr><td rowspan=1 colspan=1>Representatives of interest groups have spoken about their hope in Mr Turnchl &#x27;s ability to satisfythe public interest, to reach a political agreement and to do things well.</td></tr></table>

parse/train/r1esnoAqt7/r1esnoAqt7.md

@@ -0,0 +1,310 @@
+# MORPHO-MNIST: QUANTITATIVE ASSESSMENT AND DIAGNOSTICS FOR REPRESENTATION LEARNING
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Revealing latent structure in data is an active field of research, having introduced exciting technologies such as variational autoencoders and adversarial networks, and is essential to push machine learning towards unsupervised knowledge discovery. However, a major challenge is the lack of suitable benchmarks for an objective and quantitative evaluation of learned representations. To address this issue we introduce Morpho-MNIST, a framework that aims to answer: “to what extent has my model learned to represent specific factors of variation in the data?” We extend the popular MNIST dataset by adding a morphometric analysis enabling quantitative comparison of trained models, identification of the roles of latent variables, and characterisation of sample diversity. We further propose a set of quantifiable perturbations to assess the performance of unsupervised and supervised methods on challenging tasks such as outlier detection and domain adaptation.
+
+# 1 INTRODUCTION
+
+A key factor for progress in machine learning has been the availability of well curated, easy-to-use, standardised and sufficiently large annotated datasets for benchmarking different algorithms and models. This has led to major advances in speech recognition, computer vision, and natural language processing. A commonality between these tasks is their natural formulation as supervised learning tasks, wherein performance can be measured in terms of accuracy on a test set.
+
+The general problem of representation learning (i.e. to reveal latent structure in data) is more difficult to assess due the lack of suitable benchmarks. Although the field is very active, with many recently proposed techniques such as probabilistic autoencoders and adversarial learning, it is less clear where the field stands in terms of progress or which approaches are more expressive for specific tasks. The lack of reproducible ways to quantify performance has led to subjective means of evaluation: visualisation techniques have been used to show low-dimensional projections of the latent space and visual inspection of generated or reconstructed samples are popular to provide subjective measures of descriptiveness. On the other hand, the quality of sampled images generally tells us little about how well the learned representations capture known factors of variation in the training distribution. In order to advance progress, the availability of tools for objective assessment of representation learning methods seems essential yet lacking.
+
+This paper introduces Morpho-MNIST, a collection of shape metrics and perturbations, in a step towards quantitative assessment of representation learning. We build upon one of the most popular machine learning benchmarks, MNIST, which despite its shortcomings remains widely used. While MNIST was originally constructed to facilitate research in image classification, in the form of recognising handwritten digits (LeCun et al., 1998), it has found its use in representation learning, for example, to demonstrate that the learned latent space yields clusters consistent with digit labels. Methods aiming to disentangle the latent space claim success if individual latent variables capture specific style variations (e.g. stroke thickness, sidewards leaning digits and other visual characteristics).
+
+The main appeal of selecting MNIST as a benchmark for representation learning is that, while manifesting complex interactions between pixel intensities and underlying shapes, it has well understood and easily measurable factors of variation. More generally, MNIST remains popular in practice due to several factors: it allows reproducible comparisons with previous results reported in the literature; the dataset is sufficiently large for its complexity and consists of small, two-dimensional greyscale images defining a tractable ten-class classification problem; computation and memory requirements are low; most popular deep learning frameworks and libraries offer tutorials using MNIST, which makes it straightforward for new researchers to enter the field and to experiment with new ideas and explore latest developments. We take advantage of these qualities and extend MNIST in multiple ways, as summarised in the following.
+
+![](images/f303b3624281bc81731a0a3ec201e7e822f8774e613ebbd94322adac6cb3dcf9.jpg)  
+Figure 1: Left: MNIST morphometrics—stroke thickness and length (not shown), width, height and slant of digits. Right: MNIST perturbations (many more examples of each type in Appendix B).
+
+# 1.1 CONTRIBUTIONS
+
+Our aim is to bridge the gap between methodology-focused research and critical real-world applications that could benefit from latest machine learning methods. As we preserve the general properties of MNIST—such as image size, file format, numbers of training and test images, and the original ten-class classification problem—we believe this new quantitative framework for assessing representation learning will experience widespread use in the community and may inspire further extensions facilitated by a publicly available Morpho-MNIST code base.
+
+Morphometrics: We propose to describe true and generated digit images in terms of measurable shape attributes. These include stroke thickness and length, and the width, height, and slant of digits (cf. Fig. 1, left). Whereas some of these properties have been analysed qualitatively in previous work, we demonstrate that objectively quantifying each of them allows to identify the role of inferred representations. Moreover, these tools can be used to measure model samples, enabling assessment of generative performance with respect to sample diversity (Section 4.1) and disentanglement of latent variables (Section 4.2).
+
+These measurements can be directly employed to re-evaluate existing models and may be added retrospectively to previous experiments involving the original MNIST dataset. Adoption of our morphometric analysis may provide new insights into the effectiveness of representation learning methods in terms of revealing meaningful latent structures. Furthermore, for other datasets it suffices to design the relevant scalar metrics and include them in the very same evaluation framework.
+
+Perturbations: We introduce a set of parametrisable global and local perturbations, inspired by natural and pathological variability in medical images. Global changes involve overall thinning and thickening of digits, while local changes include both swelling and fractures (see examples on the right in Fig. 1 and many more in Appendix B). Injecting these perturbations into the dataset adds a new type of complexity to the data manifold and opens up a variety of interesting applications.
+
+The proposed perturbations are designed to enable a wide range of new studies and applications for both supervised and unsupervised tasks. Detection of ‘abnormalities’ (i.e. local perturbations) is an evident application, although more challenging tasks can also be defined, such as classification from noisy/corrupted data, domain adaptation, localisation of perturbations, characterising semantics of learned latent representations, and more. We explore a few supplementary examples of supervised tasks in Appendix D.
+
+# 1.2 RELATED WORK: DATASETS
+
+In this section, we provide an overview of some datasets that are related to MNIST, by either sharing its original source content, containing transformations of the original MNIST images or being distributed in the same format for easy replacement. We also mention a few prevalent datasets of images with generative factor annotations, similarly to the morphometrics proposed in this paper.
+
+NIST datasets: The MNIST (modified NIST) dataset (LeCun et al., 1998) was constructed from handwritten digits in NIST Special Databases 1 and 3, now released as Special Database 19 (Grother and Hanaoka, 2016). Cohen et al. (2017) generated a much larger dataset based on the same NIST database, containing additional upper- and lower-case letters, called EMNIST (extended MNIST).
+
+MNIST perturbations: The seminal paper by LeCun et al. (1998) employed data augmentation using planar affine transformations including translation, scaling, squeezing, and shearing. Loosli et al. (2007) employed random elastic deformations to construct the Infinite MNIST dataset. Other MNIST variations include rotations and insertion of random and structured background (Larochelle et al., 2007), and Tieleman (2013) applied spatial affine transformations and provided ground-truth transformation parameters.
+
+MNIST format: Due to the ubiquity of the MNIST dataset in machine learning research and the resulting multitude of compatible model architectures available, it is appealing to release new datasets in the same format $2 8 \times 2 8$ , 8-bit grayscale images). One such effort is Fashion-MNIST (Xiao et al., 2017), containing images of clothing articles from ten distinct classes, adapted from an online shopping catalogue. Another example is notMNIST (Bulatov, 2011), a dataset of character glyphs for letters $\mathbf { \delta A } ^ { \prime } - \mathbf { \delta J } ^ { \prime }$ (also ten classes), in a challengingly diverse collection of typefaces.
+
+Annotated datasets: Computer vision datasets that are popular for evaluating disentanglement of learned latent factors of variation include those from Paysan et al. (2009) and Aubry et al. (2014). They contain 2D renderings of 3D faces and chairs, respectively, with ground-truth pose parameters (azimuth, elevation) and lighting conditions (faces only). A further initiative in that direction is the dSprites dataset (Matthey et al., 2017), which consists of binary images containing three types of shapes with varying location, orientation and size. The availability of the ground-truth values of such attributes has motivated the accelerated adoption of these datasets in the evaluation of representation learning algorithms.
+
+# 1.3 RELATED WORK: QUANTITATIVE EVALUATION
+
+Evaluation of representation learning is a notoriously challenging task and remains an open research problem. Numerous solutions have been proposed, with many of the earlier ones focusing on the test log-likelihood under the model (Kingma and Welling, 2013) or, for likelihood-free models, under a kernel density estimate (KDE) of generated samples (Goodfellow et al., 2014; Makhzani et al., 2015)—being shown not to be reliable proxies for the true model likelihood (Theis et al., 2016).
+
+Another perspective for evaluation of generative models of images is the visual fidelity of its samples to the training data, which would normally require manual inspection. To address this issue, a successful family of metrics have been proposed, based on visual features extracted by the Inception network (Szegedy et al., 2016). The original Inception score (Salimans et al., 2016) relies on the ‘crispness’ of class predictions, whereas the Fréchet Inception distance (FID) (Heusel et al., 2017) and the kernel Inception distance (KID) (Binkowski et al. ´ , 2018) statistically compare high-level representations instead of the final network outputs.
+
+Although the approaches above can reveal vague signs of mode collapse, it may be useful to diagnose this phenomenon on its own. With this objective, Arora et al. (2018) proposed to estimate the support of the learned distribution (assumed discrete) using the birthday paradox test, by counting pairs of visual duplicates among model samples. Unfortunately, the adoption of this technique is hindered by its reliance on manual visual inspection to flag identical images.
+
+There have been several attempts at quantifying representation disentanglement performance. For example, Higgins et al. (2017) proposed to use the accuracy of a simple classifier trained to predict which factor of variation was held fixed in a simulated dataset. There exist further informationtheoretic approaches, involving the KL divergence contribution from each latent dimension (Dupont, 2018) or their mutual information with each known generative factor (Chen et al., 2018). Yet another method, explored in Kumar et al. (2018), is based on the predictive accuracy of each latent variable to each generative factor (continuous or discrete).
+
+![](images/831c9debb102f33904a8b60566a90410437acb89c20dd3973660f3453ff82753.jpg)  
+Figure 2: Stages of the image processing pipeline. Left to right: original image, upscaled image, binarised image, distance transform, skeleton, downscaled image.
+
+# 2 MORPHOMETRY
+
+Meaningful morphometrics are instrumental in characterising distributions of rasterised shapes, such as MNIST digits, and can be useful as additional data for downstream learning tasks. We begin this section by describing the image processing pipeline employed for extracting the metrics and for applying perturbations (Section 3), followed by details on the computation of each measurement.
+
+# 2.1 PROCESSING PIPELINE
+
+The original $2 8 \times 2 8$ resolution of the MNIST images is generally not high enough to enable satisfactory morphological processing: stroke properties (e.g. length, thickness) measured directly on the binarised images would likely be inaccurate and heavily quantised. To mitigate this issue and enable sub-pixel accuracy in the measurements, we propose to use the following processing steps: 1. upscale (e.g. $\times 4$ , to $1 1 2 \times 1 1 2 )$ 1; 2. binarise (e.g. threshold ${ \geq } 1 2 8$ ); 3. compute Euclidean distance transform (EDT) from boundaries; 4. skeletonise (medial axis, i.e. ridges of EDT); 5. apply perturbation (cf. Section 3); and 6. downscale to original resolution.
+
+We illustrate the pipeline in Fig. 2. The binary high-resolution digits have smooth boundaries and faithfully capture subtle variations in contour shape and stroke thickness that are only vaguely discernible in the low-resolution images. Additionally, note how the final downscaled image is almost indistinguishable from the original.
+
+All morphometric attributes described below are calculated for each digit after applying steps 1–4 of this pipeline. The distributions for the plain MNIST training set is plotted in Fig. 3, and the distributions after applying each type of perturbation can be found in Appendix A.
+
+![](images/9291516fa3d4332cbbc0c54e5fdcef063fb6dbd4fa49cbc6fc977d3ffb8b7621.jpg)  
+Figure 3: Distribution of morphological attributes per digit class in the plain MNIST training dataset
+
+# 2.2 STROKE LENGTH
+
+Here we approximate the trace of the pen tip, as a digit was being written, by the computed morphological skeleton. In this light, the total length of the skeleton is an estimate of the length of the pen stroke, which in turn is a measure of shape complexity.
+
+It can be computed in a single pass by accumulating the Euclidean distance of each skeleton pixel to its immediate neighbours, taking care to only count the individual contributions once. This approach is more robust against rotations than a naïve estimate by simply counting the pixels.
+
+# 2.3 STROKE THICKNESS
+
+A prominent factor of style variation in the MNIST digits is the overall thickness of the strokes, due to both legitimate differences in pen thickness and force applied, and also to the rescaling of the original NIST images by different factors.
+
+We estimate it by exploiting the computed distance transform. By virtue of how the image skeleton is computed, its pixels are approximately equidistant to the nearest boundaries, therefore we take twice the mean value of the EDT over all skeleton pixels as our global estimate.
+
+# 2.4 SLANT
+
+The extent by which handwritten symbols lean right or left (forward and backward slant, respectively) is a further notorious and quantifiable dimension of handwriting style. It introduces so much variation in the appearance of characters in images that it is common practice in OCR systems to ‘deslant’ them, in an attempt to reduce within-class variance (LeCun et al., 1998; Teow and Loe, 2002).
+
+We adapt the referred deslanting methodology to describe the slant angle of the handwritten digits. After estimating the second-order image moments, we define the slant based on the horizontal shear:
+
+$$
+\alpha = \arctan \left( - \frac { \sum _ { i , j } x _ { i j } ( i - \bar { i } ) ( j - \bar { j } ) } { \sum _ { i , j } x _ { i j } ( i - \bar { i } ) ^ { 2 } } \right) ,
+$$
+
+where $\boldsymbol { x } _ { i j }$ is the intensity of pixel $( i , j )$ , and $( \bar { i } , \bar { j } )$ are the centroid coordinates. The minus sign ensures that positive and negative values correspond to forward and backward slant, respectively.
+
+# 2.5 WIDTH AND HEIGHT
+
+It is useful to measure other general shape attributes, such as width, height, and aspect ratio, which also present substantial variation related to personal handwriting style.2 To this end, we propose to fit a bounding parallelogram to each digit, with horizontal and slanted sides (cf. Fig. 1).
+
+We sweep the image top-to-bottom with a horizontal boundary to compute a vertical marginal cumulative distribution function (CDF), and likewise left-to-right with a slanted boundary for a horizontal marginal CDF, with angle $\alpha$ as computed above. The bounds are then chosen based on equal-tailed intervals containing a given proportion of the image mass— $98 \%$ in both directions ( $1 \%$ from each side) proved accurate and robust in our experiments.
+
+# 3 PERTURBATIONS
+
+As discussed in Section 1, we bring forward a number of morphological perturbations for MNIST digits, to enable interesting applications and experimentation. In this section, we detail these parametrisable transformations, categorised as global or local.
+
+# 3.1 GLOBAL: THINNING AND THICKENING
+
+The first pair of transformations we present is based on simple morphological operations: the binarised image of a digit is dilated or eroded with a circular structuring element. Its radius is set proportionally to the estimated stroke thickness (Section 2.3), so that the overall thickness of each digit will decrease or increase by an approximately fixed factor (here, $- 7 0 \%$ and $+ 1 0 0 \%$ ; see Figs. B.1 and B.2).
+
+Since there is substantial thickness variability in the original MNIST data (cf. Fig. 3) and most thinned and thickened digits look very plausible, we believe that these perturbations can constitute a powerful form of data augmentation for training. For the same reason, we have not included these perturbations in the abnormality detection experiments (Appendix D).
+
+# 3.2 LOCAL: SWELLING
+
+In addition to the global transformations above, we introduce local perturbations with variable location and extent, which are harder to detect automatically. Given a radius $R$ , a centre location $\mathbf { r } _ { 0 }$ and a strength parameter $\gamma > 1$ , the coordinates $\mathbf { r }$ of pixels within distance $R$ of $\mathbf { r } _ { 0 }$ are nonlinearly warped according to a radial power transform:
+
+$$
+\mathbf { r } \mapsto \mathbf { r } _ { 0 } + \left( \mathbf { r } - \mathbf { r } _ { 0 } \right) \left( \frac { \left\| \mathbf { r } - \mathbf { r } _ { 0 } \right\| } { R } \right) ^ { \gamma - 1 } ,
+$$
+
+leaving the remaining portions of the image untouched and resampling with bicubic interpolation.
+
+In the experiments and released dataset, we set $\gamma = 7$ and $R = 3 \sqrt { \theta } / 2$ , where $\theta$ is thickness. Unlike simple linear scaling with $\theta$ , this choice for $R$ produces noticeable but not exaggerated effects across the thickness range observed in the dataset (cf. Fig. B.3). The centre location, $\mathbf { r } _ { 0 }$ , is picked uniformly at random from the pixels along the estimated skeleton.
+
+# 3.3 LOCAL: FRACTURES
+
+We describe the proposed procedure for adding fractures to an MNIST digit, where we define a fracture as a break in the continuity of a pen stroke. Because single fractures can in many cases be easily mistaken for true gaps between strokes, we add multiple fractures to each affected digit.
+
+When selecting the location for a fracture, we attempt to avoid getting too close to stroke tips (points on the skeleton with a single neighbour) or fork points (more than two neighbours). This is achieved by sampling only among those skeleton pixels above a certain distance to these detected points. In addition, we would like fractures to be transversal to the pen strokes. Local orientation is determined based on second-order moments of the skeleton inside a window centred at the chosen location, and the length of the fracture is estimated from the boundary EDT. Finally, the fracture is drawn onto the high-resolution binary image with a circular brush along the estimated normal.
+
+In practice, we found that adding three fractures with $1 . 5 \mathrm { p x }$ thickness, $2 \mathrm { p x }$ minimum distance to tips and forks and angle window of $5 \times 5 \mathrm { p x } ^ { 2 }$ (‘px’ as measured in the low resolution image) produces detectable but not too obvious perturbations (see Fig. B.4). We also extend the lines on both ends by $0 . 5 \mathrm { p x }$ to add some tolerance.
+
+# 4 EVALUATION CASE STUDIES
+
+In this section, we demonstrate potential uses of the proposed framework: using morphometrics to characterise the distribution of samples from generative models and finding associations between learned latent representations and morphometric attributes. In addition, we exemplify in Appendix D a variety of supervised tasks on the MNIST dataset augmented with perturbations.
+
+# 4.1 SAMPLE DIVERSITY
+
+Here we aim to illustrate ways in which the proposed MNIST morphometrics may be used to visualise distributions learned by generative models and to quantify their agreement with the true data distribution in terms of these semantic attributes. We also believe that extracting such measurements from model samples is a step toward diagnosing the issue of mode collapse.
+
+![](images/f32b46565fa4fee1954840d5aa9a4818dcf2b0db89912d426c3e7124d525793e.jpg)  
+Figure 4: Distribution of morphometric attributes for MNIST test dataset and samples from some generative models. Diagonals show marginal histograms and KDEs, upper-triangular plots show pairwise log-histograms and lower-triangular plots show pairwise KDEs.
+
+We exemplify this scenario with a vanilla GAN (Goodfellow et al., 2014) and a $\beta$ -VAE (Higgins et al., 2017), both with generator (resp. decoder) and discriminator architecture as used in the MNIST experiments in Chen et al. (2016), and encoder mirroring the decoder. We train a $\beta$ -VAE with $\beta = 4$ and a GAN, both with 64-dimensional latent space. To explore the behaviour of a much less expressive model, we additionally train a GAN with only two latent dimensions.
+
+Visualisation: Figure 4 illustrates the morphometric distributions of the plain MNIST test images and of 10,000 samples from each of these three models. As can be seen, morphometrics provide interpretable low-dimensional statistics which allow comparing distributions learned by generative models between each other and with true datasets. While Figs. 4b and $_ { \mathrm { 4 c } }$ show model samples roughly as diverse as the true images, the samples from the low-dimensional GAN in Fig. 4d seem concentrated on certain regions, covering a distribution that is less faithful to the true one in Fig. 4a.
+
+Statistical comparison: We argue that in this lower-dimensional space of morphometrics it is possible to statistically compare the distributions, since this was shown not to be effective directly in image space (e.g. Theis et al., 2016). To this end, we propose to use kernel two-sample tests based on maximum mean discrepancy (MMD) between morphometrics of the test data and of each of the sample distributions. Here, we performed the linear-time asymptotic test described in Gretton et al. (2012, $\ S 6$ (details and further considerations in Appendix C). The test results in Table 1 seem to confirm the mismatch of the low-dimensional GAN’s samples, whereas the $\beta$ -VAE and larger GAN do not show a significant departure from the data distribution.
+
+Table 1: Kernel two-sample tests between model samples and true test data   
+
+<table><tr><td>Test data vs.</td><td>Dims.</td><td>MMD² ± std. error (×10-3)</td><td>p</td></tr><tr><td>β-VAE</td><td>64</td><td>0.792 ± 1.569</td><td>.3068</td></tr><tr><td>GAN</td><td>64</td><td>1.458 ± 1.650</td><td>.1885</td></tr><tr><td>GAN</td><td>2</td><td>8.876 ± 1.807</td><td>.0000</td></tr></table>
+
+Table 2: Settings for InfoGAN disentanglement experiments   
+
+<table><tr><td></td><td>#Cat.</td><td># Cont.</td><td>#Bin.</td><td>Dataset</td></tr><tr><td>INFOGAN-A</td><td>10</td><td>2</td><td>0</td><td>PLAIN: plain only</td></tr><tr><td>INFOGAN-B</td><td>10</td><td>3</td><td>0</td><td>GLOBAL: plain + thinning + thickening</td></tr><tr><td>INFOGAN-C</td><td>10</td><td>2</td><td>2</td><td>LOCAL:1 plain + swelling + fractures</td></tr></table>
+
+Finding replicas: One potentially fruitful suggestion would be to use a variant of hierarchical agglomerative clustering on sample morphometric attributes (e.g. using standardised Euclidean distance, or other suitable metrics). With a low enough distance threshold, it would be possible to identify groups of near-replicas, the abundance of which would signify mode collapse. Alternatively, this could be applicable as a heuristic in the birthday paradox test for estimating the support of the learned distribution (Arora et al., 2018).
+
+# 4.2 DISENTANGLEMENT
+
+In this experiment, we demonstrate that: (a) standard MNIST can be augmented with morphometric attributes to quantitatively study representations computed by an inference model (as already possible with e.g. dSprites and 3D faces); (b) we can measure shape attributes of samples to assess disentanglement of a generative model, which is unprecedented to the best of our knowledge; and (c) this analysis can also diagnose when a model unexpectedly fails to learn a known aspect of the data.
+
+Methodology: We take MAP estimates of latent codes for each image (i.e. maximal logit for categorical codes and mean for continuous codes), as predicted by the variational recognition network. Using an approach related to the disentanglement measure introduced in Kumar et al. (2018), we study the correlation structures between known generative factors and latent codes learned by an InfoGAN. Specifically, we compute the partial correlation between each latent code variable and each morphometric attribute, controlling for the variation in the remaining latent variables (disregarding the noise vector).3 As opposed to the simple correlation, this technique allows us to study the net first-order effect of each latent code, all else being equal.
+
+Models were trained for 20 epochs using 64 images per batch, with no hyperparameter tuning. We emphasize that our goal was to illustrate how the proposed morphometrics can serve as tools to better understand whether they behave as intended and not to optimally train the models in each scenario.
+
+Inferential disentanglement: To illustrate how this methodology can be applied in practice to assess disentanglement, we consider two settings. The first is the same as in the MNIST experiment from Chen et al. (2016), with a 10-way categorical and two continuous latent codes, trained and evaluated on the plain MNIST digits, which we will refer to as INFOGAN-A.
+
+![](images/d32d4683786e004f014a1a5a6b0799a6ecd79a99545f8616e29171a78e01278b.jpg)  
+Figure 5: Partial correlations between inferred latent codes and morphometrics of test images. Circle area and colour strength are proportional to correlation magnitude, blue is positive and red is negative.
+
+![](images/2fbc5634f91ffb920a699c2ec1c66e36faa9960089866f4a7d2fcfe42ce84813.jpg)  
+Figure 6: Partial correlations between 1000 sampled latent codes and morphometrics of the corresponding generated images
+
+The second setting was designed to investigate whether the model could disentangle the concept of thickness, by including an additional continuous latent code and training on a dataset with exaggerated thickness variations. We constructed this dataset by randomly interleaving plain, thinned and thickened digit images in equal proportions. Since the perturbations were applied completely at random, we expect a trained generative model to identify that thickness should be largely independent of the other morphological attributes. We refer to this set-up as INFOGAN-B. Table 2 summarises the different experimental settings, for reference.
+
+In Fig. 5a, we see that INFOGAN-A learned to encode slant mostly in $c _ { 3 }$ , while $c _ { 1 } ^ { ( 8 ) }$ clearly relates to the ‘1’ class (much narrower digit shape and shorter pen stroke; cf. Fig. 3). Figure 5b quantitatively confirms the hypothesis that INFOGAN-B’s recognition network would learn to separate slant and thickness (in $c _ { 4 }$ and $c _ { 3 }$ , resp.), the most prominent factors of style variation in this dataset. Interestingly, it shows that $c _ { 3 }$ also associates with height, as thicker digits tend to be taller.
+
+Generative disentanglement: The evaluation methodology described above is useful to investigate the behaviour of the inference direction of a model, and can readily be used with datasets which include ground-truth generative factor annotations. On the other hand, unless we trust that the inference approximation is highly accurate, this tells us little about the generative expressiveness of the model. This is where computed metrics truly show their potential: we can measure generated samples, and see how their attributes relate to the latent variables used to create them.
+
+Figure 6 shows results for a similar analysis to Fig. 5, but now evaluated on samples from that model. As the tables are mostly indistinguishable, we may argue that in this case the inference and generator networks have learned to consistently encode and decode the digit shape attributes.
+
+As further illustration, Fig. 7 displays traversals of the latent space, obtained by varying a subset of the latent variables while holding the remaining ones (including noise) constant. With these examples, we are able to qualitatively verify the quantitative results in Fig. 6. Note that, until now, visual inspection was typically the only means of evaluating disentanglement and expressiveness of the generative direction of image models (e.g. Chen et al., 2016; Dupont, 2018).
+
+Diagnosing failure: We also attempted to detect whether an InfoGAN had learned to discover local perturbations (swelling and fractures). To this end, we extended the model formulation with additional Bernoulli latent codes, which would hopefully learn to encode presence/absence of each (a) INFOGAN-A: one-dimensional traversals of $c _ { 1 }$ (top, ‘digit type’) and $c _ { 3 }$ (bottom, ‘slant’). Samples in each row share the values of remaining latent variables and noise.
+
+![](images/5e42f2725678a0ac85ec4338f2af6a560e7aa6247529b0de692a8ecb066049b1.jpg)
+
+![](images/7635f4e12b11b4e8575ec07e891c4b262bab7c16190e9765111b9c11166e35b5.jpg)  
+Figure 7: InfoGAN latent space traversals
+
+(b) INFOGAN-B: two-dimensional traversal of $c _ { 4 } \times c _ { 3 }$ (‘thickness’ $\times$ ‘slant’)
+
+![](images/ab2209f4d0e67de166a89f6a2bf019917975e143e64dd0eefa4d7bfc46fd5a8d.jpg)  
+Figure 8: Partial correlations of inferred latent codes with test morphometrics (INFOGAN-C)
+
+type of local perturbation. The model investigated here, dubbed INFOGAN-C (cf. Table 2), had a 10-way categorical, two continuous and two binary codes, and was trained with a dataset of plain, swollen and fractured digits (randomly mixed as above).
+
+Again via inferential partial correlation analysis—now including ground-truth perturbation annotations—we can quantitatively verify that this particular model instance was unable to meaningfully capture the perturbations (Fig. 8, bottom-right block). In fact, it appears that the addition of the binary variables did not lead to more expressive representations in this case, even impairing the disentanglement of the categorical variables, if compared to Figs. 5a and 5b, for example.
+
+# 5 CONCLUSION
+
+With Morpho-MNIST we provide a number of mechanisms to quantitatively assess representation learning with respect to measurable factors of variation in the data. We believe that this is an important asset for future research on generative models, and we would like to emphasize that the proposed morphometrics can be used post hoc to evaluate already trained models, potentially revealing novel insights and interesting observations.
+
+A similar morphometry approach could be used with other datasets such as dSprites, e.g. estimating shape location and size, number of objects/connected components. Perhaps some generic image metrics may be useful for analysis on other datasets, e.g. relating to sharpness or colour diversity, or we could even consider using the output of object detectors (analogously to the Inception-based scores; e.g. number/class of objects, bounding boxes etc.). In future work we plan to include additional perturbations, for example, mimicking imaging artefacts commonly observed in medical imaging modalities to add further complexity and realism.
+
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+Loosli, G., Canu, S., and Bottou, L. (2007). Training invariant support vector machines using selective sampling. In Bottou, L., Chapelle, O., DeCoste, D., and Weston, J., editors, Large Scale Kernel Machines, pages 301–320. MIT Press, Cambridge, MA. Dataset URL http: //leon.bottou.org/projects/infimnist.   
+Makhzani, A., Shlens, J., Jaitly, N., Goodfellow, I., and Frey, B. (2015). Adversarial autoencoders, arXiv:1511.05644.   
+Matthey, L., Higgins, I., Hassabis, D., and Lerchner, A. (2017). dSprites: disentanglement testing sprites dataset. URL https://github.com/deepmind/dsprites-dataset/. [Accessed on: 2018-05-08].   
+Paysan, P., Knothe, R., Amberg, B., Romdhani, S., and Vetter, T. (2009). A 3D face model for pose and illumination invariant face recognition. In Proceedings of the Sixth IEEE International Conference on Advanced Video and Signal Based Surveillance (AVSS 2009), pages 296–301. IEEE. Dataset URL https://faces.dmi.unibas.ch/bfm/index.php?nav=1-1-1&id $\underline { { \underline { { \mathbf { \Pi } } } } } =$ scans.   
+Salimans, T., Goodfellow, I., Zaremba, W., Cheung, V., Radford, A., and Chen, X. (2016). Improved techniques for training GANs. In Advances in Neural Information Processing Systems 29 (NIPS 2016), pages 2234–2242.   
+Scott, D. W. (1992). Multivariate Density Estimation: Theory, Practice and Visualization. John Wiley & Sons, Inc., New York.   
+Sutherland, D. J., Tung, H.-Y., Strathmann, H., De, S., Ramdas, A., Smola, A. J., and Gretton, A. (2017). Generative models and model criticism via optimized maximum mean discrepancy. In International Conference on Learning Representations (ICLR 2017). arXiv:1611.04488.   
+Szegedy, C., Vanhoucke, V., Ioffe, S., Shlens, J., and Wojna, Z. (2016). Rethinking the Inception architecture for computer vision. In Proceedings of the 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR 2016), pages 2818–2826. IEEE.   
+Teow, L.-N. and Loe, K.-F. (2002). Robust vision-based features and classification schemes for off-line handwritten digit recognition. Pattern Recognition, 35(11):2355–2364.   
+Theis, L., van den Oord, A., and Bethge, M. (2016). A note on the evaluation of generative models. In International Conference on Learning Representations (ICLR 2016).   
+Tieleman, T. (2013). affNIST. URL https://www.cs.toronto.edu/\~tijmen/ affNIST/, Dataset URL https://www.cs.toronto.edu/\~tijmen/affNIST/. [Accessed on: 2018-05-08].   
+van der Walt, S., Schönberger, J. L., Nunez-Iglesias, J., Boulogne, F., Warner, J. D., Yager, N., Gouillart, E., and Yu, T. (2014). scikit-image: image processing in Python. PeerJ, 2:e453.   
+Xiao, H., Rasul, K., and Vollgraf, R. (2017). Fashion-MNIST: a novel image dataset for benchmarking machine learning algorithms, arXiv:1708.07747. Dataset URL https://github.com/ zalandoresearch/fashion-mnist.
+
+![](images/ffa7190b56389a989314c8ec3e6fddbd73b371e58519b900a129d67dfc0e8b18.jpg)  
+Figure A.1: Distribution of morphological attributes for plain MNIST digits. Top: training set; bottom: test set.
+
+![](images/d598c9e2130be67946ea51a27a874995f436842cbaa0889096a4e0bb68573299.jpg)  
+Figure A.2: Distribution of morphological attributes for thinned MNIST digits. Top: training set; bottom: test set.
+
+![](images/de03aab03fe5b60cab13c5ccce7321b45a7f88835bf12733bd4054e65b2c5f04.jpg)  
+Figure A.3: Distribution of morphological attributes for thickened MNIST digits. Top: training set; bottom: test set.
+
+![](images/2f4310af4b97400f938ede9c86778a44e19450bf71816ad6d9c3894f77567767.jpg)  
+Figure A.4: Distribution of morphological attributes for swollen MNIST digits. Top: training set; bottom: test set.
+
+![](images/c92e73abb977a83948c281abe7d2ced9e76ee83490f2b8629b17f6357165128c.jpg)  
+Figure A.5: Distribution of morphological attributes for fractured MNIST digits. Top: training set; bottom: test set.
+
+# B PERTURBATION EXAMPLES
+
+![](images/00fb4af145de41abc4d41966b1dadbd2a8b1f2dd99aef66bafa6529d44386544.jpg)  
+Figure B.1: Examples of globally thinned digits
+
+![](images/32f75a12f646c84ac453a84bf21254e8b12574bd44277634c493ca5c123e3f2a.jpg)  
+Figure B.2: Examples of globally thickened digits
+
+![](images/331052125bec109b20250a1f087cb26d807e97971fe89531644b1be31c712ebd.jpg)  
+Figure B.3: Examples of digits with local swellings
+
+![](images/cfed125a31fad651b341b4172ef6dc716c72ea85de54bfcbdb3671086bf92b31.jpg)  
+Figure B.4: Examples of digits with local fractures
+
+# C MMD DETAILS
+
+We employed a Gaussian product kernel with bandwidths derived from Scott’s rule, analogously to the KDE plots in Fig. 4. Scott’s rule of thumb defines the bandwidth for a density estimation kernel as $N ^ { - 1 / ( D + 4 ) }$ times the standard deviation in each dimension, where $N$ and $D$ denote sample size and number of dimensions (Scott, 1992, Eq. (6.42)). We determine the KDE bandwidths separately for real and sample data, then add their squares to obtain the squared bandwidth of the MMD’s Gaussian kernel, as it corresponds to the convolution of the density estimation kernels chosen for each set of data. See Gretton et al. (2012, $\ S 3 . 3 . 1 $ for further details on the relation between MMD and $L _ { 2 }$ distance of kernel density estimates.
+
+Whereas the bandwidth heuristic used here is fairly crude, much more sophisticated kernel selection procedures are available, e.g. by explicitly optimising the test power (Sutherland et al., 2017). A further analysis tool in a similar vein would be to apply a relative MMD similarity test (Bounliphone et al., 2016), to rank trained models based on sample fidelity. It would also be possible to adopt a model criticism methodology based on the MMD witness function (Lloyd and Ghahramani, 2015), to identify over- and under-represented regions in morphometric space (and corresponding generated image exemplars could be inspected as well).
+
+# D SUPERVISED TASKS
+
+Although the driving motivation for introducing Morpho-MNIST has been the lack of means for quantitative evaluation of generative models, the proposed framework may also be a valuable resource in the context of supervised learning. We conducted several experiments to demonstrate potential applications of these datasets with increased difficulty due to the injected perturbations: standard digit recognition, supervised abnormality detection, and thickness regression. Note such experiments can later serve as baselines for unsupervised tasks such as outlier detection and domain adaptation.
+
+We evaluated four different models: $k$ -nearest-neighbours $( k \mathsf { N N } )$ using $k = 5$ neighbours and $\ell _ { 1 }$ distance weighting, a support vector machine (SVM) with polynomial kernel and penalty parameter $C = 1 0 0$ , a multi-layer perceptron (MLP) with 784–200–200– $L$ architecture ( $L$ : number of outputs), and a LeNet-5 convolutional neural network (LeCun et al., 1998). Here, we use the same datasets as in the disentanglement experiments (Section 4.2): plain digits (PLAIN), plain mixed with thinned and thickened digits (GLOBAL), and plain mixed with swollen and fractured digits (LOCAL).
+
+For digit recognition, each model is trained once on PLAIN, then tested on both PLAIN and LOCAL test datasets, to investigate the effect of domain shift. All methods suffer a drop in test accuracy on LOCAL (Table 3, first two columns). kNN appears to be the most robust to the local perturbations, perhaps because they affect only a few pixels, leaving the image distance between neighbours largely unchanged. On the other hand, local patterns that LeNet-5 relies on may have changed considerably.
+
+The abnormality detection task is, using the LOCAL dataset, to predict whether a digit is normal or perturbed (swollen or fractured)—compare with lesion detection in medical scans. Table 3 (third column) indicates that LeNet-5 is able to detect abnormalities with high accuracy, likely thanks to local invariances of its convolutional architecture. Note that all scores (especially the simpler models’) are lower than digit classification accuracy, revealing the (possibly surprising) higher difficulty of this binary classification problem compared to the ten-class digit classification.
+
+Finally, we also constructed a regression task for digit thickness using the GLOBAL dataset, mimicking medical imaging tasks such as estimating brain age from cortical grey matter maps. Since this is a non-trivial task, requiring some awareness of local geometry, it is perhaps unsurprising that the convolutional model outperformed the others, which rely on holistic features (Table 3, last column).
+
+Table 3: Accuracy on supervised tasks using the proposed data perturbations   
+
+<table><tr><td rowspan="2">Model</td><td colspan="2">Digit Recognition (%)</td><td rowspan="2">Abnormality Detection (%)</td><td rowspan="2">Thickness Regression (RMSE, pixels)</td></tr><tr><td>PLAIN</td><td>LOCAL</td></tr><tr><td>kNN</td><td>96.25</td><td>95.22</td><td>65.10</td><td>0.4674</td></tr><tr><td>SVM</td><td>95.71</td><td>92.47</td><td>77.59</td><td>0.3647</td></tr><tr><td>MLP</td><td>97.97</td><td>93.15</td><td>88.25</td><td>0.3481</td></tr><tr><td>LeNet-5</td><td>98.95</td><td>95.33</td><td>97.53</td><td>0.2790</td></tr></table>

parse/train/r1esnoAqt7/r1esnoAqt7_content_list.json

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+    {
+        "type": "text",
+        "text": "MORPHO-MNIST: QUANTITATIVE ASSESSMENT AND DIAGNOSTICS FOR REPRESENTATION LEARNING ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Revealing latent structure in data is an active field of research, having introduced exciting technologies such as variational autoencoders and adversarial networks, and is essential to push machine learning towards unsupervised knowledge discovery. However, a major challenge is the lack of suitable benchmarks for an objective and quantitative evaluation of learned representations. To address this issue we introduce Morpho-MNIST, a framework that aims to answer: “to what extent has my model learned to represent specific factors of variation in the data?” We extend the popular MNIST dataset by adding a morphometric analysis enabling quantitative comparison of trained models, identification of the roles of latent variables, and characterisation of sample diversity. We further propose a set of quantifiable perturbations to assess the performance of unsupervised and supervised methods on challenging tasks such as outlier detection and domain adaptation. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "A key factor for progress in machine learning has been the availability of well curated, easy-to-use, standardised and sufficiently large annotated datasets for benchmarking different algorithms and models. This has led to major advances in speech recognition, computer vision, and natural language processing. A commonality between these tasks is their natural formulation as supervised learning tasks, wherein performance can be measured in terms of accuracy on a test set. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The general problem of representation learning (i.e. to reveal latent structure in data) is more difficult to assess due the lack of suitable benchmarks. Although the field is very active, with many recently proposed techniques such as probabilistic autoencoders and adversarial learning, it is less clear where the field stands in terms of progress or which approaches are more expressive for specific tasks. The lack of reproducible ways to quantify performance has led to subjective means of evaluation: visualisation techniques have been used to show low-dimensional projections of the latent space and visual inspection of generated or reconstructed samples are popular to provide subjective measures of descriptiveness. On the other hand, the quality of sampled images generally tells us little about how well the learned representations capture known factors of variation in the training distribution. In order to advance progress, the availability of tools for objective assessment of representation learning methods seems essential yet lacking. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This paper introduces Morpho-MNIST, a collection of shape metrics and perturbations, in a step towards quantitative assessment of representation learning. We build upon one of the most popular machine learning benchmarks, MNIST, which despite its shortcomings remains widely used. While MNIST was originally constructed to facilitate research in image classification, in the form of recognising handwritten digits (LeCun et al., 1998), it has found its use in representation learning, for example, to demonstrate that the learned latent space yields clusters consistent with digit labels. Methods aiming to disentangle the latent space claim success if individual latent variables capture specific style variations (e.g. stroke thickness, sidewards leaning digits and other visual characteristics). ",
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+    {
+        "type": "text",
+        "text": "The main appeal of selecting MNIST as a benchmark for representation learning is that, while manifesting complex interactions between pixel intensities and underlying shapes, it has well understood and easily measurable factors of variation. More generally, MNIST remains popular in practice due to several factors: it allows reproducible comparisons with previous results reported in the literature; the dataset is sufficiently large for its complexity and consists of small, two-dimensional greyscale images defining a tractable ten-class classification problem; computation and memory requirements are low; most popular deep learning frameworks and libraries offer tutorials using MNIST, which makes it straightforward for new researchers to enter the field and to experiment with new ideas and explore latest developments. We take advantage of these qualities and extend MNIST in multiple ways, as summarised in the following. ",
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+        "img_path": "images/f303b3624281bc81731a0a3ec201e7e822f8774e613ebbd94322adac6cb3dcf9.jpg",
+        "image_caption": [
+            "Figure 1: Left: MNIST morphometrics—stroke thickness and length (not shown), width, height and slant of digits. Right: MNIST perturbations (many more examples of each type in Appendix B). "
+        ],
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+    {
+        "type": "text",
+        "text": "1.1 CONTRIBUTIONS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our aim is to bridge the gap between methodology-focused research and critical real-world applications that could benefit from latest machine learning methods. As we preserve the general properties of MNIST—such as image size, file format, numbers of training and test images, and the original ten-class classification problem—we believe this new quantitative framework for assessing representation learning will experience widespread use in the community and may inspire further extensions facilitated by a publicly available Morpho-MNIST code base. ",
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+    {
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+        "text": "Morphometrics: We propose to describe true and generated digit images in terms of measurable shape attributes. These include stroke thickness and length, and the width, height, and slant of digits (cf. Fig. 1, left). Whereas some of these properties have been analysed qualitatively in previous work, we demonstrate that objectively quantifying each of them allows to identify the role of inferred representations. Moreover, these tools can be used to measure model samples, enabling assessment of generative performance with respect to sample diversity (Section 4.1) and disentanglement of latent variables (Section 4.2). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "These measurements can be directly employed to re-evaluate existing models and may be added retrospectively to previous experiments involving the original MNIST dataset. Adoption of our morphometric analysis may provide new insights into the effectiveness of representation learning methods in terms of revealing meaningful latent structures. Furthermore, for other datasets it suffices to design the relevant scalar metrics and include them in the very same evaluation framework. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Perturbations: We introduce a set of parametrisable global and local perturbations, inspired by natural and pathological variability in medical images. Global changes involve overall thinning and thickening of digits, while local changes include both swelling and fractures (see examples on the right in Fig. 1 and many more in Appendix B). Injecting these perturbations into the dataset adds a new type of complexity to the data manifold and opens up a variety of interesting applications. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The proposed perturbations are designed to enable a wide range of new studies and applications for both supervised and unsupervised tasks. Detection of ‘abnormalities’ (i.e. local perturbations) is an evident application, although more challenging tasks can also be defined, such as classification from noisy/corrupted data, domain adaptation, localisation of perturbations, characterising semantics of learned latent representations, and more. We explore a few supplementary examples of supervised tasks in Appendix D. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1.2 RELATED WORK: DATASETS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this section, we provide an overview of some datasets that are related to MNIST, by either sharing its original source content, containing transformations of the original MNIST images or being distributed in the same format for easy replacement. We also mention a few prevalent datasets of images with generative factor annotations, similarly to the morphometrics proposed in this paper. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "NIST datasets: The MNIST (modified NIST) dataset (LeCun et al., 1998) was constructed from handwritten digits in NIST Special Databases 1 and 3, now released as Special Database 19 (Grother and Hanaoka, 2016). Cohen et al. (2017) generated a much larger dataset based on the same NIST database, containing additional upper- and lower-case letters, called EMNIST (extended MNIST). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "MNIST perturbations: The seminal paper by LeCun et al. (1998) employed data augmentation using planar affine transformations including translation, scaling, squeezing, and shearing. Loosli et al. (2007) employed random elastic deformations to construct the Infinite MNIST dataset. Other MNIST variations include rotations and insertion of random and structured background (Larochelle et al., 2007), and Tieleman (2013) applied spatial affine transformations and provided ground-truth transformation parameters. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "MNIST format: Due to the ubiquity of the MNIST dataset in machine learning research and the resulting multitude of compatible model architectures available, it is appealing to release new datasets in the same format $2 8 \\times 2 8$ , 8-bit grayscale images). One such effort is Fashion-MNIST (Xiao et al., 2017), containing images of clothing articles from ten distinct classes, adapted from an online shopping catalogue. Another example is notMNIST (Bulatov, 2011), a dataset of character glyphs for letters $\\mathbf { \\delta A } ^ { \\prime } - \\mathbf { \\delta J } ^ { \\prime }$ (also ten classes), in a challengingly diverse collection of typefaces. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Annotated datasets: Computer vision datasets that are popular for evaluating disentanglement of learned latent factors of variation include those from Paysan et al. (2009) and Aubry et al. (2014). They contain 2D renderings of 3D faces and chairs, respectively, with ground-truth pose parameters (azimuth, elevation) and lighting conditions (faces only). A further initiative in that direction is the dSprites dataset (Matthey et al., 2017), which consists of binary images containing three types of shapes with varying location, orientation and size. The availability of the ground-truth values of such attributes has motivated the accelerated adoption of these datasets in the evaluation of representation learning algorithms. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1.3 RELATED WORK: QUANTITATIVE EVALUATION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Evaluation of representation learning is a notoriously challenging task and remains an open research problem. Numerous solutions have been proposed, with many of the earlier ones focusing on the test log-likelihood under the model (Kingma and Welling, 2013) or, for likelihood-free models, under a kernel density estimate (KDE) of generated samples (Goodfellow et al., 2014; Makhzani et al., 2015)—being shown not to be reliable proxies for the true model likelihood (Theis et al., 2016). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Another perspective for evaluation of generative models of images is the visual fidelity of its samples to the training data, which would normally require manual inspection. To address this issue, a successful family of metrics have been proposed, based on visual features extracted by the Inception network (Szegedy et al., 2016). The original Inception score (Salimans et al., 2016) relies on the ‘crispness’ of class predictions, whereas the Fréchet Inception distance (FID) (Heusel et al., 2017) and the kernel Inception distance (KID) (Binkowski et al. ´ , 2018) statistically compare high-level representations instead of the final network outputs. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Although the approaches above can reveal vague signs of mode collapse, it may be useful to diagnose this phenomenon on its own. With this objective, Arora et al. (2018) proposed to estimate the support of the learned distribution (assumed discrete) using the birthday paradox test, by counting pairs of visual duplicates among model samples. Unfortunately, the adoption of this technique is hindered by its reliance on manual visual inspection to flag identical images. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "There have been several attempts at quantifying representation disentanglement performance. For example, Higgins et al. (2017) proposed to use the accuracy of a simple classifier trained to predict which factor of variation was held fixed in a simulated dataset. There exist further informationtheoretic approaches, involving the KL divergence contribution from each latent dimension (Dupont, 2018) or their mutual information with each known generative factor (Chen et al., 2018). Yet another method, explored in Kumar et al. (2018), is based on the predictive accuracy of each latent variable to each generative factor (continuous or discrete). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/831c9debb102f33904a8b60566a90410437acb89c20dd3973660f3453ff82753.jpg",
+        "image_caption": [
+            "Figure 2: Stages of the image processing pipeline. Left to right: original image, upscaled image, binarised image, distance transform, skeleton, downscaled image. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "2 MORPHOMETRY ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Meaningful morphometrics are instrumental in characterising distributions of rasterised shapes, such as MNIST digits, and can be useful as additional data for downstream learning tasks. We begin this section by describing the image processing pipeline employed for extracting the metrics and for applying perturbations (Section 3), followed by details on the computation of each measurement. ",
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+    {
+        "type": "text",
+        "text": "2.1 PROCESSING PIPELINE ",
+        "text_level": 1,
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "The original $2 8 \\times 2 8$ resolution of the MNIST images is generally not high enough to enable satisfactory morphological processing: stroke properties (e.g. length, thickness) measured directly on the binarised images would likely be inaccurate and heavily quantised. To mitigate this issue and enable sub-pixel accuracy in the measurements, we propose to use the following processing steps: 1. upscale (e.g. $\\times 4$ , to $1 1 2 \\times 1 1 2 )$ 1; 2. binarise (e.g. threshold ${ \\geq } 1 2 8$ ); 3. compute Euclidean distance transform (EDT) from boundaries; 4. skeletonise (medial axis, i.e. ridges of EDT); 5. apply perturbation (cf. Section 3); and 6. downscale to original resolution. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We illustrate the pipeline in Fig. 2. The binary high-resolution digits have smooth boundaries and faithfully capture subtle variations in contour shape and stroke thickness that are only vaguely discernible in the low-resolution images. Additionally, note how the final downscaled image is almost indistinguishable from the original. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "All morphometric attributes described below are calculated for each digit after applying steps 1–4 of this pipeline. The distributions for the plain MNIST training set is plotted in Fig. 3, and the distributions after applying each type of perturbation can be found in Appendix A. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/9291516fa3d4332cbbc0c54e5fdcef063fb6dbd4fa49cbc6fc977d3ffb8b7621.jpg",
+        "image_caption": [
+            "Figure 3: Distribution of morphological attributes per digit class in the plain MNIST training dataset "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.2 STROKE LENGTH ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Here we approximate the trace of the pen tip, as a digit was being written, by the computed morphological skeleton. In this light, the total length of the skeleton is an estimate of the length of the pen stroke, which in turn is a measure of shape complexity. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "It can be computed in a single pass by accumulating the Euclidean distance of each skeleton pixel to its immediate neighbours, taking care to only count the individual contributions once. This approach is more robust against rotations than a naïve estimate by simply counting the pixels. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "2.3 STROKE THICKNESS ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "A prominent factor of style variation in the MNIST digits is the overall thickness of the strokes, due to both legitimate differences in pen thickness and force applied, and also to the rescaling of the original NIST images by different factors. ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We estimate it by exploiting the computed distance transform. By virtue of how the image skeleton is computed, its pixels are approximately equidistant to the nearest boundaries, therefore we take twice the mean value of the EDT over all skeleton pixels as our global estimate. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "2.4 SLANT ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            385,
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+            398
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The extent by which handwritten symbols lean right or left (forward and backward slant, respectively) is a further notorious and quantifiable dimension of handwriting style. It introduces so much variation in the appearance of characters in images that it is common practice in OCR systems to ‘deslant’ them, in an attempt to reduce within-class variance (LeCun et al., 1998; Teow and Loe, 2002). ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We adapt the referred deslanting methodology to describe the slant angle of the handwritten digits. After estimating the second-order image moments, we define the slant based on the horizontal shear: ",
+        "bbox": [
+            171,
+            476,
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2f85a799da9082ea09b940c13bb5839d575f781b5ef216bf83744102ce86128e.jpg",
+        "text": "$$\n\\alpha = \\arctan \\left( - \\frac { \\sum _ { i , j } x _ { i j } ( i - \\bar { i } ) ( j - \\bar { j } ) } { \\sum _ { i , j } x _ { i j } ( i - \\bar { i } ) ^ { 2 } } \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            357,
+            513,
+            638,
+            556
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\boldsymbol { x } _ { i j }$ is the intensity of pixel $( i , j )$ , and $( \\bar { i } , \\bar { j } )$ are the centroid coordinates. The minus sign ensures that positive and negative values correspond to forward and backward slant, respectively. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "2.5 WIDTH AND HEIGHT ",
+        "text_level": 1,
+        "bbox": [
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+            617,
+            359,
+            632
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "It is useful to measure other general shape attributes, such as width, height, and aspect ratio, which also present substantial variation related to personal handwriting style.2 To this end, we propose to fit a bounding parallelogram to each digit, with horizontal and slanted sides (cf. Fig. 1). ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We sweep the image top-to-bottom with a horizontal boundary to compute a vertical marginal cumulative distribution function (CDF), and likewise left-to-right with a slanted boundary for a horizontal marginal CDF, with angle $\\alpha$ as computed above. The bounds are then chosen based on equal-tailed intervals containing a given proportion of the image mass— $98 \\%$ in both directions ( $1 \\%$ from each side) proved accurate and robust in our experiments. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3 PERTURBATIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "As discussed in Section 1, we bring forward a number of morphological perturbations for MNIST digits, to enable interesting applications and experimentation. In this section, we detail these parametrisable transformations, categorised as global or local. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.1 GLOBAL: THINNING AND THICKENING ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+            118
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The first pair of transformations we present is based on simple morphological operations: the binarised image of a digit is dilated or eroded with a circular structuring element. Its radius is set proportionally to the estimated stroke thickness (Section 2.3), so that the overall thickness of each digit will decrease or increase by an approximately fixed factor (here, $- 7 0 \\%$ and $+ 1 0 0 \\%$ ; see Figs. B.1 and B.2). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Since there is substantial thickness variability in the original MNIST data (cf. Fig. 3) and most thinned and thickened digits look very plausible, we believe that these perturbations can constitute a powerful form of data augmentation for training. For the same reason, we have not included these perturbations in the abnormality detection experiments (Appendix D). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3.2 LOCAL: SWELLING ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In addition to the global transformations above, we introduce local perturbations with variable location and extent, which are harder to detect automatically. Given a radius $R$ , a centre location $\\mathbf { r } _ { 0 }$ and a strength parameter $\\gamma > 1$ , the coordinates $\\mathbf { r }$ of pixels within distance $R$ of $\\mathbf { r } _ { 0 }$ are nonlinearly warped according to a radial power transform: ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7d9bca38bd36d21de1336b9b593fabf6418a5f90837e8c77b003654fdc8567c7.jpg",
+        "text": "$$\n\\mathbf { r } \\mapsto \\mathbf { r } _ { 0 } + \\left( \\mathbf { r } - \\mathbf { r } _ { 0 } \\right) \\left( \\frac { \\left\\| \\mathbf { r } - \\mathbf { r } _ { 0 } \\right\\| } { R } \\right) ^ { \\gamma - 1 } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            377,
+            352,
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+            383
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "leaving the remaining portions of the image untouched and resampling with bicubic interpolation. ",
+        "bbox": [
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+            387,
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+            401
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In the experiments and released dataset, we set $\\gamma = 7$ and $R = 3 \\sqrt { \\theta } / 2$ , where $\\theta$ is thickness. Unlike simple linear scaling with $\\theta$ , this choice for $R$ produces noticeable but not exaggerated effects across the thickness range observed in the dataset (cf. Fig. B.3). The centre location, $\\mathbf { r } _ { 0 }$ , is picked uniformly at random from the pixels along the estimated skeleton. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.3 LOCAL: FRACTURES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            483,
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We describe the proposed procedure for adding fractures to an MNIST digit, where we define a fracture as a break in the continuity of a pen stroke. Because single fractures can in many cases be easily mistaken for true gaps between strokes, we add multiple fractures to each affected digit. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "When selecting the location for a fracture, we attempt to avoid getting too close to stroke tips (points on the skeleton with a single neighbour) or fork points (more than two neighbours). This is achieved by sampling only among those skeleton pixels above a certain distance to these detected points. In addition, we would like fractures to be transversal to the pen strokes. Local orientation is determined based on second-order moments of the skeleton inside a window centred at the chosen location, and the length of the fracture is estimated from the boundary EDT. Finally, the fracture is drawn onto the high-resolution binary image with a circular brush along the estimated normal. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In practice, we found that adding three fractures with $1 . 5 \\mathrm { p x }$ thickness, $2 \\mathrm { p x }$ minimum distance to tips and forks and angle window of $5 \\times 5 \\mathrm { p x } ^ { 2 }$ (‘px’ as measured in the low resolution image) produces detectable but not too obvious perturbations (see Fig. B.4). We also extend the lines on both ends by $0 . 5 \\mathrm { p x }$ to add some tolerance. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 EVALUATION CASE STUDIES ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In this section, we demonstrate potential uses of the proposed framework: using morphometrics to characterise the distribution of samples from generative models and finding associations between learned latent representations and morphometric attributes. In addition, we exemplify in Appendix D a variety of supervised tasks on the MNIST dataset augmented with perturbations. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4.1 SAMPLE DIVERSITY ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Here we aim to illustrate ways in which the proposed MNIST morphometrics may be used to visualise distributions learned by generative models and to quantify their agreement with the true data distribution in terms of these semantic attributes. We also believe that extracting such measurements from model samples is a step toward diagnosing the issue of mode collapse. ",
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/f32b46565fa4fee1954840d5aa9a4818dcf2b0db89912d426c3e7124d525793e.jpg",
+        "image_caption": [
+            "Figure 4: Distribution of morphometric attributes for MNIST test dataset and samples from some generative models. Diagonals show marginal histograms and KDEs, upper-triangular plots show pairwise log-histograms and lower-triangular plots show pairwise KDEs. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We exemplify this scenario with a vanilla GAN (Goodfellow et al., 2014) and a $\\beta$ -VAE (Higgins et al., 2017), both with generator (resp. decoder) and discriminator architecture as used in the MNIST experiments in Chen et al. (2016), and encoder mirroring the decoder. We train a $\\beta$ -VAE with $\\beta = 4$ and a GAN, both with 64-dimensional latent space. To explore the behaviour of a much less expressive model, we additionally train a GAN with only two latent dimensions. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Visualisation: Figure 4 illustrates the morphometric distributions of the plain MNIST test images and of 10,000 samples from each of these three models. As can be seen, morphometrics provide interpretable low-dimensional statistics which allow comparing distributions learned by generative models between each other and with true datasets. While Figs. 4b and $_ { \\mathrm { 4 c } }$ show model samples roughly as diverse as the true images, the samples from the low-dimensional GAN in Fig. 4d seem concentrated on certain regions, covering a distribution that is less faithful to the true one in Fig. 4a. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Statistical comparison: We argue that in this lower-dimensional space of morphometrics it is possible to statistically compare the distributions, since this was shown not to be effective directly in image space (e.g. Theis et al., 2016). To this end, we propose to use kernel two-sample tests based on maximum mean discrepancy (MMD) between morphometrics of the test data and of each of the sample distributions. Here, we performed the linear-time asymptotic test described in Gretton et al. (2012, $\\ S 6$ (details and further considerations in Appendix C). The test results in Table 1 seem to confirm the mismatch of the low-dimensional GAN’s samples, whereas the $\\beta$ -VAE and larger GAN do not show a significant departure from the data distribution. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/73d5046374f952e57c062b9aab978edd13324b07cb533c927195b7e09f57f019.jpg",
+        "table_caption": [
+            "Table 1: Kernel two-sample tests between model samples and true test data "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Test data vs.</td><td>Dims.</td><td>MMD² ± std. error (×10-3)</td><td>p</td></tr><tr><td>β-VAE</td><td>64</td><td>0.792 ± 1.569</td><td>.3068</td></tr><tr><td>GAN</td><td>64</td><td>1.458 ± 1.650</td><td>.1885</td></tr><tr><td>GAN</td><td>2</td><td>8.876 ± 1.807</td><td>.0000</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/f1c0524abcb32ebcd0ef6ae6c2d9ae6934437505eed993fe0e712b78ef6d92ba.jpg",
+        "table_caption": [
+            "Table 2: Settings for InfoGAN disentanglement experiments "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>#Cat.</td><td># Cont.</td><td>#Bin.</td><td>Dataset</td></tr><tr><td>INFOGAN-A</td><td>10</td><td>2</td><td>0</td><td>PLAIN: plain only</td></tr><tr><td>INFOGAN-B</td><td>10</td><td>3</td><td>0</td><td>GLOBAL: plain + thinning + thickening</td></tr><tr><td>INFOGAN-C</td><td>10</td><td>2</td><td>2</td><td>LOCAL:1 plain + swelling + fractures</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Finding replicas: One potentially fruitful suggestion would be to use a variant of hierarchical agglomerative clustering on sample morphometric attributes (e.g. using standardised Euclidean distance, or other suitable metrics). With a low enough distance threshold, it would be possible to identify groups of near-replicas, the abundance of which would signify mode collapse. Alternatively, this could be applicable as a heuristic in the birthday paradox test for estimating the support of the learned distribution (Arora et al., 2018). ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.2 DISENTANGLEMENT ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this experiment, we demonstrate that: (a) standard MNIST can be augmented with morphometric attributes to quantitatively study representations computed by an inference model (as already possible with e.g. dSprites and 3D faces); (b) we can measure shape attributes of samples to assess disentanglement of a generative model, which is unprecedented to the best of our knowledge; and (c) this analysis can also diagnose when a model unexpectedly fails to learn a known aspect of the data. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Methodology: We take MAP estimates of latent codes for each image (i.e. maximal logit for categorical codes and mean for continuous codes), as predicted by the variational recognition network. Using an approach related to the disentanglement measure introduced in Kumar et al. (2018), we study the correlation structures between known generative factors and latent codes learned by an InfoGAN. Specifically, we compute the partial correlation between each latent code variable and each morphometric attribute, controlling for the variation in the remaining latent variables (disregarding the noise vector).3 As opposed to the simple correlation, this technique allows us to study the net first-order effect of each latent code, all else being equal. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Models were trained for 20 epochs using 64 images per batch, with no hyperparameter tuning. We emphasize that our goal was to illustrate how the proposed morphometrics can serve as tools to better understand whether they behave as intended and not to optimally train the models in each scenario. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Inferential disentanglement: To illustrate how this methodology can be applied in practice to assess disentanglement, we consider two settings. The first is the same as in the MNIST experiment from Chen et al. (2016), with a 10-way categorical and two continuous latent codes, trained and evaluated on the plain MNIST digits, which we will refer to as INFOGAN-A. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/d32d4683786e004f014a1a5a6b0799a6ecd79a99545f8616e29171a78e01278b.jpg",
+        "image_caption": [
+            "Figure 5: Partial correlations between inferred latent codes and morphometrics of test images. Circle area and colour strength are proportional to correlation magnitude, blue is positive and red is negative. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/2fbc5634f91ffb920a699c2ec1c66e36faa9960089866f4a7d2fcfe42ce84813.jpg",
+        "image_caption": [
+            "Figure 6: Partial correlations between 1000 sampled latent codes and morphometrics of the corresponding generated images "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The second setting was designed to investigate whether the model could disentangle the concept of thickness, by including an additional continuous latent code and training on a dataset with exaggerated thickness variations. We constructed this dataset by randomly interleaving plain, thinned and thickened digit images in equal proportions. Since the perturbations were applied completely at random, we expect a trained generative model to identify that thickness should be largely independent of the other morphological attributes. We refer to this set-up as INFOGAN-B. Table 2 summarises the different experimental settings, for reference. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In Fig. 5a, we see that INFOGAN-A learned to encode slant mostly in $c _ { 3 }$ , while $c _ { 1 } ^ { ( 8 ) }$ clearly relates to the ‘1’ class (much narrower digit shape and shorter pen stroke; cf. Fig. 3). Figure 5b quantitatively confirms the hypothesis that INFOGAN-B’s recognition network would learn to separate slant and thickness (in $c _ { 4 }$ and $c _ { 3 }$ , resp.), the most prominent factors of style variation in this dataset. Interestingly, it shows that $c _ { 3 }$ also associates with height, as thicker digits tend to be taller. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Generative disentanglement: The evaluation methodology described above is useful to investigate the behaviour of the inference direction of a model, and can readily be used with datasets which include ground-truth generative factor annotations. On the other hand, unless we trust that the inference approximation is highly accurate, this tells us little about the generative expressiveness of the model. This is where computed metrics truly show their potential: we can measure generated samples, and see how their attributes relate to the latent variables used to create them. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Figure 6 shows results for a similar analysis to Fig. 5, but now evaluated on samples from that model. As the tables are mostly indistinguishable, we may argue that in this case the inference and generator networks have learned to consistently encode and decode the digit shape attributes. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "As further illustration, Fig. 7 displays traversals of the latent space, obtained by varying a subset of the latent variables while holding the remaining ones (including noise) constant. With these examples, we are able to qualitatively verify the quantitative results in Fig. 6. Note that, until now, visual inspection was typically the only means of evaluating disentanglement and expressiveness of the generative direction of image models (e.g. Chen et al., 2016; Dupont, 2018). ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Diagnosing failure: We also attempted to detect whether an InfoGAN had learned to discover local perturbations (swelling and fractures). To this end, we extended the model formulation with additional Bernoulli latent codes, which would hopefully learn to encode presence/absence of each (a) INFOGAN-A: one-dimensional traversals of $c _ { 1 }$ (top, ‘digit type’) and $c _ { 3 }$ (bottom, ‘slant’). Samples in each row share the values of remaining latent variables and noise. ",
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/5e42f2725678a0ac85ec4338f2af6a560e7aa6247529b0de692a8ecb066049b1.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "image",
+        "img_path": "images/7635f4e12b11b4e8575ec07e891c4b262bab7c16190e9765111b9c11166e35b5.jpg",
+        "image_caption": [
+            "Figure 7: InfoGAN latent space traversals "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "(b) INFOGAN-B: two-dimensional traversal of $c _ { 4 } \\times c _ { 3 }$ (‘thickness’ $\\times$ ‘slant’) ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "image",
+        "img_path": "images/ab2209f4d0e67de166a89f6a2bf019917975e143e64dd0eefa4d7bfc46fd5a8d.jpg",
+        "image_caption": [
+            "Figure 8: Partial correlations of inferred latent codes with test morphometrics (INFOGAN-C) "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "type of local perturbation. The model investigated here, dubbed INFOGAN-C (cf. Table 2), had a 10-way categorical, two continuous and two binary codes, and was trained with a dataset of plain, swollen and fractured digits (randomly mixed as above). ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Again via inferential partial correlation analysis—now including ground-truth perturbation annotations—we can quantitatively verify that this particular model instance was unable to meaningfully capture the perturbations (Fig. 8, bottom-right block). In fact, it appears that the addition of the binary variables did not lead to more expressive representations in this case, even impairing the disentanglement of the categorical variables, if compared to Figs. 5a and 5b, for example. ",
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+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+            318,
+            729
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "With Morpho-MNIST we provide a number of mechanisms to quantitatively assess representation learning with respect to measurable factors of variation in the data. We believe that this is an important asset for future research on generative models, and we would like to emphasize that the proposed morphometrics can be used post hoc to evaluate already trained models, potentially revealing novel insights and interesting observations. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "A similar morphometry approach could be used with other datasets such as dSprites, e.g. estimating shape location and size, number of objects/connected components. Perhaps some generic image metrics may be useful for analysis on other datasets, e.g. relating to sharpness or colour diversity, or we could even consider using the output of object detectors (analogously to the Inception-based scores; e.g. number/class of objects, bounding boxes etc.). In future work we plan to include additional perturbations, for example, mimicking imaging artefacts commonly observed in medical imaging modalities to add further complexity and realism. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/ffa7190b56389a989314c8ec3e6fddbd73b371e58519b900a129d67dfc0e8b18.jpg",
+        "image_caption": [
+            "Figure A.1: Distribution of morphological attributes for plain MNIST digits. Top: training set; bottom: test set. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
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+        "img_path": "images/d598c9e2130be67946ea51a27a874995f436842cbaa0889096a4e0bb68573299.jpg",
+        "image_caption": [
+            "Figure A.2: Distribution of morphological attributes for thinned MNIST digits. Top: training set; bottom: test set. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/de03aab03fe5b60cab13c5ccce7321b45a7f88835bf12733bd4054e65b2c5f04.jpg",
+        "image_caption": [
+            "Figure A.3: Distribution of morphological attributes for thickened MNIST digits. Top: training set; bottom: test set. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/2f4310af4b97400f938ede9c86778a44e19450bf71816ad6d9c3894f77567767.jpg",
+        "image_caption": [
+            "Figure A.4: Distribution of morphological attributes for swollen MNIST digits. Top: training set; bottom: test set. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/c92e73abb977a83948c281abe7d2ced9e76ee83490f2b8629b17f6357165128c.jpg",
+        "image_caption": [
+            "Figure A.5: Distribution of morphological attributes for fractured MNIST digits. Top: training set; bottom: test set. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "B PERTURBATION EXAMPLES ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/00fb4af145de41abc4d41966b1dadbd2a8b1f2dd99aef66bafa6529d44386544.jpg",
+        "image_caption": [
+            "Figure B.1: Examples of globally thinned digits "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            251,
+            143,
+            745,
+            449
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/32f75a12f646c84ac453a84bf21254e8b12574bd44277634c493ca5c123e3f2a.jpg",
+        "image_caption": [
+            "Figure B.2: Examples of globally thickened digits "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            251,
+            507,
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/331052125bec109b20250a1f087cb26d807e97971fe89531644b1be31c712ebd.jpg",
+        "image_caption": [
+            "Figure B.3: Examples of digits with local swellings "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            251,
+            133,
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+            445
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/cfed125a31fad651b341b4172ef6dc716c72ea85de54bfcbdb3671086bf92b31.jpg",
+        "image_caption": [
+            "Figure B.4: Examples of digits with local fractures "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            251,
+            550,
+            745,
+            858
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "C MMD DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            338,
+            117
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "We employed a Gaussian product kernel with bandwidths derived from Scott’s rule, analogously to the KDE plots in Fig. 4. Scott’s rule of thumb defines the bandwidth for a density estimation kernel as $N ^ { - 1 / ( D + 4 ) }$ times the standard deviation in each dimension, where $N$ and $D$ denote sample size and number of dimensions (Scott, 1992, Eq. (6.42)). We determine the KDE bandwidths separately for real and sample data, then add their squares to obtain the squared bandwidth of the MMD’s Gaussian kernel, as it corresponds to the convolution of the density estimation kernels chosen for each set of data. See Gretton et al. (2012, $\\ S 3 . 3 . 1 $ for further details on the relation between MMD and $L _ { 2 }$ distance of kernel density estimates. ",
+        "bbox": [
+            174,
+            133,
+            825,
+            246
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Whereas the bandwidth heuristic used here is fairly crude, much more sophisticated kernel selection procedures are available, e.g. by explicitly optimising the test power (Sutherland et al., 2017). A further analysis tool in a similar vein would be to apply a relative MMD similarity test (Bounliphone et al., 2016), to rank trained models based on sample fidelity. It would also be possible to adopt a model criticism methodology based on the MMD witness function (Lloyd and Ghahramani, 2015), to identify over- and under-represented regions in morphometric space (and corresponding generated image exemplars could be inspected as well). ",
+        "bbox": [
+            174,
+            252,
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+            351
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "D SUPERVISED TASKS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 17
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+        "type": "text",
+        "text": "Although the driving motivation for introducing Morpho-MNIST has been the lack of means for quantitative evaluation of generative models, the proposed framework may also be a valuable resource in the context of supervised learning. We conducted several experiments to demonstrate potential applications of these datasets with increased difficulty due to the injected perturbations: standard digit recognition, supervised abnormality detection, and thickness regression. Note such experiments can later serve as baselines for unsupervised tasks such as outlier detection and domain adaptation. ",
+        "bbox": [
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+            401,
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+    {
+        "type": "text",
+        "text": "We evaluated four different models: $k$ -nearest-neighbours $( k \\mathsf { N N } )$ using $k = 5$ neighbours and $\\ell _ { 1 }$ distance weighting, a support vector machine (SVM) with polynomial kernel and penalty parameter $C = 1 0 0$ , a multi-layer perceptron (MLP) with 784–200–200– $L$ architecture ( $L$ : number of outputs), and a LeNet-5 convolutional neural network (LeCun et al., 1998). Here, we use the same datasets as in the disentanglement experiments (Section 4.2): plain digits (PLAIN), plain mixed with thinned and thickened digits (GLOBAL), and plain mixed with swollen and fractured digits (LOCAL). ",
+        "bbox": [
+            174,
+            492,
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+    },
+    {
+        "type": "text",
+        "text": "For digit recognition, each model is trained once on PLAIN, then tested on both PLAIN and LOCAL test datasets, to investigate the effect of domain shift. All methods suffer a drop in test accuracy on LOCAL (Table 3, first two columns). kNN appears to be the most robust to the local perturbations, perhaps because they affect only a few pixels, leaving the image distance between neighbours largely unchanged. On the other hand, local patterns that LeNet-5 relies on may have changed considerably. ",
+        "bbox": [
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+        "type": "text",
+        "text": "The abnormality detection task is, using the LOCAL dataset, to predict whether a digit is normal or perturbed (swollen or fractured)—compare with lesion detection in medical scans. Table 3 (third column) indicates that LeNet-5 is able to detect abnormalities with high accuracy, likely thanks to local invariances of its convolutional architecture. Note that all scores (especially the simpler models’) are lower than digit classification accuracy, revealing the (possibly surprising) higher difficulty of this binary classification problem compared to the ten-class digit classification. ",
+        "bbox": [
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+            660,
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+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "Finally, we also constructed a regression task for digit thickness using the GLOBAL dataset, mimicking medical imaging tasks such as estimating brain age from cortical grey matter maps. Since this is a non-trivial task, requiring some awareness of local geometry, it is perhaps unsurprising that the convolutional model outperformed the others, which rely on holistic features (Table 3, last column). ",
+        "bbox": [
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+            751,
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+        "page_idx": 17
+    },
+    {
+        "type": "table",
+        "img_path": "images/ab1deafbd938ac16c233a359d2698b17d0594999c4513a3dabbb3b3750c2548c.jpg",
+        "table_caption": [
+            "Table 3: Accuracy on supervised tasks using the proposed data perturbations "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Model</td><td colspan=\"2\">Digit Recognition (%)</td><td rowspan=\"2\">Abnormality Detection (%)</td><td rowspan=\"2\">Thickness Regression (RMSE, pixels)</td></tr><tr><td>PLAIN</td><td>LOCAL</td></tr><tr><td>kNN</td><td>96.25</td><td>95.22</td><td>65.10</td><td>0.4674</td></tr><tr><td>SVM</td><td>95.71</td><td>92.47</td><td>77.59</td><td>0.3647</td></tr><tr><td>MLP</td><td>97.97</td><td>93.15</td><td>88.25</td><td>0.3481</td></tr><tr><td>LeNet-5</td><td>98.95</td><td>95.33</td><td>97.53</td><td>0.2790</td></tr></table>",
+        "bbox": [
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+        "page_idx": 18
+    }
+]

parse/train/rJgMlhRctm/rJgMlhRctm.md

@@ -0,0 +1,572 @@
+# THE NEURO-SYMBOLIC CONCEPT LEARNER: INTERPRETING SCENES, WORDS, AND SENTENCES FROM NATURAL SUPERVISION
+
+Jiayuan Mao MIT CSAIL and IIIS, Tsinghua University mjy14@mails.tsinghua.edu.cn
+
+Chuang Gan MIT-IBM Watson AI Lab ganchuang@csail.mit.edu
+
+Pushmeet Kohli   
+Deepmind   
+pushmeet@google.com
+
+Joshua B. TenenbaumMIT BCS, CBMM, CSAILjbt@mit.edu
+
+Jiajun Wu   
+MIT CSAIL   
+jiajunwu@mit.edu
+
+# ABSTRACT
+
+We propose the Neuro-Symbolic Concept Learner (NS-CL), a model that learns visual concepts, words, and semantic parsing of sentences without explicit supervision on any of them; instead, our model learns by simply looking at images and reading paired questions and answers. Our model builds an object-based scene representation and translates sentences into executable, symbolic programs. To bridge the learning of two modules, we use a neuro-symbolic reasoning module that executes these programs on the latent scene representation. Analogical to human concept learning, the perception module learns visual concepts based on the language description of the object being referred to. Meanwhile, the learned visual concepts facilitate learning new words and parsing new sentences. We use curriculum learning to guide the searching over the large compositional space of images and language. Extensive experiments demonstrate the accuracy and efficiency of our model on learning visual concepts, word representations, and semantic parsing of sentences. Further, our method allows easy generalization to new object attributes, compositions, language concepts, scenes and questions, and even new program domains. It also empowers applications including visual question answering and bidirectional image-text retrieval.
+
+# 1 INTRODUCTION
+
+Humans are capable of learning visual concepts by jointly understanding vision and language (Fazly et al., 2010; Chrupała et al., 2015; Gauthier et al., 2018). Consider the example shown in Figure 1-I. Imagine that someone with no prior knowledge of colors is presented with the images of the red and green cubes, paired with the questions and answers. They can easily identify the difference in objects’ visual appearance (in this case, color), and align it to the corresponding words in the questions and answers (Red and Green). Other object attributes (e.g., shape) can be learned in a similar fashion. Starting from there, humans are able to inductively learn the correspondence between visual concepts and word semantics (e.g., spatial relations and referential expressions, Figure 1-II), and unravel compositional logic from complex questions assisted by the learned visual concepts (Figure 1-III, also see Abend et al. (2017)).
+
+Motivated by this, we propose the neuro-symbolic concept learner (NS-CL), which jointly learns visual perception, words, and semantic language parsing from images and question-answer pairs. NS-CL has three modules: a neural-based perception module that extracts object-level representations from the scene, a visually-grounded semantic parser for translating questions into executable programs, and a symbolic program executor that reads out the perceptual representation of objects, classifies their attributes/relations, and executes the program to obtain an answer.
+
+I. Learning basic, object-based concepts.
+
+II. Learning relational concepts based on referential expressions.
+
+Q: How many objects are right of the red object?   
+A: 2.   
+Q: How many objects have the same material as the cube? A: 2
+
+![](images/754d584daf82ca5122bc2b77184a4443fab2a3eaf5007558a93a3e3db77cfce7.jpg)
+
+III. Interpret complex questions from visual cues.
+
+![](images/c4848607df288bff029e7d65ae74b6720106fd4c4a52cc290362dd12782968a1.jpg)  
+Figure 1: Humans learn visual concepts, words, and semantic parsing jointly and incrementally. I. Learning visual concepts (red vs. green) starts from looking at simple scenes, reading simple questions, and reasoning over contrastive examples (Fazly et al., 2010). II. Afterwards, we can interpret referential expressions based on the learned object-based concepts, and learn relational concepts (e.g., on the right of, the same material as). III Finally, we can interpret complex questions from visual cues by exploiting the compositional structure.
+
+![](images/ab06c9a73fe344dd0ba5bf2c7a8f688b7a572fd8a7b6c489814776ee3fe672ef.jpg)
+
+Q: How many objects are both right of the green cylinder and have the same material as the small blue ball? A: 3
+
+NS-CL learns from natural supervision (i.e., images and QA pairs), requiring no annotations on images or semantic programs for sentences. Instead, analogical to human concept learning, it learns via curriculum learning. NS-CL starts by learning representations/concepts of individual objects from short questions (e.g., What’s the color of the cylinder?) on simple scenes $\leq 3$ objects). By doing so, it learns object-based concepts such as colors and shapes. NS-CL then learns relational concepts by leveraging these object-based concepts to interpret object referrals (e.g., Is there a box right of a cylinder?). The model iteratively adapts to more complex scenes and highly compositional questions.
+
+NS-CL’s modularized design enables interpretable, robust, and accurate visual reasoning: it achieves state-of-the-art performance on the CLEVR dataset (Johnson et al., 2017a). More importantly, it naturally learns disentangled visual and language concepts, enabling combinatorial generalization w.r.t. both visual scenes and semantic programs. In particular, we demonstrate four forms of generalization. First, NS-CL generalizes to scenes with more objects and longer semantic programs than those in the training set. Second, it generalizes to new visual attribute compositions, as demonstrated on the CLEVR-CoGenT (Johnson et al., 2017a) dataset. Third, it enables fast adaptation to novel visual concepts, such as learning a new color. Finally, the learned visual concepts transfer to new tasks, such as image-caption retrieval, without any extra fine-tuning.
+
+# 2 RELATED WORK
+
+Our model is related to research on joint learning of vision and natural language. In particular, there are many papers that learn visual concepts from descriptive languages, such as image-captioning or visually-grounded question-answer pairs (Kiros et al., 2014; Shi et al., 2018; Mao et al., 2016; Vendrov et al., 2016; Ganju et al., 2017), dense language descriptions for scenes (Johnson et al., 2016), video-captioning (Donahue et al., 2015) and video-text alignment (Zhu et al., 2015).
+
+Visual question answering (VQA) stands out as it requires understanding both visual content and language. The state-of-the-art approaches usually use neural attentions (Malinowski & Fritz, 2014; Chen et al., 2015; Yang et al., 2016; Xu & Saenko, 2016). Beyond question answering, Johnson et al. (2017a) proposed the CLEVR (VQA) dataset to diagnose reasoning models. CLEVR contains synthetic visual scenes and questions generated from latent programs. Table 1 compares our model with state-of-the-art visual reasoning models (Andreas et al., 2016; Suarez et al., 2018; Santoro et al., 2017) along four directions: visual features, semantics, inference, and the requirement of extra labels.
+
+For visual representations, Johnson et al. (2017b) encoded visual scenes into a convolutional feature map for program operators. Mascharka et al. (2018); Hudson & Manning (2018) used attention as intermediate representations for transparent program execution. Recently, Yi et al. (2018) explored an interpretable, object-based visual representation for visual reasoning. It performs well, but requires fully-annotated scenes during training. Our model also adopts an object-based visual representation, but the representation is learned only based on natural supervision (questions and answers).
+
+Anderson et al. (2018) also proposed to represent the image as a collection of convolutional object features and gained substantial improvements on VQA. Their model encodes questions with neural networks and answers the questions by question-conditioned attention over the object features. In contrast, NS-CL parses question inputs into programs and executes them on object features to get the answer. This makes the reasoning process interpretable and supports combinatorial generalization over quantities (e.g., counting objects). Our model also learns general visual concepts and their association with symbolic representations of language. These learned concepts can then be explicitly interpreted and deployed in other vision-language applications such as image caption retrieval.
+
+Table 1: Comparison with other frameworks on the CLEVR VQA dataset, w.r.t. visual features, implicit or explicit semantics and supervisions.   
+
+<table><tr><td rowspan="2">Models</td><td rowspan="2">Visual Features</td><td rowspan="2">Semantics</td><td colspan="2">Extra Labels</td><td rowspan="2">Inference</td></tr><tr><td>#Prog.</td><td>Attr.</td></tr><tr><td rowspan="2">FiLM (Perez et al.,2018) IEP (Johnson et al., 2017b)</td><td>Convolutional</td><td>Implicit</td><td>0</td><td>No</td><td>Feature Manipulation</td></tr><tr><td>Convolutional</td><td>Explicit</td><td>700K</td><td>No</td><td>Feature Manipulation</td></tr><tr><td rowspan="3">MAC (Hudson &amp; Manning,2018) Stack-NMN (Hu et al., 2018)</td><td>Attentional</td><td>Implicit</td><td>0</td><td>No</td><td>Feature Manipulation</td></tr><tr><td>Attentional</td><td>Implicit</td><td>0</td><td>No</td><td>Attention Manipulation</td></tr><tr><td>Attentional</td><td>Explicit</td><td>700K</td><td>No</td><td>Attention Manipulation</td></tr><tr><td rowspan="2">NS-VQA (Yi et al., 2018) NS-CL</td><td>Object-Based</td><td>Explicit</td><td>0.2K</td><td>Yes</td><td>Symbolic Execution</td></tr><tr><td>Object-Based</td><td>Explicit</td><td>0</td><td>No</td><td>Symbolic Execution</td></tr></table>
+
+![](images/0eaed17800ac769a2eb58bcbf300d45270d4bafff0d76149de06668192032643.jpg)  
+Figure 2: We propose to use neural symbolic reasoning as a bridge to jointly learn visual concepts, words, and semantic parsing of sentences.
+
+There are two types of approaches in semantic sentence parsing for visual reasoning: implicit programs as conditioned neural operations (e.g., conditioned convolution and dual attention) (Perez et al., 2018; Hudson & Manning, 2018) and explicit programs as sequences of symbolic tokens (Andreas et al., 2016; Johnson et al., 2017b; Mascharka et al., 2018). As a representative, Andreas et al. (2016) build modular and structured neural architectures based on programs for answering questions. Explicit programs gain better interpretability, but usually require extra supervision such as groundtruth program annotations for training. This restricts their application. We propose to use visual grounding as distant supervision to parse questions in natural languages into explicit programs, with zero program annotations. Given the semantic parsing of questions into programs, Yi et al. (2018) proposed a purely symbolic executor for the inference of the answer in the logic space. Compared with theirs, we propose a quasi-symbolic executor for VQA.
+
+Our work is also related to learning interpretable and disentangled representations for visual scenes using neural networks. Kulkarni et al. (2015) proposed convolutional inverse graphics networks for learning and inferring pose of faces, while Yang et al. (2015) learned disentangled representation of pose of chairs from images. Wu et al. (2017) proposed the neural scene de-rendering framework as an inverse process of any rendering process. Siddharth et al. (2017); Higgins et al. (2018) learned disentangled representations using deep generative models. In contrast, we propose an alternative representation learning approach through joint reasoning with language.
+
+# 3 NEURO-SYMBOLIC CONCEPT LEARNER
+
+We present our neuro-symbolic concept learner, which uses a symbolic reasoning process to bridge the learning of visual concepts, words, and semantic parsing of sentences without explicit annotations for any of them. We first use a visual perception module to construct an object-based representation for a scene, and run a semantic parsing module to translate a question into an executable program. We then apply a quasi-symbolic program executor to infer the answer based on the scene representation. We use paired images, questions, and answers to jointly train the visual and language modules.
+
+![](images/b1b0ead8f89439fa1d33ca543287d03354f46bf86a5379c32ab8a6a80eef0f6d.jpg)  
+Figure 3: We treat attributes such as Shape and Color as neural operators. The operators map object representations into a visual-semantic space. We use similarity-based metric to classify objects.
+
+Shown in Figure 2, given an input image, the visual perception module detects objects in the scene and extracts a deep, latent representation for each of them. The semantic parsing module translates an input question in natural language into an executable program given a domain specific language (DSL). The generated programs have a hierarchical structure of symbolic, functional modules, each fulfilling a specific operation over the scene representation. The explicit program semantics enjoys compositionality, interpretability, and generalizability.
+
+The program executor executes the program upon the derived scene representation and answers the question. Our program executor works in a symbolic and deterministic manner. This feature ensures a transparent execution trace of the program. Our program executor has a fully differentiable design w.r.t. the visual representations and the concept representations, which supports gradient-based optimization during training.
+
+# 3.1 MODEL DETAILS
+
+Visual perception. Shown in Figure 2, given the input image, we use a pretrained Mask R-CNN (He et al., 2017) to generate object proposals for all objects. The bounding box for each single object paired with the original image is then sent to a ResNet-34 (He et al., 2015) to extract the region-based (by RoI Align) and image-based features respectively. We concatenate them to represent each object. Here, the inclusion of the representation of the full scene adds the contextual information, which is essential for the inference of relative attributes such as size or spatial position.
+
+Concept quantization. Visual reasoning requires determining an object’s attributes (e.g., its color or shape). We assume each visual attribute (e.g., shape) contains a set of visual concept (e.g., Cube). In NS-CL, visual attributes are implemented as neural operators , mapping the object representation into an attribute-specific embedding space. Figure 3 shows an inference an object’s shape. Visual concepts that belong to the shape attribute, including Cube, Sphere and Cylinder, are represented as vectors in the shape embedding space. These concept vectors are also learned along the process. We measure the cosine distances $\langle \cdot , \cdot \rangle$ between these vectors to determine the shape of the object. Specifically, we compute the probability that an object $o _ { i }$ is a cube by $\sigma \left( \langle { \mathrm { S h a p e O f } } ( o _ { i } ) , \check { v } ^ { \mathrm { C u b e } } \rangle \dot { - } \gamma \right) / \tau$ , where ShapeOf(·) denotes the neural operator, $v ^ { \scriptscriptstyle \mathrm { C u b e } }$ the concept embedding of Cube and $\sigma$ the Sigmoid function. $\gamma$ and $\tau$ are scalar constants for scaling and shifting the values of similarities. We classify relational concepts (e.g., Left) between a pair of objects similarly, except that we concatenate the visual representations for both objects to form the representation of their relation.
+
+DSL and semantic parsing. The semantic parsing module translates a natural language question into an executable program with a hierarchy of primitive operations, represented in a domain-specific language (DSL) designed for VQA. The DSL covers a set of fundamental operations for visual reasoning, such as filtering out objects with certain concepts or query the attribute of an object. The operations share the same input and output interface, and thus can be compositionally combined to form programs of any complexity. We include a complete specification of the DSL used by our framework in the Appendix A.
+
+![](images/5126e8459da4b737ebdbf49d0c8c2a50ec116ad879f9a86a9af00be83a8d36a0.jpg)  
+Figure 4: A. Demonstration of the curriculum learning of visual concepts, words, and semantic parsing of sentences by watching images and reading paired questions and answers. Scenes and questions of different complexities are illustrated to the learner in an incremental manner. B. Illustration of our neuro-symbolic inference model for VQA. The perception module begins with parsing visual scenes into object-based deep representations, while the semantic parser parse sentences into executable programs. A symbolic execution process bridges two modules.
+
+Our semantic parser generates the hierarchies of latent programs in a sequence to tree manner (Dong & Lapata, 2016). We use a bidirectional GRU (Cho et al., 2014) to encode an input question, which outputs a fixed-length embedding of the question. A decoder based on GRU cells is applied to the embedding, and recovers the hierarchy of operations as the latent program. Some operations takes concepts their parameters, such as Filter( Red ) and Query( Shape ). These concepts are chosen from all concepts appeared in the input question. Figure 4(B) shows an example, while more details can be found in Appendix B.
+
+Quasi-symbolic program execution. Given the latent program recovered from the question in natural language, a symbolic program executor executes the program and derives the answer based on the object-based visual representation. Our program executor is a collection of deterministic functional modules designed to realize all logic operations specified in the DSL. Figure 4(B) shows an illustrative execution trace of a program.
+
+To make the execution differentiable w.r.t. visual representations, we represent the intermediate results in a probabilistic manner: a set of objects is represented by a vector, as the attention mask over all objects in the scene. Each element, $\mathrm { { M a s k } } _ { i } \in [ \bar { 0 } , 1 ]$ denotes the probability that the $i$ -th object of the scene belongs to the set. For example, shown in Figure 4(B), the first Filter operation outputs a mask of length 4 (there are in total 4 objects in the scene), with each element representing the probability that the corresponding object is selected out (i.e., the probability that each object is a green cube). The output “mask” on the objects will be fed into the next module (Relate in this case) as input and the execution of programs continues. The last module outputs the final answer to the question. We refer interested readers to Appendix C for the implementation of all operators.
+
+# 3.2 TRAINING PARADIGM
+
+Optimization objective. The optimization objective of NS-CL is composed of two parts: concept learning and language understanding. Our goal is to find the optimal parameters $\Theta _ { v }$ of the visual
+
+perception module Perception (including the ResNet-34 for extracting object features, attribute operators. and concept embeddings) and $\Theta _ { s }$ of the semantic parsing module SemanticParse, to maximize the likelihood of answering the question $Q$ correctly:
+
+$$
+\Theta _ { v } , \Theta _ { s } \gets \arg \operatorname* { m a x } _ { \Theta _ { v } , \Theta _ { s } } \mathbb { E } _ { P } [ \operatorname* { P r } [ A = \mathrm { E x e c u t o r } ( \operatorname* { P e r c e p t i o n } ( S ; \Theta _ { v } ) , P ) ] ] ,
+$$
+
+where $P$ denotes the program, $A$ the answer, $S$ the scene, and Executor the quasi-symbolic executor.   
+The expectation is taken over $P \sim$ SemanticParse $\left( Q ; \Theta _ { s } \right)$ .
+
+Recall the program executor is fully differentiable w.r.t. the visual representation. We compute the gradient w.r.t. $\Theta _ { v }$ as $\nabla _ { \Theta _ { v } } \mathbb { E } _ { P } [ D _ { \mathrm { K L } } ( \mathrm { E x e c u t o r } ( \mathrm { P e r c e p t i o n } ( S ; \Theta _ { v } ) , P ) \lVert A ) ]$ . We use REINFORCE (Williams, 1992) to optimize the semantic parser $\Theta _ { s }$ : $\nabla _ { \Theta _ { s } } ~ = ~ \mathbb { E } _ { P } [ r \cdot \log \mathrm { P r } [ P ~ =$ SemanticParse $\left( Q ; \Theta _ { s } \right) ] ]$ , where the reward $r = 1$ if the answer is correct and 0 otherwise. We also use off-policy search to reduce the variance of REINFORCE, the detail of which can be found in Appendix D.
+
+Curriculum visual concept learning. Motivated by human concept learning as in Figure 1, we employ a curriculum learning approach to help joint optimization. We heuristically split the training samples into four stages (Figure 4(A)): first, learning object-level visual concepts; second, learning relational questions; third, learning more complex questions with perception modules fixed; fourth, joint fine-tuning of all modules. We found that this is essential to the learning of our neuro-symbolic concept learner. We include more technical details in Appendix E.
+
+# 4 EXPERIMENTS
+
+We demonstrate the following advantages of our NS-CL. First, it learns visual concepts with remarkable accuracy; second, it allows data-efficient visual reasoning on the CLEVR dataset (Johnson et al., 2017a); third, it generalizes well to new attributes, visual composition, and language domains.
+
+We train NS-CL on 5K images ( $\textless 1 0 \%$ of CLEVR’s 70K training images). We generate 20 questions for each image for the entire curriculum learning process. The Mask R-CNN module is pretrained on 4K generated CLEVR images with bounding box annotations, following Yi et al. (2018).
+
+# 4.1 VISUAL CONCEPT LEARNING
+
+Classification-based concept evaluation. Our model treats attributes as neural operators that map latent object representations into an attribute-specific embedding space (Figure 3). We evaluate the concept quantization of objects in the CLEVR validation split. Our model can achieve near perfect classification accuracy $( \sim 9 9 \% )$ for all object properties, suggesting it effectively learns generic concept representations. The result for spatial relations is relatively lower, because CLEVR does not have direct queries on the spatial relation between objects. Thus, spatial relation concepts can only be learned indirectly.
+
+Count-based concept evaluation. The SOTA methods do not provide interpretable representation on individual objects (Johnson et al., 2017a; Hudson & Manning, 2018; Mascharka et al., 2018) . To evaluate the visual concepts learned by such models, we generate a synthetic question set. The diagnostic question set contains simple questions as the following form: “How many red objects are there?”. We evaluate the performance on all concepts appeared in the CLEVR dataset.
+
+Table 2 summarizes the results compared with strong baselines, including methods based on convolutional features (Johnson et al., 2017b) and those based on neural attentions (Mascharka et al., 2018; Hudson & Manning, 2018). Our approach outperforms IEP by a significant margin $( 8 \% )$ and attention-based baselines by ${ > } 2 \%$ , suggesting object-based visual representations and symbolic reasoning helps to interpret visual concepts.
+
+# 4.2 DATA-EFFICIENT AND INTERPRETABLE VISUAL REASONING
+
+NS-CL jointly learns visual concepts, words and semantic parsing by watching images and reading paired questions and answers. It can be directly applied to VQA.
+
+<table><tr><td>Visual Mean Color Mat. Shape Size</td></tr><tr><td>IEP Conv. 90.6 91.0 90.0 89.9 90.6</td></tr><tr><td>MAC Attn. 95.9 98.0 91.4 94.4 94.2</td></tr><tr><td>TbD (hres.） Attn. 96.5 96.6 92.2 95.4 92.6</td></tr><tr><td>NS-CL Obj. 98.7 99.0 98.7 98.1 99.1</td></tr></table>
+
+Table 2: We also evaluate the learned visual concepts using a diagnostic question set containing simple questions such as “How many red objects are there?”. NS-CL outperforms both convolutional and attentional baselines. The suggested object-based visual representation and symbolic reasoning approach perceives better interpretation of visual concepts.
+
+Table 3: We compare different variants of baselines for a systematic study on visual features and data efficiency. Using only $10 \%$ of the training images, our model is able to achieve a comparable results with the baselines trained on the full dataset. See the text for details.   
+
+<table><tr><td>Model</td><td>Visual</td><td>Accuracy (100% Data) (10% Data)</td><td>Accuracy</td></tr><tr><td>TbD</td><td>Attn.</td><td>99.1</td><td>54.2</td></tr><tr><td>TbD-Object</td><td>Obj.</td><td>84.1</td><td>52.6</td></tr><tr><td>TbD-Mask</td><td>Attn.</td><td>99.0</td><td>55.0</td></tr><tr><td>MAC</td><td>Attn.</td><td>98.9</td><td>67.3</td></tr><tr><td>MAC-Object</td><td>Obj.</td><td>79.5</td><td>51.2</td></tr><tr><td>MAC-Mask</td><td>Attn.</td><td>98.7</td><td>68.4</td></tr><tr><td>NS-CL</td><td>Obj.</td><td>99.2</td><td>98.9</td></tr></table>
+
+Table 4 summarizes results on the CLEVR validation split. Our model achieves the state-of-theart performance among all baselines using zero program annotations, including MAC (Hudson & Manning, 2018) and FiLM (Perez et al., 2018). Our model achieves comparable performance with the strong baseline TbD-Nets (Mascharka et al., 2018), whose semantic parser is trained using 700K programs in CLEVR (ours need 0). The recent NS-VQA model from Yi et al. (2018) achieves better performance on CLEVR; however, their system requires annotated visual attributes and program traces during training, while our NS-CL needs no extra labels.
+
+Here, the visual perception module is pre-trained on ImageNet (Deng et al., 2009). Without pretraining, the concept learning accuracies drop by $0 . 2 \%$ on average and the QA accuracy drops by $0 . 5 \%$ . Meanwhile, NS-CL recovers the underlying programs of questions accurately $( > 9 9 . 9 \%$ accuracy). NS-CL can also detect ambiguous or invalid programs and indicate exceptions. Please see Appendix F for more details. NS-CL can also be applied to other visual reasoning testbeds. Please refer to Appendix G.1 for our results on the Minecraft dataset (Yi et al., 2018).
+
+For a systematic study on visual features and data efficiency, we implement two variants of the baseline models: TbD-Object and MAC-Object. Inspired by (Anderson et al., 2018), instead of the input image, TbD-Object and MAC-Object take a stack of object features as input. TbD-Mask and MAC-Mask integrate the masks of objects by using them to guide the attention over the images.
+
+Table 3 summarizes the results. Our model outperforms all baselines on data efficiency. This comes from the full disentanglement of visual concept learning and symbolic reasoning: how to execute program instructions based on the learned concepts is programmed. TbD-Object and MAC-Object demonstrate inferior results in our experiments. We attribute this to the design of model architectures and have a detailed analysis in Appendix F.3. Although TbD-Mask and MAC-Mask do not perform better than the originals, we find that using masks to guide attentions speeds up the training.
+
+Besides achieving a competitive performance on the visual reasoning testbeds, by leveraging both object-based representation and symbolic reasoning, out model learns fully interpretable visual concepts: see Appendix H for qualitative results on various datasets.
+
+# 4.3 GENERALIZATION TO NEW ATTRIBUTES AND COMPOSITIONS
+
+Generalizing to new visual compositions. The CLEVR-CoGenT dataset is designed to evaluate models’ ability to generalize to new visual compositions. It has two splits: Split A only contains gray, blue, brown and yellow cubes, but red, green, purple, and cyan cylinders; split B imposes the opposite color constraints on cubes and cylinders. If we directly learn visual concepts on split A, it overfits to classify shapes based on the color, leading to a poor generalization to split B.
+
+Our solution is based on the idea of seeing attributes as operators. Specifically, we jointly train the concept embeddings (e.g., Red, Cube, etc.) as well as the semantic parser on split A, keeping pretrained, frozen attribute operators. As we learn distinct representation spaces for different attributes, our model achieves an accuracy of $9 8 . 8 \%$ on split A and $9 8 . 9 \%$ on split B.
+
+Table 4: Our model outperforms all baselines using no program annotations. It achieves comparable results with models trained by full program annotations such as TbD.
+
+Figure 5: We test the combinatorial generalization w.r.t. the number of objects in scenes and the complexity of questions (i.e. the depth of the program trees). We makes four split of the data containing various complexities of scenes and questions. Our object-based visual representation and explicit program semantics enjoys the best (and almost-perfect) combinatorial generalization compared with strong baselines.   
+
+<table><tr><td>Model</td><td>Prog. Overall ( Anno.</td><td>Count</td><td>Cmp. Num.</td><td>Exist</td><td>Query Cmp. Attr. Attr.</td></tr><tr><td>Human</td><td>N/A 92.6</td><td>86.7</td><td>86.4</td><td>96.6</td><td>95.0 96.0</td></tr><tr><td>NMN</td><td>700K 72.1</td><td>52.5</td><td>72.7</td><td>79.3</td><td>79.0 78.0</td></tr><tr><td>N2NMN</td><td>700K 88.8</td><td>68.5</td><td>84.9</td><td>85.7</td><td>90.0 88.8</td></tr><tr><td>IEP</td><td>700K 96.9</td><td>92.7</td><td>98.7</td><td>97.1 98.1</td><td>98.9</td></tr><tr><td>DDRprog</td><td>700K 98.3</td><td>96.5</td><td>98.4</td><td>98.8 99.1</td><td>99.0</td></tr><tr><td>TbD</td><td>700K 99.1</td><td>97.6</td><td>99.4</td><td>99.2 99.5</td><td>99.6</td></tr><tr><td>RN</td><td>0 95.5</td><td>90.1</td><td>93.6</td><td>97.8</td><td>97.1</td></tr><tr><td>FiLM</td><td>0 97.6</td><td>94.5</td><td>93.8</td><td>99.2</td><td>97.9 99.2 99.0</td></tr><tr><td>MAC</td><td>0 98.9</td><td>97.2</td><td>99.4</td><td>99.5 99.3</td><td>99.5</td></tr><tr><td>NS-CL</td><td>0 98.9</td><td>98.2</td><td>99.0</td><td>98.8</td><td>99.3 99.1</td></tr></table>
+
+<table><tr><td rowspan="2">Model</td><td colspan="2">Test</td></tr><tr><td> Split A Split B Split C Split D</td><td></td></tr><tr><td>MAC</td><td>97.3 N/A</td><td>92.9 N/A</td></tr><tr><td>IEP</td><td>96.1 92.1</td><td>91.5 90.9</td></tr><tr><td>TbD</td><td>98.8 94.5</td><td>94.3 91.9</td></tr><tr><td>NS-CL</td><td>98.9 98.9</td><td>98.7 98.8</td></tr></table>
+
+![](images/0cee5f79d08130efce3b8ae5f36aa85c14b9dc1eddfb44e85e7e77fa8f614a15.jpg)  
+Figure 6: Samples collected from four splits in Section 4.3 for illustration. Models are trained on split A but evaluated on all splits for testing the combinatorial generalization.
+
+Generalizing to new visual concepts. We expect the process of concept learning can take place in an incremental manner: having learned 7 different colors, humans can learn the 8-th color incrementally and efficiently. To this end, we build a synthetic split of the CLEVR dataset to replicate the setting of incremental concept learning. Split A contains only images without any purple objects, while split B contains images with at least one purple object. We train all the models on split A first, and finetune them on 100 images from split B. We report the final QA performance on split B’s validation set. All models use a pre-trained semantic parser on the full CLEVR dataset.
+
+Our model performs a $9 3 . 9 \%$ accuracy on the QA test in Split B, outperforming the convolutional baseline IEP (Johnson et al., 2017b) and the attentional baseline TbD (Mascharka et al., 2018) by $4 . 6 \%$ and $6 . 1 \%$ respectively. The acquisition of Color operator brings more efficient learning of new visual concepts.
+
+# 4.4 COMBINATORIAL GENERALIZATION TO NEW SCENES AND QUESTIONS
+
+Having learned visual concepts on small-scale scenes (containing only few objects) and simple questions (only single-hop questions), we humans can easily generalize the knowledge to larger-scale scenes and to answer complex questions. To evaluate this, we split the CLEVR dataset into four parts: Split A contains only scenes with less than 6 objects, and questions whose latent programs having a depth less than 5; Split B contains scenes with less than 6 objects, but arbitrary questions; Split C contains arbitrary scenes, but restricts the program depth being less than 5; Split D contains arbitrary scenes and questions. Figure 6 shows some illustrative samples.
+
+As VQA baselines are unable to count a set of objects of arbitrary size, for a fair comparison, all programs containing the “count” operation over $> 6$ objects are removed from the set. For
+
+<table><tr><td>Model</td><td>Retrieval Accuracy</td></tr><tr><td>IEP</td><td>95.5</td></tr><tr><td>TbD</td><td>97.0</td></tr><tr><td>NS-CL</td><td>96.9</td></tr></table>
+
+![](images/752661ebf459d5bc64ae25b27dfc350c885cc29bf729cddaafb347f3a2a8b6ad.jpg)  
+Caption: There is a big yellow cylinder in front of a gray object.
+
+<table><tr><td>Model</td><td>Retrieval Accuracy</td></tr><tr><td>CNN-LSTM</td><td>68.9</td></tr><tr><td>NS-CL</td><td>97.0</td></tr></table>
+
+(c) Image-caption retrieval accuracy on the full dataset. Our model outperforms baselines and requires no extra training or fine-tuning of the visual perception module.
+
+(a) An illustrative pair of image and caption in our synthetic dataset.
+
+(b) Image-caption retrieval accuracy on a subset of data. Our model archives comparable results with VQA baselines.
+
+Table 5: We introduce a new simple DSL for image-caption retrieval to evaluate how well the learned visual concepts transfer. Due to the difference between VQA and caption retrieval, VQA baselines are only able to infer the result on a partial set of data. The learned object-based visual concepts can be directly transferred into the new domain for free.
+
+methods using explicit program semantics, the semantic parser is pre-trained on the full dataset and fixed. Methods with implicit program semantics (Hudson & Manning, 2018) learn an entangled representation for perception and reasoning, and cannot trivially generalize to more complex programs. We only use the training data from the Split A and then quantify the generalization ability on other three splits. Shown in Table 5, our NS-CL leads to almost-perfect generalization to larger scenes and more complex questions, outperforming all baselines by at least $4 \%$ in QA accuracy.
+
+# 4.5 EXTENDING TO OTHER PROGRAM DOMAIN
+
+The learned visual concepts can also be used in other domains such as image retrieval. With the visual scenes fixed, the learned visual concepts can be directly transferred into the new domain. We only need to learn the semantic parsing of natural language into the new DSL.
+
+We build a synthetic dataset for image retrieval and adopt a DSL from scene graph–based image retrieval (Johnson et al., 2015). The dataset contains only simple captions: “There is an <object $\mathbf { A } >$ <relation $>$ <object $\mathbf { B } >$ .” (e.g., There is a box right of a cylinder). The semantic parser learns to extract corresponding visual concepts (e.g., box, right, and cylinder) from the sentence. The program can then be executed on the visual representation to determine if the visual scene contains such relational triples.
+
+For simplicity, we treat retrieval as classifying whether a relational triple exists in the image. This functionality cannot be directly implemented on the CLEVR VQA program domain, because questions such as “Is there a box right of a cylinder” can be ambiguous if there exist multiple cylinders in the scene. Due to the entanglement of the visual representation with the specific DSL, baselines trained on CLEVR QA can not be directly applied to this task. For a fair comparison with them, we show the result in Table 5b on a subset of the generated image-caption pairs where the underlying programs have no ambiguity regarding the reference of object B. A separate semantic parser is trained for the VQA baselines, which translates captions into a CLEVR QA-compatible program (e.g., Exist(Filter(Box, Relate(Right, Filter(Cylinder))).
+
+Table 5c compares our NS-CL against typical image-text retrieval baselines on the full image-caption dataset. Without any annotations of the sentence semantics, our model learns to parse the captions into the programs in the new DSL. It outperforms the CNN-LSTM baseline by $30 \%$ .
+
+# 4.6 EXTENDING TO NATURAL IMAGES AND LANGUAGE
+
+We further conduct experiments on MS-COCO (Lin et al., 2014) images. Results are presented on the VQS dataset (Gan et al., 2017). VQS contains a subset of images and questions from the original VQA 1.0 dataset (Antol et al., 2015). All questions in the VQS dataset can be visually grounded: each question is associated with multiple image regions, annotated by humans as essential for answering the question. Figure 7 illustrates an execution trace of NS-CL on VQS.
+
+We use a syntactic dependency parser to extract programs and concepts from language (Andreas et al., 2016; Schuster et al., 2015). The object proposals and features are extracted from models pre-trained on the MS-COCO dataset and the ImageNet dataset, respectively. Illustrated in Figure 7, our model
+
+![](images/19fd3f84d85da7a459ed9dbe0d64d4c35b031d72e2bf7a72f24aec75ebf37290.jpg)  
+Figure 7: Left: An example image-question pair from the VQS dataset and the corresponding execution trace of NS-CL. Right: Results on the VQS test set. Our model achieves a comparable results with the baselines.
+
+Concept: Person On a Skateboard   
+
+<table><tr><td>Model</td><td>Accuracy</td></tr><tr><td>MLP</td><td>43.9</td></tr><tr><td>MAC</td><td>46.2</td></tr><tr><td>NS-CL</td><td>44.3</td></tr></table>
+
+Concept: Horse
+
+![](images/493a2ca6633ac207e347e40dabb3732fcc89c7b08c3274148a700ab88607e95b.jpg)  
+Figure 8: Concepts learned from VQS, including object categories, attributes, and relations.
+
+shows competitive performance on QA accuracy, comparable with the MLP baseline (Jabri et al., 2016) and the MAC network (Hudson & Manning, 2018). Additional illustrative execution traces of NS-CL are in Appendix H. Beyond answering questions, NS-CL effectively learns visual concepts from data. Figure 8 shows examples of the learned visual concepts, including object categories, attributes, and relations. Experiment setup and implementation details are in Appendix G.2.
+
+In this paper, we focus on a neuro-symbolic framework that learns visual concepts about object properties and relations. Indeed, visual question answering requires AI systems to reason about more general concepts such as events or activities (Levin, 1993). We leave the extension of NS-CL along this direction and its application to general VQA datasets (Antol et al., 2015) as future work.
+
+# 5 DISCUSSION AND FUTURE WORK
+
+We presented a method that jointly learns visual concepts, words, and semantic parsing of sentences from natural supervision. The proposed framework, NS-CL, learns by looking at images and reading paired questions and answers, without any explicit supervision such as class labels for objects. Our model learns visual concepts with remarkable accuracy. Based upon the learned concepts, our model achieves good results on question answering, and more importantly, generalizes well to new visual compositions, new visual concepts, and new domain specific languages.
+
+The design of NS-CL suggests multiple research directions. First, constructing 3D object-based representations for realistic scenes needs further exploration (Anderson et al., 2018; Baradel et al., 2018). Second, our model assumes a domain-specific language for describing formal semantics. The integration of formal semantics into the processing of complex natural language would be meaningful future work (Artzi & Zettlemoyer, 2013; Oh et al., 2017). We hope our paper could motivate future research in visual concept learning, language learning, and compositionality.
+
+Our framework can also be extended to other domains such as video understanding and robotic manipulation. Here, we would need to discover semantic representations for actions and interactions (e.g., push) beyond static spatial relations. Along this direction, researchers have studied building symbolic representations for skills (Konidaris et al., 2018) and learning instruction semantics from interaction (Oh et al., 2017) in constrained setups. Applying neuro-symbolic learning frameworks for concepts and skills would be meaningful future work toward robotic learning in complex interactive environments.
+
+Acknowledgements. We thank Kexin Yi, Haoyue Shi, and Jon Gauthier for helpful discussions and suggestions. This work was supported in part by the Center for Brains, Minds and Machines (NSF STC award CCF-1231216), ONR MURI N00014-16-1-2007, MIT-IBM Watson AI Lab, and Facebook.
+
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+
+# A CLEVR DOMAIN-SPECIFIC LANGUAGE AND IMPLEMENTATIONS
+
+We first introduce the domain-specific language (DSL) designed for the CLEVR VQA dataset (Johnson et al., 2017a). Table 6 shows the available operations in the DSL, while Table 7 explains the type system.
+
+Table 6: All operations in the domain-specific language for CLEVR VQA.   
+
+<table><tr><td>Operation</td><td> Signature</td><td>Semantics</td></tr><tr><td>Scene</td><td>O→ObjectSet</td><td>Return all objects in the scene.</td></tr><tr><td>Filter</td><td>(ObjectSet, ObjConcept) -&gt; ObjectSet</td><td>Filter out a set of objects having the object-level concept (e.g., red) from the input object set.</td></tr><tr><td>Relate</td><td>(Object, RelConcept) -→ObjectSet</td><td>Filter out a set of objects that have the relational concept (e.g.,left) with the input object.</td></tr><tr><td>AERelate</td><td>(Object, Attribute) -→ ObjectSet</td><td>(Attribute-Equality Relate） Filter out a set of objects that have the same atribute value (e.g.， same color) as the input object.</td></tr><tr><td>Intersection</td><td>(ObjectSet, ObjectSet） -→ObjectSet</td><td>Return the intersection of two ob- ject sets.</td></tr><tr><td>Union</td><td>(ObjectSet, ObjectSet) -→ ObjectSet</td><td>Return the union of two object sets.</td></tr><tr><td>Query</td><td>(Object,Attribute） -&gt;ObjConcept</td><td>Query the attribute (e.g., color) of the input object.</td></tr><tr><td>AEQuery</td><td>(Object, Object, Attribute) -→Bool</td><td>(Attribute-Equality Query） Query if two input objects have the same attribute value (e.g., same color).</td></tr><tr><td>Exist</td><td>(ObjectSet) -→Bool</td><td>Query if the set is empty.</td></tr><tr><td>Count</td><td>(ObjectSet) -→ Integer</td><td>Query the number of objects in the input set.</td></tr><tr><td>CLessThan</td><td>(ObjectSet, ObjectSet) -→Bool</td><td>(Counting LessThan) Query if the number of objects in the first input set is less than the one of the second set.</td></tr><tr><td></td><td>CGreaterThan (ObjectSet, ObjectSet) -→Bool</td><td>(Counting GreaterThan) Query if the number of objects in the first input set is greater than the one of the second set.</td></tr><tr><td>CEqual</td><td>(ObjectSet, ObjectSet) -&gt;Bool</td><td>(Counting Equal) Query if the num- ber of objects in the first input set is the same as the one of the second set.</td></tr></table>
+
+We note that some function takes Object as its input instead of ObjectSet. These functions require the uniqueness of the referral object. For example, to answer the question “What’s the color of the red object?”, there should be one and only one red object in the scene. During the program execution, the input object set will be implicitly cast to the single object (if the set is non-empty and there is only one object in the set). Such casting is named Unique in related works (Johnson et al., 2017b).
+
+Table 7: The type system of the domain-specific language for CLEVR VQA.   
+
+<table><tr><td>Type</td><td>Example</td><td>Semantics</td></tr><tr><td>ObjConcept</td><td>Red, Cube,etc.</td><td>Object-level concepts.</td></tr><tr><td>Attribute</td><td>Color, Shape,etc.</td><td>Object-level attributes.</td></tr><tr><td>RelConcept</td><td>Left,Front,etc.</td><td>Relational concepts.</td></tr><tr><td>Object</td><td>：</td><td>A single object in the scene.</td></tr><tr><td>ObjectSet</td><td>{</td><td>A set of objects in the scene.</td></tr><tr><td>Integer</td><td>0,1,2,·</td><td>A single integer.</td></tr><tr><td>Bool</td><td>True,False</td><td>A single boolean value.</td></tr></table>
+
+# B SEMANTIC PARSING
+
+As shown in Appendix A, a program can be viewed as a hierarchy of operations which take concepts as their parameters. Thus, NS-CL generates the hierarchies of latent programs in a sequence to tree manner (Dong & Lapata, 2016). The semantic parser adopts an encoder-decoder architecture, which contains four neural modules: (1) a bidirectional GRU encoder IEncoder (Cho et al., 2014) to encode an input question into a fixed-length embedding, (2) an operation decoder OpDecoder that determines the operation tokens, such as Filter, in the program based on the sentence embedding, (3) a concept decoder ConceptDecoder that selects concepts appeared in the input question as the parameters for certain operations (e.g., Filter takes an object-level concept parameter while Query takes an attribute), and (4) a set of output encoders $\{ \mathsf { O E n c o d e r } _ { i } \}$ which encode the decoded operations by OpDecoder and output the latent embedding for decoding the next operation. The operation decoder, the concept decoder, and the output encoders work jointly and recursively to generate the hierarchical program layout. Algorithm 1 illustrates the algorithmic outline of the semantic parser.
+
+# Algorithm 1: The String-to-Tree Semantic Parser.
+
+Function parse $( f , \left\{ c _ { i } \right\} )$ ): program $\gets$ EmptyProgram(); program $. o p \gets \mathsf { O p D e c o d e r } ( f )$ ; if program.op requires a concept parameter then program.concept $\gets$ ConceptDecoder $\left( f , \left\{ c _ { i } \right\} \right)$ ; for $i = 0 , 1 , \cdots$ number of non-concept inputs of program.op do $\mathsf { \ L \ p r o g r a m . i n p u t { [ i ] }  p a r s e }$ ( OEncoder $_ i ( f$ , program.op) , $\left\{ c _ { i } \right\}$ ); return program
+
+The function parse takes two inputs: the current decoding state $f$ and all concepts appeared in the question, as a set $\left\{ c _ { i } \right\}$ . The parsing procedure begins with encoding the input question by IEncoder as $f _ { 0 }$ , extracting the concept set $\left\{ c _ { i } \right\}$ from the input question, and invoking parse $\left( f _ { 0 } , \left\{ c _ { i } \right\} \right)$ .
+
+The concept set $\left\{ c _ { i } \right\}$ is extracted using hand-coded rules. We assume that each concept (including object-level concepts, relational concepts, and attributes) is associated with a single word in the question. For example, the word “red” is associated with the object-level concept Red, while the word “shape” is associated with the attribute Shape. Informally, we call these words concept words. For a given question $Q$ , the corresponding concept set $\left\{ c _ { i } \right\}$ is composed of all occurrences of the concept words in $Q$ . The set of concept words is known for the CLEVR dataset. For natural language questions, one could run POS tagging to find all concept words (Andreas et al., 2016; Schuster et al., 2015). We leave the automatic discovery of concept words as a future work (Gauthier et al., 2018). We use the word embedding of the concept words as the representation for the concepts $\left\{ c _ { i } \right\}$ . Note that, these “concept embeddings” are only for the program parsing. The visual module has separate concept embeddings for aligning object features with concepts in the visual-semantic space.
+
+We now delve into the main function $p a r s e ( f , \{ c _ { i } \} )$ : we first decode the root operation $o p$ of the hierarchy by $\mathtt { O p D e c o d e r } ( f )$ . If op requires a concept parameter (an object-level concept, a relational concept, or an attribute), ConceptDecoder will be invoked to choose a concept from all concepts $\left\{ c _ { i } \right\}$ . Assuming op takes two non-concept inputs (e.g., the operation Intersection takes two object sets as its input), there will be two branches for this root node. Thus, two output encoders OEncode $\boldsymbol { \Sigma } _ { 0 }$ and OEncoder1 will be applied to transform the current state $f$ into two sub-states $f _ { 1 }$ and $f _ { 2 }$ . parse will be recursively invoked based on $f _ { 1 }$ and $f _ { 2 }$ to generate the two branches respectively. In the DSL, the number of non-concept inputs for any operation is at most 2.
+
+In our implementation, the input encoder IEncoder first maps each word in the question into an embedding space. The word embeddings are composed of two parts: a randomly initialized word embedding of dimension 256 and a positional embedding of dimension 128 (Gehring et al., 2017). For a concept word, its word embedding only depends on which type it belongs to (i.e. object-level, relational or attribute). Thus, after being trained on a fixed dataset, the semantic parser can parse questions with novel (unseen) concept words. The sequence of word embeddings is then encoded by a two-layer GRU with a hidden dimension of $2 5 6 * 2$ (bidirectional). The function parse starts from the last hidden state of the GRU, and works recursively to generate the hierarchical program layout. Both OpDecoder and ConceptDecoder are feed-forward networks. ConceptDecoder performs attentions over the representations of all concepts $\left\{ c _ { i } \right\}$ to select the concepts. Output encoders OEncode $\boldsymbol { \Sigma } _ { 0 }$ and OEncoder1 are implemented as GRU cells.
+
+Another pre-processing of the sentence is to group consecutive object-level concept words into a group and treat them together as a single concept, inspired by the notion of “noun phrases” in natural languages. The computational intuition behind this grouping is that, the latent programs of CLEVR questions usually contain multiple consecutive Filter tokens. During the program parsing and execution, we aim to fuse all such Filters into a single Filter operation that takes multiple concepts as its parameter.
+
+A Running Example As a running example, consider again the question “What is the color of the cube right of the red matte object?”. We first process the sentence (by rules) as: “What is the <Attribute 1 (color) $>$ of the $<$ <(ObjConcept 1 (cube) $>$ <RelConcept 1 (right) $\mid >$ of the <ObjConcept 2 (red matte object) $> ? ^ { \prime }$ . The expected parsing result of this sentence is:
+
+Query( $<$ Attribute $1 >$ , Filter(<ObjConcept $1 >$ , Relate( $<$ RelConcept $1 >$ , Filter( $<$ <ObjConcept $2 >$ , Scene) ) )   
+).
+
+The semantic parser encode the word embeddings with IEncoder. The last hidden state of the GRU will be used as $f _ { 0 }$ . The word embeddings of the concept words form the set $\left\{ c _ { i } \right\} =$ {Attribute 1, ObjConcept 1, RelConcept 1, ObjConcept $2 \}$ . The function parse is then invoked recursively to generate the hierarchical program layout. Table 8 illustrates the decoding process step-by-step.
+
+# C PROGRAM EXECUTION
+
+In this section, we present the implementation of all operations listed in Table 6. We start from the implementation of Object-typed and ObjectSet-typed variables. Next, we discuss how to classify objects by object-level concepts or relational concept, followed by the implementation details of all operations.
+
+Object-typed and ObjectSet-typed variables. We consider a scene with $n$ objects. An Objecttyped variable can be represented as a vector Object of length $n$ , where $\mathrm { O b j e c t } _ { i } \in [ 0 , 1 ]$ and $\bar { \Sigma _ { i } } \mathrm { O b j e c t } _ { i } = 1$ . ${ \mathrm { O b j e c t } } _ { i }$ can be interpreted as the probability that the $i$ -th object of the scene is being referred to. Similarly, an ObjectSet-typed variable can be represented as a vector ObjectSet of length $n$ , where $\mathrm { O b j e c t S e t } _ { i } \in [ 0 , 1 ]$ . ObjectSeti can be interpreted as the probability that the $i$ -the object is in the set. To cast an ObjectSet-typed variable ObjectSet as an Object-typed variable
+
+Table 8: A step-by-step running example of the recursive parsing procedure. The parameter $\left\{ c _ { i } \right\}$ is omitted for better visualization.   
+
+<table><tr><td></td><td></td><td>StepInputsOutputs</td><td>Recursive Invocation</td></tr><tr><td rowspan="3">1</td><td rowspan="3">f</td><td>OpDecoder(fo)→ Query;</td><td rowspan="3">parse(f1)</td></tr><tr><td>ConceptDecoder(fo) →&lt; Attribute 1 &gt;;</td></tr><tr><td>OEncodero(fo,Query) →fi</td></tr><tr><td rowspan="3">2</td><td rowspan="3">f</td><td>OpDecoder(fi) →Filter;</td><td rowspan="3">parse(f2)</td></tr><tr><td>ConceptDecoder(fi) →&lt; ObjConcept 1 &gt;;</td></tr><tr><td>OEncodero(fi,Filter) →f2</td></tr><tr><td rowspan="3">3</td><td rowspan="3">f</td><td>OpDecoder(f2) →Relate;</td><td rowspan="3">parse(f3)</td></tr><tr><td>ConceptDecoder(f2) →&lt;RelConcept1 &gt;;</td></tr><tr><td>OEncodero(f2,Relate) →f3</td></tr><tr><td rowspan="3">4</td><td rowspan="3">f</td><td>OpDecoder(fs)→Filter;</td><td rowspan="3">parse(f4)</td></tr><tr><td>ConceptDecoder(fs) →&lt; ObjConcept 2 &gt;;</td></tr><tr><td>OEncodero(fs,Filter) → f4</td></tr><tr><td>5</td><td>f4</td><td>OpDecoder(f3)→Scene;</td><td>(End of branch.)</td></tr></table>
+
+Object (i.e., the Unique operation), we compute: ${ \mathrm { O b j e c t } } = \operatorname { s o f t m a x } ( \sigma ^ { - 1 } ( { \mathrm { O b j e c t S e t } } ) )$ , where $\sigma ^ { - \tilde { 1 } } ( x ) = \log ( x / ( 1 - \bar { x } ) )$ is the logit function.
+
+Concept quantization. Denote $o _ { i }$ as the visual representation of the $i$ -th object, $O C$ the set of all object-level concepts, and $A$ the set of all object-level attributes. Each object-level concept $o c$ (e.g., Red) is associated with a vector embedding $v ^ { o c }$ and a L1-normalized vector $b ^ { o c }$ of length $| { \cal { A } } | . \ b ^ { o c }$ represents which attribute does this object-level concept belong to (e.g., the concept Red belongs to the attribute $_ { \mathsf { C O 1 0 r } }$ ). All attributes $a \in A$ are implemented as neural operators, denoted as $u ^ { a }$ (e.g., $u ^ { \scriptscriptstyle \mathrm { C o l o r } }$ ). To classify the objects as being Red or not, we compute:
+
+$$
+\mathrm { P r } [ 0 \mathsf { b j e c t } i \mathrm { i s \mathsf { R e d } } ] = \sigma \left( \sum _ { a \in A } \left( b _ { a } ^ { \scriptscriptstyle \mathrm { R e d } } \cdot \frac { \langle u ^ { a } ( o _ { i } ) , v _ { \scriptscriptstyle \mathrm { R e d } } \rangle - \gamma } { \tau } \right) \right) ,
+$$
+
+where $\sigma$ denotes the Sigmoid function, $\langle \cdot , \cdot \rangle$ the cosine distance between two vectors. $\gamma$ and $\tau$ are scalar constants for scaling and shifting the values of similarities. By applying this classifier on all objects we will obtain a vector of length $n$ , denoted as ObjClassify(Red). Similarly, such classification can be done for relational concepts such as Left. This will result in an $n \times n$ matrix RelClassify(Left), where RelClassify $\left( \mathtt { I } \mathtt { e f t } \right) _ { j , i }$ is the probability that the object $i$ is left of the object $j$ .
+
+To classify whether two objects have the same attribute (e.g., have the same Color), we compute:
+
+Pr[object i has the same Color as object $j ] = \sigma \left( \frac { \langle u ^ { \mathrm { C o l o r } } ( o _ { i } ) , u ^ { \mathrm { C o l o r } } ( o _ { j } ) \rangle - \gamma } { \tau } \right) ,$
+
+We can obtain a matrix AEClassify(Color) by applying this classifier on all pairs of objects, where AEClassifier $( \mathsf { C o l o r } ) _ { j , i }$ is the probability that the object $i$ and $j$ have the same Color.
+
+Quasi-symbolic program execution. Finally, Table 9 summarizes the implementation of all operators. In practice, all probabilities are stored in the log space for better numeric stability.
+
+# D OPTIMIZATION OF THE SEMANTIC PARSER
+
+To tackle the optimization in a non-smooth program space, we apply an off-policy program search process (Sutton et al., 2000) to facilitate the learning of the semantic parser. Denote $\mathbb { P } ( s )$ as the set of all valid programs in the CLEVR DSL for the input question $s$ . We want to compute the gradient w.r.t. $\Theta _ { s }$ , the parameters of the semantic parser:
+
+$$
+\nabla _ { \Theta _ { s } } = \nabla _ { \Theta _ { s } } \mathbb { E } _ { P } [ r \cdot \log \mathrm { P r } [ P ] ] ,
+$$
+
+Table 9: All operations in the domain-specific language for CLEVR VQA. $\gamma _ { c } = 0 . 5$ and $\tau _ { c } = 0 . 2 5$ are constants for scaling and shift the probability. During inference, one can quantify all operations as Yi et al. (2018).   
+
+<table><tr><td>Signature</td><td>Implementation</td></tr><tr><td>Scene() →out: ObjectSet</td><td>outi :=1</td></tr><tr><td>Filter(in: ObjectSet,oc: ObjConcept) → out: ObjectSet</td><td>outi := min(ini,ObjClassify(oc)i)</td></tr><tr><td>Relate(in: Object,rc: RelConcept) → out:ObjectSet</td><td>outi :=∑;(inj · RelClassify(rc)j,i))</td></tr><tr><td>AERelate(in: Object, a: Attribute)→ out: ObjectSet</td><td>outi :=∑j(inj ·AEClassify(a)j,i))</td></tr><tr><td>Intersection(in(1):ObjectSet, in(2): ObjectSet) → out: ObjectSet</td><td>outi := min(in(1),in(2))</td></tr><tr><td>Union(in(1): ObjectSet,in(2): ObjectSet) → outi := max(in(1),in(2) out: ObjectSet</td><td></td></tr><tr><td>Query(in: Object, a: Attribute) → out: ObjConcept</td><td>ObjClassify(oc)i : bac Pr[out = oc] :=∑iini · ∑oc ObjClassify(oc&#x27;) · ba</td></tr><tr><td>AEQuery(in(1): Object, in(2): Object, a: Attribute) →b: Bool</td><td>b:=∑∑,(in(1) .in2) .AEClassify(@）)j))</td></tr><tr><td>Exist(in: ObjectSet) →b: Bool</td><td>b := maxi ini</td></tr><tr><td>Count(in: ObjectSet) →i: Integer</td><td>i:=∑ini</td></tr><tr><td>CLes sThan(in(1): ObjectSet, in(2): ObjectSet)→ b: Bool</td><td>b:=σ((∑in(2）-∑in(1）-1+γc)/Tc)</td></tr><tr><td>CGreaterThan(in(1): ObjectSet, in(2): ObjectSet) → b: Bool</td><td>b:=g((∑in(1)-∑in(2)-1+γc)/Tc)</td></tr><tr><td>CEqual(in(1): ObjectSet, in(2): ObjectSet)→b: Bool</td><td>b:=σ(-1∑in(1）-∑in(2)|+γc)/(γc:Te))</td></tr></table>
+
+where $P \sim \mathrm { S e m a n t i c P a r s e } ( s ; \Theta _ { s } )$ . In REINFORCE, we approximate this gradient via Monte Carlo sampling.
+
+An alternative solution is to exactly compute the gradient. Note that in the definition of the reward $r$ , only the set of programs $\mathbb { Q } ( s )$ leading to the correct answer will contribute to the gradient term. With the perception module fixed, the set $\mathbb { Q }$ can be efficiently determined by an off-policy exhaustive search of all possible programs $\mathbb { P } ( s )$ . In the third stage of the curriculum learning, we search for the set $\mathbb { Q }$ offline based on the quantified results of concept classification and compute the exact gradient $\nabla \Theta _ { s }$ . An intuitive explanation of the off-policy search is that, we enumerate all possible programs, execute them on the visual representation, and find the ones leading to the correct answer. We use $\mathbb { Q } ( s )$ as the “groundtruth” program annotation for the question, to supervise the learning, instead of running the Monte Carlo sampling-based REINFORCE.
+
+Spurious program suppression. However, directly using $\mathbb { Q } ( s )$ as the supervision by computing $\begin{array} { r } { \ell = \sum _ { p \in \mathbb { Q } ( S ) } - \log \operatorname* { P r } ( p ) } \end{array}$ can be problematic, due to the spuriousness or the ambiguity of the programs. This comes from two aspects:
+
+1) intrinsic ambiguity: two programs are different but equivalent. For example
+
+P1: AEQuery(Color, Filter(Cube), Filter(Sphere)) and P2: Exist(Filter(Sphere, AERelate(Color, Filter(Cube))))
+
+are equivalent.
+
+2) extrinsic spuriousness: one of the program is incorrect, but also leads to the correct answer in a
+
+specific scene. For example,
+
+$$
+\begin{array} { r l } & { \mathtt { P l } \colon \mathtt { F i } 1 \mathsf { t e r } \big ( \mathtt { R e d } , \mathtt { R e l } \mathsf { a t e } \big ( \mathtt { L e f t } , \mathtt { F i } 1 \mathsf { t e r } \big ( \mathtt { S p h e r e } \big ) \big ) \big ) } \\ & { \mathtt { P 2 } \colon \mathtt { F i } 1 \mathsf { t e r } \big ( \mathtt { R e d } , \mathtt { R e l } \mathsf { a t e } \big ( \mathtt { L e f t } , \mathtt { F i } 1 \mathsf { t e r } \big ( \mathtt { C u b e } \big ) \big ) \big ) } \end{array}
+$$
+
+may refer to the same red object in a specific scene. Motivated by the REINFORCE process, to suppress such spurious programs, we use the loss function:
+
+$$
+\ell = \sum _ { p \in \mathbb { Q } } \operatorname { s t o p - g r a d i e n t } ( \operatorname* { P r } [ p ] ) \cdot ( - \log \operatorname* { P r } [ p ] ) .
+$$
+
+The corresponding gradient $\nabla _ { \Theta _ { s } }$ is,
+
+$$
+\nabla \Theta _ { s } = \sum _ { p \in \Theta } \operatorname* { P r } [ p ] \cdot \nabla \Theta _ { s } \left( r \cdot \log \operatorname* { P r } [ P ] \right) = \nabla \Theta _ { s } \left( \sum _ { p \in \Theta } r \cdot \operatorname* { P r } [ p ] \right) .
+$$
+
+The key observation is that, given a sufficiently large set of scenes, a program can be identified as spurious if there exists at least one scene where the program leads to a wrong answer. As the training goes, spurious programs will get less update due to the sampling importance term $\mathrm { P r } [ p ]$ which weights the likelihood maximization term.
+
+# E CURRICULUM LEARNING SETUP
+
+During the whole training process, we gradually add more visual concepts and more complex question examples into the model. Summarized in Figure 4(A), in general, the whole training process is split into 3 stages. First, we only use questions from lesson 1 to let the model learn object-level visual concepts. Second, we train the model to parse simple questions and to learn relational concepts. In this step, we freeze the neural operators and concept embeddings of object-level concepts. Third, the model gets trained on the full question set (lesson 3), learning to understand questions of different complexities and various format. For the first several iterations in this step, we freeze the parameters in the perception modules. In addition, during the training of all stages, we gradually increase the number of objects in the scene: from 3 to 10.
+
+We select questions for each lesson in the curriculum learning by their depth of the latent program layout. For eaxmple, the program “Query(Shape, Filter(Red, Scene))” has the depth of 3, while the program “Query(Shape, Filter(Cube, Relate(Left, Filter(Red, Scene))))” has the depth of 5. Since we have fused consecutive Filter operations into a single one, the maximum depth of all programs is 9 on the CLEVR dataset. We now present the detailed split of our curriculum learning lessons:
+
+For lesson 1, we use only programs of depth 3. It contains three types of questions: querying an attribute of the object, querying the existence of a certain type of objects, count a certain type of objects, and querying if two objects have the same attribute (e.g., of the same color). These questions are almost about fundamental object-based visual concepts. For each image, we generate 5 questions of lesson 1.
+
+For lesson 2, we use programs of depth less than 5, containing a number of questions regarding relations, such as querying the attribute of an object that is left of another object. We found that in the original CLEVR dataset, all Relate operations are followed by a Filter operation. This setup degenerates the performance of the learning of relational concepts such as Left. Thus, we add a new question template into the original template set: Count(Relate( · , Filter( · Scene))) (e.g., “What’s the number of objects that are left of the cube?”). For each image, we generate 5 questions of lesson 2.
+
+For lesson 3, we use the full CLEVR question set.
+
+Curriculum learning is crucial for the learning of our neuro-symbolic concept learner. We found that by removing the curriculum setup w.r.t. the number of object in the scenes, the visual perception module will get stuck at an accuracy that is similar to a random-guess model, even if we only use stage-1 questions. If we remove the curriculum setup w.r.t. the complexity of the programs, the joint training of the visual perception module and the semantic parser can not converge.
+
+# F ABLATION STUDY
+
+We conduct ablation studies on the accuracy of semantic parsing, the impacts of the ImageNet pretraining of visual perception modules, the data efficiency of our model, and the usage of object-based representations.
+
+# F.1 SEMANTIC PARSING ACCURACY.
+
+We evaluate how well our model recovers the underlying programs of questions. Due to the intrinsic equivalence of different programs, we evaluate the accuracy of programs by executing them on the ground-truth annotations of objects. Invalid or ambiguous programs are also considered as incorrect. Our semantic parser archives $> 9 9 . 9 \%$ QA accuracy on the validation split.
+
+# F.2 IMPACTS OF THE IMAGENET PRE-TRAINING.
+
+The only extra supervision of the visual perception module comes from the pre-training of the perception modules on ImageNet (Deng et al., 2009). To quantify the influence of this pre-training, we conduct ablation experiments where we randomly initialize the perception module following He et al. (2015). The classification accuracies of the learned concepts almost remain the same except for Shape. The classification accuracy of Shape drops from 98.7 to 97.5 on the validation set while the overall QA accuracy on the CLEVR dataset drops to 98.2 from 98.9. We speculate that large-scale image recognition dataset can provide prior knowledge of shape.
+
+# F.3 DATA EFFICIENCY AND OBJECT-BASED REPRESENTATIONS
+
+In this section, we study whether and how the number of training samples and feature representations affect the overall performance of various models on the CLEVR dataset. Specifically, we compare the proposed NS-CL against two strong baselines: TbD (Mascharka et al., 2018) and MAC (Hudson & Manning, 2018).
+
+Baselines. For comparison, we implement two variants of the baseline models: TbD-Object and MAC-Object. Inspired by Anderson et al. (2018), instead of using a 2D convolutional feature map, TbD-Object and MAC-Object take a stack of object features as inputs, whose shape is $k \times d _ { o b j }$ . $k$ is the number of objects in the scene, and $d _ { o b j }$ is the feature dimension for a single object. In our experiments, we fix $k = 1 2$ as a constant value. If there are fewer than 12 objects in the scene, we add “null” objects whose features are all-zero vectors.
+
+We extract object features in the same way as NS-CL. Features are extracted from a pre-trained ResNet-34 network before the last residual block for a feature map with high resolution. For each object, its feature is composed of two parts: region-based (by RoI Align) and image-based features. We concatenate them to represent each object. As discussed, the inclusion of the representation of the full scene is essential for the inference of relative attributes such as size or spatial position on the CLEVR domain.
+
+TbD and MAC networks are originally designed to use image-level attention for reasoning. Thus, we implement two more baselines: TbD-Mask and MAC-Mask. Specifically, we replace the original attention module on images with a mask-guided attention. Denotes the union of all object masks as $M$ . Before the model applies the attention on the input image, we multiply the original attention map computed by the model with this mask $M$ . The multiplication silences the attention on pixels that are not part of any objects.
+
+Results. Table 3 summarizes the results. We found that TbD-Object and MAC-Object approach show inferior results compared with the original model. We attribute this to the design of the network architectures. Take the Relate operation (e.g., finds all objects left of a specific object $x$ ) as an example. TbD uses a stack of dilated convolutional layers to propagate the attention from object $x$ to others. In TbD-Object, we replace the stack of 2D convolutions by several 1D convolution layers, operating over the $k \times d _ { o b j }$ object features. This ignores the equivalence of objects (the order of objects should not affect the results). In contrast, MAC networks always use the attention mechanism to extract information from the image representation. This operation is invariant to the order of objects, but is not suitable for handling quantities (e.g., counting objects).
+
+As for TbD-Mask and MAC-Mask, although the mask-guided attention does not improve the overall performance, we have observed noticeably faster convergence during model training. TbD-Mask and MAC-Mask leverage the prior knowledge of object masks to facilitate the attention. Such prior has also been verified to be effective in the original TbD model: TbD employs an attention regularization during training, which encourages the model to attend to smaller regions.
+
+In general, NS-CL is more data-efficient than MAC networks and TbD. Recall that NS-CL answers questions by executing symbolic programs on the learned visual concepts. Only visual concepts (such as Red and Left) and the interpretation of questions (how to translate questions into executable programs) need to be learned from data. In contrast, both TbD and MAC networks need to additionally learn to execute (implicit or explicit) programs such as counting.
+
+For the experiments on the full CLEVR training set, we split 3,500 images $5 \%$ of the training data) as the hold-out validation set to tune the hyperparameters and select the best model. We then apply this model to the CLEVR validation split and report the testing performance. Our model reaches an accuracy of $9 9 . 2 \%$ using the CLEVR training set.
+
+# G EXTENDING TO OTHER SCENE AND LANGUAGE DOMAINS
+
+# G.1 MINECRAFT DATASET
+
+We also extend the experiments to a new reasoning testbed: Minecraft worlds (Yi et al., 2018).   
+The Minecraft reasoning dataset differs from CLEVR in both visual appearance and question types.   
+Figure 9 gives an example instance from the dataset.   
+Q: What direction is the closest creature facing?   
+A: Left.   
+P: Query(Direction, FilterMost(Closest, Filter(Creature) ))
+
+![](images/3fb31d80d725fccb01a1d6f0f5f365a0d1a3a1643e5a2731d94d6b7b9fa43a77.jpg)
+
+Figure 9: An example image and a related question-answering pair from the Minecraft dataset.
+
+Setup. Following Yi et al. (2018), we generate 10,000 Minecraft scenes using the officially opensourced tools by Wu et al. (2017). Each image contains 3 to 6 objects. The objects are chosen from 12 categories, with 4 different facing directions (front, back, left and right). They stand on a 2D plane.
+
+Besides different 3D visual appearance and image contexts, the Minecraft reasoning dataset introduces two new types of reasoning operations. We add them to our domain-specific language:
+
+1. FilterMost(ObjectSet, Concept) ObjectSet: Given a set of objects, finds the “most” one. For example, FilterMost(Closest, set) locates the object in the input set that is cloest to the camera (e.g., what is the direction of the closest animal?) 2. BelongTo(Object, ObjectSet) Bool: Query if the input object belongs to a set.
+
+Results. Table 10 summarizes the results and Figure 12 shows sample execution traces. We compare our method against the NS-VQA baseline (Yi et al., 2018), which uses strong supervision for both scene representation (e.g., object categories and positions) and program traces. In contrast, our method learns both by looking at images and reading question-answering pairs. NS-CL outperforms NS-VQA by $5 \%$ in overall accuracy. We attribute the inferior results of NS-VQA to its derendering module. Because objects in the Minecraft world usually occlude with each other, the detected object bounding boxes are inevitably noisy. During the training of the derendering module, each detected bounding box is matched with one of the ground-truth bounding boxes and uses its class and pose as supervision. Poorly localized bounding boxes lead to noisy labels and hurt the accuracy of the derendering module. This further influences the overall performance of NS-VQA.
+
+<table><tr><td>Model</td><td>Overall</td><td>Count</td><td>Exist</td><td>Belong</td><td>Query</td></tr><tr><td>NS-VQA</td><td>87.7</td><td>83.3</td><td>91.5</td><td>91.1</td><td>86.4</td></tr><tr><td>NS-CL</td><td>93.3</td><td>91.3</td><td>95.6</td><td>93.9</td><td>94.3</td></tr></table>
+
+Table 10: Our model achieves comparable results on the Minecraft dataset with baselines trained by full program annotations.
+
+# G.2 VQS DATASET
+
+We conduct experiments on the VQS dataset (Gan et al., 2017). VQS is a subset of the VQA 1.0 dataset (Antol et al., 2015). It contains questions that can be visually grounded: each question is associated with multiple image regions, annotated by humans as necessary for answering the question.
+
+Q: Does this man have any pens on him?
+
+A: Yes.   
+P: Exist(Filter(Man, Relate(Have, Filter(Pen)) ))
+
+![](images/85fbe824127773980a6930aca7fcb797319561f29dc0912c43218d7a2c961a6f.jpg)  
+Figure 10: An example image from the VQS dataset. The orange bounding boxes are object proposals. On the right, we show the original question and answer in natural language, as well as the latent program recovered by our parser. To answer this question, models are expected to attend to the man and his pen in the pocket.
+
+Setup. All models are trained on the first 63,509 images of the training set, and tested on the test split. For hyper-parameter tuning and model selection, the rest 5,000 images from the training set are used for validation. We use the multiple-choice setup for VQA: the models choose their most confident answer from 18 candidate answers for each question.
+
+To obtain the latent programs from natural languages, we use a pre-trained syntactic dependency parser (Andreas et al., 2016; Schuster et al., 2015) for extracting programs and concepts that need to be learned. A sample question and the program obtained by our parser is shown in Figure 10. The concept embeddings are initialized by the bag of words (BoW) over the GloVe word embeddings (Pennington et al., 2014).
+
+Baselines. We compare our model against two representative baselines: MLP (Jabri et al., 2016) and MAC (Hudson & Manning, 2018).
+
+MLP is a standard baseline for visual-question answering, which treats the multiple-choice task as a ranking problem. For a specific candidate answer, a multi-layer perceptron (MLP) model is used to encode a tuple of the image, the question, and the candidate answer. The MLP outputs a score for each tuple, and the answer to the question is the candidate with the highest score. We encode the image with a ResNet-34 pre-trained on ImageNet and use BoW over the GloVe word embeddings for the question and option encoding.
+
+We slightly modify the MAC network for the VQS dataset. For each candidate answer, we concatenate the question and the answer as the input to the model. The MAC model outputs a score from 0 to 1 and the answer to the question is the candidate with the highest score. The image features are extracted from the same ResNet-34 model.
+
+Results. Table 7 summarizes the results. NS-CL achieves comparable results with the MLP baseline and the MAC network designed for visual reasoning. Our model also brings transparent reasoning over natural images and language. Example execution traces generated by NS-CL are shown in Figure 13. Besides, the symbolic reasoning process helps us to inspect the model and diagnose the error sources. See the caption for details.
+
+# H VISUALIZATION OF EXECUTION TRACES AND VISUAL CONCEPTS
+
+Another appealing benefit is that our reasoning model enjoys full interpretability. Figure 11, Figure 12, and Figure 13 show NS-CL’s execution traces on CLEVR, Minecraft, and VQS, respectively. As a side product, our system detects ambiguous and invalid programs and throws out exceptions. As an example (Figure 11), the question “What’s the color of the cylinder?” can be ambiguous if there are multiple cylinders or even invalid if there are no cylinders.
+
+Figure 14 and Figure 15 include qualitative visualizations of the concepts learned from the CLEVR and Minecraft datasets, including object categories, attributes, and relations. We choose samples from the validation or test split of each dataset by generating queries of the corresponding concepts. We set a threshold to filter the returned images and objects. For quantitative evaluations of the learned concepts on the CLEVR dataset, please refer to Table 2 and Table 5.
+
+![](images/9a269c2a169535e4c1e1c3dca22a2d4c2c1feee3839ce9514a7ea2b46318498f.jpg)  
+Figure 11: Visualization of the execution trace generated by our Neuro-Symbolic Concept Learner on the CLEVR dataset. Example A and B are successful executions that generate correct answers. In example C, the execution aborts at the first operator. To inspect the reason why the execution engine fails to find the corresponding object, we can read out the visual representation of the object, and locate the error source as the misclassification of the object material. Example D shows how our symbolic execution engine can detect invalid or ambiguous programs during the execution by performing sanity checks.
+
+![](images/b8c8597133e39d31114546cdc416fa17b72f1975f4ddcc3f4d340b3b051a2703.jpg)  
+Figure 12: Exemplar execution trace generated by our Neuro-Symbolic Concept Learner on the Minecraft reasoning dataset. Example A, B and C are successful execution. Example C demonstrates the semantics of the FilterMost operation. Example D shows a failure case: the detection model fails to detect a pig hiding behind the big tree.
+
+![](images/1a090a03dcbf65d8e88296e0931f24d4866ff994df8bfc144f958e7ca4348eab.jpg)  
+Figure 13: Illustrative execution trace generated by our Neuro-Symbolic Concept Learner on the VQS dataset. Execution traces A and B shown in the figure leads to the correct answer to the question. Our model effectively learns visual concepts from data. The symbolic reasoning process brings transparent execution trace and can easily handle quantities (e.g., object counting in Example A). In Example C, although NS-CL answers the question correctly, it locates the wrong object during reasoning: a dish instead of the cake. In Example D, our model misclassifies the sport as frisbee.
+
+![](images/c8f88d5f2976254305e7cc184a2f8aa617524f48962b10a6ab16f621e80535d9.jpg)  
+Concept: Cylinder   
+Figure 14: Concepts learned on the CLEVR dataset.
+
+![](images/3da802edb0ae73b091c23f2ba2efdf8bc201843aaeda3d0974e5685c4c53a63f.jpg)  
+Figure 15: Concepts learned on the Minecraft dataset.

parse/train/yJqcM36Qvnu/yJqcM36Qvnu.md

@@ -0,0 +1,303 @@
+# Federated Graph Classification over Non-IID Graphs
+
+Han Xie, Jing Ma, Li Xiong, Carl Yang⇤ Department of Computer Science, Emory University {han.xie, jing.ma, lxiong, j.carlyang}@emory.edu
+
+# Abstract
+
+Federated learning has emerged as an important paradigm for training machine learning models in different domains. For graph-level tasks such as graph classification, graphs can also be regarded as a special type of data samples, which can be collected and stored in separate local systems. Similar to other domains, multiple local systems, each holding a small set of graphs, may benefit from collaboratively training a powerful graph mining model, such as the popular graph neural networks (GNNs). To provide more motivation towards such endeavors, we analyze real-world graphs from different domains to confirm that they indeed share certain graph properties that are statistically significant compared with random graphs. However, we also find that different sets of graphs, even from the same domain or same dataset, are non-IID regarding both graph structures and node features. To handle this, we propose a graph clustered federated learning (GCFL) framework that dynamically finds clusters of local systems based on the gradients of GNNs, and theoretically justify that such clusters can reduce the structure and feature heterogeneity among graphs owned by the local systems. Moreover, we observe the gradients of GNNs to be rather fluctuating in GCFL which impedes high-quality clustering, and design a gradient sequence-based clustering mechanism based on dynamic time warping $\mathrm { ( G C F L + ) }$ ). Extensive experimental results and in-depth analysis demonstrate the effectiveness of our proposed frameworks.
+
+# 1 Introduction
+
+Federated learning (FL) as a distributed learning paradigm that trains centralized models on decentralized data has attracted much attention recently [28, 53, 25, 18, 17]. FL allows local systems to benefit from each other while keeping their own data private. Especially, for local systems with scarce training data or lack of diverse distributions, FL provides them with the potentiality to leverage the power of data from others, in order to facilitate the performance on their own local tasks. One important problem FL concerns is data distribution heterogeneity, since the decentralized data, collected by different institutes using different methods and aiming at different tasks, are highly likely to follow non-identical distributions. Prior works approach this problem from different aspects, including optimization process [25, 18], personalized FL [13, 6, 8], clustered FL [9, 15, 2], etc.
+
+As more advanced techniques are developed for learning with graph data, using graphs to model and solve real-world problems becomes more popular. One important scenario of graph learning is graph classification, where models such as graph kernels [44, 34, 36, 45] and graph neural networks [21, 43, 49, 46, 47, 48] are used to predict graph-level labels based on the features and structures of graphs. One real scenario of graph classification is molecular property prediction, which is an important task in cheminformatics and AI medicine. In the area of bioinformatics, graph classification can be used to learn the representation of proteins and classify them into enzymes or non-enzymes. For collaboration networks, sub-networks can be classified regarding the information of research areas, topics, genre, etc. More applicable scenarios include geographic networks, temporal networks, etc.
+
+Since the key idea of $\mathrm { F L }$ is the sharing of underlying common information, as $\mathbb { \left[ \left[ 2 3 \right] \right] }$ discusses that real-world graphs preserve many common properties, we become curious about the question, whether real-world graphs from heterogeneous sources (e.g., different datasets or even divergent domains) can provide useful common information among each other? To understand this question, we first conduct preliminary data analysis to explore real-world graph properties, and try to find clues about common patterns shared among graphs across datasets. As shown in Table $\bigstar$ we analyze four typical datasets from different domains, i.e., PTC_MR (molecular structures), ENZYMES (protein structures), IMDB-BINARY (social communities), and MSRC_21 (superpixel networks). We find them to indeed share certain properties that are statistically significant compared to random graphs with the same numbers of nodes and links (generated with the Erdos–Rényi model [ ˝ 7, 10]). Such observations confirm the claim about common patterns underlying real-world graphs, which can largely influence the graph mining models and motivates us to consider the FL of graph classification across datasets and even domains. More details and discussion about Table 1 can be found in Appendix A.
+
+Table 1: Data analysis on important graph properties shared among real-world graphs across different domains. For example, large Kurtosis values $\textcircled { 1 3 2 } \textcircled { 1 }$ indicate long-tail distribution of node degrees, which is observed in ENZYMES, IMDB-BINARY, and MSRC_21; similar average shortest path lengths are observed in PTC_MR, ENZYMES, and MSRC_21, although their actual graph sizes are rather different; large CC are observed in ENZYMES, IMDB-BINARY, and MSRC_21 and large LC are observed in almost all graphs.   
+
+<table><tr><td>Property</td><td colspan="3">kurtosis of degree distribution</td><td colspan="3">avg. shortest path length</td><td colspan="3">largest component size (LC,%)</td><td colspan="3">clustering coefficient (CC)</td></tr><tr><td></td><td>real</td><td>random</td><td>p-value</td><td>real</td><td>random</td><td>p-value</td><td>real</td><td>random</td><td>p-value</td><td>real</td><td>random</td><td>p-value</td></tr><tr><td>PTC_MR (molecules)</td><td>2.1535</td><td>2.4424</td><td>0.9999</td><td>3.36</td><td>2.42</td><td>~0</td><td>100</td><td>82.68</td><td>~0</td><td>0.0095</td><td>0.1201</td><td>~0</td></tr><tr><td>ENZYMES (proteins)</td><td>3.0106</td><td>2.8243</td><td>0.0027</td><td>4.44</td><td>2.56</td><td>~0</td><td>98.24</td><td>97.69</td><td>0.2054</td><td>0.4516</td><td>0.1425</td><td>~0</td></tr><tr><td>IMDB-BINARY (social)</td><td>8.9262</td><td>2.2791</td><td>~0</td><td>1.48</td><td>1.54</td><td>~0</td><td>100</td><td>99.93</td><td>0.0023</td><td>0.9471</td><td>0.5187</td><td>~0</td></tr><tr><td>MSRC_21 (superpixel)</td><td>3.6959</td><td>2.9714</td><td>~0</td><td>4.09</td><td>2.81</td><td>~0</td><td>100</td><td>99.43</td><td>~0</td><td>0.5147</td><td>0.0655</td><td>~0</td></tr></table>
+
+Although common patterns exist among graph datasets, we can still observe certain heterogeneity. In fact, the detailed graph structure distributions and node feature distributions can both diverge due to various reasons. To demonstrate this, we design and evaluate a structure heterogeneity measure and a feature heterogeneity measure in different scenarios (c.f. Section $\boxed { 4 . 1 }$ . We refer to the graphs possibly with significant heterogeneity in our cross-dataset FL setting as non-IID graphs, which concerns both structure non-IID and feature non-IID, where naïve FL algorithms like FedAvg $\left. \boldsymbol { \widetilde { 2 8 } } \right.$ can fail and even backfire (c.f. Section $6 . 2 )$ . Moreover, as the heterogeneity varies from case to case, a dynamic FL algorithm is needed to keep track of such heterogeneity of non-IID graphs while conducting collaborative model training.
+
+Due to the observations that the graphs in one client can be similar to those in some clients but not the others, we get motivated by $\bar { \mathbb { D } }$ and find it intuitive to consider a clustered FL framework, which assigns local clients to multiple clusters with less data heterogeneity. To this end, we propose a novel graph-level clustered FL framework (termed GCFL) through integrating the powerful graph neural networks (GNNs) such as GIN $\mathbb { \lVert \underline { { 4 3 } } \rVert }$ into clustered FL, where the server can dynamically cluster the clients based on the gradients of GNN without additional prior knowledge, while collaboratively training multiple GNNs as necessary for homogeneous clusters of clients. We theoretically analyze that the model parameters of GNN indeed reflect the structures and features of graphs, and thus using the gradients of GNN for clustering in principle can yield clusters with reduced heterogeneity of both structures and features. In addition, we conduct empirical analysis to support the motivation of clustered FL across heterogeneous graph datasets in Appendix A.
+
+Although GCFL can theoretically achieve homogeneous clusters, during its training, we observe that the gradients transmitted at each communication round fluctuate a lot (c.f. Section $\bar { 5 } . 1 )$ , which could be caused by the complicated interactions among clients regarding both structure and feature heterogeneity, making the local gradients towards divergent directions. In the vanilla GCFL framework, the server calculates a matrix for clustering only based on the last transmitted gradients, which ignores the client’s multi-round behaviors. Therefore, we further propose an improved version of GCFL with gradient-series-based clustering (termed ${ \mathrm { G C F L } } +$ ).
+
+We conduct extensive experiments with various settings to demonstrate the effectiveness of our frameworks. Moreover, we provide in-depth analysis on the capability of them on reducing both structure and feature heterogeneity of clients through clustering. Lastly, we analyze the convergence of our frameworks. The experimental results show surprisingly positive results brought by our novel setting of cross-dataset/cross-domain FL for graph classification, where our ${ \mathrm { G C F L } } +$ framework can effectively and consistently outperform other straightforward baselines.
+
+# 2 Related works
+
+Federated Learning Federated learning (FL) has gained increasing attention as a training paradigm under the setting where data are distributed at remote devices and models are collaboratively trained under the coordination of a central server. FedAvg was first proposed by $ { \mathbb { \left[ \left[ 2 7 \right] \right] } }$ which illustrates the general setting of an FL framework. Since the original FedAvg relies on the optimization by SGD, data of non-IID distribution will not guarantee the stochastic gradients to be an unbiased estimation of the full gradients, thus hurting the convergence of FL. In fact, multiple experiments $\pm \Sigma \boxed { 2 5 } \boxed { 1 8 }$ have shown that the convergence will be slow and unstable, and the accuracy will degrade with FedAvg when data at each client are statistically heterogeneous (non-IID). [53, 15, 12] proposed different data sharing strategies to tackle the data heterogeneity problem by sharing the local device data or serverside proxy data, which still requires certain public common data, whereas other studies explored the convergence guarantee under the non-IID setting by assuming bounded gradients $\textcircled { 1 3 9 } , \textcircled { 5 1 }$ or additional noise $\mathbb { \lVert \rVert }$ . There are also works seeking to reduce the variance of the clients $\mathbb { P } \bot \bot \bot \bot \bot \bot $ . Furthermore, multiple works have been proposed to explore the connection between model-agnostic meta-learning (MAML) and personalized federated learning [8, 4]. They aim to learn a generalizable global model and then fine-tune it on local clients, which may still fail when data on local clients are from divergent domains with high heterogeneity. Some personalized FL works $\mathbb { H } \mathbb { L } \mathbb { 4 } \mathbb { I }$ studied the bi-level problem of optimization which decouples the local and global optimization, but having each client maintain its own model can lead to high communication cost among the clients and the server. While personalization in the FL setting can address the client heterogeneity problem to some extent, the clustered FL framework [33] can incorporate personalization at the group level to keep the benefits of personalized FL and reduce the communication cost simultaneously.
+
+Federated Learning on Graphs Although FL has been intensively studied with Euclidean data such as images, there exist few studies about FL for graph data. $\lVert 2 2 \rVert$ first introduced FL on graph data, by regarding each client as a node in a graph. [3] studied the cross-domain heterogeneity problem in FL by leveraging Graph Convolutional Networks (GCNs) to model the interaction between domains. $\mathbb { \left. 2 9 \right. }$ studied spatio-temporal data modeling in the FL setting by leveraging a Graph Neural Network (GNN) based model to capture the spatial relationship among clients. [5] proposed a generalized federated knowledge graph embedding framework that can be applied for multiple knowledge graph embedding algorithms. Moreover, there are several works exploring the GNNs under the FL setting: [16, 54, 40] focused on the privacy issue of federated GNNs; $\bar { \big [ } \bar { \big | } 3 7 \big | \big ]$ incorporated model-agnostic metalearning (MAML) into graph FL, which handled non-IID graph data while also preserving the model generalizability; $\pmb { \Vert 5 2 \Vert }$ studied the missing neighbor generation problem in the subgraph FL setting; $ { \bar { \mathbf { \delta } } }$ proposed a computationally efficient way of GCN architecture search with FL; $\bar { \mathbb { m } } ^ { \bar { | } }$ implemented an open FL benchmark system for GNNs. Most existing works consider node classification and link prediction on graphs, which cannot be trivially applied to our graph classification setting.
+
+# 3 Preliminaries
+
+# 3.1 Graph Neural Networks (GNNs)
+
+$\lVert \rVert 2 \rVert$ provides a taxonomy that categorizes Graph Neural Networks (GNNs) into recurrent GNNs, convolutional GNNs, graph autoencoders, and spatial-temporal GNNs. In general, given the structure and feature information of a graph $G = ( V , E , X )$ , where $V$ , $E$ , $X$ denote nodes, links and node features, GNNs target to learn the representations of graphs, such as a node embedding $h _ { v } \in \mathbb { R } ^ { d _ { v } }$ , or a graph embedding $\boldsymbol { h } _ { G } \in \mathbb { R } ^ { d _ { G } }$ . A GNN typically consists of message propagation and neighborhood aggregation, in which each node iteratively gathers the information propagated by its neighbors, and aggregates them with its own information to update its representation. Generally, an $L$ -layer GNN can be formulated as
+
+$$
+h _ { v } ^ { ( l + 1 ) } = \sigma ( h _ { v } ^ { ( l ) } , a g g ( \{ h _ { u } ^ { ( l ) } ; u \in \mathcal { N } _ { v } \} ) ) , \forall l \in [ L ] ,
+$$
+
+where $h _ { v } ^ { ( l ) }$ is the representation of node $v$ at the $l ^ { t h }$ layer, and $h _ { v } ^ { ( 0 ) } = x _ { v }$ is the node feature. $\mathcal { N } _ { v }$ is neighbors of node $v$ , $a g g ( \cdot )$ is an aggregation function that can vary for different GNN variants, and $\sigma$ represents an activation function.
+
+For a graph-level representation $h _ { G }$ , it can be pooled from the representations of all nodes, as
+
+$$
+h _ { G } = r e a d o u t ( \{ h _ { v } ; v \in V \} ) ,
+$$
+
+where readout $\left( \cdot \right)$ can be implemented as mean pooling, sum pooling, etc, which essentially aggregates the embeddings of all nodes on the graph into a single embedding vector to achieve tasks like graph classification and regression.
+
+# 3.2 The FedAvg algorithm
+
+McMahan et al. $\left[ \left[ 2 8 \right] \right]$ proposed an SGD-based aggregating algorithm, FedAvg, based on the fact that SGD is widely used and powerful for optimization. FedAvg is the first basic FL algorithm and is commonly used as the starting point for more advance FL framework design.
+
+The key idea of FedAvg is to aggregate the updated model parameters transmitted from local clients and then re-distribute the averaged parameters back to each client. Specifically, given $m$ clients in total, at each communication round $t$ , the server first samples a partition of clients $\{ \mathbb { S } _ { i } \} ^ { ( t ) }$ . For each client $\mathbb { S } _ { i }$ in $\{ \mathbb { S } _ { i } \} ^ { ( t ) }$ , it trains the model downloaded from the server locally with its own data distribution $\mathcal { D } _ { i }$ for $E _ { l o c a l }$ epochs. The client $\mathbb { S } _ { i }$ then transmits its updated parameters $w _ { i } ^ { ( t ) }$ to the server, and the server will aggregate these updates by
+
+$$
+w ^ { ( t + 1 ) } = \sum _ { i = 1 } ^ { m } \frac { | D _ { i } | } { | D | } w _ { i } ^ { ( t ) } ,
+$$
+
+where $| D _ { i } |$ is the size of data samples in client $\mathbb { S } _ { i }$ and $| D |$ is the total size of samples over all clients. After generating the aggregated parameters (the global model updates), the server broadcasts the new parameters $w ^ { ( t + 1 ) }$ to remote clients, and at the $( t + 1 )$ round clients use $w ^ { ( t + 1 ) }$ to start their local training for another $E _ { l o c a l }$ epochs.
+
+# 4 The GCFL framework
+
+# 4.1 Non-IID structures and features across clients
+
+From Table 1 we notice that real-world graphs tend to share certain general properties across different graphs, datasets and even domains, which motivates the graph-level FL framework. However, there still exist differences when the detailed graph structures and node features are being considered. In Table $^ { 2 , }$ we present the average pair-wise structure heterogeneity and feature heterogeneity among graphs in a single dataset, a single domain, and across different domains. Specifically, for structure heterogeneity, we use the Anonymous Walk Embeddings (AWEs) $[ \left| 1 4 \right| ]$ to generate a representation for each graph, and compute the Jensen-Shannon distance between the AWEs of each pair of graphs; for feature heterogeneity, we calculate the empirical distribution of feature similarity between all pairs of linked nodes in each graph, and compute the Jensen-Shannon divergence between the feature similarity distributions of each pair of graphs.
+
+As we can observe in Table $\boxed { 2 } ,$ both graph structures and features demonstrate different levels of heterogeneity within a single dataset, a single domain, and across different domains. We refer to graphs with such structure and feature heterogeneity as non-IID graphs. Intuitively, directly applying naïve FL algorithms like FedAvg on clients with non-IID graphs can be ineffective and even backfiring. To be specific, structure heterogeneity makes it difficult for a model to capture the universally important graph structure patterns across different clients, whereas feature heterogeneity makes it hard for a model to learn the universally appropriate message propagation functions across different clients. How can we leverage the shared graph properties among clients while addressing the non-IID structures and features across clients?
+
+# 4.2 Problem formulation
+
+Motivated by our real graph data analysis in Tables 1 and 2, we propose a novel framework of Graph Clustered Federated Learning (GCFL). The main idea of GCFL is to jointly find clusters of clients with graphs of similar structures and features, and train the graph mining models with FedAvg among clients in the same clusters.
+
+Specifically, we are inspired by the Clustered Federated Learning (CFL) framework on Euclidean data $\mathbb { \lVert 3 3 \rVert }$ and consider a clustered FL setting with one central server and a set of $n$ local clients $\{ \mathbb { S } _ { 1 } , \mathbb { S } _ { 2 } , \ldots , \mathbb { S } _ { n } \}$ . Different from the traditional FL setting, the server can dynamically cluster the clients into a set of clusters $\{ \mathbb { C } _ { 1 } , \mathbb { C } _ { 2 } , \ldots \}$ and maintain $m$ cluster-wise models. In our GCFL setting, each local client $\mathbb { S } _ { i }$ owns a set of graphs $\mathcal { \bar { G } } _ { i } = \{ G _ { 1 } , G _ { 2 } , . . . \}$ , where each $G _ { j } = ( V _ { j } , E _ { j } , X _ { j } , y _ { j } ) \in \bar { \mathcal { G } } _ { i }$ is a graph data sample with a set of nodes $V _ { j }$ , a set of edges $E _ { j }$ , node features $X _ { j }$ , and a graph class label $y _ { j }$ . The task on each local client $\mathbb { S } _ { i }$ is graph classification that predicts the class label $\hat { y } _ { j } ~ = ~ h _ { k } ^ { * } ( G _ { j } )$ for each graph $G _ { j } ~ \in ~ \mathcal { G } _ { i }$ , where $h _ { k } ^ { * }$ is the collaboratively learned optimal graph mining model for cluster $\mathbb { C } _ { k }$ to which $\mathbb { S } _ { i }$ belongs. Our goal is to minimize the loss function $F ( \Theta _ { k } ) : = \operatorname { E } _ { \mathbb { S } _ { i } \in \mathbb { C } _ { k } } [ f ( \theta _ { k , i } ; \mathcal { G } _ { i } ) ]$ , for all clusters $\{ \mathbb { C } _ { k } \bar  \}$ . The function $f ( \theta _ { k , i } ; \mathcal { G } _ { i } )$ is a local loss function for client $\mathbb { S } _ { i }$ which belongs to cluster $\mathbb { C } _ { k }$ . In the meantime, we also aim to maintain a dynamic cluster assignment $\Gamma ( \mathbb { S } _ { i } ) \to \{ \mathbb { C } _ { k } \}$ based on the $\mathrm { F L }$ process.
+
+Table 2: Summary of the average heterogeneity of features and structures for some datasets. In general, the structure heterogeneity increases from the settings of one dataset to across-dataset, and to across-domain. However, the feature heterogeneity is more case-by-case, and the high variances indicate that graphs could have large feature divergence even within the same dataset. Additionally, it is not necessarily true that one dataset itself should be more homogeneous (e.g., IMDB-BINARY).   
+
+<table><tr><td>dataset</td><td>IMDB-BINARY (social)</td><td>CoX2 (molecules)</td><td>Cox2 (molecules) PTC_MR (molecules)</td><td>Cox2 (molecules) ENZYMES (proteins)</td><td>Cox2 (molecules) IMDB-BINARY (sOCial)</td></tr><tr><td>avg. struc. hetero.</td><td>0.4406 (±0.0397)</td><td>0.3246 (±0.0145)</td><td>0.3689 (±0.0540)</td><td>0.5082 (±0.0399)</td><td>0.6079 (±0.0331)</td></tr><tr><td>avg. feat. hetero.</td><td>0.1785 (±0.1226)</td><td>0.0427 (±0.0314)</td><td>0.1837(±0.1065)</td><td>0.1912 (±0.1000)</td><td>0.1642 (±0.1006)</td></tr></table>
+
+# 4.3 Technical design
+
+GNNs are demonstrated to be powerful for learning graph representations and have been wildly used in graph mining. More importantly, the model parameters and their gradients of GNNs can reflect the graph structure and feature information (more details in Section $\bar { 4 . 4 ) }$ . Thus, we use GNNs as the graph mining model in our GCFL framework.
+
+Specifically, our GCFL framework can dynamically cluster clients by leveraging their transmitted gradients $\{ \Delta \theta _ { i } \} _ { i = 1 } ^ { n }$ , in order to maximize the collaboration among more homogeneous clients and eliminate the harm from heterogeneous clients. According to $\mathbb { B } 3 \mathbb { I }$ , if the data distribution of clients are highly heterogeneous, the general FL that trains the clients together cannot jointly optimize all their local loss functions. In this case, after some rounds of communication, the general FL will be close to the stationary point, and the norm of clients’ transmitted gradients will not all tend towards zero. Therefore, clustering clients is needed as the general FL approaches to the stationary point. Here, we first introduce a hyper-parameter $\varepsilon _ { 1 }$ as a criterion to decide whether to stop the general FL based on whether a stationary point is approached, that is,
+
+$$
+\delta _ { m e a n } = \| \sum _ { i \in [ n ] } \frac { | \mathcal { G } _ { i } | } { | \mathcal { G } | } \Delta \theta _ { i } \| < \varepsilon _ { 1 } .
+$$
+
+In the meantime, if there exist some clients still with large norms of transmitted gradients, it means that clients in the group are highly heterogeneous, and thus clustering is needed to eliminate the negative influence among them. We then introduce the second criterion with a hyper-parameter $\varepsilon _ { 2 }$ to split the clusters when
+
+$$
+\delta _ { m a x } = \operatorname* { m a x } ( \| \Delta \theta _ { i } \| ) > \varepsilon _ { 2 } > 0 .
+$$
+
+The GCFL framework follows a top-down bi-partitioning mechanism. At each communication round $t$ , the server receives $m$ sets of gradients $\{ \{ \hat { \Delta } \theta _ { i _ { 1 } } \} , \{ \tilde { \Delta \theta } _ { i _ { 2 } } \} , \dots , \{ \Delta \theta _ { i _ { m } } \} \}$ from clients in clusters $\{ \mathbb { C } _ { 1 } , \mathbb { C } _ { 2 } , \ldots , \mathbb { C } _ { m } \}$ . For a cluster $\mathbb { C } _ { k }$ , if $\delta _ { m e a n } ^ { k }$ and $\delta _ { m a x } ^ { k }$ satisfy the Eqs. 4 and $\boxed { 5 } ,$ the server will calculate a cluster-wise cosine similarity matrix $\alpha _ { k }$ , and its entries are used as weights for building a full-connected graph with nodes being all clients within the cluster. The Stoer–Wagner minimum cut algorithm $\pmb { \Vert 3 5 \Vert }$ is then applied to the constructed graph, which bi-partitions the graph and divides the cluster $\mathbb { C } _ { k } \to \{ \mathbb { C } _ { k 1 } , \mathbb { C } _ { k 2 } \}$ . The clustering mechanism based on Eqs. $\boxed { 4 }$ and $\textcircled { 5 }$ can automatically and dynamically determine the number of clusters along the FL, while the two hyper-parameters $\varepsilon _ { 1 }$ and $\varepsilon _ { 2 }$ can be easily set through some simple experiments on the validation sets following $\mathbb { \lVert 3 3 \rVert }$ .
+
+For a client $\mathbb { S } _ { i }$ in cluster $\mathbb { C } _ { k }$ , it tries to find $\widehat { \theta } _ { k , i }$ that is close to the real solution $\theta _ { k , i } ^ { * } ~ =$ $\mathrm { a r g m i n } _ { \theta _ { i } \in \Theta _ { k } } f ( \theta _ { k , i } ; \mathcal { G } _ { i } )$ . At a communication round $t$ , the client $\mathbb { S } _ { k }$ transmits its gradient to the server
+
+$$
+\Delta \theta _ { k , i } ^ { t } = \hat { \theta } _ { k , i } ^ { t } - \theta _ { k , i } ^ { t - 1 } .
+$$
+
+Since the server maintains the cluster assignments, it can aggregate the gradients cluster-wise by
+
+$$
+\theta _ { k } ^ { t + 1 } = \theta _ { k } ^ { t } + \sum _ { i \in [ n _ { k } ] } \Delta \theta _ { k , i } ^ { t } .
+$$
+
+# 4.4 Theoretical analysis
+
+We investigate the problem of graph FL with multi-domain data distribution, and use the gradientbased FL paradigm $ { \mathbb { \left[ \left[ 2 7 \right] \right] } }$ to facilitate the model training. We theoretically analyze that the gradientbased FL algorithm on GNNs can in principle reduce the structure and feature heterogeneity in clusters, along with the task difference between data from different domains. We study two general problems in order to prove that the gradients can reflect the feature, structure, and task information.
+
+Definition 4.1 Let a function $f : \mathcal { X }  \mathcal { Y }$ which maps from the metric space $( \mathcal { X } , d )$ to $( \boldsymbol { \mathcal { V } } , \boldsymbol { d ^ { \prime } } )$ , the function $f$ is considered to have $\delta$ distortion $i f \forall u , v \in \mathcal { X }$ , ${ \textstyle \frac { 1 } { \delta } } d ( u , v ) \leq d ^ { \prime } ( f ( u ) , f ( v ) ) \leq d ( u , \dot { v } )$ .
+
+Theorem 4.1 (Bourgain theorem $\pmb { I I I }$ ) Given an $n$ -point metric space $( \mathcal { X } , d )$ and an embedding function $f$ as defined above, $\forall u , v \in \mathcal { X }$ , there exist an embedding mapped from $( \mathcal { X } , d )$ to $\mathbb { R } ^ { k }$ with the distortion of the embedding being $O ( \log n )$ .
+
+Problem 1. In GCFL which involves the communication of the gradients between graphs with heterogeneous structures distributed among different clients, the structure and feature difference can be captured by the GNN gradients.
+
+For simplicity, we solve Problem 1 with the GNN of Simple Graph Convolutions (SGC) [41], through the following two propositions.
+
+Proposition 4.1 Given a graph $G$ with fixed structure represented by the normalized graph Laplacian $\mathcal { L } = \widetilde { D } ^ { - \frac { 1 } { 2 } } \widetilde { A } \widetilde { D } ^ { - \frac { 1 } { 2 } }$ , feature represented with $X$ , and an SGC $f ( \mathcal { L } , X ) = s o f t m a x ( \mathcal { L } ^ { K } X \Theta )$ with weights $\Theta$ trained on graph $G$ . If we have another graph $G ^ { \prime }$ with different structure $\mathcal { L } ^ { \prime }$ , the weight difference $| | \Theta ^ { \prime } - \Theta | | _ { 2 }$ is bounded with the structure difference.
+
+Proposition 4.2 Given a graph $G$ with fixed structure represented by the normalized graph Laplacian $\mathcal { L } = \widetilde { D } ^ { - \frac { 1 } { 2 } } \widetilde { A } \widetilde { D } ^ { - \frac { 1 } { 2 } }$ , feature represented with $X$ , and an SGC $f ( \mathcal { L } , X ) = s o f t m a x ( \mathcal { L } ^ { K } X \Theta )$ with weights $\Theta$ trained on graph $G$ . If we have another graph $G ^ { \prime }$ with different feature $\mathcal { X } ^ { \prime }$ , the weight difference $| | \Theta ^ { \prime } - \Theta | | _ { 2 }$ is bounded with the feature difference.
+
+We prove proposition 4.1 and $\boxed { 4 . 2 }$ in Appendix B. We use the Bourgain theorem to bound the difference between embeddings generated with different graph structures/features, and prove that the feature and structure information of a graph is incorporated into the model weights (gradients). By proving that the model weights (gradients) are bounded with the structure/feature difference, we show that the gradients will change with the structure and feature. This further justifies that our proposed gradient based clustering framework GCFL is able to capture the structure and feature information. In addition, we also study the following problem, which allows our GCFL framework to be further extended to cross-task graph-level federated learning in the future.
+
+Problem 2. The communicated gradients in GCFL can also capture the task heterogeneity.
+
+Proposition 4.3 Given a graph $G$ with structure represented by the normalized graph Laplacian $\mathcal { L } = \widetilde { D } ^ { - \frac { 1 } { 2 } } \widetilde { A } \widetilde { D } ^ { - \frac { 1 } { 2 } }$ , and feature represented with $X$ , if trained with different tasks, we will get the Simple SGC with bounded weights.
+
+The proof of proposition $4 . 3$ can be found in Appendix B.
+
+# 5 $\mathrm { G C F L + : }$ improved GCFL based on observation sequences of gradients
+
+# 5.1 Fluctuation of gradient norms
+
+When observing the norm of gradients for each communication round in GCFL, as shown in Figure 1, we notice that: 1) the norm of gradients continuously fluctuates; 2) different clients can have divergent scales of gradient norms. The fluctuation of gradient norms and different scales indicate that the updating directions and distances of gradients for clients are diverse, which manifests the structure and feature heterogeneity in our setting again. In our vanilla GCFL framework, the server calculates a cosine similarity matrix based on the last transmitted gradients once the clustering criteria are satisfied. However, with the observation that the norm of gradients fluctuates along the communication round, albeit with the constraints of clustering criteria, GCFL clustering based on gradient-point could omit important client behaviors and be misled by noises. For example, in Figure $^ 1$ (a), GCFL performs clustering at round 119 based on the gradients at that round, which does not effectively find graphs with lower heterogeneity.
+
+![](images/43e3be229661b6ca6d5ff130859c4fcb00ca0059a1b30b42311269fedc051031.jpg)  
+Figure 1: Norm of gradients versus communication round with six clients across datasets. Clients with datasets colored the same are split to the same cluster.
+
+# 5.2 Technical design
+
+Motivated by these observations, we propose an improved version of GCFL, named ${ \mathrm { G C F L } } +$ , which conducts clustering by taking series of gradient norms into consideration. In the ${ \mathrm { G C F L } } +$ framework, the server maintains a multi-variant time-series matrix $Q \in \mathbb { R } ^ { \{ n , d \} }$ , where $n$ is the number of clients and $d$ is the length of a gradient series being tracked. At each communication round $t$ , the server updates $Q$ by adding in the norm of gradients $\| \Delta \theta _ { i } ^ { t } \|$ to $Q ( i , : ) \in \mathbb { R } ^ { d }$ and remove the out-of-date one. ${ \mathrm { G C F L } } +$ uses the same clustering criteria as GCFL (Eqs. 4 and 5). If the clustering criteria are satisfied, the server will calculate a distance matrix $\beta$ in which each cell is the pair-wise distance of two series of gradients. Here, we use a technique called dynamic time warping (DTW) $\textcircled { \scriptsize { 1 3 1 } }$ t o measure the similarity between two data sequences. For a cluster $\mathbb { C } _ { k }$ , the server calculates its distance matrix as
+
+$$
+\beta _ { k } ( p , q ) = d i s t ( Q ( p , : ) , Q ( q , : ) ) , p , q \in i d x ( \{ \mathbb { S } _ { i } \} ) ,
+$$
+
+where $i d x ( \{ \mathbb { S } _ { i } \} )$ is the indices of all clients $\{ \mathbb { S } _ { i } \}$ in cluster $\mathbb { C } _ { k }$ . With the distance matrix $\beta$ , the server can perform bi-partitioning for clusters who meet the clustering criteria. As a result, in Figure ${ \bf 1 } \left( { \bf b } \right)$ ${ \mathrm { G C F L } } +$ performs clustering at round 118 based on the gradient sequence of length 10, which captures the longer-range behaviors of clients and effectively more homogeneous clusters.
+
+# 6 Experiments
+
+# 6.1 Experimental settings
+
+Datasets We use a total of 13 graph classification datasets $\pmb { \mathbb { B } } 0 \|$ from three domains including seven molecule datasets (MUTAG, BZR, COX2, DHFR, PTC_MR, AIDS, NCI1), three protein datasets (ENZYMES, DD, PROTEINS), and three social network datasets (COLLAB, IMDB-BINARY, IMDBMULTI), each with a set of graphs. Node features are available in some datasets, and graph labels are either binary or multi-class. Details of the datasets are presented in Appendix C.
+
+We design two settings that follow different data partitioning mechanisms, and the example real scenarios of the two settings can be found in Appendix $\boxed { \mathrm { A } }$ The first setting (i.e., single-dataset) is to randomly distribute graphs from a single dataset to a number of clients, with each client holding a distinct set of about 100 graphs, among which $10 \%$ are held out for testing. In the second setting (i.e., multi-dataset), we use multiple datasets either from a single domain or multiple domains. Each client holds a graph dataset, among which $10 \%$ are held out for testing. In the first setting, we use NCI1, PROTEINS, and IMDB-BINARY from three domains and distribute them to 30, 10, 10 clients, respectively. In the second setting, we create three data groups including MOLECULES which consists of seven datasets from the molecule domain distributed into seven clients, BIOCHEM where we add three datasets from the protein domain into MOLECULES and distribute them into 10 clients, MIX where we add three datasets from the social domain into BIOCHEM and distribute them into 13 clients.
+
+Baselines We use self-train2 as the first baseline to test whether FL can bring improvements to each client through collaborative training. In self-train, each client firstly downloads the same randomly initialized model from the server and then trains locally without any communications. Then we implement two widely used FL baselines FedAvg $[ [ 2 7 ] ]$ and FedProx $\vec { \mathbb { B } } \vec { \mathsf { S } } \vec { \mathbb { I } }$ , the latter of which can deal with data and system heterogeneity in non-graph FL. For the graph classification model, we use the same GIN $\mathbb { \lVert \boldsymbol { 4 3 } \rVert }$ design, which represents the state-of-the-art GNN for graph-level tasks. We fix the GIN architecture and hyper-parameters through all baselines in order to control the experiments across different settings.
+
+Parameter settings We use the three-layer GINs with hidden size of 64. We use a batch size of 128, and an Adam $\mathbb { \ m }$ optimizer with learning rate 0.001 and weight decay $5 e ^ { - 4 }$ . The $\mu$ for FedProx is set to 0.01. For all FL methods, the local epoch $E$ is set to 1. The two important hyper-parameters $\varepsilon _ { 1 }$ and $\varepsilon _ { 2 }$ as clustering criteria vary in different groups of data, which are set through offline training for about 50 rounds following $\pmb { \mathbb { B 3 } } \|$ . We run all experiments for five random repetitions on a server with 8 24GB NVIDIA TITAN RTX GPUs.
+
+# 6.2 Experimental results
+
+Federated graph classification within single datasets Conceptually, clients in this setting are more homogeneous. As can be seen from the results in Table ${ \bar { 3 } } ,$ our framework can obviously improve the performance of graph classification over local clients. For the NCI1 dataset distributed on 30 clients, GCFL and ${ \mathrm { G C F L } } +$ achieve $1 3 . 2 7 \%$ and $1 4 . 7 5 \%$ performance gains over self-train on average, and GCFL and ${ \mathrm { G C F L } } +$ help 10-14 more clients than FedAvg and FedProx who fail to improve about half of clients. For the PROTEINS dataset on the total 10 clients, GCFL and ${ \mathrm { G C F L } } +$ achieve $7 . 2 9 \%$ and $7 . 8 1 \%$ average performance gains compared to self-train. For IMDB-BINARY on 10 clients, FedAvg and FedProx fail to help 5/10 and 4/10 clients respectively, while both GCFL and ${ \mathrm { G C F L } } +$ are able to improve all 10 clients. Overall, FedAvg can only help around half of the clients, which demonstrates that FedAvg can be ineffective even for decenrtalized graphs from a single dataset, because of the graph non-IIDness as shown in Table $2 .$ In addition, in all three datasets, the minimum performance gain of clients over self-train using GCFL or ${ \mathrm { G C F L } } +$ is obviously larger than the minimum performance gain using FedAvg and FedProx. It indicates that even when some clients do not improve from self-train by using GCFL or ${ \mathrm { G C F L } } +$ , they can achieve more comparable performance as self-train than using FedAvg and FedProx. These experimental results demonstrate that our frameworks are effective on the single-dataset multi-client FL setting.
+
+Federated graph classification across multiple datasets According to our data analysis in Tables 1 and $2 ,$ clients in such a setting are more heterogeneous. We conduct experiments with multiple datasets in two settings: single domain (using the data group MOLECULES), and across domains (using the data groups BIOCHEM and MIX). As can be seen from the results in Table 4, our frameworks GCFL and ${ \mathrm { G C F L } } +$ can effectively improve the performance of clients with distinct datasets. The results show $1 . 7 \% - 2 . 7 \%$ improvements of our frameworks compared to self-train. In all three data groups, our ${ \mathrm { G C F L } } +$ framework can improve twice as many as clients than FedAvg, and it achieves a ratio of $1 0 0 \%$ in MOLECULES to improve all clients’ performance. The FedAvg failed to improve around $6 0 \%$ clients, which further demonstrates its ineffectiveness facing graph non-IIDness. Additionally, the ${ \mathrm { G C F L } } +$ framework also outperforms GCFL. In MIX, although GCFL can achieve the same ratio of improved clients as ${ \mathrm { G C F L } } +$ , the ${ \mathrm { G C F L } } +$ framework has a much larger minimum gain of clients than GCFL. It indicates that by ${ \mathrm { G C F L } } +$ few clients that cannot benefit from others will not be degraded through the collaborating. These results indicate that graphs across datasets or even across domains are able to help each other through proper FL, which is a surprising and interesting start point for further study.
+
+Effects of hyper-parameters $\varepsilon _ { 1 }$ and $\varepsilon _ { 2 }$ The hyper-parameter $\varepsilon _ { 1 }$ is a stopping criterion for checking whether a general FL on the current set of clients is near the stationary point. Theoretically, $\varepsilon _ { 1 }$ should be set as small as possible. The hyper-parameter $\varepsilon _ { 2 }$ is more dependent on the number of clients and the heterogeneity among them. A smaller $\varepsilon _ { 2 }$ will make the clients more likely to be clustered. When $\varepsilon _ { 1 }$ and $\varepsilon _ { 2 }$ are in the feasible ranges, their small variation can have little effect on the performance because the clustering results would largely remain the same. When $\varepsilon _ { 2 }$ is set too large, the performance will be similar as applying a basic FL algorithm directly (i.e. with a single cluster). When $\varepsilon _ { 2 }$ is set too small, more clusters with smaller sizes or even single clients will be generated. We provide additional experimental results regarding the performance of GCFL and ${ \mathrm { G C F L } } +$ w.r.t. varying $\varepsilon _ { 1 }$ and $\varepsilon _ { 2 }$ in Figure $2 .$ As shown in the subfigures, some points that represent varying $\varepsilon _ { 2 }$ overlap for each $\varepsilon _ { 1 }$ , which indicates that varying $\varepsilon _ { 2 }$ in a certain range w.r.t. the fixed $\varepsilon _ { 1 }$ leads to similar performance. Looking at a $\varepsilon _ { 2 }$ , within a certain range of $\varepsilon _ { 1 }$ , we can find the performance often fluctuating within a 0.01 variance. The results show that the performance of GCFL and ${ \mathrm { G C F L } } +$ are not very sensitive to the changes of $\varepsilon _ { 1 }$ and $\varepsilon _ { 2 }$ in reasonable ranges.
+
+Table 3: Performance on the single-dataset-multi-client setting. We present the average accuracy and minimum gain over self-train on all clients, as well as the ratio of clients which get improved.   
+
+<table><tr><td rowspan="2">Dataset (# clients) Accuracy</td><td colspan="3">NCI1 (30)</td><td colspan="3">PROTEINS (10)</td><td colspan="3">IMDB-BINARY (10)</td></tr><tr><td>average</td><td>min gain</td><td>ratio</td><td>average</td><td>min gain</td><td>ratio</td><td>average</td><td>min gain</td><td>ratio</td></tr><tr><td>self-train</td><td>0.6468(±0.053)</td><td></td><td></td><td>0.7213(±0.058)</td><td></td><td></td><td>0.7654(±0.057)</td><td></td><td></td></tr><tr><td>FedAvg</td><td>0.6474(±0.076)</td><td>-0.1333</td><td>14/30</td><td>0.7490(±0.034)</td><td>-0.0615</td><td>6/10</td><td>0.7596(±0.049)</td><td>-0.0800</td><td>5/10</td></tr><tr><td>FedProx</td><td>0.6437(±0.072)</td><td>-0.2400</td><td>16/30</td><td>0.7556(±0.036)</td><td>-0.0923</td><td>7/10</td><td>0.7746(±0.048)</td><td>-0.0600</td><td>6/10</td></tr><tr><td>GCFL</td><td>0.7326(±0.052)</td><td>-0.0462</td><td>26/30</td><td>0.7739(±0.043)</td><td>-0.0545</td><td>8/10</td><td>0.8256(±0.059)</td><td>0.0182</td><td>10/10</td></tr><tr><td>GCFL+</td><td>0.7422(±0.053)</td><td>-0.1143</td><td>28/30</td><td>0.7776(±0.037)</td><td>-0.0154</td><td>9/10</td><td>0.8299(±0.052)</td><td>0.0167</td><td>10/10</td></tr></table>
+
+Table 4: Performance on the multi-dataset-multi-client setting. Metrics are the same as Table 3.   
+
+<table><tr><td rowspan="2">Dataset (# domains) Accuracy</td><td colspan="3">MOLECULES (1)</td><td colspan="3">BIOCHEM (2)</td><td colspan="3">MIX (3)</td></tr><tr><td>average</td><td>min gain</td><td>ratio</td><td>average</td><td>min gain</td><td>ratio</td><td>average</td><td>min gain</td><td>ratio</td></tr><tr><td>self-train</td><td>0.7543(±0.017)</td><td></td><td></td><td>0.7129(±0.016)</td><td></td><td></td><td>0.7001(±0.034)</td><td>一</td><td>一</td></tr><tr><td>FedAvg</td><td>0.7524(±0.026)</td><td>-0.0132</td><td>3/7</td><td>0.6944(±0.027)</td><td>-0.1467</td><td>4/10</td><td>0.6886(±0.023)</td><td>-0.1233</td><td>5/13</td></tr><tr><td>FedProx</td><td>0.7668(±0.032)</td><td>-0.0054</td><td>5/7</td><td>0.7053(±0.026)</td><td>-0.1000</td><td>5/10</td><td>0.6897(±0.026)</td><td>-0.1367</td><td>5/13</td></tr><tr><td>GCFL</td><td>0.7661(±0.016)</td><td>0.0010</td><td>77</td><td>0.7172(±0.019)</td><td>-0.0700</td><td>7/10</td><td>0.7056(±0.019)</td><td>-0.1400</td><td>10/13</td></tr><tr><td>GCFL+</td><td>0.7745(±0.030)</td><td>0.0010</td><td>77</td><td>0.7312(±0.031)</td><td>-0.0300</td><td>8/10</td><td>0.7121(±0.021)</td><td>-0.0233</td><td>10/13</td></tr></table>
+
+# 6.3 Structure and feature analysis in clusters
+
+We conduct an in-depth analysis to explore the clustering results of GCFL and ${ \mathrm { G C F L } } +$ . As can be seen in Figure $^ { 3 , }$ after being clustered by GCFL and ${ \mathrm { G C F L } } +$ , the overall structure and feature heterogeneity of clients’ graphs within clusters are reduced significantly compared to the original values, especially for the multiple dataset setting (Figure 3c and $3 \mathrm { \check { d } } )$ . For the one dataset setting (Figure 3a and $\textcircled { 3 6 }$ , since features all fall in the same space, pairs of clients tend to have more homogeneous features. Therefore, the feature heterogeneity only gets reduced slightly after clustering. Unlike feature heterogeneity, the structure heterogeneity within clusters decreases significantly. In the setting of multiple datasets, as shown in Figure $3 \mathrm { c }$ and 3d, both structure and feature heterogeneity decrease significantly, which is intuitive since datasets across domains usually tend to have higher heterogeneity, as discussed in 4.1. We also look into the clusters and find that datasets from the same domains are more likely to be clustered together, while datasets from different domains also constantly get clustered together and benefit each other. For example, the clustering of ${ \mathrm { G C F L } } +$ corresponding to Figure 3d groups two social networks COLLAB and IMDB-BINARY together with PROTEINS and also several molecules datasets, and there is also a cluster of NCI1, DD, and IMDB-MULTI which are molecules, proteins and social networks, respectively. These analysis manifests that domains of datasets can verify the sanity of clusters to some extent, but one cannot solely rely on such prior knowledge to determine the optimal clusters, which demonstrates the necessity of our frameworks with the ability of performance-driven dynamic clustering along the process of FL.
+
+![](images/85b86777d3218d7a7a70c6401a5de233e241f3c00ebc553bfb92b5118e64d8b9.jpg)  
+Figure 2: Performance of GCFL and ${ \mathrm { G C F L } } +$ on MOLECULES w.r.t varying $\varepsilon _ { 1 }$ and $\varepsilon _ { 2 }$
+
+![](images/01e172976109936df459cfe72fffef5f8e9d533391f5838ef187596f36c6d3f9.jpg)  
+Figure 3: Structure (blue) and feature (red) heterogeneity within clusters found by GCFL and ${ \mathrm { G C F L } } +$ Dashed lines denote the heterogeneity over all clients before clustering.
+
+![](images/cfe40ccd78fb6af12529ae3711b0d04adfd9ecad8ee25983bbffc6ebe00bc656.jpg)  
+Figure 4: Average with standard deviation of the training curves of all clients.
+
+# 6.4 Convergence analysis
+
+We visualize the testing loss with respect to the communication round to show the convergence of GCFL and ${ \mathrm { G C F L } } +$ compared with the standard federated learning baselines. Figure $\boxed { 4 }$ shows the training curves on two settings, which illustrates that GCFL and ${ \mathrm { G C F L } } +$ achieves similar convergence rate as FedProx, which is the state-of-the-art FL framework dealing with non-IID Euclidean data. We also notice that both GCFL, ${ \mathrm { G C F L } } +$ and FedProx can converge to a lower loss compared with FedAvg, which corroborates our consideration of the non-IID problem in our setting.
+
+# 6.5 More results in Appendix
+
+In Table 3 and 4, we averaged the accuracy across all clients for presentation simplicity. To understand the detailed performance by clients and clusters, we present different Violin plots in Appendix D. Besides, we also show more results regarding various settings (overlapping clients, real vs. synthetic node features, standardized gradient-sequence matrix in ${ \mathrm { G C F L } } +$ , etc) in Appendix D.
+
+# 7 Conclusion
+
+In this work, we propose a novel setting of cross-dataset and cross-domain federated graph classification. The techniques (GCFL and $\mathrm { G C F L + }$ ) we develop allow multiple data owners holding structure and feature non-IID graphs to collaboratively train powerful graph classification neural networks without the need of direct data sharing. As the first trial, we focus on the effectiveness of FL in this setting and have not carefully studied other issues such as data privacy, although it is intuitive to preserve the privacy of clients by introducing an encryption mechanism (e.g. applying orthonormal transformations), and to prevent from adversarial scenarios by clustering out the malicious clients. Due to its evident motivations and proofs on the effective FL in a new setting, we believe this work can serve as a stepping stone for many interesting future studies.
+
+# Acknowledgments and Disclosure of Funding
+
+The work is partially supported by National Science Foundation (NSF) under CNS-2124104, CNS2125530, CNS-1952192, and IIS-1838200, National Institute of Health (NIH) under R01GM118609 and UL1TR002378, and the internal funding and GPU servers provided by the Computer Science Department of Emory University.
+
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+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] Mainly see Sections 4-6   
+(b) Did you describe the limitations of your work? [Yes] See Section 7   
+(c) Did you discuss any potential negative societal impacts of your work? [N/A] See Section 7   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] See Section 4.4 (b) Did you include complete proofs of all theoretical results? [Yes] See Appendix
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] See Section 6.1 and the supplemental material   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Section 6.1   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] For example see Tables 3 and 4   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Section 6.1
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes] Wherever we mention them for the first time   
+(b) Did you mention the license of the assets? [N/A] They are public   
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] We include our models and code in the supplemental material   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/train/yJqcM36Qvnu/yJqcM36Qvnu_content_list.json

@@ -0,0 +1,1299 @@
+[
+    {
+        "type": "text",
+        "text": "Federated Graph Classification over Non-IID Graphs ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            122,
+            820,
+            147
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Han Xie, Jing Ma, Li Xiong, Carl Yang⇤ Department of Computer Science, Emory University {han.xie, jing.ma, lxiong, j.carlyang}@emory.edu ",
+        "bbox": [
+            294,
+            200,
+            705,
+            243
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            279,
+            535,
+            295
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Federated learning has emerged as an important paradigm for training machine learning models in different domains. For graph-level tasks such as graph classification, graphs can also be regarded as a special type of data samples, which can be collected and stored in separate local systems. Similar to other domains, multiple local systems, each holding a small set of graphs, may benefit from collaboratively training a powerful graph mining model, such as the popular graph neural networks (GNNs). To provide more motivation towards such endeavors, we analyze real-world graphs from different domains to confirm that they indeed share certain graph properties that are statistically significant compared with random graphs. However, we also find that different sets of graphs, even from the same domain or same dataset, are non-IID regarding both graph structures and node features. To handle this, we propose a graph clustered federated learning (GCFL) framework that dynamically finds clusters of local systems based on the gradients of GNNs, and theoretically justify that such clusters can reduce the structure and feature heterogeneity among graphs owned by the local systems. Moreover, we observe the gradients of GNNs to be rather fluctuating in GCFL which impedes high-quality clustering, and design a gradient sequence-based clustering mechanism based on dynamic time warping $\\mathrm { ( G C F L + ) }$ ). Extensive experimental results and in-depth analysis demonstrate the effectiveness of our proposed frameworks. ",
+        "bbox": [
+            233,
+            310,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            598,
+            310,
+            616
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Federated learning (FL) as a distributed learning paradigm that trains centralized models on decentralized data has attracted much attention recently [28, 53, 25, 18, 17]. FL allows local systems to benefit from each other while keeping their own data private. Especially, for local systems with scarce training data or lack of diverse distributions, FL provides them with the potentiality to leverage the power of data from others, in order to facilitate the performance on their own local tasks. One important problem FL concerns is data distribution heterogeneity, since the decentralized data, collected by different institutes using different methods and aiming at different tasks, are highly likely to follow non-identical distributions. Prior works approach this problem from different aspects, including optimization process [25, 18], personalized FL [13, 6, 8], clustered FL [9, 15, 2], etc. ",
+        "bbox": [
+            174,
+            630,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "As more advanced techniques are developed for learning with graph data, using graphs to model and solve real-world problems becomes more popular. One important scenario of graph learning is graph classification, where models such as graph kernels [44, 34, 36, 45] and graph neural networks [21, 43, 49, 46, 47, 48] are used to predict graph-level labels based on the features and structures of graphs. One real scenario of graph classification is molecular property prediction, which is an important task in cheminformatics and AI medicine. In the area of bioinformatics, graph classification can be used to learn the representation of proteins and classify them into enzymes or non-enzymes. For collaboration networks, sub-networks can be classified regarding the information of research areas, topics, genre, etc. More applicable scenarios include geographic networks, temporal networks, etc. ",
+        "bbox": [
+            173,
+            761,
+            825,
+            898
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Since the key idea of $\\mathrm { F L }$ is the sharing of underlying common information, as $\\mathbb { \\left[ \\left[ 2 3 \\right] \\right] }$ discusses that real-world graphs preserve many common properties, we become curious about the question, whether real-world graphs from heterogeneous sources (e.g., different datasets or even divergent domains) can provide useful common information among each other? To understand this question, we first conduct preliminary data analysis to explore real-world graph properties, and try to find clues about common patterns shared among graphs across datasets. As shown in Table $\\bigstar$ we analyze four typical datasets from different domains, i.e., PTC_MR (molecular structures), ENZYMES (protein structures), IMDB-BINARY (social communities), and MSRC_21 (superpixel networks). We find them to indeed share certain properties that are statistically significant compared to random graphs with the same numbers of nodes and links (generated with the Erdos–Rényi model [ ˝ 7, 10]). Such observations confirm the claim about common patterns underlying real-world graphs, which can largely influence the graph mining models and motivates us to consider the FL of graph classification across datasets and even domains. More details and discussion about Table 1 can be found in Appendix A. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "table",
+        "img_path": "images/402319ddd33b9b3ddb5de7924418353c4ece9d264329765991b4241a782eec45.jpg",
+        "table_caption": [
+            "Table 1: Data analysis on important graph properties shared among real-world graphs across different domains. For example, large Kurtosis values $\\textcircled { 1 3 2 } \\textcircled { 1 }$ indicate long-tail distribution of node degrees, which is observed in ENZYMES, IMDB-BINARY, and MSRC_21; similar average shortest path lengths are observed in PTC_MR, ENZYMES, and MSRC_21, although their actual graph sizes are rather different; large CC are observed in ENZYMES, IMDB-BINARY, and MSRC_21 and large LC are observed in almost all graphs. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Property</td><td colspan=\"3\">kurtosis of degree distribution</td><td colspan=\"3\">avg. shortest path length</td><td colspan=\"3\">largest component size (LC,%)</td><td colspan=\"3\">clustering coefficient (CC)</td></tr><tr><td></td><td>real</td><td>random</td><td>p-value</td><td>real</td><td>random</td><td>p-value</td><td>real</td><td>random</td><td>p-value</td><td>real</td><td>random</td><td>p-value</td></tr><tr><td>PTC_MR (molecules)</td><td>2.1535</td><td>2.4424</td><td>0.9999</td><td>3.36</td><td>2.42</td><td>~0</td><td>100</td><td>82.68</td><td>~0</td><td>0.0095</td><td>0.1201</td><td>~0</td></tr><tr><td>ENZYMES (proteins)</td><td>3.0106</td><td>2.8243</td><td>0.0027</td><td>4.44</td><td>2.56</td><td>~0</td><td>98.24</td><td>97.69</td><td>0.2054</td><td>0.4516</td><td>0.1425</td><td>~0</td></tr><tr><td>IMDB-BINARY (social)</td><td>8.9262</td><td>2.2791</td><td>~0</td><td>1.48</td><td>1.54</td><td>~0</td><td>100</td><td>99.93</td><td>0.0023</td><td>0.9471</td><td>0.5187</td><td>~0</td></tr><tr><td>MSRC_21 (superpixel)</td><td>3.6959</td><td>2.9714</td><td>~0</td><td>4.09</td><td>2.81</td><td>~0</td><td>100</td><td>99.43</td><td>~0</td><td>0.5147</td><td>0.0655</td><td>~0</td></tr></table>",
+        "bbox": [
+            176,
+            362,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Although common patterns exist among graph datasets, we can still observe certain heterogeneity. In fact, the detailed graph structure distributions and node feature distributions can both diverge due to various reasons. To demonstrate this, we design and evaluate a structure heterogeneity measure and a feature heterogeneity measure in different scenarios (c.f. Section $\\boxed { 4 . 1 }$ . We refer to the graphs possibly with significant heterogeneity in our cross-dataset FL setting as non-IID graphs, which concerns both structure non-IID and feature non-IID, where naïve FL algorithms like FedAvg $\\left. \\boldsymbol { \\widetilde { 2 8 } } \\right.$ can fail and even backfire (c.f. Section $6 . 2 )$ . Moreover, as the heterogeneity varies from case to case, a dynamic FL algorithm is needed to keep track of such heterogeneity of non-IID graphs while conducting collaborative model training. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Due to the observations that the graphs in one client can be similar to those in some clients but not the others, we get motivated by $\\bar { \\mathbb { D } }$ and find it intuitive to consider a clustered FL framework, which assigns local clients to multiple clusters with less data heterogeneity. To this end, we propose a novel graph-level clustered FL framework (termed GCFL) through integrating the powerful graph neural networks (GNNs) such as GIN $\\mathbb { \\lVert \\underline { { 4 3 } } \\rVert }$ into clustered FL, where the server can dynamically cluster the clients based on the gradients of GNN without additional prior knowledge, while collaboratively training multiple GNNs as necessary for homogeneous clusters of clients. We theoretically analyze that the model parameters of GNN indeed reflect the structures and features of graphs, and thus using the gradients of GNN for clustering in principle can yield clusters with reduced heterogeneity of both structures and features. In addition, we conduct empirical analysis to support the motivation of clustered FL across heterogeneous graph datasets in Appendix A. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Although GCFL can theoretically achieve homogeneous clusters, during its training, we observe that the gradients transmitted at each communication round fluctuate a lot (c.f. Section $\\bar { 5 } . 1 )$ , which could be caused by the complicated interactions among clients regarding both structure and feature heterogeneity, making the local gradients towards divergent directions. In the vanilla GCFL framework, the server calculates a matrix for clustering only based on the last transmitted gradients, which ignores the client’s multi-round behaviors. Therefore, we further propose an improved version of GCFL with gradient-series-based clustering (termed ${ \\mathrm { G C F L } } +$ ). ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We conduct extensive experiments with various settings to demonstrate the effectiveness of our frameworks. Moreover, we provide in-depth analysis on the capability of them on reducing both structure and feature heterogeneity of clients through clustering. Lastly, we analyze the convergence of our frameworks. The experimental results show surprisingly positive results brought by our novel setting of cross-dataset/cross-domain FL for graph classification, where our ${ \\mathrm { G C F L } } +$ framework can effectively and consistently outperform other straightforward baselines. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Related works ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            89,
+            325,
+            106
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Federated Learning Federated learning (FL) has gained increasing attention as a training paradigm under the setting where data are distributed at remote devices and models are collaboratively trained under the coordination of a central server. FedAvg was first proposed by $ { \\mathbb { \\left[ \\left[ 2 7 \\right] \\right] } }$ which illustrates the general setting of an FL framework. Since the original FedAvg relies on the optimization by SGD, data of non-IID distribution will not guarantee the stochastic gradients to be an unbiased estimation of the full gradients, thus hurting the convergence of FL. In fact, multiple experiments $\\pm \\Sigma \\boxed { 2 5 } \\boxed { 1 8 }$ have shown that the convergence will be slow and unstable, and the accuracy will degrade with FedAvg when data at each client are statistically heterogeneous (non-IID). [53, 15, 12] proposed different data sharing strategies to tackle the data heterogeneity problem by sharing the local device data or serverside proxy data, which still requires certain public common data, whereas other studies explored the convergence guarantee under the non-IID setting by assuming bounded gradients $\\textcircled { 1 3 9 } , \\textcircled { 5 1 }$ or additional noise $\\mathbb { \\lVert \\rVert }$ . There are also works seeking to reduce the variance of the clients $\\mathbb { P } \\bot \\bot \\bot \\bot \\bot \\bot $ . Furthermore, multiple works have been proposed to explore the connection between model-agnostic meta-learning (MAML) and personalized federated learning [8, 4]. They aim to learn a generalizable global model and then fine-tune it on local clients, which may still fail when data on local clients are from divergent domains with high heterogeneity. Some personalized FL works $\\mathbb { H } \\mathbb { L } \\mathbb { 4 } \\mathbb { I }$ studied the bi-level problem of optimization which decouples the local and global optimization, but having each client maintain its own model can lead to high communication cost among the clients and the server. While personalization in the FL setting can address the client heterogeneity problem to some extent, the clustered FL framework [33] can incorporate personalization at the group level to keep the benefits of personalized FL and reduce the communication cost simultaneously. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Federated Learning on Graphs Although FL has been intensively studied with Euclidean data such as images, there exist few studies about FL for graph data. $\\lVert 2 2 \\rVert$ first introduced FL on graph data, by regarding each client as a node in a graph. [3] studied the cross-domain heterogeneity problem in FL by leveraging Graph Convolutional Networks (GCNs) to model the interaction between domains. $\\mathbb { \\left. 2 9 \\right. }$ studied spatio-temporal data modeling in the FL setting by leveraging a Graph Neural Network (GNN) based model to capture the spatial relationship among clients. [5] proposed a generalized federated knowledge graph embedding framework that can be applied for multiple knowledge graph embedding algorithms. Moreover, there are several works exploring the GNNs under the FL setting: [16, 54, 40] focused on the privacy issue of federated GNNs; $\\bar { \\big [ } \\bar { \\big | } 3 7 \\big | \\big ]$ incorporated model-agnostic metalearning (MAML) into graph FL, which handled non-IID graph data while also preserving the model generalizability; $\\pmb { \\Vert 5 2 \\Vert }$ studied the missing neighbor generation problem in the subgraph FL setting; $ { \\bar { \\mathbf { \\delta } } }$ proposed a computationally efficient way of GCN architecture search with FL; $\\bar { \\mathbb { m } } ^ { \\bar { | } }$ implemented an open FL benchmark system for GNNs. Most existing works consider node classification and link prediction on graphs, which cannot be trivially applied to our graph classification setting. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 Preliminaries ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 Graph Neural Networks (GNNs) ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            666,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "$\\lVert \\rVert 2 \\rVert$ provides a taxonomy that categorizes Graph Neural Networks (GNNs) into recurrent GNNs, convolutional GNNs, graph autoencoders, and spatial-temporal GNNs. In general, given the structure and feature information of a graph $G = ( V , E , X )$ , where $V$ , $E$ , $X$ denote nodes, links and node features, GNNs target to learn the representations of graphs, such as a node embedding $h _ { v } \\in \\mathbb { R } ^ { d _ { v } }$ , or a graph embedding $\\boldsymbol { h } _ { G } \\in \\mathbb { R } ^ { d _ { G } }$ . A GNN typically consists of message propagation and neighborhood aggregation, in which each node iteratively gathers the information propagated by its neighbors, and aggregates them with its own information to update its representation. Generally, an $L$ -layer GNN can be formulated as ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6b17c5f65c103cdf9a6ad11798f1f72bb97e7a19f4d803d36724062fced1802c.jpg",
+        "text": "$$\nh _ { v } ^ { ( l + 1 ) } = \\sigma ( h _ { v } ^ { ( l ) } , a g g ( \\{ h _ { u } ^ { ( l ) } ; u \\in \\mathcal { N } _ { v } \\} ) ) , \\forall l \\in [ L ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            336,
+            804,
+            658,
+            824
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $h _ { v } ^ { ( l ) }$ is the representation of node $v$ at the $l ^ { t h }$ layer, and $h _ { v } ^ { ( 0 ) } = x _ { v }$ is the node feature. $\\mathcal { N } _ { v }$ is neighbors of node $v$ , $a g g ( \\cdot )$ is an aggregation function that can vary for different GNN variants, and $\\sigma$ represents an activation function. ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "For a graph-level representation $h _ { G }$ , it can be pooled from the representations of all nodes, as ",
+        "bbox": [
+            171,
+            877,
+            784,
+            893
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/68e9d5c2e8434c7b1e934257fc6b543ddc0d58023efe24fe0a4f2659fd9418f8.jpg",
+        "text": "$$\nh _ { G } = r e a d o u t ( \\{ h _ { v } ; v \\in V \\} ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            400,
+            895,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where readout $\\left( \\cdot \\right)$ can be implemented as mean pooling, sum pooling, etc, which essentially aggregates the embeddings of all nodes on the graph into a single embedding vector to achieve tasks like graph classification and regression. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 The FedAvg algorithm ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "McMahan et al. $\\left[ \\left[ 2 8 \\right] \\right]$ proposed an SGD-based aggregating algorithm, FedAvg, based on the fact that SGD is widely used and powerful for optimization. FedAvg is the first basic FL algorithm and is commonly used as the starting point for more advance FL framework design. ",
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+        "type": "text",
+        "text": "The key idea of FedAvg is to aggregate the updated model parameters transmitted from local clients and then re-distribute the averaged parameters back to each client. Specifically, given $m$ clients in total, at each communication round $t$ , the server first samples a partition of clients $\\{ \\mathbb { S } _ { i } \\} ^ { ( t ) }$ . For each client $\\mathbb { S } _ { i }$ in $\\{ \\mathbb { S } _ { i } \\} ^ { ( t ) }$ , it trains the model downloaded from the server locally with its own data distribution $\\mathcal { D } _ { i }$ for $E _ { l o c a l }$ epochs. The client $\\mathbb { S } _ { i }$ then transmits its updated parameters $w _ { i } ^ { ( t ) }$ to the server, and the server will aggregate these updates by ",
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+        "img_path": "images/199af2920a89ed7de5bd5a20c8a6c83b9223e89467ca055372420db20de09384.jpg",
+        "text": "$$\nw ^ { ( t + 1 ) } = \\sum _ { i = 1 } ^ { m } \\frac { | D _ { i } | } { | D | } w _ { i } ^ { ( t ) } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $| D _ { i } |$ is the size of data samples in client $\\mathbb { S } _ { i }$ and $| D |$ is the total size of samples over all clients. After generating the aggregated parameters (the global model updates), the server broadcasts the new parameters $w ^ { ( t + 1 ) }$ to remote clients, and at the $( t + 1 )$ round clients use $w ^ { ( t + 1 ) }$ to start their local training for another $E _ { l o c a l }$ epochs. ",
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+    {
+        "type": "text",
+        "text": "4 The GCFL framework ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "4.1 Non-IID structures and features across clients ",
+        "text_level": 1,
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+        "text": "From Table 1 we notice that real-world graphs tend to share certain general properties across different graphs, datasets and even domains, which motivates the graph-level FL framework. However, there still exist differences when the detailed graph structures and node features are being considered. In Table $^ { 2 , }$ we present the average pair-wise structure heterogeneity and feature heterogeneity among graphs in a single dataset, a single domain, and across different domains. Specifically, for structure heterogeneity, we use the Anonymous Walk Embeddings (AWEs) $[ \\left| 1 4 \\right| ]$ to generate a representation for each graph, and compute the Jensen-Shannon distance between the AWEs of each pair of graphs; for feature heterogeneity, we calculate the empirical distribution of feature similarity between all pairs of linked nodes in each graph, and compute the Jensen-Shannon divergence between the feature similarity distributions of each pair of graphs. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "As we can observe in Table $\\boxed { 2 } ,$ both graph structures and features demonstrate different levels of heterogeneity within a single dataset, a single domain, and across different domains. We refer to graphs with such structure and feature heterogeneity as non-IID graphs. Intuitively, directly applying naïve FL algorithms like FedAvg on clients with non-IID graphs can be ineffective and even backfiring. To be specific, structure heterogeneity makes it difficult for a model to capture the universally important graph structure patterns across different clients, whereas feature heterogeneity makes it hard for a model to learn the universally appropriate message propagation functions across different clients. How can we leverage the shared graph properties among clients while addressing the non-IID structures and features across clients? ",
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+        "type": "text",
+        "text": "4.2 Problem formulation ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Motivated by our real graph data analysis in Tables 1 and 2, we propose a novel framework of Graph Clustered Federated Learning (GCFL). The main idea of GCFL is to jointly find clusters of clients with graphs of similar structures and features, and train the graph mining models with FedAvg among clients in the same clusters. ",
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+        "text": "Specifically, we are inspired by the Clustered Federated Learning (CFL) framework on Euclidean data $\\mathbb { \\lVert 3 3 \\rVert }$ and consider a clustered FL setting with one central server and a set of $n$ local clients $\\{ \\mathbb { S } _ { 1 } , \\mathbb { S } _ { 2 } , \\ldots , \\mathbb { S } _ { n } \\}$ . Different from the traditional FL setting, the server can dynamically cluster the clients into a set of clusters $\\{ \\mathbb { C } _ { 1 } , \\mathbb { C } _ { 2 } , \\ldots \\}$ and maintain $m$ cluster-wise models. In our GCFL setting, each local client $\\mathbb { S } _ { i }$ owns a set of graphs $\\mathcal { \\bar { G } } _ { i } = \\{ G _ { 1 } , G _ { 2 } , . . . \\}$ , where each $G _ { j } = ( V _ { j } , E _ { j } , X _ { j } , y _ { j } ) \\in \\bar { \\mathcal { G } } _ { i }$ is a graph data sample with a set of nodes $V _ { j }$ , a set of edges $E _ { j }$ , node features $X _ { j }$ , and a graph class label $y _ { j }$ . The task on each local client $\\mathbb { S } _ { i }$ is graph classification that predicts the class label $\\hat { y } _ { j } ~ = ~ h _ { k } ^ { * } ( G _ { j } )$ for each graph $G _ { j } ~ \\in ~ \\mathcal { G } _ { i }$ , where $h _ { k } ^ { * }$ is the collaboratively learned optimal graph mining model for cluster $\\mathbb { C } _ { k }$ to which $\\mathbb { S } _ { i }$ belongs. Our goal is to minimize the loss function $F ( \\Theta _ { k } ) : = \\operatorname { E } _ { \\mathbb { S } _ { i } \\in \\mathbb { C } _ { k } } [ f ( \\theta _ { k , i } ; \\mathcal { G } _ { i } ) ]$ , for all clusters $\\{ \\mathbb { C } _ { k } \\bar  \\}$ . The function $f ( \\theta _ { k , i } ; \\mathcal { G } _ { i } )$ is a local loss function for client $\\mathbb { S } _ { i }$ which belongs to cluster $\\mathbb { C } _ { k }$ . In the meantime, we also aim to maintain a dynamic cluster assignment $\\Gamma ( \\mathbb { S } _ { i } ) \\to \\{ \\mathbb { C } _ { k } \\}$ based on the $\\mathrm { F L }$ process. ",
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+    {
+        "type": "table",
+        "img_path": "images/83581d31a6fc5d4637e7dc60a7fa6b822274f784170aa014b5e250822b84dfb3.jpg",
+        "table_caption": [
+            "Table 2: Summary of the average heterogeneity of features and structures for some datasets. In general, the structure heterogeneity increases from the settings of one dataset to across-dataset, and to across-domain. However, the feature heterogeneity is more case-by-case, and the high variances indicate that graphs could have large feature divergence even within the same dataset. Additionally, it is not necessarily true that one dataset itself should be more homogeneous (e.g., IMDB-BINARY). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>dataset</td><td>IMDB-BINARY (social)</td><td>CoX2 (molecules)</td><td>Cox2 (molecules) PTC_MR (molecules)</td><td>Cox2 (molecules) ENZYMES (proteins)</td><td>Cox2 (molecules) IMDB-BINARY (sOCial)</td></tr><tr><td>avg. struc. hetero.</td><td>0.4406 (±0.0397)</td><td>0.3246 (±0.0145)</td><td>0.3689 (±0.0540)</td><td>0.5082 (±0.0399)</td><td>0.6079 (±0.0331)</td></tr><tr><td>avg. feat. hetero.</td><td>0.1785 (±0.1226)</td><td>0.0427 (±0.0314)</td><td>0.1837(±0.1065)</td><td>0.1912 (±0.1000)</td><td>0.1642 (±0.1006)</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4.3 Technical design ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "GNNs are demonstrated to be powerful for learning graph representations and have been wildly used in graph mining. More importantly, the model parameters and their gradients of GNNs can reflect the graph structure and feature information (more details in Section $\\bar { 4 . 4 ) }$ . Thus, we use GNNs as the graph mining model in our GCFL framework. ",
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+        "text": "Specifically, our GCFL framework can dynamically cluster clients by leveraging their transmitted gradients $\\{ \\Delta \\theta _ { i } \\} _ { i = 1 } ^ { n }$ , in order to maximize the collaboration among more homogeneous clients and eliminate the harm from heterogeneous clients. According to $\\mathbb { B } 3 \\mathbb { I }$ , if the data distribution of clients are highly heterogeneous, the general FL that trains the clients together cannot jointly optimize all their local loss functions. In this case, after some rounds of communication, the general FL will be close to the stationary point, and the norm of clients’ transmitted gradients will not all tend towards zero. Therefore, clustering clients is needed as the general FL approaches to the stationary point. Here, we first introduce a hyper-parameter $\\varepsilon _ { 1 }$ as a criterion to decide whether to stop the general FL based on whether a stationary point is approached, that is, ",
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+        "img_path": "images/f5596ff082ed21eca4f0a1c7505c317eb7b5bd05029f2ffe8cb6be490a257367.jpg",
+        "text": "$$\n\\delta _ { m e a n } = \\| \\sum _ { i \\in [ n ] } \\frac { | \\mathcal { G } _ { i } | } { | \\mathcal { G } | } \\Delta \\theta _ { i } \\| < \\varepsilon _ { 1 } .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "In the meantime, if there exist some clients still with large norms of transmitted gradients, it means that clients in the group are highly heterogeneous, and thus clustering is needed to eliminate the negative influence among them. We then introduce the second criterion with a hyper-parameter $\\varepsilon _ { 2 }$ to split the clusters when ",
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+        "img_path": "images/e3b8ee151eff0375f25625c00bce5a14b56ad4446d807944d923ecfc01660e97.jpg",
+        "text": "$$\n\\delta _ { m a x } = \\operatorname* { m a x } ( \\| \\Delta \\theta _ { i } \\| ) > \\varepsilon _ { 2 } > 0 .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The GCFL framework follows a top-down bi-partitioning mechanism. At each communication round $t$ , the server receives $m$ sets of gradients $\\{ \\{ \\hat { \\Delta } \\theta _ { i _ { 1 } } \\} , \\{ \\tilde { \\Delta \\theta } _ { i _ { 2 } } \\} , \\dots , \\{ \\Delta \\theta _ { i _ { m } } \\} \\}$ from clients in clusters $\\{ \\mathbb { C } _ { 1 } , \\mathbb { C } _ { 2 } , \\ldots , \\mathbb { C } _ { m } \\}$ . For a cluster $\\mathbb { C } _ { k }$ , if $\\delta _ { m e a n } ^ { k }$ and $\\delta _ { m a x } ^ { k }$ satisfy the Eqs. 4 and $\\boxed { 5 } ,$ the server will calculate a cluster-wise cosine similarity matrix $\\alpha _ { k }$ , and its entries are used as weights for building a full-connected graph with nodes being all clients within the cluster. The Stoer–Wagner minimum cut algorithm $\\pmb { \\Vert 3 5 \\Vert }$ is then applied to the constructed graph, which bi-partitions the graph and divides the cluster $\\mathbb { C } _ { k } \\to \\{ \\mathbb { C } _ { k 1 } , \\mathbb { C } _ { k 2 } \\}$ . The clustering mechanism based on Eqs. $\\boxed { 4 }$ and $\\textcircled { 5 }$ can automatically and dynamically determine the number of clusters along the FL, while the two hyper-parameters $\\varepsilon _ { 1 }$ and $\\varepsilon _ { 2 }$ can be easily set through some simple experiments on the validation sets following $\\mathbb { \\lVert 3 3 \\rVert }$ . ",
+        "bbox": [
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+        "type": "text",
+        "text": "For a client $\\mathbb { S } _ { i }$ in cluster $\\mathbb { C } _ { k }$ , it tries to find $\\widehat { \\theta } _ { k , i }$ that is close to the real solution $\\theta _ { k , i } ^ { * } ~ =$ $\\mathrm { a r g m i n } _ { \\theta _ { i } \\in \\Theta _ { k } } f ( \\theta _ { k , i } ; \\mathcal { G } _ { i } )$ . At a communication round $t$ , the client $\\mathbb { S } _ { k }$ transmits its gradient to the server ",
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+        "img_path": "images/c6c6ff501f7e7fa080b442225fa0225a0dd6662a67e02080d332c1a3535ee258.jpg",
+        "text": "$$\n\\Delta \\theta _ { k , i } ^ { t } = \\hat { \\theta } _ { k , i } ^ { t } - \\theta _ { k , i } ^ { t - 1 } .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Since the server maintains the cluster assignments, it can aggregate the gradients cluster-wise by ",
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+    {
+        "type": "equation",
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+        "text": "$$\n\\theta _ { k } ^ { t + 1 } = \\theta _ { k } ^ { t } + \\sum _ { i \\in [ n _ { k } ] } \\Delta \\theta _ { k , i } ^ { t } .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "4.4 Theoretical analysis ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We investigate the problem of graph FL with multi-domain data distribution, and use the gradientbased FL paradigm $ { \\mathbb { \\left[ \\left[ 2 7 \\right] \\right] } }$ to facilitate the model training. We theoretically analyze that the gradientbased FL algorithm on GNNs can in principle reduce the structure and feature heterogeneity in clusters, along with the task difference between data from different domains. We study two general problems in order to prove that the gradients can reflect the feature, structure, and task information. ",
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+        "type": "text",
+        "text": "Definition 4.1 Let a function $f : \\mathcal { X }  \\mathcal { Y }$ which maps from the metric space $( \\mathcal { X } , d )$ to $( \\boldsymbol { \\mathcal { V } } , \\boldsymbol { d ^ { \\prime } } )$ , the function $f$ is considered to have $\\delta$ distortion $i f \\forall u , v \\in \\mathcal { X }$ , ${ \\textstyle \\frac { 1 } { \\delta } } d ( u , v ) \\leq d ^ { \\prime } ( f ( u ) , f ( v ) ) \\leq d ( u , \\dot { v } )$ . ",
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+    {
+        "type": "text",
+        "text": "Theorem 4.1 (Bourgain theorem $\\pmb { I I I }$ ) Given an $n$ -point metric space $( \\mathcal { X } , d )$ and an embedding function $f$ as defined above, $\\forall u , v \\in \\mathcal { X }$ , there exist an embedding mapped from $( \\mathcal { X } , d )$ to $\\mathbb { R } ^ { k }$ with the distortion of the embedding being $O ( \\log n )$ . ",
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+    {
+        "type": "text",
+        "text": "Problem 1. In GCFL which involves the communication of the gradients between graphs with heterogeneous structures distributed among different clients, the structure and feature difference can be captured by the GNN gradients. ",
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+        "type": "text",
+        "text": "For simplicity, we solve Problem 1 with the GNN of Simple Graph Convolutions (SGC) [41], through the following two propositions. ",
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+        "text": "Proposition 4.1 Given a graph $G$ with fixed structure represented by the normalized graph Laplacian $\\mathcal { L } = \\widetilde { D } ^ { - \\frac { 1 } { 2 } } \\widetilde { A } \\widetilde { D } ^ { - \\frac { 1 } { 2 } }$ , feature represented with $X$ , and an SGC $f ( \\mathcal { L } , X ) = s o f t m a x ( \\mathcal { L } ^ { K } X \\Theta )$ with weights $\\Theta$ trained on graph $G$ . If we have another graph $G ^ { \\prime }$ with different structure $\\mathcal { L } ^ { \\prime }$ , the weight difference $| | \\Theta ^ { \\prime } - \\Theta | | _ { 2 }$ is bounded with the structure difference. ",
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+    {
+        "type": "text",
+        "text": "Proposition 4.2 Given a graph $G$ with fixed structure represented by the normalized graph Laplacian $\\mathcal { L } = \\widetilde { D } ^ { - \\frac { 1 } { 2 } } \\widetilde { A } \\widetilde { D } ^ { - \\frac { 1 } { 2 } }$ , feature represented with $X$ , and an SGC $f ( \\mathcal { L } , X ) = s o f t m a x ( \\mathcal { L } ^ { K } X \\Theta )$ with weights $\\Theta$ trained on graph $G$ . If we have another graph $G ^ { \\prime }$ with different feature $\\mathcal { X } ^ { \\prime }$ , the weight difference $| | \\Theta ^ { \\prime } - \\Theta | | _ { 2 }$ is bounded with the feature difference. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We prove proposition 4.1 and $\\boxed { 4 . 2 }$ in Appendix B. We use the Bourgain theorem to bound the difference between embeddings generated with different graph structures/features, and prove that the feature and structure information of a graph is incorporated into the model weights (gradients). By proving that the model weights (gradients) are bounded with the structure/feature difference, we show that the gradients will change with the structure and feature. This further justifies that our proposed gradient based clustering framework GCFL is able to capture the structure and feature information. In addition, we also study the following problem, which allows our GCFL framework to be further extended to cross-task graph-level federated learning in the future. ",
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+        "type": "text",
+        "text": "Problem 2. The communicated gradients in GCFL can also capture the task heterogeneity. ",
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+    {
+        "type": "text",
+        "text": "Proposition 4.3 Given a graph $G$ with structure represented by the normalized graph Laplacian $\\mathcal { L } = \\widetilde { D } ^ { - \\frac { 1 } { 2 } } \\widetilde { A } \\widetilde { D } ^ { - \\frac { 1 } { 2 } }$ , and feature represented with $X$ , if trained with different tasks, we will get the Simple SGC with bounded weights. ",
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+    {
+        "type": "text",
+        "text": "The proof of proposition $4 . 3$ can be found in Appendix B. ",
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+    {
+        "type": "text",
+        "text": "5 $\\mathrm { G C F L + : }$ improved GCFL based on observation sequences of gradients ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "5.1 Fluctuation of gradient norms ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "When observing the norm of gradients for each communication round in GCFL, as shown in Figure 1, we notice that: 1) the norm of gradients continuously fluctuates; 2) different clients can have divergent scales of gradient norms. The fluctuation of gradient norms and different scales indicate that the updating directions and distances of gradients for clients are diverse, which manifests the structure and feature heterogeneity in our setting again. In our vanilla GCFL framework, the server calculates a cosine similarity matrix based on the last transmitted gradients once the clustering criteria are satisfied. However, with the observation that the norm of gradients fluctuates along the communication round, albeit with the constraints of clustering criteria, GCFL clustering based on gradient-point could omit important client behaviors and be misled by noises. For example, in Figure $^ 1$ (a), GCFL performs clustering at round 119 based on the gradients at that round, which does not effectively find graphs with lower heterogeneity. ",
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+        "img_path": "images/43e3be229661b6ca6d5ff130859c4fcb00ca0059a1b30b42311269fedc051031.jpg",
+        "image_caption": [
+            "Figure 1: Norm of gradients versus communication round with six clients across datasets. Clients with datasets colored the same are split to the same cluster. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "5.2 Technical design ",
+        "text_level": 1,
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "Motivated by these observations, we propose an improved version of GCFL, named ${ \\mathrm { G C F L } } +$ , which conducts clustering by taking series of gradient norms into consideration. In the ${ \\mathrm { G C F L } } +$ framework, the server maintains a multi-variant time-series matrix $Q \\in \\mathbb { R } ^ { \\{ n , d \\} }$ , where $n$ is the number of clients and $d$ is the length of a gradient series being tracked. At each communication round $t$ , the server updates $Q$ by adding in the norm of gradients $\\| \\Delta \\theta _ { i } ^ { t } \\|$ to $Q ( i , : ) \\in \\mathbb { R } ^ { d }$ and remove the out-of-date one. ${ \\mathrm { G C F L } } +$ uses the same clustering criteria as GCFL (Eqs. 4 and 5). If the clustering criteria are satisfied, the server will calculate a distance matrix $\\beta$ in which each cell is the pair-wise distance of two series of gradients. Here, we use a technique called dynamic time warping (DTW) $\\textcircled { \\scriptsize { 1 3 1 } }$ t o measure the similarity between two data sequences. For a cluster $\\mathbb { C } _ { k }$ , the server calculates its distance matrix as ",
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+    {
+        "type": "equation",
+        "img_path": "images/ed0e0bd16c5117babdb17214290385cde13b2877458f720320925be37e260a1f.jpg",
+        "text": "$$\n\\beta _ { k } ( p , q ) = d i s t ( Q ( p , : ) , Q ( q , : ) ) , p , q \\in i d x ( \\{ \\mathbb { S } _ { i } \\} ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $i d x ( \\{ \\mathbb { S } _ { i } \\} )$ is the indices of all clients $\\{ \\mathbb { S } _ { i } \\}$ in cluster $\\mathbb { C } _ { k }$ . With the distance matrix $\\beta$ , the server can perform bi-partitioning for clusters who meet the clustering criteria. As a result, in Figure ${ \\bf 1 } \\left( { \\bf b } \\right)$ ${ \\mathrm { G C F L } } +$ performs clustering at round 118 based on the gradient sequence of length 10, which captures the longer-range behaviors of clients and effectively more homogeneous clusters. ",
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+        "type": "text",
+        "text": "6 Experiments ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "6.1 Experimental settings ",
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+        "text": "Datasets We use a total of 13 graph classification datasets $\\pmb { \\mathbb { B } } 0 \\|$ from three domains including seven molecule datasets (MUTAG, BZR, COX2, DHFR, PTC_MR, AIDS, NCI1), three protein datasets (ENZYMES, DD, PROTEINS), and three social network datasets (COLLAB, IMDB-BINARY, IMDBMULTI), each with a set of graphs. Node features are available in some datasets, and graph labels are either binary or multi-class. Details of the datasets are presented in Appendix C. ",
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+        "type": "text",
+        "text": "We design two settings that follow different data partitioning mechanisms, and the example real scenarios of the two settings can be found in Appendix $\\boxed { \\mathrm { A } }$ The first setting (i.e., single-dataset) is to randomly distribute graphs from a single dataset to a number of clients, with each client holding a distinct set of about 100 graphs, among which $10 \\%$ are held out for testing. In the second setting (i.e., multi-dataset), we use multiple datasets either from a single domain or multiple domains. Each client holds a graph dataset, among which $10 \\%$ are held out for testing. In the first setting, we use NCI1, PROTEINS, and IMDB-BINARY from three domains and distribute them to 30, 10, 10 clients, respectively. In the second setting, we create three data groups including MOLECULES which consists of seven datasets from the molecule domain distributed into seven clients, BIOCHEM where we add three datasets from the protein domain into MOLECULES and distribute them into 10 clients, MIX where we add three datasets from the social domain into BIOCHEM and distribute them into 13 clients. ",
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+        "type": "text",
+        "text": "Baselines We use self-train2 as the first baseline to test whether FL can bring improvements to each client through collaborative training. In self-train, each client firstly downloads the same randomly initialized model from the server and then trains locally without any communications. Then we implement two widely used FL baselines FedAvg $[ [ 2 7 ] ]$ and FedProx $\\vec { \\mathbb { B } } \\vec { \\mathsf { S } } \\vec { \\mathbb { I } }$ , the latter of which can deal with data and system heterogeneity in non-graph FL. For the graph classification model, we use the same GIN $\\mathbb { \\lVert \\boldsymbol { 4 3 } \\rVert }$ design, which represents the state-of-the-art GNN for graph-level tasks. We fix the GIN architecture and hyper-parameters through all baselines in order to control the experiments across different settings. ",
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+        "type": "text",
+        "text": "Parameter settings We use the three-layer GINs with hidden size of 64. We use a batch size of 128, and an Adam $\\mathbb { \\ m }$ optimizer with learning rate 0.001 and weight decay $5 e ^ { - 4 }$ . The $\\mu$ for FedProx is set to 0.01. For all FL methods, the local epoch $E$ is set to 1. The two important hyper-parameters $\\varepsilon _ { 1 }$ and $\\varepsilon _ { 2 }$ as clustering criteria vary in different groups of data, which are set through offline training for about 50 rounds following $\\pmb { \\mathbb { B 3 } } \\|$ . We run all experiments for five random repetitions on a server with 8 24GB NVIDIA TITAN RTX GPUs. ",
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+        "type": "text",
+        "text": "6.2 Experimental results ",
+        "text_level": 1,
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+        "text": "Federated graph classification within single datasets Conceptually, clients in this setting are more homogeneous. As can be seen from the results in Table ${ \\bar { 3 } } ,$ our framework can obviously improve the performance of graph classification over local clients. For the NCI1 dataset distributed on 30 clients, GCFL and ${ \\mathrm { G C F L } } +$ achieve $1 3 . 2 7 \\%$ and $1 4 . 7 5 \\%$ performance gains over self-train on average, and GCFL and ${ \\mathrm { G C F L } } +$ help 10-14 more clients than FedAvg and FedProx who fail to improve about half of clients. For the PROTEINS dataset on the total 10 clients, GCFL and ${ \\mathrm { G C F L } } +$ achieve $7 . 2 9 \\%$ and $7 . 8 1 \\%$ average performance gains compared to self-train. For IMDB-BINARY on 10 clients, FedAvg and FedProx fail to help 5/10 and 4/10 clients respectively, while both GCFL and ${ \\mathrm { G C F L } } +$ are able to improve all 10 clients. Overall, FedAvg can only help around half of the clients, which demonstrates that FedAvg can be ineffective even for decenrtalized graphs from a single dataset, because of the graph non-IIDness as shown in Table $2 .$ In addition, in all three datasets, the minimum performance gain of clients over self-train using GCFL or ${ \\mathrm { G C F L } } +$ is obviously larger than the minimum performance gain using FedAvg and FedProx. It indicates that even when some clients do not improve from self-train by using GCFL or ${ \\mathrm { G C F L } } +$ , they can achieve more comparable performance as self-train than using FedAvg and FedProx. These experimental results demonstrate that our frameworks are effective on the single-dataset multi-client FL setting. ",
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+    {
+        "type": "text",
+        "text": "Federated graph classification across multiple datasets According to our data analysis in Tables 1 and $2 ,$ clients in such a setting are more heterogeneous. We conduct experiments with multiple datasets in two settings: single domain (using the data group MOLECULES), and across domains (using the data groups BIOCHEM and MIX). As can be seen from the results in Table 4, our frameworks GCFL and ${ \\mathrm { G C F L } } +$ can effectively improve the performance of clients with distinct datasets. The results show $1 . 7 \\% - 2 . 7 \\%$ improvements of our frameworks compared to self-train. In all three data groups, our ${ \\mathrm { G C F L } } +$ framework can improve twice as many as clients than FedAvg, and it achieves a ratio of $1 0 0 \\%$ in MOLECULES to improve all clients’ performance. The FedAvg failed to improve around $6 0 \\%$ clients, which further demonstrates its ineffectiveness facing graph non-IIDness. Additionally, the ${ \\mathrm { G C F L } } +$ framework also outperforms GCFL. In MIX, although GCFL can achieve the same ratio of improved clients as ${ \\mathrm { G C F L } } +$ , the ${ \\mathrm { G C F L } } +$ framework has a much larger minimum gain of clients than GCFL. It indicates that by ${ \\mathrm { G C F L } } +$ few clients that cannot benefit from others will not be degraded through the collaborating. These results indicate that graphs across datasets or even across domains are able to help each other through proper FL, which is a surprising and interesting start point for further study. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Effects of hyper-parameters $\\varepsilon _ { 1 }$ and $\\varepsilon _ { 2 }$ The hyper-parameter $\\varepsilon _ { 1 }$ is a stopping criterion for checking whether a general FL on the current set of clients is near the stationary point. Theoretically, $\\varepsilon _ { 1 }$ should be set as small as possible. The hyper-parameter $\\varepsilon _ { 2 }$ is more dependent on the number of clients and the heterogeneity among them. A smaller $\\varepsilon _ { 2 }$ will make the clients more likely to be clustered. When $\\varepsilon _ { 1 }$ and $\\varepsilon _ { 2 }$ are in the feasible ranges, their small variation can have little effect on the performance because the clustering results would largely remain the same. When $\\varepsilon _ { 2 }$ is set too large, the performance will be similar as applying a basic FL algorithm directly (i.e. with a single cluster). When $\\varepsilon _ { 2 }$ is set too small, more clusters with smaller sizes or even single clients will be generated. We provide additional experimental results regarding the performance of GCFL and ${ \\mathrm { G C F L } } +$ w.r.t. varying $\\varepsilon _ { 1 }$ and $\\varepsilon _ { 2 }$ in Figure $2 .$ As shown in the subfigures, some points that represent varying $\\varepsilon _ { 2 }$ overlap for each $\\varepsilon _ { 1 }$ , which indicates that varying $\\varepsilon _ { 2 }$ in a certain range w.r.t. the fixed $\\varepsilon _ { 1 }$ leads to similar performance. Looking at a $\\varepsilon _ { 2 }$ , within a certain range of $\\varepsilon _ { 1 }$ , we can find the performance often fluctuating within a 0.01 variance. The results show that the performance of GCFL and ${ \\mathrm { G C F L } } +$ are not very sensitive to the changes of $\\varepsilon _ { 1 }$ and $\\varepsilon _ { 2 }$ in reasonable ranges. ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/cf3c47d7e32cf93fcb981c885f6a4293e8f31e0e03c72c959557169dd3f47981.jpg",
+        "table_caption": [
+            "Table 3: Performance on the single-dataset-multi-client setting. We present the average accuracy and minimum gain over self-train on all clients, as well as the ratio of clients which get improved. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Dataset (# clients) Accuracy</td><td colspan=\"3\">NCI1 (30)</td><td colspan=\"3\">PROTEINS (10)</td><td colspan=\"3\">IMDB-BINARY (10)</td></tr><tr><td>average</td><td>min gain</td><td>ratio</td><td>average</td><td>min gain</td><td>ratio</td><td>average</td><td>min gain</td><td>ratio</td></tr><tr><td>self-train</td><td>0.6468(±0.053)</td><td></td><td></td><td>0.7213(±0.058)</td><td></td><td></td><td>0.7654(±0.057)</td><td></td><td></td></tr><tr><td>FedAvg</td><td>0.6474(±0.076)</td><td>-0.1333</td><td>14/30</td><td>0.7490(±0.034)</td><td>-0.0615</td><td>6/10</td><td>0.7596(±0.049)</td><td>-0.0800</td><td>5/10</td></tr><tr><td>FedProx</td><td>0.6437(±0.072)</td><td>-0.2400</td><td>16/30</td><td>0.7556(±0.036)</td><td>-0.0923</td><td>7/10</td><td>0.7746(±0.048)</td><td>-0.0600</td><td>6/10</td></tr><tr><td>GCFL</td><td>0.7326(±0.052)</td><td>-0.0462</td><td>26/30</td><td>0.7739(±0.043)</td><td>-0.0545</td><td>8/10</td><td>0.8256(±0.059)</td><td>0.0182</td><td>10/10</td></tr><tr><td>GCFL+</td><td>0.7422(±0.053)</td><td>-0.1143</td><td>28/30</td><td>0.7776(±0.037)</td><td>-0.0154</td><td>9/10</td><td>0.8299(±0.052)</td><td>0.0167</td><td>10/10</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/0066cb938d8f2dcc681f721dccfa0d721185e00dd89245c1673ce8bf06c0baef.jpg",
+        "table_caption": [
+            "Table 4: Performance on the multi-dataset-multi-client setting. Metrics are the same as Table 3. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Dataset (# domains) Accuracy</td><td colspan=\"3\">MOLECULES (1)</td><td colspan=\"3\">BIOCHEM (2)</td><td colspan=\"3\">MIX (3)</td></tr><tr><td>average</td><td>min gain</td><td>ratio</td><td>average</td><td>min gain</td><td>ratio</td><td>average</td><td>min gain</td><td>ratio</td></tr><tr><td>self-train</td><td>0.7543(±0.017)</td><td></td><td></td><td>0.7129(±0.016)</td><td></td><td></td><td>0.7001(±0.034)</td><td>一</td><td>一</td></tr><tr><td>FedAvg</td><td>0.7524(±0.026)</td><td>-0.0132</td><td>3/7</td><td>0.6944(±0.027)</td><td>-0.1467</td><td>4/10</td><td>0.6886(±0.023)</td><td>-0.1233</td><td>5/13</td></tr><tr><td>FedProx</td><td>0.7668(±0.032)</td><td>-0.0054</td><td>5/7</td><td>0.7053(±0.026)</td><td>-0.1000</td><td>5/10</td><td>0.6897(±0.026)</td><td>-0.1367</td><td>5/13</td></tr><tr><td>GCFL</td><td>0.7661(±0.016)</td><td>0.0010</td><td>77</td><td>0.7172(±0.019)</td><td>-0.0700</td><td>7/10</td><td>0.7056(±0.019)</td><td>-0.1400</td><td>10/13</td></tr><tr><td>GCFL+</td><td>0.7745(±0.030)</td><td>0.0010</td><td>77</td><td>0.7312(±0.031)</td><td>-0.0300</td><td>8/10</td><td>0.7121(±0.021)</td><td>-0.0233</td><td>10/13</td></tr></table>",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "6.3 Structure and feature analysis in clusters ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We conduct an in-depth analysis to explore the clustering results of GCFL and ${ \\mathrm { G C F L } } +$ . As can be seen in Figure $^ { 3 , }$ after being clustered by GCFL and ${ \\mathrm { G C F L } } +$ , the overall structure and feature heterogeneity of clients’ graphs within clusters are reduced significantly compared to the original values, especially for the multiple dataset setting (Figure 3c and $3 \\mathrm { \\check { d } } )$ . For the one dataset setting (Figure 3a and $\\textcircled { 3 6 }$ , since features all fall in the same space, pairs of clients tend to have more homogeneous features. Therefore, the feature heterogeneity only gets reduced slightly after clustering. Unlike feature heterogeneity, the structure heterogeneity within clusters decreases significantly. In the setting of multiple datasets, as shown in Figure $3 \\mathrm { c }$ and 3d, both structure and feature heterogeneity decrease significantly, which is intuitive since datasets across domains usually tend to have higher heterogeneity, as discussed in 4.1. We also look into the clusters and find that datasets from the same domains are more likely to be clustered together, while datasets from different domains also constantly get clustered together and benefit each other. For example, the clustering of ${ \\mathrm { G C F L } } +$ corresponding to Figure 3d groups two social networks COLLAB and IMDB-BINARY together with PROTEINS and also several molecules datasets, and there is also a cluster of NCI1, DD, and IMDB-MULTI which are molecules, proteins and social networks, respectively. These analysis manifests that domains of datasets can verify the sanity of clusters to some extent, but one cannot solely rely on such prior knowledge to determine the optimal clusters, which demonstrates the necessity of our frameworks with the ability of performance-driven dynamic clustering along the process of FL. ",
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+    {
+        "type": "image",
+        "img_path": "images/85b86777d3218d7a7a70c6401a5de233e241f3c00ebc553bfb92b5118e64d8b9.jpg",
+        "image_caption": [
+            "Figure 2: Performance of GCFL and ${ \\mathrm { G C F L } } +$ on MOLECULES w.r.t varying $\\varepsilon _ { 1 }$ and $\\varepsilon _ { 2 }$ "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/01e172976109936df459cfe72fffef5f8e9d533391f5838ef187596f36c6d3f9.jpg",
+        "image_caption": [
+            "Figure 3: Structure (blue) and feature (red) heterogeneity within clusters found by GCFL and ${ \\mathrm { G C F L } } +$ Dashed lines denote the heterogeneity over all clients before clustering. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
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+        "type": "image",
+        "img_path": "images/cfe40ccd78fb6af12529ae3711b0d04adfd9ecad8ee25983bbffc6ebe00bc656.jpg",
+        "image_caption": [
+            "Figure 4: Average with standard deviation of the training curves of all clients. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "6.4 Convergence analysis ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We visualize the testing loss with respect to the communication round to show the convergence of GCFL and ${ \\mathrm { G C F L } } +$ compared with the standard federated learning baselines. Figure $\\boxed { 4 }$ shows the training curves on two settings, which illustrates that GCFL and ${ \\mathrm { G C F L } } +$ achieves similar convergence rate as FedProx, which is the state-of-the-art FL framework dealing with non-IID Euclidean data. We also notice that both GCFL, ${ \\mathrm { G C F L } } +$ and FedProx can converge to a lower loss compared with FedAvg, which corroborates our consideration of the non-IID problem in our setting. ",
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+    },
+    {
+        "type": "text",
+        "text": "6.5 More results in Appendix ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In Table 3 and 4, we averaged the accuracy across all clients for presentation simplicity. To understand the detailed performance by clients and clusters, we present different Violin plots in Appendix D. Besides, we also show more results regarding various settings (overlapping clients, real vs. synthetic node features, standardized gradient-sequence matrix in ${ \\mathrm { G C F L } } +$ , etc) in Appendix D. ",
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+    },
+    {
+        "type": "text",
+        "text": "7 Conclusion ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this work, we propose a novel setting of cross-dataset and cross-domain federated graph classification. The techniques (GCFL and $\\mathrm { G C F L + }$ ) we develop allow multiple data owners holding structure and feature non-IID graphs to collaboratively train powerful graph classification neural networks without the need of direct data sharing. As the first trial, we focus on the effectiveness of FL in this setting and have not carefully studied other issues such as data privacy, although it is intuitive to preserve the privacy of clients by introducing an encryption mechanism (e.g. applying orthonormal transformations), and to prevent from adversarial scenarios by clustering out the malicious clients. Due to its evident motivations and proofs on the effective FL in a new setting, we believe this work can serve as a stepping stone for many interesting future studies. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgments and Disclosure of Funding ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            823,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "The work is partially supported by National Science Foundation (NSF) under CNS-2124104, CNS2125530, CNS-1952192, and IIS-1838200, National Institute of Health (NIH) under R01GM118609 and UL1TR002378, and the internal funding and GPU servers provided by the Computer Science Department of Emory University. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "References   \n[1] Jean Bourgain. On lipschitz embedding of finite metric spaces in hilbert space. Israel Journal of Mathematics, 52 (1):46–52, 1985.   \n[2] Christopher Briggs, Zhong Fan, and Peter Andras. Federated learning with hierarchical clustering of local updates to improve training on non-iid data. In IJCNN, 2020.   \n[3] Debora Caldarola, Massimiliano Mancini, Fabio Galasso, Marco Ciccone, Emanuele Rodolà, and Barbara Caputo. Cluster-driven graph federated learning over multiple domains. In CVPRW, 2021.   \n[4] Fei Chen, Mi Luo, Zhenhua Dong, Zhenguo Li, and Xiuqiang He. Federated meta-learning with fast convergence and efficient communication. arXiv preprint arXiv:1802.07876, 2018.   \n[5] Mingyang Chen, Wen Zhang, Zonggang Yuan, Yantao Jia, and Huajun Chen. Fede: Embedding knowledge graphs in federated setting. arXiv preprint arXiv:2010.12882, 2020.   \n[6] Canh T. Dinh, Nguyen H. Tran, and Tuan Dung Nguyen. 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