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+# IMPROVING NEURAL LANGUAGE MODELS WITH A CONTINUOUS CACHE
+
+Edouard Grave, Armand Joulin, Nicolas Usunier Facebook AI Research {egrave,ajoulin,usunier}@fb.com
+
+# ABSTRACT
+
+We propose an extension to neural network language models to adapt their prediction to the recent history. Our model is a simplified version of memory augmented networks, which stores past hidden activations as memory and accesses them through a dot product with the current hidden activation. This mechanism is very efficient and scales to very large memory sizes. We also draw a link between the use of external memory in neural network and cache models used with count based language models. We demonstrate on several language model datasets that our approach performs significantly better than recent memory augmented networks.
+
+# 1 INTRODUCTION
+
+Language models, which are probability distributions over sequences of words, have many applications such as machine translation (Brown et al., 1993), speech recognition (Bahl et al., 1983) or dialogue agents (Stolcke et al., 2000). While traditional neural networks language models have obtained state-of-the-art performance in this domain (Jozefowicz et al., 2016; Mikolov et al., 2010), they lack the capacity to adapt to their recent history, limiting their application to dynamic environments (Dodge et al., 2015). A recent approach to solve this problem is to augment these networks with an external memory (Graves et al., 2014; Grefenstette et al., 2015; Joulin & Mikolov, 2015; Sukhbaatar et al., 2015). These models can potentially use their external memory to store new information and adapt to a changing environment.
+
+While these networks have obtained promising results on language modeling datasets (Sukhbaatar et al., 2015), they are quite computationally expensive. Typically, they have to learn a parametrizable mechanism to read or write to memory cells (Graves et al., 2014; Joulin & Mikolov, 2015). This may limit both the size of their usable memory as well as the quantity of data they can be trained on. In this work, we propose a very light-weight alternative that shares some of the properties of memory augmented networks, notably the capability to dynamically adapt over time. By minimizing the computation burden of the memory, we are able to use larger memory and scale to bigger datasets. We observe in practice that this allows us to surpass the perfomance of memory augmented networks on different language modeling tasks.
+
+Our model share some similarities with a model proposed by Kuhn (1988), called the cache model. A cache model stores a simple representation of the recent past, often in the form of unigrams, and uses them for prediction (Kuhn & De Mori, 1990). This contextual information is quite cheap to store and can be accessed efficiently. It also does not need any training and can be appplied on top of any model. This makes this model particularly interesting for domain adaptation (Kneser & Steinbiss, 1993).
+
+Our main contribution is to propose a continuous version of the cache model, called Neural Cache Model, that can be adapted to any neural network language model. We store recent hidden activations and use them as representation for the context. Using simply a dot-product with the current hidden activations, they turn out to be extremely informative for prediction. Our model requires no training and can be used on any pre-trained neural networks. It also scales effortlessly to thousands of memory cells. We demonstrate the quality of the Neural Cache models on several language model tasks and the LAMBADA dataset (Paperno et al., 2016).
+
+# 2 LANGUAGE MODELING
+
+A language model is a probability distribution over sequences of words. Let $V$ be the size of the vocabulary; each word is represented by a one-hot encoding vector $x$ in $\mathbb { R } ^ { V } = \nu$ , corresponding to its index in the vocabulary. Using the chain rule, the probability assigned to a sequence of words $x _ { 1 } , \ldots , x _ { T }$ can be factorized as
+
+$$
+p ( x _ { 1 } , . . . , x _ { T } ) = \prod _ { t = 1 } ^ { T } p ( x _ { t } \mid x _ { t - 1 } , . . . , x _ { 1 } ) .
+$$
+
+Language modeling is often framed as learning the conditional probability over words, given the history (Bahl et al., 1983).
+
+This conditional probability is traditionally approximated with non-parameteric models based on counting statistics (Goodman, 2001). In particular, smoothed N-gram models (Katz, 1987; Kneser & Ney, 1995) achieve good performance in practice (Mikolov et al., 2011). Parametrized alternatives are either maximum entropy language models (Rosenfeld, 1996), feedforward networks (Bengio et al., 2003) or recurrent networks (Mikolov et al., 2010). In particular, recurrent networks are currently the best solution to approximate this conditional probability, achieving state-of-the-arts performance on standard language modeling benchmarks (Jozefowicz et al., 2016; Zilly et al., 2016).
+
+Recurrent networks. Assuming that we have a vector $h _ { t } \in \mathbb { R } ^ { d }$ encoding the history $x _ { t } , . . . , x _ { 1 }$ , the conditional probability of a word $w$ can be parametrized as
+
+$$
+p _ { v o c a b } ( w \mid x _ { t } , . . . , x _ { 1 } ) \propto \exp ( h _ { t } ^ { \top } o _ { w } ) .
+$$
+
+The history vector $h _ { t }$ is computed by a recurrent network by recursively applying an equation of the form
+
+$$
+h _ { t } = \Phi \left( x _ { t } , h _ { t - 1 } \right) ,
+$$
+
+where $\Phi$ is a function depending on the architecture of the network. Several architecture for recurrent networks have been proposed, such as the Elman network (Elman, 1990), the long short-term memory (LSTM) (Hochreiter & Schmidhuber, 1997) or the gated recurrent unit (GRU) (Chung et al., 2014). One of the simplest recurrent networks is the Elman network (Elman, 1990), where
+
+$$
+h _ { t } = \sigma \left( L x _ { t } + R h _ { t - 1 } \right) ,
+$$
+
+where $\sigma$ is a non-linearity such as the logistic or tanh functions, $L \in \mathbb { R } ^ { d \times V }$ is a word embedding matrix and $R \in \mathbb { R } ^ { d \times d }$ is the recurrent matrix. The LSTM architecture is particularly interesting in the context of language modelling (Jozefowicz et al., 2016) and we refer the reader to Graves et al. (2013) for details on this architecture.
+
+The parameters of recurrent neural network language models are learned by minimizing the negative log-likelihood of the training data. This objective function is usually minimized by using the stochastic gradient descent algorithm, or variants such as Adagrad (Duchi et al., 2011). The gradient is computed using the truncated backpropagation through time algorithm (Werbos, 1990; Williams & Peng, 1990).
+
+Cache model. After a word appears once in a document, it is much more likely to appear again. As an example, the frequency of the word tiger on the Wikipedia page of the same name is $2 . 8 \%$ , compared to $0 . 0 0 3 7 \%$ over the whole Wikipedia. Cache models exploit this simple observation to improve $n$ -gram language models by capturing long-range dependencies in documents. More precisely, these models have a cache component, which contains the words that appeared in the recent history (either the document or a fixed number of words). A simple language model, such as a unigram or smoothed bigram model, is fitted on the words of the cache and interpolated with the static language model (trained over a larger dataset). This technique has many advantages. First, this is a very efficient way to adapt a language model to a new domain. Second, such models can predict out-of-vocabulary words (OOV words), after seeing them once. Finally, this helps capture long-range dependencies in documents, in order to generate more coherent text.
+
+![](images/cf4652e06ce92d3dc252856a2cd9705b2ca2a4147643ec3cc2c62263c4eaeb03.jpg)  
+Figure 1: The neural cache stores the previous hidden states in memory cells. They are then used as keys to retrieve their corresponding word, that is the next word. There is no transformation applied to the storage during writing and reading.
+
+# 3 NEURAL CACHE MODEL
+
+The Neural Cache Model adds a cache-like memory to neural network language models. It exploits the hidden representations $h _ { t }$ to define a probability distribution over the words in the cache. As illustrated Figure 1, the cache stores pairs $\left( h _ { i } , x _ { i + 1 } \right)$ of a hidden representation, and the word which was generated based on this representation (we remind the reader that the vector $h _ { i }$ encodes the history $x _ { i } , . . . , x _ { 1 } )$ . At time $t$ , we then define a probability distribution over words stored in the cache based on the stored hidden representations and the current one $h _ { t }$ as
+
+$$
+p _ { c a c h e } ( w \mid h _ { 1 . . t } , ~ x _ { 1 . . t } ) \propto \sum _ { i = 1 } ^ { t - 1 } \mathbb { 1 } _ { \{ w = x _ { i + 1 } \} } \exp ( \theta h _ { t } ^ { \top } h _ { i } )
+$$
+
+where the scalar $\theta$ is a parameter which controls the flatness of the distribution. When $\theta$ is equal to zero, the probability distribution over the history is uniform, and our model is equivalent to a unigram cache model (Kuhn & De Mori, 1990).
+
+From the point of view of memory-augmented neural networks, the probability $p _ { c a c h e } ( w \mid h _ { 1 . . t } , \ x _ { 1 . . t } )$ given by the neural cache model can be interpreted as the probability to retrieve the word $w$ from the memory given the query $h _ { t }$ , where the desired answer is the next word $x _ { t + 1 }$ . Using previous hidden states as keys for the words in the memory, the memory lookup operator can be implemented with simple dot products between the keys and the query. In contrast to existing memory-augmented neural networks, the neural cache model avoids the need to learn the memory lookup operator. Such a cache can thus be added to a pre-trained recurrent neural language model without fine tuning of the parameters, and large cache size can be used with negligible impact on the computational cost of a prediction.
+
+Neural cache language model. Following the standard practice in n-gram cache-based language models, the final probability of a word is given by the linear interpolation of the cache language model with the regular language model, obtaining:
+
+$$
+p ( w \mid h _ { 1 . . t } , ~ x _ { 1 . . t } ) = ( 1 - \lambda ) p _ { v o c a b } ( w \mid h _ { t } ) + \lambda p _ { c a c h e } ( w \mid h _ { 1 . . t } , x _ { 1 . . t } ) .
+$$
+
+Instead of taking a linear interpolation between the two distribution with a fixed $\lambda$ , we also consider a global normalization over the two distribution:
+
+$$
+p ( w \mid h _ { 1 . . t } , ~ x _ { 1 . . t } ) \propto \left( \exp ( h _ { t } ^ { \top } o _ { w } ) + \sum _ { i = 1 } ^ { t - 1 } \mathbb { 1 } _ { \{ w = x _ { i + 1 } \} } \exp ( \theta h _ { t } ^ { \top } h _ { i } + \alpha ) \right) .
+$$
+
+This corresponds to taking a softmax over the vocabulary and the words in the cache. The parameter $\alpha$ controls the weight of the cache component, and is the counterpart of the $\lambda$ parameter for linear interpolation.
+
+The addition of the neural cache to a recurrent neural language model inherits the advantages of $n$ - gram caches in usual cache-based models: The probability distribution over words is updated online depending on the context, and out-of-vocabulary words can be predicted as soon as they have been seen at least once in the recent history. The neural cache also inherits the ability of the hidden states of recurrent neural networks to model longer-term contexts than small $n$ -grams, and thus allows for a finer modeling of the current context than e.g., unigram caches.
+
+![](images/89dee18a863e0919f58a31c0b0fced0db90e513f3d196f09ed8d2380bb79b7fb.jpg)  
+Figure 2: Perplexity on the validation set of Penn Tree Bank for linear interpolation (left) and global normalization (right), for various values of hyperparameters $\theta$ , $\lambda$ and $\alpha$ . We use a cache model of size 500. The base model has a validation perplexity of 86.9. The best linear interpolation has a perplexity of 74.6, while the best global normalization has a perplexity of 74.9.
+
+Table 1: Test perplexity on the Penn Tree Bank.   
+
+<table><tr><td>Model</td><td>Test PPL</td></tr><tr><td>RNN+LSA+KN5+cache (Mikolov &amp; Zweig, 2012)</td><td>90.3</td></tr><tr><td>LSTM (Zaremba et al., 2014)</td><td>78.4</td></tr><tr><td>Variational LSTM (Gal &amp; Ghahramani, 2015)</td><td>73.4</td></tr><tr><td>Recurrent Highway Network (Zilly et al., 2016)</td><td>66.0</td></tr><tr><td>Pointer Sentinel LSTM (Merity et al.,2016)</td><td>70.9</td></tr><tr><td>LSTM (our implem.)</td><td>82.3</td></tr><tr><td>Neural cache model</td><td>72.1</td></tr></table>
+
+Training procedure. For now, we first train the (recurrent) neural network language model, without the cache component. We only apply the cache model at test time, and choose the hyperparameters $\theta$ and $\lambda$ (or $\alpha$ ) on the validation set. A big advantage of our method is that it is very easy and cheap to apply, with already trained neural models. There is no need to perform backpropagation over large contexts, and we can thus apply our method with large cache sizes (larger than one thousand).
+
+# 4 RELATED WORK
+
+Cache model. Adding a cache to a language model was intoducted in the context of speech recognition(Kuhn, 1988; Kupiec, 1989; Kuhn & De Mori, 1990). These models were further extended by Jelinek et al. (1991) into a smoothed trigram language model, reporting reduction in both perplexity and word error rates. Della Pietra et al. (1992) adapt the cache to a general $n$ -gram model such that it satisfies marginal constraints obtained from the current document.
+
+Adaptive language models. Other adaptive language models have been proposed in the past: Kneser & Steinbiss (1993) and Iyer & Ostendorf (1999) dynamically adapt the parameters of their model to the recent history using different weight interpolation schemes. Bellegarda (2000) and Coccaro & Jurafsky (1998) use latent semantic analysis to adapt their models to the current context. Similarly, topic features have been used with either maximum entropy models (Khudanpur & Wu, 2000) or recurrent networks (Mikolov & Zweig, 2012; Wang & Cho, 2015). Finally, Lau et al. (1993) proposes to use pairs of distant of words to capture long-range dependencies.
+
+Memory augmented neural networks. In the context of sequence prediction, several memory augmented neural networks have obtained promising results (Sukhbaatar et al., 2015; Graves et al., 2014; Grefenstette et al., 2015; Joulin & Mikolov, 2015). In particular, Sukhbaatar et al. (2015) stores a representation of the recent past and accesses it using an attention mechanism Bahdanau et al. (2014). Sukhbaatar et al. (2015) shows that this reduces the perplexity for language modeling.
+
+![](images/1f2579aa6e9ffc10db91676682d66b0fb2f6c4f3c580061eb001161b8e41d356.jpg)  
+Figure 3: Perplexity on the validation set of wikitext2 for linear interpolation (left) and global normalization (right), for various values of hyperparameters $\theta$ , $\lambda$ and $\alpha$ . We use a cache model of size 2000. The base model has a validation perplexity of 104.2. The best linear interpolation has a perplexity of 72.1, while the best global normalization has a perplexity of 73.5.
+
+<table><tr><td>Model</td><td>wikitext2</td><td>wikitext103</td></tr><tr><td>Zoneout + Variational LSTM (Merity et al.,2016)</td><td>100.9</td><td></td></tr><tr><td>Pointer Sentinel LSTM (Merity et al., 2016)</td><td>80.8</td><td>1</td></tr><tr><td>LSTM (our implementation)</td><td>99.3</td><td>48.7</td></tr><tr><td>Neural cache model (size = 100)</td><td>81.6</td><td>44.8</td></tr><tr><td>Neural cache model (size = 2,000)</td><td>68.9</td><td>40.8</td></tr></table>
+
+Table 2: Test perplexity on the wikitext datasets. The two datasets share the same validation and test sets, making all the results comparable.
+
+This approach has been successfully applied to question answering, when the answer is contained in a given paragraph (Chen et al., 2016; Hermann et al., 2015; Kadlec et al., 2016; Sukhbaatar et al., 2015). Similarly, Vinyals et al. (2015) explores the use of this mechanism to reorder sequences of tokens. Their network uses an attention (or “pointer”) over the input sequence to predict which element should be selected as the next output. Gulcehre et al. (2016) have shown that a similar mechanism called pointer softmax could be used in the context of machine translation, to decide which word to copy from the source to target.
+
+Independently of our work, Merity et al. (2016) apply the same mechanism to recurrent network. Unlike our work, they uses the current hidden activation as a representation of the current input (while we use it to represent the output). This requires additional learning of a transformation between the current representation and those in the past. The advantage of our approach is that we can scale to very large caches effortlessly.
+
+# 5 EXPERIMENTS
+
+In this section, we evaluate our method on various language modeling datasets, which have different sizes and characteristics. On all datasets, we train a static recurrent neural network language model with LSTM units. We then use the hidden representations from this model to obtain our cache, which is interpolated with the static LSTM model. We also evaluate a unigram cache model interpolated with the static model as another baseline.
+
+# 5.1 SMALL SCALE EXPERIMENTS
+
+Datasets. In this section, we describe experiments performed on two small datasets: the Penn Tree Bank (Marcus et al., 1993) and the wikitext2 (Merity et al., 2016) datasets. The Penn Tree Bank dataset is made of articles from the Wall Street Journal, contains $9 2 9 \mathrm { k }$ training tokens and has a vocabulary size of 10k. The wikitext2 dataset is derived from Wikipedia articles, contains 2M training tokens and has a vocabulary size of 33k. These datasets contain non-shuffled documents, therefore requiring models to capture inter-sentences dependencies to perform well.
+
+![](images/239b837a7406cfdca609786eda77fbccc139af2404a2727b6d8c5998ce090737.jpg)  
+Figure 4: Test perplexity as a function of the number of words in the cache, for our method and a unigram cache baseline. We observe that our approach can uses larger caches than the baseline.
+
+Implementation details. We train recurrent neural network language models with 1024 LSTM units, regularized with dropout (probability of dropping out units equals to 0.65). We use the Adagrad algorithm, with a learning rate of 0.2, a batchsize of 20 and initial weight uniformly sampled in the range $[ - 0 . 0 5 , 0 . 0 5 ]$ . We clip the norm of the gradient to 0.1 and unroll the network for 30 steps. We consider cache sizes on a logarithmic scale, from 50 to 10, 000, and fit the cache hyperparameters on the validation set.
+
+Results. We report the perplexity on the validation sets in Figures 2 and 3, for various values of hyperparameters, for linear interpolation and global normalization. First, we observe that on both datasets, the linear interpolation method performs slightly better than the global normalization approach. It is also easier to apply in practice, and we thus use this method in the remainder of this paper. In Tables 1 and 2, we report the test perplexity of our approach and state-of-the-art models. Our approach is competitive with previous models, in particular with the pointer sentinel LSTM model of Merity et al. (2016). On Penn Tree Bank, we note that the improvement over the base model is similar for both methods. On the wikitext2 dataset, both methods obtain similar results when using the same cache size (100 words). Since our method is computationally cheap, it is easy to increase the cache to larger values (2, 000 words), leading to dramatic improvements ( $3 0 \%$ over the baseline, $1 2 \%$ over a small cache of 100 words).
+
+# 5.2 MEDIUM SCALE EXPERIMENTS
+
+Datasets and implementation details. In this section, we describe experiments performed over two medium scale datasets: text8 and wikitext103. Both datasets are derived from Wikipedia, but different pre-processing were applied. The text8 dataset contains 17M training tokens and has a vocabulary size of 44k words, while the wikitext103 dataset has a training set of size 103M, and a vocabulary size of 267k words. We use the same setting as in the previous section, except for the batchsize (we use 128) and dropout parameters (we use 0.45 for text8 and 0.25 for wikitext103). Since both datasets have large vocabularies, we use the adaptive softmax (Grave et al., 2016) for faster training.
+
+Results. We report the test perplexity as a function of the cache size in Figure 4, for the neural cache model and a unigram cache baseline. We observe that our approach can exploits larger cache sizes, compared to the baseline. In Table 2, we observe that the improvement in perplexity of our method over the LSTM baseline on wikitext103 is smaller than for wikitext2 (approx. $1 6 \%$ v.s. $3 0 \%$ ). The fact that improvements obtained with more advanced techniques decrease when the size of training data increases has already been observed by Goodman (2001). Both wikitext datasets sharing the same test set, we also observe that the LSTM baseline, trained on 103M tokens (wikitext103), strongly outperforms more sophisticated methods, trained on 2M tokens (wikitext2). For these two reasons, we believe that it is important to evaluate and compare methods on relatively large datasets.
+
+<table><tr><td>Model</td><td>Test</td></tr><tr><td>LSTM-500 (Mikolov et al., 2014) SCRNN (Mikolov et al., 2014)</td><td>156</td></tr><tr><td>MemNN (Sukhbaatar et al., 2015)</td><td>161 147</td></tr><tr><td>LSTM-1024 (our implem.)</td><td>121.8</td></tr><tr><td>Neural cache model</td><td>99.9</td></tr></table>
+
+(a) text8
+
+<table><tr><td>Model</td><td>Dev</td><td>Ctrl</td></tr><tr><td>WB5 (Paperno et al., 2016) WB5+cache (Paperno et al., 2016)</td><td>3125 768</td><td>285</td></tr><tr><td>LSTM-512 (Paperno et al., 2016)</td><td>5357</td><td>270 149</td></tr><tr><td>LSTM-1024 (our implem.)</td><td>4088</td><td>94</td></tr><tr><td>Neural cache model</td><td>138</td><td>129</td></tr></table>
+
+(b) lambada
+
+![](images/3927384ae624d65f7ffe727b8d722a996fbce28ae8ec55bfda11221b9c005b0a.jpg)  
+Table 3: Perplexity on the text8 and lambada datasets. WB5 stands for 5-gram language model with Witten-Bell smoothing.   
+Figure 5: Perplexity on the development and control sets of lambada, as a function of the interpolation parameters $\lambda$ .
+
+# 5.3 EXPERIMENTS ON THE LAMBADA DATASET
+
+Finally, we report experiments carried on the lambada dataset, introduced by Paperno et al. (2016). This is a dataset of short passages extracted from novels. The goal is to predict the last word of the excerpt. This dataset was built so that human subjects solve the task perfectly when given the full context (approx. 4.6 sentences), but fail to do so when only given the sentence with the target word. Thus, most state-of-the-art language models fail on this dataset. The lambada training set contains approximately 200M tokens and has a vocabulary size of 93, 215. We report results for our method in Table 3, as well the performance of baselines from Paperno et al. (2016). Adding a neural cache model to the LSTM baseline strongly improves the performance on the lambada dataset. We also observe in Figure 5 that the best interpolation parameter between the static model and the cache is not the same for the development and control sets. This is due to the fact that more than $83 \%$ of passages of the development set include the target word, while this is true for only $14 \%$ of the control set. Ideally, a model should have strong results on both sets. One possible generalization of our model would be to adapt the interpolation parameter based on the current vector representation of the history $h _ { t }$ .
+
+# 6 CONCLUSION
+
+We presented the neural cache model to augment neural language models with a longer-term memory that dynamically updates the word probablilities based on the long-term context. A neural cache can be added on top of a pre-trained language model at negligible cost. Our experiments on both language modeling tasks and the challenging LAMBADA dataset shows that significant performance gains can be expected by adding this external memory component.
+
+Technically, the neural cache models is similar to some recent memory-augmented neural networks such as pointer networks. However, its specific design makes it possible to avoid learning the memory lookup component. This makes the neural cache appealing since it can use larger cache sizes than memory-augment networks and can be applied as easily as traditional count-based caches.
+
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parse/train/B184E5qee/B184E5qee_content_list.json

@@ -0,0 +1,1343 @@
+[
+    {
+        "type": "text",
+        "text": "IMPROVING NEURAL LANGUAGE MODELS WITH A CONTINUOUS CACHE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Edouard Grave, Armand Joulin, Nicolas Usunier Facebook AI Research {egrave,ajoulin,usunier}@fb.com ",
+        "bbox": [
+            184,
+            170,
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+            212
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            250,
+            544,
+            263
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We propose an extension to neural network language models to adapt their prediction to the recent history. Our model is a simplified version of memory augmented networks, which stores past hidden activations as memory and accesses them through a dot product with the current hidden activation. This mechanism is very efficient and scales to very large memory sizes. We also draw a link between the use of external memory in neural network and cache models used with count based language models. We demonstrate on several language model datasets that our approach performs significantly better than recent memory augmented networks. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            438,
+            336,
+            454
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Language models, which are probability distributions over sequences of words, have many applications such as machine translation (Brown et al., 1993), speech recognition (Bahl et al., 1983) or dialogue agents (Stolcke et al., 2000). While traditional neural networks language models have obtained state-of-the-art performance in this domain (Jozefowicz et al., 2016; Mikolov et al., 2010), they lack the capacity to adapt to their recent history, limiting their application to dynamic environments (Dodge et al., 2015). A recent approach to solve this problem is to augment these networks with an external memory (Graves et al., 2014; Grefenstette et al., 2015; Joulin & Mikolov, 2015; Sukhbaatar et al., 2015). These models can potentially use their external memory to store new information and adapt to a changing environment. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "While these networks have obtained promising results on language modeling datasets (Sukhbaatar et al., 2015), they are quite computationally expensive. Typically, they have to learn a parametrizable mechanism to read or write to memory cells (Graves et al., 2014; Joulin & Mikolov, 2015). This may limit both the size of their usable memory as well as the quantity of data they can be trained on. In this work, we propose a very light-weight alternative that shares some of the properties of memory augmented networks, notably the capability to dynamically adapt over time. By minimizing the computation burden of the memory, we are able to use larger memory and scale to bigger datasets. We observe in practice that this allows us to surpass the perfomance of memory augmented networks on different language modeling tasks. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Our model share some similarities with a model proposed by Kuhn (1988), called the cache model. A cache model stores a simple representation of the recent past, often in the form of unigrams, and uses them for prediction (Kuhn & De Mori, 1990). This contextual information is quite cheap to store and can be accessed efficiently. It also does not need any training and can be appplied on top of any model. This makes this model particularly interesting for domain adaptation (Kneser & Steinbiss, 1993). ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Our main contribution is to propose a continuous version of the cache model, called Neural Cache Model, that can be adapted to any neural network language model. We store recent hidden activations and use them as representation for the context. Using simply a dot-product with the current hidden activations, they turn out to be extremely informative for prediction. Our model requires no training and can be used on any pre-trained neural networks. It also scales effortlessly to thousands of memory cells. We demonstrate the quality of the Neural Cache models on several language model tasks and the LAMBADA dataset (Paperno et al., 2016). ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "2 LANGUAGE MODELING ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "A language model is a probability distribution over sequences of words. Let $V$ be the size of the vocabulary; each word is represented by a one-hot encoding vector $x$ in $\\mathbb { R } ^ { V } = \\nu$ , corresponding to its index in the vocabulary. Using the chain rule, the probability assigned to a sequence of words $x _ { 1 } , \\ldots , x _ { T }$ can be factorized as ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/acc07e4a6c3e15f3bbe4cfe94568aefbe99a3059e2e8b12efb4c02cd4c8ea96d.jpg",
+        "text": "$$\np ( x _ { 1 } , . . . , x _ { T } ) = \\prod _ { t = 1 } ^ { T } p ( x _ { t } \\mid x _ { t - 1 } , . . . , x _ { 1 } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Language modeling is often framed as learning the conditional probability over words, given the history (Bahl et al., 1983). ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "This conditional probability is traditionally approximated with non-parameteric models based on counting statistics (Goodman, 2001). In particular, smoothed N-gram models (Katz, 1987; Kneser & Ney, 1995) achieve good performance in practice (Mikolov et al., 2011). Parametrized alternatives are either maximum entropy language models (Rosenfeld, 1996), feedforward networks (Bengio et al., 2003) or recurrent networks (Mikolov et al., 2010). In particular, recurrent networks are currently the best solution to approximate this conditional probability, achieving state-of-the-arts performance on standard language modeling benchmarks (Jozefowicz et al., 2016; Zilly et al., 2016). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Recurrent networks. Assuming that we have a vector $h _ { t } \\in \\mathbb { R } ^ { d }$ encoding the history $x _ { t } , . . . , x _ { 1 }$ , the conditional probability of a word $w$ can be parametrized as ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4c8c85ce02c3fed812b8ffc9cc7d49261d2e29dd724165813cc8b860431d7c7d.jpg",
+        "text": "$$\np _ { v o c a b } ( w \\mid x _ { t } , . . . , x _ { 1 } ) \\propto \\exp ( h _ { t } ^ { \\top } o _ { w } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The history vector $h _ { t }$ is computed by a recurrent network by recursively applying an equation of the form ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/01898cc49588f79d99cdd30f741fc05992c36dd8afbe5747ca8db0bb2f582f20.jpg",
+        "text": "$$\nh _ { t } = \\Phi \\left( x _ { t } , h _ { t - 1 } \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            434,
+            479,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $\\Phi$ is a function depending on the architecture of the network. Several architecture for recurrent networks have been proposed, such as the Elman network (Elman, 1990), the long short-term memory (LSTM) (Hochreiter & Schmidhuber, 1997) or the gated recurrent unit (GRU) (Chung et al., 2014). One of the simplest recurrent networks is the Elman network (Elman, 1990), where ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ab2c3d9dd446a4bab91926eca8341d3f0283146eb28bcd4fe45ec19154a40635.jpg",
+        "text": "$$\nh _ { t } = \\sigma \\left( L x _ { t } + R h _ { t - 1 } \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            416,
+            563,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $\\sigma$ is a non-linearity such as the logistic or tanh functions, $L \\in \\mathbb { R } ^ { d \\times V }$ is a word embedding matrix and $R \\in \\mathbb { R } ^ { d \\times d }$ is the recurrent matrix. The LSTM architecture is particularly interesting in the context of language modelling (Jozefowicz et al., 2016) and we refer the reader to Graves et al. (2013) for details on this architecture. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The parameters of recurrent neural network language models are learned by minimizing the negative log-likelihood of the training data. This objective function is usually minimized by using the stochastic gradient descent algorithm, or variants such as Adagrad (Duchi et al., 2011). The gradient is computed using the truncated backpropagation through time algorithm (Werbos, 1990; Williams & Peng, 1990). ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Cache model. After a word appears once in a document, it is much more likely to appear again. As an example, the frequency of the word tiger on the Wikipedia page of the same name is $2 . 8 \\%$ , compared to $0 . 0 0 3 7 \\%$ over the whole Wikipedia. Cache models exploit this simple observation to improve $n$ -gram language models by capturing long-range dependencies in documents. More precisely, these models have a cache component, which contains the words that appeared in the recent history (either the document or a fixed number of words). A simple language model, such as a unigram or smoothed bigram model, is fitted on the words of the cache and interpolated with the static language model (trained over a larger dataset). This technique has many advantages. First, this is a very efficient way to adapt a language model to a new domain. Second, such models can predict out-of-vocabulary words (OOV words), after seeing them once. Finally, this helps capture long-range dependencies in documents, in order to generate more coherent text. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/cf4652e06ce92d3dc252856a2cd9705b2ca2a4147643ec3cc2c62263c4eaeb03.jpg",
+        "image_caption": [
+            "Figure 1: The neural cache stores the previous hidden states in memory cells. They are then used as keys to retrieve their corresponding word, that is the next word. There is no transformation applied to the storage during writing and reading. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 NEURAL CACHE MODEL ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            291,
+            411,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The Neural Cache Model adds a cache-like memory to neural network language models. It exploits the hidden representations $h _ { t }$ to define a probability distribution over the words in the cache. As illustrated Figure 1, the cache stores pairs $\\left( h _ { i } , x _ { i + 1 } \\right)$ of a hidden representation, and the word which was generated based on this representation (we remind the reader that the vector $h _ { i }$ encodes the history $x _ { i } , . . . , x _ { 1 } )$ . At time $t$ , we then define a probability distribution over words stored in the cache based on the stored hidden representations and the current one $h _ { t }$ as ",
+        "bbox": [
+            173,
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f1b7d288810d0584e3e99ae63c335fa222c148f5a440e358a3c773da9d5e6dba.jpg",
+        "text": "$$\np _ { c a c h e } ( w \\mid h _ { 1 . . t } , ~ x _ { 1 . . t } ) \\propto \\sum _ { i = 1 } ^ { t - 1 } \\mathbb { 1 } _ { \\{ w = x _ { i + 1 } \\} } \\exp ( \\theta h _ { t } ^ { \\top } h _ { i } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            320,
+            409,
+            676,
+            452
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where the scalar $\\theta$ is a parameter which controls the flatness of the distribution. When $\\theta$ is equal to zero, the probability distribution over the history is uniform, and our model is equivalent to a unigram cache model (Kuhn & De Mori, 1990). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "From the point of view of memory-augmented neural networks, the probability $p _ { c a c h e } ( w \\mid h _ { 1 . . t } , \\ x _ { 1 . . t } )$ given by the neural cache model can be interpreted as the probability to retrieve the word $w$ from the memory given the query $h _ { t }$ , where the desired answer is the next word $x _ { t + 1 }$ . Using previous hidden states as keys for the words in the memory, the memory lookup operator can be implemented with simple dot products between the keys and the query. In contrast to existing memory-augmented neural networks, the neural cache model avoids the need to learn the memory lookup operator. Such a cache can thus be added to a pre-trained recurrent neural language model without fine tuning of the parameters, and large cache size can be used with negligible impact on the computational cost of a prediction. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Neural cache language model. Following the standard practice in n-gram cache-based language models, the final probability of a word is given by the linear interpolation of the cache language model with the regular language model, obtaining: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fba678248f6ba9ae5246555d1723736cb0a90f83870a6551296f554bd6da8797.jpg",
+        "text": "$$\np ( w \\mid h _ { 1 . . t } , ~ x _ { 1 . . t } ) = ( 1 - \\lambda ) p _ { v o c a b } ( w \\mid h _ { t } ) + \\lambda p _ { c a c h e } ( w \\mid h _ { 1 . . t } , x _ { 1 . . t } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Instead of taking a linear interpolation between the two distribution with a fixed $\\lambda$ , we also consider a global normalization over the two distribution: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/403e26b4e6492678125e29b85dc856092417258cba2a732365346b6598ee6fac.jpg",
+        "text": "$$\np ( w \\mid h _ { 1 . . t } , ~ x _ { 1 . . t } ) \\propto \\left( \\exp ( h _ { t } ^ { \\top } o _ { w } ) + \\sum _ { i = 1 } ^ { t - 1 } \\mathbb { 1 } _ { \\{ w = x _ { i + 1 } \\} } \\exp ( \\theta h _ { t } ^ { \\top } h _ { i } + \\alpha ) \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            256,
+            746,
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+            790
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "This corresponds to taking a softmax over the vocabulary and the words in the cache. The parameter $\\alpha$ controls the weight of the cache component, and is the counterpart of the $\\lambda$ parameter for linear interpolation. ",
+        "bbox": [
+            174,
+            791,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The addition of the neural cache to a recurrent neural language model inherits the advantages of $n$ - gram caches in usual cache-based models: The probability distribution over words is updated online depending on the context, and out-of-vocabulary words can be predicted as soon as they have been seen at least once in the recent history. The neural cache also inherits the ability of the hidden states of recurrent neural networks to model longer-term contexts than small $n$ -grams, and thus allows for a finer modeling of the current context than e.g., unigram caches. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/89dee18a863e0919f58a31c0b0fced0db90e513f3d196f09ed8d2380bb79b7fb.jpg",
+        "image_caption": [
+            "Figure 2: Perplexity on the validation set of Penn Tree Bank for linear interpolation (left) and global normalization (right), for various values of hyperparameters $\\theta$ , $\\lambda$ and $\\alpha$ . We use a cache model of size 500. The base model has a validation perplexity of 86.9. The best linear interpolation has a perplexity of 74.6, while the best global normalization has a perplexity of 74.9. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            207,
+            107,
+            794,
+            265
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "table",
+        "img_path": "images/543adbe3d8e87f4f1b8b8e6d36b015312b70af92a7c10e2eb7aff3c69e291e5d.jpg",
+        "table_caption": [
+            "Table 1: Test perplexity on the Penn Tree Bank. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Test PPL</td></tr><tr><td>RNN+LSA+KN5+cache (Mikolov &amp; Zweig, 2012)</td><td>90.3</td></tr><tr><td>LSTM (Zaremba et al., 2014)</td><td>78.4</td></tr><tr><td>Variational LSTM (Gal &amp; Ghahramani, 2015)</td><td>73.4</td></tr><tr><td>Recurrent Highway Network (Zilly et al., 2016)</td><td>66.0</td></tr><tr><td>Pointer Sentinel LSTM (Merity et al.,2016)</td><td>70.9</td></tr><tr><td>LSTM (our implem.)</td><td>82.3</td></tr><tr><td>Neural cache model</td><td>72.1</td></tr></table>",
+        "bbox": [
+            299,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Training procedure. For now, we first train the (recurrent) neural network language model, without the cache component. We only apply the cache model at test time, and choose the hyperparameters $\\theta$ and $\\lambda$ (or $\\alpha$ ) on the validation set. A big advantage of our method is that it is very easy and cheap to apply, with already trained neural models. There is no need to perform backpropagation over large contexts, and we can thus apply our method with large cache sizes (larger than one thousand). ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            621,
+            341,
+            637
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Cache model. Adding a cache to a language model was intoducted in the context of speech recognition(Kuhn, 1988; Kupiec, 1989; Kuhn & De Mori, 1990). These models were further extended by Jelinek et al. (1991) into a smoothed trigram language model, reporting reduction in both perplexity and word error rates. Della Pietra et al. (1992) adapt the cache to a general $n$ -gram model such that it satisfies marginal constraints obtained from the current document. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Adaptive language models. Other adaptive language models have been proposed in the past: Kneser & Steinbiss (1993) and Iyer & Ostendorf (1999) dynamically adapt the parameters of their model to the recent history using different weight interpolation schemes. Bellegarda (2000) and Coccaro & Jurafsky (1998) use latent semantic analysis to adapt their models to the current context. Similarly, topic features have been used with either maximum entropy models (Khudanpur & Wu, 2000) or recurrent networks (Mikolov & Zweig, 2012; Wang & Cho, 2015). Finally, Lau et al. (1993) proposes to use pairs of distant of words to capture long-range dependencies. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Memory augmented neural networks. In the context of sequence prediction, several memory augmented neural networks have obtained promising results (Sukhbaatar et al., 2015; Graves et al., 2014; Grefenstette et al., 2015; Joulin & Mikolov, 2015). In particular, Sukhbaatar et al. (2015) stores a representation of the recent past and accesses it using an attention mechanism Bahdanau et al. (2014). Sukhbaatar et al. (2015) shows that this reduces the perplexity for language modeling. ",
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+    {
+        "type": "image",
+        "img_path": "images/1f2579aa6e9ffc10db91676682d66b0fb2f6c4f3c580061eb001161b8e41d356.jpg",
+        "image_caption": [
+            "Figure 3: Perplexity on the validation set of wikitext2 for linear interpolation (left) and global normalization (right), for various values of hyperparameters $\\theta$ , $\\lambda$ and $\\alpha$ . We use a cache model of size 2000. The base model has a validation perplexity of 104.2. The best linear interpolation has a perplexity of 72.1, while the best global normalization has a perplexity of 73.5. "
+        ],
+        "image_footnote": [],
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+    },
+    {
+        "type": "table",
+        "img_path": "images/7ccf2bc92c623cb2c0431afc3d1302f5b0c39f3e48ebc23a341a2a6c56f498bb.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>wikitext2</td><td>wikitext103</td></tr><tr><td>Zoneout + Variational LSTM (Merity et al.,2016)</td><td>100.9</td><td></td></tr><tr><td>Pointer Sentinel LSTM (Merity et al., 2016)</td><td>80.8</td><td>1</td></tr><tr><td>LSTM (our implementation)</td><td>99.3</td><td>48.7</td></tr><tr><td>Neural cache model (size = 100)</td><td>81.6</td><td>44.8</td></tr><tr><td>Neural cache model (size = 2,000)</td><td>68.9</td><td>40.8</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "Table 2: Test perplexity on the wikitext datasets. The two datasets share the same validation and test sets, making all the results comparable. ",
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+        "type": "text",
+        "text": "This approach has been successfully applied to question answering, when the answer is contained in a given paragraph (Chen et al., 2016; Hermann et al., 2015; Kadlec et al., 2016; Sukhbaatar et al., 2015). Similarly, Vinyals et al. (2015) explores the use of this mechanism to reorder sequences of tokens. Their network uses an attention (or “pointer”) over the input sequence to predict which element should be selected as the next output. Gulcehre et al. (2016) have shown that a similar mechanism called pointer softmax could be used in the context of machine translation, to decide which word to copy from the source to target. ",
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+    },
+    {
+        "type": "text",
+        "text": "Independently of our work, Merity et al. (2016) apply the same mechanism to recurrent network. Unlike our work, they uses the current hidden activation as a representation of the current input (while we use it to represent the output). This requires additional learning of a transformation between the current representation and those in the past. The advantage of our approach is that we can scale to very large caches effortlessly. ",
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In this section, we evaluate our method on various language modeling datasets, which have different sizes and characteristics. On all datasets, we train a static recurrent neural network language model with LSTM units. We then use the hidden representations from this model to obtain our cache, which is interpolated with the static LSTM model. We also evaluate a unigram cache model interpolated with the static model as another baseline. ",
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+    {
+        "type": "text",
+        "text": "5.1 SMALL SCALE EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Datasets. In this section, we describe experiments performed on two small datasets: the Penn Tree Bank (Marcus et al., 1993) and the wikitext2 (Merity et al., 2016) datasets. The Penn Tree Bank dataset is made of articles from the Wall Street Journal, contains $9 2 9 \\mathrm { k }$ training tokens and has a vocabulary size of 10k. The wikitext2 dataset is derived from Wikipedia articles, contains 2M training tokens and has a vocabulary size of 33k. These datasets contain non-shuffled documents, therefore requiring models to capture inter-sentences dependencies to perform well. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/239b837a7406cfdca609786eda77fbccc139af2404a2727b6d8c5998ce090737.jpg",
+        "image_caption": [
+            "Figure 4: Test perplexity as a function of the number of words in the cache, for our method and a unigram cache baseline. We observe that our approach can uses larger caches than the baseline. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Implementation details. We train recurrent neural network language models with 1024 LSTM units, regularized with dropout (probability of dropping out units equals to 0.65). We use the Adagrad algorithm, with a learning rate of 0.2, a batchsize of 20 and initial weight uniformly sampled in the range $[ - 0 . 0 5 , 0 . 0 5 ]$ . We clip the norm of the gradient to 0.1 and unroll the network for 30 steps. We consider cache sizes on a logarithmic scale, from 50 to 10, 000, and fit the cache hyperparameters on the validation set. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Results. We report the perplexity on the validation sets in Figures 2 and 3, for various values of hyperparameters, for linear interpolation and global normalization. First, we observe that on both datasets, the linear interpolation method performs slightly better than the global normalization approach. It is also easier to apply in practice, and we thus use this method in the remainder of this paper. In Tables 1 and 2, we report the test perplexity of our approach and state-of-the-art models. Our approach is competitive with previous models, in particular with the pointer sentinel LSTM model of Merity et al. (2016). On Penn Tree Bank, we note that the improvement over the base model is similar for both methods. On the wikitext2 dataset, both methods obtain similar results when using the same cache size (100 words). Since our method is computationally cheap, it is easy to increase the cache to larger values (2, 000 words), leading to dramatic improvements ( $3 0 \\%$ over the baseline, $1 2 \\%$ over a small cache of 100 words). ",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "5.2 MEDIUM SCALE EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Datasets and implementation details. In this section, we describe experiments performed over two medium scale datasets: text8 and wikitext103. Both datasets are derived from Wikipedia, but different pre-processing were applied. The text8 dataset contains 17M training tokens and has a vocabulary size of 44k words, while the wikitext103 dataset has a training set of size 103M, and a vocabulary size of 267k words. We use the same setting as in the previous section, except for the batchsize (we use 128) and dropout parameters (we use 0.45 for text8 and 0.25 for wikitext103). Since both datasets have large vocabularies, we use the adaptive softmax (Grave et al., 2016) for faster training. ",
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+    },
+    {
+        "type": "text",
+        "text": "Results. We report the test perplexity as a function of the cache size in Figure 4, for the neural cache model and a unigram cache baseline. We observe that our approach can exploits larger cache sizes, compared to the baseline. In Table 2, we observe that the improvement in perplexity of our method over the LSTM baseline on wikitext103 is smaller than for wikitext2 (approx. $1 6 \\%$ v.s. $3 0 \\%$ ). The fact that improvements obtained with more advanced techniques decrease when the size of training data increases has already been observed by Goodman (2001). Both wikitext datasets sharing the same test set, we also observe that the LSTM baseline, trained on 103M tokens (wikitext103), strongly outperforms more sophisticated methods, trained on 2M tokens (wikitext2). For these two reasons, we believe that it is important to evaluate and compare methods on relatively large datasets. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/e4401a09807758628869e7fe073e98e2b07cdc8d5546f26b4f8537e4f91d3d75.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "(a) text8 "
+        ],
+        "table_body": "<table><tr><td>Model</td><td>Test</td></tr><tr><td>LSTM-500 (Mikolov et al., 2014) SCRNN (Mikolov et al., 2014)</td><td>156</td></tr><tr><td>MemNN (Sukhbaatar et al., 2015)</td><td>161 147</td></tr><tr><td>LSTM-1024 (our implem.)</td><td>121.8</td></tr><tr><td>Neural cache model</td><td>99.9</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/21c8a1001b5deed64f6dc04ca863863ad060c729118e38a15501690e04b07843.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "(b) lambada "
+        ],
+        "table_body": "<table><tr><td>Model</td><td>Dev</td><td>Ctrl</td></tr><tr><td>WB5 (Paperno et al., 2016) WB5+cache (Paperno et al., 2016)</td><td>3125 768</td><td>285</td></tr><tr><td>LSTM-512 (Paperno et al., 2016)</td><td>5357</td><td>270 149</td></tr><tr><td>LSTM-1024 (our implem.)</td><td>4088</td><td>94</td></tr><tr><td>Neural cache model</td><td>138</td><td>129</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/3927384ae624d65f7ffe727b8d722a996fbce28ae8ec55bfda11221b9c005b0a.jpg",
+        "image_caption": [
+            "Table 3: Perplexity on the text8 and lambada datasets. WB5 stands for 5-gram language model with Witten-Bell smoothing. ",
+            "Figure 5: Perplexity on the development and control sets of lambada, as a function of the interpolation parameters $\\lambda$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.3 EXPERIMENTS ON THE LAMBADA DATASET ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Finally, we report experiments carried on the lambada dataset, introduced by Paperno et al. (2016). This is a dataset of short passages extracted from novels. The goal is to predict the last word of the excerpt. This dataset was built so that human subjects solve the task perfectly when given the full context (approx. 4.6 sentences), but fail to do so when only given the sentence with the target word. Thus, most state-of-the-art language models fail on this dataset. The lambada training set contains approximately 200M tokens and has a vocabulary size of 93, 215. We report results for our method in Table 3, as well the performance of baselines from Paperno et al. (2016). Adding a neural cache model to the LSTM baseline strongly improves the performance on the lambada dataset. We also observe in Figure 5 that the best interpolation parameter between the static model and the cache is not the same for the development and control sets. This is due to the fact that more than $83 \\%$ of passages of the development set include the target word, while this is true for only $14 \\%$ of the control set. Ideally, a model should have strong results on both sets. One possible generalization of our model would be to adapt the interpolation parameter based on the current vector representation of the history $h _ { t }$ . ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We presented the neural cache model to augment neural language models with a longer-term memory that dynamically updates the word probablilities based on the long-term context. A neural cache can be added on top of a pre-trained language model at negligible cost. Our experiments on both language modeling tasks and the challenging LAMBADA dataset shows that significant performance gains can be expected by adding this external memory component. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Technically, the neural cache models is similar to some recent memory-augmented neural networks such as pointer networks. However, its specific design makes it possible to avoid learning the memory lookup component. This makes the neural cache appealing since it can use larger cache sizes than memory-augment networks and can be applied as easily as traditional count-based caches. ",
+        "bbox": [
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+    {
+        "type": "text",
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parse/train/ByuI-mW0W/ByuI-mW0W.md

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+# TOWARDS A TESTABLE NOTION OF GENERALISATION FOR GENERATIVE ADVERSARIAL NETWORKS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+We consider the question of how to assess generative adversarial networks, in particular with respect to whether or not they generalise beyond memorising the training data. We propose a simple procedure for assessing generative adversarial network performance based on a principled consideration of what the actual goal of generalisation is. Our approach involves using a test set to estimate the Wasserstein distance between the generative distribution produced by our procedure, and the underlying data distribution. We use this procedure to assess the performance of several modern generative adversarial network architectures. We find that this procedure is sensitive to the choice of ground metric on the underlying data space, and suggest a choice of ground metric that substantially improves performance. We finally suggest that attending to the ground metric used in Wasserstein generative adversarial network training may be fruitful, and provide a concrete formulation for doing so.
+
+# 1 INTRODUCTION
+
+Generative adversarial networks (GANs) (Goodfellow et al., 2014) have attracted significant interest as a means for generative modelling. However, recently concerns have been raised about their ability to generalise from training data and their capacity to overfit (Arora & Zhang, 2017; Arora et al., 2017). Moreover, techniques for evaluating the quality of GAN output are either ad hoc, lack theoretical rigor, or are not suitably objective – often times “visual inspection” of samples is the main tool of choice for the practitioner. More fundamentally, it is sometimes unclear exactly what we want a GAN to do: what is the learning task that we are trying to achieve?
+
+In this paper, we provide a simple formulation of the GAN training framework, which consists of using a finite dataset to estimate an underlying data distribution. The quality of GAN output is measured precisely in terms of a statistical distance $D$ between the estimated and true distribution. Within this context, we propose an intuitive notion of what it means for a GAN to generalise.
+
+We also show how our notion of performance can be measured empirically for any GAN architecture when $D$ is chosen to be a Wasserstein distance, which – unlike other methods such as the inception score (Salimans et al., 2016) – requires no density assumptions about the data-generating distribution. We investigate this choice of $D$ empirically, finding that its performance is heavily dependent on the choice of ground metric underlying the Wasserstein distance. We suggest a novel choice of ground metric that we show performs well, and also discuss how we might otherwise use this observation to improve the design of Wasserstein GANs (WGANs) (Arjovsky et al., 2017).
+
+# 2 THE OBJECTIVE OF GENERATIVE MODELLING
+
+GANs promise a means for learning complex probability distributions in an unsupervised fashion. In order to assess their effectiveness, we must first define precisely what we mean by this. We seek to do so in this section, presenting a formulation of the broader goal of generative modelling that we believe is widely compatible with much present work in this area. We also provide a natural notion of generalisation that arises in our framework.
+
+Our setup consists of the following components. We assume some distribution $\pi$ on a set $\mathcal { X }$ . For instance, $\mathcal { X }$ may be the set of $3 2 \mathrm { x } 3 2 $ colour images, and $\pi$ the distribution from which the CIFAR-10 dataset was sampled. We assume that $\pi$ is completely intractable: we do not know its density (or even if it has one), and we do not have a procedure to draw novel samples from it. However, we do suppose that we have a fixed dataset $X$ consisting of samples $x _ { 1 } , \cdot \cdot \cdot , x _ { | X | } \overset { \mathrm { i i d } } { \sim } \pi$ . Equivalently, we have the empirical distribution
+
+$$
+\hat { X } : = \frac { 1 } { | X | } \sum _ { n = 1 } ^ { | X | } \delta _ { x _ { n } } ,
+$$
+
+where $\delta$ denotes the Dirac mass.
+
+Let $\mathcal { P } ( \mathcal { X } )$ denote the set of probability distributions on $\mathcal { X }$ . Our aim is to use $X$ to produce a distribution in $\mathcal { P } ( \mathcal { X } )$ that is as “close” as possible to $\pi$ . We choose to measure closeness in terms of a function $D : \mathcal { P } ( \mathcal { X } ) \times \mathcal { P } ( \mathcal { X } )  \mathbb { R }$ . Usually $D$ will be chosen to be a statistical divergence, which means that $D ( P , Q ) \geq 0$ for all $P$ and $Q$ , with equality iff $P = Q$ . The task of a learning algorithm $\alpha$ in this context is then as follows:
+
+$$
+\boxed { \mathrm { S e l e c t } \alpha ( X ) \in \mathcal { P } ( \mathcal { X } ) \mathrm { ~ s u c h ~ t h a t ~ } D ( \alpha ( X ) , \pi ) \mathrm { ~ i s ~ a s ~ s m a l l ~ a s ~ p o s s i b l e . } }
+$$
+
+We believe (1) constitutes an intuitive and useful formulation of the problem of generative modelling that is largely in keeping with present research efforts.
+
+Now, we can immediately see that one possibility is simply to choose $\alpha ( X ) : = { \hat { X } }$ . Moreover, in the case that $D$ is a metric for the weak topology on $\mathcal { P } ( \mathcal { X } )$ such as a Wasserstein distance, we have that $D ( \hat { X } , \pi ) \to 0$ almost surely as $| X | \to \infty$ , so that, assuming $| X |$ is large enough, we can already expect $D ( \hat { X } , \pi )$ to be small. This then suggests the following natural notion of generalisation:
+
+$$
+\Bigl | \mathrm { A ~ c h o i c e ~ o f } \alpha \mathrm { g e n e r a l i s e s ~ f o r ~ a ~ g i v e n } X \mathrm { i f } \ D ( \alpha ( X ) , \pi ) < D ( \hat { X } , \pi ) . \Bigr |
+$$
+
+In other words, using $\alpha$ here has actually achieved something: perhaps through some process of smoothing or interpolation, it has injected additional information into $\hat { X }$ that has moved us closer to $\pi$ than we were a priori.
+
+# 3 GENERALISATION IN GANS
+
+The previous section presented (1) as a general goal of generative modelling. In this section, we turn specifically to GANs. We begin by providing a general model for how many of the existing varieties of GAN operate, at least ideally. We then show how this model fits into our framework above, before considering the issue of generalisation in this context.
+
+Most GAN algorithms in widespread use adhere to the following template: they take as input a distribution $P$ , from which we assume we can sample, and compute (or approximate)
+
+$$
+\Gamma ( P ) : = \underset { Q \in \mathcal { Q } } { \arg \operatorname* { m i n } } D _ { \Gamma } ( P , Q )
+$$
+
+for some choices of $\mathcal Q \subseteq \mathcal P ( \mathcal X )$ and $D _ { \Gamma } : \mathcal { P } ( \mathcal { X } ) \times \mathcal { P } ( \mathcal { X } )  \mathbb { R }$ . In other words, in the ideal case, a GAN maps $P$ to its $D _ { \Gamma }$ -projection onto $\mathcal { Q }$ . Note that we will not necessarily have that $D _ { \Gamma } = D$ : $D _ { \Gamma }$ is fixed given a particular GAN architecture, whereas the choice of $D$ is simply a feature of our problem definition (1) and is essentially ours to make.
+
+In practice, $\mathcal { Q }$ is the set of pushforward measures $\nu \circ G ^ { - 1 }$ obtained from a fixed noise distribution $\nu$ on a noise space $\mathcal { Z }$ and some set $\mathcal { G }$ of functions $G : { \mathcal { Z } }  { \mathcal { X } }$ . Precisely, $\mathcal { Q } = \left\{ \nu \circ G ^ { - 1 } : G \in \mathcal { G } \right\}$ . $\mathcal { G }$ itself usually corresponds to the set of functions realisable by some neural network architecture, and $\nu$ is some multivariate uniform or Gaussian distribution. However, numerous choices of $D _ { \Gamma }$ have been proposed: the original GAN formulation (Goodfellow et al., 2014) took $D _ { \Gamma }$ to be the Jenson-Shannon divergence, whereas the $f$ -GAN (Nowozin et al., 2016) generalised this to arbitrary $f$ -divergences, and the Wasserstein GAN (Arjovsky et al., 2017) advocated the Wasserstein distance. Many of the results proved in these papers involve showing that (usually under some assumptions, such as sufficient discriminator capacity) a proposed objective for $G$ is in fact equivalent to $\bar { D _ { \Gamma } ( P , \nu \circ }$ $G ^ { - 1 }$ ).
+
+In terms of our framework in the previous section, using a GAN $\Gamma$ amounts to choosing
+
+$$
+\alpha ( X ) : = \Gamma ( \hat { X } ) = \underset { Q \in \mathcal { Q } } { \arg \operatorname* { m i n } } D _ { \Gamma } ( \hat { X } , Q ) .
+$$
+
+We emphasise again the important distinction between $D$ and $D _ { \Gamma }$ . In our setup, minimising $D$ defines our ultimate goal, whereas minimising $D _ { \Gamma }$ (over $\mathcal { Q }$ ) defines how we will attempt to achieve that goal. Even if $D \neq D _ { \Gamma }$ , it is still at least conceivable that $D ( \Gamma ( { \hat { X } } ) , \pi )$ might be small, and therefore this choice of $\alpha$ might be sensible. Also note that, crucially, $\Gamma$ receives $\hat { X }$ as input rather than $\pi$ itself. We only have access to a fixed number of CIFAR-10 samples, for example, not an infinite stream. Moreover, training GANs usually involves making many passes over the same dataset, so that, in effect, sampling from $P$ will repeatedly yield the same data points. We would not expect this to occur with nonzero probability if $P = \pi$ for most $\pi$ of interest.
+
+The observation that $P$ is $\hat { X }$ rather than $\pi$ was also recently made by Arora et al. (2017). The authors argue that this introduces a problem for the ability of GANs to generalise, since, if $D _ { \Gamma }$ is a divergence (which is almost always the case), and if $\mathcal { Q }$ is too big (in particular, if it is big enough that $\hat { X } \in \mathcal { Q } ,$ ), then we trivially have that $\Gamma ( { \hat { X } } ) = { \hat { X } }$ . In other words, the GAN objective appears actively to encourage $\Gamma ( { \hat { X } } )$ to memorise the dataset $X$ , and never to produce novel samples from outside of it. The authors’ proposed remedy involves trying to find a better choice of $D _ { \Gamma }$ . The problem, they argue, is that popular choices of $D _ { \Gamma }$ do not satisfy the condition
+
+$D _ { \Gamma } ( \pi , Q ) \approx D _ { \Gamma } ( { \hat { X } } , Q )$ with high probability given a modest number of samples in $X$ .
+
+They point out that this is certainly violated when $D _ { \Gamma }$ is the Jensen-Shannon divergence JS, since
+
+$$
+\operatorname { J S } ( P \parallel Q ) = \log 2
+$$
+
+when one of $P$ and $Q$ is discrete and the other continuous, and give a similar result for $D _ { \Gamma }$ a Wasserstein distance in the case that $\pi$ is Gaussian. As a solution, they introduce the neural network distance $D _ { \mathrm { N N } }$ defined by
+
+$$
+D _ { \mathrm { N N } } ( P , Q ) : = \operatorname* { m a x } _ { f \in \mathcal { F } } \mathbb { E } _ { x \sim P } \left[ f ( x ) \right] - \mathbb { E } _ { x \sim Q } \left[ f ( x ) \right]
+$$
+
+for some choice of a class of functions $\mathcal { F }$ . They show that, assuming some smoothness conditions on the members of $\mathcal { F }$ , the choice $D _ { \Gamma } = D _ { \mathrm { N N } }$ satisfies (3), which means that if we minimise $D _ { \mathrm { N N } } ( \hat { X } , Q )$ in $Q$ then we can be confident that the value of $D _ { \mathrm { N N } } ( \pi , Q )$ is small also.
+
+However, we do not believe that (3) is sufficient to ensure good generalisation behaviour for GANs. What we care about ultimately is not the value of $D _ { \Gamma }$ , but rather of $D$ , and (3) invites choosing $D _ { \Gamma }$ in such a way that gives no guarantees about $D$ at all. We see, for instance, that the degenerate choice $D _ { 0 } ( P , Q ) : = 0$ trivially satisfies (3), and indeed is also a pseudometric, just like $D _ { \mathrm { N N } }$ . It is therefore unclear what mathematical properties of $D _ { \mathrm { N N } }$ render it more suitable for estimating $\pi$ than the obviously undesirable $D _ { 0 }$ . The authors do acknowledge this shortcoming of $D _ { \mathrm { N N } }$ , pointing out that $D _ { \mathrm { N N } } ( P , Q )$ can be small even if $P$ and $Q$ are “quite different” in some sense.
+
+The problematic consequences of the fact that $P = { \hat { X } }$ apply only in the case that $\mathcal { Q }$ is too large. In practice, however, $\mathcal { Q }$ is heavily restricted, since $\mathcal { G }$ is restricted via a choice of neural network architecture; hence we do not know $a$ priori whether $\boldsymbol { \Gamma } ( \boldsymbol { \hat { X } } ) = \boldsymbol { \hat { X } }$ is even possible. As such, we do not see the choice of $\alpha ( X ) = \Gamma ( \hat { X } )$ as necessarily a bad idea, and instead believe that it is an open empirical question as to how well GANs perform the task (1). In fact, this $\alpha$ falls perfectly within the framework of minimum distance estimation (Wolfowitz, 1957; Basu et al., 2011), which involves estimating an underlying distribution by minimising a distance measure to a given empirical distribution.
+
+# 4 TESTING GANS
+
+Our goal in this section is to assess how well GANs achieve (1) by estimating $D ( \Gamma ( X ) , \pi )$ for various $\Gamma$ and $\pi$ . This raises some difficulties, given that $\pi$ is intractable. Our approach is to take $D$ to be the first Wasserstein distance $W _ { d _ { x } }$ defined by
+
+$$
+W _ { d x } ( P , Q ) = \operatorname* { i n f } _ { \gamma \in \Pi ( P , Q ) } \int _ { \mathcal { X } \times \mathcal { X } } d _ { \mathcal { X } } ( x , y ) \mathrm { d } \gamma ( x , y ) ,
+$$
+
+where $d _ { \mathcal { X } }$ is a metric on $\mathcal { X }$ referred to as the ground metric, and $\Pi ( P , Q )$ denotes the set of joint distributions on the product space $\mathcal { X } \times \mathcal { X }$ with marginals $P$ and $Q$ . The Wasserstein distance is appealing since it is sensitive to the topology of the underlying set $\mathcal { X }$ , which we control by our choice of $d _ { \mathcal { X } }$ . Moreover, $W _ { d x }$ metricises weak convergence for the Wasserstein space $\mathcal { P } _ { d _ { \mathcal { X } } } ( \mathcal { X } )$ defined by
+
+$$
+{ \mathcal { P } } _ { d _ { \mathcal { X } } } ( \mathcal { X } ) : = \left\{ P \in { \mathcal { P } } ( \mathcal { X } ) : \int _ { \mathcal { X } } d _ { \mathcal { X } } ( x _ { 0 } , y ) \mathrm { d } P ( y ) < \infty { \mathrm { ~ f o r ~ s o m e ~ } } x _ { 0 } \in \mathcal { X } \right\}
+$$
+
+(see (Villani, 2008)). Consequently, if we denote by $A$ a set of samples $a _ { 1 } , \cdots , a _ { | A | }$ iid∼ $\alpha ( X )$ and by $Y$ a set of samples (separate from $X$ ) $y _ { 1 } , \cdots , y _ { | Y | } \overset { \mathrm { i i d } } { \sim } \pi .$ , with $\hat { A }$ and $\hat { Y }$ the corresponding empirical distributions, then, provided
+
+$$
+\alpha ( X ) , \pi \in { \mathcal { P } } _ { d _ { \mathcal { X } } } ( { \mathcal { X } } ) ,
+$$
+
+we have that $D ( { \hat { A } } , { \hat { Y } } ) \to D ( \alpha ( X ) , \pi )$ almost surely as $\operatorname* { m i n } \{ | A | , | Y | \}  \infty$ . Note that condition (4) holds automatically in the case that $( \mathcal { X } , d _ { \mathcal { X } } )$ is compact, since then $\mathcal { P } _ { d _ { \mathcal { X } } } ( \mathcal { X } ) = \mathcal { P } ( \mathcal { X } )$ .
+
+As such, to estimate $D ( \alpha ( X ) , \pi )$ , for $D = W _ { d _ { x } }$ , we propose the following. Before training, we move some of our samples from $X$ into a testing set $Y$ . We next train our GAN on $X$ , obtaining $\alpha ( X )$ . We then take samples $A$ from $\alpha ( X )$ , and obtain the estimate
+
+$$
+W _ { d _ { \mathcal { X } } } ( \hat { A } , \hat { Y } ) \approx W _ { d _ { \mathcal { X } } } ( \alpha ( X ) , \pi ) ,
+$$
+
+where the left-hand side can be computed exactly by solving a linear program since both $\hat { A }$ and $\hat { Y }$ are discrete (Villani, 2003). We can also use the same methodology to estimate $W _ { d _ { \mathcal { X } } } ( \hat { X } , \pi )$ by $W _ { d x } ( \hat { X } , \hat { Y } )$ , which suggests testing if
+
+$$
+W _ { d _ { \mathcal { X } } } ( \hat { A } , \hat { Y } ) < W _ { d _ { \mathcal { X } } } ( \hat { X } , \hat { Y } ) .
+$$
+
+as a proxy for determining whether (2) holds. A summary of our procedure is given in Algorithm 1.
+
+# Algorithm 1 Procedure for testing GANs
+
+<table><tr><td>1: Split samples from π into a training set X and a testing set Y</td></tr><tr><td>2: Compute α(X) by training a GAN on X 3: Obtain a sample A from α(X)</td></tr></table>
+
+# 4.1 RESULTS
+
+We applied our methodology to test two popular GAN varieties – the DCGAN (Radford et al., 2015) and the Improved Wasserstein GAN (I-WGAN) (Gulrajani et al., 2017) – on the MNIST and CIFAR-10 datasets. In all cases when computing the relevant Wasserstein distances, our empirical distributions consisted of 10000 samples.
+
+# 4.1.1 $L ^ { 2 }$ AS GROUND METRIC
+
+We initially took our ground metric $d _ { \mathcal { X } }$ to be the $L ^ { 2 }$ distance. This has the appealing property of making $( \boldsymbol { \mathcal { X } } , d _ { \boldsymbol { \mathcal { X } } } )$ compact when $\mathcal { X }$ is a space of RGB or greyscale images, therefore ensuring that (4) holds.
+
+Figure 1 shows $W _ { L ^ { 2 } } ( \hat { A } , \hat { Y } )$ plotted over multiple training runs for MNIST and CIFAR-10 when $A$ is obtained from an I-WGAN, with typical samples shown in Figures 6 and 7 in the appendix. For both datasets, $W _ { L ^ { 2 } } ( \hat { A } , \hat { Y } )$ decays towards an asymptote in a way that nicely corresponds to the visual quality of the samples produced. Moreover, $W _ { L ^ { 2 } } ( \hat { A } , \hat { Y } )$ is much closer to $W _ { L ^ { 2 } } ( \hat { X } , \hat { Y } )$ towards the tail end of training for MNIST than for CIFAR-10. This seems to reflect the fact that, visually, the eventual I-WGAN MNIST samples do seem quite close to true MNIST samples, whereas the eventual I-WGAN CIFAR-10 samples are easily identified as fake.
+
+![](images/a00baa7a2f8c1338c34dc16ac2b9d21542acedfe8552ef8248e5e4e39486957e.jpg)  
+Figure 1: Output of Algorithm 1 with $d _ { \mathcal { X } } = L ^ { 2 }$ for I-WGAN trained on MNIST (left) and CIFAR-10 (right)
+
+![](images/c579bf2b3cf37f9b358f5494cc533abd1bcd6e975256427480f8283fbd348944.jpg)  
+Figure 2: Output of Algorithm 1 with $d _ { \mathcal { X } } = L ^ { 2 }$ for DCGAN trained on CIFAR-10
+
+However, when we re-ran the same experiment using a DCGAN on CIFAR-10, we obtained the $W _ { L ^ { 2 } } ( \hat { A } , \hat { Y } )$ trajectories shown in Figure 2. Typical examples are shown in Figure 8. Strangely, we observe that $W _ { L ^ { 2 } } ( \hat { A } , \hat { Y } ) < W _ { L ^ { 2 } } ( \hat { X } , \hat { Y } )$ very early on in training – at around batch 500 – when the samples resemble the heavily blurry Figure 8b. This raises some obvious concerns about the appropriateness of $W _ { L ^ { 2 } }$ as a metric for GAN quality.
+
+We therefore sought to understand this strange behaviour. Motivated by the visual blurriness of the samples in Figure 8b, we explored the effect on $W _ { L ^ { p } } ( \hat { X } , \hat { Y } )$ of blurring the CIFAR-10 training set $X$ . In particular, we let $X$ and $Y$ each consist of 10000 distinct CIFAR-10 samples in $X$ . We then independently convolved each channel of each image with a Gaussian kernel having standard deviation $\sigma$ , obtaining a blurred dataset $\beta _ { \sigma } ( X )$ and corresponding empirical distribution $\hat { \beta } _ { \sigma } ( X )$ . The visual effect of this procedure is shown in Figure 9 in the appendix. We then computed $W _ { L ^ { p } } ( \hat { \beta } _ { \sigma } ( X ) , \hat { Y } )$ with $\sigma$ ranging between 0 and 10 for a variety of values of $p$ . The results of this experiment in the case $p = 2$ are shown in Figure 3, and similar results were observed for other values of $p$ : in all cases, we found that
+
+$$
+W _ { L ^ { p } } ( \hat { X } , \hat { Y } ) > W _ { L ^ { p } } ( \hat { \beta } _ { \sigma } ( X ) , \hat { Y } )
+$$
+
+whenever $\sigma > 0$ . That is, blurring $X$ by any amount brings $\hat { X }$ closer to $\hat { Y }$ in $\boldsymbol { W } _ { L ^ { p } }$ than not. This occurs even though $X$ is distributed identically to $Y$ (both being drawn from $\pi$ ), while $\beta _ { \sigma } ( X )$ (presumably) is not when $\sigma > 0$ .
+
+![](images/45f392895d2f6ffc598a181c84f164d1b14ef60a793bd79e85f92972b0cf924d.jpg)  
+Figure 3: Effect of blurring CIFAR-10 on $W _ { L ^ { 2 } }$ (left) and $W _ { L ^ { 2 } \circ \eta }$ (right)
+
+# 4.1.2 EMBEDDED $L ^ { 2 }$ AS GROUND METRIC
+
+To remedy these issues, we sought to replace $L ^ { 2 }$ with a choice of $d _ { \mathcal { X } }$ that is more naturally suited to the space of images in question. To this end we tried mapping $\mathcal { X }$ through a fixed pre-trained neural network $\eta$ into a feature space $\mathcal { V }$ , and then computing distances using some metric $d _ { \mathcal { Y } }$ on $\mathcal { V }$ , rather than in $\mathcal { X }$ directly. It is easily seen that, provided $\eta$ is injective, $d _ { \mathcal { V } } \circ \eta : \mathcal { X } \times \mathcal { X } \to \mathbb { R }$ defined by
+
+$$
+( d _ { \mathcal { V } } \circ \eta ) ( x , x ^ { \prime } ) : = d _ { \mathcal { V } } ( \eta ( x ) , \eta ( x ^ { \prime } ) )
+$$
+
+is a metric. It also holds that, when $\eta$ is $( d _ { \mathcal { X } } , d _ { \mathcal { Y } } )$ -continuous, $( \mathcal { X } , d _ { \mathcal { Y } } \circ \eta )$ is compact when $( \mathcal { X } , d _ { \mathcal { X } } )$ is: given a sequence $x _ { i } \subseteq { \mathcal { X } }$ , there exists a subsequence $x _ { i ^ { \prime } }$ that converges in $d _ { \mathcal { X } }$ to some $x$ (by compactness), so that $( d \ / y \circ \eta ) ( x _ { i ^ { \prime } } , x ) = d \ / y ( \eta ( x _ { i ^ { \prime } } ) , \eta ( x ) )  0$ by continuity of $\eta$ . Neither of these conditions on $\eta$ – injectivity and $( d _ { \mathcal { X } } , d _ { \mathcal { Y } } )$ -continuity – are unreasonable to expect from a typical neural network trained using stochastic gradient descent (at least, when $d _ { \mathcal { X } }$ and $d _ { \mathcal { Y } }$ are typical metrics such as $L ^ { p }$ distances). Consequently, $W _ { d _ { y } \circ \eta }$ constitutes a valid metric on $\mathcal { P } _ { d _ { y } \circ \eta } ( \mathcal { X } )$ , and (4) is automatically satisfied.
+
+To test the performance of $W _ { d y \circ \eta }$ , we repeated the blurring experiment described above. We took $\eta ( x )$ to be the result of scaling $x \in \mathcal { X }$ to size $2 2 4 \mathbf { x } 2 2 4$ , mapping the result through a DenseNet-121 (Huang et al., 2016) pre-trained on ImageNet (Deng et al., 2009), and extracting features immediately before the linear output layer. Under the same experimental setup as above otherwise, we obtained the plot of $W _ { L ^ { 2 } \circ \eta } ( \beta _ { \sigma } ( \hat { X } ) , \hat { Y } )$ shown in Figure 3. Happily, we now see that this curve increases monotonically as $\sigma$ grows in accordance with the declining visual quality of $\beta _ { \sigma } ( X )$ shown in Figure 9.
+
+Next, we computed $W _ { L ^ { 2 } \circ \eta } ( \hat { A } , \hat { Y } )$ over the course of GAN training. For the I-WGAN we obtained the results on MNIST and CIFAR-10 shown in Figure 4;1 for the DCGAN on CIFAR-10, we obtained the curve shown in Figure 5. In all cases we see that $W _ { L ^ { 2 } \circ \eta } ( \hat { A } , \hat { Y } )$ decreases monotonically towards an asymptote in a way that accurately summarises the visual quality of the samples throughout the training run. Moreover, there is always a large gap between the eventual value of $W _ { L ^ { 2 } \circ \eta } ( \hat { A } , \hat { Y } )$ and $W _ { L ^ { 2 } \circ \eta } ( \hat { X } , \hat { Y } )$ , which reflects the fact that the GAN samples are still visually distinguishable from real $\pi$ samples. In particular, we see an improvement in this respect for the I-WGAN on MNIST: in Figure 1 the asymptotic value of $W _ { L ^ { 2 } } ( \hat { \hat { A } } , \hat { Y } )$ for MNIST was barely discernible from $W _ { L ^ { 2 } } ( \hat { X } , \hat { Y } )$ , despite the fact that it is still quite easy to tell real samples from generated ones (see e.g. the various mistakes present in Figure 6).
+
+# 5 DISCUSSION AND FUTURE WORK
+
+We believe our work reveals two promising avenues of future inquiry. First, we suggest that $W _ { L ^ { p } \circ \eta }$ is an appealing choice of $D$ , both due to its nice theoretical properties – it metricises weak convergence, and does not require us to make any density assumptions about $\pi$ – and due to its sound empirical performance demonstrated above. It would be very interesting to use this $D$ to produce a systematic and objective comparison of the performance of all current major GAN implementations, and indeed to use this as a metric for guiding future GAN design. We also view the test (5) as potentially useful for determining whether our algorithms are overfitting. This would be particularly so if applied via a cross-validation procedure: if we consistently observe that (5) holds when training a GAN according to many different $X$ and $Y$ partitions of our total $\pi$ samples, then it seems reasonable to infer that $\alpha ( X )$ has indeed learnt something useful about $\pi$ .
+
+![](images/570e844b51dbee20d95f6a30d2c3878bb1d1165315288d1f52be0133d3933878.jpg)  
+Figure 4: Output of Algorithm 1 with $d _ { \mathcal { X } } = L ^ { 2 } \circ \eta$ for I-WGAN trained on MNIST (left) and CIFAR-10 (right)
+
+![](images/852317811ce1e44b66dd32c17d799d4f81b0b3e03c4c07617a867f42293c9ab6.jpg)  
+Figure 5: Output of Algorithm 1 with $d _ { \mathcal { X } } = L ^ { 2 } \circ \eta$ for DCGAN trained on CIFAR-10
+
+We also believe that the empirical inadequacy of $W _ { L ^ { 2 } }$ that we observed suggests a path towards a better WGAN architecture. At present, WGAN implementations implicitly use $W _ { L ^ { 2 } }$ for their choice of $D _ { \Gamma }$ . We suspect that altering this to our suggested $W _ { L ^ { 2 } \circ \eta }$ may yield better quality samples. We briefly give here one possible way to do so that is largely compatible with existing WGAN setups. In particular, following Arjovsky et al. (2017), we take
+
+$$
+D _ { \Gamma } ( P , Q ) = \underset { f \in \mathcal { F } } { \operatorname* { m a x } } \mathbb { E } _ { x \sim P } \left[ f ( x ) \right] - \mathbb { E } _ { x \sim Q } \left[ f ( x ) \right]
+$$
+
+for a class $\mathcal { F }$ of functions $f : \mathcal { X }  \mathbb { R }$ that are all $\left( L ^ { 2 } \circ \eta , d _ { \mathbb { R } } \right)$ -Lipschitz for some fixed Lipschitz constant $K$ . Here $d _ { \mathbb { R } }$ denotes the usual distance on $\mathbb { R }$ . To optimise over such an $\mathcal { F }$ in practice, we can require our discriminator $f : \mathcal { X } \to \mathbb { R }$ to have the form ${ \bar { f } } ( x ) : = h ( \eta ( x ) )$ , where $h : \mathcal { V } \to \mathbb { R }$ is $( d _ { \mathcal { Y } } , d _ { \mathbb { R } } )$ -Lipschitz, which entails that $f$ is Lipschitz provided $\eta$ is (which is almost always the case in practice). In other words, we compute
+
+$$
+D _ { \Gamma } ( P , Q ) = \operatorname* { m a x } _ { h \in \mathcal { F } ^ { \prime } } \mathbb { E } _ { x \sim P } \left[ h ( \eta ( x ) ) \right] - \mathbb { E } _ { x \sim Q } \left[ h ( \eta ( x ) ) \right] ,
+$$
+
+where ${ \mathcal { F } } ^ { \prime }$ is a class of $( d _ { \mathcal { Y } } , d _ { \mathbb { R } } )$ -Lipschitz functions. Optimising over this objective may now proceed as usual via weight-clipping like (Arjovsky et al., 2017), or via a gradient penalty like (Gulrajani et al., 2017). Note that this suggestion may be understood as training a standard WGAN with the initial layers of the discriminator fixed to the embedding $\eta$ ; our analysis here shows that this is equivalent to optimising with respect to $W _ { L ^ { 2 } \circ \eta }$ instead of $W _ { L ^ { 2 } }$ . We have begun some experimentation in this area but leave a more detailed empirical inquiry to future work.
+
+It is also clearly important to establish better theoretical guarantees for our method. At present, we have no guarantee that the number of samples in $A$ and $Y$ are enough to ensure that
+
+$$
+D ( \hat { A } , \hat { Y } ) \approx D ( \alpha ( X ) , \hat { Y } )
+$$
+
+(perhaps with some fixed bias that is fairly independent of $\alpha$ , so that it is valid to use the value of $D ( \hat { A } , \hat { Y } )$ to compare different choices of $\alpha$ ), or that (5) entails (2) with high probability. We do however note that some recent theoretical work on the convergence rate of empirical Wasserstein estimations (Weed & Bach, 2017) does suggest that it is plausible to hope for fast convergence of $D ( \hat { A } , \hat { Y } )$ to $D ( \alpha ( X ) , { \hat { Y } } )$ . We also believe that the convincing empirical behaviour of $W _ { L ^ { 2 } \circ \eta }$ does suggest that it is possible to say something more substantial about our approach, which we leave to future work.
+
+# 6 RELATED WORK
+
+The maximum mean discrepancy (MMD) is another well-known notion of distance on probability distributions, which has been used for testing whether two distributions are the same or not (Gretton et al., 2012) and also for learning generative models in the style of GAN (Li et al., 2015; Dziugaite et al., 2015; Sutherland et al., 2016; Li et al., 2017). It is parameterised by a characteristic kernel $k$ , and defines the distance between probability distributions by means of the distance of $k$ ’s reproducing kernel Hilbert space (RKHS). The MMD induces the same weak topology on distributions as the one of the Wasserstein distance. Under a mild condition, the MMD between distributions $P$ and $Q$ under a kernel $k$ can be understood as the outcome of the following two-step calculation. First, we pushforward $P$ and $Q$ from their original space $\mathcal { X }$ to a Hilbert space $\mathcal { H }$ (isomorphic to $k$ ’s RKHS) using a feature function $\phi : \mathcal { X } \to \mathcal { H }$ induced by the kernel $k$ . Typically, $\mathcal { H }$ is an infinite-dimensional space, such as the set $\ell _ { 2 }$ of square-summable sequences in ${ \mathcal { R } } ^ { \infty }$ as in Mercer’s theorem. Let $P ^ { \prime }$ and $Q ^ { \prime }$ be the resulting distributions on the feature space. Second, we compute the supremum of $\mathbb { E } _ { P ^ { \prime } ( X ) } [ f ( X ) ] - \mathbb { E } _ { Q ^ { \prime } ( X ) } [ f ( X ) ]$ over all linear 1-Lipschitz functions $f : \mathcal { H } \to \mathbb { R }$ . The result of this calculation is the MMD between $P$ and $Q$ .
+
+This two-step calculation shows the key difference between MMD and our use of Wasserstein distance and neural embedding $\eta$ . While the co-domain of the feature function $\phi$ is an infinitedimensional space (e.g. $\ell _ { 2 }$ ) in most cases, its counterpart $\eta$ in our setting uses a finite-dimensional space as co-domain. This means that the MMD possibly uses a richer feature space than our approach. On the other hand, the MMD takes the supremum over only linear functions $f$ among all 1-Lipschitz functions, whereas the Wasserstein distance considers all these 1-Lipschitz functions. These different balancing acts between the expressiveness of features and that of functions taken in the supremum affect the learning and testing of various GAN approaches as observed experimentally in the literature. One interesting future direction is to carry out systematic study on the theoretical and practical implications of these differences.
+
+# REFERENCES
+
+Martin Arjovsky, Soumith Chintala, and Leon Bottou. Wasserstein gan. ´ arXiv preprint arXiv:1701.07875, 2017.   
+Sanjeev Arora and Yi Zhang. Do gans actually learn the distribution? an empirical study. arXiv preprint arXiv:1706.08224, 2017.   
+Sanjeev Arora, Rong Ge, Yingyu Liang, Tengyu Ma, and Yi Zhang. Generalization and Equilibrium in Generative Adversarial Nets (GANs). arXiv preprint arXiv:1703.00573, 2017.   
+Ayanendranath Basu, Hiroyuki Shioya, and Chanseok Park. Statistical inference: the minimum distance approach. CRC Press, 2011.   
+Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hierarchical image database. In Computer Vision and Pattern Recognition, 2009. CVPR 2009. IEEE Conference on, pp. 248–255. IEEE, 2009.
+
+Gintare Karolina Dziugaite, Daniel M. Roy, and Zoubin Ghahramani. Training generative neural networks via maximum mean discrepancy optimization. In Proceedings of the Thirty-First Conference on Uncertainty in Artificial Intelligence, UAI 2015, July 12-16, 2015, Amsterdam, The Netherlands, pp. 258–267, 2015.
+
+Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Advances in neural information processing systems, pp. 2672–2680, 2014.
+
+Arthur Gretton, Karsten M. Borgwardt, Malte J. Rasch, Bernhard Scholkopf, and Alexander J.¨ Smola. A kernel two-sample test. Journal of Machine Learning Research, 13:723–773, 2012.
+
+Ishaan Gulrajani, Faruk Ahmed, Martin Arjovsky, Vincent Dumoulin, and Aaron Courville. Improved training of wasserstein gans. arXiv preprint arXiv:1704.00028, 2017.
+
+Gao Huang, Zhuang Liu, Kilian Q Weinberger, and Laurens van der Maaten. Densely connected convolutional networks. arXiv preprint arXiv:1608.06993, 2016.
+
+Chun-Liang Li, Wei-Cheng Chang, Yu Cheng, Yiming Yang, and Barnabas P ´ oczos. MMD GAN: ´ towards deeper understanding of moment matching network. arXiv preprint arXiv:1705.08584, 2017.
+
+Yujia Li, Kevin Swersky, and Richard S. Zemel. Generative moment matching networks. In Proceedings of the 32nd International Conference on Machine Learning, ICML 2015, Lille, France, 6-11 July 2015, pp. 1718–1727, 2015.
+
+Sebastian Nowozin, Botond Cseke, and Ryota Tomioka. f-gan: Training generative neural samplers using variational divergence minimization. In Advances in Neural Information Processing Systems, pp. 271–279, 2016.
+
+Alec Radford, Luke Metz, and Soumith Chintala. Unsupervised representation learning with deep convolutional generative adversarial networks. arXiv preprint arXiv:1511.06434, 2015.
+
+Tim Salimans, Ian Goodfellow, Wojciech Zaremba, Vicki Cheung, Alec Radford, and Xi Chen. Improved techniques for training gans. In Advances in Neural Information Processing Systems, pp. 2234–2242, 2016.
+
+Dougal J. Sutherland, Hsiao-Yu Fish Tung, Heiko Strathmann, Soumyajit De, Aaditya Ramdas, Alexander J. Smola, and Arthur Gretton. Generative models and model criticism via optimized maximum mean discrepancy. arXiv preprint arXiv:1611.04488, 2016.
+
+Cedric Villani. ´ Topics in optimal transportation. Number 58. American Mathematical Soc., 2003.
+
+Cedric Villani. ´ Optimal transport: old and new, volume 338. Springer Science & Business Media, 2008.
+
+Jonathan Weed and Francis Bach. Sharp asymptotic and finite-sample rates of convergence of empirical measures in wasserstein distance. arXiv preprint arXiv:1707.00087, 2017.
+
+Jacob Wolfowitz. The minimum distance method. The Annals of Mathematical Statistics, pp. 75–88, 1957.
+
+A EXAMPLE I-WGAN SAMPLES
+
+![](images/299b991f8908ee02ee348dccccbb828ad8fbf9b2fe032078582a7571aa052396.jpg)  
+Figure 6: Samples from I-WGAN trained on MNIST
+
+![](images/065b6587704900cb66c309c2e9ca20f09a0a7d9d4494f91a33ebe8b5024c26b7.jpg)  
+Figure 7: Samples from I-WGAN trained on CIFAR-10
+
+B EXAMPLE DCGAN SAMPLES
+
+![](images/b16a225d78db6b6de2a7cb0b97ef2efaa0d11a8461afc63ce1f1e2360a25339b.jpg)  
+Figure 8: Samples from DCGAN trained on CIFAR-10 test set
+
+![](images/914e843c27d498e9477ae0e3631b8a88f4623f1ce1a32b22549f33ca9bb534c5.jpg)  
+Figure 9: Effect of different $\sigma$ on $\beta _ { \sigma } ( X )$ , for $X$ the CIFAR-10 training set

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+        "type": "text",
+        "text": "TOWARDS A TESTABLE NOTION OF GENERALISATION FOR GENERATIVE ADVERSARIAL NETWORKS ",
+        "text_level": 1,
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+        "text": "Anonymous authors Paper under double-blind review ",
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+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+        "text": "We consider the question of how to assess generative adversarial networks, in particular with respect to whether or not they generalise beyond memorising the training data. We propose a simple procedure for assessing generative adversarial network performance based on a principled consideration of what the actual goal of generalisation is. Our approach involves using a test set to estimate the Wasserstein distance between the generative distribution produced by our procedure, and the underlying data distribution. We use this procedure to assess the performance of several modern generative adversarial network architectures. We find that this procedure is sensitive to the choice of ground metric on the underlying data space, and suggest a choice of ground metric that substantially improves performance. We finally suggest that attending to the ground metric used in Wasserstein generative adversarial network training may be fruitful, and provide a concrete formulation for doing so. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Generative adversarial networks (GANs) (Goodfellow et al., 2014) have attracted significant interest as a means for generative modelling. However, recently concerns have been raised about their ability to generalise from training data and their capacity to overfit (Arora & Zhang, 2017; Arora et al., 2017). Moreover, techniques for evaluating the quality of GAN output are either ad hoc, lack theoretical rigor, or are not suitably objective – often times “visual inspection” of samples is the main tool of choice for the practitioner. More fundamentally, it is sometimes unclear exactly what we want a GAN to do: what is the learning task that we are trying to achieve? ",
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+        "text": "In this paper, we provide a simple formulation of the GAN training framework, which consists of using a finite dataset to estimate an underlying data distribution. The quality of GAN output is measured precisely in terms of a statistical distance $D$ between the estimated and true distribution. Within this context, we propose an intuitive notion of what it means for a GAN to generalise. ",
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+        "text": "We also show how our notion of performance can be measured empirically for any GAN architecture when $D$ is chosen to be a Wasserstein distance, which – unlike other methods such as the inception score (Salimans et al., 2016) – requires no density assumptions about the data-generating distribution. We investigate this choice of $D$ empirically, finding that its performance is heavily dependent on the choice of ground metric underlying the Wasserstein distance. We suggest a novel choice of ground metric that we show performs well, and also discuss how we might otherwise use this observation to improve the design of Wasserstein GANs (WGANs) (Arjovsky et al., 2017). ",
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+        "type": "text",
+        "text": "2 THE OBJECTIVE OF GENERATIVE MODELLING ",
+        "text_level": 1,
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+        "text": "GANs promise a means for learning complex probability distributions in an unsupervised fashion. In order to assess their effectiveness, we must first define precisely what we mean by this. We seek to do so in this section, presenting a formulation of the broader goal of generative modelling that we believe is widely compatible with much present work in this area. We also provide a natural notion of generalisation that arises in our framework. ",
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+        "text": "Our setup consists of the following components. We assume some distribution $\\pi$ on a set $\\mathcal { X }$ . For instance, $\\mathcal { X }$ may be the set of $3 2 \\mathrm { x } 3 2 $ colour images, and $\\pi$ the distribution from which the CIFAR-10 dataset was sampled. We assume that $\\pi$ is completely intractable: we do not know its density (or even if it has one), and we do not have a procedure to draw novel samples from it. However, we do suppose that we have a fixed dataset $X$ consisting of samples $x _ { 1 } , \\cdot \\cdot \\cdot , x _ { | X | } \\overset { \\mathrm { i i d } } { \\sim } \\pi$ . Equivalently, we have the empirical distribution ",
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+        "img_path": "images/72db4c497a8087ecefbcaf7229a670d5054b3f17f444272d311852990706b3eb.jpg",
+        "text": "$$\n\\hat { X } : = \\frac { 1 } { | X | } \\sum _ { n = 1 } ^ { | X | } \\delta _ { x _ { n } } ,\n$$",
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+        "text": "where $\\delta$ denotes the Dirac mass. ",
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+    {
+        "type": "text",
+        "text": "Let $\\mathcal { P } ( \\mathcal { X } )$ denote the set of probability distributions on $\\mathcal { X }$ . Our aim is to use $X$ to produce a distribution in $\\mathcal { P } ( \\mathcal { X } )$ that is as “close” as possible to $\\pi$ . We choose to measure closeness in terms of a function $D : \\mathcal { P } ( \\mathcal { X } ) \\times \\mathcal { P } ( \\mathcal { X } )  \\mathbb { R }$ . Usually $D$ will be chosen to be a statistical divergence, which means that $D ( P , Q ) \\geq 0$ for all $P$ and $Q$ , with equality iff $P = Q$ . The task of a learning algorithm $\\alpha$ in this context is then as follows: ",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "equation",
+        "img_path": "images/009a7d84dbeac365386a34b8c98073bce2ae289f8a352dd982f03a19d0f96efd.jpg",
+        "text": "$$\n\\boxed { \\mathrm { S e l e c t } \\alpha ( X ) \\in \\mathcal { P } ( \\mathcal { X } ) \\mathrm { ~ s u c h ~ t h a t ~ } D ( \\alpha ( X ) , \\pi ) \\mathrm { ~ i s ~ a s ~ s m a l l ~ a s ~ p o s s i b l e . } }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "We believe (1) constitutes an intuitive and useful formulation of the problem of generative modelling that is largely in keeping with present research efforts. ",
+        "bbox": [
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+        "type": "text",
+        "text": "Now, we can immediately see that one possibility is simply to choose $\\alpha ( X ) : = { \\hat { X } }$ . Moreover, in the case that $D$ is a metric for the weak topology on $\\mathcal { P } ( \\mathcal { X } )$ such as a Wasserstein distance, we have that $D ( \\hat { X } , \\pi ) \\to 0$ almost surely as $| X | \\to \\infty$ , so that, assuming $| X |$ is large enough, we can already expect $D ( \\hat { X } , \\pi )$ to be small. This then suggests the following natural notion of generalisation: ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/8496faf9b293258ae7817b5b4fa65822a3552d0c9ffa65b8c6d5fa78df952a06.jpg",
+        "text": "$$\n\\Bigl | \\mathrm { A ~ c h o i c e ~ o f } \\alpha \\mathrm { g e n e r a l i s e s ~ f o r ~ a ~ g i v e n } X \\mathrm { i f } \\ D ( \\alpha ( X ) , \\pi ) < D ( \\hat { X } , \\pi ) . \\Bigr |\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In other words, using $\\alpha$ here has actually achieved something: perhaps through some process of smoothing or interpolation, it has injected additional information into $\\hat { X }$ that has moved us closer to $\\pi$ than we were a priori. ",
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+    {
+        "type": "text",
+        "text": "3 GENERALISATION IN GANS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The previous section presented (1) as a general goal of generative modelling. In this section, we turn specifically to GANs. We begin by providing a general model for how many of the existing varieties of GAN operate, at least ideally. We then show how this model fits into our framework above, before considering the issue of generalisation in this context. ",
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+    {
+        "type": "text",
+        "text": "Most GAN algorithms in widespread use adhere to the following template: they take as input a distribution $P$ , from which we assume we can sample, and compute (or approximate) ",
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+        "type": "equation",
+        "img_path": "images/e6287e635295b75a0440bbaa9f6ed0ac5714dc97966173e3f942b20fec91fa61.jpg",
+        "text": "$$\n\\Gamma ( P ) : = \\underset { Q \\in \\mathcal { Q } } { \\arg \\operatorname* { m i n } } D _ { \\Gamma } ( P , Q )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "for some choices of $\\mathcal Q \\subseteq \\mathcal P ( \\mathcal X )$ and $D _ { \\Gamma } : \\mathcal { P } ( \\mathcal { X } ) \\times \\mathcal { P } ( \\mathcal { X } )  \\mathbb { R }$ . In other words, in the ideal case, a GAN maps $P$ to its $D _ { \\Gamma }$ -projection onto $\\mathcal { Q }$ . Note that we will not necessarily have that $D _ { \\Gamma } = D$ : $D _ { \\Gamma }$ is fixed given a particular GAN architecture, whereas the choice of $D$ is simply a feature of our problem definition (1) and is essentially ours to make. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In practice, $\\mathcal { Q }$ is the set of pushforward measures $\\nu \\circ G ^ { - 1 }$ obtained from a fixed noise distribution $\\nu$ on a noise space $\\mathcal { Z }$ and some set $\\mathcal { G }$ of functions $G : { \\mathcal { Z } }  { \\mathcal { X } }$ . Precisely, $\\mathcal { Q } = \\left\\{ \\nu \\circ G ^ { - 1 } : G \\in \\mathcal { G } \\right\\}$ . $\\mathcal { G }$ itself usually corresponds to the set of functions realisable by some neural network architecture, and $\\nu$ is some multivariate uniform or Gaussian distribution. However, numerous choices of $D _ { \\Gamma }$ have been proposed: the original GAN formulation (Goodfellow et al., 2014) took $D _ { \\Gamma }$ to be the Jenson-Shannon divergence, whereas the $f$ -GAN (Nowozin et al., 2016) generalised this to arbitrary $f$ -divergences, and the Wasserstein GAN (Arjovsky et al., 2017) advocated the Wasserstein distance. Many of the results proved in these papers involve showing that (usually under some assumptions, such as sufficient discriminator capacity) a proposed objective for $G$ is in fact equivalent to $\\bar { D _ { \\Gamma } ( P , \\nu \\circ }$ $G ^ { - 1 }$ ). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In terms of our framework in the previous section, using a GAN $\\Gamma$ amounts to choosing ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/42b4cbe19df6ad9bb9f3e4c8af185b6f7569dd5ae396bf7c3c2b38e3abfcaf17.jpg",
+        "text": "$$\n\\alpha ( X ) : = \\Gamma ( \\hat { X } ) = \\underset { Q \\in \\mathcal { Q } } { \\arg \\operatorname* { m i n } } D _ { \\Gamma } ( \\hat { X } , Q ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We emphasise again the important distinction between $D$ and $D _ { \\Gamma }$ . In our setup, minimising $D$ defines our ultimate goal, whereas minimising $D _ { \\Gamma }$ (over $\\mathcal { Q }$ ) defines how we will attempt to achieve that goal. Even if $D \\neq D _ { \\Gamma }$ , it is still at least conceivable that $D ( \\Gamma ( { \\hat { X } } ) , \\pi )$ might be small, and therefore this choice of $\\alpha$ might be sensible. Also note that, crucially, $\\Gamma$ receives $\\hat { X }$ as input rather than $\\pi$ itself. We only have access to a fixed number of CIFAR-10 samples, for example, not an infinite stream. Moreover, training GANs usually involves making many passes over the same dataset, so that, in effect, sampling from $P$ will repeatedly yield the same data points. We would not expect this to occur with nonzero probability if $P = \\pi$ for most $\\pi$ of interest. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The observation that $P$ is $\\hat { X }$ rather than $\\pi$ was also recently made by Arora et al. (2017). The authors argue that this introduces a problem for the ability of GANs to generalise, since, if $D _ { \\Gamma }$ is a divergence (which is almost always the case), and if $\\mathcal { Q }$ is too big (in particular, if it is big enough that $\\hat { X } \\in \\mathcal { Q } ,$ ), then we trivially have that $\\Gamma ( { \\hat { X } } ) = { \\hat { X } }$ . In other words, the GAN objective appears actively to encourage $\\Gamma ( { \\hat { X } } )$ to memorise the dataset $X$ , and never to produce novel samples from outside of it. The authors’ proposed remedy involves trying to find a better choice of $D _ { \\Gamma }$ . The problem, they argue, is that popular choices of $D _ { \\Gamma }$ do not satisfy the condition ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "$D _ { \\Gamma } ( \\pi , Q ) \\approx D _ { \\Gamma } ( { \\hat { X } } , Q )$ with high probability given a modest number of samples in $X$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "They point out that this is certainly violated when $D _ { \\Gamma }$ is the Jensen-Shannon divergence JS, since ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/72f371f44c438deeaf78eed103e3ced7af6c2aa5a15ecd2a7c0f70ae5a4861db.jpg",
+        "text": "$$\n\\operatorname { J S } ( P \\parallel Q ) = \\log 2\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "when one of $P$ and $Q$ is discrete and the other continuous, and give a similar result for $D _ { \\Gamma }$ a Wasserstein distance in the case that $\\pi$ is Gaussian. As a solution, they introduce the neural network distance $D _ { \\mathrm { N N } }$ defined by ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5e03cdcb6194087161d9f8eb6ddbd3167d4cb14dd86d2d360728b63f4a0ff0eb.jpg",
+        "text": "$$\nD _ { \\mathrm { N N } } ( P , Q ) : = \\operatorname* { m a x } _ { f \\in \\mathcal { F } } \\mathbb { E } _ { x \\sim P } \\left[ f ( x ) \\right] - \\mathbb { E } _ { x \\sim Q } \\left[ f ( x ) \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "for some choice of a class of functions $\\mathcal { F }$ . They show that, assuming some smoothness conditions on the members of $\\mathcal { F }$ , the choice $D _ { \\Gamma } = D _ { \\mathrm { N N } }$ satisfies (3), which means that if we minimise $D _ { \\mathrm { N N } } ( \\hat { X } , Q )$ in $Q$ then we can be confident that the value of $D _ { \\mathrm { N N } } ( \\pi , Q )$ is small also. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "However, we do not believe that (3) is sufficient to ensure good generalisation behaviour for GANs. What we care about ultimately is not the value of $D _ { \\Gamma }$ , but rather of $D$ , and (3) invites choosing $D _ { \\Gamma }$ in such a way that gives no guarantees about $D$ at all. We see, for instance, that the degenerate choice $D _ { 0 } ( P , Q ) : = 0$ trivially satisfies (3), and indeed is also a pseudometric, just like $D _ { \\mathrm { N N } }$ . It is therefore unclear what mathematical properties of $D _ { \\mathrm { N N } }$ render it more suitable for estimating $\\pi$ than the obviously undesirable $D _ { 0 }$ . The authors do acknowledge this shortcoming of $D _ { \\mathrm { N N } }$ , pointing out that $D _ { \\mathrm { N N } } ( P , Q )$ can be small even if $P$ and $Q$ are “quite different” in some sense. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The problematic consequences of the fact that $P = { \\hat { X } }$ apply only in the case that $\\mathcal { Q }$ is too large. In practice, however, $\\mathcal { Q }$ is heavily restricted, since $\\mathcal { G }$ is restricted via a choice of neural network architecture; hence we do not know $a$ priori whether $\\boldsymbol { \\Gamma } ( \\boldsymbol { \\hat { X } } ) = \\boldsymbol { \\hat { X } }$ is even possible. As such, we do not see the choice of $\\alpha ( X ) = \\Gamma ( \\hat { X } )$ as necessarily a bad idea, and instead believe that it is an open empirical question as to how well GANs perform the task (1). In fact, this $\\alpha$ falls perfectly within the framework of minimum distance estimation (Wolfowitz, 1957; Basu et al., 2011), which involves estimating an underlying distribution by minimising a distance measure to a given empirical distribution. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4 TESTING GANS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our goal in this section is to assess how well GANs achieve (1) by estimating $D ( \\Gamma ( X ) , \\pi )$ for various $\\Gamma$ and $\\pi$ . This raises some difficulties, given that $\\pi$ is intractable. Our approach is to take $D$ to be the first Wasserstein distance $W _ { d _ { x } }$ defined by ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7fec5d376faa6f9358aea4622c5ee4c74a4f729ecfa67147dbd8189208f07fb6.jpg",
+        "text": "$$\nW _ { d x } ( P , Q ) = \\operatorname* { i n f } _ { \\gamma \\in \\Pi ( P , Q ) } \\int _ { \\mathcal { X } \\times \\mathcal { X } } d _ { \\mathcal { X } } ( x , y ) \\mathrm { d } \\gamma ( x , y ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $d _ { \\mathcal { X } }$ is a metric on $\\mathcal { X }$ referred to as the ground metric, and $\\Pi ( P , Q )$ denotes the set of joint distributions on the product space $\\mathcal { X } \\times \\mathcal { X }$ with marginals $P$ and $Q$ . The Wasserstein distance is appealing since it is sensitive to the topology of the underlying set $\\mathcal { X }$ , which we control by our choice of $d _ { \\mathcal { X } }$ . Moreover, $W _ { d x }$ metricises weak convergence for the Wasserstein space $\\mathcal { P } _ { d _ { \\mathcal { X } } } ( \\mathcal { X } )$ defined by ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c3ff49cd3aea245490dc11874730261c84af2d718a5f3e1ae26a6f1d5ec35c57.jpg",
+        "text": "$$\n{ \\mathcal { P } } _ { d _ { \\mathcal { X } } } ( \\mathcal { X } ) : = \\left\\{ P \\in { \\mathcal { P } } ( \\mathcal { X } ) : \\int _ { \\mathcal { X } } d _ { \\mathcal { X } } ( x _ { 0 } , y ) \\mathrm { d } P ( y ) < \\infty { \\mathrm { ~ f o r ~ s o m e ~ } } x _ { 0 } \\in \\mathcal { X } \\right\\}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "(see (Villani, 2008)). Consequently, if we denote by $A$ a set of samples $a _ { 1 } , \\cdots , a _ { | A | }$ iid∼ $\\alpha ( X )$ and by $Y$ a set of samples (separate from $X$ ) $y _ { 1 } , \\cdots , y _ { | Y | } \\overset { \\mathrm { i i d } } { \\sim } \\pi .$ , with $\\hat { A }$ and $\\hat { Y }$ the corresponding empirical distributions, then, provided ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1d8b0d0baf7751fc094514a7eb0cb14a1eb1eb94c44d3cc3807c763fecb19097.jpg",
+        "text": "$$\n\\alpha ( X ) , \\pi \\in { \\mathcal { P } } _ { d _ { \\mathcal { X } } } ( { \\mathcal { X } } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "we have that $D ( { \\hat { A } } , { \\hat { Y } } ) \\to D ( \\alpha ( X ) , \\pi )$ almost surely as $\\operatorname* { m i n } \\{ | A | , | Y | \\}  \\infty$ . Note that condition (4) holds automatically in the case that $( \\mathcal { X } , d _ { \\mathcal { X } } )$ is compact, since then $\\mathcal { P } _ { d _ { \\mathcal { X } } } ( \\mathcal { X } ) = \\mathcal { P } ( \\mathcal { X } )$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As such, to estimate $D ( \\alpha ( X ) , \\pi )$ , for $D = W _ { d _ { x } }$ , we propose the following. Before training, we move some of our samples from $X$ into a testing set $Y$ . We next train our GAN on $X$ , obtaining $\\alpha ( X )$ . We then take samples $A$ from $\\alpha ( X )$ , and obtain the estimate ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8858e5e950845cbf28c9685545c49709087cba2b02ba92305a5b9e3282c6df45.jpg",
+        "text": "$$\nW _ { d _ { \\mathcal { X } } } ( \\hat { A } , \\hat { Y } ) \\approx W _ { d _ { \\mathcal { X } } } ( \\alpha ( X ) , \\pi ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            393,
+            392,
+            602,
+            412
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where the left-hand side can be computed exactly by solving a linear program since both $\\hat { A }$ and $\\hat { Y }$ are discrete (Villani, 2003). We can also use the same methodology to estimate $W _ { d _ { \\mathcal { X } } } ( \\hat { X } , \\pi )$ by $W _ { d x } ( \\hat { X } , \\hat { Y } )$ , which suggests testing if ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/378bd542877e3aeae416ae3234fee2ba0cfeb82dbc4c5507c4142253f3932341.jpg",
+        "text": "$$\nW _ { d _ { \\mathcal { X } } } ( \\hat { A } , \\hat { Y } ) < W _ { d _ { \\mathcal { X } } } ( \\hat { X } , \\hat { Y } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "as a proxy for determining whether (2) holds. A summary of our procedure is given in Algorithm 1. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 Procedure for testing GANs ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "table",
+        "img_path": "images/29945c3e790cf57aaba3e0d8dca70348a692f6df9498e926a06d75773838c9a7.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>1: Split samples from π into a training set X and a testing set Y</td></tr><tr><td>2: Compute α(X) by training a GAN on X 3: Obtain a sample A from α(X)</td></tr></table>",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.1 RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We applied our methodology to test two popular GAN varieties – the DCGAN (Radford et al., 2015) and the Improved Wasserstein GAN (I-WGAN) (Gulrajani et al., 2017) – on the MNIST and CIFAR-10 datasets. In all cases when computing the relevant Wasserstein distances, our empirical distributions consisted of 10000 samples. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.1.1 $L ^ { 2 }$ AS GROUND METRIC ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We initially took our ground metric $d _ { \\mathcal { X } }$ to be the $L ^ { 2 }$ distance. This has the appealing property of making $( \\boldsymbol { \\mathcal { X } } , d _ { \\boldsymbol { \\mathcal { X } } } )$ compact when $\\mathcal { X }$ is a space of RGB or greyscale images, therefore ensuring that (4) holds. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Figure 1 shows $W _ { L ^ { 2 } } ( \\hat { A } , \\hat { Y } )$ plotted over multiple training runs for MNIST and CIFAR-10 when $A$ is obtained from an I-WGAN, with typical samples shown in Figures 6 and 7 in the appendix. For both datasets, $W _ { L ^ { 2 } } ( \\hat { A } , \\hat { Y } )$ decays towards an asymptote in a way that nicely corresponds to the visual quality of the samples produced. Moreover, $W _ { L ^ { 2 } } ( \\hat { A } , \\hat { Y } )$ is much closer to $W _ { L ^ { 2 } } ( \\hat { X } , \\hat { Y } )$ towards the tail end of training for MNIST than for CIFAR-10. This seems to reflect the fact that, visually, the eventual I-WGAN MNIST samples do seem quite close to true MNIST samples, whereas the eventual I-WGAN CIFAR-10 samples are easily identified as fake. ",
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+        "page_idx": 3
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+    {
+        "type": "image",
+        "img_path": "images/a00baa7a2f8c1338c34dc16ac2b9d21542acedfe8552ef8248e5e4e39486957e.jpg",
+        "image_caption": [
+            "Figure 1: Output of Algorithm 1 with $d _ { \\mathcal { X } } = L ^ { 2 }$ for I-WGAN trained on MNIST (left) and CIFAR-10 (right) "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/c579bf2b3cf37f9b358f5494cc533abd1bcd6e975256427480f8283fbd348944.jpg",
+        "image_caption": [
+            "Figure 2: Output of Algorithm 1 with $d _ { \\mathcal { X } } = L ^ { 2 }$ for DCGAN trained on CIFAR-10 "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "However, when we re-ran the same experiment using a DCGAN on CIFAR-10, we obtained the $W _ { L ^ { 2 } } ( \\hat { A } , \\hat { Y } )$ trajectories shown in Figure 2. Typical examples are shown in Figure 8. Strangely, we observe that $W _ { L ^ { 2 } } ( \\hat { A } , \\hat { Y } ) < W _ { L ^ { 2 } } ( \\hat { X } , \\hat { Y } )$ very early on in training – at around batch 500 – when the samples resemble the heavily blurry Figure 8b. This raises some obvious concerns about the appropriateness of $W _ { L ^ { 2 } }$ as a metric for GAN quality. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We therefore sought to understand this strange behaviour. Motivated by the visual blurriness of the samples in Figure 8b, we explored the effect on $W _ { L ^ { p } } ( \\hat { X } , \\hat { Y } )$ of blurring the CIFAR-10 training set $X$ . In particular, we let $X$ and $Y$ each consist of 10000 distinct CIFAR-10 samples in $X$ . We then independently convolved each channel of each image with a Gaussian kernel having standard deviation $\\sigma$ , obtaining a blurred dataset $\\beta _ { \\sigma } ( X )$ and corresponding empirical distribution $\\hat { \\beta } _ { \\sigma } ( X )$ . The visual effect of this procedure is shown in Figure 9 in the appendix. We then computed $W _ { L ^ { p } } ( \\hat { \\beta } _ { \\sigma } ( X ) , \\hat { Y } )$ with $\\sigma$ ranging between 0 and 10 for a variety of values of $p$ . The results of this experiment in the case $p = 2$ are shown in Figure 3, and similar results were observed for other values of $p$ : in all cases, we found that ",
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+    {
+        "type": "equation",
+        "img_path": "images/49eb788026668dbe23179ba106fe5f1a7b89c7ecef90561da83199fbf9fe3b52.jpg",
+        "text": "$$\nW _ { L ^ { p } } ( \\hat { X } , \\hat { Y } ) > W _ { L ^ { p } } ( \\hat { \\beta } _ { \\sigma } ( X ) , \\hat { Y } )\n$$",
+        "text_format": "latex",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "whenever $\\sigma > 0$ . That is, blurring $X$ by any amount brings $\\hat { X }$ closer to $\\hat { Y }$ in $\\boldsymbol { W } _ { L ^ { p } }$ than not. This occurs even though $X$ is distributed identically to $Y$ (both being drawn from $\\pi$ ), while $\\beta _ { \\sigma } ( X )$ (presumably) is not when $\\sigma > 0$ . ",
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+        "page_idx": 4
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+    {
+        "type": "image",
+        "img_path": "images/45f392895d2f6ffc598a181c84f164d1b14ef60a793bd79e85f92972b0cf924d.jpg",
+        "image_caption": [
+            "Figure 3: Effect of blurring CIFAR-10 on $W _ { L ^ { 2 } }$ (left) and $W _ { L ^ { 2 } \\circ \\eta }$ (right) "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.1.2 EMBEDDED $L ^ { 2 }$ AS GROUND METRIC ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To remedy these issues, we sought to replace $L ^ { 2 }$ with a choice of $d _ { \\mathcal { X } }$ that is more naturally suited to the space of images in question. To this end we tried mapping $\\mathcal { X }$ through a fixed pre-trained neural network $\\eta$ into a feature space $\\mathcal { V }$ , and then computing distances using some metric $d _ { \\mathcal { Y } }$ on $\\mathcal { V }$ , rather than in $\\mathcal { X }$ directly. It is easily seen that, provided $\\eta$ is injective, $d _ { \\mathcal { V } } \\circ \\eta : \\mathcal { X } \\times \\mathcal { X } \\to \\mathbb { R }$ defined by ",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "equation",
+        "img_path": "images/fc986f31825ce97f27c30f6bea83a93cab4570a43cdf5ae563a88b9b46e2ffac.jpg",
+        "text": "$$\n( d _ { \\mathcal { V } } \\circ \\eta ) ( x , x ^ { \\prime } ) : = d _ { \\mathcal { V } } ( \\eta ( x ) , \\eta ( x ^ { \\prime } ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "is a metric. It also holds that, when $\\eta$ is $( d _ { \\mathcal { X } } , d _ { \\mathcal { Y } } )$ -continuous, $( \\mathcal { X } , d _ { \\mathcal { Y } } \\circ \\eta )$ is compact when $( \\mathcal { X } , d _ { \\mathcal { X } } )$ is: given a sequence $x _ { i } \\subseteq { \\mathcal { X } }$ , there exists a subsequence $x _ { i ^ { \\prime } }$ that converges in $d _ { \\mathcal { X } }$ to some $x$ (by compactness), so that $( d \\ / y \\circ \\eta ) ( x _ { i ^ { \\prime } } , x ) = d \\ / y ( \\eta ( x _ { i ^ { \\prime } } ) , \\eta ( x ) )  0$ by continuity of $\\eta$ . Neither of these conditions on $\\eta$ – injectivity and $( d _ { \\mathcal { X } } , d _ { \\mathcal { Y } } )$ -continuity – are unreasonable to expect from a typical neural network trained using stochastic gradient descent (at least, when $d _ { \\mathcal { X } }$ and $d _ { \\mathcal { Y } }$ are typical metrics such as $L ^ { p }$ distances). Consequently, $W _ { d _ { y } \\circ \\eta }$ constitutes a valid metric on $\\mathcal { P } _ { d _ { y } \\circ \\eta } ( \\mathcal { X } )$ , and (4) is automatically satisfied. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To test the performance of $W _ { d y \\circ \\eta }$ , we repeated the blurring experiment described above. We took $\\eta ( x )$ to be the result of scaling $x \\in \\mathcal { X }$ to size $2 2 4 \\mathbf { x } 2 2 4$ , mapping the result through a DenseNet-121 (Huang et al., 2016) pre-trained on ImageNet (Deng et al., 2009), and extracting features immediately before the linear output layer. Under the same experimental setup as above otherwise, we obtained the plot of $W _ { L ^ { 2 } \\circ \\eta } ( \\beta _ { \\sigma } ( \\hat { X } ) , \\hat { Y } )$ shown in Figure 3. Happily, we now see that this curve increases monotonically as $\\sigma$ grows in accordance with the declining visual quality of $\\beta _ { \\sigma } ( X )$ shown in Figure 9. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Next, we computed $W _ { L ^ { 2 } \\circ \\eta } ( \\hat { A } , \\hat { Y } )$ over the course of GAN training. For the I-WGAN we obtained the results on MNIST and CIFAR-10 shown in Figure 4;1 for the DCGAN on CIFAR-10, we obtained the curve shown in Figure 5. In all cases we see that $W _ { L ^ { 2 } \\circ \\eta } ( \\hat { A } , \\hat { Y } )$ decreases monotonically towards an asymptote in a way that accurately summarises the visual quality of the samples throughout the training run. Moreover, there is always a large gap between the eventual value of $W _ { L ^ { 2 } \\circ \\eta } ( \\hat { A } , \\hat { Y } )$ and $W _ { L ^ { 2 } \\circ \\eta } ( \\hat { X } , \\hat { Y } )$ , which reflects the fact that the GAN samples are still visually distinguishable from real $\\pi$ samples. In particular, we see an improvement in this respect for the I-WGAN on MNIST: in Figure 1 the asymptotic value of $W _ { L ^ { 2 } } ( \\hat { \\hat { A } } , \\hat { Y } )$ for MNIST was barely discernible from $W _ { L ^ { 2 } } ( \\hat { X } , \\hat { Y } )$ , despite the fact that it is still quite easy to tell real samples from generated ones (see e.g. the various mistakes present in Figure 6). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "5 DISCUSSION AND FUTURE WORK ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We believe our work reveals two promising avenues of future inquiry. First, we suggest that $W _ { L ^ { p } \\circ \\eta }$ is an appealing choice of $D$ , both due to its nice theoretical properties – it metricises weak convergence, and does not require us to make any density assumptions about $\\pi$ – and due to its sound empirical performance demonstrated above. It would be very interesting to use this $D$ to produce a systematic and objective comparison of the performance of all current major GAN implementations, and indeed to use this as a metric for guiding future GAN design. We also view the test (5) as potentially useful for determining whether our algorithms are overfitting. This would be particularly so if applied via a cross-validation procedure: if we consistently observe that (5) holds when training a GAN according to many different $X$ and $Y$ partitions of our total $\\pi$ samples, then it seems reasonable to infer that $\\alpha ( X )$ has indeed learnt something useful about $\\pi$ . ",
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/570e844b51dbee20d95f6a30d2c3878bb1d1165315288d1f52be0133d3933878.jpg",
+        "image_caption": [
+            "Figure 4: Output of Algorithm 1 with $d _ { \\mathcal { X } } = L ^ { 2 } \\circ \\eta$ for I-WGAN trained on MNIST (left) and CIFAR-10 (right) "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/852317811ce1e44b66dd32c17d799d4f81b0b3e03c4c07617a867f42293c9ab6.jpg",
+        "image_caption": [
+            "Figure 5: Output of Algorithm 1 with $d _ { \\mathcal { X } } = L ^ { 2 } \\circ \\eta$ for DCGAN trained on CIFAR-10 "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "We also believe that the empirical inadequacy of $W _ { L ^ { 2 } }$ that we observed suggests a path towards a better WGAN architecture. At present, WGAN implementations implicitly use $W _ { L ^ { 2 } }$ for their choice of $D _ { \\Gamma }$ . We suspect that altering this to our suggested $W _ { L ^ { 2 } \\circ \\eta }$ may yield better quality samples. We briefly give here one possible way to do so that is largely compatible with existing WGAN setups. In particular, following Arjovsky et al. (2017), we take ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/648ca1bda995ec0da1b7e1edb3e784a917d564764576aa4e14043f1c4f1d29f0.jpg",
+        "text": "$$\nD _ { \\Gamma } ( P , Q ) = \\underset { f \\in \\mathcal { F } } { \\operatorname* { m a x } } \\mathbb { E } _ { x \\sim P } \\left[ f ( x ) \\right] - \\mathbb { E } _ { x \\sim Q } \\left[ f ( x ) \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "for a class $\\mathcal { F }$ of functions $f : \\mathcal { X }  \\mathbb { R }$ that are all $\\left( L ^ { 2 } \\circ \\eta , d _ { \\mathbb { R } } \\right)$ -Lipschitz for some fixed Lipschitz constant $K$ . Here $d _ { \\mathbb { R } }$ denotes the usual distance on $\\mathbb { R }$ . To optimise over such an $\\mathcal { F }$ in practice, we can require our discriminator $f : \\mathcal { X } \\to \\mathbb { R }$ to have the form ${ \\bar { f } } ( x ) : = h ( \\eta ( x ) )$ , where $h : \\mathcal { V } \\to \\mathbb { R }$ is $( d _ { \\mathcal { Y } } , d _ { \\mathbb { R } } )$ -Lipschitz, which entails that $f$ is Lipschitz provided $\\eta$ is (which is almost always the case in practice). In other words, we compute ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e1465655a529e7cf7d05755c5e7b30696835d8cec970be3979d83b287dac9eaa.jpg",
+        "text": "$$\nD _ { \\Gamma } ( P , Q ) = \\operatorname* { m a x } _ { h \\in \\mathcal { F } ^ { \\prime } } \\mathbb { E } _ { x \\sim P } \\left[ h ( \\eta ( x ) ) \\right] - \\mathbb { E } _ { x \\sim Q } \\left[ h ( \\eta ( x ) ) \\right] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where ${ \\mathcal { F } } ^ { \\prime }$ is a class of $( d _ { \\mathcal { Y } } , d _ { \\mathbb { R } } )$ -Lipschitz functions. Optimising over this objective may now proceed as usual via weight-clipping like (Arjovsky et al., 2017), or via a gradient penalty like (Gulrajani et al., 2017). Note that this suggestion may be understood as training a standard WGAN with the initial layers of the discriminator fixed to the embedding $\\eta$ ; our analysis here shows that this is equivalent to optimising with respect to $W _ { L ^ { 2 } \\circ \\eta }$ instead of $W _ { L ^ { 2 } }$ . We have begun some experimentation in this area but leave a more detailed empirical inquiry to future work. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "It is also clearly important to establish better theoretical guarantees for our method. At present, we have no guarantee that the number of samples in $A$ and $Y$ are enough to ensure that ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ada0393c46af90c360ac7efbf614a765263589887854562610a2da8297df75a3.jpg",
+        "text": "$$\nD ( \\hat { A } , \\hat { Y } ) \\approx D ( \\alpha ( X ) , \\hat { Y } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "(perhaps with some fixed bias that is fairly independent of $\\alpha$ , so that it is valid to use the value of $D ( \\hat { A } , \\hat { Y } )$ to compare different choices of $\\alpha$ ), or that (5) entails (2) with high probability. We do however note that some recent theoretical work on the convergence rate of empirical Wasserstein estimations (Weed & Bach, 2017) does suggest that it is plausible to hope for fast convergence of $D ( \\hat { A } , \\hat { Y } )$ to $D ( \\alpha ( X ) , { \\hat { Y } } )$ . We also believe that the convincing empirical behaviour of $W _ { L ^ { 2 } \\circ \\eta }$ does suggest that it is possible to say something more substantial about our approach, which we leave to future work. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "6 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            292,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The maximum mean discrepancy (MMD) is another well-known notion of distance on probability distributions, which has been used for testing whether two distributions are the same or not (Gretton et al., 2012) and also for learning generative models in the style of GAN (Li et al., 2015; Dziugaite et al., 2015; Sutherland et al., 2016; Li et al., 2017). It is parameterised by a characteristic kernel $k$ , and defines the distance between probability distributions by means of the distance of $k$ ’s reproducing kernel Hilbert space (RKHS). The MMD induces the same weak topology on distributions as the one of the Wasserstein distance. Under a mild condition, the MMD between distributions $P$ and $Q$ under a kernel $k$ can be understood as the outcome of the following two-step calculation. First, we pushforward $P$ and $Q$ from their original space $\\mathcal { X }$ to a Hilbert space $\\mathcal { H }$ (isomorphic to $k$ ’s RKHS) using a feature function $\\phi : \\mathcal { X } \\to \\mathcal { H }$ induced by the kernel $k$ . Typically, $\\mathcal { H }$ is an infinite-dimensional space, such as the set $\\ell _ { 2 }$ of square-summable sequences in ${ \\mathcal { R } } ^ { \\infty }$ as in Mercer’s theorem. Let $P ^ { \\prime }$ and $Q ^ { \\prime }$ be the resulting distributions on the feature space. Second, we compute the supremum of $\\mathbb { E } _ { P ^ { \\prime } ( X ) } [ f ( X ) ] - \\mathbb { E } _ { Q ^ { \\prime } ( X ) } [ f ( X ) ]$ over all linear 1-Lipschitz functions $f : \\mathcal { H } \\to \\mathbb { R }$ . The result of this calculation is the MMD between $P$ and $Q$ . ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "This two-step calculation shows the key difference between MMD and our use of Wasserstein distance and neural embedding $\\eta$ . While the co-domain of the feature function $\\phi$ is an infinitedimensional space (e.g. $\\ell _ { 2 }$ ) in most cases, its counterpart $\\eta$ in our setting uses a finite-dimensional space as co-domain. This means that the MMD possibly uses a richer feature space than our approach. On the other hand, the MMD takes the supremum over only linear functions $f$ among all 1-Lipschitz functions, whereas the Wasserstein distance considers all these 1-Lipschitz functions. These different balancing acts between the expressiveness of features and that of functions taken in the supremum affect the learning and testing of various GAN approaches as observed experimentally in the literature. One interesting future direction is to carry out systematic study on the theoretical and practical implications of these differences. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            690,
+            285,
+            704
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
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+        "text": "A EXAMPLE I-WGAN SAMPLES ",
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+    },
+    {
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+        "img_path": "images/299b991f8908ee02ee348dccccbb828ad8fbf9b2fe032078582a7571aa052396.jpg",
+        "image_caption": [
+            "Figure 6: Samples from I-WGAN trained on MNIST "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
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+        "img_path": "images/065b6587704900cb66c309c2e9ca20f09a0a7d9d4494f91a33ebe8b5024c26b7.jpg",
+        "image_caption": [
+            "Figure 7: Samples from I-WGAN trained on CIFAR-10 "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "B EXAMPLE DCGAN SAMPLES ",
+        "bbox": [
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+    {
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+        "image_caption": [
+            "Figure 8: Samples from DCGAN trained on CIFAR-10 test set "
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+        "bbox": [
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+    {
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+        "image_caption": [
+            "Figure 9: Effect of different $\\sigma$ on $\\beta _ { \\sigma } ( X )$ , for $X$ the CIFAR-10 training set "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            235,
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+        "page_idx": 12
+    }
+]

parse/train/HJx0U64FwS/HJx0U64FwS.md

@@ -0,0 +1,652 @@
+# A MECHANISM OF IMPLICIT REGULARIZATION IN DEEP LEARNING
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Despite a lot of theoretical efforts, very little is known about mechanisms of implicit regularization by which the low complexity contributes to generalization in deep learning. In particular, causality between the generalization performance, implicit regularization and nonlinearity of activation functions is one of the basic mysteries of deep neural networks (DNNs). In this work, we introduce a novel technique for DNNs called “random walk analysis” and reveal a mechanism of the implicit regularization caused by nonlinearity of ReLU activation. Surprisingly, our theoretical results suggest that the learned DNNs interpolate almost linearly between data points, which leads to the low complexity solutions in the over-parameterized regime. As a result, we prove that stochastic gradient descent can learn a class of continuously differentiable functions with generalization bounds of the order of $O ( n ^ { - 2 } )$ $\scriptstyle { n }$ : the number of samples). Furthermore, our analysis is independent of the kernel methods, including neural tangent kernels.
+
+# 1 INTRODUCTION
+
+Deep Neural Networks (DNNs) have demonstrated dominating performance in numerous machine learning tasks, and it shows great generalization performance in the over-parameterized regime. Theoretically, mechanisms of implicit regularization, which is considered as an important factor of such generalization performance in over-parameterized DNNs, still remain unknown (Zhang et al., 2016; Neyshabur, 2017). In the over-parameterized regime, recent studies report that generalization bounds for DNNs can be obtained by replacing the neural network with its linear approximation model with respect to weight parameters at initialization. Most of these studies rely on the connection between deep learning and neural tangent kernels (NTKs) (Daniely et al., 2016; Jacot et al., 2018; Arora et al., 2019b), which characterizes the dynamics of network outputs throughout gradient descent training in the infinite width limit. However, a source of implicit regularization in over-parameterized DNNs has not been identified. Recent empirical and theoretical results indicate that generalization performance and implicit regularization of over-parameterized DNNs cannot be captured by NTK analysis (Wei et al., 2018; Allen-Zhu & Li, 2019; Woodworth et al., 2019; Geiger et al., 2019). Understanding how implicit regularization properly controls the superfluous expressive power of over-parameterized DNNs gives us new insights into the theoretical analysis in deep learning. This leads to the first question:
+
+# Question 1. What kind of low complexity is caused by implicit regularization in deep learning?
+
+Linear networks without activation functions are important subject, and there are a number of theoretical works on the implicit regularization in over-parameterized neural networks mainly focusing on linear models (Ji & Telgarsky, 2018; Gidel et al., 2019; Arora et al., 2019a). In contrast, whole properties of over-parameterized DNNs that may result from nonlinearity of activation functions cannot be captured by the approximated linear models, specifically, the kernel regression predictor using the NTK. However, some mechanisms of the implicit regularization can depend on nonlinearity. This leads to the next question:
+
+# Question 2. Can we identify a mechanism of implicit regularization that depends on nonlinearity of activation functions?
+
+Now for optimization algorithms, the training dynamics of full batch gradient descent (GD) is better understood although GD is too expensive for most applications and one often uses stochastic gradient decent (SGD) instead. In most cases, the authors used GD to derive their results by the NTK analysis. Recent work showed that without any structural assumptions about the data distribution, two-layer or three-layer over-parameterized networks trained by SGD can learn $C ^ { \infty }$ -class functions (Allen-Zhu et al., 2018a; Arora et al., 2019c). However, it is not clear the relation between generalization and implicit regularization for DNNs optimized by SGD. Towards this end, the following question is also unsolved:
+
+![](images/9f9cbbdd08d6e1887780e2fcf92f642c78f14744f8f0f7ba7fe82cfad0b64f15.jpg)  
+Figure 1: Random walks having a step size distributed according to $\mathcal { N } ( 0 , 2 / m )$ $\mathbf { \zeta } _ { m } = 1 0 ^ { 3 }$ ) after $\alpha$ steps. The smaller the number of steps $\alpha$ is, the more straight. For visibility, we transformed the random walk sequence $\{ \mathrm { y } _ { i } \} _ { i = 0 } ^ { \alpha }$ into $\{ \bar { z _ { i } } \} _ { i = 0 } ^ { \alpha }$ defined by $\mathbf { z } _ { i } : = \bar { ( } \mathbf { y } _ { i } - \mathbf { y } _ { 0 } ) - ( i / \alpha ) ( \mathbf { y } _ { \alpha } - \mathbf { y } _ { 0 } )$ .
+
+# Question 3. Is it possible to obtain provable generalization bounds based on the implicit regularization for DNNs optimized by SGD?
+
+To answer these questions, we introduce a novel analysis for DNNs and characterize a mechanism of implicit regularization that caused by nonlinearity of ReLU activation. Our results indicate that the DNNs (trained by SGD) interpolate almost linearly between data points, which leads to the low complexity solutions in the over-parameterized regime. Accordingly, we prove that SGD with random initialization can learn a class of continuously differentiable functions with generalization error bounds of the order of $O ( n ^ { - 2 } )$ ( $\scriptstyle { n }$ : the number of samples), which is independent of the NTK analysis.
+
+In order to introduce our analysis for implicit regularization, let us focus on the DNN output on a one-dimensional linear path between training points. We define it as $\pmb { x } ( s ) : = ( 1 - s ) \pmb { x } ^ { ( p ) } + s \pmb { x } ^ { ( q ) }$ $( s \in [ 0 , 1 ] )$ , where $\pmb { x } ^ { ( p ) }$ and $\pmb { x } ^ { ( q ) }$ are training points. The corresponding function of the DNN with ReLU (ReLU DNN) is continuous piecewise linear on the linear path $\pmb { x } ( s )$ . In the hidden layer, the corresponding function $g ^ { l } ( \pmb { x } )$ of each unit in the $l$ -th layer is also continuous piecewise linear on the linear path $\pmb { x } ( s )$ , that is, a composite function $( g ^ { l } \circ \pmb { x } ) ( \bar { s } )$ is continuous piecewise linear. Since DNN nonlinearity is linked to breakpoints, which is caused by ReLU activation, the set of the breakpoints plays a key role in the network behavior. These kind of breakpoints are also known as kinks or knots (Steinwart, 2019). Since $( g ^ { l } \circ \pmb { x } ) ( s )$ is continuous piecewise linear, the gradient of $( g ^ { l } \circ \pmb { x } ) ( s )$ has gaps at the breakpoints, which we call gradient gaps. We focus on gradient gaps with respect to the parameter $s$ .1 Our key finding is that the gradient gaps are a constant multiple of independent Gaussian random variables according to $\mathcal { N } ( 0 , \bar { 2 } / m )$ , where $m$ is the number of units in the hidden layer. Since the gradient of the unit $( g ^ { l } \circ \pmb { x } ) ( s )$ is the sum of the gradient gaps, it is a “Gaussian random walk”2 (see Step 2 in the proof of Lemma 3). The relation between the step size variance and the number of steps determines the Gaussian random walk behavior. In the corresponding function of the DNN, the step size variance is the variance of weights, and the number of steps is the number of breakpoints. In a regime where ( the number of steps $\times$ the step size variance) $\le \ { \cal O } ( 1 )$ , the Gaussian random walk strolls little from the origin with high probability (see Figure 1). Our results show that the ReLU DNN is in the same regime and “simplicity” of the gradient depends crucially on the number of breakpoints. Hanin & Rolnick (2019) proves that the average number of breakpoints is linear in the number of hidden units at initialization, and we also give a proof of an upper bound on the number of breakpoints even after training (see $\ S 3 . 1 \rrangle$ .
+
+In this work, we prove a priori generalization estimates, which is independent of the posterior data distribution, by analyzing the behavior of DNNs on the linear path $\pmb { x } ( s )$ . In other words, we show that in Theorem 2, the difference between the network output function and its linear interpolation is evaluated only from the amount of the weight change. We note that in Theorem 2, we dose not use the properties of the trained neural networks. Our key analysis is that the gradient of unit $( g ^ { l } \circ \pmb { x } ) ( s )$ is a “Gaussian random walk” and approximately equal to a straight line on the linear path $\pmb { x } ( s )$ , which we call random walk analysis. Our technique is based on a priori estimates that variation of the gradients between data points is extremely small and depends essentially on the value of the number of breakpoints times the variance of weights.
+
+Difference from NTK. Interestingly, our analysis idea is different from other previous work based on NTK. Our findings are some novel aspects as follows:
+
+• The NTK is defined using the gradient of the DNN output with respect to weight parameter space. In contrast, the linear approximation (Lemma 3 in this paper) is defined using the gradient of the DNN output with respect to input parameter space. In other words, the variables to be differentiated are different. The random walk analysis indicates that over-parameterized ReLU DNNs interpolate almost linearly between the data points. For ReLU activation, since the NTK kernel mapping is not Lipschitz but $1 / 2$ -Holder, it is difficult to obtain such a result in the NTK analysis ¨ without a tradeoff between smoothness and approximation (Bietti & Mairal, 2019).
+
+Our Contributions. In this work, we consider an $L$ -layer over-parameterized ReLU neural network with $l ^ { 2 }$ regression task using SGD from random initialization. We show the constitutive relation between implicit regularization and generalization, which enables us to provide new insights on the role of implicit regularization in deep learning. Our main contributions are as follows:
+
+Our random walk analysis provides a priori estimates of low complexity in overparameterized deep neural networks, and directly indicates that the unit output between data points is properly controlled by weight initialization and SGD to keep connecting the data points almost straight, which is one of the underlying mechanisms of implicit regularization in the over-parameterized regime.   
+• Our result suggests that implicit regularization is attributed to the nonlinearity of ReLU DNN, which is indicated by the fact that variance of weight and the number of breakpoints determine the Gaussian random walk behavior. We also prove that in one-dimensional input case, SGD with random initialization can learn $C ^ { 1 }$ -class functions. Our generalization estimates are based on the implicit regularization.
+
+# 2 PRELIMINARIES AND NOTATION
+
+Notation. For $n \in \mathbb N$ , we let $[ n ] = \{ 1 , 2 , \dots , n \}$ . We use ${ \mathcal { N } } ( \mu , \sigma ^ { 2 } )$ to denote the Gaussian distribution of mean $\mu$ and variance $\sigma ^ { 2 }$ . We use $\lVert \pmb { v } \rVert _ { 2 }$ to denote the Euclidean norm of a vector $\pmb { v }$ , use $\| \pmb { v } \| _ { F }$ to denote the Frobenius norm of a vector $\textbf {  { v } }$ . When $\mathbf { Z }$ is a sub-Gaussian random variable, we let $\| \mathbf { z } \| _ { \psi _ { 2 } } = \operatorname* { i n f } \{ t > 0 \mid \mathbb { E } [ \exp ( \mathbf { z } ^ { 2 } / t ^ { 2 } ) ] \leq 2 \}$ to denote sub-Gaussian norm (Vershynin, 2018). 2For a vector $\textbf {  { v } }$ , we denote by $[ { \pmb v } ] _ { i }$ or $v _ { i }$ the $i$ -th element of $\textbf {  { v } }$ . For a matrix $M$ , we denote by $[ M ] _ { i , j }$ or $M _ { i , j }$ the entry in the $i$ -th row and $j$ -th column of $M$ , and we denote by $[ M ] _ { i }$ the $i$ -th row vector of $M$ . We denote by $\mathbf { 1 } _ { \{ E \} }$ the indicator function for the event $E$ . ReLU activation is given by $\phi ( x ) = \operatorname* { m a x } \{ 0 , x \}$ , and for a vector $\mathbf { \pmb { a } } \in \mathbb { R } ^ { n }$ , we define $\boldsymbol { \phi } ( \mathbf { a } ) = \bigl ( \phi ( a _ { 1 } ) , \phi ( a _ { 2 } ) , \dots , \phi ( a _ { n } ) \bigr )$ .
+
+Network structure. In this work, an $L$ -layer fully-connected feed-forward ReLU neural network with $m$ units in each hidden layer is given by $f : \mathbb { R } ^ { \dot { d } }  \mathbb { R } ^ { c }$ . For input $\pmb { x } \in \mathbb { R } ^ { d }$ , unit output $g ^ { l } ( \pmb { x } )$ of each layer $l \in [ L ]$ and network output $f ( { \pmb x } )$ are given by the following functions:
+
+$$
+\begin{array} { r } { g ^ { 0 } ( \pmb { x } ) : = \mathbf { A } \pmb { x } , \qquad g ^ { l } ( \pmb { x } ) : = \mathbf { W } ^ { l } \phi ( g ^ { l - 1 } ( \pmb { x } ) ) \quad ( l \in [ L ] ) , \qquad f ( \pmb { x } ) : = \mathbf { B } \phi ( g ^ { L } ( \pmb { x } ) ) , } \end{array}
+$$
+
+where $\mathbf { A } \in \mathbb { R } ^ { m \times d }$ , $\mathbf { W } ^ { l } \in \mathbb { R } ^ { m \times m }$ $( l \in L )$ , and $\mathbf { B } \in \mathbb { R } ^ { c \times m }$ are weight matrices. We assume the following Gaussian initialization: $\mathbf { A } _ { i , j } \sim { \mathcal { N } } ( 0 , 2 / m )$ , $\mathbf { W } _ { i , j } ^ { l } \sim { \mathcal { N } } ( 0 , 2 / m )$ , and $\mathbf { B } _ { i , j } \sim \mathcal { N } ( 0 , 2 / c )$ . For input $_ { \textbf { \em x } }$ and weight matrices $\overrightarrow { \mathbf { W } } : = ( \mathbf { W } ^ { 1 } , \mathbf { W } ^ { 2 } , \ldots , \mathbf { W } ^ { L } )$ , the network output $f ( { \pmb x } )$ is also denoted by $f ( \overrightarrow { \mathbf { W } } , \boldsymbol { x } )$ . In this work, we only update weights in $\overrightarrow { \bf W }$ and leave $\mathbf { A }$ and $\mathbf { B }$ at the random initialization. For input $\textbf { \em x } \in \mathbb { R } ^ { d }$ and $l \in \ [ L ]$ , we denote by $\mathbf { G } ^ { l } ( \pmb { x } )$ a diagonal matrix, which represents the activation pattern of the $l .$ -th layer, which we call an indicator matrix. More precisely, we define the $i$ -th diagonal element as $[ \mathbf { G } ^ { l } ( \pmb { x } ) ] _ { i , i } : = \mathbf { 1 } _ { \{ g _ { i } ^ { l } ( \pmb { x } ) \geq 0 \} }$ . Since the ReLU activation is positive homogeneous, we obtain the following equality: $\partial ^ { \aa } ( { \pmb x } ) = { \bf \tilde { W } } ^ { l } { \bf G } ^ { l - 1 } ( { \pmb x } ) g ^ { l - 1 } ( { \pmb x } )$ .
+
+Dataset and loss function. The data are generated from an unknown distribution $\mathcal { D }$ over $( { \pmb x } , { \pmb y } ) \in$ $\mathbb { R } ^ { d } \times \mathbb { R } ^ { c }$ , where $_ { \textbf { \em x } }$ is the input data point and $\textbf {  { y } }$ is the label associated with this data point. We assume without loss of generality that for each input $\pmb { x } = ( x _ { 1 } , \dots , x _ { d } )$ , using additional coordinates $( x _ { d + 1 } , x _ { d + 2 } )$ , the replacement input √ $\pmb { x } ^ { \prime \prime } : = ( x _ { 1 } , \ldots , x _ { d } , x _ { d + 1 } , x _ { d + 2 } )$ is normalized so that $\| x ^ { \prime \prime } \| _ { 2 } =$ 1 and its last coordinate $x _ { d + 2 } = 1 / \sqrt { 2 } .$ . 3 we also use $_ { \textbf { \em x } }$ to denote the replacement input $\pmb { x } ^ { \prime \prime } \in \mathbb { R } ^ { d + 2 }$ . The training data $\mathbb { Z } : = \{ ( \pmb { x } ^ { ( 1 ) } , \pmb { y } ^ { ( 1 ) } ) , \ldots , ( \pmb { x } ^ { ( n ) } , \pmb { y } ^ { ( n ) } ) \}$ is given as $n$ i.i.d. samples from $\mathcal { D }$ . We define the minimum distance of the training data: $\delta : = \operatorname* { m i n } \{ \| { \pmb x } ^ { ( i ) } - { \pmb x } ^ { ( j ) } \| _ { 2 } : \forall i , j \in [ n ] , i \neq j \} > 0$ . For the $l ^ { 2 }$ regression loss $\begin{array} { r } { \ell ( \hat { \pmb y } , \pmb y ) : = \frac { 1 } { 2 } \| \hat { \pmb y } - \pmb y \| _ { 2 } ^ { 2 } } \end{array}$ and a subset of training data $\mathbb { Z } ^ { ( \tau ) } \subset \mathbb { Z }$ , we define our regression objective as follows: $\bar { \mathcal { L } } _ { \mathbb { Z } ^ { ( \tau ) } } ( \overrightarrow { \mathbf { W } } ) : = \mathbb { E } _ { ( \mathbf { x } , \mathbf { y } ) \sim \mathbb { Z } ^ { ( \tau ) } } [ \ell ( f ( \overrightarrow { \mathbf { W } } , \mathbf { x } ) , \mathbf { y } ) ]$ .
+
+Stochastic gradient descent with Gaussian initialization. We use mini-batch SGD to train the network with a constant learning rate $\eta ~ > ~ 0$ , a batch size $b$ and iteration number $T$ . Let $\begin{array} { r } { \vec { \mathbf { W } } ^ { ( 0 ) } : = ( \mathbf { W } ^ { 1 , ( 0 ) } , \mathbf { W } ^ { 2 , ( 0 ) } , \dots , \mathbf { W } ^ { L , ( 0 ) } ) , } \end{array}$ , A, $\mathbf { B }$ be weight matrices generated from the above Gaussian initialization. Suppose we start at $\overrightarrow { \mathbf { W } ^ { ( 0 ) } }$ and for each $l \in [ L ]$ and $t = 0 , 1 , \dots , T - 1$ ,
+
+$$
+\mathbf { W } ^ { l , ( t + 1 ) } = \mathbf { W } ^ { l , ( t ) } - \eta \cdot \nabla _ { \mathbf { W } ^ { l } } \mathcal { L } _ { \mathbb { Z } ^ { ( t ) } } ( \overrightarrow { \mathbf { W } } ^ { ( t ) } ) ,
+$$
+
+where $\mathbb { Z } ^ { ( t ) } \subset \mathbb { Z }$ is a mini-batch of size $b$ . For input $_ { \textbf { \em x } }$ , each layer $l \in [ L ]$ and each step $t \in [ T ]$ , we denote by $g ^ { l , ( t ) } ( \pmb { x } )$ the unit output, $f ^ { ( t ) } ( { \pmb x } )$ the network output, $\mathbf { W } ^ { l , ( t ) }$ the weight matrix, and $G ^ { l , ( t ) } ( { \pmb x } )$ the indicator matrix.
+
+In the above setting, recent paper (Allen-Zhu et al., 2018b) shows that SGD can allow an overparameterized multi-layer network to attain arbitrarily low training error as follows:
+
+(Co and of , let 0, $ \varepsilon ~ \in ~ ( 0 , 1 ] , ~ \delta ~ \in ~$ $( 0 , { \cal O } ( 1 / L ) ]$ $b \in [ n ]$ $\begin{array} { r } { m \ \geq \ \widetilde \Omega \left( \frac { \mathsf { p o l y } ( n , L , \delta ^ { - 1 } ) d } { b } \right) } \end{array}$ $\begin{array} { r } { \eta \ : = \ \Theta \left( \frac { b \delta d } { \mathsf { p o l y } ( n , L ) m \log ^ { 2 } m } \right) ^ { \cdot } , } \end{array}$ $T =$ $\begin{array} { r } { \Theta \left( \frac { \mathsf { p o l y } ( n , L ) \log ^ { 2 } m } { b \delta ^ { 2 } } \log \left( \frac { n \log m } { \varepsilon } \right) \right) } \end{array}$ $\overrightarrow { \mathbf { W } } ^ { ( 0 ) } , \mathbf { A } , \mathbf { B }$ are at r zation. Then, it satisfies $1 - e ^ { - \Omega ( \log ^ { 2 } m ) }$ $\mathbb { Z } ^ { ( 1 ) } , \ldots \mathbb { Z } ^ { ( T ) }$ $\begin{array} { r } { ( \dot { \overrightarrow { \mathbf { W } } } ) \leq \varepsilon , \| \mathbf { W } ^ { l , ( t ) } - \mathbf { W } ^ { l , ( 0 ) } \| _ { F } \leq O \left( \lambda \frac { \log m } { \sqrt { m } } \right) , \| [ \mathbf { W } ^ { l , ( t ) } ] _ { i } - [ \mathbf { W } ^ { l , ( 0 ) } ] _ { i } \| _ { 2 } \leq O \left( \lambda \frac { \log m } { m } \right) } \end{array}$ (∀t ∈ [T ]), where λ := n3.5√cδ√b .
+
+Our results on generalization also crucially depend on this analysis.
+
+# 3 RANDOM WALK ANALYSIS
+
+In this work, we consider the behavior of the unit output in each layer on a one-dimensional linear path between two data points, which we denote by $\pmb { x } ( s )$ $( s \in [ 0 , 1 ] )$ ). In this section, for notational simplicity, we drop the superscript with respect to $( t )$ , which is the $t$ -th iteration.
+
+Definition 3.1 (One-dimensional Linear Path). For each pair of data points $\pmb { x } ^ { ( p ) }$ and $\pmb { x } ^ { ( q ) } \left( p \neq q \right)$ , We define $\pmb { x } ( s ) : = ( 1 - s ) \pmb { x } ^ { ( p ) } + s \pmb { x } ^ { ( q ) } , s \in [$ $s \in [ 0 , 1 ]$ , and denote by $\pmb { v } : = \pmb { x } ^ { ( q ) } - \pmb { x } ^ { ( p ) }$ the direction vector.
+
+Note that the unit output $g _ { i } ^ { l } ( { \pmb x } ( s ) )$ and the network output $f _ { i } ( { \pmb x } ( s ) )$ are continuous piecewise linear functions on $s \in [ 0 , 1 ]$ , and the network output can be expressed as follows:
+
+$$
+f ( \pmb { x } ( s ) ) = \mathbf { B } \mathbf { G } ^ { L } ( \pmb { x } ( s ) ) \mathbf { W } ^ { L } \mathbf { G } ^ { L - 1 } ( \pmb { x } ( s ) ) \mathbf { W } ^ { L - 1 } \cdot \cdot \mathbf { G } ^ { 1 } ( \pmb { x } ( s ) ) \mathbf { W } ^ { 1 } \mathbf { G } ^ { 0 } ( \pmb { x } ( s ) ) \mathbf { A } \pmb { x } ( s ) .
+$$
+
+Note that $g _ { i } ^ { l } ( { \pmb x } ( s ) )$ and $f _ { i } ( { \pmb x } ( s ) )$ have linear approximations at the point $s = 0$ as follows4:
+
+$$
+\tilde { g } _ { i } ^ { l } ( s ) : = g _ { i } ^ { l } ( { \pmb x } ( 0 ) ) + s \cdot \left[ \frac { d } { d s } g _ { i } ^ { l } ( { \pmb x } ( s ) ) \right] _ { s = 0 } , \tilde { f } _ { i } ( s ) : = f _ { i } ( { \pmb x } ( 0 ) ) + s \cdot \left[ \frac { d } { d s } f _ { i } ( { \pmb x } ( s ) ) \right] _ { s = 0 } .
+$$
+
+We prove that these piecewise linear functions are almost straight, which contributes to the generalization of NN. In other words, the unit output $g _ { i } ^ { l } ( { \pmb x } ( s ) )$ can be approximated by a linear function with high accuracy. Moreover, the small difference between $g _ { i } ^ { l } ( { \pmb x } ( s ) )$ and $\widetilde { g } _ { i } ^ { l } ( s )$ indicates the low complexity of the unit output.
+
+Theorem 2 (A Priori Estimates for Implicit Regularization). Under the same setting as Theorem $^ { l }$ , with probability at least $1 - e ^ { - \Omega ( \log ^ { 2 } { \bar { m } } ) }$ , for every $\pmb { x } ^ { ( p ) }$ , $\pmb { x } ^ { ( q ) }$ $p , q \in [ n ]$ , $p \neq q$ ), $t \in [ T ] , l \in [ L ]$ $i \in [ m ]$ and $j \in [ c ]$ , we have
+
+$$
+\begin{array} { r l r } & { } & { \underset { 0 \leq s \leq 1 } { \operatorname* { s u p } } \Big \vert g _ { i } ^ { l , ( t ) } ( { \pmb x } ( s ) ) - \tilde { g } _ { i } ^ { l , ( t ) } ( s ) \Big \vert \leq { \cal O } \left( \frac { \log ^ { l } m } { \sqrt { m } } \right) \| { \pmb v } \| _ { 2 } , } \\ & { } & { \underset { 0 \leq s \leq 1 } { \operatorname* { s u p } } \Big \vert f _ { j } ^ { ( t ) } ( { \pmb x } ( s ) ) - \tilde { f } _ { j } ^ { ( t ) } ( s ) \Big \vert \leq { \cal O } \left( \frac { \log ^ { L + 1 } m } { \sqrt { c } } \right) \| { \pmb v } \| _ { 2 } . } \end{array}
+$$
+
+# 3.1 INTUITION BEHIND IMPLICIT REGULARIZATION
+
+To prove Theorem 2, we introduce our key analysis that the gradient of $g ^ { l , ( t ) } ( \pmb { x } ( s ) )$ is “Gaussian random walk” and nearly equal to a straight line on the linear path $\pmb { x } ( s )$ , which we call random walk analysis. For simplicity, we explain the outline of the proof for the initial state of the network (i.e. $t = 0$ ). All proofs are given in the Supplementary Material (including $0 \leq t \leq T$ ).
+
+To state our key Lemma, we define the following function: $\breve { g } ^ { l } ( s ) : = \mathbf { W } ^ { l } \phi ( \tilde { g } ^ { l - 1 } ( s ) )$ , which is an analogue of the unit output $g ^ { l } ( \pmb { x } ( s ) ) = \mathbf { W } ^ { l } \phi ( g ^ { l - \tilde { 1 } } ( \pmb { x } ( s ) ) )$ . In other words, $\breve { g } ^ { l } ( s )$ is the one in which $g ^ { l - 1 } ( \mathbf { \bar { x } } ( s ) )$ is replaced by $\tilde { g } ^ { l - 1 } ( \boldsymbol { s } )$ . Next, Lemma 3 shows that each $\breve { g } _ { i } ^ { l }$ can be well approximated by a linear function $\tilde { g } _ { i } ^ { l } ( s )$ with high probability.
+
+Lemma 3 (Linear Approximation Analogue). With probability at least $1 - e ^ { - \Omega ( \log ^ { 2 } m ) }$ , for every $\pmb { x } ^ { ( p ) } , \pmb { x } ^ { ( q ) } \ ( p , q \in [ n ] , p \neq q )$ , $t \in [ T ]$ and $l \in [ L ]$ , we have
+
+$$
+\begin{array} { r l r } { \displaystyle \operatorname* { s u p } _ { 0 \leq s \leq 1 } \left. \check { g } _ { i } ^ { l , ( t ) } ( s ) - \tilde { g } _ { i } ^ { l , ( t ) } ( s ) \right. \leq O \left( \frac { \log ^ { 2 } m } { \sqrt { m } } \right) \| { \pmb v } \| _ { 2 } , } & { ( \forall i \in [ m ] ) , } & \\ { \displaystyle \operatorname* { s u p } _ { 0 \leq s \leq 1 } \left. \check { f } _ { j } ^ { ( t ) } ( s ) - \tilde { f } _ { j } ^ { ( t ) } ( s ) \right. \leq O \left( \frac { \log ^ { 2 } m } { \sqrt { c } } \right) \| { \pmb v } \| _ { 2 } , } & { ( \forall j \in [ c ] ) . } & \end{array}
+$$
+
+The purpose of this subsection is to give an intuitive explanation of the proof of Lemma 3, which can be divided into two steps. The first step gives estimates of the number of breakpoints. We show that the number of breakpoints of the piecewise linear function $\breve { g } _ { i } ^ { l } ( s )$ is less than or equal to the number of units in the layer (i.e. $m$ ).
+
+The second step gives estimates of gradient gaps of $\breve { g } _ { i } ^ { l } ( s )$ . We show that the gradient gaps are independent Gaussian random variables and the gradient of $\breve { g } ^ { l } ( s )$ is the sum of the gradient gaps, which indicates that the gradient is a “Gaussian random walk”. Note that in this setting, the number of breakpoints is equal to the total number of steps of the random walk.
+
+Step 1. By the definition of $\breve { g } ^ { l } ( s )$ , we may write
+
+$$
+\check { g } _ { i } ^ { \ l } ( s ) = \sum _ { j = 1 } ^ { m } [ \mathbf { W } ^ { l } ] _ { i , j } \ \phi ( \tilde { g } _ { j } ^ { \ l - 1 } ( s ) ) \ .
+$$
+
+Note that the input to the above $\phi$ , that is $\tilde { g } _ { j } ^ { \ l - 1 } ( s )$ , is a linear function on $s \in [ 0 , 1 ]$ . For each $j \in [ m ]$ , the linear equation $\tilde { g } _ { j } ^ { l - 1 } ( s ) = 0$ has at most one solution $\boldsymbol { s } = \boldsymbol { s } ^ { * }$ ). If the solution $s = s ^ { * }$ satisfies $0 ~ < ~ s ^ { * } ~ < ~ 1$ , then $\breve { g } _ { i } { } ^ { l } ( s )$ has a breakpoint at $s \ = \ s ^ { * }$ . This shows that the number of breakpoints of $\breve { g } _ { i } { } ^ { l } ( s )$ is bounded by the number of the linear equations $\{ \widetilde { g } _ { j } ^ { \ l - 1 } ( s ) = 0 \} _ { j \in [ m ] }$ , which is equal to $m$ . We denote by $0 < s _ { 1 } < s _ { 2 } < \dots < s _ { \alpha } < 1$ breakpoints of $\breve { g } _ { i } { } ^ { l } ( s )$ , where $\alpha$ is the number of breakpoints. For $\beta \in [ \alpha ]$ , we set an open interval of breakpoints $\mathbb { I } _ { \beta } : = \left( s _ { \beta } , s _ { \beta + 1 } \right)$ .
+
+Step 2. To give an intuition that the gradient of $\smile \breve { g } _ { i } ^ { l } ( s ) - \tilde { g } _ { i } ^ { \ l } ( s )$ is “Gaussian random walk”, we fix some notation. For $\beta \in [ \alpha ]$ , we define the gradient $\nabla _ { \beta }$ :
+
+$$
+\nabla _ { \beta } : = \frac { d } { d s } \Big ( \check { g } _ { i } ^ { l } ( s ) - \tilde { g } _ { i } ^ { l } ( s ) \Big ) \bigg | _ { s \in \mathbb { I } _ { \beta } } .
+$$
+
+Note that $\tilde { g } _ { i } { } ^ { l } ( s )$ is a linear function, and there is no breakpoints for $\breve { g } _ { i } ^ { l } ( s )$ on $\mathbb { I } _ { \beta }$ . Note also that since $\smile \breve { g } _ { i } ^ { l } ( s ) - \tilde { g } _ { i } ^ { \ l } ( s )$ is linear on $\mathbb { I } _ { \beta }$ , the gradient is constant on $\mathbb { I } _ { \beta }$ . For $\beta \in [ \alpha ]$ , we define a gradient gap: $\mathbf { x } _ { \beta } : = \nabla _ { \beta } - \nabla _ { \beta - 1 }$ , and we have the following estimate with probability at least $1 - e ^ { - \Omega ( \log ^ { 2 } m ) }$ :
+
+$$
+\mathbf { x } _ { \beta } = S \omega _ { \beta } \mathrm { w h e r e } \omega _ { \beta } \sim \mathcal { N } ( 0 , 2 / m ) , \quad | S | \leq \frac { \log m } { \sqrt { m } } \| \pmb { v } \| _ { 2 } .
+$$
+
+Note that $\omega _ { \beta }$ is an element of the weight matrix $\mathbf { W } ^ { l }$ . Thus, $\{ \mathbf { { x } } _ { \beta } \}$ are independent Gaussian random variables, and $\nabla _ { \beta }$ is the sum of $\mathbf { X } _ { \beta }$ , that is
+
+$$
+\nabla _ { \beta } = \sum _ { \gamma = 1 } ^ { \beta } \mathbf { x } _ { \gamma } , ~ ( \beta \in [ \alpha ] ) .
+$$
+
+This shows that the gradient $\nabla _ { \beta }$ is a “Gaussian random walk” and depends essentially on the number of breakpoints $\alpha$ and the variance of weights $2 / m$ . Using the randomness of $\mathbf { W } ^ { l }$ and general Hoeffding’s inequality (Vershynin, 2018), we have
+
+$$
+\mathbb { P } \bigg [ \Big \vert \sum _ { \gamma = 1 } ^ { \beta } S \omega _ { \gamma } \Big \vert \leq \varepsilon \bigg ] \geq 1 - e ^ { - \Omega \left( \frac { m ^ { 2 } \varepsilon ^ { 2 } } { \alpha \| v \| _ { 2 } ^ { 2 } \log ^ { 2 } m } \right) } .
+$$
+
+According to Step 1, the number of breakpoints $\alpha$ is less than or equal to $m$ . Thus, $\nabla _ { \beta }$ is bounded by hav ${ \frac { \log ^ { 2 } m } { \sqrt { m } } } \| \pmb { v } \| _ { 2 }$ $1 - e ^ { - \Omega ( \log ^ { 2 } m ) }$ . Therefore, with the same probabilihus, integrating this inequality from $\begin{array} { r } { \left| \frac { d } { d s } \big ( \check { g } _ { i } ^ { l } ( s ) - \tilde { g } _ { i } ^ { l } ( s ) \big ) \right| \leq \frac { \log m } { \sqrt { m } } \| \pmb { v } \| _ { 2 } , ( \forall s \in [ 0 , 1 ] ) . } \end{array}$ $s = 0$ to $s = 1$ shows the estimates of (7) in Lemma 3. Note that the proof of (8) in Lemma 3 is idential to the above proof, except the fact that each entry of $\mathbf { B }$ follows from $\mathcal { N } ( 0 , 2 / c )$ insted of $\mathcal { N } ( 0 , 2 / m )$ .
+
+Note that as illustrated in Figure 1, for sufficiently large $m$ , the gradient $\nabla _ { \beta }$ can be made arbitrarily small, which means that the networks interpolate almost linearly between the data points.
+
+# 3.2 PROOF THEOREM 2
+
+The purpose of this subsection is to give the proof of Theorem 2. We proceed by induction on the layer $l$ . Note that since $g ^ { 0 } ( { \pmb x } ( s ) ) = { \bf A } { \pmb x } ( s )$ is linear and $\tilde { g } ^ { 0 } ( s ) = g ^ { \dot { 0 } } ( { \pmb x } ( s ) ) , \breve { g } ^ { \dot { 1 } } ( s )$ is identically equal to $g ^ { 1 } ( { \pmb x } ( s ) ) = \mathbf { W } ^ { 1 } \phi ( A { \pmb x } ( s ) )$ . Thus, the case $l = 1$ is true. Assume the theorem holds for any layer $l = k$ , and let us prove it for $l = k + 1$ . Using the triangle inequality, we have
+
+$$
+\left| g _ { i } ^ { k + 1 } ( { \pmb x } ( s ) ) - \tilde { g } _ { i } ^ { k + 1 } ( s ) \right| \leq \left| g _ { i } ^ { k + 1 } ( { \pmb x } ( s ) ) - \check { g } _ { i } ^ { k + 1 } ( s ) \right| + \left| \check { g } _ { i } ^ { k + 1 } ( s ) - \tilde { g } _ { i } ^ { k + 1 } ( s ) \right| .
+$$
+
+By Proposition 11.3 of (Allen-Zhu et al., 2018b), for any $y , \tilde { y } \in \mathbb { R } ^ { m }$ , there exists a diagonal matrix $\vec { D } \in \bar { \mathbb { R } ^ { m \times m } }$ such that $| D _ { j , j } | \leq 1$ and $\phi ( { \pmb y } ) - \phi ( \tilde { \pmb y } ) = D ( { \pmb y } - \tilde { \pmb y } )$ . Thus, for the first term on the RHS, we may write
+
+$$
+\begin{array} { r l } & { \quad \displaystyle \left| g _ { i } ^ { k + 1 } ( \pmb { x } ( s ) ) - \check { g } _ { i } ^ { k + 1 } ( s ) \right| = \left| \left[ \mathbf { W } ^ { k + 1 } \phi \big ( g ^ { k } ( \pmb { x } ( s ) ) \big ) \right] _ { i } - \left[ \mathbf { W } ^ { k + 1 } \phi \big ( \tilde { g } ^ { k } ( s ) \big ) \right] _ { i } \right| } \\ & { = \displaystyle \left| \left[ \mathbf { W } ^ { k + 1 } D \big ( g ^ { k } ( \pmb { x } ( s ) ) - \tilde { g } ^ { k } ( s ) \big ) \right] _ { i } \right| = \displaystyle \left| \sum _ { \rho = 1 } ^ { m } [ \mathbf { W } ^ { k } ] _ { i , \rho } D _ { \rho , \rho } \big ( g _ { \rho } ^ { k } ( \pmb { x } ( s ) ) - \tilde { g } _ { \rho } ^ { k } ( s ) \big ) \right| } \\ & { \leq \displaystyle \left\| g ^ { k } ( \pmb { x } ( s ) ) - \tilde { g } ^ { k } ( s ) \right\| _ { 2 } \frac { \log m } { \sqrt { m } } . } \end{array}
+$$
+
+In the last inequality, we use general Hoeffding’s inequality (Vershynin, 2018), and this inequality holds with probability at least $1 - e ^ { - \Omega ( \log ^ { 2 } m ) }$ . We can now apply our induction hypothesis to the above estimate. Thus, applying (7) in Lemma 3 to the second term on the RHS of (14) shows the estimates of (5) in Theorem 2. Next, Using the triangle inequality for $f _ { j } ^ { ( t ) }$ , we have
+
+$$
+\left| f _ { j } ^ { ( t ) } ( \pmb { x } ( s ) ) - \tilde { f } _ { j } ^ { ( t ) } ( s ) \right| \leq \left| f _ { j } ^ { ( t ) } ( \pmb { x } ( s ) ) - \check { f } _ { j } ^ { ( t ) } ( s ) \right| + \left| \check { f } _ { j } ^ { ( t ) } ( s ) - \tilde { f } _ { j } ^ { ( t ) } ( s ) \right| .
+$$
+
+In a similar way, using the randomness of $\mathbf { B }$ (recall each entry of $\mathbf { B }$ follows from $\mathcal { N } ( 0 , 2 / c ) )$ , with probability at least $1 - e ^ { - \Omega ( \log ^ { 2 } m ) }$ , we have
+
+$$
+\Big | f _ { j } ^ { ( t ) } ( x ( s ) ) - \check { f } _ { j } ^ { ( t ) } ( s ) \Big | \le \big \| g ^ { L } ( x ( s ) ) - \tilde { g } ^ { L } ( s ) \big \| _ { 2 } \sqrt { \frac { m } { c } } \log m \le O \left( \frac { \log ^ { 3 } m } { \sqrt { c } } \right) \| v \| _ { 2 } .
+$$
+
+In the last inequality, we use (7) in Lemma 3. Thus, applying (8) in Lemma 3 to the second term on the RHS of (18) shows the estimates of (6) in Theorem 2. We complete the proof.
+
+# 4 GENERALIZATION
+
+In this section, considering the mechanism of implicit regularization revealed by random walk analysis, we provide a priori estimates for the generalization performance of over-parameterized deep neural networks, in an $l ^ { 2 }$ regression task on a one-dimensional input-space. Since the network is over-parameterized, the expressive power of the network is rich enough to considerably overfit the data. Nevertheless, it is known empirically that properly initialized over-parameterized deep neural networks can achieve the good generalization performance while fitting all training data. We estimate the low complexity of over-parameterized deep networks and show that the error between a trained over-parameterized neural network and the target function can be uniformly bounded by an arbitrarily small positive number.
+
+Setting. We propose a new type of generalization bounds (Theorem 4), and prove this theorem in models with one dimensional input: $d = 1$ . In the following, we restrict ourselves to a onedimensional regression task on an interval $[ 0 , \nu ] \subset \mathbb { R }$ . The training dataset $\{ \left( \boldsymbol { x } ^ { ( i ) } , \boldsymbol { y } ^ { ( i ) } \right) \} _ { i \in [ n ] }$ is given as $n$ i.i.d. samples from some unknown distribution $\mathcal { D }$ . We assume that the corresponding target function for the regression task $f ^ { * } : [ 0 , \nu ] \ \to \ \mathbb { R } ^ { c }$ is a $C ^ { 1 }$ - class function. This implies that $\pmb { y } ^ { ( i ) } = f ^ { * } ( \pmb { x } ^ { ( i ) } )$ . Without loss of generality, we may assume that the input data $\{ x ^ { ( i ) } \} _ { i \in [ n ] }$ follow a uniform distribution on $[ 0 , \nu ]$ , and after relabeling, we may assume that the data points $\{ x ^ { ( 1 ) } , \ldots , x ^ { ( n ) } \}$ are ordered by index: $0 < x ^ { ( 1 ) } < \cdots < x ^ { ( n ) } < \nu$ . We define $( x ^ { ( 0 ) } , y ^ { ( 0 ) } ) : =$ $( 0 , f ^ { * } ( x ^ { ( 0 ) } ) )$ , $( \bar { x } ^ { ( n + 1 ) } , y ^ { ( n + 1 ) } ) : = ( \nu , f ^ { * } ( \nu ) )$ and $\begin{array} { r } { \mathcal { T } : = \operatorname* { s u p } _ { k \in [ n + 1 ] } \left( x ^ { ( k ) } - x ^ { ( k - 1 ) } \right) } \end{array}$ . We denote by ${ \hat { f } } : [ 0 , \nu ] \to \mathbb { R }$ the linear interpolation of the data points $\{ \left( x ^ { ( i ) } , f ^ { * } ( x ^ { ( i ) } ) \right) \} _ { i \in [ n ] }$ .
+
+A priori generalization bounds. Now, we introduce a novel approach for $a$ priori generalization bounds, which is based on random walk analysis in the over-parameterized regime. This can be interpreted as the significant expressive power of over-parameterized neural networks is controlled by implicit regularization.
+
+Theorem 4. Suppose $f ^ { * } ( x )$ is a $C ^ { 1 }$ - class function on $[ 0 , \nu ]$ . Under the same setting as Theorem 1, for $\delta \in ( 0 , 1 / 2 ]$ , then with probability at least $1 - ( \delta + e ^ { - \Omega ( \log ^ { 2 } m ) } )$ , we have
+
+$$
+\mathbb { E } _ { ( x , y ) \sim \mathcal { D } } \left[ \ell \left( f ^ { ( T ) } ( x ) , y \right) \right] \leq O \left( \frac { \nu ^ { 2 } c } { n ^ { 2 } \delta ^ { 2 } } \right) .
+$$
+
+Proof sketch of generalization. The purpose of this paragraph is to give an intuitive explanation of the proof of Theorem 4 for one-dimensional output $c = 1$ . All proofs are given in the Supplementary Material.
+
+To estimate $\mathbb { E } _ { ( x , y ) \sim \mathcal { D } } \left[ \ell \left( f ^ { ( T ) } ( x ) , y \right) \right]$ , we evaluate $\left\| f ^ { ( T ) } ( x ) - f ^ { * } ( x ) \right\| _ { 2 }$ , which we may write as
+
+$$
+\left\| f ^ { ( T ) } ( x ) - f ^ { * } ( x ) \right\| _ { 2 } \leq \left| f ^ { ( T ) } ( x ) - { \hat { f } } ( x ) \right| + \left| { \hat { f } } ( x ) - f ^ { * } ( x ) \right|
+$$
+
+The first term of the RHS of (21) represents an error between the trained network output $f ^ { ( T ) } ( x )$ and the piecewise linear function ${ \hat { f } } ( x )$ . We can use the results of random walk analysis to evaluate this critical term as follows. Theorem 2 provides that the linear approximation error of the network output, which we denote by $ { \varepsilon } _ { f ^ { ( T ) } }$ , is small in each interval $[ x ^ { ( k - 1 ) } , x ^ { ( k ) } ]$ , and hence in each interval $[ x ^ { ( k - 1 ) } , x ^ { ( k ) } ]$ , the difference between $f ^ { ( T ) } ( x )$ and ${ \hat { f } } ( x )$ falls within the error $\mathcal { E } _ { f ^ { ( T ) } }$ . This suggests that the network output between adjacent training points is properly controlled by weight initialization and SGD to keep connecting the points almost straight, which results in low complexity of over-parameterized neural networks. This statement can be extended to the interval $[ 0 , \nu ]$ . Fix $\varepsilon > 0$ , and suppose $m \geq \widetilde { O } \left( \mathcal { T } ^ { 6 } / \varepsilon ^ { 3 } \right)$ . Then with high probability at least $1 - e ^ { - \Omega ( \log ^ { 2 } m ) }$ , we have
+
+$$
+\operatorname* { s u p } _ { x \in [ 0 , \nu ] } \left| f ^ { ( T ) } ( x ) - { \hat { f } } ( x ) \right| \leq { \sqrt { \frac { \varepsilon } { 2 } } }
+$$
+
+The second term of RHS of (21) is the error of the piecewise linear approximation of $f ^ { * } ( x )$ by ${ \hat { f } } ( x )$ on the interval $[ 0 , \nu ]$ . Note that the error can be reduced by increasing $n$ . Hence, for any fixed $\varepsilon > 0$ , there exists $\delta > 0$ such that if $\begin{array} { r } { n \geq O \left( \frac { \nu } { \sqrt { \varepsilon } \delta } \right) } \end{array}$ , with probability at least $1 - \delta$ , we have $\begin{array} { r } { \operatorname* { s u p } _ { x \in [ 0 , \nu ] } | \hat { f } ( x ) - f ^ { * } ( x ) | \leq \sqrt { \frac { \varepsilon } { 2 } } , } \end{array}$ . Thus, from (21), for $m$ and $n$ sufficiently large, the error between the trained network output and the target function is uniformly bounded by $\varepsilon$ on the interval $[ 0 , \nu ]$ with probability at least $1 - \left( e ^ { - \Omega ( \log ^ { 2 } m ) } + \delta \right)$ . This gives a priori estimates for the generalization performance of over-parameterized neural networks.
+
+# 5 RELATED WORK
+
+Implicit regularization in neural networks has recently become an active area of research in machine learning. A number of works have focused on the behavior of gradient descent on overparameterized neural networks (Neyshabur et al., 2014; Lin et al., 2016; Zhang et al., 2016; Soudry et al., 2018; Rahaman et al., 2018). In order to get a handle on implicit regularization in deep neural networks, the majority of theoretical attention has been devoted to linear neural networks (Ji & Telgarsky, 2018; Gidel et al., 2019; Arora et al., 2019a).
+
+Many works try to explain generalization of over-parameterized neural networks. Recent works have shown that on sufficiently over-parameterized neural networks, the learning dynamics of gradient descent are governed by the NTK (Daniely et al., 2016; Jacot et al., 2018; Du et al., 2018; AllenZhu et al., 2018b; Lee et al., 2019; Arora et al., 2019b). In these settings, the implicit regularization and the generalization error of the resulting network can be analyzed via NTK and the reproducing kernel Hilbert space (RKHS) (Bietti & Mairal, 2019; Nakkiran et al., 2019). To extend it to SGD, Hayou et al. (2019) introduce a stochastic differential equation dependent on the NTK. Closely related work Cao & Gu (2019) showed that the expected 0-1 loss of a wide enough ReLU network trained with SGD and random initialization can be bounded by the training loss of a random feature model induced by the network gradient at initialization.
+
+This contrasts with other recent results that show a provable separation between the generalization error obtained by neural networks and kernel methods (Wei et al., 2018; Allen-Zhu et al., 2018a;
+
+Allen-Zhu & Li, 2019). Several papers suggest that training deep models with gradient descent can behave differently from kernel methods, and have much richer implicit regularization (Chizat et al., 2019; Woodworth et al., 2019; Yehudai & Shamir, 2019; Geiger et al., 2019).
+
+# 6 CONCLUSION
+
+In this work, probability estimates for the network output behavior (i.e. random walk analysis) provide a priori generalization estimates for $l ^ { 2 }$ regression problems. We prove that even after training, network gradients between the data points are approximately Gaussian random walks, and the variation of the gradients between the data points is extremely small and depends essentially on the number of breakpoints and the variance of weights. To the best of our knowledge, this paper is the first to show a mechanism of implicit regularization and to prove the generalization bounds by using the implicit regularization for deep (three or more layer) neural networks with ReLU activation. As a result, we also show that over-parameterized deep neural networks can learn $C ^ { 1 }$ - class functions. Importantly, our analysis is independent of the kernel generalization analysis, and the generalization bounds are different from the NTK inductive bias of the RKHS norm.
+
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+
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+
+# A PROOFS
+
+A.1 PROOF OF LEMMA 3
+
+Lemma 3 (reshown, see $\ S 3 . 1 \ r .$ ). With probability at least $1 - e ^ { - \Omega ( \log ^ { 2 } m ) }$ , for every $\pmb { x } ^ { ( p ) }$ , $\pmb { x } ^ { ( q ) } \left( p , q \in \right.$ $[ n ] , p \neq q ) , t \in [ T ]$ and $l \in [ L ]$ , we have
+
+$$
+\begin{array} { r l r } { \displaystyle \operatorname* { s u p } _ { 0 \leq s \leq 1 } \left. \check { g } _ { i } ^ { l , ( t ) } ( s ) - \tilde { g } _ { i } ^ { l , ( t ) } ( s ) \right. \leq O \left( \frac { \log ^ { 2 } m } { \sqrt { m } } \right) \| { \pmb v } \| _ { 2 } , } & { ( \forall i \in [ m ] ) , } & \\ { \displaystyle \operatorname* { s u p } _ { 0 \leq s \leq 1 } \left. \check { f } _ { j } ^ { ( t ) } ( s ) - \tilde { f } _ { j } ^ { ( t ) } ( s ) \right. \leq O \left( \frac { \log ^ { 2 } m } { \sqrt { c } } \right) \| { \pmb v } \| _ { 2 } , } & { ( \forall j \in [ c ] ) . } & \end{array}
+$$
+
+Proof. We prove Lemma 3 for a fixed $l \in [ L ]$ and $t \in [ T ]$ every pair of data points ${ \pmb x } ^ { ( p ) } , { \pmb x } ^ { ( q ) }$ $( p , q \in [ n ] , \mathbf { \bar { \psi } } p \neq q )$ , because we can apply union bound at the end.
+
+Recall the definition $g ^ { l , ( t ) } ( x )$ (the unit output), $\mathbf { G } ^ { l , ( t ) } ( x )$ (the indicator matrix function) and $\pmb { x } ( s ) =$ $( 1 - s ) \pmb { x } ^ { ( p ) } + s \pmb { x } ^ { ( q ) }$ , $s \in [ 0 , 1 ]$ (a linearly interpolating path between data points $\pmb { x } ^ { ( p ) } , \pmb { x } ^ { ( q ) } )$ , we have
+
+$$
+g ^ { l , ( t ) } ( \pmb { x } ( s ) ) = \mathbf { W } ^ { l , ( t ) } \mathbf { G } ^ { l - 1 , ( t ) } ( \pmb { x } ( s ) ) \mathbf { W } ^ { l - 1 , ( t ) } \cdot \cdot \mathbf { \nabla } \mathbf { W } ^ { 1 , ( t ) } \mathbf { G } ^ { 0 , ( t ) } ( \pmb { x } ( s ) ) \mathbf { A } \pmb { x } ( s ) .
+$$
+
+Using the definition of $\widetilde { g } ^ { l , ( t ) } ( s )$ and $\breve { g } ^ { \ l , ( t ) } ( s )$ , we also have
+
+$$
+\begin{array} { r } { \widetilde { g } ^ { l , ( t ) } ( s ) = \mathbf { W } ^ { l , ( t ) } \mathbf { G } ^ { l - 1 , ( t ) } ( \boldsymbol { x } ( 0 ) ) \mathbf { W } ^ { l - 1 , ( t ) } \cdot \cdot \cdot \mathbf { W } ^ { 1 , ( t ) } \mathbf { G } ^ { 0 , ( t ) } ( \boldsymbol { x } ( 0 ) ) \mathbf { A } \boldsymbol { x } ( s ) , } \end{array}
+$$
+
+$$
+\begin{array} { r l r } & { \check { g } ^ { \mathit { l } , ( t ) } ( s ) = \mathbf { W } ^ { l , ( t ) } \phi ( \widetilde { g } ^ { \mathit { l } - 1 , ( t ) } ( s ) ) } & { ( 2 ) } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad ( 2 ) } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad = \mathbf { W } ^ { l , ( t ) } \widetilde { \mathbf { G } } ^ { \mathit { l } - 1 , ( t ) } ( s ) \mathbf { W } ^ { l - 1 , ( t ) } \mathbf { G } ^ { l - 2 , ( t ) } ( { \pmb x } ( 0 ) ) \mathbf { W } ^ { l - 2 , ( t ) } \cdots \mathbf { W } ^ { 1 , ( t ) } \mathbf { G } ^ { 0 , ( t ) } ( { \pmb x } ( 0 ) ) \mathbf { A } { \pmb x } ( s ) , } & \end{array}
+$$
+
+where $\widetilde { \mathbf { G } } ^ { l - 1 , ( t ) } ( s )$ is an indicator diagonal matrix for $s \in [ 0 , 1 ]$ as follows:
+
+$$
+[ \widetilde { \mathbf { G } } ^ { l - 1 , ( t ) } ( s ) ] _ { i , i } : = \mathbf { 1 } _ { \left\{ \widetilde { g } _ { i } ^ { \iota - 1 , ( t ) } ( s ) \geq 0 \right\} } , \quad [ \widetilde { \mathbf { G } } ^ { l - 1 , ( t ) } ( s ) ] _ { i , j } : = 0 \ ( i \neq j ) .
+$$
+
+We begin by proving that the number of breakpoints of the function $\breve { g } _ { i } ^ { ~ l , ( t ) } ( s )$ in the interval [0.1] is bounded by $m$ . For fixed $i \in [ m ]$ , we have
+
+$$
+\begin{array} { l } { \check { g } _ { i } ^ { \ l , ( t ) } ( s ) = [ \mathbf { W } ^ { l , ( t ) } \phi ( \widetilde { g } ^ { \ l - 1 , ( t ) } ( s ) ) ] _ { i } } \\ { = \displaystyle \sum _ { j = 1 } ^ { m } [ \mathbf { W } ^ { l , ( t ) } ] _ { i , j } \ \phi ( \widetilde { g } _ { j } ^ { l - 1 , ( t ) } ( s ) ) } \\ { = \displaystyle \sum _ { j = 1 } ^ { m } [ \mathbf { W } ^ { l , ( t ) } ] _ { i , j } \ \widetilde { g } _ { j } ^ { \ l - 1 , ( t ) } ( s ) \mathbf { 1 } _ { \left\{ \widetilde { g } _ { j } ^ { \ l - 1 , ( t ) } ( s ) \geq 0 \right\} } } \end{array}
+$$
+
+Note that the input to the above $\phi$ , that is g  l−1,(t)j (s), is a linear function on s ∈ [0, 1]. For each $j \in [ m ]$ , the linear equation $\widetilde { g } _ { j } ^ { l - 1 , ( t ) } ( s ) = 0$ has at most one solution $s = s ^ { * }$ .5 If the solution $s = s ^ { * }$ satisfies inequality $0 < s ^ { * } < 1$ , then $\breve { g } _ { i } ^ { ~ l , ( t ) } ( s )$ has a breakpoint at $s = s ^ { * }$ . This shows that the number of breakpoints of $\breve { g } _ { i } ^ { ~ l , ( t ) } ( s )$ is bounded by the number of the linear equations
+
+$$
+\begin{array} { r } { \widetilde { g } _ { j } ^ { l - 1 , ( t ) } ( s ) = 0 ( j \in [ m ] ) , } \end{array}
+$$
+
+which is clearly equal to $m$ .
+
+We denote by 0 < s1 < s2 < · · · < sα < 1 all breakpoints of g˘ l,(t)i (s), where α is the number of breakpoints. For notational simplicity, we set $s _ { 0 } = 0$ and $s _ { \alpha + 1 } = 1$ . Note that, with probability 1, breakpoints are all distinct from each other. Each breakpoint $s _ { \beta }$ $\beta \in [ \alpha ] )$ corresponds to a linear equation g  l−1,(t)j (s) = 0 for some j ∈ [m]. In other words, for each breakpoint sβ (β ∈ [α]), there exists a unique element $j \in [ m ]$ such that g  l−1,(t)j (s) = 0. Therefore, for s = sβ , we set j = jβ (β ∈ [α]) then we have g l−jβ 1,(t)(sβ ) = 0.
+
+It is easy to verify $\begin{array} { r } { \frac { d } { d s } \pmb { x } ( s ) = \pmb { x } ^ { ( q ) } - \pmb { x } ^ { ( p ) } = : \pmb { v } } \end{array}$ . Therefore,
+
+$$
+\begin{array} { r l } & { \cfrac { d } { d s } \widetilde { g } ^ { l , ( t ) } ( s ) = \mathbf { W } ^ { l , ( t ) } \mathbf { G } ^ { l - 1 , ( t ) } ( { \pmb x } ( 0 ) ) \mathbf { W } ^ { l - 1 , ( t ) } \mathbf { G } ^ { l - 2 , ( t ) } ( { \pmb x } ( 0 ) ) \mathbf { W } ^ { l - 2 , ( t ) } \cdot \cdot \cdot \mathbf { W } ^ { 1 , ( t ) } \mathbf { G } ^ { 0 , ( t ) } ( { \pmb x } ( 0 ) ) \mathbf { A } { \pmb v } } \\ & { \qquad ( 3 5 } \\ & { \cfrac { d } { d s } \widetilde { g } ^ { l , ( t ) } ( s ) = \mathbf { W } ^ { l , ( t ) } \widetilde { \mathbf { G } } ^ { l - 1 , ( t ) } ( s ) \mathbf { W } ^ { l - 1 , ( t ) } \mathbf { G } ^ { l - 2 , ( t ) } ( { \pmb x } ( 0 ) ) \mathbf { W } ^ { l - 2 , ( t ) } \cdot \cdot \cdot \mathbf { W } ^ { 1 , ( t ) } \mathbf { G } ^ { 0 , ( t ) } ( { \pmb x } ( 0 ) ) \mathbf { A } { \pmb v } . } \end{array}
+$$
+
+This implies,
+
+$$
+\begin{array} { r l r } {  { \frac { d } { d s } \check { g } ^ { l , ( t ) } ( s ) - \frac { d } { d s } \widetilde { g } ^ { l , ( t ) } ( s ) } } \\ & { } & { = \mathbf { W } ^ { l , ( t ) } ( \widetilde { \mathbf { G } } ^ { l - 1 , ( t ) } ( s ) - \mathbf { G } ^ { l - 1 , ( t ) } ( { \boldsymbol x } ( 0 ) ) ) \mathbf { W } ^ { l - 1 , ( t ) } \mathbf { G } ^ { l - 2 , ( t ) } ( { \boldsymbol x } ( 0 ) ) \cdot \cdot \cdot \mathbf { W } ^ { 1 , ( t ) } \mathbf { G } ^ { 0 , ( t ) } ( { \boldsymbol x } ( 0 ) ) \mathbf { A } \boldsymbol { v } . } \end{array}
+$$
+
+Note that from the definition of the indicator matrix, [Gl−1,(t)(x(0))]i,i = 1{[gl−1,(t)(x(0))]i≥0}, and from the definition of the linear approximation, $\widetilde { g } ^ { l - 1 , ( t ) } ( 0 ) = g ^ { l - 1 , ( t ) } ( \pmb { x } ( 0 ) )$ , which says that $\widetilde { \mathbf { G } } ^ { l - 1 , ( t ) } ( 0 )$ equals $\mathbf { G } ^ { l - 1 , ( t ) } ( \pmb { x } ( 0 ) )$ .
+
+For each $i \in [ m ]$ , we may write
+
+$$
+\frac { d } { d s } \check { g } _ { i } ^ { \ l , ( t ) } ( s ) - \frac { d } { d s } \widetilde { g } _ { i } ^ { \ l , ( t ) } ( s ) = \sum _ { j = 1 } ^ { m } [ \mathbf { W } ^ { l , ( t ) } ] _ { i , j } [ \mathbf { M } ^ { ( t ) } \pmb { v } ] _ { j } \widetilde { \lambda } _ { j } ,
+$$
+
+where $\mathbf { M } ^ { ( t ) }$ is a matrix:
+
+$$
+\mathbf { M } ^ { ( t ) } : = \mathbf { W } ^ { l - 1 , ( t ) } \mathbf { G } ^ { l - 2 , ( t ) } ( \pmb { x } ( 0 ) ) \mathbf { W } ^ { l - 2 , ( t ) } \cdot \cdot \mathbf { \nabla } \mathbf { \cdot } \mathbf { W } ^ { 1 , ( t ) } \mathbf { G } ^ { 0 , ( t ) } ( \pmb { x } ( 0 ) ) \mathbf { A }
+$$
+
+and $\widetilde { \lambda } _ { j }$ is the difference of indicator functions:
+
+$$
+\widetilde { \lambda } _ { j } : = \mathbf { 1 } _ { \left\{ \widetilde { g } _ { j } ^ { \ l - 1 , ( t ) } ( s ) \geq 0 \right\} } - \mathbf { 1 } _ { \left\{ \widetilde { g } _ { j } ^ { \ l - 1 , ( t ) } ( 0 ) \geq 0 \right\} } .
+$$
+
+For $\beta \in [ \alpha ]$ , we set an open interval of breakpoints $\mathbb { I } _ { \beta } : = ( s _ { \beta } , s _ { \beta + 1 } )$ , and recall the notation of the gradient $\nabla _ { \beta }$
+
+$$
+\nabla _ { \beta } : = \left. \frac { d } { d s } \left( \check { g } _ { i } ^ { l , ( t ) } ( s ) - \tilde { g } _ { i } ^ { l , ( t ) } ( s ) \right) \right. _ { s \in \mathbb { I } _ { \beta } } ,
+$$
+
+and the gradient gap $\mathbf { x } _ { \beta } : = \nabla _ { \beta } - \nabla _ { \beta - 1 }$ . Note that since there is not a breakpoint in $\mathbb { I } _ { \beta }$ , the gradient $\nabla _ { \beta }$ is constant in $\mathbb { I } _ { \beta }$ .
+
+Hence, for $\beta \in [ \alpha ]$ and $s \in \mathbb { I } _ { \beta }$ , we may write
+
+$$
+\frac { d } { d s } \check { g } _ { i } ^ { \ l , ( t ) } ( s ) - \frac { d } { d s } \widetilde { g } _ { i } ^ { \ l , ( t ) } ( s ) = \sum _ { \gamma = 1 } ^ { \beta } \mathbf { x } _ { \gamma } .
+$$
+
+Note that the gradient gap $\mathbf { X } _ { \gamma }$ $( \gamma \in [ \alpha ] )$ can be rewritten as
+
+$$
+\mathbf { x } _ { \gamma } = [ \mathbf { W } ^ { l , ( t ) } ] _ { i j _ { \gamma } } [ \mathbf { M } ^ { ( t ) } v ] _ { j _ { \gamma } } \widetilde { \zeta } _ { \gamma } ,
+$$
+
+where $\widetilde { \zeta } _ { \gamma }$ is a difference of indicator functions:
+
+$$
+\widetilde { \zeta } _ { \gamma } : = \mathbf { 1 } _ { \left\{ \widetilde { g } _ { j _ { \gamma } } ^ { \imath - 1 , ( \mathfrak { t } ) } ( d _ { \gamma } ) \geq 0 \right\} } - \mathbf { 1 } _ { \left\{ \widetilde { g } _ { j _ { \gamma } } ^ { \imath - 1 , ( \mathfrak { t } ) } ( d _ { \gamma - 1 } ) \geq 0 \right\} } , \quad \ \forall d _ { \gamma - 1 } \in \mathbb { I } _ { \gamma - 1 } \mathrm { a n d } \forall d _ { \gamma } \in \mathbb { I } _ { \gamma } .
+$$
+
+Note that $\widetilde { \zeta } _ { \gamma }$ is independent of the choice of $d _ { \gamma - 1 }$ and $d _ { \gamma }$ . Note also that since the linear function g l−1,(t)jγ (s) switches the sign at s = sγ, we have ζeγ = ±1.
+
+We begin by proving the upper bound on $\scriptstyle \sum _ { \gamma = 1 } ^ { \beta } \mathbf { x } _ { \gamma }$ . For notational simplicity, we use $\mathbf { W } ^ { l }$ to denote $\mathbf { W } ^ { l , ( 0 ) }$ , and for $l \in [ L ]$ we set
+
+$$
+\mathbf { d W } ^ { l } : = \mathbf { W } ^ { l , ( t ) } - \mathbf { W } ^ { l , ( 0 ) } = \mathbf { W } ^ { l , ( t ) } - \mathbf { W } ^ { l } ,
+$$
+
+then we may write for $\gamma \in [ \alpha ]$ ,
+
+$$
+\begin{array} { r l } { \displaystyle [ \mathbf { M } ^ { ( t ) } \pmb { v } ] _ { j _ { \gamma } } = [ \mathbf { W } ^ { l - 1 , ( t ) } \mathbf { G } ^ { l - 2 , ( t ) } ( \pmb { x } ( 0 ) ) \mathbf { W } ^ { l - 2 , ( t ) } \cdot \cdot \mathbf { W } ^ { 1 , ( t ) } \mathbf { G } ^ { 0 , ( t ) } ( \pmb { x } ( 0 ) ) \mathbf { A } \pmb { v } ] _ { j _ { \gamma } } } & { } \\ { = \displaystyle \sum _ { \rho = 1 } ^ { m } [ \mathbf { W } ^ { l - 1 , ( t ) } ] _ { j _ { \gamma } , \rho } [ \mathbf { N } \pmb { v } ] _ { \rho } } & { } \\ { = \displaystyle \sum _ { \rho = 1 } ^ { m } [ \mathbf { W } ^ { l - 1 } ] _ { j _ { \gamma } , \rho } [ \mathbf { N } \pmb { v } ] _ { \rho } + \displaystyle \sum _ { \rho = 1 } ^ { m } [ \mathbf { d } \mathbf { W } ^ { l - 1 } ] _ { j _ { \gamma } , \rho } [ \mathbf { N } \pmb { v } ] _ { \rho } } & { } \\ { = : \mathbf { h } _ { \gamma } + \mathbf { d } \mathbf { h } _ { \gamma } , } \end{array}
+$$
+
+Therefore, we have
+
+$$
+\begin{array} { r l } & { \mathbf { x } _ { \gamma } \ = \ \widetilde { \zeta } _ { \gamma } [ \mathbf { W } ^ { l , ( t ) } ] _ { i , j _ { \gamma } } [ \mathbf { M } ^ { ( t ) } \pmb { v } ] _ { j _ { \gamma } } } \\ & { \quad = \ \widetilde { \zeta } _ { \gamma } \left( [ \mathbf { W } ^ { l } ] _ { i , j _ { \gamma } } + [ \mathbf { d W } ^ { l } ] _ { i , j _ { \gamma } } \right) \left( \mathbf { h } _ { \gamma } + \mathbf { d h } _ { \gamma } \right) } \\ & { \quad = \ \widetilde { \zeta } _ { \gamma } [ \mathbf { W } ^ { l } ] _ { i , j _ { \gamma } } \mathbf { h } _ { \gamma } + \widetilde { \zeta } _ { \gamma } [ \mathbf { d W } ^ { l } ] _ { i , j _ { \gamma } } \mathbf { h } _ { \gamma } + \widetilde { \zeta } _ { \gamma } [ \mathbf { W } ^ { l } ] _ { i , j _ { \gamma } } \mathbf { d h } _ { \gamma } + \widetilde { \zeta } _ { \gamma } [ \mathbf { d W } ^ { l } ] _ { i , j _ { \gamma } } \mathbf { d h } _ { \gamma } } \\ & { \quad = : \ \mathbf { x } _ { \gamma } ^ { ( 0 ) } + \mathbf { x } _ { \gamma } ^ { ( 1 ) } + \mathbf { x } _ { \gamma } ^ { ( 2 ) } + \mathbf { x } _ { \gamma } ^ { ( 3 ) } . } \end{array}
+$$
+
+In order to estimate $\mathbf { N v }$ , we use the following estimate (see Lemma 7.1 and Claim 11.2 in (AllenZhu et al., 2018b)):
+
+Lemma. If $\varepsilon \in ( 0 , 1 ]$ , with probability at least $1 - e ^ { - \Omega ( m \varepsilon ^ { 2 } / L ) }$ , for a fixed vector $_ z$ and $l \in [ L ]$ , we have
+
+$$
+\| \mathbf { G } ^ { l , ( t ) } ( \pmb { x } ( 0 ) ) \mathbf { W } ^ { l , ( t ) } \cdot \cdot \cdot \mathbf { W } ^ { 1 , ( t ) } \mathbf { G } ^ { 0 , ( t ) } ( \pmb { x } ( 0 ) ) \mathbf { A } \pmb { z } \| _ { 2 } \le ( 1 + \varepsilon ) \| \pmb { z } \| _ { 2 } .
+$$
+
+This Lemma implies that with probability at least $1 - e ^ { - \Omega ( m / L ) }$ , we have
+
+$$
+\| \mathbf { N } \pmb { v } \| _ { 2 } = \| \mathbf { G } ^ { l - 2 , ( t ) } ( \pmb { x } ( 0 ) ) \mathbf { W } ^ { l - 2 , ( t ) } \cdot \cdot \cdot \mathbf { W } ^ { 1 , ( t ) } \mathbf { G } ^ { 0 , ( t ) } ( \pmb { x } ( 0 ) ) \mathbf { A } \pmb { v } \| _ { 2 } \le 2 \| \pmb { v } \| _ { 2 } .
+$$
+
+We will estimate the 1st term $\mathbf { x } _ { \gamma } ^ { ( 0 ) }$ . Conditioning on this event (56) happens, using the randomness of $\mathbf { W } ^ { l - 1 }$ and general Hoeffding’s inequality (Vershynin, 2018), for each fixed vector $\textbf {  { v } }$ , we have
+
+$$
+\mathbb { P } \left\{ | \mathbf { h } _ { \gamma } | > \tau \right\} \leq 2 \exp \left( - \frac { C \tau ^ { 2 } } { ( 2 / m ) \| \mathbf { N } \pmb { v } \| _ { 2 } ^ { 2 } } \right) .
+$$
+
+Choose $\begin{array} { r } { \tau : = \frac { \log m } { \sqrt { m } } \| \mathbf { N } \pmb { v } \| _ { 2 } } \end{array}$ . Then, with probability at least $1 - e ^ { - \Omega ( \log ^ { 2 } m ) }$ , we have
+
+$$
+| \mathbf { h } _ { \gamma } | \leq \frac { \log m } { \sqrt { m } } \| \mathbf { N } \pmb { v } \| _ { 2 } .
+$$
+
+We set $\mathbf { u } _ { \gamma } ^ { ( 0 ) } : = \widetilde { \zeta } _ { \gamma } \mathbf { h } _ { \gamma }$ and ${ \bf u } ^ { ( 0 ) } = ( { \bf u } _ { 1 } ^ { ( 0 ) } , \ldots , { \bf u } _ { \beta } ^ { ( 0 ) } )$ . Note that since $\widetilde { \zeta } _ { \gamma } = \pm 1$ , we have $\left| \mathbf { u } _ { \gamma } ^ { ( 0 ) } \right| =$ $\begin{array} { r } { \left| \widetilde { \zeta } _ { \gamma } \mathbf { h } _ { \gamma } \right| \leq \frac { \log m } { \sqrt { m } } \left\| \mathbf { N } \pmb { v } \right\| _ { 2 } } \end{array}$ . The set of the gradient gaps $\{ \mathbf { x } _ { \gamma } ^ { ( 0 ) } \} = \{ \mathbf { u } _ { \gamma } ^ { ( 0 ) } [ \mathbf { W } ^ { l } ] _ { i , j _ { \gamma } } \}$ are a sequence of independent Gaussian random variables. Hence, $\textstyle \sum _ { \gamma = 1 } ^ { \beta } \mathbf { x } _ { \gamma } ^ { ( 0 ) }$ is a “Gaussian random walk” (12). Thus, using the randomness of $\mathbf { W } ^ { l }$ , conditioning on the above event (58), fixing any $\textbf {  { v } }$ , with probability at
+
+least $1 - e ^ { - \Omega ( \log ^ { 2 } m ) }$ , we have
+
+$$
+\begin{array} { r l r } {  { | \sum _ { \gamma = 1 } ^ { \beta } \mathbf { x } _ { \gamma } ^ { ( 0 ) } | \leq \frac { \log m } { \sqrt { m } } \| \mathbf { u } ^ { ( 0 ) } \| _ { 2 } } } \\ & { } & { \leq \frac { \log m } { \sqrt { m } } ( \sum _ { \gamma = 1 } ^ { \beta } ( \frac { \log m } { \sqrt { m } } \| \mathbf { N } v \| _ { 2 } ) ^ { 2 } ) ^ { 1 / 2 } } \\ & { } & { \leq 2 \frac { \log ^ { 2 } m } { \sqrt { m } } \| v \| _ { 2 } . } \end{array}
+$$
+
+Therefore, we have the estimate of $\mathbf { X } _ { \gamma } ^ { ( 0 ) }$
+
+Next, we will estimate the 2nd term $\mathbf { X } _ { \gamma } ^ { ( 1 ) }$ . By definition, we may write
+
+$$
+\sum _ { \gamma = 1 } ^ { \beta } \mathbf { x } _ { \gamma } ^ { ( 1 ) } = \sum _ { \gamma = 1 } ^ { \beta } \widetilde { \zeta } _ { \gamma } [ \mathbf { d W } ^ { l } ] _ { i , j _ { \gamma } } \mathbf { h } _ { \gamma } .
+$$
+
+Note that we use Theorem 1 (the convergence theorem) to write
+
+$$
+\| [ \mathbf { d W } ^ { l } ] _ { i } \| _ { 2 } = \| [ \mathbf { W } ^ { l , ( t ) } ] _ { i } - [ \mathbf { W } ^ { l } ] _ { i } \| _ { 2 } \leq O \left( \lambda \frac { \log m } { m } \right) .
+$$
+
+Thus, applying the estimate of $\mathsf { h } _ { \gamma }$ to eq. (58), we have
+
+$$
+\begin{array} { r l r } {  {  \sum _ { \gamma = 1 } ^ { \beta } \mathbf { x } _ { \gamma } ^ { ( 1 ) }  \leq \ 2 \sum _ { \gamma = 1 } ^ { \beta } ( \frac { \log m } { \sqrt { m } } \| \pmb { v } \| _ { 2 } ) [ \mathbf { d } \mathbf { W } ^ { l } ] _ { i , j _ { \gamma } } } } \\ & { } & { \leq \ \frac { \log m } { \sqrt { m } } \| \pmb { v } \| _ { 2 } \cdot \sqrt { \beta } \ \| [ \mathbf { d } \mathbf { W } ^ { l } ] _ { i } \| _ { 2 } } \\ & { } & { \leq C \lambda \frac { \log ^ { 2 } m } { m } \| \pmb { v } \| _ { 2 } \leq \frac { \log ^ { 2 } m } { \sqrt { m } } \| \pmb { v } \| _ { 2 } . } \end{array}
+$$
+
+The last inequality uses $\lambda \leq o ( { \sqrt { m } } )$ , which is indicated by the result of Theorem 1, therefore we have the estimate of $\mathbf { X } _ { \gamma } ^ { ( 1 ) }$ .
+
+Next, we will estimate the 3rd term $\mathbf { X } _ { \gamma } ^ { ( 2 ) }$ . By definition, we may write
+
+$$
+\sum _ { \gamma = 1 } ^ { \beta } \mathbf { x } _ { \gamma } ^ { ( 2 ) } = \sum _ { \gamma = 1 } ^ { \beta } \widetilde { \zeta } _ { \gamma } [ \mathbf { W } ^ { l } ] _ { i , j _ { \gamma } } \mathbf { d } \mathbf { h } _ { \gamma } .
+$$
+
+We set ${ \bf u } _ { \gamma } ^ { ( 2 ) } : = \tilde { \zeta } _ { \gamma } { \bf d } { \bf h } _ { \gamma }$ and $\mathbf { u } ^ { ( 2 ) } : = ( \mathbf { u } _ { 1 } ^ { ( 2 ) } , \ldots , \mathbf { u } _ { \beta } ^ { ( 2 ) } )$ . Using the randomness of $\mathbf { W } ^ { l }$ and general Hoeffding’s inequality (Vershynin, 2018), for each fixed vector $\textbf {  { v } }$ , we have
+
+$$
+\begin{array} { r } { \mathbb { P } \left\{ \left| \displaystyle \sum _ { \gamma = 1 } ^ { \beta } \mathbf { u } _ { \gamma } ^ { ( 2 ) } [ \mathbf { W } ^ { l } ] _ { i , j _ { \gamma } } \right| > \tau \right\} \leq ~ 2 \exp \left( - \frac { C \tau ^ { 2 } } { \| \mathbf { u } ^ { ( 2 ) } \| _ { 2 } ^ { 2 } ~ \operatorname* { m a x } _ { \gamma } ^ { 2 } \| [ \mathbf { W } ^ { l } ] _ { i , j _ { \gamma } } \| _ { \psi _ { 2 } } } \right) } \\ { \leq ~ 2 \exp \left( - \frac { C m \tau ^ { 2 } } { 2 \| \mathbf { u } ^ { ( 2 ) } \| _ { 2 } ^ { 2 } } \right) ~ } \end{array}
+$$
+
+Note that
+
+$$
+\mathrm { \bf d h } _ { \gamma } = \sum _ { \rho = 1 } ^ { m } [ { \bf d W } ^ { l - 1 } ] _ { j _ { \gamma } , \rho } [ { \bf N } { \pmb v } ] _ { \rho } ,
+$$
+
+and choose $\begin{array} { r } { \tau : = \frac { \log m } { \sqrt { m } } \| \mathbf { u } ^ { ( 2 ) } \| _ { 2 } } \end{array}$ , with probability $1 - e ^ { - \Omega ( \log ^ { 2 } m ) }$ , we have
+
+$$
+\begin{array} { r l r } {  { | \sum _ { \gamma = 1 } ^ { \beta } \mathbf { x } _ { \gamma } ^ { ( 2 ) } | \le \frac { \log m } { \sqrt { m } } \| \mathbf { u } ^ { ( 2 ) } \| _ { 2 } \le \frac { \log m } { \sqrt { m } } \{ \displaystyle \sum _ { \gamma = 1 } ^ { \beta } ( \sum _ { \rho = 1 } ^ { m } [ \mathbf { d W } ^ { l - 1 } ] _ { j _ { \gamma , \rho } } [ \mathbf { N } v ] _ { \rho } ) ^ { 2 } \} ^ { 1 / 2 } } } \\ & { } & { \le \frac { \log m } { \sqrt { m } } ( \displaystyle \sum _ { \gamma = 1 } ^ { \beta } \| [ \mathbf { d W } ^ { l - 1 } ] _ { j _ { \gamma } } \| _ { 2 } ^ { 2 } \| v \| _ { 2 } ^ { 2 } ) ^ { 1 / 2 } \le \frac { \log m } { \sqrt { m } } \| \mathbf { d W } ^ { l - 1 } \| _ { F } \| v \| _ { 2 } } \\ & { } & { \le \frac { \log ^ { 2 } m } { \sqrt { m } } \| v \| _ { 2 } . } \end{array}
+$$
+
+The last inequality uses $\lambda \leq o ( { \sqrt { m } } )$ and the result of Theorem 1:
+
+$$
+\| \mathbf { d W } ^ { l - 1 } \| _ { F } = \| \mathbf { W } ^ { l - 1 , ( t ) } - \mathbf { W } ^ { l - 1 } \| _ { F } \leq O \left( \lambda { \frac { \log m } { \sqrt { m } } } \right) .
+$$
+
+Therefore, we have the estimate of $\mathbf { X } _ { \gamma } ^ { ( 2 ) }$
+
+Next, we will estimate the last term $\mathbf { X } _ { \gamma } ^ { ( 3 ) }$ . Note that since $\widetilde { \zeta } _ { \gamma } = \pm 1$ , we have $| \widetilde { \zeta } _ { \gamma } | = 1$ , and conditioning on this event (56) happens, we may write
+
+$$
+\begin{array} { r l } & { \displaystyle \sum _ { \gamma = 1 } ^ { \beta } \mathbf { x } _ { \gamma } ^ { ( 3 ) } \bigg \vert = \left. \displaystyle \sum _ { \gamma = 1 } ^ { \beta } \widetilde { \zeta } _ { \gamma } [ \mathbf { d W } ^ { l } ] _ { : , j _ { \gamma } } \mathbf { d h } _ { \gamma } \right. } \\ & { \qquad \leq \| [ \mathbf { d W } ^ { l } ] _ { : } \| _ { 2 } \left\{ \displaystyle \sum _ { \gamma = 1 } ^ { \beta } \left( \displaystyle \sum _ { \rho = 1 } ^ { m } [ \mathbf { d W } ^ { l - 1 } ] _ { j _ { \gamma } , \rho } [ \mathbf { N } \mathbf { v } ] _ { \rho } \right) ^ { 2 } \right\} ^ { 1 / 2 } } \\ & { \qquad \leq \| [ \mathbf { d W } ^ { l } ] _ { i } \| _ { 2 } \left( \displaystyle \sum _ { \gamma = 1 } ^ { \beta } \| [ \mathbf { d W } ^ { l - 1 } ] _ { j _ { \gamma } } \| _ { 2 } ^ { 2 } \| \mathbf { v } \| _ { 2 } ^ { 2 } \right) ^ { 1 / 2 } } \\ & { \qquad \leq \| [ \mathbf { d W } ^ { l } ] _ { i } \| _ { 2 } \| _ { 2 } \| \mathbf { d W } ^ { l - 1 } \| _ { F } \| \mathbf { v } \| _ { 2 } \leq \ \frac { \log ^ { 2 } m } { \sqrt { m } } \| \mathbf { v } \| _ { 2 } . } \end{array}
+$$
+
+Putting this all together, with probability $1 - e ^ { - \Omega ( \log ^ { 2 } m ) }$ , we have
+
+$$
+\left| { \frac { d } { d s } } { \check { g } } _ { i } ^ { \iota } ( s ) - { \frac { d } { d s } } { \widetilde { g } } _ { i } ^ { \iota } ( s ) \right| = \left| \sum _ { \gamma = 1 } ^ { \beta } \mathbf { x } _ { \gamma } \right| \leq 4 { \frac { \log ^ { 2 } m } { \sqrt { m } } } \| { \pmb v } \| _ { 2 } .
+$$
+
+Therefore, we have
+
+$$
+\begin{array} { r l r } & { } & { \displaystyle \left. \check { g } _ { i } ^ { l } ( s ) - \widetilde { g } _ { i } ^ { l } ( s ) \right. \le \displaystyle \int _ { 0 } ^ { s } \left. \frac { d } { d s } \check { g } _ { i } ^ { l } ( s ) - \frac { d } { d s } \widetilde { g } _ { i } ^ { l } ( s ) \right. d s } \\ & { } & { \displaystyle \le 4 s \frac { \log ^ { 2 } m } { \sqrt { m } } \lVert \pmb { v } \rVert _ { 2 } \quad ( 0 \le \forall s \le 1 ) . } \end{array}
+$$
+
+This finishes the proof of (7) in Lemma 3. Finally, the proof of (8) in Lemma 3 is idential to the above proof, except the fact that each entry of $\mathbf { B }$ follows from $\mathcal { N } ( 0 , 2 / c )$ insted of $\mathcal { N } ( 0 , 2 / m )$ . We complete the proof.
+
+# A.2 PROOF OF GENERALIZATION THEOREM
+
+Theorem 2 (reshown, see $\ S 3$ ). Under the same setting as Theorem $^ { l }$ , with probability at least $1 - e ^ { - \Omega ( \log ^ { 2 } m ) }$ , for every $\pmb { x } ^ { ( p ) }$ , $\pmb { x } ^ { ( q ) }$ ( $p , q \in \left[ n \right] p \neq q ,$ ), $t \in [ T ]$ , $l \in [ L ]$ , $i \in [ m ]$ and $j \in [ c ]$ , we
+
+have
+
+$$
+\begin{array} { r l r } & { } & { \underset { 0 \leq s \leq 1 } { \operatorname* { s u p } } \Big \vert g _ { i } ^ { l , ( t ) } ( { \pmb x } ( s ) ) - \tilde { g } _ { i } ^ { l , ( t ) } ( s ) \Big \vert \leq { \cal O } \left( \frac { \log ^ { l } m } { \sqrt { m } } \right) \| { \pmb v } \| _ { 2 } , } \\ & { } & { \underset { 0 \leq s \leq 1 } { \operatorname* { s u p } } \Big \vert f _ { j } ^ { ( t ) } ( { \pmb x } ( s ) ) - \tilde { f } _ { j } ^ { ( t ) } ( s ) \Big \vert \leq { \cal O } \left( \frac { \log ^ { L + 1 } m } { \sqrt { c } } \right) \| { \pmb v } \| _ { 2 } . } \end{array}
+$$
+
+In order to give provable guarantees for the generalization performance of over-parameterized deep networks in the ${ \bar { l } } ^ { 2 }$ regression task on the one-dimensional input-space, we apply the above Theorem 2 to bound the error between the trained network output $f ^ { ( T ) } ( x )$ and the piecewise linear function $\hat { f } ( { \pmb x } )$ . Note that if we apply Theorem 1 to this case (one-dimensional models), simple preprocessing is needed: It just consider a mapping $[ 0 , \nu ]  S _ { + } ^ { 1 }$ (a half circle).
+
+$$
+S _ { + } ^ { 1 } = \{ x = ( x _ { 1 } , x _ { 2 } ) \in \mathbb { R } ^ { 2 } : \| x \| _ { 2 } = 1 , x _ { 2 } \geq 1 \}
+$$
+
+It is apparent that there exists a $C ^ { \infty }$ diffeomorphism from $[ 0 , \nu ]$ to $S _ { + } ^ { 1 }$ . Under this setting, the input from $S _ { + } ^ { 1 }$ is the two-dimension. More precisely, the input needs another axis of coordinates to express the biases term. This procedure is written in the subsection of Dataset and loss function in the paper p.4. As a result, we deal with not the case of $d = 1$ but the case of $d = 3$ in Theorem 1.
+
+Lemma 5. Under the same setting as Theorem $^ { l }$ and 2, for $\delta \in ( 0 , 1 / 2 ]$ , then with probability at least $1 - ( \delta + e ^ { - \Omega ( \log ^ { 2 } m ) } )$ , we have
+
+$$
+\operatorname* { s u p } _ { x \in [ 0 , \nu ] } \left. f _ { j } ^ { ( T ) } ( x ) - \hat { f } _ { j } ( x ) \right. \leq O \left( \frac { \log ^ { L + 1 } m } { m ^ { 1 / 6 } } \cdot \frac { \nu } { n \delta } \right) .
+$$
+
+Proof. Without loss of generality, we may assume that the entry of the weight matrix $\begin{array} { r l } { \mathrm { B } _ { i , j } \sim } \end{array}$ $\mathcal { N } ( 0 , 2 / \mathfrak { c } )$ , where ${ \mathfrak { c } } = \Theta \left( m ^ { 1 / 3 } \right)$ . 6
+
+For $k \in [ n + 1 ]$ , we denote by $\left| \mathbb { I } _ { k } \right|$ the length of the interval $\mathbb { I } _ { k } = \left[ \boldsymbol { x } ^ { ( k - 1 ) } , \boldsymbol { x } ^ { ( k ) } \right]$ . We set $c = { \mathfrak { c } } =$ $m ^ { 1 / 3 }$ in Theorem 2, with probability at least $1 - e ^ { - \Omega ( \log ^ { 2 } m ) }$ , we have
+
+$$
+\operatorname* { s u p } _ { x \in \mathbb { I } _ { k } } \left. f _ { j } ^ { ( T ) } ( x ) - \tilde { f } _ { j } ^ { ( T ) } ( x ) \right. \leq C _ { j } \frac { \log ^ { L + 1 } m } { m ^ { 1 / 6 } } \left( \operatorname* { s u p } _ { k \in [ n + 1 ] } \left. x ^ { ( k ) } - x ^ { ( k - 1 ) } \right. \right) = C _ { j } \frac { \log ^ { L + 1 } m } { m ^ { 1 / 6 } } \mathcal { T }
+$$
+
+where $C _ { j }$ is a constant, and $\begin{array} { r } { \mathcal { T } : = \operatorname* { s u p } _ { k \in [ n + 1 ] } \left| x ^ { ( k ) } - x ^ { ( k - 1 ) } \right| } \end{array}$ . Here we set $\begin{array} { r } { \varepsilon : = C _ { j } \frac { \log ^ { L + 1 } m } { m ^ { 1 / 6 } } \mathcal { T } } \end{array}$
+
+Note that $f _ { j } ^ { ( T ) } ( x ^ { ( k - 1 ) } ) = \hat { f } _ { j } ( x ^ { ( k - 1 ) } )$ and $f _ { j } ^ { ( T ) } ( x ^ { ( k ) } ) = \hat { f } _ { j } ( x ^ { ( k ) } )$ , by eq. (85) we have
+
+$$
+\left| \tilde { f } _ { j } ^ { ( T ) } ( x ^ { ( k - 1 ) } ) - \hat { f } _ { j } ( x ^ { ( k - 1 ) } ) \right| \le \varepsilon \quad \mathrm { ~ a n d ~ } \quad \left| \tilde { f } _ { j } ^ { ( T ) } ( x ^ { ( k ) } ) - \hat { f } _ { j } ( x ^ { ( k ) } ) \right| \le \varepsilon .
+$$
+
+Note that the piecewise linear function $\hat { f } _ { j }$ linearly connects $\left( x ^ { ( k - 1 ) } , \hat { f } _ { j } ( x ^ { ( k - 1 ) } ) \right)$ and $\Big ( x ^ { ( k ) } , \hat { f } _ { j } ( x ^ { ( k ) } ) \Big )$ . Thus, the following inequality holds.
+
+$$
+\operatorname* { s u p } _ { x \in \mathbb { I } _ { k } } \left. \tilde { f } _ { j } ^ { ( T ) } ( x ) - \hat { f } _ { j } ( x ) \right. \leq \varepsilon
+$$
+
+We use eq. (85) and eq. (87) to see that
+
+$$
+\operatorname* { s u p } _ { x \in \mathbb { I } _ { k } } \left. f _ { j } ^ { ( T ) } ( x ) - \hat { f } _ { j } ( x ) \right. \leq \operatorname* { s u p } _ { x \in \mathbb { I } _ { k } } \left. f _ { j } ^ { ( T ) } ( x ) - \tilde { f } _ { j } ^ { ( T ) } ( x ) \right. + \operatorname* { s u p } _ { x \in \mathbb { I } _ { k } } \left. \tilde { f } _ { j } ^ { ( T ) } ( x ) - \hat { f } _ { j } ( x ) \right. \leq 2 \varepsilon .
+$$
+
+Note that $\varepsilon$ is not dependent on $k$ . Thus, we have
+
+$$
+\operatorname* { s u p } _ { x \in [ 0 , \nu ] } \left. f _ { j } ^ { ( T ) } ( x ) - \hat { f } _ { j } ( x ) \right. \leq 2 \varepsilon .
+$$
+
+Next, we evaluate $\begin{array} { r } { \mathcal { T } = \operatorname* { s u p } _ { k \in [ n + 1 ] } \left| x ^ { ( k ) } - x ^ { ( k - 1 ) } \right| . } \end{array}$ Recall that the training dataset $\{ x ^ { ( k ) } \} _ { k \in [ n ] }$ is generated from the uniform distribution in the interval $[ 0 , \nu ]$ . We denote by $\mu : = n / \nu$ the density of training samples. In this setting, it is well known that for every $r > 0$ $( r < \nu )$ , the number of training samples in the interval $[ 0 , r ]$ follows the Poisson distribution with mean $\mu r$ , then the sequence of inter-sample distances is independent and identically distributed exponential random variables having mean $1 / \mu \left( = \textstyle { \frac { \nu } { n } } \right)$ (Daley & Vere-Jones, 2007). Define $\Delta$ to be the above exponential distribution having mean $\nu / n$ , which is the inter-sample distance. Then, using Markov’s inequality, for $\tau > 0$ , we have
+
+$$
+\mathbb { P } [ \Delta \ge \tau ] \le \frac { \mathbb { E } [ \Delta ] } { \tau } = \frac { \nu } { n \tau } .
+$$
+
+$\textstyle { \tau : = { \frac { \nu } { n \tau } } }$ $1 - \delta$
+
+$$
+\mathcal { T } \leq \frac { \nu } { n \delta } .
+$$
+
+Combining this with eq. (89), with probability at least $1 - ( \delta + e ^ { - \Omega ( \log ^ { 2 } m ) } )$ , we have
+
+$$
+\operatorname* { s u p } _ { x \in [ 0 , \nu ] } \left. f _ { j } ^ { ( T ) } ( x ) - \hat { f } _ { j } ( x ) \right. \leq 2 C _ { j } \frac { \log ^ { L + 1 } m } { m ^ { 1 / 6 } } \mathcal { T } \leq 2 C _ { j } \frac { \log ^ { L + 1 } m } { m ^ { 1 / 6 } } \frac { \nu } { n \delta } .
+$$
+
+We complete the proof of the Lemma 5.
+
+Lemma 6. Suppose $f _ { j } ^ { * } ( x )$ is a $C ^ { 1 }$ - class function on $[ 0 , \nu ]$ and $\hat { f } _ { j } ( x )$ is the linear interpolation of the data points $\{ \left( \boldsymbol { x } ^ { ( i ) } , \boldsymbol { f } _ { j } ^ { * } ( \boldsymbol { x } ^ { ( i ) } ) \right) \} _ { i \in [ n ] }$ . For $\delta \in ( 0 , 1 / 2 ]$ , with probability at least $1 - \delta$ , we have
+
+$$
+\operatorname* { s u p } _ { x \in [ 0 , \nu ] } \left| \hat { f } _ { j } ( x ) - f _ { j } ^ { * } ( x ) \right| \leq O \left( \frac { \nu } { n \delta } \right) .
+$$
+
+Proof. We use $f _ { j } ^ { * } { ' } ( x )$ to denote the first derivative of $f _ { j } ^ { * } ( x )$ . Since $f _ { j } ^ { * }$ is a $C ^ { 1 }$ - class function on the compact set $[ 0 , \nu ]$ , $f _ { j } ^ { * \prime }$ is a bounded function. Thus, we can define $\mathcal { S } ^ { ( k ) }$ and $s$ as follows:
+
+$$
+S ^ { ( k ) } : = \operatorname* { s u p } _ { x \in \mathbb { I } _ { k } } \left| f _ { j } ^ { * \prime } ( x ) \right| , \quad S : = \operatorname* { s u p } _ { k \in [ n + 1 ] } S ^ { ( k ) } .
+$$
+
+Since $f _ { j } ^ { * } ( x )$ and $\hat { f } _ { j } ( x )$ are continuous functions, there exists ${ \mathfrak { h } } ^ { ( k ) } ( 0 \leq { \mathfrak { h } } ^ { ( k ) } \leq x ^ { ( k ) } - x ^ { ( k - 1 ) } )$ such that
+
+$$
+\operatorname* { s u p } _ { x \in I _ { k } } \left| f _ { j } ^ { * } ( x ) - \hat { f } _ { j } ( x ) \right| = \left| f _ { j } ^ { * } ( x ^ { ( k - 1 ) } + \mathfrak { h } _ { k } ) - \hat { f } _ { j } ( x ^ { ( k - 1 ) } + \mathfrak { h } _ { k } ) \right| .
+$$
+
+By the mean value theorem, there exists a real number ${ \mathfrak { c } } ^ { ( k ) } \in \mathbb { I } _ { k }$ such that,
+
+$$
+\hat { f } _ { j } ( x ^ { ( k - 1 ) } + \mathfrak { h } ^ { ( k ) } ) = f _ { j } ^ { * \prime } ( \mathfrak { c } ^ { ( k ) } ) \mathfrak { h } ^ { ( k ) } + f _ { j } ^ { * } ( x ^ { ( k - 1 ) } ) .
+$$
+
+Note that by the definition of $\hat { f } _ { j }$ , we have $f _ { j } ^ { * } ( x ^ { ( k - 1 ) } ) = { \hat { f } } _ { j } ( x ^ { ( k - 1 ) } )$ , therefore, we also have that
+
+$$
+\begin{array} { r l } { \Big | f _ { j } ^ { * } \big ( x ^ { ( k - 1 ) } + \mathfrak { h } ^ { ( k ) } \big ) - \hat { f } _ { j } \big ( x ^ { ( k - 1 ) } + \mathfrak { h } ^ { ( k ) } \big ) \Big | = \Big | f _ { j } ^ { * } \big ( x ^ { ( k - 1 ) } + \mathfrak { h } ^ { ( k ) } \big ) - f _ { j } ^ { * } \big ( x ^ { ( k - 1 ) } \big ) - f _ { j } ^ { * \prime } \big ( \mathfrak { c } ^ { ( k ) } \big ) \mathfrak { h } ^ { ( k ) } \Big | ^ { 2 } } & { \quad \mathrm { ~ f ~ o ~ r ~ } \quad \mathfrak { z } _ { j } ^ { * } = 0 , } \\ & { \le \displaystyle \int _ { x ^ { ( k - 1 ) } } ^ { x ^ { ( k - 1 ) } + \mathfrak { h } ^ { ( k ) } } \Big | f _ { j } ^ { * \prime } \big ( x \big ) - f _ { j } ^ { * \prime } \big ( \mathfrak { c } ^ { ( k ) } \big ) \Big | d \mathfrak { x } } \\ & { \le 2 \mathfrak { h } ^ { ( k ) } \mathcal { S } ^ { ( k ) } . } \end{array}
+$$
+
+By considering whole interval $[ 0 , \nu ]$ , we have
+
+$$
+\operatorname* { s u p } _ { x \in [ 0 , \nu ] } \left| \hat { f } _ { j } ( x ) - f _ { j } ^ { * } ( x ) \right| = \operatorname* { s u p } _ { k \in [ n + 1 ] } \bigg ( \operatorname* { s u p } _ { x \in I _ { k } } \left| f _ { j } ^ { * } ( x ) - \hat { f } _ { j } ( x ) \right| \bigg ) \leq 2 7 \mathcal S
+$$
+
+Now we apply the estimate of $\tau$ from eq. (91). We may therefore rewrite eq. (100) as
+
+$$
+\operatorname* { s u p } _ { x \in [ 0 , \nu ] } \left| \hat { f } _ { j } ( x ) - f _ { j } ^ { * } ( x ) \right| \leq O \left( \frac { \nu } { n \delta } \right) .
+$$
+
+We complete the proof of Lemma 6.
+
+# Proof of Theorem 4 (Generalization)
+
+Proof. From eq. (89) and eq. (93), with probability at least $1 - ( \delta + e ^ { - \Omega ( \log ^ { 2 } m ) } )$ , we have for every $j = 1 , 2 , \cdots , c$ ,
+
+$$
+\begin{array} { r l r } & { } & { \bigg \vert f _ { j } ^ { ( T ) } ( x ) - f _ { j } ^ { * } ( x ) \bigg \vert \leq \bigg \vert f _ { j } ^ { ( T ) } ( x ) - \hat { f } _ { j } ( x ) \bigg \vert + \bigg \vert \hat { f } _ { j } ( x ) - f _ { j } ^ { * } ( x ) \bigg \vert } \\ & { } & { \leq C _ { j } \frac { \nu } { n \delta } \left( 1 + \frac { \log ^ { L + 1 } m } { m ^ { 1 / 6 } } \right) \qquad ( \forall x \in [ 0 , \nu ] ) . } \end{array}
+$$
+
+Using the triangle inequality, this gives
+
+$$
+\begin{array} { r l r } & { } & { \ell \left( f _ { j } ^ { ( T ) } ( x ) , y \right) = \displaystyle \frac { 1 } { 2 } \left\| f ^ { ( T ) } ( x ) - f ^ { * } ( x ) \right\| _ { 2 } ^ { 2 } = \displaystyle \frac { 1 } { 2 } \sum _ { j = 1 } ^ { c } \left\| f _ { j } ^ { ( T ) } ( x ) - f _ { j } ^ { * } ( x ) \right\| ^ { 2 } } \\ & { } & { \leq \displaystyle \frac { 1 } { 2 } C ^ { \prime 2 } c \frac { \nu ^ { 2 } } { n ^ { 2 } \delta ^ { 2 } } \left( 1 + \frac { \log ^ { L + 1 } m } { m ^ { 1 / 6 } } \right) ^ { 2 } \quad \mathrm { ( } \forall x \in [ 0 , \nu ] \mathrm { ) } . } \end{array}
+$$
+
+where $C ^ { \prime } : = \operatorname* { m a x } _ { j = 1 , \cdots , c } C _ { j }$
+
+Thus, we have the following estimate:
+
+With probability at least $1 - ( \delta + e ^ { - \Omega ( \log ^ { 2 } m ) } )$ , we have
+
+$$
+\begin{array} { r l } { \mathbb { E } _ { ( x , y ) \sim \mathcal { D } } \left[ \ell \left( f ^ { ( T ) } ( x ) , y \right) \right] = } & { \frac { 1 } { \nu } \int _ { 0 } ^ { \nu } \frac { 1 } { 2 } \left\| f ^ { ( T ) } ( x ) - f ^ { * } ( x ) \right\| ^ { 2 } d x } \\ { \leq } & { \frac { 1 } { 2 } C ^ { \prime 2 } c \frac { \nu ^ { 2 } } { n ^ { 2 } \delta ^ { 2 } } \left( 1 + \frac { \log ^ { L + 1 } m } { m ^ { 1 / 6 } } \right) ^ { 2 } . } \end{array}
+$$
+
+We complete the proof.

parse/train/RmuXDtjDhG/RmuXDtjDhG.md

@@ -0,0 +1,292 @@
+# Causal Abstractions of Neural Networks
+
+Atticus Geiger∗, Hanson Lu∗, Thomas Icard, and Christopher Potts Stanford Stanford, CA 94305-2150 {atticusg, hansonlu, icard, cgpotts}@stanford.edu
+
+# Abstract
+
+Structural analysis methods (e.g., probing and feature attribution) are increasingly important tools for neural network analysis. We propose a new structural analysis method grounded in a formal theory of causal abstraction that provides rich characterizations of model-internal representations and their roles in input/output behavior. In this method, neural representations are aligned with variables in interpretable causal models, and then interchange interventions are used to experimentally verify that the neural representations have the causal properties of their aligned variables. We apply this method in a case study to analyze neural models trained on Multiply Quantified Natural Language Inference (MQNLI) corpus, a highly complex NLI dataset that was constructed with a tree-structured natural logic causal model. We discover that a BERT-based model with state-of-the-art performance successfully realizes parts of the natural logic model’s causal structure, whereas a simpler baseline model fails to show any such structure, demonstrating that BERT representations encode the compositional structure of MQNLI.
+
+# 1 Introduction
+
+Explainability and interpretability have long been central issues for neural networks, and they have taken on renewed importance as such models are now ubiquitous in research and technology. Recent structural evaluation methods seek to reveal the internal structure of these “black box” models. Structural methods include probes, attributions (feature importance methods), and interventions (manipulations of model-internal states). These methods can complement standard behavioral techniques (e.g., performance on gold evaluation sets), and they can yield insights into how and why models make the predictions they do. However, these tools have their limitations, and it has often been assumed that more ambitious and systematic causal analysis of such models is beyond reach.
+
+Although there is a sense in which neural networks are “black boxes”, they have the virtue of being completely closed and controlled systems. This means that standard empirical challenges of causal inference due to lack of observability simply do not arise. The challenge is rather to identify high-level causal regularities that abstract away from irrelevant (but arbitrarily observable and manipulable) low-level details. Our contribution in this paper is to show that this challenge can be met. Drawing on recent innovations in the formal theory of causal abstraction [1, 2, 5, 22], we offer a methodology for meaningful causal explanations of neural network behavior.
+
+Our methodology causal abstraction analysis2 consists of three stages. (1) Formulate a hypothesis by defining a causal model that might explain network behavior. Candidate causal models can be naturally adapted from theoretical and empirical modeling work in linguistics and cognitive sciences. (2) Search for an alignment between neural representations in the network and variables in the high-level causal model. (3) Verify experimentally that the neural representations have the same causal properties as their aligned high-level variables using the interchange intervention method of Geiger et al. [11].
+
+As a case study, we apply this methodology to LSTM-based and BERT-based natural language inference (NLI) models trained on the logically complex Multiply Quantified NLI (MQNLI) dataset of Geiger et al. [10]. This challenging dataset was constructed with a tree-structured natural logic causal model [17, 29, 14]. Our BERT-based model has the structure of a standard NLI classifier, and yet it is able to perform well on MQNLI $( 8 8 \% )$ , a result Geiger et al. achieved only with highly customized task-specific models. By contrast, our LSTM-based model is much less successful $(46 \% )$ .
+
+The obvious scientific question in this case study is what drives the success of the BERT-based model on this challenging task. To answer this we employ our methodology. (1) We formulate hypotheses by defining simplified variants of the natural logic causal model. (2) We search over potential alignments between neural representations in BERT and variables in our high-level causal models. (3) We perform interchange interventions on the BERT model for each alignment. We find that our BERT model partially realizes the causal structure of the natural logic causal model; crucially, the LSTM model does not. High-level causal explanation for system behavior is often considered a gold standard for interpretability, one that may be thought quixotic for complex neural models [16]. The point of our case study is to show that this high standard can be achieved.
+
+We conclude by comparing our methodology to probing and the attribution method of integrated gradients [27]. We argue probing is unable to provide a causal characterization of models. We show formally that attribution methods do measure causal properties, and in that way they are similar to the tool of interchange interventions. However, our methodology of causal abstraction analysis provides a framework for systematically measuring and aggregating such causal properties in order to evaluate a precise hypothesis about abstract causal structure.
+
+# 2 Related Work
+
+Probes Probes are generally supervised models trained on the internal representations of networks with the goal of determining what those internal representations encode [7, 13, 20, 28]. Probes are fundamentally unable to directly measure causal properties of neural representations, and Ravichander et al. [21], Elazar et al. [9], and Geiger et al. [11] have argued that probes are limited in their ability to provide even indirect evidence of causal properties.
+
+We now present an analytic example in which probing identifies seemingly crucial information in representations that have no causal impact on behavior. We assume the structure of the simple addition network $N _ { + }$ in Figure 1. For our embedding, we simply map every integer $i$ in $\mathbb { N } _ { 9 }$ to the 1-dimensional vector [i]. The weight matrices are
+
+$$
+W _ { 1 } = { \left( \begin{array} { l } { 1 } \\ { 1 } \\ { 0 } \end{array} \right) } \quad W _ { 2 } = { \left( \begin{array} { l } { 1 } \\ { 1 } \\ { 1 } \end{array} \right) } \quad W _ { 3 } = { \left( \begin{array} { l } { 0 } \\ { 0 } \\ { 1 } \end{array} \right) } \quad \mathbf { w } = { \left( \begin{array} { l } { 0 } \\ { 1 } \\ { 0 } \end{array} \right) }
+$$
+
+The output for an input sequence $\mathbf { x } = ( i , j , k )$ is given by $\left( \mathbf { x } W _ { 1 } ; \mathbf { x } W _ { 2 } ; \mathbf { x } W _ { 3 } \right) \mathbf { w } .$ .
+
+In this network, $\mathbf { x } W _ { 1 }$ perfectly encodes $i + j$ , and $\mathbf { x } W _ { 3 }$ perfectly encodes $k$ . Thus, the identity model probe will be perfect in probing those representations for this information. However, neither representation plays a causal role in the network behavior; only $\mathbf { x } W _ { 2 }$ contributes to the output.
+
+Attribution Methods Attribution methods aim to quantify the degree to which a network representation contributes to the output prediction of the model, for a specific example or set of examples [3, 24, 26, 27, 32]. In contrast to probing, the well known integrated gradients method (IG) can be given an unambiguous causal interpretation. Following [27] we define the vector $I G ( \mathbf { x } )$ , for an input $\mathbf { x }$ relative to a baseline b, to have ith component $I G _ { i } ( \mathbf { x } )$ given by the expression on the left:
+
+$$
+\left( x _ { i } - b _ { i } \right) \cdot \int _ { \alpha = 0 } ^ { 1 } \frac { \partial F ( \alpha \mathbf { x } + ( 1 - \alpha ) \mathbf { b } ) } { \partial x _ { i } } d \alpha = \left( x _ { i } - b _ { i } \right) \cdot \int _ { \alpha = 0 } ^ { 1 } \operatorname* { l i m } _ { \epsilon \to 0 } \frac { F ( \mathbf { x } ^ { \alpha , \epsilon } ) - F ( \mathbf { x } ^ { \alpha } ) } { \epsilon } d \alpha
+$$
+
+Abbreviating the weighted average $\alpha { \mathbf { x } } + ( 1 - \alpha ) { \mathbf { b } }$ by $\mathbf { x } ^ { \alpha }$ , letting $\mathbf { x } ^ { \alpha , \epsilon }$ be the vector that differs from $\mathbf { x } ^ { \alpha }$ in that the ith coordinate is increased by $\epsilon$ , and then expanding the definition of partial derivative, this can be written in the form given on the right. The difference $F ( \mathbf { x } ^ { \alpha , \epsilon } ) - F ( \mathbf { x } ^ { \alpha } )$ is known in the (a) The causal model $C _ { + }$ (right) that first computes $S _ { 1 } = X + Y$ and $W = Z$ , before computing the final output $S _ { 2 } = W + S _ { 1 }$ aligned with the neural network $N _ { + }$ (left) with $L _ { 1 }$ highlighted as the hypothesized location encoding $S _ { 1 } = X + Y$ and $L _ { 2 }$ as the location encoding $W = Z$ .
+
+![](images/e49109d1609a9c954cfc856da49b3385d339e102b0552dcca5c4500592d22457.jpg)
+
+![](images/e335832fbeda3931c2fa44e5cf03db667825ddc14116bfb99e3fc5af8c82b463.jpg)
+
+(b) Low-level neural network interchange intervention. The network processes two different input sequences. The neural representation created at $L _ { 1 }$ for input sequence $( 1 , 2 , { \bar { 3 } } )$ is replaced by the corresponding representation created for input sequence $( 4 , 5 , 6 )$ .
+
+(c) High-level symbolic computation interchange intervention. The computation processes two different input sequences. The sum at $S _ { 1 }$ for input sequence $( 1 , 2 , 3 )$ is replaced by the corresponding representation created for input sequence $( 4 , 5 , 6 )$ .
+
+Figure 1: Our motivating example where we hypothesis that a symbolic computation $C _ { + }$ is a causal abstraction of a neural network $N _ { + }$ under a particular alignment (top). We can experimentally confirm this hypothesis by conducting an interchange intervention on both the network and the computation with every pair of inputs and evaluating whether the intervened network and intervened computation have the same counterfactual output behavior. We schematically depict an interchange intervention on the network $N _ { + }$ (bottom left) and the computation $C _ { + }$ (bottom right) with the base input $( 1 , 2 , 3 )$ and the source input $( 4 , 5 , 6 )$ . Observe that the output of the intervened neural network matches the output of the intervened symbolic computation, so we have success for this pair of inputs.
+
+causal literature as the (individual) causal effect on the output (e.g., [15]) of increasing neuron $i$ by $\epsilon$ relative to the fixed input $\mathbf { x } ^ { \alpha }$ . So, essentially, $I G _ { i } ( \mathbf { x } )$ is measuring the average “limiting” causal effect of increasing neuron $i$ along the straight line from the baseline vector to the input vector $\mathbf { x }$ , weighted by the difference at $i$ between input and baseline. More recently, Chattopadhyay et al. [6] develop an attribution method that explicitly treats neural models as structured causal models and directly computes the individual causal effect of a feature to determine its attribution.
+
+Attribution methods can measure causal properties, and, in that way, they are similar to the tool of interchange interventions. However, our methodology of causal abstraction analysis provides a framework for systematically measuring and aggregating such causal properties in order to evaluate a precise hypothesis about abstract causal structure.
+
+Causal Abstraction Our goal is to evaluate whether the internal structure of a neural network realizes an abstract causal process. To concretize this, we turn to formal, broadly interventionist theories of causality [25, 19], in which causal processes are characterized by effects of interventions, and theories of abstraction [1, 2, 5, 22] where relationships between two causal processes are determined by the presence of systematic correspondences between the effects of interventions.
+
+The notion of abstraction that we employ here is a relatively simple one called constructive abstraction [1]. Informally, a high-level model is a constructive abstraction of a low-level model if there is a way to partition the variables in the low-level model where each high-level variable can be assigned to a low-level partition cell, such that there is a systematic correspondence between interventions on the low-level partition cells and interventions on the high-level variables.
+
+There are two properties of constructive abstraction that make it ideal for neural network analysis. First, the information content of partition cells of low-level variables can be determined by the high-level variables that they correspond to. For neural networks, the partition cells of low-level variables are sets of neurons, and our method supports reasoning at the level of vector representations (sets of neurons). Second, the causal dependencies between partitions of low-level variables are not necessarily preserved as causal dependencies between the high-level variables corresponding to these partitions. For example, the low-level model might be a fully connected neural network, whereas the high-level model might have much sparser connections. For neural network analysis, this means we can find causal abstractions that have far simpler causal structures than the underlying neural networks. We provide an example in the next section.
+
+# 3 Causal Abstraction Analysis of Neural Networks
+
+We now describe our methodology in more detail, illustrating the relevant concepts with an example of a neural network performing basic arithmetic. Specifically, suppose that we have a neural network $N _ { + }$ that takes in three vector representations $D _ { x } , D _ { y } , D _ { z }$ representing the integers $x , y .$ , and $z$ , and outputs the sum of the three inputs: $N _ { + } ( D _ { x } , D _ { y } , D _ { z } ) = x + y + z$ . We seek an informative causal explanation of this network’s behavior.
+
+Formulating a Hypothesis A human performing this task might follow an algorithm in which they add together the first two numbers and then add that sum to the third number. We can hypothesize that the behavior of $N _ { + }$ is explained by this symbolic computation. Specifically, the network combines $D _ { x }$ and $D _ { y }$ to create an internal representation at some location $L _ { 1 }$ encoding $x + y$ ; it encodes $z$ at some location $L _ { 2 }$ ; and $L _ { 1 }$ and $L _ { 2 }$ are composed to encode $a + z$ at the location of the output representation. This hypothesis is given schematically in Figure 1a.
+
+Following our methodology, we first define the causal model $C _ { + }$ in Figure 1a. Our informal hypothesis that a neural network’s behavior is explained by a simple algorithm can then be restated more formally: $C _ { + }$ is a constructive abstraction of the neural network $N _ { + }$ .
+
+Alignment Search Now that we have hypothesized that the causal model $C _ { + }$ is a causal abstraction of the network $N _ { + }$ , the next step is to align the neural representations in $N _ { + }$ with the variables in $C _ { + }$ . The input embeddings $D _ { x }$ , $D _ { y }$ , and $D _ { z }$ must be aligned with the input variables $X , Y$ , and $Z$ and the output neuron $O$ must be aligned with the output variable $S _ { 2 }$ . That leaves the intermediate variables $S _ { 1 }$ and $W$ to be aligned with neural representations at some undetermined locations $L _ { 1 }$ and $L _ { 2 }$ . If this were an actual experiment (see below), we would perform an alignment search to consider many possible values for $L _ { 1 }$ and $L _ { 2 }$ . Each alignment is a hypothesis about where the network $N _ { + }$ stores and uses the values of $S _ { 1 }$ and $W$ . For the example, we assume the alignment in Figure 1a.
+
+Interchange Interventions Finally, for a given alignment, we experimentally determine whether the neural representations at $L _ { 1 }$ and $L _ { 2 }$ have the same causal properties as $S _ { 1 }$ and $W$ . The basic experimental technique is an interchange intervention, in which a neural representation created during prediction on a “base” input is interchanged with the representation created for a “source” input [11]. We now show informally that this method can be used to prove that the causal model $C _ { + }$ is a constructive abstraction of the neural network $N _ { + }$ (Appendix $\mathbf { G }$ has formal details).
+
+We first intervene on the causal model. Consider two inputs a, $\mathbf { a } ^ { \prime } \in ( \mathbb { N } _ { 9 } ) ^ { 3 }$ where $\mathbb { N } _ { 9 }$ is the set of integers 0–9. Let $\mathbf { a } = ( x , y , z )$ be the base input and $\mathbf { a } ^ { \prime } = \left( x ^ { \prime } , y ^ { \prime } , z ^ { \prime } \right)$ be the source input. Define
+
+$$
+C _ { + } ^ { S _ { 1 }  \mathbf { a } ^ { \prime } } ( \mathbf { a } ) = x ^ { \prime } + y ^ { \prime } + z
+$$
+
+to be the output provided by $C _ { + }$ when $S _ { 1 }$ , the variable representing the intermediate sum, is intervened on and set to the value $x ^ { \prime } + y ^ { \prime }$ . Thus, for example, if the base input is $C _ { + } ( 1 , 2 , 3 ) = 6$ , and the source input is ${ \bf a } ^ { \prime } = ( 4 , 5 , 6 )$ , then $C _ { + } ^ { S _ { 1 }  \mathbf { a } ^ { \prime } } ( 1 , 2 , 3 ) = 4 + 5 + 3 = 1 2$ . This process is depicted in Figure 1c. Next, we intervene on the neural network $N _ { + }$ . Let $\mathbf { D }$ be an embedding space that provides unique representations for $\mathbb { N } _ { 9 }$ , and consider two inputs $\boldsymbol { D } = ( D _ { x } , D _ { y } , D _ { z } )$ and $D ^ { \prime } = \bar { ( } D _ { x ^ { \prime } } , D _ { y ^ { \prime } } , \bar { D _ { z ^ { \prime } } } )$ , where all $D _ { i }$ and $D _ { i ^ { \prime } }$ are drawn from $\mathbf { D }$ . In parallel with (1), define
+
+$$
+N _ { + } ^ { L _ { 1 }  D ^ { \prime } } ( D )
+$$
+
+to be the output provided by $N _ { + }$ processing the input $D$ when the representation at location $L _ { 1 }$ is replaced with the representation at location $L _ { 1 }$ created when $N _ { + }$ is processing the input $D ^ { \prime }$ . This process is depicted in Figure 1b.
+
+With these two definitions, we can define what it means to test the hypothesis that $N _ { + }$ computes $x + y$ at position $L _ { 1 }$ . Where $D _ { \mathbf { a } }$ is an embedding for a and $D _ { \mathbf { a ^ { \prime } } }$ is an embedding for $\mathbf { a } ^ { \prime }$ , we test:
+
+$$
+C _ { + } ^ { S _ { 1 }  \mathbf { a } ^ { \prime } } ( \mathbf { a } ) = N _ { + } ^ { L _ { 1 }  D _ { \mathbf { a } ^ { \prime } } } ( D _ { \mathbf { a } } )
+$$
+
+If this equality holds for all source and base inputs a and $\mathbf { a } ^ { \prime }$ , then we can conclude that, for every intervention on $S _ { 1 }$ , there is an equivalent intervention on $L _ { 1 }$ . If we can establish a corresponding claim for $W$ and $L _ { 2 }$ , then we have shown that $C _ { + }$ is a constructive abstraction of $N _ { + }$ , since the inputs’ relationships are established by our embedding and there are no other interventions on $C _ { + }$ to test.
+
+Analysis Suppose that all of our intervention experiments verify our hypothesis that $C _ { + }$ is a constructive abstraction of $N _ { + }$ with variables $S _ { 1 }$ and $W$ aligned to neural representations at $L _ { 1 }$ and $L _ { 2 }$ . This explains network behavior by resolving two crucial questions.
+
+First, we learn what information is encoded in the representations $L _ { 1 }$ and $L _ { 2 }$ . Neural representations encode the values of the high-level variables they are aligned with. The location $L _ { 1 }$ encodes the variable $S _ { 1 }$ and the location $L _ { 2 }$ encodes the variable $W$ . This is similar to what probing achieves. However, our method is crucially different from probing. In probing, information content is established through purely correlational properties, meaning a neural representation with no causal role in network behavior can be successfully probed, as we showed in Section 2. In causal abstraction analysis, information content is established through purely causal properties, ensuring that the neural representation is actually implicated in model behavior.
+
+Second, we learn what causal role $L _ { 1 }$ and $L _ { 2 }$ play in network behavior. Neural representations play a parallel causal role to their aligned high-level variables. At the location $L _ { 1 }$ , $D _ { x }$ and $D _ { y }$ are composed to form a neural representation with content $x + y$ that is then composed with $L _ { 2 }$ to create an output. The fact that $S _ { 1 }$ doesn’t depend on $z$ tells us that while $L _ { 1 }$ depends on $D _ { z }$ and representations at $L _ { 1 }$ may even correlate with $z$ , the information about $z$ is not causally represented at $L _ { 1 }$ . At the location $L _ { 2 }$ , the value of $z$ is simply repeated and then composed with $L _ { 1 }$ to create a final output.
+
+Our method assigns causally impactful information content, but also identifies the abstract causal structure along which representations are composed. It thus encompasses and improves on both correlational (probing) and attribution methods.
+
+# 4 The Natural Language Inference Task and Models
+
+Multiply Quantified NLI Dataset The Multiply Quantified NLI (MQNLI) dataset of Geiger et al. [10] contains templatically generated English-language NLI examples that involve very complex interactions between quantifiers, negation, and modifiers. We provide a few examples in Figure 2b; the empty-string symbol $\varepsilon$ ensures perfect alignments at the token level both between premises and hypotheses and across all examples.
+
+The MQNLI examples are labeled using an algorithmic implementation of the natural logic of MacCartney and Manning [18] over tree structures, and MQNLI has train/dev/test splits that vary in their difficulty. In the hardest setting, the train set is provably the minimal set of examples required to ensure that the dev and test sets can be perfectly solved by a simple symbolic model; in the easier settings, the train set redundantly encodes necessary information, which might allow a model to perform perfectly in assessment by memorization despite not having found a truly general solution. For a fuller review of the dataset, see Appendix A.
+
+MQNLI is a fitting benchmark given our goals for a few reasons. First, we can focus on the hardest splits that can be generated, which will stress-test our NLI architectures in a standard behavioral way. Second, the MQNLI labeling algorithm itself suggests an appropriate causal model of the data-generating process. Figure 2a summarizes this model in tree form, and it is presented in full detail in Geiger et al. [10]. This allows us to rigorously assess whether a neural network has learned to implement variants of this causal model. The complexity of the MQNLI examples creates many opportunities to do this in linguistically interesting ways.
+
+![](images/0cb3d60b262f4095536b9b0c886f7ffd2120462b64ededc66c7b1b8302c8ce7b.jpg)  
+(a) The causal structure of the high-level natural logic causal model $C _ { N a t L o g }$ that performs inference on MQNLI. The superscripts $P$ and $H$ stand for ‘premise’ and ‘hypothesis’ and the subscripts ‘Subj’ and ‘Obj’ stand for ‘Subject’ and ‘Object’. The node labels are used to explain the experimental results in Section 5
+
+$\varepsilon$ every $\varepsilon$ baker $\varepsilon \varepsilon \varepsilon$ eats $\varepsilon$ no $\varepsilon$ bread   
+contradiction   
+$\varepsilon$ no angry baker $\varepsilon \varepsilon \varepsilon$ eats $\varepsilon$ no $\varepsilon$ bread   
+$\varepsilon$ every silly professor $\varepsilon \varepsilon \varepsilon$ sells not every $\varepsilon$ book   
+neutral   
+$\varepsilon$ every silly professor $\varepsilon \varepsilon \varepsilon$ sells not every $\varepsilon$ chair   
+not every sad baker $\varepsilon \varepsilon$ fairly admits not every odd idea   
+entailment   
+$\varepsilon$ some $\varepsilon$ baker does not $\varepsilon$ admits $\varepsilon$ no $\varepsilon$ idea
+
+(b) MQNLI examples. The $\varepsilon$ token serves as padding (but still attended to by the model) and ensures a perfect alignment between both premises and hypotheses and across all examples. It is semantically an identity element.
+
+<table><tr><td>Model</td><td>Train</td><td>Dev</td><td>Test</td></tr><tr><td>CBoW</td><td>88.04</td><td>54.18</td><td>53.99</td></tr><tr><td>TreeNN</td><td>67.01</td><td>54.01</td><td>53.73</td></tr><tr><td>CompTreeNN</td><td>99.65</td><td>80.17</td><td>80.21</td></tr><tr><td>BiLSTM</td><td>99.42</td><td>46.41</td><td>46.32</td></tr><tr><td>BERT</td><td>99.99</td><td>88.25</td><td>88.50</td></tr></table>
+
+(c) MQNLI results. The first three models are from Geiger et al. 10, where the CompTreeNN is a taskspecific model not suitable for general NLI and functions as an idealized upperbound. Our results show that BERT-based models can surpass this without such alignments.
+
+Figure 2: The natural logic causal model (top), MQNLI examples (left) and MQNLI results (right).
+
+Models We evaluated two models on MQNLI: a randomly initialized multilayered Bidirectional LSTM (BiLSTM; [23]) and a BERT-based classifier model in which the English bert-base parameters [8] are fine-tuned on the MQNLI train set. Output predictions are computed using the final representation above the [CLS] token. Models are trained to predict the relation of every pair of aligned phrases in Figure 2a. Additional model and training details are given in Appendix B.
+
+Results Figure 2c summarizes the results of our BERT and BiLSTM models on the hardest fair generalization task Geiger et al. [10] creates with MQNLI. We find that our BiLSTM model is not able to learn this task, and that our BERT model is able to achieve high accuracy. The only models in Geiger et al. [10] able to achieve above $50 \%$ accuracy were task-specific tree-structured models with the structure of the tree in Figure 2a. Thus, our BERT-based model is the first general-purpose model able to achieve good performance on this hard generalization task. Without pretraining, the BERT-based model achieves ${ \approx } 4 9 . 1 \%$ , confirming that pretraining is essential, as expected.
+
+A natural hypothesis is that the BERT-based model achieves this high performance because it has in effect induced some approximation to the tree-like structure of the data-generating process in its own internal layers. With causal abstraction analysis, we are actually in a position to test this hypothesis.
+
+# 5 A Case Study in Structural Neural Network Analysis
+
+# 5.1 Causal Abstractions of Neural NLI models
+
+Formulating Our Hypotheses We proceed just as we did for the simple motivating example in Section 3, except that we are now seeking to assess the extent to which the natural logic algebra in Figure 2a is a causal abstraction of the trained neural models in the above section.
+
+The hallmark of Figure 2a is that it defines an alignment between premise and hypothesis at both lexical and phrasal levels. This pway. For a given non-leaf node $N$ mits us to run intein Figure 2a, let $C _ { N a t L o g } ^ { N }$ interventions in a nbe a submodel of $\boldsymbol { C } _ { N a t L o g }$ compositionalthat computes the relation between the aligned phrases under and uses them to compute the final output relation between premise and hypothesis. For example, let $C _ { N a t L o g } ^ { \mathrm { N P _ { O b j } } }$ be the submodel of $\boldsymbol { C } _ { N a t L o g }$ that computes the relation between the two aligned object noun phrases and then uses that relation in computing the final output relation between premise and hypothesis (see Figure 3 right). We would like to ask whether our trained neural models also compute this relation between object noun phrases and use it to make a final prediction. We can pose this same question for other nodes which correspond to a pair of aligned subphrases.
+
+![](images/091a108d0acc3c4c953e97f27659d8f183813696f3f905ddb228de930349f9aa.jpg)  
+Figure 3: A BERT-based NLI model (left) aligned with the natural logic causal model $C _ { N a t L o g } ^ { \mathrm { N P _ { O b j } } }$ (right), where the fourth vector representation above the $\mathrm { A d j } _ { \mathrm { O b j } } ^ { P }$ token in the network is aligned with $\mathrm { N P _ { O b j } }$ , the variable representing the relation between the object noun phrases. When analyzing a sample of 1000 examples, we found a subset of 383 where $C _ { N a t L o g } ^ { \mathrm { N P _ { O b j } } }$ is an abstraction of $N _ { N L I }$ under this alignment.
+
+nd the variable  causal model., we consider a $N$ . In principle, any location in the network could bry hypothesis in this space would be intractable. t of hidden representations based on the identity of he right oneus, for each. The BERT $N _ { N L I }$ $N$ $C _ { N a t L o g } ^ { N }$ ${ C _ { N a t L o g } ^ { N } } ^ { N }$ $N$ model we use has 12 Transformer layers [30], meaning that there are 12 hidden representations for each input token. Each alignment search considers aligning the intermediate high-level variable with dozens of possible locations in the grid of BERT representations. Specifically, the following locations were considered for each $N$ :
+
+$\mathrm { Q _ { S u b j } }$ , $\mathrm { \Delta \ A j _ { \mathrm { S u b j } } }$ , $\mathrm { N _ { S u b j } }$ , Neg, Adv, V, $\mathrm { Q _ { O b j } }$ , $\mathrm { \Delta A d j _ { O b j } }$ , $\mathrm { N _ { O b j } }$ : hidden representations above the two descendant leaf tokens. • $\mathrm { N P _ { S u b j } }$ , VP, and $\mathrm { N P _ { O b j } }$ : same but above the four descendant leaf tokens.
+
+$\mathrm { Q P _ { O b j } }$ idden representations above . $\mathrm { Q } _ { \mathrm { 0 b j } } ^ { P }$ $\mathrm { Q } _ { \mathrm { O b j } } ^ { H }$   
+. NegP: same but above $\mathrm { N e g } ^ { P }$ and $\mathrm { N e g } ^ { H }$ .   
+All nodes (for BERT): same but above   
+[CLS] and [SEP].
+
+For each alignment considered, we performed a full causal abstraction analysis. We report the results from the best alignments in Table 1, and we summarize the results from all alignments in Appendix D.
+
+Interchange Interventions We first focus on our high-level causal models. Consider a non-leaf node $N$ from Figure 2a and two input token sequences $e$ and $e ^ { \prime }$ from MQNLI. Define
+
+$$
+C _ { N a t L o g } ^ { N  e ^ { \prime } } ( e )
+$$
+
+to be the output provided by the causal mode $C _ { N a t L o g } ^ { N }$ when processing input $e$ where the relation $N$ is changed to the relation between those subphrases   
+in $e ^ { \prime }$ . For example, simplifying for the sake of exposition, suppose $e$ is (some happy baker, no $\epsilon$   
+baker), which has output label contradiction, and suppose $e ^ { \prime }$ is (every happy person, some happy   
+aker), which has output label , the noun phrase relation is e ntailment.ailment; in e wish to intervene on the noun, it is reverse entailment. Thus, $N = \mathrm { N P }$ . In the   
+$e$ $e ^ { \prime }$ $C _ { N a t L o g } ^ { \mathrm { N P }  e ^ { \prime } } ( e )$ $e$ to entailment while holding everything else about $e$ constant. This   
+results in the output label for the example (some happy person, no $\epsilon$ baker), which is neutral.
+
+Next, we consider interventions in a neural model $N _ { N L I }$ . Define
+
+$$
+N _ { N L I } ^ { L  e ^ { \prime } } ( e )
+$$
+
+Table 1: Largest subsets of examples on which specific models $C _ { N a t L o g } ^ { N }$ are abstractions of an LSTM and BERT model trained on MQNLI. We record the size of such subsets as a percentage of the total 1000 examples. On this subset, we know that the neural models compute a representation of the relation between the aligned subphrases under $N$ and use this information to make a final prediction.   
+
+<table><tr><td>Nodesadded</td><td>BERT</td></tr><tr><td>Adsubj</td><td>30.5</td></tr><tr><td>Nsub</td><td>37.2</td></tr><tr><td>Negp</td><td>14.9</td></tr><tr><td>AdvP</td><td>26.9</td></tr><tr><td>VP</td><td>35.6</td></tr><tr><td>Q</td><td>16.2</td></tr><tr><td>Adjsubj H</td><td>13.4</td></tr><tr><td>Nsub</td><td>12.0</td></tr><tr><td>NegH</td><td>34.4</td></tr><tr><td>AdvH</td><td>16.2</td></tr><tr><td>VH</td><td>13.4</td></tr><tr><td>Qbi</td><td>12.0</td></tr></table>
+
+<table><tr><td>Causal Model</td><td>LSTM</td><td>BERT</td></tr><tr><td>Qsubj</td><td>0.7</td><td>13.1</td></tr><tr><td>Qobj</td><td>0.9</td><td>7.3</td></tr><tr><td>Neg</td><td>0.7</td><td>21.4</td></tr><tr><td>Adjsubj</td><td>2.5</td><td>6.7</td></tr><tr><td>Nsubj</td><td>1.2</td><td>5.5</td></tr><tr><td>Adjobj</td><td>0.9</td><td>14.1</td></tr><tr><td>Nobj</td><td>0.7</td><td>8.8</td></tr><tr><td>V</td><td>0.4</td><td>11.4</td></tr><tr><td>Adv</td><td>1.4</td><td>7.9</td></tr><tr><td>NPsubj</td><td>1.0</td><td>6.7</td></tr><tr><td>NPobj</td><td>0.7</td><td>38.3</td></tr><tr><td>VP</td><td>0.4</td><td>11.4</td></tr><tr><td>NegP</td><td>0.9</td><td>11.8</td></tr></table>
+
+<table><tr><td>Nodes removed</td><td>BERT</td></tr><tr><td></td><td>31.9</td></tr><tr><td>A N</td><td>15.7</td></tr><tr><td></td><td>33.8</td></tr><tr><td>A</td><td>15.8 31.9</td></tr><tr><td></td><td>14.1</td></tr><tr><td>A</td><td>32.2</td></tr><tr><td></td><td>31.6</td></tr><tr><td></td><td>8.8</td></tr><tr><td></td><td></td></tr><tr><td>,Abj</td><td>32.1</td></tr></table>
+
+(a) Main results (clique sizes) for nonleaf nodes of the tree in Figure 2a. The hypothesis we have most evidence for is that the BERT model computes a representation of the $\mathrm { N P _ { O b j } }$ node with the alignment shown in Figure 3. Remarkably, with 1000 examples sampled, we found a subset of 383 examples where $C _ { N a t L o g } ^ { \mathrm { N P _ { O b j } } }$ is an abstraction of BERT.
+
+(b) Detailed results (clique sizes) for Altein a “neighborhood” around the model $C _ { N a t L o g } ^ { \mathrm { N P _ { O b j } } }$ causal models, which has a single intermediate variable composed of four lexical items (See Figure 3). At left, we have alternative causal models where one or two of those lexical items are removed from the composition. At right, we have alternatives obtained by adding one lexical item to the composition. We observe that no alternative hypothesis about causal structure considered has more evidence.
+
+to be the output provided by $N _ { N L I }$ processing the input $e$ when the representation at location $L$ is replaced with the representation at location $L$ created when $N _ { N L I }$ is processing $e ^ { \prime }$ . This is exactly the process depicted in Figure 1, except now the networks are the complex trained networks of Section 4.
+
+Our hypothesis linking Figure 2a with a model $N _ { N L I }$ takes the same form as (3). The causal model $C _ { N a t L o g } ^ { N }$ is a constructive abstraction of $N _ { N L I }$ when, for some representation location $L$ , it is the case that, for all MQNLI examples $e$ and $e ^ { \prime }$ , we have
+
+$$
+C _ { N a t L o g } ^ { N  e ^ { \prime } } ( e ) = N _ { N L I } ^ { L  e ^ { \prime } } ( e )
+$$
+
+This asserts a correspondence between interventions on the representations at $L$ in network $N _ { N L I }$ and interventions on the variable $N$ in the causal model $C _ { N a t L o g } ^ { N }$ . If it holds, then $N _ { N L I }$ computes the relation between the aligned phrases under the node $N$ and uses this information to compute the relation between the premise and hypothesis.
+
+We call a pair of examples $( e , e ^ { \prime } )$ successful if it satisfies equation (6), i.e., interventions in both the target causal model and neural model produce equal results. In addition, to isolate the causal impact of our interventions, we specifically focus on pairs $( e , e ^ { \prime } )$ for which performing the intervention produces a different output value than without the intervention. We call a pair $( e , e ^ { \prime } )$ impactful if:
+
+$$
+C _ { N a t L o g } ^ { N  e ^ { \prime } } ( e ) \neq C _ { N a t L o g } ^ { N } ( e )
+$$
+
+Quantifying Partial Success Equation (6) universally quantifies over all examples. We do not expect this kind of perfect correspondence to emerge in practice for real problems: neural network training is often approximate and variable in nature, and even our best model does not achieve perfect performance. However, we can still ask how to find the largest subset of MQNLI on which $C _ { N a t L o g } ^ { N }$ (6) holds for a given model. To do this, we seekis an abstraction of our neural models, for each non-leaf node N in CNatLog.
+
+More specifically, considering each example in MQNLI as a vertex in a graph, we add an undirected edge between two examples $e _ { i }$ and $e _ { j }$ if and only if both the ordered pairs $( e _ { i } , e _ { j } )$ and $( e _ { j } , e _ { i } )$ satisfy (6). In other words, if and only if all ex $C _ { N a t L o g } ^ { N }$ isin n abstraction of a neural model on a subset of examples form a clique. $S$ of MQNLI $S$
+
+The number of interventions we need to run scales quadratically with the number of inputs we consider, so we sample 1000 MQNLI examples, producing a total of $1 0 0 0 ^ { 2 } = 1 \mathbf { M }$ ordered pairs. We only consider examples for which the neural network outputs a correct label. For each node $N$ and each of its corresponding neural network locations $L$ , we perform interventions on all of these pairs.
+
+We choose to measure the largest clique with at least one impactful edge, because (1) the causal abstraction relation holds with full force on that clique, but other measures such as the total number of connections lack this theoretical grounding, and (2) if a clique has at least one impactful edge, that guarantees the high-level variable is being used.
+
+Results and Analysis For each target causal model node $N$ and neural network representation location $L$ , we construct a graph as described above with 1000 examples as vertices and add an edge between two examples $e _ { i }$ and $e _ { j }$ if and only if both $( e _ { i } , e _ { j } )$ and $( e _ { j } , e _ { i } )$ are successful. We then find the largest clique in this graph with at least one impactful edge and record its size.
+
+Table 1a shows, for each causal model node $N$ , the maximum size of cliques found among all neural locations. With this stricter impactful criterion (as opposed to simply using intervention success), our results show that, for almost all nodes abstraction of BERT on a significant numb $N$ , our target causal model of examples in our datase $C _ { N a t L o g } ^ { \widetilde { N } }$ is indeed a causale subsets are much smaller for the BiLSTM model.
+
+We also investigated alternative high-level causal structures that are not variants of $\boldsymbol { C } _ { N a t L o g }$ from Figure 2a. Specifically, we consider alternative models in a “neighborhood” around the model CNPObjNatLog that can be obtained by adding one leaf, or by removing one or two leaves to the composition. These results are in Table 1b. Remarkably, all of these alternative models result in smaller clique sizes, significantly so for many of them. This further supports the significance of our results.
+
+This analysis is simtwo crucial differenis an abstraction of o the analysis of our hirst, for each variable , whereas in the addit $N$ othetical addition example in Section 3, ex, we are hypothesizing that the causal model  example there was only one model. To inv ${ \dot { C } } _ { N a t L o g } ^ { N }$ $N _ { N L I }$   
+this difference, we take $N = \mathrm { N P _ { O b j } }$ as a paradigm case, as it is the model with the strongest results. (The results for other nodes are in Appendix D.) Second, we only achieved partial experimental success, whereas in the addition example we assumed complete success. Crucially, this means that the following analysis will be valid only on subsets of the input space on which the abstraction relation holds between NNLI and CNPObjNatLog.
+
+We visualize the results of our intervention experiments for the node $\mathrm { N P _ { O b j } }$ in Figure 4. The alignment with the largest subset of inputs aligns the NPObj variable in CNPObjNatLog with the neural representation on the fourth layer of BERT above the $\mathrm { A d j } _ { \mathrm { O b j } } ^ { P }$ token (see Figure 3). Because neural representations encode the value of their aligned variables and play a parallel causal role to their high-level variables, we know that, on this subset of input examples, at the fourth neural representation above the $\mathrm { A d j } _ { \mathrm { O b j } } ^ { P }$ token, the four input embeddings for the object nouns and adjectives in the premise and hypothesis are composed to form a neural representation with information content of the relation between the object noun phrases in the premise and hypothesis. Then this representation is composed with the other input-embeddings to create an output representing the relation between the premise and hypothesis.
+
+# 5.2 Comparison with Other Structural Analysis Methods
+
+Probes We probed neural representation locations for the relation between aligned subexpressions on a subset of 12,800 randomly selected MQNLI examples. For a pair of aligned subexpressions below a node $N$ in Figure 2a, we probe the columns above the same set of restricted class of tokens as described in Section 5.1.
+
+To evaluate these probes, we report accuracy as well as selectivity as defined by Hewitt and Liang [12]: probe accuracy minus control accuracy, where control accuracy is the train set accuracy of a probe with the same architecture but trained on a control task to factor out probe success that can be attributed to the probe model itself. Our control task is to learn a random mapping from node types to semantic relations; see Appendix C for full details on how this task was constructed.
+
+![](images/c313e3750ef3c4775c2304d720db7884b4f4d011c838d444e19be0b735a6a5ed.jpg)  
+Figure 4: Interchange intervention and probing results for the $\mathrm { N P _ { O b j } }$ position. Vertical axes denote layers of BERT and horizontal axes denote the token position of hidden representations. The intervention success rates reported here are calculated based on intervention experiments with a change in the output label. Clique sizes are reported as $\%$ of 1000 examples.
+
+Figure 4 summarizes our probing results for $N = \mathrm { N P _ { O b j } }$ , along with corresponding interchange intervention results for comparison. Probes tell us that information about the relation between the aligned noun phrases is encoded in nearly all of the locations we considered, and using the selectivity metric does not result in any qualitative change. In contrast, our intervention heatmaps indicate only a small number of locations store this information in a causally relevant way. Clearly, our intervention experiments are far more discriminating than probes. Appendix D provides examples involving other variables along with the intervention experiments, where the general trend of interchange interventions being more discriminating holds.
+
+Integrated Gradients Attribution methods that estimate feature importance can measure causal properties of neural representations, but a single feature importance method is an impoverished characterization of a representation’s role in network behavior. Whereas our interchange interventions gave us high-level information about how a neural representation is composed and what it is composed into, attribution methods simply tell us “how much” a representation contributes to the network output on a give input. Moreover, intervention interchanges provide a rich, high-level characterization of causal structure on a space of inputs.
+
+We use integrated gradients on our models to verify the intuitive hypothesis that if a premise and hypothesis differ by a single token, then the neural representations above that token should be more causally responsible for the network output than other representations. For example, given premise ‘Every sleepy cat meows’ and hypothesis ‘Some hungry cat meows’, the attributive modifier position is different and the rest are matched. The neural representations above the adjective tokens sleepy and hungry should be more important for the network output than others, because if those adjectives were the same, the example label would change from neutral to entailment. We summarize the results of our integrated gradient experiments in Appendix E, where we confirm our intuitive hypothesis.
+
+# 6 Conclusion
+
+We have introduced a methodology for deriving interpretable causal explanations of neural network behaviors, grounded in a formal theory of causal abstraction. The methodology involves first formulating a hypothesis in the form of a high-level, interpretable causal model, then searching for an alignment between the neural network and the causal model, and finally verifying experimentally that the neural representations encode the same causal properties and information content as the corresponding components of the high-level causal model. As a case study demonstrating the feasibility of the approach, we analyzed neural models trained on the semantically formidable MQNLI dataset. Guided by the intuition that success on this challenging task may call for a way of recapitulating the causal structure of the natural logic model that generates the MQNLI data, we were able to verify the hypothesis that a state-of-the-art BERT-based model partially realizes this structure, whereas baseline models that do not perform as well fail to do so. This suggestive case study demonstrates that our theoretically grounded methodology can work in practice.
+
+# Acknowledgments and Disclosure of Funding
+
+Our thanks to Amir Feder, Noah Goodman, Elisa Kreiss, Josh Rozner, Zhengxuan Wu, and our anonymous reviewers. This research is supported in part by grants from Facebook and Google.
+
+References   
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+        "text": "Causal Abstractions of Neural Networks ",
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+        "text": "Atticus Geiger∗, Hanson Lu∗, Thomas Icard, and Christopher Potts Stanford Stanford, CA 94305-2150 {atticusg, hansonlu, icard, cgpotts}@stanford.edu ",
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+        "text": "Structural analysis methods (e.g., probing and feature attribution) are increasingly important tools for neural network analysis. We propose a new structural analysis method grounded in a formal theory of causal abstraction that provides rich characterizations of model-internal representations and their roles in input/output behavior. In this method, neural representations are aligned with variables in interpretable causal models, and then interchange interventions are used to experimentally verify that the neural representations have the causal properties of their aligned variables. We apply this method in a case study to analyze neural models trained on Multiply Quantified Natural Language Inference (MQNLI) corpus, a highly complex NLI dataset that was constructed with a tree-structured natural logic causal model. We discover that a BERT-based model with state-of-the-art performance successfully realizes parts of the natural logic model’s causal structure, whereas a simpler baseline model fails to show any such structure, demonstrating that BERT representations encode the compositional structure of MQNLI. ",
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+        "text": "1 Introduction ",
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+        "text": "Explainability and interpretability have long been central issues for neural networks, and they have taken on renewed importance as such models are now ubiquitous in research and technology. Recent structural evaluation methods seek to reveal the internal structure of these “black box” models. Structural methods include probes, attributions (feature importance methods), and interventions (manipulations of model-internal states). These methods can complement standard behavioral techniques (e.g., performance on gold evaluation sets), and they can yield insights into how and why models make the predictions they do. However, these tools have their limitations, and it has often been assumed that more ambitious and systematic causal analysis of such models is beyond reach. ",
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+        "text": "Although there is a sense in which neural networks are “black boxes”, they have the virtue of being completely closed and controlled systems. This means that standard empirical challenges of causal inference due to lack of observability simply do not arise. The challenge is rather to identify high-level causal regularities that abstract away from irrelevant (but arbitrarily observable and manipulable) low-level details. Our contribution in this paper is to show that this challenge can be met. Drawing on recent innovations in the formal theory of causal abstraction [1, 2, 5, 22], we offer a methodology for meaningful causal explanations of neural network behavior. ",
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+        "text": "Our methodology causal abstraction analysis2 consists of three stages. (1) Formulate a hypothesis by defining a causal model that might explain network behavior. Candidate causal models can be naturally adapted from theoretical and empirical modeling work in linguistics and cognitive sciences. (2) Search for an alignment between neural representations in the network and variables in the high-level causal model. (3) Verify experimentally that the neural representations have the same causal properties as their aligned high-level variables using the interchange intervention method of Geiger et al. [11]. ",
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+        "type": "text",
+        "text": "As a case study, we apply this methodology to LSTM-based and BERT-based natural language inference (NLI) models trained on the logically complex Multiply Quantified NLI (MQNLI) dataset of Geiger et al. [10]. This challenging dataset was constructed with a tree-structured natural logic causal model [17, 29, 14]. Our BERT-based model has the structure of a standard NLI classifier, and yet it is able to perform well on MQNLI $( 8 8 \\% )$ , a result Geiger et al. achieved only with highly customized task-specific models. By contrast, our LSTM-based model is much less successful $(46 \\% )$ . ",
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+        "type": "text",
+        "text": "The obvious scientific question in this case study is what drives the success of the BERT-based model on this challenging task. To answer this we employ our methodology. (1) We formulate hypotheses by defining simplified variants of the natural logic causal model. (2) We search over potential alignments between neural representations in BERT and variables in our high-level causal models. (3) We perform interchange interventions on the BERT model for each alignment. We find that our BERT model partially realizes the causal structure of the natural logic causal model; crucially, the LSTM model does not. High-level causal explanation for system behavior is often considered a gold standard for interpretability, one that may be thought quixotic for complex neural models [16]. The point of our case study is to show that this high standard can be achieved. ",
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+        "text": "We conclude by comparing our methodology to probing and the attribution method of integrated gradients [27]. We argue probing is unable to provide a causal characterization of models. We show formally that attribution methods do measure causal properties, and in that way they are similar to the tool of interchange interventions. However, our methodology of causal abstraction analysis provides a framework for systematically measuring and aggregating such causal properties in order to evaluate a precise hypothesis about abstract causal structure. ",
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+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
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+        "text": "Probes Probes are generally supervised models trained on the internal representations of networks with the goal of determining what those internal representations encode [7, 13, 20, 28]. Probes are fundamentally unable to directly measure causal properties of neural representations, and Ravichander et al. [21], Elazar et al. [9], and Geiger et al. [11] have argued that probes are limited in their ability to provide even indirect evidence of causal properties. ",
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+        "text": "We now present an analytic example in which probing identifies seemingly crucial information in representations that have no causal impact on behavior. We assume the structure of the simple addition network $N _ { + }$ in Figure 1. For our embedding, we simply map every integer $i$ in $\\mathbb { N } _ { 9 }$ to the 1-dimensional vector [i]. The weight matrices are ",
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+        "text": "$$\nW _ { 1 } = { \\left( \\begin{array} { l } { 1 } \\\\ { 1 } \\\\ { 0 } \\end{array} \\right) } \\quad W _ { 2 } = { \\left( \\begin{array} { l } { 1 } \\\\ { 1 } \\\\ { 1 } \\end{array} \\right) } \\quad W _ { 3 } = { \\left( \\begin{array} { l } { 0 } \\\\ { 0 } \\\\ { 1 } \\end{array} \\right) } \\quad \\mathbf { w } = { \\left( \\begin{array} { l } { 0 } \\\\ { 1 } \\\\ { 0 } \\end{array} \\right) }\n$$",
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+        "type": "text",
+        "text": "The output for an input sequence $\\mathbf { x } = ( i , j , k )$ is given by $\\left( \\mathbf { x } W _ { 1 } ; \\mathbf { x } W _ { 2 } ; \\mathbf { x } W _ { 3 } \\right) \\mathbf { w } .$ . ",
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+        "type": "text",
+        "text": "In this network, $\\mathbf { x } W _ { 1 }$ perfectly encodes $i + j$ , and $\\mathbf { x } W _ { 3 }$ perfectly encodes $k$ . Thus, the identity model probe will be perfect in probing those representations for this information. However, neither representation plays a causal role in the network behavior; only $\\mathbf { x } W _ { 2 }$ contributes to the output. ",
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+        "text": "Attribution Methods Attribution methods aim to quantify the degree to which a network representation contributes to the output prediction of the model, for a specific example or set of examples [3, 24, 26, 27, 32]. In contrast to probing, the well known integrated gradients method (IG) can be given an unambiguous causal interpretation. Following [27] we define the vector $I G ( \\mathbf { x } )$ , for an input $\\mathbf { x }$ relative to a baseline b, to have ith component $I G _ { i } ( \\mathbf { x } )$ given by the expression on the left: ",
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+        "img_path": "images/39732885457ef5cb11d4d2c4eafaf50b2afb79af77929088cc288e40fc0ff5ab.jpg",
+        "text": "$$\n\\left( x _ { i } - b _ { i } \\right) \\cdot \\int _ { \\alpha = 0 } ^ { 1 } \\frac { \\partial F ( \\alpha \\mathbf { x } + ( 1 - \\alpha ) \\mathbf { b } ) } { \\partial x _ { i } } d \\alpha = \\left( x _ { i } - b _ { i } \\right) \\cdot \\int _ { \\alpha = 0 } ^ { 1 } \\operatorname* { l i m } _ { \\epsilon \\to 0 } \\frac { F ( \\mathbf { x } ^ { \\alpha , \\epsilon } ) - F ( \\mathbf { x } ^ { \\alpha } ) } { \\epsilon } d \\alpha\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Abbreviating the weighted average $\\alpha { \\mathbf { x } } + ( 1 - \\alpha ) { \\mathbf { b } }$ by $\\mathbf { x } ^ { \\alpha }$ , letting $\\mathbf { x } ^ { \\alpha , \\epsilon }$ be the vector that differs from $\\mathbf { x } ^ { \\alpha }$ in that the ith coordinate is increased by $\\epsilon$ , and then expanding the definition of partial derivative, this can be written in the form given on the right. The difference $F ( \\mathbf { x } ^ { \\alpha , \\epsilon } ) - F ( \\mathbf { x } ^ { \\alpha } )$ is known in the (a) The causal model $C _ { + }$ (right) that first computes $S _ { 1 } = X + Y$ and $W = Z$ , before computing the final output $S _ { 2 } = W + S _ { 1 }$ aligned with the neural network $N _ { + }$ (left) with $L _ { 1 }$ highlighted as the hypothesized location encoding $S _ { 1 } = X + Y$ and $L _ { 2 }$ as the location encoding $W = Z$ . ",
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+        "text": "(b) Low-level neural network interchange intervention. The network processes two different input sequences. The neural representation created at $L _ { 1 }$ for input sequence $( 1 , 2 , { \\bar { 3 } } )$ is replaced by the corresponding representation created for input sequence $( 4 , 5 , 6 )$ . ",
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+        "type": "text",
+        "text": "(c) High-level symbolic computation interchange intervention. The computation processes two different input sequences. The sum at $S _ { 1 }$ for input sequence $( 1 , 2 , 3 )$ is replaced by the corresponding representation created for input sequence $( 4 , 5 , 6 )$ . ",
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+        "type": "text",
+        "text": "Figure 1: Our motivating example where we hypothesis that a symbolic computation $C _ { + }$ is a causal abstraction of a neural network $N _ { + }$ under a particular alignment (top). We can experimentally confirm this hypothesis by conducting an interchange intervention on both the network and the computation with every pair of inputs and evaluating whether the intervened network and intervened computation have the same counterfactual output behavior. We schematically depict an interchange intervention on the network $N _ { + }$ (bottom left) and the computation $C _ { + }$ (bottom right) with the base input $( 1 , 2 , 3 )$ and the source input $( 4 , 5 , 6 )$ . Observe that the output of the intervened neural network matches the output of the intervened symbolic computation, so we have success for this pair of inputs. ",
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+    {
+        "type": "text",
+        "text": "causal literature as the (individual) causal effect on the output (e.g., [15]) of increasing neuron $i$ by $\\epsilon$ relative to the fixed input $\\mathbf { x } ^ { \\alpha }$ . So, essentially, $I G _ { i } ( \\mathbf { x } )$ is measuring the average “limiting” causal effect of increasing neuron $i$ along the straight line from the baseline vector to the input vector $\\mathbf { x }$ , weighted by the difference at $i$ between input and baseline. More recently, Chattopadhyay et al. [6] develop an attribution method that explicitly treats neural models as structured causal models and directly computes the individual causal effect of a feature to determine its attribution. ",
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+        "type": "text",
+        "text": "Attribution methods can measure causal properties, and, in that way, they are similar to the tool of interchange interventions. However, our methodology of causal abstraction analysis provides a framework for systematically measuring and aggregating such causal properties in order to evaluate a precise hypothesis about abstract causal structure. ",
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+        "text": "Causal Abstraction Our goal is to evaluate whether the internal structure of a neural network realizes an abstract causal process. To concretize this, we turn to formal, broadly interventionist theories of causality [25, 19], in which causal processes are characterized by effects of interventions, and theories of abstraction [1, 2, 5, 22] where relationships between two causal processes are determined by the presence of systematic correspondences between the effects of interventions. ",
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+        "type": "text",
+        "text": "The notion of abstraction that we employ here is a relatively simple one called constructive abstraction [1]. Informally, a high-level model is a constructive abstraction of a low-level model if there is a way to partition the variables in the low-level model where each high-level variable can be assigned to a low-level partition cell, such that there is a systematic correspondence between interventions on the low-level partition cells and interventions on the high-level variables. ",
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+        "type": "text",
+        "text": "There are two properties of constructive abstraction that make it ideal for neural network analysis. First, the information content of partition cells of low-level variables can be determined by the high-level variables that they correspond to. For neural networks, the partition cells of low-level variables are sets of neurons, and our method supports reasoning at the level of vector representations (sets of neurons). Second, the causal dependencies between partitions of low-level variables are not necessarily preserved as causal dependencies between the high-level variables corresponding to these partitions. For example, the low-level model might be a fully connected neural network, whereas the high-level model might have much sparser connections. For neural network analysis, this means we can find causal abstractions that have far simpler causal structures than the underlying neural networks. We provide an example in the next section. ",
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+        "type": "text",
+        "text": "3 Causal Abstraction Analysis of Neural Networks ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We now describe our methodology in more detail, illustrating the relevant concepts with an example of a neural network performing basic arithmetic. Specifically, suppose that we have a neural network $N _ { + }$ that takes in three vector representations $D _ { x } , D _ { y } , D _ { z }$ representing the integers $x , y .$ , and $z$ , and outputs the sum of the three inputs: $N _ { + } ( D _ { x } , D _ { y } , D _ { z } ) = x + y + z$ . We seek an informative causal explanation of this network’s behavior. ",
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+        "text": "Formulating a Hypothesis A human performing this task might follow an algorithm in which they add together the first two numbers and then add that sum to the third number. We can hypothesize that the behavior of $N _ { + }$ is explained by this symbolic computation. Specifically, the network combines $D _ { x }$ and $D _ { y }$ to create an internal representation at some location $L _ { 1 }$ encoding $x + y$ ; it encodes $z$ at some location $L _ { 2 }$ ; and $L _ { 1 }$ and $L _ { 2 }$ are composed to encode $a + z$ at the location of the output representation. This hypothesis is given schematically in Figure 1a. ",
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+    {
+        "type": "text",
+        "text": "Following our methodology, we first define the causal model $C _ { + }$ in Figure 1a. Our informal hypothesis that a neural network’s behavior is explained by a simple algorithm can then be restated more formally: $C _ { + }$ is a constructive abstraction of the neural network $N _ { + }$ . ",
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+        "type": "text",
+        "text": "Alignment Search Now that we have hypothesized that the causal model $C _ { + }$ is a causal abstraction of the network $N _ { + }$ , the next step is to align the neural representations in $N _ { + }$ with the variables in $C _ { + }$ . The input embeddings $D _ { x }$ , $D _ { y }$ , and $D _ { z }$ must be aligned with the input variables $X , Y$ , and $Z$ and the output neuron $O$ must be aligned with the output variable $S _ { 2 }$ . That leaves the intermediate variables $S _ { 1 }$ and $W$ to be aligned with neural representations at some undetermined locations $L _ { 1 }$ and $L _ { 2 }$ . If this were an actual experiment (see below), we would perform an alignment search to consider many possible values for $L _ { 1 }$ and $L _ { 2 }$ . Each alignment is a hypothesis about where the network $N _ { + }$ stores and uses the values of $S _ { 1 }$ and $W$ . For the example, we assume the alignment in Figure 1a. ",
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+        "type": "text",
+        "text": "Interchange Interventions Finally, for a given alignment, we experimentally determine whether the neural representations at $L _ { 1 }$ and $L _ { 2 }$ have the same causal properties as $S _ { 1 }$ and $W$ . The basic experimental technique is an interchange intervention, in which a neural representation created during prediction on a “base” input is interchanged with the representation created for a “source” input [11]. We now show informally that this method can be used to prove that the causal model $C _ { + }$ is a constructive abstraction of the neural network $N _ { + }$ (Appendix $\\mathbf { G }$ has formal details). ",
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+        "type": "text",
+        "text": "We first intervene on the causal model. Consider two inputs a, $\\mathbf { a } ^ { \\prime } \\in ( \\mathbb { N } _ { 9 } ) ^ { 3 }$ where $\\mathbb { N } _ { 9 }$ is the set of integers 0–9. Let $\\mathbf { a } = ( x , y , z )$ be the base input and $\\mathbf { a } ^ { \\prime } = \\left( x ^ { \\prime } , y ^ { \\prime } , z ^ { \\prime } \\right)$ be the source input. Define ",
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+        "text": "$$\nC _ { + } ^ { S _ { 1 }  \\mathbf { a } ^ { \\prime } } ( \\mathbf { a } ) = x ^ { \\prime } + y ^ { \\prime } + z\n$$",
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+    {
+        "type": "text",
+        "text": "to be the output provided by $C _ { + }$ when $S _ { 1 }$ , the variable representing the intermediate sum, is intervened on and set to the value $x ^ { \\prime } + y ^ { \\prime }$ . Thus, for example, if the base input is $C _ { + } ( 1 , 2 , 3 ) = 6$ , and the source input is ${ \\bf a } ^ { \\prime } = ( 4 , 5 , 6 )$ , then $C _ { + } ^ { S _ { 1 }  \\mathbf { a } ^ { \\prime } } ( 1 , 2 , 3 ) = 4 + 5 + 3 = 1 2$ . This process is depicted in Figure 1c. Next, we intervene on the neural network $N _ { + }$ . Let $\\mathbf { D }$ be an embedding space that provides unique representations for $\\mathbb { N } _ { 9 }$ , and consider two inputs $\\boldsymbol { D } = ( D _ { x } , D _ { y } , D _ { z } )$ and $D ^ { \\prime } = \\bar { ( } D _ { x ^ { \\prime } } , D _ { y ^ { \\prime } } , \\bar { D _ { z ^ { \\prime } } } )$ , where all $D _ { i }$ and $D _ { i ^ { \\prime } }$ are drawn from $\\mathbf { D }$ . In parallel with (1), define ",
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+        "text": "$$\nN _ { + } ^ { L _ { 1 }  D ^ { \\prime } } ( D )\n$$",
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+        "type": "text",
+        "text": "to be the output provided by $N _ { + }$ processing the input $D$ when the representation at location $L _ { 1 }$ is replaced with the representation at location $L _ { 1 }$ created when $N _ { + }$ is processing the input $D ^ { \\prime }$ . This process is depicted in Figure 1b. ",
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+    {
+        "type": "text",
+        "text": "With these two definitions, we can define what it means to test the hypothesis that $N _ { + }$ computes $x + y$ at position $L _ { 1 }$ . Where $D _ { \\mathbf { a } }$ is an embedding for a and $D _ { \\mathbf { a ^ { \\prime } } }$ is an embedding for $\\mathbf { a } ^ { \\prime }$ , we test: ",
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+        "text": "$$\nC _ { + } ^ { S _ { 1 }  \\mathbf { a } ^ { \\prime } } ( \\mathbf { a } ) = N _ { + } ^ { L _ { 1 }  D _ { \\mathbf { a } ^ { \\prime } } } ( D _ { \\mathbf { a } } )\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "If this equality holds for all source and base inputs a and $\\mathbf { a } ^ { \\prime }$ , then we can conclude that, for every intervention on $S _ { 1 }$ , there is an equivalent intervention on $L _ { 1 }$ . If we can establish a corresponding claim for $W$ and $L _ { 2 }$ , then we have shown that $C _ { + }$ is a constructive abstraction of $N _ { + }$ , since the inputs’ relationships are established by our embedding and there are no other interventions on $C _ { + }$ to test. ",
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+        "type": "text",
+        "text": "Analysis Suppose that all of our intervention experiments verify our hypothesis that $C _ { + }$ is a constructive abstraction of $N _ { + }$ with variables $S _ { 1 }$ and $W$ aligned to neural representations at $L _ { 1 }$ and $L _ { 2 }$ . This explains network behavior by resolving two crucial questions. ",
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+        "type": "text",
+        "text": "First, we learn what information is encoded in the representations $L _ { 1 }$ and $L _ { 2 }$ . Neural representations encode the values of the high-level variables they are aligned with. The location $L _ { 1 }$ encodes the variable $S _ { 1 }$ and the location $L _ { 2 }$ encodes the variable $W$ . This is similar to what probing achieves. However, our method is crucially different from probing. In probing, information content is established through purely correlational properties, meaning a neural representation with no causal role in network behavior can be successfully probed, as we showed in Section 2. In causal abstraction analysis, information content is established through purely causal properties, ensuring that the neural representation is actually implicated in model behavior. ",
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+        "type": "text",
+        "text": "Second, we learn what causal role $L _ { 1 }$ and $L _ { 2 }$ play in network behavior. Neural representations play a parallel causal role to their aligned high-level variables. At the location $L _ { 1 }$ , $D _ { x }$ and $D _ { y }$ are composed to form a neural representation with content $x + y$ that is then composed with $L _ { 2 }$ to create an output. The fact that $S _ { 1 }$ doesn’t depend on $z$ tells us that while $L _ { 1 }$ depends on $D _ { z }$ and representations at $L _ { 1 }$ may even correlate with $z$ , the information about $z$ is not causally represented at $L _ { 1 }$ . At the location $L _ { 2 }$ , the value of $z$ is simply repeated and then composed with $L _ { 1 }$ to create a final output. ",
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+        "type": "text",
+        "text": "Our method assigns causally impactful information content, but also identifies the abstract causal structure along which representations are composed. It thus encompasses and improves on both correlational (probing) and attribution methods. ",
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+        "type": "text",
+        "text": "4 The Natural Language Inference Task and Models ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Multiply Quantified NLI Dataset The Multiply Quantified NLI (MQNLI) dataset of Geiger et al. [10] contains templatically generated English-language NLI examples that involve very complex interactions between quantifiers, negation, and modifiers. We provide a few examples in Figure 2b; the empty-string symbol $\\varepsilon$ ensures perfect alignments at the token level both between premises and hypotheses and across all examples. ",
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+        "type": "text",
+        "text": "The MQNLI examples are labeled using an algorithmic implementation of the natural logic of MacCartney and Manning [18] over tree structures, and MQNLI has train/dev/test splits that vary in their difficulty. In the hardest setting, the train set is provably the minimal set of examples required to ensure that the dev and test sets can be perfectly solved by a simple symbolic model; in the easier settings, the train set redundantly encodes necessary information, which might allow a model to perform perfectly in assessment by memorization despite not having found a truly general solution. For a fuller review of the dataset, see Appendix A. ",
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+        "text": "MQNLI is a fitting benchmark given our goals for a few reasons. First, we can focus on the hardest splits that can be generated, which will stress-test our NLI architectures in a standard behavioral way. Second, the MQNLI labeling algorithm itself suggests an appropriate causal model of the data-generating process. Figure 2a summarizes this model in tree form, and it is presented in full detail in Geiger et al. [10]. This allows us to rigorously assess whether a neural network has learned to implement variants of this causal model. The complexity of the MQNLI examples creates many opportunities to do this in linguistically interesting ways. ",
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+        "img_path": "images/0cb3d60b262f4095536b9b0c886f7ffd2120462b64ededc66c7b1b8302c8ce7b.jpg",
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+            "(a) The causal structure of the high-level natural logic causal model $C _ { N a t L o g }$ that performs inference on MQNLI. The superscripts $P$ and $H$ stand for ‘premise’ and ‘hypothesis’ and the subscripts ‘Subj’ and ‘Obj’ stand for ‘Subject’ and ‘Object’. The node labels are used to explain the experimental results in Section 5 "
+        ],
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+        "type": "text",
+        "text": "$\\varepsilon$ every $\\varepsilon$ baker $\\varepsilon \\varepsilon \\varepsilon$ eats $\\varepsilon$ no $\\varepsilon$ bread   \ncontradiction   \n$\\varepsilon$ no angry baker $\\varepsilon \\varepsilon \\varepsilon$ eats $\\varepsilon$ no $\\varepsilon$ bread   \n$\\varepsilon$ every silly professor $\\varepsilon \\varepsilon \\varepsilon$ sells not every $\\varepsilon$ book   \nneutral   \n$\\varepsilon$ every silly professor $\\varepsilon \\varepsilon \\varepsilon$ sells not every $\\varepsilon$ chair   \nnot every sad baker $\\varepsilon \\varepsilon$ fairly admits not every odd idea   \nentailment   \n$\\varepsilon$ some $\\varepsilon$ baker does not $\\varepsilon$ admits $\\varepsilon$ no $\\varepsilon$ idea ",
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+    {
+        "type": "text",
+        "text": "(b) MQNLI examples. The $\\varepsilon$ token serves as padding (but still attended to by the model) and ensures a perfect alignment between both premises and hypotheses and across all examples. It is semantically an identity element. ",
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+        "type": "table",
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+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Train</td><td>Dev</td><td>Test</td></tr><tr><td>CBoW</td><td>88.04</td><td>54.18</td><td>53.99</td></tr><tr><td>TreeNN</td><td>67.01</td><td>54.01</td><td>53.73</td></tr><tr><td>CompTreeNN</td><td>99.65</td><td>80.17</td><td>80.21</td></tr><tr><td>BiLSTM</td><td>99.42</td><td>46.41</td><td>46.32</td></tr><tr><td>BERT</td><td>99.99</td><td>88.25</td><td>88.50</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "(c) MQNLI results. The first three models are from Geiger et al. 10, where the CompTreeNN is a taskspecific model not suitable for general NLI and functions as an idealized upperbound. Our results show that BERT-based models can surpass this without such alignments. ",
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+        "text": "Figure 2: The natural logic causal model (top), MQNLI examples (left) and MQNLI results (right). ",
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+        "type": "text",
+        "text": "Models We evaluated two models on MQNLI: a randomly initialized multilayered Bidirectional LSTM (BiLSTM; [23]) and a BERT-based classifier model in which the English bert-base parameters [8] are fine-tuned on the MQNLI train set. Output predictions are computed using the final representation above the [CLS] token. Models are trained to predict the relation of every pair of aligned phrases in Figure 2a. Additional model and training details are given in Appendix B. ",
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+        "type": "text",
+        "text": "Results Figure 2c summarizes the results of our BERT and BiLSTM models on the hardest fair generalization task Geiger et al. [10] creates with MQNLI. We find that our BiLSTM model is not able to learn this task, and that our BERT model is able to achieve high accuracy. The only models in Geiger et al. [10] able to achieve above $50 \\%$ accuracy were task-specific tree-structured models with the structure of the tree in Figure 2a. Thus, our BERT-based model is the first general-purpose model able to achieve good performance on this hard generalization task. Without pretraining, the BERT-based model achieves ${ \\approx } 4 9 . 1 \\%$ , confirming that pretraining is essential, as expected. ",
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+        "text": "A natural hypothesis is that the BERT-based model achieves this high performance because it has in effect induced some approximation to the tree-like structure of the data-generating process in its own internal layers. With causal abstraction analysis, we are actually in a position to test this hypothesis. ",
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+    {
+        "type": "text",
+        "text": "5 A Case Study in Structural Neural Network Analysis ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "5.1 Causal Abstractions of Neural NLI models ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Formulating Our Hypotheses We proceed just as we did for the simple motivating example in Section 3, except that we are now seeking to assess the extent to which the natural logic algebra in Figure 2a is a causal abstraction of the trained neural models in the above section. ",
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+        "text": "The hallmark of Figure 2a is that it defines an alignment between premise and hypothesis at both lexical and phrasal levels. This pway. For a given non-leaf node $N$ mits us to run intein Figure 2a, let $C _ { N a t L o g } ^ { N }$ interventions in a nbe a submodel of $\\boldsymbol { C } _ { N a t L o g }$ compositionalthat computes the relation between the aligned phrases under and uses them to compute the final output relation between premise and hypothesis. For example, let $C _ { N a t L o g } ^ { \\mathrm { N P _ { O b j } } }$ be the submodel of $\\boldsymbol { C } _ { N a t L o g }$ that computes the relation between the two aligned object noun phrases and then uses that relation in computing the final output relation between premise and hypothesis (see Figure 3 right). We would like to ask whether our trained neural models also compute this relation between object noun phrases and use it to make a final prediction. We can pose this same question for other nodes which correspond to a pair of aligned subphrases. ",
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+        "img_path": "images/091a108d0acc3c4c953e97f27659d8f183813696f3f905ddb228de930349f9aa.jpg",
+        "image_caption": [
+            "Figure 3: A BERT-based NLI model (left) aligned with the natural logic causal model $C _ { N a t L o g } ^ { \\mathrm { N P _ { O b j } } }$ (right), where the fourth vector representation above the $\\mathrm { A d j } _ { \\mathrm { O b j } } ^ { P }$ token in the network is aligned with $\\mathrm { N P _ { O b j } }$ , the variable representing the relation between the object noun phrases. When analyzing a sample of 1000 examples, we found a subset of 383 where $C _ { N a t L o g } ^ { \\mathrm { N P _ { O b j } } }$ is an abstraction of $N _ { N L I }$ under this alignment. "
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+        "type": "text",
+        "text": "nd the variable  causal model., we consider a $N$ . In principle, any location in the network could bry hypothesis in this space would be intractable. t of hidden representations based on the identity of he right oneus, for each. The BERT $N _ { N L I }$ $N$ $C _ { N a t L o g } ^ { N }$ ${ C _ { N a t L o g } ^ { N } } ^ { N }$ $N$ model we use has 12 Transformer layers [30], meaning that there are 12 hidden representations for each input token. Each alignment search considers aligning the intermediate high-level variable with dozens of possible locations in the grid of BERT representations. Specifically, the following locations were considered for each $N$ : ",
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+    {
+        "type": "text",
+        "text": "$\\mathrm { Q _ { S u b j } }$ , $\\mathrm { \\Delta \\ A j _ { \\mathrm { S u b j } } }$ , $\\mathrm { N _ { S u b j } }$ , Neg, Adv, V, $\\mathrm { Q _ { O b j } }$ , $\\mathrm { \\Delta A d j _ { O b j } }$ , $\\mathrm { N _ { O b j } }$ : hidden representations above the two descendant leaf tokens. • $\\mathrm { N P _ { S u b j } }$ , VP, and $\\mathrm { N P _ { O b j } }$ : same but above the four descendant leaf tokens. ",
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+    {
+        "type": "text",
+        "text": "$\\mathrm { Q P _ { O b j } }$ idden representations above . $\\mathrm { Q } _ { \\mathrm { 0 b j } } ^ { P }$ $\\mathrm { Q } _ { \\mathrm { O b j } } ^ { H }$   \n. NegP: same but above $\\mathrm { N e g } ^ { P }$ and $\\mathrm { N e g } ^ { H }$ .   \nAll nodes (for BERT): same but above   \n[CLS] and [SEP]. ",
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+        "type": "text",
+        "text": "For each alignment considered, we performed a full causal abstraction analysis. We report the results from the best alignments in Table 1, and we summarize the results from all alignments in Appendix D. ",
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+        "type": "text",
+        "text": "Interchange Interventions We first focus on our high-level causal models. Consider a non-leaf node $N$ from Figure 2a and two input token sequences $e$ and $e ^ { \\prime }$ from MQNLI. Define ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/d4c7228daaaa32940d8648a0c44c008433ce8d5493441c18f3f3e1a97148d235.jpg",
+        "text": "$$\nC _ { N a t L o g } ^ { N  e ^ { \\prime } } ( e )\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "to be the output provided by the causal mode $C _ { N a t L o g } ^ { N }$ when processing input $e$ where the relation $N$ is changed to the relation between those subphrases   \nin $e ^ { \\prime }$ . For example, simplifying for the sake of exposition, suppose $e$ is (some happy baker, no $\\epsilon$   \nbaker), which has output label contradiction, and suppose $e ^ { \\prime }$ is (every happy person, some happy   \naker), which has output label , the noun phrase relation is e ntailment.ailment; in e wish to intervene on the noun, it is reverse entailment. Thus, $N = \\mathrm { N P }$ . In the   \n$e$ $e ^ { \\prime }$ $C _ { N a t L o g } ^ { \\mathrm { N P }  e ^ { \\prime } } ( e )$ $e$ to entailment while holding everything else about $e$ constant. This   \nresults in the output label for the example (some happy person, no $\\epsilon$ baker), which is neutral. ",
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+        "type": "text",
+        "text": "Next, we consider interventions in a neural model $N _ { N L I }$ . Define ",
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+    {
+        "type": "equation",
+        "img_path": "images/d559ae0772c879936317fe1c73a5c69e9586ef874de37d00f537ebfe86448777.jpg",
+        "text": "$$\nN _ { N L I } ^ { L  e ^ { \\prime } } ( e )\n$$",
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+    {
+        "type": "table",
+        "img_path": "images/bbd2a8666e8a0bbd6ba2f39edda9103a8eb96b0868b41a8f823e0ed3852f7fc8.jpg",
+        "table_caption": [
+            "Table 1: Largest subsets of examples on which specific models $C _ { N a t L o g } ^ { N }$ are abstractions of an LSTM and BERT model trained on MQNLI. We record the size of such subsets as a percentage of the total 1000 examples. On this subset, we know that the neural models compute a representation of the relation between the aligned subphrases under $N$ and use this information to make a final prediction. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Nodesadded</td><td>BERT</td></tr><tr><td>Adsubj</td><td>30.5</td></tr><tr><td>Nsub</td><td>37.2</td></tr><tr><td>Negp</td><td>14.9</td></tr><tr><td>AdvP</td><td>26.9</td></tr><tr><td>VP</td><td>35.6</td></tr><tr><td>Q</td><td>16.2</td></tr><tr><td>Adjsubj H</td><td>13.4</td></tr><tr><td>Nsub</td><td>12.0</td></tr><tr><td>NegH</td><td>34.4</td></tr><tr><td>AdvH</td><td>16.2</td></tr><tr><td>VH</td><td>13.4</td></tr><tr><td>Qbi</td><td>12.0</td></tr></table>",
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+    },
+    {
+        "type": "table",
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+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Causal Model</td><td>LSTM</td><td>BERT</td></tr><tr><td>Qsubj</td><td>0.7</td><td>13.1</td></tr><tr><td>Qobj</td><td>0.9</td><td>7.3</td></tr><tr><td>Neg</td><td>0.7</td><td>21.4</td></tr><tr><td>Adjsubj</td><td>2.5</td><td>6.7</td></tr><tr><td>Nsubj</td><td>1.2</td><td>5.5</td></tr><tr><td>Adjobj</td><td>0.9</td><td>14.1</td></tr><tr><td>Nobj</td><td>0.7</td><td>8.8</td></tr><tr><td>V</td><td>0.4</td><td>11.4</td></tr><tr><td>Adv</td><td>1.4</td><td>7.9</td></tr><tr><td>NPsubj</td><td>1.0</td><td>6.7</td></tr><tr><td>NPobj</td><td>0.7</td><td>38.3</td></tr><tr><td>VP</td><td>0.4</td><td>11.4</td></tr><tr><td>NegP</td><td>0.9</td><td>11.8</td></tr></table>",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/12cb2c08af89e5c77bc8f6afb5e129ff597b6b75a2c7d3b302f5078d9f7c7d6e.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Nodes removed</td><td>BERT</td></tr><tr><td></td><td>31.9</td></tr><tr><td>A N</td><td>15.7</td></tr><tr><td></td><td>33.8</td></tr><tr><td>A</td><td>15.8 31.9</td></tr><tr><td></td><td>14.1</td></tr><tr><td>A</td><td>32.2</td></tr><tr><td></td><td>31.6</td></tr><tr><td></td><td>8.8</td></tr><tr><td></td><td></td></tr><tr><td>,Abj</td><td>32.1</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "(a) Main results (clique sizes) for nonleaf nodes of the tree in Figure 2a. The hypothesis we have most evidence for is that the BERT model computes a representation of the $\\mathrm { N P _ { O b j } }$ node with the alignment shown in Figure 3. Remarkably, with 1000 examples sampled, we found a subset of 383 examples where $C _ { N a t L o g } ^ { \\mathrm { N P _ { O b j } } }$ is an abstraction of BERT. ",
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+    {
+        "type": "text",
+        "text": "(b) Detailed results (clique sizes) for Altein a “neighborhood” around the model $C _ { N a t L o g } ^ { \\mathrm { N P _ { O b j } } }$ causal models, which has a single intermediate variable composed of four lexical items (See Figure 3). At left, we have alternative causal models where one or two of those lexical items are removed from the composition. At right, we have alternatives obtained by adding one lexical item to the composition. We observe that no alternative hypothesis about causal structure considered has more evidence. ",
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+    {
+        "type": "text",
+        "text": "to be the output provided by $N _ { N L I }$ processing the input $e$ when the representation at location $L$ is replaced with the representation at location $L$ created when $N _ { N L I }$ is processing $e ^ { \\prime }$ . This is exactly the process depicted in Figure 1, except now the networks are the complex trained networks of Section 4. ",
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+        "type": "text",
+        "text": "Our hypothesis linking Figure 2a with a model $N _ { N L I }$ takes the same form as (3). The causal model $C _ { N a t L o g } ^ { N }$ is a constructive abstraction of $N _ { N L I }$ when, for some representation location $L$ , it is the case that, for all MQNLI examples $e$ and $e ^ { \\prime }$ , we have ",
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+    {
+        "type": "equation",
+        "img_path": "images/4744fd6ee432b685118e8febf69bcb388251fe1bfc500dfbd6c1a2214e68ab27.jpg",
+        "text": "$$\nC _ { N a t L o g } ^ { N  e ^ { \\prime } } ( e ) = N _ { N L I } ^ { L  e ^ { \\prime } } ( e )\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "This asserts a correspondence between interventions on the representations at $L$ in network $N _ { N L I }$ and interventions on the variable $N$ in the causal model $C _ { N a t L o g } ^ { N }$ . If it holds, then $N _ { N L I }$ computes the relation between the aligned phrases under the node $N$ and uses this information to compute the relation between the premise and hypothesis. ",
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+        "type": "text",
+        "text": "We call a pair of examples $( e , e ^ { \\prime } )$ successful if it satisfies equation (6), i.e., interventions in both the target causal model and neural model produce equal results. In addition, to isolate the causal impact of our interventions, we specifically focus on pairs $( e , e ^ { \\prime } )$ for which performing the intervention produces a different output value than without the intervention. We call a pair $( e , e ^ { \\prime } )$ impactful if: ",
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+        "img_path": "images/eb795cef2be9c9e3b11f7e1634161a259d5eb3445bc58ececca30bc45818a55e.jpg",
+        "text": "$$\nC _ { N a t L o g } ^ { N  e ^ { \\prime } } ( e ) \\neq C _ { N a t L o g } ^ { N } ( e )\n$$",
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+        "type": "text",
+        "text": "Quantifying Partial Success Equation (6) universally quantifies over all examples. We do not expect this kind of perfect correspondence to emerge in practice for real problems: neural network training is often approximate and variable in nature, and even our best model does not achieve perfect performance. However, we can still ask how to find the largest subset of MQNLI on which $C _ { N a t L o g } ^ { N }$ (6) holds for a given model. To do this, we seekis an abstraction of our neural models, for each non-leaf node N in CNatLog. ",
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+    {
+        "type": "text",
+        "text": "More specifically, considering each example in MQNLI as a vertex in a graph, we add an undirected edge between two examples $e _ { i }$ and $e _ { j }$ if and only if both the ordered pairs $( e _ { i } , e _ { j } )$ and $( e _ { j } , e _ { i } )$ satisfy (6). In other words, if and only if all ex $C _ { N a t L o g } ^ { N }$ isin n abstraction of a neural model on a subset of examples form a clique. $S$ of MQNLI $S$ ",
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+    {
+        "type": "text",
+        "text": "The number of interventions we need to run scales quadratically with the number of inputs we consider, so we sample 1000 MQNLI examples, producing a total of $1 0 0 0 ^ { 2 } = 1 \\mathbf { M }$ ordered pairs. We only consider examples for which the neural network outputs a correct label. For each node $N$ and each of its corresponding neural network locations $L$ , we perform interventions on all of these pairs. ",
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+    {
+        "type": "text",
+        "text": "We choose to measure the largest clique with at least one impactful edge, because (1) the causal abstraction relation holds with full force on that clique, but other measures such as the total number of connections lack this theoretical grounding, and (2) if a clique has at least one impactful edge, that guarantees the high-level variable is being used. ",
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+    {
+        "type": "text",
+        "text": "Results and Analysis For each target causal model node $N$ and neural network representation location $L$ , we construct a graph as described above with 1000 examples as vertices and add an edge between two examples $e _ { i }$ and $e _ { j }$ if and only if both $( e _ { i } , e _ { j } )$ and $( e _ { j } , e _ { i } )$ are successful. We then find the largest clique in this graph with at least one impactful edge and record its size. ",
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+    {
+        "type": "text",
+        "text": "Table 1a shows, for each causal model node $N$ , the maximum size of cliques found among all neural locations. With this stricter impactful criterion (as opposed to simply using intervention success), our results show that, for almost all nodes abstraction of BERT on a significant numb $N$ , our target causal model of examples in our datase $C _ { N a t L o g } ^ { \\widetilde { N } }$ is indeed a causale subsets are much smaller for the BiLSTM model. ",
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+    {
+        "type": "text",
+        "text": "We also investigated alternative high-level causal structures that are not variants of $\\boldsymbol { C } _ { N a t L o g }$ from Figure 2a. Specifically, we consider alternative models in a “neighborhood” around the model CNPObjNatLog that can be obtained by adding one leaf, or by removing one or two leaves to the composition. These results are in Table 1b. Remarkably, all of these alternative models result in smaller clique sizes, significantly so for many of them. This further supports the significance of our results. ",
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+    {
+        "type": "text",
+        "text": "This analysis is simtwo crucial differenis an abstraction of o the analysis of our hirst, for each variable , whereas in the addit $N$ othetical addition example in Section 3, ex, we are hypothesizing that the causal model  example there was only one model. To inv ${ \\dot { C } } _ { N a t L o g } ^ { N }$ $N _ { N L I }$   \nthis difference, we take $N = \\mathrm { N P _ { O b j } }$ as a paradigm case, as it is the model with the strongest results. (The results for other nodes are in Appendix D.) Second, we only achieved partial experimental success, whereas in the addition example we assumed complete success. Crucially, this means that the following analysis will be valid only on subsets of the input space on which the abstraction relation holds between NNLI and CNPObjNatLog. ",
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+    },
+    {
+        "type": "text",
+        "text": "We visualize the results of our intervention experiments for the node $\\mathrm { N P _ { O b j } }$ in Figure 4. The alignment with the largest subset of inputs aligns the NPObj variable in CNPObjNatLog with the neural representation on the fourth layer of BERT above the $\\mathrm { A d j } _ { \\mathrm { O b j } } ^ { P }$ token (see Figure 3). Because neural representations encode the value of their aligned variables and play a parallel causal role to their high-level variables, we know that, on this subset of input examples, at the fourth neural representation above the $\\mathrm { A d j } _ { \\mathrm { O b j } } ^ { P }$ token, the four input embeddings for the object nouns and adjectives in the premise and hypothesis are composed to form a neural representation with information content of the relation between the object noun phrases in the premise and hypothesis. Then this representation is composed with the other input-embeddings to create an output representing the relation between the premise and hypothesis. ",
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+    {
+        "type": "text",
+        "text": "5.2 Comparison with Other Structural Analysis Methods ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Probes We probed neural representation locations for the relation between aligned subexpressions on a subset of 12,800 randomly selected MQNLI examples. For a pair of aligned subexpressions below a node $N$ in Figure 2a, we probe the columns above the same set of restricted class of tokens as described in Section 5.1. ",
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+    },
+    {
+        "type": "text",
+        "text": "To evaluate these probes, we report accuracy as well as selectivity as defined by Hewitt and Liang [12]: probe accuracy minus control accuracy, where control accuracy is the train set accuracy of a probe with the same architecture but trained on a control task to factor out probe success that can be attributed to the probe model itself. Our control task is to learn a random mapping from node types to semantic relations; see Appendix C for full details on how this task was constructed. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/c313e3750ef3c4775c2304d720db7884b4f4d011c838d444e19be0b735a6a5ed.jpg",
+        "image_caption": [
+            "Figure 4: Interchange intervention and probing results for the $\\mathrm { N P _ { O b j } }$ position. Vertical axes denote layers of BERT and horizontal axes denote the token position of hidden representations. The intervention success rates reported here are calculated based on intervention experiments with a change in the output label. Clique sizes are reported as $\\%$ of 1000 examples. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Figure 4 summarizes our probing results for $N = \\mathrm { N P _ { O b j } }$ , along with corresponding interchange intervention results for comparison. Probes tell us that information about the relation between the aligned noun phrases is encoded in nearly all of the locations we considered, and using the selectivity metric does not result in any qualitative change. In contrast, our intervention heatmaps indicate only a small number of locations store this information in a causally relevant way. Clearly, our intervention experiments are far more discriminating than probes. Appendix D provides examples involving other variables along with the intervention experiments, where the general trend of interchange interventions being more discriminating holds. ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Integrated Gradients Attribution methods that estimate feature importance can measure causal properties of neural representations, but a single feature importance method is an impoverished characterization of a representation’s role in network behavior. Whereas our interchange interventions gave us high-level information about how a neural representation is composed and what it is composed into, attribution methods simply tell us “how much” a representation contributes to the network output on a give input. Moreover, intervention interchanges provide a rich, high-level characterization of causal structure on a space of inputs. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We use integrated gradients on our models to verify the intuitive hypothesis that if a premise and hypothesis differ by a single token, then the neural representations above that token should be more causally responsible for the network output than other representations. For example, given premise ‘Every sleepy cat meows’ and hypothesis ‘Some hungry cat meows’, the attributive modifier position is different and the rest are matched. The neural representations above the adjective tokens sleepy and hungry should be more important for the network output than others, because if those adjectives were the same, the example label would change from neutral to entailment. We summarize the results of our integrated gradient experiments in Appendix E, where we confirm our intuitive hypothesis. ",
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "6 Conclusion ",
+        "text_level": 1,
+        "bbox": [
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+            297,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We have introduced a methodology for deriving interpretable causal explanations of neural network behaviors, grounded in a formal theory of causal abstraction. The methodology involves first formulating a hypothesis in the form of a high-level, interpretable causal model, then searching for an alignment between the neural network and the causal model, and finally verifying experimentally that the neural representations encode the same causal properties and information content as the corresponding components of the high-level causal model. As a case study demonstrating the feasibility of the approach, we analyzed neural models trained on the semantically formidable MQNLI dataset. Guided by the intuition that success on this challenging task may call for a way of recapitulating the causal structure of the natural logic model that generates the MQNLI data, we were able to verify the hypothesis that a state-of-the-art BERT-based model partially realizes this structure, whereas baseline models that do not perform as well fail to do so. This suggestive case study demonstrates that our theoretically grounded methodology can work in practice. ",
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+    },
+    {
+        "type": "text",
+        "text": "Acknowledgments and Disclosure of Funding ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Our thanks to Amir Feder, Noah Goodman, Elisa Kreiss, Josh Rozner, Zhengxuan Wu, and our anonymous reviewers. This research is supported in part by grants from Facebook and Google. ",
+        "bbox": [
+            174,
+            121,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
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parse/train/Rx9dBZaV_IP/Rx9dBZaV_IP.md

@@ -0,0 +1,263 @@
+# Conformal Time-Series Forecasting
+
+Kamile Stankevi˙ ciˇ ut¯ e˙ University of Oxford University of Cambridge ks830@cam.ac.uk
+
+Ahmed M. Alaa University of California, Los Angeles ahmedmalaa@ucla.edu
+
+Mihaela van der Schaar University of Cambridge University of California, Los Angeles The Alan Turing Institute mv472@cam.ac.uk
+
+# Abstract
+
+Current approaches for (multi-horizon) time-series forecasting using recurrent neural networks (RNNs) focus on issuing point estimates, which are insufficient for informing decision-making in critical application domains wherein uncertainty estimates are also required. Existing methods for uncertainty quantification in RNNbased time-series forecasts are limited as they may require significant alterations to the underlying architecture, may be computationally complex, may be difficult to calibrate, may incur high sample complexity, and may not provide theoretical validity guarantees for the issued uncertainty intervals. In this work, we extend the inductive conformal prediction framework to the time-series forecasting setup, and propose a lightweight uncertainty estimation procedure to address the above limitations. With minimal exchangeability assumptions, our approach provides uncertainty intervals with theoretical guarantees on frequentist coverage for any multi-horizon forecast predictor and any dataset. We demonstrate the effectiveness of the conformal forecasting framework by comparing it with existing baselines on a variety of synthetic and real-world datasets.
+
+# 1 Introduction
+
+Time-series forecasting tasks are central to a broad range of application domains, including stock price predictions [1, 2], service demand forecasting [3, 4], and medical prognoses [5–7]. Recurrent neural networks (RNNs) and their variants (e.g., LSTM, GRU, etc.) constitute an instrumental class of models that are most commonly used to carry out time-series forecasting tasks [8, 9]. These models, however, are usually used to issue point predictions—i.e., singular estimates of the future values of a time-series. In many high-stakes applications—such as finance and medicine—these are not enough; estimates of uncertainty are also required for accurate risk assessment and decision-making [10]. For example, clinical practitioners need to make treatment decisions accounting for all potential scenarios, where less likely scenarios may have graver consequences and require more care compared to the more likely scenarios [11, 12].
+
+While various methods for uncertainty estimation in standard feed-forward neural networks have been recently proposed [13–15], equivalent methods for RNN-based time-series models are still under-explored. Existing solutions include Bayesian recurrent neural networks [16–18], quantile regression models [3, 19], latent variable models with deep state-space architectures [6, 20], and post-hoc uncertainty estimates using bootstrapping, jackknife or other ensembling procedures [21– 23]. Each of these solutions has its own limitations: Bayesian models may be difficult to calibrate, quantile predictors may “overfit” their uncertainty estimates, and bootstrapping methods scale poorly for RNNs with large number of parameters. Almost all existing methods share at least one of the two major drawbacks: (1) they require substantial modifications to the underlying model architecture, and (2) they provide no theoretical guarantees on frequentist coverage, any of the exceptions being computationally intractable.
+
+We aim to address the above limitations by adapting conformal prediction (CP) [24, 25]—a framework used to derive prediction intervals with guaranteed finite-sample frequentist coverage—to the timeseries forecasting setup. CP has originally been designed to construct prediction intervals for scalar targets; on the other hand, observations and predictions in time-series forecasting involve temporally dependent, potentially multivariate sequences that are not, in general, directly comparable due to differences in observation lengths, irregular frequencies, non-stationarity, and other variations in temporal dynamics (comparison between training points being a key step in CP). We extend CP to a novel, computationally efficient conformal forecasting framework that can leverage any underlying point forecasting model to produce multi-step prediction intervals with coverage guarantees across the prediction horizon. We focus on RNN-based conformal forecasting architectures, which we call conformal forecasting RNNs (CF-RNNs), and explore their effectiveness in providing valid and efficient coverage intervals.
+
+# 2 Related Work
+
+Most previous work in the area of uncertainty quantification in deep learning focuses on feed-forward neural network models. Much less work has been done on uncertainty estimation for time-series models. In what follows, we discuss previous methods developed for uncertainty estimation for RNNs, which we also summarise in Table 1.
+
+Table 1: Overview of the most popular RNN-based probabilistic forecasting methods.   
+
+<table><tr><td>Method</td><td>Paradigm</td><td>Architecture</td><td>Time-series observations</td><td>Frequentist coverage</td></tr><tr><td>Bayesian RNNs [16-18]</td><td>Bayesian</td><td>Built-in</td><td>Multiple</td><td></td></tr><tr><td>Monte Carlo dropout [26]</td><td>Bayesian (approx.)</td><td>Built-in</td><td>Multiple</td><td></td></tr><tr><td>MQ-[3], SQF-RNN [19]</td><td></td><td>Built-in</td><td>Multiple</td><td>一</td></tr><tr><td>BJ-RNN [21]</td><td>Frequentist</td><td>Post-hoc</td><td>Multiple</td><td>1-2α</td></tr><tr><td>EnbPI [27]</td><td>Frequentist (approx.)</td><td>Ensemble</td><td>Single</td><td>1-α</td></tr><tr><td>CF-RNN (proposed)</td><td>Frequentist</td><td>Post-hoc</td><td>Multiple</td><td>1-α</td></tr></table>
+
+Bayesian RNNs [16–18] extend the ideas of Bayesian inference to RNN models, expressing the model (epistemic1) uncertainty through distributions on model parameters [28, 29]. Exact Bayesian inference quickly becomes infeasible, however; various approximations based on Markov chain Monte Carlo [30–33] or variational inference [34–37] are needed. Bayesian neural networks depend on significant changes in the underlying model architecture (at least doubling the number of parameters), and rely on a good choice of prior (which may be challenging in practice). While simplifying techniques such as Monte Carlo dropout [26] (with RNN-specific adaptation in Gal and Ghahramani [38]) exist, they are often difficult to calibrate [21].
+
+Quantile RNNs can be viewed as a deep neural network extension of quantile regression [39] for sequential data: instead of returning a series of point estimates across the prediction horizon, quantile RNNs learn the prediction intervals directly, with upper and lower bounds of the forecast as separate prediction targets. The standard approach to achieve this is to use the appropriate pinball loss function as the objective. While successful applications of this approach in time-series forecasting exist [3], naively learning individual bounds may have problems such as quantile crossing; more recent approaches Gasthaus et al. [19] resolve this by fitting the entire quantile function. Quantile RNNs are additionally at risk of quantile overfitting due to poor sample complexity [21].
+
+Ensembles are based on the principle of training and combining multiple models, e.g. deep ensembles trained on different random initialisations [40, 41], or models retrained on partial datasets (jackknife or bootstrap resampling-based RNNs, [22, 21]). Deep neural network ensembles are in general not mathematically principled for uncertainty quantification [42]; while resampling-based models resolve this and provide post-hoc frequentist coverage guarantees, they are instead limited in their time and space complexity. For example, exact inference on the state-of-the-art blockwise jackknife RNN (BJ-RNN) model [21] takes ${ \cal O } ( P ^ { 3 } )$ time for $P$ parameters, and even with simplifying approximations—which in turn deteriorate accuracy—the model does not scale beyond small datasets.
+
+![](images/2485dd58fe846731adacaec3932cb398c702ae29f62851100eddf4b91bd8c219.jpg)  
+Figure 1: Time-series observation paradigms. (Left) The dataset is assumed to comprise a single timeseries, with observations being individual time-steps within the time-series. These observations are temporally dependent. (Right) The dataset consists of a set of independent time-series, where the entire series is treated as an observation. Independence of time-series implies their exchangeability.
+
+Conformal prediction (CP) For a given significance level (error rate) $\alpha$ , the goal of CP [24, 25] is to return a prediction region $\varGamma ^ { \alpha }$ that is guaranteed to contain the true value with probability of at least $( 1 - \alpha )$ . In regression problems (such as time-series forecasting), CP is modified to work inductively using an additional calibration set and an underlying model—an approach called inductive conformal prediction (ICP) [43, 44]. Little work has been done applying (I)CP methods for time-series forecasting; the main challenge is that CP assumes exchangeability, where any permutation of the dataset observations is equiprobable. However, the time-steps within a time-series are inherently non-exchangeable due to temporal dependencies (Figure 1, left); naively applying CP to derive forecast intervals from a given time-series is therefore not methodologically valid and lacks the validity guarantees. One notable exception is the EnbPI model [27], which bypasses the exchangeability assumption (introducing some others) and uses an ensemble of bootstrapped estimators to provide approximately valid intervals. However, we argue that learning from a single time-series—while useful in cases where indeed only one time-series is available—may not be optimal in settings where datasets contain multiple time-series, the shared patterns of which could potentially be exploited (Figure 1, right). To the best of our knowledge, no existing method has applied CP to the latter forecasting setting (despite it being more methodologically grounded); yet the datasets of multiple time-series are increasingly common and useful [45].
+
+# 3 Conformal forecasting RNNs (CF-RNNs)
+
+In this Section, we introduce the conformal forecasting RNN (CF-RNN) model. We start off by formalizing the multi-horizon time-series forecasting problem in Section 3.1, and providing the necessary background on inductive conformal prediction (ICP) for regression tasks in Section 3.2. We introduce the details of the conformal forecasting procedure in Section 3.3.
+
+# 3.1 Multi-horizon time-series forecasting
+
+Let $y _ { t : t ^ { \prime } } = ( y _ { t } , y _ { t + 1 } \ldots , y _ { t ^ { \prime } } )$ be a time-series of $d$ -dimensional observations $y _ { t } , \ldots , y _ { t ^ { \prime } } \in \mathbb R ^ { d }$ that start at time step $t$ and end at time step $t ^ { \prime }$ . A multi-horizon time-series forecast predicts future values
+
+$$
+\hat { y } _ { ( t ^ { \prime } + 1 ) : ( t ^ { \prime } + H ) } = \left( \hat { y } _ { t ^ { \prime } + 1 } , \ldots , \hat { y } _ { t ^ { \prime } + H } \right) \in \mathbb { R } ^ { H \times d } ,
+$$
+
+given the history of observed values $y _ { 1 : t ^ { \prime } }$ , where $H$ is the number of steps to be predicted (the prediction horizon). For critical applications, we are interested in the uncertainty associated with the forecast—for each time step $h$ in the prediction horizon, we would like to obtain prediction intervals of the form $[ \hat { y } _ { t + h } ^ { L } , \hat { y } _ { t + h } ^ { U } ]$ , $h \in \{ 1 , \ldots , H \}$ , so that the ground truth value $y _ { t + h }$ is contained in the interval with a sufficiently high probability. We fix a desired significance level (or error rate) $\alpha$ , such that the ground-truth values of the entire time-series trajectory are contained within the intervals; i.e.,
+
+$$
+\mathbb { P } \left[ y _ { t + h } \in [ \hat { y } _ { t + h } ^ { L } , \hat { y } _ { t + h } ^ { U } ] , \forall h \in \{ 1 , \dots , H \} \right] \geq 1 - \alpha .
+$$
+
+# 3.2 Inductive conformal prediction (ICP)
+
+Given a set of observations $\mathcal { D } = \{ ( \mathbf { x } ^ { ( i ) } , y ^ { ( i ) } ) \} _ { i = 1 } ^ { l }$ and a new example $\mathbf { x } ^ { ( l + 1 ) }$ , the ICP procedure [24, 46, 47] returns a prediction interval $\varGamma ^ { \alpha }$ such that the property of validity is satisfied:
+
+Property 1. (Validity) Under the exhangeability assumption, any conformal predictor will return the prediction region $\begin{array} { r } { \dot { T } ^ { \alpha } ( \mathbf { x } ^ { ( i ) } ) } \end{array}$ such that the probability of error $y ^ { ( l + 1 ) } \not \in { \Gamma } ^ { \alpha } ( \dot { \mathbf { x } } ^ { ( l + 1 ) } )$ is not greater than $\alpha$ . Alternatively:
+
+$$
+\mathbb { P } [ y ^ { ( l + 1 ) } \in { \Gamma } ^ { \alpha } ( \mathbf { x } ^ { ( l + 1 ) } ) \mid \mathcal { D } ] \geq 1 - \alpha .
+$$
+
+The conformal prediction framework is distribution-free (i.e. it does not have any assumptions on the distribution of the underlying data $\mathcal { D }$ ), and applies to any underlying predictive model as long as the exchangeability assumption is satisfied:
+
+Assumption 1. (Exchangeability) In a dataset of l observations $\{ ( \mathbf { x } ^ { ( i ) } , y ^ { ( i ) } ) \} _ { i = 1 } ^ { l }$ , any of its $l !$ permutations are equiprobable. Note that independent identically distributed (iid) observations satisfy exchangeability.
+
+The inductive2 variant of CP operates by splitting the training set into the proper training set of size $n$ and a calibration set of size $m$ : $\mathcal { D } = \mathcal { D } _ { \mathrm { t r a i n } } \cup \mathcal { D } _ { \mathrm { c a l } }$ . The proper training set is used to train the underlying (auxiliary) model $M$ , and the calibration set is used to obtain the nonconformity scores, which measure how unusual is the given example compared to previously observed data. While CP guarantees validity for any nonconformity score (including a random number generator), the most commonly used nonconformity score in regression is of the form
+
+$$
+R _ { i } = A ( \mathcal D , ( { \mathbf x } ^ { ( i ) } , y ^ { ( i ) } ) ) = A ( M ( { \mathbf x } ^ { ( i ) } | \mathcal D ) , y ^ { ( i ) } ) ,
+$$
+
+where $\varDelta$ is some distance metric. While any choice for $M$ is valid, the best architecture depends on the dataset and the problem. When $\varDelta ( \hat { y } , y ) = | \hat { y } - y |$ , the nonconformity score $R _ { i } = | \hat { y } ^ { ( i ) } - y ^ { ( i ) } |$ corresponds to the residual error between the prediction of the underlying model and the true label.
+
+The resulting empirical nonconformity score distribution $\{ R _ { i } \} _ { i = 1 } ^ { l }$ is used to compute a critical nonconformity score $\hat { \varepsilon }$ , which corresponds to the $\lceil ( m + 1 ) ( \mathrm { i } - \bar { \alpha } ) \rceil$ -th smallest residual [48]. For a new example $\mathbf { x } ^ { ( l + 1 ) }$ , the prediction interval is then:
+
+$$
+\Gamma ^ { \alpha } ( \mathbf { x } ^ { ( l + 1 ) } ) = [ \hat { y } ^ { ( l + 1 ) } - \hat { \varepsilon } , \hat { y } ^ { ( l + 1 ) } + \hat { \varepsilon } ] ,
+$$
+
+with $\hat { y } ^ { ( l + 1 ) } = M ( \mathbf { x } ^ { ( l + 1 ) } )$ .
+
+# 3.3 CF-RNN: ICP for multi-horizon RNNs
+
+So far we have considered the case when the labels $y \in \mathbb R$ are scalar, but multi-horizon time-series forecasts return $H$ $d$ -dimensional) values (in this work, we focus on $d = 1$ ; extending the results to multivariate time-series is left for future work). We extend the ICP framework to handle the multi-horizon forecasting setup, while maintaining the validity of the resulting multi-horizon forecast intervals—we call this the conformal forecasting framework.
+
+![](images/ca21b6032dc890cbc6c6b27dcd47cf9ac7c544e4d236e7afc42c06fe26532770.jpg)  
+Figure 2: CF-RNN uncertainty estimation procedure. (a) The calibration set is used to obtain the empirical distribution of nonconformity scores $\hat { \varepsilon } _ { h }$ , and its appropriate quantile is selected depending on the desired target coverage level. (b) Critical nonconformity scores are used to obtain the prediction interval.
+
+Let $\mathcal { D }$ be the set of exchangeable observations of the form $\left( y _ { 1 : T } , y _ { T + 1 : T + H } \right)$ , where $y _ { 1 : T }$ is the time-series consisting of $T$ observed steps, and $y _ { T + 1 : T + H }$ is the $H$ -step forecast. Note that the label $y _ { T + 1 : T + H }$ is now an $H$ -dimensional value, in contrast with the scalar $y$ value from before. Due to the sequential nature of the task, we will use an RNN as the underlying model $M$ . We set $M$ to produce multi-horizon forecasts directly (where at each time step $t$ , all values of the $H$ -step target $y _ { t + 1 : t + H }$ are predicted at the same time from a single embedding) rather than recursively (where a single prediction is obtained at a time, and successive values are obtained by iteratively feeding them back into the RNN). We motivate our choice of the direct strategy by its robustness to error accumulation [50, 3], and conditionally independent predictions given the state of $M$ (which will be important for theoretical guarantees as discussed below). We now replace the single-dimensional nonconformity score defined earlier by its $H$ -dimensional counterpart,
+
+$$
+R _ { i } = \left[ | y _ { t + 1 } ^ { ( i ) } - \hat { y } _ { t + 1 } ^ { ( i ) } | , \dots , | y _ { t + H } ^ { ( i ) } - \hat { y } _ { t + H } ^ { ( i ) } | \right] ^ { \top } ,
+$$
+
+where $\left[ \widehat { y } _ { t + 1 } ^ { ( i ) } , \ldots , \widehat { y } _ { t + H } ^ { ( i ) } \right] ^ { \top } = M ( y _ { 1 : t } ^ { ( i ) } )$ . Since the $H$ conditionally independent predictions are obtained from the same embedding, we apply Bonferroni correction to the critical calibration scores in order to maintain the desired error rate $\alpha$ . In particular, the original $\alpha$ is divided by $H$ , so that the critical nonconformity scores $\hat { \varepsilon } _ { 1 } , \dots , \hat { \varepsilon } _ { H }$ become the $\lceil ( m + 1 ) ( 1 - \alpha / H ) \rceil$ -th smallest residuals in the corresponding nonconformity score distributions. The resulting set of prediction intervals is therefore
+
+$$
+\begin{array} { r } { \int _ { 1 } ^ { \alpha } \left( y _ { ( 1 : t ) } ^ { ( l + 1 ) } \right) , \ldots , \int _ { H } ^ { \alpha } \left( y _ { ( 1 : t ) } ^ { ( l + 1 ) } \right) , } \end{array}
+$$
+
+where
+
+$$
+\begin{array} { r } { T _ { h } ^ { \alpha } \left( y _ { ( 1 : t ) } ^ { ( l + 1 ) } \right) = \left[ \hat { y } _ { t + h } ^ { ( l + 1 ) } - \hat { \varepsilon } _ { h } , \hat { y } _ { t + h } ^ { ( l + 1 ) } + \hat { \varepsilon } _ { h } \right] \quad \forall h \in \{ 1 , \ldots , H \} . } \end{array}
+$$
+
+In summary, the conformal forecasting RNN (CF-RNN) model consists of an RNN issuing point forecasts, and a conformal forecasting procedure to derive the uncertainty. The entire procedure for constructing prediction intervals in CF-RNN is illustrated in Figure 2 and summarized in Algorithm 1. Finally, we show the theoretical motivations behind our approach via the following Theorem, which provides validity for intervals obtained with the conformal forecasting procedure.
+
+Theorem 1. (Conformal forecasting validity) Let $\mathcal { D } = \left\{ \left( y _ { 1 : t } ^ { ( i ) } , y _ { t + 1 : t + H } ^ { ( i ) } \right) \right\} _ { i = 1 } ^ { l }$ be the dataset of exchangeable time-series observations and their i=1-step forecasts obtained from the same underlying probability distribution. Let M be the recurrent neural network predicting $H$ -step forecasts using the direct strategy. For any significance level $\alpha \in [ 0 , 1 ]$ , the intervals obtained with the ICP-based conformal forecasting algorithm will have the error rate of at most $\alpha$ ; alternatively,
+
+$$
+\mathbb { P } \left( \forall h \in \{ 1 , \dots , H \} . y _ { t + h } \in \left[ \hat { y } _ { t + h } - \hat { \varepsilon } _ { h } , \hat { y } _ { t + h } + \hat { \varepsilon } _ { h } \right] \right) \geq 1 - \alpha .
+$$
+
+The proof follows from conditional validity of ICP in Vovk [51] and Boole’s inequality. The full statement and detailed proof is provided in Appendix A.
+
+# Algorithm 1 Conformal forecasting RNN (CF-RNN)
+
+<table><tr><td>1:Input: A trained model M producing H-step forecasts,</td><td>m</td></tr><tr><td>2: 3:</td><td>,target error rate α. i=1 Output: Critical nonconformity scores ε1,., H.</td></tr><tr><td>4:</td><td>Initializeε1={,...,εH={.</td></tr><tr><td>5:</td><td>fori=1 to m do</td></tr><tr><td>6:</td><td>yt+1:t+H ← M(yi). (i)</td></tr><tr><td>7:</td><td>forh=1 to H do</td></tr><tr><td>8:</td><td>(i) (i)</td></tr><tr><td>9:</td><td>end for</td></tr><tr><td>10: 11:</td><td>end for for h=1toHdo</td></tr><tr><td>12:</td><td>(Bonferroni and finite sample correction)</td></tr><tr><td>13:</td><td>εh ←[(m+1)(1-α/H)]-th smallest residual in εh.</td></tr><tr><td>14:</td><td>end for</td></tr><tr><td>15:</td><td>return ε1,...,€H.</td></tr><tr><td>16:</td><td>For a new time-series example y*:t:</td></tr><tr><td></td><td></td></tr><tr><td>17:</td><td>Yt+1:t+H ←M(yi:t).</td></tr><tr><td>18:</td><td>return intervals 9t+1 ±ε1,... 9t+H ±εH.</td></tr></table>
+
+# 4 Experiments
+
+In this section, we showcase the performance of the conformal forecasting RNN (CF-RNN) model against three baselines: the frequentist blockwise jackknife RNN (BJ-RNN) [21], the multi-quantile RNN (MQ-RNN) [3], and the Monte Carlo dropout-based RNN (DP-RNN) [26]. We chose these baselines as the most popular and representative examples of the different paradigms for uncertainty estimation (frequentist, quantile prediction and Bayesian uncertainty estimation for BJ-RNN, MQRNN and DP-RNN respectively). All architectures use LSTM as the underlying recurrent neural network, and are adapted to produce direct multi-horizon forecasts. We first present the performance of CF-RNNs on synthetic data with controlled properties. Since BJ-RNNs do not scale to larger real datasets, we use smaller synthetic datasets to provide the comparison of BJ-RNNs with the other methods. Finally, we compare the performance of CF-RNNs with the remaining two baselines on three real-world medical datasets. The code is available at github.com/kamilest/conformal-rnn.
+
+# 4.1 Synthetic data
+
+We first generate the synthetic time-series consisting of two components: the autoregressive process determining the trend of the time-series, and the noise process representing the inherent uncertainty of the dataset.3 For a time-series of length $T$ , this is expressed mathematically as:
+
+$$
+y _ { t } = \sum _ { k = 0 } ^ { t } a ^ { k } \cdot x _ { k } + \epsilon _ { t } , \forall k \in \{ 1 , \ldots , T \} ,
+$$
+
+where $x _ { t } \sim \mathcal { N } ( \mu _ { x } , \sigma _ { x } ^ { 2 } )$ , $a = 0 . 9$ is the memory parameter and $\epsilon _ { t } \sim \mathcal { N } ( 0 , \sigma _ { t } ^ { 2 } )$ is the noise process. We consider five time-dependent noise variance profiles, $\sigma _ { t } ^ { 2 } = 0 . 1 t n$ and five static noise variance profiles $\sigma _ { t } ^ { 2 } = 0 . 1 n$ , for $\bar { n = \{ 1 , \ldots , 5 \} }$ .
+
+# 4.2 Results
+
+We train the models on 2000 training sequences (with CF-RNNs splitting this dataset into 1000 true training and 1000 calibration sequences) for the two noise variance profiles. We aim to forecast prediction intervals for $H$ future values $y _ { T + 1 : T + H }$ for a default coverage rate of $90 \%$ $\alpha = 0 . 1 $ ). Here, $T = 1 5$ and $H = 5$ . The RNN hyperparameters for the networks underlying the uncertainty estimation models are fixed in order to ensure fair comparison, and largely follow those provided in previous work [21]. These are detailed in the Appendix B along with the time-series model parameters. Where possible,4 we repeat the experiments five times with a new randomly generated dataset, reporting the variation in empirical joint coverage over the different realisations.
+
+Table 2: Comparison of joint coverages produced by CF-RNNs and competing baselines on autoregressive series with static or time-dependent noise profiles. Where possible, empirical joint coverages are aggregated over repeated trials with randomly generated datasets.   
+
+<table><tr><td rowspan="2" colspan="2">Noise mode</td><td colspan="4">Empirical joint coverage</td></tr><tr><td>CF-RNN</td><td>BJ-RNN</td><td>MQ-RNN</td><td>DP-RNN</td></tr><tr><td rowspan="5">Static =0.1n</td><td>n=1</td><td>92.8 ± 0.8%</td><td>100%</td><td>65.0 ± 2.7%</td><td>5.4 ± 0.5%</td></tr><tr><td>n=2</td><td>94.0 ± 0.4%</td><td>100%</td><td>65.6 ± 3.4%</td><td>5.6 ± 1.0%</td></tr><tr><td>n=3</td><td>94.6 ± 1.6%</td><td>100%</td><td>66.4 ± 1.9%</td><td>5.0 ± 0.9%</td></tr><tr><td>n=4</td><td>94.3 ± 1.4%</td><td>100%</td><td>65.2 ± 4.4%</td><td>4.7 ± 1.0%</td></tr><tr><td>n=5</td><td>94.3 ± 1.4%</td><td>100%</td><td>67.2 ± 1.6%</td><td>4.2 ± 1.0%</td></tr><tr><td rowspan="5">Time-dependent ²=0.1tn</td><td>n=1</td><td>92.7 ± 1.3%</td><td>99.4%</td><td>63.4 ± 1.5%</td><td>2.5 ± 1.1%</td></tr><tr><td>n=2</td><td>92.4 ± 0.9%</td><td>100%</td><td>60.9 ± 1.9%</td><td>0.4 ± 0.2%</td></tr><tr><td>n=3</td><td>90.9 ± 1.3%</td><td>100%</td><td>57.2 ± 2.1%</td><td>0.3 ± 0.2%</td></tr><tr><td>n=4</td><td>90.6 ± 1.2%</td><td>97.0%</td><td>57.1 ± 3.7%</td><td>0.0 ± 0.1%</td></tr><tr><td>n=5</td><td>91.1 ± 0.7%</td><td>99.4%</td><td>58.6 ± 2.1%</td><td>0.1 ± 0.1%</td></tr></table>
+
+Table 3: Prediction interval widths of CF-RNNs and competing baselines on synthetic datasets of autoregressive time-series with static or time-dependent noise profiles. The mean and standard deviation are reported over all prediction horizons and random seeds.   
+
+<table><tr><td rowspan="2" colspan="2">Noise mode</td><td colspan="4">Interval widths</td></tr><tr><td>CF-RNN</td><td>BJ-RNN</td><td>MQ-RNN</td><td>DP-RNN</td></tr><tr><td rowspan="5">Static 8=0.1n</td><td>n=1</td><td></td><td>16.45 ± 3.69</td><td>98.45 ± 25.95</td><td>9.47 ± 1.99</td><td>2.82 ± 0.33</td></tr><tr><td></td><td>n=2</td><td>16.97 ± 3.34</td><td>32.53 ± 2.92</td><td>9.63 ± 1.85</td><td>2.95± 0.37</td></tr><tr><td></td><td>n=3</td><td>17.12 ± 3.50</td><td>35.82 ± 1.59</td><td>9.72 ± 1.92</td><td>2.77 ± 0.37</td></tr><tr><td></td><td>n=4</td><td>17.34 ± 3.77</td><td>33.83 ± 2.49</td><td>9.71 ± 1.80</td><td>2.87 ± 0.35</td></tr><tr><td>n=5</td><td></td><td>16.97 ± 3.27</td><td>51.23 ± 3.21</td><td>9.84 ± 1.99</td><td>2.85 ± 0.38</td></tr><tr><td rowspan="5">Time-dependent ²=0.1tn</td><td></td><td>n=1</td><td>19.80 ± 3.61</td><td>27.09 ± 1.16</td><td>11.50 ± 1.66</td><td>3.01 ± 0.35</td></tr><tr><td>n=2</td><td></td><td>25.74 ± 3.32</td><td>104.85 ± 5.68</td><td>15.45 ± 1.68</td><td>3.15 ± 0.37</td></tr><tr><td>n=3</td><td></td><td>32.70 ± 3.97</td><td>36.45 ± 1.25</td><td>20.05 ± 2.02</td><td>3.62 ± 0.33</td></tr><tr><td></td><td>n=4</td><td>40.74 ± 4.10</td><td>33.24 ± 2.32</td><td>25.02 ± 2.11</td><td>3.91 ± 0.45</td></tr><tr><td>n=5</td><td></td><td>49.00 ± 5.58</td><td>51.45 ± 5.37</td><td>30.55 ± 2.54</td><td>4.15 ± 0.57</td></tr></table>
+
+Tables 2 and 3 compare the joint coverage uncertainty intervals of the models. Both CF-RNN and BJ-RNN empirically surpass the target joint coverage of $90 \%$ $\alpha = 0 . 1$ ) in both static and timedependent noise settings, satisfying the finite-sample frequentist coverage guarantees as required.
+
+![](images/346fc75b4c5b7c33b8e95f69509eb0ef5ede033153eb203a54f9a8a5fc222b26.jpg)  
+Figure 3: Trade-offs between the dataset size, joint coverage and interval widths. (Left and Middle) The relationship between the training dataset size, joint coverage rate (left) and average interval width (middle) for CF-RNN, MQ-RNN and DP-RNN baselines. (Right) The trade-off between the coverage rate and prediction horizon for a fixed prediction interval width in CF-RNN models with different types of the underlying RNN.
+
+Table 3 additionally illustrates that CF-RNN intervals adapt to the properties of the temporal dynamics of the dataset: when the noise is static (and the time-series more predictable), CF-RNN prediction interval widths do not change much with increasing base variance; on the other hand, when the noise profile is time-dependent—so that inherent noise of the time-series accumulates in addition to uncertainty of the model itself—the average intervals get wider with increasing dataset uncertainty. The other frequentist baseline—BJ-RNN—has markedly wider intervals than those of CF-RNN. This might be important to maintain the perfect coverage; however, we argue that as long as the coverage rate surpasses target coverage, the intervals should be as efficient (narrow) as possible to be the most informative for decision making. (Consider that infinite intervals would have perfect coverage but would not be informative.) In addition, BJ-RNNs take prohibitively long to compute (the reason for which they contain only a single seed and will also be excluded from the comparisons on real data). Conversely, the ICP procedure only requires running the trained RNN model on a calibration set (where the model can be calibrated for any desired coverage simultaneously, with no additional computational cost), at which point adding uncertainty intervals to a prediction takes constant time. On the other hand, baselines following the alternative (non-frequentist) paradigms—MQ-RNN and DP-RNN—both fail to achieve target coverage, sometimes reporting coverage rates as low as zero. For this reason, while the two models also come with narrower (more efficient) intervals, lack of coverage guarantees makes them less useful in high stakes real-world applications.
+
+Experiments on the data with controlled properties provide additional insight on the trade-offs between the desired coverage rate and how far into the future can the predictions be reliably made. These trade-offs are shown in Figure 3 as applied to the datasets with time-dependent noise variance profile $\sigma _ { t } ^ { 2 } = 0 . 1 t$ . The left and middle panels show the average performance of the CF-RNN, MQ-RNN and DP-RNN baselines depending on the training dataset size. CF-RNN is the only model to achieve and maintain the required joint coverage rate with finite number of examples; additionally, with more data (larger calibration datasets), the distribution of nonconformity scores can be specified more accurately, so the width of the intervals decreases. Finally, in the panel on the right we fix the prediction interval width and for each horizon $H$ compute the largest coverage level $1 - \alpha$ maintained by CF-RNN. As shown in the Figure, low target coverage levels allow us to make valid predictions far into the future, and ideal coverage levels can only be achieved with horizons near the prediction point. The overall trend is maintained for every recurrent neural network model $M$ .
+
+# 4.3 Experiments on real data
+
+We now demonstrate the effectiveness of our procedure on real-world time-series. We train the proposed CF-RNN architecture as well as the MQ-RNN and DP-RNN baselines on three datasets summarised in Table 4. For the first task, we use the data from the Medical Information Mart for Intensive Care (MIMIC-III) [52] dataset, where we forecast daily observations of white blood cell counts of varying lengths. For the second task, we use the electroencephalography (EEG) dataset from the UCI machine learning repository [53], where we forecast trajectories of downsampled EEG signals obtained from healthy subjects exposed to three types of visual stimuli. For the final task, we forecast daily COVID-19 cases within the United Kingdom local authority districts. All datasets are publicly available and the medical data is anonymised. We selected these datasets to represent a variety of scenarios of real time-series: the numbers of available training instances span different orders of magnitude (from hundreds to tens of thousands), the datasets have varying observation sequence lengths, different stationarity properties (e.g. the COVID-19 time-series are synchronous—each time step representing the same point in time—the others are not), different noise profiles (e.g. EEG signal data will inherently have higher frequencies than MIMIC-III white blood cell count data), and different target prediction horizons. We note that COVID-19 dataset is especially challenging as the forecasts contain a wave of infections and lockdowns, with the wave starting at different points for every region. Details on datasets and their preprocessing are provided in Appendix C.
+
+Table 4: Dataset properties. The number in parentheses under the training example column indicates how many examples were used for training when a calibration set was required, such as in the case of CF-RNNs.   
+
+<table><tr><td>Dataset</td><td># Training sequences</td><td>Window length T</td><td>Prediction horizon H</td></tr><tr><td>MIMIC-III [52]</td><td>3823 (2000)</td><td>[3,47]</td><td>2</td></tr><tr><td>EEG [53]</td><td>19200 (15360)</td><td>40</td><td>10</td></tr><tr><td>COVID-19 [54]</td><td>300 (200)</td><td>100</td><td>50</td></tr></table>
+
+Table 5: Uncertainty forecasting model performance on three real-world datasets. Coverage refers to joint coverage (higher is better), and is averaged over the random splits of the dataset and training seeds. Prediction interval lengths (lower is better) are averaged over the prediction horizons and random seeds.   
+
+<table><tr><td rowspan="2">Model</td><td colspan="2">MIMIC-III</td><td colspan="2">EEG</td><td colspan="2">COVID-19</td></tr><tr><td>Coverage</td><td>CI/PI lengths</td><td>Coverage</td><td>CI/PI lengths</td><td>Coverage</td><td>CI/PI lengths</td></tr><tr><td>DP-RNN</td><td>40.2 ± 13.9%</td><td>3.59 ± 0.90</td><td>3.3 ± 0.7%</td><td>7.39 ± 0.74</td><td>0.0 ±0.0%</td><td>61.18 ± 32.37</td></tr><tr><td>MQ-RNN</td><td>89.3 ± 1.2%</td><td>16.16 ± 3.92</td><td>48.0 ±4.0%</td><td>21.39 ± 2.36</td><td>15.0 ± 5.9%</td><td>136.56 ± 63.32</td></tr><tr><td>CF-RNN</td><td>94.0 ± 1.2%</td><td>20.59 ± 3.10</td><td>96.5 ± 1.0%</td><td>61.86 ± 18.02</td><td>89.7±5.3%</td><td>733.95 ± 582.52</td></tr></table>
+
+Performance of the models is summarised in Table 5. We note that the underlying LSTM model of the proposed CF-RNN architecture had the same hyperparameters as the competing baselines, yet fewer training instances (as some of the examples are used for the calibration procedure). Despite this, CF-RNN obtains the highest coverage for all datasets, and is the only model to empirically achieve the target joint coverage rates. While this seems to disproportionately affect the efficiency of CF-RNN intervals (as these are the widest), the predictions are indeed reliable across the range of datasets and scenarios. On the other hand, the baseline models seem to have competitive coverage with better efficiency in some settings (e.g. MIMIC-III), yet revert to unreliable predictions in less certain scenarios (e.g. COVID-19). In other words, CF-RNN adapts its prediction interval widths to reliably match the required target coverage, increasing the width for unpredictable datasets.
+
+Finally, we briefly explore the importance of and motivation behind the Bonferroni correction of the error rate in the CF-RNN calibration procedure. Table 6 shows that calibration scores without Bonferroni correction generally lead to poor joint coverage, even though independent coverage rates normally achieve the target coverage (most notable exception being the COVID-19 dataset selected for its forecasting difficulty).
+
+Table 6: Bonferroni-corrected and uncorrected empirical coverages of the CF-RNN model. Joint coverage is aggregated over the different random seeds; independent coverages present the range of observed values across all horizons and all random seeds.   
+
+<table><tr><td rowspan="2">Model</td><td colspan="2">MIMIC-III</td><td colspan="2">EEG</td><td colspan="2">COVID-19</td></tr><tr><td>Joint</td><td>Independent</td><td>Joint</td><td>Independent</td><td>Joint</td><td>Independent</td></tr><tr><td>CF-RNN</td><td>94.0 ± 1.2%</td><td>[93.8%, 96.8%]</td><td>96.5 ± 1.0%</td><td>[98.3%,99.8%]</td><td>89.7 ± 5.3%</td><td>[87.5%,100.0%]</td></tr><tr><td>Uncorrected</td><td>89.0 ± 1.4%</td><td>[89.0%,91.4%]</td><td>59.4 ± 2.4%</td><td>[85.5%,91.6%]</td><td>55.5±8.0%</td><td>[77.5%,98.8%]</td></tr></table>
+
+# 5 Conclusion
+
+In this paper, we extended the ICP framework to the multi-horizon time-series forecasting problem, providing a lightweight algorithm with theoretical guarantees for frequentist coverage. Extending from the initial investigation presented in Appendix D, future work would focus on increasing the overall efficiency of prediction intervals by reducing their width, which would involve making them more adaptive to individual observations.
+
+# Acknowledgments
+
+The authors would like to thank the reviewers for their helpful comments. This work was supported by AstraZeneca, the US Office of Naval Research (ONR) and the National Science Foundation (NSF, grant number 1722516).
+
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+        "text": "Conformal Time-Series Forecasting ",
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+    {
+        "type": "text",
+        "text": "Kamile Stankevi˙ ciˇ ut¯ e˙ University of Oxford University of Cambridge ks830@cam.ac.uk ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Ahmed M. Alaa University of California, Los Angeles ahmedmalaa@ucla.edu ",
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+        ],
+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "Mihaela van der Schaar University of Cambridge University of California, Los Angeles The Alan Turing Institute mv472@cam.ac.uk ",
+        "bbox": [
+            374,
+            277,
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+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Current approaches for (multi-horizon) time-series forecasting using recurrent neural networks (RNNs) focus on issuing point estimates, which are insufficient for informing decision-making in critical application domains wherein uncertainty estimates are also required. Existing methods for uncertainty quantification in RNNbased time-series forecasts are limited as they may require significant alterations to the underlying architecture, may be computationally complex, may be difficult to calibrate, may incur high sample complexity, and may not provide theoretical validity guarantees for the issued uncertainty intervals. In this work, we extend the inductive conformal prediction framework to the time-series forecasting setup, and propose a lightweight uncertainty estimation procedure to address the above limitations. With minimal exchangeability assumptions, our approach provides uncertainty intervals with theoretical guarantees on frequentist coverage for any multi-horizon forecast predictor and any dataset. We demonstrate the effectiveness of the conformal forecasting framework by comparing it with existing baselines on a variety of synthetic and real-world datasets. ",
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
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+            643,
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+        ],
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+        "text": "Time-series forecasting tasks are central to a broad range of application domains, including stock price predictions [1, 2], service demand forecasting [3, 4], and medical prognoses [5–7]. Recurrent neural networks (RNNs) and their variants (e.g., LSTM, GRU, etc.) constitute an instrumental class of models that are most commonly used to carry out time-series forecasting tasks [8, 9]. These models, however, are usually used to issue point predictions—i.e., singular estimates of the future values of a time-series. In many high-stakes applications—such as finance and medicine—these are not enough; estimates of uncertainty are also required for accurate risk assessment and decision-making [10]. For example, clinical practitioners need to make treatment decisions accounting for all potential scenarios, where less likely scenarios may have graver consequences and require more care compared to the more likely scenarios [11, 12]. ",
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+        "type": "text",
+        "text": "While various methods for uncertainty estimation in standard feed-forward neural networks have been recently proposed [13–15], equivalent methods for RNN-based time-series models are still under-explored. Existing solutions include Bayesian recurrent neural networks [16–18], quantile regression models [3, 19], latent variable models with deep state-space architectures [6, 20], and post-hoc uncertainty estimates using bootstrapping, jackknife or other ensembling procedures [21– 23]. Each of these solutions has its own limitations: Bayesian models may be difficult to calibrate, quantile predictors may “overfit” their uncertainty estimates, and bootstrapping methods scale poorly for RNNs with large number of parameters. Almost all existing methods share at least one of the two major drawbacks: (1) they require substantial modifications to the underlying model architecture, and (2) they provide no theoretical guarantees on frequentist coverage, any of the exceptions being computationally intractable. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We aim to address the above limitations by adapting conformal prediction (CP) [24, 25]—a framework used to derive prediction intervals with guaranteed finite-sample frequentist coverage—to the timeseries forecasting setup. CP has originally been designed to construct prediction intervals for scalar targets; on the other hand, observations and predictions in time-series forecasting involve temporally dependent, potentially multivariate sequences that are not, in general, directly comparable due to differences in observation lengths, irregular frequencies, non-stationarity, and other variations in temporal dynamics (comparison between training points being a key step in CP). We extend CP to a novel, computationally efficient conformal forecasting framework that can leverage any underlying point forecasting model to produce multi-step prediction intervals with coverage guarantees across the prediction horizon. We focus on RNN-based conformal forecasting architectures, which we call conformal forecasting RNNs (CF-RNNs), and explore their effectiveness in providing valid and efficient coverage intervals. ",
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+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
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+    {
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+        "text": "Most previous work in the area of uncertainty quantification in deep learning focuses on feed-forward neural network models. Much less work has been done on uncertainty estimation for time-series models. In what follows, we discuss previous methods developed for uncertainty estimation for RNNs, which we also summarise in Table 1. ",
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+    {
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+        "img_path": "images/a6a69d10e31e7a2dd6634f491c13b0db50f7733b32f3d37ea46813179edc6c6f.jpg",
+        "table_caption": [
+            "Table 1: Overview of the most popular RNN-based probabilistic forecasting methods. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Paradigm</td><td>Architecture</td><td>Time-series observations</td><td>Frequentist coverage</td></tr><tr><td>Bayesian RNNs [16-18]</td><td>Bayesian</td><td>Built-in</td><td>Multiple</td><td></td></tr><tr><td>Monte Carlo dropout [26]</td><td>Bayesian (approx.)</td><td>Built-in</td><td>Multiple</td><td></td></tr><tr><td>MQ-[3], SQF-RNN [19]</td><td></td><td>Built-in</td><td>Multiple</td><td>一</td></tr><tr><td>BJ-RNN [21]</td><td>Frequentist</td><td>Post-hoc</td><td>Multiple</td><td>1-2α</td></tr><tr><td>EnbPI [27]</td><td>Frequentist (approx.)</td><td>Ensemble</td><td>Single</td><td>1-α</td></tr><tr><td>CF-RNN (proposed)</td><td>Frequentist</td><td>Post-hoc</td><td>Multiple</td><td>1-α</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Bayesian RNNs [16–18] extend the ideas of Bayesian inference to RNN models, expressing the model (epistemic1) uncertainty through distributions on model parameters [28, 29]. Exact Bayesian inference quickly becomes infeasible, however; various approximations based on Markov chain Monte Carlo [30–33] or variational inference [34–37] are needed. Bayesian neural networks depend on significant changes in the underlying model architecture (at least doubling the number of parameters), and rely on a good choice of prior (which may be challenging in practice). While simplifying techniques such as Monte Carlo dropout [26] (with RNN-specific adaptation in Gal and Ghahramani [38]) exist, they are often difficult to calibrate [21]. ",
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+    {
+        "type": "text",
+        "text": "Quantile RNNs can be viewed as a deep neural network extension of quantile regression [39] for sequential data: instead of returning a series of point estimates across the prediction horizon, quantile RNNs learn the prediction intervals directly, with upper and lower bounds of the forecast as separate prediction targets. The standard approach to achieve this is to use the appropriate pinball loss function as the objective. While successful applications of this approach in time-series forecasting exist [3], naively learning individual bounds may have problems such as quantile crossing; more recent approaches Gasthaus et al. [19] resolve this by fitting the entire quantile function. Quantile RNNs are additionally at risk of quantile overfitting due to poor sample complexity [21]. ",
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+        "text": "Ensembles are based on the principle of training and combining multiple models, e.g. deep ensembles trained on different random initialisations [40, 41], or models retrained on partial datasets (jackknife or bootstrap resampling-based RNNs, [22, 21]). Deep neural network ensembles are in general not mathematically principled for uncertainty quantification [42]; while resampling-based models resolve this and provide post-hoc frequentist coverage guarantees, they are instead limited in their time and space complexity. For example, exact inference on the state-of-the-art blockwise jackknife RNN (BJ-RNN) model [21] takes ${ \\cal O } ( P ^ { 3 } )$ time for $P$ parameters, and even with simplifying approximations—which in turn deteriorate accuracy—the model does not scale beyond small datasets. ",
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+            "Figure 1: Time-series observation paradigms. (Left) The dataset is assumed to comprise a single timeseries, with observations being individual time-steps within the time-series. These observations are temporally dependent. (Right) The dataset consists of a set of independent time-series, where the entire series is treated as an observation. Independence of time-series implies their exchangeability. "
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+        "text": "Conformal prediction (CP) For a given significance level (error rate) $\\alpha$ , the goal of CP [24, 25] is to return a prediction region $\\varGamma ^ { \\alpha }$ that is guaranteed to contain the true value with probability of at least $( 1 - \\alpha )$ . In regression problems (such as time-series forecasting), CP is modified to work inductively using an additional calibration set and an underlying model—an approach called inductive conformal prediction (ICP) [43, 44]. Little work has been done applying (I)CP methods for time-series forecasting; the main challenge is that CP assumes exchangeability, where any permutation of the dataset observations is equiprobable. However, the time-steps within a time-series are inherently non-exchangeable due to temporal dependencies (Figure 1, left); naively applying CP to derive forecast intervals from a given time-series is therefore not methodologically valid and lacks the validity guarantees. One notable exception is the EnbPI model [27], which bypasses the exchangeability assumption (introducing some others) and uses an ensemble of bootstrapped estimators to provide approximately valid intervals. However, we argue that learning from a single time-series—while useful in cases where indeed only one time-series is available—may not be optimal in settings where datasets contain multiple time-series, the shared patterns of which could potentially be exploited (Figure 1, right). To the best of our knowledge, no existing method has applied CP to the latter forecasting setting (despite it being more methodologically grounded); yet the datasets of multiple time-series are increasingly common and useful [45]. ",
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+    {
+        "type": "text",
+        "text": "3 Conformal forecasting RNNs (CF-RNNs) ",
+        "text_level": 1,
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+        "text": "In this Section, we introduce the conformal forecasting RNN (CF-RNN) model. We start off by formalizing the multi-horizon time-series forecasting problem in Section 3.1, and providing the necessary background on inductive conformal prediction (ICP) for regression tasks in Section 3.2. We introduce the details of the conformal forecasting procedure in Section 3.3. ",
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+        "type": "text",
+        "text": "3.1 Multi-horizon time-series forecasting ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Let $y _ { t : t ^ { \\prime } } = ( y _ { t } , y _ { t + 1 } \\ldots , y _ { t ^ { \\prime } } )$ be a time-series of $d$ -dimensional observations $y _ { t } , \\ldots , y _ { t ^ { \\prime } } \\in \\mathbb R ^ { d }$ that start at time step $t$ and end at time step $t ^ { \\prime }$ . A multi-horizon time-series forecast predicts future values ",
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+        "img_path": "images/8da51189ca7397e7a2eb219be1e7ea8c476e9bf96454cc959f8ca832528be0b7.jpg",
+        "text": "$$\n\\hat { y } _ { ( t ^ { \\prime } + 1 ) : ( t ^ { \\prime } + H ) } = \\left( \\hat { y } _ { t ^ { \\prime } + 1 } , \\ldots , \\hat { y } _ { t ^ { \\prime } + H } \\right) \\in \\mathbb { R } ^ { H \\times d } ,\n$$",
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+        "type": "text",
+        "text": "given the history of observed values $y _ { 1 : t ^ { \\prime } }$ , where $H$ is the number of steps to be predicted (the prediction horizon). For critical applications, we are interested in the uncertainty associated with the forecast—for each time step $h$ in the prediction horizon, we would like to obtain prediction intervals of the form $[ \\hat { y } _ { t + h } ^ { L } , \\hat { y } _ { t + h } ^ { U } ]$ , $h \\in \\{ 1 , \\ldots , H \\}$ , so that the ground truth value $y _ { t + h }$ is contained in the interval with a sufficiently high probability. We fix a desired significance level (or error rate) $\\alpha$ , such that the ground-truth values of the entire time-series trajectory are contained within the intervals; i.e., ",
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+        "text": "$$\n\\mathbb { P } \\left[ y _ { t + h } \\in [ \\hat { y } _ { t + h } ^ { L } , \\hat { y } _ { t + h } ^ { U } ] , \\forall h \\in \\{ 1 , \\dots , H \\} \\right] \\geq 1 - \\alpha .\n$$",
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+        "type": "text",
+        "text": "3.2 Inductive conformal prediction (ICP) ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Given a set of observations $\\mathcal { D } = \\{ ( \\mathbf { x } ^ { ( i ) } , y ^ { ( i ) } ) \\} _ { i = 1 } ^ { l }$ and a new example $\\mathbf { x } ^ { ( l + 1 ) }$ , the ICP procedure [24, 46, 47] returns a prediction interval $\\varGamma ^ { \\alpha }$ such that the property of validity is satisfied: ",
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+        "text": "Property 1. (Validity) Under the exhangeability assumption, any conformal predictor will return the prediction region $\\begin{array} { r } { \\dot { T } ^ { \\alpha } ( \\mathbf { x } ^ { ( i ) } ) } \\end{array}$ such that the probability of error $y ^ { ( l + 1 ) } \\not \\in { \\Gamma } ^ { \\alpha } ( \\dot { \\mathbf { x } } ^ { ( l + 1 ) } )$ is not greater than $\\alpha$ . Alternatively: ",
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+        "text": "$$\n\\mathbb { P } [ y ^ { ( l + 1 ) } \\in { \\Gamma } ^ { \\alpha } ( \\mathbf { x } ^ { ( l + 1 ) } ) \\mid \\mathcal { D } ] \\geq 1 - \\alpha .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "The conformal prediction framework is distribution-free (i.e. it does not have any assumptions on the distribution of the underlying data $\\mathcal { D }$ ), and applies to any underlying predictive model as long as the exchangeability assumption is satisfied: ",
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+        "text": "Assumption 1. (Exchangeability) In a dataset of l observations $\\{ ( \\mathbf { x } ^ { ( i ) } , y ^ { ( i ) } ) \\} _ { i = 1 } ^ { l }$ , any of its $l !$ permutations are equiprobable. Note that independent identically distributed (iid) observations satisfy exchangeability. ",
+        "bbox": [
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+        "type": "text",
+        "text": "The inductive2 variant of CP operates by splitting the training set into the proper training set of size $n$ and a calibration set of size $m$ : $\\mathcal { D } = \\mathcal { D } _ { \\mathrm { t r a i n } } \\cup \\mathcal { D } _ { \\mathrm { c a l } }$ . The proper training set is used to train the underlying (auxiliary) model $M$ , and the calibration set is used to obtain the nonconformity scores, which measure how unusual is the given example compared to previously observed data. While CP guarantees validity for any nonconformity score (including a random number generator), the most commonly used nonconformity score in regression is of the form ",
+        "bbox": [
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+        "img_path": "images/58b2fdc6ed98ad07fd2c015e4209e3ee04c0a1cf34a885146d451224aa368cdd.jpg",
+        "text": "$$\nR _ { i } = A ( \\mathcal D , ( { \\mathbf x } ^ { ( i ) } , y ^ { ( i ) } ) ) = A ( M ( { \\mathbf x } ^ { ( i ) } | \\mathcal D ) , y ^ { ( i ) } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $\\varDelta$ is some distance metric. While any choice for $M$ is valid, the best architecture depends on the dataset and the problem. When $\\varDelta ( \\hat { y } , y ) = | \\hat { y } - y |$ , the nonconformity score $R _ { i } = | \\hat { y } ^ { ( i ) } - y ^ { ( i ) } |$ corresponds to the residual error between the prediction of the underlying model and the true label. ",
+        "bbox": [
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+        "type": "text",
+        "text": "The resulting empirical nonconformity score distribution $\\{ R _ { i } \\} _ { i = 1 } ^ { l }$ is used to compute a critical nonconformity score $\\hat { \\varepsilon }$ , which corresponds to the $\\lceil ( m + 1 ) ( \\mathrm { i } - \\bar { \\alpha } ) \\rceil$ -th smallest residual [48]. For a new example $\\mathbf { x } ^ { ( l + 1 ) }$ , the prediction interval is then: ",
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+    {
+        "type": "equation",
+        "img_path": "images/05ae12ab301dba8a2e60ae4ffe6faa4986e645efdb1677ca5346bc8fc84d13fc.jpg",
+        "text": "$$\n\\Gamma ^ { \\alpha } ( \\mathbf { x } ^ { ( l + 1 ) } ) = [ \\hat { y } ^ { ( l + 1 ) } - \\hat { \\varepsilon } , \\hat { y } ^ { ( l + 1 ) } + \\hat { \\varepsilon } ] ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "with $\\hat { y } ^ { ( l + 1 ) } = M ( \\mathbf { x } ^ { ( l + 1 ) } )$ . ",
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+        "type": "text",
+        "text": "3.3 CF-RNN: ICP for multi-horizon RNNs ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "So far we have considered the case when the labels $y \\in \\mathbb R$ are scalar, but multi-horizon time-series forecasts return $H$ $d$ -dimensional) values (in this work, we focus on $d = 1$ ; extending the results to multivariate time-series is left for future work). We extend the ICP framework to handle the multi-horizon forecasting setup, while maintaining the validity of the resulting multi-horizon forecast intervals—we call this the conformal forecasting framework. ",
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+        "img_path": "images/ca21b6032dc890cbc6c6b27dcd47cf9ac7c544e4d236e7afc42c06fe26532770.jpg",
+        "image_caption": [
+            "Figure 2: CF-RNN uncertainty estimation procedure. (a) The calibration set is used to obtain the empirical distribution of nonconformity scores $\\hat { \\varepsilon } _ { h }$ , and its appropriate quantile is selected depending on the desired target coverage level. (b) Critical nonconformity scores are used to obtain the prediction interval. "
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+        "type": "text",
+        "text": "Let $\\mathcal { D }$ be the set of exchangeable observations of the form $\\left( y _ { 1 : T } , y _ { T + 1 : T + H } \\right)$ , where $y _ { 1 : T }$ is the time-series consisting of $T$ observed steps, and $y _ { T + 1 : T + H }$ is the $H$ -step forecast. Note that the label $y _ { T + 1 : T + H }$ is now an $H$ -dimensional value, in contrast with the scalar $y$ value from before. Due to the sequential nature of the task, we will use an RNN as the underlying model $M$ . We set $M$ to produce multi-horizon forecasts directly (where at each time step $t$ , all values of the $H$ -step target $y _ { t + 1 : t + H }$ are predicted at the same time from a single embedding) rather than recursively (where a single prediction is obtained at a time, and successive values are obtained by iteratively feeding them back into the RNN). We motivate our choice of the direct strategy by its robustness to error accumulation [50, 3], and conditionally independent predictions given the state of $M$ (which will be important for theoretical guarantees as discussed below). We now replace the single-dimensional nonconformity score defined earlier by its $H$ -dimensional counterpart, ",
+        "bbox": [
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+        "page_idx": 4
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+    {
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+        "img_path": "images/eda5553a131e2f0ad32af957b2965c12a3266706b8c8153f06114aec3c645c3d.jpg",
+        "text": "$$\nR _ { i } = \\left[ | y _ { t + 1 } ^ { ( i ) } - \\hat { y } _ { t + 1 } ^ { ( i ) } | , \\dots , | y _ { t + H } ^ { ( i ) } - \\hat { y } _ { t + H } ^ { ( i ) } | \\right] ^ { \\top } ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\left[ \\widehat { y } _ { t + 1 } ^ { ( i ) } , \\ldots , \\widehat { y } _ { t + H } ^ { ( i ) } \\right] ^ { \\top } = M ( y _ { 1 : t } ^ { ( i ) } )$ . Since the $H$ conditionally independent predictions are obtained from the same embedding, we apply Bonferroni correction to the critical calibration scores in order to maintain the desired error rate $\\alpha$ . In particular, the original $\\alpha$ is divided by $H$ , so that the critical nonconformity scores $\\hat { \\varepsilon } _ { 1 } , \\dots , \\hat { \\varepsilon } _ { H }$ become the $\\lceil ( m + 1 ) ( 1 - \\alpha / H ) \\rceil$ -th smallest residuals in the corresponding nonconformity score distributions. The resulting set of prediction intervals is therefore ",
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+        "text": "$$\n\\begin{array} { r } { \\int _ { 1 } ^ { \\alpha } \\left( y _ { ( 1 : t ) } ^ { ( l + 1 ) } \\right) , \\ldots , \\int _ { H } ^ { \\alpha } \\left( y _ { ( 1 : t ) } ^ { ( l + 1 ) } \\right) , } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where ",
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+        "text": "$$\n\\begin{array} { r } { T _ { h } ^ { \\alpha } \\left( y _ { ( 1 : t ) } ^ { ( l + 1 ) } \\right) = \\left[ \\hat { y } _ { t + h } ^ { ( l + 1 ) } - \\hat { \\varepsilon } _ { h } , \\hat { y } _ { t + h } ^ { ( l + 1 ) } + \\hat { \\varepsilon } _ { h } \\right] \\quad \\forall h \\in \\{ 1 , \\ldots , H \\} . } \\end{array}\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "In summary, the conformal forecasting RNN (CF-RNN) model consists of an RNN issuing point forecasts, and a conformal forecasting procedure to derive the uncertainty. The entire procedure for constructing prediction intervals in CF-RNN is illustrated in Figure 2 and summarized in Algorithm 1. Finally, we show the theoretical motivations behind our approach via the following Theorem, which provides validity for intervals obtained with the conformal forecasting procedure. ",
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+        "text": "Theorem 1. (Conformal forecasting validity) Let $\\mathcal { D } = \\left\\{ \\left( y _ { 1 : t } ^ { ( i ) } , y _ { t + 1 : t + H } ^ { ( i ) } \\right) \\right\\} _ { i = 1 } ^ { l }$ be the dataset of exchangeable time-series observations and their i=1-step forecasts obtained from the same underlying probability distribution. Let M be the recurrent neural network predicting $H$ -step forecasts using the direct strategy. For any significance level $\\alpha \\in [ 0 , 1 ]$ , the intervals obtained with the ICP-based conformal forecasting algorithm will have the error rate of at most $\\alpha$ ; alternatively, ",
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+        "type": "text",
+        "text": "",
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+        "type": "equation",
+        "img_path": "images/8f99c0c718d9daf997cce4b36b75c87403443f44e2d40bf8be39e08f35d1334f.jpg",
+        "text": "$$\n\\mathbb { P } \\left( \\forall h \\in \\{ 1 , \\dots , H \\} . y _ { t + h } \\in \\left[ \\hat { y } _ { t + h } - \\hat { \\varepsilon } _ { h } , \\hat { y } _ { t + h } + \\hat { \\varepsilon } _ { h } \\right] \\right) \\geq 1 - \\alpha .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The proof follows from conditional validity of ICP in Vovk [51] and Boole’s inequality. The full statement and detailed proof is provided in Appendix A. ",
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+    {
+        "type": "text",
+        "text": "Algorithm 1 Conformal forecasting RNN (CF-RNN) ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "table",
+        "img_path": "images/6ef8b73f34343949b2945823d9748960cfe7114470bda97bfb2ab7e4888ab3ab.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>1:Input: A trained model M producing H-step forecasts,</td><td>m</td></tr><tr><td>2: 3:</td><td>,target error rate α. i=1 Output: Critical nonconformity scores ε1,., H.</td></tr><tr><td>4:</td><td>Initializeε1={,...,εH={.</td></tr><tr><td>5:</td><td>fori=1 to m do</td></tr><tr><td>6:</td><td>yt+1:t+H ← M(yi). (i)</td></tr><tr><td>7:</td><td>forh=1 to H do</td></tr><tr><td>8:</td><td>(i) (i)</td></tr><tr><td>9:</td><td>end for</td></tr><tr><td>10: 11:</td><td>end for for h=1toHdo</td></tr><tr><td>12:</td><td>(Bonferroni and finite sample correction)</td></tr><tr><td>13:</td><td>εh ←[(m+1)(1-α/H)]-th smallest residual in εh.</td></tr><tr><td>14:</td><td>end for</td></tr><tr><td>15:</td><td>return ε1,...,€H.</td></tr><tr><td>16:</td><td>For a new time-series example y*:t:</td></tr><tr><td></td><td></td></tr><tr><td>17:</td><td>Yt+1:t+H ←M(yi:t).</td></tr><tr><td>18:</td><td>return intervals 9t+1 ±ε1,... 9t+H ±εH.</td></tr></table>",
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+        "type": "text",
+        "text": "4 Experiments ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we showcase the performance of the conformal forecasting RNN (CF-RNN) model against three baselines: the frequentist blockwise jackknife RNN (BJ-RNN) [21], the multi-quantile RNN (MQ-RNN) [3], and the Monte Carlo dropout-based RNN (DP-RNN) [26]. We chose these baselines as the most popular and representative examples of the different paradigms for uncertainty estimation (frequentist, quantile prediction and Bayesian uncertainty estimation for BJ-RNN, MQRNN and DP-RNN respectively). All architectures use LSTM as the underlying recurrent neural network, and are adapted to produce direct multi-horizon forecasts. We first present the performance of CF-RNNs on synthetic data with controlled properties. Since BJ-RNNs do not scale to larger real datasets, we use smaller synthetic datasets to provide the comparison of BJ-RNNs with the other methods. Finally, we compare the performance of CF-RNNs with the remaining two baselines on three real-world medical datasets. The code is available at github.com/kamilest/conformal-rnn. ",
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+        "type": "text",
+        "text": "4.1 Synthetic data ",
+        "text_level": 1,
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+        "text": "We first generate the synthetic time-series consisting of two components: the autoregressive process determining the trend of the time-series, and the noise process representing the inherent uncertainty of the dataset.3 For a time-series of length $T$ , this is expressed mathematically as: ",
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+        "type": "equation",
+        "img_path": "images/fa18ffaa78c773ea9920d7e7acfb11e3b75334cd3cae2c729249a5fd27ad0232.jpg",
+        "text": "$$\ny _ { t } = \\sum _ { k = 0 } ^ { t } a ^ { k } \\cdot x _ { k } + \\epsilon _ { t } , \\forall k \\in \\{ 1 , \\ldots , T \\} ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $x _ { t } \\sim \\mathcal { N } ( \\mu _ { x } , \\sigma _ { x } ^ { 2 } )$ , $a = 0 . 9$ is the memory parameter and $\\epsilon _ { t } \\sim \\mathcal { N } ( 0 , \\sigma _ { t } ^ { 2 } )$ is the noise process. We consider five time-dependent noise variance profiles, $\\sigma _ { t } ^ { 2 } = 0 . 1 t n$ and five static noise variance profiles $\\sigma _ { t } ^ { 2 } = 0 . 1 n$ , for $\\bar { n = \\{ 1 , \\ldots , 5 \\} }$ . ",
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+    {
+        "type": "text",
+        "text": "4.2 Results ",
+        "text_level": 1,
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+        "text": "We train the models on 2000 training sequences (with CF-RNNs splitting this dataset into 1000 true training and 1000 calibration sequences) for the two noise variance profiles. We aim to forecast prediction intervals for $H$ future values $y _ { T + 1 : T + H }$ for a default coverage rate of $90 \\%$ $\\alpha = 0 . 1 $ ). Here, $T = 1 5$ and $H = 5$ . The RNN hyperparameters for the networks underlying the uncertainty estimation models are fixed in order to ensure fair comparison, and largely follow those provided in previous work [21]. These are detailed in the Appendix B along with the time-series model parameters. Where possible,4 we repeat the experiments five times with a new randomly generated dataset, reporting the variation in empirical joint coverage over the different realisations. ",
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+        "type": "table",
+        "img_path": "images/0331e56ed22881bfbee26a2114badde042dc7044f8a0bb5f357ec781d9bd5707.jpg",
+        "table_caption": [
+            "Table 2: Comparison of joint coverages produced by CF-RNNs and competing baselines on autoregressive series with static or time-dependent noise profiles. Where possible, empirical joint coverages are aggregated over repeated trials with randomly generated datasets. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\" colspan=\"2\">Noise mode</td><td colspan=\"4\">Empirical joint coverage</td></tr><tr><td>CF-RNN</td><td>BJ-RNN</td><td>MQ-RNN</td><td>DP-RNN</td></tr><tr><td rowspan=\"5\">Static =0.1n</td><td>n=1</td><td>92.8 ± 0.8%</td><td>100%</td><td>65.0 ± 2.7%</td><td>5.4 ± 0.5%</td></tr><tr><td>n=2</td><td>94.0 ± 0.4%</td><td>100%</td><td>65.6 ± 3.4%</td><td>5.6 ± 1.0%</td></tr><tr><td>n=3</td><td>94.6 ± 1.6%</td><td>100%</td><td>66.4 ± 1.9%</td><td>5.0 ± 0.9%</td></tr><tr><td>n=4</td><td>94.3 ± 1.4%</td><td>100%</td><td>65.2 ± 4.4%</td><td>4.7 ± 1.0%</td></tr><tr><td>n=5</td><td>94.3 ± 1.4%</td><td>100%</td><td>67.2 ± 1.6%</td><td>4.2 ± 1.0%</td></tr><tr><td rowspan=\"5\">Time-dependent ²=0.1tn</td><td>n=1</td><td>92.7 ± 1.3%</td><td>99.4%</td><td>63.4 ± 1.5%</td><td>2.5 ± 1.1%</td></tr><tr><td>n=2</td><td>92.4 ± 0.9%</td><td>100%</td><td>60.9 ± 1.9%</td><td>0.4 ± 0.2%</td></tr><tr><td>n=3</td><td>90.9 ± 1.3%</td><td>100%</td><td>57.2 ± 2.1%</td><td>0.3 ± 0.2%</td></tr><tr><td>n=4</td><td>90.6 ± 1.2%</td><td>97.0%</td><td>57.1 ± 3.7%</td><td>0.0 ± 0.1%</td></tr><tr><td>n=5</td><td>91.1 ± 0.7%</td><td>99.4%</td><td>58.6 ± 2.1%</td><td>0.1 ± 0.1%</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/73d7fd1aa0dda7f2aae8ebc66d9557a3a8828a67967e15473314a458527d0ade.jpg",
+        "table_caption": [
+            "Table 3: Prediction interval widths of CF-RNNs and competing baselines on synthetic datasets of autoregressive time-series with static or time-dependent noise profiles. The mean and standard deviation are reported over all prediction horizons and random seeds. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\" colspan=\"2\">Noise mode</td><td colspan=\"4\">Interval widths</td></tr><tr><td>CF-RNN</td><td>BJ-RNN</td><td>MQ-RNN</td><td>DP-RNN</td></tr><tr><td rowspan=\"5\">Static 8=0.1n</td><td>n=1</td><td></td><td>16.45 ± 3.69</td><td>98.45 ± 25.95</td><td>9.47 ± 1.99</td><td>2.82 ± 0.33</td></tr><tr><td></td><td>n=2</td><td>16.97 ± 3.34</td><td>32.53 ± 2.92</td><td>9.63 ± 1.85</td><td>2.95± 0.37</td></tr><tr><td></td><td>n=3</td><td>17.12 ± 3.50</td><td>35.82 ± 1.59</td><td>9.72 ± 1.92</td><td>2.77 ± 0.37</td></tr><tr><td></td><td>n=4</td><td>17.34 ± 3.77</td><td>33.83 ± 2.49</td><td>9.71 ± 1.80</td><td>2.87 ± 0.35</td></tr><tr><td>n=5</td><td></td><td>16.97 ± 3.27</td><td>51.23 ± 3.21</td><td>9.84 ± 1.99</td><td>2.85 ± 0.38</td></tr><tr><td rowspan=\"5\">Time-dependent ²=0.1tn</td><td></td><td>n=1</td><td>19.80 ± 3.61</td><td>27.09 ± 1.16</td><td>11.50 ± 1.66</td><td>3.01 ± 0.35</td></tr><tr><td>n=2</td><td></td><td>25.74 ± 3.32</td><td>104.85 ± 5.68</td><td>15.45 ± 1.68</td><td>3.15 ± 0.37</td></tr><tr><td>n=3</td><td></td><td>32.70 ± 3.97</td><td>36.45 ± 1.25</td><td>20.05 ± 2.02</td><td>3.62 ± 0.33</td></tr><tr><td></td><td>n=4</td><td>40.74 ± 4.10</td><td>33.24 ± 2.32</td><td>25.02 ± 2.11</td><td>3.91 ± 0.45</td></tr><tr><td>n=5</td><td></td><td>49.00 ± 5.58</td><td>51.45 ± 5.37</td><td>30.55 ± 2.54</td><td>4.15 ± 0.57</td></tr></table>",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Tables 2 and 3 compare the joint coverage uncertainty intervals of the models. Both CF-RNN and BJ-RNN empirically surpass the target joint coverage of $90 \\%$ $\\alpha = 0 . 1$ ) in both static and timedependent noise settings, satisfying the finite-sample frequentist coverage guarantees as required. ",
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+    {
+        "type": "image",
+        "img_path": "images/346fc75b4c5b7c33b8e95f69509eb0ef5ede033153eb203a54f9a8a5fc222b26.jpg",
+        "image_caption": [
+            "Figure 3: Trade-offs between the dataset size, joint coverage and interval widths. (Left and Middle) The relationship between the training dataset size, joint coverage rate (left) and average interval width (middle) for CF-RNN, MQ-RNN and DP-RNN baselines. (Right) The trade-off between the coverage rate and prediction horizon for a fixed prediction interval width in CF-RNN models with different types of the underlying RNN. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Table 3 additionally illustrates that CF-RNN intervals adapt to the properties of the temporal dynamics of the dataset: when the noise is static (and the time-series more predictable), CF-RNN prediction interval widths do not change much with increasing base variance; on the other hand, when the noise profile is time-dependent—so that inherent noise of the time-series accumulates in addition to uncertainty of the model itself—the average intervals get wider with increasing dataset uncertainty. The other frequentist baseline—BJ-RNN—has markedly wider intervals than those of CF-RNN. This might be important to maintain the perfect coverage; however, we argue that as long as the coverage rate surpasses target coverage, the intervals should be as efficient (narrow) as possible to be the most informative for decision making. (Consider that infinite intervals would have perfect coverage but would not be informative.) In addition, BJ-RNNs take prohibitively long to compute (the reason for which they contain only a single seed and will also be excluded from the comparisons on real data). Conversely, the ICP procedure only requires running the trained RNN model on a calibration set (where the model can be calibrated for any desired coverage simultaneously, with no additional computational cost), at which point adding uncertainty intervals to a prediction takes constant time. On the other hand, baselines following the alternative (non-frequentist) paradigms—MQ-RNN and DP-RNN—both fail to achieve target coverage, sometimes reporting coverage rates as low as zero. For this reason, while the two models also come with narrower (more efficient) intervals, lack of coverage guarantees makes them less useful in high stakes real-world applications. ",
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+    {
+        "type": "text",
+        "text": "Experiments on the data with controlled properties provide additional insight on the trade-offs between the desired coverage rate and how far into the future can the predictions be reliably made. These trade-offs are shown in Figure 3 as applied to the datasets with time-dependent noise variance profile $\\sigma _ { t } ^ { 2 } = 0 . 1 t$ . The left and middle panels show the average performance of the CF-RNN, MQ-RNN and DP-RNN baselines depending on the training dataset size. CF-RNN is the only model to achieve and maintain the required joint coverage rate with finite number of examples; additionally, with more data (larger calibration datasets), the distribution of nonconformity scores can be specified more accurately, so the width of the intervals decreases. Finally, in the panel on the right we fix the prediction interval width and for each horizon $H$ compute the largest coverage level $1 - \\alpha$ maintained by CF-RNN. As shown in the Figure, low target coverage levels allow us to make valid predictions far into the future, and ideal coverage levels can only be achieved with horizons near the prediction point. The overall trend is maintained for every recurrent neural network model $M$ . ",
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+    {
+        "type": "text",
+        "text": "4.3 Experiments on real data ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We now demonstrate the effectiveness of our procedure on real-world time-series. We train the proposed CF-RNN architecture as well as the MQ-RNN and DP-RNN baselines on three datasets summarised in Table 4. For the first task, we use the data from the Medical Information Mart for Intensive Care (MIMIC-III) [52] dataset, where we forecast daily observations of white blood cell counts of varying lengths. For the second task, we use the electroencephalography (EEG) dataset from the UCI machine learning repository [53], where we forecast trajectories of downsampled EEG signals obtained from healthy subjects exposed to three types of visual stimuli. For the final task, we forecast daily COVID-19 cases within the United Kingdom local authority districts. All datasets are publicly available and the medical data is anonymised. We selected these datasets to represent a variety of scenarios of real time-series: the numbers of available training instances span different orders of magnitude (from hundreds to tens of thousands), the datasets have varying observation sequence lengths, different stationarity properties (e.g. the COVID-19 time-series are synchronous—each time step representing the same point in time—the others are not), different noise profiles (e.g. EEG signal data will inherently have higher frequencies than MIMIC-III white blood cell count data), and different target prediction horizons. We note that COVID-19 dataset is especially challenging as the forecasts contain a wave of infections and lockdowns, with the wave starting at different points for every region. Details on datasets and their preprocessing are provided in Appendix C. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "table",
+        "img_path": "images/c338bc044a1540abb85ab945cca85c32be381a6942b1b198c0600d3eca7122ff.jpg",
+        "table_caption": [
+            "Table 4: Dataset properties. The number in parentheses under the training example column indicates how many examples were used for training when a calibration set was required, such as in the case of CF-RNNs. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td># Training sequences</td><td>Window length T</td><td>Prediction horizon H</td></tr><tr><td>MIMIC-III [52]</td><td>3823 (2000)</td><td>[3,47]</td><td>2</td></tr><tr><td>EEG [53]</td><td>19200 (15360)</td><td>40</td><td>10</td></tr><tr><td>COVID-19 [54]</td><td>300 (200)</td><td>100</td><td>50</td></tr></table>",
+        "bbox": [
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+    },
+    {
+        "type": "table",
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+        "table_caption": [
+            "Table 5: Uncertainty forecasting model performance on three real-world datasets. Coverage refers to joint coverage (higher is better), and is averaged over the random splits of the dataset and training seeds. Prediction interval lengths (lower is better) are averaged over the prediction horizons and random seeds. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Model</td><td colspan=\"2\">MIMIC-III</td><td colspan=\"2\">EEG</td><td colspan=\"2\">COVID-19</td></tr><tr><td>Coverage</td><td>CI/PI lengths</td><td>Coverage</td><td>CI/PI lengths</td><td>Coverage</td><td>CI/PI lengths</td></tr><tr><td>DP-RNN</td><td>40.2 ± 13.9%</td><td>3.59 ± 0.90</td><td>3.3 ± 0.7%</td><td>7.39 ± 0.74</td><td>0.0 ±0.0%</td><td>61.18 ± 32.37</td></tr><tr><td>MQ-RNN</td><td>89.3 ± 1.2%</td><td>16.16 ± 3.92</td><td>48.0 ±4.0%</td><td>21.39 ± 2.36</td><td>15.0 ± 5.9%</td><td>136.56 ± 63.32</td></tr><tr><td>CF-RNN</td><td>94.0 ± 1.2%</td><td>20.59 ± 3.10</td><td>96.5 ± 1.0%</td><td>61.86 ± 18.02</td><td>89.7±5.3%</td><td>733.95 ± 582.52</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "Performance of the models is summarised in Table 5. We note that the underlying LSTM model of the proposed CF-RNN architecture had the same hyperparameters as the competing baselines, yet fewer training instances (as some of the examples are used for the calibration procedure). Despite this, CF-RNN obtains the highest coverage for all datasets, and is the only model to empirically achieve the target joint coverage rates. While this seems to disproportionately affect the efficiency of CF-RNN intervals (as these are the widest), the predictions are indeed reliable across the range of datasets and scenarios. On the other hand, the baseline models seem to have competitive coverage with better efficiency in some settings (e.g. MIMIC-III), yet revert to unreliable predictions in less certain scenarios (e.g. COVID-19). In other words, CF-RNN adapts its prediction interval widths to reliably match the required target coverage, increasing the width for unpredictable datasets. ",
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+    {
+        "type": "text",
+        "text": "Finally, we briefly explore the importance of and motivation behind the Bonferroni correction of the error rate in the CF-RNN calibration procedure. Table 6 shows that calibration scores without Bonferroni correction generally lead to poor joint coverage, even though independent coverage rates normally achieve the target coverage (most notable exception being the COVID-19 dataset selected for its forecasting difficulty). ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/91ef56010eff78edfeec48263082ff30ca55c7158e5aa579f432e95950d0033a.jpg",
+        "table_caption": [
+            "Table 6: Bonferroni-corrected and uncorrected empirical coverages of the CF-RNN model. Joint coverage is aggregated over the different random seeds; independent coverages present the range of observed values across all horizons and all random seeds. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Model</td><td colspan=\"2\">MIMIC-III</td><td colspan=\"2\">EEG</td><td colspan=\"2\">COVID-19</td></tr><tr><td>Joint</td><td>Independent</td><td>Joint</td><td>Independent</td><td>Joint</td><td>Independent</td></tr><tr><td>CF-RNN</td><td>94.0 ± 1.2%</td><td>[93.8%, 96.8%]</td><td>96.5 ± 1.0%</td><td>[98.3%,99.8%]</td><td>89.7 ± 5.3%</td><td>[87.5%,100.0%]</td></tr><tr><td>Uncorrected</td><td>89.0 ± 1.4%</td><td>[89.0%,91.4%]</td><td>59.4 ± 2.4%</td><td>[85.5%,91.6%]</td><td>55.5±8.0%</td><td>[77.5%,98.8%]</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "5 Conclusion ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we extended the ICP framework to the multi-horizon time-series forecasting problem, providing a lightweight algorithm with theoretical guarantees for frequentist coverage. Extending from the initial investigation presented in Appendix D, future work would focus on increasing the overall efficiency of prediction intervals by reducing their width, which would involve making them more adaptive to individual observations. ",
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+    },
+    {
+        "type": "text",
+        "text": "Acknowledgments ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "The authors would like to thank the reviewers for their helpful comments. This work was supported by AstraZeneca, the US Office of Naval Research (ONR) and the National Science Foundation (NSF, grant number 1722516). ",
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+    {
+        "type": "text",
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