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+    {
+        "type": "text",
+        "text": "Learning Debiased Representation via Disentangled Feature Augmentation ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Jungsoo Lee\\*1,2 Eungyeup $\\mathbf { K i m } ^ { * 1 , 2 }$ Juyoung Lee2 Jihyeon Lee1 Jaegul Choo1 ",
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+        "text": "1KAIST AI, 2Kakao Enterprise, South Korea 1{bebeto, eykim94, gina3833, jchoo}@kaist.ac.kr, 2{bebeto.lee, josh.ey, michael.jy}@kakaoenterprise.com ",
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+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+        "text": "Image classification models tend to make decisions based on peripheral attributes of data items that have strong correlation with a target variable (i.e., dataset bias). These biased models suffer from the poor generalization capability when evaluated on unbiased datasets. Existing approaches for debiasing often identify and emphasize those samples with no such correlation (i.e., bias-conflicting) without defining the bias type in advance. However, such bias-conflicting samples are significantly scarce in biased datasets, limiting the debiasing capability of these approaches. This paper first presents an empirical analysis revealing that training with “diverse” bias-conflicting samples beyond a given training set is crucial for debiasing as well as the generalization capability. Based on this observation, we propose a novel feature-level data augmentation technique in order to synthesize diverse bias-conflicting samples. To this end, our method learns the disentangled representation of (1) the intrinsic attributes (i.e., those inherently defining a certain class) and (2) bias attributes (i.e., peripheral attributes causing the bias), from a large number of bias-aligned samples, the bias attributes of which have strong correlation with the target variable. Using the disentangled representation, we synthesize bias-conflicting samples that contain the diverse intrinsic attributes of bias-aligned samples by swapping their latent features. By utilizing these diversified bias-conflicting features during the training, our approach achieves superior classification accuracy and debiasing results against the existing baselines on synthetic and real-world datasets. ",
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+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Despite the recent advancement of deep neural networks, they often rely overly on the correlation between peripheral attributes and labels, referred to as dataset bias [1], especially when such strong bias is found in a given dataset. A majority of samples in the biased dataset exhibit visual attributes that are not innate but frequently co-occur with target labels (i.e., bias attributes). For example, most of the bird images in the training dataset may contain the background as the blue sky, while the birds may still be found in different places. Thus, the model trained with such a biased dataset is likely to learn the bias attributes more than intrinsic attributes, the innate visual attributes that inherently define a certain class, e.g., the wings of birds. This causes the model to learn shortcuts for classification [2], failing to generalize on the images with no such correlations (e.g., birds on grounds or grass) during the test phase. Throughout the paper, bias-aligned samples correspond to data items containing a strong correlation between bias attributes and labels (e.g., birds in the sky), while bias-conflicting samples indicate the other cases that are rarely found (e.g., birds on grounds). ",
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+    {
+        "type": "text",
+        "text": "To tackle such a task, previous studies often define a specific bias type (e.g., color and texture) in advance [3, 4, 5, 6, 7, 8, 9, 10], which enables them to design a debiasing network tailored for the predefined bias type. For example, Bahng et al. [6] leverage BagNet [11], which has limited size of receptive fields, to focus on learning color and texture. However, defining a bias type in advance 1) limits the capability of debiasing in other bias types and 2) requires expensive labor to manually identify the bias type. To handle such an issue, a recent approach [12] defines a bias based on an intuitive observation that the bias attributes are often easier to learn than the intrinsic attributes for neural networks. In this regard, they re-weight bias-conflicting samples while de-emphasizing the bias-aligned ones. However, we point out that the reason behind the limited generalization capability of existing debiasing approaches lies in the significant scarcity of bias-conflicting samples compared to the bias-aligned ones in a given training set. In other words, it is challenging to learn the debiased representation from these scarce bias-conflicting samples because the models are prone to memorize (thus being overfitted to) these samples, failing to learn the intrinsic attributes. Therefore, we claim that a neural network can learn properly debiased representation when these data items are diversified during training. ",
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+        "text": "",
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+    {
+        "type": "text",
+        "text": "We conduct a brief experiment to demonstrate the importance of diversity in debiasing. Diversity in our work indicates the different valid realization of intrinsic attributes in a certain class (e.g., thick, narrow, tilted, and scribbled digit shapes in MNIST [13]). Our observation is that training a model with diverse bias-conflicting samples beyond a given training set is crucial for learning debiased representation (Section 3.2). In this regard, synthesizing bias-conflicting samples is one of the straightforward approaches to increase the diversity of such samples. In fact, a large amount of bias-aligned samples in a given training set already contain diverse intrinsic attributes, which can work as informative sources for increasing the diversity. However, as bias and intrinsic attributes are highly entangled in their embedding space, it is difficult to extract the intrinsic ones from these bias-aligned samples. Therefore, disentangling these correlations enables to synthesize diversified bias-conflicting samples that originate from bias-aligned samples. ",
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+    {
+        "type": "text",
+        "text": "In this paper, we propose a novel feature augmentation approach via disentangled representation for debiasing. We first train two different encoders to embed images into the disentangled representation of their intrinsic and bias attributes. With the disentangled representation, we randomly swap the latent vectors extracted from different images, most of which are bias-aligned samples in our training set. These swapped features thus contain both bias and intrinsic attributes without the correlation between them, which, in turn, can work as augmented bias-conflicting samples in our training. These features include intrinsic features of bias-aligned ones, increasing the diversity of a given training set, especially for bias-conflicting data items. Furthermore, to enhance the quality of diversified features, we propose a scheduling strategy of feature augmentation which enables to utilize the representation disentangled to a certain degree. In summary, the main contributions of our work include: ",
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+        "text": "• Through our preliminary experiment, we reveal that increasing the diversity of biasconflicting samples is crucial for debiasing.   \n• Based on such an observation, we propose a novel feature augmentation method via disentangled representation for diversifying the bias-conflicting samples.   \n• We achieve the state-of-the-art performances in two synthetic datasets (i.e., Colored MNIST and Corrupted CIFAR-10) and one real-world dataset (i.e., Biased FFHQ) against existing baselines. ",
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+    {
+        "type": "text",
+        "text": "2 Related Work ",
+        "text_level": 1,
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+        "text": "Debiasing predefined bias Several existing approaches mitigate the bias by pre-defining a certain bias type, either explicitly [3, 4, 5] or implicitly [6, 7, 8, 9, 10, 14]. For example, Bahng et al. [6] and Wang et al. [7] design a color- and texture-oriented network to adversarially learn a debiased model against the biased one. However, as these methods still require a specific bias type such as texture in advance, they lack the general applicability to the datasets where the bias types are demanding to recognize. ",
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+        "type": "text",
+        "text": "Instead of defining certain types of bias, recent approaches [12, 15, 16] rely on the straightforward assumption that networks are prone to exploit the bias when it acts as a shortcut [2], i.e., easy to learn in the early training phase. Nam et al. [12] emphasize the bias-conflicting samples during training by using generalized cross-entropy loss [17]. Darlow et al. [15] and Huang et al. [16] presume that high gradient of latent vectors accounts for the shortcuts that model learns. In the line with the recent studies, we tackle debiasing without pre-defining a certain bias type. ",
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+    {
+        "type": "table",
+        "img_path": "images/83d3fedd9fc22820a4ff2be918d1b3a117cfeef5fcf061495324c81548bb9256.jpg",
+        "table_caption": [
+            "Table 1: The classification accuracy on the unbiased test sets. The diversity ratio indicates the ratio of bias-conflicting samples in the dataset pooled for each experiment. The sampling ratio refers to the ratio of bias-conflicting samples included in each mini-batch. We report the averaged accuracy over three independent trials with the standard deviation. In both datasets, we observe that the bias can be mitigated with diverse bias-conflicting samples even with a small sampling ratio. Bold and underlined values indicate the best and second best accuracy, respectively. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td>Diversity ratio</td><td>Sampling ratio</td><td>Accuracy (%)</td></tr><tr><td rowspan=\"4\">Colored MNIST</td><td>5%</td><td>50%</td><td>83.77±2.03</td></tr><tr><td>1%</td><td>50%</td><td>67.19±1.99</td></tr><tr><td>5%</td><td>1%</td><td>77.97±6.00</td></tr><tr><td>1%</td><td>1%</td><td>49.91±4.22</td></tr><tr><td rowspan=\"4\">Corrupted CIFAR-10</td><td>5%</td><td>50%</td><td>46.99±0.82</td></tr><tr><td>1%</td><td>50%</td><td>33.08±0.80</td></tr><tr><td>5%</td><td>1%</td><td>36.66±0.55</td></tr><tr><td>1%</td><td>1%</td><td>23.98±0.00</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Data augmentation for debiasing Geirhos et al. [10] mitigate the texture bias by utilizing additional training images with their styles being transferred by adaptive instance normalization (AdaIN) [18]. Minderer et al. [19] train an image-to-image translation network for removing shortcut cues in the self-supervised task. However, such image-level data augmentation is limited to resolving the predefined texture bias which can not be adopted to other general types of bias. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "One alternative is to exploit the latent space for data augmentation. For example, Darlow et al. [15] adversarially perturb the latent vectors corresponding to the high gradients to generate the samples against bias. Zhou et al. [20] mix the style of different source domains by AdaIN [18] to increase the domain generalization ability. Despite the effectiveness of the augmentation in the latent space, the strong unwanted correlation between bias attributes and labels prevents from obtaining the desirable intrinsic features. We resolve this issue by leveraging the disentangled representation in debiasing, which is widely used in image-to-image translation task [21, 22, 23]. To the best of our knowledge, no previous work in debiasing leverage this disentangled representation for the purpose of feature augmentation. For the rest of the paper, we elaborate how we perform the feature augmentation based on the disentangled representation. ",
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+    {
+        "type": "text",
+        "text": "3 Importance of Diversity in Debiasing ",
+        "text_level": 1,
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+        "text": "This section describes the details of a toy-set experiment in which we observe the importance of diversity in learning debiased representation. In Section 3.1, we first introduce the two synthetic datasets, Colored MNIST and Corrupted CIFAR-10, that we utilize for the observation. Then, we elaborate the results of the experiments in Section 3.2. ",
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+    {
+        "type": "text",
+        "text": "3.1 Dataset ",
+        "text_level": 1,
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+        "text": "Colored MNIST is a modified MNIST dataset [13] with the color bias. We select ten distinct colors and inject each color on the foreground of each digit to create color bias. By adjusting the number of bias-conflicting data samples in the training set, we obtain four different datasets with the ratio of bias-conflicting samples of $0 . 5 \\%$ , $1 \\%$ , $2 \\%$ , and $5 \\%$ . ",
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+    {
+        "type": "text",
+        "text": "Corrupted CIFAR-10 has ten different types of texture bias applied in CIFAR-10 [24] dataset, constructed by following the design protocol of Hendrycks and Dietterich [25]. Each class is highly correlated with a certain texture (e.g., frost and brightness). Corrupted CIFAR-10 also has four different datasets with their correlation ratios as in Colored MNIST. ",
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+    {
+        "type": "text",
+        "text": "3.2 Increasing diversity outperforms oversampling ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "To confirm the significance of diversity of bias-conflicting samples in debiasing, we train four different settings: oversampling bias-conflicting samples by $50 \\%$ in each mini-batch (i.e., 128 from a batch size of 256), from the pool of i) $5 \\%$ dataset and ii) $1 \\%$ dataset, sampling bias-conflicting samples by $1 \\%$ in each mini-batch (i.e., 2 from a batch size of 256) from the pool of iii) $5 \\%$ dataset and iv) $1 \\%$ dataset. Oversampling provides the same amount of bias-conflicting samples as the aligned ones to the model in every training step. Bias-conflicting images sampled from the pool of $5 \\%$ dataset have more diverse appearances of bias-conflicting samples compared to those from $1 \\%$ dataset. ",
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+        "text": "Table 1 shows the image classification accuracy of each setting validated on the unbiased test images. Apparently, oversampling diverse bias-conflicting samples (first row) outperforms the other three methods. Similarly, sampling a small amount of bias-conflicting samples with the least diversity (fourth row) shows the lowest classification accuracy. The interesting finding is that sampling fewer but diverse conflicting samples in each mini-batch (third row) outperforms oversampling bias-conflicting samples with limited diversity (second row). These results lead to the conclusion that the diversity of bias-conflicting samples is a more crucial factor for learning debiased representation than the ratio of sampling in the training. As the diversity is limited (the latter case), the model can be easily overfitted to the given bias-conflicting samples, thus less likely to learn the generalized intrinsic attributes. With the Colored MNIST as an example, the shape of digits may vary. To be more specific, the digit shape may be thick, narrow, tilted, scribbled, and etc. If the bias-conflicting samples do not include certain visual facets (e.g., not including scribbled digit images) due to the limited number of samples, the model may imperfectly learn the intrinsic attributes of digit shapes. On the other hand, in the former case (third row), the model can learn multiple facets of intrinsic attributes when they are sampled from the diverse pool of datasets, resulting in learning intrinsic attributes even without oversampling the bias-conflicting images. ",
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+    {
+        "type": "text",
+        "text": "4 Debiasing via disentangled feature augmentation ",
+        "text_level": 1,
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+        "text": "Motivated by such an observation in Section 3.2, we propose a feature-level augmentation strategy for synthesizing additional bias-conflicting samples, as illustrated in Fig. 1. First, we train the two separate encoders which embed an image into disentangled latent vectors corresponding to the intrinsic and bias attributes, respectively (Section 4.1). Swapping these feature vectors among training samples enables to augment the bias-conflicting samples which no more contain a correlation between two attributes (Section 4.2). To further enhance the effectiveness, we schedule the feature augmentation after the representation is disentangled at a certain degree (Section 4.3). ",
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+    {
+        "type": "text",
+        "text": "4.1 Learning disentangled representation ",
+        "text_level": 1,
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+        "text": "In contrast to the bias-conflicting samples, a large amount of bias-aligned images have diverse appearances of their intrinsic attributes. By leveraging these attributes for augmentation, we can naturally obtain the diversified bias-conflicting samples containing the diverse intrinsic attributes. However, it remains challenging in that these attributes are strongly correlated with the bias attributes in the bias-aligned samples. Therefore, we propose to design two encoders with their linear classifiers to extract the disentangled latent vectors from the input images. ",
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+    {
+        "type": "text",
+        "text": "As shown in Fig. 1, encoders $E _ { i }$ and $E _ { b }$ embed an image $x$ into intrinsic feature vectors $z _ { i } = E _ { i } ( x )$ and bias feature vectors $z _ { b } = E _ { b } ( x )$ , respectively. Afterward, linear classifiers $C _ { i }$ and $C _ { b }$ take the concatenated vector $z = [ z _ { i } ; z _ { b } ]$ as input to predict the target label $y$ . To train $E _ { i }$ and $C _ { i }$ as intrinsic feature extractor and $E _ { b }$ and $C _ { b }$ as bias extractor, we utilize the relative difficulty score of each data sample, proposed in the previous work of Nam et al. [12]. More specifically, we train $E _ { b }$ and $C _ { b }$ to be overfitted to the bias attributes by utilizing the generalized cross entropy (GCE) [17], while $E _ { i }$ and $C _ { i }$ are trained with the cross entropy (CE) loss. Then, the samples with high CE loss from $C _ { b }$ can be regarded as the bias-conflicting samples compared to the samples with low CE loss. In this regard, we obtain the relative difficulty score of each data sample as ",
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+        "type": "equation",
+        "img_path": "images/bbbe47c3a4a358548e4ad83e7af80b437263cacd63d1efee67850dec87318fbc.jpg",
+        "text": "$$\nW ( z ) = \\frac { C E ( C _ { b } ( z ) , y ) } { C E ( C _ { i } ( z ) , y ) + C E ( C _ { b } ( z ) , y ) } .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "As bias-conflicting samples obtain high values of $W$ , we emphasize the loss of these samples for ",
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+    {
+        "type": "text",
+        "text": "training $E _ { i }$ and $C _ { i }$ , enforcing them to learn the intrinsic attributes. Therefore, the objective function for disentanglement can be written as ",
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+        "type": "equation",
+        "img_path": "images/591988491dc4b7ed8280c6eb29fd063777f6d0c62f091cb472ec8467565a1a85.jpg",
+        "text": "$$\nL _ { \\mathrm { d i s } } = W ( z ) C E ( C _ { i } ( z ) , y ) + \\lambda _ { \\mathrm { d i s } } G C E ( C _ { b } ( z ) , y ) .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "To ensure that $C _ { i }$ and $C _ { b }$ predicts target labels mainly based on $z _ { i }$ and $z _ { b }$ , respectively, the loss ",
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+        "type": "text",
+        "text": "$C _ { i }$ is not backpropagated to $E _ { b }$ , and vice versa. ",
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+    {
+        "type": "image",
+        "img_path": "images/eedd1538e5f1b3d233b423f13d2940012c778d468a152b8756c79289fdabf7bd.jpg",
+        "image_caption": [
+            "Figure 1: The overview of our proposed debiasing approach. $( E _ { i } , C _ { i } )$ and $( E _ { b } , C _ { b } )$ are pairs of an encoder and a linear classifier trained to learn the disentangled representation of intrinsic attributes and bias attributes, respectively. With the disentangled features $z _ { i }$ and $z _ { b }$ , the feature augmentation is performed by swapping these latent vectors among different training samples, after certain iterations of training. $R$ refers to the re-weighting algorithm which implicitly differentiates bias-aligned samples and bias-conflicting samples. Each color indicates the different data samples. "
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+    {
+        "type": "text",
+        "text": "Algorithm 1 Debiasing with disentangled feature augmentation ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Input: image $x$ , label $y$ , iteration $t$ , augment iteration $t _ { \\mathrm { s w a p } }$   \nInitialize two networks $( E _ { i } , C _ { i } )$ , $( E _ { b } , C _ { b } )$   \nwhile not converged do Extract $z _ { i } , z _ { b }$ from $E _ { i } ( x )$ , $E _ { b } ( x )$ Concatenate $\\boldsymbol { z } = [ z _ { i } ; z _ { b } ]$ Update $( E _ { i } , C _ { i } )$ , $( E _ { b } , C _ { b } )$ with $L _ { \\mathrm { d i s } } = W ( z ) C E ( C _ { i } ( z ) , y ) + G C E ( C _ { b } ( z ) , y )$ if $t > t _ { \\mathrm { s w a p } }$ : Randomly permute $\\boldsymbol { z } = [ z _ { i } , z _ { b } ]$ into $z _ { \\mathrm { s w a p } } = [ z _ { i } ; \\tilde { z _ { b } } ]$ Calculate Ls $\\ L _ { \\mathrm { v a p } } = W ( z ) C E ( C _ { i } ( z _ { \\mathrm { s w a p } } ) , \\dot { y } ) + G C E ( C _ { b } ( z _ { \\mathrm { s w a p } } ) , \\tilde { y }$ ) Update $( E _ { i } , C _ { i } ^ { \\dot { \\mathbf { \\alpha } } } )$ , $( E _ { b } , C _ { b } )$ with $L _ { \\mathrm { t o t a l } } = L _ { \\mathrm { d i s } } + \\lambda _ { \\mathrm { s w a p } } L _ { \\mathrm { s w a p } }$   \nend ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.2 Feature swapping for augmentation ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "While such an architecture disentangles the intrinsic features and bias features, $E _ { i }$ and $C _ { i }$ are still mainly trained with an excessively small amount of bias-conflicting samples. Therefore, $E _ { i }$ and $C _ { i }$ fail to fully acquire the intrinsic representation of a target class. To promote further improvement in learning intrinsic feature vectors, we diversify the bias-conflicting samples by swapping the disentangled latent vectors among the training sets. In other words, we randomly permute the intrinsic features and bias features in each mini-batch and obtain $z _ { \\mathrm { s w a p } } ~ = ~ [ z _ { i } ; \\tilde { z _ { b } } ]$ where $\\tilde { z _ { b } }$ denotes the randomly permuted bias attributes of $z _ { b }$ . As the intrinsic and bias attributes in ${ z _ { \\mathrm { s w a p } } }$ are obtained from two different images, they certainly have less correlation compared to $\\boldsymbol { z } = [ z _ { i } ; \\dot { z _ { b } } ]$ where both are from the same image. Since the biased dataset is mostly composed of bias-aligned samples, these vectors are likely from the bias-aligned samples, highly diversified compared to the bias-conflicting ones. Then, $z _ { \\mathrm { s w a p } } = [ z _ { i } ; \\tilde { z _ { b } } ]$ act as augmented bias-conflicting latent vectors with diversity inherited from the bias-aligned samples. Along with $L _ { \\mathrm { d i s } }$ , we add the following loss function to train two neural networks with the augmented features ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/25f4e7d4303a98eac47c89a2ec40cb70946cff0d0f962c7a16d9f80f3b5f9cd3.jpg",
+        "text": "$$\nL _ { \\mathrm { s w a p } } = W ( z ) C E ( C _ { i } ( z _ { \\mathrm { s w a p } } ) , y ) + \\lambda _ { \\mathrm { s w a p } _ { b } } G C E ( C _ { b } ( z _ { \\mathrm { s w a p } } ) , \\tilde { y } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            757,
+            702,
+            775
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\tilde { y }$ denotes target labels for permute bias attributes $\\tilde { z }$ . Thus, total loss function is described as ",
+        "bbox": [
+            176,
+            779,
+            820,
+            796
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d84005a2615fa7be5750b598978baba859fbbf9923113d2c68f0b103a1204685.jpg",
+        "text": "$$\nL _ { \\mathrm { t o t a l } } = L _ { \\mathrm { d i s } } + \\lambda _ { \\mathrm { s w a p } } L _ { \\mathrm { s w a p } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            413,
+            809,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\lambda _ { \\mathrm { s w a p } }$ is adjusted for weighting the importance of the feature augmentation. ",
+        "bbox": [
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+            830,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.3 Scheduling the feature augmentation ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            864,
+            470,
+            880
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "While training with additional synthesized features helps to mitigate the unwanted correlation, utilizing them from the beginning of training does not improve the debiasing performance. To be more specific, in the early stage of training, the representations of $z _ { i }$ and $z _ { b }$ are imperfectly disentangled to be used as the sources of feature augmentation. Feature augmentation should be conducted after two features are disentangled at a certain degree. Without the disentangled representation, the augmented features work as noisy samples which may aggravate the debiasing performances. We verify the importance of scheduling the feature augmentation in Table 3. Our approach can be summarized with Algorithm 1. ",
+        "bbox": [
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+            883,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/ba755777f4dbc4501ef0965a443c6e1994c4a532a25d0df3ee04f0f6e490293c.jpg",
+        "image_caption": [
+            "Figure 2: Example images of datasets utilized in our work. In each dataset, the images above the dotted line indicate the bias-aligned samples while the ones below the dotted line are the bias-conflicting samples. For Colored MNIST and Corrupted CIFAR-10, each column indicates each class. For BFFHQ, the group of three columns indicates each class. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5 Experiment ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            452,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "This section demonstrates the effectiveness of feature augmentation based on disentangled representation in debiasing with both quantitative and qualitative evaluation. We compare our method with the previous approaches in debiasing with three different datasets with varied bias ratios. Then, we conduct the ablation study which demonstrates the importance of 1) learning disentangled representation, 2) feature augmentation, and 3) scheduling feature augmentation. For the qualitative evaluation, we verify how our approach disentangles the intrinsic features and bias features by visualizing them on 2D embedding space via t-SNE [26] and reconstructing images from them. ",
+        "bbox": [
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+            482,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5.1 Experiment details ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            595,
+            344,
+            611
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Baselines Our baselines consist of vanilla network, HEX [7], EnD [27], ReBias [6] and LfF [12]. Vanilla denotes the classification model trained only with the original cross-entropy (CE) loss, without any debiasing strategies. EnD explicitly leverages the bias labels (e.g., the color label in Colored MNIST) during the training phase. HEX and ReBias explicitly presume the texture of an image as a bias type, while LfF requires no prior knowledge on it. ",
+        "bbox": [
+            174,
+            621,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Datasets As shown in Fig. 2, we use two synthetic datasets (Colored MNIST and Corrputed CIFAR10) and one real-world dataset (Biased FFHQ) to evaluate the generalization of debiasing baselines over various domains. Biased FFHQ (BFFHQ) is curated from FFHQ dataset [28] which contains human face images annotated with their facial attributes. Among the facial attributes, we select age and gender as the intrinsic and bias attribute, respectively, and construct the dataset with images of high correlation between them. More specifically, most of the females are ‘young’ (i.e., age ranging from 10 to 29) and males are ‘old’ (i.e., age ranging from 40 to 59). Therefore, bias-aligned samples which compose the majority of the dataset are young women and old men. ",
+        "bbox": [
+            174,
+            696,
+            825,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For each dataset, we set the degree of correlation by adjusting the number of bias-conflicting samples among the training dataset. The ratio of bias-conflicting samples are $0 . 5 \\%$ , $1 \\%$ , $2 \\%$ and $5 \\%$ for both Colored MNIST and Corrupted CIFAR-10, respectively, and $0 . 5 \\%$ for BFFHQ. For the evaluation of Colored MNIST and Corrupted CIFAR-10, we construct an unbiased test set which includes images without the high correlation existing in the training set. For the BFFHQ, we construct a bias-conflicting test set which excludes the bias-aligned samples from the unbiased test set. The reason is as following. The bias-aligned images consist a half of the unbiased test set in BFFHQ which may still be correctly classified by the biased classifier. This inflates the accuracy of the unbiased test set which is not our original intention. Therefore, we intentionally use the bias-conflicting test set for the BFFHQ. ",
+        "bbox": [
+            174,
+            814,
+            823,
+            911
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/67ff2bf0aae2bcc42a0e601bb03460c58770220d546fe0a4923a3f852246afb1.jpg",
+        "table_caption": [
+            "Table 2: Image classification accuracy evaluated on unbiased test sets of Colored MNIST and Corrupted CIFAR-10, and the bias-conflicting test set of BFFHQ with varying ratio of bias-conflicting samples. We denote whether the model requires a bias type in advance by cross mark (i.e., not required), and check mark (i.e., required). Best performing results are marked in bold, while secondbest results are denoted with underlines. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Dataset</td><td rowspan=\"2\">Ratio (%)</td><td>Vanilla [29]</td><td>HEX [7]</td><td>EnD [27]</td><td>ReBias [6]</td><td>LfF[12]</td><td>Ours</td></tr><tr><td>X</td><td>√</td><td>√</td><td>√</td><td>×</td><td>X</td></tr><tr><td rowspan=\"4\">Colored MNIST</td><td>0.5</td><td>35.19±3.49</td><td>30.33±0.76</td><td>34.28±1.20</td><td>70.47±1.84</td><td>52.50±2.43</td><td>65.22±4.41</td></tr><tr><td>1.0</td><td>52.09±2.88</td><td>43.73±5.50</td><td>49.50±2.51</td><td>87.4±0.78</td><td>61.89±4.97</td><td>81.73±2.34</td></tr><tr><td>2.0</td><td>65.86±3.59</td><td>56.85±2.58</td><td>68.45±2.16</td><td>92.91±0.15</td><td>71.03±2.44</td><td>84.79±0.95</td></tr><tr><td>5.0</td><td>82.17±0.74</td><td>74.62±3.20</td><td>81.15±1.43</td><td>96.96±0.04</td><td>80.57±3.84</td><td>89.66±1.09</td></tr><tr><td rowspan=\"4\">Corrupted CIFAR-10</td><td>0.5</td><td>23.08±1.25</td><td>13.87±0.06</td><td>22.89±0.27</td><td>22.27±0.41</td><td>28.57±1.30</td><td>29.95±0.71</td></tr><tr><td>1.0</td><td>25.82±0.33</td><td>14.81±0.42</td><td>25.46±0.41</td><td>25.72±0.20</td><td>33.07±0.77</td><td>36.49±1.79</td></tr><tr><td>2.0</td><td>30.06±0.71</td><td>15.20±0.54</td><td>31.31±0.35</td><td>31.66±0.43</td><td>39.91±0.30</td><td>41.78±2.29</td></tr><tr><td>5.0</td><td>39.42±0.64</td><td>16.04±0.63</td><td>40.26±0.85</td><td>43.43±0.41</td><td>50.27±1.56</td><td>51.13±1.28</td></tr><tr><td>BFFHQ</td><td>0.5</td><td>56.87±2.69</td><td>52.83±0.90</td><td>56.87±1.42</td><td>59.46±0.64</td><td>62.2±1.0</td><td>63.87±0.31</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Implementation details We use multi-layer perceptron (MLP) with three hidden layers for Colored MNIST, and ResNet-18 [29] for the remaining datasets. To accommodate the disentangled vectors, we double the number of hidden units in the last fully-connected layer of each network. During the inference phase, we use $C _ { i } ( z )$ for the final prediction, where $z = [ z _ { i } ; z _ { b } ]$ . For the training, we set the batch size of 256 for Colored MNIST and Corrupted CIFAR-10, respectively, and 64 for BFFHQ. Bias-conflicting augmentation is scheduled to be applied after 10K iterations for all datasets. We report the averaged accuracy of the unbiased test sets over three independent trials with the mean and the standard deviation. We include the remaining implementation details in Section D in the supplementary material. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.2 Quantitative evaluation ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            537,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Comparison on test sets Table 2 shows the comparisons of image classification accuracy evaluated on the test sets. In general, our approach demonstrates the superior performance in both synthetic and real-world datasets against the baselines with large gaps. Especially, compared to the baselines which do not define the bias types in advance (vanilla [29] and LfF [12]), our approach achieves the stateof-the-art performance across all datasets. This indicates that utilizing the diversified bias-conflicting samples through our augmentation plays a pivotal role in learning debiased representation regardless of the bias types. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Regarding the real-world dataset, our approach also outperforms HEX [7] and ReBias [6] which utilize the tailored modules for a specific bias type (e.g., color and texture), and EnD [27] that uses the explicit bias labels. We even show superior performance compared to HEX in Colored MNIST without defining the bias type beforehand. While ReBias achieves the best accuracy in Colored MNIST, they utilize BagNet [11] in order to focus on the color bias. Even without using such an architecture, we achieve the second best performance which is comparable to ReBias. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Ablation studies Table 3 demonstrates the importance of each module in our approach through ablation studies: 1) disentangled representation learning, 2) feature augmentation, and 3) scheduling feature augmentation. We set the ratio of bias-conflicting samples to $1 \\%$ for Colored MNIST and Corrupted CIFAR10, and $0 . 5 \\%$ for BFFHQ. We also compare each module with the vanilla network (first row). We observe that performing the scheduled feature augmentation shows the best classification accuracy on the test sets across all datasets. We also show that performing feature augmentation at the early stage of training does not guarantee the effectiveness of debiasing. Performing feature augmentation at the beginning of training rather aggravates the performance. That is, when the representation of intrinsic attributes and bias attributes are not disentangled at a certain degree, augmented features may act as noisy samples. Training with these additional noisy features prevents models from achieving further improvement. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/640cd15be1f3922387516897449916cc0552d848b6fd0e50dfd742c4bbe68396.jpg",
+        "table_caption": [
+            "Table 3: Ablation studies on 1) disentangled representation learning, 2) feature augmentation, and 3) scheduling feature augmentation. Each row indicates the different training settings with check mark denoting the setting applied. We average the accuracy of each training over three independent trials. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Disentangle</td><td>Augment</td><td>Scheduled Augment</td><td>Colored MNIST</td><td>Corrupted CIFAR10</td><td>BFFHQ</td></tr><tr><td></td><td></td><td></td><td>52.09±2.88</td><td>25.82±0.33</td><td>56.87±2.69</td></tr><tr><td>&lt;&gt;&gt;</td><td></td><td></td><td>74.03±2.40</td><td>27.73±1.02</td><td>59.4±2.46</td></tr><tr><td></td><td></td><td></td><td>72.29±3.82</td><td>32.81±2.47</td><td>61.27±3.26</td></tr><tr><td></td><td>?</td><td>√</td><td>81.73±2.34</td><td>52.31±1.00</td><td>63.87±0.31</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.3 Analysis ",
+        "text_level": 1,
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "2D Projection of Disentangled Representation Fig. 3 shows the projection of latent vectors $z _ { i }$ and $z _ { b }$ extracted from the intrinsic encoder $E _ { i }$ and bias encoder $E _ { b }$ , respectively, on a 2D space using Colored MNIST. We show projection of $z _ { i }$ and $z _ { b }$ in Fig. 3(a) and Fig. 3(b), respectively. The colors of projected dots in the first row (i) and the second row (ii) indicate the target labels and bias labels, respectively. We observe that $z _ { i }$ are clustered according to the target labels while $z _ { b }$ are clustered with the bias labels. The results represent that our method successfully learns the disentangled intrinsic and bias attributes. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Prediction with Disentangled Representation In Table 4, we report the 1) original and 2) swapping accuracy of $C _ { i }$ and $C _ { b }$ , the linear classifiers of the intrinsic and the bias encoder, respectively. To be specific, for the original accuracy, we extract the two disentangled vectors, $z _ { i }$ and $z _ { b }$ , from the same image, concatenate them to make $z = [ z _ { i } ; z _ { b } ]$ , and forward them into each linear classifier. For the swapping accuracy, however, we first permute $z _ { b }$ and concatenate $z _ { i }$ with the permuted $z _ { b }$ (i.e., denoted as $\\tilde { z _ { b } }$ in Section 4.2) to make $z _ { \\mathrm { s w a p } } = [ z _ { i } ; \\tilde { z _ { b } } ]$ . Then, we pass these concatenated latent vectors to each linear classifier. Afterward, we evaluate the accuracy of predicted labels of 1) $C _ { i } ( z )$ and $C _ { i } ( z _ { \\mathrm { { s w a p } } } )$ with intrinsic labels and 2) $C _ { b } ( z )$ and $C _ { b } ( z _ { \\mathrm { s w a p } } )$ with bias labels. The Intrinsic and Bias columns in Table 4 denote the accuracy with respect to the target labels and bias labels, respectively. Even the feature vectors of bias attributes are randomly swapped, our method maintains a reasonable classification accuracy. This indicates that our model well disentangles between $z _ { i }$ and $z _ { b }$ , and $C _ { i }$ robustly utilizes $z _ { i }$ to predict target labels even when $z _ { b }$ is taken from the different image, and vice versa. Note that we utilized the parameters of the model trained on each dataset after converging at a certain degree. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/518a65caee93a7bacd73aa33ae86be3e48d6ec3f4a7026d8b5d9f262d7aad24c.jpg",
+        "image_caption": [
+            "Figure 3: Each row (i and ii) include 2D projection of $z _ { i }$ and $z _ { b }$ with the colors encoded by their labels (i.e., groundtruth labels in row i and bias labels in row ii) in Colored MNIST. We observe that $z _ { i }$ and $z _ { b }$ are well clustered according to the target and bias labels, respectively. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            272,
+            604,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/4b6fa74e2300f766b4c8d14c9e199ec7082888863873462ec3799e2dcd6af230.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"3\">Accuracy(%)</td><td colspan=\"2\">Colored MNIST</td><td colspan=\"2\">Corrupted CIFAR10</td><td colspan=\"2\">BFFHQ</td></tr><tr><td>Intrinsic</td><td>Bias</td><td>Intrinsic</td><td>Bias</td><td>Intrinsic</td><td>Bias</td></tr><tr><td>Original</td><td>76.08</td><td>98.07</td><td>35.63</td><td>74.16</td><td>57.40</td><td>49.00</td></tr><tr><td>Swapping</td><td>71.40</td><td>94.29</td><td>35.14</td><td>76.46</td><td>58.40</td><td>51.60</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Table 4: Accuracy from disentangled representation. The ratio of bias-conflicting samples in Colored MNIST, Corrupted CIFAR-10, and BFFHQ are $1 \\%$ , $1 \\%$ , and $0 . 5 \\%$ , respectively. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/2b4e9d878f240ee01a845a45ee2022674a4d89c928c89f547b0b14a52c57dfe1.jpg",
+        "image_caption": [
+            "Figure 4: Reconstructed images from disentangled representation in Colored MNIST. Each column and row indicate the samples where the bias attribute (color) and the intrinsic attribute (digit) are extracted, respectively. By swapping the bias features with a given intrinsic feature, we observe that the color changes while maintaining the digit. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Reconstruction of Disentangled Representation Fig. 4 shows the reconstructed images of Colored MNIST by using the disentangled representation of intrinsic features and bias features. Images in the first row and column indicate the images used for extracting the bias attribute (i.e., color) and intrinsic attribute (i.e., digit), respectively. We train an auxiliary decoder by providing the latent vector $z$ from our pre-trained models as input in order to visualize the disentangled representations at the pixel level. By changing the bias attributes (as the column changes), the color of digit changes while maintaining the digit shape. This demonstrates that the bias features and intrinsic features independently contain color and digit information, respectively. Note that the reconstruction loss for updating the decoder is not backpropagated to our pre-trained classification models. Due to this fact, the reconstructed images may lack qualities such as showing blurry images. Further implementation details are included in Section D in the supplementary material. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6 Conclusions ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this work, we propose a feature augmentation method based on the disentangled representation of intrinsic and bias attributes. The main intuition behind our work is that increasing the diversity of bias-conflicting samples beyond a given training set is crucial for debiasing. Since the biased dataset strongly correlates the bias attributes and labels, we intentionally train two different encoders and extract bias features and intrinsic features. After the representations are disentangled to a certain degree, we proliferate the bias-conflicting samples by randomly swapping the vectors. We demonstrate the effectiveness of feature augmentation via extensive experiments, ablation studies, and qualitative evaluation of the disentangled representation. We believe our work inspires the future work of learning debiased representation with the improved generalization capability. ",
+        "bbox": [
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+            646,
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+    },
+    {
+        "type": "text",
+        "text": "Acknowledgements This work was supported by the Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korean government(MSIT) (No. 2019-0-00075, Artificial Intelligence Graduate School Program(KAIST), No. 2021-0-01778, Development of human image synthesis and discrimination technology below the perceptual threshold), the Air Force Research Laboratory, under agreement number FA9550-18-S-0003, and Kakao Enterprise. This material is based on research sponsored by The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. ",
+        "bbox": [
+            174,
+            795,
+            825,
+            885
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "References   \n[1] A. Torralba and A. A. Efros. Unbiased look at dataset bias. CVPR ’11, 2011.   \n[2] Robert Geirhos, Jörn-Henrik Jacobsen, Claudio Michaelis, Richard Zemel, Wieland Brendel, Matthias Bethge, and Felix A Wichmann. Shortcut learning in deep neural networks. Nature Machine Intelligence, 2(11):665–673, 2020.   \n[3] Byungju Kim, Hyunwoo Kim, Kyungsu Kim, Sungjin Kim, and Junmo Kim. Learning not to learn: Training deep neural networks with biased data. In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2019.   \n[4] Yi Li and Nuno Vasconcelos. Repair: Removing representation bias by dataset resampling. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 9572–9581, 2019.   \n[5] Shiori Sagawa, Pang Wei Koh, Tatsunori B Hashimoto, and Percy Liang. 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Arbitrary style transfer in real-time with adaptive instance normalization. In ICCV, 2017.   \n[19] Matthias Minderer, Olivier Bachem, N. Houlsby, and M. Tschannen. Automatic shortcut removal for self-supervised representation learning. In ICML, 2020.   \n[20] Kaiyang Zhou, Yongxin Yang, Yu Qiao, and Tao Xiang. Domain generalization with mixstyle. In ICLR, 2021.   \n[21] Hsin-Ying Lee, Hung-Yu Tseng, Jia-Bin Huang, Maneesh Kumar Singh, and Ming-Hsuan Yang. Diverse image-to-image translation via disentangled representations. In European Conference on Computer Vision, 2018.   \n[22] Xun Huang, Ming-Yu Liu, Serge Belongie, and Jan Kautz. Multimodal unsupervised image-to-image translation. In ECCV, 2018.   \n[23] Taesung Park, Jun-Yan Zhu, Oliver Wang, Jingwan Lu, Eli Shechtman, Alexei A. Efros, and Richard Zhang. Swapping autoencoder for deep image manipulation. In Advances in Neural Information Processing Systems, 2020.   \n[24] A. Krizhevsky and G. Hinton. Learning multiple layers of features from tiny images. Master’s thesis, Department of Computer Science, University of Toronto, 2009.   \n[25] Dan Hendrycks and Thomas Dietterich. Benchmarking neural network robustness to common corruptions and perturbations. In International Conference on Learning Representations, 2019.   \n[26] Laurens van der Maaten and Geoffrey Hinton. Visualizing data using t-SNE. Journal of Machine Learning Research, 9:2579–2605, 2008.   \n[27] Enzo Tartaglione, Carlo Alberto Barbano, and Marco Grangetto. End: Entangling and disentangling deep representations for bias correction. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 13508–13517, June 2021.   \n[28] Tero Karras, Samuli Laine, and Timo Aila. A style-based generator architecture for generative adversarial networks. 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+        "bbox": [
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+    }
+]

parse/train/QkRbdiiEjM/QkRbdiiEjM.md

@@ -0,0 +1,416 @@
+# ADAGCN: ADABOOSTING GRAPH CONVOLUTIONAL NETWORKS INTO DEEP MODELS
+
+Ke Sun   
+Zhejiang Lab   
+Key Lab. of Machine Perception (MoE), School of EECS, Peking University   
+ajksunke@pku.edu.cn
+
+Zhanxing Zhu\* Beijing Institute of Big Data Research, Beijing, China zhanxing.zhu@pku.edu.cn
+
+Zhouchen Lin∗   
+Key Lab. of Machine Perception (MoE), School of EECS, Peking University   
+Pazhou Lab, Guangzhou, China   
+zlin@pku.edu.cn
+
+# ABSTRACT
+
+The design of deep graph models still remains to be investigated and the crucial part is how to explore and exploit the knowledge from different hops of neighbors in an efficient way. In this paper, we propose a novel RNN-like deep graph neural network architecture by incorporating AdaBoost into the computation of network; and the proposed graph convolutional network called AdaGCN (Adaboosting Graph Convolutional Network) has the ability to efficiently extract knowledge from high-order neighbors of current nodes and then integrates knowledge from different hops of neighbors into the network in an Adaboost way. Different from other graph neural networks that directly stack many graph convolution layers, AdaGCN shares the same base neural network architecture among all “layers” and is recursively optimized, which is similar to an RNN. Besides, We also theoretically established the connection between AdaGCN and existing graph convolutional methods, presenting the benefits of our proposal. Finally, extensive experiments demonstrate the consistent state-of-the-art prediction performance on graphs across different label rates and the computational advantage of our approach AdaGCN 1.
+
+# 1 INTRODUCTION
+
+Recently, research related to learning on graph structural data has gained considerable attention in machine learning community. Graph neural networks (Gori et al., 2005; Hamilton et al., 2017; Velickovi ˇ c et al., 2018), particularly graph convolutional networks (Kipf & Welling, 2017; Deffer- ´ rard et al., 2016; Bruna et al., 2014) have demonstrated their remarkable ability on node classification (Kipf & Welling, 2017), link prediction (Zhu et al., 2016) and clustering tasks (Fortunato, 2010). Despite their enormous success, almost all of these models have shallow model architectures with only two or three layers. The shallow design of GCN appears counterintuitive as deep versions of these models, in principle, have access to more information, but perform worse. Oversmoothing (Li et al., 2018) has been proposed to explain why deep GCN fails, showing that by repeatedly applying Laplacian smoothing, GCN may mix the node features from different clusters and makes them indistinguishable. This also indicates that by stacking too many graph convolutional layers, the embedding of each node in GCN is inclined to converge to certain value (Li et al., 2018), making it harder for classification. These shallow model architectures restricted by oversmoothing issue limit their ability to extract the knowledge from high-order neighbors, i.e., features from remote hops of neighbors for current nodes. Therefore, it is crucial to design deep graph models such that high-order information can be aggregated in an effective way for better predictions.
+
+There are some works (Xu et al., 2018b; Liao et al., 2019; Klicpera et al., 2018; Li et al., 2019; Liu et al., 2020) that tried to address this issue partially, and the discussion can refer to Appendix A.1. By contrast, we argue that a key direction of constructing deep graph models lies in the efficient exploration and effective combination of information from different orders of neighbors. Due to the apparent sequential relationship between different orders of neighbors, it is a natural choice to incorporate boosting algorithm into the design of deep graph models. As an important realization of boosting theory, AdaBoost (Freund et al., 1999) is extremely easy to implement and keeps competitive in terms of both practical performance and computational cost (Hastie et al., 2009). Moreover, boosting theory has been used to analyze the success of ResNets in computer vision (Huang et al., 2018) and AdaGAN (Tolstikhin et al., 2017) has already successfully incorporated boosting algorithm into the training of GAN (Goodfellow et al., 2014).
+
+In this work, we focus on incorporating AdaBoost into the design of deep graph convolutional networks in a non-trivial way. Firstly, in pursuit of the introduction of AdaBoost framework, we refine the type of graph convolutions and thus obtain a novel RNN-like GCN architecture called AdaGCN. Our approach can efficiently extract knowledge from different orders of neighbors and then combine these information in an AdaBoost manner with iterative updating of the node weights. Also, we compare our AdaGCN with existing methods from the perspective of both architectural difference and feature representation power to show the benefits of our method. Finally, we conduct extensive experiments to demonstrate the consistent state-of-the-art performance of our approach across different label rates and computational advantage over other alternatives.
+
+# 2 OUR APPROACH: ADAGCN
+
+# 2.1 ESTABLISHMENT OF ADAGCN
+
+Consider an undirected graph $\mathcal { G } = ( \nu , \mathcal { E } )$ with $N$ nodes $v _ { i } \in \mathcal V$ , edges $( v _ { i } , v _ { j } ) \in \mathcal { E }$ . $A \in \mathbb { R } ^ { N \times N }$ is the adjacency matrix with corresponding degree matrix $\begin{array} { r } { D _ { i i } = \sum _ { j } \dot { A } _ { i j } } \end{array}$ . In the vanilla GCN model (Kipf & Welling, 2017) for semi-supervised node classification, the graph embedding of nodes with two convolutional layers is formulated as:
+
+$$
+Z = \hat { A } \mathrm { R e L U } ( \hat { A } X W ^ { ( 0 ) } ) W ^ { ( 1 ) }
+$$
+
+where $Z \in \mathbb { R } ^ { N \times K }$ is the final embedding matrix (output logits) of nodes before softmax and $K$ is the number of classes. $X \in \mathbb { R } ^ { N \times C }$ denotes the feature matrix where $C$ is the input dimension. $\hat { A } = \tilde { D } ^ { - \frac { 1 } { 2 } } \tilde { A } \tilde { D } ^ { - \frac { 1 } { 2 } }$ where ${ \tilde { A } } = A + I$ and $\tilde { D }$ is the degree matrix of $\tilde { A }$ . In addition, $W ^ { ( 0 ) } \in \mathbb { R } ^ { C \times H }$ is the input-to-hidden weight matrix for a hidden layer with $H$ feature maps and $W ^ { ( 1 ) } \in \mathbb { R } ^ { H \times K }$ is the hidden-to-output weight matrix.
+
+Our key motivation of constructing deep graph models is to efficiently explore information of highorder neighbors and then combine these messages from different orders of neighbors in an AdaBoost way. Nevertheless, if we naively extract information from high-order neighbors based on GCN, we are faced with stacking $l$ layers’ parameter matrix $W ^ { ( i ) } , i = 0 , . . . , l - 1$ , which is definitely costly in computation. Besides, Multi-Scale Deep Graph Convolutional Networks (Luan et al., 2019) also theoretically demonstrated that the output can only contain the stationary information of graph structure and loses all the local information in nodes for being smoothed if we simply deepen GCN. Intuitively, the desirable representation of node features does not necessarily need too many nonlinear transformation $f$ applied on them. This is simply due to the fact that the feature of each node is normally one-dimensional sparse vector rather than multi-dimensional data structures, e.g., images, that intuitively need deep convolution network to extract high-level representation for vision tasks. This insight has been empirically demonstrated in many recent works (Wu et al., 2019; Klicpera et al., 2018; Xu et al., 2018a), showing that a two-layer fully-connected neural networks is a better choice in the implementation. Similarly, our AdaGCN also follows this direction by choosing an appropriate $f$ in each layer rather than directly deepen GCN layers.
+
+Thus, we propose to remove ReLU to avoid the expensive joint optimization of multiple parameter matrices. Similarly, Simplified Graph Convolution (SGC) (Wu et al., 2019) also adopted this practice, arguing that nonlinearity between GCN layers is not crucial and the majority of the benefits arises from local weighting of neighboring features. Then the simplified graph convolution is:
+
+![](images/1a73cdd96f289c3170574f4224643074e89e71dc26fb8af0e704be928232997c.jpg)  
+Figure 1: The RNN-like architecture of AdaGCN with each base classifier $f _ { \theta } ^ { ( l ) }$ sharing the same neural network architecture $f _ { \theta }$ . $w ^ { l }$ and $\theta _ { l }$ denote node weights and parameters computed after the $l$ -th base classifier, respectively.
+
+$$
+Z = \hat { A } ^ { l } X W ^ { ( 0 ) } W ^ { ( 1 ) } \cdots W ^ { ( l - 1 ) } = \hat { A } ^ { l } X \tilde { W } ,
+$$
+
+where we collapse $W ^ { ( 0 ) } W ^ { ( 1 ) } \cdot \cdot \cdot W ^ { ( l - 1 ) }$ as $\tilde { W }$ and $\hat { A } ^ { l }$ denotes $\hat { A }$ to the $l$ -th power. In particular, one crucial impact of ReLU in GCN is to accelerate the convergence of matrix multiplication since the ReLU is a contraction mapping intuitively. Thus, the removal of ReLU operation could also alleviate the oversmoothing issue, i.e. slowering the convergence of node embedding to indistinguishable ones (Li et al., 2018). Additionally, without ReLU this simplified graph convolution is also able to avoid the aforementioned joint optimization over multiple parameter matrices, resulting in computational benefits. Nevertheless, we find that this type of stacked linear transformation from graph convolution has insufficient power in representing information of high-order neighbors, which is revealed in our experiment described in Appendix A.2. Therefore, we propose to utilize an appropriate nonlinear function $f _ { \theta }$ , e.g., a two-layer fully-connected neural network, to replace the linear transformation $\tilde { W }$ in Eq. 2 and enhance the representation ability of each base classifier in AdaGCN as follows:
+
+$$
+Z ^ { ( l ) } = f _ { \theta } ( \hat { A } ^ { l } X ) ,
+$$
+
+where $Z ^ { ( l ) }$ represents the final embedding matrix (output logits before Softmax) after the $l$ -th base classifier in AdaGCN. This formulation also implies that the $l$ -th base classifier in AdaGCN is extracting knowledge from features of current nodes and their $l$ -th hop of neighbors. Due to the fact that the function of $l$ -th base classifier in AdaGCN is similar to that of the $l$ -th layer in other traditional GCN-based methods that directly stack many graph convolutional layers, we regard the whole part of l-th base classifier as the $l$ -th layers in AdaGCN. As for the realization of Multi-class AdaBoost, we apply SAMME (Stagewise Additive Modeling using a Multi-class Exponential loss function) algorithm (Hastie et al., 2009), a natural and clean multi-class extension of the two-class AdaBoost adaptively combining weak classifiers.
+
+As illustrated in Figure 1, we apply base classifie r f (l) to extract knowledge from current node feature and $l$ -th hop of neighbors by minimizing current weighted loss. Then we directly compute the weighted error rate $e r r ^ { ( l ) }$ and corresponding weight $\alpha ^ { ( l ) }$ of current base classifier $f _ { \theta } ^ { ( l ) }$ as follows:
+
+$$
+\begin{array} { c } { { e r r ^ { ( l ) } = \displaystyle \sum _ { i = 1 } ^ { n } w _ { i } \mathbb { I } \left( c _ { i } \neq f _ { \theta } ^ { ( l ) } \left( x _ { i } \right) \right) / \sum _ { i = 1 } ^ { n } w _ { i } } } \\ { { \displaystyle \alpha ^ { ( l ) } = \log \frac { 1 - e r r ^ { ( l ) } } { e r r ^ { ( l ) } } + \log ( K - 1 ) , } } \end{array}
+$$
+
+where $w _ { i }$ denotes the weight of $i$ -th node and $c _ { i }$ represents the category of current $i$ -th node. To attain a positive $\alpha ^ { ( l ) }$ , we only need $( 1 - e r r ^ { ( l ) } ) > 1 / K$ , i.e., the accuracy of each weak classifier
+
+should be better than random guess (Hastie et al., 2009). This can be met easily to guarantee the weights to be updated in the right direction. Then we adjust nodes’ weights by increasing weights on incorrectly classified ones:
+
+$$
+w _ { i } \gets w _ { i } \cdot \exp \left( \alpha ^ { ( l ) } \cdot \mathbb { I } \left( c _ { i } \neq f _ { \theta } ^ { ( l ) } \left( x _ { i } \right) \right) \right) , i = 1 , \dots , n
+$$
+
+After re-normalizing the weights, we then compute $\hat { A } ^ { l + 1 } X = \hat { A } \cdot ( \hat { A } ^ { l } X )$ to sequentially extract knowledge from of AdaGCN is t $l { + } 1$ -th hop of neighbors in the following base classifier different from traditional AdaBoost, we only define $f _ { \theta } ^ { ( l + 1 ) }$ One crucial point, e.g. a two-layer $f _ { \theta }$ fully connected neural network, which in practice is recursively optimized in each base classifier just similar to a recurrent neural network. This also indicates that the parameters from last base classifier are leveraged as the initialization of next base classifier, which coincides with our intuition that $l + 1$ -th hop of neighbors are directly connected from $l$ -th hop of neighbors. The efficacy of this kind of layer-wise training has been similarly verified in (Belilovsky et al., 2018) recently. Further, we combine the predictions from different orders of neighbors in an Adaboost way to obtain the final prediction $C ( A , X )$ :
+
+$$
+C ( A , X ) = \arg \operatorname* { m a x } _ { k } \sum _ { l = 0 } ^ { L } \alpha ^ { ( l ) } f _ { \theta } ^ { ( l ) } ( \hat { A } ^ { l } X )
+$$
+
+Finally, we obtain the concise form of AdaGCN in the following:
+
+$$
+\begin{array} { r l } & { \hat { A } ^ { l } X = \hat { A } \cdot ( \hat { A } ^ { l - 1 } X ) } \\ & { Z ^ { ( l ) } = f _ { \theta } ^ { ( l ) } ( \hat { A } ^ { l } X ) } \\ & { Z = \mathrm { A d a B o o s t } ( Z ^ { ( l ) } ) } \end{array}
+$$
+
+Note that $f _ { \theta }$ is non-linear, rather than linear in SGC (Wu et al., 2019), to guarantee the representation power. As shown in Figure 1, the architecture of AdaGCN is a variant of RNN with synchronous sequence input and output. Although the same classifier architecture is adopted for $f _ { \theta } ^ { ( \bar { l } ) }$ , their parameters are different, which is different from vanilla RNN. We provide a detailed description of the our algorithm in Section 3.
+
+# 2.2 COMPARISON WITH EXISTING METHODS
+
+Architectural Difference. As illustrated in Figure 1 and 2, there is an apparent difference among the architectures of GCN (Kipf & Welling, 2017), SGC (Wu et al., 2019), Jumping Knowledge (JK) (Xu et al., 2018b) and AdaGCN. Compared with these existing graph convolutional approaches that sequentially convey intermediate result $Z ^ { ( l ) }$ to compute final prediction, our AdaGCN transmits weights of nodes $w ^ { i }$ , aggregated features of different hops of neighbors ${ \hat { A } } ^ { l } X$ . More importantly, in AdaGCN the embedding $Z ^ { ( l ) }$ is independent of the flow of computation in the network and the sparse adjacent matrix $\hat { A }$ is also not directly involved in the computation of individual network because we compute
+
+![](images/2d1ea197072ef3f76ecf7d5e86939afa479f716a2e4be601ef8cb8d2f3fec644.jpg)  
+Figure 2: Comparison of the graph model architectures. $f _ { a }$ in JK network denotes one aggregation layer with aggregation function such as concatenation or max pooling.
+
+${ \hat { A } } ^ { ( l + 1 ) } X$ in advance and then feed it instead of $\hat { A }$ into the classifier $f _ { \theta } ^ { ( l + 1 ) }$ , thus yielding significant computation reduction, which will be discussed further in Section 3.
+
+Connection with PPNP and APPNP. We also established a strong connection between AdaGCN and previous state-of-the-art Personalized Propagation of Neural Predictions (PPNP) and Approximate PPNP (APPNP) (Klicpera et al., 2018) method that leverages personalized pagerank to reconstruct graph convolutions in order to use information from a large and adjustable neighborhood. The analysis can be summarized in the following Proposition 1. Proof can refer to Appendix A.3.
+
+Proposition 1. Suppose that $\gamma$ is the teleport factor. Let matrix sequence $\{ Z ^ { ( l ) } \}$ be from the output of each layer l in AdaGCN, then PPNP is equivalent to the Exponential Moving Average (EMA) with exponentially decreasing factor $\gamma$ on $\{ Z ^ { ( l ) } \}$ in a sharing parameters version, and its approximate version APPNP can be viewed as the approximated form of EMA with a limited number of terms.
+
+Proposition 1 illustrates that AdaGCN can be viewed as an adaptive form of APPNP, formulated as:
+
+$$
+Z = \sum _ { l = 0 } ^ { L } \alpha ^ { ( l ) } f _ { \theta } ^ { ( l ) } ( \hat { A } ^ { l } X )
+$$
+
+Specifically, the first discrepancy between AdaGCN and APPNP lies in the adaptive coefficient $\alpha ^ { ( l ) }$ in AdaGCN determined by the error of $l$ -th base classifier $f _ { \theta } ^ { ( l ) }$ rather than fixed exponentially decreased weights in APPNP. In addition, AdaGCN employs classifier $f _ { \theta } ^ { ( l ) }$ with different parameters to learn the embedding of different orders of neighbors, while APPNP shares these parameters in its form. We verified this benefit of our approach in our experiments shown in Section 4.2.
+
+Comparison with MixHop MixHop (Abu-El-Haija et al., 2019) applied the similar way of graph convolution by repeatedly mixing feature representations of neighbors at various distance. Proposition 2 proves that both AdaGCN and MixHop are able to represent feature differences among neighbors while previous GCNs-based methods cannot. Proof can refer to Appendix A.4. Recap the definition of general layer-wise Neighborhood Mixing (Abu-El-Haija et al., 2019) as follows:
+
+Definition 1. General layer-wise Neighborhood Mixing: $A$ graph convolution network has the ability to represent the layer-wise neighborhood mixing if for any $b _ { 0 } , b _ { 1 } , . . . , b _ { L } ,$ , there exists an injective mapping $f$ with a setting of its parameters, such that the output of this graph convolution network can express the following formula:
+
+$$
+f \left( \sum _ { l = 0 } ^ { L } b _ { l } \sigma \left( \hat { A } ^ { l } X \right) \right)
+$$
+
+Proposition 2. AdaGCNs defined by our proposed approach (Eq. equation 7) are capable of representing general layer-wise neighborhood mixing, i.e., can meet the Definition $^ { l }$ .
+
+Albeit the similarity, AdaGCN distinguishes from MixHop in many aspects. Firstly, MixHop concatenates all outputs from each order of neighbors while we combines these predictions in an Adaboost way, which has theoretical generalization guarantee based on boosting theory Hastie et al. (2009). Oono & Suzuki (2020) have recently derived the optimization and generalization guarantees of multi-scale GNNs, serving as the theoretical backbone of AdaGCN. Meantime, MixHop allows full linear mixing of different orders of neighboring features, while AdaGCN utilizes different nonlinear transformation $f _ { \theta } ^ { ( l ) }$ among all layers, enjoying stronger expressive power.
+
+# 3 ALGORITHM
+
+In practice, we employ SAMME.R (Hastie et al., 2009), the soft version of SAMME, in AdaGCN. SAMME.R (R for Real) algorithm (Hastie et al., 2009) leverages real-valued confidence-rated predictions, i.e., weighted probability estimates, rather than predicted hard labels in SAMME, in the prediction combination, which has demonstrated a better generalization and faster convergence than SAMME. We elaborate the final version of AdaGCN in Algorithm 1. We provide the analysis on the choice of model depth $L$ in Appendix A.7, and then we elaborate the computational advantage of AdaGCN in the following.
+
+Analysis of Computational Advantage. Due to the similarity of graph convolution in MixHop (Abu-El-Haija et al., 2019), AdaGCN also requires no additional memory or computational complexity compared with previous GCN models. Meanwhile, our approach enjoys huge computational advantage compared with GCN-based models, e.g., PPNP and APPNP, stemming from excluding the additional computation involved in sparse tensors, such as the sparse tensor multiplication between $\hat { A }$ and other dense tensors, in the forward and backward propagation of the neural network. Specifically, there are only $L$ times sparse tensor operations for an AdaGCN model with $L$ layers, i.e., $\hat { A } ^ { l } X = \overset { \cdot } { A } \cdot ( \hat { A } ^ { l - 1 } X )$ for each layer $l$ . This operation in each layer yields a dense tensor
+
+# Algorithm 1 AdaGCN based on SAMME.R Algorithm
+
+Input: Features Matrix $X$ , normalized adjacent matrix $\hat { A }$ , a two-layer fully connected network $f _ { \theta }$ , number of layers $L$ and number of classes $K$ .
+
+Output: Final combined prediction $C ( A , X )$
+
+1: Initialize the node weights $w _ { i } = 1 / n , i = 1 , 2 , . . . , n$ on training set, neighbors feature matrix ${ \hat { X } } ^ { ( 0 ) } = X$ and classifier $f _ { \theta } ^ { ( - 1 ) }$ .
+
+2: for $l = 0$ to do
+
+3: Fit the graph convolutional classifier f (l) on neighbor feature matrix $\hat { X } ^ { ( l ) }$ based on $f _ { \theta } ^ { ( l - 1 ) }$ by minimizing current weighted loss.
+
+4: Obtain the weighted probability estimates $p ^ { ( l ) } ( \hat { X } ^ { ( l ) } )$ for $f _ { \theta } ^ { ( l ) }$
+
+5: Compute the individual prediction $h _ { k } ^ { ( l ) } ( x )$ for the current graph convolutional classifier $f _ { \theta } ^ { ( l ) }$
+
+$$
+h _ { k } ^ { ( l ) } ( \hat { X } ^ { ( l ) } ) \gets ( K - 1 ) \left( \log p _ { k } ^ { ( l ) } ( \hat { X } ^ { ( l ) } ) - \frac { 1 } { K } \sum _ { k ^ { \prime } } \log p _ { k ^ { \prime } } ^ { ( l ) } ( \hat { X } ^ { ( l ) } ) \right)
+$$
+
+where $k = 1 , \ldots , K$
+
+6: Adjust the node weights $w _ { i }$ for each node $x _ { i }$ with label $y _ { i }$ on training set:
+
+$$
+w _ { i }  w _ { i } \cdot \exp ( - \frac { K - 1 } { K } y _ { i } ^ { \top } \log p ^ { ( l ) } ( x _ { i } ) ) , i = 1 , \ldots , n
+$$
+
+7: Re-normalize all weights $w _ { i }$
+
+8: Update $l { + } 1$ -hop neighbor feature matrix $\hat { X } ^ { ( l + 1 ) }$ :
+
+$$
+\hat { X } ^ { ( l + 1 ) } = \hat { A } \hat { X } ^ { ( l ) }
+$$
+
+# 9: end for
+
+10: Combine all predictions $h _ { k } ^ { ( l ) } ( \hat { X } ^ { ( l ) } )$ for $l = 0 , . . . , L$
+
+$$
+C ( A , X ) = \arg \operatorname* { m a x } _ { k } \sum _ { l = 0 } ^ { L } h _ { k } ^ { ( l ) } ( \hat { X } ^ { ( l ) } )
+$$
+
+11: return Final combined prediction $C ( A , X )$ .
+
+$B ^ { l } = \hat { A } ^ { l } X$ for the $l$ -th layer, which is then fed into the computation in a two-layer fully-connected network, i.e., $f _ { \theta } ^ { ( l ) } ( B ^ { l } ) = \mathrm { \bar { R e L U } } ( B ^ { l } W ^ { ( 0 ) } ) W ^ { ( 1 ) }$ . Due to the fact that dense tensor $B ^ { l }$ has been computed in advance, there is no other computation related to sparse tensors in the multiple forward and backward propagation procedures while training the neural network. By contrast, this multiple computation involved in sparse tensors in the GCN-based models, e.g., GCN: $\hat { A } \mathrm { R e L U } ( \hat { A } X W ^ { ( 0 ) } ) W ^ { ( 1 ) }$ , is highly expensive. AdaGCN avoids these additional sparse tensor operations in the neural network and then attains huge computational efficiency. We demonstrate this viewpoint in the Section 4.3.
+
+# 4 EXPERIMENTS
+
+Experimental Setup. We select five commonly used graphs: CiteSeer, Cora-ML (Bojchevski & Gunnemann, 2018; McCallum et al., 2000), PubMed (Sen et al., 2008), MS-Academic (Shchur ¨ et al., 2018) and Reddit. Dateset statistics are summarized in Table 1. Recent graph neural networks suffer from overfitting to a single splitting of training, validation and test datasets (Klicpera et al., 2018). To address this problem, inspired by (Klicpera et al., 2018), we test all approaches on multiple random splits and initialization to conduct a rigorous study. Detailed dataset splittings are provided in Appendix A.6.
+
+Table 1: Dateset statistics   
+
+<table><tr><td>Dateset</td><td>Nodes</td><td>Edges</td><td>Classes</td><td>Features</td><td>LabelRate</td></tr><tr><td>CiteSeer</td><td>3,327</td><td>4,732</td><td>6</td><td>3,703</td><td>3.6%</td></tr><tr><td>Cora</td><td>2,708</td><td>5,429</td><td>7</td><td>1,433</td><td>5.2%</td></tr><tr><td>PubMed</td><td>19,717</td><td>44,338</td><td>3</td><td>500</td><td>0.3%</td></tr><tr><td>MS Academic</td><td>18,333</td><td>81,894</td><td>15</td><td>6,805</td><td>1.6%</td></tr><tr><td>Reddit</td><td>232,965</td><td>11,606,919</td><td>41</td><td>602</td><td>65.9%</td></tr></table>
+
+![](images/08c70065488901989e7be342ddee25f0d0fd482f68fda0d5da2a451fb95c7178.jpg)  
+Figure 3: Comparison of test accuracy of different models as the layer increases. We regard the $l$ -th base classifier as the $l$ -th layer in AdaGCN as both of them are leveraged to exploit the information from $l$ -th order of neighbors for current nodes.
+
+Basic Setting of Baselines and AdaGCN. We compare AdaGCN with GCN (Kipf & Welling, 2017) and Simple Graph Convolution (SGC) (Wu et al., 2019) in Figure 3. In Table 2, we employ the same baselines as (Klicpera et al., 2018): V.GCN (vanilla GCN) (Kipf & Welling, 2017) and GCN with our early stopping, N-GCN (network of GCN) (Abu-El-Haija et al., 2018a), GAT (Graph Attention Networks) (Velickovi ˇ c et al., 2018), BT.FP (bootstrapped feature propagation) (Buchnik ´ & Cohen, 2018) and JK (jumping knowledge networks with concatenation) (Xu et al., 2018b). In the computation part, we additionally compare AdaGCN with FastGCN (Chen et al., 2018) and GraphSAGE (Hamilton et al., 2017). We refer to the result of baselines from (Klicpera et al., 2018) and the implementation of AdaGCN is adapted from APPNP. For AdaGCN, after the line search on hyper-parameters, we set $h = 5 0 0 0$ hidden units for the first four datasets except Ms-academic with $h = 3 0 0 0$ , and 15, 12, 20 and 5 layers respectively due to the different graph structures. In addition, we set dropout rate to 0 for Citeseer and Cora-ML datasets and 0.2 for the other datasets and $5 \times 1 0 ^ { - 3 } L _ { 2 }$ regularization on the first linear layer. We set weight decay as $1 \times 1 0 ^ { - 3 }$ for Citeseer while $1 \times 1 0 ^ { - 4 }$ for others. More detailed model parameters and analysis about our early stopping mechanism can be referred from Appendix A.6.
+
+# 4.1 DESIGN OF DEEP GRAPH MODELS TO CIRCUMVENT OVERSMOOTHING EFFECT
+
+It is well-known that GCN suffers from oversmoothing (Li et al., 2018) with the stacking of more graph convolutions. However, combination of knowledge from each layer to design deep graph models is a reasonable method to circumvent oversmoothing issue. In our experiment, we aim to explore the prediction performance of GCN, GCN with residual connection (Kipf & Welling, 2017), SGC and our AdaGCN with a growing number of layers.
+
+Table 2: Average accuracy under 100 runs with uncertainties showing the $95 \%$ confidence level calculated by bootstrapping. OOM denotes “out of memory”. “(ours)” denotes the results based on our implementation, which are slight lower than numbers above from original literature (Klicpera et al., 2018). P values of paired t test between APPNP (ours) and AdaGCN are provided in the last row.   
+
+<table><tr><td>Model</td><td>Citeseer</td><td>Cora-ML</td><td>Pubmed</td><td>MSAcademic</td></tr><tr><td>V.GCN</td><td>73.51±0.48</td><td>82.30±0.34</td><td>77.65±0.40</td><td>91.65±0.09</td></tr><tr><td>GCN</td><td>75.40±0.30</td><td>83.41±0.39</td><td>78.68±0.38</td><td>92.10±0.08</td></tr><tr><td>N-GCN</td><td>74.25±0.40</td><td>82.25±0.30</td><td>77.43±0.42</td><td>92.86±0.11</td></tr><tr><td>GAT</td><td>75.39±0.27</td><td>84.37±0.24</td><td>77.76±0.44</td><td>91.22±0.07</td></tr><tr><td>JK</td><td>73.03±0.47</td><td>82.69±0.35</td><td>77.88±0.38</td><td>91.71±0.10</td></tr><tr><td>BT.FP</td><td>73.55±0.57</td><td>80.84±0.97</td><td>72.94±1.00</td><td>91.61±0.24</td></tr><tr><td>PPNP</td><td>75.83±0.27</td><td>85.29±0.25</td><td>OOM</td><td>OOM</td></tr><tr><td>APPNP</td><td>75.73±0.30</td><td>85.09±0.25</td><td>79.73±0.31</td><td>93.27±0.08</td></tr><tr><td>PPNP (ours)</td><td>75.53±0.32</td><td>84.39±0.28</td><td>OOM</td><td>OOM</td></tr><tr><td>APPNP (ours)</td><td>75.41±0.35</td><td>84.28±0.28</td><td>79.41±0.34</td><td>92.98±0.07</td></tr><tr><td>AdaGCN</td><td>76.68±0.20</td><td>85.97±0.20</td><td>79.95±0.21</td><td>93.17±0.07</td></tr><tr><td>P value</td><td>1.8×10-15</td><td>2.2×10-16</td><td>1.1×10-5</td><td>2.1×10-9</td></tr></table>
+
+<table><tr><td></td><td>Citeseer</td><td>Cora-ML</td><td>Pubmed</td><td>MSAcademic</td></tr><tr><td>Label Rates</td><td>1.0% / 2.0%</td><td>2.0% / 4.0%</td><td>0.1% / 0.2%</td><td>0.6% / 1.2%</td></tr><tr><td>V.GCN</td><td>67.6±1.4/70.8±1.4</td><td>76.4±1.3/81.7±0.8</td><td>70.1±1.4/74.6±1.6</td><td>89.7±0.4/91.1±0.2</td></tr><tr><td>GCN</td><td>70.3±0.9/72.7±1.1</td><td>80.0±0.7/82.8±0.9</td><td>71.1±1.1/75.2±1.0</td><td>89.8±0.4/91.2±0.3</td></tr><tr><td>PPNP APPNP</td><td>72.5±0.9/74.7±0.7 72.2±1.3/74.2±1.1</td><td>80.1±0.7/83.0±0.6</td><td>OOM</td><td>OOM</td></tr><tr><td>AdaGCN</td><td></td><td>80.1±0.7/83.2±0.6</td><td>74.0±1.5/77.2±1.2</td><td>91.7±0.2/92.6±0.2</td></tr><tr><td></td><td>74.2±0.3/75.5±0.3</td><td>83.7±0.3/85.3±0.2</td><td>77.1±0.5/79.3±0.3</td><td>92.1±0.1/92.7±0.1</td></tr></table>
+
+Table 3: Average accuracy across different label rates with 20 splittings of datasets under 100 runs.
+
+From Figure 3, it can be easily observed that oversmoothing leads to the rapid decreasing of accuracy for GCN (blue line) as the layer increases. In contrast, the speed of smoothing (green line) of SGC is much slower than GCN due to the lack of ReLU analyzed in Section 2.1. Similarly, GCN with residual connection (yellow line) partially mitigates the oversmoothing effect of original GCN but fails to take advantage of information from different orders of neighbors to improve the prediction performance constantly. Remarkably, AdaGCN (red line) is able to consistently enhance the performance with the increasing of layers across the three datasets. This implies that AdaGCN can efficiently incorporate knowledge from different orders of neighbors and circumvent oversmoothing of original GCN in the process of constructing deep graph models. In addition, the fluctuation of performance for AdaGCN is much lower than GCN especially when the number of layer is large.
+
+# 4.2 PREDICTION PERFORMANCE
+
+We conduct a rigorous study of AdaGCN on four datasets under multiple splittings of dataset. The results from Table 2 suggest the state-of-the-art performance of our approach and the improvement compared with APPNP validates the benefit of adaptive form for our AdaGCN. More rigorously, p values under paired t test demonstrate the significance of improvement for our method.
+
+In the realistic setting, graphs usually have different labeled nodes and thus it is necessary to investigate the robust performance of methods on different number of labeled nodes. Here we utilize label rates to measure the different numbers of labeled nodes and then sample corresponding labeled nodes per class on graphs respectively. Table 3 presents the consistent state-of-the-art performance of AdaGCN under different label rates. An interesting manifestation from Table 3 is that AdaGCN yields more improvement on fewer label rates compared with APPNP, showing more efficiency on graphs with few labeled nodes. Inspired by the Layer Effect on graphs (Sun et al., 2019), we argue that the increase of layers in AdaGCN can result in more benefits on the efficient propagation of label signals especially on graphs with limited labeled nodes.
+
+More rigorously, we additionally conduct the comparison on a larger dataset, i.e., Reddit. We choose the best layer as 4 due to the fact that AdaGCN with larger number of layers tends to suffer from overfitting on this relatively simple dataset (with high label rate $6 5 . 9 \%$ ). Table 4 suggests that AdaGCN can still outperform other typical baselines, including V.GCN, PPNP and APPNP. More experimental details can be referred from Appendix A.6.
+
+Table 4: Average F1-scores and per-epoch training time of typical methods on Reddit dataset under 5 runs.   
+
+<table><tr><td>Reddit</td><td>F1-Score</td><td>Per-epoch training time</td></tr><tr><td>V.GCN</td><td>94.46±0.06</td><td>5627.46ms</td></tr><tr><td>PPNP</td><td>OOM</td><td>OOM</td></tr><tr><td>APPNP</td><td>95.04±0.07</td><td>29489.81ms</td></tr><tr><td>AdaGCN</td><td>95.39±0.13</td><td>32.29ms</td></tr></table>
+
+# 4.3 COMPUTATIONAL EFFICIENCY
+
+Without the additional computational cost involved in sparse tensors in the propagation of the neural network, AdaGCN presents huge computational efficiency. From the left part of Figure 4, it exhibits that AdaGCN has the fastest speed of per-epoch training time in comparison with other methods except the comparative performance with FastGCN in Pubmed. In addition, there is a somewhat inconsistency in computation of FastGCN, with fastest speed in Pubmed but slower than
+
+![](images/89a94feea60c1c911e184d58c6f81cdb9979c2377be74b696a65f7feb1710e99.jpg)  
+Figure 4: Left: Per-epoch training time of AdaGCN vs other methods under 5 runs on four datasets. Right: Per-epoch training time of AdaGCN compared with GCN and SGC with the increasing of layers and the digit after $\mathbf { \bar { \Sigma } } ^ { 6 } = \mathbf { \bar { \Sigma } } ^ { 5 }$ denotes the slope in a fitted linear regression.
+
+GCN on Cora-ML and MS-Academic datasets. Furthermore, with multiple power iterations involved in sparse tensors, APPNP unfortunately has relatively expensive computation cost. It should be noted that this computational advantage of AdaGCN is more significant when it comes to large datasets, e.g., Reddit. Table 4 demonstrates AdaGCN has the potential to perform much faster on larger datasets.
+
+Besides, we explore the computational cost of ReLU and sparse adjacency tensor with respect to the number of layers in the right part of Figure 4. We focus on comparing AdaGCN with SGC and GCN as other GCN-based methods, such as GraphSAGE and APPNP, behave similarly with GCN. Particularly, we can easily observe that both SGC (green line) and GCN (red line) show a linear increasing tendency and GCN yields a larger slope arises from ReLU and more parameters. For SGC, stacking more layers directly is undesirable regarding the computation. Thus, a limited number of SGC layers is preferable with more advanced optimization techniques Wu et al. (2019). It also shows that the computational cost involved sparse matrices in neural networks plays a dominant role in all the cost especially when the layer is large enough. In contrast, our AdaGCN (pink line) displays an almost constant trend as the layer increases simply because it excludes the extra computation involved in sparse tensors $\hat { A }$ , such as $\cdot \cdot \cdot \hat { A } \mathrm { ~ R e L U } ( \hat { A } X W ^ { ( 0 ) } ) W ^ { ( 1 ) } \cdot \cdot \cdot ,$ , in the process of training neural networks. AdaGCN maintains the updating of parameters in the $f _ { \theta } ^ { ( l ) }$ with a fixed architecture in each layer while the layer-wise optimization, therefore displaying a nearly constant computation cost within each epoch although more epochs are normally needed in the entire layer-wise training. We leave the analysis of exact time and memory complexity of AdaGCN as future works, but boosting-based algorithms including AdaGCN is memory-efficient (Oono & Suzuki, 2020).
+
+# 5 DISCUSSIONS AND CONCLUSION
+
+One potential concern is that AdaBoost (Hastie et al., 2009; Freund et al., 1999) is established on i.i.d. hypothesis while graphs have inherent data-dependent property. Fortunately, the statistical convergence and consistency of boosting (Lugosi & Vayatis, 2001; Mannor et al., 2003) can still be preserved when the samples are weakly dependent (Lozano et al., 2013). More discussion can refer to Appendix A.5. In this paper, we propose a novel RNN-like deep graph neural network architecture called AdaGCNs. With the delicate architecture design, our approach AdaGCN can effectively explore and exploit knowledge from different orders of neighbors in an Adaboost way. Our work paves a way towards better combining different-order neighbors to design deep graph models rather than only stacking on specific type of graph convolution.
+
+# ACKNOWLEDGMENTS
+
+Z. Lin is supported by NSF China (grant no.s 61625301 and 61731018), Major Scientific Research Project of Zhejiang Lab (grant no.s 2019KB0AC01 and 2019KB0AB02), Beijing Academy of Artificial Intelligence, and Qualcomm.
+
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+
+# A APPENDIX
+
+# A.1 RELATED WORKS ON DEEP GRAPH MODELS
+
+A straightforward solution (Kipf & Welling, 2017; Xu et al., 2018b) inspired by ResNets (He et al., 2016) was by adding residual connections, but this practice was unsatisfactory both in prediction performance and computational efficiency towards building deep graph models, as shown in our experiments in Section 4.1 and 4.3. More recently, JK (Jumping Knowledge Networks (Xu et al., 2018b)) introduced jumping connections into final aggregation mechanism in order to extract knowledge from different layers of graph convolutions. However, this straightforward change of GCN architecture exhibited inconsistent empirical performance for different aggregation operators, which cannot demonstrate the successful construction of deep layers. In addition, Graph powering-based method (Jin et al., 2019) implicitly leveraged more spatial information by extending classical spectral graph theory to robust graph theory, but they concentrated on defending adversarial attacks rather than model depth. LanczosNet (Liao et al., 2019) utilized Lanczos algorithm to construct low rank approximations of the graph Laplacian and then can exploit multi-scale information. Moreover, APPNP (Approximate Personalized Propagation of Neural Predictions, (Klicpera et al., 2018)) leveraged the relationship between GCN and personalized PageRank to derive an improved global propagation scheme. Beyond these, DeepGCNs (Li et al., 2019) directly adapted residual, dense connection and dilated convolutions to GCN architecture, but it mainly focused on the task of point cloud semantic segmentation and has not demonstrated its effectiveness in typical graph tasks. Similar to our work, Deep Adaptive Graph Neural Network (DAGNN) (Liu et al., 2020) also focused on incorporating information from large receptive fields through the entanglement of representation transformation and propagation, while our work efficiently ensembles knowledge from large receptive fields in an Adaboost manner. Other related works based on global attention models (Puny et al., 2020) and sample-based methods (Zeng et al., 2019) are also helpful to construct deep graph models.
+
+# A.2 INSUFFICIENT REPRESENTATION POWER OF ADASGC
+
+As illustrated in Figure 5, with the increasing of layers, AdaSGC with only linear transformation has insufficient representation power both in extracting knowledge from high-order neighbors and combining information from different orders of neighbors while AdaGCN exhibits a consistent improvement of performance as the layer increases.
+
+![](images/9104e19081c34688617941b936149f4c8b5052597b8f8093558b211df918728a.jpg)  
+Figure 5: AdaSGC vs AdaGCN.
+
+# A.3 PROOF OF PROPOSITION 1
+
+Firstly, we further elaborate the Proposition 1 as follows, then we provide the proof.
+
+Suppose that $\gamma$ is the teleport factor. Consider the output $Z _ { \mathrm { P P N P } } = \gamma ( \mathbb { I } - ( 1 - \gamma ) \hat { A } ) ^ { - 1 } f _ { \theta } ( X )$ in PPNP and $Z _ { \mathrm { A P P N P } }$ from its approxminated version APPNP. Let matrix sequence $\{ Z ^ { ( l ) } \}$ be from the output of each layer $l$ in AdaGCN, then PPNP is equivalent to the Exponential Moving Average (EMA) with exponentially decreasing factor In add $\gamma$ , a first-order infinite impulse response filter, on ion, APPNP, which we reformulate in Eq. 10, c $\{ Z ^ { ( l ) } \}$ in a sharing parameters version, i.e., iewed as the approximated form of E $f _ { \theta } ^ { ( l ) } \equiv f _ { \theta }$
+
+limited number of terms.
+
+$$
+Z _ { \mathrm { A P P N P } } = ( \gamma \sum _ { l = 0 } ^ { L - 1 } ( 1 - \gamma ) ^ { l } \hat { A } ^ { l } + ( 1 - \gamma ) ^ { L } \hat { A } ^ { L } ) f _ { \theta } ( X )
+$$
+
+Proof. According to Neumann Theorem, $Z _ { \mathrm { P P N P } }$ can be expanded as a Neumann series:
+
+$$
+\begin{array} { r l } {  { Z _ { \mathrm { P P N P } } = \gamma ( \mathbb { I } - ( 1 - \gamma ) \hat { A } ) ^ { - 1 } f _ { \theta } ( X ) } } \\ & { = \gamma \sum _ { l = 0 } ^ { \infty } ( 1 - \gamma ) ^ { l } \hat { A } ^ { l } f _ { \theta } ( X ) , } \end{array}
+$$
+
+where feature embedding matrix sequence $\{ Z ^ { ( l ) } \}$ for each order of neighbors share the same parameters $f _ { \theta }$ . If we relax this sharing nature to the adaptive form with respect to the layer and put $\hat { A } ^ { l }$ into $f _ { \theta }$ , then the output $Z$ can be approximately formulated as:
+
+$$
+Z _ { \mathrm { P P N P } } \approx \gamma \sum _ { l = 0 } ^ { \infty } ( 1 - \gamma ) ^ { l } f _ { \theta } ^ { ( l ) } ( \hat { A } ^ { l } X )
+$$
+
+This relaxed version from PPNP is the Exponential Moving Average form of matrix sequence $\{ Z ^ { ( l ) } \}$ with exponential decreasing factor $\gamma$ . Moreover, if we approximate the EMA by truncating it after $L - 1$ items, then the weight omitted by stopping after $L - 1$ items is $( 1 - \gamma ) ^ { L }$ . Thus, the approximated EMA is exactly the APPNP form:
+
+$$
+Z _ { \mathrm { A P P N P } } = ( \gamma \sum _ { l = 0 } ^ { L - 1 } ( 1 - \gamma ) ^ { l } \hat { A } ^ { l } + ( 1 - \gamma ) ^ { L } \hat { A } ^ { L } ) f _ { \theta } ( X )
+$$
+
+# A.4 PROOF OF PROPOSITION 2
+
+Proof. We consider a two layers fully-connected neural network as $f$ in Eq. 8, then the output of AdaGCN can be formulated as:
+
+$$
+Z = \sum _ { l = 0 } ^ { L } \alpha ^ { ( l ) } \sigma ( \hat { A } ^ { l } X W ^ { ( 0 ) } ) W ^ { ( 1 ) }
+$$
+
+Particularly, we set W (0) $\begin{array} { r } { W ^ { ( 0 ) } = \frac { b _ { l } } { \mathrm { s i g n } ( b _ { l } ) \alpha ^ { ( l ) } } \mathbb { I } } \end{array}$ blsign(b )α(l) I and W (1) = sign(bl)I where sign(bl) is the signed incidence scalar w.r.t $b _ { l }$ . Then the output of AdaGCN can be presented as:
+
+$$
+\begin{array} { l } { { \displaystyle Z = \sum _ { l = 0 } ^ { L } \alpha ^ { ( l ) } \sigma ( \hat { A } ^ { l } X \frac { b _ { l } } { \mathrm { s i g n } ( b _ { l } ) \alpha ^ { ( l ) } } \mathbb { I } ) \mathrm { s i g n } ( b _ { l } ) \mathbb { I } } } \\ { ~ } \\ { { \displaystyle ~ = \sum _ { l = 0 } ^ { L } \alpha ^ { ( l ) } \sigma ( \hat { A } ^ { l } X ) \frac { b _ { l } } { \mathrm { s i g n } ( b _ { l } ) \alpha ^ { ( l ) } } \mathrm { s i g n } ( b _ { l } ) } } \\ { { \displaystyle ~ = \sum _ { l = 0 } ^ { L } b _ { l } \sigma \left( \hat { A } ^ { l } X \right) } } \end{array}
+$$
+
+The proof that GCNs-based methods are not capable of representing general layer-wise neighborhood mixing has been demonstrated in MixHop (Abu-El-Haija et al., 2019). Proposition 2 proved. □
+
+# A.5 EXPLANATION ABOUT CONSISTENCY OF BOOSTING ON DEPENDENT DATA
+
+Definition 2. ( $\beta$ -mixing sequences.) Let $\sigma _ { i } ^ { j } = \sigma ( W ) = \sigma ( W _ { i } , W _ { i + 1 } , . . . , W _ { j } )$ be the $\sigma$ -field generated by $a$ strictly stationary sequence of random variables $W = ( W _ { i } , W _ { i + 1 } , . . . , W _ { j } )$ . The $\beta$ -mixing coefficient is defined by:
+
+$$
+\beta _ { W } ( n ) = \operatorname* { s u p } _ { k } \mathbb { E } \operatorname* { s u p } \left\{ \left| \mathbb { P } \left( A | \sigma _ { 1 } ^ { k } \right) - \mathbb { P } ( A ) \right| : A \in \sigma _ { k + n } ^ { \infty } \right\}
+$$
+
+Then a sequence $W$ is called $\beta$ -mixing if $l i m _ { n  \infty } \beta _ { W } ( n ) = 0 .$ . Further, $i t$ is algebraically $\beta$ -mixing if there is a positive constant $r _ { \beta }$ such that $\beta _ { W } ( n ) = \mathcal { O } ( n ^ { - r _ { \beta } } )$ .
+
+Definition 3. (Consistency) $A$ classification rule is consistent for a certain distribution $P$ if $E ( L ( h _ { n } ) ) =$ $P \{ h _ { n } ( X ) ~ = ~ Y \} ~ \to ~ a$ as $n  \infty$ where $a$ is a constant. It is strongly Bayes-risk consistent $i f$ $l i m _ { n  \infty } L ( h _ { n } ) = a$ almost surely.
+
+Under these definitions, the convergence and consistence of regularized boosting method on stationary $\beta$ - mixing sequences can be proved under mild assumptions. More details can be referred from (Lozano et al., 2013).
+
+# A.6 EXPERIMENTAL DETAILS
+
+Early Stopping on AdaGCN. We apply the same early stopping mechanism across all the methods as (Klicpera et al., 2018) for fair comparison. Furthermore, boosting theory also has the capacity to perfectly incorporate early stopping and it has been shown that for several boosting algorithms including AdaBoost, this regularization via early stopping can provide guarantees of consistency (Zhang et al., 2005; Jiang et al., 2004; Buhlmann ¨ & Yu, 2003).
+
+Dataset Splitting. We choose a training set of a fixed nodes per class, an early stopping set of 500 nodes and test set of remained nodes. Each experiment is run with 5 random initialization on each data split, leading to a total of 100 runs per experiment. On a standard setting, we randomly select 20 nodes per class. For the two different label rates on each graph, we select 6, 11 nodes per class on citeseer, 8, 16 nodes per class on Cora-ML, 7, 14 nodes per class on Pubmed and 8, 15 nodes per class on MS-Academic dataset.
+
+Model parameters. For all GCN-based approaches, we use the same hyper-parameters in the original paper: learning rate of 0.01, 0.5 dropout rate, $5 \times \mathrm { \dot { 1 } 0 ^ { - 4 } ~ } L _ { 2 }$ regularization weight, and 16 hidden units. For FastGCN, we adopt the officially released code to conduct our experiments. PPNP and APPNP are adapted with best setting: $K = 1 0$ power iteration steps for APPNP, teleport probability $\gamma = 0 . 1$ on Cora-ML, Citeseer and Pubmed, $\gamma = 0 . 2$ on Ms-Academic. In addition, we use two layers with $h = 6 4$ hidden units and apply L2 regularization with $\lambda = 5 \times 1 0 ^ { - 3 }$ on the weights of the first layer and use dropout with dropout rate $d = 0 . 5$ on both layers and the adjacency matrix. The early stopping criterion uses a patience of $p = 1 0 0$ and an (unreachably high) maximum of $n = 1 0 0 0 0$ epochs.The implementation of AdaGCN is adapted from PPNP and APPNP. Corresponding patience $p = 3 0 0$ and $n = 5 0 0$ in the early stopping of AdaGCN. Moreover, SGC is re-implemented in a straightforward way without incorporating advanced optimization for better illustration and comparison. Other baselines are adopted the same parameters described in PPNP and APPNP.
+
+Settings on Reddit dataset. By repeatedly tuning the parameters of these typical methods on Reddit, we finally choose weight decay rate as $1 \dot { 0 } ^ { - 4 }$ , hidden layer size 100 and epoch 20000 for AdaGCN. For APPNP, we opt weight decay rate as $1 0 ^ { - 5 }$ , dropout rate as 0 and epoch 500. V.GCN applies the same parameters in (Kipf & Welling, 2017) and we choose epoch as 500. All approaches have not deployed early stopping due to the expensive computational cost on the large Reddit dataset, which is also a fair comparison.
+
+# A.7 CHOICE OF THE NUMBER OF LAYERS
+
+Different from the “forcible” behaviors in CNNs that directly stack many convolution layers, in our AdaGCN there is a theoretical guidance on the choice of model depth $L$ , i.e., the number of base classifiers or layers, derived from boosting theory. Specifically, according to the boosting theory, the increasing of $L$ can exponentially decreases the empirical loss, however, from the perspective of VC-dimension, an overly large $L$ can yield overfitting of AdaGCN. It should be noted that the deeper graph convolution layers in AdaGCN are not always better, which indeed heavily depends on the the complexity of data. In practice, $L$ can be determined via cross-validation. Specifically, we start a VC-dimension-based analysis to illustrate that too large $L$ can yield overfitting of AdaGCN. For $L$ layers of AdaGCN, its hypothesis set is
+
+$$
+\mathcal { F } _ { L } = \left\{ \underset { k } { \arg \operatorname* { m a x } } \left( \sum _ { l = 1 } ^ { L } \alpha ^ { ( l ) } f _ { \theta } ^ { ( l ) } \right) : \alpha ^ { ( l ) } \in \mathbb { R } , l \in [ 1 , L ] \right\}
+$$
+
+Then the VC-dimension of ${ \mathcal { F } } _ { T }$ can be bounded as follows in terms of the VC-dimension $d$ of the family of base hypothesis:
+
+$$
+\begin{array} { r } { \mathrm { V C d i m } \left( \mathcal { F } _ { L } \right) \leq 2 ( d + 1 ) ( L + 1 ) \log _ { 2 } ( ( L + 1 ) e ) , } \end{array}
+$$
+
+where $e$ is a constant and the upper bounds grows as $L$ increases. Combined with VC-dimension generalization bounds, these results imply that larger values of $L$ can lead to overfitting of AdaBoost. This situation also happens in AdaGCN, which inspires us that there is no need to stack too many layers on AdaGCN in order to avoid overfitting. In practice, $L$ is typically determined via cross-validation.

parse/train/QkRbdiiEjM/QkRbdiiEjM_content_list.json

@@ -0,0 +1,2090 @@
+[
+    {
+        "type": "text",
+        "text": "ADAGCN: ADABOOSTING GRAPH CONVOLUTIONAL NETWORKS INTO DEEP MODELS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
+            820,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Ke Sun   \nZhejiang Lab   \nKey Lab. of Machine Perception (MoE), School of EECS, Peking University   \najksunke@pku.edu.cn ",
+        "bbox": [
+            184,
+            171,
+            687,
+            226
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Zhanxing Zhu\\* Beijing Institute of Big Data Research, Beijing, China zhanxing.zhu@pku.edu.cn ",
+        "bbox": [
+            183,
+            246,
+            539,
+            289
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Zhouchen Lin∗   \nKey Lab. of Machine Perception (MoE), School of EECS, Peking University   \nPazhou Lab, Guangzhou, China   \nzlin@pku.edu.cn ",
+        "bbox": [
+            184,
+            310,
+            686,
+            366
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            404,
+            544,
+            417
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The design of deep graph models still remains to be investigated and the crucial part is how to explore and exploit the knowledge from different hops of neighbors in an efficient way. In this paper, we propose a novel RNN-like deep graph neural network architecture by incorporating AdaBoost into the computation of network; and the proposed graph convolutional network called AdaGCN (Adaboosting Graph Convolutional Network) has the ability to efficiently extract knowledge from high-order neighbors of current nodes and then integrates knowledge from different hops of neighbors into the network in an Adaboost way. Different from other graph neural networks that directly stack many graph convolution layers, AdaGCN shares the same base neural network architecture among all “layers” and is recursively optimized, which is similar to an RNN. Besides, We also theoretically established the connection between AdaGCN and existing graph convolutional methods, presenting the benefits of our proposal. Finally, extensive experiments demonstrate the consistent state-of-the-art prediction performance on graphs across different label rates and the computational advantage of our approach AdaGCN 1. ",
+        "bbox": [
+            233,
+            433,
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+            654
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            678,
+            336,
+            694
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recently, research related to learning on graph structural data has gained considerable attention in machine learning community. Graph neural networks (Gori et al., 2005; Hamilton et al., 2017; Velickovi ˇ c et al., 2018), particularly graph convolutional networks (Kipf & Welling, 2017; Deffer- ´ rard et al., 2016; Bruna et al., 2014) have demonstrated their remarkable ability on node classification (Kipf & Welling, 2017), link prediction (Zhu et al., 2016) and clustering tasks (Fortunato, 2010). Despite their enormous success, almost all of these models have shallow model architectures with only two or three layers. The shallow design of GCN appears counterintuitive as deep versions of these models, in principle, have access to more information, but perform worse. Oversmoothing (Li et al., 2018) has been proposed to explain why deep GCN fails, showing that by repeatedly applying Laplacian smoothing, GCN may mix the node features from different clusters and makes them indistinguishable. This also indicates that by stacking too many graph convolutional layers, the embedding of each node in GCN is inclined to converge to certain value (Li et al., 2018), making it harder for classification. These shallow model architectures restricted by oversmoothing issue limit their ability to extract the knowledge from high-order neighbors, i.e., features from remote hops of neighbors for current nodes. Therefore, it is crucial to design deep graph models such that high-order information can be aggregated in an effective way for better predictions. ",
+        "bbox": [
+            174,
+            709,
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+            890
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "There are some works (Xu et al., 2018b; Liao et al., 2019; Klicpera et al., 2018; Li et al., 2019; Liu et al., 2020) that tried to address this issue partially, and the discussion can refer to Appendix A.1. By contrast, we argue that a key direction of constructing deep graph models lies in the efficient exploration and effective combination of information from different orders of neighbors. Due to the apparent sequential relationship between different orders of neighbors, it is a natural choice to incorporate boosting algorithm into the design of deep graph models. As an important realization of boosting theory, AdaBoost (Freund et al., 1999) is extremely easy to implement and keeps competitive in terms of both practical performance and computational cost (Hastie et al., 2009). Moreover, boosting theory has been used to analyze the success of ResNets in computer vision (Huang et al., 2018) and AdaGAN (Tolstikhin et al., 2017) has already successfully incorporated boosting algorithm into the training of GAN (Goodfellow et al., 2014). ",
+        "bbox": [
+            174,
+            152,
+            825,
+            305
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this work, we focus on incorporating AdaBoost into the design of deep graph convolutional networks in a non-trivial way. Firstly, in pursuit of the introduction of AdaBoost framework, we refine the type of graph convolutions and thus obtain a novel RNN-like GCN architecture called AdaGCN. Our approach can efficiently extract knowledge from different orders of neighbors and then combine these information in an AdaBoost manner with iterative updating of the node weights. Also, we compare our AdaGCN with existing methods from the perspective of both architectural difference and feature representation power to show the benefits of our method. Finally, we conduct extensive experiments to demonstrate the consistent state-of-the-art performance of our approach across different label rates and computational advantage over other alternatives. ",
+        "bbox": [
+            174,
+            311,
+            825,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 OUR APPROACH: ADAGCN",
+        "text_level": 1,
+        "bbox": [
+            176,
+            457,
+            436,
+            473
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 ESTABLISHMENT OF ADAGCN ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            488,
+            428,
+            502
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Consider an undirected graph $\\mathcal { G } = ( \\nu , \\mathcal { E } )$ with $N$ nodes $v _ { i } \\in \\mathcal V$ , edges $( v _ { i } , v _ { j } ) \\in \\mathcal { E }$ . $A \\in \\mathbb { R } ^ { N \\times N }$ is the adjacency matrix with corresponding degree matrix $\\begin{array} { r } { D _ { i i } = \\sum _ { j } \\dot { A } _ { i j } } \\end{array}$ . In the vanilla GCN model (Kipf & Welling, 2017) for semi-supervised node classification, the graph embedding of nodes with two convolutional layers is formulated as: ",
+        "bbox": [
+            174,
+            512,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/479bfd1676720b5e088e236474529cd0bb37dd3aed121757c761d9d38ed36dfc.jpg",
+        "text": "$$\nZ = \\hat { A } \\mathrm { R e L U } ( \\hat { A } X W ^ { ( 0 ) } ) W ^ { ( 1 ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            397,
+            574,
+            601,
+            593
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $Z \\in \\mathbb { R } ^ { N \\times K }$ is the final embedding matrix (output logits) of nodes before softmax and $K$ is the number of classes. $X \\in \\mathbb { R } ^ { N \\times C }$ denotes the feature matrix where $C$ is the input dimension. $\\hat { A } = \\tilde { D } ^ { - \\frac { 1 } { 2 } } \\tilde { A } \\tilde { D } ^ { - \\frac { 1 } { 2 } }$ where ${ \\tilde { A } } = A + I$ and $\\tilde { D }$ is the degree matrix of $\\tilde { A }$ . In addition, $W ^ { ( 0 ) } \\in \\mathbb { R } ^ { C \\times H }$ is the input-to-hidden weight matrix for a hidden layer with $H$ feature maps and $W ^ { ( 1 ) } \\in \\mathbb { R } ^ { H \\times K }$ is the hidden-to-output weight matrix. ",
+        "bbox": [
+            173,
+            597,
+            823,
+            672
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our key motivation of constructing deep graph models is to efficiently explore information of highorder neighbors and then combine these messages from different orders of neighbors in an AdaBoost way. Nevertheless, if we naively extract information from high-order neighbors based on GCN, we are faced with stacking $l$ layers’ parameter matrix $W ^ { ( i ) } , i = 0 , . . . , l - 1$ , which is definitely costly in computation. Besides, Multi-Scale Deep Graph Convolutional Networks (Luan et al., 2019) also theoretically demonstrated that the output can only contain the stationary information of graph structure and loses all the local information in nodes for being smoothed if we simply deepen GCN. Intuitively, the desirable representation of node features does not necessarily need too many nonlinear transformation $f$ applied on them. This is simply due to the fact that the feature of each node is normally one-dimensional sparse vector rather than multi-dimensional data structures, e.g., images, that intuitively need deep convolution network to extract high-level representation for vision tasks. This insight has been empirically demonstrated in many recent works (Wu et al., 2019; Klicpera et al., 2018; Xu et al., 2018a), showing that a two-layer fully-connected neural networks is a better choice in the implementation. Similarly, our AdaGCN also follows this direction by choosing an appropriate $f$ in each layer rather than directly deepen GCN layers. ",
+        "bbox": [
+            174,
+            679,
+            825,
+            888
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Thus, we propose to remove ReLU to avoid the expensive joint optimization of multiple parameter matrices. Similarly, Simplified Graph Convolution (SGC) (Wu et al., 2019) also adopted this practice, arguing that nonlinearity between GCN layers is not crucial and the majority of the benefits arises from local weighting of neighboring features. Then the simplified graph convolution is: ",
+        "bbox": [
+            174,
+            895,
+            820,
+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/1a73cdd96f289c3170574f4224643074e89e71dc26fb8af0e704be928232997c.jpg",
+        "image_caption": [
+            "Figure 1: The RNN-like architecture of AdaGCN with each base classifier $f _ { \\theta } ^ { ( l ) }$ sharing the same neural network architecture $f _ { \\theta }$ . $w ^ { l }$ and $\\theta _ { l }$ denote node weights and parameters computed after the $l$ -th base classifier, respectively. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            233,
+            99,
+            761,
+            282
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            171,
+            366,
+            821,
+            396
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/839a02831d4beab41d811224bbd33d7d392453f93054338f36f39704ad0c934d.jpg",
+        "text": "$$\nZ = \\hat { A } ^ { l } X W ^ { ( 0 ) } W ^ { ( 1 ) } \\cdots W ^ { ( l - 1 ) } = \\hat { A } ^ { l } X \\tilde { W } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            352,
+            402,
+            643,
+            421
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where we collapse $W ^ { ( 0 ) } W ^ { ( 1 ) } \\cdot \\cdot \\cdot W ^ { ( l - 1 ) }$ as $\\tilde { W }$ and $\\hat { A } ^ { l }$ denotes $\\hat { A }$ to the $l$ -th power. In particular, one crucial impact of ReLU in GCN is to accelerate the convergence of matrix multiplication since the ReLU is a contraction mapping intuitively. Thus, the removal of ReLU operation could also alleviate the oversmoothing issue, i.e. slowering the convergence of node embedding to indistinguishable ones (Li et al., 2018). Additionally, without ReLU this simplified graph convolution is also able to avoid the aforementioned joint optimization over multiple parameter matrices, resulting in computational benefits. Nevertheless, we find that this type of stacked linear transformation from graph convolution has insufficient power in representing information of high-order neighbors, which is revealed in our experiment described in Appendix A.2. Therefore, we propose to utilize an appropriate nonlinear function $f _ { \\theta }$ , e.g., a two-layer fully-connected neural network, to replace the linear transformation $\\tilde { W }$ in Eq. 2 and enhance the representation ability of each base classifier in AdaGCN as follows: ",
+        "bbox": [
+            173,
+            429,
+            825,
+            599
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/082d6d2dbd42a145074cf69433f19616c8a6686a5d99b6ea295ba2e0768fb0a1.jpg",
+        "text": "$$\nZ ^ { ( l ) } = f _ { \\theta } ( \\hat { A } ^ { l } X ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            441,
+            598,
+            555,
+            617
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $Z ^ { ( l ) }$ represents the final embedding matrix (output logits before Softmax) after the $l$ -th base classifier in AdaGCN. This formulation also implies that the $l$ -th base classifier in AdaGCN is extracting knowledge from features of current nodes and their $l$ -th hop of neighbors. Due to the fact that the function of $l$ -th base classifier in AdaGCN is similar to that of the $l$ -th layer in other traditional GCN-based methods that directly stack many graph convolutional layers, we regard the whole part of l-th base classifier as the $l$ -th layers in AdaGCN. As for the realization of Multi-class AdaBoost, we apply SAMME (Stagewise Additive Modeling using a Multi-class Exponential loss function) algorithm (Hastie et al., 2009), a natural and clean multi-class extension of the two-class AdaBoost adaptively combining weak classifiers. ",
+        "bbox": [
+            173,
+            622,
+            825,
+            750
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "As illustrated in Figure 1, we apply base classifie r f (l) to extract knowledge from current node feature and $l$ -th hop of neighbors by minimizing current weighted loss. Then we directly compute the weighted error rate $e r r ^ { ( l ) }$ and corresponding weight $\\alpha ^ { ( l ) }$ of current base classifier $f _ { \\theta } ^ { ( l ) }$ as follows: ",
+        "bbox": [
+            174,
+            756,
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+            806
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f7829c9b64ccc92a3fe9034cea924801f0734ff9088c6dc3f7f8392a38704a0f.jpg",
+        "text": "$$\n\\begin{array} { c } { { e r r ^ { ( l ) } = \\displaystyle \\sum _ { i = 1 } ^ { n } w _ { i } \\mathbb { I } \\left( c _ { i } \\neq f _ { \\theta } ^ { ( l ) } \\left( x _ { i } \\right) \\right) / \\sum _ { i = 1 } ^ { n } w _ { i } } } \\\\ { { \\displaystyle \\alpha ^ { ( l ) } = \\log \\frac { 1 - e r r ^ { ( l ) } } { e r r ^ { ( l ) } } + \\log ( K - 1 ) , } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            356,
+            813,
+            640,
+            891
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $w _ { i }$ denotes the weight of $i$ -th node and $c _ { i }$ represents the category of current $i$ -th node. To attain a positive $\\alpha ^ { ( l ) }$ , we only need $( 1 - e r r ^ { ( l ) } ) > 1 / K$ , i.e., the accuracy of each weak classifier ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "should be better than random guess (Hastie et al., 2009). This can be met easily to guarantee the weights to be updated in the right direction. Then we adjust nodes’ weights by increasing weights on incorrectly classified ones: ",
+        "bbox": [
+            174,
+            103,
+            825,
+            146
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c529f1de4a4a32909d74f4fbd21895802d097185ff169dfe41f483bef48eab06.jpg",
+        "text": "$$\nw _ { i } \\gets w _ { i } \\cdot \\exp \\left( \\alpha ^ { ( l ) } \\cdot \\mathbb { I } \\left( c _ { i } \\neq f _ { \\theta } ^ { ( l ) } \\left( x _ { i } \\right) \\right) \\right) , i = 1 , \\dots , n\n$$",
+        "text_format": "latex",
+        "bbox": [
+            313,
+            150,
+            684,
+            178
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "After re-normalizing the weights, we then compute $\\hat { A } ^ { l + 1 } X = \\hat { A } \\cdot ( \\hat { A } ^ { l } X )$ to sequentially extract knowledge from of AdaGCN is t $l { + } 1$ -th hop of neighbors in the following base classifier different from traditional AdaBoost, we only define $f _ { \\theta } ^ { ( l + 1 ) }$ One crucial point, e.g. a two-layer $f _ { \\theta }$ fully connected neural network, which in practice is recursively optimized in each base classifier just similar to a recurrent neural network. This also indicates that the parameters from last base classifier are leveraged as the initialization of next base classifier, which coincides with our intuition that $l + 1$ -th hop of neighbors are directly connected from $l$ -th hop of neighbors. The efficacy of this kind of layer-wise training has been similarly verified in (Belilovsky et al., 2018) recently. Further, we combine the predictions from different orders of neighbors in an Adaboost way to obtain the final prediction $C ( A , X )$ : ",
+        "bbox": [
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+            181,
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+            327
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/440625521e489a931bcb76881b131e45dcc997b3d0ecd270627f3c4c591403f0.jpg",
+        "text": "$$\nC ( A , X ) = \\arg \\operatorname* { m a x } _ { k } \\sum _ { l = 0 } ^ { L } \\alpha ^ { ( l ) } f _ { \\theta } ^ { ( l ) } ( \\hat { A } ^ { l } X )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            367,
+            330,
+            632,
+            375
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Finally, we obtain the concise form of AdaGCN in the following: ",
+        "bbox": [
+            173,
+            377,
+            601,
+            393
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f64fb090e595476d374bb89d9937c13e86b9d9200615161e186419bcd55c51da.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\hat { A } ^ { l } X = \\hat { A } \\cdot ( \\hat { A } ^ { l - 1 } X ) } \\\\ & { Z ^ { ( l ) } = f _ { \\theta } ^ { ( l ) } ( \\hat { A } ^ { l } X ) } \\\\ & { Z = \\mathrm { A d a B o o s t } ( Z ^ { ( l ) } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            426,
+            396,
+            570,
+            462
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Note that $f _ { \\theta }$ is non-linear, rather than linear in SGC (Wu et al., 2019), to guarantee the representation power. As shown in Figure 1, the architecture of AdaGCN is a variant of RNN with synchronous sequence input and output. Although the same classifier architecture is adopted for $f _ { \\theta } ^ { ( \\bar { l } ) }$ , their parameters are different, which is different from vanilla RNN. We provide a detailed description of the our algorithm in Section 3. ",
+        "bbox": [
+            173,
+            470,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.2 COMPARISON WITH EXISTING METHODS ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            560,
+            500,
+            575
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Architectural Difference. As illustrated in Figure 1 and 2, there is an apparent difference among the architectures of GCN (Kipf & Welling, 2017), SGC (Wu et al., 2019), Jumping Knowledge (JK) (Xu et al., 2018b) and AdaGCN. Compared with these existing graph convolutional approaches that sequentially convey intermediate result $Z ^ { ( l ) }$ to compute final prediction, our AdaGCN transmits weights of nodes $w ^ { i }$ , aggregated features of different hops of neighbors ${ \\hat { A } } ^ { l } X$ . More importantly, in AdaGCN the embedding $Z ^ { ( l ) }$ is independent of the flow of computation in the network and the sparse adjacent matrix $\\hat { A }$ is also not directly involved in the computation of individual network because we compute ",
+        "bbox": [
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+            587,
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+            815
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/2d1ea197072ef3f76ecf7d5e86939afa479f716a2e4be601ef8cb8d2f3fec644.jpg",
+        "image_caption": [
+            "Figure 2: Comparison of the graph model architectures. $f _ { a }$ in JK network denotes one aggregation layer with aggregation function such as concatenation or max pooling. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            488,
+            595,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "${ \\hat { A } } ^ { ( l + 1 ) } X$ in advance and then feed it instead of $\\hat { A }$ into the classifier $f _ { \\theta } ^ { ( l + 1 ) }$ , thus yielding significant computation reduction, which will be discussed further in Section 3. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Connection with PPNP and APPNP. We also established a strong connection between AdaGCN and previous state-of-the-art Personalized Propagation of Neural Predictions (PPNP) and Approximate PPNP (APPNP) (Klicpera et al., 2018) method that leverages personalized pagerank to reconstruct graph convolutions in order to use information from a large and adjustable neighborhood. The analysis can be summarized in the following Proposition 1. Proof can refer to Appendix A.3. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Proposition 1. Suppose that $\\gamma$ is the teleport factor. Let matrix sequence $\\{ Z ^ { ( l ) } \\}$ be from the output of each layer l in AdaGCN, then PPNP is equivalent to the Exponential Moving Average (EMA) with exponentially decreasing factor $\\gamma$ on $\\{ Z ^ { ( l ) } \\}$ in a sharing parameters version, and its approximate version APPNP can be viewed as the approximated form of EMA with a limited number of terms. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Proposition 1 illustrates that AdaGCN can be viewed as an adaptive form of APPNP, formulated as: ",
+        "bbox": [
+            173,
+            170,
+            821,
+            186
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0f4c2e3a4600e1f73493536f3ec57debe8c70138685eeab72bfcb189f2fd8317.jpg",
+        "text": "$$\nZ = \\sum _ { l = 0 } ^ { L } \\alpha ^ { ( l ) } f _ { \\theta } ^ { ( l ) } ( \\hat { A } ^ { l } X )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            419,
+            193,
+            578,
+            236
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Specifically, the first discrepancy between AdaGCN and APPNP lies in the adaptive coefficient $\\alpha ^ { ( l ) }$ in AdaGCN determined by the error of $l$ -th base classifier $f _ { \\theta } ^ { ( l ) }$ rather than fixed exponentially decreased weights in APPNP. In addition, AdaGCN employs classifier $f _ { \\theta } ^ { ( l ) }$ with different parameters to learn the embedding of different orders of neighbors, while APPNP shares these parameters in its form. We verified this benefit of our approach in our experiments shown in Section 4.2. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Comparison with MixHop MixHop (Abu-El-Haija et al., 2019) applied the similar way of graph convolution by repeatedly mixing feature representations of neighbors at various distance. Proposition 2 proves that both AdaGCN and MixHop are able to represent feature differences among neighbors while previous GCNs-based methods cannot. Proof can refer to Appendix A.4. Recap the definition of general layer-wise Neighborhood Mixing (Abu-El-Haija et al., 2019) as follows: ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Definition 1. General layer-wise Neighborhood Mixing: $A$ graph convolution network has the ability to represent the layer-wise neighborhood mixing if for any $b _ { 0 } , b _ { 1 } , . . . , b _ { L } ,$ , there exists an injective mapping $f$ with a setting of its parameters, such that the output of this graph convolution network can express the following formula: ",
+        "bbox": [
+            173,
+            398,
+            825,
+            457
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0725158346d191445306954723b8c3434105ca25993ef4b7e96a823c6690c091.jpg",
+        "text": "$$\nf \\left( \\sum _ { l = 0 } ^ { L } b _ { l } \\sigma \\left( \\hat { A } ^ { l } X \\right) \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Proposition 2. AdaGCNs defined by our proposed approach (Eq. equation 7) are capable of representing general layer-wise neighborhood mixing, i.e., can meet the Definition $^ { l }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Albeit the similarity, AdaGCN distinguishes from MixHop in many aspects. Firstly, MixHop concatenates all outputs from each order of neighbors while we combines these predictions in an Adaboost way, which has theoretical generalization guarantee based on boosting theory Hastie et al. (2009). Oono & Suzuki (2020) have recently derived the optimization and generalization guarantees of multi-scale GNNs, serving as the theoretical backbone of AdaGCN. Meantime, MixHop allows full linear mixing of different orders of neighboring features, while AdaGCN utilizes different nonlinear transformation $f _ { \\theta } ^ { ( l ) }$ among all layers, enjoying stronger expressive power. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3 ALGORITHM ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            672,
+            312,
+            689
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In practice, we employ SAMME.R (Hastie et al., 2009), the soft version of SAMME, in AdaGCN. SAMME.R (R for Real) algorithm (Hastie et al., 2009) leverages real-valued confidence-rated predictions, i.e., weighted probability estimates, rather than predicted hard labels in SAMME, in the prediction combination, which has demonstrated a better generalization and faster convergence than SAMME. We elaborate the final version of AdaGCN in Algorithm 1. We provide the analysis on the choice of model depth $L$ in Appendix A.7, and then we elaborate the computational advantage of AdaGCN in the following. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Analysis of Computational Advantage. Due to the similarity of graph convolution in MixHop (Abu-El-Haija et al., 2019), AdaGCN also requires no additional memory or computational complexity compared with previous GCN models. Meanwhile, our approach enjoys huge computational advantage compared with GCN-based models, e.g., PPNP and APPNP, stemming from excluding the additional computation involved in sparse tensors, such as the sparse tensor multiplication between $\\hat { A }$ and other dense tensors, in the forward and backward propagation of the neural network. Specifically, there are only $L$ times sparse tensor operations for an AdaGCN model with $L$ layers, i.e., $\\hat { A } ^ { l } X = \\overset { \\cdot } { A } \\cdot ( \\hat { A } ^ { l - 1 } X )$ for each layer $l$ . This operation in each layer yields a dense tensor ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 AdaGCN based on SAMME.R Algorithm ",
+        "text_level": 1,
+        "bbox": [
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+            102,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Input: Features Matrix $X$ , normalized adjacent matrix $\\hat { A }$ , a two-layer fully connected network $f _ { \\theta }$ , number of layers $L$ and number of classes $K$ . ",
+        "bbox": [
+            171,
+            122,
+            823,
+            150
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Output: Final combined prediction $C ( A , X )$ ",
+        "bbox": [
+            174,
+            151,
+            475,
+            165
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "1: Initialize the node weights $w _ { i } = 1 / n , i = 1 , 2 , . . . , n$ on training set, neighbors feature matrix ${ \\hat { X } } ^ { ( 0 ) } = X$ and classifier $f _ { \\theta } ^ { ( - 1 ) }$ . ",
+        "bbox": [
+            181,
+            167,
+            821,
+            200
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "2: for $l = 0$ to do ",
+        "bbox": [
+            181,
+            199,
+            312,
+            212
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3: Fit the graph convolutional classifier f (l) on neighbor feature matrix $\\hat { X } ^ { ( l ) }$ based on $f _ { \\theta } ^ { ( l - 1 ) }$ by minimizing current weighted loss. ",
+        "bbox": [
+            179,
+            210,
+            826,
+            241
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4: Obtain the weighted probability estimates $p ^ { ( l ) } ( \\hat { X } ^ { ( l ) } )$ for $f _ { \\theta } ^ { ( l ) }$ ",
+        "bbox": [
+            181,
+            242,
+            619,
+            258
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5: Compute the individual prediction $h _ { k } ^ { ( l ) } ( x )$ for the current graph convolutional classifier $f _ { \\theta } ^ { ( l ) }$ ",
+        "bbox": [
+            178,
+            279,
+            820,
+            296
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1652506aac2e1fda8a6e4347de31c9a78ca75177e0250d97983e15c1b7936a76.jpg",
+        "text": "$$\nh _ { k } ^ { ( l ) } ( \\hat { X } ^ { ( l ) } ) \\gets ( K - 1 ) \\left( \\log p _ { k } ^ { ( l ) } ( \\hat { X } ^ { ( l ) } ) - \\frac { 1 } { K } \\sum _ { k ^ { \\prime } } \\log p _ { k ^ { \\prime } } ^ { ( l ) } ( \\hat { X } ^ { ( l ) } ) \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            305,
+            295,
+            735,
+            337
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $k = 1 , \\ldots , K$ ",
+        "bbox": [
+            212,
+            334,
+            356,
+            347
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "6: Adjust the node weights $w _ { i }$ for each node $x _ { i }$ with label $y _ { i }$ on training set: ",
+        "bbox": [
+            178,
+            347,
+            699,
+            361
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2ef9bceab49eeb331bc8abb1fb6a4671dc1074c695721ca37a30f167c752bedf.jpg",
+        "text": "$$\nw _ { i }  w _ { i } \\cdot \\exp ( - \\frac { K - 1 } { K } y _ { i } ^ { \\top } \\log p ^ { ( l ) } ( x _ { i } ) ) , i = 1 , \\ldots , n\n$$",
+        "text_format": "latex",
+        "bbox": [
+            333,
+            359,
+            709,
+            393
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "7: Re-normalize all weights $w _ { i }$ ",
+        "bbox": [
+            183,
+            391,
+            408,
+            405
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "8: Update $l { + } 1$ -hop neighbor feature matrix $\\hat { X } ^ { ( l + 1 ) }$ : ",
+        "bbox": [
+            181,
+            406,
+            535,
+            421
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/eb7fdde56262c65f7b127d62bf0c5a882398005636785be5975bd7f98149bed9.jpg",
+        "text": "$$\n\\hat { X } ^ { ( l + 1 ) } = \\hat { A } \\hat { X } ^ { ( l ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            464,
+            420,
+            578,
+            436
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "9: end for ",
+        "text_level": 1,
+        "bbox": [
+            181,
+            433,
+            254,
+            445
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "10: Combine all predictions $h _ { k } ^ { ( l ) } ( \\hat { X } ^ { ( l ) } )$ for $l = 0 , . . . , L$ ",
+        "bbox": [
+            176,
+            444,
+            537,
+            463
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/985151f356da72a96c21c69bccded2fb92227a8e741883df321c2b87af85a19a.jpg",
+        "text": "$$\nC ( A , X ) = \\arg \\operatorname* { m a x } _ { k } \\sum _ { l = 0 } ^ { L } h _ { k } ^ { ( l ) } ( \\hat { X } ^ { ( l ) } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            393,
+            460,
+            632,
+            503
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "11: return Final combined prediction $C ( A , X )$ . ",
+        "bbox": [
+            174,
+            502,
+            491,
+            517
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "$B ^ { l } = \\hat { A } ^ { l } X$ for the $l$ -th layer, which is then fed into the computation in a two-layer fully-connected network, i.e., $f _ { \\theta } ^ { ( l ) } ( B ^ { l } ) = \\mathrm { \\bar { R e L U } } ( B ^ { l } W ^ { ( 0 ) } ) W ^ { ( 1 ) }$ . Due to the fact that dense tensor $B ^ { l }$ has been computed in advance, there is no other computation related to sparse tensors in the multiple forward and backward propagation procedures while training the neural network. By contrast, this multiple computation involved in sparse tensors in the GCN-based models, e.g., GCN: $\\hat { A } \\mathrm { R e L U } ( \\hat { A } X W ^ { ( 0 ) } ) W ^ { ( 1 ) }$ , is highly expensive. AdaGCN avoids these additional sparse tensor operations in the neural network and then attains huge computational efficiency. We demonstrate this viewpoint in the Section 4.3. ",
+        "bbox": [
+            173,
+            547,
+            825,
+            654
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            674,
+            326,
+            690
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Experimental Setup. We select five commonly used graphs: CiteSeer, Cora-ML (Bojchevski & Gunnemann, 2018; McCallum et al., 2000), PubMed (Sen et al., 2008), MS-Academic (Shchur ¨ et al., 2018) and Reddit. Dateset statistics are summarized in Table 1. Recent graph neural networks suffer from overfitting to a single splitting of training, validation and test datasets (Klicpera et al., 2018). To address this problem, inspired by (Klicpera et al., 2018), we test all approaches on multiple random splits and initialization to conduct a rigorous study. Detailed dataset splittings are provided in Appendix A.6. ",
+        "bbox": [
+            173,
+            705,
+            826,
+            804
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "table",
+        "img_path": "images/c273d40b4c6674d2909ce8352add45db4321a173aad01914f1aaccd56b68cc7a.jpg",
+        "table_caption": [
+            "Table 1: Dateset statistics "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dateset</td><td>Nodes</td><td>Edges</td><td>Classes</td><td>Features</td><td>LabelRate</td></tr><tr><td>CiteSeer</td><td>3,327</td><td>4,732</td><td>6</td><td>3,703</td><td>3.6%</td></tr><tr><td>Cora</td><td>2,708</td><td>5,429</td><td>7</td><td>1,433</td><td>5.2%</td></tr><tr><td>PubMed</td><td>19,717</td><td>44,338</td><td>3</td><td>500</td><td>0.3%</td></tr><tr><td>MS Academic</td><td>18,333</td><td>81,894</td><td>15</td><td>6,805</td><td>1.6%</td></tr><tr><td>Reddit</td><td>232,965</td><td>11,606,919</td><td>41</td><td>602</td><td>65.9%</td></tr></table>",
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+        "image_caption": [
+            "Figure 3: Comparison of test accuracy of different models as the layer increases. We regard the $l$ -th base classifier as the $l$ -th layer in AdaGCN as both of them are leveraged to exploit the information from $l$ -th order of neighbors for current nodes. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Basic Setting of Baselines and AdaGCN. We compare AdaGCN with GCN (Kipf & Welling, 2017) and Simple Graph Convolution (SGC) (Wu et al., 2019) in Figure 3. In Table 2, we employ the same baselines as (Klicpera et al., 2018): V.GCN (vanilla GCN) (Kipf & Welling, 2017) and GCN with our early stopping, N-GCN (network of GCN) (Abu-El-Haija et al., 2018a), GAT (Graph Attention Networks) (Velickovi ˇ c et al., 2018), BT.FP (bootstrapped feature propagation) (Buchnik ´ & Cohen, 2018) and JK (jumping knowledge networks with concatenation) (Xu et al., 2018b). In the computation part, we additionally compare AdaGCN with FastGCN (Chen et al., 2018) and GraphSAGE (Hamilton et al., 2017). We refer to the result of baselines from (Klicpera et al., 2018) and the implementation of AdaGCN is adapted from APPNP. For AdaGCN, after the line search on hyper-parameters, we set $h = 5 0 0 0$ hidden units for the first four datasets except Ms-academic with $h = 3 0 0 0$ , and 15, 12, 20 and 5 layers respectively due to the different graph structures. In addition, we set dropout rate to 0 for Citeseer and Cora-ML datasets and 0.2 for the other datasets and $5 \\times 1 0 ^ { - 3 } L _ { 2 }$ regularization on the first linear layer. We set weight decay as $1 \\times 1 0 ^ { - 3 }$ for Citeseer while $1 \\times 1 0 ^ { - 4 }$ for others. More detailed model parameters and analysis about our early stopping mechanism can be referred from Appendix A.6. ",
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+    {
+        "type": "text",
+        "text": "4.1 DESIGN OF DEEP GRAPH MODELS TO CIRCUMVENT OVERSMOOTHING EFFECT ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "It is well-known that GCN suffers from oversmoothing (Li et al., 2018) with the stacking of more graph convolutions. However, combination of knowledge from each layer to design deep graph models is a reasonable method to circumvent oversmoothing issue. In our experiment, we aim to explore the prediction performance of GCN, GCN with residual connection (Kipf & Welling, 2017), SGC and our AdaGCN with a growing number of layers. ",
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+        "type": "table",
+        "img_path": "images/a9a59100763a2527fb6a96b9a9df7bd81c7b9a8506c5c36277cc12f82fbbf294.jpg",
+        "table_caption": [
+            "Table 2: Average accuracy under 100 runs with uncertainties showing the $95 \\%$ confidence level calculated by bootstrapping. OOM denotes “out of memory”. “(ours)” denotes the results based on our implementation, which are slight lower than numbers above from original literature (Klicpera et al., 2018). P values of paired t test between APPNP (ours) and AdaGCN are provided in the last row. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Citeseer</td><td>Cora-ML</td><td>Pubmed</td><td>MSAcademic</td></tr><tr><td>V.GCN</td><td>73.51±0.48</td><td>82.30±0.34</td><td>77.65±0.40</td><td>91.65±0.09</td></tr><tr><td>GCN</td><td>75.40±0.30</td><td>83.41±0.39</td><td>78.68±0.38</td><td>92.10±0.08</td></tr><tr><td>N-GCN</td><td>74.25±0.40</td><td>82.25±0.30</td><td>77.43±0.42</td><td>92.86±0.11</td></tr><tr><td>GAT</td><td>75.39±0.27</td><td>84.37±0.24</td><td>77.76±0.44</td><td>91.22±0.07</td></tr><tr><td>JK</td><td>73.03±0.47</td><td>82.69±0.35</td><td>77.88±0.38</td><td>91.71±0.10</td></tr><tr><td>BT.FP</td><td>73.55±0.57</td><td>80.84±0.97</td><td>72.94±1.00</td><td>91.61±0.24</td></tr><tr><td>PPNP</td><td>75.83±0.27</td><td>85.29±0.25</td><td>OOM</td><td>OOM</td></tr><tr><td>APPNP</td><td>75.73±0.30</td><td>85.09±0.25</td><td>79.73±0.31</td><td>93.27±0.08</td></tr><tr><td>PPNP (ours)</td><td>75.53±0.32</td><td>84.39±0.28</td><td>OOM</td><td>OOM</td></tr><tr><td>APPNP (ours)</td><td>75.41±0.35</td><td>84.28±0.28</td><td>79.41±0.34</td><td>92.98±0.07</td></tr><tr><td>AdaGCN</td><td>76.68±0.20</td><td>85.97±0.20</td><td>79.95±0.21</td><td>93.17±0.07</td></tr><tr><td>P value</td><td>1.8×10-15</td><td>2.2×10-16</td><td>1.1×10-5</td><td>2.1×10-9</td></tr></table>",
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+        "type": "table",
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+        "table_caption": [],
+        "table_footnote": [
+            "Table 3: Average accuracy across different label rates with 20 splittings of datasets under 100 runs. "
+        ],
+        "table_body": "<table><tr><td></td><td>Citeseer</td><td>Cora-ML</td><td>Pubmed</td><td>MSAcademic</td></tr><tr><td>Label Rates</td><td>1.0% / 2.0%</td><td>2.0% / 4.0%</td><td>0.1% / 0.2%</td><td>0.6% / 1.2%</td></tr><tr><td>V.GCN</td><td>67.6±1.4/70.8±1.4</td><td>76.4±1.3/81.7±0.8</td><td>70.1±1.4/74.6±1.6</td><td>89.7±0.4/91.1±0.2</td></tr><tr><td>GCN</td><td>70.3±0.9/72.7±1.1</td><td>80.0±0.7/82.8±0.9</td><td>71.1±1.1/75.2±1.0</td><td>89.8±0.4/91.2±0.3</td></tr><tr><td>PPNP APPNP</td><td>72.5±0.9/74.7±0.7 72.2±1.3/74.2±1.1</td><td>80.1±0.7/83.0±0.6</td><td>OOM</td><td>OOM</td></tr><tr><td>AdaGCN</td><td></td><td>80.1±0.7/83.2±0.6</td><td>74.0±1.5/77.2±1.2</td><td>91.7±0.2/92.6±0.2</td></tr><tr><td></td><td>74.2±0.3/75.5±0.3</td><td>83.7±0.3/85.3±0.2</td><td>77.1±0.5/79.3±0.3</td><td>92.1±0.1/92.7±0.1</td></tr></table>",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "From Figure 3, it can be easily observed that oversmoothing leads to the rapid decreasing of accuracy for GCN (blue line) as the layer increases. In contrast, the speed of smoothing (green line) of SGC is much slower than GCN due to the lack of ReLU analyzed in Section 2.1. Similarly, GCN with residual connection (yellow line) partially mitigates the oversmoothing effect of original GCN but fails to take advantage of information from different orders of neighbors to improve the prediction performance constantly. Remarkably, AdaGCN (red line) is able to consistently enhance the performance with the increasing of layers across the three datasets. This implies that AdaGCN can efficiently incorporate knowledge from different orders of neighbors and circumvent oversmoothing of original GCN in the process of constructing deep graph models. In addition, the fluctuation of performance for AdaGCN is much lower than GCN especially when the number of layer is large. ",
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+    {
+        "type": "text",
+        "text": "4.2 PREDICTION PERFORMANCE ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We conduct a rigorous study of AdaGCN on four datasets under multiple splittings of dataset. The results from Table 2 suggest the state-of-the-art performance of our approach and the improvement compared with APPNP validates the benefit of adaptive form for our AdaGCN. More rigorously, p values under paired t test demonstrate the significance of improvement for our method. ",
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+    {
+        "type": "text",
+        "text": "In the realistic setting, graphs usually have different labeled nodes and thus it is necessary to investigate the robust performance of methods on different number of labeled nodes. Here we utilize label rates to measure the different numbers of labeled nodes and then sample corresponding labeled nodes per class on graphs respectively. Table 3 presents the consistent state-of-the-art performance of AdaGCN under different label rates. An interesting manifestation from Table 3 is that AdaGCN yields more improvement on fewer label rates compared with APPNP, showing more efficiency on graphs with few labeled nodes. Inspired by the Layer Effect on graphs (Sun et al., 2019), we argue that the increase of layers in AdaGCN can result in more benefits on the efficient propagation of label signals especially on graphs with limited labeled nodes. ",
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+    {
+        "type": "text",
+        "text": "More rigorously, we additionally conduct the comparison on a larger dataset, i.e., Reddit. We choose the best layer as 4 due to the fact that AdaGCN with larger number of layers tends to suffer from overfitting on this relatively simple dataset (with high label rate $6 5 . 9 \\%$ ). Table 4 suggests that AdaGCN can still outperform other typical baselines, including V.GCN, PPNP and APPNP. More experimental details can be referred from Appendix A.6. ",
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+    {
+        "type": "table",
+        "img_path": "images/7a0ee394464230ce42f72c85a3a14812da82333c9c748b6c8abe075e1e646d43.jpg",
+        "table_caption": [
+            "Table 4: Average F1-scores and per-epoch training time of typical methods on Reddit dataset under 5 runs. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Reddit</td><td>F1-Score</td><td>Per-epoch training time</td></tr><tr><td>V.GCN</td><td>94.46±0.06</td><td>5627.46ms</td></tr><tr><td>PPNP</td><td>OOM</td><td>OOM</td></tr><tr><td>APPNP</td><td>95.04±0.07</td><td>29489.81ms</td></tr><tr><td>AdaGCN</td><td>95.39±0.13</td><td>32.29ms</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4.3 COMPUTATIONAL EFFICIENCY ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Without the additional computational cost involved in sparse tensors in the propagation of the neural network, AdaGCN presents huge computational efficiency. From the left part of Figure 4, it exhibits that AdaGCN has the fastest speed of per-epoch training time in comparison with other methods except the comparative performance with FastGCN in Pubmed. In addition, there is a somewhat inconsistency in computation of FastGCN, with fastest speed in Pubmed but slower than ",
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+    {
+        "type": "image",
+        "img_path": "images/89a94feea60c1c911e184d58c6f81cdb9979c2377be74b696a65f7feb1710e99.jpg",
+        "image_caption": [
+            "Figure 4: Left: Per-epoch training time of AdaGCN vs other methods under 5 runs on four datasets. Right: Per-epoch training time of AdaGCN compared with GCN and SGC with the increasing of layers and the digit after $\\mathbf { \\bar { \\Sigma } } ^ { 6 } = \\mathbf { \\bar { \\Sigma } } ^ { 5 }$ denotes the slope in a fitted linear regression. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "GCN on Cora-ML and MS-Academic datasets. Furthermore, with multiple power iterations involved in sparse tensors, APPNP unfortunately has relatively expensive computation cost. It should be noted that this computational advantage of AdaGCN is more significant when it comes to large datasets, e.g., Reddit. Table 4 demonstrates AdaGCN has the potential to perform much faster on larger datasets. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Besides, we explore the computational cost of ReLU and sparse adjacency tensor with respect to the number of layers in the right part of Figure 4. We focus on comparing AdaGCN with SGC and GCN as other GCN-based methods, such as GraphSAGE and APPNP, behave similarly with GCN. Particularly, we can easily observe that both SGC (green line) and GCN (red line) show a linear increasing tendency and GCN yields a larger slope arises from ReLU and more parameters. For SGC, stacking more layers directly is undesirable regarding the computation. Thus, a limited number of SGC layers is preferable with more advanced optimization techniques Wu et al. (2019). It also shows that the computational cost involved sparse matrices in neural networks plays a dominant role in all the cost especially when the layer is large enough. In contrast, our AdaGCN (pink line) displays an almost constant trend as the layer increases simply because it excludes the extra computation involved in sparse tensors $\\hat { A }$ , such as $\\cdot \\cdot \\cdot \\hat { A } \\mathrm { ~ R e L U } ( \\hat { A } X W ^ { ( 0 ) } ) W ^ { ( 1 ) } \\cdot \\cdot \\cdot ,$ , in the process of training neural networks. AdaGCN maintains the updating of parameters in the $f _ { \\theta } ^ { ( l ) }$ with a fixed architecture in each layer while the layer-wise optimization, therefore displaying a nearly constant computation cost within each epoch although more epochs are normally needed in the entire layer-wise training. We leave the analysis of exact time and memory complexity of AdaGCN as future works, but boosting-based algorithms including AdaGCN is memory-efficient (Oono & Suzuki, 2020). ",
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+    {
+        "type": "text",
+        "text": "5 DISCUSSIONS AND CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "One potential concern is that AdaBoost (Hastie et al., 2009; Freund et al., 1999) is established on i.i.d. hypothesis while graphs have inherent data-dependent property. Fortunately, the statistical convergence and consistency of boosting (Lugosi & Vayatis, 2001; Mannor et al., 2003) can still be preserved when the samples are weakly dependent (Lozano et al., 2013). More discussion can refer to Appendix A.5. In this paper, we propose a novel RNN-like deep graph neural network architecture called AdaGCNs. With the delicate architecture design, our approach AdaGCN can effectively explore and exploit knowledge from different orders of neighbors in an Adaboost way. Our work paves a way towards better combining different-order neighbors to design deep graph models rather than only stacking on specific type of graph convolution. ",
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Z. Lin is supported by NSF China (grant no.s 61625301 and 61731018), Major Scientific Research Project of Zhejiang Lab (grant no.s 2019KB0AC01 and 2019KB0AB02), Beijing Academy of Artificial Intelligence, and Qualcomm. ",
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+    {
+        "type": "text",
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            297,
+            117
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.1 RELATED WORKS ON DEEP GRAPH MODELS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            135,
+            529,
+            150
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A straightforward solution (Kipf & Welling, 2017; Xu et al., 2018b) inspired by ResNets (He et al., 2016) was by adding residual connections, but this practice was unsatisfactory both in prediction performance and computational efficiency towards building deep graph models, as shown in our experiments in Section 4.1 and 4.3. More recently, JK (Jumping Knowledge Networks (Xu et al., 2018b)) introduced jumping connections into final aggregation mechanism in order to extract knowledge from different layers of graph convolutions. However, this straightforward change of GCN architecture exhibited inconsistent empirical performance for different aggregation operators, which cannot demonstrate the successful construction of deep layers. In addition, Graph powering-based method (Jin et al., 2019) implicitly leveraged more spatial information by extending classical spectral graph theory to robust graph theory, but they concentrated on defending adversarial attacks rather than model depth. LanczosNet (Liao et al., 2019) utilized Lanczos algorithm to construct low rank approximations of the graph Laplacian and then can exploit multi-scale information. Moreover, APPNP (Approximate Personalized Propagation of Neural Predictions, (Klicpera et al., 2018)) leveraged the relationship between GCN and personalized PageRank to derive an improved global propagation scheme. Beyond these, DeepGCNs (Li et al., 2019) directly adapted residual, dense connection and dilated convolutions to GCN architecture, but it mainly focused on the task of point cloud semantic segmentation and has not demonstrated its effectiveness in typical graph tasks. Similar to our work, Deep Adaptive Graph Neural Network (DAGNN) (Liu et al., 2020) also focused on incorporating information from large receptive fields through the entanglement of representation transformation and propagation, while our work efficiently ensembles knowledge from large receptive fields in an Adaboost manner. Other related works based on global attention models (Puny et al., 2020) and sample-based methods (Zeng et al., 2019) are also helpful to construct deep graph models. ",
+        "bbox": [
+            174,
+            161,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.2 INSUFFICIENT REPRESENTATION POWER OF ADASGC ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            436,
+            591,
+            450
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "As illustrated in Figure 5, with the increasing of layers, AdaSGC with only linear transformation has insufficient representation power both in extracting knowledge from high-order neighbors and combining information from different orders of neighbors while AdaGCN exhibits a consistent improvement of performance as the layer increases. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/9104e19081c34688617941b936149f4c8b5052597b8f8093558b211df918728a.jpg",
+        "image_caption": [
+            "Figure 5: AdaSGC vs AdaGCN. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            243,
+            532,
+            743,
+            742
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.3 PROOF OF PROPOSITION 1 ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            806,
+            398,
+            820
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Firstly, we further elaborate the Proposition 1 as follows, then we provide the proof. ",
+        "bbox": [
+            176,
+            833,
+            666,
+            847
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Suppose that $\\gamma$ is the teleport factor. Consider the output $Z _ { \\mathrm { P P N P } } = \\gamma ( \\mathbb { I } - ( 1 - \\gamma ) \\hat { A } ) ^ { - 1 } f _ { \\theta } ( X )$ in PPNP and $Z _ { \\mathrm { A P P N P } }$ from its approxminated version APPNP. Let matrix sequence $\\{ Z ^ { ( l ) } \\}$ be from the output of each layer $l$ in AdaGCN, then PPNP is equivalent to the Exponential Moving Average (EMA) with exponentially decreasing factor In add $\\gamma$ , a first-order infinite impulse response filter, on ion, APPNP, which we reformulate in Eq. 10, c $\\{ Z ^ { ( l ) } \\}$ in a sharing parameters version, i.e., iewed as the approximated form of E $f _ { \\theta } ^ { ( l ) } \\equiv f _ { \\theta }$ ",
+        "bbox": [
+            174,
+            854,
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+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "limited number of terms. ",
+        "bbox": [
+            173,
+            104,
+            321,
+            117
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/10d5214253e0645ceb5d29adc18773a3a80c52b544128358cdf7d7ccc7ee2b37.jpg",
+        "text": "$$\nZ _ { \\mathrm { A P P N P } } = ( \\gamma \\sum _ { l = 0 } ^ { L - 1 } ( 1 - \\gamma ) ^ { l } \\hat { A } ^ { l } + ( 1 - \\gamma ) ^ { L } \\hat { A } ^ { L } ) f _ { \\theta } ( X )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            341,
+            116,
+            656,
+            156
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Proof. According to Neumann Theorem, $Z _ { \\mathrm { P P N P } }$ can be expanded as a Neumann series: ",
+        "bbox": [
+            173,
+            169,
+            684,
+            183
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/151d0d1efe50cd748f79bb5e87c438614d6cd7aaf86369bb1cb154a702941439.jpg",
+        "text": "$$\n\\begin{array} { r l } {  { Z _ { \\mathrm { P P N P } } = \\gamma ( \\mathbb { I } - ( 1 - \\gamma ) \\hat { A } ) ^ { - 1 } f _ { \\theta } ( X ) } } \\\\ & { = \\gamma \\sum _ { l = 0 } ^ { \\infty } ( 1 - \\gamma ) ^ { l } \\hat { A } ^ { l } f _ { \\theta } ( X ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            390,
+            184,
+            607,
+            241
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "where feature embedding matrix sequence $\\{ Z ^ { ( l ) } \\}$ for each order of neighbors share the same parameters $f _ { \\theta }$ . If we relax this sharing nature to the adaptive form with respect to the layer and put $\\hat { A } ^ { l }$ into $f _ { \\theta }$ , then the output $Z$ can be approximately formulated as: ",
+        "bbox": [
+            173,
+            242,
+            826,
+            285
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9a9adc20053375dbcedb0f1412f8231082310e3a9b289da3475853e746c7c92a.jpg",
+        "text": "$$\nZ _ { \\mathrm { P P N P } } \\approx \\gamma \\sum _ { l = 0 } ^ { \\infty } ( 1 - \\gamma ) ^ { l } f _ { \\theta } ^ { ( l ) } ( \\hat { A } ^ { l } X )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            395,
+            286,
+            602,
+            324
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "This relaxed version from PPNP is the Exponential Moving Average form of matrix sequence $\\{ Z ^ { ( l ) } \\}$ with exponential decreasing factor $\\gamma$ . Moreover, if we approximate the EMA by truncating it after $L - 1$ items, then the weight omitted by stopping after $L - 1$ items is $( 1 - \\gamma ) ^ { L }$ . Thus, the approximated EMA is exactly the APPNP form: ",
+        "bbox": [
+            173,
+            327,
+            825,
+            376
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fec00de48a9ebb2279abbd19ceacf54ce4453ed9b429fd034726117a508ec678.jpg",
+        "text": "$$\nZ _ { \\mathrm { A P P N P } } = ( \\gamma \\sum _ { l = 0 } ^ { L - 1 } ( 1 - \\gamma ) ^ { l } \\hat { A } ^ { l } + ( 1 - \\gamma ) ^ { L } \\hat { A } ^ { L } ) f _ { \\theta } ( X )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            343,
+            373,
+            655,
+            412
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.4 PROOF OF PROPOSITION 2 ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            440,
+            400,
+            454
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Proof. We consider a two layers fully-connected neural network as $f$ in Eq. 8, then the output of AdaGCN can be formulated as: ",
+        "bbox": [
+            178,
+            465,
+            823,
+            489
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/996bf3eabe21cbdba2c7100d8a6a7675146392da3f66f3f8bd961b3ba9ba5dcd.jpg",
+        "text": "$$\nZ = \\sum _ { l = 0 } ^ { L } \\alpha ^ { ( l ) } \\sigma ( \\hat { A } ^ { l } X W ^ { ( 0 ) } ) W ^ { ( 1 ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            400,
+            487,
+            598,
+            526
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Particularly, we set W (0) $\\begin{array} { r } { W ^ { ( 0 ) } = \\frac { b _ { l } } { \\mathrm { s i g n } ( b _ { l } ) \\alpha ^ { ( l ) } } \\mathbb { I } } \\end{array}$ blsign(b )α(l) I and W (1) = sign(bl)I where sign(bl) is the signed incidence scalar w.r.t $b _ { l }$ . Then the output of AdaGCN can be presented as: ",
+        "bbox": [
+            173,
+            527,
+            823,
+            558
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/86f7c445d1874b8d19c360d700757c6ca0c38a680317ca3b57615d1efe166bfb.jpg",
+        "text": "$$\n\\begin{array} { l } { { \\displaystyle Z = \\sum _ { l = 0 } ^ { L } \\alpha ^ { ( l ) } \\sigma ( \\hat { A } ^ { l } X \\frac { b _ { l } } { \\mathrm { s i g n } ( b _ { l } ) \\alpha ^ { ( l ) } } \\mathbb { I } ) \\mathrm { s i g n } ( b _ { l } ) \\mathbb { I } } } \\\\ { ~ } \\\\ { { \\displaystyle ~ = \\sum _ { l = 0 } ^ { L } \\alpha ^ { ( l ) } \\sigma ( \\hat { A } ^ { l } X ) \\frac { b _ { l } } { \\mathrm { s i g n } ( b _ { l } ) \\alpha ^ { ( l ) } } \\mathrm { s i g n } ( b _ { l } ) } } \\\\ { { \\displaystyle ~ = \\sum _ { l = 0 } ^ { L } b _ { l } \\sigma \\left( \\hat { A } ^ { l } X \\right) } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            364,
+            559,
+            632,
+            679
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The proof that GCNs-based methods are not capable of representing general layer-wise neighborhood mixing has been demonstrated in MixHop (Abu-El-Haija et al., 2019). Proposition 2 proved. □ ",
+        "bbox": [
+            169,
+            679,
+            823,
+            705
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.5 EXPLANATION ABOUT CONSISTENCY OF BOOSTING ON DEPENDENT DATA",
+        "text_level": 1,
+        "bbox": [
+            176,
+            719,
+            735,
+            734
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Definition 2. ( $\\beta$ -mixing sequences.) Let $\\sigma _ { i } ^ { j } = \\sigma ( W ) = \\sigma ( W _ { i } , W _ { i + 1 } , . . . , W _ { j } )$ be the $\\sigma$ -field generated by $a$ strictly stationary sequence of random variables $W = ( W _ { i } , W _ { i + 1 } , . . . , W _ { j } )$ . The $\\beta$ -mixing coefficient is defined by: ",
+        "bbox": [
+            173,
+            743,
+            826,
+            782
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7a68943e19cc6439832e844d30e286d4f6decb2c90f9a1be908c5f79741fd8e7.jpg",
+        "text": "$$\n\\beta _ { W } ( n ) = \\operatorname* { s u p } _ { k } \\mathbb { E } \\operatorname* { s u p } \\left\\{ \\left| \\mathbb { P } \\left( A | \\sigma _ { 1 } ^ { k } \\right) - \\mathbb { P } ( A ) \\right| : A \\in \\sigma _ { k + n } ^ { \\infty } \\right\\}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            325,
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+            808
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Then a sequence $W$ is called $\\beta$ -mixing if $l i m _ { n  \\infty } \\beta _ { W } ( n ) = 0 .$ . Further, $i t$ is algebraically $\\beta$ -mixing if there is a positive constant $r _ { \\beta }$ such that $\\beta _ { W } ( n ) = \\mathcal { O } ( n ^ { - r _ { \\beta } } )$ . ",
+        "bbox": [
+            171,
+            809,
+            823,
+            835
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Definition 3. (Consistency) $A$ classification rule is consistent for a certain distribution $P$ if $E ( L ( h _ { n } ) ) =$ $P \\{ h _ { n } ( X ) ~ = ~ Y \\} ~ \\to ~ a$ as $n  \\infty$ where $a$ is a constant. It is strongly Bayes-risk consistent $i f$ $l i m _ { n  \\infty } L ( h _ { n } ) = a$ almost surely. ",
+        "bbox": [
+            174,
+            837,
+            825,
+            876
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Under these definitions, the convergence and consistence of regularized boosting method on stationary $\\beta$ - mixing sequences can be proved under mild assumptions. More details can be referred from (Lozano et al., 2013). ",
+        "bbox": [
+            174,
+            885,
+            825,
+            924
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.6 EXPERIMENTAL DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            393,
+            117
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Early Stopping on AdaGCN. We apply the same early stopping mechanism across all the methods as (Klicpera et al., 2018) for fair comparison. Furthermore, boosting theory also has the capacity to perfectly incorporate early stopping and it has been shown that for several boosting algorithms including AdaBoost, this regularization via early stopping can provide guarantees of consistency (Zhang et al., 2005; Jiang et al., 2004; Buhlmann ¨ & Yu, 2003). ",
+        "bbox": [
+            173,
+            128,
+            823,
+            193
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Dataset Splitting. We choose a training set of a fixed nodes per class, an early stopping set of 500 nodes and test set of remained nodes. Each experiment is run with 5 random initialization on each data split, leading to a total of 100 runs per experiment. On a standard setting, we randomly select 20 nodes per class. For the two different label rates on each graph, we select 6, 11 nodes per class on citeseer, 8, 16 nodes per class on Cora-ML, 7, 14 nodes per class on Pubmed and 8, 15 nodes per class on MS-Academic dataset. ",
+        "bbox": [
+            173,
+            199,
+            825,
+            262
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Model parameters. For all GCN-based approaches, we use the same hyper-parameters in the original paper: learning rate of 0.01, 0.5 dropout rate, $5 \\times \\mathrm { \\dot { 1 } 0 ^ { - 4 } ~ } L _ { 2 }$ regularization weight, and 16 hidden units. For FastGCN, we adopt the officially released code to conduct our experiments. PPNP and APPNP are adapted with best setting: $K = 1 0$ power iteration steps for APPNP, teleport probability $\\gamma = 0 . 1$ on Cora-ML, Citeseer and Pubmed, $\\gamma = 0 . 2$ on Ms-Academic. In addition, we use two layers with $h = 6 4$ hidden units and apply L2 regularization with $\\lambda = 5 \\times 1 0 ^ { - 3 }$ on the weights of the first layer and use dropout with dropout rate $d = 0 . 5$ on both layers and the adjacency matrix. The early stopping criterion uses a patience of $p = 1 0 0$ and an (unreachably high) maximum of $n = 1 0 0 0 0$ epochs.The implementation of AdaGCN is adapted from PPNP and APPNP. Corresponding patience $p = 3 0 0$ and $n = 5 0 0$ in the early stopping of AdaGCN. Moreover, SGC is re-implemented in a straightforward way without incorporating advanced optimization for better illustration and comparison. Other baselines are adopted the same parameters described in PPNP and APPNP. ",
+        "bbox": [
+            173,
+            268,
+            825,
+            409
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Settings on Reddit dataset. By repeatedly tuning the parameters of these typical methods on Reddit, we finally choose weight decay rate as $1 \\dot { 0 } ^ { - 4 }$ , hidden layer size 100 and epoch 20000 for AdaGCN. For APPNP, we opt weight decay rate as $1 0 ^ { - 5 }$ , dropout rate as 0 and epoch 500. V.GCN applies the same parameters in (Kipf & Welling, 2017) and we choose epoch as 500. All approaches have not deployed early stopping due to the expensive computational cost on the large Reddit dataset, which is also a fair comparison. ",
+        "bbox": [
+            174,
+            415,
+            825,
+            479
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "A.7 CHOICE OF THE NUMBER OF LAYERS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            496,
+            477,
+            510
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Different from the “forcible” behaviors in CNNs that directly stack many convolution layers, in our AdaGCN there is a theoretical guidance on the choice of model depth $L$ , i.e., the number of base classifiers or layers, derived from boosting theory. Specifically, according to the boosting theory, the increasing of $L$ can exponentially decreases the empirical loss, however, from the perspective of VC-dimension, an overly large $L$ can yield overfitting of AdaGCN. It should be noted that the deeper graph convolution layers in AdaGCN are not always better, which indeed heavily depends on the the complexity of data. In practice, $L$ can be determined via cross-validation. Specifically, we start a VC-dimension-based analysis to illustrate that too large $L$ can yield overfitting of AdaGCN. For $L$ layers of AdaGCN, its hypothesis set is ",
+        "bbox": [
+            173,
+            520,
+            825,
+            622
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5f9a31cb13b73057b65e49197ef1fd4274c52684341bc79b131e3f08336c6a2e.jpg",
+        "text": "$$\n\\mathcal { F } _ { L } = \\left\\{ \\underset { k } { \\arg \\operatorname* { m a x } } \\left( \\sum _ { l = 1 } ^ { L } \\alpha ^ { ( l ) } f _ { \\theta } ^ { ( l ) } \\right) : \\alpha ^ { ( l ) } \\in \\mathbb { R } , l \\in [ 1 , L ] \\right\\}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            325,
+            628,
+            671,
+            667
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Then the VC-dimension of ${ \\mathcal { F } } _ { T }$ can be bounded as follows in terms of the VC-dimension $d$ of the family of base hypothesis: ",
+        "bbox": [
+            168,
+            674,
+            821,
+            699
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ad16f8ac4713f15a0aee730c09d5ebcb1d68a9781f0020f1c59bea0781eb2549.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathrm { V C d i m } \\left( \\mathcal { F } _ { L } \\right) \\leq 2 ( d + 1 ) ( L + 1 ) \\log _ { 2 } ( ( L + 1 ) e ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            348,
+            698,
+            648,
+            713
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "where $e$ is a constant and the upper bounds grows as $L$ increases. Combined with VC-dimension generalization bounds, these results imply that larger values of $L$ can lead to overfitting of AdaBoost. This situation also happens in AdaGCN, which inspires us that there is no need to stack too many layers on AdaGCN in order to avoid overfitting. In practice, $L$ is typically determined via cross-validation. ",
+        "bbox": [
+            174,
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+            768
+        ],
+        "page_idx": 14
+    }
+]

parse/train/S1lyyANYwr/S1lyyANYwr.md

@@ -0,0 +1,718 @@
+# CONSTRAINED MARKOV DECISION PROCESSES VIA BACKWARD VALUE FUNCTIONS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Although Reinforcement Learning (RL) algorithms have found tremendous success in simulated domains, they often cannot directly be applied to physical systems, especially in cases where there are hard constraints to satisfy (e.g. on safety or resources). In standard RL, the agent is incentivized to explore any behavior as long as it maximizes rewards, but in the real world undesired behavior can damage either the system or the agent in a way that breaks the learning process itself. In this work, we model the problem of learning with constraints as a Constrained Markov Decision Process, and provide a new on-policy formulation for solving it. A key contribution of our approach is to translate cumulative cost constraints into state-based constraints. Through this, we define a safe policy improvement method which maximizes returns while ensuring that the constraints are satisfied at every step. We provide theoretical guarantees under which the agent converges while ensuring safety over the course of training. We also highlight computational advantages of this approach. The effectiveness of our approach is demonstrated on safe navigation tasks and in safety-constrained versions of MuJoCo environments, with deep neural networks.
+
+# 1 INTRODUCTION
+
+Reinforcement Learning (RL) provides a sound decision-theoretic framework to optimize the behavior of learning agents in an interactive setting (Sutton & Barto, 2018). Recently, the field of RL has found success in many high-dimensional domains, like video games, Go, robot locomotion and navigation. However, most of the success of RL algorithms has been limited to simulators, where the learning algorithm has the ability to reset the simulator. In the physical world, an agent will need to avoid harmful behavior (e.g. damaging the environment or the agent’s hardware) while learning to explore behaviors that maximize the reward.
+
+A few popular approaches for avoiding undesired behaviors for high-dimensional systems include reward-shaping (Moldovan & Abbeel, 2012), reachability-preserving algorithms (Mitchell, 2003; Eysenbach et al., 2017), state-level surrogate constraint satisfaction algorithms (Dalal et al., 2018), risk-sensitive algorithms (Tamar et al., 2013; Chow et al., 2015) and apprenticeship learning (Abbeel & Ng, 2004). There also exists model-based Bayesian approaches that are focused on imposing the constraints via the dynamics (such as classifying parts of state space as unsafe) and then using model predictive control to incorporate the constraints in the policy optimization and planning (Turchetta et al., 2016; Berkenkamp et al., 2017; Wachi et al., 2018; Koller et al., 2018). A natural way to model safety is via constraint satisfaction. A standard formulation for adding constraints to RL problems is the Constrained Markov Decision Process (CMDP) framework (Altman, 1999), wherein the environment is extended to also provide feedback on constraint costs. The agent must then attempt to maximize its expected return while also satisfying cumulative constraints.
+
+A few algorithms have been proposed to solve CMDPs for high-dimensional domains with continuous action spaces - however they come with their own caveats. Reward Constrained Policy Optimization (Tessler et al., 2018) and Primal Dual Policy Optimization (Chow et al., 2015) do not guarantee constraint satisfaction during the learning procedure, only on the final policy. Constrained Policy Optimization (Achiam et al., 2017) provides monotonic policy improvement but is computationally expensive due to requiring a backtracking line-search procedure and conjugate gradient algorithm for approximating the Fisher Information Matrix. Lyapunov-based Safe Policy Optimization (Chow et al., 2019) requires solving a Linear Program (LP) at every step of policy evaluation, although they show that there exists heuristics which can be substituted for the LP at the expense of theoretical guarantees.
+
+In this work, we propose an alternate formulation for solving CMDPs that transforms trajectory-level constraints into localized state-dependent constraints, through which a safe policy improvement step can be defined. In our approach, we define a notion of Backward Value Functions, which act as an estimator of the expected cost collected by the agent so far and can be learned via standard RL bootstrap techniques. We provide conditions under which this new formulation is able to solve CMDPs without violating the constraints during the learning process. Our formulation allows us to define state-level constraints without explicitly solving a LP or the Dual problem at every iteration. Our method is implemented as a reduction to any model-free on-policy bootstrap based RL algorithm, both for deterministic and stochastic policies, and discrete and continuous action spaces. We provide the empirical evidence of our approach with Deep RL methods on various safety benchmarks, including 2D navigation grid worlds (Leike et al., 2017; Chow et al., 2018), and MuJoCo tasks (Achiam et al., 2017; Chow et al., 2019).
+
+# 2 CONSTRAINED MARKOV DECISION PROCESSES
+
+We write ${ \mathcal { P } } ( Y )$ for the set of probability distributions on a space $Y$ . A Markov Decision Process (MDP) (Puterman, 2014) is a tuple $( \mathcal { X } , \mathcal { A } , \mathcal { P } , r , x _ { 0 } )$ , where $\mathcal { X }$ is a set of states, $\mathcal { A }$ is a set of actions, $r : \mathcal { X } \times \mathcal { A }  [ 0 , R _ { M A X } ]$ is a reward function, $\mathcal { P } : \mathcal { X } \times \mathcal { A }  \mathcal { P } ( \mathcal { X } )$ is a transition probability function, and $x _ { 0 }$ is a fixed starting state. For simplicity we assume a deterministic reward function and starting state, but our results generalize.
+
+A Constrained Markov Decision Process (CMDP) (Altman, 1999) is a MDP with additional constraints that restrict the set of permissible policies for the MDP. Formally, a CMDP is a tuple $( \mathcal { X } , \mathcal { A } , \mathcal { P } , r , x _ { 0 } , d , d _ { 0 } )$ , where $d : \mathcal { X }  [ 0 , D _ { M A X } ]$ is the cost function1 and $\mathbf { \Phi } _ { M _ { 0 } } \in \mathbb { R } ^ { \geq 0 }$ is the maximum allowed cumulative cost. The set of feasible policies that satisfy the CMDP is the subset of stationary policies $\begin{array} { r } { \Pi _ { \mathcal { D } } : = \{ \pi : \mathcal { X }  \mathcal { P } ( \mathcal { A } ) \mid \mathbb { E } [ \sum _ { t = 0 } ^ { T } d ( x _ { t } ) \mid x _ { 0 } , \pi ] \leq d _ { 0 } \} } \end{array}$ . We consider a finite time horizon $T$ after which the episode terminates. The expected sum of rewards following a policy $\pi$ from an initial state $x$ is given by the value function $\begin{array} { r } { V ^ { \pi } ( x ) = \mathbb { E } [ \sum _ { t = 0 } ^ { T } r ( x _ { t } , a _ { t } ) \mid \pi , x ] } \end{array}$ . Analogously, the expected sum of costs is given by the cost value function $V _ { \mathcal { D } } ^ { \pi } ( x ) = \mathbb { E } [ \sum _ { t = 0 } ^ { T } d ( x _ { t } ) \mid \pi , x ]$ . The RL problem in the CMDP is to find the feasible policy which maximizes expected returns from the initial state $x _ { 0 }$ , i.e.
+
+$$
+\pi ^ { * } = \arg \operatorname* { m a x } _ { \pi \in \Pi _ { \mathcal { D } } } V ^ { \pi } ( x _ { 0 } )
+$$
+
+An important point to note about CMDPs is that, in the original formulation, the cost function depends on immediate states but the constraint is cumulative and thus depends on the entire trajectory.
+
+In the case of MDPs, where a model of the environment is not known or is not easily obtained, it is still possible for the agent to find the optimal policy using Temporal Difference (TD) methods (Sutton, 1988). Broadly, these methods update the estimates of the value functions via bootstraps of previous estimates on sampled transitions (we refer the reader to Sutton & Barto (2018) for more information). In the on-policy setting, we alternate between estimating the state-action value function $Q ^ { \pi }$ for a given $\pi$ and updating the policy to be greedy with respect to the value function.
+
+# 3 SAFE POLICY ITERATION VIA BACKWARD VALUE FUNCTIONS
+
+Our approach proposes to convert the trajectory-level constraints of the CMDP into single-step state-wise constraints in such a way that satisfying the state-wise formulation will entail satisfying the original trajectory-level problem. The advantages of this approach are twofold: i) working with single-step state-wise constraints allows us to obtain analytical solutions to the optimization problem, and ii) the state-wise constraints can be defined via value-function-like quantities and can thus be estimated with well-studied value-based methods. The state-wise constraints are defined via Backward Value Functions, in Section 3.2, and in Section 3.3 we provide a safe policy iteration procedure which satisfies said constraints (and thus the original problem).
+
+# 3.1 BACKWARD MARKOV CHAIN
+
+Unlike in traditional RL, in the CMDP setting the agent needs to take into account the constraints which it has accumulated so far in order to plan accordingly for the future. Intuitively, the accumulated cost so far can be estimated via the cost value function $V _ { \mathcal { D } }$ running “backward in time”. Before giving the details of our approach and formally introducing the Backward Value Functions, we review the main ideas, which are built upon the work of Morimura et al. (2010), who also considered time-reversed Markov chains but from the standpoint of estimating the gradient of the log stationary distribution; we extend these ideas to TD methods.
+
+Assumption 3.1 (Stationarity). The MDP is ergodic for any policy $\pi$ , i.e., the Markov chain characterized by the transition probability $\begin{array} { r } { \mathcal { P } ^ { \pi } ( x _ { t + 1 } \vert \bar { x } _ { t } ) = \sum _ { a _ { t } \in \mathcal { A } } \mathcal { P } ( \bar { x } _ { t + 1 } \vert x _ { t } , a _ { t } ) \pi ( a _ { t } \vert x _ { t } ) } \end{array}$ is irreducible and aperiodic.
+
+Let $\mathcal { M } ( \pi )$ denote the Markov chain characterized by transition probability $\mathcal { P } ^ { \pi } ( x _ { t + 1 } \vert x _ { t } )$ . The above assumption implies that there exists a unique stationary distribution $\eta ^ { \pi }$ associated with $\pi$ , such that it satisfies: $\begin{array} { r } { \bar { \eta } ^ { \pi } ( x _ { t + 1 } ) = \sum _ { x _ { t } \in \mathcal { X } } \mathcal { P } ^ { \pi } ( x _ { t + 1 } \vert x _ { t } ) \eta ^ { \pi } ( x _ { t } ) } \end{array}$ . We abuse the notation and denote $\mathcal { P } ^ { \pi } ( x _ { t + 1 } , a _ { t } | x _ { t } ) = \mathcal { P } ( x _ { t + 1 } | x _ { t } , a _ { t } ) \pi ( a _ { t } | \bar { x } _ { t } )$ .
+
+According to Bayes’ rule, the probability $q ( x _ { t - 1 } , a _ { t - 1 } | x _ { t } )$ of a previous state-action pair $( x _ { t - 1 } , a _ { t - 1 } )$ leading to the current state $x _ { t }$ is given by:
+
+$$
+q ( x _ { t - 1 } , a _ { t - 1 } | x _ { t } ) = \frac { \mathcal { P } ( x _ { t } | x _ { t - 1 } , a _ { t - 1 } ) P r ( x _ { t - 1 } , a _ { t - 1 } ) } { \sum _ { x _ { t - 1 } \in \mathcal { X } } \sum _ { a _ { t - 1 } \in \mathcal { A } } \mathcal { P } ( x _ { t } | x _ { t - 1 } , a _ { t - 1 } ) P r ( x _ { t - 1 } , a _ { t - 1 } ) } .
+$$
+
+From Assumption 3.1, we have that $\begin{array} { r l r } { P r ( x _ { t - 1 } , a _ { t - 1 } ) } & { { } = } & { \eta ^ { \pi } ( x _ { t - 1 } ) \pi ( a _ { t - 1 } | x _ { t - 1 } ) } \end{array}$ , and $\begin{array} { r l r } { \sum _ { x _ { t - 1 } \in \mathcal { X } } \sum _ { a _ { t - 1 } \in \mathcal { A } } \mathcal { P } ( x _ { t } | x _ { t - 1 } , a _ { t - 1 } ) P r ( x _ { t - 1 } , a _ { t - 1 } ) } & { = } & { \eta ^ { \pi } ( x _ { t } ) } \end{array}$ . We denote the posterior $q ( x _ { t - 1 } , a _ { t - 1 } | x _ { t } )$ as backward (or time-reversed) probability $\mathbf {  } \pi ^ { \pi } ( x _ { t - 1 } , a _ { t - 1 } | x _ { t } )$ , and we have:
+
+$$
+\begin{array} { r l } & { \overleftarrow { \boldsymbol { \mathcal { P } } } ^ { \pi } ( x _ { t - 1 } , a _ { t - 1 } | x _ { t } ) = \frac { \mathcal { P } ( x _ { t } | x _ { t - 1 } , a _ { t - 1 } ) \eta ^ { \pi } ( x _ { t - 1 } ) \pi ( a _ { t - 1 } | x _ { t - 1 } ) } { \eta ^ { \pi } ( x _ { t } ) } } \\ & { \phantom { \quad \quad \ } = \frac { \mathcal { P } ^ { \pi } ( x _ { t } , a _ { t - 1 } | x _ { t - 1 } ) \eta ^ { \pi } ( x _ { t - 1 } ) } { \eta ^ { \pi } ( x _ { t } ) } . } \end{array}
+$$
+
+The forward Markov chain, characterized by the transition matrix $\mathscr { P } ^ { \pi } ( x _ { t + 1 } | x _ { t } )$ , runs forward in time, i.e., it gives the probability of the next state in which the agent will end up. Analogously, a backward Markov chain is denoted by the transition matrix $\begin{array} { r } { \mathbf { et { } { ' } { \mathcal { P } } } ^ { \pi } ( x _ { t - 1 } \vert x _ { t } ) = \sum _ { a _ { t - 1 } \in A } \mathbf { et { } { ' } { \mathcal { P } } ^ { \pi } } ( x _ { t - 1 } , a _ { t - 1 } \vert x _ { t } ) . } \end{array}$ , and describes the state and action the agent took to reach the current state.
+
+Definition 3.1 (Backward Markov Chain). A backward Markov chain associated with $\mathcal { M } ( \pi )$ is denoted by $\overleftarrow { B } ( \pi )$ and is characterized by the transition probability $\overleftarrow { \mathcal { P } } ^ { \pi } ( x _ { t - 1 } | x _ { t } )$ .
+
+# 3.2 BACKWARD VALUE FUNCTION
+
+We define the Backward Value Function (BVF) to be a value function running on the backward Markov chain $\overleftarrow { B } ( \pi )$ . A BVF is the expected sum of returns or costs collected by the agent so far. We are mainly interested in maintaining estimates of the cumulative cost incurred at a state in order to express the total constraint state-wise.
+
+We note that, since every Markov chain $\mathcal { M } ( \pi )$ is ergodic by Assumption 3.1, the corresponding backward Markov chain $B ( \pi )$ is also ergodic (Morimura et al., 2010, Prop. B.1). In particular, every policy $\pi$ can reach the initial state via some path in the transition graph of the backward Markov chain. Thus, the backwards Markov chain are also finite-horizon for some $T _ { B }$ , with $x _ { 0 }$ corresponding to the terminal state. We define a finite-horizon Backward Value Function for cost as:
+
+$$
+\overleftarrow { V } _ { \mathcal { D } } ^ { \pi } ( x _ { t } ) = \mathbb { E } _ { \overleftarrow { \mathcal { B } } ( \pi ) } \left[ \sum _ { k = 0 } ^ { T _ { \mathcal { B } } } d ( x _ { t - k } ) | x _ { t } \right] .
+$$
+
+Proposition 3.1 (Sampling). Samples from the forward Markov chain $\mathcal { M } ( \pi )$ can be used directly to estimate the statistics of the backward Markov chain $\overleftarrow { B } ( \pi )$ (or the Backward Value Function). We
+
+have:
+
+$$
+\begin{array} { r l r } {  { \mathbb { E } _ { \overleftarrow { \mathcal { B } } ( \pi ) } [ \sum _ { k = 0 } ^ { K } d ( x _ { t - k } ) | x _ { t } ] = \mathbb { E } _ { \mathcal { M } ( \pi ) } [ \sum _ { k = 0 } ^ { K } d ( x _ { t - k } ) | x _ { t } , \eta ^ { \pi } ( x _ { t - K } ) ] , } } \\ & { } & { = \mathbb { E } _ { \mathcal { M } ( \pi ) } [ \sum _ { k = 0 } ^ { K } d ( x _ { t + k } ) | x _ { t + K } , \eta ^ { \pi } ( x _ { t } ) ] , } \end{array}
+$$
+
+where EM(π) and E #»B(π) are expectations over the forward and backward chains respectively. The Equation (3) holds true even in the limit $K  \infty$ .
+
+The proof is given in Appendix B.1. Using the above proposition, we get an interchangeability property that removes the need to sample from the backward chain. We can use the traditional RL setting and draw samples from the forward chain and still estimate the BVFs. Equation (2) can be written recursively as:
+
+$$
+\begin{array} { r } { \overleftarrow { V } _ { \mathcal { D } } ^ { \pi } ( x _ { t } ) = \mathbb { E } _ { \overleftarrow { B } ( \pi ) } \left[ d ( x _ { t } ) + \overleftarrow { V } _ { \mathcal { D } } ^ { \pi } ( x _ { t - 1 } ) \right] . } \end{array}
+$$
+
+In operator form, the above equation can also be written as:
+
+$$
+( \overleftarrow { \boldsymbol { T } } ^ { \pi } \overleftarrow { \boldsymbol { V } } _ { \mathcal { D } } ^ { \pi } ) ( \boldsymbol { x } _ { t } ) = \mathbb { E } _ { \boldsymbol { x } _ { t - 1 } \sim \overleftarrow { \boldsymbol { P } } ^ { \pi } } \left[ d ( \boldsymbol { x } _ { t } ) + \overleftarrow { \boldsymbol { V } } _ { \mathcal { D } } ^ { \pi } ( \boldsymbol { x } _ { t - 1 } ) \right] .
+$$
+
+Proposition 3.2 (Fixed point). For a policy $\pi$ , the associated Backward Value Function vector, $\overleftarrow { V } ^ { \pi }$ , satisfies $\begin{array} { r } { \operatorname* { l i m } _ { k  \infty } ( \overleftarrow { T } ^ { \pi } ) ^ { k } \overleftarrow { V } = \overleftarrow { V } ^ { \pi } } \end{array}$ for every vector , and $\overleftarrow { V } ^ { \pi }$ is the unique solution of the equation $\overleftarrow { V } ^ { \pi } = \overleftarrow { T } ^ { \pi } \overleftarrow { V } ^ { \pi }$ .
+
+The proof is given in Appendix B.2. The above proposition allows us to soundly extend the RL methods based on Bellman operators for the estimation of BVFs.
+
+# 3.3 SAFE POLICY IMPROVEMENT VIA BVF-BASED CONSTRAINTS
+
+With the Backward Value Function framework, the trajectory-level optimization problem associated with a CMDP can be rewritten in state-wise form. Recall that a feasible policy must satisfy the constraint:
+
+$$
+\mathbb { E } _ { \mathcal { M } ( \pi ) } \left[ \sum _ { k = 0 } ^ { T } d ( x _ { k } ) \mid x _ { 0 } \right] \leq d _ { 0 } .
+$$
+
+Alternatively, for each timestep $t \in [ 0 , T ]$ of a trajectory:
+
+$$
+\mathbb { E } \left[ \sum _ { k = 0 } ^ { t } d ( x _ { k } ) \mid x _ { 0 } , \pi \right] + \mathbb { E } \left[ \sum _ { k = t } ^ { T } d ( x _ { k } ) \mid x _ { 0 } , \pi \right] - \mathbb { E } \left[ d ( x _ { t } ) \mid x _ { 0 } \right] \leq d _ { 0 } .
+$$
+
+Via the identities $\begin{array} { r } { \mathbb { E } [ \sum _ { k = t } ^ { T } d ( x _ { k } ) \mid x _ { 0 } , \pi ] \leq \mathbb { E } _ { x _ { t } \sim \delta _ { x _ { 0 } } ( P ^ { \pi } ) ^ { t } } [ V _ { D } ^ { \pi } ( x _ { t } ) ] } \end{array}$ and $\begin{array} { r } { \mathbb { E } [ \sum _ { k = 0 } ^ { t } d ( x _ { k } ) \mid x _ { 0 } , \pi ] \le } \end{array}$ $\mathbb { E } _ { \boldsymbol { x } _ { k } \sim \delta _ { \boldsymbol { x } _ { 0 } } ( \boldsymbol { P } ^ { \pi } ) ^ { t } } [ \overleftarrow { V } _ { \boldsymbol { D } } ^ { \pi } ( \boldsymbol { x } _ { t } ) ]$ (derived in Appendix $\mathrm { C } ) ^ { 2 }$ , we remark that the quantity on the LHS is less than the expectation over $k$ -step trajectories of $\overleftarrow { V } _ { \mathcal { D } } ^ { \pi } ( x _ { t } ) + V _ { \mathcal { D } } ^ { \pi } ( x _ { t } ) - d ( x _ { t } )$ . In other words, for each $t \in [ 0 , T ]$ :
+
+$$
+\mathbb { E } _ { \mathcal { M } ( \pi ) } \left[ \sum _ { k = 0 } ^ { T } d ( x _ { k } ) \mid x _ { 0 } \right] \le \mathbb { E } _ { x _ { t } \sim \delta _ { x _ { 0 } } ( P ^ { \pi } ) ^ { t } } \left[ \overleftarrow { V } _ { \mathcal { D } } ^ { \pi } ( x _ { t } ) + V _ { \mathcal { D } } ^ { \pi } ( x _ { t } ) - d ( x _ { t } ) \right] \le d _ { 0 } .
+$$
+
+These are the state-wise constraints that should hold at each step in a given trajectory - we refer to them as the value-based constraints. Satisfying the value-based constraints will automatically satisfy the given CMDP constraints.
+
+This formulation allows us to introduce a policy improvement step, which maintains a safe feasible policy at every iteration by using the previous estimates of the forward and backward value functions3. The policy improvement step is defined by a linear program, which performs a greedy update with respect to the current state-action value function subject to the value-based constraints:
+
+$$
+\begin{array} { r l } & { \pi _ { k + 1 } ( \cdot | x ) = \underset { \pi \in \Pi } { \arg \operatorname* { m a x } } \big \langle \pi ( \cdot | x ) , Q ^ { \pi _ { k } } ( x , \cdot ) \big \rangle , } \\ & { \quad s . t . \left. \pi ( \cdot | x ) , Q _ { \mathcal { D } } ^ { \pi _ { k } } ( x , \cdot ) \right. + \overleftarrow { V } _ { \mathcal { D } } ^ { \pi _ { k } } ( x ) - d ( x ) \leq d _ { 0 } , \quad \forall x \in \mathcal { X } . } \end{array}
+$$
+
+Our first result is that the policies obtained by the policy improvement step will satisfy the safety constraints. We write $\mathrm { T V } ( \cdot , \cdot )$ for the total variation metric between distributions.
+
+Theorem 3.1 (Consistent Feasibility). Assume that successive policies are updated sufficiently slowly, i.e. $\begin{array} { r } { \mathrm { T V } \big ( \pi _ { k + 1 } \big ( \cdot | x \big ) , \pi _ { k } \big ( \cdot | x \big ) \big ) \leq \frac { d _ { 0 } - V _ { \mathcal { D } } ^ { \pi _ { k } } ( x _ { 0 } ) } { 2 D _ { \mathrm { M A X } } T ^ { 2 } } } \end{array}$ .4 Then the policy iteration step given by (SPI) is consistently feasible, i.e. if $\pi _ { k }$ Mis feasible at $x _ { 0 }$ then so is $\pi _ { k + 1 }$ .
+
+It is also possible to consider larger neighbourhoods for updates of successive policies, but at the cost of everywhere-feasibility. For want of space, we present that result in Appendix D.
+
+Next we show that the policy iteration step given by (SPI) leads to monotonic improvement.
+
+Theorem 3.2 (Policy Improvement). Let $\pi _ { n }$ and $\pi _ { n + 1 }$ be successive policies generated by the policy iteration step of (SPI). Then $V ^ { \pi _ { n + 1 } } ( x ) \geq V ^ { \pi _ { n } } ( x ) \forall x \in \mathcal { X }$ . In particular, the sequence of value functions $\{ V ^ { \pi _ { n } } \} _ { n \geq 0 }$ given by (SPI) monotonically converges.
+
+Proofs for Theorems 3.1 and 3.2 are given in Appendix D. Finding the sub-optimality gap (if any) remains an interesting question left for future work.
+
+# 4 PRACTICAL IMPLEMENTATION CONSIDERATIONS
+
+# 4.1 DISCRETE ACTION SPACE
+
+In discrete action spaces, the problem in (SPI) can be solved exactly as a Linear Programming problem. It is possible to approximate its analytical solution by casting it into the corresponding entropy-regularized counterpart (Neu et al., 2017; Chow et al., 2018). The details of the closed form solution can be found in Appendix E.
+
+Furthermore, if we restrict the set of policies to be deterministic, then it is possible to have an in-graph solution as well. The procedure then closely resembles the Action Elimination Procedure (Puterman, 2014, Chapter 6), where non-optimal actions are identified as being those which violate the constraints.
+
+# 4.2 EXTENSION TO CONTINUOUS CONTROL
+
+For MDPs with only state-dependent costs, Dalal et al. (2018) proposed the use of safety layers, a constraint projection approach, that enables action correction at each step. At any given state, an unconstrained action is selected and is passed to the safety layer, which projects the action to the nearest action (in Euclidean norm) satisfying the necessary constraints. We extend this approach to stochastic policies to handle the corrections for the actions generated by stochastic policies. When the policy is parameterized with a Gaussian distribution, then the safety-layer can still be used by projecting both the mean and standard-deviation vector to the constraint-satisfying hyper-plane5. In most cases, the standard-deviation vector is kept fixed or independent of the state (Kostrikov, 2018; Dhariwal et al., 2017), which allows us to formulate the problem as solving the following $L 2$ -projection of the mean of the Gaussian in Euclidean space. For $\mu _ { \pi } ( . ; \theta )$ , at any given state $x \in \mathcal { X }$ , the safety layer solves the following projection problem:
+
+$$
+\begin{array} { l } { \displaystyle \arg \operatorname* { m i n } _ { \mu } [ \frac { 1 } { 2 } \| ( \mu - \mu _ { \pi } ( x ) \| ^ { 2 } ] ,  } \\ { \displaystyle  \mathrm { s . t . } \quad Q _ { \mathcal { D } } ^ { \pi } ( x , \mu ) + \overleftarrow { V } _ { \mathcal { D } } ^ { \pi } ( x ) - d ( x ) \leq d _ { 0 } . } \end{array}
+$$
+
+As shown in Dalal et al. (2018); Chow et al. (2019), if the constraints have linear nature then an analytical solution exists. In order to get a linearized version of the constraints (and simplify the projection), we can approximate the constraint with its first-order Taylor series at $\mu = \mu _ { \pi } ( x )$ :
+
+$$
+\begin{array} { r l } & { \underset { \mu } { \arg \operatorname* { m i n } } [ \frac { 1 } { 2 } \| ( \mu - \mu _ { \pi } ( x ) \| ^ { 2 } ] , } \\ & { \mathrm { s . t . } \quad \overset {  } { V } _ { \mathcal { D } } ^ { \pi } ( x ) - d ( x ) + \underset { \mathrm { r e s . } \mu _ { \pi } ( x ) ) } { \underbrace { Q _ { \mathcal { D } } ^ { \pi } ( x , \mu _ { \pi } ( x ) ) + ( \mu - \mu _ { \pi } ( x ) ) ^ { T } ( \nabla Q _ { \mathcal { D } } ^ { \pi } ( x , \mu ) | _ { \mu = \mu _ { \pi } ( x ) } ) } } \leq d _ { 0 } . } \end{array}
+$$
+
+The above objective function is positive-definite and quadratic, and the constraints are linear. Though this problem can be solved by an in-graph QP solver, there exists an analytical solution (see Appendix G):
+
+Proposition 4.1. At a given state $x \in \mathcal { X }$ , the solution to the Eq. (5), $\mu ^ { * }$ is:
+
+where,
+
+$$
+\begin{array} { c } { { \mu ^ { * } = \mu _ { \pi } ( x ) - \lambda ^ { * } ( x ) \cdot g _ { \mu , D } ( x ) , } } \\ { { g _ { \mu , \mathcal { D } } ( x ) = \nabla Q _ { \mathcal { D } } ^ { \pi } ( x , \mu ) | _ { \mu = \mu _ { \pi } ( x ) } , } } \\ { { \lambda ^ { * } ( x ) = \left( \frac { - ( d _ { 0 } + d ( x ) - \overleftarrow { V } _ { \mathcal { D } } ^ { \pi } ( x ) - Q _ { \mathcal { D } } ^ { \pi } ( x , \mu _ { \pi } ( x ) ) ) } { g _ { \mu , \mathcal { D } } ( x ) ^ { T } g _ { \mu , \mathcal { D } } ( x ) } \right) ^ { + } . } } \end{array}
+$$
+
+# 5 RELATED WORK
+
+Lagrangian-based methods: Initially introduced in Altman (1999), more scalable versions of the Lagrangian based methods have been proposed over the years (Moldovan & Abbeel, 2012; Tessler et al., 2018; Chow et al., 2015). The general form of the Lagrangian methods is to convert the problem to an unconstrained problem via Langrange multipliers. If the policy parameters are denoted by $\theta$ , then Lagrangian formulation becomes: $\begin{array} { r l } { \operatorname* { m i n } _ { \lambda \geq 0 } \operatorname* { m a x } _ { \theta } ( L ( \theta , \lambda ) } & { = } \end{array}$ $\begin{array} { r } { \operatorname* { m i n } _ { \lambda \geq 0 } \operatorname* { m a x } _ { \theta } \left[ V ^ { \pi _ { \theta } } ( x _ { 0 } ) \right. \stackrel { . } { - } \left. \lambda ( V _ { \mathcal { D } } ^ { \pi _ { \theta } } ( x _ { 0 } ) \right. \stackrel { . } { - } \left. \bar { d _ { 0 } } ) ) \right] } \end{array}$ , where $L$ is the Lagrangian and $\lambda$ is the Lagrange multiplier (penalty coefficient). The main problems of the Lagrangian methods are that the Lagrangian multiplier is either a hyper-parameter (without much intuition), or is solved on a lower time-scale. That makes the unconstrained RL problem a three time-scale 6 problem, which makes it very difficult to optimize in practice. Another problem is that during the optimization, this procedure can violate the constraints. Ideally, we want a method that can respect the constraint throughout the training and not just at the final optimal policy.
+
+Lyapunov-based methods: In control theory, the stability of the system under a fixed policy is computed using Lyapunov functions (Khalil, 1996). A Lyapunov function is a type of scalar potential function that keeps track of the energy that a system continually dissipates. Recently, Chow et al. (2018; 2019) provide a method of constructing the Lyapunov functions to guarantee global safety of a behavior policy using a set of local linear constraints. Their method requires the knowledge of $T V ( \pi , \pi ^ { * } )$ to guarantee the theoretical claims. They substitute the ideally required Lyapunov function with an approximate solution that requires solving a LP problem at every iteration. For the practical scalable versions, they use a heuristic, a constant Lyapunov function for all states that only depends on the initial state and the horizon. While our methods also constructs state-wise constraints, there are two notable differences: a) our assumption only rely on the current policy candidate and the baseline policy, instead of the baseline and the optimal policy, b) our method does not require solving an LP at every update step to construct the constraint and as such the only approximation error that is introduced comes from the function approximation.
+
+Conservative Policy Improvement: Constrained Policy Optimization (CPO) (Achiam et al., 2017) extends the trust-region policy optimization (Schulman et al., 2015) algorithm to satisfy constraints during training as well as after convergence. CPO is computationally expensive as it uses an approximation to the Fisher Information Matrix which requires many steps of conjugate gradient descent $\cdot n _ { c g }$ steps) followed by a backtracking line-search procedure ${ \bf \nabla } _ { n _ { l s } }$ steps) for each iteration, so it is more expensive by $\mathcal { O } ( n _ { c g } + n _ { l s } )$ per update. Furthermore, accurately estimating the curvature requires a large number of samples in each batch (Wu et al., 2017).
+
+# 6 EXPERIMENTS
+
+We empirically validate our approach on RL benchmarks to measure the performance of the agent with respect to the accumulated return and cost during training in the presence of neural-networks based function approximators. We compare our approach with the respective Unconstrained versions, and the Lyapunov-based approach (Chow et al., 2018; 2019) in each setting. Even though our formulation is based on the undiscounted case, we use discounting with $\gamma = 0 . 9 9$ for estimating the value functions in order to be consistent with the baselines.
+
+# 6.1 STOCHASTIC GRID WORLD
+
+Motivated by the safety in navigation tasks, we first consider a stochastic 2D grid world (Leike et al., 2017; Chow et al., 2018). The agent (green cell in Fig. 1a) starts in the bottom-right corner, the safe region, and the objective is to move to the goal on the other side of the grid (blue cell). The agent can only move in the adjoining cells in the cardinal directions. It gets a reward of $+ 1 0 0 0$ on reaching the goal, and a penalty of $- 1$ at every timestep. Thus, the task is to reach the goal in the shortest amount of time. There are a number of pits in the terrain (red cells) that represent the safety constraint and the agent gets a cost of 10 on passing through any pit cell. Occasionally, with probability $p = 0 . 0 5$ , a random action will be executed instead of the one selected by the agent. Thus, the task is to reach to the goal in the shortest amount of time, while passing through the red grids at most $d _ { 0 } / 1 0$ times. The size of the grid is $1 2 \times 1 2$ cells, and the pits are randomly generated for each grid with probability $\rho = 0 . 3$ . The agent starts at (12, 12) and the goal is selected uniformly on $( \alpha , 0 )$ , where $\alpha \sim U ( 0 , 1 2 )$ . The threshold $d _ { 0 } = 2 0$ implies the agent can pass at most two pits. The maximum horizon is 200 steps, after which the episode terminates.
+
+We use the action elimination procedure described in Sec 4.1 in combination with $n$ -step SARSA (Rummery & Niranjan, 1994; Peng & Williams, 1994) using neural networks and multiple synchronous agents as in Mnih et al. (2016). We use $\epsilon$ -greedy exploration. The results are shown in Fig. 1 (more experimental details can be found in Appendix H). We observe that the agent is able to respect the safety constraints more adequately than the Lyapunov-based method, albeit at the expense of some decrease in return, which is the expected trade-off for satisfying the constraints.
+
+# 6.2 MUJOCO BENCHMARKS
+
+Based on the safety experiments in Achiam et al. (2017); Chow et al. (2019), we design three simulated robot locomotion continuous control tasks using the MuJoCo simulator (Todorov et al., 2012) and OpenAI Gym (Brockman et al., 2016): (1) Point-Gather: A point-mass agent $( S \subseteq \mathbb { R } ^ { 9 } , A \subseteq \mathbb { R } ^ { 2 } )$ is rewarded for collecting the green apples and constrained to avoid the red bombs; (2) Safe-Cheetah: A bi-pedal agent $( S \subseteq \mathbb { R } ^ { 1 8 } , A \subseteq \bar { \mathbb { R } } ^ { \bar { 6 } } )$ is rewarded for running at high speed, but at the same time constrained by a speed limit; (3) Point-Circle: The point-mass agent $( S \subseteq \mathbb { R } ^ { 9 } , A \subseteq \mathbb { R } ^ { 2 } )$ is rewarded for running along the circumference of a circle in counter-clockwise direction, but is constrained to stay within a safe region smaller than the radius of the circle.
+
+We integrate our method on top of the A2C algorithms (Mnih et al., 2016) and PPO (Schulman et al., 2017), using the procedure described in Section 4.2. More details about the tasks and network architecture can be found in the Appendix I. Algorithmic details can be found in Appendix J. The results with A2C are shown in Fig. 2 and the results with PPO are shown in Fig. 3. We observe that our Safe method is able to respect the safety constraint throughout most of the learning, and with much greater degree of compliance than the Lyapunov-based method, especially when combined with A2C. The one case where the Safe method fails to respect the constraint is in Point-Circle with PPO (Fig. 3(c)). Upon further examination, we note that the training in this scenario has one of two outcomes: some runs end with the learner in an infeasible set of states from which it cannot recover; other runs end in a good policy that respects the constraint. We discuss solutions to overcome this in the final section.
+
+![](images/3b4052a22ce8846d401fa4615e7efc0de148e614012456de72475e384e75a14d.jpg)  
+Figure 1: (a) Example of a gridworld environment. (b,c) Performance over the training for Unconstrained (red), Lyapunov-based (green), and our method (blue) all trained with n-step SARSA on 2D GridWorld task over 20 random seeds. The $\mathbf { X }$ -axis is the number of episodes in thousands. The dotted black line in (c) denotes the constraint threshold, $d _ { 0 }$ . The bold line represents mean, and the shaded region denotes $80 \%$ confidence-intervals.
+
+# 7 DISCUSSION
+
+We present a method for solving constrained MDPs that respects trajectory-level constraints by converting them into state dependent value-based constraints, and show how the method can be used to handle safety limitations in both discrete and continuous spaces. The main advantage of our approach is that the optimization problem is more easily solved with value-based constraints, while providing similar guarantees and requiring less approximations. The empirical results presented show that our approach is able to solve the tasks with good performance while maintaining safety throughout training. It is important to note that there is a fundamental trade-off between exploration and safety. It is impossible to be $100 \%$ safe without some knowledge; in cases where that knowledge is not provided a priori, it must be acquired through exploration. We see this in some of our results (Gridworld, Point-Circle) where our safe policy goes above the constraint in the very early phases of training (all our experiments started from a random policy). We note that the other methods also suffer from this shortcoming. An open question is how to provide initial conditions or a priori knowledge, to avoid this burn-in phase. Another complementary strategy to explore is for cases where an agent is stuck in an unsafe or infeasible policy space, where a recovery method (trained by purely minimizing the constraints) could be useful to help the agent recover (Achiam et al., 2017; Chow et al., 2019).
+
+![](images/c1f70449eff4a9665f06b9ffe485a3c12e833d5172e744ca166dd7aed6222a77.jpg)  
+Figure 2: A2C Performance over the training for Unconstrained (red), Lyapunov-based (green), and our method (blue) all trained with A2C on MuJoCo tasks over 10 random seeds. The $\mathbf { X }$ -axis is the number of episodes in thousands. The dotted black line denotes $d _ { 0 }$ . The bold line represents the mean, and the shaded region denotes the $80 \%$ confidence-intervals.
+
+![](images/4338cd35459809c48e19e12d0c1097bf16b5d282705518666c09c5ef42cfe47e.jpg)  
+Figure 3: PPO Performance over the training for Unconstrained (red), Lyapunov-based (green), and our method (blue) all trained with PPO on MuJoCo tasks over 10 random seeds. The $\mathbf { X }$ -axis is the number of episodes in thousands, and y-axis denotes the undiscounted accumulated returns. The dotted black line denotes $d _ { 0 }$ . The bold line represents the mean, and the shaded region denotes the $80 \%$ confidence-intervals.
+
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+
+# A REPRODUCIBILITY CHECKLIST
+
+We follow the reproducibility checklist (Pineau, 2018) and point to relevant sections explaining them here. For all algorithms presented, check if you include:
+
+• A clear description of the algorithm. The algorithms are explained in Sec. J. Any additional details for Discrete methods are provided in Sec. 4.1, and for continuous Sec. 4.2.   
+• An analysis of the complexity (time, space, sample size) of the algorithm. The analytical solution in Eq. (5) consists of a few vector arithmetic and relu operator and as such has the same complexity as the baselines. For the discrete case, with deterministic policies the solution again can be implemented as part of the computation graph, consisting of basic vector arithmetic operations, and has very little additional overhead. For discrete actions with stochastic policies, one needs to sovle the LP problem in (SPI). In that case the complexity is same as the baseline safe-methods (Lyapunov), and is higher than the unconstrained versions. In terms of computation time (for Deep-RL experiments) the newly proposed algorithms are almost identical to the baselines due to its parallelizable nature. We do not make any claims about the sample complexity.   
+• A link to a downloadable source code, including all dependencies. The code will be made available after the acceptance of the paper.
+
+For any theoretical claim, check if you include:
+
+• A statement of the result. See the main paper for all the claims we make. Additional details are provided in the Appendix.   
+• A clear explanation of any assumptions. See the main paper for all the assumptions.   
+• A complete proof of the claim. See the main paper. The cross references to the proofs in the Appendix have been included in the main paper.
+
+For all figures and tables that present empirical results, check if you include:
+
+• A complete description of the data collection process, including sample size. For the base agent we standard benchmarks provided in OpenAI Gym (Brockman et al., 2016), and rllab (Duan et al., 2016). We use the code from Achiam et al. (2017) for building the Point-Circle and Point-Gather environments.
+
+• A link to downloadable version of the dataset or simulation environment. See: github.com/openai/gym for OpenAI Gym benchmarks, github.com/jachiam/cpo for rllab based Circle and Gather environments.
+
+• An explanation of how samples were allocated for training / validation / testing. We do not use a split as we run multiple runs over random seeds to examine the optimization performance.
+
+• An explanation of any data that were excluded. NA • The range of hyper-parameters considered, method to select the best hyper-parameter configuration, and specification of all hyper-parameters used to generate results. The default hyper-parameters for the MuJoCo baselines are taken from Kostrikov (2018). The ranges and parameters for Grid experiments are described in Sec. H, and for MuJoCo are described in Sec. I.
+
+• The exact number of evaluation runs. The number of evaluation runs is mentioned in the caption corresponding to each result.
+
+• A description of how experiments were run. See Experiments Sec. 6 in the main paper and in the Appendix Sec. H and Sec. I.
+
+• A clear definition of the specific measure or statistics used to report results. Undiscounted return and cost using the current policy over the horizon are plotted after every 1000 episodes are plotted. We use a linear-filter with 0.7 weight for smoothing. We use the smooting algorithm provided by TensorBoard (https://github.com/tensorflow/ tensorboard).
+
+• Clearly defined error bars. Standard error used in all cases.   
+• A description of results with central tendency (e.g. mean) and variation (e.g. stddev). The bold lines in the figure represent the mean, and the shaded region denotes the $8 0 \%$ confidence interval.   
+• A description of the computing infrastructure used. We distribute all runs across 10 CPU nodes (Intel(R) Xeon(R) CPU E5-2650 v4) and 1 GPU (GP 100) per run for experiments.
+
+# B BACKWARD VALUE FUNCTIONS
+
+We have the following result from Proposition 1 from Morimura et al. (2010). We give the proof too for the sake of completeness.
+
+Proposition B.1. Let the forward Markov chain $\mathcal { M } ( \pi )$ be irreducible and ergodic, i.e., has a stationary distribution. Then the associated backward Markov chain $\overleftarrow { B } ( \pi )$ is also ergodic and has the same unique stationary distribution as $\mathcal { M } ( \pi )$ :
+
+$$
+\eta ^ { \pi } ( x ) = \overleftarrow { \eta } ^ { \pi } ( x ) ,
+$$
+
+$$
+( \forall x \in { \mathcal { X } } )
+$$
+
+where $\eta ^ { \pi } ( x )$ and $\overleftarrow { \eta } ^ { \pi } ( x )$ are the stationary distributions of $\mathcal { M } ( \pi )$ and $\overleftarrow { B } ( \pi )$ .
+
+Proof. Multiply both sides of Eq. (1) by $\eta ^ { \pi } ( x _ { t } )$ and sum over all actions $a _ { t - 1 } \in { \mathcal { A } }$ we obtain detailed balance like equations (with respect to time):
+
+$$
+\begin{array} { r } { \overleftarrow { \mathcal { P } } ^ { \pi } ( x _ { t - 1 } | x _ { t } ) \eta ^ { \pi } ( x _ { t } ) = \mathcal { P } ^ { \pi } ( x _ { t } , a _ { t - 1 } | x _ { t - 1 } ) \eta ^ { \pi } ( x _ { t - 1 } ) . \qquad ( \forall x _ { t - 1 } \in \mathcal { X } , x _ { t } \in \mathcal { X } ) } \end{array}
+$$
+
+Sum over all possible $x _ { t }$ we have:
+
+$$
+\sum _ { x _ { t } \in \mathcal { X } } \overleftarrow { \mathcal { P } } ^ { \pi } ( x _ { t - 1 } | x _ { t } ) \eta ^ { \pi } ( x _ { t } ) = \eta ^ { \pi } ( x _ { t - 1 } ) .
+$$
+
+The above equation indicates that $\overleftarrow { B } ( \pi )$ has same stationary distribution as $\mathcal { M } ( \pi )$ . In the matrix form the above equation can be written as $\eta \overleftarrow { P } ^ { \pi } = \eta$ , that implies that $\eta$ is stationary distribution with $\scriptstyle \overleftarrow { P } ^ { \pi }$ transition matrix.
+
+# B.1 RELATION BETWEEN FORWARD AND BACKWARD MARKOV CHAINS AND BACKWARD VALUE FUNCTIONS
+
+Proof. We use the technique of Proposition 2 of Morimura et al. (2010) to prove this. Using the Markov property and then substituting Eq. (1) for each term we have:
+
+$$
+\begin{array} { r l } & { \overline { { \mathfrak { p } } } ^ { \pi } ( x _ { t - 1 } , a _ { t - 1 } , \dots , x _ { t - K } , a _ { t - K } | x _ { t } ) = \overline { { \mathcal { P } } } ^ { \pi } ( x _ { t - 1 } , a _ { t - 1 } | x _ { t } ) \dots \overleftarrow { \mathcal { P } } ^ { \pi } ( x _ { t - K } , a _ { t - K } | x _ { t - K + 1 } ) , } \\ & { \qquad = \frac { \mathcal { P } ^ { \pi } ( x _ { t } , a _ { t - 1 } | x _ { t - 1 } ) \dots \mathcal { P } ^ { \pi } ( x _ { t - K + 1 } , a _ { t - K } | x _ { t - K } ) \eta ^ { \pi } ( x _ { t - K } ) } { \eta ^ { \pi } ( x _ { t } ) } , } \\ & { \qquad \propto \mathcal { P } ^ { \pi } ( x _ { t } , a _ { t - 1 } | x _ { t - 1 } ) \dots \mathcal { P } ^ { \pi } ( x _ { t - K + 1 } , a _ { t - K } | x _ { t - K } ) \eta ^ { \pi } ( x _ { t - K } ) . } \end{array}
+$$
+
+This proves the proposition for finite $K$ . Using the Prop. B.1, $K  \infty$ case is proven too:
+
+$$
+\begin{array} { l } { \displaystyle \underset { K \to \infty } { \operatorname* { l i m } } \mathbb { E } _ { \overleftarrow { \mathcal { B } } ( \pi ) } \left[ \displaystyle \sum _ { k = 0 } ^ { K } d ( x _ { t - k } ) | x _ { t } \right] = \displaystyle \operatorname* { l i m } _ { K \to \infty } \mathbb { E } _ { \mathcal { M } ( \pi ) } \left[ \displaystyle \sum _ { k = 0 } ^ { K } d ( x _ { t - k } ) | x _ { t } , \eta ^ { \pi } ( x _ { t - K } ) \right] , \medskip } \\ { \displaystyle = \displaystyle \sum _ { x \in \mathcal { X } } \sum _ { a \in \mathcal { A } } \pi ( a | x ) \eta ^ { \pi } ( x ) d ( x ) . } \end{array}
+$$
+
+# B.2 TD FOR BVF
+
+Proof. We use the same technique from Stochastic Shortest Path dynamic programming (Bertsekas et al., 1995, Vol 2, Proposition 1.1) to prove the above proposition. The general outline of the proof is given below, for more details we refer the reader to the textbook.
+
+We have,
+
+$$
+{ \mathrm { } } ^ {  \pi } { } ^ {  } { \mathrm { } } V = d + { \overleftarrow { P } } ^ { \pi } { \overleftarrow { V } } .
+$$
+
+(Eq. (4) in matrix notation)
+
+Using induction argument, we have for all $\mathbf { \overline { { V } } } \in \mathbb { R } ^ { n }$ and $k \geq 1$ , we have:
+
+$$
+\left( \overleftarrow { \boldsymbol { \mathcal { T } } } ^ { \pi } \right) ^ { k } \overleftarrow { \boldsymbol { V } } = \left( \overleftarrow { \boldsymbol { P } } ^ { \pi } \right) ^ { k } \overleftarrow { \boldsymbol { V } } + \sum _ { m = 0 } ^ { k - 1 } { \left( \overleftarrow { \boldsymbol { P } } ^ { \pi } \right) ^ { m } } d ,
+$$
+
+Taking the limit, and using the result, $\begin{array} { r } { \operatorname* { l i m } _ { k \to \infty } \left( \overleftarrow { P } ^ { \pi } \right) ^ { k } \overleftarrow { V } = 0 } \end{array}$ , regarding proper policies from Bertsekas et al. (1995, Vol 2, Equation 1.2), we have:
+
+$$
+\operatorname* { l i m } _ { k \to \infty } \left( { \overleftarrow { \mathcal { T } } } ^ { \pi } \right) ^ { k } { \overleftarrow { V } } = \operatorname* { l i m } _ { k \to \infty } \sum _ { m = 0 } ^ { k - 1 } { \left( { \overleftarrow { P } } ^ { \pi } \right) } ^ { m } d = { \overleftarrow { V } } ^ { \pi } ,
+$$
+
+Also we have by definition:
+
+$$
+\left( \overleftarrow { T } ^ { \pi } \right) ^ { k + 1 } \overleftarrow { V } = d + \overleftarrow { P } ^ { \pi } \left( \overleftarrow { T } ^ { \pi } \right) ^ { k } \overleftarrow { V } ,
+$$
+
+and by taking the limit $k \to \infty$ , we have:
+
+$$
+\begin{array} { r } { \overleftarrow { V } ^ { \pi } = d + \overleftarrow { P } ^ { \pi } \overleftarrow { V } ^ { \pi } , } \end{array}
+$$
+
+which is equivalent to,
+
+$$
+\mathbf { \Sigma } ^ {  \pi } = \mathbf { \Sigma } ^ {  \pi } \mathbf { \Sigma } ^ {  \pi } .
+$$
+
+To show uniqueness, note that if $\mathbf { \Sigma } _ { \overline { { V } } } ^ {  } = \mathbf { \Sigma } ^ {  \pi } \mathbf { \Sigma } _ { \overline { { V } } } ^ {  }$ , then $\overleftarrow { V } = \left( \overleftarrow { \mathcal { T } } ^ { \pi } \right) ^ { k } \overleftarrow { V }$ for all $k$ and letting $k \to \infty$ we get $\overleftarrow { V } = \overleftarrow { V } ^ { \pi }$ .
+
+# C VALUE-BASED CONSTRAINT LEMMA
+
+Lemma C.1. $\begin{array} { r } { \mathbb { E } \left[ \sum _ { k = t } ^ { T } d ( x _ { k } ) \mid x _ { 0 } , \pi \right] \leq \mathbb { E } _ { x _ { t } \sim \delta _ { x _ { 0 } } ( P ^ { \pi } ) ^ { t } } \left[ V _ { D } ^ { \pi } ( x _ { t } ) \right] \mathrm { a n d } \mathbb { E } \left[ \sum _ { k = 0 } ^ { t } d ( x _ { k } ) \mid \operatorname { c o r s c o s } ( x _ { k } ) \mid \operatorname { c o r s c o s } ( x _ { k } ) \mid \operatorname { c o r s c o s } ( x _ { k } ) \right] } \end{array}$ $\begin{array} { r } { \mathbb { E } \left[ \sum _ { k = 0 } ^ { t } d ( x _ { k } ) \mid x _ { 0 } , \pi \right] \leq } \end{array}$ $\mathbb { E } _ { \boldsymbol { x } _ { k } \sim \delta _ { \boldsymbol { x } _ { 0 } } ( { P } ^ { \pi } ) ^ { t } } \left[ \overleftarrow { V } _ { \mathcal { D } } ^ { \bar { \pi } } ( \boldsymbol { x } _ { k } ) \right]$
+
+Proof. Follows since adding more steps to the trajectory (from $T \mathrm { ~ - ~ } t$ steps to $T$ ) can only increase the expected total cost. $\begin{array} { r c l } { \mathbb { E } \left[ \sum _ { k = t } ^ { T } d ( x _ { k } ) \mid x _ { 0 } , \pi \right] } & { = } & { \delta _ { x _ { 0 } } ( P ^ { \pi } ) ^ { t } \left( \sum _ { k = t } ^ { T } ( P ^ { \pi } ) ^ { k } \right) d } \end{array} \leq$ $\begin{array} { r } { \delta _ { x _ { 0 } } ( P ^ { \pi } ) ^ { t } \left( \sum _ { k = t } ^ { T + t } ( P ^ { \pi } ) ^ { k } \right) d = \mathbb { E } _ { x _ { t } \sim \delta _ { x _ { 0 } } ( P ^ { \pi } ) ^ { t } } \left[ V _ { \mathcal { D } } ^ { \pi } ( x _ { t } ) \right] . } \end{array}$ . The backward case is analogous. □
+
+# D PROPERTIES OF THE POLICY ITERATION (SPI)
+
+Theorem D.1. Let $\begin{array} { r } { \sigma ( x ) : = \mathrm { T V } ( \pi _ { k + 1 } ( \cdot | x ) , \pi _ { k } ( \cdot | x ) ) = ( 1 / 2 ) \sum _ { a } \left| \pi _ { k + 1 } ( a | x ) - \pi _ { k } ( a | x ) \right| } \end{array}$ denote the total variation between policies $\pi _ { k } ( \cdot | x )$ and $\pi _ { k + 1 } ( \cdot | x )$ . If the policies are updated sufficiently slowly and $\pi _ { k }$ is feasible, then so is $\pi _ { k + 1 }$ . More specifically:
+
+(I) If $\pi _ { k }$ is feasible at $x _ { 0 }$ and $\begin{array} { r } { \sigma ( x ) \leq \frac { d _ { 0 } - V _ { \mathcal { D } } ^ { \pi _ { k } } ( x _ { 0 } ) } { 2 T ^ { 2 } D _ { \mathrm { M A X } } } \forall x } \end{array}$ ) ∀x then πk+1 is feasible at x0.
+
+$\mathbf { \Pi } ^ { ( \mathbf { I I } ) }$ If $\pi _ { k }$ is feasible everywhere (i.e. $\begin{array} { r l r } { V _ { \mathcal { D } } ^ { \pi _ { k } } ( x ) } & { { } \le } & { d _ { 0 } \forall x ) } \end{array}$ and $\sigma ( x )$ ≤ $\frac { d _ { 0 } - V _ { \mathcal { D } } ^ { \pi _ { k } } \left( x \right) } { 2 T \operatorname* { m a x } _ { x ^ { \prime } } \left\{ d _ { 0 } - \overline { { V } } _ { \mathcal { D } } ^ { \pi _ { k } } \left( x ^ { \prime } \right) - d ( x ^ { \prime } ) \right\} } \forall x$ then $\pi _ { k + 1 }$ is feasible everywhere.
+
+We note that the second case allows the policies to be updated in a larger neighborhood but requires $\pi _ { k }$ to be feasible everywhere. By contrast the first item updates policies in a smaller neighbourhood but only requires feasibility at the starting state.
+
+Proof. Similar to the analysis in Chow et al. (2018). We aim to show that $V _ { \mathcal { D } } ^ { \pi _ { k + 1 } } ( x _ { 0 } ) \leq d _ { 0 }$ . For simplicity we consider $k = 0$ , and by induction the other cases will follow. We write $P _ { 0 } =$ $P ^ { \pi _ { 0 } } , P _ { 1 } = P ^ { \pi _ { 1 } }$ , $\Delta ( a | x ) = \pi _ { 1 } ( a | x ) - \pi _ { 0 } ( a | x )$ , and $\begin{array} { r } { P _ { \Delta } = \left[ \sum _ { a \in A } \Delta ( a | x ) P ( x ^ { \prime } | x , a ) \right] _ { \{ x ^ { \prime } , x \} } } \end{array}$ . Note that $( I - P _ { 0 } ) = ( I - P _ { 1 } + P _ { \Delta } )$ , and therefore $( I - P _ { 1 } + P _ { \Delta } ) ( I - P _ { 0 } ) ^ { - 1 } = I _ { | \mathcal { X } | \times | \mathcal { X } | }$ . Thus, we find
+
+$$
+( I - P _ { 0 } ) ^ { - 1 } = ( I - P _ { 1 } ) ^ { - 1 } ( I _ { | \mathcal { X } | \times | \mathcal { X } | } + P _ { \Delta } ( I - P _ { 0 } ) ^ { - 1 } ) .
+$$
+
+Multiplying both sides by the cost vector $d$ one has
+
+$$
+V _ { \mathcal { D } } ^ { \pi _ { 0 } } ( x ) = \mathbb { E } \left[ \sum _ { t = 0 } ^ { T } d ( x _ { t } ) + \varepsilon ( x _ { t } ) \mid \pi _ { 1 } , x \right] ,
+$$
+
+for each $x$ , where $\begin{array} { r } { \varepsilon ( x ) = \sum _ { a \in A } \Delta ( a | x ) \sum _ { x ^ { \prime } \in \mathcal { X } } P ( x ^ { \prime } | x , a ) V _ { \mathcal { D } } ^ { \pi _ { 0 } } ( x ^ { \prime } ) } \end{array}$ . Splitting the expectation, we have
+
+$$
+V _ { \mathcal { D } } ^ { \pi _ { 1 } } ( x ) = V _ { \mathcal { D } } ^ { \pi _ { 0 } } ( x ) - \mathbb { E } \left[ \sum _ { t = 0 } ^ { T } \varepsilon ( x _ { t } ) \mid \pi _ { 1 } , x \right]
+$$
+
+For case (I) we note that $V _ { \mathcal { D } } ^ { \pi _ { 0 } } ( x ^ { \prime } ) \le D _ { \mathrm { M A X } } T$ and so $- 2 \sigma ( x _ { t } ) D _ { \mathrm { M A X } } T \le \varepsilon ( x _ { t } ) \forall x _ { t }$ . Using $\sigma ( x _ { t } ) \leq$ $( d _ { 0 } - V _ { \mathcal { D } } ^ { \pi _ { k } } ) / 2 D _ { \mathrm { M A X } } T ^ { 2 }$ gives $V _ { \mathcal { D } } ^ { \pi _ { 1 } } ( x _ { 0 } ) \leq V _ { \mathcal { D } } ^ { \pi _ { 0 } } ( x _ { 0 } ) - 2 D _ { \mathrm { M A X } } T ^ { 2 } ( d _ { 0 } - V _ { \mathcal { D } } ^ { \pi _ { 0 } } ( x _ { 0 } ) ) / ( 2 D _ { \mathrm { M A X } } T ^ { 2 } ) = d _ { 0 }$ , i.e. $\pi _ { 0 }$ is feasible at $x _ { 0 }$ .
+
+For case ${ \bf ( I I ) }$ we note that $V _ { \mathcal { D } } ^ { \pi _ { 0 } } ( x ) \leq \operatorname* { m a x } _ { x ^ { \prime } } \{ d _ { 0 } - \overleftarrow { V } _ { \mathcal { D } } ^ { \pi _ { 0 } } ( x ^ { \prime } ) - d ( x ^ { \prime } ) \} = : \Theta$ since $\pi _ { 0 }$ is feasible at every $x$ . As before, we have $- 2 \sigma ( x _ { t } ) \Theta \le \varepsilon ( x _ { t } ) \ \forall x _ { t }$ and so $V _ { \mathcal { D } } ^ { \pi _ { 1 } } ( x ) \le V _ { \mathcal { D } } ^ { \pi _ { 0 } } ( x ) - 2 \Theta T ( d _ { 0 } -$ $V _ { \mathcal { D } } ^ { \pi _ { 0 } } ( x ) ) / ( 2 \Theta T ) = d _ { 0 } \forall x$ , i.e. $\pi _ { 1 }$ is feasible everywhere.
+
+Theorem D.2. Let $\pi _ { n }$ and $\pi _ { n + 1 }$ be successive policies generated be the policy iteration algorithm of (SPI). Then $V ^ { \pi _ { n + 1 } } \geq V ^ { \pi _ { n } }$ .
+
+Proof. Note that $\pi _ { n + 1 }$ and $\pi _ { n }$ are both feasible solutions of the LP (SPI). Since $\pi _ { n + 1 }$ maximizes $V ^ { \pi }$ over all feasible solutions, the result follows.
+
+# E ANALYTICAL SOLUTION OF THE UPDATE - DISCRETE CASE
+
+We follow the same procedure as (Chow et al., 2018, Section E.1) to convert the problem to its Shannon entropy regularized version:
+
+$$
+\begin{array} { r l } { \underset { \pi \in \Delta } { \operatorname* { m a x } } } & { \pi ( . | x ) ^ { T } ( Q ( x , . ) + \tau \log \pi ( . | x ) ) , } \\ { \mathrm { s . t . } } & { \pi ( . | x ) ^ { T } Q _ { \mathcal { D } } ( x , . ) + \overleftarrow { V } _ { \mathcal { D } } ^ { \pi } ( x ) - d ( x ) \leq d _ { 0 } , } \end{array}
+$$
+
+where $\tau > 0$ is a regularization constant. Consider the Lagrangian problem for optimization:
+
+$$
+\operatorname* { m a x } _ { \lambda \geq 0 } \operatorname* { m a x } _ { \pi \in \Delta } \Gamma _ { x } ( \pi , \lambda ) = \pi ( . | x ) ^ { T } ( Q ( x , . ) + \lambda Q _ { \mathcal { D } } ( x , . ) + \tau \log \pi ( . | x ) ) + \lambda ( d _ { 0 } + d ( x ) - \overleftarrow { V } ( x ) )
+$$
+
+From entropy-regularized literature (Neu et al., 2017), the inner $\lambda$ -solution policy has the form:
+
+$$
+\pi _ { \Gamma , \lambda } ^ { * } ( . | x ) \propto \exp { \left( - \frac { Q ( x , . ) + \lambda Q _ { \mathcal { D } } ( x , . ) } { \tau } \right) }
+$$
+
+We now need to solve for the optimal lagrange multiplier $\lambda ^ { * }$ at $x$ .
+
+$$
+\operatorname* { m a x } _ { \lambda \geq 0 } - \tau \log - \mathrm { s u m - e x p } \left( - \frac { Q ( x , . ) + \lambda Q _ { \mathcal { D } } ( x , . ) } { \tau } \right) + \lambda ( d _ { 0 } + d ( x ) - \overleftarrow { V } _ { \mathcal { D } } ( x ) ) ,
+$$
+
+where log-sum- $\begin{array} { r } { \exp ( y ) = \log \sum _ { a } e x p ( y _ { a } ) } \end{array}$ is a convex function in $y$ , and objective is a concave function of $\lambda$ . Using KKT conditions, the $\nabla _ { \lambda }$ gives the solution:
+
+$$
+\left( d _ { 0 } + d ( x ) - \overleftarrow { V } _ { \mathcal { D } } ( x ) \right) - \frac { \sum _ { a } Q _ { \mathcal { D } } ( x , a ) \exp ( \left( - \frac { Q ( x , a ) + \lambda Q _ { \mathcal { D } } ( x , a ) } { \tau } \right) ) } { \sum _ { a } \exp ( \left( - \frac { Q ( x , a ) + \lambda Q _ { \mathcal { D } } ( x , a ) } { \tau } \right) ) } = 0
+$$
+
+Using parameterization of $z = \exp ( - \lambda )$ , the above condition can be written as polynomial equation in $z$ :
+
+$$
+\sum _ { a } \left( d _ { 0 } + d ( x ) - \overleftarrow { V } _ { \mathcal { D } } ( x ) - Q _ { \mathcal { D } } ( x , a ) \right) \cdot \left( \exp ( - \frac { Q ( x , a ) } { \tau } ) \right) z ^ { \frac { Q _ { \mathcal { D } } ( x , a ) } { \tau } } = 0
+$$
+
+The roots to this polynomial will give $0 ~ \leq ~ z ^ { * } ( x ) ~ \leq ~ 1$ , using which one can find $\lambda ^ { * } ( x ) \ =$ $- \log ( z ^ { * } ( x ) )$ . The roots can be found using the Newton’s method. The final optimal policy of the entropy-regularized process is then:
+
+$$
+\pi _ { \Gamma } ^ { * } \propto \exp \left( - \frac { Q ( x , \cdot ) + \lambda ^ { * } Q _ { \mathcal { D } } ( x , \cdot ) } { \tau } \right)
+$$
+
+# F EXTENSION OF SAFETY LAYER TO STOCHASTIC POLICIES WITH GAUSSIAN PARAMTERIZATION
+
+Consider stochastic gaussian policies parameterized by mean $\mu ( x ; \theta )$ and standard-deviation $\sigma ( x ; \phi )$ , and the actions sampled have the form $\mu ( x ; \theta ) + \sigma ( x ; \phi ) \epsilon$ , where $\epsilon \sim \mathcal { N } ( 0 , I )$ is the noise. Here, $< \mu ( x ; \theta ) , \sigma ( x ; \phi ) >$ are both deterministic w.r.t. the parameters $\theta , \phi$ and $x$ , and as such both of them together can be treated in the same way as deterministic policy $( \pi ( x ) = < \mu ( x ) , \sigma ( x ) > )$ . The actual action sampled and executed in the environment is still stochastic, but we have moved the stochasticity fron the policy to the environment. This allows us to define and work with action-value functions $Q _ { \mathcal { D } } ( x , \mu _ { \pi } ( x ) , \sigma _ { \pi } ( x ) )$ . In this case, the corresponding projected actions have the form $\mu ^ { \prime } + \sigma ^ { \prime } \epsilon$ . The main objective of the safety layer (without the constraints) can be further simplified as:
+
+$$
+\begin{array} { r l } & { \quad \mathrm { c r r o n s i n } \ : \sum _ { k = 0 } ^ { \infty } \operatorname* { m i n } _ { \rho \to \infty } \rho _ { k \to \infty , \infty , 0 \leq n \leq n } \left. \frac { 1 } { 2 } \left\| \left( \rho ^ { \prime } \right. - \rho ( k _ { \star } ( \sigma ) ) - ( \rho _ { \star \star } ( \sigma ) ) \right. \sigma _ { \star } ( \sigma ) \mathrm { c l } \rho \right. \right\| ^ { 2 } } \\ & { \quad \times \quad \mathrm { c r o n s i n } \ : \sum _ { k = 0 } ^ { \infty } \operatorname* { s u p } _ { \rho \to \infty , 0 \leq n \leq t _ { \star } } \left. \frac { 1 } { 2 } \left\| \left( \rho ^ { \prime } - \rho _ { \star \star } ( \sigma ) \right) + ( ( \sigma ^ { \prime } - \sigma _ { \star } ( \sigma ) ) ) \zeta \right\| ^ { 2 } \right. } \\ & { \quad \times \quad \mathrm { c r o n s i n } \ : \frac { 1 } { 2 } \sum _ { k = 0 } ^ { \infty } \operatorname* { c r o n s i n } _ { \rho \to \infty , 0 \leq n \leq t _ { \star } } \left. \frac { 1 } { 2 } \left\| \left( \rho ^ { \prime } - \rho _ { \star \star } ( \sigma ) \right) \right\| ^ { 2 } + \left\| ( \sigma ^ { \prime } - \sigma _ { \star } ( \sigma ) ) \right\| \mathrm { c r o s i n } \ : \rho ^ { \prime } + \frac { 2 } { \lambda } \leq \rho ^ { \prime } - \rho _ { \star \star } ( \sigma ) \mathrm { c r o s i n } \ : \rho ^ { \prime } \right. \leq \sigma _ { \star } ^ { \prime } } \\ & { \quad \times \frac { \mathrm { c r o n s i n } } { \rho ^ { \prime } \rho ^ { \prime } \rho ^ { \prime } } \frac { 1 } { 2 } \left( \left\| \partial ^ { 2 } - \mu ( \sigma ) \right\| ^ { 2 } + \Xi _ { \infty < \infty , \infty , 0 \leq n } \left. \left. ( \sigma ^ { \prime } - \sigma _ { \star } ( \sigma ) ) \right. \right\| ^ { 2 } \right) } \\ &  \quad \times \frac { \mathrm { c r o s i n } } { \rho ^ { \prime } \rho ^ { \prime } } \frac { 1 } { 2 } \left( \left\| \partial ^ { 2 } - \mu _ { \star } ( \sigma ) \right\| ^ { 2 } + \left\| ( \sigma ^ { \prime } - \sigma _ { \star } ( \sigma ) ) \right\| ^ { 2 }  \end{array}
+$$
+
+As both $\mu _ { \pi } ( . ; \theta )$ and $\sigma _ { \pi } ( . ; \phi )$ are modelled by independent set of parameters (different neural networks, usually) we can solve each of the safety layer problem independently, w.r.t. only those parameters.
+
+# G ANALYTICAL SOLUTION IN SAFETY LAYER
+
+The proof is similar to the proof of the Proposition 1 of Dalal et al. (2018). We have the following optimization problem:
+
+$$
+\begin{array} { r l } & { \displaystyle \arg \operatorname* { m i n } _ { \mu } [ \frac { 1 } { 2 } \| ( \mu - \mu _ { \pi } ( x ) \| ^ { 2 } ] ,  } \\ & { \displaystyle  s . \mathrm { ~ t ~ . ~ } \quad \mathbf { \Sigma } _ { V } ^ { \pi } ( x ) - d ( x ) + Q _ { D } ^ { \pi } ( x , \mu _ { \pi } ( x ) ) + ( \mu - \mu _ { \pi } ( x ) ) ^ { T } ( \nabla Q _ { D } ^ { \pi } ( x , \mu ) | _ { \mu = \mu _ { \pi } ( x ) } ) \leq d _ { 0 } } \end{array}
+$$
+
+As the objective function and constraints are convex, and the feasible solution, $\mu ^ { * } , \lambda ^ { * }$ , should satisfy the KKT conditions. We define $\epsilon ( x ) = ( d _ { 0 } + d ( x ) - \overleftarrow { V } _ { \mathcal { D } } ^ { \pi } ( x ) - Q _ { \mathcal { D } } ^ { \pi } ( x , \mu _ { \pi } ( \overleftarrow { \lambda } ) ) )$ , and $g _ { \mu , \mathcal { D } } ( x ) =$ $\nabla Q _ { \mathcal { D } } ^ { \pi } ( x , u ) | _ { u = \mu _ { \pi } ( x ) }$ . Thus, we can write the Lagrangian as:
+
+$$
+L ( \mu , \lambda ) = \frac { 1 } { 2 } \left\| ( \mu - \mu _ { \pi } ( x ) \| ^ { 2 } + \lambda ( ( \mu - \mu _ { \pi } ( x ) ) ^ { T } g _ { \mu , \mathcal { D } } ( x ) - \epsilon ( x ) ) \right.
+$$
+
+From the KKT conditions, we get:
+
+$$
+\begin{array} { r } { \nabla _ { \mu } L = \mu - \mu _ { \pi } ( x ) + \lambda g _ { \mu , \mathcal { D } } ( x ) = 0 } \\ { ( \mu - \mu _ { \pi } ( x ) ) ^ { T } g _ { \mu , \mathcal { D } } ( x ) - \epsilon ( x ) = 0 } \end{array}
+$$
+
+From Eq. (7), we have:
+
+$$
+\mu ^ { * } = \mu _ { \pi } ( x ) - \lambda ^ { * } ( x ) \cdot g _ { \mu , D } ( x )
+$$
+
+Substituting Eq. (9) in Eq. (8), we get:
+
+$$
+\begin{array} { r } { - \lambda ^ { * } ( x ) \cdot g _ { \mu , D } ( x ) ^ { T } g _ { \mu , D } ( x ) - \epsilon ( x ) = 0 } \\ { \lambda ^ { * } = \frac { - \epsilon ( x ) } { g _ { \mu , D } ( x ) ^ { T } g _ { \mu , D } ( x ) } } \end{array}
+$$
+
+When the constraints are satisfied $( \epsilon ( x ) > 0 )$ , the $\lambda$ should be inactive, and hence we have $( ) ^ { + }$ operator, that is 0 for negative values.
+
+# H DETAILS OF GRID-WORLD EXPERIMENTS
+
+# H.1 ARCHITECTURE AND TRAINING DETAILS
+
+We use one-hot encoding of the agent’s location in the grid as the observation, i.e. $x$ is a binary vector of dimension $\mathbb { R } ^ { 1 2 \times 1 2 }$ . The agent is trained for $2 0 0 \mathrm { k }$ episodes, and the current policy’s performance is evaluated after every 1k episodes.
+
+The same three layer neural network with the architecture is used for state encoding for all the different the estimators. The feed-forward neural network has hidden layers of size 64, 64, 64, and relu activations. For the state-action value based estimators, the last layer is a linear layer with 4 outputs, for each action. For value function based estimators the last layer is linear layer with a single output.
+
+We use Adam Optimizer for training all the estimators. A learning rate of 1e-3 was selected for all the reward based estimators and a learning rate of 5e-4 was selected for all the cost based estimators. The same range of learning rate parameters for considered for all estimators i.e. {1e-5, 5e-5, 1e-4, 5e-4, 1e-3, 5e-3, 1e-2, 5e-2, 1e-1}.
+
+We use $\mathbf { n }$ -step trajectory length in A2C with $n = 4$ , i.e., trajectories of length $n$ were collected and the estimators were updated to used via the td-errors based on that. We use the number of parallel agents 20 in all the experiments. The range of parameters considered was $n \in \{ 1 , 4 , 2 0 \}$ . The same value of $n$ was used for all the baselines.
+
+![](images/d4d99889d39bf9a1060e808ee6bfbf2853dc4aa096bb03498c9b4191d742b1e8.jpg)  
+Figure 4: MuJoCo Safety Environments
+
+# I DETAILS OF THE MUJOCO EXPERIMENTS
+
+# I.1 ENVIRONMENTS DESCRIPTION
+
+• Point-Gather: The environment (Fig.4c) is taken from Achiam et al. (2017), where the point mass agent gets a reward of $+ 1 0 . 0$ for collecting a green apple, and a cost of 1 for collecting a red bomb. Two apples and eight bombs are spawned randomly at the start of each episode. The constraints are defined over the nmber of bombs collected over the episode. Episode horizon is 15 and threshold $d _ { 0 } = 4$ .
+
+• Safe-Cheetah: This environment (Fig.4b) is taken from Chow et al. (2019). A bi-pedal agent (HalfCheetah-v0) is augmented with speed safety constraints. The agent gets the reward based on the speed with which it runs, and the constrain is define on the speed to be less than 1, i.e., it gets a constraint cost based on $\mathbb { 1 } [ | v | > 1 ]$ , where $v$ is the velocity at the state. The maximum length of the episode is 200 and the constraint threshold is $d 0 = 5 0$ .
+
+• Point-Circle: This environment (Fig.4a) is taken from Achiam et al. (2017). The pointmass agent is rewarded for running along the circumference of a circle of radius 15 in counter-clockwise direction, with the reward and cost function:
+
+$$
+\begin{array} { l } { \displaystyle { R ( s ) = \frac { v ^ { T } [ - y , x ] } { 1 + | \| [ x , y ] \| _ { 2 } - 1 5 | } , } } \\ { \displaystyle { C ( s ) = \mathbb { 1 } [ | x | > 2 . 5 ] , } } \end{array}
+$$
+
+where $x , y$ are coordinates in the plane and $v$ is the velocity. The length of the episode is 65 and the constraint threshold $d _ { 0 } = 1 0 . 0$ .
+
+# I.2 NETWORK ARCHITECTURE AND TRAINING DETAILS
+
+The architecture and the training procedure is based on the open-source implementations (Kostrikov, 2018). All the value based estimators use a network architecture of 2 hidden layers of size 200, 50 hidden units with tanh non-linearity, followed by a linear layer with single output. For the actor, we model mean using a network architecture of 2 hidden layers of size 100, 50 hidden units with tanh non-linearity, followed by a linear layer with dimensions of the action-space and tanh non-linearity. For the $Q ( x , \mu )$ we also a 2 layer neural network with 200, ( $5 0 +$ action-dimension) hidden units and tanh non-linearity. We concatenate the mean in the second layer, and add a linear layer with single output in the end.
+
+Entropy regularization with $\beta = 0 . 0 0 1$ was used for all the experiments and the baselines. The trajectory length for different environments. For PPO GAE with $\lambda = 0 . 9 5$ was used for every algorithm. 20 parallel actors were used for every algorithm for each experiment. We searched the trajectory length hyper-parameter in the range 5,20,100 for every environment. We used the trajectory length of 1000 over which the samples are collected for PPO, for all environments. For the A2C experiments, for SafeCheetah trajectory length of 5 is used and for the rest 20 is used.
+
+We use Adam Optimizer for training all the estimators. The learning rate of the critic is always 0.5 the learning rate of the actor. For the cost estimators, the same learning rate was used for forward and backward estimators. The same range of learning rate parameters for considered for all estimators i.e. {1e-5, 5e-5, 1e-4, 5e-4, 1e-3, 5e-3, 1e-2, 5e-2, 1e-1}.
+
+# I.3 OTHER DETAILS
+
+As we mentioned in Sec. 7, due to exploration the agent can potentially end up being in an infeasible policy space. To prevent that from happening a recovery policy (or safe-guard policy) (Achiam et al., 2017; Chow et al., 2019) is used to recover back to the feasible policy space. We run the experiments with and without the use of recovery policies (in the same procedure as the baselines), and chose the run that performs the best. We noticed that, empirically, for our approach recovery policies are only required for Point-Circle environments, as the agent has much more probability of being stuck in the constraint space.
+
+In order to take error due to function approximation into account, Achiam et al. (2017) use costshaping to smooth out the sparse constraint, and Chow et al. (2019) use a relaxed threshold, i.e. $d _ { 0 } \cdot ( 1 ^ { - } \delta )$ , instead of $d _ { 0 }$ , where $\delta \in ( 0 , 1 )$ . We run experiments with $\delta = \{ 0 . 0 , 0 . 2 \}$ for each algorithms, and use the best among them. We found that empirically, only for Safe-Cheetah $\delta = 0 . 2$ works better compared to $\delta = 0 . 0$ .
+
+# J ALGORITHM DETAILS
+
+# J.1 N-STEP SYNCHRONOUS SARSA
+
+The algorithm for n-step Synchronous SARSA is similar to the n-step Asynchronous Q-learning of Mnih et al. (2016), except that it uses SARSA instead of Q-learning, is synchronous, and instead of greedy maximization step of $\epsilon$ -greedy we use (SPI). When working with discrete actions and deterministic policies, this can be solved as part of the computation-graph itself. The algorithm is presented in Alg. 1.
+
+# J.2 A2C
+
+In Actor Critic (Konda & Tsitsiklis, 2000) algorithms, the parameterized policy (actor) is denoted by $\pi ( a | x ; \theta )$ , and is updated to minimizing the following loss:
+
+$$
+L ( \theta ) = \mathbb { E } [ - \log \pi ( a _ { t } | x _ { t } ; \theta ) ( r _ { t } + \gamma V ^ { \pi } ( x _ { t + 1 } - V _ { x _ { t } } ) ) ]
+$$
+
+The algorithm for A2C with Safety Layer given by Eq. (5) is similar to the Synchronous version of Actor-Critic (Mnih et al., 2016), except that it has estimates for the costs and safety-layer. Note that due to the projection property of the safety layer, it is possible to sample directly from the projected mean. Also, as the projection is a result of vector products and max, it is differentiable and and computed in-graph (via relu). The algorithm is presented in Alg. 2.
+
+# J.3 PPO
+
+The PPO algorithm build on top of the Actor-Critic algorithm and is very similar to Algorithm 2. The main difference is how the PPO loss for the actor is defined as:
+
+$$
+L ^ { C L I P } ( \boldsymbol { \theta } ) = \mathbb { E } [ \operatorname* { m i n } ( \rho _ { t } ( \boldsymbol { \theta } ) A _ { t } , c l i p ( \rho _ { t } ( \boldsymbol { \theta } ) , 1 - \epsilon , 1 + \epsilon ) A _ { t } ) ] ,
+$$
+
+where the likelihood ration is $\begin{array} { r } { \rho _ { t } ( \theta ) = \frac { \pi _ { \theta } \left( a _ { t } \vert x _ { t } \right) } { \pi _ { \theta _ { o l d } } \left( a _ { t } \vert x _ { t } \right) } } \end{array}$ πθ(at|xt)πθ (at|xt) , with πold being the policy parameters before the update, $\epsilon < 1$ is a hyper-parameters that controls the clipping and $A _ { t }$ is the generalized advantage estimator:
+
+$$
+A _ { t } ^ { G A E ( \lambda , \gamma ) } = \sum _ { k = 0 } ^ { T - 1 } ( \lambda \gamma ) ^ { k } \delta _ { t + k } ^ { V ^ { \pi } } ,
+$$
+
+# Algorithm 1 Synchronous n-step SARSA
+
+Input: $\theta$ parameters for $Q ( x , . ; \theta ) , \theta _ { \mathcal { D } }$ parameters for $Q _ { \mathcal { D } } ( x , . ; \theta _ { \mathcal { D } } ) , \phi _ { \mathcal { D } }$ parameters for $ _ {  } ( x ; \phi _ { \mathcal { D } } )$ $\pi _ { 0 }$ initial feasible policy.
+
+for episode $e \in { 1 , . . . , M }$ do
+
+Add the initial state to the trajectory buffer $\tau  \{ x _ { 0 } \}$   
+$t \gets 1$   
+while $t < T$ do: $t _ { s t a r t } \gets t$ while $t < t + n$ or $t = = T$ do Select $a _ { t }$ using (SPI), execute $a _ { t }$ , observe $x _ { t + 1 }$ and reward $r _ { t }$ and cost $d _ { t }$ . Add experiences to a buffer, i.e., $\tau \gets ( a _ { t } , r _ { t } , d _ { t } , x _ { t + 1 } )$ . $t \gets t + 1$
+
+# end while
+
+Calculate the next action for $x _ { t + 1 }$ using the current policy estimates, $a _ { t + 1 }$ Bootstrap the targets:
+
+$$
+\begin{array} { r l } & { R \gets \left\{ \begin{array} { l l } { 0 } & { \mathrm { i f ~ } t = = T } \\ { Q ( x _ { t + 1 } , a _ { t + 1 } ; \theta ) } & { \mathrm { o t h e r w i s e } } \end{array} \right. } \\ & { R _ { \mathcal { D } } \gets \left\{ \begin{array} { l l } { 0 } & { \mathrm { i f ~ } t = = T } \\ { Q _ { \mathcal { D } } ( x _ { t + 1 } , a _ { t + 1 } ; \theta _ { \mathcal { D } } ) } & { \mathrm { o t h e r w i s e } } \end{array} \right. } \\ & { \overleftarrow { R } _ { \mathcal { D } } \gets \left\{ \begin{array} { l l } { 0 } & { \mathrm { i f ~ } t = = 0 } \\ { \overleftarrow { V } ( x _ { t _ { s t a r t - 1 } ; \phi _ { \mathcal { D } } } ) } & { \mathrm { o t h e r w i s e } } \end{array} \right. } \end{array}
+$$
+
+$\triangleright$ Calculate the targets for the transitions in buffer for $i \in \{ t - 1 , \ldots , t _ { s t a r t } \}$ do $R  r _ { i } + \gamma R$ $R _ { \mathcal { D } }  d _ { i } + \gamma R _ { \mathcal { D } }$ Accumulate the gradients wrt $\theta , \theta _ { \mathcal { D } }$ :
+
+$$
+\begin{array} { c } { { d \theta  d \theta + \frac { \partial ( R - Q ( x _ { i } , a _ { i } ; \theta ) ) ^ { 2 } } { \partial \theta } } } \\ { { d \theta _ { \mathcal { D } }  d \theta _ { \mathcal { D } } + \frac { \partial ( R _ { \mathcal { D } } - Q _ { \mathcal { D } } ( x _ { i } , a _ { i } ; \theta _ { \mathcal { D } } ) ) ^ { 2 } } { \partial \theta _ { \mathcal { D } } } } } \end{array}
+$$
+
+end for
+
+for $i \in \{ t _ { s t a r t } , . . . , t \}$ do $\mathbf { \overleftarrow { R } } _ { \mathcal { D } } \gets d _ { i } + \gamma \mathbf { \overleftarrow { R } } _ { \mathcal { D } }$ Accumulate the gradients wrt $\phi _ { \mathcal { D } }$ :
+
+$$
+d \phi _ { \mathcal { D } } \gets d \phi _ { \mathcal { D } } + \frac { \partial ( \overleftarrow { R } _ { \mathcal { D } } - \overleftarrow { V } _ { \mathcal { D } } ( x _ { i } ; \phi _ { \mathcal { D } } ) ) ^ { 2 } } { \partial \phi _ { \mathcal { D } } }
+$$
+
+# end for
+
+Do synchronous batch update with the accumulated gradients to update $\theta , \theta _ { \mathcal { D } } , \phi _ { \mathcal { D } }$ using $d \theta , d \theta _ { \mathcal { D } } , d \phi _ { \mathcal { D } }$ .
+
+# end while
+
+Empty the trajectory buffer, $\tau$
+
+end for
+
+# Algorithm 2 Synchronous A2C with Safety Layer
+
+Input: $\theta$ parameters for $\pi ( x ; \theta )$ , $\phi$ the parameters for $V ( x ; \phi )$ , $\theta _ { \mathcal { D } }$ parameters for $Q _ { \mathcal { D } } ( x , \mu ; \theta _ { \mathcal { D } } )$ ,   
+$\phi _ { \mathcal { D } }$ parameters for $ _ { \overline { { \cal V } } _ { \mathcal { D } } ( x ; \phi _ { \mathcal { D } } ) }$ ;   
+for episode $e \in { 1 , . . . , M }$ do Add the initial state to the trajectory buffer $\tau  \{ x _ { 0 } \}$ $t \gets 1$ while $t < T$ do: $t _ { s t a r t } \gets t$ while $t < t + n$ or $t = = T$ do Select $a _ { t }$ using sampling from the projected mean $\mu _ { t }$ via the safety layer Eq.(5), execute   
+$a _ { t }$ , observe $x _ { t + 1 }$ and reward $r _ { t }$ and cost $d _ { t }$ . Add experiences to a buffer, i.e., $\tau \gets ( a _ { t } , \mu _ { t } , r _ { t } , d _ { t } , x _ { t + 1 } )$ . $t \gets t + 1$
+
+# end while
+
+Calculate the next mean for xt+1 using the current policy estimates, µt+1 Bootstrap the targets:
+
+$$
+\begin{array} { r l } & { R \gets \left\{ \begin{array} { l l } { 0 } & { \mathrm { i f ~ } t = = T } \\ { V ( x _ { t + 1 } , a _ { t + 1 } ; \phi ) } & { \mathrm { o t h e r w i s e } } \end{array} \right. } \\ & { R _ { \mathcal { D } } \gets \left\{ \begin{array} { l l } { 0 } & { \mathrm { i f ~ } t = = T } \\ { Q _ { \mathcal { D } } ( x _ { t + 1 } , \mu _ { t + 1 } ; \theta _ { \mathcal { D } } ) } & { \mathrm { o t h e r w i s e } } \end{array} \right. } \\ & { \overleftarrow { R } _ { \mathcal { D } } \gets \left\{ \begin{array} { l l } { 0 } & { \mathrm { i f ~ } t = = 0 } \\ { \overleftarrow { V } ( x _ { t _ { s t a r t - 1 } ; \phi _ { \mathcal { D } } } ) } & { \mathrm { o t h e r w i s e } } \end{array} \right. } \end{array}
+$$
+
+$\triangleright$ Calculate the targets for the transitions in buffer
+
+for $i \in \{ t - 1 , \ldots , t _ { s t a r t } \}$ do $R \gets r _ { i } + \gamma R$ $R _ { \mathcal { D } }  d _ { i } + \gamma R _ { \mathcal { D } }$
+
+Accumulate the gradients w.r.t. $\theta , \phi , \theta _ { \mathcal { D } }$ :
+
+$$
+\begin{array} { r l } & { \quad d \theta  d \theta + \nabla _ { \theta } \log \pi ( a _ { i } \mid x _ { i } ; \theta ) ( R - V ( x _ { i } ; \phi ) ) } \\ & { \quad d \phi  d \phi + \frac { \partial ( R - V ( x _ { i } \phi ) ) ^ { 2 } } { \partial \phi } } \\ & { \quad d \theta _ { \mathcal { D } }  d \theta _ { \mathcal { D } } + \frac { \partial ( R _ { \mathcal { D } } - Q _ { \mathcal { D } } ( x _ { i } , \mu _ { i } ; \theta _ { \mathcal { D } } ) ) ^ { 2 } } { \partial \theta _ { \mathcal { D } } } } \end{array}
+$$
+
+end for
+
+for $i \in \{ t _ { s t a r t } , . . . , t \}$ do $\mathbf { \overleftarrow { R } } _ { \mathcal { D } } \gets d _ { i } + \gamma \mathbf { \overleftarrow { R } } _ { \mathcal { D } }$ Accumulate the gradients wrt $\phi _ { \mathcal { D } }$ :
+
+$$
+d \phi _ { \mathcal { D } } \gets d \phi _ { \mathcal { D } } + \frac { \partial ( \overleftarrow { R } _ { \mathcal { D } } - \overleftarrow { V } _ { \mathcal { D } } ( x _ { i } ; \phi _ { \mathcal { D } } ) ) ^ { 2 } } { \partial \phi _ { \mathcal { D } } }
+$$
+
+# end for
+
+Do synchronous batch update with the accumulated gradients to update $\theta , \phi , \theta _ { \mathcal { D } } , \phi _ { \mathcal { D } }$ using $d \theta , d \phi , d \theta _ { \mathcal { D } } , d \phi _ { \mathcal { D } }$ .
+
+# end while
+
+Empty the trajectory buffer, $\tau$
+
+# end for
+
+where $T$ is the maxmimum number of timestamps in an episode trajectory, and $\delta _ { j }$ denotes the TD error at $j$ . The value function is updated using the $\gamma \lambda$ -returns from the GAE:
+
+$$
+L ( \phi ) = \mathbb { E } [ ( V ^ { \pi } ( x ; \phi ) - ( V ^ { \pi } ( x ; \phi _ { o l d } ) + A _ { t } ) ) ^ { 2 } ] .
+$$
+
+Similar to the the forward value estimates the backward value estimates are defined in the similar sense. One way to think of it is to assume the trajectories are reversed and we are doing the regular GAE estimation for the value functions.
+
+The GAE updates for the regular value function can be seen in the $\lambda$ -operator form as:
+
+$$
+\begin{array} { r } { \mathcal { T } _ { \lambda } ^ { \pi } \boldsymbol { v } ^ { \pi } = ( I - \gamma \lambda P ^ { \pi } ) ^ { - 1 } ( \boldsymbol { r } ^ { \pi } + \gamma P ^ { \pi } \boldsymbol { v } ^ { \pi } - \boldsymbol { v } ^ { \pi } ) + \boldsymbol { v } ^ { \pi } . } \end{array}
+$$
+
+In similar spirit it can be shown that the $\lambda$ -operator for SARSA has the form:
+
+$$
+\begin{array} { r } { \mathcal { T } _ { \lambda } ^ { \pi } q ^ { \pi } = ( I - \lambda \gamma P ^ { \pi } ) ^ { - 1 } ( \mathcal { T } ^ { \pi } q ^ { \pi } - q ^ { \pi } ) + q ^ { \pi } , } \end{array}
+$$
+
+where $( T ^ { \pi } q ^ { \pi } - q ^ { \pi } )$ denotes the TD error. Thus, the GAE estimates can be applied for the Q-functions in the similar form, i.e.
+
+$$
+\begin{array} { r l r } {  { B _ { t } ^ { G A E ( \lambda , \gamma ) } = \sum _ { k = 0 } ^ { T - 1 } ( \lambda \gamma ) ^ { k } \delta _ { t + k } ^ { Q _ { D } ^ { \pi } } , } } \\ & { } & { L ( \theta _ { \mathcal { D } } ) = \mathbb { E } [ ( Q _ { \mathcal { D } } ^ { \pi } ( x , a ; \theta _ { \mathcal { D } } ) - ( Q _ { \mathcal { D } } ^ { \theta _ { \mathcal { D } } } ( x , a ; \theta _ { \mathcal { D } _ { o l d } } ) + B _ { t } ) ) ^ { 2 } ] . } \end{array}
+$$

parse/train/SkB-_mcel/SkB-_mcel.md

@@ -0,0 +1,338 @@
+# CENTRAL MOMENT DISCREPANCY (CMD) FOR DOMAIN-INVARIANT REPRESENTATION LEARNING
+
+Werner Zellinger, Edwin Lughofer & Susanne Saminger-Platz∗
+
+Department of Knowledge-Based Mathematical Systems   
+Johannes Kepler University Linz, Austria   
+{werner.zellinger, edwin.lughofer, susanne.saminger-platz}@jku.at
+
+Thomas Grubinger & Thomas Natschlager ¨ †
+
+Data Analysis Systems Software Competence Center Hagenberg, Austria {thomas.grubinger, thomas.natschlaeger}@scch.at
+
+# ABSTRACT
+
+The learning of domain-invariant representations in the context of domain adaptation with neural networks is considered. We propose a new regularization method that minimizes the domain-specific latent feature representations directly in the hidden activation space. Although some standard distribution matching approaches exist that can be interpreted as the matching of weighted sums of moments, e.g. Maximum Mean Discrepancy, an explicit order-wise matching of higher order moments has not been considered before. We propose to match the higher order central moments of probability distributions by means of order-wise moment differences. Our model does not require computationally expensive distance and kernel matrix computations. We utilize the equivalent representation of probability distributions by moment sequences to define a new distance function, called Central Moment Discrepancy (CMD). We prove that CMD is a metric on the set of probability distributions on a compact interval. We further prove that convergence of probability distributions on compact intervals w. r. t. the new metric implies convergence in distribution of the respective random variables. We test our approach on two different benchmark data sets for object recognition (Office) and sentiment analysis of product reviews (Amazon reviews). CMD achieves a new state-of-the-art performance on most domain adaptation tasks of Office and outperforms networks trained with Maximum Mean Discrepancy, Variational Fair Autoencoders and Domain Adversarial Neural Networks on Amazon reviews. In addition, a post-hoc parameter sensitivity analysis shows that the new approach is stable w. r. t. parameter changes in a certain interval. The source code of the experiments is publicly available1.
+
+# 1 INTRODUCTION
+
+The collection and preprocessing of large amounts of data for new domains is often time consuming and expensive. This in turn limits the application of state-of-the-art methods like deep neural network architectures, that require large amounts of data. However, often data from related domains can be used to improve the prediction model in the new domain. This paper addresses the particularly important and challenging domain-invariant representation learning task of unsupervised domain adaptation (Glorot et al., 2011; Li et al., 2014; Pan et al., 2011; Ganin et al., 2016). In unsupervised domain adaptation, the training data consists of labeled data from the source domain(s) and unlabeled data from the target domain. In practice, this setting is quite common, as in many applications the collection of input data is cheap, but the collection of labels is expensive. Typical examples include image analysis tasks and sentiment analysis, where labels have to be collected manually.
+
+Recent research shows that domain adaptation approaches work particularly well with (deep) neural networks, which produce outstanding results on some domain adaptation data sets (Ganin et al., 2016; Sun & Saenko, 2016; Li et al., 2016; Aljundi et al., 2015; Long et al., 2015; Li et al., 2015; Zhuang et al., 2015; Louizos et al., 2016). The most successful methods have in common that they encourage similarity between the latent network representations w. r. t. the different domains. This similarity is often enforced by minimizing a certain distance between the networks’ domainspecific hidden activations. Three outstanding approaches for the choice of the distance function are the Proxy $\mathcal { A }$ -distance (Ben-David et al., 2010), the Kullback-Leibler (KL) divergence Kullback & Leibler (1951), applied to the mean of the activations (Zhuang et al., 2015), and the Maximum Mean Discrepancy (Gretton et al., 2006, MMD).
+
+Two of them, the MMD and the KL-divergence approach, can be viewed as the matching of statistical moments. The KL-divergence approach is based on mean (first raw moment) matching. Using the Taylor expansion of the Gaussian kernel, most MMD-based approaches can be viewed as minimizing a certain distance between weighted sums of all raw moments (Li et al., 2015).
+
+The interpretation of the KL-divergence approaches and MMD-based approaches as moment matching procedures motivate us to match the higher order moments of the domain-specific activation distributions directly in the hidden activation space. The matching of the higher order moments is performed explicitly for each moment order and each hidden coordinate. Compared to KL-divergencebased approaches, which only match the first moment, our approach also matches higher order moments. In comparison to MMD-based approaches, our method explicitly matches the moments for each order, and it does not require any computationally expensive distance- and kernel matrix computations.
+
+The proposed distribution matching method induces a metric between probability distributions. This is possible since distributions on compact intervals have an equivalent representation by means of their moment sequences. We utilize central moments due to their translation invariance and natural geometric interpretation. We call the new metric Central Moment Discrepancy (CMD).
+
+The contributions of this paper are as follows:
+
+• We propose to match the domain-specific hidden representations by explicitly minimizing differences of higher order central moments for each moment order. We utilize the equivalent representation of probability distributions by moment sequences to define a new distance function, which we call Central Moment Discrepancy (CMD). Probability theoretic analysis is used to prove that CMD is a metric on the set of probability distributions on a compact interval. We additionally prove that convergence of probability distributions on compact intervals w. r. t. to the new metric implies convergence in distribution of the respective random variables. This means that minimizing the CMD metric between probability distributions leads to convergence of the cumulative distribution functions of the random variables.   
+• In contrast to MMD-based approaches our method does not require computationally expensive kernel matrix computations. We achieve a new state-of-the-art performance on most domain adaptation tasks of Office and outperform networks trained with MMD, variational fair autoencoders and domain adversarial neural networks on Amazon reviews.   
+• A parameter sensitivity analysis shows that CMD is insensitive to parameter changes within a certain interval. Consequently, no additional hyper-parameter search has to be performed.
+
+# 2 HIDDEN ACTIVATION MATCHING
+
+We consider the unsupervised domain adaptation setting (Glorot et al., 2011; Li et al., 2014; Pan et al., 2011; Ganin et al., 2016) with an input space $\mathcal { X }$ and a label space $\mathcal { V }$ . Two distributions over $\mathcal { X } \times \mathcal { V }$ are given: the labeled source domain $D _ { S }$ and the unlabeled target domain $D _ { T }$ . Two corresponding samples are given: the source sample $S = ( X _ { S } , Y _ { S } ) = \{ ( x _ { i } , y _ { i } ) \} _ { i = 1 } ^ { n } \stackrel { \mathrm { i . i . d . } } { \sim } ( D _ { S } ) ^ { n }$ and the target sample $T = X _ { T } = \{ x _ { i } \} _ { i = 1 } ^ { m } \stackrel { \mathrm { i . i . d . } } { \sim } ( D _ { T } ) ^ { m }$ . The goal of the unsupervised domain adaptation setting is to build a classifier $f : \mathcal { X }  \mathcal { Y }$ with a low target risk $R _ { T } ( \bar { f } ) = \operatorname* { P r } _ { ( x , y ) \sim D _ { T } } ( f ( x ) \bar { \neq } y )$ , while no information about the labels in $D _ { T }$ is given.
+
+![](images/d949f49b236dc8cf762beb46551e710cdcadbc0def130f064a8a3079eb35f3e0.jpg)  
+Figure 1: Schematic sketch of a three layer neural network trained with backpropagation based on objective (2). $\nabla _ { \theta }$ refers to the gradient w. r. t. $\theta$ .
+
+We focus our studies on neural network classifiers $f _ { \theta } : \mathcal { X }  \mathcal { Y }$ with parameters $\theta \in \Theta$ , the input space $\mathcal { X } ~ = ~ \mathbb { R } ^ { I }$ with input dimension $I$ , and the label space $\mathcal { Y } ~ = ~ [ 0 , 1 ] ^ { | C | }$ with the cardinality $| C |$ of the set of classes $C$ . We further assume a network output $f _ { \theta } ( x ) \in [ 0 , 1 ] ^ { | C | }$ of an example $x \in \mathbb { R } ^ { I }$ to be normalized by the softmax-function $\sigma : \mathbb { R } ^ { | C | }  [ 0 , 1 ] ^ { | C | }$ with $\begin{array} { r } { \sigma ( \dot { z } ) _ { j } = \frac { e ^ { z _ { j } } } { \sum _ { k = 1 } ^ { | C | } e ^ { z _ { k } } } } \end{array}$ for $z = \{ z _ { 1 } , \ldots , z _ { | C | } \}$ . We focus on bounded activation functions $g _ { H } : \mathbb { R } \to [ a , b ] ^ { N }$ for the hidden layer $H$ with $N$ hidden nodes, e.g. the hyperbolic tangent or the sigmoid function. Unbounded activation functions, e.g. rectified linear units or exponential linear units, can be used if the output is clipped or normalized to be bounded. Using the loss function $l : \Theta \times \mathcal { X } \times \mathcal { Y }  \mathbb { R }$ , e.g. cross-entropy $\begin{array} { r } { l } { ( \theta , x , y ) = - \sum _ { i \in C } y _ { i } \log ( f _ { \theta } ( x ) _ { i } ) } \end{array}$ , and the sample set $( X , Y ) \subset \mathbb { R } ^ { I } \times [ 0 , 1 ] ^ { | C | }$ , we define the objective function as
+
+$$
+\operatorname* { m i n } _ { \theta \in \Theta } \mathbf { E } ( l ( \theta , X , Y ) )
+$$
+
+where $\mathbf { E }$ denotes the empirical expectation, i.e. ${ \bf E } ( l ( \theta , X , Y ) ) = { \textstyle { \frac { 1 } { | ( X , Y ) | } } } \sum _ { ( x , y ) \in ( X , Y ) } l ( \theta , x , y )$ . Let us denote the source hidden activations by $A _ { H } ( \theta , X _ { S } ) = g _ { H } ( \theta _ { H } ^ { T } A _ { H ^ { \prime } } ( \theta , X _ { S } ) ) \subset [ a , b ] ^ { N }$ and the target hidden activations by $A _ { H } ( \theta , X _ { T } ) = g _ { H } ( \theta _ { H } ^ { T } A _ { H ^ { \prime } } ( \theta , X _ { T } ) ) \subset [ a , b ] ^ { N }$ for the hidden layer $H$ with $N$ hidden nodes and parameter $\theta _ { H }$ , and the hidden layer $H ^ { \prime }$ before $H$ .
+
+One fundamental assumption of most unsupervised domain adaptation networks is that the source risk $R _ { S } ( f )$ is a good indicator for the target risk $R _ { T } ( f )$ , when the domain-specific latent space representations are similar (Ganin et al., 2016). This similarity can be enforced by matching the distributions of the hidden activations $A _ { H } ( \theta , X _ { S } )$ and $A _ { H } ( \theta , X _ { T } )$ of higher layers $H$ . Recent stateof-the-art approaches define a domain regularizer $d : ( [ a , b ] ^ { N } ) ^ { \acute { n } } \times ( [ \bar { a } , b ] ^ { N } ) ^ { \acute { m } }  [ 0 , \infty )$ , which gives a measure for the domain discrepancy in the activation space $[ a , b ] ^ { N }$ . The domain regularizer is added to the objective by means of an additional weighting parameter $\lambda$ .
+
+$$
+\begin{array} { r l } { \underset { \theta \in \Theta } { \operatorname* { m i n } } } & { { } \mathbf { E } ( l ( \theta , X _ { S } , Y _ { S } ) ) + \lambda \cdot d ( A _ { H } ( \theta , X _ { S } ) , A _ { H } ( \theta , X _ { T } ) ) } \end{array}
+$$
+
+Fig. 1 shows a sketch of the described architecture and fig. 2 shows the hidden activations of a simple neural network optimized by eq. (1) (left) and eq. (2) (right). It can be seen that similar activation distributions are obtained when being optimized on the basis of the domain regularized objective.
+
+# 3 RELATED WORK
+
+Recently, several measures $d$ for objective (2) have been proposed. One approach is the Proxy $\mathcal { A }$ - distance, given by $\hat { d } _ { A } = 2 ( 1 - 2 \epsilon )$ , where $\epsilon$ is the generalization error on the problem of discriminating between source and target samples (Ben-David et al., 2010). Ganin et al. (2016) compute the value $\epsilon$ with a neural network classifier that is simultaneously trained with the original network by means of a gradient reversal layer. They call their approach domain-adversarial neural networks. Unfortunately, a new classifier has to be trained in this approach including the need of new parameters, additional computation times and validation procedures.
+
+![](images/8b400c4378773bbe409c62ea1b808deadde1a9b1e7c140f6e543b12d0797353f.jpg)  
+Figure 2: Hidden activation distributions for a simple one-layer classification network with sigmoid activation functions and five hidden nodes trained with the standard objective (1) (left) and objective (2) that includes the domain discrepancy minimization (right). The approach of this paper was used as domain regularizer. Dark gray: activations of the source domain, light gray: activations of the target domain.
+
+Another approach is to make use of the MMD (Gretton et al., 2006) as domain regularizer.
+
+where $\begin{array} { r } { \mathbf { E } ( K ( X , Y ) ) = \frac { 1 } { | X | \cdot | Y | } \sum _ { k \in K ( X , Y ) } k } \end{array}$ is the empirical expectation of the kernel products $k$ between all examples in $X$ and $Y$ stored by the kernel matrix $K ( X , Y )$ . A suitable choice of the kernel seems to be the Gaussian kernel $e ^ { - \beta \| x - y \| ^ { 2 } }$ (Louizos et al., 2016; Li et al., 2015; Tzeng et al., 2014). This approach has two major drawbacks: (a) the need of tuning an additional kernel parameter $\beta$ , and (b) the need of the kernel matrix computation $K ( X , Y )$ (computational complexity $\mathcal { O } ( n ^ { 2 } +$ $n m + m ^ { 2 } )$ ), which becomes inefficient (resource-intensive) in case of large data sets. Concerning (a), the tuning of $\beta$ is sophisticated since no target samples are available in the domain adaptation setting. Suitable tuning procedures are transfer learning specific cross-validation methods (Zhong et al., 2010). More general methods that don’t utilize source labels include heuristics that are based on kernel space properties (Sriperumbudur et al., 2009; Gretton et al., 2012), combinations of multiple kernels (Li et al., 2015), and kernel choices that maximize the MMD test power (Sutherland et al., 2016). The drawback (b) of the kernel matrix computation can be handled by approximating the MMD (Zhao & Meng, 2015), or by using linear time estimators (Gretton et al., 2012). In this work we focus on the quadratic-time MMD with the Gaussian kernel (Gretton et al., 2012; Tzeng et al., 2014) and transfer learning specific cross-validation for parameter tuning (Zhong et al., 2010; Ganin et al., 2016).
+
+The two approaches MMD and the Proxy $\mathcal { A }$ -distance have in common that they do not minimize the domain discrepancy explicitly in the hidden activation space. In contrast, the authors in Zhuang et al. (2015) do so by minimizing a modified version of the Kullback-Leibler divergence of the mean activations (MKL). That is, for samples $X , Y \subset \mathbb { R } ^ { N }$ ,
+
+$$
+\operatorname { M K L } ( X , Y ) = \sum _ { i = 1 } ^ { N } \mathbf { E } ( X ) _ { i } \log { \frac { \mathbf { E } ( X ) _ { i } } { \mathbf { E } ( Y ) _ { i } } } + \mathbf { E } ( Y ) _ { i } \log { \frac { \mathbf { E } ( Y ) _ { i } } { \mathbf { E } ( X ) _ { i } } }
+$$
+
+with $\mathbf { E } ( X ) _ { i }$ being the $i ^ { \mathrm { { t h } } }$ coordinate of the empirical expectation $\begin{array} { r } { \mathbf { E } ( X ) = \frac { 1 } { | X | } \sum _ { x \in X } x } \end{array}$ . This approach is fast to compute and has an explicit interpretation in the activation space. Our empirical observations (section Experiments) show that minimizing the distance between only the first moment (mean) of the activation distributions can be improved by also minimizing the distance between higher order moments.
+
+As noted in the introduction, our approach is motivated by the fact that the MMD and the KLdivergence approach can be seen as the matching of statistical moments of the hidden activations $A _ { H } ( \bar { \theta } , X _ { S } )$ and $A _ { H } ( \theta , X _ { T } )$ . In particular, MMD-based approaches that use the Gaussian kernel are equivalent to minimizing a certain distance between weighted sums of all moments of the hidden activation distributions (Li et al., 2015).
+
+We propose to minimize differences of higher order central moments of the activations $A _ { H } ( \theta , X _ { S } )$ and $\bar { \bf A } _ { H } ^ { - } ( \theta , X _ { T } )$ . The difference minimization is performed explicitly for each moment order. Our approach utilizes the equivalent representation of probability distributions in terms of its moment series. We further utilize central moments due to their translation invariance and natural geometric interpretation. Our approach contrasts with other moment-based approaches, as they either match only the first moment (MKL) or they don’t explicitly match the moments for each order (MMD). As a result, our approach improves over MMD-based approaches in terms of computational complexity with $\mathcal { O } \left( N ( n \overline { { + } } m ) \right)$ for CMD and $\mathcal { O } \left( N ( n ^ { 2 } + n m + m ^ { 2 } ) \right)$ for MMD. In contrast to MKL-based approaches more accurate distribution matching characteristics are obtained. In addition, CMD achieves a new state-of-the-art performance on most domain adaptation tasks of Office and outperforms networks trained with MMD, variational fair autoencoders and domain adversarial neural networks on Amazon reviews.
+
+# 4 CENTRAL MOMENT DISCREPANCY (CMD)
+
+In this section we first propose a new distance function CMD on probability distributions on compact intervals. The definition is extended by two theorems that identify CMD as a metric and analyze a convergence property. The final domain regularizer is then defined as an empirical estimate of CMD. The proofs of the theorems are given in the appendix.
+
+Definition 1 (CMD metric). Let $X = ( X _ { 1 } , \ldots , X _ { n } ) $ and $Y = ( Y _ { 1 } , \ldots , Y _ { n } )$ be bounded random vectors independent and identically distributed from two probability distributions $p$ and $q$ on the compact interval $[ a , b ] ^ { N }$ . The central moment discrepancy metric (CMD) is defined by
+
+$$
+C M D ( p , q ) = \frac { 1 } { \left| b - a \right| } \left\| \mathbb { E } ( X ) - \mathbb { E } ( Y ) \right\| _ { 2 } + \sum _ { k = 2 } ^ { \infty } \frac { 1 } { \left| b - a \right| ^ { k } } \left\| c _ { k } ( X ) - c _ { k } ( Y ) \right\| _ { 2 }
+$$
+
+where $\mathbb { E } ( X )$ is the expectation of $X$ , and
+
+$$
+c _ { k } ( \boldsymbol { X } ) = \left( \mathbb { E } \Big ( \prod _ { i = 1 } ^ { N } \left( \boldsymbol { X } _ { i } - \mathbb { E } ( \boldsymbol { X } _ { i } ) \right) ^ { r _ { i } } \Big ) \right) _ { r _ { 1 } + \ldots + r _ { N } = k }
+$$
+
+is the central moment vector of order $k$ .
+
+The first order central moments are zero, the second order central moments are related to variance, and the third and fourth order central moments are related to the skewness and the kurtosis of probability distributions. It is easy to see that ${ \bf C M D } ( p , q ) \ge 0$ , $\mathrm { C M D } ( p , q ) = \mathrm { C M D } ( q , p )$ , $\mathrm { C M D } ( p , q ) \leq \mathrm { C M D } ( p , r ) + \mathrm { C M D } ( r , q )$ and $p = q \Rightarrow { \bf C M D } ( p , q ) = 0$ . The following theorem shows the remaining property for CMD to be a metric on the set of probability distributions on a compact interval.
+
+Theorem 1. Let $p$ and $q$ be two probability distributions on a compact interval and let CMD be defined as in (5), then
+
+$$
+C M D ( p , q ) = 0 \Rightarrow p = q
+$$
+
+Our approach is to minimize the discrepancy between the domain-specific hidden activation distributions by minimizing the CMD. Thus, in the optimization procedure, we increasingly expect to see the domain-specific cumulative distribution functions approach each other. This characteristic can be expressed by the concept of convergence in distribution and it is shown in the following theorem.
+
+Theorem 2. Let $p _ { n }$ and $p$ be probability distributions on a compact interval and let CMD be defined as in (5), then
+
+$$
+C M D ( p _ { n } , p ) \to 0 \Rightarrow p _ { n } \stackrel { d } { \to } p
+$$
+
+where $\xrightarrow { d }$ denotes convergence in distribution.
+
+We define the final central moment discrepancy regularizer as an empirical estimate of the CMD metric. Only the central moments that correspond to the marginal distributions are computed. The number of central moments is limited by a new parameter $K$ and the expectation is sampled by the empirical expectation.
+
+Definition 2 (CMD regularizer). Let $X$ and $Y$ be bounded random samples with respective probability distributions $p$ and $q$ on the interval $[ a , b ] ^ { N }$ . The central moment discrepancy regularizer $C M D _ { K }$ is defined as an empirical estimate of the CMD metric, by
+
+$$
+C M D _ { K } ( X , Y ) = { \frac { 1 } { | b - a | } } \| \mathbf { E } ( X ) - \mathbf { E } ( Y ) \| _ { 2 } + \sum _ { k = 2 } ^ { K } { \frac { 1 } { | b - a | ^ { k } } } \| C _ { k } ( X ) - C _ { k } ( Y ) \| _ { 2 }
+$$
+
+where $\begin{array} { r } { \mathbf { E } ( X ) = \frac { 1 } { | X | } \sum _ { x \in X } x } \end{array}$ is the empirical expectation vector computed on the sample $X$ and $C _ { k } ( X ) = \mathbf { E } ( ( x - \mathbf { E } ( X ) ) ^ { k } )$ is the vector of all $k ^ { t h }$ order sample central moments of the coordinates of $X$ .
+
+This definition includes three approximation steps: (a) the computation of only marginal central moments, (b) the bound on the order of central moment terms via parameter $K$ , and (c) the sampling of the probability distributions by the replacement of the expected value with the empirical expectation.
+
+Applying approximation (a) and assuming independent marginal distributions, a zero CMD distance value still implies equal joint distributions (thm. 1) but convergence in distribution (thm. 2) applies only to the marginals. In the case of dependent marginal distributions, zero CMD distance implies equal marginals and convergence in CMD implies convergence in distribution of the marginals. However, the matching properties for the joint distributions are not obtained with dependent marginals and approximation (a). The computational complexity is reduced to be linear w. r. t. the number of samples.
+
+Concerning (b), proposition 1 shows that the marginal distribution specific CMD terms have an upper bound that is strictly decreasing with increasing moment order. This bound is convergent to zero. That is, higher CMD terms can contribute less to the overall distance value. This observation is experimentally strengthened in subsection Parameter Sensitivity.
+
+Proposition 1. Let $X$ and $Y$ be bounded random vectors with respective probability distributions $p$ and $q$ on the compact interval $[ a , b ] ^ { N }$ . Then
+
+$$
+\frac { 1 } { | b - a | ^ { k } } \| c _ { k } ( X ) - c _ { k } ( Y ) \| _ { 2 } \leq 2 \sqrt { N } \left( \frac { 1 } { k + 1 } \left( \frac { k } { k + 1 } \right) ^ { k } + \frac { 1 } { 2 ^ { 1 + k } } \right)
+$$
+
+where $c _ { k } ( X ) \ : = \ : \mathbb { E } ( ( X - \mathbb { E } ( X ) ) ^ { k } )$ is the vector of all $k ^ { t h }$ order sample central moments of the marginal distributions of $p$ .
+
+Concerning approximation (c), the joint application of the weak law of large numbers (Billingsley, 2008) with the continuous mapping theorem (Billingsley, 2013) proves that this approximation creates a consistent estimate.
+
+We would like to underline that the training of neural networks with eq. (2) and the CMD regularizer in eq. (6) can be easily realized by gradient descent algorithms. The gradients of the CMD regularizer are simple aggregations of derivatives of the standard functions $g _ { H } , x ^ { k }$ and $\lVert . \rVert _ { 2 }$ .
+
+# 5 EXPERIMENTS
+
+Our experimental evaluations are based on two benchmark datasets for domain adaptation, Amazon reviews and Office, described in subsection Datasets. The experimental setup is discussed in subsection Experimental Setup and our classification accuracy results are discussed in subsection Results. Subsection Parameter Sensitivity analysis the accuracy sensitivity w. r. t. parameter changes of $K$ for CMD and $\beta$ for MMD.
+
+# 5.1 DATASETS
+
+Amazon reviews: For our first experiment we use the Amazon reviews data set with the same preprocessing as used by Chen et al. (2012); Ganin et al. (2016); Louizos et al. (2016). The data set contains product reviews of four different product categories: books, DVDs, kitchen appliances and electronics. Reviews are encoded in 5000 dimensional feature vectors of bag-of-words unigrams and bigrams with binary labels: 0 if the product is ranked by $1 - 3$ stars and 1 if the product is ranked by 4 or 5 stars. From the four categories we obtain twelve domain adaptation tasks (each category serves once as source category and once as target category).
+
+Office: The second experiment is based on the computer vision classification data set from Saenko et al. (2010) with images from three distinct domains: amazon (A), webcam (W) and dslr (D). This data set is a de facto standard for domain adaptation algorithms in computer vision. Amazon, the largest domain, is a composition of 2817 images and its corresponding 31 classes. Following previous works we assess the performance of our method across all six possible transfer tasks.
+
+# 5.2 EXPERIMENTAL SETUP
+
+# Amazon Reviews:
+
+For the Amazon reviews experiment, we use the same data splits as previous works for every task. Thus we have 2000 labeled source examples and 2000 unlabeled target examples for training, and between 3000 and 6000 examples for testing.
+
+We use a similar architecture as Ganin et al. (2016) with one dense hidden layer with 50 hidden nodes, sigmoid activation functions and softmax output function. Three neural networks are trained by means of eq. (2): (a) a base model without domain regularization $\lambda = 0$ ), (b) with the MMD as domain regularizer and (c) with CMD as domain regularizer. These models are additionally compared with the state-of-the-art models VFAE (Louizos et al., 2016) and DANN (Ganin et al., 2016). The models (a),(b) and (c) are trained with similar setup as in Louizos et al. (2016) and Ganin et al. (2016).
+
+For the CMD regularizer, the $\lambda$ parameter of eq. (2) is set to 1, i.e. the weighting parameter $\lambda$ is neglected. The parameter $K$ is heuristically set to five, as the first five moments capture rich geometric information about the shape of a distribution and $K = 5$ is small enough to be computationally efficient. However, the experiments in subsection Parameter Sensitivity show that similar results are obtained for $K \geq 3$ .
+
+For the MMD regularizer we use the Gaussian kernel with parameter $\beta$ . We performed a hyperparameter search for $\beta$ and $\lambda$ , which has to be performed in an unsupervised way (no labels in the target domain). We use a variant of the reverse cross-validation approach proposed by Zhong et al. (2010), in which we initialize the model weights of the reverse classifier by the weights of the first learned classifier (see Ganin et al. (2016) for details). Thereby, the parameter $\lambda$ is tuned on 10 values between 0.1 and 500 on a logarithmic scale. The parameter $\beta$ is tuned on 10 values between 0.01 and 10 on a logarithmic scale. Without this parameter search, no competitive prediction accuracy results could be obtained.
+
+Since we have to deal with sparse data, we rely on the Adagrad optimizer (Duchi et al., 2011). For all evaluations, the default parametrization is used as implemented in Keras (Chollet, 2015). All evaluations are repeated 10 times based on different shuffles of the data, and the mean accuracies and standard deviations are analyzed.
+
+Office: Since the office dataset is rather small with only 2817 images in its largest domain, we use the latent representations of the convolution neural network VGG16 of Simonyan & Zisserman (2014). In particular we train a classifier with one hidden layer, 256 hidden nodes and sigmoid activation function on top of the output of the first dense layer in the network. We again train one base model without domain regularization and a CMD regularized version with $K = 5$ and $\lambda = 1$ .
+
+We follow the standard training protocol for this data set and use all available source and target examples during training. Using this ”fully-transductive” protocol, we compare our method with other state-of-the-art approaches including DLID (Chopra et al., 2013), DDC (Tzeng et al., 2014), DAN (Long et al., 2015), Deep CORAL (Sun & Saenko, 2016), and DANN (Ganin et al., 2016), based on fine-tuning of the baseline model AlexNet (Krizhevsky et al., 2012). We further compare our method to LSSA (Aljundi et al., 2015), CORAL (Sun et al., 2016), and AdaBN (Li et al., 2016), based on the fine-tuning of InceptionBN (Ioffe & Szegedy, 2015).
+
+As an alternative to Adagrad for non-sparse data, we use the Adadelta optimizer from Zeiler (2012). Again, the default parametrization from Keras is used. We handle unbalances between source and target sample by randomly down-sampling (up-sampling) the source sample. In addition, we ensure a sub-sampled source batch that is balanced w. r. t. the class labels.
+
+Since all hyper-parameters are set a-priori, no hyper-parameter search has to be performed.
+
+All experiments are repeated 10 times with randomly shuffled data sets and random initializations.
+
+# 5.3 RESULTS
+
+Amazon Reviews: Table 1 shows the classification accuracies of four models: The Source Only model is the non domain regularized neural network trained with objective (1), and serves as a base model for the domain adaptation improvements. The models MMD and CMD are trained with the same architecture and objective (2) with $d$ as the domain regularizer MMD and CMD, respectively. VFAE refers to the Variational Fair Autoencoder of Louizos et al. (2016), including a slightly modified version of the MMD regularizer for faster computations, and DANN refers to the domainadversarial neural networks model of Ganin et al. (2016). The last two columns are taken directly from these publications.
+
+As one can observe in table 1, our accuracy of the CMD-based model is the highest in 9 out of 12 domain adaptation tasks, whereas on the remaining 3 it is the second best method. However, the difference in accuracy compared to the best method is smaller than the standard deviation over all data shuffles.
+
+Table 1: Prediction accuracy $\pm$ standard deviation on the Amazon reviews dataset. The last two columns are taken directly from Louizos et al. (2016) and Ganin et al. (2016).   
+
+<table><tr><td rowspan=1 colspan=1>Source-&gt;Target</td><td rowspan=1 colspan=1>Source Only</td><td rowspan=1 colspan=1>MMD</td><td rowspan=1 colspan=1>CMD</td><td rowspan=1 colspan=1>VFAE</td><td rowspan=1 colspan=1>DANN</td></tr><tr><td rowspan=1 colspan=1>books-→dvd</td><td rowspan=1 colspan=1>.787 ± .004</td><td rowspan=1 colspan=1>.796 ± .008</td><td rowspan=1 colspan=1>.805 ± .007</td><td rowspan=1 colspan=1>.799</td><td rowspan=1 colspan=1>.784</td></tr><tr><td rowspan=1 colspan=1>books-→electronics</td><td rowspan=1 colspan=1>.714± .009</td><td rowspan=1 colspan=1>.758 ± .018</td><td rowspan=1 colspan=1>.787 ± .007</td><td rowspan=1 colspan=1>.792</td><td rowspan=1 colspan=1>.733</td></tr><tr><td rowspan=1 colspan=1>books-→kitchen</td><td rowspan=1 colspan=1>.745 ± .006</td><td rowspan=1 colspan=1>.787 ± .019</td><td rowspan=1 colspan=1>.813 ± .008</td><td rowspan=1 colspan=1>.816</td><td rowspan=1 colspan=1>.779</td></tr><tr><td rowspan=1 colspan=1>dvd-&gt;books</td><td rowspan=1 colspan=1>.746 ± .019</td><td rowspan=1 colspan=1>.780 ± .018</td><td rowspan=1 colspan=1>.795 ± .005</td><td rowspan=1 colspan=1>.755</td><td rowspan=1 colspan=1>.723</td></tr><tr><td rowspan=1 colspan=1>dvd-&gt;electronics</td><td rowspan=1 colspan=1>.724 ± .011</td><td rowspan=1 colspan=1>.766 ± .025</td><td rowspan=1 colspan=1>.797 ± .010</td><td rowspan=1 colspan=1>.786</td><td rowspan=1 colspan=1>.754</td></tr><tr><td rowspan=1 colspan=1>dvd-&gt;kitchen</td><td rowspan=1 colspan=1>.765 ± .012</td><td rowspan=1 colspan=1>.796 ± .019</td><td rowspan=1 colspan=1>.830 ± .012</td><td rowspan=1 colspan=1>.822</td><td rowspan=1 colspan=1>.783</td></tr><tr><td rowspan=1 colspan=1> electronics-→books</td><td rowspan=1 colspan=1>.711 ± .006</td><td rowspan=1 colspan=1>.733 ± .017</td><td rowspan=1 colspan=1>.744 ± .008</td><td rowspan=1 colspan=1>.727</td><td rowspan=1 colspan=1>.713</td></tr><tr><td rowspan=1 colspan=1>electronics-→dvd</td><td rowspan=1 colspan=1>.719 ± .009</td><td rowspan=1 colspan=1>.748 ± .013</td><td rowspan=1 colspan=1>.763 ± .006</td><td rowspan=1 colspan=1>.765</td><td rowspan=1 colspan=1>.738</td></tr><tr><td rowspan=1 colspan=1>electronics-→kitchen</td><td rowspan=1 colspan=1>.844 ± .005</td><td rowspan=1 colspan=1>.857 ± .007</td><td rowspan=1 colspan=1>.860 ± .004</td><td rowspan=1 colspan=1>.850</td><td rowspan=1 colspan=1>.854</td></tr><tr><td rowspan=1 colspan=1>kitchen-→books</td><td rowspan=1 colspan=1>.699 ± .014</td><td rowspan=1 colspan=1>.740 ± .017</td><td rowspan=1 colspan=1>.756 ± .006</td><td rowspan=1 colspan=1>.720</td><td rowspan=1 colspan=1>.709</td></tr><tr><td rowspan=1 colspan=1>kitchen-&gt;dvd</td><td rowspan=1 colspan=1>.734 ± .011</td><td rowspan=1 colspan=1>.763 ± .011</td><td rowspan=1 colspan=1>.775 ± .005</td><td rowspan=1 colspan=1>.733</td><td rowspan=1 colspan=1>.740</td></tr><tr><td rowspan=1 colspan=1>kitchen-→electronics</td><td rowspan=1 colspan=1>.833 ± .004</td><td rowspan=1 colspan=1>.844 ± .007</td><td rowspan=1 colspan=1>.854 ± .003</td><td rowspan=1 colspan=1>.838</td><td rowspan=1 colspan=1>.843</td></tr><tr><td rowspan=1 colspan=1> average</td><td rowspan=1 colspan=1>.752 ± .009</td><td rowspan=1 colspan=1>.781 ± .015</td><td rowspan=1 colspan=1>.798 ± .007</td><td rowspan=1 colspan=1>.784</td><td rowspan=1 colspan=1>.763</td></tr></table>
+
+Office: Table 2 shows the classification accuracy of different models trained on the Office dataset. Note that some of the methods (LSSA, CORAL and AdaBN) are evaluated based on the InceptionBN model, which shows higher accuracy than the base model (VGG16) of our method in most tasks. However, our method outperforms related state-of-the-art methods on all except two tasks, on which it performs similar. We improve the previous state-of-the-art method AdaBN (Li et al., 2016) by more than $3 . 2 \%$ in average accuracy.
+
+# 5.4 PARAMETER SENSITIVITY
+
+The first sensitivity experiment aims at providing evidence regarding the accuracy sensitivity of the CMD regularizer w. r. t. parameter changes of $K$ . That is, the contribution of higher terms in the CMD regularizer are analyzed. The claim is that the accuracy of CMD-based networks does not depend strongly on the choice of $K$ in a range around its default value 5.
+
+In fig. 3 on the upper left we analyze the classification accuracy of a CMD-based network trained on all tasks of the Amazon reviews experiment. We perform a grid search for the two regularization hyper-parameters $\lambda$ and $K$ . We empirically choose a representative stable region for each parameter, [0.3, 3] for $\lambda$ and $\{ 1 , . . . , 7 \}$ for $K$ . Since we want to analyze the sensitivity w. r. t. $K$ , we averaged over the $\lambda$ -dimension, resulting in one accuracy value per $K$ for each of the 12 tasks. Each accuracy is transformed into an accuracy ratio value by dividing it with the accuracy of $K = 5$ . Thus, for each $K$ and task we get one value representing the ratio between the obtained accuracy (for this $K$ and task) and the accuracy of $K = 5$ . The results are shown in fig. 3 (upper left). The accuracy ratios between $K = 5$ and $K \in \{ 3 , 4 , 6 , 7 \}$ are lower than $0 . 5 \%$ , which underpins the claim that the accuracy of CMD-based networks does not depend strongly on the choice of $K$ in a range around its default value 5. For $K = 1$ and $K = 2$ higher ratio values are obtained. In addition, for these two values many tasks show worse accuracy than obtained by $K \in \{ 3 , 4 , 5 , 6 , 7 \}$ . From this we additionally conclude that higher values of $K$ are preferable to $K = 1$ and $K = 2$ .
+
+Table 2: Prediction accuracy $\pm$ standard deviation on the Office dataset. The first 10 rows are taken directly from the papers of Ganin et al. (2016) and Li et al. (2016). The models DLID –DANN are based on the AlexNet model, LSSA –AdaBN are based on the InceptionBN model, and our method (CMD) is based on the VGG16 model.   
+
+<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>A→W</td><td rowspan=1 colspan=1>D→W</td><td rowspan=1 colspan=1>W→D</td><td rowspan=1 colspan=1>A→D</td><td rowspan=1 colspan=1>D→A</td><td rowspan=1 colspan=1>W→A</td><td rowspan=1 colspan=1>average</td></tr><tr><td rowspan=1 colspan=1>AlexNet</td><td rowspan=1 colspan=1>.616</td><td rowspan=1 colspan=1>.954</td><td rowspan=1 colspan=1>.990</td><td rowspan=1 colspan=1>.638</td><td rowspan=1 colspan=1>.511</td><td rowspan=1 colspan=1>.498</td><td rowspan=1 colspan=1>.701</td></tr><tr><td rowspan=1 colspan=1>DLID</td><td rowspan=1 colspan=1>.519</td><td rowspan=1 colspan=1>.782</td><td rowspan=1 colspan=1>.899</td><td rowspan=1 colspan=1>■</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=1 colspan=1>DDC</td><td rowspan=1 colspan=1>.618</td><td rowspan=1 colspan=1>.950</td><td rowspan=1 colspan=1>.985</td><td rowspan=1 colspan=1>.644</td><td rowspan=1 colspan=1>.521</td><td rowspan=1 colspan=1>.522</td><td rowspan=1 colspan=1>.707</td></tr><tr><td rowspan=1 colspan=1>Deep CORAL</td><td rowspan=1 colspan=1>.664</td><td rowspan=1 colspan=1>.957</td><td rowspan=1 colspan=1>.992</td><td rowspan=1 colspan=1>.668</td><td rowspan=1 colspan=1>.528</td><td rowspan=1 colspan=1>.515</td><td rowspan=1 colspan=1>.721</td></tr><tr><td rowspan=1 colspan=1>DAN</td><td rowspan=1 colspan=1>.685</td><td rowspan=1 colspan=1>.960</td><td rowspan=1 colspan=1>.990</td><td rowspan=1 colspan=1>.670</td><td rowspan=1 colspan=1>.540</td><td rowspan=1 colspan=1>.531</td><td rowspan=1 colspan=1>.729</td></tr><tr><td rowspan=1 colspan=1>DANN</td><td rowspan=1 colspan=1>.730</td><td rowspan=1 colspan=1>.964</td><td rowspan=1 colspan=1>.992</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=1 colspan=1>InceptionBN</td><td rowspan=1 colspan=1>.703</td><td rowspan=1 colspan=1>.943</td><td rowspan=1 colspan=1>1.00</td><td rowspan=1 colspan=1>.705</td><td rowspan=1 colspan=1>.601</td><td rowspan=1 colspan=1>.579</td><td rowspan=1 colspan=1>.755</td></tr><tr><td rowspan=1 colspan=1>LSSA</td><td rowspan=1 colspan=1>.677</td><td rowspan=1 colspan=1>.961</td><td rowspan=1 colspan=1>.984</td><td rowspan=1 colspan=1>.713</td><td rowspan=1 colspan=1>.578</td><td rowspan=1 colspan=1>.578</td><td rowspan=1 colspan=1>.749</td></tr><tr><td rowspan=1 colspan=1>CORAL</td><td rowspan=1 colspan=1>.709</td><td rowspan=1 colspan=1>.957</td><td rowspan=1 colspan=1>.998</td><td rowspan=1 colspan=1>.719</td><td rowspan=1 colspan=1>.590</td><td rowspan=1 colspan=1>.602</td><td rowspan=1 colspan=1>.763</td></tr><tr><td rowspan=1 colspan=1>AdaBN</td><td rowspan=1 colspan=1>.742</td><td rowspan=1 colspan=1>.957</td><td rowspan=1 colspan=1>.998</td><td rowspan=1 colspan=1>.731</td><td rowspan=1 colspan=1>.598</td><td rowspan=1 colspan=1>.574</td><td rowspan=1 colspan=1>.767</td></tr><tr><td rowspan=1 colspan=1>VGG16</td><td rowspan=1 colspan=1>.676 ± .006</td><td rowspan=1 colspan=1>.961 ± .003</td><td rowspan=1 colspan=1>.992 ± .002</td><td rowspan=1 colspan=1>.739 ± .009</td><td rowspan=1 colspan=1>.582 ± .005</td><td rowspan=1 colspan=1>.578 ± .004</td><td rowspan=1 colspan=1>.755</td></tr><tr><td rowspan=1 colspan=1>CMD</td><td rowspan=1 colspan=1>.770± .006</td><td rowspan=1 colspan=1>.963 ± .004</td><td rowspan=1 colspan=1>.992 ± .002</td><td rowspan=1 colspan=1>.796 ± .006</td><td rowspan=1 colspan=1>.638 ± .007</td><td rowspan=1 colspan=1>.633 ± .006</td><td rowspan=1 colspan=1>.799</td></tr></table>
+
+The same experimental procedure is performed with MMD regularization wighted by $\lambda \in [ 5 , 4 5 ]$ and Gaussian kernel parameter $\beta \in [ 0 . 3 , 1 . 7 ]$ . We calculate the ratio values w. r. t. the accuracy of $\beta = 1 . 2$ , since this value of $\beta$ shows the highest mean accuracy of all tasks. Fig. 3 (upper right) shows the results. It can be seen that the accuracy of the MMD network is more sensitive to parameter changes than the CMD regularized version. Note that the problem of finding the best settings for the parameter $\beta$ of the Gaussian kernel is a well known problem (Hsu et al., 2003).
+
+The default number of hidden nodes in all our experiments is 256 because of the high classification accuracy of the networks without domain regularization (Source Only) on the source domains. The question arises if the accuracy of the CMD is lower for higher numbers of hidden nodes. That is, if the accuracy ratio between the accuracy, of the CMD regularized networks compared to the accuracy of the Source Only models, decreases with increasing hidden activation dimension. In order to answer this question we calculate these ratio values for each task of the Amazon reviews data set for different number of hidden nodes $( 1 2 8 , 2 5 6 , 3 8 4 , \dots , 1 6 6 4 )$ . For higher numbers of hidden nodes our Source Only models don’t converge with the optimization settings under consideration. For the parameters $\lambda$ and $K$ we use our default setting $\lambda = 1$ and $K = 5$ . Fig. 3 on the lower left shows the ratio values (vertical axis) for every number of hidden nodes (horizontal axis) and every task (colored lines). It can be seen that the accuracy improvement of the CMD domain regularizer varies between $4 \%$ and $6 \%$ . However, no accuracy ratio decrease can be observed.
+
+Please note that we use a default setting for $K$ and $\lambda$ . Thus, fig. 3 shows that our default setting $( \lambda = 1 , K = 5 )$ ) can be used independently of the number of hidden nodes. This is an additional result.
+
+The same procedure is performed with the MMD weighted by parameter $\lambda = 9$ and $\beta = 1 . 2$ as these values show the highest classification accuracy for 256 hidden nodes. Fig. 3 on the lower right shows that the accuracy improvement using the MMD decreases with increasing number of hidden nodes for this parameter setting. That is, for accurate performance of the MMD, additional parameter tuning procedures for $\lambda$ and $\beta$ need to be performed. Note that the problem of finding the best setting for the parameter $\beta$ of the Gaussian kernel is a well known problem (Hsu et al., 2003).
+
+# 6 CONCLUSION AND OUTLOOK
+
+In this paper we proposed the central moment discrepancy (CMD) for domain-invariant representation learning, a distance function between probability distributions. Similar to other state-of-the-art approaches (MMD, KL-divergence, Proxy $\mathcal { A }$ -distance), the CMD function can be used to minimize the domain discrepancy of latent feature representations. This is achieved by order-wise differences of central moments. By using probability theoretic analysis, we proved that CMD is a metric and that convergence in CMD implies convergence in distribution for probability distributions on compact intervals. Our method yields state-of-the-art performance on most tasks of the Office benchmark data set and outperforms Gaussian kernel based MMD, VFAE and DANN on most tasks of the Amazon reviews benchmark data set. These results are achieved with the default parameter setting of $K = 5$ . In addition, we experimentally underpinned the claim that the classification accuracy is not sensitive to the particular choice of $K$ for $K \geq 3$ . Therefore, no computationally expensive hyper-parameter selection is required.
+
+![](images/148b54999e298debe226db502ce3049a16ec505493a70cbdc28e5e3458045499.jpg)  
+Figure 3: Sensitivity of classification accuracy w. r. t. different parameters of CMD (left) and MMD (right) on the Amazon reviews dataset. The horizontal axes show parameter values and the vertical axes show accuracy ratio values. Each line represents accuracy ratio values for one specific task. The ratio values are computed w. r. t. the default accuracy for CMD (upper left), w. r. t. the best obtainable accuracy for MMD (upper right) and w. r. t. the non domain regularized network accuracies (lower left and lower right).
+
+In our experimental analysis we compared our approach to different other state-of-the-art distribution matching methods like the Maximum Mean Discrepancy (MMD) based on the Gaussian kernel using a quadratic time estimate. In the future we want to extend our experimental analysis to other MMD approaches including other kernels, parameter selection procedures and linear time estimators. In addition, we plan to use the CMD for training generative models and to further investigate the approximation quality of the proposed empirical estimate.
+
+# A THEOREM PROOFS
+
+Theorem 1. Let $p$ and $q$ be two probability distributions on a compact interval and let CMD be defined as in (5), then
+
+$$
+C M D ( p , q ) = 0 \Rightarrow p = q
+$$
+
+Proof. Let $X$ and $Y$ be two random vectors that have probability distributions $p$ and $q$ , respectively. Let ${ \hat { X } } = X - \mathbb { E } ( X )$ and $\hat { Y } = Y - \mathbb { E } ( Y )$ be the mean centered random variables. From $\mathbf { C M D } ( p , q ) = 0$ it follows that all moments of the bounded random variables $\hat { X }$ and $\hat { Y }$ are equal. Therefore, the joint moment generating functions of $\hat { X }$ and $\hat { Y }$ are equal. Using the property that $p$ and $q$ have compact support, we obtain the equality of the joint distribution functions of $\hat { X }$ and $\hat { Y }$ . Since $\mathbb { E } ( X ) = \mathbb { E } ( Y )$ , it follows that $X = Y$ . □
+
+Theorem 2. Let $p _ { n }$ and $p$ be probability distributions on a compact interval and let CMD be defined as in (5), then
+
+$$
+C M D ( p _ { n } , p ) \to 0 \Rightarrow p _ { n } \stackrel { d } { \to } p
+$$
+
+where $\xrightarrow { d }$ denotes convergence in distribution.
+
+Proof. Let $X _ { n }$ and $X$ be random vectors that have probability distributions $p _ { n }$ and $p$ respectively. Let ${ \hat { X } } = X - \mathbb { E } ( X )$ and ${ \hat { X } } _ { n } \ = \ X _ { n } - \mathbb { E } ( X _ { n } )$ be the mean centered random variables. From $\mathrm { C M D } ( X _ { n } , X ) \to 0$ it follows that the moments of $\hat { X } _ { n }$ converge to the moments of $\hat { X }$ . Therefore, the joint moment generating functions of $\hat { X } _ { n }$ converge to the joint moment generating function of $\hat { X }$ , which implies convergence in distribution of the mean centered random variables. Using $\mathbb { E } ( X _ { n } )  \mathbb { E } ( X )$ we obtain $p _ { n } \stackrel { d } { \to } p$ . □
+
+Proposition 1. Let $X$ and $Y$ be bounded random vectors with respective probability distributions $p$ and $q$ on the compact interval $[ a , b ] ^ { N }$ . Then
+
+$$
+\frac { 1 } { | b - a | ^ { k } } \| c _ { k } ( X ) - c _ { k } ( Y ) \| _ { 2 } \leq 2 \sqrt { N } \left( \frac { 1 } { k + 1 } \left( \frac { k } { k + 1 } \right) ^ { k } + \frac { 1 } { 2 ^ { 1 + k } } \right)
+$$
+
+where $c _ { k } ( X ) \ : = \ : \mathbb { E } ( ( X - \mathbb { E } ( X ) ) ^ { k } )$ is the vector of all $k ^ { t h }$ order sample central moments of the marginal distributions of $p$ .
+
+Proof. Let $\textstyle { \mathcal { X } } ( [ a , b ] )$ be the set of all random variables with values in $[ a , b ]$ . Then it follows that
+
+$$
+\begin{array} { r l } { \frac { 1 } { \left. \hat { b } - a \right. ^ { k } } \left. \epsilon _ { k } ( X ) - c _ { k } ( Y ) \right. _ { 2 } = \left. \frac { c _ { k } ( X ) } { \left. \hat { b } - a \right. ^ { k } } - \frac { c _ { k } ( Y ) } { \left. \hat { b } - a \right. ^ { k } } \right. _ { 2 } } & { } \\ { \leq \left. \frac { c _ { k } ( X ) } { \left. \hat { b } - a \right. ^ { k } } \right. _ { 2 } + \left. \frac { c _ { k } ( Y ) } { \left. \hat { b } - a \right. ^ { k } } \right. _ { 2 } } & { } \\ { = \left. \mathbb { E } \left( \left( \frac { X - \mathbb { E } ( X ) } { \left. b - a \right. } \right) ^ { k } \right) \right. _ { 2 } + \left. \mathbb { E } \left( \left( \frac { Y - \mathbb { E } ( Y ) } { \left. b - a \right. } \right) ^ { k } \right) \right. _ { 2 } } & { } \\ { \leq \left. \mathbb { E } \left( \left. \frac { X - \mathbb { E } ( X ) } { b - a } \right. ^ { k } \right) \right. _ { 2 } + \left. \mathbb { E } \left( \left. \frac { Y - \mathbb { E } ( Y ) } { b - a } \right. ^ { k } \right) \right. _ { 2 } } & { } \\ { \leq 2 \sqrt { N } \underset { X \in \mathcal { X } ( \{ a , b \} ) } { \overset { \mathrm { u p } } { \sum } } \mathbb { E } \left( \left. \frac { X - \mathbb { E } ( X ) } { b - a } \right. ^ { k } \right) } & { } \end{array}
+$$
+
+The latter term refers to the absolute central moment of order $k$ , for which the smallest upper bound is known (Egozcue et al., 2012):
+
+$$
+{ \frac { 1 } { | b - a | ^ { k } } } \left\| c _ { k } ( X ) - c _ { k } ( Y ) \right\| _ { 2 } \leq 2 { \sqrt { N } } \operatorname* { s u p } _ { x \in [ 0 , 1 ] } x ( 1 - x ) ^ { k } + ( 1 - x ) x ^ { k }
+$$
+
+Egozcue et al. (2012) also give a more explicit bound:
+
+$$
+\frac { 1 } { | b - a | ^ { k } } \left\| c _ { k } ( X ) - c _ { k } ( Y ) \right\| _ { 2 } \leq 2 \sqrt { N } \left( \frac { 1 } { k + 1 } \left( \frac { k } { k + 1 } \right) ^ { k } + \frac { 1 } { 2 ^ { 1 + k } } \right)
+$$
+
+# ACKNOWLEDGEMENTS
+
+The research reported in this paper has been supported by the Austrian Ministry for Transport, Innovation and Technology, the Federal Ministry of Science, Research and Economy, and the Province of Upper Austria in the frame of the COMET center SCCH.
+
+We would like to thank Bernhard Moser and Florian Sobieczky for fruitful discussions on metric spaces.
+
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+Fuzhen Zhuang, Xiaohu Cheng, Ping Luo, Sinno Jialin Pan, and Qing He. Supervised representation learning: Transfer learning with deep autoencoders. In International Joint Conference on Artificial Intelligence, 2015.

parse/train/SkB-_mcel/SkB-_mcel_content_list.json

@@ -0,0 +1,1797 @@
+[
+    {
+        "type": "text",
+        "text": "CENTRAL MOMENT DISCREPANCY (CMD) FOR DOMAIN-INVARIANT REPRESENTATION LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Werner Zellinger, Edwin Lughofer & Susanne Saminger-Platz∗ ",
+        "bbox": [
+            184,
+            170,
+            624,
+            185
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Department of Knowledge-Based Mathematical Systems   \nJohannes Kepler University Linz, Austria   \n{werner.zellinger, edwin.lughofer, susanne.saminger-platz}@jku.at ",
+        "bbox": [
+            184,
+            185,
+            813,
+            226
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Thomas Grubinger & Thomas Natschlager ¨ † ",
+        "bbox": [
+            186,
+            247,
+            488,
+            262
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Data Analysis Systems Software Competence Center Hagenberg, Austria {thomas.grubinger, thomas.natschlaeger}@scch.at ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            340,
+            544,
+            354
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The learning of domain-invariant representations in the context of domain adaptation with neural networks is considered. We propose a new regularization method that minimizes the domain-specific latent feature representations directly in the hidden activation space. Although some standard distribution matching approaches exist that can be interpreted as the matching of weighted sums of moments, e.g. Maximum Mean Discrepancy, an explicit order-wise matching of higher order moments has not been considered before. We propose to match the higher order central moments of probability distributions by means of order-wise moment differences. Our model does not require computationally expensive distance and kernel matrix computations. We utilize the equivalent representation of probability distributions by moment sequences to define a new distance function, called Central Moment Discrepancy (CMD). We prove that CMD is a metric on the set of probability distributions on a compact interval. We further prove that convergence of probability distributions on compact intervals w. r. t. the new metric implies convergence in distribution of the respective random variables. We test our approach on two different benchmark data sets for object recognition (Office) and sentiment analysis of product reviews (Amazon reviews). CMD achieves a new state-of-the-art performance on most domain adaptation tasks of Office and outperforms networks trained with Maximum Mean Discrepancy, Variational Fair Autoencoders and Domain Adversarial Neural Networks on Amazon reviews. In addition, a post-hoc parameter sensitivity analysis shows that the new approach is stable w. r. t. parameter changes in a certain interval. The source code of the experiments is publicly available1. ",
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The collection and preprocessing of large amounts of data for new domains is often time consuming and expensive. This in turn limits the application of state-of-the-art methods like deep neural network architectures, that require large amounts of data. However, often data from related domains can be used to improve the prediction model in the new domain. This paper addresses the particularly important and challenging domain-invariant representation learning task of unsupervised domain adaptation (Glorot et al., 2011; Li et al., 2014; Pan et al., 2011; Ganin et al., 2016). In unsupervised domain adaptation, the training data consists of labeled data from the source domain(s) and unlabeled data from the target domain. In practice, this setting is quite common, as in many applications the collection of input data is cheap, but the collection of labels is expensive. Typical examples include image analysis tasks and sentiment analysis, where labels have to be collected manually. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Recent research shows that domain adaptation approaches work particularly well with (deep) neural networks, which produce outstanding results on some domain adaptation data sets (Ganin et al., 2016; Sun & Saenko, 2016; Li et al., 2016; Aljundi et al., 2015; Long et al., 2015; Li et al., 2015; Zhuang et al., 2015; Louizos et al., 2016). The most successful methods have in common that they encourage similarity between the latent network representations w. r. t. the different domains. This similarity is often enforced by minimizing a certain distance between the networks’ domainspecific hidden activations. Three outstanding approaches for the choice of the distance function are the Proxy $\\mathcal { A }$ -distance (Ben-David et al., 2010), the Kullback-Leibler (KL) divergence Kullback & Leibler (1951), applied to the mean of the activations (Zhuang et al., 2015), and the Maximum Mean Discrepancy (Gretton et al., 2006, MMD). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Two of them, the MMD and the KL-divergence approach, can be viewed as the matching of statistical moments. The KL-divergence approach is based on mean (first raw moment) matching. Using the Taylor expansion of the Gaussian kernel, most MMD-based approaches can be viewed as minimizing a certain distance between weighted sums of all raw moments (Li et al., 2015). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The interpretation of the KL-divergence approaches and MMD-based approaches as moment matching procedures motivate us to match the higher order moments of the domain-specific activation distributions directly in the hidden activation space. The matching of the higher order moments is performed explicitly for each moment order and each hidden coordinate. Compared to KL-divergencebased approaches, which only match the first moment, our approach also matches higher order moments. In comparison to MMD-based approaches, our method explicitly matches the moments for each order, and it does not require any computationally expensive distance- and kernel matrix computations. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The proposed distribution matching method induces a metric between probability distributions. This is possible since distributions on compact intervals have an equivalent representation by means of their moment sequences. We utilize central moments due to their translation invariance and natural geometric interpretation. We call the new metric Central Moment Discrepancy (CMD). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The contributions of this paper are as follows: ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• We propose to match the domain-specific hidden representations by explicitly minimizing differences of higher order central moments for each moment order. We utilize the equivalent representation of probability distributions by moment sequences to define a new distance function, which we call Central Moment Discrepancy (CMD). Probability theoretic analysis is used to prove that CMD is a metric on the set of probability distributions on a compact interval. We additionally prove that convergence of probability distributions on compact intervals w. r. t. to the new metric implies convergence in distribution of the respective random variables. This means that minimizing the CMD metric between probability distributions leads to convergence of the cumulative distribution functions of the random variables.   \n• In contrast to MMD-based approaches our method does not require computationally expensive kernel matrix computations. We achieve a new state-of-the-art performance on most domain adaptation tasks of Office and outperform networks trained with MMD, variational fair autoencoders and domain adversarial neural networks on Amazon reviews.   \n• A parameter sensitivity analysis shows that CMD is insensitive to parameter changes within a certain interval. Consequently, no additional hyper-parameter search has to be performed. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 HIDDEN ACTIVATION MATCHING ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We consider the unsupervised domain adaptation setting (Glorot et al., 2011; Li et al., 2014; Pan et al., 2011; Ganin et al., 2016) with an input space $\\mathcal { X }$ and a label space $\\mathcal { V }$ . Two distributions over $\\mathcal { X } \\times \\mathcal { V }$ are given: the labeled source domain $D _ { S }$ and the unlabeled target domain $D _ { T }$ . Two corresponding samples are given: the source sample $S = ( X _ { S } , Y _ { S } ) = \\{ ( x _ { i } , y _ { i } ) \\} _ { i = 1 } ^ { n } \\stackrel { \\mathrm { i . i . d . } } { \\sim } ( D _ { S } ) ^ { n }$ and the target sample $T = X _ { T } = \\{ x _ { i } \\} _ { i = 1 } ^ { m } \\stackrel { \\mathrm { i . i . d . } } { \\sim } ( D _ { T } ) ^ { m }$ . The goal of the unsupervised domain adaptation setting is to build a classifier $f : \\mathcal { X }  \\mathcal { Y }$ with a low target risk $R _ { T } ( \\bar { f } ) = \\operatorname* { P r } _ { ( x , y ) \\sim D _ { T } } ( f ( x ) \\bar { \\neq } y )$ , while no information about the labels in $D _ { T }$ is given. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/d949f49b236dc8cf762beb46551e710cdcadbc0def130f064a8a3079eb35f3e0.jpg",
+        "image_caption": [
+            "Figure 1: Schematic sketch of a three layer neural network trained with backpropagation based on objective (2). $\\nabla _ { \\theta }$ refers to the gradient w. r. t. $\\theta$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
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+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We focus our studies on neural network classifiers $f _ { \\theta } : \\mathcal { X }  \\mathcal { Y }$ with parameters $\\theta \\in \\Theta$ , the input space $\\mathcal { X } ~ = ~ \\mathbb { R } ^ { I }$ with input dimension $I$ , and the label space $\\mathcal { Y } ~ = ~ [ 0 , 1 ] ^ { | C | }$ with the cardinality $| C |$ of the set of classes $C$ . We further assume a network output $f _ { \\theta } ( x ) \\in [ 0 , 1 ] ^ { | C | }$ of an example $x \\in \\mathbb { R } ^ { I }$ to be normalized by the softmax-function $\\sigma : \\mathbb { R } ^ { | C | }  [ 0 , 1 ] ^ { | C | }$ with $\\begin{array} { r } { \\sigma ( \\dot { z } ) _ { j } = \\frac { e ^ { z _ { j } } } { \\sum _ { k = 1 } ^ { | C | } e ^ { z _ { k } } } } \\end{array}$ for $z = \\{ z _ { 1 } , \\ldots , z _ { | C | } \\}$ . We focus on bounded activation functions $g _ { H } : \\mathbb { R } \\to [ a , b ] ^ { N }$ for the hidden layer $H$ with $N$ hidden nodes, e.g. the hyperbolic tangent or the sigmoid function. Unbounded activation functions, e.g. rectified linear units or exponential linear units, can be used if the output is clipped or normalized to be bounded. Using the loss function $l : \\Theta \\times \\mathcal { X } \\times \\mathcal { Y }  \\mathbb { R }$ , e.g. cross-entropy $\\begin{array} { r } { l } { ( \\theta , x , y ) = - \\sum _ { i \\in C } y _ { i } \\log ( f _ { \\theta } ( x ) _ { i } ) } \\end{array}$ , and the sample set $( X , Y ) \\subset \\mathbb { R } ^ { I } \\times [ 0 , 1 ] ^ { | C | }$ , we define the objective function as ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/924c2513d7142168627f490eb6dc2636c8768b0fb121c2334fa60e09f27b3737.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\theta \\in \\Theta } \\mathbf { E } ( l ( \\theta , X , Y ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            429,
+            556,
+            563,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbf { E }$ denotes the empirical expectation, i.e. ${ \\bf E } ( l ( \\theta , X , Y ) ) = { \\textstyle { \\frac { 1 } { | ( X , Y ) | } } } \\sum _ { ( x , y ) \\in ( X , Y ) } l ( \\theta , x , y )$ . Let us denote the source hidden activations by $A _ { H } ( \\theta , X _ { S } ) = g _ { H } ( \\theta _ { H } ^ { T } A _ { H ^ { \\prime } } ( \\theta , X _ { S } ) ) \\subset [ a , b ] ^ { N }$ and the target hidden activations by $A _ { H } ( \\theta , X _ { T } ) = g _ { H } ( \\theta _ { H } ^ { T } A _ { H ^ { \\prime } } ( \\theta , X _ { T } ) ) \\subset [ a , b ] ^ { N }$ for the hidden layer $H$ with $N$ hidden nodes and parameter $\\theta _ { H }$ , and the hidden layer $H ^ { \\prime }$ before $H$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "One fundamental assumption of most unsupervised domain adaptation networks is that the source risk $R _ { S } ( f )$ is a good indicator for the target risk $R _ { T } ( f )$ , when the domain-specific latent space representations are similar (Ganin et al., 2016). This similarity can be enforced by matching the distributions of the hidden activations $A _ { H } ( \\theta , X _ { S } )$ and $A _ { H } ( \\theta , X _ { T } )$ of higher layers $H$ . Recent stateof-the-art approaches define a domain regularizer $d : ( [ a , b ] ^ { N } ) ^ { \\acute { n } } \\times ( [ \\bar { a } , b ] ^ { N } ) ^ { \\acute { m } }  [ 0 , \\infty )$ , which gives a measure for the domain discrepancy in the activation space $[ a , b ] ^ { N }$ . The domain regularizer is added to the objective by means of an additional weighting parameter $\\lambda$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b1dfa7693843aef8ae529ee0bdaa996e475a2f9220054e9d56f288fb3686e105.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\underset { \\theta \\in \\Theta } { \\operatorname* { m i n } } } & { { } \\mathbf { E } ( l ( \\theta , X _ { S } , Y _ { S } ) ) + \\lambda \\cdot d ( A _ { H } ( \\theta , X _ { S } ) , A _ { H } ( \\theta , X _ { T } ) ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Fig. 1 shows a sketch of the described architecture and fig. 2 shows the hidden activations of a simple neural network optimized by eq. (1) (left) and eq. (2) (right). It can be seen that similar activation distributions are obtained when being optimized on the basis of the domain regularized objective. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Recently, several measures $d$ for objective (2) have been proposed. One approach is the Proxy $\\mathcal { A }$ - distance, given by $\\hat { d } _ { A } = 2 ( 1 - 2 \\epsilon )$ , where $\\epsilon$ is the generalization error on the problem of discriminating between source and target samples (Ben-David et al., 2010). Ganin et al. (2016) compute the value $\\epsilon$ with a neural network classifier that is simultaneously trained with the original network by means of a gradient reversal layer. They call their approach domain-adversarial neural networks. Unfortunately, a new classifier has to be trained in this approach including the need of new parameters, additional computation times and validation procedures. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/8b400c4378773bbe409c62ea1b808deadde1a9b1e7c140f6e543b12d0797353f.jpg",
+        "image_caption": [
+            "Figure 2: Hidden activation distributions for a simple one-layer classification network with sigmoid activation functions and five hidden nodes trained with the standard objective (1) (left) and objective (2) that includes the domain discrepancy minimization (right). The approach of this paper was used as domain regularizer. Dark gray: activations of the source domain, light gray: activations of the target domain. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Another approach is to make use of the MMD (Gretton et al., 2006) as domain regularizer. ",
+        "bbox": [
+            173,
+            361,
+            766,
+            377
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\begin{array} { r } { \\mathbf { E } ( K ( X , Y ) ) = \\frac { 1 } { | X | \\cdot | Y | } \\sum _ { k \\in K ( X , Y ) } k } \\end{array}$ is the empirical expectation of the kernel products $k$ between all examples in $X$ and $Y$ stored by the kernel matrix $K ( X , Y )$ . A suitable choice of the kernel seems to be the Gaussian kernel $e ^ { - \\beta \\| x - y \\| ^ { 2 } }$ (Louizos et al., 2016; Li et al., 2015; Tzeng et al., 2014). This approach has two major drawbacks: (a) the need of tuning an additional kernel parameter $\\beta$ , and (b) the need of the kernel matrix computation $K ( X , Y )$ (computational complexity $\\mathcal { O } ( n ^ { 2 } +$ $n m + m ^ { 2 } )$ ), which becomes inefficient (resource-intensive) in case of large data sets. Concerning (a), the tuning of $\\beta$ is sophisticated since no target samples are available in the domain adaptation setting. Suitable tuning procedures are transfer learning specific cross-validation methods (Zhong et al., 2010). More general methods that don’t utilize source labels include heuristics that are based on kernel space properties (Sriperumbudur et al., 2009; Gretton et al., 2012), combinations of multiple kernels (Li et al., 2015), and kernel choices that maximize the MMD test power (Sutherland et al., 2016). The drawback (b) of the kernel matrix computation can be handled by approximating the MMD (Zhao & Meng, 2015), or by using linear time estimators (Gretton et al., 2012). In this work we focus on the quadratic-time MMD with the Gaussian kernel (Gretton et al., 2012; Tzeng et al., 2014) and transfer learning specific cross-validation for parameter tuning (Zhong et al., 2010; Ganin et al., 2016). ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The two approaches MMD and the Proxy $\\mathcal { A }$ -distance have in common that they do not minimize the domain discrepancy explicitly in the hidden activation space. In contrast, the authors in Zhuang et al. (2015) do so by minimizing a modified version of the Kullback-Leibler divergence of the mean activations (MKL). That is, for samples $X , Y \\subset \\mathbb { R } ^ { N }$ , ",
+        "bbox": [
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+        "page_idx": 3
+    },
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+        "type": "equation",
+        "img_path": "images/6ffe21e012178da1ae5f5f2bfb5f71dc683cef2dca6ad5e45905258e07c0a3b8.jpg",
+        "text": "$$\n\\operatorname { M K L } ( X , Y ) = \\sum _ { i = 1 } ^ { N } \\mathbf { E } ( X ) _ { i } \\log { \\frac { \\mathbf { E } ( X ) _ { i } } { \\mathbf { E } ( Y ) _ { i } } } + \\mathbf { E } ( Y ) _ { i } \\log { \\frac { \\mathbf { E } ( Y ) _ { i } } { \\mathbf { E } ( X ) _ { i } } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            300,
+            693,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "with $\\mathbf { E } ( X ) _ { i }$ being the $i ^ { \\mathrm { { t h } } }$ coordinate of the empirical expectation $\\begin{array} { r } { \\mathbf { E } ( X ) = \\frac { 1 } { | X | } \\sum _ { x \\in X } x } \\end{array}$ . This approach is fast to compute and has an explicit interpretation in the activation space. Our empirical observations (section Experiments) show that minimizing the distance between only the first moment (mean) of the activation distributions can be improved by also minimizing the distance between higher order moments. ",
+        "bbox": [
+            174,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As noted in the introduction, our approach is motivated by the fact that the MMD and the KLdivergence approach can be seen as the matching of statistical moments of the hidden activations $A _ { H } ( \\bar { \\theta } , X _ { S } )$ and $A _ { H } ( \\theta , X _ { T } )$ . In particular, MMD-based approaches that use the Gaussian kernel are equivalent to minimizing a certain distance between weighted sums of all moments of the hidden activation distributions (Li et al., 2015). ",
+        "bbox": [
+            173,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We propose to minimize differences of higher order central moments of the activations $A _ { H } ( \\theta , X _ { S } )$ and $\\bar { \\bf A } _ { H } ^ { - } ( \\theta , X _ { T } )$ . The difference minimization is performed explicitly for each moment order. Our approach utilizes the equivalent representation of probability distributions in terms of its moment series. We further utilize central moments due to their translation invariance and natural geometric interpretation. Our approach contrasts with other moment-based approaches, as they either match only the first moment (MKL) or they don’t explicitly match the moments for each order (MMD). As a result, our approach improves over MMD-based approaches in terms of computational complexity with $\\mathcal { O } \\left( N ( n \\overline { { + } } m ) \\right)$ for CMD and $\\mathcal { O } \\left( N ( n ^ { 2 } + n m + m ^ { 2 } ) \\right)$ for MMD. In contrast to MKL-based approaches more accurate distribution matching characteristics are obtained. In addition, CMD achieves a new state-of-the-art performance on most domain adaptation tasks of Office and outperforms networks trained with MMD, variational fair autoencoders and domain adversarial neural networks on Amazon reviews. ",
+        "bbox": [
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+        "type": "text",
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "4 CENTRAL MOMENT DISCREPANCY (CMD) ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "In this section we first propose a new distance function CMD on probability distributions on compact intervals. The definition is extended by two theorems that identify CMD as a metric and analyze a convergence property. The final domain regularizer is then defined as an empirical estimate of CMD. The proofs of the theorems are given in the appendix. ",
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+        "page_idx": 4
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+        "type": "text",
+        "text": "Definition 1 (CMD metric). Let $X = ( X _ { 1 } , \\ldots , X _ { n } ) $ and $Y = ( Y _ { 1 } , \\ldots , Y _ { n } )$ be bounded random vectors independent and identically distributed from two probability distributions $p$ and $q$ on the compact interval $[ a , b ] ^ { N }$ . The central moment discrepancy metric (CMD) is defined by ",
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+        "text": "$$\nC M D ( p , q ) = \\frac { 1 } { \\left| b - a \\right| } \\left\\| \\mathbb { E } ( X ) - \\mathbb { E } ( Y ) \\right\\| _ { 2 } + \\sum _ { k = 2 } ^ { \\infty } \\frac { 1 } { \\left| b - a \\right| ^ { k } } \\left\\| c _ { k } ( X ) - c _ { k } ( Y ) \\right\\| _ { 2 }\n$$",
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+        "type": "text",
+        "text": "where $\\mathbb { E } ( X )$ is the expectation of $X$ , and ",
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+        "type": "equation",
+        "img_path": "images/8494f559263fe8d687c834833d57649f58722ba2d55c25c5b4090519e2f5bf25.jpg",
+        "text": "$$\nc _ { k } ( \\boldsymbol { X } ) = \\left( \\mathbb { E } \\Big ( \\prod _ { i = 1 } ^ { N } \\left( \\boldsymbol { X } _ { i } - \\mathbb { E } ( \\boldsymbol { X } _ { i } ) \\right) ^ { r _ { i } } \\Big ) \\right) _ { r _ { 1 } + \\ldots + r _ { N } = k }\n$$",
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+    {
+        "type": "text",
+        "text": "is the central moment vector of order $k$ . ",
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+    {
+        "type": "text",
+        "text": "The first order central moments are zero, the second order central moments are related to variance, and the third and fourth order central moments are related to the skewness and the kurtosis of probability distributions. It is easy to see that ${ \\bf C M D } ( p , q ) \\ge 0$ , $\\mathrm { C M D } ( p , q ) = \\mathrm { C M D } ( q , p )$ , $\\mathrm { C M D } ( p , q ) \\leq \\mathrm { C M D } ( p , r ) + \\mathrm { C M D } ( r , q )$ and $p = q \\Rightarrow { \\bf C M D } ( p , q ) = 0$ . The following theorem shows the remaining property for CMD to be a metric on the set of probability distributions on a compact interval. ",
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+        "type": "text",
+        "text": "Theorem 1. Let $p$ and $q$ be two probability distributions on a compact interval and let CMD be defined as in (5), then ",
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+        "type": "equation",
+        "img_path": "images/5eb212ee82d158d888e32ec52a1fdbc23d754c4aae8665ed68692b0dc2c8127a.jpg",
+        "text": "$$\nC M D ( p , q ) = 0 \\Rightarrow p = q\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Our approach is to minimize the discrepancy between the domain-specific hidden activation distributions by minimizing the CMD. Thus, in the optimization procedure, we increasingly expect to see the domain-specific cumulative distribution functions approach each other. This characteristic can be expressed by the concept of convergence in distribution and it is shown in the following theorem. ",
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+        "type": "text",
+        "text": "Theorem 2. Let $p _ { n }$ and $p$ be probability distributions on a compact interval and let CMD be defined as in (5), then ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/e7e1accce301dc46d32a8a7d6d95c2d0e269952514a8d4791eee60d41febf4d4.jpg",
+        "text": "$$\nC M D ( p _ { n } , p ) \\to 0 \\Rightarrow p _ { n } \\stackrel { d } { \\to } p\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $\\xrightarrow { d }$ denotes convergence in distribution. ",
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+    {
+        "type": "text",
+        "text": "We define the final central moment discrepancy regularizer as an empirical estimate of the CMD metric. Only the central moments that correspond to the marginal distributions are computed. The number of central moments is limited by a new parameter $K$ and the expectation is sampled by the empirical expectation. ",
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+    {
+        "type": "text",
+        "text": "Definition 2 (CMD regularizer). Let $X$ and $Y$ be bounded random samples with respective probability distributions $p$ and $q$ on the interval $[ a , b ] ^ { N }$ . The central moment discrepancy regularizer $C M D _ { K }$ is defined as an empirical estimate of the CMD metric, by ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/e7b0919001e30ae4e74d2344cc6b9d0039382f7f0b89246b16241651b0c1009c.jpg",
+        "text": "$$\nC M D _ { K } ( X , Y ) = { \\frac { 1 } { | b - a | } } \\| \\mathbf { E } ( X ) - \\mathbf { E } ( Y ) \\| _ { 2 } + \\sum _ { k = 2 } ^ { K } { \\frac { 1 } { | b - a | ^ { k } } } \\| C _ { k } ( X ) - C _ { k } ( Y ) \\| _ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\begin{array} { r } { \\mathbf { E } ( X ) = \\frac { 1 } { | X | } \\sum _ { x \\in X } x } \\end{array}$ is the empirical expectation vector computed on the sample $X$ and $C _ { k } ( X ) = \\mathbf { E } ( ( x - \\mathbf { E } ( X ) ) ^ { k } )$ is the vector of all $k ^ { t h }$ order sample central moments of the coordinates of $X$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "This definition includes three approximation steps: (a) the computation of only marginal central moments, (b) the bound on the order of central moment terms via parameter $K$ , and (c) the sampling of the probability distributions by the replacement of the expected value with the empirical expectation. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Applying approximation (a) and assuming independent marginal distributions, a zero CMD distance value still implies equal joint distributions (thm. 1) but convergence in distribution (thm. 2) applies only to the marginals. In the case of dependent marginal distributions, zero CMD distance implies equal marginals and convergence in CMD implies convergence in distribution of the marginals. However, the matching properties for the joint distributions are not obtained with dependent marginals and approximation (a). The computational complexity is reduced to be linear w. r. t. the number of samples. ",
+        "bbox": [
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+        "type": "text",
+        "text": "Concerning (b), proposition 1 shows that the marginal distribution specific CMD terms have an upper bound that is strictly decreasing with increasing moment order. This bound is convergent to zero. That is, higher CMD terms can contribute less to the overall distance value. This observation is experimentally strengthened in subsection Parameter Sensitivity. ",
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+    {
+        "type": "text",
+        "text": "Proposition 1. Let $X$ and $Y$ be bounded random vectors with respective probability distributions $p$ and $q$ on the compact interval $[ a , b ] ^ { N }$ . Then ",
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/ea48d72771a29b1c86cd92b4a09a1eeaf6e2880c7fc2fdcf50c3cd62f8990f35.jpg",
+        "text": "$$\n\\frac { 1 } { | b - a | ^ { k } } \\| c _ { k } ( X ) - c _ { k } ( Y ) \\| _ { 2 } \\leq 2 \\sqrt { N } \\left( \\frac { 1 } { k + 1 } \\left( \\frac { k } { k + 1 } \\right) ^ { k } + \\frac { 1 } { 2 ^ { 1 + k } } \\right)\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $c _ { k } ( X ) \\ : = \\ : \\mathbb { E } ( ( X - \\mathbb { E } ( X ) ) ^ { k } )$ is the vector of all $k ^ { t h }$ order sample central moments of the marginal distributions of $p$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Concerning approximation (c), the joint application of the weak law of large numbers (Billingsley, 2008) with the continuous mapping theorem (Billingsley, 2013) proves that this approximation creates a consistent estimate. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We would like to underline that the training of neural networks with eq. (2) and the CMD regularizer in eq. (6) can be easily realized by gradient descent algorithms. The gradients of the CMD regularizer are simple aggregations of derivatives of the standard functions $g _ { H } , x ^ { k }$ and $\\lVert . \\rVert _ { 2 }$ . ",
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Our experimental evaluations are based on two benchmark datasets for domain adaptation, Amazon reviews and Office, described in subsection Datasets. The experimental setup is discussed in subsection Experimental Setup and our classification accuracy results are discussed in subsection Results. Subsection Parameter Sensitivity analysis the accuracy sensitivity w. r. t. parameter changes of $K$ for CMD and $\\beta$ for MMD. ",
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+    {
+        "type": "text",
+        "text": "5.1 DATASETS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Amazon reviews: For our first experiment we use the Amazon reviews data set with the same preprocessing as used by Chen et al. (2012); Ganin et al. (2016); Louizos et al. (2016). The data set contains product reviews of four different product categories: books, DVDs, kitchen appliances and electronics. Reviews are encoded in 5000 dimensional feature vectors of bag-of-words unigrams and bigrams with binary labels: 0 if the product is ranked by $1 - 3$ stars and 1 if the product is ranked by 4 or 5 stars. From the four categories we obtain twelve domain adaptation tasks (each category serves once as source category and once as target category). ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Office: The second experiment is based on the computer vision classification data set from Saenko et al. (2010) with images from three distinct domains: amazon (A), webcam (W) and dslr (D). This data set is a de facto standard for domain adaptation algorithms in computer vision. Amazon, the largest domain, is a composition of 2817 images and its corresponding 31 classes. Following previous works we assess the performance of our method across all six possible transfer tasks. ",
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+    {
+        "type": "text",
+        "text": "5.2 EXPERIMENTAL SETUP ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Amazon Reviews: ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "For the Amazon reviews experiment, we use the same data splits as previous works for every task. Thus we have 2000 labeled source examples and 2000 unlabeled target examples for training, and between 3000 and 6000 examples for testing. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We use a similar architecture as Ganin et al. (2016) with one dense hidden layer with 50 hidden nodes, sigmoid activation functions and softmax output function. Three neural networks are trained by means of eq. (2): (a) a base model without domain regularization $\\lambda = 0$ ), (b) with the MMD as domain regularizer and (c) with CMD as domain regularizer. These models are additionally compared with the state-of-the-art models VFAE (Louizos et al., 2016) and DANN (Ganin et al., 2016). The models (a),(b) and (c) are trained with similar setup as in Louizos et al. (2016) and Ganin et al. (2016). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For the CMD regularizer, the $\\lambda$ parameter of eq. (2) is set to 1, i.e. the weighting parameter $\\lambda$ is neglected. The parameter $K$ is heuristically set to five, as the first five moments capture rich geometric information about the shape of a distribution and $K = 5$ is small enough to be computationally efficient. However, the experiments in subsection Parameter Sensitivity show that similar results are obtained for $K \\geq 3$ . ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "For the MMD regularizer we use the Gaussian kernel with parameter $\\beta$ . We performed a hyperparameter search for $\\beta$ and $\\lambda$ , which has to be performed in an unsupervised way (no labels in the target domain). We use a variant of the reverse cross-validation approach proposed by Zhong et al. (2010), in which we initialize the model weights of the reverse classifier by the weights of the first learned classifier (see Ganin et al. (2016) for details). Thereby, the parameter $\\lambda$ is tuned on 10 values between 0.1 and 500 on a logarithmic scale. The parameter $\\beta$ is tuned on 10 values between 0.01 and 10 on a logarithmic scale. Without this parameter search, no competitive prediction accuracy results could be obtained. ",
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+    {
+        "type": "text",
+        "text": "Since we have to deal with sparse data, we rely on the Adagrad optimizer (Duchi et al., 2011). For all evaluations, the default parametrization is used as implemented in Keras (Chollet, 2015). All evaluations are repeated 10 times based on different shuffles of the data, and the mean accuracies and standard deviations are analyzed. ",
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+        "type": "text",
+        "text": "Office: Since the office dataset is rather small with only 2817 images in its largest domain, we use the latent representations of the convolution neural network VGG16 of Simonyan & Zisserman (2014). In particular we train a classifier with one hidden layer, 256 hidden nodes and sigmoid activation function on top of the output of the first dense layer in the network. We again train one base model without domain regularization and a CMD regularized version with $K = 5$ and $\\lambda = 1$ . ",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "We follow the standard training protocol for this data set and use all available source and target examples during training. Using this ”fully-transductive” protocol, we compare our method with other state-of-the-art approaches including DLID (Chopra et al., 2013), DDC (Tzeng et al., 2014), DAN (Long et al., 2015), Deep CORAL (Sun & Saenko, 2016), and DANN (Ganin et al., 2016), based on fine-tuning of the baseline model AlexNet (Krizhevsky et al., 2012). We further compare our method to LSSA (Aljundi et al., 2015), CORAL (Sun et al., 2016), and AdaBN (Li et al., 2016), based on the fine-tuning of InceptionBN (Ioffe & Szegedy, 2015). ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "As an alternative to Adagrad for non-sparse data, we use the Adadelta optimizer from Zeiler (2012). Again, the default parametrization from Keras is used. We handle unbalances between source and target sample by randomly down-sampling (up-sampling) the source sample. In addition, we ensure a sub-sampled source batch that is balanced w. r. t. the class labels. ",
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "Since all hyper-parameters are set a-priori, no hyper-parameter search has to be performed. ",
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "All experiments are repeated 10 times with randomly shuffled data sets and random initializations. ",
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+    {
+        "type": "text",
+        "text": "5.3 RESULTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Amazon Reviews: Table 1 shows the classification accuracies of four models: The Source Only model is the non domain regularized neural network trained with objective (1), and serves as a base model for the domain adaptation improvements. The models MMD and CMD are trained with the same architecture and objective (2) with $d$ as the domain regularizer MMD and CMD, respectively. VFAE refers to the Variational Fair Autoencoder of Louizos et al. (2016), including a slightly modified version of the MMD regularizer for faster computations, and DANN refers to the domainadversarial neural networks model of Ganin et al. (2016). The last two columns are taken directly from these publications. ",
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+    {
+        "type": "text",
+        "text": "As one can observe in table 1, our accuracy of the CMD-based model is the highest in 9 out of 12 domain adaptation tasks, whereas on the remaining 3 it is the second best method. However, the difference in accuracy compared to the best method is smaller than the standard deviation over all data shuffles. ",
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+    {
+        "type": "table",
+        "img_path": "images/c7778f6e9f52c56d0935368a5ed28e5723a0f216cb7ce940279b481fda34cd7a.jpg",
+        "table_caption": [
+            "Table 1: Prediction accuracy $\\pm$ standard deviation on the Amazon reviews dataset. The last two columns are taken directly from Louizos et al. (2016) and Ganin et al. (2016). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Source-&gt;Target</td><td rowspan=1 colspan=1>Source Only</td><td rowspan=1 colspan=1>MMD</td><td rowspan=1 colspan=1>CMD</td><td rowspan=1 colspan=1>VFAE</td><td rowspan=1 colspan=1>DANN</td></tr><tr><td rowspan=1 colspan=1>books-→dvd</td><td rowspan=1 colspan=1>.787 ± .004</td><td rowspan=1 colspan=1>.796 ± .008</td><td rowspan=1 colspan=1>.805 ± .007</td><td rowspan=1 colspan=1>.799</td><td rowspan=1 colspan=1>.784</td></tr><tr><td rowspan=1 colspan=1>books-→electronics</td><td rowspan=1 colspan=1>.714± .009</td><td rowspan=1 colspan=1>.758 ± .018</td><td rowspan=1 colspan=1>.787 ± .007</td><td rowspan=1 colspan=1>.792</td><td rowspan=1 colspan=1>.733</td></tr><tr><td rowspan=1 colspan=1>books-→kitchen</td><td rowspan=1 colspan=1>.745 ± .006</td><td rowspan=1 colspan=1>.787 ± .019</td><td rowspan=1 colspan=1>.813 ± .008</td><td rowspan=1 colspan=1>.816</td><td rowspan=1 colspan=1>.779</td></tr><tr><td rowspan=1 colspan=1>dvd-&gt;books</td><td rowspan=1 colspan=1>.746 ± .019</td><td rowspan=1 colspan=1>.780 ± .018</td><td rowspan=1 colspan=1>.795 ± .005</td><td rowspan=1 colspan=1>.755</td><td rowspan=1 colspan=1>.723</td></tr><tr><td rowspan=1 colspan=1>dvd-&gt;electronics</td><td rowspan=1 colspan=1>.724 ± .011</td><td rowspan=1 colspan=1>.766 ± .025</td><td rowspan=1 colspan=1>.797 ± .010</td><td rowspan=1 colspan=1>.786</td><td rowspan=1 colspan=1>.754</td></tr><tr><td rowspan=1 colspan=1>dvd-&gt;kitchen</td><td rowspan=1 colspan=1>.765 ± .012</td><td rowspan=1 colspan=1>.796 ± .019</td><td rowspan=1 colspan=1>.830 ± .012</td><td rowspan=1 colspan=1>.822</td><td rowspan=1 colspan=1>.783</td></tr><tr><td rowspan=1 colspan=1> electronics-→books</td><td rowspan=1 colspan=1>.711 ± .006</td><td rowspan=1 colspan=1>.733 ± .017</td><td rowspan=1 colspan=1>.744 ± .008</td><td rowspan=1 colspan=1>.727</td><td rowspan=1 colspan=1>.713</td></tr><tr><td rowspan=1 colspan=1>electronics-→dvd</td><td rowspan=1 colspan=1>.719 ± .009</td><td rowspan=1 colspan=1>.748 ± .013</td><td rowspan=1 colspan=1>.763 ± .006</td><td rowspan=1 colspan=1>.765</td><td rowspan=1 colspan=1>.738</td></tr><tr><td rowspan=1 colspan=1>electronics-→kitchen</td><td rowspan=1 colspan=1>.844 ± .005</td><td rowspan=1 colspan=1>.857 ± .007</td><td rowspan=1 colspan=1>.860 ± .004</td><td rowspan=1 colspan=1>.850</td><td rowspan=1 colspan=1>.854</td></tr><tr><td rowspan=1 colspan=1>kitchen-→books</td><td rowspan=1 colspan=1>.699 ± .014</td><td rowspan=1 colspan=1>.740 ± .017</td><td rowspan=1 colspan=1>.756 ± .006</td><td rowspan=1 colspan=1>.720</td><td rowspan=1 colspan=1>.709</td></tr><tr><td rowspan=1 colspan=1>kitchen-&gt;dvd</td><td rowspan=1 colspan=1>.734 ± .011</td><td rowspan=1 colspan=1>.763 ± .011</td><td rowspan=1 colspan=1>.775 ± .005</td><td rowspan=1 colspan=1>.733</td><td rowspan=1 colspan=1>.740</td></tr><tr><td rowspan=1 colspan=1>kitchen-→electronics</td><td rowspan=1 colspan=1>.833 ± .004</td><td rowspan=1 colspan=1>.844 ± .007</td><td rowspan=1 colspan=1>.854 ± .003</td><td rowspan=1 colspan=1>.838</td><td rowspan=1 colspan=1>.843</td></tr><tr><td rowspan=1 colspan=1> average</td><td rowspan=1 colspan=1>.752 ± .009</td><td rowspan=1 colspan=1>.781 ± .015</td><td rowspan=1 colspan=1>.798 ± .007</td><td rowspan=1 colspan=1>.784</td><td rowspan=1 colspan=1>.763</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Office: Table 2 shows the classification accuracy of different models trained on the Office dataset. Note that some of the methods (LSSA, CORAL and AdaBN) are evaluated based on the InceptionBN model, which shows higher accuracy than the base model (VGG16) of our method in most tasks. However, our method outperforms related state-of-the-art methods on all except two tasks, on which it performs similar. We improve the previous state-of-the-art method AdaBN (Li et al., 2016) by more than $3 . 2 \\%$ in average accuracy. ",
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+    {
+        "type": "text",
+        "text": "5.4 PARAMETER SENSITIVITY ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The first sensitivity experiment aims at providing evidence regarding the accuracy sensitivity of the CMD regularizer w. r. t. parameter changes of $K$ . That is, the contribution of higher terms in the CMD regularizer are analyzed. The claim is that the accuracy of CMD-based networks does not depend strongly on the choice of $K$ in a range around its default value 5. ",
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+    {
+        "type": "text",
+        "text": "In fig. 3 on the upper left we analyze the classification accuracy of a CMD-based network trained on all tasks of the Amazon reviews experiment. We perform a grid search for the two regularization hyper-parameters $\\lambda$ and $K$ . We empirically choose a representative stable region for each parameter, [0.3, 3] for $\\lambda$ and $\\{ 1 , . . . , 7 \\}$ for $K$ . Since we want to analyze the sensitivity w. r. t. $K$ , we averaged over the $\\lambda$ -dimension, resulting in one accuracy value per $K$ for each of the 12 tasks. Each accuracy is transformed into an accuracy ratio value by dividing it with the accuracy of $K = 5$ . Thus, for each $K$ and task we get one value representing the ratio between the obtained accuracy (for this $K$ and task) and the accuracy of $K = 5$ . The results are shown in fig. 3 (upper left). The accuracy ratios between $K = 5$ and $K \\in \\{ 3 , 4 , 6 , 7 \\}$ are lower than $0 . 5 \\%$ , which underpins the claim that the accuracy of CMD-based networks does not depend strongly on the choice of $K$ in a range around its default value 5. For $K = 1$ and $K = 2$ higher ratio values are obtained. In addition, for these two values many tasks show worse accuracy than obtained by $K \\in \\{ 3 , 4 , 5 , 6 , 7 \\}$ . From this we additionally conclude that higher values of $K$ are preferable to $K = 1$ and $K = 2$ . ",
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+        "type": "table",
+        "img_path": "images/b909242f35b2ecf57dd32a326f5845543670c558447f8fdc49a7e9d05ef5c171.jpg",
+        "table_caption": [
+            "Table 2: Prediction accuracy $\\pm$ standard deviation on the Office dataset. The first 10 rows are taken directly from the papers of Ganin et al. (2016) and Li et al. (2016). The models DLID –DANN are based on the AlexNet model, LSSA –AdaBN are based on the InceptionBN model, and our method (CMD) is based on the VGG16 model. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>A→W</td><td rowspan=1 colspan=1>D→W</td><td rowspan=1 colspan=1>W→D</td><td rowspan=1 colspan=1>A→D</td><td rowspan=1 colspan=1>D→A</td><td rowspan=1 colspan=1>W→A</td><td rowspan=1 colspan=1>average</td></tr><tr><td rowspan=1 colspan=1>AlexNet</td><td rowspan=1 colspan=1>.616</td><td rowspan=1 colspan=1>.954</td><td rowspan=1 colspan=1>.990</td><td rowspan=1 colspan=1>.638</td><td rowspan=1 colspan=1>.511</td><td rowspan=1 colspan=1>.498</td><td rowspan=1 colspan=1>.701</td></tr><tr><td rowspan=1 colspan=1>DLID</td><td rowspan=1 colspan=1>.519</td><td rowspan=1 colspan=1>.782</td><td rowspan=1 colspan=1>.899</td><td rowspan=1 colspan=1>■</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=1 colspan=1>DDC</td><td rowspan=1 colspan=1>.618</td><td rowspan=1 colspan=1>.950</td><td rowspan=1 colspan=1>.985</td><td rowspan=1 colspan=1>.644</td><td rowspan=1 colspan=1>.521</td><td rowspan=1 colspan=1>.522</td><td rowspan=1 colspan=1>.707</td></tr><tr><td rowspan=1 colspan=1>Deep CORAL</td><td rowspan=1 colspan=1>.664</td><td rowspan=1 colspan=1>.957</td><td rowspan=1 colspan=1>.992</td><td rowspan=1 colspan=1>.668</td><td rowspan=1 colspan=1>.528</td><td rowspan=1 colspan=1>.515</td><td rowspan=1 colspan=1>.721</td></tr><tr><td rowspan=1 colspan=1>DAN</td><td rowspan=1 colspan=1>.685</td><td rowspan=1 colspan=1>.960</td><td rowspan=1 colspan=1>.990</td><td rowspan=1 colspan=1>.670</td><td rowspan=1 colspan=1>.540</td><td rowspan=1 colspan=1>.531</td><td rowspan=1 colspan=1>.729</td></tr><tr><td rowspan=1 colspan=1>DANN</td><td rowspan=1 colspan=1>.730</td><td rowspan=1 colspan=1>.964</td><td rowspan=1 colspan=1>.992</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td><td rowspan=1 colspan=1>-</td></tr><tr><td rowspan=1 colspan=1>InceptionBN</td><td rowspan=1 colspan=1>.703</td><td rowspan=1 colspan=1>.943</td><td rowspan=1 colspan=1>1.00</td><td rowspan=1 colspan=1>.705</td><td rowspan=1 colspan=1>.601</td><td rowspan=1 colspan=1>.579</td><td rowspan=1 colspan=1>.755</td></tr><tr><td rowspan=1 colspan=1>LSSA</td><td rowspan=1 colspan=1>.677</td><td rowspan=1 colspan=1>.961</td><td rowspan=1 colspan=1>.984</td><td rowspan=1 colspan=1>.713</td><td rowspan=1 colspan=1>.578</td><td rowspan=1 colspan=1>.578</td><td rowspan=1 colspan=1>.749</td></tr><tr><td rowspan=1 colspan=1>CORAL</td><td rowspan=1 colspan=1>.709</td><td rowspan=1 colspan=1>.957</td><td rowspan=1 colspan=1>.998</td><td rowspan=1 colspan=1>.719</td><td rowspan=1 colspan=1>.590</td><td rowspan=1 colspan=1>.602</td><td rowspan=1 colspan=1>.763</td></tr><tr><td rowspan=1 colspan=1>AdaBN</td><td rowspan=1 colspan=1>.742</td><td rowspan=1 colspan=1>.957</td><td rowspan=1 colspan=1>.998</td><td rowspan=1 colspan=1>.731</td><td rowspan=1 colspan=1>.598</td><td rowspan=1 colspan=1>.574</td><td rowspan=1 colspan=1>.767</td></tr><tr><td rowspan=1 colspan=1>VGG16</td><td rowspan=1 colspan=1>.676 ± .006</td><td rowspan=1 colspan=1>.961 ± .003</td><td rowspan=1 colspan=1>.992 ± .002</td><td rowspan=1 colspan=1>.739 ± .009</td><td rowspan=1 colspan=1>.582 ± .005</td><td rowspan=1 colspan=1>.578 ± .004</td><td rowspan=1 colspan=1>.755</td></tr><tr><td rowspan=1 colspan=1>CMD</td><td rowspan=1 colspan=1>.770± .006</td><td rowspan=1 colspan=1>.963 ± .004</td><td rowspan=1 colspan=1>.992 ± .002</td><td rowspan=1 colspan=1>.796 ± .006</td><td rowspan=1 colspan=1>.638 ± .007</td><td rowspan=1 colspan=1>.633 ± .006</td><td rowspan=1 colspan=1>.799</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The same experimental procedure is performed with MMD regularization wighted by $\\lambda \\in [ 5 , 4 5 ]$ and Gaussian kernel parameter $\\beta \\in [ 0 . 3 , 1 . 7 ]$ . We calculate the ratio values w. r. t. the accuracy of $\\beta = 1 . 2$ , since this value of $\\beta$ shows the highest mean accuracy of all tasks. Fig. 3 (upper right) shows the results. It can be seen that the accuracy of the MMD network is more sensitive to parameter changes than the CMD regularized version. Note that the problem of finding the best settings for the parameter $\\beta$ of the Gaussian kernel is a well known problem (Hsu et al., 2003). ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The default number of hidden nodes in all our experiments is 256 because of the high classification accuracy of the networks without domain regularization (Source Only) on the source domains. The question arises if the accuracy of the CMD is lower for higher numbers of hidden nodes. That is, if the accuracy ratio between the accuracy, of the CMD regularized networks compared to the accuracy of the Source Only models, decreases with increasing hidden activation dimension. In order to answer this question we calculate these ratio values for each task of the Amazon reviews data set for different number of hidden nodes $( 1 2 8 , 2 5 6 , 3 8 4 , \\dots , 1 6 6 4 )$ . For higher numbers of hidden nodes our Source Only models don’t converge with the optimization settings under consideration. For the parameters $\\lambda$ and $K$ we use our default setting $\\lambda = 1$ and $K = 5$ . Fig. 3 on the lower left shows the ratio values (vertical axis) for every number of hidden nodes (horizontal axis) and every task (colored lines). It can be seen that the accuracy improvement of the CMD domain regularizer varies between $4 \\%$ and $6 \\%$ . However, no accuracy ratio decrease can be observed. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Please note that we use a default setting for $K$ and $\\lambda$ . Thus, fig. 3 shows that our default setting $( \\lambda = 1 , K = 5 )$ ) can be used independently of the number of hidden nodes. This is an additional result. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The same procedure is performed with the MMD weighted by parameter $\\lambda = 9$ and $\\beta = 1 . 2$ as these values show the highest classification accuracy for 256 hidden nodes. Fig. 3 on the lower right shows that the accuracy improvement using the MMD decreases with increasing number of hidden nodes for this parameter setting. That is, for accurate performance of the MMD, additional parameter tuning procedures for $\\lambda$ and $\\beta$ need to be performed. Note that the problem of finding the best setting for the parameter $\\beta$ of the Gaussian kernel is a well known problem (Hsu et al., 2003). ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSION AND OUTLOOK ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this paper we proposed the central moment discrepancy (CMD) for domain-invariant representation learning, a distance function between probability distributions. Similar to other state-of-the-art approaches (MMD, KL-divergence, Proxy $\\mathcal { A }$ -distance), the CMD function can be used to minimize the domain discrepancy of latent feature representations. This is achieved by order-wise differences of central moments. By using probability theoretic analysis, we proved that CMD is a metric and that convergence in CMD implies convergence in distribution for probability distributions on compact intervals. Our method yields state-of-the-art performance on most tasks of the Office benchmark data set and outperforms Gaussian kernel based MMD, VFAE and DANN on most tasks of the Amazon reviews benchmark data set. These results are achieved with the default parameter setting of $K = 5$ . In addition, we experimentally underpinned the claim that the classification accuracy is not sensitive to the particular choice of $K$ for $K \\geq 3$ . Therefore, no computationally expensive hyper-parameter selection is required. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/148b54999e298debe226db502ce3049a16ec505493a70cbdc28e5e3458045499.jpg",
+        "image_caption": [
+            "Figure 3: Sensitivity of classification accuracy w. r. t. different parameters of CMD (left) and MMD (right) on the Amazon reviews dataset. The horizontal axes show parameter values and the vertical axes show accuracy ratio values. Each line represents accuracy ratio values for one specific task. The ratio values are computed w. r. t. the default accuracy for CMD (upper left), w. r. t. the best obtainable accuracy for MMD (upper right) and w. r. t. the non domain regularized network accuracies (lower left and lower right). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In our experimental analysis we compared our approach to different other state-of-the-art distribution matching methods like the Maximum Mean Discrepancy (MMD) based on the Gaussian kernel using a quadratic time estimate. In the future we want to extend our experimental analysis to other MMD approaches including other kernels, parameter selection procedures and linear time estimators. In addition, we plan to use the CMD for training generative models and to further investigate the approximation quality of the proposed empirical estimate. ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "A THEOREM PROOFS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            771,
+            366,
+            787
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Theorem 1. Let $p$ and $q$ be two probability distributions on a compact interval and let CMD be defined as in (5), then ",
+        "bbox": [
+            173,
+            803,
+            823,
+            830
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2da06e0a83a897d5766eeb4571672f03a9421eae873745ba329d5a5be6e39520.jpg",
+        "text": "$$\nC M D ( p , q ) = 0 \\Rightarrow p = q\n$$",
+        "text_format": "latex",
+        "bbox": [
+            406,
+            829,
+            591,
+            847
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Proof. Let $X$ and $Y$ be two random vectors that have probability distributions $p$ and $q$ , respectively. Let ${ \\hat { X } } = X - \\mathbb { E } ( X )$ and $\\hat { Y } = Y - \\mathbb { E } ( Y )$ be the mean centered random variables. From $\\mathbf { C M D } ( p , q ) = 0$ it follows that all moments of the bounded random variables $\\hat { X }$ and $\\hat { Y }$ are equal. Therefore, the joint moment generating functions of $\\hat { X }$ and $\\hat { Y }$ are equal. Using the property that $p$ and $q$ have compact support, we obtain the equality of the joint distribution functions of $\\hat { X }$ and $\\hat { Y }$ . Since $\\mathbb { E } ( X ) = \\mathbb { E } ( Y )$ , it follows that $X = Y$ . □ ",
+        "bbox": [
+            174,
+            859,
+            825,
+            924
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            102,
+            825,
+            132
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Theorem 2. Let $p _ { n }$ and $p$ be probability distributions on a compact interval and let CMD be defined as in (5), then ",
+        "bbox": [
+            171,
+            138,
+            825,
+            167
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6db989fde41b39b0606374f39ff4032520648ef7028a2f24938d4dcb4dcba899.jpg",
+        "text": "$$\nC M D ( p _ { n } , p ) \\to 0 \\Rightarrow p _ { n } \\stackrel { d } { \\to } p\n$$",
+        "text_format": "latex",
+        "bbox": [
+            392,
+            166,
+            606,
+            188
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "where $\\xrightarrow { d }$ denotes convergence in distribution. ",
+        "bbox": [
+            176,
+            190,
+            475,
+            208
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Proof. Let $X _ { n }$ and $X$ be random vectors that have probability distributions $p _ { n }$ and $p$ respectively. Let ${ \\hat { X } } = X - \\mathbb { E } ( X )$ and ${ \\hat { X } } _ { n } \\ = \\ X _ { n } - \\mathbb { E } ( X _ { n } )$ be the mean centered random variables. From $\\mathrm { C M D } ( X _ { n } , X ) \\to 0$ it follows that the moments of $\\hat { X } _ { n }$ converge to the moments of $\\hat { X }$ . Therefore, the joint moment generating functions of $\\hat { X } _ { n }$ converge to the joint moment generating function of $\\hat { X }$ , which implies convergence in distribution of the mean centered random variables. Using $\\mathbb { E } ( X _ { n } )  \\mathbb { E } ( X )$ we obtain $p _ { n } \\stackrel { d } { \\to } p$ . □ ",
+        "bbox": [
+            173,
+            220,
+            826,
+            321
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Proposition 1. Let $X$ and $Y$ be bounded random vectors with respective probability distributions $p$ and $q$ on the compact interval $[ a , b ] ^ { N }$ . Then ",
+        "bbox": [
+            173,
+            327,
+            825,
+            356
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1f222f0a96bd872cbf3b7c995c9e1bcb67723273d4cca7c7866cd8ecebdc085d.jpg",
+        "text": "$$\n\\frac { 1 } { | b - a | ^ { k } } \\| c _ { k } ( X ) - c _ { k } ( Y ) \\| _ { 2 } \\leq 2 \\sqrt { N } \\left( \\frac { 1 } { k + 1 } \\left( \\frac { k } { k + 1 } \\right) ^ { k } + \\frac { 1 } { 2 ^ { 1 + k } } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            274,
+            358,
+            723,
+            400
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "where $c _ { k } ( X ) \\ : = \\ : \\mathbb { E } ( ( X - \\mathbb { E } ( X ) ) ^ { k } )$ is the vector of all $k ^ { t h }$ order sample central moments of the marginal distributions of $p$ . ",
+        "bbox": [
+            174,
+            402,
+            823,
+            433
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Proof. Let $\\textstyle { \\mathcal { X } } ( [ a , b ] )$ be the set of all random variables with values in $[ a , b ]$ . Then it follows that ",
+        "bbox": [
+            173,
+            446,
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+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4dbbad3cd3a3dd81e91127c5b2d26fa92e3fe18da9bb89ab0aef7a0ad2bbfda8.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\frac { 1 } { \\left. \\hat { b } - a \\right. ^ { k } } \\left. \\epsilon _ { k } ( X ) - c _ { k } ( Y ) \\right. _ { 2 } = \\left. \\frac { c _ { k } ( X ) } { \\left. \\hat { b } - a \\right. ^ { k } } - \\frac { c _ { k } ( Y ) } { \\left. \\hat { b } - a \\right. ^ { k } } \\right. _ { 2 } } & { } \\\\ { \\leq \\left. \\frac { c _ { k } ( X ) } { \\left. \\hat { b } - a \\right. ^ { k } } \\right. _ { 2 } + \\left. \\frac { c _ { k } ( Y ) } { \\left. \\hat { b } - a \\right. ^ { k } } \\right. _ { 2 } } & { } \\\\ { = \\left. \\mathbb { E } \\left( \\left( \\frac { X - \\mathbb { E } ( X ) } { \\left. b - a \\right. } \\right) ^ { k } \\right) \\right. _ { 2 } + \\left. \\mathbb { E } \\left( \\left( \\frac { Y - \\mathbb { E } ( Y ) } { \\left. b - a \\right. } \\right) ^ { k } \\right) \\right. _ { 2 } } & { } \\\\ { \\leq \\left. \\mathbb { E } \\left( \\left. \\frac { X - \\mathbb { E } ( X ) } { b - a } \\right. ^ { k } \\right) \\right. _ { 2 } + \\left. \\mathbb { E } \\left( \\left. \\frac { Y - \\mathbb { E } ( Y ) } { b - a } \\right. ^ { k } \\right) \\right. _ { 2 } } & { } \\\\ { \\leq 2 \\sqrt { N } \\underset { X \\in \\mathcal { X } ( \\{ a , b \\} ) } { \\overset { \\mathrm { u p } } { \\sum } } \\mathbb { E } \\left( \\left. \\frac { X - \\mathbb { E } ( X ) } { b - a } \\right. ^ { k } \\right) } & { } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            210,
+            464,
+            789,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "The latter term refers to the absolute central moment of order $k$ , for which the smallest upper bound is known (Egozcue et al., 2012): ",
+        "bbox": [
+            173,
+            666,
+            823,
+            694
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/56be9b802973a29fab8366079f09a9f857fc103141f36fbb20f2be19b1014b09.jpg",
+        "text": "$$\n{ \\frac { 1 } { | b - a | ^ { k } } } \\left\\| c _ { k } ( X ) - c _ { k } ( Y ) \\right\\| _ { 2 } \\leq 2 { \\sqrt { N } } \\operatorname* { s u p } _ { x \\in [ 0 , 1 ] } x ( 1 - x ) ^ { k } + ( 1 - x ) x ^ { k }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            274,
+            696,
+            723,
+            731
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Egozcue et al. (2012) also give a more explicit bound: ",
+        "bbox": [
+            174,
+            733,
+            529,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6e5a59cda511989521d0f2aa04ede361ea66132be08bf133ac1b86a7a66cdc99.jpg",
+        "text": "$$\n\\frac { 1 } { | b - a | ^ { k } } \\left\\| c _ { k } ( X ) - c _ { k } ( Y ) \\right\\| _ { 2 } \\leq 2 \\sqrt { N } \\left( \\frac { 1 } { k + 1 } \\left( \\frac { k } { k + 1 } \\right) ^ { k } + \\frac { 1 } { 2 ^ { 1 + k } } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            748,
+            723,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGEMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            823,
+            334,
+            837
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "The research reported in this paper has been supported by the Austrian Ministry for Transport, Innovation and Technology, the Federal Ministry of Science, Research and Economy, and the Province of Upper Austria in the frame of the COMET center SCCH. ",
+        "bbox": [
+            174,
+            845,
+            823,
+            888
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "We would like to thank Bernhard Moser and Florian Sobieczky for fruitful discussions on metric spaces. ",
+        "bbox": [
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+        ],
+        "page_idx": 10
+    },
+    {
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parse/train/SylVNerFvr/SylVNerFvr.md

@@ -0,0 +1,290 @@
+# PERMUTATION EQUIVARIANT MODELS FOR COMPOSITIONAL GENERALIZATION IN LANGUAGE
+
+Jonathan Gordon∗ University of Cambridge jg801@cam.ac.uk
+
+David Lopez-Paz, Marco Baroni, Diane Bouchacourt Facebook AI Research {dlp, mbaroni, dianeb}@fb.com
+
+# ABSTRACT
+
+Humans understand novel sentences by composing meanings and roles of core language components. In contrast, neural network models for natural language modeling fail when such compositional generalization is required. The main contribution of this paper is to hypothesize that language compositionality is a form of group-equivariance. Based on this hypothesis, we propose a set of tools for constructing equivariant sequence-to-sequence models. Throughout a variety of experiments on the SCAN tasks, we analyze the behavior of existing models under the lens of equivariance, and demonstrate that our equivariant architecture is able to achieve the type compositional generalization required in human language understanding.
+
+# 1 INTRODUCTION
+
+When using language, humans recombine known concepts to understand novel sentences. For instance, if one understands the meaning of “run”, “jump”, and “jump twice”, then one understands the meaning of “run twice”, even if such sentence was never heard before. This relies on the notion of language compositionality, which states that the meaning of a sentence (“jump twice”) is to be obtained by the meaning of its constituents (e.g. the verb “jump" and the quantifying adverb “twice") and the use of algebraic computation (a verb combined with a quantifying adverb $m$ results in doing that verb $m$ times) (Kratzer & Heim, 1998).
+
+In the realm of machines, deep learning has achieved unprecedented results in language modeling tasks (Bahdanau et al., 2015; Vaswani et al., 2017). However, these models are sample inefficient, and do not generalize to examples that require the use of language compositionality (Lake & Baroni, 2018; Loula et al., 2018; Dessì & Baroni, 2019). This result suggests that deep language models fail to leverage compositionality; a failure remaining to this day a roadblock towards true natural language understanding.
+
+Focusing on this issue, Lake & Baroni (2018) proposed the Simplified version of the CommAI Navigation (SCAN), a dataset to benchmark the compositional generalization capabilities of state-ofthe-art sequence-to-sequence (seq2seq) translation models (Sutskever et al., 2014; Bahdanau et al., 2015). In a nutshell, the SCAN dataset contains compositional navigation commands such as JUMP TWICE AFTER RUN LEFT, to be translated into the sequence of actions LTURN RUN JUMP JUMP.
+
+Using SCAN, Lake & Baroni (2018) demonstrated that seq2seq models fail spectacularly at tasks requiring the use of language compositionality. Following our introductory example, models trained on the three commands JUMP, RUN and JUMP TWICE fail to generalize to RUN TWICE. Most recently, Dessì & Baroni (2019) showed that architectures based on temporal convolutions meet the same fate.
+
+SCAN did not only reveal the lack of compositionality in language models, but it also became the blueprint to build novel language models able to handle language compositionality. On the one hand, Russin et al. (2019) proposed a seq2seq model where semantic and syntactic information are represented separately, in a hope that such disentanglement would elicit compositional rules. However, their model was not able to solve all of the compositional tasks comprising SCAN. On the other hand, Lake (2019) introduced a meta-learning approach with excellent performance in multiple
+
+SCAN tasks. However, their method requires substantial amounts of additional supervision, and a complex meta-learning procedure hand-engineered for each task.
+
+In this paper, we take a holistic look at the problem and connect language compositionality in SCAN to the disparate literature in models equivariant to certain group symmetries (Kondor, 2008; Cohen & Welling, 2016; Ravanbakhsh et al., 2017; Kondor & Trivedi, 2018). Interesting links have recently been proposed between group symmetries and the areas of causality (Arjovsky et al., 2019) and disentangled representation learning (Higgins et al., 2018), and this work proceeds in a similar fashion. In particular, the main contribution of this work is not to chase performance numbers, but to put forward the novel hypothesis that language compositionality can be understood as a form of group-equivariance (Section 3). To sustain our hypothesis, we provide tools to construct seq2seq models equivariant when the group symmetries are known (Section 4), and demonstrate that these models solve all SCAN tasks, except length generalization (Section 6).1
+
+# 2 THE SCAN COMPOSITIONAL TASKS
+
+The purpose of the Simplified version of the CommAI Navigation (SCAN) tasks (Lake & Baroni, 2018) is to benchmark the abilities of machine translation models for compositional generalization. Following prior literature (Lake & Baroni, 2018; Baroni, 2019; Russin et al., 2019; Andreas, 2019), compositional generalization is understood as the ability to translate novel families of sentences, when this requires leveraging the compositional structure in language.
+
+The SCAN dataset contains compositional navigation commands in English (the input-language) paired with a desired action sequence (the output-language). For instance, the input-language sentence JUMP TWICE AND RUN LEFT is paired to the output-language actions sequence JUMP JUMP LTURN RUN. The rest of our exposition uses SMALL CAPS to denote examples in the input-language, and LARGE CAPS to denote examples in the output-language. Appendix A contains a full description of the grammar generating the SCAN language.
+
+To evaluate the compositional generalization abilities of sequence-to-sequence (seq2seq) machine translation models (Sutskever et al., 2014; Bahdanau et al., 2015), Lake & Baroni (2018) proposes four main tasks based on SCAN:
+
+1. Simple task: data pairs are randomly split into training and test sets. No compositional generalization is required.   
+2. Add jump task: the only command in the training set containing the verb JUMP is the command JUMP. All commands not containing JUMP are in the training set (for instance, RUN TWICE, and WALK RIGHT THRICE AND LOOK LEFT). The test set contains all commands containing JUMP (for instance, JUMP TWICE, and RUN LEFT AND JUMP RIGHT). To succeed in this task, models must learn that JUMP is a verb, and that any verb can be composed with an adverbial number to be repeated a number of times.   
+3. Around right task: the phrase AROUND RIGHT is held out from the training set; however, both AROUND and RIGHT are shown in all other contexts (for example, AROUND LEFT or OPPOSITE RIGHT). To succeed at this task, models must learn that both RIGHT and LEFT are directions, and can be combined with AROUND and OPPOSITE.   
+4. Length generalization task: the training set contains pairs such that the length of the action sequence in the output-language is shorter than 24 actions. The test set contains all pairs with action sequences of a length greater or equal than 24 actions. The type of compositional ability required to succeed at this task is more difficult to sketch out, as we discuss in Section   
+6.2.
+
+Lake & Baroni (2018) use these four tasks to demonstrate that state-of-the-art seq2seq translation models (Bahdanau et al., 2015) succeed at Simple task, but fail at the other three tasks requiring compositional generalization. Convolutional architectures (Dessì & Baroni, 2019) achieve only slightly better performance, and state-of-the-art methods specially developed to address SCAN tasks fall short from the best achievable performance (Russin et al., 2019), or call for substantial amounts of additional supervision (Lake, 2019).
+
+In the following, we take a holistic look at the language compositionality problems in SCAN, and highlight their connection to equivariant maps in group theory.
+
+# 3 SCAN COMPOSITIONALITY AS GROUP EQUIVARIANCE
+
+This section puts forward the hypothesis that:
+
+Models achieving the compositional generalization required in certain SCAN tasks are equivariant with respect to permutation group operations2 in the input and output languages.
+
+To unfold the meaning of our hypothesis, we must revisit some basic concepts in group theory. A discrete group $G$ is a set of elements $\{ g _ { 1 } , \dotsc , g _ { | G | } \}$ , equipped with a binary group operation “·” satisfying the four group axioms (closure, associativity, identity, and invertibility). The sequel focuses on permutation groups $G$ , whose elements are permutations of a set $\mathcal { X }$ , and whose binary group operation composes the permutations contained in $G$ . The set of all permutations of $\mathcal { X }$ is a group, but not all subsets of permutations of $\mathcal { X }$ satisfy the four group axioms, and therefore they do not form a group. For each element $g \in G$ , we define the group operation $T _ { g } : \mathcal { X }  \mathcal { X }$ as the map applying the permutation $g$ to the element $x \in \mathcal { X }$ , to obtain $T _ { g } x$ . Armed with these definitions, we are ready to introduce the main object of study in this paper: equivariant maps.
+
+Definition 1 (Equivariant map). Let $\mathcal { X }$ and $\mathcal { V }$ be two sets. Let $G$ be a group whose group operation on $\mathcal { X }$ is denoted by $T _ { g } : \mathcal { X }  \mathcal { X }$ , and whose group operation on $\mathcal { V }$ is denoted by $T _ { g } ^ { \prime } : \mathcal { V } \to \mathcal { V }$ . Then, $\Phi : \mathcal { X }  \mathcal { Y }$ is an equivariant map if and only if $\Phi \left( T _ { g } x \right) = T _ { g } ^ { \prime } \Phi ( x )$ for all $x \in \mathcal { X }$ and $g \in G$ .
+
+The operation groups $( T _ { g } , T _ { g } ^ { \prime } )$ defined above operate on entire sequences, an enormous space when we consider those sequences to be language sentences. In the following two definitions, we relax group operations and equivariant maps to operate at a word level.
+
+Definition 2 (Local group operations). Let $\mathcal { X }$ be a set of sequences (or sentences), where each sequence $x \in \mathcal { X }$ contains elements $x _ { i } \in \mathcal V$ from a vocabulary set $\nu$ , for all $x _ { i } \in x$ . Let $G$ be a group with associated group operation $T _ { g } : \mathcal { X }  \mathcal { X }$ . Then, we say that $T _ { g }$ is a local group operation if there exists a group operation $T _ { g _ { w } } : \mathcal { V } \stackrel { \cdot } { \times } \mathcal { V }$ such that $T _ { g } x = ( T _ { g _ { w } } x _ { 1 } , \ldots , T _ { g _ { w } } x _ { L _ { x } } )$ for all $x \in \mathcal { X }$ .
+
+When understanding sequences as language sentences, the group operation $T _ { g _ { w } }$ would be a permutation of the words from the language vocabulary. Such operation can be implemented in terms of a permutation matrix, a $| \nu | \times | \nu |$ matrix with zero/one entries where each row and each column sum to one. Finally, we leverage the definition of local group operations to define locally equivariant maps.
+
+Definition 3 (Locally equivariant map). Let $\mathcal { X }$ and $\mathcal { V }$ be two sets of sequences. Let $G$ be a group whose group operation on $\mathcal { X }$ is local in its vocabulary, denoted by $T _ { g } : \mathcal { X }  \mathcal { X }$ , and whose group operation on $\mathcal { V }$ is local in its vocabulary and denoted by $T _ { g } ^ { \prime } : \mathcal { V } \times \overset { \cdot } { \mathcal { V } }$ . Then, we say that $\Phi : \mathcal { X }  \mathcal { Y }$ is an equivariant map if and only if $\Phi ( T _ { g } x ) = T _ { g } ^ { \prime } \Phi ( x )$ for all $x \in \mathcal { X }$ and $g \in G$ .
+
+![](images/62214294913b795f13f25fa3b0bb3772768f35b141b293fc27293c3c40b75421.jpg)  
+Figure 1: (a) Commutative diagram for equivariance. (b) Local equivariance enables generalization to verb replacement in SCAN. (c) Local equivariance does not enable generalization to conjunction replacement in SCAN.
+
+Now, how do equivariances and local equivariances manifest themselves in the world of SCAN? To assist our examples, the commutative diagram in Figure 1a summarizes the group theory notations introduced so far. In Figure 1b and Figure 1c, we parallel these notations to two different examples of compositional skills required to solve SCAN: verb and conjunction replacement. In the SCAN domain, $\mathcal { X }$ is the set of sentences in the input-language, and $\mathcal { V }$ is the set of sentences in the outputlanguage. Furthermore, let $\Phi$ be a locally equivariant SCAN translation model, and let $G$ be a group with associated local group operations that permutes words in the input- and output- languages.
+
+On the one hand, we observe in Figure 1b that local equivariance enables compositional generalization in the case of verb replacement. This is because replacing one verb in the input-language can be implemented in terms of a local group operation. In turn, this input-verb replacement corresponds deterministically to a second local group operation that replaces the corresponding verb in the output-language. The same would apply to a SCAN task where we are interested in generalizing to the replacement of LEFT and RIGHT. As such, a translation model $\Phi$ with these compositional generalization capabilities must be locally equivariant.
+
+On the other hand, we observe in Figure 1c that local equivariance is insufficient to enable compositional generalization in the case of conjunction replacement. This is because no local group operation in the output-language would be able to implement the necessary changes induced by the replacement of AND by AFTER in the input-language. In such cases, we refer to the equivariance as global equivariance. In particular, we can see how blocks of multiple words in the output-language swap their relative location. Local equivariances are also insufficient to enable compositional generalization in the Length generalization SCAN task and we elaborate on this in Section 6.2.
+
+In the following section, we propose a set of tools to implement equivariant seq2seq translation models, and propose a particular architecture with which we conduct our experiments.
+
+# 4 IMPLEMENTING AN EQUIVARIANT SEQUENCE-TO-SEQUENCE MODEL
+
+We now implement our proposed equivariant seq2seq model, following the encoder-decoder architecture illustrated in Figure 2. Readers unfamiliar with group theory may parse Figure 2 by temporarily discarding the $^ { 6 6 } G - '$ ” prefixes, and realize that each depicted module is a well-known building block of recurrent neural network models.
+
+![](images/de7e289be9fe669ae8e8d3f107fe9665a01367089f8c4a00e8f1c4e9a5d75256.jpg)  
+Figure 2: Architecture of our fully-equivariant seq2seq model. Variables shaded in gray are mappings $G \to \mathbb { R } ^ { K }$ , implemented as $| G | \times K$ matrices. Encoder and decoder meet at $\tilde { h } _ { 0 } : = h _ { L _ { x } }$ .
+
+To make our model equivariant, we will make intense use of group convolutions.
+
+Definition 4 (Group convolution (Kondor & Trivedi, 2018)). Let $G$ be a discrete group. Let $f : G $ $\mathbb { R } ^ { K }$ be an input function. Let $\psi = \{ \psi ^ { i } : G \to \mathbb { R } ^ { K } \} _ { i = 1 } ^ { K ^ { \prime } }$ be a set of learnable filter functions. Then, each scalar real entry from the result of $G$ -convolving $f$ and $\psi$ is given by a $| G | \times K ^ { \prime }$ matrix with entries
+
+$$
+G \mathrm { - } \mathrm { C o n v } ( f ; \psi ) _ { g , i } = \sum _ { h \in \mathrm { d o m } ( f ) } \sum _ { k = 1 } ^ { K } f _ { k } ( h ) \psi _ { k } ^ { i } ( g ^ { - 1 } h ) ,
+$$
+
+for all $g \in G$ and $i \in \{ 1 , \ldots , K ^ { \prime } \}$ . As shown by (Kondor $\&$ Trivedi, 2018), the $G$ -Conv layer is equivariant wrt the operations of $G$ . We apply this definition in two ways: (i) “convolving” words with learnable filters to generate equivariant embeddings. Later, when we introduce our notations, we discuss how words may be viewed as functions so as to fit the definition. And (ii) convolving two group representations, in which case $\operatorname { d o m } ( f ) = G$ .
+
+We note that there are several additional methods proposed in the literature for constructing permutation equivariant layers (e.g. , Zaheer et al., 2017; Ravanbakhsh et al., 2017). However, as demonstrated by Kondor & Trivedi (2018); Bloem-Reddy & Teh (2019), the above form is very general and subsumes most alternatives. Further, while layers based on weight-sharing may be more efficient than the general form of Definition 4, the parameter tying restricts the capacity of the layer. For example, the permutation equivariant layer of Zaheer et al. (2017) requires weight matrices that are restricted to a form $\lambda I + { \overset { \cdot } { \gamma } } ( \mathbf { 1 1 } ) ^ { T }$ , with learnable parameters $\lambda$ and $\gamma$ . This layer has fewer learnable parameters than the convolutional form of Definition 4. Thus, for reasons of generality and capacity, we employ the general and expressive convolutional form of Definition 4 for our permutation equivariant layers.
+
+Equivariant with respect what group? The previous $G$ -Conv layer requires choosing a discrete group $G$ . As hinted in Section 3, we will choose $G$ to contain $| G |$ permutations of language vocabularies, e.g. products of cyclic groups on sets of words. Note that for a vocabulary size of $| V |$ , the set of all permutations has a size of $n !$ . However, it suffices to consider subgroups containing permutations such that every word can be reached by composing elements of the subgroup. For example, while the group of permutations on the four verbs in SCAN consists of 24 elements, it will suffice to choose $G$ as the circular shift group on the four verbs, which is a subgroup of four elements. Following standard notation in group theory, we write $g \cdot h$ to denote the composition of two group elements $g , h \in G$ , and $g ^ { - 1 }$ to denote the inverse element of $g$ .
+
+As final preliminaries, denoting $[ V ] = \{ 1 , \dots , | V | \}$ , we represent a word $w$ in the input-language by the function $w : [ V ] \to \{ 0 , 1 \}$ , where $\begin{array} { r } { \sum _ { v \in [ V ] } w ( v ) = 1 } \end{array}$ , and similarly by using $\tilde { w } \in \tilde { V }$ for the output-language. These notations are functional representations of word one-hot encodings that will play well with our notations. Note that this representation is equivalent to one-hot vectors, and in what follows we use the shorthand $w$ for the one-hot vector representation of words.
+
+To avoid notational clutter, we use $g$ to denote the permutation-matrix-representation of the corresponding group element. Thus, the group operation on a word gw can be implemented as matrix multiplication between the permutation matrix $g$ and the one-hot vector $w$ . Note that this operation results in another one-hot vector, i.e. another word in the vocabulary. Similarly, the binary group operation can be written as matrix multiplication $g h$ between two group members $g , h \in G$ . Here too, multiplication of permutation matrices results in permutation matrix, so $g h \in G$ .
+
+We now describe each of the components in our $G$ -equivariant translation model, by following the transformation process of an input sequence $x = ( w _ { 1 } , \ldots , w _ { L _ { x } } )$ (in SCAN, a navigation command in English) into its output translation $y = ( \tilde { w } _ { 1 } , \dots , \tilde { w } _ { L _ { y } } )$ (in SCAN, a sequence of actions).
+
+# 4.1 G-EQUIVARIANT ENCODER
+
+Upon arrival, the input-language sentence $x = ( w _ { 1 } , \dots , w _ { L _ { x } } )$ is sent to a $G$ -equivariant encoder. The first step in the encoding process is to transform each input word $w _ { t }$ into a permutation equivariant embedding $e ( w _ { t } )$ . As mentioned before, each word $w _ { t }$ is represented by the one-hot vector $w _ { t } :$ $[ V ]  \{ 0 , 1 \}$ . The corresponding embedding is obtained by applying a set of $K$ 1-dimensional learnable filter functions $\{ \bar { \psi } ^ { i } : [ V ] \stackrel { - } { \to } \mathbb { R } \} _ { i = 1 } ^ { K }$ in a group convolution (throughout the section, we use $K$ everywhere to ease notation). Using Definition 4, the embedding, which we call $G$ -Embed, is then represented as a matrix $\mathbb { R } ^ { | G | \times K }$ , where
+
+$$
+e ( w ) _ { g , i } = G \mathrm { - E m b e d } ( w ; \psi ) _ { g , i } = \psi ^ { i } ( g ^ { - 1 } w ) ,
+$$
+
+for all $g \in G$ and $i = \{ 1 , \ldots , K \}$ . Note that since $w$ is a one-hot vector, $G$ -Embed is a particularly simple instantiation of Definition 4, as summation over $\operatorname { d o m } ( f )$ consists of only a single term. The corresponding embedding is a function $e ( w _ { t } ) : G \to \mathbb { R } ^ { K }$ , which can be represented as a $| G | \times K$ matrix, where each row corresponds to the embedding of the word $g w$ for a particular $g \in G$ . This layer can be implemented by defining $\psi$ with standard deep learning embedding modules.3
+
+Importantly, we note that for this layer, both $\psi$ and $w$ are functions on $[ V ]$ . However, the resulting embedding $e ( w )$ is a function on the group $G$ . Therefore, in all subsequent computations we will require the learnable filters $\psi$ to also be functions on $G$ .
+
+We illustrate this layer with an example. Let $G$ be the cyclic group that permutes the words LEFT and RIGHT. We can think of $g _ { 1 }$ as the identity, and $g _ { 2 }$ as permuting the words LEFT and RIGHT (leaving all other words unchanged). In this case, embedding LEFT results in the $2 \times K$ matrix $[ \psi ( \mathrm { L E F T } ) ^ { T } , \psi ( \mathrm { R I G H T } ) ^ { T } ] ^ { T }$ , while embedding JUMP results in $\mathsf { \bar { \Psi } } ( \mathbf { J } \mathbf { U } \mathbf { M } \mathbf { P } ) ^ { T } , \psi ( \mathbf { J } \mathbf { U } \mathbf { M } \mathbf { P } ) ^ { T } ] ^ { T }$ , since both $g _ { 1 }$ and $g _ { 2 }$ act as the identity permutation for JUMP.
+
+Next, the word embedding $e ( w _ { t } )$ is sent to a permutation equivariant Recurrent Neural Network ( $G$ -RNN). The cells of a $G$ -RNN mimic those of a standard RNN, where linear transformations are replaced by $G$ -Convs (Definition 4). This cell receives two inputs (the word embedding $e ( w _ { t } )$ and the previous hidden state $h _ { t - 1 }$ ) and returns one output (the current hidden state $h _ { t }$ ), all three being functions $G \to \mathbb { R } ^ { K }$ , parametrized as $| G | \times K$ matrices. More specifically:
+
+$$
+h _ { t } = G \mathrm { - R N N } ( e ( w _ { t } ) , h _ { t - 1 } ) = \sigma ( G \mathrm { - C o n v } ( h _ { t - 1 } ; \psi _ { h } ) + G \mathrm { - C o n v } ( e ( w _ { t } ) ; \psi _ { e } ) ) ,
+$$
+
+where $\psi _ { h } , \psi _ { e } \colon G \to \mathbb { R } ^ { K }$ are learnable filters (represented as $| G | \times K$ matrices), and $\sigma$ is a point-wise activation function.
+
+The cell $G$ -RNN is equivariant because the sum of two equivariant representations is equivariant (Cohen & Welling, 2016), and the pointwise transformation of an equivariant representation is also equivariant. To initialize the hidden state, we set $h _ { 0 } = \vec { 0 }$ . We note that our experiments use the equivariant analog of LSTM cells (Hochreiter & Schmidhuber, 1997), which we denote $G$ -LSTM, since these achieved the best performance. We include the architecture of $G$ -LSTM cells in Appendix B.
+
+This completes the description of our equivariant encoder, illustrated in Figure 2a.
+
+# 4.2 G-EQUIVARIANT DECODER
+
+Once the entire input-language sentence $x = ( w _ { 1 } , \ldots , w _ { L _ { x } } )$ has been encoded into the hidden representations $h = ( h _ { 1 } , \ldots , h _ { L _ { x } } )$ , we are ready to start the decoding process that will produce the output-language translation $y = ( \tilde { w } _ { 1 } , \dots , \tilde { w } _ { L _ { y } } )$ .
+
+As illustrated in Figure 2b, our equivariant decoder is also run by an equivariant recurrent cell $G$ -RNN. We denote the hidden states of the recurrent decoding process by $\tilde { h } _ { t }$ , where $\tilde { h } _ { 0 } = h _ { L _ { x } }$ . At time $t$ , the two inputs to the decoding $G$ -RNN cell are the previous hidden state $\tilde { h } _ { t - 1 }$ as well as an attention ${ { \bar { a } } _ { t } }$ over all the encoding hidden states $h$ . (Once again, all variables are mappings $G \to \mathbb { R } ^ { K }$ implemented as $| G | \times K$ matrices.)
+
+Attention mechanisms (Bahdanau et al., 2015; Vaswani et al., 2017) have emerged as a central tool in language modelling. Fortunately, attention mechanisms are typically implemented as linear combinations, and a linear combination of equivariant representations is itself an equivariant representation. We now leverage this fact to develop an equivariant attention mechanism. Given all the encoder hidden states $h$ , as well as the previous decoding hidden state $\tilde { h } _ { t - 1 }$ , we propose the equivariant analog of dot-product attention (Luong et al., 2015) as
+
+$$
+\begin{array} { r l } & { \bar { a _ { t } } = G \mathrm { - } \mathrm { A t t e n t i o n } ( \tilde { h } _ { t - 1 } , h ) = \displaystyle \sum _ { j = 1 } ^ { L _ { x } } \alpha _ { t , j } h _ { j } \mathrm { , ~ w h e r e ~ } } \\ & { \alpha _ { t , j } = \displaystyle \frac { \exp \beta _ { t , j } } { \sum _ { k = 1 } ^ { L _ { x } } \exp \beta _ { t , k } } \mathrm { , ~ a n d ~ } \beta _ { t , j } = \displaystyle \sum _ { g \in G } \tilde { h } _ { t - 1 } ( g ) ^ { \top } h _ { j } ( g ) . } \end{array}
+$$
+
+Following Figure 2b, the attention $\bar { a _ { t } }$ and a $G$ -embedding $e ( \tilde { w } _ { t - 1 } )$ for the previous output word are concatenated and sent to a $G$ -Convolution.4 The concatenation with $e ( \tilde { w } _ { t - 1 } )$ provides the decoder with information regarding the previously embedded word. In practice, during training we use teacher-forcing (Williams $\&$ Zipser, 1989) to provide the decoder with information about the correct output sequences. This process returns a final hidden representation $\phi : G \to \mathbb { R } ^ { K }$ .
+
+As a final step in the decoding process, we need to convert $\phi$ into a collection of logits over the output-language vocabulary. Then, sampling from the categorical distribution induced by these logits at time $t$ (or taking the maximimum) will produce the word $\tilde { w } _ { t }$ , to be appended in the output-language translation, $y$ . This final decoding module can be implemented as follows:
+
+$$
+G \mathrm { - D e c o d e } ( \phi ; \psi ) _ { \tilde { w } } = \sum _ { h \in G } \sum _ { k = 1 } ^ { K } \phi _ { k } ( h ) \psi _ { k } ( h ^ { - 1 } \tilde { w } ) ,
+$$
+
+where $\psi = [ \tilde { V } ]  \mathbb { R } ^ { k }$ are the learnable parameters of this layer (represented by a $| \tilde { V } | \times K$ matrix).
+
+Recall that $\phi ( h ) \in \mathbb { R } ^ { K }$ is the final-layer representation for the group element $h$ , and that $h ^ { - 1 } \tilde { w }$ is the inverse element of $h \in G$ applied to the output word $\tilde { w }$ (represented as a one-hot vector), which results in another word in the output language. Thus, $\psi$ is a learnable embedding of the output words into $\mathbb { R } ^ { K }$ . This layer is evaluated at every $\tilde { w }$ in the output vocabulary to produce a scalar. The resulting vector of logits represents a categorical distribution over the output vocabulary. While similar, this layer is not a group convolution (Definition 4). Rather, equivariance for this module is achieved via parameter-sharing (Ravanbakhsh et al., 2017).
+
+This completes the description of our equivariant decoder, illustrated in Figure 2b. Composing the equivariant encoder and decoder results in our complete sequence-2-sequence model. Importantly, since all operations in this model are equivariant, the complete model is itself also equivariant to the group $G$ (Kondor & Trivedi, 2018). In Section 6, we provide further implementation details for our model, and detail our empirical evaluation of its equivariant properties and their relation to the SCAN tasks described in Section 2.
+
+# 5 RELATED WORK
+
+In this section we review state-of-the-art methods to address SCAN compositional tasks. We focus on two recent models that we will compare to in our experiments.
+
+On the one hand, the syntactic attention model of Russin et al. (2019) builds on the idea that compositional generalization can be achieved by language models given the correct architectural organization. Borrowing inspiration from neuroscience, Russin et al. (2019) argue that compositionality might arise when using separate processing channels for semantic and syntactic information. In their model, the attention weights depend on a recurrent encoding of the input sequence, which they refer to as the syntactic representation. The attention weights are then applied to separate, context-independent embeddings of the words in the input sequence, which intend to model a semantic representation. We find (Russin et al., 2019) interesting from a group equivariance perspective, since one way to enforce equivariance is to use an invariant representation (about syntax) together with an additional representation (about semantics) that maintains the information about the original “sentence pose”.
+
+On the other hand, the meta-learning (Thrun & Pratt, 2012; Schmidhuber, 1987) approach of Lake (2019) is a model that learns to generalize. In particular, Lake (2019) designs one specific and complex meta-learning procedure for each SCAN task, where a distribution over tasks is provided to the learner (Finn et al., 2017; Gordon et al., 2018). For example, in the Add jump and Around right tasks, the meta-learning procedure of Lake (2019) samples permutations from the relevant groups (the permutation groups on the verbs and set of directions, respectively). This is interpreted as data-augmentation, a valid procedure for encouraging equivariance (Cohen & Welling, 2016; Andreas, 2019; Weiler et al., 2018). However, at test-time, Lake (2019) sets the context set to the correct mapping between the permuted commands and their corresponding actions. For example, in the Add jump task, the context set for meta-testing would consist of the following pairs: {(WALK, WALK), (RUN, RUN), (LOOK, LOOK), (JUMP, JUMP) }. This is equivalent to providing the model with one-to-one information regarding the correct command-to-action mapping for the permuted words.
+
+# 6 EXPERIMENTS
+
+We now evaluate the empirical performance of our equivariant seq2seq model (described in Section 4) on the four SCAN tasks (described in Section 2). We compare our equivariant seq2seq to regular seq2seq models (Lake & Baroni, 2018), convolutional models (Dessì & Baroni, 2019), the syntactic attention model of Russin et al. (2019), and the meta-learning approach of Lake (2019). The compared seq2seq models use bi-directional, single-layer LSTM cells with 64 hidden units. For the equivariant seq2seq models, we use the cyclic permutation group on the verbs for the Add jump task, and the cyclic permutation group on directions for the Around right task. For Length, we use the product of those groups. Our model knows that the same group operates on both the input- and output- languages. However, it does not receive information regarding the correspondence between commands and actions in the set of words being permuted in the input / output languages. This is in contrast to Lake (2019), where (as stated in Section 5), it is necessary to provide the model with explicit information regarding the correct command-to-action mapping at test-time.
+
+<table><tr><td>Model</td><td>Simple</td><td>Add Jump</td><td>Around Right</td><td>Length</td></tr><tr><td>seq2seq (Lake &amp; Baroni, 2018)</td><td>99.7</td><td>1.2</td><td>NA</td><td>13.8</td></tr><tr><td>CNN (Dessi &amp; Baroni, 2019)</td><td>100.0</td><td>69.2 ± 9.2</td><td>56.7±10.2</td><td>0.0</td></tr><tr><td>Syntactic Attention (Russin et al., 2019)</td><td>100.0</td><td>91.0± 27.4</td><td>28.9±34.8</td><td>15.2 ± 0.7</td></tr><tr><td>Meta seq2seq (Lake, 2019)</td><td>NA</td><td>99.9</td><td>99.9*</td><td>16.64</td></tr><tr><td>seq2seq (comparable architecture)</td><td>100.0</td><td>0.0±0.0</td><td>0.02 ±2e-2</td><td>12.4 ± 2.3</td></tr><tr><td>Equivariant seq2seq (ours)</td><td>100.0</td><td>99.1 ± 0.04</td><td>92.0 ± 0.24</td><td>15.9 ± 3.2</td></tr></table>
+
+Table 1: Test accuracies for four SCAN tasks, comparing our equivariant seq2seq to the state-of-the-art.
+
+Training procedures match those of Lake & Baroni (2018) where possible. We train models for $2 0 0 k$ iterations, where each iteration consists of a minibatch of size 1, using the Adam optimizer to perform parameter updates with default parameters (Kingma & Ba, 2015) with a learning rate of 1e-4. We use teacher-forcing (Williams & Zipser, 1989) with a ratio of 0.5, and early-stopping based on a validation set consisting on $1 0 \%$ of the training examples. As in previous works, we compute test accuracies by counting how many exact translations each model provides, across the test set associated to each task.
+
+# 6.1 RESULTS
+
+Table 1 summarizes the results of our experiments. First and as expected, all models achieve excellent performance on the Simple task, which does not require any form of compositional generalization.
+
+Second, our equivariant seq2seq model performs very well at the Add jump and Around right SCAN tasks, which are the two tasks satisfying our local equivariance assumption from Definition 3. Our equivariant seq2seq model significantly outperforms the regular seq2seq (Lake & Baroni, 2018) and convolutional (Dessì & Baroni, 2019) models, as well as the state-of-the-art methods of Russin et al. (2019) and Lake (2019). This result is an encouraging piece of evidence supporting our main hypothesis from Section 3. Next, let us compare the results of our equivariant seq2seq model with the previous state-of-the-art Russin et al. (2019); Lake (2019) in more detail.
+
+On the one hand, the syntactic attention model of Russin et al. (2019) achieves significant improvements over baselines methods at the Add jump SCAN task. However, it does not fare so well on the Around right task. Furthermore, its performance has high variance. Although we here report the numbers from Russin et al. (2019), we observed such high variance in our own implementation as well, where the model often achieved $0 \%$ test accuracy. We hypothesize that modeling the invariance of the syntactic attention directly would result in improved performance and stability. This can be achieved, for instance, by replacing all verbs in the syntactic module by a shared word. As expected, by explicitly exploiting equivariance, our model outperforms Russin et al. (2019) on the Add jump and Around right SCAN tasks, also being much more robust.
+
+On the other hand, the meta-learning model of Lake (2019) achieves excellent performance on the local equivariance tasks Add jump and Around right. This is additional evidence supporting the usefulness of local equivariance. In contrast to our model, Lake (2019) requires (i) a complicated model and training procedure tailored to each task, (ii) providing the model with the correct permutation of words, equivalent to telling the model the “true” mappings between the input and output words, and (iii) augmenting the set of words being permuted, to ensure enough diversity in the training distribution (for instance, adding additional directions beyond RIGHT and LEFT).
+
+As seen in Table 1, length generalization remains a tough challenge in SCAN. While generating long sequences is a known challenge in seq2seq models (Bahdanau et al., 2015), we believe that this is not the main issue with our equivariant seq2seq model, as it is able to produce long translations when these appear in the training set (as are the other models). Therefore, this is not a capacity problem, but one of not being able to express the Length generalization SCAN task in terms of local equivariances on both input- and output- languages. We hypothesize that this is the very reason why (Russin et al., 2019; Lake, 2019) also fail on this task.
+
+However, we suspect that some forms of local equivariance on the input language, but global equivariance on the output language, may help. For example, RUN TWICE, RUN THRICE and RUN AROUND LEFT TWICE are all input commands contained in the training set of the length task. A trained seq2seq model is able to execute them, but fails on the unseen test command RUN AROUND LEFT THRICE, suggesting that the network did not correctly understand the relationship between TWICE and THRICE. Using a network that is explicitly equivariant to the permutation of TWICE and THRICE should be able to generalize correctly on RUN AROUND LEFT THRICE. However, while the TWICE-THRICE permutation is a local group operation (Definition 2), the corresponding operation on the output language, which is to repeat the same action sequence multiple times, is a global group operation. Similarly, permuting AND and AFTER in the input sequence using a local group operation, while operating globally on the output language by permuting the order of the associated actions, should help succeed on the Length generalization SCAN task. How to formalize the aforementioned global operations on the output language and build the desired equivariant network remains a fascinating open research question that we leave for future work.
+
+# 7 DISCUSSION AND FUTURE WORK
+
+This work has introduced hypothesis linking between group equivariance and compositional generalization in language. Motivated by this hypothesis, we have proposed an equivariant seq2seq translation model, which achieves state-of-the-art performance on a variety of SCAN tasks.
+
+Our work has several points for improvement. Most importantly, our model requires knowing the permutation symmetries of interest, to be provided by some domain expert. While this is simple to do in the synthetic language of SCAN, it may prove more difficult in real-world tasks. We propose three directions to attack this problem. (i) Group words by their parts-of-speech (e.g., nouns, verbs, etc.), which can be done automatically by standard part-of-speech taggers (Màrquez & Rodríguez, 1998); (ii) Learn such groupings of words from corpora, for example using the recent work of Andreas (2019); (iii) Most appealingly, parameterize the symmetry group and learn operations end-to-end while enforcing the group structure. For permutation symmetries, the group elements can be parameterized by permutation matrices, and learned from data (Lyu et al., 2019). Our preliminary work in this direction hints that this is a fruitful avenue for future research.
+
+A further consideration to address is that of computational overhead. In particular, for the convolutional form we use in this work (Definition 4), computational complexity scales linearly with the size of the group, $\mathcal { O } ( | G | )$ . This arises from the need to sum over group elements when the representation is a function on $G$ , and may be prohibitive when considering large groups. One way of addressing this issue when large symmetry groups are of interest is to consider more efficient computational layers for permutation equivariance (e.g Zaheer et al., 2017; Ravanbakhsh et al., 2017). These methods incur less computational overhead at the cost of restricting the layer capacity. Another interesting option for future research is to consider sub-sampling group elements when performing the summation in Definition 4, which requires further consideration of the consequences of doing so.
+
+Another exciting direction for future research is to consider global equivariances. Many operations of interest, e.g. groups operating directly on parse trees, can only be expressed as global equivariances. Modeling these equivariances holds exciting possibilities for capturing non-trivial symmetries in language tasks, but also requires more sophisticated machinery than is proposed in this work.
+
+Finally, in further theoretical work, we would like to explore the relation between our equivariance framework and the idea of compositionality in formal semantics (Kratzer & Heim, 1998). On the one hand, the classic idea of compositionality as an isomorphism between syntax and semantics is intuitively related to the notion of group equivariance. On the other hand, as shown by the failures at the length generalization example, it is still unclear how to apply our ideas to more sophisticated forms of permutation, such as those involving grammatical phrases rather than words. This would also require to extend our approach to account for the context-sensitivity that pervades linguistic composition (c.f., the natural interpretation of “run” in “run the marathon” vs. ”run the code”).
+
+# ACKNOWLEDGMENTS
+
+We thank Emmanuel Dupoux and Clara Vania for helpful feedback and discussions.
+
+# REFERENCES
+
+Jacob Andreas. Good-enough compositional data augmentation, 2019.
+
+Martin Arjovsky, Léon Bottou, Ishaan Gulrajani, and David Lopez-Paz. Invariant risk minimization. arXiv preprint arXiv:1907.02893, 2019.
+
+Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. Neural machine translation by jointly learning to align and translate. In Proceedings of ICLR Conference Track, San Diego, CA, 2015. Published online: http://www.iclr.cc/doku.php?id $\underline { { \underline { { \mathbf { \Pi } } } } } =$ iclr2015:main.
+
+Marco Baroni. Linguistic generalization and compositionality in modern artificial neural networks. arXiv preprint arXiv:1904.00157, 2019.
+
+Benjamin Bloem-Reddy and Yee Whye Teh. Probabilistic symmetry and invariant neural networks. arXiv preprint arXiv:1901.06082, 2019.
+
+Taco Cohen and Max Welling. Group equivariant convolutional networks. In International conference on machine learning, pp. 2990–2999, 2016.
+
+Roberto Dessì and Marco Baroni. CNNs found to jump around more skillfully than RNNs: Compositional generalization in seq2seq convolutional networks. arXiv preprint arXiv:1905.08527, 2019.
+
+Chelsea Finn, Pieter Abbeel, and Sergey Levine. Model-agnostic meta-learning for fast adaptation of deep networks. In Proceedings of the 34th International Conference on Machine Learning-Volume 70, pp. 1126–1135. JMLR. org, 2017.
+
+Jonathan Gordon, John Bronskill, Matthias Bauer, Sebastian Nowozin, and Richard E Turner. Metalearning probabilistic inference for prediction. arXiv preprint arXiv:1805.09921, 2018.
+
+Irina Higgins, David Amos, David Pfau, Sebastien Racaniere, Loic Matthey, Danilo Rezende, and Alexander Lerchner. Towards a definition of disentangled representations. arXiv preprint arXiv:1812.02230, 2018.
+
+Sepp Hochreiter and Jürgen Schmidhuber. Long short-term memory. Neural computation, 9(8): 1735–1780, 1997.
+
+Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In In International Conference on Learning Representations (ICLR), 2015.
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+Imre Risi Kondor. Group theoretical methods in machine learning. Columbia University, 2008.
+
+Risi Kondor and Shubhendu Trivedi. On the generalization of equivariance and convolution in neural networks to the action of compact groups. arXiv preprint arXiv:1802.03690, 2018.
+
+Angelika Kratzer and Irene Heim. Semantics in generative grammar, volume 1185. Blackwell Oxford, 1998.
+
+Brenden M Lake. Compositional generalization through meta sequence-to-sequence learning. arXiv preprint arXiv:1906.05381, 2019.
+
+Brenden M. Lake and Marco Baroni. Generalization without systematicity: On the compositional skills of sequence-to-sequence recurrent networks. In Proceedings of the 35th International Conference on Machine Learning, ICML 2018, Stockholm, Sweden, pp. 2879–2888, 2018.
+
+João Loula, Marco Baroni, and Brenden Lake. Rearranging the familiar: Testing compositional generalization in recurrent networks. pp. 108–114, 01 2018. doi: 10.18653/v1/W18-5413.
+
+Minh-Thang Luong, Hieu Pham, and Christopher D Manning. Effective approaches to attention-based neural machine translation. arXiv preprint arXiv:1508.04025, 2015.
+
+Jiancheng Lyu, Shuai Zhang, Yingyong Qi, and Jack Xin. AutoShuffleNet: Learning permutation matrices via an exact lipschitz continuous penalty in deep convolutional neural networks. arXiv preprint arXiv:1901.08624, 2019.
+
+Lluís Màrquez and Horacio Rodríguez. Part-of-speech tagging using decision trees. In European Conference on Machine Learning, pp. 25–36. Springer, 1998.
+
+Siamak Ravanbakhsh, Jeff Schneider, and Barnabas Poczos. Equivariance through parameter-sharing. In Proceedings of the 34th International Conference on Machine Learning-Volume 70, pp. 2892– 2901. JMLR. org, 2017.
+
+Jake Russin, Jason Jo, and Randall C O’Reilly. Compositional generalization in a deep seq2seq model by separating syntax and semantics. arXiv preprint arXiv:1904.09708, 2019.
+
+Jürgen Schmidhuber. Evolutionary principles in self-referential learning, or on learning how to learn: the meta-meta-... hook. PhD thesis, Technische Universität München, 1987.
+
+Ilya Sutskever, Oriol Vinyals, and Quoc V Le. Sequence to sequence learning with neural networks. In Advances in neural information processing systems, pp. 3104–3112, 2014.
+
+Sebastian Thrun and Lorien Pratt. Learning to learn. Springer Science & Business Media, 2012.
+
+Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in neural information processing systems, pp. 5998–6008, 2017.
+
+Maurice Weiler, Fred A Hamprecht, and Martin Storath. Learning steerable filters for rotation equivariant cnns. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 849–858, 2018.
+
+Ronald J Williams and David Zipser. A learning algorithm for continually running fully recurrent neural networks. Neural computation, 1(2):270–280, 1989.
+
+Manzil Zaheer, Satwik Kottur, Siamak Ravanbakhsh, Barnabas Poczos, Ruslan R Salakhutdinov, and Alexander J Smola. Deep sets. In Advances in Neural Information Processing Systems, pp. 3394–3404, 2017.
+
+# A DETAILS ON THE SCAN DATASET
+
+SCAN is composed from a non-recursive grammar, as shown in Figure 3. In particular, SCAN consists of all commands that can be generated from this grammar (20,910 command sequences), with their deterministic mapping into actions, as detailed by Figure 4
+
+$$
+{ \begin{array} { r l r l } & { \mathbf { C } \to \mathbf { S } { \mathrm { ~ a n d ~ } } \mathbf { S } } & & { \mathbf { V } \to \mathbf { D } [ 1 ] { \mathrm { ~ o p p o s i t e ~ } } \mathbf { D } [ 2 ] } & { \mathbf { D } \to { \mathrm { t u r n ~ l e f t } } } \\ & { \mathbf { C } \to \mathbf { S } { \mathrm { ~ a f t e r ~ } } \mathbf { S } } & & { \mathbf { V } \to \mathbf { D } [ 1 ] { \mathrm { ~ a r o u n d ~ } } \mathbf { D } [ 2 ] } & { \mathbf { D } \to { \mathrm { t u r n ~ r i g h t } } } \\ & { \mathbf { C } \to \mathbf { S } } & & { \mathbf { V } \to \mathbf { D } } & & { \mathbf { U } \to \mathbf { w a l k } } \\ & { \mathbf { S } \to \mathbf { V } { \mathrm { ~ t w i c e } } } & & { \mathbf { V } \to \mathbf { U } } & { \mathbf { U } \to \mathrm { l o o k } } \\ & { \mathbf { S } \to \mathbf { V } { \mathrm { ~ t h r i c e } } } & & { \mathbf { D } \to \mathbf { U } \mathrm { l e f t } } & { \mathbf { U } \to \mathrm { r u n } } \\ & { \mathbf { S } \to \mathbf { V } } & & { \mathbf { D } \to \mathbf { U } { \mathrm { ~ r i g h t } } } & { \mathbf { U } \to \mathbf { j u m p } } \end{array} }
+$$
+
+Figure 3: The grammar used to generate commands in the SCAN domain. Indexing notation is used to allow infixing: read ${ \bf \widehat { \mathbf { } } } D [ i ]$ as “the $_ { i }$ -th element directly dominated by category $D '$ . Image borrowed from Lake & Baroni (2018).
+
+![](images/b2b3db604764fa21c60f32c6a50a99b4b1e8272cd16c2c2509857634fbf767cf.jpg)  
+Figure 4: The SCAN translation mapping. Double brackets denote the interpretation function translating SCAN’s command (input language) into the action (output) language (which are denoted by upper-case strings. Image borrowed from Lake & Baroni (2018).
+
+# B G-LSTM DETAILS
+
+We provide the equations for implementing our G-LSTM. Given $h _ { t - 1 } , c _ { t - 1 }$ (hidden state and cellstate, respectively), and $e ( w ) _ { t }$ (all of which are of the form $G \mapsto \mathbb { R } ^ { K }$ , we can describe the G-LSTM cell as follows:
+
+$$
+\begin{array} { r l r l } & { i _ { t } = \sigma \left( { \pmb x } _ { t } \ast \psi _ { i i } + s _ { t - 1 } \ast \psi _ { i h } \right) ; \quad \quad \quad } & & { f _ { t } = \sigma \left( { \pmb x } _ { t } \ast \psi _ { f i } + s _ { t - 1 } \ast \psi _ { f h } \right) } \\ & { g _ { t } = \operatorname { t a n h } \left( { \pmb x } _ { t } \ast \psi _ { g i } + s _ { t - 1 } \ast \psi _ { g h } \right) ; \quad \quad \quad } & & { o _ { t } = \sigma \left( { \pmb x } _ { t } \ast \psi _ { o i } + s _ { t - 1 } \ast \psi _ { o h } \right) } \\ & { c _ { t } = f _ { t } \circ { \pmb c } _ { t - 1 } + i _ { t } \circ g _ { t } ; \quad \quad \quad } & & { h _ { t } = o _ { t } \operatorname { c a n h } ( { \pmb c } _ { t } ) , } \end{array}
+$$
+
+where $\{ \psi _ { j k } : G \mapsto \mathbb { R } ^ { K } ; j \in \{ i , f , g , o \} ; k \in \{ i , h \} \}$ are the learnable filters of the cell. Here we have used the shorthand
+
+$$
+\pmb { f } \ast \pmb { \psi } : = \left[ \pmb { f } \ast \pmb { \psi } \right] ( \pmb { g } ) \quad \forall \pmb { g } \in G
+$$
+
+for two functions on the group.

parse/train/SylVNerFvr/SylVNerFvr_content_list.json

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+    {
+        "type": "text",
+        "text": "PERMUTATION EQUIVARIANT MODELS FOR COMPOSITIONAL GENERALIZATION IN LANGUAGE ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "Jonathan Gordon∗ University of Cambridge jg801@cam.ac.uk ",
+        "bbox": [
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+            212
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "David Lopez-Paz, Marco Baroni, Diane Bouchacourt Facebook AI Research {dlp, mbaroni, dianeb}@fb.com ",
+        "bbox": [
+            401,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            250,
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+            263
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Humans understand novel sentences by composing meanings and roles of core language components. In contrast, neural network models for natural language modeling fail when such compositional generalization is required. The main contribution of this paper is to hypothesize that language compositionality is a form of group-equivariance. Based on this hypothesis, we propose a set of tools for constructing equivariant sequence-to-sequence models. Throughout a variety of experiments on the SCAN tasks, we analyze the behavior of existing models under the lens of equivariance, and demonstrate that our equivariant architecture is able to achieve the type compositional generalization required in human language understanding. ",
+        "bbox": [
+            233,
+            284,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            448,
+            336,
+            463
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "When using language, humans recombine known concepts to understand novel sentences. For instance, if one understands the meaning of “run”, “jump”, and “jump twice”, then one understands the meaning of “run twice”, even if such sentence was never heard before. This relies on the notion of language compositionality, which states that the meaning of a sentence (“jump twice”) is to be obtained by the meaning of its constituents (e.g. the verb “jump\" and the quantifying adverb “twice\") and the use of algebraic computation (a verb combined with a quantifying adverb $m$ results in doing that verb $m$ times) (Kratzer & Heim, 1998). ",
+        "bbox": [
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+            473,
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+            570
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In the realm of machines, deep learning has achieved unprecedented results in language modeling tasks (Bahdanau et al., 2015; Vaswani et al., 2017). However, these models are sample inefficient, and do not generalize to examples that require the use of language compositionality (Lake & Baroni, 2018; Loula et al., 2018; Dessì & Baroni, 2019). This result suggests that deep language models fail to leverage compositionality; a failure remaining to this day a roadblock towards true natural language understanding. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Focusing on this issue, Lake & Baroni (2018) proposed the Simplified version of the CommAI Navigation (SCAN), a dataset to benchmark the compositional generalization capabilities of state-ofthe-art sequence-to-sequence (seq2seq) translation models (Sutskever et al., 2014; Bahdanau et al., 2015). In a nutshell, the SCAN dataset contains compositional navigation commands such as JUMP TWICE AFTER RUN LEFT, to be translated into the sequence of actions LTURN RUN JUMP JUMP. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Using SCAN, Lake & Baroni (2018) demonstrated that seq2seq models fail spectacularly at tasks requiring the use of language compositionality. Following our introductory example, models trained on the three commands JUMP, RUN and JUMP TWICE fail to generalize to RUN TWICE. Most recently, Dessì & Baroni (2019) showed that architectures based on temporal convolutions meet the same fate. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "SCAN did not only reveal the lack of compositionality in language models, but it also became the blueprint to build novel language models able to handle language compositionality. On the one hand, Russin et al. (2019) proposed a seq2seq model where semantic and syntactic information are represented separately, in a hope that such disentanglement would elicit compositional rules. However, their model was not able to solve all of the compositional tasks comprising SCAN. On the other hand, Lake (2019) introduced a meta-learning approach with excellent performance in multiple ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "SCAN tasks. However, their method requires substantial amounts of additional supervision, and a complex meta-learning procedure hand-engineered for each task. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we take a holistic look at the problem and connect language compositionality in SCAN to the disparate literature in models equivariant to certain group symmetries (Kondor, 2008; Cohen & Welling, 2016; Ravanbakhsh et al., 2017; Kondor & Trivedi, 2018). Interesting links have recently been proposed between group symmetries and the areas of causality (Arjovsky et al., 2019) and disentangled representation learning (Higgins et al., 2018), and this work proceeds in a similar fashion. In particular, the main contribution of this work is not to chase performance numbers, but to put forward the novel hypothesis that language compositionality can be understood as a form of group-equivariance (Section 3). To sustain our hypothesis, we provide tools to construct seq2seq models equivariant when the group symmetries are known (Section 4), and demonstrate that these models solve all SCAN tasks, except length generalization (Section 6).1 ",
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+    {
+        "type": "text",
+        "text": "2 THE SCAN COMPOSITIONAL TASKS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            290,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The purpose of the Simplified version of the CommAI Navigation (SCAN) tasks (Lake & Baroni, 2018) is to benchmark the abilities of machine translation models for compositional generalization. Following prior literature (Lake & Baroni, 2018; Baroni, 2019; Russin et al., 2019; Andreas, 2019), compositional generalization is understood as the ability to translate novel families of sentences, when this requires leveraging the compositional structure in language. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The SCAN dataset contains compositional navigation commands in English (the input-language) paired with a desired action sequence (the output-language). For instance, the input-language sentence JUMP TWICE AND RUN LEFT is paired to the output-language actions sequence JUMP JUMP LTURN RUN. The rest of our exposition uses SMALL CAPS to denote examples in the input-language, and LARGE CAPS to denote examples in the output-language. Appendix A contains a full description of the grammar generating the SCAN language. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To evaluate the compositional generalization abilities of sequence-to-sequence (seq2seq) machine translation models (Sutskever et al., 2014; Bahdanau et al., 2015), Lake & Baroni (2018) proposes four main tasks based on SCAN: ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "1. Simple task: data pairs are randomly split into training and test sets. No compositional generalization is required.   \n2. Add jump task: the only command in the training set containing the verb JUMP is the command JUMP. All commands not containing JUMP are in the training set (for instance, RUN TWICE, and WALK RIGHT THRICE AND LOOK LEFT). The test set contains all commands containing JUMP (for instance, JUMP TWICE, and RUN LEFT AND JUMP RIGHT). To succeed in this task, models must learn that JUMP is a verb, and that any verb can be composed with an adverbial number to be repeated a number of times.   \n3. Around right task: the phrase AROUND RIGHT is held out from the training set; however, both AROUND and RIGHT are shown in all other contexts (for example, AROUND LEFT or OPPOSITE RIGHT). To succeed at this task, models must learn that both RIGHT and LEFT are directions, and can be combined with AROUND and OPPOSITE.   \n4. Length generalization task: the training set contains pairs such that the length of the action sequence in the output-language is shorter than 24 actions. The test set contains all pairs with action sequences of a length greater or equal than 24 actions. The type of compositional ability required to succeed at this task is more difficult to sketch out, as we discuss in Section   \n6.2. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Lake & Baroni (2018) use these four tasks to demonstrate that state-of-the-art seq2seq translation models (Bahdanau et al., 2015) succeed at Simple task, but fail at the other three tasks requiring compositional generalization. Convolutional architectures (Dessì & Baroni, 2019) achieve only slightly better performance, and state-of-the-art methods specially developed to address SCAN tasks fall short from the best achievable performance (Russin et al., 2019), or call for substantial amounts of additional supervision (Lake, 2019). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In the following, we take a holistic look at the language compositionality problems in SCAN, and highlight their connection to equivariant maps in group theory. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3 SCAN COMPOSITIONALITY AS GROUP EQUIVARIANCE",
+        "text_level": 1,
+        "bbox": [
+            178,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "This section puts forward the hypothesis that: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Models achieving the compositional generalization required in certain SCAN tasks are equivariant with respect to permutation group operations2 in the input and output languages. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To unfold the meaning of our hypothesis, we must revisit some basic concepts in group theory. A discrete group $G$ is a set of elements $\\{ g _ { 1 } , \\dotsc , g _ { | G | } \\}$ , equipped with a binary group operation “·” satisfying the four group axioms (closure, associativity, identity, and invertibility). The sequel focuses on permutation groups $G$ , whose elements are permutations of a set $\\mathcal { X }$ , and whose binary group operation composes the permutations contained in $G$ . The set of all permutations of $\\mathcal { X }$ is a group, but not all subsets of permutations of $\\mathcal { X }$ satisfy the four group axioms, and therefore they do not form a group. For each element $g \\in G$ , we define the group operation $T _ { g } : \\mathcal { X }  \\mathcal { X }$ as the map applying the permutation $g$ to the element $x \\in \\mathcal { X }$ , to obtain $T _ { g } x$ . Armed with these definitions, we are ready to introduce the main object of study in this paper: equivariant maps. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Definition 1 (Equivariant map). Let $\\mathcal { X }$ and $\\mathcal { V }$ be two sets. Let $G$ be a group whose group operation on $\\mathcal { X }$ is denoted by $T _ { g } : \\mathcal { X }  \\mathcal { X }$ , and whose group operation on $\\mathcal { V }$ is denoted by $T _ { g } ^ { \\prime } : \\mathcal { V } \\to \\mathcal { V }$ . Then, $\\Phi : \\mathcal { X }  \\mathcal { Y }$ is an equivariant map if and only if $\\Phi \\left( T _ { g } x \\right) = T _ { g } ^ { \\prime } \\Phi ( x )$ for all $x \\in \\mathcal { X }$ and $g \\in G$ . ",
+        "bbox": [
+            174,
+            380,
+            825,
+            425
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The operation groups $( T _ { g } , T _ { g } ^ { \\prime } )$ defined above operate on entire sequences, an enormous space when we consider those sequences to be language sentences. In the following two definitions, we relax group operations and equivariant maps to operate at a word level. ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Definition 2 (Local group operations). Let $\\mathcal { X }$ be a set of sequences (or sentences), where each sequence $x \\in \\mathcal { X }$ contains elements $x _ { i } \\in \\mathcal V$ from a vocabulary set $\\nu$ , for all $x _ { i } \\in x$ . Let $G$ be a group with associated group operation $T _ { g } : \\mathcal { X }  \\mathcal { X }$ . Then, we say that $T _ { g }$ is a local group operation if there exists a group operation $T _ { g _ { w } } : \\mathcal { V } \\stackrel { \\cdot } { \\times } \\mathcal { V }$ such that $T _ { g } x = ( T _ { g _ { w } } x _ { 1 } , \\ldots , T _ { g _ { w } } x _ { L _ { x } } )$ for all $x \\in \\mathcal { X }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "When understanding sequences as language sentences, the group operation $T _ { g _ { w } }$ would be a permutation of the words from the language vocabulary. Such operation can be implemented in terms of a permutation matrix, a $| \\nu | \\times | \\nu |$ matrix with zero/one entries where each row and each column sum to one. Finally, we leverage the definition of local group operations to define locally equivariant maps. ",
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+        "type": "text",
+        "text": "Definition 3 (Locally equivariant map). Let $\\mathcal { X }$ and $\\mathcal { V }$ be two sets of sequences. Let $G$ be a group whose group operation on $\\mathcal { X }$ is local in its vocabulary, denoted by $T _ { g } : \\mathcal { X }  \\mathcal { X }$ , and whose group operation on $\\mathcal { V }$ is local in its vocabulary and denoted by $T _ { g } ^ { \\prime } : \\mathcal { V } \\times \\overset { \\cdot } { \\mathcal { V } }$ . Then, we say that $\\Phi : \\mathcal { X }  \\mathcal { Y }$ is an equivariant map if and only if $\\Phi ( T _ { g } x ) = T _ { g } ^ { \\prime } \\Phi ( x )$ for all $x \\in \\mathcal { X }$ and $g \\in G$ . ",
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+    {
+        "type": "image",
+        "img_path": "images/62214294913b795f13f25fa3b0bb3772768f35b141b293fc27293c3c40b75421.jpg",
+        "image_caption": [
+            "Figure 1: (a) Commutative diagram for equivariance. (b) Local equivariance enables generalization to verb replacement in SCAN. (c) Local equivariance does not enable generalization to conjunction replacement in SCAN. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Now, how do equivariances and local equivariances manifest themselves in the world of SCAN? To assist our examples, the commutative diagram in Figure 1a summarizes the group theory notations introduced so far. In Figure 1b and Figure 1c, we parallel these notations to two different examples of compositional skills required to solve SCAN: verb and conjunction replacement. In the SCAN domain, $\\mathcal { X }$ is the set of sentences in the input-language, and $\\mathcal { V }$ is the set of sentences in the outputlanguage. Furthermore, let $\\Phi$ be a locally equivariant SCAN translation model, and let $G$ be a group with associated local group operations that permutes words in the input- and output- languages. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "On the one hand, we observe in Figure 1b that local equivariance enables compositional generalization in the case of verb replacement. This is because replacing one verb in the input-language can be implemented in terms of a local group operation. In turn, this input-verb replacement corresponds deterministically to a second local group operation that replaces the corresponding verb in the output-language. The same would apply to a SCAN task where we are interested in generalizing to the replacement of LEFT and RIGHT. As such, a translation model $\\Phi$ with these compositional generalization capabilities must be locally equivariant. ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "On the other hand, we observe in Figure 1c that local equivariance is insufficient to enable compositional generalization in the case of conjunction replacement. This is because no local group operation in the output-language would be able to implement the necessary changes induced by the replacement of AND by AFTER in the input-language. In such cases, we refer to the equivariance as global equivariance. In particular, we can see how blocks of multiple words in the output-language swap their relative location. Local equivariances are also insufficient to enable compositional generalization in the Length generalization SCAN task and we elaborate on this in Section 6.2. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In the following section, we propose a set of tools to implement equivariant seq2seq translation models, and propose a particular architecture with which we conduct our experiments. ",
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+    {
+        "type": "text",
+        "text": "4 IMPLEMENTING AN EQUIVARIANT SEQUENCE-TO-SEQUENCE MODEL ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We now implement our proposed equivariant seq2seq model, following the encoder-decoder architecture illustrated in Figure 2. Readers unfamiliar with group theory may parse Figure 2 by temporarily discarding the $^ { 6 6 } G - '$ ” prefixes, and realize that each depicted module is a well-known building block of recurrent neural network models. ",
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+        "img_path": "images/de7e289be9fe669ae8e8d3f107fe9665a01367089f8c4a00e8f1c4e9a5d75256.jpg",
+        "image_caption": [
+            "Figure 2: Architecture of our fully-equivariant seq2seq model. Variables shaded in gray are mappings $G \\to \\mathbb { R } ^ { K }$ , implemented as $| G | \\times K$ matrices. Encoder and decoder meet at $\\tilde { h } _ { 0 } : = h _ { L _ { x } }$ . "
+        ],
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+    {
+        "type": "text",
+        "text": "To make our model equivariant, we will make intense use of group convolutions. ",
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+        "type": "text",
+        "text": "Definition 4 (Group convolution (Kondor & Trivedi, 2018)). Let $G$ be a discrete group. Let $f : G $ $\\mathbb { R } ^ { K }$ be an input function. Let $\\psi = \\{ \\psi ^ { i } : G \\to \\mathbb { R } ^ { K } \\} _ { i = 1 } ^ { K ^ { \\prime } }$ be a set of learnable filter functions. Then, each scalar real entry from the result of $G$ -convolving $f$ and $\\psi$ is given by a $| G | \\times K ^ { \\prime }$ matrix with entries ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "equation",
+        "img_path": "images/1f0589ce8705876a6603955b6afe47d8fbbb5e74b187909fe69677514213d699.jpg",
+        "text": "$$\nG \\mathrm { - } \\mathrm { C o n v } ( f ; \\psi ) _ { g , i } = \\sum _ { h \\in \\mathrm { d o m } ( f ) } \\sum _ { k = 1 } ^ { K } f _ { k } ( h ) \\psi _ { k } ^ { i } ( g ^ { - 1 } h ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "for all $g \\in G$ and $i \\in \\{ 1 , \\ldots , K ^ { \\prime } \\}$ . As shown by (Kondor $\\&$ Trivedi, 2018), the $G$ -Conv layer is equivariant wrt the operations of $G$ . We apply this definition in two ways: (i) “convolving” words with learnable filters to generate equivariant embeddings. Later, when we introduce our notations, we discuss how words may be viewed as functions so as to fit the definition. And (ii) convolving two group representations, in which case $\\operatorname { d o m } ( f ) = G$ . ",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We note that there are several additional methods proposed in the literature for constructing permutation equivariant layers (e.g. , Zaheer et al., 2017; Ravanbakhsh et al., 2017). However, as demonstrated by Kondor & Trivedi (2018); Bloem-Reddy & Teh (2019), the above form is very general and subsumes most alternatives. Further, while layers based on weight-sharing may be more efficient than the general form of Definition 4, the parameter tying restricts the capacity of the layer. For example, the permutation equivariant layer of Zaheer et al. (2017) requires weight matrices that are restricted to a form $\\lambda I + { \\overset { \\cdot } { \\gamma } } ( \\mathbf { 1 1 } ) ^ { T }$ , with learnable parameters $\\lambda$ and $\\gamma$ . This layer has fewer learnable parameters than the convolutional form of Definition 4. Thus, for reasons of generality and capacity, we employ the general and expressive convolutional form of Definition 4 for our permutation equivariant layers. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Equivariant with respect what group? The previous $G$ -Conv layer requires choosing a discrete group $G$ . As hinted in Section 3, we will choose $G$ to contain $| G |$ permutations of language vocabularies, e.g. products of cyclic groups on sets of words. Note that for a vocabulary size of $| V |$ , the set of all permutations has a size of $n !$ . However, it suffices to consider subgroups containing permutations such that every word can be reached by composing elements of the subgroup. For example, while the group of permutations on the four verbs in SCAN consists of 24 elements, it will suffice to choose $G$ as the circular shift group on the four verbs, which is a subgroup of four elements. Following standard notation in group theory, we write $g \\cdot h$ to denote the composition of two group elements $g , h \\in G$ , and $g ^ { - 1 }$ to denote the inverse element of $g$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "As final preliminaries, denoting $[ V ] = \\{ 1 , \\dots , | V | \\}$ , we represent a word $w$ in the input-language by the function $w : [ V ] \\to \\{ 0 , 1 \\}$ , where $\\begin{array} { r } { \\sum _ { v \\in [ V ] } w ( v ) = 1 } \\end{array}$ , and similarly by using $\\tilde { w } \\in \\tilde { V }$ for the output-language. These notations are functional representations of word one-hot encodings that will play well with our notations. Note that this representation is equivalent to one-hot vectors, and in what follows we use the shorthand $w$ for the one-hot vector representation of words. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To avoid notational clutter, we use $g$ to denote the permutation-matrix-representation of the corresponding group element. Thus, the group operation on a word gw can be implemented as matrix multiplication between the permutation matrix $g$ and the one-hot vector $w$ . Note that this operation results in another one-hot vector, i.e. another word in the vocabulary. Similarly, the binary group operation can be written as matrix multiplication $g h$ between two group members $g , h \\in G$ . Here too, multiplication of permutation matrices results in permutation matrix, so $g h \\in G$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We now describe each of the components in our $G$ -equivariant translation model, by following the transformation process of an input sequence $x = ( w _ { 1 } , \\ldots , w _ { L _ { x } } )$ (in SCAN, a navigation command in English) into its output translation $y = ( \\tilde { w } _ { 1 } , \\dots , \\tilde { w } _ { L _ { y } } )$ (in SCAN, a sequence of actions). ",
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+    {
+        "type": "text",
+        "text": "4.1 G-EQUIVARIANT ENCODER ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Upon arrival, the input-language sentence $x = ( w _ { 1 } , \\dots , w _ { L _ { x } } )$ is sent to a $G$ -equivariant encoder. The first step in the encoding process is to transform each input word $w _ { t }$ into a permutation equivariant embedding $e ( w _ { t } )$ . As mentioned before, each word $w _ { t }$ is represented by the one-hot vector $w _ { t } :$ $[ V ]  \\{ 0 , 1 \\}$ . The corresponding embedding is obtained by applying a set of $K$ 1-dimensional learnable filter functions $\\{ \\bar { \\psi } ^ { i } : [ V ] \\stackrel { - } { \\to } \\mathbb { R } \\} _ { i = 1 } ^ { K }$ in a group convolution (throughout the section, we use $K$ everywhere to ease notation). Using Definition 4, the embedding, which we call $G$ -Embed, is then represented as a matrix $\\mathbb { R } ^ { | G | \\times K }$ , where ",
+        "bbox": [
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+    {
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+        "img_path": "images/0b67eb27e276a793517b61c490e06930b84bcdc017d503e68e20d5b8236b2e71.jpg",
+        "text": "$$\ne ( w ) _ { g , i } = G \\mathrm { - E m b e d } ( w ; \\psi ) _ { g , i } = \\psi ^ { i } ( g ^ { - 1 } w ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "for all $g \\in G$ and $i = \\{ 1 , \\ldots , K \\}$ . Note that since $w$ is a one-hot vector, $G$ -Embed is a particularly simple instantiation of Definition 4, as summation over $\\operatorname { d o m } ( f )$ consists of only a single term. The corresponding embedding is a function $e ( w _ { t } ) : G \\to \\mathbb { R } ^ { K }$ , which can be represented as a $| G | \\times K$ matrix, where each row corresponds to the embedding of the word $g w$ for a particular $g \\in G$ . This layer can be implemented by defining $\\psi$ with standard deep learning embedding modules.3 ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Importantly, we note that for this layer, both $\\psi$ and $w$ are functions on $[ V ]$ . However, the resulting embedding $e ( w )$ is a function on the group $G$ . Therefore, in all subsequent computations we will require the learnable filters $\\psi$ to also be functions on $G$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We illustrate this layer with an example. Let $G$ be the cyclic group that permutes the words LEFT and RIGHT. We can think of $g _ { 1 }$ as the identity, and $g _ { 2 }$ as permuting the words LEFT and RIGHT (leaving all other words unchanged). In this case, embedding LEFT results in the $2 \\times K$ matrix $[ \\psi ( \\mathrm { L E F T } ) ^ { T } , \\psi ( \\mathrm { R I G H T } ) ^ { T } ] ^ { T }$ , while embedding JUMP results in $\\mathsf { \\bar { \\Psi } } ( \\mathbf { J } \\mathbf { U } \\mathbf { M } \\mathbf { P } ) ^ { T } , \\psi ( \\mathbf { J } \\mathbf { U } \\mathbf { M } \\mathbf { P } ) ^ { T } ] ^ { T }$ , since both $g _ { 1 }$ and $g _ { 2 }$ act as the identity permutation for JUMP. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Next, the word embedding $e ( w _ { t } )$ is sent to a permutation equivariant Recurrent Neural Network ( $G$ -RNN). The cells of a $G$ -RNN mimic those of a standard RNN, where linear transformations are replaced by $G$ -Convs (Definition 4). This cell receives two inputs (the word embedding $e ( w _ { t } )$ and the previous hidden state $h _ { t - 1 }$ ) and returns one output (the current hidden state $h _ { t }$ ), all three being functions $G \\to \\mathbb { R } ^ { K }$ , parametrized as $| G | \\times K$ matrices. More specifically: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/e3476b264cba61d32d922f690ed925fb3b11a8b76b3f3ce0b6570ff64c5f1983.jpg",
+        "text": "$$\nh _ { t } = G \\mathrm { - R N N } ( e ( w _ { t } ) , h _ { t - 1 } ) = \\sigma ( G \\mathrm { - C o n v } ( h _ { t - 1 } ; \\psi _ { h } ) + G \\mathrm { - C o n v } ( e ( w _ { t } ) ; \\psi _ { e } ) ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\psi _ { h } , \\psi _ { e } \\colon G \\to \\mathbb { R } ^ { K }$ are learnable filters (represented as $| G | \\times K$ matrices), and $\\sigma$ is a point-wise activation function. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The cell $G$ -RNN is equivariant because the sum of two equivariant representations is equivariant (Cohen & Welling, 2016), and the pointwise transformation of an equivariant representation is also equivariant. To initialize the hidden state, we set $h _ { 0 } = \\vec { 0 }$ . We note that our experiments use the equivariant analog of LSTM cells (Hochreiter & Schmidhuber, 1997), which we denote $G$ -LSTM, since these achieved the best performance. We include the architecture of $G$ -LSTM cells in Appendix B. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This completes the description of our equivariant encoder, illustrated in Figure 2a. ",
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+    {
+        "type": "text",
+        "text": "4.2 G-EQUIVARIANT DECODER ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Once the entire input-language sentence $x = ( w _ { 1 } , \\ldots , w _ { L _ { x } } )$ has been encoded into the hidden representations $h = ( h _ { 1 } , \\ldots , h _ { L _ { x } } )$ , we are ready to start the decoding process that will produce the output-language translation $y = ( \\tilde { w } _ { 1 } , \\dots , \\tilde { w } _ { L _ { y } } )$ . ",
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+    {
+        "type": "text",
+        "text": "As illustrated in Figure 2b, our equivariant decoder is also run by an equivariant recurrent cell $G$ -RNN. We denote the hidden states of the recurrent decoding process by $\\tilde { h } _ { t }$ , where $\\tilde { h } _ { 0 } = h _ { L _ { x } }$ . At time $t$ , the two inputs to the decoding $G$ -RNN cell are the previous hidden state $\\tilde { h } _ { t - 1 }$ as well as an attention ${ { \\bar { a } } _ { t } }$ over all the encoding hidden states $h$ . (Once again, all variables are mappings $G \\to \\mathbb { R } ^ { K }$ implemented as $| G | \\times K$ matrices.) ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Attention mechanisms (Bahdanau et al., 2015; Vaswani et al., 2017) have emerged as a central tool in language modelling. Fortunately, attention mechanisms are typically implemented as linear combinations, and a linear combination of equivariant representations is itself an equivariant representation. We now leverage this fact to develop an equivariant attention mechanism. Given all the encoder hidden states $h$ , as well as the previous decoding hidden state $\\tilde { h } _ { t - 1 }$ , we propose the equivariant analog of dot-product attention (Luong et al., 2015) as ",
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+    {
+        "type": "equation",
+        "img_path": "images/12fa98e6e771a8a681c53d3100b6c2d6a72d8510d18c4a31184a30e3904cf1d0.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\bar { a _ { t } } = G \\mathrm { - } \\mathrm { A t t e n t i o n } ( \\tilde { h } _ { t - 1 } , h ) = \\displaystyle \\sum _ { j = 1 } ^ { L _ { x } } \\alpha _ { t , j } h _ { j } \\mathrm { , ~ w h e r e ~ } } \\\\ & { \\alpha _ { t , j } = \\displaystyle \\frac { \\exp \\beta _ { t , j } } { \\sum _ { k = 1 } ^ { L _ { x } } \\exp \\beta _ { t , k } } \\mathrm { , ~ a n d ~ } \\beta _ { t , j } = \\displaystyle \\sum _ { g \\in G } \\tilde { h } _ { t - 1 } ( g ) ^ { \\top } h _ { j } ( g ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Following Figure 2b, the attention $\\bar { a _ { t } }$ and a $G$ -embedding $e ( \\tilde { w } _ { t - 1 } )$ for the previous output word are concatenated and sent to a $G$ -Convolution.4 The concatenation with $e ( \\tilde { w } _ { t - 1 } )$ provides the decoder with information regarding the previously embedded word. In practice, during training we use teacher-forcing (Williams $\\&$ Zipser, 1989) to provide the decoder with information about the correct output sequences. This process returns a final hidden representation $\\phi : G \\to \\mathbb { R } ^ { K }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "As a final step in the decoding process, we need to convert $\\phi$ into a collection of logits over the output-language vocabulary. Then, sampling from the categorical distribution induced by these logits at time $t$ (or taking the maximimum) will produce the word $\\tilde { w } _ { t }$ , to be appended in the output-language translation, $y$ . This final decoding module can be implemented as follows: ",
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+        "page_idx": 6
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+    {
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+        "img_path": "images/045665848c47c1dae87624eae604f28ae808ca1a5a24513f6e6a3abdb866b1b3.jpg",
+        "text": "$$\nG \\mathrm { - D e c o d e } ( \\phi ; \\psi ) _ { \\tilde { w } } = \\sum _ { h \\in G } \\sum _ { k = 1 } ^ { K } \\phi _ { k } ( h ) \\psi _ { k } ( h ^ { - 1 } \\tilde { w } ) ,\n$$",
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+    {
+        "type": "text",
+        "text": "where $\\psi = [ \\tilde { V } ]  \\mathbb { R } ^ { k }$ are the learnable parameters of this layer (represented by a $| \\tilde { V } | \\times K$ matrix). ",
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+    {
+        "type": "text",
+        "text": "Recall that $\\phi ( h ) \\in \\mathbb { R } ^ { K }$ is the final-layer representation for the group element $h$ , and that $h ^ { - 1 } \\tilde { w }$ is the inverse element of $h \\in G$ applied to the output word $\\tilde { w }$ (represented as a one-hot vector), which results in another word in the output language. Thus, $\\psi$ is a learnable embedding of the output words into $\\mathbb { R } ^ { K }$ . This layer is evaluated at every $\\tilde { w }$ in the output vocabulary to produce a scalar. The resulting vector of logits represents a categorical distribution over the output vocabulary. While similar, this layer is not a group convolution (Definition 4). Rather, equivariance for this module is achieved via parameter-sharing (Ravanbakhsh et al., 2017). ",
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+        "type": "text",
+        "text": "This completes the description of our equivariant decoder, illustrated in Figure 2b. Composing the equivariant encoder and decoder results in our complete sequence-2-sequence model. Importantly, since all operations in this model are equivariant, the complete model is itself also equivariant to the group $G$ (Kondor & Trivedi, 2018). In Section 6, we provide further implementation details for our model, and detail our empirical evaluation of its equivariant properties and their relation to the SCAN tasks described in Section 2. ",
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+        "type": "text",
+        "text": "5 RELATED WORK ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section we review state-of-the-art methods to address SCAN compositional tasks. We focus on two recent models that we will compare to in our experiments. ",
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+        "text": "On the one hand, the syntactic attention model of Russin et al. (2019) builds on the idea that compositional generalization can be achieved by language models given the correct architectural organization. Borrowing inspiration from neuroscience, Russin et al. (2019) argue that compositionality might arise when using separate processing channels for semantic and syntactic information. In their model, the attention weights depend on a recurrent encoding of the input sequence, which they refer to as the syntactic representation. The attention weights are then applied to separate, context-independent embeddings of the words in the input sequence, which intend to model a semantic representation. We find (Russin et al., 2019) interesting from a group equivariance perspective, since one way to enforce equivariance is to use an invariant representation (about syntax) together with an additional representation (about semantics) that maintains the information about the original “sentence pose”. ",
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+        "text": "On the other hand, the meta-learning (Thrun & Pratt, 2012; Schmidhuber, 1987) approach of Lake (2019) is a model that learns to generalize. In particular, Lake (2019) designs one specific and complex meta-learning procedure for each SCAN task, where a distribution over tasks is provided to the learner (Finn et al., 2017; Gordon et al., 2018). For example, in the Add jump and Around right tasks, the meta-learning procedure of Lake (2019) samples permutations from the relevant groups (the permutation groups on the verbs and set of directions, respectively). This is interpreted as data-augmentation, a valid procedure for encouraging equivariance (Cohen & Welling, 2016; Andreas, 2019; Weiler et al., 2018). However, at test-time, Lake (2019) sets the context set to the correct mapping between the permuted commands and their corresponding actions. For example, in the Add jump task, the context set for meta-testing would consist of the following pairs: {(WALK, WALK), (RUN, RUN), (LOOK, LOOK), (JUMP, JUMP) }. This is equivalent to providing the model with one-to-one information regarding the correct command-to-action mapping for the permuted words. ",
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+    {
+        "type": "text",
+        "text": "6 EXPERIMENTS ",
+        "text_level": 1,
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+        "text": "We now evaluate the empirical performance of our equivariant seq2seq model (described in Section 4) on the four SCAN tasks (described in Section 2). We compare our equivariant seq2seq to regular seq2seq models (Lake & Baroni, 2018), convolutional models (Dessì & Baroni, 2019), the syntactic attention model of Russin et al. (2019), and the meta-learning approach of Lake (2019). The compared seq2seq models use bi-directional, single-layer LSTM cells with 64 hidden units. For the equivariant seq2seq models, we use the cyclic permutation group on the verbs for the Add jump task, and the cyclic permutation group on directions for the Around right task. For Length, we use the product of those groups. Our model knows that the same group operates on both the input- and output- languages. However, it does not receive information regarding the correspondence between commands and actions in the set of words being permuted in the input / output languages. This is in contrast to Lake (2019), where (as stated in Section 5), it is necessary to provide the model with explicit information regarding the correct command-to-action mapping at test-time. ",
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+    {
+        "type": "table",
+        "img_path": "images/075a010d5029c17e4a8823f73a0d8ceb50bf41954fc12d3338a8bf5a59137709.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 1: Test accuracies for four SCAN tasks, comparing our equivariant seq2seq to the state-of-the-art. "
+        ],
+        "table_body": "<table><tr><td>Model</td><td>Simple</td><td>Add Jump</td><td>Around Right</td><td>Length</td></tr><tr><td>seq2seq (Lake &amp; Baroni, 2018)</td><td>99.7</td><td>1.2</td><td>NA</td><td>13.8</td></tr><tr><td>CNN (Dessi &amp; Baroni, 2019)</td><td>100.0</td><td>69.2 ± 9.2</td><td>56.7±10.2</td><td>0.0</td></tr><tr><td>Syntactic Attention (Russin et al., 2019)</td><td>100.0</td><td>91.0± 27.4</td><td>28.9±34.8</td><td>15.2 ± 0.7</td></tr><tr><td>Meta seq2seq (Lake, 2019)</td><td>NA</td><td>99.9</td><td>99.9*</td><td>16.64</td></tr><tr><td>seq2seq (comparable architecture)</td><td>100.0</td><td>0.0±0.0</td><td>0.02 ±2e-2</td><td>12.4 ± 2.3</td></tr><tr><td>Equivariant seq2seq (ours)</td><td>100.0</td><td>99.1 ± 0.04</td><td>92.0 ± 0.24</td><td>15.9 ± 3.2</td></tr></table>",
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+        "type": "text",
+        "text": "",
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+        "type": "text",
+        "text": "Training procedures match those of Lake & Baroni (2018) where possible. We train models for $2 0 0 k$ iterations, where each iteration consists of a minibatch of size 1, using the Adam optimizer to perform parameter updates with default parameters (Kingma & Ba, 2015) with a learning rate of 1e-4. We use teacher-forcing (Williams & Zipser, 1989) with a ratio of 0.5, and early-stopping based on a validation set consisting on $1 0 \\%$ of the training examples. As in previous works, we compute test accuracies by counting how many exact translations each model provides, across the test set associated to each task. ",
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+    {
+        "type": "text",
+        "text": "6.1 RESULTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Table 1 summarizes the results of our experiments. First and as expected, all models achieve excellent performance on the Simple task, which does not require any form of compositional generalization. ",
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+        "type": "text",
+        "text": "Second, our equivariant seq2seq model performs very well at the Add jump and Around right SCAN tasks, which are the two tasks satisfying our local equivariance assumption from Definition 3. Our equivariant seq2seq model significantly outperforms the regular seq2seq (Lake & Baroni, 2018) and convolutional (Dessì & Baroni, 2019) models, as well as the state-of-the-art methods of Russin et al. (2019) and Lake (2019). This result is an encouraging piece of evidence supporting our main hypothesis from Section 3. Next, let us compare the results of our equivariant seq2seq model with the previous state-of-the-art Russin et al. (2019); Lake (2019) in more detail. ",
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+        "type": "text",
+        "text": "On the one hand, the syntactic attention model of Russin et al. (2019) achieves significant improvements over baselines methods at the Add jump SCAN task. However, it does not fare so well on the Around right task. Furthermore, its performance has high variance. Although we here report the numbers from Russin et al. (2019), we observed such high variance in our own implementation as well, where the model often achieved $0 \\%$ test accuracy. We hypothesize that modeling the invariance of the syntactic attention directly would result in improved performance and stability. This can be achieved, for instance, by replacing all verbs in the syntactic module by a shared word. As expected, by explicitly exploiting equivariance, our model outperforms Russin et al. (2019) on the Add jump and Around right SCAN tasks, also being much more robust. ",
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+    {
+        "type": "text",
+        "text": "On the other hand, the meta-learning model of Lake (2019) achieves excellent performance on the local equivariance tasks Add jump and Around right. This is additional evidence supporting the usefulness of local equivariance. In contrast to our model, Lake (2019) requires (i) a complicated model and training procedure tailored to each task, (ii) providing the model with the correct permutation of words, equivalent to telling the model the “true” mappings between the input and output words, and (iii) augmenting the set of words being permuted, to ensure enough diversity in the training distribution (for instance, adding additional directions beyond RIGHT and LEFT). ",
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+    {
+        "type": "text",
+        "text": "As seen in Table 1, length generalization remains a tough challenge in SCAN. While generating long sequences is a known challenge in seq2seq models (Bahdanau et al., 2015), we believe that this is not the main issue with our equivariant seq2seq model, as it is able to produce long translations when these appear in the training set (as are the other models). Therefore, this is not a capacity problem, but one of not being able to express the Length generalization SCAN task in terms of local equivariances on both input- and output- languages. We hypothesize that this is the very reason why (Russin et al., 2019; Lake, 2019) also fail on this task. ",
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+    {
+        "type": "text",
+        "text": "However, we suspect that some forms of local equivariance on the input language, but global equivariance on the output language, may help. For example, RUN TWICE, RUN THRICE and RUN AROUND LEFT TWICE are all input commands contained in the training set of the length task. A trained seq2seq model is able to execute them, but fails on the unseen test command RUN AROUND LEFT THRICE, suggesting that the network did not correctly understand the relationship between TWICE and THRICE. Using a network that is explicitly equivariant to the permutation of TWICE and THRICE should be able to generalize correctly on RUN AROUND LEFT THRICE. However, while the TWICE-THRICE permutation is a local group operation (Definition 2), the corresponding operation on the output language, which is to repeat the same action sequence multiple times, is a global group operation. Similarly, permuting AND and AFTER in the input sequence using a local group operation, while operating globally on the output language by permuting the order of the associated actions, should help succeed on the Length generalization SCAN task. How to formalize the aforementioned global operations on the output language and build the desired equivariant network remains a fascinating open research question that we leave for future work. ",
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+    {
+        "type": "text",
+        "text": "7 DISCUSSION AND FUTURE WORK ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "This work has introduced hypothesis linking between group equivariance and compositional generalization in language. Motivated by this hypothesis, we have proposed an equivariant seq2seq translation model, which achieves state-of-the-art performance on a variety of SCAN tasks. ",
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+    {
+        "type": "text",
+        "text": "Our work has several points for improvement. Most importantly, our model requires knowing the permutation symmetries of interest, to be provided by some domain expert. While this is simple to do in the synthetic language of SCAN, it may prove more difficult in real-world tasks. We propose three directions to attack this problem. (i) Group words by their parts-of-speech (e.g., nouns, verbs, etc.), which can be done automatically by standard part-of-speech taggers (Màrquez & Rodríguez, 1998); (ii) Learn such groupings of words from corpora, for example using the recent work of Andreas (2019); (iii) Most appealingly, parameterize the symmetry group and learn operations end-to-end while enforcing the group structure. For permutation symmetries, the group elements can be parameterized by permutation matrices, and learned from data (Lyu et al., 2019). Our preliminary work in this direction hints that this is a fruitful avenue for future research. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "A further consideration to address is that of computational overhead. In particular, for the convolutional form we use in this work (Definition 4), computational complexity scales linearly with the size of the group, $\\mathcal { O } ( | G | )$ . This arises from the need to sum over group elements when the representation is a function on $G$ , and may be prohibitive when considering large groups. One way of addressing this issue when large symmetry groups are of interest is to consider more efficient computational layers for permutation equivariance (e.g Zaheer et al., 2017; Ravanbakhsh et al., 2017). These methods incur less computational overhead at the cost of restricting the layer capacity. Another interesting option for future research is to consider sub-sampling group elements when performing the summation in Definition 4, which requires further consideration of the consequences of doing so. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Another exciting direction for future research is to consider global equivariances. Many operations of interest, e.g. groups operating directly on parse trees, can only be expressed as global equivariances. Modeling these equivariances holds exciting possibilities for capturing non-trivial symmetries in language tasks, but also requires more sophisticated machinery than is proposed in this work. ",
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+    {
+        "type": "text",
+        "text": "Finally, in further theoretical work, we would like to explore the relation between our equivariance framework and the idea of compositionality in formal semantics (Kratzer & Heim, 1998). On the one hand, the classic idea of compositionality as an isomorphism between syntax and semantics is intuitively related to the notion of group equivariance. On the other hand, as shown by the failures at the length generalization example, it is still unclear how to apply our ideas to more sophisticated forms of permutation, such as those involving grammatical phrases rather than words. This would also require to extend our approach to account for the context-sensitivity that pervades linguistic composition (c.f., the natural interpretation of “run” in “run the marathon” vs. ”run the code”). ",
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We thank Emmanuel Dupoux and Clara Vania for helpful feedback and discussions. ",
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+    },
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+        "text": "Maurice Weiler, Fred A Hamprecht, and Martin Storath. Learning steerable filters for rotation equivariant cnns. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 849–858, 2018. ",
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+        "text": "Ronald J Williams and David Zipser. A learning algorithm for continually running fully recurrent neural networks. Neural computation, 1(2):270–280, 1989. ",
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+            171,
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+        "text": "Manzil Zaheer, Satwik Kottur, Siamak Ravanbakhsh, Barnabas Poczos, Ruslan R Salakhutdinov, and Alexander J Smola. Deep sets. In Advances in Neural Information Processing Systems, pp. 3394–3404, 2017. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "A DETAILS ON THE SCAN DATASET ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            495,
+            118
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "SCAN is composed from a non-recursive grammar, as shown in Figure 3. In particular, SCAN consists of all commands that can be generated from this grammar (20,910 command sequences), with their deterministic mapping into actions, as detailed by Figure 4 ",
+        "bbox": [
+            176,
+            126,
+            825,
+            170
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ffdefaaec04300f9cd5a56d492bbfcce0be28894a68c47774f9d6551bf8e4d38.jpg",
+        "text": "$$\n{ \\begin{array} { r l r l } & { \\mathbf { C } \\to \\mathbf { S } { \\mathrm { ~ a n d ~ } } \\mathbf { S } } & & { \\mathbf { V } \\to \\mathbf { D } [ 1 ] { \\mathrm { ~ o p p o s i t e ~ } } \\mathbf { D } [ 2 ] } & { \\mathbf { D } \\to { \\mathrm { t u r n ~ l e f t } } } \\\\ & { \\mathbf { C } \\to \\mathbf { S } { \\mathrm { ~ a f t e r ~ } } \\mathbf { S } } & & { \\mathbf { V } \\to \\mathbf { D } [ 1 ] { \\mathrm { ~ a r o u n d ~ } } \\mathbf { D } [ 2 ] } & { \\mathbf { D } \\to { \\mathrm { t u r n ~ r i g h t } } } \\\\ & { \\mathbf { C } \\to \\mathbf { S } } & & { \\mathbf { V } \\to \\mathbf { D } } & & { \\mathbf { U } \\to \\mathbf { w a l k } } \\\\ & { \\mathbf { S } \\to \\mathbf { V } { \\mathrm { ~ t w i c e } } } & & { \\mathbf { V } \\to \\mathbf { U } } & { \\mathbf { U } \\to \\mathrm { l o o k } } \\\\ & { \\mathbf { S } \\to \\mathbf { V } { \\mathrm { ~ t h r i c e } } } & & { \\mathbf { D } \\to \\mathbf { U } \\mathrm { l e f t } } & { \\mathbf { U } \\to \\mathrm { r u n } } \\\\ & { \\mathbf { S } \\to \\mathbf { V } } & & { \\mathbf { D } \\to \\mathbf { U } { \\mathrm { ~ r i g h t } } } & { \\mathbf { U } \\to \\mathbf { j u m p } } \\end{array} }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            266,
+            183,
+            727,
+            291
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "",
+        "image_caption": [
+            "Figure 3: The grammar used to generate commands in the SCAN domain. Indexing notation is used to allow infixing: read ${ \\bf \\widehat { \\mathbf { } } } D [ i ]$ as “the $_ { i }$ -th element directly dominated by category $D '$ . Image borrowed from Lake & Baroni (2018). "
+        ],
+        "image_footnote": [],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/b2b3db604764fa21c60f32c6a50a99b4b1e8272cd16c2c2509857634fbf767cf.jpg",
+        "image_caption": [
+            "Figure 4: The SCAN translation mapping. Double brackets denote the interpretation function translating SCAN’s command (input language) into the action (output) language (which are denoted by upper-case strings. Image borrowed from Lake & Baroni (2018). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            363,
+            816,
+            496
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B G-LSTM DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            568,
+            366,
+            584
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We provide the equations for implementing our G-LSTM. Given $h _ { t - 1 } , c _ { t - 1 }$ (hidden state and cellstate, respectively), and $e ( w ) _ { t }$ (all of which are of the form $G \\mapsto \\mathbb { R } ^ { K }$ , we can describe the G-LSTM cell as follows: ",
+        "bbox": [
+            174,
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+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6cb197d2e522b3a4032a6e3b05869b1988c56fc4a3e7f148648ca0c7ed08f118.jpg",
+        "text": "$$\n\\begin{array} { r l r l } & { i _ { t } = \\sigma \\left( { \\pmb x } _ { t } \\ast \\psi _ { i i } + s _ { t - 1 } \\ast \\psi _ { i h } \\right) ; \\quad \\quad \\quad } & & { f _ { t } = \\sigma \\left( { \\pmb x } _ { t } \\ast \\psi _ { f i } + s _ { t - 1 } \\ast \\psi _ { f h } \\right) } \\\\ & { g _ { t } = \\operatorname { t a n h } \\left( { \\pmb x } _ { t } \\ast \\psi _ { g i } + s _ { t - 1 } \\ast \\psi _ { g h } \\right) ; \\quad \\quad \\quad } & & { o _ { t } = \\sigma \\left( { \\pmb x } _ { t } \\ast \\psi _ { o i } + s _ { t - 1 } \\ast \\psi _ { o h } \\right) } \\\\ & { c _ { t } = f _ { t } \\circ { \\pmb c } _ { t - 1 } + i _ { t } \\circ g _ { t } ; \\quad \\quad \\quad } & & { h _ { t } = o _ { t } \\operatorname { c a n h } ( { \\pmb c } _ { t } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            228,
+            641,
+            766,
+            698
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "where $\\{ \\psi _ { j k } : G \\mapsto \\mathbb { R } ^ { K } ; j \\in \\{ i , f , g , o \\} ; k \\in \\{ i , h \\} \\}$ are the learnable filters of the cell. Here we have used the shorthand ",
+        "bbox": [
+            169,
+            703,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a04ca58b5e03b7bc6c123d18106db087b0cf79f43e89b247d5ef508f30b19169.jpg",
+        "text": "$$\n\\pmb { f } \\ast \\pmb { \\psi } : = \\left[ \\pmb { f } \\ast \\pmb { \\psi } \\right] ( \\pmb { g } ) \\quad \\forall \\pmb { g } \\in G\n$$",
+        "text_format": "latex",
+        "bbox": [
+            395,
+            731,
+            602,
+            748
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "for two functions on the group. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 11
+    }
+]

parse/train/c8P9NQVtmnO/c8P9NQVtmnO.md

@@ -0,0 +1,382 @@
+# FOURIER NEURAL OPERATOR FORPARAMETRIC PARTIAL DIFFERENTIAL EQUATIONS
+
+Zongyi Li zongyili $@$ caltech.edu
+
+Nikola Kovachki nkovachki@caltech.edu
+
+Kamyar Azizzadenesheli kamyar $@$ purdue.edu
+
+Burigede Liu bgl@caltech.edu
+
+Kaushik Bhattacharya bhatta@caltech.edu
+
+Andrew Stuart astuart@caltech.edu
+
+Anima Anandkumar anima@caltech.edu
+
+# ABSTRACT
+
+The classical development of neural networks has primarily focused on learning mappings between finite-dimensional Euclidean spaces. Recently, this has been generalized to neural operators that learn mappings between function spaces. For partial differential equations (PDEs), neural operators directly learn the mapping from any functional parametric dependence to the solution. Thus, they learn an entire family of PDEs, in contrast to classical methods which solve one instance of the equation. In this work, we formulate a new neural operator by parameterizing the integral kernel directly in Fourier space, allowing for an expressive and efficient architecture. We perform experiments on Burgers’ equation, Darcy flow, and Navier-Stokes equation. The Fourier neural operator is the first ML-based method to successfully model turbulent flows with zero-shot super-resolution. It is up to three orders of magnitude faster compared to traditional PDE solvers. Additionally, it achieves superior accuracy compared to previous learning-based solvers under fixed resolution.
+
+# 1 INTRODUCTION
+
+Many problems in science and engineering involve solving complex partial differential equation (PDE) systems repeatedly for different values of some parameters. Examples arise in molecular dynamics, micro-mechanics, and turbulent flows. Often such systems require fine discretization in order to capture the phenomenon being modeled. As a consequence, traditional numerical solvers are slow and sometimes inefficient. For example, when designing materials such as airfoils, one needs to solve the associated inverse problem where thousands of evaluations of the forward model are needed. A fast method can make such problems feasible.
+
+Conventional solvers vs. Data-driven methods. Traditional solvers such as finite element methods (FEM) and finite difference methods (FDM) solve the equation by discretizing the space. Therefore, they impose a trade-off on the resolution: coarse grids are fast but less accurate; fine grids are accurate but slow. Complex PDE systems, as described above, usually require a very fine discretization, and therefore very challenging and time-consuming for traditional solvers. On the other hand, data-driven methods can directly learn the trajectory of the family of equations from the data. As a result, the learning-based method can be orders of magnitude faster than the conventional solvers.
+
+Machine learning methods may hold the key to revolutionizing scientific disciplines by providing fast solvers that approximate or enhance traditional ones (Raissi et al., 2019; Jiang et al., 2020; Greenfeld et al., 2019; Kochkov et al., 2021). However, classical neural networks map between finite-dimensional spaces and can therefore only learn solutions tied to a specific discretization. This is often a limitation for practical applications and therefore the development of mesh-invariant neural networks is required. We first outline two mainstream neural network-based approaches for PDEs – the finite-dimensional operators and Neural-FEM.
+
+Finite-dimensional operators. These approaches parameterize the solution operator as a deep convolutional neural network between finite-dimensional Euclidean spaces Guo et al. (2016); Zhu
+
+![](images/47713b0420c585850977bd3b16156952782e947ca3562ab3b8e9a44fed87d0ed.jpg)  
+Zero-shot super-resolution: Navier-Stokes Equation with Reynolds number 10000; Ground truth on top and prediction on bottom; trained on $6 4 \times 6 4 \times 2 0$ dataset; evaluated on $2 5 6 \times 2 5 6 \times 8 0$ (see Section 5.4).
+
+Figure 1: top: The architecture of the Fourier layer; bottom: Example flow from Navier-Stokes.
+
+& Zabaras (2018); Adler & Oktem (2017); Bhatnagar et al. (2019); Khoo et al. (2017). Such approaches are, by definition, mesh-dependent and will need modifications and tuning for different resolutions and discretizations in order to achieve consistent error (if at all possible). Furthermore, these approaches are limited to the discretization size and geometry of the training data and hence, it is not possible to query solutions at new points in the domain. In contrast, we show, for our method, both invariance of the error to grid resolution, and the ability to transfer the solution between meshes.
+
+Neural-FEM. The second approach directly parameterizes the solution function as a neural network (E & Yu, 2018; Raissi et al., 2019; Bar & Sochen, 2019; Smith et al., 2020; Pan & Duraisamy, 2020). This approach is designed to model one specific instance of the PDE, not the solution operator. It is mesh-independent and accurate, but for any given new instance of the functional parameter/coefficient, it requires training a new neural network. The approach closely resembles classical methods such as finite elements, replacing the linear span of a finite set of local basis functions with the space of neural networks. The Neural-FEM approach suffers from the same computational issue as classical methods: the optimization problem needs to be solved for every new instance. Furthermore, the approach is limited to a setting in which the underlying PDE is known.
+
+Neural Operators. Recently, a new line of work proposed learning mesh-free, infinitedimensional operators with neural networks (Lu et al., 2019; Bhattacharya et al., 2020; Nelsen & Stuart, 2020; Li et al., 2020b;a; Patel et al., 2021). The neural operator remedies the mesh-dependent nature of the finite-dimensional operator methods discussed above by producing a single set of network parameters that may be used with different discretizations. It has the ability to transfer solutions between meshes. Furthermore, the neural operator needs to be trained only once. Obtaining a solution for a new instance of the parameter requires only a forward pass of the network, alleviating the major computational issues incurred in Neural-FEM methods. Lastly, the neural operator requires no knowledge of the underlying PDE, only data. Thus far, neural operators have not yielded efficient numerical algorithms that can parallel the success of convolutional or recurrent neural networks in the finite-dimensional setting due to the cost of evaluating integral operators. Through the fast Fourier transform, our work alleviates this issue.
+
+Fourier Transform. The Fourier transform is frequently used in spectral methods for solving differential equations, since differentiation is equivalent to multiplication in the Fourier domain. Fourier transforms have also played an important role in the development of deep learning. In theory, they appear in the proof of the universal approximation theorem (Hornik et al., 1989) and, empirically, they have been used to speed up convolutional neural networks (Mathieu et al., 2013). Neural network architectures involving the Fourier transform or the use of sinusoidal activation functions have also been proposed and studied (Bengio et al., 2007; Mingo et al., 2004; Sitzmann et al., 2020). Recently, some spectral methods for PDEs have been extended to neural networks (Fan et al., 2019a;b; Kashinath et al., 2020). We build on these works by proposing a neural operator architecture defined directly in Fourier space with quasi-linear time complexity and state-of-the-art approximation capabilities.
+
+Our Contributions. We introduce the Fourier neural operator, a novel deep learning architecture able to learn mappings between infinite-dimensional spaces of functions; the integral operator is restricted to a convolution, and instantiated through a linear transformation in the Fourier domain.
+
+• The Fourier neural operator is the first work that learns the resolution-invariant solution operator for the family of Navier-Stokes equation in the turbulent regime, where previous graph-based neural operators do not converge. • By construction, the method shares the same learned network parameters irrespective of the discretization used on the input and output spaces. It can do zero-shot super-resolution: trained on a lower resolution directly evaluated on a higher resolution, as shown in Figure 1. • The proposed method consistently outperforms all existing deep learning methods even when fixing the resolution to be $6 4 \times 6 4$ . It achieves error rates that are $3 0 \%$ lower on Burgers’ Equation, $6 0 \%$ lower on Darcy Flow, and $3 0 \%$ lower on Navier Stokes (turbulent regime with Reynolds number 10000). When learning the mapping for the entire time series, the method achieves $< 1 \%$ error with Reynolds number 1000 and $8 \%$ error with Reynolds number 10000. • On a $2 5 6 \times 2 5 6$ grid, the Fourier neural operator has an inference time of only 0.005s compared to the $2 . 2 s$ of the pseudo-spectral method used to solve Navier-Stokes. Despite its tremendous speed advantage, the method does not suffer from accuracy degradation when used in downstream applications such as solving the Bayesian inverse problem, as shown in Figure 6.
+
+We observed that the proposed framework can approximate complex operators raising in PDEs that are highly non-linear, with high frequency modes and slow energy decay. The power of neural operators comes from combining linear, global integral operators (via the Fourier transform) and nonlinear, local activation functions. Similar to the way standard neural networks approximate highly non-linear functions by combining linear multiplications with non-linear activations, the proposed neural operators can approximate highly non-linear operators.
+
+# 2 LEARNING OPERATORS
+
+Our methodology learns a mapping between two infinite dimensional spaces from a finite collection of observed input-output pairs. Let $D \subset \mathbb { R } ^ { d }$ be a bounded, open set and $\mathcal { A } = \mathcal { A } ( D ; \mathbb { R } ^ { d _ { a } } )$ and $\mathcal { U } = \mathcal { U } ( D ; \mathbb { R } ^ { d _ { u } } )$ be separable Banach spaces of function taking values in $\mathbb { R } ^ { d _ { a } }$ and $\mathbb { R } ^ { d _ { u } }$ respectively. Furthermore let $G ^ { \dagger } : { \mathcal { A } }  { \mathcal { U } }$ be a (typically) non-linear map. We study maps $G ^ { \dagger }$ which arise as the solution operators of parametric PDEs – see Section 5 for examples. Suppose we have observations $\{ a _ { j } , u _ { j } \} _ { j = 1 } ^ { N }$ where $a _ { j } \sim \mu$ is an i.i.d. sequence from the probability measure $\mu$ supported on $\mathcal { A }$ and $u _ { j } = G ^ { \dagger } ( a _ { j } )$ is possibly corrupted with noise. We aim to build an approximation of $G ^ { \dagger }$ by constructing a parametric map
+
+$$
+G : { \mathcal { A } } \times \Theta \to { \mathcal { U } } \qquad { \mathrm { o r ~ e q u i v a l e n t l y , } } \qquad G _ { \theta } : { \mathcal { A } } \to { \mathcal { U } } , \quad \theta \in \Theta
+$$
+
+for some finite-dimensional parameter space $\Theta$ by choosing $\theta ^ { \dagger } \in \Theta$ so that $G ( \cdot , \theta ^ { \dagger } ) = G _ { \theta ^ { \dagger } } \approx G ^ { \dagger }$ . This is a natural framework for learning in infinite-dimensions as one could define a cost functional $C : \mathcal { U } \times \mathcal { U }  \mathbb { R }$ and seek a minimizer of the problem
+
+$$
+\operatorname* { m i n } _ { \theta \in \Theta } \mathbb { E } _ { a \sim \mu } [ C ( G ( a , \theta ) , G ^ { \dagger } ( a ) ) ]
+$$
+
+which directly parallels the classical finite-dimensional setting (Vapnik, 1998). Showing the existence of minimizers, in the infinite-dimensional setting, remains a challenging open problem. We will approach this problem in the test-train setting by using a data-driven empirical approximation to the cost used to determine $\theta$ and to test the accuracy of the approximation. Because we conceptualize our methodology in the infinite-dimensional setting, all finite-dimensional approximations share a common set of parameters which are consistent in infinite dimensions. A table of notation is shown in Appendix 3.
+
+Learning the Operator. Approximating the operator $G ^ { \dagger }$ is a different and typically much more challenging task than finding the solution $u \in \mathcal { U }$ of a PDE for a single instance of the parameter $a \in { \mathcal { A } }$ . Most existing methods, ranging from classical finite elements, finite differences, and finite volumes to modern machine learning approaches such as physics-informed neural networks
+
+(a)
+
+![](images/24f2ccbe3bc2d8627081eba7adb6f66640a082a7d8e5c2af70cd7899d53c7623.jpg)  
+Figure 2: top: The architecture of the neural operators; bottom: Fourier layer.
+
+(a) The full architecture of neural operator: start from input $a$ . 1. Lift to a higher dimension channel space by a neural network $P$ . 2. Apply four layers of integral operators and activation functions. 3. Project back to the target dimension by a neural network $Q$ . Output $u$ . (b) Fourier layers: Start from input $v$ . On top: apply the Fourier transform $\mathcal { F }$ ; a linear transform $R$ on the lower Fourier modes and filters out the higher modes; then apply the inverse Fourier transform $\mathcal { F } ^ { - 1 }$ . On the bottom: apply a local linear transform $W$ .
+
+(PINNs) (Raissi et al., 2019) aim at the latter and can therefore be computationally expensive. This makes them impractical for applications where a solution to the PDE is required for many different instances of the parameter. On the other hand, our approach directly approximates the operator and is therefore much cheaper and faster, offering tremendous computational savings when compared to traditional solvers. For an example application to Bayesian inverse problems, see Section 5.5.
+
+Discretization. Since our data $a _ { j }$ and $u _ { j }$ are, in general, functions, to work with them numerically, we assume access only to point-wise evaluations. Let $D _ { j } \ = \ \{ x _ { 1 } , \ldots , x _ { n } \} \ \subset \ D$ be a $n$ -point discretization of the domain $D$ and assume we have observations $a _ { j } | _ { D _ { j } } \in \mathbb { R } ^ { n \times d _ { a } }$ , $u _ { j } | _ { D _ { j } } \in \mathbb { R } ^ { n \times d _ { v } }$ , for a finite collection of input-output pairs indexed by $j$ . To be discretization-invariant, the neural operator can produce an answer $u ( x )$ for any $x \in D$ , potentially $x \notin D _ { j }$ . Such a property is highly desirable as it allows a transfer of solutions between different grid geometries and discretizations.
+
+# 3 NEURAL OPERATOR
+
+The neural operator, proposed in (Li et al., 2020b), is formulated as an iterative architecture $v _ { 0 } \mapsto$ $v _ { 1 } \mapsto . . . \mapsto v _ { T }$ where $v _ { j }$ for $j = 0 , 1 , \ldots , T - 1$ is a sequence of functions each taking values in $\mathbb { R } ^ { d _ { v } }$ . As shown in Figure 2 (a), the input $a \in { \mathcal { A } }$ is first lifted to a higher dimensional representation $v _ { 0 } ( x ) = P ( a ( x ) )$ by the local transformation $P$ which is usually parameterized by a shallow fullyconnected neural network. Then we apply several iterations of updates $v _ { t } \mapsto v _ { t + 1 }$ (defined below). The output $u ( x ) = Q ( v _ { T } ( x ) )$ is the projection of $v _ { T }$ by the local transformation $Q : \mathbb { R } ^ { d _ { v } }  \mathbb { R } ^ { d _ { u } }$ . In each iteration, the update $v _ { t } \mapsto v _ { t + 1 }$ is defined as the composition of a non-local integral operator $\kappa$ and a local, nonlinear activation function $\sigma$ .
+
+Definition 1 (Iterative updates) Define the update to the representation $v _ { t } \mapsto v _ { t + 1 }$ by
+
+$$
+v _ { t + 1 } ( x ) : = \sigma \Bigl ( W v _ { t } ( x ) + \bigl ( K ( a ; \phi ) v _ { t } \bigr ) ( x ) \Bigr ) , \qquad \forall x \in D
+$$
+
+where $\mathcal { K } : \mathcal { A } \times \Theta _ { \mathcal { K } } \to \mathcal { L } ( \mathcal { U } ( D ; \mathbb { R } ^ { d _ { v } } ) , \mathcal { U } ( D ; \mathbb { R } ^ { d _ { v } } ) )$ maps to bounded linear operators on $\mathcal { U } ( D ; \mathbb { R } ^ { d _ { v } } )$ and is parameterized by $\phi \in \Theta \kappa$ , $W : \mathbb { R } ^ { d _ { v } }  \mathbb { R } ^ { d _ { v } }$ is a linear transformation, and $\sigma : \mathbb { R }  \mathbb { R }$ is $a$ non-linear activation function whose action is defined component-wise.
+
+We choose $\textstyle \mathcal { K } ( a ; \phi )$ to be a kernel integral transformation parameterized by a neural network.
+
+Definition 2 (Kernel integral operator $\kappa$ ) Define the kernel integral operator mapping in (2) by
+
+$$
+\big ( \mathcal { K } ( a ; \phi ) v _ { t } \big ) ( x ) : = \int _ { D } \kappa \big ( x , y , a ( x ) , a ( y ) ; \phi \big ) v _ { t } ( y ) \mathrm { d } y , \qquad \forall x \in D
+$$
+
+where $\kappa _ { \phi } : \mathbb { R } ^ { 2 ( d + d _ { a } ) }  \mathbb { R } ^ { d _ { v } \times d _ { v } }$ is a neural network parameterized by $\phi \in \Theta \kappa$
+
+Here $\kappa _ { \phi }$ plays the role of a kernel function which we learn from data. Together definitions 1 and 2 constitute a generalization of neural networks to infinite-dimensional spaces as first proposed in Li et al. (2020b). Notice even the integral operator is linear, the neural operator can learn highly non-linear operators by composing linear integral operators with non-linear activation functions, analogous to standard neural networks.
+
+If we remove the dependence on the function $a$ and impose $\kappa _ { \phi } ( x , y ) = \kappa _ { \phi } ( x - y )$ , we obtain that (3) is a convolution operator, which is a natural choice from the perspective of fundamental solutions. We exploit this fact in the following section by parameterizing $\kappa _ { \phi }$ directly in Fourier space and using the Fast Fourier Transform (FFT) to efficiently compute (3). This leads to a fast architecture that obtains state-of-the-art results for PDE problems.
+
+# 4 FOURIER NEURAL OPERATOR
+
+We propose replacing the kernel integral operator in (3), by a convolution operator defined in Fourier space. Let $\mathcal { F }$ denote the Fourier transform of a function $\dot { f } : D  \mathbb { R } ^ { d _ { v } }$ and $\scriptstyle { \mathcal { F } } ^ { - 1 }$ its inverse then
+
+$$
+( { \mathcal { F } } f ) _ { j } ( k ) = \int _ { D } f _ { j } ( x ) e ^ { - 2 i \pi \langle x , k \rangle } \mathrm { d } x , \qquad ( { \mathcal { F } } ^ { - 1 } f ) _ { j } ( x ) = \int _ { D } f _ { j } ( k ) e ^ { 2 i \pi \langle x , k \rangle } \mathrm { d } k
+$$
+
+for $j = 1 , \ldots , d _ { v }$ where $i = \sqrt { - 1 }$ is the imaginary unit. By letting $\kappa _ { \phi } ( x , y , a ( x ) , a ( y ) ) = \kappa _ { \phi } ( x - y )$ in (3) and applying the convolution theorem, we find that
+
+$$
+\bigl ( \mathcal { K } ( a ; \phi ) v _ { t } \bigr ) ( x ) = \mathcal { F } ^ { - 1 } \bigl ( \mathcal { F } ( \kappa _ { \phi } ) \cdot \mathcal { F } ( v _ { t } ) \bigr ) ( x ) , \qquad \forall x \in D .
+$$
+
+We, therefore, propose to directly parameterize $\kappa _ { \phi }$ in Fourier space.
+
+Definition 3 (Fourier integral operator $\kappa$ ) Define the Fourier integral operator
+
+$$
+\bigl ( \mathcal { K } ( \phi ) v _ { t } \bigr ) ( x ) = \mathcal { F } ^ { - 1 } \Bigl ( R _ { \phi } \cdot ( \mathcal { F } v _ { t } ) \Bigr ) ( x ) \qquad \forall x \in D
+$$
+
+where $R _ { \phi }$ is the Fourier transform of a periodic function $\kappa : \bar { D }  \mathbb R ^ { d _ { v } \times d _ { v } }$ parameterized by $\phi \in \Theta \kappa$ . An illustration is given in Figure 2 (b).
+
+For frequency mode $k \in D$ , we have $( \mathcal { F } v _ { t } ) ( k ) \in \mathbb { C } ^ { d _ { v } }$ and $R _ { \phi } ( k ) \in \mathbb { C } ^ { d _ { v } \times d _ { v } }$ . Notice that since we assume $\kappa$ is periodic, it admits a Fourier series expansion, so we may work with the discrete modes $k \in  { \mathbb { Z } ^ { d } }$ . We pick a finite-dimensional parameterization by truncating the Fourier series at a maximal number of modes $k _ { \operatorname* { m a x } } = | Z _ { k _ { \operatorname* { m a x } } } | = | \{ k \in \mathbb { Z } ^ { d } : | k _ { j } | \leq k _ { \operatorname* { m a x } , j }$ , for $j = 1 , \ldots , d \} |$ . We thus parameterize $R _ { \phi }$ directly as complex-valued $( k _ { \operatorname* { m a x } } \times d _ { v } \times d _ { v } )$ -tensor comprising a collection of truncated Fourier modes and therefore drop $\phi$ from our notation. Since $\kappa$ is real-valued, we impose conjugate symmetry. We note that the set $Z _ { k _ { \mathrm { m a x } } }$ is not the canonical choice for the low frequency modes of $v _ { t }$ . Indeed, the low frequency modes are usually defined by placing an upper-bound on the $\ell _ { 1 }$ -norm of $k \in  { \mathbb { Z } ^ { d } }$ . We choose $Z _ { k _ { \mathrm { m a x } } }$ as above since it allows for an efficient implementation.
+
+The discrete case and the FFT. Assuming the domain $D$ is discretized with $n \in \mathbb N$ points, we have that $v _ { t } \in \mathbb { R } ^ { n \times d _ { v } }$ and $\mathcal { F } ( v _ { t } ) \in \mathbb { C } ^ { n \times d _ { v } ^ { \smile } }$ . Since we convolve $v _ { t }$ with a function which only has $k _ { \mathrm { m a x } }$ Fourier modes, we may simply truncate the higher modes to obtain $\mathcal { F } ( v _ { t } ) \in \mathbb { C } ^ { k _ { \operatorname* { m a x } } \times d _ { v } }$ . Multiplication by the weight tensor $R \in \mathbf { \bar { \mathbb { C } } } ^ { k _ { \operatorname* { m a x } } \times d _ { v } \times d _ { v } }$ is then
+
+$$
+\big ( R \cdot ( \mathcal { F } v _ { t } ) \big ) _ { k , l } = \sum _ { j = 1 } ^ { d _ { v } } R _ { k , l , j } ( \mathcal { F } v _ { t } ) _ { k , j } , \qquad k = 1 , \dots , k _ { \operatorname* { m a x } } , \quad j = 1 , \dots , d _ { v } .
+$$
+
+When the discretization is uniform with resolution $s _ { 1 } \times \cdot \cdot \cdot \times s _ { d } = n$ , $\mathcal { F }$ can be replaced by the Fast Fourier Transform. For $f \in \mathbb { R } ^ { n \times d _ { v } }$ , $k = ( k _ { 1 } , \ldots , k _ { d } ) \in \mathbb { Z } _ { s _ { 1 } } \times \cdot \cdot \cdot \times \mathbb { Z } _ { s _ { d } }$ , and $x = ( x _ { 1 } , \ldots , x _ { d } ) \in D$ , the FFT $\hat { \mathcal { F } }$ and its inverse $\hat { \mathcal { F } } ^ { - 1 }$ are defined as
+
+$$
+\begin{array} { r l } & { ( \hat { \mathcal { F } } f ) _ { l } ( k ) = \displaystyle \sum _ { x _ { 1 } = 0 } ^ { s _ { 1 } - 1 } \cdots \sum _ { x _ { d } = 0 } ^ { s _ { d } - 1 } f _ { l } ( x _ { 1 } , \ldots , x _ { d } ) e ^ { - 2 i \pi \sum _ { j = 1 } ^ { d } \frac { x _ { j } k _ { j } } { s _ { j } } } , } \\ & { ( \hat { \mathcal { F } } ^ { - 1 } f ) _ { l } ( x ) = \displaystyle \sum _ { k _ { 1 } = 0 } ^ { s _ { 1 } - 1 } \cdots \sum _ { k _ { d } = 0 } ^ { s _ { d } - 1 } f _ { l } ( k _ { 1 } , \ldots , k _ { d } ) e ^ { 2 i \pi \sum _ { j = 1 } ^ { d } \frac { x _ { j } k _ { j } } { s _ { j } } } } \end{array}
+$$
+
+for $l = 1 , \ldots , d _ { v }$ . In this case, the set of truncated modes becomes
+
+$$
+Z _ { k _ { \mathrm { m a x } } } = \{ ( k _ { 1 } , \ldots , k _ { d } ) \in \mathbb { Z } _ { s _ { 1 } } \times \cdot \cdot \cdot \times \mathbb { Z } _ { s _ { d } } \mid k _ { j } \leq k _ { \operatorname* { m a x } , j }
+$$
+
+$$
+s _ { j } - k _ { j } \leq k _ { \operatorname* { m a x } , j } , \mathrm { f o r } j = 1 , \ldots , d \} .
+$$
+
+When implemented, $R$ is treated as a $( s _ { 1 } \times \cdot \cdot \cdot \times s _ { d } \times d _ { v } \times d _ { v } )$ -tensor and the above definition of $Z _ { k _ { \mathrm { m a x } } }$ corresponds to the “corners” of $R$ , which allows for a straight-forward parallel implementation of (5) via matrix-vector multiplication. In practice, we have found that choosing $k _ { \operatorname* { m a x } , j } = 1 2$ which yields $k _ { \operatorname* { m a x } } = 1 2 ^ { d }$ parameters per channel to be sufficient for all the tasks that we consider.
+
+Parameterizations of $R$ . In general, $R$ can be defined to depend on $( \mathcal { F } a )$ to parallel (3). Indeed, we can define $R _ { \phi } : \mathbb { Z } ^ { d } \times \mathbb { R } ^ { d _ { v } }  \mathbb { R } ^ { d _ { v } \times d _ { v } }$ as a parametric function that maps $\big ( k , ( \mathcal { F } a ) ( k ) \big )$ to the values of the appropriate Fourier modes. We have experimented with linear as well as neural network parameterizations of $R _ { \phi }$ . We find that the linear parameterization has a similar performance to the previously described direct parameterization, while neural networks have worse performance. This is likely due to the discrete structure of the space $\mathbb { Z } ^ { d }$ . Our experiments in this work focus on the direct parameterization presented above.
+
+Invariance to discretization. The Fourier layers are discretization-invariant because they can learn from and evaluate functions which are discretized in an arbitrary way. Since parameters are learned directly in Fourier space, resolving the functions in physical space simply amounts to projecting on the basis $e ^ { 2 \pi i \left. x , k \right. }$ which are well-defined everywhere on $\mathbb { R } ^ { d }$ . This allows us to achieve zero-shot super-resolution as shown in Section 5.4. Furthermore, our architecture has a consistent error at any resolution of the inputs and outputs. On the other hand, notice that, in Figure 3, the standard CNN methods we compare against have an error that grows with the resolution.
+
+Quasi-linear complexity. The weight tensor $R$ contains $k _ { \operatorname* { m a x } { } } < n$ modes, so the inner multiplication has complexity $O ( k _ { \operatorname* { m a x } } )$ . Therefore, the majority of the computational cost lies in computing the Fourier transform $\mathcal { F } ( v _ { t } )$ and its inverse. General Fourier transforms have complexity ${ \dot { O } } ( n ^ { 2 } )$ , however, since we truncate the series the complexity is in fact $O ( n k _ { \operatorname* { m a x } } )$ , while the FFT has complexity $O ( n \log n )$ . Generally, we have found using FFTs to be very efficient. However a uniform discretization is required.
+
+# 5 NUMERICAL EXPERIMENTS
+
+In this section, we compare the proposed Fourier neural operator with multiple finite-dimensional architectures as well as operator-based approximation methods on the 1-d Burgers’ equation, the 2-d Darcy Flow problem, and 2-d Navier-Stokes equation. The data generation processes are discussed in Appendices A.3.1, A.3.2, and A.3.3 respectively. We do not compare against traditional solvers (FEM/FDM) or neural-FEM type methods since our goal is to produce an efficient operator approximation that can be used for downstream applications. We demonstrate one such application to the Bayesian inverse problem in Section 5.5.
+
+We construct our Fourier neural operator by stacking four Fourier integral operator layers as specified in (2) and (4) with the ReLU activation as well as batch normalization. Unless otherwise specified, we use $N = 1 0 0 0$ training instances and 200 testing instances. We use Adam optimizer to train for 500 epochs with an initial learning rate of 0.001 that is halved every 100 epochs. We set $k _ { \operatorname* { m a x } , j } = 1 6 , d _ { v } = 6 4$ for the 1-d problem and $k _ { \operatorname* { m a x } , j } = 1 2 , d _ { v } = 3 2$ for the 2-d problems. Lower resolution data are downsampled from higher resolution. All the computation is carried on a single Nvidia V100 GPU with 16GB memory.
+
+Remark on Resolution. Traditional PDE solvers such as FEM and FDM approximate a single function and therefore their error to the continuum decreases as the resolution is increased. On the other hand, operator approximation is independent of the ways its data is discretized as long as all relevant information is resolved. Resolution-invariant operators have consistent error rates among different resolutions as shown in Figure 3. Further, resolution-invariant operators can do zero-shot super-resolution, as shown in Section 5.4.
+
+Benchmarks for time-independent problems (Burgers and Darcy): NN: a simple point-wise feedforward neural network. RBM: the classical Reduced Basis Method (using a POD basis) (De
+
+![](images/ccd83d87eb17dedce6b2605c5a253578b14747d2dce6282d84e35bec0901f81f.jpg)  
+Figure 3: Benchmark on Burger’s equation, Darcy Flow, and Navier-Stokes
+
+Left: benchmarks on Burgers equation; Mid: benchmarks on Darcy Flow for different resolutions; Right: the learning curves on Navier-Stokes $\nu = 1 \mathrm { e } - 3$ with different benchmarks. Train and test on the same resolution. For acronyms, see Section 5; details in Tables 1, 3, 4.
+
+Vore, 2014). FCN: a the-state-of-the-art neural network architecture based on Fully Convolution Networks (Zhu & Zabaras, 2018). PCANN: an operator method using PCA as an autoencoder on both the input and output data and interpolating the latent spaces with a neural network (Bhattacharya et al., 2020). GNO: the original graph neural operator (Li et al., 2020b). MGNO: the multipole graph neural operator (Li et al., 2020a). LNO: a neural operator method based on the low-rank decomposition of the kernel $\begin{array} { r } { \kappa ( x , y ) : = \sum _ { j = 1 } ^ { r } \phi _ { j } ( x ) \psi _ { j } ( y ) } \end{array}$ , similar to the unstacked DeepONet proposed in (Lu et al., 2019). FNO: the newly purposed Fourier neural operator.
+
+Benchmarks for time-dependent problems (Navier-Stokes): ResNet: 18 layers of 2-d convolution with residual connections (He et al., 2016). U-Net: A popular choice for image-to-image regression tasks consisting of four blocks with 2-d convolutions and deconvolutions (Ronneberger et al., 2015). TF-Net: A network designed for learning turbulent flows based on a combination of spatial and temporal convolutions (Wang et al., 2020). FNO-2d: 2-d Fourier neural operator with a RNN structure in time. FNO-3d: 3-d Fourier neural operator that directly convolves in space-time.
+
+# 5.1 BURGERS’ EQUATION
+
+The 1-d Burgers’ equation is a non-linear PDE with various applications including modeling the one dimensional flow of a viscous fluid. It takes the form
+
+$$
+\begin{array} { r l } { \partial _ { t } u ( x , t ) + \partial _ { x } ( u ^ { 2 } ( x , t ) / 2 ) = \nu \partial _ { x x } u ( x , t ) , \quad } & { x \in ( 0 , 1 ) , t \in ( 0 , 1 ] } \\ { u ( x , 0 ) = u _ { 0 } ( x ) , \quad } & { x \in ( 0 , 1 ) } \end{array}
+$$
+
+with periodic boundary conditions where $u _ { 0 } \in L _ { \mathrm { p e r } } ^ { 2 } ( ( 0 , 1 ) ; \mathbb { R } )$ is the initial condition and $\nu \in \mathbb { R } _ { + }$ is the viscosity coefficient. We aim to learn the operator mapping the initial condition to the solution at time one, $G ^ { \dagger } : L _ { \mathrm { p e r } } ^ { 2 } ( ( 0 , 1 ) ; \mathbb { R } ) \to H _ { \mathrm { p e r } } ^ { r } ( ( 0 , 1 ) ; \mathbb { R } )$ defined by $u _ { 0 } \mapsto u ( \cdot , 1 )$ for any $r > 0$ .
+
+The results of our experiments are shown in Figure 3 (a) and Table 3 (Appendix A.3.1). Our proposed method obtains the lowest relative error compared to any of the benchmarks. Further, the error is invariant with the resolution, while the error of convolution neural network based methods (FCN) grows with the resolution. Compared to other neural operator methods such as GNO and MGNO that use Nystrom sampling in physical space, the Fourier neural operator is both more accurate and ¨ more computationally efficient.
+
+# 5.2 DARCY FLOW
+
+We consider the steady-state of the 2-d Darcy Flow equation on the unit box which is the second order, linear, elliptic PDE
+
+$$
+\begin{array} { r l r l } { - \nabla \cdot ( a ( x ) \nabla u ( x ) ) = f ( x ) } & { } & { x \in ( 0 , 1 ) ^ { 2 } } \\ { u ( x ) = 0 } & { } & { x \in \partial ( 0 , 1 ) ^ { 2 } } \end{array}
+$$
+
+with a Dirichlet boundary where $a \ \in \ L ^ { \infty } ( ( 0 , 1 ) ^ { 2 } ; \mathbb { R } _ { + } )$ is the diffusion coefficient and $f \in$ $L ^ { 2 } ( ( 0 , 1 ) ^ { 2 } ; \mathbb { R } )$ is the forcing function. This PDE has numerous applications including modeling the pressure of subsurface flow, the deformation of linearly elastic materials, and the electric potential in conductive materials. We are interested in learning the operator mapping the diffusion coefficient to the solution, $G ^ { \dag } : L ^ { \infty } ( ( 0 , 1 ) _ { : } ^ { 2 } ; \mathbb { R } _ { + } ) \to H _ { 0 } ^ { 1 } ( ( 0 , \bar { 1 } ) ^ { 2 } ; \mathbb { R } _ { + } )$ defined by $a \mapsto u$ . Note that although the PDE is linear, the operator $G ^ { \dagger }$ is not.
+
+The results of our experiments are shown in Figure 3 (b) and Table 4 (Appendix A.3.2). The proposed Fourier neural operator obtains nearly one order of magnitude lower relative error compared to any benchmarks. We again observe the invariance of the error with respect to the resolution.
+
+# 5.3 NAVIER-STOKES EQUATION
+
+We consider the 2-d Navier-Stokes equation for a viscous, incompressible fluid in vorticity form on the unit torus:
+
+$$
+\begin{array} { r l r l } { \partial _ { t } w ( x , t ) + u ( x , t ) \cdot \nabla w ( x , t ) = \nu \Delta w ( x , t ) + f ( x ) , } & { } & { x \in ( 0 , 1 ) ^ { 2 } , t \in ( 0 , T ] } \\ { \nabla \cdot u ( x , t ) = 0 , } & { } & & { x \in ( 0 , 1 ) ^ { 2 } , t \in [ 0 , T ] } \\ { w ( x , 0 ) = w _ { 0 } ( x ) , } & { } & & { x \in ( 0 , 1 ) ^ { 2 } } \end{array}
+$$
+
+where $u \in C ( [ 0 , T ] ; H _ { \mathrm { p e r } } ^ { r } ( ( 0 , 1 ) ^ { 2 } ; \mathbb { R } ^ { 2 } ) )$ for any $r > 0$ is the velocity field, $w = \nabla \times u$ is the vorticity, $w _ { 0 } \in L _ { \mathrm { p e r } } ^ { 2 } ( ( 0 , 1 ) ^ { 2 } ; \mathbb { R } )$ is the initial vorticity, $\nu \in \mathbb { R } _ { + }$ is the viscosity coefficient, and $f \in$ $L _ { \mathrm { p e r } } ^ { 2 } ( ( 0 , 1 ) ^ { 2 } ; \mathbb { R } )$ is the forcing function. We are interested in learning the operator mapping the vorticity up to time 10 to the vorticity up to some later time $T > 1 0$ $1 0 , G ^ { \dagger } : C ( [ 0 , 1 0 ] ; H _ { \mathrm { p e r } } ^ { r } ( ( 0 , 1 ) ^ { 2 } ; \mathbb { R } ) ) \to$ $C ( ( 1 0 , T ] ; H _ { \mathrm { p e r } } ^ { r } ( ( 0 , 1 ) ^ { 2 } ; \mathbb { R } ) )$ defined by $w | _ { ( 0 , 1 ) ^ { 2 } \times [ 0 , 1 0 ] } \mapsto w | _ { ( 0 , 1 ) ^ { 2 } \times ( 1 0 , T ] }$ . Given the vorticity it is easy to derive the velocity. While vorticity is harder to model compared to velocity, it provides more information. By formulating the problem on vorticity, the neural network models mimic the pseudospectral method. We experiment with the viscosities $\nu = { 1 } \mathrm { e } \mathrm { - } 3 , { 1 } \mathrm { e } \mathrm { - } 4 , { 1 } \mathrm { e } \mathrm { - } 5$ , decreasing the final time $T$ as the dynamic becomes chaotic. Since the baseline methods are not resolution-invariant, we fix the resolution to be $6 4 \times 6 4$ for both training and testing.
+
+Table 1: Benchmarks on Navier Stokes (fixing resolution $6 4 \times 6 4$ for both training and testing)   
+
+<table><tr><td>Config</td><td>Parameters</td><td>Time per</td><td>v=le-3 T=50</td><td>v=le-4 T=30 N = 1000</td><td>V=le-4 T=30 N = 10000</td><td>v=le-5 T=20</td></tr><tr><td>FNO-3D</td><td>6,558,537</td><td>epoch 38.99s</td><td>N = 1000 0.0086</td><td>0.1918</td><td>0.0820</td><td>N = 1000 0.1893</td></tr><tr><td>FNO-2D</td><td>414,517</td><td>127.80s</td><td>0.0128</td><td>0.1559</td><td>0.0834</td><td>0.1556</td></tr><tr><td>U-Net</td><td>24,950,491</td><td>48.67s</td><td>0.0245</td><td>0.2051</td><td>0.1190</td><td>0.1982</td></tr><tr><td>TF-Net</td><td>7,451,724</td><td>47.21s</td><td>0.0225</td><td>0.2253</td><td>0.1168</td><td>0.2268</td></tr><tr><td>ResNet</td><td>266,641</td><td>78.47s</td><td>0.0701</td><td>0.2871</td><td>0.2311</td><td>0.2753</td></tr></table>
+
+As shown in Table 1, the FNO-3D has the best performance when there is sufficient data $( \nu ~ =$ 1e−3, $N = 1 0 0 0$ and $\nu = 1 \mathrm { e } { - } 4 , N = 1 0 0 0 0 )$ . For the configurations where the amount of data is insufficient $\langle \nu = 1 \mathrm { e - } 4$ , $N = 1 0 0 0$ and $\nu = 1 \mathrm { e } { - } 5 , N = 1 0 0 0 )$ , all methods have $> 1 5 \%$ error with FNO-2D achieving the lowest. Note that we only present results for spatial resolution $6 4 \times 6 4$ since all benchmarks we compare against are designed for this resolution. Increasing it degrades their performance while FNO achieves the same errors.
+
+2D and 3D Convolutions. FNO-2D, U-Net, TF-Net, and ResNet all do 2D-convolution in the spatial domain and recurrently propagate in the time domain $( 2 \mathrm { D } \mathrm { + R N N } )$ . The operator maps the solution at the previous 10 time steps to the next time step (2D functions to 2D functions). On the other hand, FNO-3D performs convolution in space-time. It maps the initial time steps directly to the full trajectory (3D functions to 3D functions). The $2 \mathrm { D } { + } \mathrm { R } \mathrm { N } \mathrm { N }$ structure can propagate the solution to any arbitrary time $T$ in increments of a fixed interval length $\Delta t$ , while the Conv3D structure is fixed to the interval $[ 0 , T ]$ but can transfer the solution to an arbitrary time-discretization. We find the 3-d method to be more expressive and easier to train compared to its RNN-structured counterpart.
+
+# 5.4 ZERO-SHOT SUPER-RESOLUTION.
+
+The neural operator is mesh-invariant, so it can be trained on a lower resolution and evaluated at a higher resolution, without seeing any higher resolution data (zero-shot super-resolution). Figure 1 shows an example where we train the FNO-3D model on $6 4 \times 6 4 \times 2 0$ resolution data in the setting above with $( \nu = 1 \mathrm { e } { - } 4 , N = 1 0 0 0 0 )$ and transfer to $2 5 6 \times 2 5 6 \times 8 0$ resolution, demonstrating super-resolution in space-time. Fourier neural operator is the only model among the benchmarks (FNO-2D, U-Net, TF-Net, and ResNet) that can do zero-shot super-resolution. And surprisingly, it can do super-resolution not only in the spatial domain but also in the temporal domain.
+
+# 5.5 BAYESIAN INVERSE PROBLEM
+
+In this experiment, we use a function space Markov chain Monte Carlo (MCMC) method (Cotter et al., 2013) to draw samples from the posterior distribution of the initial vorticity in Navier-Stokes given sparse, noisy observations at time $T = 5 0$ . We compare the Fourier neural operator acting as a surrogate model with the traditional solvers used to generate our train-test data (both run on GPU). We generate 25,000 samples from the posterior (with a 5,000 sample burn-in period), requiring 30,000 evaluations of the forward operator.
+
+As shown in Figure 6 (Appendix A.5), FNO and the traditional solver recover almost the same posterior mean which, when pushed forward, recovers well the late-time dynamic of Navier Stokes. In sharp contrast, FNO takes $0 . 0 0 5 s$ to evaluate a single instance while the traditional solver, after being optimized to use the largest possible internal time-step which does not lead to blow-up, takes $2 . 2 s$ . This amounts to 2.5 minutes for the MCMC using FNO and over 18 hours for the traditional solver. Even if we account for data generation and training time (offline steps) which take 12 hours, using FNO is still faster! Once trained, FNO can be used to quickly perform multiple MCMC runs for different initial conditions and observations, while the traditional solver will take 18 hours for every instance. Furthermore, since FNO is differentiable, it can easily be applied to PDE-constrained optimization problems without the need for the adjoint method.
+
+Spectral analysis. Due to the way we parameterize $R _ { \phi }$ , the function output by (4) has at most $k _ { \mathrm { m a x } , j }$ Fourier modes per channel. This, however, does not mean that the Fourier neural operator can only approximate functions up to $k _ { \operatorname* { m a x } , j }$ modes. Indeed, the activation functions which occur between integral operators and the final decoder network $Q$ recover the high frequency modes. As an example, consider a solution to the Navier-Stokes equation with viscosity $\nu = 1 \mathrm { e } - 3$ . Truncating this function at 20 Fourier modes yields an error around $2 \%$ while our Fourier neural operator learns the parametric dependence and produces approximations to an error of $\leq 1 \%$ with only $k _ { \operatorname* { m a x } , j } = 1 2$ parameterized modes.
+
+Non-periodic boundary condition. Traditional Fourier methods work only with periodic boundary conditions. However, the Fourier neural operator does not have this limitation. This is due to the linear transform $W$ (the bias term) which keeps the track of non-periodic boundary. As an example, the Darcy Flow and the time domain of Navier-Stokes have non-periodic boundary conditions, and the Fourier neural operator still learns the solution operator with excellent accuracy.
+
+# 6 DISCUSSION AND CONCLUSION
+
+Requirements on Data. Data-driven methods rely on the quality and quantity of data. To learn Navier-Stokes equation with Reynolds number $R e \ = \ 1 \mathrm { e } { + } 4$ , we need to generate $N \ = \ 1 0 0 0 0$ training pairs $\{ a _ { j } , u _ { j } \}$ with the numerical solver. However, for more challenging PDEs, generating a few training samples can be already very expensive. A future direction is to combine neural operators with numerical solvers to levitate the requirements on data. Recurrent structure. The neural operator has an iterative structure that can naturally be formulated as a recurrent network where all layers share the same parameters without sacrificing performance. (We did not impose this restriction in the experiments.) Computer vision. Operator learning is not restricted to PDEs. Images can naturally be viewed as real-valued functions on 2-d domains and videos simply add a temporal structure. Our approach is therefore a natural choice for problems in computer vision where invariance to discretization crucial is important (Chi et al., 2020).
+
+# ACKNOWLEDGEMENTS
+
+The authors want to thank Ray Wang and Rose Yu for meaningful discussions. Z. Li gratefully acknowledges the financial support from the Kortschak Scholars Program. A. Anandkumar is supported in part by Bren endowed chair, LwLL grants, Beyond Limits, Raytheon, Microsoft, Google, Adobe faculty fellowships, and DE Logi grant. K. Bhattacharya, N. B. Kovachki, B. Liu, and A. M. Stuart gratefully acknowledge the financial support of the Army Research Laboratory through the Cooperative Agreement Number W911NF-12-0022. Research was sponsored by the Army Research Laboratory and was accomplished under Cooperative Agreement Number W911NF-12-2- 0022. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.
+
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+
+Zongyi Li, Nikola Kovachki, Kamyar Azizzadenesheli, Burigede Liu, Kaushik Bhattacharya, Andrew Stuart, and Anima Anandkumar. Neural operator: Graph kernel network for partial differential equations. arXiv preprint arXiv:2003.03485, 2020b.
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+
+Yinhao Zhu and Nicholas Zabaras. Bayesian deep convolutional encoder–decoder networks for surrogate modeling and uncertainty quantification. Journal of Computational Physics, 2018. ISSN 0021-9991. doi: https://doi.org/10.1016/j.jcp.2018.04.018. URL http://www. sciencedirect.com/science/article/pii/S0021999118302341.
+
+# A APPENDIX
+
+# A.1 TABLE OF NOTATIONS
+
+A table of notations is given in Table 2.
+
+Table 2: table of notations   
+
+<table><tr><td rowspan=1 colspan=2>Notation                                          Meaning</td></tr><tr><td rowspan=1 colspan=1>Operator learningDcRdxEDa ∈ A= (D;Rda)u ∈U= (D;Rdu)DjG² :A→Uμ</td><td rowspan=1 colspan=1>The spatial domain for the PDEPoints in the the spatial domainThe input coefficient functionsThe target solution functionsThe discretization of (aj,uj)The operator mapping the coefficients to the solutionsA probability measure where aj sampled from.</td></tr><tr><td rowspan=1 colspan=1>Neural operatorU(x)∈RdudaduduK : R2(d+1) →Rdu×du中t=0,...,T0</td><td rowspan=1 colspan=1>The neural network representation of u(x)Dimension of the input a(x).Dimension of the output u(x).The dimension of the representation v(x)The kernel maps (x,y,a(x),a(y)) to a dy × d matrixThe parameters of the kernel network KThe time steps (layers)The activation function</td></tr><tr><td rowspan=1 colspan=1>Fourier operatorF,F-1RWkkmax</td><td rowspan=1 colspan=1>Fourier transformation and its inverse.The linear transformation applied on the lower Fourier modes.The linear transformation (bias term) applied on the spatial domain.Fourier modes /wave numbers.The max Fourier modes used in the Fourier layer.</td></tr><tr><td rowspan=1 colspan=1>HyperparametersNnSVT</td><td rowspan=1 colspan=1>The number of training pairs.The size of the discretization.The resolution of the discretization (sd = n).The viscosity.The time interval [O,T] for time-dependent equation.</td></tr></table>
+
+# A.2 SPECTRAL ANALYSIS
+
+The spectral decay of the Navier Stokes equation data is shown in Figure 4. The spectrum decay has a slope $k ^ { - 5 / 3 }$ , matching the energy spectrum in the turbulence region. And we notice the energy spectrum does not decay along with time.
+
+We also present the spectral decay in term of the truncation $k _ { m a x }$ defined in 4 as shown in Figure5. We note all equations (Burgers, Darcy, and Navier-Stokes with $\nu \leq 1 \mathrm { e } { - 4 }$ ) exhibit high frequency modes. Even we truncate at $k _ { m a x } = 1 2$ in the Fourier layer, the Fourier neural operator can recover the high frequency modes.
+
+# A.3 DATA GENERATION
+
+In this section, we provide the details of data generator for the three equation we used in Section 5.
+
+![](images/935eb5d09a2858b9d89d2f07d454ba4a50ddfeaa6db73e62950aa4080022dbfb.jpg)  
+Figure 4: Spectral Decay of Navier-Stokes equations
+
+The spectral decay of the Navier-stokes equation data we used in section 5.3. The y-axis is the spectrum; the $\mathbf { X }$ -axis is the wavenumber $| k | = k _ { 1 } + k _ { 2 }$ .
+
+![](images/bd9a3b97b63e751f0867517ae5ededa124bd7af9e103c3f18efc52feb56cdf6a.jpg)  
+Figure 5: Spectral Decay in term of $k _ { m a x }$
+
+The error of truncation in one single Fourier layer without applying the linear transform $R$ . The y-axis is the normalized truncation error; the $\mathbf { X }$ -axis is the truncation mode $k _ { m a x }$ .
+
+# A.3.1 BURGERS EQUATION
+
+Recall the 1-d Burger’s equation on the unit torus:
+
+$$
+\begin{array} { r l } { \partial _ { t } u ( x , t ) + \partial _ { x } ( u ^ { 2 } ( x , t ) / 2 ) = \nu \partial _ { x x } u ( x , t ) , \quad } & { x \in ( 0 , 1 ) , t \in ( 0 , 1 ] } \\ { u ( x , 0 ) = u _ { 0 } ( x ) , \quad } & { x \in ( 0 , 1 ) . } \end{array}
+$$
+
+The initial condition $u _ { 0 } ( x )$ is generated according to $u _ { 0 } \sim \mu$ where $\mu = \mathcal { N } ( 0 , 6 2 5 ( - \Delta + 2 5 I ) ^ { - 2 } )$ with periodic boundary conditions. We set the viscosity to $\nu = 0 . 1$ and solve the equation using a split step method where the heat equation part is solved exactly in Fourier space then the non-linear part is advanced, again in Fourier space, using a very fine forward Euler method. We solve on a spatial mesh with resolution $2 ^ { 1 3 } = \bar { 8 1 9 2 }$ and use this dataset to subsample other resolutions.
+
+# A.3.2 DARCY FLOW
+
+The 2-d Darcy Flow is a second-order linear elliptic equation of the form
+
+$$
+\begin{array} { r l } { - \nabla \cdot ( a ( x ) \nabla u ( x ) ) = f ( x ) \quad } & { x \in ( 0 , 1 ) ^ { 2 } } \\ { u ( x ) = 0 \quad } & { x \in \partial ( 0 , 1 ) ^ { 2 } . } \end{array}
+$$
+
+The coefficients $a ( x )$ are generated according to $a \sim \mu$ where $\mu = \psi _ { \# } \mathcal N ( 0 , ( - \Delta + 9 I ) ^ { - 2 } )$ with zero Neumann boundary conditions on the Laplacian. The mapping $\psi : \mathbb { R }  \mathbb { R }$ takes the value 12 on the positive part of the real line and 3 on the negative and the push-forward is defined pointwise. The forcing is kept fixed $f ( x ) = 1$ . Such constructions are prototypical models for many physical systems such as permeability in subsurface flows and material microstructures in elasticity. Solutions $u$ are obtained by using a second-order finite difference scheme on a $4 2 1 \times 4 2 1$ grid. Different resolutions are downsampled from this dataset.
+
+# A.3.3 NAVIER-STOKES EQUATION
+
+Recall the 2-d Navier-Stokes equation for a viscous, incompressible fluid in vorticity form on the unit torus:
+
+$$
+\begin{array} { r l r l } & { \partial _ { t } w ( x , t ) + u ( x , t ) \cdot \nabla w ( x , t ) = \nu \Delta w ( x , t ) + f ( x ) , } & & { x \in ( 0 , 1 ) ^ { 2 } , t \in ( 0 , T ] } \\ & { \qquad \nabla \cdot u ( x , t ) = 0 , } & & { x \in ( 0 , 1 ) ^ { 2 } , t \in [ 0 , T ] } \\ & { \qquad w ( x , 0 ) = w _ { 0 } ( x ) , } & & { x \in ( 0 , 1 ) ^ { 2 } . } \end{array}
+$$
+
+The initial condition $w _ { 0 } ( x )$ is generated according to $w _ { 0 } \sim \mu$ where $\mu = \mathcal { N } ( 0 , 7 ^ { 3 / 2 } ( - \Delta { + } 4 9 I ) ^ { - 2 . 5 } )$ with periodic boundary conditions. The forcing is kept fixed $f ( x ) = 0 . 1 ( \sin ( 2 \pi ( x _ { 1 } + x _ { 2 } ) ) +$ $\cos ( { \bar { 2 \pi } } ( x _ { 1 } + x _ { 2 } ) ) )$ ). The equation is solved using the stream-function formulation with a pseudospectral method. First a Poisson equation is solved in Fourier space to find the velocity field. Then the vorticity is differentiated and the non-linear term is computed is physical space after which it is dealiased. Time is advanced with a Crank–Nicolson update where the non-linear term does not enter the implicit part. All data are generated on a $2 5 6 \times 2 5 6$ grid and are downsampled to $6 4 \times 6 4$ . We use a time-step of $\mathrm { 1 e { - } 4 }$ for the Crank–Nicolson scheme in the data-generated process where we record the solution every $t = 1$ time units. The step is increased to $2 \mathrm { e } { - 2 }$ when used in MCMC for the Bayesian inverse problem.
+
+# A.4 RESULTS OF BURGERS’ EQUATION AND DARCY FLOW
+
+The details error rate on Burgers’ equation and Darcy Flow are listed in Table 3 and Table 4.
+
+Table 3: Benchmarks on 1-d Burgers’ equation   
+
+<table><tr><td>Networks</td><td>s= 256</td><td>s= 512</td><td>s= 1024</td><td>s = 2048</td><td>s = 4096</td><td>s= 8192</td></tr><tr><td>NN</td><td>0.4714</td><td>0.4561</td><td>0.4803</td><td>0.4645</td><td>0.4779</td><td>0.4452</td></tr><tr><td>GCN</td><td>0.3999</td><td>0.4138</td><td>0.4176</td><td>0.4157</td><td>0.4191</td><td>0.4198</td></tr><tr><td>FCN</td><td>0.0958</td><td>0.1407</td><td>0.1877</td><td>0.2313</td><td>0.2855</td><td>0.3238</td></tr><tr><td>PCANN</td><td>0.0398</td><td>0.0395</td><td>0.0391</td><td>0.0383</td><td>0.0392</td><td>0.0393</td></tr><tr><td>GNO</td><td>0.0555</td><td>0.0594</td><td>0.0651</td><td>0.0663</td><td>0.0666</td><td>0.0699</td></tr><tr><td>LNO</td><td>0.0212</td><td>0.0221</td><td>0.0217</td><td>0.0219</td><td>0.0200</td><td>0.0189</td></tr><tr><td>MGNO</td><td>0.0243</td><td>0.0355</td><td>0.0374</td><td>0.0360</td><td>0.0364</td><td>0.0364</td></tr><tr><td>FNO</td><td>0.0149</td><td>0.0158</td><td>0.0160</td><td>0.0146</td><td>0.0142</td><td>0.0139</td></tr></table>
+
+Table 4: Benchmarks on 2-d Darcy Flow   
+
+<table><tr><td>Networks</td><td>s=85</td><td>s=141</td><td>s=211</td><td>s= 421</td></tr><tr><td>NN</td><td>0.1716</td><td>0.1716</td><td>0.1716</td><td>0.1716</td></tr><tr><td>FCN</td><td>0.0253</td><td>0.0493</td><td>0.0727</td><td>0.1097</td></tr><tr><td>PCANN</td><td>0.0299</td><td>0.0298</td><td>0.0298</td><td>0.0299</td></tr><tr><td>RBM</td><td>0.0244</td><td>0.0251</td><td>0.0255</td><td>0.0259</td></tr><tr><td>GNO</td><td>0.0346</td><td>0.0332</td><td>0.0342</td><td>0.0369</td></tr><tr><td>LNO</td><td>0.0520</td><td>0.0461</td><td>0.0445</td><td></td></tr><tr><td>MGNO</td><td>0.0416</td><td>0.0428</td><td>0.0428</td><td>0.0420</td></tr><tr><td>FNO</td><td>0.0108</td><td>0.0109</td><td>0.0109</td><td>0.0098</td></tr></table>
+
+# A.5 BAYESIAN INVERSE PROBLEM
+
+Results of the Bayesian inverse problem for the Navier-Stokes equation are shown in Figure 6. It can be seen that the result using Fourier neural operator as a surrogate is as good as the result of the traditional solver.
+
+![](images/a7207702dccdb90c87f83495300ee949d84c7d72cff3999894b46e8c49e78d4d.jpg)
+
+The top left panel shows the true initial vorticity while bottom left panel shows the true observed vorticity at $T = 5 0$ with black dots indicating the locations of the observation points placed on a $7 \times 7$ grid. The top middle panel shows the posterior mean of the initial vorticity given the noisy observations estimated with MCMC using the traditional solver, while the top right panel shows the same thing but using FNO as a surrogate model. The bottom middle and right panels show the vorticity at $T = 5 0$ when the respective approximate posterior means are used as initial conditions.
+
+Figure 6: Results of the Bayesian inverse problem for the Navier-Stokes equation.

parse/train/c8P9NQVtmnO/c8P9NQVtmnO_content_list.json

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+    {
+        "type": "text",
+        "text": "FOURIER NEURAL OPERATOR FORPARAMETRIC PARTIAL DIFFERENTIAL EQUATIONS",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Zongyi Li zongyili $@$ caltech.edu ",
+        "bbox": [
+            183,
+            172,
+            328,
+            202
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Nikola Kovachki nkovachki@caltech.edu ",
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+            562,
+            200
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Kamyar Azizzadenesheli kamyar $@$ purdue.edu ",
+        "bbox": [
+            638,
+            172,
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+    },
+    {
+        "type": "text",
+        "text": "Burigede Liu bgl@caltech.edu ",
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Kaushik Bhattacharya bhatta@caltech.edu ",
+        "bbox": [
+            318,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Andrew Stuart astuart@caltech.edu ",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Anima Anandkumar anima@caltech.edu ",
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+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "The classical development of neural networks has primarily focused on learning mappings between finite-dimensional Euclidean spaces. Recently, this has been generalized to neural operators that learn mappings between function spaces. For partial differential equations (PDEs), neural operators directly learn the mapping from any functional parametric dependence to the solution. Thus, they learn an entire family of PDEs, in contrast to classical methods which solve one instance of the equation. In this work, we formulate a new neural operator by parameterizing the integral kernel directly in Fourier space, allowing for an expressive and efficient architecture. We perform experiments on Burgers’ equation, Darcy flow, and Navier-Stokes equation. The Fourier neural operator is the first ML-based method to successfully model turbulent flows with zero-shot super-resolution. It is up to three orders of magnitude faster compared to traditional PDE solvers. Additionally, it achieves superior accuracy compared to previous learning-based solvers under fixed resolution. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Many problems in science and engineering involve solving complex partial differential equation (PDE) systems repeatedly for different values of some parameters. Examples arise in molecular dynamics, micro-mechanics, and turbulent flows. Often such systems require fine discretization in order to capture the phenomenon being modeled. As a consequence, traditional numerical solvers are slow and sometimes inefficient. For example, when designing materials such as airfoils, one needs to solve the associated inverse problem where thousands of evaluations of the forward model are needed. A fast method can make such problems feasible. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Conventional solvers vs. Data-driven methods. Traditional solvers such as finite element methods (FEM) and finite difference methods (FDM) solve the equation by discretizing the space. Therefore, they impose a trade-off on the resolution: coarse grids are fast but less accurate; fine grids are accurate but slow. Complex PDE systems, as described above, usually require a very fine discretization, and therefore very challenging and time-consuming for traditional solvers. On the other hand, data-driven methods can directly learn the trajectory of the family of equations from the data. As a result, the learning-based method can be orders of magnitude faster than the conventional solvers. ",
+        "bbox": [
+            174,
+            679,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Machine learning methods may hold the key to revolutionizing scientific disciplines by providing fast solvers that approximate or enhance traditional ones (Raissi et al., 2019; Jiang et al., 2020; Greenfeld et al., 2019; Kochkov et al., 2021). However, classical neural networks map between finite-dimensional spaces and can therefore only learn solutions tied to a specific discretization. This is often a limitation for practical applications and therefore the development of mesh-invariant neural networks is required. We first outline two mainstream neural network-based approaches for PDEs – the finite-dimensional operators and Neural-FEM. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Finite-dimensional operators. These approaches parameterize the solution operator as a deep convolutional neural network between finite-dimensional Euclidean spaces Guo et al. (2016); Zhu ",
+        "bbox": [
+            174,
+            895,
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/47713b0420c585850977bd3b16156952782e947ca3562ab3b8e9a44fed87d0ed.jpg",
+        "image_caption": [
+            "Zero-shot super-resolution: Navier-Stokes Equation with Reynolds number 10000; Ground truth on top and prediction on bottom; trained on $6 4 \\times 6 4 \\times 2 0$ dataset; evaluated on $2 5 6 \\times 2 5 6 \\times 8 0$ (see Section 5.4). "
+        ],
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+    {
+        "type": "text",
+        "text": "Figure 1: top: The architecture of the Fourier layer; bottom: Example flow from Navier-Stokes. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "& Zabaras (2018); Adler & Oktem (2017); Bhatnagar et al. (2019); Khoo et al. (2017). Such approaches are, by definition, mesh-dependent and will need modifications and tuning for different resolutions and discretizations in order to achieve consistent error (if at all possible). Furthermore, these approaches are limited to the discretization size and geometry of the training data and hence, it is not possible to query solutions at new points in the domain. In contrast, we show, for our method, both invariance of the error to grid resolution, and the ability to transfer the solution between meshes. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Neural-FEM. The second approach directly parameterizes the solution function as a neural network (E & Yu, 2018; Raissi et al., 2019; Bar & Sochen, 2019; Smith et al., 2020; Pan & Duraisamy, 2020). This approach is designed to model one specific instance of the PDE, not the solution operator. It is mesh-independent and accurate, but for any given new instance of the functional parameter/coefficient, it requires training a new neural network. The approach closely resembles classical methods such as finite elements, replacing the linear span of a finite set of local basis functions with the space of neural networks. The Neural-FEM approach suffers from the same computational issue as classical methods: the optimization problem needs to be solved for every new instance. Furthermore, the approach is limited to a setting in which the underlying PDE is known. ",
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+    {
+        "type": "text",
+        "text": "Neural Operators. Recently, a new line of work proposed learning mesh-free, infinitedimensional operators with neural networks (Lu et al., 2019; Bhattacharya et al., 2020; Nelsen & Stuart, 2020; Li et al., 2020b;a; Patel et al., 2021). The neural operator remedies the mesh-dependent nature of the finite-dimensional operator methods discussed above by producing a single set of network parameters that may be used with different discretizations. It has the ability to transfer solutions between meshes. Furthermore, the neural operator needs to be trained only once. Obtaining a solution for a new instance of the parameter requires only a forward pass of the network, alleviating the major computational issues incurred in Neural-FEM methods. Lastly, the neural operator requires no knowledge of the underlying PDE, only data. Thus far, neural operators have not yielded efficient numerical algorithms that can parallel the success of convolutional or recurrent neural networks in the finite-dimensional setting due to the cost of evaluating integral operators. Through the fast Fourier transform, our work alleviates this issue. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Fourier Transform. The Fourier transform is frequently used in spectral methods for solving differential equations, since differentiation is equivalent to multiplication in the Fourier domain. Fourier transforms have also played an important role in the development of deep learning. In theory, they appear in the proof of the universal approximation theorem (Hornik et al., 1989) and, empirically, they have been used to speed up convolutional neural networks (Mathieu et al., 2013). Neural network architectures involving the Fourier transform or the use of sinusoidal activation functions have also been proposed and studied (Bengio et al., 2007; Mingo et al., 2004; Sitzmann et al., 2020). Recently, some spectral methods for PDEs have been extended to neural networks (Fan et al., 2019a;b; Kashinath et al., 2020). We build on these works by proposing a neural operator architecture defined directly in Fourier space with quasi-linear time complexity and state-of-the-art approximation capabilities. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our Contributions. We introduce the Fourier neural operator, a novel deep learning architecture able to learn mappings between infinite-dimensional spaces of functions; the integral operator is restricted to a convolution, and instantiated through a linear transformation in the Fourier domain. ",
+        "bbox": [
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+        "type": "text",
+        "text": "• The Fourier neural operator is the first work that learns the resolution-invariant solution operator for the family of Navier-Stokes equation in the turbulent regime, where previous graph-based neural operators do not converge. • By construction, the method shares the same learned network parameters irrespective of the discretization used on the input and output spaces. It can do zero-shot super-resolution: trained on a lower resolution directly evaluated on a higher resolution, as shown in Figure 1. • The proposed method consistently outperforms all existing deep learning methods even when fixing the resolution to be $6 4 \\times 6 4$ . It achieves error rates that are $3 0 \\%$ lower on Burgers’ Equation, $6 0 \\%$ lower on Darcy Flow, and $3 0 \\%$ lower on Navier Stokes (turbulent regime with Reynolds number 10000). When learning the mapping for the entire time series, the method achieves $< 1 \\%$ error with Reynolds number 1000 and $8 \\%$ error with Reynolds number 10000. • On a $2 5 6 \\times 2 5 6$ grid, the Fourier neural operator has an inference time of only 0.005s compared to the $2 . 2 s$ of the pseudo-spectral method used to solve Navier-Stokes. Despite its tremendous speed advantage, the method does not suffer from accuracy degradation when used in downstream applications such as solving the Bayesian inverse problem, as shown in Figure 6. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We observed that the proposed framework can approximate complex operators raising in PDEs that are highly non-linear, with high frequency modes and slow energy decay. The power of neural operators comes from combining linear, global integral operators (via the Fourier transform) and nonlinear, local activation functions. Similar to the way standard neural networks approximate highly non-linear functions by combining linear multiplications with non-linear activations, the proposed neural operators can approximate highly non-linear operators. ",
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+    {
+        "type": "text",
+        "text": "2 LEARNING OPERATORS",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Our methodology learns a mapping between two infinite dimensional spaces from a finite collection of observed input-output pairs. Let $D \\subset \\mathbb { R } ^ { d }$ be a bounded, open set and $\\mathcal { A } = \\mathcal { A } ( D ; \\mathbb { R } ^ { d _ { a } } )$ and $\\mathcal { U } = \\mathcal { U } ( D ; \\mathbb { R } ^ { d _ { u } } )$ be separable Banach spaces of function taking values in $\\mathbb { R } ^ { d _ { a } }$ and $\\mathbb { R } ^ { d _ { u } }$ respectively. Furthermore let $G ^ { \\dagger } : { \\mathcal { A } }  { \\mathcal { U } }$ be a (typically) non-linear map. We study maps $G ^ { \\dagger }$ which arise as the solution operators of parametric PDEs – see Section 5 for examples. Suppose we have observations $\\{ a _ { j } , u _ { j } \\} _ { j = 1 } ^ { N }$ where $a _ { j } \\sim \\mu$ is an i.i.d. sequence from the probability measure $\\mu$ supported on $\\mathcal { A }$ and $u _ { j } = G ^ { \\dagger } ( a _ { j } )$ is possibly corrupted with noise. We aim to build an approximation of $G ^ { \\dagger }$ by constructing a parametric map ",
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+        "img_path": "images/ce8bfa45892a89d37ad73c14678d95b7f3aab54f4ec674b46d752fc6f05ca8b4.jpg",
+        "text": "$$\nG : { \\mathcal { A } } \\times \\Theta \\to { \\mathcal { U } } \\qquad { \\mathrm { o r ~ e q u i v a l e n t l y , } } \\qquad G _ { \\theta } : { \\mathcal { A } } \\to { \\mathcal { U } } , \\quad \\theta \\in \\Theta\n$$",
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+    {
+        "type": "text",
+        "text": "for some finite-dimensional parameter space $\\Theta$ by choosing $\\theta ^ { \\dagger } \\in \\Theta$ so that $G ( \\cdot , \\theta ^ { \\dagger } ) = G _ { \\theta ^ { \\dagger } } \\approx G ^ { \\dagger }$ . This is a natural framework for learning in infinite-dimensions as one could define a cost functional $C : \\mathcal { U } \\times \\mathcal { U }  \\mathbb { R }$ and seek a minimizer of the problem ",
+        "bbox": [
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+    {
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+        "img_path": "images/63a588da887b93f0fdf6742589fde5e0d967662adcc7509b52c54e2aa4ecf15a.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\theta \\in \\Theta } \\mathbb { E } _ { a \\sim \\mu } [ C ( G ( a , \\theta ) , G ^ { \\dagger } ( a ) ) ]\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "which directly parallels the classical finite-dimensional setting (Vapnik, 1998). Showing the existence of minimizers, in the infinite-dimensional setting, remains a challenging open problem. We will approach this problem in the test-train setting by using a data-driven empirical approximation to the cost used to determine $\\theta$ and to test the accuracy of the approximation. Because we conceptualize our methodology in the infinite-dimensional setting, all finite-dimensional approximations share a common set of parameters which are consistent in infinite dimensions. A table of notation is shown in Appendix 3. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Learning the Operator. Approximating the operator $G ^ { \\dagger }$ is a different and typically much more challenging task than finding the solution $u \\in \\mathcal { U }$ of a PDE for a single instance of the parameter $a \\in { \\mathcal { A } }$ . Most existing methods, ranging from classical finite elements, finite differences, and finite volumes to modern machine learning approaches such as physics-informed neural networks ",
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "(a) ",
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+    {
+        "type": "image",
+        "img_path": "images/24f2ccbe3bc2d8627081eba7adb6f66640a082a7d8e5c2af70cd7899d53c7623.jpg",
+        "image_caption": [
+            "Figure 2: top: The architecture of the neural operators; bottom: Fourier layer. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "(a) The full architecture of neural operator: start from input $a$ . 1. Lift to a higher dimension channel space by a neural network $P$ . 2. Apply four layers of integral operators and activation functions. 3. Project back to the target dimension by a neural network $Q$ . Output $u$ . (b) Fourier layers: Start from input $v$ . On top: apply the Fourier transform $\\mathcal { F }$ ; a linear transform $R$ on the lower Fourier modes and filters out the higher modes; then apply the inverse Fourier transform $\\mathcal { F } ^ { - 1 }$ . On the bottom: apply a local linear transform $W$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "(PINNs) (Raissi et al., 2019) aim at the latter and can therefore be computationally expensive. This makes them impractical for applications where a solution to the PDE is required for many different instances of the parameter. On the other hand, our approach directly approximates the operator and is therefore much cheaper and faster, offering tremendous computational savings when compared to traditional solvers. For an example application to Bayesian inverse problems, see Section 5.5. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Discretization. Since our data $a _ { j }$ and $u _ { j }$ are, in general, functions, to work with them numerically, we assume access only to point-wise evaluations. Let $D _ { j } \\ = \\ \\{ x _ { 1 } , \\ldots , x _ { n } \\} \\ \\subset \\ D$ be a $n$ -point discretization of the domain $D$ and assume we have observations $a _ { j } | _ { D _ { j } } \\in \\mathbb { R } ^ { n \\times d _ { a } }$ , $u _ { j } | _ { D _ { j } } \\in \\mathbb { R } ^ { n \\times d _ { v } }$ , for a finite collection of input-output pairs indexed by $j$ . To be discretization-invariant, the neural operator can produce an answer $u ( x )$ for any $x \\in D$ , potentially $x \\notin D _ { j }$ . Such a property is highly desirable as it allows a transfer of solutions between different grid geometries and discretizations. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3 NEURAL OPERATOR",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The neural operator, proposed in (Li et al., 2020b), is formulated as an iterative architecture $v _ { 0 } \\mapsto$ $v _ { 1 } \\mapsto . . . \\mapsto v _ { T }$ where $v _ { j }$ for $j = 0 , 1 , \\ldots , T - 1$ is a sequence of functions each taking values in $\\mathbb { R } ^ { d _ { v } }$ . As shown in Figure 2 (a), the input $a \\in { \\mathcal { A } }$ is first lifted to a higher dimensional representation $v _ { 0 } ( x ) = P ( a ( x ) )$ by the local transformation $P$ which is usually parameterized by a shallow fullyconnected neural network. Then we apply several iterations of updates $v _ { t } \\mapsto v _ { t + 1 }$ (defined below). The output $u ( x ) = Q ( v _ { T } ( x ) )$ is the projection of $v _ { T }$ by the local transformation $Q : \\mathbb { R } ^ { d _ { v } }  \\mathbb { R } ^ { d _ { u } }$ . In each iteration, the update $v _ { t } \\mapsto v _ { t + 1 }$ is defined as the composition of a non-local integral operator $\\kappa$ and a local, nonlinear activation function $\\sigma$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Definition 1 (Iterative updates) Define the update to the representation $v _ { t } \\mapsto v _ { t + 1 }$ by ",
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/42a764afbb2e9c0534e5e536e6dc0579ef0d1c4eba0a8f84e4982896ec56a874.jpg",
+        "text": "$$\nv _ { t + 1 } ( x ) : = \\sigma \\Bigl ( W v _ { t } ( x ) + \\bigl ( K ( a ; \\phi ) v _ { t } \\bigr ) ( x ) \\Bigr ) , \\qquad \\forall x \\in D\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\mathcal { K } : \\mathcal { A } \\times \\Theta _ { \\mathcal { K } } \\to \\mathcal { L } ( \\mathcal { U } ( D ; \\mathbb { R } ^ { d _ { v } } ) , \\mathcal { U } ( D ; \\mathbb { R } ^ { d _ { v } } ) )$ maps to bounded linear operators on $\\mathcal { U } ( D ; \\mathbb { R } ^ { d _ { v } } )$ and is parameterized by $\\phi \\in \\Theta \\kappa$ , $W : \\mathbb { R } ^ { d _ { v } }  \\mathbb { R } ^ { d _ { v } }$ is a linear transformation, and $\\sigma : \\mathbb { R }  \\mathbb { R }$ is $a$ non-linear activation function whose action is defined component-wise. ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "We choose $\\textstyle \\mathcal { K } ( a ; \\phi )$ to be a kernel integral transformation parameterized by a neural network. ",
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+    {
+        "type": "text",
+        "text": "Definition 2 (Kernel integral operator $\\kappa$ ) Define the kernel integral operator mapping in (2) by ",
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+        "type": "equation",
+        "img_path": "images/f9af6bb2641ee5a8e63d78627d529568e58edf11f3e4176b85e3e63b28392231.jpg",
+        "text": "$$\n\\big ( \\mathcal { K } ( a ; \\phi ) v _ { t } \\big ) ( x ) : = \\int _ { D } \\kappa \\big ( x , y , a ( x ) , a ( y ) ; \\phi \\big ) v _ { t } ( y ) \\mathrm { d } y , \\qquad \\forall x \\in D\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $\\kappa _ { \\phi } : \\mathbb { R } ^ { 2 ( d + d _ { a } ) }  \\mathbb { R } ^ { d _ { v } \\times d _ { v } }$ is a neural network parameterized by $\\phi \\in \\Theta \\kappa$ ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Here $\\kappa _ { \\phi }$ plays the role of a kernel function which we learn from data. Together definitions 1 and 2 constitute a generalization of neural networks to infinite-dimensional spaces as first proposed in Li et al. (2020b). Notice even the integral operator is linear, the neural operator can learn highly non-linear operators by composing linear integral operators with non-linear activation functions, analogous to standard neural networks. ",
+        "bbox": [
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+        "type": "text",
+        "text": "If we remove the dependence on the function $a$ and impose $\\kappa _ { \\phi } ( x , y ) = \\kappa _ { \\phi } ( x - y )$ , we obtain that (3) is a convolution operator, which is a natural choice from the perspective of fundamental solutions. We exploit this fact in the following section by parameterizing $\\kappa _ { \\phi }$ directly in Fourier space and using the Fast Fourier Transform (FFT) to efficiently compute (3). This leads to a fast architecture that obtains state-of-the-art results for PDE problems. ",
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+    {
+        "type": "text",
+        "text": "4 FOURIER NEURAL OPERATOR",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "We propose replacing the kernel integral operator in (3), by a convolution operator defined in Fourier space. Let $\\mathcal { F }$ denote the Fourier transform of a function $\\dot { f } : D  \\mathbb { R } ^ { d _ { v } }$ and $\\scriptstyle { \\mathcal { F } } ^ { - 1 }$ its inverse then ",
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+        "img_path": "images/2655492bf00d2d93d030205bbeda98484965342cb48acf08c82b6b938a44fe35.jpg",
+        "text": "$$\n( { \\mathcal { F } } f ) _ { j } ( k ) = \\int _ { D } f _ { j } ( x ) e ^ { - 2 i \\pi \\langle x , k \\rangle } \\mathrm { d } x , \\qquad ( { \\mathcal { F } } ^ { - 1 } f ) _ { j } ( x ) = \\int _ { D } f _ { j } ( k ) e ^ { 2 i \\pi \\langle x , k \\rangle } \\mathrm { d } k\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "for $j = 1 , \\ldots , d _ { v }$ where $i = \\sqrt { - 1 }$ is the imaginary unit. By letting $\\kappa _ { \\phi } ( x , y , a ( x ) , a ( y ) ) = \\kappa _ { \\phi } ( x - y )$ in (3) and applying the convolution theorem, we find that ",
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+        "img_path": "images/2200b45319742785ffa15f6358ec8be8d0e3533d23e34967098405085c2d736a.jpg",
+        "text": "$$\n\\bigl ( \\mathcal { K } ( a ; \\phi ) v _ { t } \\bigr ) ( x ) = \\mathcal { F } ^ { - 1 } \\bigl ( \\mathcal { F } ( \\kappa _ { \\phi } ) \\cdot \\mathcal { F } ( v _ { t } ) \\bigr ) ( x ) , \\qquad \\forall x \\in D .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "We, therefore, propose to directly parameterize $\\kappa _ { \\phi }$ in Fourier space. ",
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+    {
+        "type": "text",
+        "text": "Definition 3 (Fourier integral operator $\\kappa$ ) Define the Fourier integral operator ",
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+        "img_path": "images/8cf5c65e0b01909811ec997b5ccb7ac364eea4474026d154e642dc8a290de891.jpg",
+        "text": "$$\n\\bigl ( \\mathcal { K } ( \\phi ) v _ { t } \\bigr ) ( x ) = \\mathcal { F } ^ { - 1 } \\Bigl ( R _ { \\phi } \\cdot ( \\mathcal { F } v _ { t } ) \\Bigr ) ( x ) \\qquad \\forall x \\in D\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $R _ { \\phi }$ is the Fourier transform of a periodic function $\\kappa : \\bar { D }  \\mathbb R ^ { d _ { v } \\times d _ { v } }$ parameterized by $\\phi \\in \\Theta \\kappa$ . An illustration is given in Figure 2 (b). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For frequency mode $k \\in D$ , we have $( \\mathcal { F } v _ { t } ) ( k ) \\in \\mathbb { C } ^ { d _ { v } }$ and $R _ { \\phi } ( k ) \\in \\mathbb { C } ^ { d _ { v } \\times d _ { v } }$ . Notice that since we assume $\\kappa$ is periodic, it admits a Fourier series expansion, so we may work with the discrete modes $k \\in  { \\mathbb { Z } ^ { d } }$ . We pick a finite-dimensional parameterization by truncating the Fourier series at a maximal number of modes $k _ { \\operatorname* { m a x } } = | Z _ { k _ { \\operatorname* { m a x } } } | = | \\{ k \\in \\mathbb { Z } ^ { d } : | k _ { j } | \\leq k _ { \\operatorname* { m a x } , j }$ , for $j = 1 , \\ldots , d \\} |$ . We thus parameterize $R _ { \\phi }$ directly as complex-valued $( k _ { \\operatorname* { m a x } } \\times d _ { v } \\times d _ { v } )$ -tensor comprising a collection of truncated Fourier modes and therefore drop $\\phi$ from our notation. Since $\\kappa$ is real-valued, we impose conjugate symmetry. We note that the set $Z _ { k _ { \\mathrm { m a x } } }$ is not the canonical choice for the low frequency modes of $v _ { t }$ . Indeed, the low frequency modes are usually defined by placing an upper-bound on the $\\ell _ { 1 }$ -norm of $k \\in  { \\mathbb { Z } ^ { d } }$ . We choose $Z _ { k _ { \\mathrm { m a x } } }$ as above since it allows for an efficient implementation. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The discrete case and the FFT. Assuming the domain $D$ is discretized with $n \\in \\mathbb N$ points, we have that $v _ { t } \\in \\mathbb { R } ^ { n \\times d _ { v } }$ and $\\mathcal { F } ( v _ { t } ) \\in \\mathbb { C } ^ { n \\times d _ { v } ^ { \\smile } }$ . Since we convolve $v _ { t }$ with a function which only has $k _ { \\mathrm { m a x } }$ Fourier modes, we may simply truncate the higher modes to obtain $\\mathcal { F } ( v _ { t } ) \\in \\mathbb { C } ^ { k _ { \\operatorname* { m a x } } \\times d _ { v } }$ . Multiplication by the weight tensor $R \\in \\mathbf { \\bar { \\mathbb { C } } } ^ { k _ { \\operatorname* { m a x } } \\times d _ { v } \\times d _ { v } }$ is then ",
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+        "img_path": "images/a50987d5675235f466eda99223cb0c54fead1145b68e9570a6af6f47a2c7b93c.jpg",
+        "text": "$$\n\\big ( R \\cdot ( \\mathcal { F } v _ { t } ) \\big ) _ { k , l } = \\sum _ { j = 1 } ^ { d _ { v } } R _ { k , l , j } ( \\mathcal { F } v _ { t } ) _ { k , j } , \\qquad k = 1 , \\dots , k _ { \\operatorname* { m a x } } , \\quad j = 1 , \\dots , d _ { v } .\n$$",
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+        "text": "When the discretization is uniform with resolution $s _ { 1 } \\times \\cdot \\cdot \\cdot \\times s _ { d } = n$ , $\\mathcal { F }$ can be replaced by the Fast Fourier Transform. For $f \\in \\mathbb { R } ^ { n \\times d _ { v } }$ , $k = ( k _ { 1 } , \\ldots , k _ { d } ) \\in \\mathbb { Z } _ { s _ { 1 } } \\times \\cdot \\cdot \\cdot \\times \\mathbb { Z } _ { s _ { d } }$ , and $x = ( x _ { 1 } , \\ldots , x _ { d } ) \\in D$ , the FFT $\\hat { \\mathcal { F } }$ and its inverse $\\hat { \\mathcal { F } } ^ { - 1 }$ are defined as ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0b740c497b76ac2bead63723ca2b0c6bf201fdffb1976f8cce7d0bea5e6d35e0.jpg",
+        "text": "$$\n\\begin{array} { r l } & { ( \\hat { \\mathcal { F } } f ) _ { l } ( k ) = \\displaystyle \\sum _ { x _ { 1 } = 0 } ^ { s _ { 1 } - 1 } \\cdots \\sum _ { x _ { d } = 0 } ^ { s _ { d } - 1 } f _ { l } ( x _ { 1 } , \\ldots , x _ { d } ) e ^ { - 2 i \\pi \\sum _ { j = 1 } ^ { d } \\frac { x _ { j } k _ { j } } { s _ { j } } } , } \\\\ & { ( \\hat { \\mathcal { F } } ^ { - 1 } f ) _ { l } ( x ) = \\displaystyle \\sum _ { k _ { 1 } = 0 } ^ { s _ { 1 } - 1 } \\cdots \\sum _ { k _ { d } = 0 } ^ { s _ { d } - 1 } f _ { l } ( k _ { 1 } , \\ldots , k _ { d } ) e ^ { 2 i \\pi \\sum _ { j = 1 } ^ { d } \\frac { x _ { j } k _ { j } } { s _ { j } } } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "for $l = 1 , \\ldots , d _ { v }$ . In this case, the set of truncated modes becomes ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/5a4eed44151b76398d92d7fba1b62f7ab67440a7a5bc0774f3281bf8409d6ade.jpg",
+        "text": "$$\nZ _ { k _ { \\mathrm { m a x } } } = \\{ ( k _ { 1 } , \\ldots , k _ { d } ) \\in \\mathbb { Z } _ { s _ { 1 } } \\times \\cdot \\cdot \\cdot \\times \\mathbb { Z } _ { s _ { d } } \\mid k _ { j } \\leq k _ { \\operatorname* { m a x } , j }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/244111bc0504a38eca980f4221813a64321cba0d617f2f1246b2a982b3756176.jpg",
+        "text": "$$\ns _ { j } - k _ { j } \\leq k _ { \\operatorname* { m a x } , j } , \\mathrm { f o r } j = 1 , \\ldots , d \\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "When implemented, $R$ is treated as a $( s _ { 1 } \\times \\cdot \\cdot \\cdot \\times s _ { d } \\times d _ { v } \\times d _ { v } )$ -tensor and the above definition of $Z _ { k _ { \\mathrm { m a x } } }$ corresponds to the “corners” of $R$ , which allows for a straight-forward parallel implementation of (5) via matrix-vector multiplication. In practice, we have found that choosing $k _ { \\operatorname* { m a x } , j } = 1 2$ which yields $k _ { \\operatorname* { m a x } } = 1 2 ^ { d }$ parameters per channel to be sufficient for all the tasks that we consider. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Parameterizations of $R$ . In general, $R$ can be defined to depend on $( \\mathcal { F } a )$ to parallel (3). Indeed, we can define $R _ { \\phi } : \\mathbb { Z } ^ { d } \\times \\mathbb { R } ^ { d _ { v } }  \\mathbb { R } ^ { d _ { v } \\times d _ { v } }$ as a parametric function that maps $\\big ( k , ( \\mathcal { F } a ) ( k ) \\big )$ to the values of the appropriate Fourier modes. We have experimented with linear as well as neural network parameterizations of $R _ { \\phi }$ . We find that the linear parameterization has a similar performance to the previously described direct parameterization, while neural networks have worse performance. This is likely due to the discrete structure of the space $\\mathbb { Z } ^ { d }$ . Our experiments in this work focus on the direct parameterization presented above. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Invariance to discretization. The Fourier layers are discretization-invariant because they can learn from and evaluate functions which are discretized in an arbitrary way. Since parameters are learned directly in Fourier space, resolving the functions in physical space simply amounts to projecting on the basis $e ^ { 2 \\pi i \\left. x , k \\right. }$ which are well-defined everywhere on $\\mathbb { R } ^ { d }$ . This allows us to achieve zero-shot super-resolution as shown in Section 5.4. Furthermore, our architecture has a consistent error at any resolution of the inputs and outputs. On the other hand, notice that, in Figure 3, the standard CNN methods we compare against have an error that grows with the resolution. ",
+        "bbox": [
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "Quasi-linear complexity. The weight tensor $R$ contains $k _ { \\operatorname* { m a x } { } } < n$ modes, so the inner multiplication has complexity $O ( k _ { \\operatorname* { m a x } } )$ . Therefore, the majority of the computational cost lies in computing the Fourier transform $\\mathcal { F } ( v _ { t } )$ and its inverse. General Fourier transforms have complexity ${ \\dot { O } } ( n ^ { 2 } )$ , however, since we truncate the series the complexity is in fact $O ( n k _ { \\operatorname* { m a x } } )$ , while the FFT has complexity $O ( n \\log n )$ . Generally, we have found using FFTs to be very efficient. However a uniform discretization is required. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5 NUMERICAL EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In this section, we compare the proposed Fourier neural operator with multiple finite-dimensional architectures as well as operator-based approximation methods on the 1-d Burgers’ equation, the 2-d Darcy Flow problem, and 2-d Navier-Stokes equation. The data generation processes are discussed in Appendices A.3.1, A.3.2, and A.3.3 respectively. We do not compare against traditional solvers (FEM/FDM) or neural-FEM type methods since our goal is to produce an efficient operator approximation that can be used for downstream applications. We demonstrate one such application to the Bayesian inverse problem in Section 5.5. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We construct our Fourier neural operator by stacking four Fourier integral operator layers as specified in (2) and (4) with the ReLU activation as well as batch normalization. Unless otherwise specified, we use $N = 1 0 0 0$ training instances and 200 testing instances. We use Adam optimizer to train for 500 epochs with an initial learning rate of 0.001 that is halved every 100 epochs. We set $k _ { \\operatorname* { m a x } , j } = 1 6 , d _ { v } = 6 4$ for the 1-d problem and $k _ { \\operatorname* { m a x } , j } = 1 2 , d _ { v } = 3 2$ for the 2-d problems. Lower resolution data are downsampled from higher resolution. All the computation is carried on a single Nvidia V100 GPU with 16GB memory. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Remark on Resolution. Traditional PDE solvers such as FEM and FDM approximate a single function and therefore their error to the continuum decreases as the resolution is increased. On the other hand, operator approximation is independent of the ways its data is discretized as long as all relevant information is resolved. Resolution-invariant operators have consistent error rates among different resolutions as shown in Figure 3. Further, resolution-invariant operators can do zero-shot super-resolution, as shown in Section 5.4. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Benchmarks for time-independent problems (Burgers and Darcy): NN: a simple point-wise feedforward neural network. RBM: the classical Reduced Basis Method (using a POD basis) (De",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/ccd83d87eb17dedce6b2605c5a253578b14747d2dce6282d84e35bec0901f81f.jpg",
+        "image_caption": [
+            "Figure 3: Benchmark on Burger’s equation, Darcy Flow, and Navier-Stokes "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Left: benchmarks on Burgers equation; Mid: benchmarks on Darcy Flow for different resolutions; Right: the learning curves on Navier-Stokes $\\nu = 1 \\mathrm { e } - 3$ with different benchmarks. Train and test on the same resolution. For acronyms, see Section 5; details in Tables 1, 3, 4. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Vore, 2014). FCN: a the-state-of-the-art neural network architecture based on Fully Convolution Networks (Zhu & Zabaras, 2018). PCANN: an operator method using PCA as an autoencoder on both the input and output data and interpolating the latent spaces with a neural network (Bhattacharya et al., 2020). GNO: the original graph neural operator (Li et al., 2020b). MGNO: the multipole graph neural operator (Li et al., 2020a). LNO: a neural operator method based on the low-rank decomposition of the kernel $\\begin{array} { r } { \\kappa ( x , y ) : = \\sum _ { j = 1 } ^ { r } \\phi _ { j } ( x ) \\psi _ { j } ( y ) } \\end{array}$ , similar to the unstacked DeepONet proposed in (Lu et al., 2019). FNO: the newly purposed Fourier neural operator. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Benchmarks for time-dependent problems (Navier-Stokes): ResNet: 18 layers of 2-d convolution with residual connections (He et al., 2016). U-Net: A popular choice for image-to-image regression tasks consisting of four blocks with 2-d convolutions and deconvolutions (Ronneberger et al., 2015). TF-Net: A network designed for learning turbulent flows based on a combination of spatial and temporal convolutions (Wang et al., 2020). FNO-2d: 2-d Fourier neural operator with a RNN structure in time. FNO-3d: 3-d Fourier neural operator that directly convolves in space-time. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.1 BURGERS’ EQUATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The 1-d Burgers’ equation is a non-linear PDE with various applications including modeling the one dimensional flow of a viscous fluid. It takes the form ",
+        "bbox": [
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+            604,
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/35059570958dc92136c92d64b959e399aa1d8948c3f4d274058c5610596bc403.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\partial _ { t } u ( x , t ) + \\partial _ { x } ( u ^ { 2 } ( x , t ) / 2 ) = \\nu \\partial _ { x x } u ( x , t ) , \\quad } & { x \\in ( 0 , 1 ) , t \\in ( 0 , 1 ] } \\\\ { u ( x , 0 ) = u _ { 0 } ( x ) , \\quad } & { x \\in ( 0 , 1 ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "with periodic boundary conditions where $u _ { 0 } \\in L _ { \\mathrm { p e r } } ^ { 2 } ( ( 0 , 1 ) ; \\mathbb { R } )$ is the initial condition and $\\nu \\in \\mathbb { R } _ { + }$ is the viscosity coefficient. We aim to learn the operator mapping the initial condition to the solution at time one, $G ^ { \\dagger } : L _ { \\mathrm { p e r } } ^ { 2 } ( ( 0 , 1 ) ; \\mathbb { R } ) \\to H _ { \\mathrm { p e r } } ^ { r } ( ( 0 , 1 ) ; \\mathbb { R } )$ defined by $u _ { 0 } \\mapsto u ( \\cdot , 1 )$ for any $r > 0$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The results of our experiments are shown in Figure 3 (a) and Table 3 (Appendix A.3.1). Our proposed method obtains the lowest relative error compared to any of the benchmarks. Further, the error is invariant with the resolution, while the error of convolution neural network based methods (FCN) grows with the resolution. Compared to other neural operator methods such as GNO and MGNO that use Nystrom sampling in physical space, the Fourier neural operator is both more accurate and ¨ more computationally efficient. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.2 DARCY FLOW ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We consider the steady-state of the 2-d Darcy Flow equation on the unit box which is the second order, linear, elliptic PDE ",
+        "bbox": [
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+            851,
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e6b078b5b47a2516f9680a7f778fd5682099e289884fa5c4f3a25cfe6e970d7f.jpg",
+        "text": "$$\n\\begin{array} { r l r l } { - \\nabla \\cdot ( a ( x ) \\nabla u ( x ) ) = f ( x ) } & { } & { x \\in ( 0 , 1 ) ^ { 2 } } \\\\ { u ( x ) = 0 } & { } & { x \\in \\partial ( 0 , 1 ) ^ { 2 } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            882,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "with a Dirichlet boundary where $a \\ \\in \\ L ^ { \\infty } ( ( 0 , 1 ) ^ { 2 } ; \\mathbb { R } _ { + } )$ is the diffusion coefficient and $f \\in$ $L ^ { 2 } ( ( 0 , 1 ) ^ { 2 } ; \\mathbb { R } )$ is the forcing function. This PDE has numerous applications including modeling the pressure of subsurface flow, the deformation of linearly elastic materials, and the electric potential in conductive materials. We are interested in learning the operator mapping the diffusion coefficient to the solution, $G ^ { \\dag } : L ^ { \\infty } ( ( 0 , 1 ) _ { : } ^ { 2 } ; \\mathbb { R } _ { + } ) \\to H _ { 0 } ^ { 1 } ( ( 0 , \\bar { 1 } ) ^ { 2 } ; \\mathbb { R } _ { + } )$ defined by $a \\mapsto u$ . Note that although the PDE is linear, the operator $G ^ { \\dagger }$ is not. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The results of our experiments are shown in Figure 3 (b) and Table 4 (Appendix A.3.2). The proposed Fourier neural operator obtains nearly one order of magnitude lower relative error compared to any benchmarks. We again observe the invariance of the error with respect to the resolution. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.3 NAVIER-STOKES EQUATION ",
+        "text_level": 1,
+        "bbox": [
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+            252,
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+            267
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We consider the 2-d Navier-Stokes equation for a viscous, incompressible fluid in vorticity form on the unit torus: ",
+        "bbox": [
+            173,
+            277,
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+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9b12661f6d77b0927b2c18fb479c5ca7ebe952833dbecf438abc760481cbfa40.jpg",
+        "text": "$$\n\\begin{array} { r l r l } { \\partial _ { t } w ( x , t ) + u ( x , t ) \\cdot \\nabla w ( x , t ) = \\nu \\Delta w ( x , t ) + f ( x ) , } & { } & { x \\in ( 0 , 1 ) ^ { 2 } , t \\in ( 0 , T ] } \\\\ { \\nabla \\cdot u ( x , t ) = 0 , } & { } & & { x \\in ( 0 , 1 ) ^ { 2 } , t \\in [ 0 , T ] } \\\\ { w ( x , 0 ) = w _ { 0 } ( x ) , } & { } & & { x \\in ( 0 , 1 ) ^ { 2 } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            233,
+            306,
+            764,
+            368
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "where $u \\in C ( [ 0 , T ] ; H _ { \\mathrm { p e r } } ^ { r } ( ( 0 , 1 ) ^ { 2 } ; \\mathbb { R } ^ { 2 } ) )$ for any $r > 0$ is the velocity field, $w = \\nabla \\times u$ is the vorticity, $w _ { 0 } \\in L _ { \\mathrm { p e r } } ^ { 2 } ( ( 0 , 1 ) ^ { 2 } ; \\mathbb { R } )$ is the initial vorticity, $\\nu \\in \\mathbb { R } _ { + }$ is the viscosity coefficient, and $f \\in$ $L _ { \\mathrm { p e r } } ^ { 2 } ( ( 0 , 1 ) ^ { 2 } ; \\mathbb { R } )$ is the forcing function. We are interested in learning the operator mapping the vorticity up to time 10 to the vorticity up to some later time $T > 1 0$ $1 0 , G ^ { \\dagger } : C ( [ 0 , 1 0 ] ; H _ { \\mathrm { p e r } } ^ { r } ( ( 0 , 1 ) ^ { 2 } ; \\mathbb { R } ) ) \\to$ $C ( ( 1 0 , T ] ; H _ { \\mathrm { p e r } } ^ { r } ( ( 0 , 1 ) ^ { 2 } ; \\mathbb { R } ) )$ defined by $w | _ { ( 0 , 1 ) ^ { 2 } \\times [ 0 , 1 0 ] } \\mapsto w | _ { ( 0 , 1 ) ^ { 2 } \\times ( 1 0 , T ] }$ . Given the vorticity it is easy to derive the velocity. While vorticity is harder to model compared to velocity, it provides more information. By formulating the problem on vorticity, the neural network models mimic the pseudospectral method. We experiment with the viscosities $\\nu = { 1 } \\mathrm { e } \\mathrm { - } 3 , { 1 } \\mathrm { e } \\mathrm { - } 4 , { 1 } \\mathrm { e } \\mathrm { - } 5$ , decreasing the final time $T$ as the dynamic becomes chaotic. Since the baseline methods are not resolution-invariant, we fix the resolution to be $6 4 \\times 6 4$ for both training and testing. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/71d51cc690696b945b4850656f8b74e980438b1c65d893d870d195af9cea91c3.jpg",
+        "table_caption": [
+            "Table 1: Benchmarks on Navier Stokes (fixing resolution $6 4 \\times 6 4$ for both training and testing) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Config</td><td>Parameters</td><td>Time per</td><td>v=le-3 T=50</td><td>v=le-4 T=30 N = 1000</td><td>V=le-4 T=30 N = 10000</td><td>v=le-5 T=20</td></tr><tr><td>FNO-3D</td><td>6,558,537</td><td>epoch 38.99s</td><td>N = 1000 0.0086</td><td>0.1918</td><td>0.0820</td><td>N = 1000 0.1893</td></tr><tr><td>FNO-2D</td><td>414,517</td><td>127.80s</td><td>0.0128</td><td>0.1559</td><td>0.0834</td><td>0.1556</td></tr><tr><td>U-Net</td><td>24,950,491</td><td>48.67s</td><td>0.0245</td><td>0.2051</td><td>0.1190</td><td>0.1982</td></tr><tr><td>TF-Net</td><td>7,451,724</td><td>47.21s</td><td>0.0225</td><td>0.2253</td><td>0.1168</td><td>0.2268</td></tr><tr><td>ResNet</td><td>266,641</td><td>78.47s</td><td>0.0701</td><td>0.2871</td><td>0.2311</td><td>0.2753</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "As shown in Table 1, the FNO-3D has the best performance when there is sufficient data $( \\nu ~ =$ 1e−3, $N = 1 0 0 0$ and $\\nu = 1 \\mathrm { e } { - } 4 , N = 1 0 0 0 0 )$ . For the configurations where the amount of data is insufficient $\\langle \\nu = 1 \\mathrm { e - } 4$ , $N = 1 0 0 0$ and $\\nu = 1 \\mathrm { e } { - } 5 , N = 1 0 0 0 )$ , all methods have $> 1 5 \\%$ error with FNO-2D achieving the lowest. Note that we only present results for spatial resolution $6 4 \\times 6 4$ since all benchmarks we compare against are designed for this resolution. Increasing it degrades their performance while FNO achieves the same errors. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "2D and 3D Convolutions. FNO-2D, U-Net, TF-Net, and ResNet all do 2D-convolution in the spatial domain and recurrently propagate in the time domain $( 2 \\mathrm { D } \\mathrm { + R N N } )$ . The operator maps the solution at the previous 10 time steps to the next time step (2D functions to 2D functions). On the other hand, FNO-3D performs convolution in space-time. It maps the initial time steps directly to the full trajectory (3D functions to 3D functions). The $2 \\mathrm { D } { + } \\mathrm { R } \\mathrm { N } \\mathrm { N }$ structure can propagate the solution to any arbitrary time $T$ in increments of a fixed interval length $\\Delta t$ , while the Conv3D structure is fixed to the interval $[ 0 , T ]$ but can transfer the solution to an arbitrary time-discretization. We find the 3-d method to be more expressive and easier to train compared to its RNN-structured counterpart. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.4 ZERO-SHOT SUPER-RESOLUTION. ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The neural operator is mesh-invariant, so it can be trained on a lower resolution and evaluated at a higher resolution, without seeing any higher resolution data (zero-shot super-resolution). Figure 1 shows an example where we train the FNO-3D model on $6 4 \\times 6 4 \\times 2 0$ resolution data in the setting above with $( \\nu = 1 \\mathrm { e } { - } 4 , N = 1 0 0 0 0 )$ and transfer to $2 5 6 \\times 2 5 6 \\times 8 0$ resolution, demonstrating super-resolution in space-time. Fourier neural operator is the only model among the benchmarks (FNO-2D, U-Net, TF-Net, and ResNet) that can do zero-shot super-resolution. And surprisingly, it can do super-resolution not only in the spatial domain but also in the temporal domain. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "5.5 BAYESIAN INVERSE PROBLEM ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this experiment, we use a function space Markov chain Monte Carlo (MCMC) method (Cotter et al., 2013) to draw samples from the posterior distribution of the initial vorticity in Navier-Stokes given sparse, noisy observations at time $T = 5 0$ . We compare the Fourier neural operator acting as a surrogate model with the traditional solvers used to generate our train-test data (both run on GPU). We generate 25,000 samples from the posterior (with a 5,000 sample burn-in period), requiring 30,000 evaluations of the forward operator. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "As shown in Figure 6 (Appendix A.5), FNO and the traditional solver recover almost the same posterior mean which, when pushed forward, recovers well the late-time dynamic of Navier Stokes. In sharp contrast, FNO takes $0 . 0 0 5 s$ to evaluate a single instance while the traditional solver, after being optimized to use the largest possible internal time-step which does not lead to blow-up, takes $2 . 2 s$ . This amounts to 2.5 minutes for the MCMC using FNO and over 18 hours for the traditional solver. Even if we account for data generation and training time (offline steps) which take 12 hours, using FNO is still faster! Once trained, FNO can be used to quickly perform multiple MCMC runs for different initial conditions and observations, while the traditional solver will take 18 hours for every instance. Furthermore, since FNO is differentiable, it can easily be applied to PDE-constrained optimization problems without the need for the adjoint method. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Spectral analysis. Due to the way we parameterize $R _ { \\phi }$ , the function output by (4) has at most $k _ { \\mathrm { m a x } , j }$ Fourier modes per channel. This, however, does not mean that the Fourier neural operator can only approximate functions up to $k _ { \\operatorname* { m a x } , j }$ modes. Indeed, the activation functions which occur between integral operators and the final decoder network $Q$ recover the high frequency modes. As an example, consider a solution to the Navier-Stokes equation with viscosity $\\nu = 1 \\mathrm { e } - 3$ . Truncating this function at 20 Fourier modes yields an error around $2 \\%$ while our Fourier neural operator learns the parametric dependence and produces approximations to an error of $\\leq 1 \\%$ with only $k _ { \\operatorname* { m a x } , j } = 1 2$ parameterized modes. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Non-periodic boundary condition. Traditional Fourier methods work only with periodic boundary conditions. However, the Fourier neural operator does not have this limitation. This is due to the linear transform $W$ (the bias term) which keeps the track of non-periodic boundary. As an example, the Darcy Flow and the time domain of Navier-Stokes have non-periodic boundary conditions, and the Fourier neural operator still learns the solution operator with excellent accuracy. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6 DISCUSSION AND CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Requirements on Data. Data-driven methods rely on the quality and quantity of data. To learn Navier-Stokes equation with Reynolds number $R e \\ = \\ 1 \\mathrm { e } { + } 4$ , we need to generate $N \\ = \\ 1 0 0 0 0$ training pairs $\\{ a _ { j } , u _ { j } \\}$ with the numerical solver. However, for more challenging PDEs, generating a few training samples can be already very expensive. A future direction is to combine neural operators with numerical solvers to levitate the requirements on data. Recurrent structure. The neural operator has an iterative structure that can naturally be formulated as a recurrent network where all layers share the same parameters without sacrificing performance. (We did not impose this restriction in the experiments.) Computer vision. Operator learning is not restricted to PDEs. Images can naturally be viewed as real-valued functions on 2-d domains and videos simply add a temporal structure. Our approach is therefore a natural choice for problems in computer vision where invariance to discretization crucial is important (Chi et al., 2020). ",
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+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGEMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "The authors want to thank Ray Wang and Rose Yu for meaningful discussions. Z. Li gratefully acknowledges the financial support from the Kortschak Scholars Program. A. Anandkumar is supported in part by Bren endowed chair, LwLL grants, Beyond Limits, Raytheon, Microsoft, Google, Adobe faculty fellowships, and DE Logi grant. K. Bhattacharya, N. B. Kovachki, B. Liu, and A. M. Stuart gratefully acknowledge the financial support of the Army Research Laboratory through the Cooperative Agreement Number W911NF-12-0022. Research was sponsored by the Army Research Laboratory and was accomplished under Cooperative Agreement Number W911NF-12-2- 0022. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
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+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            299,
+            117
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.1 TABLE OF NOTATIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            138,
+            370,
+            154
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A table of notations is given in Table 2. ",
+        "bbox": [
+            176,
+            169,
+            431,
+            184
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/36a13449f0046a3ff9f1ac75f7e3f076fde5013a7f27d4c32177a153dea584aa.jpg",
+        "table_caption": [
+            "Table 2: table of notations "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=2>Notation                                          Meaning</td></tr><tr><td rowspan=1 colspan=1>Operator learningDcRdxEDa ∈ A= (D;Rda)u ∈U= (D;Rdu)DjG² :A→Uμ</td><td rowspan=1 colspan=1>The spatial domain for the PDEPoints in the the spatial domainThe input coefficient functionsThe target solution functionsThe discretization of (aj,uj)The operator mapping the coefficients to the solutionsA probability measure where aj sampled from.</td></tr><tr><td rowspan=1 colspan=1>Neural operatorU(x)∈RdudaduduK : R2(d+1) →Rdu×du中t=0,...,T0</td><td rowspan=1 colspan=1>The neural network representation of u(x)Dimension of the input a(x).Dimension of the output u(x).The dimension of the representation v(x)The kernel maps (x,y,a(x),a(y)) to a dy × d matrixThe parameters of the kernel network KThe time steps (layers)The activation function</td></tr><tr><td rowspan=1 colspan=1>Fourier operatorF,F-1RWkkmax</td><td rowspan=1 colspan=1>Fourier transformation and its inverse.The linear transformation applied on the lower Fourier modes.The linear transformation (bias term) applied on the spatial domain.Fourier modes /wave numbers.The max Fourier modes used in the Fourier layer.</td></tr><tr><td rowspan=1 colspan=1>HyperparametersNnSVT</td><td rowspan=1 colspan=1>The number of training pairs.The size of the discretization.The resolution of the discretization (sd = n).The viscosity.The time interval [O,T] for time-dependent equation.</td></tr></table>",
+        "bbox": [
+            179,
+            238,
+            816,
+            666
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.2 SPECTRAL ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            715,
+            367,
+            729
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The spectral decay of the Navier Stokes equation data is shown in Figure 4. The spectrum decay has a slope $k ^ { - 5 / 3 }$ , matching the energy spectrum in the turbulence region. And we notice the energy spectrum does not decay along with time. ",
+        "bbox": [
+            174,
+            746,
+            823,
+            790
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We also present the spectral decay in term of the truncation $k _ { m a x }$ defined in 4 as shown in Figure5. We note all equations (Burgers, Darcy, and Navier-Stokes with $\\nu \\leq 1 \\mathrm { e } { - 4 }$ ) exhibit high frequency modes. Even we truncate at $k _ { m a x } = 1 2$ in the Fourier layer, the Fourier neural operator can recover the high frequency modes. ",
+        "bbox": [
+            174,
+            796,
+            825,
+            853
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.3 DATA GENERATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            878,
+            351,
+            893
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In this section, we provide the details of data generator for the three equation we used in Section 5. ",
+        "bbox": [
+            171,
+            909,
+            820,
+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/935eb5d09a2858b9d89d2f07d454ba4a50ddfeaa6db73e62950aa4080022dbfb.jpg",
+        "image_caption": [
+            "Figure 4: Spectral Decay of Navier-Stokes equations "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            99,
+            813,
+            233
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The spectral decay of the Navier-stokes equation data we used in section 5.3. The y-axis is the spectrum; the $\\mathbf { X }$ -axis is the wavenumber $| k | = k _ { 1 } + k _ { 2 }$ . ",
+        "bbox": [
+            178,
+            236,
+            821,
+            263
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/bd9a3b97b63e751f0867517ae5ededa124bd7af9e103c3f18efc52feb56cdf6a.jpg",
+        "image_caption": [
+            "Figure 5: Spectral Decay in term of $k _ { m a x }$ "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            357,
+            310,
+            622,
+            465
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The error of truncation in one single Fourier layer without applying the linear transform $R$ . The y-axis is the normalized truncation error; the $\\mathbf { X }$ -axis is the truncation mode $k _ { m a x }$ . ",
+        "bbox": [
+            178,
+            469,
+            816,
+            498
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.3.1 BURGERS EQUATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            550,
+            379,
+            565
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Recall the 1-d Burger’s equation on the unit torus: ",
+        "bbox": [
+            174,
+            574,
+            501,
+            590
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6e266443ee3297c09e01877fb89f2283484014036e582f35080d5cdfcf4d1dd8.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\partial _ { t } u ( x , t ) + \\partial _ { x } ( u ^ { 2 } ( x , t ) / 2 ) = \\nu \\partial _ { x x } u ( x , t ) , \\quad } & { x \\in ( 0 , 1 ) , t \\in ( 0 , 1 ] } \\\\ { u ( x , 0 ) = u _ { 0 } ( x ) , \\quad } & { x \\in ( 0 , 1 ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            272,
+            597,
+            725,
+            636
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The initial condition $u _ { 0 } ( x )$ is generated according to $u _ { 0 } \\sim \\mu$ where $\\mu = \\mathcal { N } ( 0 , 6 2 5 ( - \\Delta + 2 5 I ) ^ { - 2 } )$ with periodic boundary conditions. We set the viscosity to $\\nu = 0 . 1$ and solve the equation using a split step method where the heat equation part is solved exactly in Fourier space then the non-linear part is advanced, again in Fourier space, using a very fine forward Euler method. We solve on a spatial mesh with resolution $2 ^ { 1 3 } = \\bar { 8 1 9 2 }$ and use this dataset to subsample other resolutions. ",
+        "bbox": [
+            174,
+            643,
+            826,
+            714
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.3.2 DARCY FLOW ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            729,
+            330,
+            744
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The 2-d Darcy Flow is a second-order linear elliptic equation of the form ",
+        "bbox": [
+            176,
+            755,
+            651,
+            770
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cd85e3d5bdf7b7766601c9186c999d1c867cfcf4297553b1b4e41d1ce1ba8b83.jpg",
+        "text": "$$\n\\begin{array} { r l } { - \\nabla \\cdot ( a ( x ) \\nabla u ( x ) ) = f ( x ) \\quad } & { x \\in ( 0 , 1 ) ^ { 2 } } \\\\ { u ( x ) = 0 \\quad } & { x \\in \\partial ( 0 , 1 ) ^ { 2 } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            343,
+            776,
+            650,
+            819
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The coefficients $a ( x )$ are generated according to $a \\sim \\mu$ where $\\mu = \\psi _ { \\# } \\mathcal N ( 0 , ( - \\Delta + 9 I ) ^ { - 2 } )$ with zero Neumann boundary conditions on the Laplacian. The mapping $\\psi : \\mathbb { R }  \\mathbb { R }$ takes the value 12 on the positive part of the real line and 3 on the negative and the push-forward is defined pointwise. The forcing is kept fixed $f ( x ) = 1$ . Such constructions are prototypical models for many physical systems such as permeability in subsurface flows and material microstructures in elasticity. Solutions $u$ are obtained by using a second-order finite difference scheme on a $4 2 1 \\times 4 2 1$ grid. Different resolutions are downsampled from this dataset. ",
+        "bbox": [
+            173,
+            825,
+            825,
+            924
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.3.3 NAVIER-STOKES EQUATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            426,
+            118
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Recall the 2-d Navier-Stokes equation for a viscous, incompressible fluid in vorticity form on the unit torus: ",
+        "bbox": [
+            171,
+            127,
+            825,
+            156
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5c98ee54fd750f00edcc776df323d812aa3d76cffb3aff694ff3b2f98e6e5d60.jpg",
+        "text": "$$\n\\begin{array} { r l r l } & { \\partial _ { t } w ( x , t ) + u ( x , t ) \\cdot \\nabla w ( x , t ) = \\nu \\Delta w ( x , t ) + f ( x ) , } & & { x \\in ( 0 , 1 ) ^ { 2 } , t \\in ( 0 , T ] } \\\\ & { \\qquad \\nabla \\cdot u ( x , t ) = 0 , } & & { x \\in ( 0 , 1 ) ^ { 2 } , t \\in [ 0 , T ] } \\\\ & { \\qquad w ( x , 0 ) = w _ { 0 } ( x ) , } & & { x \\in ( 0 , 1 ) ^ { 2 } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            233,
+            159,
+            764,
+            222
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "The initial condition $w _ { 0 } ( x )$ is generated according to $w _ { 0 } \\sim \\mu$ where $\\mu = \\mathcal { N } ( 0 , 7 ^ { 3 / 2 } ( - \\Delta { + } 4 9 I ) ^ { - 2 . 5 } )$ with periodic boundary conditions. The forcing is kept fixed $f ( x ) = 0 . 1 ( \\sin ( 2 \\pi ( x _ { 1 } + x _ { 2 } ) ) +$ $\\cos ( { \\bar { 2 \\pi } } ( x _ { 1 } + x _ { 2 } ) ) )$ ). The equation is solved using the stream-function formulation with a pseudospectral method. First a Poisson equation is solved in Fourier space to find the velocity field. Then the vorticity is differentiated and the non-linear term is computed is physical space after which it is dealiased. Time is advanced with a Crank–Nicolson update where the non-linear term does not enter the implicit part. All data are generated on a $2 5 6 \\times 2 5 6$ grid and are downsampled to $6 4 \\times 6 4$ . We use a time-step of $\\mathrm { 1 e { - } 4 }$ for the Crank–Nicolson scheme in the data-generated process where we record the solution every $t = 1$ time units. The step is increased to $2 \\mathrm { e } { - 2 }$ when used in MCMC for the Bayesian inverse problem. ",
+        "bbox": [
+            173,
+            228,
+            826,
+            367
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "A.4 RESULTS OF BURGERS’ EQUATION AND DARCY FLOW ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            383,
+            594,
+            398
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "The details error rate on Burgers’ equation and Darcy Flow are listed in Table 3 and Table 4. ",
+        "bbox": [
+            173,
+            410,
+            777,
+            425
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/c56b8d0a4c0505fc677446ae10285c6e11a6c1586b4c9b61a11421432108ecbf.jpg",
+        "table_caption": [
+            "Table 3: Benchmarks on 1-d Burgers’ equation "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Networks</td><td>s= 256</td><td>s= 512</td><td>s= 1024</td><td>s = 2048</td><td>s = 4096</td><td>s= 8192</td></tr><tr><td>NN</td><td>0.4714</td><td>0.4561</td><td>0.4803</td><td>0.4645</td><td>0.4779</td><td>0.4452</td></tr><tr><td>GCN</td><td>0.3999</td><td>0.4138</td><td>0.4176</td><td>0.4157</td><td>0.4191</td><td>0.4198</td></tr><tr><td>FCN</td><td>0.0958</td><td>0.1407</td><td>0.1877</td><td>0.2313</td><td>0.2855</td><td>0.3238</td></tr><tr><td>PCANN</td><td>0.0398</td><td>0.0395</td><td>0.0391</td><td>0.0383</td><td>0.0392</td><td>0.0393</td></tr><tr><td>GNO</td><td>0.0555</td><td>0.0594</td><td>0.0651</td><td>0.0663</td><td>0.0666</td><td>0.0699</td></tr><tr><td>LNO</td><td>0.0212</td><td>0.0221</td><td>0.0217</td><td>0.0219</td><td>0.0200</td><td>0.0189</td></tr><tr><td>MGNO</td><td>0.0243</td><td>0.0355</td><td>0.0374</td><td>0.0360</td><td>0.0364</td><td>0.0364</td></tr><tr><td>FNO</td><td>0.0149</td><td>0.0158</td><td>0.0160</td><td>0.0146</td><td>0.0142</td><td>0.0139</td></tr></table>",
+        "bbox": [
+            217,
+            468,
+            779,
+            599
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/e22806d5de75d4a273cc28b37c33ef587312a47d10797fdd464221cead993e68.jpg",
+        "table_caption": [
+            "Table 4: Benchmarks on 2-d Darcy Flow "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Networks</td><td>s=85</td><td>s=141</td><td>s=211</td><td>s= 421</td></tr><tr><td>NN</td><td>0.1716</td><td>0.1716</td><td>0.1716</td><td>0.1716</td></tr><tr><td>FCN</td><td>0.0253</td><td>0.0493</td><td>0.0727</td><td>0.1097</td></tr><tr><td>PCANN</td><td>0.0299</td><td>0.0298</td><td>0.0298</td><td>0.0299</td></tr><tr><td>RBM</td><td>0.0244</td><td>0.0251</td><td>0.0255</td><td>0.0259</td></tr><tr><td>GNO</td><td>0.0346</td><td>0.0332</td><td>0.0342</td><td>0.0369</td></tr><tr><td>LNO</td><td>0.0520</td><td>0.0461</td><td>0.0445</td><td></td></tr><tr><td>MGNO</td><td>0.0416</td><td>0.0428</td><td>0.0428</td><td>0.0420</td></tr><tr><td>FNO</td><td>0.0108</td><td>0.0109</td><td>0.0109</td><td>0.0098</td></tr></table>",
+        "bbox": [
+            312,
+            656,
+            686,
+            787
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "A.5 BAYESIAN INVERSE PROBLEM ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            816,
+            429,
+            832
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Results of the Bayesian inverse problem for the Navier-Stokes equation are shown in Figure 6. It can be seen that the result using Fourier neural operator as a surrogate is as good as the result of the traditional solver. ",
+        "bbox": [
+            176,
+            843,
+            823,
+            886
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/a7207702dccdb90c87f83495300ee949d84c7d72cff3999894b46e8c49e78d4d.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
+            194,
+            340,
+            803,
+            570
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "The top left panel shows the true initial vorticity while bottom left panel shows the true observed vorticity at $T = 5 0$ with black dots indicating the locations of the observation points placed on a $7 \\times 7$ grid. The top middle panel shows the posterior mean of the initial vorticity given the noisy observations estimated with MCMC using the traditional solver, while the top right panel shows the same thing but using FNO as a surrogate model. The bottom middle and right panels show the vorticity at $T = 5 0$ when the respective approximate posterior means are used as initial conditions. ",
+        "bbox": [
+            181,
+            579,
+            818,
+            662
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Figure 6: Results of the Bayesian inverse problem for the Navier-Stokes equation. ",
+        "bbox": [
+            228,
+            672,
+            766,
+            688
+        ],
+        "page_idx": 15
+    }
+]