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+# SELF-SUPERVISED VIDEO REPRESENTATION LEARNING WITH CONSTRAINED SPATIOTEMPORAL JIGSAW
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+This paper proposes a novel pretext task for self-supervised video representation learning by exploiting spatiotemporal continuity in videos. It is motivated by the fact that videos are spatiotemporal by nature and a representation learned to detect spatiotemporal continuity/discontinuity is thus beneficial for downstream video content analysis tasks. A natural choice of such a pretext task is to construct spatiotemporal (3D) jigsaw puzzles and learn to solve them. However, this task turns out to be intractable. We thus propose Constrained Spatiotemporal Jigsaw (CSJ) whereby the 3D jigsaws are formed in a constrained manner to ensure that large continuous spatiotemporal cuboids exist in a shuffled clip to provide sufficient cues for the model to reason about the continuity. With the constrained jigsaw puzzles, instead of solving them directly, which could still be extremely hard, we carefully design four surrogate tasks that are more solvable but meanwhile still ensure that the learned representation is sensitive to spatiotemporal continuity at both the local and global levels. Extensive experiments show that our CSJ achieves state-of-the-art on two downstream tasks across various benchmarks.
+
+# 1 INTRODUCTION
+
+Self-supervised learning (SSL) has achieved tremendous successes recently for static images (He et al., 2020; Chen et al., 2020) and shown to be able to outperform supervised learning on a wide range of downstream image understanding tasks. However, such successes have not yet been reproduced for videos. Since different SSL models differ mostly on the pretext tasks employed on the unlabeled training data, designing pretext tasks more suitable for videos is the current focus for self-supervised video representation learning (Han et al., 2020; Wang et al., 2020).
+
+Videos are spatiotemporal data and spatiotemporal analysis is the key to many video content understanding tasks. A good video representation learned from the self-supervised pretext task should therefore capture discriminative information jointly along both spatial and temporal dimensions. It is thus somewhat counter-intuitive to note that most existing SSL pretext tasks for videos do not explicitly require joint spatiotemporal video understanding. For example, some spatial pretext tasks have been borrowed from images without any modification (Jing et al., 2018), ignoring the temporal dimension. On the other hand, many recent video-specific pretext tasks typically involve speed or temporal order prediction (Lee et al., 2017; Wei et al., 2018; Benaim et al., 2020; Wang et al., 2020), i.e., operating predominately along the temporal axis.
+
+A natural choice for a spatiotemporal pretext task is to solve 3D jigsaw puzzles, whose 2D counterpart has been successfully used for images (Noroozi & Favaro, 2016). Indeed, solving 3D puzzles requires the learned model to understand spatiotemporal continuity, a key step towards video content understanding. However, directly solving a 3D puzzle turns out to be intractable: a puzzle of $3 \times 3 \times 3$ pieces (the same size as a Rubik’s cube) can have 27! possible permutations. Video volume even in a short clip is much larger than that. Nevertheless, the latest neural sorting models (Paumard et al., 2020; Du et al., 2020) can only handle permutations a few orders of magnitude less, so offer no solution. This is hardly surprising because such a task is daunting even for humans: Most people would struggle with a standard Rubik’s cube, let alone a much larger one.
+
+In this paper, we propose a novel Constrained Spatiotemporal Jigsaw (CSJ) pretext task for selfsupervised video representation learning. The key idea is to form 3D jigsaw puzzles in a constrained manner so that it becomes solvable. This is achieved by factorizing the permutations (shuffling)
+
+![](images/0e2765ac1fc37cb3ed7161a49676d6807733f039ef6007ddb1e38232e0101f70.jpg)  
+Figure 1: Illustration of our constrained jigsaw and the surrogate pretext tasks using an image example (only spatial for clarity). Our constrained jigsaw can be easily extended to the spatiotemporal domain as done in this work. (a): The raw image. (b),(c): Comparing an unconstrained puzzle (b) and our constrained one (c), it is clear that ours is much more continuous (hence interpretable) reflected by the size of the largest continuous cuboids (LCCs, rectangles in images here) shown in red. (d),(e): Illustration of the importance of the relative order of the top-2 LCCs for determining the global continuity level of the shuffled image. (d) and (e) have the same top-2 LCCs, but only (d) keeps the correct relative order between them. Locating these LCCs and predicting their relative order are thus the key objectives of our surrogate tasks.
+
+into the three spatiotemporal dimensions and then applying them sequentially. This ensures that for a given video clip, large continuous spatiotemporal cuboids exist after the constrained shuffling to provide sufficient cues for the model to reason about spatiotemporal continuity (see Fig. 1(b)(c)). Such large continuous cuboids are also vital for human understanding of video as revealed in neuroscience and visual studies (Stringer et al., 2006; Chen et al., 2019). Even with the constrained puzzles, solving them directly could still be extremely hard. Consequently, instead of directly solving the puzzles (i.e., recovering the permutation matrix so that each piece can be put back), four surrogate tasks are carefully designed. They are more solvable but meanwhile still ensure that the learned representation is sensitive to spatiotemporal continuity at both the local and global levels. Concretely, given a video clip shuffled with our constrained permutations, we make sure that the top-2 largest continuous cuboids (LCCs) dominate the clip volume. The level of continuity in the shuffle clip as a whole is thus determined mainly by the volumes of these LCCs, and whether they are at the right order (see Fig. 1(d)(e)) both spatially and temporally. Our surrogate tasks are thus designed to locate these LCCs and predict their order so that the model learned with these tasks can be sensitive to spatiotemporal continuity both locally and globally.
+
+Our main contributions are three-fold: (1) We introduce a new pretext task for self-supervised video representation learning called Constrained Spatiotemporal Jigsaw (CSJ). To our best knowledge, this is the first work on self-supervised video representation learning that leverages spatiotemporal jigsaw understanding. (2) We propose a novel constrained shuffling method to construct easy 3D jigsaws containing large LCCs. Four surrogate tasks are then formulated in place of the original jigsaw solving tasks. They are much more solvable yet remain effective in learning spatiotemporal discriminative representations. (3) Extensive experiments show that our approach achieves state-ofthe-art on two downstream tasks across various benchmarks.
+
+# 2 RELATED WORK
+
+Self-supervised Learning with Pretext Tasks Self-supervised learning (SSL) typically employs a pretext task to generate pseudo-labels for unlabeled data via some forms of data transformation. According to the transformations used by the pretext task, existing SSL methods for video presentation learning can be divided into three categories: (1) Spatial-Only Transformations: Derived from the original image domain (Gidaris et al., 2018), Jing et al. (2018) leveraged the spatial-only transformations for self-supervised video presentation learning. (2) Temporal-Only Transformations: Misra et al. (2016); Fernando et al. (2017); Lee et al. (2017); Wei et al. (2018) obtained shuffled video frames with the temporal-only transformations and then distinguished whether the shuffled frames are in chronological order. Xu et al. (2019) chose to shuffle video clips instead of frames. Benaim et al. (2020); Yao et al. (2020); Jenni et al. (2020) exploited the speed transformation via determining whether one video clip is accelerated. (3) Spatiotemporal Transformations: There are only a few recent approaches (Ahsan et al., 2019; Kim et al., 2019) that leveraged both spatial and temporal transformations by permuting 3D spatiotemporal cuboids. However, due to the aforementioned intractability of solving the spatiotemporal jigsaw puzzles, they only leveraged either temporal or spatial permutations as training signals, i.e., they exploited the two domains independently. Therefore, no true spatiotemporal permutations have been considered in Ahsan et al. (2019); Kim et al. (2019). In contrast, given that both spatial appearances and temporal relations are important cues for video representation learning, the focus of this work is on investigating how to exploit the spatial and temporal continuity jointly for self-supervised video presentation learning. To that end, our Constrained Spatiotemporal Jigsaw (CSJ) presents the first spatiotemporal continuity based pretext task for video SSL, thanks to a novel constrained 3D jigsaw and four surrogate tasks to reason about the continuity in the 3D jigsaw puzzles without solving them directly.
+
+Self-supervised Learning with Contrastive Learning Contrastive learning is another selfsupervised learning approach that has become increasingly popular in the image domain (Misra & Maaten, 2020; He et al., 2020; Chen et al., 2020). Recently, it has been incorporated into video SSL as well. Contrastive learning and transformation based pretext tasks are orthogonal to each other and often combined in that different transformed versions of a data sample form the positive set used in contrastive learning. In El-Nouby et al. (2019); Knights et al. (2020); Qian et al. (2020); Wang et al. (2020); Yang et al. (2020), the positive/negative samples were generated based on temporal transformations only. In contrast, some recent works (Han et al., 2019; 2020; Zhuang et al., 2020) leveraged features from the future frame embeddings or with the memory bank (Wu et al., 2018). They modeled spatiotemporal representations using only contrastive learning without transformations. Contrastive learning is also exploited in one of our surrogate pretext tasks. Different from existing works, we explore the spatiotemporal transformations in the form of CSJ and employ contrastive learning to distinguish different levels of spatiotemporal continuity in shuffled jigsaws. This enables us to learn more discriminative spatiotemporal representations.
+
+# 3 CONSTRAINED SPATIOTEMPORAL JIGSAW
+
+# 3.1 PROBLEM DEFINITION
+
+The main goal of self-supervised video representation learning is to learn a video feature representation function $f ( \cdot )$ without using any human annotations. A general approach to achieving this goal is to generate a supervisory signal $\textbf {  { y } }$ from an unlabeled video clip $_ { \textbf { \em x } }$ and construct a pretext task $P$ to predict $\textbf {  { y } }$ from $f ( { \pmb x } )$ . The process of solving the pretext task $P$ encourages $f ( \cdot )$ to learn discriminative spatiotemporal representations.
+
+The pretext task $P$ is constructed typically by applying to a video clip a transformation function $t ( \cdot ; \pmb \theta )$ parameterized by $\pmb { \theta }$ and then automatically deriving $\textbf {  { y } }$ from $\pmb \theta$ , e.g., $\textbf {  { y } }$ can be the type of the transformation. Based on this premise, $P$ is defined as the prediction of $\textbf {  { y } }$ using the feature map of the transformed video clip $f ( \widetilde { \pmb x } )$ , i.e., $P : f ( { \widetilde { \mathbf { x } } } ) \to { \pmb y }$ , where $\widetilde { \pmb { x } } = t ( \pmb { x } ; \pmb { \theta } )$ . For example, in Lee et al. (2017), $t ( \cdot ; \pmb \theta )$ e e e denotes a temporal transformation that permutes the four frames of video clip $_ { \textbf { \em x } }$ in a temporal order $\pmb \theta$ , $\widetilde { \pmb { x } } = t ( \pmb { x } ; \pmb { \theta } )$ is the shuffled clip, and the pseudo-label $\textbf {  { y } }$ is defined as the permutation order $\pmb \theta$ e(e.g., 1324, 4312, etc.). The pretext task $P$ is then a classification problem of 24 categories because there are $4 ! = 2 4$ possible orders.
+
+# 3.2 CONSTRAINED PERMUTATIONS
+
+Solving spatiotemporal video jigsaw puzzles seems to be an ideal pretext task for learning discriminative representation as it requires an understanding of spatiotemporal continuity. After shuffling the pixels in a video clip using a 3D permutation matrix, the pretext task is to recover the permutation matrix. However, as explained earlier, this task is intractable given even moderate video clip sizes. Our solution is to introduce constraints on the permutations. As a result, a new pretext task $P _ { \mathrm { C S J } }$ based on Constrained Spatiotemporal Jigsaw (see Fig. 2(a)) is formulated, which is much easier to solve than a random/unconstrained jigsaw.
+
+Specifically, our goal is to introduce constraints to the permutations so that the resultant shuffled video clip is guaranteed to have large continuous cuboids (see Fig. 2(a)). Similar to humans (Stringer et al., 2006), having large continuous cuboids is key for a model to understand a 3D jigsaw and therefore to have any chance to solve it. Formally, the volume of a shuffled video clip $\widetilde { \pmb x }$ are denoted as $\{ T , H , W \}$ e, measuring its sizes along the temporal, height, and width dimensions, respectively. A cuboid is defined as a crop of $\widetilde { \pmb x }$ : $\pmb { c } = \widetilde { \pmb { x } } _ { t _ { 1 } : t _ { 2 } , h _ { 1 } : h _ { 2 } , w _ { 1 } : w _ { 2 } }$ , where $t _ { 1 } , t _ { 2 } \in \{ 1 , 2 , \dots , T \} , h _ { 1 } , h _ { 2 } \in$ $\{ 1 , 2 , \ldots , H \} , w _ { 1 } , w _ { 2 } \in \{ 1 , 2 , \ldots , W \}$ . If all the jigsaw pieces (smallest video clip unit, e.g. a pixel or a 3D pixel block) in $^ c$ keep the same relative order as they were in $_ { \textbf { \em x } }$ (before being shuffled), we call the cuboid $^ c$ as a continuous cuboid $c ^ { \mathrm { c o n t } }$ . The cuboid’s volume equals $\left( t _ { 2 } - t _ { 1 } \right) \mathbf { \bar { \times } } \left( h _ { 2 } - h _ { 1 } \right) \times \mathbf { \bar { \Sigma } }$ $( w _ { 2 } - w _ { 1 } )$ , and the largest continuous cuboid (LCC) $c _ { \mathrm { m a x } } ^ { \mathrm { c o n t } }$ is the $c ^ { \mathrm { c o n t } }$ with the largest volume.
+
+![](images/1078e24fffb3ba13c20ebafabd92aee164ef40d676a91283df6ddb5e42457aac.jpg)  
+Figure 2: (a) Illustration of our Constrained Spatiotemporal Jigsaw (CSJ) (see Sec. 3.2). (b) The pipeline of our proposed framework for self-supervised video representation learning (see Sec. 3.3). A raw video clip is transformed into 8 shuffled clips with our Constrained Spatiotemporal Jigsaw (CSJ), and a 3D CNN sharing weights extracts the feature representations from them. The model is then trained by solving four self-supervised tasks jointly.
+
+We introduce two permutation strategies to ensure that the volumes of LCCs are large in relation to the whole video clip volume after our shuffling transformation $t ( \cdot ; \theta _ { \mathrm { C S J } } )$ . First, instead of shuffling $_ { \textbf { \em x } }$ in three spatiotemporal dimensions simultaneously, $t ( \cdot ; \theta _ { \mathrm { C S J } } )$ factorizes the permutations into the three spatiotemporal dimensions and then utilizes them sequentially to generate shuffled clips, e.g., in the order of $T , W , H$ and only once. Note that the volume of the generated $\widetilde { \pmb x }$ stays the same with different permutation orders (e.g., $T W H$ and $H T W$ e). Second, we shuffle a group of jigsaw pieces together instead of each piece individually along each dimension. Taking spatial shuffling as an example, if there are 8 pieces per frame (along each of the two spatial dimensions), $\theta _ { \mathrm { C S J } }$ could be represented as the permutation from $\{ 1 2 3 4 5 6 7 8 \}$ to $\{ 8 4 5 6 7 1 2 3 \}$ . The longest and the secondlongest index ranges are: [2, 5] for coordinates $\{ 4 5 6 7 \}$ , and [6, 8] for coordinates $\lbrace 1 2 3 \rbrace$ . With these two permutation strategies, not only do we have large LCCs, but also they are guaranteed to have clearly separable boundaries (see Fig. 2(b)) with surrounding pieces due to the factorized and grouped permutation design. This means that they are easily detectable.
+
+# 3.3 SURROGATE TASKS
+
+Having permutation constraints preserves more spatiotemporal continuity in the shuffled clip and reduces the amount of possible permutations. But exploiting these constraints to make a neural sorting model tractable is still far from trivial. Instead of solving the jigsaw directly, our $P _ { \mathrm { C S J } }$ is thus formulated as four surrogate tasks: Largest Continuous Cuboid Detection (LCCD), Clip Shuffling Pattern Classification (CSPC), Contrastive Learning over Shuffled Clips (CLSC), and Clip Continuity Measure Regression (CCMR). As illustrated in Fig. 2(b), given an unlabeled clip $_ { \textbf { \em x } }$ , we first construct a mini-batch of 8 clips $\{ \widetilde { \pmb { x } } _ { 1 } , \widetilde { \pmb { x } } _ { 2 } , . . . , \widetilde { \pmb { x } } _ { 8 } \}$ by shuffling $_ { \textbf { \em x } }$ with different but related constrained permutae e etions (to be detailed later). These shuffled clips and the raw clip $_ { \textbf { \em x } }$ are then fed into a 3D CNN model $f ( \cdot )$ for spatiotemporal representation learning with a non-local operation (Wang et al., 2018):
+
+$$
+f _ { \mathrm { N L } } ( \widetilde { \mathbf { x } } _ { i } ) = \mathrm { N L } ( f ( \widetilde { \mathbf { x } } _ { i } ) , f ( \pmb { x } ) ) ,
+$$
+
+where $\operatorname { N L } ( \cdot , \cdot )$ denotes the non-local operator, and $f ( \widetilde { \pmb x } _ { i } )$ and $f ( { \pmb x } )$ denote the feature map of $\widetilde { \pmb { x } } _ { i }$ and $_ { \textbf { \em x } }$ from the last convolutional layer of $f ( \cdot )$ e, respectively. The resultant feature map $f _ { \mathrm { N L } } ( \widetilde { \pmb x } _ { i } )$ eis further passed through a spatial pooling layer followed by a separately fully-connected layer for each surrogate task. Note that the raw video feature map $f ( { \pmb x } )$ is used as guidance through the nonlocal based attention mechanism to help fulfill the tasks. This is similar to humans needing to see the completed jigsaw picture to help solve the puzzle.
+
+Before we detail the four tasks, we first explain how the eight permutations from the same raw clip are generated. First, the factorized and grouped permutations are applied to $_ { \textbf { \em x } }$ to create one shuffled clip. By examining the largest and the second-largest continuous puzzle piece numbers of each dimension $( \{ T , H , \bar { W } \} )$ , we can easily identify the top-2 largest continuous cuboids (LCCs). Next, by varying the relative order of the top-2 LCCs either in the correct (original) order or the reverse order in each dimension, $2 \times 2 \times 2 { = } 8$ permutations are obtained. By controlling the group size in permutation, we can make sure that the top-2 LCCs account for a large proportion, saying $80 \%$ of the total clip volume. Our four tasks are thus centered around these two LCCs as they largely determine the overall spatiotemporal continuity of the shuffled clip.
+
+The first task LCCD is to locate the top-2 LCCs $\{ c _ { \mathrm { m a x } } ^ { \mathrm { c o n t } } ( j ) : j = 1 , 2 \}$ and formulated as a regression problem. Given a ground-truth LCC $c _ { \mathrm { m a x } } ^ { \mathrm { c o n t } } ( j )$ , a Gaussian kernel is applied to its center to depict the possibility of each pixel in $\widetilde { \pmb x }$ belonging to the LCC. This leads to a soft mask $M _ { \mathrm { L C C D } } ^ { j }$ with the same size of $\widetilde { \pmb x }$ : $M _ { \mathrm { L C C D } } ^ { j }$ is all 0 outside the region of $c _ { \mathrm { m a x } } ^ { \mathrm { c o n t } } ( j )$ , and $\exp \bigl ( - \frac { | | \boldsymbol { a } - \boldsymbol { a } _ { \mathrm { c } } | | ^ { 2 } } { 2 \sigma _ { g } ^ { 2 } } \bigr )$ inside the region, where ${ \mathbf { } } a , a _ { \mathrm { { c } } }$ denote any pixel and the center point, respectively. $\sigma _ { g }$ is the hyper-parameter which is set as 1 empirically. In the training stage, FPN (Lin et al., 2017) is used for multi-level feature fusion. LCCD is optimized using the MSE loss in each point:
+
+$$
+{ \cal L } _ { \mathrm { L C C D } } = \sum _ { j \in \{ 1 , 2 \} } \sum _ { { \bf { a } } \in \widetilde { { \pmb x } } } \mathrm { M S E } ( M _ { \mathrm { L C C D } } ^ { j } ( { \pmb a } ) , M _ { \mathrm { L C C D } } ^ { j } ( { \pmb a } ) ^ { ' } ) ,
+$$
+
+where $\mathrm { M S E } ( \cdot , \cdot )$ denotes the MSE loss function, and $M _ { \mathrm { L C C D } } ^ { j } ( \pmb { a } ) ^ { \prime }$ 0 is the prediction of each pixel $^ { a }$
+
+CSPC is designed to recognize the shuffling pattern of a shuffled clip. As mentioned early, the eight shuffled clips in each mini-batch are created from the same raw clip and differ only in the relative order of the top-2 LCCs along each of the three dimensions. There are thus eight permutations depending on the order (correct or reverse) in each dimension. Based on this understanding, CSPC is formulated as a multi-class classification task to recognize each shuffled clip into one of these eight classes, which is optimized using the Cross-Entropy (CE) loss:
+
+$$
+L _ { \mathrm { C S P C } } = \sum _ { i \in \{ 0 , 1 , . . . , 7 \} } { \bf C E } ( l _ { \mathrm { C S P C } } [ i ] , l _ { \mathrm { C S P C } } ^ { ' } [ i ] ) ,
+$$
+
+where $\operatorname { C E } ( \cdot , \cdot )$ denotes the CE loss function and $l _ { \mathrm { C S P C } } ^ { ' } [ i ]$ is the predicted class label of $i$ -th sample (shuffled clip) in each mini-batch.
+
+The two tasks above emphasize on local spatiotemporal continuity understanding. In contrast, CLSC leverages the contrastive loss to encourage global continuity understanding. In particular, since the top-2 LCCs dominate the volume of a clip, it is safe to assume that if their relative order is correct in all three dimensions, the shuffled clip largely preserve continuity compared to the original clip, while all other 7 permutations feature large discontinuity in at least one dimension. We thus form a contrastive learning task with the original video $_ { \textbf { \em x } }$ and the most continuous shuffled video $\widetilde { \pmb { x } } _ { i }$ as a positive pair, and $_ { \textbf { \em x } }$ and the rest $\widetilde { \pmb { x } } _ { j }$ $( j \neq i )$ ) as negative pairs. CLSC is optimized using the e eNoise Contrastive Estimation (NCE) (Tian et al., 2020) loss:
+
+$$
+\begin{array} { r } { L _ { \mathrm { C L S C } } = - \log \frac { \exp ( \sin ( f ( \pmb { x } ) , f ( \widetilde { \pmb { x } } _ { i } ) ) / \tau ) } { \exp ( \sin ( f ( \pmb { x } ) , f ( \widetilde { \pmb { x } } _ { i } ) ) / \tau ) + \sum _ { j } \exp ( \sin ( f ( \pmb { x } ) , f ( \widetilde { \pmb { x } } _ { j } ) ) / \tau ) } , } \end{array}
+$$
+
+where $\sin ( \cdot , \cdot )$ is defined by the dot product: $f ( \pmb { x } ) ^ { \top } f ( \widetilde { \pmb { x } } _ { i } )$ , and $\tau$ is the temperature hyper-parameter.   
+Note that the non-local operator is not used in CLSC.
+
+CCMR is similar to CLSC in that it also enforces global continuity understanding, but differs in that it is a regression task aimed at predicting a global continuity measure. We consider two such measures. Since the total size of the top-2 LCCs $\{ c _ { \mathrm { m a x } } ^ { \mathrm { c o n t } } ( j ) : j = 1 , 2 \}$ is a good indicator of how continuous a shuffle video clip is, the first measure $l _ { l d }$ directly measures the relative total size of the top-2 LCCs: $l _ { l d } = \frac { \mathbf { v } ( { c } _ { \mathrm { m a x } } ^ { \mathrm { c o n t } } ( 1 ) ) + \mathbf { v } ( { c } _ { \mathrm { m a x } } ^ { \mathrm { c o n t } } ( 2 ) ) } { \mathbf { v } ( \widetilde { \pmb { x } } ) }$ , where $\mathbf { v } ( \cdot )$ represents the volume of a clip/cuboid.
+
+The second measure $l _ { \mathrm { h d } } ^ { \mathrm { t / h / w } }$ examines the shuffling degree of $\widetilde { \pmb x }$ in each dimension, computed as the normalized hamming distance: $\frac { \mathrm { h a m m i n g } ( \widetilde { \pmb { x } } ) } { N _ { c } ( N _ { c } - 1 ) / 2 }$ where hamming $( \cdot )$ denotes the hamming distance in each dimension between the original piece sequence and the permuted one, and $N _ { c }$ represents the number of pieces in each dimension so that $N _ { c } ( \bar { N } _ { c } { - } 1 ) / 2$ indicates the maximum possible hamming distance in the dimension. CCMR is optimized using the Mean Squared Error (MSE) loss:
+
+$$
+\begin{array} { r } { L _ { \mathrm { C C M R } } = \mathbf { M S E } \big ( \big [ l _ { \mathrm { l d } } , l _ { \mathrm { h d } } ^ { \mathrm { t } } , l _ { \mathrm { h d } } ^ { \mathrm { h } } , l _ { \mathrm { h d } } ^ { \mathrm { w } } \big ] , \big [ l _ { \mathrm { l d } } ^ { ' } , l _ { \mathrm { h d } } ^ { \mathrm { t } ^ { \prime } } , l _ { \mathrm { h d } } ^ { \mathrm { h } ^ { \prime } } , l _ { \mathrm { h d } } ^ { \mathrm { w } ^ { \prime } } \big ] \big ) , } \end{array}
+$$
+
+where $l _ { \mathrm { l d } } ^ { ' } , l _ { \mathrm { h d } } ^ { \mathrm { t } ^ { \prime } } , l _ { \mathrm { h d } } ^ { \mathrm { h } ^ { \prime } } , l _ { \mathrm { h d } } ^ { \mathrm { w } ^ { \prime } }$ are the prediction of the model.
+
+# 3.4 OVERALL LEARNING OBJECTIVE
+
+Our entire CSJ framework is optimized end-to-end with the learning objective defined as:
+
+$$
+L = \sigma _ { 1 } L _ { \mathrm { L C C D } } + \sigma _ { 2 } L _ { \mathrm { C S P C } } + \sigma _ { 3 } L _ { \mathrm { C L S C } } + \sigma _ { 4 } L _ { \mathrm { C C M R } } ,
+$$
+
+where $\sigma _ { 1 } , \sigma _ { 2 } , \sigma _ { 3 } , \sigma _ { 4 }$ denote the weights for the four losses. We deploy the adaptive weighting mechanism (Kendall et al., 2018) to weight these tasks, and thus there is no free hyper-parameters to tune. We also adopt curriculum learning (Bengio et al., 2009; Korbar et al., 2018) to train our network by shuffling clips from easy to hard. More details are presented in Appendix. A.1 and A.2.
+
+# 4 EXPERIMENTS
+
+# 4.1 DATASETS AND SETTINGS
+
+We select three benchmark datasets for performance evaluation: UCF101 (Soomro et al., 2012), HMDB51 (Kuehne et al., 2011), and Kinetics-400 (K400) (Kay et al., 2017), containing 13K/7K/306K video clips from 101/51/400 action classes, respectively. In the self-supervised pretraining stage, we utilize the first training split of UCF101/HMDB51 and the training split of K400 without using their labels. As in Han et al. (2020), we adopt R2D3D as the backbone network, which is modified from R3D (Hara et al., 2018) with fewer parameters. By fine-tuning the pre-trained model, we can evaluate the SSL performance on a downstream task (i.e., action classification). Following Han et al. (2019); He et al. (2020), two evaluation protocols are used: comparisons against state-of-the-arts follow the more popular fully fine-tuning evaluation protocol, but ablation analysis takes both the linear evaluation and fully fine-tuning protocols. For the experiments on supervised learning, we report top-1 accuracy on the first test split of UCF101/HMDB51 as the standard (Han et al., 2020). More details of the datasets are provided in Appendix B.
+
+# 4.2 IMPLEMENTATION DETAILS
+
+Raw videos in these datasets are decoded at a frame rate of 24-30 fps. From each raw video, we start from a randomly selected frame index and sample a consecutive 16-frame video clip with a temporal stride of 4. For data augmentation, we first resize the video frames to $1 2 8 \times 1 7 1$ pixels, from which we extract random crops of size $1 1 2 \times 1 1 2$ pixels. We also apply random horizontal flipping and random color jittering to the video frames during training. We exploit only the raw RGB video frames as input, and do not leverage optical flow or other auxiliary signals for self-supervised pretraining. We adopt the Adam optimizer with a weight decay of $1 0 ^ { - 3 }$ and a batch size of 8 per GPU (with a total of 32 GPUs). We deploy cosine annealing learning rate with an initial value of $\bar { 1 } 0 ^ { - 4 }$ and 100 epochs. The jigsaw puzzle piece sizes of $\{ T , H , { \bar { W } } \}$ dimensions are set as $1 , 4 , 4$ , respectively. A $1 6 \times 1 1 2 \times 1 1 2$ video clip thus contains $1 6 \times 2 8 \times 2 8$ pieces. We set the temperature hyper-parameter $\tau$ to 0.07. A dropout of 0.5 is applied to the final layer of each task. More implementation details of the fine-tuning and test evaluation stages can be found in Appendix B.
+
+# 4.3 MAIN RESULTS
+
+Comparison in Action Recognition A standard way to evaluate a self-supervised video representation learning model is to use it to initialize an action recognition model on a small dataset. Specifically, after self-supervised pre-training on UCF101/HMDB51/K400, we exploit the learned backbone for fully fine-tuning on UCF101 and HMDB51, following Han et al. (2020); Wang et al. (2020).
+
+Table 1: Comparison to the state-of-the-art on UCF101(U) and HMDB51 $\mathrm { ( H ) }$ . All models are pretrained with the RGB modality only. $\dagger$ : Methods with temporal-only transformations. $^ \ddag$ : Methods with both spatial and temporal transformations. $* 1$ : Methods that leverage spatiotemporal representations. HT: HowTo100M. The underline represents the second-best result.   
+
+<table><tr><td>Methods</td><td>Backbone</td><td>Pre-trained Datasets</td><td>Input size (T*H)</td><td>UCF101</td><td>HMDB51</td></tr><tr><td>CMC (ECCV&#x27;20) (Tian et al.,2020)</td><td>C2D</td><td>U</td><td>1</td><td>55.3</td><td></td></tr><tr><td>Skip-Clip† (El-Nouby et al.,2019)</td><td>R3D-18</td><td>U/H</td><td>16*112</td><td>64.4</td><td>一</td></tr><tr><td>VCOPt (CVPR&#x27;19) (Xu et al.,2019)</td><td>R3D-18</td><td>U/H</td><td>16*112</td><td>64.9</td><td>29.5</td></tr><tr><td>VCP† (AAAI&#x27;20) (Luo et al.,2020b)</td><td>R3D-18</td><td>U/H</td><td>16*112</td><td>66.0</td><td>31.5</td></tr><tr><td>PRPt (CVPR&#x27;20)(Yao et al.,2020)</td><td>R3D-18</td><td>U/H</td><td>16*112</td><td>66.5</td><td>29.7</td></tr><tr><td>MemDPC* (ECCV&#x27;20) (Han et al.,2020) CSJ*(ours)</td><td>R2D3D-18</td><td>U/H</td><td>40*128</td><td>69.2</td><td>一</td></tr><tr><td>Video-Jigsaw* (WACV&#x27;19) (Ahsan et al.,2019)</td><td>R2D3D-18</td><td>U/H</td><td>16*112</td><td>70.4</td><td>36.0</td></tr><tr><td>Statisics*(CVPR&#x27;19) (Wang et al.,2019)</td><td>C2D C3D</td><td>K400</td><td>25*224</td><td>55.4</td><td>27.0</td></tr><tr><td>ST-Puzzle‡ (AAAI&#x27;19) (Kim et al.,2019)</td><td>R3D-18</td><td>K400</td><td>16*112</td><td>61.2</td><td>33.4</td></tr><tr><td>DPC*(ICCVW&#x27;19) (Han et al.,2019)</td><td>R2D3D-18</td><td>K400</td><td>16*112</td><td>63.9</td><td>33.7</td></tr><tr><td>SpeedNet (CVPR&#x27;20) (Benaim et al.,2020)</td><td></td><td>K400</td><td>40*128</td><td>68.2</td><td>34.5</td></tr><tr><td>VIE*(CVPR&#x27;20) (Zhuang et al.,2020)</td><td>I3D</td><td>K400</td><td>16*224</td><td>66.7</td><td>43.7</td></tr><tr><td>Pace† (ECCV&#x27;20) (Wang et al.,2020)</td><td>R3D-18</td><td>K400</td><td>16*112</td><td>75.5</td><td>44.6</td></tr><tr><td>CSJ* (ours)</td><td>R(2+1)D-18</td><td>K400</td><td>16*112</td><td>77.1</td><td>36.6</td></tr><tr><td></td><td>R2D3D-18</td><td>K400</td><td>16*112</td><td>76.2</td><td>46.7</td></tr><tr><td>MemDPC* (ECCV&#x27;20) (Han et al.,2020) CBT (Sun et al.,2019)</td><td>R2D3D-34</td><td>K400</td><td>40*224</td><td>78.1</td><td>41.2</td></tr><tr><td>CSJ* (ours)</td><td>S3D R2D3D-34</td><td>K600+HT K400</td><td>一 16*224</td><td>79.5</td><td>44.6</td></tr><tr><td></td><td></td><td></td><td></td><td>79.5</td><td>50.9</td></tr><tr><td>Upper Bound:Fully-Supervised</td><td>R3D-34</td><td>K400</td><td>16*224</td><td>87.7</td><td>59.1</td></tr></table>
+
+Table 2: Comparison with state-of-the-art self-supervised learning methods for nearest neighbor video retrieval (top- $k$ recall) on UCF101. The underline represents the second-best result.   
+
+<table><tr><td>Methods</td><td>Backbone</td><td>Top1</td><td>Top5</td><td>Top10</td><td>Top20</td><td>Top50</td></tr><tr><td>VCOP (CVPR&#x27;19) (Xu et al., 2019)</td><td>R3D-18</td><td>14.1</td><td>30.3</td><td>40.0</td><td>51.1</td><td>66.5</td></tr><tr><td>VCP (AAAI&#x27;20) (Luo et al., 2020b)</td><td>R3D-18</td><td>18.6</td><td>33.6</td><td>42.5</td><td>53.5</td><td>68.1</td></tr><tr><td>SpeedNet (CVPR&#x27;20) (Benaim et al., 2020)</td><td>S3D-G</td><td>13.1</td><td>28.1</td><td>37.5</td><td>49.5</td><td>65.0</td></tr><tr><td>PRP (CVPR&#x27;2O) (Yao et al.,2020)</td><td>R3D-18</td><td>22.8</td><td>38.5</td><td>46.7</td><td>55.2</td><td>69.1</td></tr><tr><td>Pace (ECCV&#x27;2O) (Wang et al., 2020)</td><td>R3D-18</td><td>19.9</td><td>36.2</td><td>46.1</td><td>55.6</td><td>69.2</td></tr><tr><td>ERUV (Luo et al., 2020a)</td><td>R3D-18</td><td>21.4</td><td>35.2</td><td>43.8</td><td>53.1</td><td>68.3</td></tr><tr><td>MemDPC (ECCV&#x27;20) (Han et al., 2020)</td><td>R2D3D-18</td><td>20.2</td><td>40.4</td><td>52.4</td><td>64.7</td><td>1</td></tr><tr><td>CSJ (ours)</td><td>R2D3D-18</td><td>21.5</td><td>40.5</td><td>53.2</td><td>64.9</td><td>70.0</td></tr></table>
+
+We consider one baseline: fully-supervised learning with pre-training on K400. Note that this baseline is commonly regarded as the upper bound of self-supervised representation learning (Alwassel et al., 2019). From Table 1, we have the following observations: (1) Our CSJ achieves state-of-theart performance on both UCF101 and HMDB51. Particularly, with the backbone R2D3D-18 that is weaker than $\mathrm { R } ( 2 + 1 ) \mathrm { D } { - } 1 8 $ , our CSJ performs comparably w.r.t. Pace on UCF101 but achieves a $10 \%$ improvement over Pace on HMDB51. (2) By exploiting spatiotemporal transformations for self-supervised representation learning, our CSJ beats either methods with only temporal transformations $( \dag )$ or methods with both spatial and temporal transformations $( \ddagger )$ , as well as those learning spatiotemporal representations $( \ast )$ via only contrastive learning (w./o. spatiotemporal transformations). (3) Our CSJ also outperforms CBT (Sun et al., 2019), which used ten-times more massive datasets (K600 (Carreira et al., 20 $ { \left| 8 \right. } +  { \mathrm { H } }$ owto100M (Miech et al., 2019)) and multiple modalities (RGB $+ .$ Audio). (4) Our CSJ is the closest to the fully-supervised one (upper bound), validating its effectiveness in self-supervised video representation learning.
+
+Comparison in Video Retrieval We evaluate our CSJ method in the video retrieval task. Following Xu et al. (2019), we extract each video clips’ embeddings with the pre-training model and use each clip in the test set to query the $k$ nearest clips in the training set. The comparative results in Table 2 show that our method outperforms all other self-supervised methods and achieves new state-of-the-art in video retrieval on UCF101. Particularly, our method beats the latest competitor
+
+Table 3: Evaluation of pre-training tasks with the backbone R2D3D-18 under linear probe and fully fine-tuning protocols on UCF101. AW: Adaptive Weighting. CL: Curriculum Learning.   
+
+<table><tr><td>Tasks</td><td>Linear Probe</td><td>Fully Fine-tuning</td></tr><tr><td>Random Initialization</td><td>8.3</td><td>63.6</td></tr><tr><td>LCCD</td><td>21.8</td><td>67.8</td></tr><tr><td>CSPC</td><td>22.6</td><td>68.1</td></tr><tr><td>CLSC</td><td>18.9</td><td>68.1</td></tr><tr><td>CCMR</td><td>22.7</td><td>68.1</td></tr><tr><td>CCMR+CSPC</td><td>24.7</td><td>69.2</td></tr><tr><td>CCMR+CSPC+CLSC</td><td>25.5</td><td>69.3</td></tr><tr><td>CCMR+CSPC+CLSC+LCCD</td><td>27.9</td><td>69.5</td></tr><tr><td>CCMR+CSPC+CLSC+LCCD+AW</td><td>28.2</td><td>70.0</td></tr><tr><td>CCMR+CSPC+CLSC+LCCD+AW+CL</td><td>28.5</td><td>70.4</td></tr></table>
+
+![](images/05879f14857b6a2ef744f82417429a2c36c8fe64aa824955368e18b252cb5f3a.jpg)  
+Figure 3: Attention visualization of the last feature maps from the fine-tuned models on UCF101. The first row denotes the raw frames from videos, and the last two rows correspond to fine-tuning from random initialization and our self-supervised pre-trained model, respectively.
+
+PRP (Yao et al., 2020) on four out of five metrics. This indicates that our proposed CSJ is also effective for video representation learning in video retrieval.
+
+# 4.4 FURTHER EVALUATIONS
+
+Ablation Study We conduct ablative experiments to validate the effectiveness of four CSJ surrogate tasks and two additional learning strategies. From Table 3, we can observe that: (1) Selfsupervised learning with each of the four tasks shows better generalization than fine-tuning the network from scratch (random initialization). (2) By training over all the four tasks jointly, we can achieve large performance gains (see ‘+LCCD’ vs. ‘CCMR’). (3) Each additional learning strategy (i.e., adaptive weighting or curriculum learning) leads to a small boost to the performance by 0.3- $0 . 5 \%$ . (4) Our full model achieves a remarkable classification accuracy of $7 0 . 4 \%$ , demonstrating the effectiveness of our proposed CSJ with only the RGB video stream (without additional optical flow, audio, or text modalities). More ablative analysis can be found in Appendix D.
+
+Visualization of Attention Maps Fig. 3 visualizes the attention map of the last feature maps from two models fine-tuned on UCF101 with or without adopting our self-supervised pre-training. Since each frame’s attention map involves four adjacent frames, it actually contains spatiotemporal semantic features. We can see that our self-supervised pre-training with CSJ indeed helps to better capture meaningful spatiotemporal information and thus recognize the action categories more correctly.
+
+![](images/364822a46bd24076097579693c7a18c390267468a5a5de153032980e75a3b9f3.jpg)  
+Figure 4: Visualization of the LCCD predictions from the pre-trained models. Each row denotes the frames at time stamp $= ( 0 , 4 , 8 , 1 2 )$ from one video clip. (a) raw frames (with color jittering); (b) shuffled frames; (c) the ground truth of LCCD; (d) network’s prediction.
+
+Visualization of LCCD Predictions We also demonstrate the visualization of the LCCD predictions from the pre-trained models in Fig. 4. We can observe that solving the LCCD task indeed enables the model to learn the locations of LCCs and understand spatiotemporal continuity, which is a key step towards video content understanding.
+
+# 5 CONCLUSION
+
+We have introduced a novel self-supervised video representation learning method named Constrained Spatiotemporal Jigsaw (CSJ). By introducing constrained permutations, our proposed CSJ is the first to leverage spatiotemporal jigsaw in self-supervised video representation learning. We also propose four surrogate tasks based on our constrained spatiotemporal jigsaws. They are designed to encourage a video representation model to understand the spatiotemporal continuity, a key building block towards video content analysis. Extensive experiments were carried out to validate the effectiveness of each of the four CSJ tasks and also show that our approach achieves the state-of-the-art on two downstream tasks across various benchmarks.
+
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+
+# A ADDITIONAL LEARNING STRATEGIES
+
+# A.1 ADAPTIVE WEIGHT
+
+Formally, our CSJ has two continuous outputs $\mathrm { y } _ { 1 } , \mathrm { y } _ { 4 }$ from LCCD and CCMR, and two discrete outputs $\mathrm { y } _ { 2 } , \mathrm { y } _ { 3 }$ from CSPC and CLSC, modeled with Gaussian likelihoods and softmax likelihoods, respectively. The joint loss for these four tasks $L ( \mathbf { W } , \sigma _ { 1 } , \sigma _ { 2 } , \sigma _ { 3 } , \sigma _ { 4 } )$ is:
+
+$$
+\begin{array} { r l } & { L ( \mathbf { W } , \sigma _ { 1 } , \sigma _ { 2 } , \sigma _ { 3 } , \sigma _ { 4 } ) } \\ & { = - \log \mathcal { N } ( \mathbf { y } _ { 1 } ; \mathbf { f } ^ { \mathbf { W } } ( \mathbf { x } ) , \sigma _ { 1 } ^ { 2 } ) \cdot - \log \mathcal { N } ( \mathbf { y } _ { 4 } ; \mathbf { f } ^ { \mathbf { W } } ( \mathbf { x } ) , \sigma _ { 4 } ^ { 2 } ) } \\ & { \quad \cdot \ \mathrm { s o f t m a x } ( \mathbf { y } _ { 2 } = \sigma ; \mathbf { f } ^ { \mathbf { W } } ( \mathbf { x } ) , \sigma _ { 2 } ) \cdot \mathrm { s o f t m a x } ( \mathbf { y } _ { 3 } = \sigma ; \mathbf { f } ^ { \mathbf { W } } ( \mathbf { x } ) , \sigma _ { 3 } ) } \\ & { = \frac { 1 } { 2 \sigma _ { 1 } ^ { 2 } } | | \mathbf { y } _ { 1 } - \mathbf { f } ^ { \mathbf { W } } ( \mathbf { x } ) | | ^ { 2 } + \log \sigma _ { 1 } - \frac { 1 } { 2 \sigma _ { 4 } ^ { 2 } } | | \mathbf { y } _ { 4 } - \mathbf { f } ^ { \mathbf { W } } ( \mathbf { x } ) | | ^ { 2 } + \log \sigma _ { 4 } } \\ & { \quad - \log p ( \mathbf { y } _ { 2 } | \mathbf { f } ^ { \mathbf { W } } ( \mathbf { x } ) , \sigma _ { 2 } ) - \log p ( \mathbf { y } _ { 3 } | \mathbf { f } ^ { \mathbf { W } } ( \mathbf { x } ) , \sigma _ { 3 } ) } \\ & { \approx \frac { 1 } { 2 \sigma _ { 1 } ^ { 2 } } L ( \mathbf { W } ) + \frac { 1 } { \sigma _ { 2 } ^ { 2 } } L _ { 2 } ( \mathbf { W } ) + \frac { 1 } { \sigma _ { 3 } ^ { 2 } } L _ { 3 } ( \mathbf { W } ) + \frac { 1 } { 2 \sigma _ { 4 } ^ { 2 } } L _ { 4 } ( \mathbf { W } ) } \\ & { \quad + \log \sigma _ { 1 } + \log \sigma _ { 2 } + \log \sigma _ { 3 } + \log \sigma _ { 4 } , } \end{array}
+$$
+
+where $\sigma$ is the weight factor that can be automatically learned from the network, and the log likelihood for the output y is defined as:
+
+$$
+\log p ( \mathrm { y } = c | \mathbf { f } ^ { \mathbf { W } } ( \mathbf { x } ) , \boldsymbol { \sigma } ) = \frac { 1 } { \sigma ^ { 2 } } f _ { c } ^ { \mathbf { W } } ( \mathbf { x } ) - \log \sum _ { c ^ { \prime } } \exp ( \frac { 1 } { \sigma ^ { 2 } } f _ { c ^ { \prime } } ^ { \mathbf { W } } ( \mathbf { x } ) ) .
+$$
+
+# A.2 CURRICULUM LEARNING
+
+We adopt curriculum learning (Korbar et al., 2018) to train our network by shuffling clips from easy to hard. Let $d$ be the shuffle degree of a shuffled clip $\widetilde { \pmb x }$ , representing the number of continuous cuboids in each dimension. We gradually increase $d$ efrom 3 to 5 during the training phase to produce more permuted clips. Note that when the video content is ambiguous in one dimension, e.g., a static video clip inflated from an image, there is no temporal variance to learn the transformation. Kim et al. (2019); Noroozi & Favaro (2016) also mentioned this problem as similar-looking ambiguity. To solve this problem, we calculate the variance on each dimension and set a threshold. If the variance is lower than the threshold, we decrease $d$ from 3 to 1 so that the pieces are not shuffled in the corresponding dimension.
+
+# B DATASETS AND IMPLEMENTATION
+
+# B.1 DETAILS OF DATASETS
+
+UCF101 (Soomro et al., 2012) is a widely-used dataset in the action recognition task, which contains 13,320 videos with 101 action classes. The dataset is divided into three training/testing splits. In this paper, following prior works (Wang et al., 2020; Han et al., 2020), we use the first training split as the pre-training dataset and the first testing split for evaluation.
+
+HMDB51 (Kuehne et al., 2011) is a relatively small action recognition dataset, consisting of 6,766 videos with 51 categories. It is also divided into three training/testing splits. Following Wang et al. (2020); Han et al. (2020), we use the first training split as the pre-training dataset and the first testing split for evaluation.
+
+Kinetics-400 (K400) (Kay et al., 2017) is a very large action recognition dataset consisting of 400 human action classes and around 306k videos. In this work, we use the training split of K400 as the pre-training dataset.
+
+# B.2 IMPLEMENTATION DETAILS
+
+In the fine-tuning stage, weights of convolutional layers are initialized with self-supervised pretraining, but weights of fully-connected layers are randomly initialized. The whole network is then trained with the cross-entropy loss. The pre-processing and training strategies are the same as in the
+
+Table 4: The structure of the encoding function $f ( \cdot )$ . R2D3D-18 is used as an example.   
+
+<table><tr><td rowspan=1 colspan=1>stage</td><td rowspan=1 colspan=3>detail</td><td rowspan=1 colspan=1>output sizeT×HW×C</td></tr><tr><td rowspan=1 colspan=1> input data</td><td rowspan=1 colspan=3>1</td><td rowspan=1 colspan=1>16×112²×3</td></tr><tr><td rowspan=1 colspan=1>conV1</td><td rowspan=1 colspan=3>1 × 7²,64stride 1, 22</td><td rowspan=1 colspan=1>16 × 56² × 64</td></tr><tr><td rowspan=1 colspan=1>pool1</td><td rowspan=1 colspan=3>1 × 3²,64stride 1, 22</td><td rowspan=1 colspan=1>16 × 28² × 64</td></tr><tr><td rowspan=1 colspan=1>res2</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>[1 ×3²,641 × 3²,64</td><td rowspan=1 colspan=1>×2</td><td rowspan=1 colspan=1>16 × 28² × 64</td></tr><tr><td rowspan=1 colspan=1>res3</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>1 × 3²,1281× 3²,128</td><td rowspan=1 colspan=1>×2</td><td rowspan=1 colspan=1>16 × 14² × 128</td></tr><tr><td rowspan=1 colspan=1>res4</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>3× 3²,2563 × 32,256</td><td rowspan=1 colspan=1>×2</td><td rowspan=1 colspan=1>8×7² × 256</td></tr><tr><td rowspan=1 colspan=1>res5</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>3 × 3²,512[3 × 3²,512</td><td rowspan=1 colspan=1>×2</td><td rowspan=1 colspan=1>4 × 4² × 512</td></tr><tr><td rowspan=1 colspan=1>Avgpool</td><td rowspan=1 colspan=3>4 × 4²,512stride 1,12</td><td rowspan=1 colspan=1>1 ×1² × 512</td></tr></table>
+
+self-supervised pre-training stage, except that the total epochs are 300 and the initial learning rate is $1 0 ^ { - 3 }$ . We use a batch size of 64 per GPU and a total of 8 GPUs for fine-tuning.
+
+We follow the standard evaluation protocol (Han et al., 2020) during inference and use ten-crop to take the same sequence length as training from the video. The predicted label of each video is calculated by averaging the softmax probabilities of all clips in the video.
+
+# C NETWORK ARCHITECTURE
+
+We deploy the same network backbone R2D3D as Han et al. (2019; 2020), which is a 3D-ResNet (R3D) similar to Hara et al. (2018). The only difference between R2D3D and R3D lies in that: R2D3D keeps the first two residual blocks as 2D convolutional blocks while R3D uses 3D blocks. Therefore, the modified R2D3D has fewer parameters (only the last two blocks are 3D convolutions). We present the CNN structure of R2D3D in Table 4.
+
+# D ADDITIONAL ABLATION STUDIES
+
+Table 5: Evaluation of pre-training tasks under different designs of LCCD on UCF101.   
+
+<table><tr><td rowspan=1 colspan=1>Methods</td><td rowspan=1 colspan=1>LCCS</td><td rowspan=1 colspan=1>LCCD+MLccs</td><td rowspan=1 colspan=1>LCCD + L1</td><td rowspan=1 colspan=1>LCCD+MSE</td></tr><tr><td rowspan=1 colspan=1>Top-1 Acc</td><td rowspan=1 colspan=1>66.5</td><td rowspan=1 colspan=1>64.0</td><td rowspan=1 colspan=1>66.5</td><td rowspan=1 colspan=1>67.8</td></tr></table>
+
+# D.1 LCCD
+
+Instead of predicting center points using the detection method, we also design a segmentation method – largest continuous cuboid segmentation (LCCS) to predicts the location of top-2 LCCs $\{ c _ { \operatorname* { m a x } } ^ { \mathrm { c o n t } } ( j ) : j = 1 , 2 \}$ . The difference between LCCD and LCCS lies in that: LCCS is formulated as a segmentation task to discriminate whether a pixel is in the region of $c _ { \mathrm { m a x } } ^ { \mathrm { c o n t } } ( j )$ . Concretely, LCCS predicts a binary mask $M _ { \mathrm { L C C S } } ^ { j }$ where only points in the region of $\{ c _ { \mathrm { m a x } } ^ { \mathrm { c o n t } } ( j )$ are set to be 1, otherwise 0. As a result, LCCS is optimized using the Cross Entropy (CE) loss at each point:
+
+$$
+{ \cal L } _ { \mathrm { L C C S } } = \sum _ { j \in \{ 1 , 2 \} } \sum _ { { \pmb { a } } \in { \pmb { \widetilde x } } } \mathrm { C E } ( M _ { \mathrm { L C C S } } ^ { j } ( { \pmb { a } } ) , M _ { \mathrm { L C C S } } ^ { j } ( { \pmb { a } } ) ^ { ' } ) ,
+$$
+
+where $\operatorname { C E } ( \cdot , \cdot )$ denotes the CE loss function, and $M _ { \mathrm { L C C S } } ^ { j } ( \pmb { a } ) ^ { \prime }$ is the predicted class of pixel $\textbf { \em a }$ .
+
+We report the performance of four different designs of LCCD in Table 5: (1) LCCS: LCCS is used instead of LCCD. (2) $\mathrm { L C C D } { + } M _ { \mathrm { L C C S } }$ : The Gaussian mask $M _ { \mathrm { L C C D } }$ is substituted by the binary mask $M _ { \mathrm { L C C S } }$ , but the LCCD task is optimized using the MSE loss. (3) $\mathrm { L C C D + L 1 }$ : The LCCD task is optimized by the L1 loss. (4) $\mathrm { L C C D + M S E }$ : The LCCD task is optimized by the MSE loss. From Table 5, it can be seen that the segmentation task also helps self-supervised representation learning but doesn’t perform as well as LCCD. Also, under the three different settings of LCCD, the MSE loss with the Gaussian map performs the best.
+
+Table 6: Evaluation of different temperature $\tau$ for CLSC on UCF101.   
+
+<table><tr><td rowspan=1 colspan=1>Methods</td><td rowspan=1 colspan=1>T=1</td><td rowspan=1 colspan=1>T =0.1</td><td rowspan=1 colspan=1>T = 0.07</td></tr><tr><td rowspan=1 colspan=1>Top-1 Acc</td><td rowspan=1 colspan=1>60.9</td><td rowspan=1 colspan=1>66.3</td><td rowspan=1 colspan=1>68.1</td></tr></table>
+
+# D.2 CLSC
+
+Table 6 above shows the accuracies obtained with different temperatures $\tau$ used in contrastive learning. We can observe that: (1) When $\tau$ is in the range $1 \sim 0 . 0 7$ , the accuracy increases with smaller $\tau$ . (2) When $\tau$ is large (e.g., 1), the accuracy drops considerably. In this work, $\tau$ is set to 0.0.
+
+Table 7: Evaluation of different designs of CSPC on UCF101.   
+
+<table><tr><td rowspan=1 colspan=1>Methods</td><td rowspan=1 colspan=1>2 Categories</td><td rowspan=1 colspan=1>4 Categories</td><td rowspan=1 colspan=1>8 Categories</td></tr><tr><td rowspan=1 colspan=1>Top-1 Acc</td><td rowspan=1 colspan=1>67.0</td><td rowspan=1 colspan=1>68.0</td><td rowspan=1 colspan=1>68.1</td></tr></table>
+
+# D.3 CSPC
+
+In addition to our CSPC with 8 pattern categories (see Sec. 3.3), we consider another two designs: (1) 2 Categories: the shuffled clip is discriminated by whether it has the same relative order of the top-2 LCCs as the raw clip. It is almost the same as CLSC but is optimized by the CE loss. (2) 4 Categories: the shuffled clip is discriminated by how it differs from the raw clip: non-difference, spatial-only difference, temporal-only difference, spatiotemporal difference. From Table 7, we can see that CSPC with 8 categories outperforms the other two designs. These results support our motivation for leveraging spatiotemporal transformations.
+
+Table 8: Evaluation of different designs of CCMR on UCF101.   
+
+<table><tr><td rowspan=1 colspan=1>Methods</td><td rowspan=1 colspan=1>ld</td><td rowspan=1 colspan=1>hd</td><td rowspan=1 colspan=1>ld+hd</td></tr><tr><td rowspan=1 colspan=1>Top-1 Acc</td><td rowspan=1 colspan=1>66.9</td><td rowspan=1 colspan=1>67.2</td><td rowspan=1 colspan=1>68.1</td></tr></table>
+
+# D.4 CCMR
+
+We report the performance of three different designs of CCMR: (1) ld: the learning degree $l _ { \mathrm { l d } }$ is used as supervision, which only contains volume information. (2) hdare used, which contain only the relative order information. (3) mming distances : both ld and hd $l _ { \mathrm { h d } } ^ { \mathrm { t } } , l _ { \mathrm { h d } } ^ { \mathrm { h } } , l _ { \mathrm { h d } } ^ { \mathrm { w } }$ $\mathrm { l d } + \mathrm { h d }$   
+supervision. From Table 8, we can see that: First, both ld and hd help the model to learn continuous characteristics during pre-training, and hd outperforms ld by a small margin. Second, our CCMR learns the best representation by combining ld and hd.
+
+# D.5 RESULTS OF DIRECTLY SOLVING CSJ
+
+We also demonstrate the results of solving the CSJ task directly in Table 9. We randomly shuffle video clips into $4 \times 4 \times 4$ jigsaw puzzles. To recognize the correct permutation, the model solve a $( 4 ! \times 4 ! \times 4 ! )$ -way classification task in the pre-training stage. We compare the CSJ task with the joint LCCD+CCMR task under the same setting for fair comparison. Linear evaluation is adopted to show the effectiveness of different tasks. We can observe from the table that solving LCCD $^ +$ CCMR jointly is more effective than solving CSJ directly.
+
+# E TEMPORAL ACTION SEGMENTATION
+
+To show the effectiveness of our CSJ for solving new downstream tasks, we apply the pretrained model obtained by our CSJ to temporal action segmentation, which is more challenging than the conventional action recognition and retrieval tasks. Specifically, we choose to compare our CSJ model with the latest competitor MemDPC (Han et al., 2020) on the Breakfast dataset (Kuehne et al., 2014). For fair comparison, our CSJ model and the MemDPC model adopt the same R2D3D34 backbone. Due the time constraint, from the original Breakfast dataset, we only use a small subset of 200 long videos as the training set for fine-tuning, and select a few long videos for the test. For temporal action segmentation, we follow the overall framework of MS-TCN (Abu Farha & Gall, 2019), but changes its backbone to R2D3D-34 pretrained by our CSJ or MemDPC.
+
+![](images/91d5f16e4c3031f297cf0187ff6396d46e14d07e9f8ebd39a1ff3cfc677d685f.jpg)  
+Figure 5: Qualitative results for the temporal action segmentation task on the Breakfast dataset. Note that the notation $\varnothing$ denotes an unannotated segment in the ground truth.
+
+We present the qualitative results on two test videos in Fig. 5. We can clearly observe that our CSJ outperforms MemDPC on both test videos. Particularly, the predictions of our CSJ are much closer to the ground truth, but MemDPC tends to produce unwanted segments for temporal action segmentation: it wrongly recognizes the segment (color in yellow) in the middle part of the first video as ‘Pour Milk’, and the segment (color in black) in the last part of the second video as ‘Stir Coffee’. In conclusion, as compared to the latest SSVRL method MemDPC, our CSJ can learn more robust features for temporal action segmentation due to its ‘true’ spatiotemporal jigsaw understanding.

parse/train/4AWko4A35ss/4AWko4A35ss_content_list.json

@@ -0,0 +1,1837 @@
+[
+    {
+        "type": "text",
+        "text": "SELF-SUPERVISED VIDEO REPRESENTATION LEARNING WITH CONSTRAINED SPATIOTEMPORAL JIGSAW ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            171,
+            398,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            234,
+            544,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "This paper proposes a novel pretext task for self-supervised video representation learning by exploiting spatiotemporal continuity in videos. It is motivated by the fact that videos are spatiotemporal by nature and a representation learned to detect spatiotemporal continuity/discontinuity is thus beneficial for downstream video content analysis tasks. A natural choice of such a pretext task is to construct spatiotemporal (3D) jigsaw puzzles and learn to solve them. However, this task turns out to be intractable. We thus propose Constrained Spatiotemporal Jigsaw (CSJ) whereby the 3D jigsaws are formed in a constrained manner to ensure that large continuous spatiotemporal cuboids exist in a shuffled clip to provide sufficient cues for the model to reason about the continuity. With the constrained jigsaw puzzles, instead of solving them directly, which could still be extremely hard, we carefully design four surrogate tasks that are more solvable but meanwhile still ensure that the learned representation is sensitive to spatiotemporal continuity at both the local and global levels. Extensive experiments show that our CSJ achieves state-of-the-art on two downstream tasks across various benchmarks. ",
+        "bbox": [
+            233,
+            265,
+            764,
+            472
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            496,
+            336,
+            512
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Self-supervised learning (SSL) has achieved tremendous successes recently for static images (He et al., 2020; Chen et al., 2020) and shown to be able to outperform supervised learning on a wide range of downstream image understanding tasks. However, such successes have not yet been reproduced for videos. Since different SSL models differ mostly on the pretext tasks employed on the unlabeled training data, designing pretext tasks more suitable for videos is the current focus for self-supervised video representation learning (Han et al., 2020; Wang et al., 2020). ",
+        "bbox": [
+            174,
+            526,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Videos are spatiotemporal data and spatiotemporal analysis is the key to many video content understanding tasks. A good video representation learned from the self-supervised pretext task should therefore capture discriminative information jointly along both spatial and temporal dimensions. It is thus somewhat counter-intuitive to note that most existing SSL pretext tasks for videos do not explicitly require joint spatiotemporal video understanding. For example, some spatial pretext tasks have been borrowed from images without any modification (Jing et al., 2018), ignoring the temporal dimension. On the other hand, many recent video-specific pretext tasks typically involve speed or temporal order prediction (Lee et al., 2017; Wei et al., 2018; Benaim et al., 2020; Wang et al., 2020), i.e., operating predominately along the temporal axis. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "A natural choice for a spatiotemporal pretext task is to solve 3D jigsaw puzzles, whose 2D counterpart has been successfully used for images (Noroozi & Favaro, 2016). Indeed, solving 3D puzzles requires the learned model to understand spatiotemporal continuity, a key step towards video content understanding. However, directly solving a 3D puzzle turns out to be intractable: a puzzle of $3 \\times 3 \\times 3$ pieces (the same size as a Rubik’s cube) can have 27! possible permutations. Video volume even in a short clip is much larger than that. Nevertheless, the latest neural sorting models (Paumard et al., 2020; Du et al., 2020) can only handle permutations a few orders of magnitude less, so offer no solution. This is hardly surprising because such a task is daunting even for humans: Most people would struggle with a standard Rubik’s cube, let alone a much larger one. ",
+        "bbox": [
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+            750,
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+            875
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we propose a novel Constrained Spatiotemporal Jigsaw (CSJ) pretext task for selfsupervised video representation learning. The key idea is to form 3D jigsaw puzzles in a constrained manner so that it becomes solvable. This is achieved by factorizing the permutations (shuffling) ",
+        "bbox": [
+            176,
+            882,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/0e2765ac1fc37cb3ed7161a49676d6807733f039ef6007ddb1e38232e0101f70.jpg",
+        "image_caption": [
+            "Figure 1: Illustration of our constrained jigsaw and the surrogate pretext tasks using an image example (only spatial for clarity). Our constrained jigsaw can be easily extended to the spatiotemporal domain as done in this work. (a): The raw image. (b),(c): Comparing an unconstrained puzzle (b) and our constrained one (c), it is clear that ours is much more continuous (hence interpretable) reflected by the size of the largest continuous cuboids (LCCs, rectangles in images here) shown in red. (d),(e): Illustration of the importance of the relative order of the top-2 LCCs for determining the global continuity level of the shuffled image. (d) and (e) have the same top-2 LCCs, but only (d) keeps the correct relative order between them. Locating these LCCs and predicting their relative order are thus the key objectives of our surrogate tasks. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            178,
+            103,
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+            207
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "into the three spatiotemporal dimensions and then applying them sequentially. This ensures that for a given video clip, large continuous spatiotemporal cuboids exist after the constrained shuffling to provide sufficient cues for the model to reason about spatiotemporal continuity (see Fig. 1(b)(c)). Such large continuous cuboids are also vital for human understanding of video as revealed in neuroscience and visual studies (Stringer et al., 2006; Chen et al., 2019). Even with the constrained puzzles, solving them directly could still be extremely hard. Consequently, instead of directly solving the puzzles (i.e., recovering the permutation matrix so that each piece can be put back), four surrogate tasks are carefully designed. They are more solvable but meanwhile still ensure that the learned representation is sensitive to spatiotemporal continuity at both the local and global levels. Concretely, given a video clip shuffled with our constrained permutations, we make sure that the top-2 largest continuous cuboids (LCCs) dominate the clip volume. The level of continuity in the shuffle clip as a whole is thus determined mainly by the volumes of these LCCs, and whether they are at the right order (see Fig. 1(d)(e)) both spatially and temporally. Our surrogate tasks are thus designed to locate these LCCs and predict their order so that the model learned with these tasks can be sensitive to spatiotemporal continuity both locally and globally. ",
+        "bbox": [
+            173,
+            358,
+            825,
+            566
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our main contributions are three-fold: (1) We introduce a new pretext task for self-supervised video representation learning called Constrained Spatiotemporal Jigsaw (CSJ). To our best knowledge, this is the first work on self-supervised video representation learning that leverages spatiotemporal jigsaw understanding. (2) We propose a novel constrained shuffling method to construct easy 3D jigsaws containing large LCCs. Four surrogate tasks are then formulated in place of the original jigsaw solving tasks. They are much more solvable yet remain effective in learning spatiotemporal discriminative representations. (3) Extensive experiments show that our approach achieves state-ofthe-art on two downstream tasks across various benchmarks. ",
+        "bbox": [
+            173,
+            573,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            709,
+            344,
+            726
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Self-supervised Learning with Pretext Tasks Self-supervised learning (SSL) typically employs a pretext task to generate pseudo-labels for unlabeled data via some forms of data transformation. According to the transformations used by the pretext task, existing SSL methods for video presentation learning can be divided into three categories: (1) Spatial-Only Transformations: Derived from the original image domain (Gidaris et al., 2018), Jing et al. (2018) leveraged the spatial-only transformations for self-supervised video presentation learning. (2) Temporal-Only Transformations: Misra et al. (2016); Fernando et al. (2017); Lee et al. (2017); Wei et al. (2018) obtained shuffled video frames with the temporal-only transformations and then distinguished whether the shuffled frames are in chronological order. Xu et al. (2019) chose to shuffle video clips instead of frames. Benaim et al. (2020); Yao et al. (2020); Jenni et al. (2020) exploited the speed transformation via determining whether one video clip is accelerated. (3) Spatiotemporal Transformations: There are only a few recent approaches (Ahsan et al., 2019; Kim et al., 2019) that leveraged both spatial and temporal transformations by permuting 3D spatiotemporal cuboids. However, due to the aforementioned intractability of solving the spatiotemporal jigsaw puzzles, they only leveraged either temporal or spatial permutations as training signals, i.e., they exploited the two domains independently. Therefore, no true spatiotemporal permutations have been considered in Ahsan et al. (2019); Kim et al. (2019). In contrast, given that both spatial appearances and temporal relations are important cues for video representation learning, the focus of this work is on investigating how to exploit the spatial and temporal continuity jointly for self-supervised video presentation learning. To that end, our Constrained Spatiotemporal Jigsaw (CSJ) presents the first spatiotemporal continuity based pretext task for video SSL, thanks to a novel constrained 3D jigsaw and four surrogate tasks to reason about the continuity in the 3D jigsaw puzzles without solving them directly. ",
+        "bbox": [
+            173,
+            743,
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+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            825,
+            229
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Self-supervised Learning with Contrastive Learning Contrastive learning is another selfsupervised learning approach that has become increasingly popular in the image domain (Misra & Maaten, 2020; He et al., 2020; Chen et al., 2020). Recently, it has been incorporated into video SSL as well. Contrastive learning and transformation based pretext tasks are orthogonal to each other and often combined in that different transformed versions of a data sample form the positive set used in contrastive learning. In El-Nouby et al. (2019); Knights et al. (2020); Qian et al. (2020); Wang et al. (2020); Yang et al. (2020), the positive/negative samples were generated based on temporal transformations only. In contrast, some recent works (Han et al., 2019; 2020; Zhuang et al., 2020) leveraged features from the future frame embeddings or with the memory bank (Wu et al., 2018). They modeled spatiotemporal representations using only contrastive learning without transformations. Contrastive learning is also exploited in one of our surrogate pretext tasks. Different from existing works, we explore the spatiotemporal transformations in the form of CSJ and employ contrastive learning to distinguish different levels of spatiotemporal continuity in shuffled jigsaws. This enables us to learn more discriminative spatiotemporal representations. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 CONSTRAINED SPATIOTEMPORAL JIGSAW ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            449,
+            552,
+            465
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.1 PROBLEM DEFINITION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            479,
+            370,
+            494
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The main goal of self-supervised video representation learning is to learn a video feature representation function $f ( \\cdot )$ without using any human annotations. A general approach to achieving this goal is to generate a supervisory signal $\\textbf {  { y } }$ from an unlabeled video clip $_ { \\textbf { \\em x } }$ and construct a pretext task $P$ to predict $\\textbf {  { y } }$ from $f ( { \\pmb x } )$ . The process of solving the pretext task $P$ encourages $f ( \\cdot )$ to learn discriminative spatiotemporal representations. ",
+        "bbox": [
+            174,
+            506,
+            823,
+            575
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The pretext task $P$ is constructed typically by applying to a video clip a transformation function $t ( \\cdot ; \\pmb \\theta )$ parameterized by $\\pmb { \\theta }$ and then automatically deriving $\\textbf {  { y } }$ from $\\pmb \\theta$ , e.g., $\\textbf {  { y } }$ can be the type of the transformation. Based on this premise, $P$ is defined as the prediction of $\\textbf {  { y } }$ using the feature map of the transformed video clip $f ( \\widetilde { \\pmb x } )$ , i.e., $P : f ( { \\widetilde { \\mathbf { x } } } ) \\to { \\pmb y }$ , where $\\widetilde { \\pmb { x } } = t ( \\pmb { x } ; \\pmb { \\theta } )$ . For example, in Lee et al. (2017), $t ( \\cdot ; \\pmb \\theta )$ e e e denotes a temporal transformation that permutes the four frames of video clip $_ { \\textbf { \\em x } }$ in a temporal order $\\pmb \\theta$ , $\\widetilde { \\pmb { x } } = t ( \\pmb { x } ; \\pmb { \\theta } )$ is the shuffled clip, and the pseudo-label $\\textbf {  { y } }$ is defined as the permutation order $\\pmb \\theta$ e(e.g., 1324, 4312, etc.). The pretext task $P$ is then a classification problem of 24 categories because there are $4 ! = 2 4$ possible orders. ",
+        "bbox": [
+            174,
+            582,
+            825,
+            693
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2 CONSTRAINED PERMUTATIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            709,
+            431,
+            723
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Solving spatiotemporal video jigsaw puzzles seems to be an ideal pretext task for learning discriminative representation as it requires an understanding of spatiotemporal continuity. After shuffling the pixels in a video clip using a 3D permutation matrix, the pretext task is to recover the permutation matrix. However, as explained earlier, this task is intractable given even moderate video clip sizes. Our solution is to introduce constraints on the permutations. As a result, a new pretext task $P _ { \\mathrm { C S J } }$ based on Constrained Spatiotemporal Jigsaw (see Fig. 2(a)) is formulated, which is much easier to solve than a random/unconstrained jigsaw. ",
+        "bbox": [
+            174,
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+            833
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Specifically, our goal is to introduce constraints to the permutations so that the resultant shuffled video clip is guaranteed to have large continuous cuboids (see Fig. 2(a)). Similar to humans (Stringer et al., 2006), having large continuous cuboids is key for a model to understand a 3D jigsaw and therefore to have any chance to solve it. Formally, the volume of a shuffled video clip $\\widetilde { \\pmb x }$ are denoted as $\\{ T , H , W \\}$ e, measuring its sizes along the temporal, height, and width dimensions, respectively. A cuboid is defined as a crop of $\\widetilde { \\pmb x }$ : $\\pmb { c } = \\widetilde { \\pmb { x } } _ { t _ { 1 } : t _ { 2 } , h _ { 1 } : h _ { 2 } , w _ { 1 } : w _ { 2 } }$ , where $t _ { 1 } , t _ { 2 } \\in \\{ 1 , 2 , \\dots , T \\} , h _ { 1 } , h _ { 2 } \\in$ $\\{ 1 , 2 , \\ldots , H \\} , w _ { 1 } , w _ { 2 } \\in \\{ 1 , 2 , \\ldots , W \\}$ . If all the jigsaw pieces (smallest video clip unit, e.g. a pixel or a 3D pixel block) in $^ c$ keep the same relative order as they were in $_ { \\textbf { \\em x } }$ (before being shuffled), we call the cuboid $^ c$ as a continuous cuboid $c ^ { \\mathrm { c o n t } }$ . The cuboid’s volume equals $\\left( t _ { 2 } - t _ { 1 } \\right) \\mathbf { \\bar { \\times } } \\left( h _ { 2 } - h _ { 1 } \\right) \\times \\mathbf { \\bar { \\Sigma } }$ $( w _ { 2 } - w _ { 1 } )$ , and the largest continuous cuboid (LCC) $c _ { \\mathrm { m a x } } ^ { \\mathrm { c o n t } }$ is the $c ^ { \\mathrm { c o n t } }$ with the largest volume. ",
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+        "type": "image",
+        "img_path": "images/1078e24fffb3ba13c20ebafabd92aee164ef40d676a91283df6ddb5e42457aac.jpg",
+        "image_caption": [
+            "Figure 2: (a) Illustration of our Constrained Spatiotemporal Jigsaw (CSJ) (see Sec. 3.2). (b) The pipeline of our proposed framework for self-supervised video representation learning (see Sec. 3.3). A raw video clip is transformed into 8 shuffled clips with our Constrained Spatiotemporal Jigsaw (CSJ), and a 3D CNN sharing weights extracts the feature representations from them. The model is then trained by solving four self-supervised tasks jointly. "
+        ],
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "We introduce two permutation strategies to ensure that the volumes of LCCs are large in relation to the whole video clip volume after our shuffling transformation $t ( \\cdot ; \\theta _ { \\mathrm { C S J } } )$ . First, instead of shuffling $_ { \\textbf { \\em x } }$ in three spatiotemporal dimensions simultaneously, $t ( \\cdot ; \\theta _ { \\mathrm { C S J } } )$ factorizes the permutations into the three spatiotemporal dimensions and then utilizes them sequentially to generate shuffled clips, e.g., in the order of $T , W , H$ and only once. Note that the volume of the generated $\\widetilde { \\pmb x }$ stays the same with different permutation orders (e.g., $T W H$ and $H T W$ e). Second, we shuffle a group of jigsaw pieces together instead of each piece individually along each dimension. Taking spatial shuffling as an example, if there are 8 pieces per frame (along each of the two spatial dimensions), $\\theta _ { \\mathrm { C S J } }$ could be represented as the permutation from $\\{ 1 2 3 4 5 6 7 8 \\}$ to $\\{ 8 4 5 6 7 1 2 3 \\}$ . The longest and the secondlongest index ranges are: [2, 5] for coordinates $\\{ 4 5 6 7 \\}$ , and [6, 8] for coordinates $\\lbrace 1 2 3 \\rbrace$ . With these two permutation strategies, not only do we have large LCCs, but also they are guaranteed to have clearly separable boundaries (see Fig. 2(b)) with surrounding pieces due to the factorized and grouped permutation design. This means that they are easily detectable. ",
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+    {
+        "type": "text",
+        "text": "3.3 SURROGATE TASKS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Having permutation constraints preserves more spatiotemporal continuity in the shuffled clip and reduces the amount of possible permutations. But exploiting these constraints to make a neural sorting model tractable is still far from trivial. Instead of solving the jigsaw directly, our $P _ { \\mathrm { C S J } }$ is thus formulated as four surrogate tasks: Largest Continuous Cuboid Detection (LCCD), Clip Shuffling Pattern Classification (CSPC), Contrastive Learning over Shuffled Clips (CLSC), and Clip Continuity Measure Regression (CCMR). As illustrated in Fig. 2(b), given an unlabeled clip $_ { \\textbf { \\em x } }$ , we first construct a mini-batch of 8 clips $\\{ \\widetilde { \\pmb { x } } _ { 1 } , \\widetilde { \\pmb { x } } _ { 2 } , . . . , \\widetilde { \\pmb { x } } _ { 8 } \\}$ by shuffling $_ { \\textbf { \\em x } }$ with different but related constrained permutae e etions (to be detailed later). These shuffled clips and the raw clip $_ { \\textbf { \\em x } }$ are then fed into a 3D CNN model $f ( \\cdot )$ for spatiotemporal representation learning with a non-local operation (Wang et al., 2018): ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f32191da64c9aca571754c11df4aab1017c36080d410c7cff291db8af555d770.jpg",
+        "text": "$$\nf _ { \\mathrm { N L } } ( \\widetilde { \\mathbf { x } } _ { i } ) = \\mathrm { N L } ( f ( \\widetilde { \\mathbf { x } } _ { i } ) , f ( \\pmb { x } ) ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\operatorname { N L } ( \\cdot , \\cdot )$ denotes the non-local operator, and $f ( \\widetilde { \\pmb x } _ { i } )$ and $f ( { \\pmb x } )$ denote the feature map of $\\widetilde { \\pmb { x } } _ { i }$ and $_ { \\textbf { \\em x } }$ from the last convolutional layer of $f ( \\cdot )$ e, respectively. The resultant feature map $f _ { \\mathrm { N L } } ( \\widetilde { \\pmb x } _ { i } )$ eis further passed through a spatial pooling layer followed by a separately fully-connected layer for each surrogate task. Note that the raw video feature map $f ( { \\pmb x } )$ is used as guidance through the nonlocal based attention mechanism to help fulfill the tasks. This is similar to humans needing to see the completed jigsaw picture to help solve the puzzle. ",
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+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Before we detail the four tasks, we first explain how the eight permutations from the same raw clip are generated. First, the factorized and grouped permutations are applied to $_ { \\textbf { \\em x } }$ to create one shuffled clip. By examining the largest and the second-largest continuous puzzle piece numbers of each dimension $( \\{ T , H , \\bar { W } \\} )$ , we can easily identify the top-2 largest continuous cuboids (LCCs). Next, by varying the relative order of the top-2 LCCs either in the correct (original) order or the reverse order in each dimension, $2 \\times 2 \\times 2 { = } 8$ permutations are obtained. By controlling the group size in permutation, we can make sure that the top-2 LCCs account for a large proportion, saying $80 \\%$ of the total clip volume. Our four tasks are thus centered around these two LCCs as they largely determine the overall spatiotemporal continuity of the shuffled clip. ",
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+    {
+        "type": "text",
+        "text": "The first task LCCD is to locate the top-2 LCCs $\\{ c _ { \\mathrm { m a x } } ^ { \\mathrm { c o n t } } ( j ) : j = 1 , 2 \\}$ and formulated as a regression problem. Given a ground-truth LCC $c _ { \\mathrm { m a x } } ^ { \\mathrm { c o n t } } ( j )$ , a Gaussian kernel is applied to its center to depict the possibility of each pixel in $\\widetilde { \\pmb x }$ belonging to the LCC. This leads to a soft mask $M _ { \\mathrm { L C C D } } ^ { j }$ with the same size of $\\widetilde { \\pmb x }$ : $M _ { \\mathrm { L C C D } } ^ { j }$ is all 0 outside the region of $c _ { \\mathrm { m a x } } ^ { \\mathrm { c o n t } } ( j )$ , and $\\exp \\bigl ( - \\frac { | | \\boldsymbol { a } - \\boldsymbol { a } _ { \\mathrm { c } } | | ^ { 2 } } { 2 \\sigma _ { g } ^ { 2 } } \\bigr )$ inside the region, where ${ \\mathbf { } } a , a _ { \\mathrm { { c } } }$ denote any pixel and the center point, respectively. $\\sigma _ { g }$ is the hyper-parameter which is set as 1 empirically. In the training stage, FPN (Lin et al., 2017) is used for multi-level feature fusion. LCCD is optimized using the MSE loss in each point: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/949b6cc4924d36c2320f85c899cd367cdea3c3fb35cbfc206596dabb9032cf9e.jpg",
+        "text": "$$\n{ \\cal L } _ { \\mathrm { L C C D } } = \\sum _ { j \\in \\{ 1 , 2 \\} } \\sum _ { { \\bf { a } } \\in \\widetilde { { \\pmb x } } } \\mathrm { M S E } ( M _ { \\mathrm { L C C D } } ^ { j } ( { \\pmb a } ) , M _ { \\mathrm { L C C D } } ^ { j } ( { \\pmb a } ) ^ { ' } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            409,
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathrm { M S E } ( \\cdot , \\cdot )$ denotes the MSE loss function, and $M _ { \\mathrm { L C C D } } ^ { j } ( \\pmb { a } ) ^ { \\prime }$ 0 is the prediction of each pixel $^ { a }$ ",
+        "bbox": [
+            171,
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "CSPC is designed to recognize the shuffling pattern of a shuffled clip. As mentioned early, the eight shuffled clips in each mini-batch are created from the same raw clip and differ only in the relative order of the top-2 LCCs along each of the three dimensions. There are thus eight permutations depending on the order (correct or reverse) in each dimension. Based on this understanding, CSPC is formulated as a multi-class classification task to recognize each shuffled clip into one of these eight classes, which is optimized using the Cross-Entropy (CE) loss: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/48e20e210dbec32facf8e5632a608c134ac60fdde0191ac8ce01ea94d87dae04.jpg",
+        "text": "$$\nL _ { \\mathrm { C S P C } } = \\sum _ { i \\in \\{ 0 , 1 , . . . , 7 \\} } { \\bf C E } ( l _ { \\mathrm { C S P C } } [ i ] , l _ { \\mathrm { C S P C } } ^ { ' } [ i ] ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            359,
+            564,
+            637,
+            602
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\operatorname { C E } ( \\cdot , \\cdot )$ denotes the CE loss function and $l _ { \\mathrm { C S P C } } ^ { ' } [ i ]$ is the predicted class label of $i$ -th sample (shuffled clip) in each mini-batch. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The two tasks above emphasize on local spatiotemporal continuity understanding. In contrast, CLSC leverages the contrastive loss to encourage global continuity understanding. In particular, since the top-2 LCCs dominate the volume of a clip, it is safe to assume that if their relative order is correct in all three dimensions, the shuffled clip largely preserve continuity compared to the original clip, while all other 7 permutations feature large discontinuity in at least one dimension. We thus form a contrastive learning task with the original video $_ { \\textbf { \\em x } }$ and the most continuous shuffled video $\\widetilde { \\pmb { x } } _ { i }$ as a positive pair, and $_ { \\textbf { \\em x } }$ and the rest $\\widetilde { \\pmb { x } } _ { j }$ $( j \\neq i )$ ) as negative pairs. CLSC is optimized using the e eNoise Contrastive Estimation (NCE) (Tian et al., 2020) loss: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/eb953d201cb8d73386a8cee9aa86599758df0af996550c66f5ea9c315298fe77.jpg",
+        "text": "$$\n\\begin{array} { r } { L _ { \\mathrm { C L S C } } = - \\log \\frac { \\exp ( \\sin ( f ( \\pmb { x } ) , f ( \\widetilde { \\pmb { x } } _ { i } ) ) / \\tau ) } { \\exp ( \\sin ( f ( \\pmb { x } ) , f ( \\widetilde { \\pmb { x } } _ { i } ) ) / \\tau ) + \\sum _ { j } \\exp ( \\sin ( f ( \\pmb { x } ) , f ( \\widetilde { \\pmb { x } } _ { j } ) ) / \\tau ) } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\sin ( \\cdot , \\cdot )$ is defined by the dot product: $f ( \\pmb { x } ) ^ { \\top } f ( \\widetilde { \\pmb { x } } _ { i } )$ , and $\\tau$ is the temperature hyper-parameter.   \nNote that the non-local operator is not used in CLSC. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "CCMR is similar to CLSC in that it also enforces global continuity understanding, but differs in that it is a regression task aimed at predicting a global continuity measure. We consider two such measures. Since the total size of the top-2 LCCs $\\{ c _ { \\mathrm { m a x } } ^ { \\mathrm { c o n t } } ( j ) : j = 1 , 2 \\}$ is a good indicator of how continuous a shuffle video clip is, the first measure $l _ { l d }$ directly measures the relative total size of the top-2 LCCs: $l _ { l d } = \\frac { \\mathbf { v } ( { c } _ { \\mathrm { m a x } } ^ { \\mathrm { c o n t } } ( 1 ) ) + \\mathbf { v } ( { c } _ { \\mathrm { m a x } } ^ { \\mathrm { c o n t } } ( 2 ) ) } { \\mathbf { v } ( \\widetilde { \\pmb { x } } ) }$ , where $\\mathbf { v } ( \\cdot )$ represents the volume of a clip/cuboid. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The second measure $l _ { \\mathrm { h d } } ^ { \\mathrm { t / h / w } }$ examines the shuffling degree of $\\widetilde { \\pmb x }$ in each dimension, computed as the normalized hamming distance: $\\frac { \\mathrm { h a m m i n g } ( \\widetilde { \\pmb { x } } ) } { N _ { c } ( N _ { c } - 1 ) / 2 }$ where hamming $( \\cdot )$ denotes the hamming distance in each dimension between the original piece sequence and the permuted one, and $N _ { c }$ represents the number of pieces in each dimension so that $N _ { c } ( \\bar { N } _ { c } { - } 1 ) / 2$ indicates the maximum possible hamming distance in the dimension. CCMR is optimized using the Mean Squared Error (MSE) loss: ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4767e40c1da86d5fa8565151bed879de55d0227d5a08c9c19370d0825a27f481.jpg",
+        "text": "$$\n\\begin{array} { r } { L _ { \\mathrm { C C M R } } = \\mathbf { M S E } \\big ( \\big [ l _ { \\mathrm { l d } } , l _ { \\mathrm { h d } } ^ { \\mathrm { t } } , l _ { \\mathrm { h d } } ^ { \\mathrm { h } } , l _ { \\mathrm { h d } } ^ { \\mathrm { w } } \\big ] , \\big [ l _ { \\mathrm { l d } } ^ { ' } , l _ { \\mathrm { h d } } ^ { \\mathrm { t } ^ { \\prime } } , l _ { \\mathrm { h d } } ^ { \\mathrm { h } ^ { \\prime } } , l _ { \\mathrm { h d } } ^ { \\mathrm { w } ^ { \\prime } } \\big ] \\big ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $l _ { \\mathrm { l d } } ^ { ' } , l _ { \\mathrm { h d } } ^ { \\mathrm { t } ^ { \\prime } } , l _ { \\mathrm { h d } } ^ { \\mathrm { h } ^ { \\prime } } , l _ { \\mathrm { h d } } ^ { \\mathrm { w } ^ { \\prime } }$ are the prediction of the model. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "3.4 OVERALL LEARNING OBJECTIVE ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Our entire CSJ framework is optimized end-to-end with the learning objective defined as: ",
+        "bbox": [
+            169,
+            279,
+            759,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cfe22f00a9d2c9501eceed80a3a2982950d8de5d45231b7c6b4e07ca3e5da892.jpg",
+        "text": "$$\nL = \\sigma _ { 1 } L _ { \\mathrm { L C C D } } + \\sigma _ { 2 } L _ { \\mathrm { C S P C } } + \\sigma _ { 3 } L _ { \\mathrm { C L S C } } + \\sigma _ { 4 } L _ { \\mathrm { C C M R } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            330,
+            299,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $\\sigma _ { 1 } , \\sigma _ { 2 } , \\sigma _ { 3 } , \\sigma _ { 4 }$ denote the weights for the four losses. We deploy the adaptive weighting mechanism (Kendall et al., 2018) to weight these tasks, and thus there is no free hyper-parameters to tune. We also adopt curriculum learning (Bengio et al., 2009; Korbar et al., 2018) to train our network by shuffling clips from easy to hard. More details are presented in Appendix. A.1 and A.2. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.1 DATASETS AND SETTINGS ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+    },
+    {
+        "type": "text",
+        "text": "We select three benchmark datasets for performance evaluation: UCF101 (Soomro et al., 2012), HMDB51 (Kuehne et al., 2011), and Kinetics-400 (K400) (Kay et al., 2017), containing 13K/7K/306K video clips from 101/51/400 action classes, respectively. In the self-supervised pretraining stage, we utilize the first training split of UCF101/HMDB51 and the training split of K400 without using their labels. As in Han et al. (2020), we adopt R2D3D as the backbone network, which is modified from R3D (Hara et al., 2018) with fewer parameters. By fine-tuning the pre-trained model, we can evaluate the SSL performance on a downstream task (i.e., action classification). Following Han et al. (2019); He et al. (2020), two evaluation protocols are used: comparisons against state-of-the-arts follow the more popular fully fine-tuning evaluation protocol, but ablation analysis takes both the linear evaluation and fully fine-tuning protocols. For the experiments on supervised learning, we report top-1 accuracy on the first test split of UCF101/HMDB51 as the standard (Han et al., 2020). More details of the datasets are provided in Appendix B. ",
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+    },
+    {
+        "type": "text",
+        "text": "4.2 IMPLEMENTATION DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Raw videos in these datasets are decoded at a frame rate of 24-30 fps. From each raw video, we start from a randomly selected frame index and sample a consecutive 16-frame video clip with a temporal stride of 4. For data augmentation, we first resize the video frames to $1 2 8 \\times 1 7 1$ pixels, from which we extract random crops of size $1 1 2 \\times 1 1 2$ pixels. We also apply random horizontal flipping and random color jittering to the video frames during training. We exploit only the raw RGB video frames as input, and do not leverage optical flow or other auxiliary signals for self-supervised pretraining. We adopt the Adam optimizer with a weight decay of $1 0 ^ { - 3 }$ and a batch size of 8 per GPU (with a total of 32 GPUs). We deploy cosine annealing learning rate with an initial value of $\\bar { 1 } 0 ^ { - 4 }$ and 100 epochs. The jigsaw puzzle piece sizes of $\\{ T , H , { \\bar { W } } \\}$ dimensions are set as $1 , 4 , 4$ , respectively. A $1 6 \\times 1 1 2 \\times 1 1 2$ video clip thus contains $1 6 \\times 2 8 \\times 2 8$ pieces. We set the temperature hyper-parameter $\\tau$ to 0.07. A dropout of 0.5 is applied to the final layer of each task. More implementation details of the fine-tuning and test evaluation stages can be found in Appendix B. ",
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+    {
+        "type": "text",
+        "text": "4.3 MAIN RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Comparison in Action Recognition A standard way to evaluate a self-supervised video representation learning model is to use it to initialize an action recognition model on a small dataset. Specifically, after self-supervised pre-training on UCF101/HMDB51/K400, we exploit the learned backbone for fully fine-tuning on UCF101 and HMDB51, following Han et al. (2020); Wang et al. (2020). ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/1bff20829ec6efe1bcb747c6d0c5de7a41a48da0a5bf9690b0e99a23afd672cd.jpg",
+        "table_caption": [
+            "Table 1: Comparison to the state-of-the-art on UCF101(U) and HMDB51 $\\mathrm { ( H ) }$ . All models are pretrained with the RGB modality only. $\\dagger$ : Methods with temporal-only transformations. $^ \\ddag$ : Methods with both spatial and temporal transformations. $* 1$ : Methods that leverage spatiotemporal representations. HT: HowTo100M. The underline represents the second-best result. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Methods</td><td>Backbone</td><td>Pre-trained Datasets</td><td>Input size (T*H)</td><td>UCF101</td><td>HMDB51</td></tr><tr><td>CMC (ECCV&#x27;20) (Tian et al.,2020)</td><td>C2D</td><td>U</td><td>1</td><td>55.3</td><td></td></tr><tr><td>Skip-Clip† (El-Nouby et al.,2019)</td><td>R3D-18</td><td>U/H</td><td>16*112</td><td>64.4</td><td>一</td></tr><tr><td>VCOPt (CVPR&#x27;19) (Xu et al.,2019)</td><td>R3D-18</td><td>U/H</td><td>16*112</td><td>64.9</td><td>29.5</td></tr><tr><td>VCP† (AAAI&#x27;20) (Luo et al.,2020b)</td><td>R3D-18</td><td>U/H</td><td>16*112</td><td>66.0</td><td>31.5</td></tr><tr><td>PRPt (CVPR&#x27;20)(Yao et al.,2020)</td><td>R3D-18</td><td>U/H</td><td>16*112</td><td>66.5</td><td>29.7</td></tr><tr><td>MemDPC* (ECCV&#x27;20) (Han et al.,2020) CSJ*(ours)</td><td>R2D3D-18</td><td>U/H</td><td>40*128</td><td>69.2</td><td>一</td></tr><tr><td>Video-Jigsaw* (WACV&#x27;19) (Ahsan et al.,2019)</td><td>R2D3D-18</td><td>U/H</td><td>16*112</td><td>70.4</td><td>36.0</td></tr><tr><td>Statisics*(CVPR&#x27;19) (Wang et al.,2019)</td><td>C2D C3D</td><td>K400</td><td>25*224</td><td>55.4</td><td>27.0</td></tr><tr><td>ST-Puzzle‡ (AAAI&#x27;19) (Kim et al.,2019)</td><td>R3D-18</td><td>K400</td><td>16*112</td><td>61.2</td><td>33.4</td></tr><tr><td>DPC*(ICCVW&#x27;19) (Han et al.,2019)</td><td>R2D3D-18</td><td>K400</td><td>16*112</td><td>63.9</td><td>33.7</td></tr><tr><td>SpeedNet (CVPR&#x27;20) (Benaim et al.,2020)</td><td></td><td>K400</td><td>40*128</td><td>68.2</td><td>34.5</td></tr><tr><td>VIE*(CVPR&#x27;20) (Zhuang et al.,2020)</td><td>I3D</td><td>K400</td><td>16*224</td><td>66.7</td><td>43.7</td></tr><tr><td>Pace† (ECCV&#x27;20) (Wang et al.,2020)</td><td>R3D-18</td><td>K400</td><td>16*112</td><td>75.5</td><td>44.6</td></tr><tr><td>CSJ* (ours)</td><td>R(2+1)D-18</td><td>K400</td><td>16*112</td><td>77.1</td><td>36.6</td></tr><tr><td></td><td>R2D3D-18</td><td>K400</td><td>16*112</td><td>76.2</td><td>46.7</td></tr><tr><td>MemDPC* (ECCV&#x27;20) (Han et al.,2020) CBT (Sun et al.,2019)</td><td>R2D3D-34</td><td>K400</td><td>40*224</td><td>78.1</td><td>41.2</td></tr><tr><td>CSJ* (ours)</td><td>S3D R2D3D-34</td><td>K600+HT K400</td><td>一 16*224</td><td>79.5</td><td>44.6</td></tr><tr><td></td><td></td><td></td><td></td><td>79.5</td><td>50.9</td></tr><tr><td>Upper Bound:Fully-Supervised</td><td>R3D-34</td><td>K400</td><td>16*224</td><td>87.7</td><td>59.1</td></tr></table>",
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+        "table_caption": [
+            "Table 2: Comparison with state-of-the-art self-supervised learning methods for nearest neighbor video retrieval (top- $k$ recall) on UCF101. The underline represents the second-best result. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Methods</td><td>Backbone</td><td>Top1</td><td>Top5</td><td>Top10</td><td>Top20</td><td>Top50</td></tr><tr><td>VCOP (CVPR&#x27;19) (Xu et al., 2019)</td><td>R3D-18</td><td>14.1</td><td>30.3</td><td>40.0</td><td>51.1</td><td>66.5</td></tr><tr><td>VCP (AAAI&#x27;20) (Luo et al., 2020b)</td><td>R3D-18</td><td>18.6</td><td>33.6</td><td>42.5</td><td>53.5</td><td>68.1</td></tr><tr><td>SpeedNet (CVPR&#x27;20) (Benaim et al., 2020)</td><td>S3D-G</td><td>13.1</td><td>28.1</td><td>37.5</td><td>49.5</td><td>65.0</td></tr><tr><td>PRP (CVPR&#x27;2O) (Yao et al.,2020)</td><td>R3D-18</td><td>22.8</td><td>38.5</td><td>46.7</td><td>55.2</td><td>69.1</td></tr><tr><td>Pace (ECCV&#x27;2O) (Wang et al., 2020)</td><td>R3D-18</td><td>19.9</td><td>36.2</td><td>46.1</td><td>55.6</td><td>69.2</td></tr><tr><td>ERUV (Luo et al., 2020a)</td><td>R3D-18</td><td>21.4</td><td>35.2</td><td>43.8</td><td>53.1</td><td>68.3</td></tr><tr><td>MemDPC (ECCV&#x27;20) (Han et al., 2020)</td><td>R2D3D-18</td><td>20.2</td><td>40.4</td><td>52.4</td><td>64.7</td><td>1</td></tr><tr><td>CSJ (ours)</td><td>R2D3D-18</td><td>21.5</td><td>40.5</td><td>53.2</td><td>64.9</td><td>70.0</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "We consider one baseline: fully-supervised learning with pre-training on K400. Note that this baseline is commonly regarded as the upper bound of self-supervised representation learning (Alwassel et al., 2019). From Table 1, we have the following observations: (1) Our CSJ achieves state-of-theart performance on both UCF101 and HMDB51. Particularly, with the backbone R2D3D-18 that is weaker than $\\mathrm { R } ( 2 + 1 ) \\mathrm { D } { - } 1 8 $ , our CSJ performs comparably w.r.t. Pace on UCF101 but achieves a $10 \\%$ improvement over Pace on HMDB51. (2) By exploiting spatiotemporal transformations for self-supervised representation learning, our CSJ beats either methods with only temporal transformations $( \\dag )$ or methods with both spatial and temporal transformations $( \\ddagger )$ , as well as those learning spatiotemporal representations $( \\ast )$ via only contrastive learning (w./o. spatiotemporal transformations). (3) Our CSJ also outperforms CBT (Sun et al., 2019), which used ten-times more massive datasets (K600 (Carreira et al., 20 $ { \\left| 8 \\right. } +  { \\mathrm { H } }$ owto100M (Miech et al., 2019)) and multiple modalities (RGB $+ .$ Audio). (4) Our CSJ is the closest to the fully-supervised one (upper bound), validating its effectiveness in self-supervised video representation learning. ",
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+    {
+        "type": "text",
+        "text": "Comparison in Video Retrieval We evaluate our CSJ method in the video retrieval task. Following Xu et al. (2019), we extract each video clips’ embeddings with the pre-training model and use each clip in the test set to query the $k$ nearest clips in the training set. The comparative results in Table 2 show that our method outperforms all other self-supervised methods and achieves new state-of-the-art in video retrieval on UCF101. Particularly, our method beats the latest competitor ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/aedbac6975b7a52eaf9df1c001520e62e0a2e157f7d308adadbac66bfea6305a.jpg",
+        "table_caption": [
+            "Table 3: Evaluation of pre-training tasks with the backbone R2D3D-18 under linear probe and fully fine-tuning protocols on UCF101. AW: Adaptive Weighting. CL: Curriculum Learning. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Tasks</td><td>Linear Probe</td><td>Fully Fine-tuning</td></tr><tr><td>Random Initialization</td><td>8.3</td><td>63.6</td></tr><tr><td>LCCD</td><td>21.8</td><td>67.8</td></tr><tr><td>CSPC</td><td>22.6</td><td>68.1</td></tr><tr><td>CLSC</td><td>18.9</td><td>68.1</td></tr><tr><td>CCMR</td><td>22.7</td><td>68.1</td></tr><tr><td>CCMR+CSPC</td><td>24.7</td><td>69.2</td></tr><tr><td>CCMR+CSPC+CLSC</td><td>25.5</td><td>69.3</td></tr><tr><td>CCMR+CSPC+CLSC+LCCD</td><td>27.9</td><td>69.5</td></tr><tr><td>CCMR+CSPC+CLSC+LCCD+AW</td><td>28.2</td><td>70.0</td></tr><tr><td>CCMR+CSPC+CLSC+LCCD+AW+CL</td><td>28.5</td><td>70.4</td></tr></table>",
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/05879f14857b6a2ef744f82417429a2c36c8fe64aa824955368e18b252cb5f3a.jpg",
+        "image_caption": [
+            "Figure 3: Attention visualization of the last feature maps from the fine-tuned models on UCF101. The first row denotes the raw frames from videos, and the last two rows correspond to fine-tuning from random initialization and our self-supervised pre-trained model, respectively. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "PRP (Yao et al., 2020) on four out of five metrics. This indicates that our proposed CSJ is also effective for video representation learning in video retrieval. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.4 FURTHER EVALUATIONS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Ablation Study We conduct ablative experiments to validate the effectiveness of four CSJ surrogate tasks and two additional learning strategies. From Table 3, we can observe that: (1) Selfsupervised learning with each of the four tasks shows better generalization than fine-tuning the network from scratch (random initialization). (2) By training over all the four tasks jointly, we can achieve large performance gains (see ‘+LCCD’ vs. ‘CCMR’). (3) Each additional learning strategy (i.e., adaptive weighting or curriculum learning) leads to a small boost to the performance by 0.3- $0 . 5 \\%$ . (4) Our full model achieves a remarkable classification accuracy of $7 0 . 4 \\%$ , demonstrating the effectiveness of our proposed CSJ with only the RGB video stream (without additional optical flow, audio, or text modalities). More ablative analysis can be found in Appendix D. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Visualization of Attention Maps Fig. 3 visualizes the attention map of the last feature maps from two models fine-tuned on UCF101 with or without adopting our self-supervised pre-training. Since each frame’s attention map involves four adjacent frames, it actually contains spatiotemporal semantic features. We can see that our self-supervised pre-training with CSJ indeed helps to better capture meaningful spatiotemporal information and thus recognize the action categories more correctly. ",
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+    {
+        "type": "image",
+        "img_path": "images/364822a46bd24076097579693c7a18c390267468a5a5de153032980e75a3b9f3.jpg",
+        "image_caption": [
+            "Figure 4: Visualization of the LCCD predictions from the pre-trained models. Each row denotes the frames at time stamp $= ( 0 , 4 , 8 , 1 2 )$ from one video clip. (a) raw frames (with color jittering); (b) shuffled frames; (c) the ground truth of LCCD; (d) network’s prediction. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Visualization of LCCD Predictions We also demonstrate the visualization of the LCCD predictions from the pre-trained models in Fig. 4. We can observe that solving the LCCD task indeed enables the model to learn the locations of LCCs and understand spatiotemporal continuity, which is a key step towards video content understanding. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
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+            176,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We have introduced a novel self-supervised video representation learning method named Constrained Spatiotemporal Jigsaw (CSJ). By introducing constrained permutations, our proposed CSJ is the first to leverage spatiotemporal jigsaw in self-supervised video representation learning. We also propose four surrogate tasks based on our constrained spatiotemporal jigsaws. They are designed to encourage a video representation model to understand the spatiotemporal continuity, a key building block towards video content analysis. Extensive experiments were carried out to validate the effectiveness of each of the four CSJ tasks and also show that our approach achieves the state-of-the-art on two downstream tasks across various benchmarks. ",
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
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+        "text": "Xiaolong Wang, Ross Girshick, Abhinav Gupta, and Kaiming He. Non-local neural networks. In CVPR, pp. 7794–7803, 2018.   \nDonglai Wei, Joseph Lim, Andrew Zisserman, and William T. Freeman. Learning and using the arrow of time. In CVPR, pp. 8052–8060, 2018.   \nZhirong Wu, Yuanjun Xiong, Stella X Yu, and Dahua Lin. Unsupervised feature learning via nonparametric instance discrimination. In CVPR, pp. 3733–3742, 2018.   \nDejing Xu, Jun Xiao, Zhou Zhao, Jian Shao, Di Xie, and Yueting Zhuang. Self-supervised spatiotemporal learning via video clip order prediction. In CVPR, pp. 10334–10343, 2019.   \nCeyuan Yang, Yinghao Xu, Bo Dai, and Bolei Zhou. Video representation learning with visual tempo consistency. arXiv preprint arXiv:2006.15489, 2020.   \nYuan Yao, Chang Liu, Dezhao Luo, Yu Zhou, and Qixiang Ye. Video playback rate perception for self-supervised spatio-temporal representation learning. In CVPR, pp. 6548–6557, 2020.   \nChengxu Zhuang, Tianwei She, Alex Andonian, Max Sobol Mark, and Daniel Yamins. Unsupervised learning from video with deep neural embeddings. In CVPR, pp. 9563–9572, 2020. ",
+        "bbox": [
+            171,
+            101,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A ADDITIONAL LEARNING STRATEGIES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            522,
+            118
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.1 ADAPTIVE WEIGHT",
+        "text_level": 1,
+        "bbox": [
+            176,
+            128,
+            356,
+            142
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Formally, our CSJ has two continuous outputs $\\mathrm { y } _ { 1 } , \\mathrm { y } _ { 4 }$ from LCCD and CCMR, and two discrete outputs $\\mathrm { y } _ { 2 } , \\mathrm { y } _ { 3 }$ from CSPC and CLSC, modeled with Gaussian likelihoods and softmax likelihoods, respectively. The joint loss for these four tasks $L ( \\mathbf { W } , \\sigma _ { 1 } , \\sigma _ { 2 } , \\sigma _ { 3 } , \\sigma _ { 4 } )$ is: ",
+        "bbox": [
+            174,
+            154,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/eace98d6fd67a272c42110dcb931d7c8a8bb1b5dec0450ddff107a6c4d778bac.jpg",
+        "text": "$$\n\\begin{array} { r l } & { L ( \\mathbf { W } , \\sigma _ { 1 } , \\sigma _ { 2 } , \\sigma _ { 3 } , \\sigma _ { 4 } ) } \\\\ & { = - \\log \\mathcal { N } ( \\mathbf { y } _ { 1 } ; \\mathbf { f } ^ { \\mathbf { W } } ( \\mathbf { x } ) , \\sigma _ { 1 } ^ { 2 } ) \\cdot - \\log \\mathcal { N } ( \\mathbf { y } _ { 4 } ; \\mathbf { f } ^ { \\mathbf { W } } ( \\mathbf { x } ) , \\sigma _ { 4 } ^ { 2 } ) } \\\\ & { \\quad \\cdot \\ \\mathrm { s o f t m a x } ( \\mathbf { y } _ { 2 } = \\sigma ; \\mathbf { f } ^ { \\mathbf { W } } ( \\mathbf { x } ) , \\sigma _ { 2 } ) \\cdot \\mathrm { s o f t m a x } ( \\mathbf { y } _ { 3 } = \\sigma ; \\mathbf { f } ^ { \\mathbf { W } } ( \\mathbf { x } ) , \\sigma _ { 3 } ) } \\\\ & { = \\frac { 1 } { 2 \\sigma _ { 1 } ^ { 2 } } | | \\mathbf { y } _ { 1 } - \\mathbf { f } ^ { \\mathbf { W } } ( \\mathbf { x } ) | | ^ { 2 } + \\log \\sigma _ { 1 } - \\frac { 1 } { 2 \\sigma _ { 4 } ^ { 2 } } | | \\mathbf { y } _ { 4 } - \\mathbf { f } ^ { \\mathbf { W } } ( \\mathbf { x } ) | | ^ { 2 } + \\log \\sigma _ { 4 } } \\\\ & { \\quad - \\log p ( \\mathbf { y } _ { 2 } | \\mathbf { f } ^ { \\mathbf { W } } ( \\mathbf { x } ) , \\sigma _ { 2 } ) - \\log p ( \\mathbf { y } _ { 3 } | \\mathbf { f } ^ { \\mathbf { W } } ( \\mathbf { x } ) , \\sigma _ { 3 } ) } \\\\ & { \\approx \\frac { 1 } { 2 \\sigma _ { 1 } ^ { 2 } } L ( \\mathbf { W } ) + \\frac { 1 } { \\sigma _ { 2 } ^ { 2 } } L _ { 2 } ( \\mathbf { W } ) + \\frac { 1 } { \\sigma _ { 3 } ^ { 2 } } L _ { 3 } ( \\mathbf { W } ) + \\frac { 1 } { 2 \\sigma _ { 4 } ^ { 2 } } L _ { 4 } ( \\mathbf { W } ) } \\\\ & { \\quad + \\log \\sigma _ { 1 } + \\log \\sigma _ { 2 } + \\log \\sigma _ { 3 } + \\log \\sigma _ { 4 } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            294,
+            204,
+            705,
+            358
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "where $\\sigma$ is the weight factor that can be automatically learned from the network, and the log likelihood for the output y is defined as: ",
+        "bbox": [
+            171,
+            364,
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+            392
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/219cae6835b9b3bca98ca92291b26056a8ac279f29e4a8cfc8e10411da527631.jpg",
+        "text": "$$\n\\log p ( \\mathrm { y } = c | \\mathbf { f } ^ { \\mathbf { W } } ( \\mathbf { x } ) , \\boldsymbol { \\sigma } ) = \\frac { 1 } { \\sigma ^ { 2 } } f _ { c } ^ { \\mathbf { W } } ( \\mathbf { x } ) - \\log \\sum _ { c ^ { \\prime } } \\exp ( \\frac { 1 } { \\sigma ^ { 2 } } f _ { c ^ { \\prime } } ^ { \\mathbf { W } } ( \\mathbf { x } ) ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            289,
+            398,
+            710,
+            436
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.2 CURRICULUM LEARNING",
+        "text_level": 1,
+        "bbox": [
+            176,
+            455,
+            395,
+            469
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We adopt curriculum learning (Korbar et al., 2018) to train our network by shuffling clips from easy to hard. Let $d$ be the shuffle degree of a shuffled clip $\\widetilde { \\pmb x }$ , representing the number of continuous cuboids in each dimension. We gradually increase $d$ efrom 3 to 5 during the training phase to produce more permuted clips. Note that when the video content is ambiguous in one dimension, e.g., a static video clip inflated from an image, there is no temporal variance to learn the transformation. Kim et al. (2019); Noroozi & Favaro (2016) also mentioned this problem as similar-looking ambiguity. To solve this problem, we calculate the variance on each dimension and set a threshold. If the variance is lower than the threshold, we decrease $d$ from 3 to 1 so that the pieces are not shuffled in the corresponding dimension. ",
+        "bbox": [
+            173,
+            481,
+            825,
+            607
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B DATASETS AND IMPLEMENTATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            622,
+            493,
+            638
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B.1 DETAILS OF DATASETS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            648,
+            377,
+            664
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "UCF101 (Soomro et al., 2012) is a widely-used dataset in the action recognition task, which contains 13,320 videos with 101 action classes. The dataset is divided into three training/testing splits. In this paper, following prior works (Wang et al., 2020; Han et al., 2020), we use the first training split as the pre-training dataset and the first testing split for evaluation. ",
+        "bbox": [
+            173,
+            675,
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+            731
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "HMDB51 (Kuehne et al., 2011) is a relatively small action recognition dataset, consisting of 6,766 videos with 51 categories. It is also divided into three training/testing splits. Following Wang et al. (2020); Han et al. (2020), we use the first training split as the pre-training dataset and the first testing split for evaluation. ",
+        "bbox": [
+            174,
+            738,
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+            794
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Kinetics-400 (K400) (Kay et al., 2017) is a very large action recognition dataset consisting of 400 human action classes and around 306k videos. In this work, we use the training split of K400 as the pre-training dataset. ",
+        "bbox": [
+            174,
+            800,
+            825,
+            843
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B.2 IMPLEMENTATION DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            854,
+            408,
+            869
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In the fine-tuning stage, weights of convolutional layers are initialized with self-supervised pretraining, but weights of fully-connected layers are randomly initialized. The whole network is then trained with the cross-entropy loss. The pre-processing and training strategies are the same as in the ",
+        "bbox": [
+            176,
+            882,
+            823,
+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/6395e12e228509649ea9b03ca3209c0ad423368e2e47c7d7ea64e21a0954ee96.jpg",
+        "table_caption": [
+            "Table 4: The structure of the encoding function $f ( \\cdot )$ . R2D3D-18 is used as an example. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>stage</td><td rowspan=1 colspan=3>detail</td><td rowspan=1 colspan=1>output sizeT×HW×C</td></tr><tr><td rowspan=1 colspan=1> input data</td><td rowspan=1 colspan=3>1</td><td rowspan=1 colspan=1>16×112²×3</td></tr><tr><td rowspan=1 colspan=1>conV1</td><td rowspan=1 colspan=3>1 × 7²,64stride 1, 22</td><td rowspan=1 colspan=1>16 × 56² × 64</td></tr><tr><td rowspan=1 colspan=1>pool1</td><td rowspan=1 colspan=3>1 × 3²,64stride 1, 22</td><td rowspan=1 colspan=1>16 × 28² × 64</td></tr><tr><td rowspan=1 colspan=1>res2</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>[1 ×3²,641 × 3²,64</td><td rowspan=1 colspan=1>×2</td><td rowspan=1 colspan=1>16 × 28² × 64</td></tr><tr><td rowspan=1 colspan=1>res3</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>1 × 3²,1281× 3²,128</td><td rowspan=1 colspan=1>×2</td><td rowspan=1 colspan=1>16 × 14² × 128</td></tr><tr><td rowspan=1 colspan=1>res4</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>3× 3²,2563 × 32,256</td><td rowspan=1 colspan=1>×2</td><td rowspan=1 colspan=1>8×7² × 256</td></tr><tr><td rowspan=1 colspan=1>res5</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>3 × 3²,512[3 × 3²,512</td><td rowspan=1 colspan=1>×2</td><td rowspan=1 colspan=1>4 × 4² × 512</td></tr><tr><td rowspan=1 colspan=1>Avgpool</td><td rowspan=1 colspan=3>4 × 4²,512stride 1,12</td><td rowspan=1 colspan=1>1 ×1² × 512</td></tr></table>",
+        "bbox": [
+            264,
+            130,
+            730,
+            372
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "self-supervised pre-training stage, except that the total epochs are 300 and the initial learning rate is $1 0 ^ { - 3 }$ . We use a batch size of 64 per GPU and a total of 8 GPUs for fine-tuning. ",
+        "bbox": [
+            176,
+            387,
+            823,
+            415
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We follow the standard evaluation protocol (Han et al., 2020) during inference and use ten-crop to take the same sequence length as training from the video. The predicted label of each video is calculated by averaging the softmax probabilities of all clips in the video. ",
+        "bbox": [
+            174,
+            421,
+            825,
+            465
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C NETWORK ARCHITECTURE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            479,
+            436,
+            496
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We deploy the same network backbone R2D3D as Han et al. (2019; 2020), which is a 3D-ResNet (R3D) similar to Hara et al. (2018). The only difference between R2D3D and R3D lies in that: R2D3D keeps the first two residual blocks as 2D convolutional blocks while R3D uses 3D blocks. Therefore, the modified R2D3D has fewer parameters (only the last two blocks are 3D convolutions). We present the CNN structure of R2D3D in Table 4. ",
+        "bbox": [
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+            580
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "D ADDITIONAL ABLATION STUDIES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            597,
+            490,
+            613
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/764e80fdb434f74d6e3d9fcada3c59a93671172bab4a6591b5e11c3813f58786.jpg",
+        "table_caption": [
+            "Table 5: Evaluation of pre-training tasks under different designs of LCCD on UCF101. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Methods</td><td rowspan=1 colspan=1>LCCS</td><td rowspan=1 colspan=1>LCCD+MLccs</td><td rowspan=1 colspan=1>LCCD + L1</td><td rowspan=1 colspan=1>LCCD+MSE</td></tr><tr><td rowspan=1 colspan=1>Top-1 Acc</td><td rowspan=1 colspan=1>66.5</td><td rowspan=1 colspan=1>64.0</td><td rowspan=1 colspan=1>66.5</td><td rowspan=1 colspan=1>67.8</td></tr></table>",
+        "bbox": [
+            222,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "D.1 LCCD ",
+        "text_level": 1,
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Instead of predicting center points using the detection method, we also design a segmentation method – largest continuous cuboid segmentation (LCCS) to predicts the location of top-2 LCCs $\\{ c _ { \\operatorname* { m a x } } ^ { \\mathrm { c o n t } } ( j ) : j = 1 , 2 \\}$ . The difference between LCCD and LCCS lies in that: LCCS is formulated as a segmentation task to discriminate whether a pixel is in the region of $c _ { \\mathrm { m a x } } ^ { \\mathrm { c o n t } } ( j )$ . Concretely, LCCS predicts a binary mask $M _ { \\mathrm { L C C S } } ^ { j }$ where only points in the region of $\\{ c _ { \\mathrm { m a x } } ^ { \\mathrm { c o n t } } ( j )$ are set to be 1, otherwise 0. As a result, LCCS is optimized using the Cross Entropy (CE) loss at each point: ",
+        "bbox": [
+            173,
+            722,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/65729811390c6767f78eff6db55e89f27571a0c184151c4f5bc16fcae3376658.jpg",
+        "text": "$$\n{ \\cal L } _ { \\mathrm { L C C S } } = \\sum _ { j \\in \\{ 1 , 2 \\} } \\sum _ { { \\pmb { a } } \\in { \\pmb { \\widetilde x } } } \\mathrm { C E } ( M _ { \\mathrm { L C C S } } ^ { j } ( { \\pmb { a } } ) , M _ { \\mathrm { L C C S } } ^ { j } ( { \\pmb { a } } ) ^ { ' } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            334,
+            814,
+            661,
+            851
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "where $\\operatorname { C E } ( \\cdot , \\cdot )$ denotes the CE loss function, and $M _ { \\mathrm { L C C S } } ^ { j } ( \\pmb { a } ) ^ { \\prime }$ is the predicted class of pixel $\\textbf { \\em a }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We report the performance of four different designs of LCCD in Table 5: (1) LCCS: LCCS is used instead of LCCD. (2) $\\mathrm { L C C D } { + } M _ { \\mathrm { L C C S } }$ : The Gaussian mask $M _ { \\mathrm { L C C D } }$ is substituted by the binary mask $M _ { \\mathrm { L C C S } }$ , but the LCCD task is optimized using the MSE loss. (3) $\\mathrm { L C C D + L 1 }$ : The LCCD task is optimized by the L1 loss. (4) $\\mathrm { L C C D + M S E }$ : The LCCD task is optimized by the MSE loss. From Table 5, it can be seen that the segmentation task also helps self-supervised representation learning but doesn’t perform as well as LCCD. Also, under the three different settings of LCCD, the MSE loss with the Gaussian map performs the best. ",
+        "bbox": [
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+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
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+    {
+        "type": "table",
+        "img_path": "images/80be8980029128ba9a8eca4f8b1c7f34d9c51f776c64fcc4d46250a5b4df8ce1.jpg",
+        "table_caption": [
+            "Table 6: Evaluation of different temperature $\\tau$ for CLSC on UCF101. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Methods</td><td rowspan=1 colspan=1>T=1</td><td rowspan=1 colspan=1>T =0.1</td><td rowspan=1 colspan=1>T = 0.07</td></tr><tr><td rowspan=1 colspan=1>Top-1 Acc</td><td rowspan=1 colspan=1>60.9</td><td rowspan=1 colspan=1>66.3</td><td rowspan=1 colspan=1>68.1</td></tr></table>",
+        "bbox": [
+            238,
+            194,
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "D.2 CLSC ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            242,
+            264,
+            257
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Table 6 above shows the accuracies obtained with different temperatures $\\tau$ used in contrastive learning. We can observe that: (1) When $\\tau$ is in the range $1 \\sim 0 . 0 7$ , the accuracy increases with smaller $\\tau$ . (2) When $\\tau$ is large (e.g., 1), the accuracy drops considerably. In this work, $\\tau$ is set to 0.0. ",
+        "bbox": [
+            174,
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+            311
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/0cff68af1bd53ea6d1e635dfc61e1f70415ab164ccb41a19165abbdb75aef1cf.jpg",
+        "table_caption": [
+            "Table 7: Evaluation of different designs of CSPC on UCF101. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Methods</td><td rowspan=1 colspan=1>2 Categories</td><td rowspan=1 colspan=1>4 Categories</td><td rowspan=1 colspan=1>8 Categories</td></tr><tr><td rowspan=1 colspan=1>Top-1 Acc</td><td rowspan=1 colspan=1>67.0</td><td rowspan=1 colspan=1>68.0</td><td rowspan=1 colspan=1>68.1</td></tr></table>",
+        "bbox": [
+            241,
+            345,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "D.3 CSPC ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            395,
+            263,
+            409
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "In addition to our CSPC with 8 pattern categories (see Sec. 3.3), we consider another two designs: (1) 2 Categories: the shuffled clip is discriminated by whether it has the same relative order of the top-2 LCCs as the raw clip. It is almost the same as CLSC but is optimized by the CE loss. (2) 4 Categories: the shuffled clip is discriminated by how it differs from the raw clip: non-difference, spatial-only difference, temporal-only difference, spatiotemporal difference. From Table 7, we can see that CSPC with 8 categories outperforms the other two designs. These results support our motivation for leveraging spatiotemporal transformations. ",
+        "bbox": [
+            173,
+            420,
+            825,
+            518
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/e6076bdfe8b2af58487f62f3115a3c6e1293babc7551660dfd82985183bce4f1.jpg",
+        "table_caption": [
+            "Table 8: Evaluation of different designs of CCMR on UCF101. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Methods</td><td rowspan=1 colspan=1>ld</td><td rowspan=1 colspan=1>hd</td><td rowspan=1 colspan=1>ld+hd</td></tr><tr><td rowspan=1 colspan=1>Top-1 Acc</td><td rowspan=1 colspan=1>66.9</td><td rowspan=1 colspan=1>67.2</td><td rowspan=1 colspan=1>68.1</td></tr></table>",
+        "bbox": [
+            251,
+            551,
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+            583
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "D.4 CCMR ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            606,
+            269,
+            621
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We report the performance of three different designs of CCMR: (1) ld: the learning degree $l _ { \\mathrm { l d } }$ is used as supervision, which only contains volume information. (2) hdare used, which contain only the relative order information. (3) mming distances : both ld and hd $l _ { \\mathrm { h d } } ^ { \\mathrm { t } } , l _ { \\mathrm { h d } } ^ { \\mathrm { h } } , l _ { \\mathrm { h d } } ^ { \\mathrm { w } }$ $\\mathrm { l d } + \\mathrm { h d }$   \nsupervision. From Table 8, we can see that: First, both ld and hd help the model to learn continuous characteristics during pre-training, and hd outperforms ld by a small margin. Second, our CCMR learns the best representation by combining ld and hd. ",
+        "bbox": [
+            174,
+            632,
+            825,
+            718
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "D.5 RESULTS OF DIRECTLY SOLVING CSJ ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            734,
+            480,
+            748
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We also demonstrate the results of solving the CSJ task directly in Table 9. We randomly shuffle video clips into $4 \\times 4 \\times 4$ jigsaw puzzles. To recognize the correct permutation, the model solve a $( 4 ! \\times 4 ! \\times 4 ! )$ -way classification task in the pre-training stage. We compare the CSJ task with the joint LCCD+CCMR task under the same setting for fair comparison. Linear evaluation is adopted to show the effectiveness of different tasks. We can observe from the table that solving LCCD $^ +$ CCMR jointly is more effective than solving CSJ directly. ",
+        "bbox": [
+            174,
+            760,
+            825,
+            844
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "E TEMPORAL ACTION SEGMENTATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            864,
+            513,
+            881
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "To show the effectiveness of our CSJ for solving new downstream tasks, we apply the pretrained model obtained by our CSJ to temporal action segmentation, which is more challenging than the conventional action recognition and retrieval tasks. Specifically, we choose to compare our CSJ model with the latest competitor MemDPC (Han et al., 2020) on the Breakfast dataset (Kuehne et al., 2014). For fair comparison, our CSJ model and the MemDPC model adopt the same R2D3D34 backbone. Due the time constraint, from the original Breakfast dataset, we only use a small subset of 200 long videos as the training set for fine-tuning, and select a few long videos for the test. For temporal action segmentation, we follow the overall framework of MS-TCN (Abu Farha & Gall, 2019), but changes its backbone to R2D3D-34 pretrained by our CSJ or MemDPC. ",
+        "bbox": [
+            174,
+            895,
+            823,
+            924
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/91d5f16e4c3031f297cf0187ff6396d46e14d07e9f8ebd39a1ff3cfc677d685f.jpg",
+        "image_caption": [
+            "Figure 5: Qualitative results for the temporal action segmentation task on the Breakfast dataset. Note that the notation $\\varnothing$ denotes an unannotated segment in the ground truth. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            141,
+            818,
+            554
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            627,
+            825,
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "We present the qualitative results on two test videos in Fig. 5. We can clearly observe that our CSJ outperforms MemDPC on both test videos. Particularly, the predictions of our CSJ are much closer to the ground truth, but MemDPC tends to produce unwanted segments for temporal action segmentation: it wrongly recognizes the segment (color in yellow) in the middle part of the first video as ‘Pour Milk’, and the segment (color in black) in the last part of the second video as ‘Stir Coffee’. In conclusion, as compared to the latest SSVRL method MemDPC, our CSJ can learn more robust features for temporal action segmentation due to its ‘true’ spatiotemporal jigsaw understanding. ",
+        "bbox": [
+            174,
+            732,
+            825,
+            830
+        ],
+        "page_idx": 15
+    }
+]

parse/train/B1TTpYKgx/B1TTpYKgx.md

@@ -0,0 +1,670 @@
+# ON THE EXPRESSIVE POWER OF DEEP NEURAL NETWORKS
+
+Maithra Raghu Google Brain and Cornell University
+
+Ben Poole Stanford University and Google Brain
+
+Jon Kleinberg Cornell University
+
+Surya Ganguli Stanford University
+
+Jascha Sohl-Dickstein Google Brain
+
+# ABSTRACT
+
+We study the expressive power of deep neural networks before and after training. Considering neural nets after random initialization, we show that three natural measures of expressivity all display an exponential dependence on the depth of the network. We prove, theoretically and experimentally, that all of these measures are in fact related to a fourth quantity, trajectory length. This quantity grows exponentially in the depth of the network, and is responsible for the depth sensitivity observed. These results translate to consequences for networks during and after training. They suggest that parameters earlier in a network have greater influence on its expressive power – in particular, given a layer, its influence on expressivity is determined by the remaining depth of the network after that layer. This is verified with experiments on MNIST and CIFAR-10. We also explore the effect of training on the input-output map, and find that it trades off between the stability and expressivity of the input-output map.
+
+# 1 INTRODUCTION
+
+Neural network architectures have proven “unreasonably effective” (LeCun, 2014; Karpathy, 2015) on many tasks, including image classification (Krizhevsky et al., 2012), identifying particles in high energy physics (Baldi et al., 2014), playing Go (Silver et al., 2016), and modeling human student learning (Piech et al., 2015). Despite their power, we have limited knowledge of how and why neural networks work, and much of this understanding is qualitative and heuristic.
+
+To aim for a more precise understanding, we must disentangle factors influencing their effectiveness, trainability, or how well they can be fit to data; generalizability, or how well they perform on novel examples; and expressivity, or the set of functions they can compute.
+
+All three of these properties are crucial for understanding the performance of neural networks. Indeed, for success at a particular task, neural nets must first be effectively trained on a dataset, which has prompted investigation into properties of objective function landscapes (Dauphin et al., 2014; Goodfellow et al., 2014; Choromanska et al., 2014), and the design of optimization procedures specifically suited to neural networks (Martens and Grosse, 2015). Trained networks must also be capable of generalizing to unseen data, and understanding generalization in neural networks is also an active line of research: (Hardt et al., 2015) bounds generalization error in terms of stochastic gradient descent steps, (Sontag, 1998; Bartlett and Maass, 2003; Bartlett et al., 1998) study generalization error through VC dimension, and (Hinton et al., 2015) looks at developing smaller models with better generalization.
+
+In this paper, we focus on the third of these properties, expressivity — the capability of neural networks to accurately represent different kinds of functions. As the class of functions achievable by a neural network is dependent on properties of its architecture, e.g. depth, width, fully connected, convolutional, etc; a better understanding of expressivity may greatly inform architectural choice and inspire more tailored training methods.
+
+Prior work on expressivity has yielded many fascinating results by directly examining the achievable functions of a particular architecture. Through this, neural networks have been shown to be universal approximators (Hornik et al., 1989; Cybenko, 1989), and connections between boolean and threshold networks and ReLU networks developed in (Maass et al., 1994; Pan and Srikumar, 2015). The inherent expressivity due to increased depth has also been studied in (Eldan and Shamir, 2015; Telgarsky, 2015; Martens et al., 2013; Bianchini and Scarselli, 2014), and (Pascanu et al., 2013; Montufar et al., 2014), with the latter introducing the number of linear regions as a measure of expressivity.
+
+These results, while compelling, also highlight limitations of much of the existing work on expressivity. Much of the work examining achievable functions relies on unrealistic architectural assumptions, such as layers being exponentially wide (in the universal approximation theorem). Furthermore, architectures are often compared via ‘hardcoded’ weight values – a specific function that can be represented efficiently by one architecture is shown to only be inefficiently approximated by another.
+
+Comparing architectures in such a fashion limits the generality of the conclusions, and does not entirely address the goal of understanding expressivity — to provide characteristic properties of a typical set of networks arising from a particular architecture, and extrapolate to practical consequences.
+
+Random networks To address this, we begin our analysis of network expressivity on a family of networks arising in practice �� the behaviour of networks after random initialization. As random initialization is the starting point to most training methods, results on random networks provide natural baselines to compare trained networks with, and are also useful in highlighting properties of trained networks (see Section 3). The expressivity of these random networks is largely unexplored. In previous work (Poole et al., 2016) we studied the propagation of Riemannian curvature through random networks by developing a mean field theory approach, which quantitatively supports the conjecture that deep networks can disentangle curved manifolds in input space. Here, we take a more direct approach, exactly relating the architectural properties of the network to measures of expressivity and exploring the consequences for trained networks.
+
+Measures of Expressivity In particular, we examine the effect of the depth and width of a network architecture on three different natural measures of functional richness: number of transitions, activation patterns, and number of dichotomies.
+
+Transitions: Counting neuron transitions is introduced indirectly via linear regions in (Pascanu et al., 2013), and provides a tractable method to estimate the degree of non-linearity of the computed function.
+
+Activation Patterns: Transitions of a single neuron can be extended to the outputs of all neurons in all layers, leading to the (global) definition of a network activation pattern, also a measure of nonlinearity. Network activation patterns directly show how the network partitions input space (into convex polytopes), through connections to the theory of hyperplane arrangements.
+
+Dichotomies: We also measure the heterogeneity of a generic class of functions from a particular architecture by counting dichotomies, ‘statistically dual’ to sweeping input in some cases. This measure reveals the importance of remaining depth in expressivity, in both simulation and practice.
+
+Connection to Trajectory Length All three measures display an exponential increase with depth, but not width (most strikingly in Figure 4). We discover and prove the underlying reason for this – all three measures are directly proportional to a fourth quantity, trajectory length. In Theorem 1) we show that trajectory length grows exponentially with depth (also supported by experiments, Figure 1) which explains the depth sensitivity of the other three measures.
+
+Consequences for Trained Networks Our empirical and theoretical results connecting transitions and dichotomies to trajectory length also suggest that parameters earlier in the network should have exponentially greater influence on parameters later in the network. In other words, the influence on expressivity of parameters, and thus layers, is directly related to the remaining depth of the network after that layer. Experiments on MNIST and CIFAR-10 support this hypothesis — training only earlier layers leads to higher accuracy than training only later layers. We also find, with experiments on MNIST, that the training process trades off between the stability of the input-output map and its expressivity.
+
+![](images/9540c49cfa28a4a0970f68b2cc7aae5d214be508abe3d34f33db8a0812115d6c.jpg)  
+Figure 1: The exponential growth of trajectory length with depth, in a random deep network with hard-tanh nonlinearities. A circular trajectory is chosen between two random vectors. The image of that trajectory is taken at each layer of the network, and its length measured. $( a , b )$ The trajectory length vs. layer, in terms of the network width $k$ and weight variance $\sigma _ { w } ^ { 2 }$ , both of which determine its growth rate. $( c , d )$ The average ratio of a trajectory’s length in layer $d + 1$ relative to its length in layer (The $d$ . The solid line shows simulated data, while them 1). Growth rate is a function of layer width dashed lines show up, and weight variance and lower bounds. $k$ $\hat { \sigma } _ { w } ^ { 2 }$
+
+# 2 GROWTH OF TRAJECTORY LENGTH AND MEASURES OF EXPRESSIVITY
+
+In this section we examine random networks, proving and empirically verifing the exponential growth of trajectory length with depth. We then relate trajectory length to transitions, activation patterns and dichotomies, and show their exponential increase with depth.
+
+# 2.1 NOTATION AND DEFINITIONS
+
+Let $F _ { W }$ denote a neural network. In this section, we consider architectures with input dimension $m$ , $n$ hidden layers all of width $k$ , and (for convenience) a scalar readout layer. (So, $F _ { W } : \mathbb { R } ^ { m }  \mathbb { R } .$ ) Our results mostly examine the cases where $\phi$ is a hard-tanh (Collobert and Bengio, 2004) or ReLU nonlinearity. All hard-tanh results carry over to tanh with additional technical steps.
+
+We use v( $v _ { i } ^ { ( d ) }$ to denote the $i ^ { t h }$ neuron in hidden layer $d$ . We also let $x = z ^ { ( 0 ) }$ be an input, $h ^ { ( d ) }$ be the hidden representation at layer $d$ , and $\phi$ the non-linearity. The weights and bias are called $W ^ { ( d ) }$ and $b ^ { ( d ) }$ respectively. So we have the relations
+
+$$
+\begin{array} { r } { \boldsymbol { h } ^ { ( d ) } = \boldsymbol { W } ^ { ( d ) } \boldsymbol { z } ^ { ( d ) } + \boldsymbol { b } ^ { ( d ) } , \qquad \boldsymbol { z } ^ { ( d + 1 ) } = \phi ( \boldsymbol { h } ^ { ( d ) } ) . } \end{array}
+$$
+
+Definitions Say a neuron transitions when it switches linear region in its activation function (i.e. for ReLU, switching between zero and linear regimes, for hard-tanh, switching between negative saturation, unsaturated and positive saturation). For hard-tanh, we refer to a sign transition as the neuron switching sign, and a saturation transition as switching from being saturated between $\pm 1$ . The Activation Pattern of the entire network is defined by the output regions of every neuron. More precisely, given an input $x$ , we let ${ \mathcal { A } } ( F _ { W } , x )$ be a vector representing the activation region of every hidden neuron in the network. So for a ReLU network $F _ { W }$ , we can take $\mathcal { A } ( F _ { W } , x ) \ \in \{ - 1 , 1 \} ^ { n \bar { k } }$ with $- 1$ meaning the neuron is in the zero regime, and 1 meaning it is in the linear regime. For hard-tanh network $F _ { W }$ , we can (overloading notation slightly) take $\mathcal { A } ( F _ { W } , x ) \in \{ - 1 , 0 , 1 \} ^ { n k }$ . The use of this notation will be clear by context. Given a set of inputs $S$ , we say a dichotomy over $S$ is a labeling of each point in $S$ as $\pm 1$ .
+
+We assume the weights of our neural networks are initialized as random Gaussians, with appropriate variance scaling to account for width, i.e. $W _ { i j } ^ { ( d ) } \sim \mathcal { N } ( 0 , \sigma _ { w } ^ { 2 } / k )$ , and biases $b _ { i } ^ { ( d ) } \sim \mathcal { N } ( 0 , \sigma _ { b } ^ { 2 } )$ . In the analysis below, we sweep through a one dimensional input trajectory $x ( t )$ . The results hold for almost any such smooth $x ( t )$ , provided that at any point $x ( t )$ , the trajectory direction has some non-zero magnitude perpendicular to $x ( t )$ .
+
+# 2.2 TRAJECTORY LENGTH AND NEURON TRANSITIONS
+
+We first prove how the trajectory length grows, and relate it to neuron transitions.
+
+# 2.2.1 BOUND ON TRAJECTORY LENGTH GROWTH
+
+We prove (with a more exact lower bound in the Appendix):
+
+Theorem 1. Bound on Growth of Trajectory Length Let $F _ { W }$ be a hard tanh random neural network and $x ( t )$ a one dimensional trajectory in input space. Define $z ^ { ( d ) } ( x ( t ) ) = z ^ { ( d ) } ( t )$ to be the image of the trajectory in layer $d$ of $F _ { W }$ , and let $\begin{array} { r } { l ( z ^ { ( d ) } ( t ) ) = \int _ { t } \bigg | \bigg | \frac { d z ^ { ( d ) } ( t ) } { d t } \bigg | \bigg | \ a } \end{array}$ t be the arc length of $z ^ { ( d ) } ( t )$ . Then
+
+$$
+\mathbb { E } \left[ l ( z ^ { ( d ) } ( t ) ) \right] \geq O \left( \left( \frac { \sigma _ { w } } { ( \sigma _ { w } ^ { 2 } + \sigma _ { b } ^ { 2 } ) ^ { 1 / 4 } } \cdot \frac { \sqrt { k } } { \sqrt { \sqrt { \sigma _ { w } ^ { 2 } + \sigma _ { b } ^ { 2 } } + k } } \right) ^ { d } \right) l ( x ( t ) )
+$$
+
+This bound is tight in the limits of large $\sigma _ { w }$ and $k$ . An immediate Corollary for $\sigma _ { b } = 0$ , i.e. no bias, is
+
+Corollary 1. Bound on Growth of Trajectory Length Without Bias For $F _ { W }$ with zero bias, we have
+
+$$
+\mathbb { E } \left[ l ( z ^ { ( d ) } ( t ) ) \right] \ge O \left( \left( \frac { \sqrt { \sigma _ { w } k } } { \sqrt { \sigma _ { w } + k } } \right) ^ { d } \right) l ( x ( t ) )
+$$
+
+The theorem shows that the image of a trajectory in layer $d$ has grown exponentially in $d$ , with the scaling $\sigma _ { w }$ and width of the network $k$ determining the base. We additionally state and prove a simple $O ( \sigma _ { w } ^ { d } )$ growth upper bound in the Appendix. Figure 1 demonstrates this behavior in simulation, and compares against the bounds. Note also that if the variance of the bias is comparatively too large i.e. $\sigma _ { b } > > \sigma _ { w }$ , then we no longer see exponential growth. This corresponds to the phase transition described in (Poole et al., 2016).
+
+The proof can be found in the Appendix. A rough outline is as follows: we look at the expected growth of the difference between a point $z ^ { ( d ) } ( t )$ on the curve and a small perturbation ${ z } ^ { ( d ) } ( t + d t )$ , from layer $d$ to layer $d + 1$ . Denoting this quantity $\left| \left| \delta z ^ { ( d ) } ( t ) \right| \right|$ , we derive a recurrence relating $\left| \left| \delta z ^ { ( d + 1 ) } ( t ) \right| \right|$ and $\left| \left| \delta z ^ { ( d ) } ( t ) \right| \right|$ which can be composed to give the desired growth rate.
+
+The analysis is complicated by the statistical dependence on the image of the input $z ^ { ( d + 1 ) } ( t )$ . So we instead form a recursion by looking at the component of the difference perpendicular to the image of the input in that layer, i.e. $\left| \left| \delta z _ { \perp } ^ { ( \bar { d } + 1 ) } ( t ) \right| \right|$ . For a typical trajectory, the perpendicular component preserves a fraction $\sqrt { \frac { k - 1 } { k } }$ of the total trajectory length, and our derived growth rate thus provides a close lower bound, as demonstrated in Figure 1(c,d).
+
+# 2.2.2 RELATION TO NUMBER OF TRANSITIONS
+
+Further experiments (Figure 2) show:
+
+Observation 1. The number of sign transitions in a network $F _ { W }$ is directly proportional to the length of the latent image of the curve, $z ^ { ( n ) } ( t )$ .
+
+![](images/9cd2e88a477e6867512977c9f2ef2f08a422125962d1154b571d969f7c5b11c3.jpg)  
+Figure 2: The number of transitions is linear in trajectory length. Here we compare the empirical number of sign changes to the length of the trajectory, for images of the same trajectory at different layers of a hard-tanh network. We repeat this comparison for a variety of network architectures, with different network width k and weight variance σ2w.
+
+We intuit a reason for this observation as follows: note that for a network $F _ { W }$ with $n$ hidden layers, the linear, one dimensional, readout layer outputs a value by computing the inner product $W ^ { ( n ) } z ^ { ( n ) }$ . The sign of the output is then determined by whether this quantity is $\geq 0$ or not. In particular, the decision boundary is a hyperplane, with equation $W ^ { ( n ) } z ^ { ( n ) } = 0$ . So, the number of transitions the output neuron makes as $x ( t )$ is traced is exactly the number of times $z ^ { ( n ) } ( t )$ crosses the decision boundary. As $F _ { W }$ is a random neural network, with signs of weight entries split purely randomly between $\pm 1$ , it would suggest that points far enough away from each other would have independent signs, i.e. a direct proportionality between the length of $z ^ { ( n ) } ( t )$ and the number of times it crosses the decision boundary.
+
+We can also prove this in the special case when $\sigma _ { w }$ is very large. Note that by Theorem 1, very large $\sigma _ { w }$ results in a trajectory growth rate of
+
+$$
+g ( k , \sigma _ { w } , \sigma _ { b } , n ) = O \left( \left( \frac { \sqrt { k } } { \sqrt { 1 + \frac { \sigma _ { b } ^ { 2 } } { \sigma _ { w } ^ { 2 } } } } \right) ^ { n } \right)
+$$
+
+Large $\sigma _ { w }$ also means that for any input (bounded away from zero), almost all neurons are saturated. Furthermore, any neuron transitioning from 1 to $- 1$ (or vice versa) does so almost instantaneously. In particular, at most one neuron within a layer is transitioning for any input. We can then show that in the large $\sigma _ { w }$ limit the number of transitions matches the trajectory length (proof in the Appendix, via a reduction to magnitudes of independent Gaussians):
+
+Theorem 2. Number of transitions in large weight limit Given $F _ { W }$ , in the very large $\sigma _ { w }$ regime, the number of sign transitions of the network as an input $x ( t )$ is swept is of the order of $g ( k , \sigma _ { w } , \sigma _ { b } , n )$ .
+
+# 2.3 TRANSITIONS AND ACTIVATION PATTERNS
+
+We can generalize the ’local’ notion of expressivity of a neuron‘s sign transitions to a ’global’ measure of activation patterns over the entire network. We can formally relate network activation patterns to specific hyperplane arrangements, which allows proof of three exciting results.
+
+First, we can precisely state the effect of a neural network on input space, also visualized in Figure 3
+
+Theorem 3. Regions in Input Space Given a network $F _ { W }$ with with ReLU or hard-tanh activations, input space is partitioned into convex regions (polytopes), with $F _ { W }$ corresponding to a different linear function on each region.
+
+This results in a bijection between transitions and activation patterns for ‘well-behaved’ trajectories, see the proof of Theorem 3 and Corollary 2 in Appendix.
+
+Finally, returning to the goal of understanding expressivity, we can upper bound the expressive power of a particular architecture according to the activation patterns measure:
+
+![](images/cc1ccb96b5adb9526cfa340865b7243e35efeb151e4f6828a97b8025882ee210.jpg)  
+Figure 3: Deep networks with piecewise linear activations subdivide input space into convex polytopes. Here we plot the boundaries in input space separating unit activation and inactivation for all units in a three layer ReLU network, with four units in each layer. The left pane shows activation boundaries (corresponding to a hyperplane arrangement) in gray for the first layer only, partitioning the plane into regions. The center pane shows activation boundaries for the first two layers. Inside every first layer region, the second layer activation boundaries form a different hyperplane arrangement. The right pane shows activation boundaries for the first three layers, with different hyperplane arrangements inside all first and second layer regions. This final set of convex regions correspond to different activation patterns of the network – i.e. different linear functions.
+
+![](images/29fc7bd83b5430c43b86733b033cebda70e32de272a32fd6a8ea5b55132a59e4.jpg)  
+Figure 4: The number of functions achievable in a deep hard-tanh network by sweeping a single layer’s weights along a one dimensional trajectory is exponential in the remaining depth, but increases only slowly with network width. Here we plot the number of classification dichotomies over $s = 1 5$ input vectors achieved by sweeping the first layer weights in a hard-tanh network along a one-dimensional great circle trajectory. We show this $( a )$ as a function of remaining depth for several widths, and $( b )$ as a function of width for several remaining depths. All networks were generated with weight variance $\sigma _ { w } ^ { 2 } = 8$ , and bias variance $\sigma _ { b } ^ { 2 } = 0$ .
+
+Theorem 4. (Tight) Upper bound for Number of Activation Patterns Given a neural network $F _ { W }$ , inputs in $\mathbb { R } ^ { m }$ , with ReLU or hard-tanh activations, and with $n$ hidden layers of width $k$ , the number of activation patterns grows at most like $O ( k ^ { m n } )$ for ReLU, or $O ( ( 2 k ) ^ { \dot { m } n } )$ for hard-tanh.
+
+# 2.4 DICHOTOMIES: A NATURAL DUAL
+
+So far, we have looked at the effects of depth and width on the expressiveness (measured through transitions and activations) of a generic function computed by that network architecture. These measures are directly related to trajectory length, which is the underlying reason for exponential depth dependence.
+
+A natural extension is to study a class of functions that might arise from a particular architecture. One such class of functions is formed by sweeping the weights of a network instead of the input. More formally, we pick random matrices, $W , W ^ { \prime }$ , and consider the weight interpolation $W \cos ( t ) +$ $W ^ { \prime } \sin ( t )$ , each choice of weights giving a different function. When this process is applied to just the first layer, we have a statistical duality with sweeping a circular input.
+
+![](images/a519d502464bf5ef469d00b624caf69f3040537762773ea4e586a81abc286e54.jpg)  
+Figure 5: Expressive power depends only on remaining network depth. Here we plot the number of dichotomies achieved by sweeping the weights in different network layers through a 1-dimensional great circle trajectory, as a function of the remaining network depth. The number of achievable dichotomies does not depend on the total network depth, only on the number of layers above the layer swept. All networks had width $k = 1 2 8$ , weight variance $\sigma _ { w } ^ { 2 } \ = \ 8$ , number of datapoints $s = 1 5$ , and hard-tanh nonlinearities. The blue dashed line indicates all $2 ^ { s }$ possible dichotomies for this random dataset.
+
+![](images/bea8634915238321c7c559cd938ddfb02ce5f355809275f5d8eb55e44d1a9a4c.jpg)  
+Figure 6: Demonstration of expressive power of remaining depth on MNIST. Here we plot train and test accuracy achieved by training exactly one layer of a fully connected neural net on MNIST. The different lines are generated by varying the hidden layer chosen to train. All other layers are kept frozen after random initialization. We see that training lower hidden layers leads to better performance. The networks had width $k = 1 0 0$ , weight variance $\sigma _ { w } ^ { 2 } = 2$ , and hard-tanh nonlinearities. Note that we only train from the second hidden layer (weights $W ^ { ( 1 ) }$ ) onwards, so that the number of parameters trained remains fixed. While the theory addresses training accuracy and not generalization accuracy, the same monotonic pattern is seen for both.
+
+Given this class of functions, one useful measure of expressivity is determining how heterogeneous this class is. Inspired by classification tasks we formalize it as: given a set of inputs, $S = \{ x _ { 1 } , . . , x _ { s } \} \subset \mathbb { R } ^ { m }$ , how many of the $2 ^ { s }$ possible dichotomies does this function class produce on $S 2$
+
+For non-random inputs and non-random functions, this is a well known question upper bounded by the Sauer-Shelah lemma (Sauer, 1972). We discuss this further in Appendix D.1. In the random setting, the statistical duality of weight sweeping and input sweeping suggests a direct proportion to transitions and trajectory length for a fixed input. Furthermore, if the $x _ { i } \in S$ are sufficiently uncorrelated (e.g. random) class label transitions should occur independently for each $x _ { i }$ Indeed, we show this in Figure 4 (more figures, e.g. dichotomies vs transitions and observations, are included in the Appendix).
+
+Observation 2. Depth and Expressivity in a Function Class. Given the function class $\mathcal { F }$ as above, the number of dichotomies expressible by $\mathcal { F }$ over a set of random inputs $S$ by sweeping the first layer weights along a one dimensional trajectory $W ^ { ( 0 ) } ( t )$ is exponential in the network depth $n$ .
+
+![](images/2313ca1793045a2989e309b8086f17d5a552aae81eb646af9d6eea89db6de449.jpg)  
+Figure 7: We repeat a similar experiment in Figure 6 with a fully connected network on CIFAR-10, and mostly observe that training lower layers again leads to better performance. The networks had width $k = 2 0 0$ , weight variance $\sigma _ { w } ^ { 2 } = 1$ , and hard-tanh nonlinearities. We again only train from the second hidden layer on so that the number of parameters remains fixed.
+
+Table 1: List and location of key theoretical and experimental results.   
+
+<table><tr><td rowspan=1 colspan=1>Property</td><td rowspan=1 colspan=1>Architecture</td><td rowspan=1 colspan=1>Results</td></tr><tr><td rowspan=1 colspan=1>Trajectory length</td><td rowspan=1 colspan=1>hard-tanh</td><td rowspan=1 colspan=1>Asymptotically tight lower bound (Thm 1)Upper bound (Appendix Section A)Simulation (Fig 1)</td></tr><tr><td rowspan=1 colspan=1>Neuron transitions</td><td rowspan=1 colspan=1>hard-tanh</td><td rowspan=1 colspan=1>Expectation in large weight limit (Thm 2)Simulation (Fig 2)</td></tr><tr><td rowspan=1 colspan=1>Dichotomies</td><td rowspan=1 colspan=1>hard-tanh</td><td rowspan=1 colspan=1>Simulation (Figs 4 and 10)</td></tr><tr><td rowspan=1 colspan=1>Regions in input space</td><td rowspan=1 colspan=1>hard-tanh      andReLU</td><td rowspan=1 colspan=1>Consist of convex polytopes (Thm 3)</td></tr><tr><td rowspan=1 colspan=1>Network activation patterns</td><td rowspan=1 colspan=1>hard-tanh      andReLU</td><td rowspan=1 colspan=1>Tight upper bound (Thm 6)</td></tr><tr><td rowspan=1 colspan=1>Effect of remaining depth</td><td rowspan=1 colspan=1>hard-tanh</td><td rowspan=1 colspan=1>Simulation (Fig 5)Experiment on MNIST (Fig 6)Experiments on CIFAR-10 (Fig 7)</td></tr><tr><td rowspan=1 colspan=1>Effect of training on trajec-tory length</td><td rowspan=1 colspan=1>hard-tanh</td><td rowspan=1 colspan=1>Experiment on MNIST (Fig 8, 9)</td></tr></table>
+
+# 3 TRAINED NETWORKS
+
+Remaining Depth The results from Section 2, particularly those linking dichotomies to trajectory length, suggest that earlier layers in the network might have more expressive power. In particular, the remaining depth of the network beyond the layer might directly influence its expressive power. We see that this holds in the random network case (Figure 5), and also for networks trained on MNIST and CIFAR-10. In Figures 6, 7 we randomly initialized a neural network, and froze all the layers except for one, which we trained.
+
+Training trades off between input-output map stability and expressivity We also look at the effect of training on measures of expressivity by plotting the change in trajectory length and number of transitions (see Appendix) during the training process. We find that for a network initialized with large $\sigma _ { w }$ , the training process appears to stabilize the input-output map – monotonically decreasing trajectory length (Figure 8) except for the final few steps. Interestingly, this happens at a faster rate in the vicinity of the data than for random inputs, and is accomplished without reducing weight magnitudes.
+
+For a network closer to the boundary of the exponential regime $\sigma _ { w } ^ { 2 } = 3$ , where trajectory length growth is still exponential but with a much smaller base, the training process increases the trajectory length, enabiling greater expressivity in the resulting input-output map, Figure 9
+
+![](images/74b0e9143975f9fba87768660c0d8e0017f914fc5b16041cdf306983578f823b.jpg)  
+Figure 8: Training acts to stabilize the input-output map by decreasing trajectory length for $\sigma _ { w }$ large. The left pane plots the growth of trajectory length as a circular interpolation between two MNIST datapoints is propagated through the network, at different train steps. Red indicates the start of training, with purple the end of training. Interestingly, and supporting the observation on remaining depth, the first layer appears to increase trajectory length, in contrast with all later layers, suggesting it is being primarily used to fit the data. The right pane shows an identical plot but for an interpolation between random points, which also display decreasing trajectory length, but at a slower rate. Note the output layer is not plotted, due to artificial scaling of length through normalization. The network is initialized with $\sigma _ { w } ^ { 2 } = 1 6$ . A similar plot is observed for the number of transitions (see Appendix.)
+
+![](images/7c6562e93cdc1645c1ebb86496984e065a90bf269f5672c46788b16a5102077b.jpg)  
+Figure 9: Training increases expressivity of input-output map for $\sigma _ { w }$ small. The left pane plots the growth of trajectory length as a circular interpolation between two MNIST datapoints is propagated through the network, at different train steps. Red indicates the start of training, with purple the end of training. We see that the training process increases trajectory length, likely to increase the expressivity of the input-output map to enable greater accuracy. The right pane shows an identical plot but for an interpolation between random points, which also displays increasing trajectory length, but at a slower rate. Note the output layer is not plotted, due to artificial scaling of length through normalization. The network is initialized with $\sigma _ { w } ^ { 2 } = 3$ .
+
+# 4 CONCLUSION
+
+In this paper, we studied the expressivity of neural networks through three measures, neuron transitions, activation patterns and dichotomies, and explained the observed exponential dependence on depth of all three measures by demonstrating the underlying link to latent trajectory length. Having explored these results in the context of random networks, we then looked at the consequences for trained networks (see Table 1). We find that the remaining depth above a network layer influences its expressive power, which might inspire new pre-training or initialization schemes. Furthermore, we see that training interpolates between expressive power and better generalization. This relation between initial and final parameters might inform early stopping and warm starting rules.
+
+# ACKNOWLEDGEMENTS
+
+We thank Samy Bengio, Ian Goodfellow, Laurent Dinh, and Quoc Le for extremely helpful discussion.
+
+# REFERENCES
+
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+
+# Appendix
+
+Here we include the full proofs from sections in the paper.
+
+# A PROOFS AND ADDITIONAL RESULTS FROM SECTION 2.2
+
+Proof of Theorem 1 We prove this result for $F _ { W }$ with zero bias for technical simplicity. The result also translates over to $F _ { W }$ with bias with a couple of technical modifications.
+
+# A.1 NOTATION AND PRELIMINARY RESULTS
+
+Difference of points on trajectory Given $x ( t ) = x , x ( t + d t ) = x + \delta x$ in the trajectory, let $\delta z ^ { ( d ) } = $ $z ^ { ( d ) } ( x + \delta x ) - z ^ { ( d ) } ( x )$
+
+Parallel and Perpendicular Components: Given vectors $x , y$ , we can write $y = y _ { \perp } + y _ { \parallel }$ where $y _ { \perp }$ is the component of $y$ perpendicular to $x$ , and $y _ { \parallel }$ is the component parallel to $x$ . (Strictly speaking, these components should also have a subscript $x$ , but we suppress it as the direction with respect to which parallel and perpendicular components are being taken will be explicitly stated.)
+
+This notation can also be used with a matrix $W$ , see Lemma 1.
+
+Before stating and proving the main theorem, we need a few preliminary results.
+
+Lemma 1. Matrix Decomposition Let $x , y \in \mathbb { R } ^ { k }$ be fixed non-zero vectors, and let $W$ be a (full rank) matrix. Then, we can write
+
+$$
+W = { } ^ { \parallel } W _ { \parallel } + { } ^ { \parallel } W _ { \perp } + { } ^ { \perp } W _ { \parallel } + { } ^ { \perp } W _ { \perp }
+$$
+
+such that
+
+$$
+\begin{array} { r l r } { { ^ { \parallel } W _ { \perp } x = 0 } } & { { } \quad } & { { ^ { \perp } W _ { \perp } x = 0 } } \\ { { y ^ { T \perp } W _ { \parallel } = 0 } } & { { } \quad } & { { y ^ { T \perp } W _ { \perp } = 0 } } \end{array}
+$$
+
+i.e. the row space of $W$ is decomposed to perpendicular and parallel components with respect to $x$ (subscript on right), and the column space is decomposed to perpendicular and parallel components of $y$ (superscript on left).
+
+Proof. Let $V , U$ be rotations such that $V x = ( | | x | | , 0 . . . , 0 ) ^ { T }$ and $U y = ( | | y | | , 0 . . . 0 ) ^ { T }$ . Now let $\tilde { W } = U W V ^ { T }$ , and let $\tilde { W } = \mathbb { I } \tilde { W } _ { \parallel } + \mathbb { I } \tilde { W } _ { \perp } + { ^ \perp \tilde { W } _ { \parallel } } + { ^ \perp \tilde { W } _ { \perp } }$ , with ${ } ^ { \parallel } \tilde { W } _ { \parallel }$ having non-zero term exactly $\tilde { W } _ { 1 1 }$ , ${ \| } \tilde { W } _ { \perp }$ having non-zero entries exactly $\tilde { W } _ { 1 i }$ for $2 \leq i \leq k$ . Finally, we let $^ \perp \tilde { W } _ { \parallel }$ have non-zero entries exactly $\tilde { W } _ { i 1 }$ , with $2 \leq i \leq k$ and $^ { \perp } \tilde { W } _ { \perp }$ have the remaining entries non-zero.
+
+If we define $\tilde { x } = V x$ and $\tilde { y } = U y$ , then we see that
+
+$$
+\begin{array} { r l r } { { ^ { \parallel } \tilde { W } _ { \perp } \tilde { x } = 0 } } & { { } \quad } & { { ^ { \perp } \tilde { W } _ { \perp } \tilde { x } = 0 } } \\ { { \tilde { y } ^ { T \perp } \tilde { W } _ { \parallel } = 0 } } & { { } \quad } & { { \tilde { y } ^ { T \perp } \tilde { W } _ { \perp } = 0 } } \end{array}
+$$
+
+as $\tilde { x } , \tilde { y }$ have only one non-zero term, which does not correspond to a non-zero term in the components of $\tilde { W }$ in the equations.
+
+Then, defining $\mathbb { I } \mathbb { W } _ { \parallel } = U ^ { T \parallel } \tilde { W } _ { \parallel } V$ , and the other components analogously, we get equations of the form
+
+$$
+^ { \parallel } W _ { \perp } x = U ^ { T \parallel } \tilde { W } _ { \perp } V x = U ^ { T \parallel } \tilde { W } _ { \perp } \tilde { x } = 0
+$$
+
+Observation 3. Given $W , x$ as before, and considering $W _ { \parallel }$ , $W _ { \perp }$ with respect to $x$ (wlog a unit vector) we can express them directly in terms of $W$ as follows: Letting $W ^ { ( i ) }$ be the ith row of $W$ , we have
+
+$$
+W _ { \parallel } = \left( \begin{array} { c } { ( ( W ^ { ( 0 ) } ) ^ { T } \cdot x ) x } \\ { \vdots } \\ { ( ( W ^ { ( k ) } ) ^ { T } \cdot x ) x } \end{array} \right)
+$$
+
+i.e. the projection of each row in the direction of $x$ . And of course
+
+$$
+W _ { \perp } = W - W _ { \parallel }
+$$
+
+The motivation to consider such a decomposition of $W$ is for the resulting independence between different components, as shown in the following lemma.
+
+Lemma 2. Independence of Projections Let x be a given vector (wlog of unit norm.) If $W$ is a random matrix with $W _ { i j } \sim \mathcal { N } ( 0 , \overline { { \sigma ^ { 2 } } } )$ , then $W _ { \parallel }$ and $W _ { \perp }$ with respect to $x$ are independent random variables.
+
+Proof. There are two possible proof methods:
+
+(a) We use the rotational invariance of random Gaussian matrices, i.e. if $W$ is a Gaussian matrix, iid entries ${ \mathcal { N } } ( 0 , \sigma ^ { 2 } )$ , and $R$ is a rotation, then $R W$ is also iid Gaussian, entries ${ \mathcal { N } } ( 0 , \sigma ^ { 2 } )$ . (This follows easily from affine transformation rules for multivariate Gaussians.) Let $V$ be a rotation as in Lemma 1. Then $\tilde { W } = W V ^ { T }$ is also iid Gaussian, and furthermore, $\tilde { W } _ { \parallel }$ and $\tilde { W } _ { \perp }$ partition the entries of $\tilde { W }$ , so are evidently independent. But then $W _ { \parallel } =$ $\tilde { W } _ { \parallel } V ^ { T }$ and $W _ { \perp } = \tilde { W } _ { \perp } V ^ { T }$ are also independent.
+
+(b) From the observation note that $W _ { \parallel }$ and $W _ { \perp }$ have a centered multivariate joint Gaussian distribution (both consist of linear combinations of the entries $W _ { i j }$ in $W$ .) So it suffices to show that $W _ { \parallel }$ and $W _ { \perp }$ have covariance 0. Because both are centered Gaussians, this is equivalent to showing $\mathbb { E } ( < W _ { \parallel } , W _ { \perp } > ) = 0$ . We have that
+
+$$
+\mathbb { E } ( < W _ { \parallel } , W _ { \perp } > ) = \mathbb { E } ( W _ { \parallel } W _ { \perp } ^ { T } ) = \mathbb { E } ( W _ { \parallel } W ^ { T } ) - \mathbb { E } ( W _ { \parallel } W _ { \parallel } ^ { T } )
+$$
+
+As any two rows of $W$ are independent, we see from the observation that $\mathbb { E } ( W _ { \parallel } W ^ { T } )$ is a diagonal matrix, with the ith diagonal entry just $( ( W ^ { ( 0 ) } ) ^ { T } \cdot x ) ^ { 2 }$ . But similarly, $\mathbb { E } ( W _ { \parallel } W _ { \parallel } ^ { T } )$ is also a diagonal matrix, with the same diagonal entries - so the claim follows.
+
+In the following two lemmas, we use the rotational invariance of Gaussians as well as the chi distribution to prove results about the expected norm of a random Gaussian vector.
+
+Lemma 3. Norm of a Gaussian vector Let $X \in \mathbb { R } ^ { k }$ be a random Gaussian vector, with $X _ { i }$ iid, $\sim \mathcal { N } ( 0 , \sigma ^ { 2 } )$ . Then
+
+$$
+\mathbb { E } \left[ | | X | | \right] = \sigma \sqrt { 2 } \frac { \Gamma ( ( k + 1 ) / 2 ) } { \Gamma ( k / 2 ) }
+$$
+
+Proof. We use the fact that if $Y$ is a random Gaussian, and $Y _ { i } \sim \mathcal { N } ( 0 , 1 )$ then $| | Y | |$ follows a chi distribution. This means that $\begin{array} { r } { \mathbb { E } ( | | X / \sigma | | ) = \sqrt { 2 } \Gamma ( ( k + 1 ) / 2 ) / \Gamma ( k / 2 ) . } \end{array}$ , the mean of a chi distribution with $k$ degrees of freedom, and the result follows by noting that the expectation in the lemma is $\sigma$ multiplied by the above expectation. □
+
+We will find it useful to bound ratios of the Gamma function (as appear in Lemma 3) and so introduce the following inequality, from (Kershaw, 1983) that provides an extension of Gautschi’s Inequality.
+
+Theorem 5. An Extension of Gautschi’s Inequality For $\cdot 0 < s < 1$ , we have
+
+$$
+\left( x + { \frac { s } { 2 } } \right) ^ { 1 - s } \leq { \frac { \Gamma ( x + 1 ) } { \Gamma ( x + s ) } } \leq \left( x - { \frac { 1 } { 2 } } + \left( s + { \frac { 1 } { 4 } } \right) ^ { { \frac { 1 } { 2 } } } \right) ^ { 1 - s }
+$$
+
+We now show:
+
+Lemma 4. Norm of Projections Let $W$ be a $k$ by $k$ random Gaussian matrix with iid entries $\sim$ ${ \mathcal { N } } ( 0 , \sigma ^ { 2 } )$ , and $x , y$ two given vectors. Partition $W$ into components as in Lemma $I$ and let $x _ { \perp }$ be $a$ nonzero vector perpendicular to x. Then
+
+$$
+\mathbb { E } [ | | ^ { \perp } W _ { \perp } x _ { \perp } | ] = | | x _ { \perp } | | \sigma \sqrt { 2 } \frac { \Gamma ( k / 2 ) } { \Gamma ( ( k - 1 ) / 2 } \geq | | x _ { \perp } | | \sigma \sqrt { 2 } ( \frac { k } { 2 } - \frac { 3 } { 4 } ) ^ { 1 / 2 }
+$$
+
+$( b )$ If $\mathbb { 1 } _ { \mathcal { A } }$ is an identity matrix with non-zeros diagonal entry $i$ iff $i \in \mathcal { A } \subset [ k ]$ , and $| { \cal A } | > 2$ , then
+
+$$
+\mathbb { E } \left[ \left| \left| \mathbb { 1 } _ { \boldsymbol { A } } ^ { \bot } W _ { \bot } x _ { \bot } \right| \right| \right] \ge \left| \left| x _ { \bot } \right| \right| \sigma \sqrt { 2 } \frac { \Gamma ( \left| A \right| / 2 ) } { \Gamma ( ( \left| A \right| - 1 ) / 2 ) } \ge \left| \left| x _ { \bot } \right| \right| \sigma \sqrt { 2 } \left( \frac { \left| A \right| } { 2 } - \frac { 3 } { 4 } \right) ^ { 1 / 2 }
+$$
+
+Proof. (a) Let $U , V , \tilde { W }$ be as in Lemma 1. As $U , V$ are rotations, $\tilde { W }$ is also iid Gaussian. Furthermore for any fixed $W$ , with $\tilde { a } = V a$ , by taking inner products, and square-rooting, we see that $\left| \left| \tilde { W } \tilde { a } \right| \right| = \left| \left| W a \right| \right|$ . So in particular
+
+$$
+\mathbb { E } [ | | ^ { \perp } W _ { \perp } x _ { \perp } | | ] = \mathbb { E } [ | | ^ { \perp } \tilde { W } _ { \perp } \tilde { x } _ { \perp } | | ]
+$$
+
+But from the definition of non-zero entries of $^ { \perp } \tilde { W } _ { \perp }$ , and the form of $\tilde { x } _ { \perp }$ (a zero entry in the first coordinate), it follows that $\perp \tilde { W } _ { \perp } \tilde { x } _ { \perp }$ has exactly $k - 1$ non zero entries, each a centered Gaussian with variance $( k - 1 ) \sigma ^ { 2 } | | x _ { \perp } | | ^ { 2 }$ . By Lemma 3, the expected norm is as in the statement. We then apply Theorem 5 to get the lower bound.
+
+(b) First note we can view $\mathbb { 1 } _ { \mathcal { A } } { } ^ { \perp } W _ { \perp } = { } ^ { \perp } \mathbb { 1 } _ { \mathcal { A } } W _ { \perp }$ . (Projecting down to a random (as $W$ is random) subspace of fixed size $| { \mathcal { A } } | = m$ and then making perpendicular commutes with making perpendicular and then projecting everything down to the subspace.)
+
+So we can view $W$ as a random $m$ by $k$ matrix, and for $x , y$ as in Lemma 1 (with $y$ projected down onto $m$ dimensions), we can again define $U , V$ as $k$ by $k$ and $m$ by $m$ rotation matrices respectively, and $\tilde { W } = U W \bar { V ^ { T } }$ , with analogous properties to Lemma 1. Now we can finish as in part (a), except that $\perp _ { \tilde { W } _ { \perp } \tilde { x } }$ may have only $m - 1$ entries, (depending on whether $y$ is annihilated by projecting down by $\mathbb { 1 } _ { A }$ ) each of variance $( k - 1 ) \sigma ^ { 2 } \left| \left| x _ { \perp } \right| \right| ^ { 2 }$ .
+
+Lemma 5. Norm and Translation Let $X$ be a centered multivariate Gaussian, with diagonal covariance matrix, and $\mu$ a constant vector.
+
+$$
+\mathbb { E } ( | | X - \mu | | ) \geq \mathbb { E } ( | | X | | )
+$$
+
+Proof. The inequality can be seen intuitively geometrically: as $X$ has diagonal covariance matrix, the contours of the pdf of $\vert \vert X \vert \vert$ are circular centered at 0, decreasing radially. However, the contours of the pdf of $\left| \left| X - \mu \right| \right|$ are shifted to be centered around $\lvert \lvert \mu \rvert \rvert$ , and so shifting back $\mu$ to 0 reduces the norm.
+
+A more formal proof can be seen as follows: let the pdf of $X$ be $f _ { X } ( \cdot )$ . Then we wish to show
+
+$$
+\int _ { x } \left| | x - \mu | \right| f _ { X } ( x ) d x \geq \int _ { x } \left| | x | \right| f _ { X } ( x ) d x
+$$
+
+Now we can pair points $x , - x$ , using the fact that $f _ { X } ( x ) = f _ { X } ( - x )$ and the triangle inequality on the integrand to get
+
+$$
+\int _ { | x | } \left( | | x - \mu | | + | | - x - \mu | | \right) f _ { X } ( x ) d x \geq \int _ { | x | } | 2 x | | f _ { X } ( x ) d x = \int _ { | x | } \left( | | x | | + | | - x | | \right) f _ { X } ( x ) d x
+$$
+
+# A.2 PROOF OF THEOREM
+
+Proof. We first prove the zero bias case, Theorem 1. To do so, it is sufficient to prove that
+
+$$
+\mathbb { E } \left[ \Big | \Big | \delta z ^ { ( d + 1 ) } ( t ) \Big | \right] \geq O \left( \left( \frac { \sqrt { \sigma k } } { \sqrt { \sigma + k } } \right) ^ { d + 1 } \right) \Big | \Big | \delta z ^ { ( 0 ) } ( t ) \Big | \Big |
+$$
+
+as integrating over $t$ gives us the statement of the theorem.
+
+For ease of notation, we will suppress the $t$ in $z ^ { ( d ) } ( t )$ .
+
+We first write
+
+$$
+W ^ { ( d ) } = W _ { \bot } ^ { ( d ) } + W _ { \parallel } ^ { ( d ) }
+$$
+
+where the division is done with respect to $z ^ { ( d ) }$ . Note that this means $h ^ { ( d + 1 ) } = W _ { \parallel } ^ { ( d ) } z ^ { ( d ) }$ as the other component annihilates (maps to 0) $z ^ { ( d ) }$ .
+
+We can also define $\mathcal { A } _ { W _ { \parallel } ^ { ( d ) } } = \{ i : i \in [ k ] , | h _ { i } ^ { ( d + 1 ) } | < 1 \}$ i.e. the set of indices for which the hidden representation is not saturated. Letting $W _ { i }$ denote the $i$ th row of matrix $W$ , we now claim that:
+
+$$
+\mathbb { E } _ { W ^ { ( d ) } } \left[ \Big | \Big | \delta z ^ { ( d + 1 ) } \Big | \Big | \right] = \mathbb { E } _ { W _ { \parallel } ^ { ( d ) } } \mathbb { E } _ { W _ { \perp } ^ { ( d ) } } \left[ \left( \sum _ { i \in \mathcal { A } _ { W _ { \parallel } ^ { ( d ) } } } ( ( W _ { \perp } ^ { ( d ) } ) _ { i } \delta z ^ { ( d ) } + ( W _ { \parallel } ^ { ( d ) } ) _ { i } \delta z ^ { ( d ) } ) ^ { 2 } \right) ^ { 1 / 2 } \right]
+$$
+
+Indeed, by Lemma 2 we first split the expectation over $W ^ { ( d ) }$ into a tower of expectations over the two independent parts of $W$ to get
+
+$$
+\mathbb { E } _ { W ^ { ( d ) } } [ \Big | \Big | \delta z ^ { ( d + 1 ) } \Big | \Big | ] = \mathbb { E } _ { W _ { \parallel } ^ { ( d ) } } \mathbb { E } _ { W _ { \perp } ^ { ( d ) } } [ \Big | \Big | \phi ( W ^ { ( d ) } \delta z ^ { ( d ) } ) \Big | | ]
+$$
+
+But conditioning on $W _ { \parallel } ^ { ( d ) }$ in the inner expectation gives us $\smash { h ^ { ( d + 1 ) } }$ and $\mathcal { A } _ { W _ { \parallel } ^ { ( d ) } }$ , allowing us to replace the norm over $\phi ( W ^ { ( d ) } \delta z ^ { ( d ) } )$ with the sum in the term on the right hand side of the claim.
+
+Till now, we have mostly focused on partitioning the matrix $W ^ { ( d ) }$ . But we can also set $\delta z ^ { ( d ) } = $ $\delta z _ { \parallel } ^ { ( d ) } + \delta z _ { \perp } ^ { ( d ) }$ where the perpendicular and parallel are with respect to $z ^ { ( d ) }$ . In fact, to get the expression in $( ^ { * * } )$ , we derive a recurrence as below:
+
+$$
+\mathbb { E } _ { W ^ { ( d ) } } [ \Big | \Big | \delta z _ { \perp } ^ { ( d + 1 ) } \Big | \Big | ] \geq O ( \frac { \sqrt { \sigma k } } { \sqrt { \sigma + k } } ) \mathbb { E } _ { W ^ { ( d ) } } [ \Big | \Big | \delta z _ { \perp } ^ { ( d ) } \Big | | ]
+$$
+
+To get this, we first need to define $\tilde { z } ^ { ( d + 1 ) } = \mathbb { 1 } _ { \mathcal { A } _ { w _ { \parallel } ^ { ( d ) } } } h ^ { ( d + 1 ) }$ - the latent vector $\boldsymbol { h } ^ { ( d + 1 ) }$ with all saturated units zeroed out.
+
+We then split the column space of $W ^ { ( d ) } = { } ^ { \perp } W ^ { ( d ) } + \Vdash W ^ { ( d ) }$ , where the split is with respect to $\tilde { z } ^ { ( d + 1 ) }$ . Letting z(d+1)⊥ be the part perpendicular to z(d+1), and A the set of units that are unsaturated, we have an important relation:
+
+# Claim
+
+$$
+\left| \left| \delta z _ { \perp } ^ { ( d + 1 ) } \right| \right| \geq \left| \left| ^ { \perp } W ^ { ( d ) } \delta z ^ { ( d ) } \mathbb { 1 } _ { \mathcal { A } } \right| \right|
+$$
+
+(where the indicator in the right hand side zeros out coordinates not in the active set.)
+
+To see this, first note, by definition,
+
+$$
+\delta \boldsymbol { z } _ { \perp } ^ { ( d + 1 ) } = \boldsymbol { W } ^ { ( d ) } \delta \boldsymbol { z } ^ { ( d ) } \cdot \mathbb { 1 } _ { \boldsymbol { \mathcal { A } } } - \langle \boldsymbol { W } ^ { ( d ) } \delta \boldsymbol { z } ^ { ( d ) } \cdot \mathbb { 1 } _ { \boldsymbol { \mathcal { A } } } , \hat { \boldsymbol { z } } ^ { ( d + 1 ) } \rangle \hat { \boldsymbol { z } } ^ { ( d + 1 ) }
+$$
+
+where the ˆ· indicates a unit vector.
+
+Similarly
+
+$$
+^ { \perp } W ^ { ( d ) } \delta z ^ { ( d ) } = W ^ { ( d ) } \delta z ^ { ( d ) } - \langle W ^ { ( d ) } \delta z ^ { ( d ) } , { \hat { \tilde { z } } } ^ { ( d + 1 ) } \rangle { \hat { \tilde { z } } } ^ { ( d + 1 ) }
+$$
+
+Now note that for any index $i \in \mathcal A$ , the right hand sides of (1) and (2) are identical, and so the vectors on the left hand side agree for all $i \in \mathcal { A }$ . In particular,
+
+$$
+\delta \boldsymbol { z } _ { \perp } ^ { ( d + 1 ) } \cdot \mathbb { 1 } _ { A } = { } ^ { \perp } W ^ { ( d ) } \delta \boldsymbol { z } ^ { ( d ) } \cdot \mathbb { 1 } _ { A }
+$$
+
+Now the claim follows easily by noting that $\left| \left| \delta z _ { \perp } ^ { ( d + 1 ) } \right| \right| \geq \left| \left| \delta z _ { \perp } ^ { ( d + 1 ) } \cdot \mathbb { 1 } _ { A } \right| \right| .$
+
+Returning to $( ^ { * } )$ , we split $\delta z ^ { ( d ) } = \delta z _ { \perp } ^ { ( d ) } + \delta z _ { \parallel } ^ { ( d ) }$ , $W _ { \perp } ^ { ( d ) } = { } ^ { \parallel } W _ { \perp } ^ { ( d ) } + { } ^ { \perp } W _ { \perp } ^ { ( d ) }$ (and $W _ { \parallel } ^ { ( d ) }$ analogously), and after some cancellation, we have
+
+$$
+\ ? \ ? \complement ^ { ( d + 1 ) } \left\| \Big | \right\| = \mathbb { E } _ { W _ { \parallel } ^ { ( d ) } } \mathbb { E } _ { W _ { \perp } ^ { ( d ) } } \left[ \left( \sum _ { i \in A _ { W _ { \parallel } ^ { ( d ) } } } \left( ( ^ { \perp } W _ { \perp } ^ { ( d ) } + \mathbb { I } W _ { \perp } ^ { ( d ) } ) _ { i } \delta z _ { \perp } ^ { ( d ) } + ( ^ { \perp } W _ { \parallel } ^ { ( d ) } + \mathbb { I } W _ { \parallel } ^ { ( d ) } ) _ { i } \delta z _ { \parallel } ^ { ( d ) } \right) \right) \right]
+$$
+
+We would like a recurrence in terms of only perpendicular components however, so we first drop the ${ } ^ { \parallel } W _ { \perp } ^ { ( d ) } , { } ^ { \parallel } W _ { \parallel } ^ { ( d ) }$ (which can be done without decreasing the norm as they are perpendicular to the remaining terms) and using the above claim, have
+
+$$
+\mathbb { E } _ { W ^ { ( d ) } } \left[ \Big | \Big | \delta z _ { \perp } ^ { ( d + 1 ) } \Big | \Big | \right] \ge \mathbb { E } _ { W _ { \parallel } ^ { ( d ) } } \mathbb { E } _ { W _ { \perp } ^ { ( d ) } } \left[ \left( \sum _ { i \in A _ { W _ { \parallel } ^ { ( d ) } } } \left( \big ( ^ { \perp } W _ { \perp } ^ { ( d ) } \big ) _ { i } \delta z _ { \perp } ^ { ( d ) } + \big ( ^ { \perp } W _ { \parallel } ^ { ( d ) } \big ) _ { i } \delta z _ { \parallel } ^ { ( d ) } \right) ^ { 2 } \right) ^ { 1 / 2 } \right]
+$$
+
+But in the inner expectation, the term $^ \perp W _ { \parallel } ^ { ( d ) } \delta z _ { \parallel } ^ { ( d ) }$ is just a constant, as we are conditioning on $W _ { \parallel } ^ { ( d ) }$ . So using Lemma 5 we have
+
+$$
+\hat { \tau } _ { W _ { \perp } ^ { ( d ) } } \left[ \left( \sum _ { i \in A _ { W _ { \parallel } ^ { ( d ) } } } \left( \binom { \perp } { W _ { \perp } ^ { ( d ) } } _ { i } \delta z _ { \perp } ^ { ( d ) } + \binom { \perp } { W _ { \parallel } ^ { ( d ) } } _ { i } \delta z _ { \parallel } ^ { ( d ) } \right) ^ { 2 } \right) ^ { 1 / 2 } \right] \geq \mathbb { E } _ { W _ { \perp } ^ { ( d ) } } \left[ \left( \sum _ { i \in A _ { W _ { \parallel } ^ { ( d ) } } } \left( \binom { \perp } { W _ { \perp } ^ { ( d ) } } _ { i } \delta z _ { \perp } ^ { ( d ) } + \binom { \perp } { W _ { \parallel } ^ { ( d ) } } _ { i } \delta z _ { \perp } ^ { ( d ) } \right) ^ { 2 } \right) ^ { 1 / 2 } \right] ,
+$$
+
+We can then apply Lemma 4 to get
+
+$$
+\mathbb { E } _ { W _ { \perp } ^ { ( d ) } } \left[ \left( \sum _ { i \in A _ { w _ { \parallel } ^ { ( d ) } } } \left( ( ^ { \perp } W _ { \perp } ^ { ( d ) } ) _ { i } \delta z _ { \perp } ^ { ( d ) } \right) ^ { 2 } \right) ^ { 1 / 2 } \right] \geq \frac { \sigma } { \sqrt { k } } \sqrt { 2 \frac { \sqrt { 2 | A _ { W _ { \parallel } ^ { ( d ) } } | - 3 } } { 2 } } \mathbb { E } \left[ \left| \left| \delta z _ { \perp } ^ { ( d ) } \right| \right| \right]
+$$
+
+The outer expectation on the right hand side only affects the term in the expectation through the size of the non-saturated set of units. Letting $p = \bar { \mathbb { P } } ( | h _ { i } ^ { ( d + 1 ) } | < 1 )$ , and noting that we get a non-zero norm only if $| \mathcal { A } _ { W _ { \parallel } ^ { ( d ) } } | \geq 2$ (else we cannot project down a dimension), and for $| \mathcal { A } _ { W _ { \parallel } ^ { ( d ) } } | \geq 2$ ,
+
+$$
+\sqrt { 2 } \frac { \sqrt { 2 | { \cal A } _ { W _ { \parallel } ^ { ( d ) } } | - 3 } } { 2 } \geq \frac { 1 } { \sqrt { 2 } } \sqrt { | { \cal A } _ { W _ { \parallel } ^ { ( d ) } } | }
+$$
+
+we get
+
+$$
+\mathbb { E } _ { W ^ { ( d ) } } [ \Big | \Big | \delta z _ { \perp } ^ { ( d + 1 ) } \Big | \Big | ] \ge \frac { 1 } { \sqrt { 2 } } ( \sum _ { j = 2 } ^ { k } \binom { k } { j } p ^ { j } ( 1 - p ) ^ { k - j } \frac { \sigma } { \sqrt { k } } \sqrt { j } ) \mathbb { E } [ \Big | \Big | \delta z _ { \perp } ^ { ( d ) } \Big | | ]
+$$
+
+We use the fact that we have the probability mass function for an √ $( k , p )$ binomial random variable to bound the $\sqrt { j }$ term:
+
+$$
+\begin{array} { c } { { \displaystyle \sum _ { i = 2 } ^ { k } { \binom { k } { j } } p ^ { j } ( 1 - p ) ^ { k - j } \frac { \sigma } { \sqrt { k } } \sqrt { j } = - { \binom { k } { 1 } } p ( 1 - p ) ^ { k - 1 } \frac { \sigma } { \sqrt { k } } + \displaystyle \sum _ { j = 0 } ^ { k } { \binom { k } { j } } p ^ { j } ( 1 - p ) ^ { k - j } \frac { \sigma } { \sqrt { k } } \sqrt { j } } } \\ { { = - \sigma \sqrt { k } p ( 1 - p ) ^ { k - 1 } + k p \cdot \frac { \sigma } { \sqrt { k } } \displaystyle \sum _ { j = 1 } ^ { k } \frac { 1 } { \sqrt { j } } { \binom { k } { j - 1 } } p ^ { j - 1 } ( 1 - p ) ^ { k - j } } } \end{array}
+$$
+
+But by using Jensen’s inequality with $1 / \sqrt { x }$ , we get
+
+$$
+\sum _ { j = 1 } ^ { k } { \frac { 1 } { \sqrt { j } } } { \binom { k - 1 } { j - 1 } } p ^ { j - 1 } ( 1 - p ) ^ { k - j } \geq { \frac { 1 } { \sqrt { \sum _ { j = 1 } ^ { k } j { \binom { k - 1 } { j - 1 } } p ^ { j - 1 } ( 1 - p ) ^ { k - j } } } } = { \frac { 1 } { \sqrt { ( k - 1 ) p + 1 } } }
+$$
+
+where the last equality follows by recognising the expectation of a binomial $( k - 1 , p )$ random variable. So putting together, we get
+
+$$
+\mathbb { E } _ { W ^ { ( d ) } } \left[ \left| \left| \delta z _ { \perp } ^ { ( d + 1 ) } \right| \right] \right] \ge \frac { 1 } { \sqrt { 2 } } \left( - \sigma \sqrt { k } p ( 1 - p ) ^ { k - 1 } + \sigma \cdot \frac { \sqrt { k } p } { \sqrt { 1 + ( k - 1 ) p } } \right) \mathbb { E } \left[ \left| \left| \delta z _ { \perp } ^ { ( d ) } \right| \right| \right]
+$$
+
+To lower bound $p$ , we first note that as $h _ { i } ^ { ( d + 1 ) }$ is a normal random variable with variance $\leq \sigma ^ { 2 }$ , if $A \sim { \mathcal { N } } ( 0 , \sigma ^ { 2 } )$
+
+$$
+\mathbb { P } ( | h _ { i } ^ { ( d + 1 ) } | < 1 ) \ge \mathbb { P } ( | A | < 1 ) \ge \frac { 1 } { \sigma \sqrt { 2 \pi } }
+$$
+
+where the last inequality holds for $\sigma \geq 1$ and follows by Taylor expanding $e ^ { - x ^ { 2 } / 2 }$ around 0. Similarly, we can also show that $\textstyle p \leq { \frac { 1 } { \sigma } }$ .
+
+So this becomes
+
+$$
+\begin{array} { l } { \displaystyle \mathbb { E } [ | | \delta z ^ { ( d + 1 ) } | ] \geq ( \frac { 1 } { \sqrt { 2 } } ( \frac { 1 } { ( 2 \pi ) ^ { 1 / 4 } } \frac { \sqrt { \sigma k } } { \sqrt { \sigma \sqrt { 2 \pi } + ( k - 1 ) } } - \sqrt { k } ( 1 - \frac { 1 } { \sigma } ) ^ { k - 1 } ) ) \mathbb { E } [ | | \delta z _ { \perp } ^ { ( d ) } | ] ] } \\ { \displaystyle \qquad = O ( \frac { \sqrt { \sigma k } } { \sqrt { \sigma + k } } ) \mathbb { E } [ | | \delta z _ { \perp } ^ { ( d ) } | ] } \end{array}
+$$
+
+Finally, we can compose this, to get
+
+$$
+\mathbb { E } \left[ \Big | \Big | \delta z ^ { ( d + 1 ) } \Big | \right] \geq \left( \frac { 1 } { \sqrt { 2 } } \left( \frac { 1 } { ( 2 \pi ) ^ { 1 / 4 } } \frac { \sqrt { \sigma k } } { \sqrt { \sigma \sqrt { 2 \pi } + ( k - 1 ) } } - \sqrt { k } \left( 1 - \frac { 1 } { \sigma } \right) ^ { k - 1 } \right) \right) ^ { d + 1 } c \cdot | | \delta x ( t ) | |
+$$
+
+with the constant $c$ being the ratio of $| | \delta x ( t ) \perp | |$ to $| | \delta x ( t ) | |$ . So if our trajectory direction is almost orthogonal to $x ( t )$ (which will be the case for e.g. random circular arcs, $c$ can be seen to be $\approx 1$ by splitting into components as in Lemma 1, and using Lemmas 3, 4.)
+
+Result for non-zero bias In fact, we can easily extend the above result to the case of non-zero bias. The insight is to note that because $\delta z ^ { ( d + 1 ) }$ involves taking a difference between $z ^ { ( d + 1 ) } ( t + d t )$ and $z ^ { ( d + 1 ) } ( t )$ , the bias term does not enter at all into the expression for $\delta z ^ { ( d + 1 ) }$ . So the computations above hold, and equation (a) becomes
+
+$$
+\mathbb { E } _ { W ^ { ( d ) } } \left[ \left| \left| \delta z _ { \perp } ^ { ( d + 1 ) } \right| \right] \right] \geq \frac { 1 } { \sqrt { 2 } } \left( - \sigma _ { w } \sqrt { k } p ( 1 - p ) ^ { k - 1 } + \sigma _ { w } \cdot \frac { \sqrt { k } p } { \sqrt { 1 + ( k - 1 ) p } } \right) \mathbb { E } \left[ \left| \left| \delta z _ { \perp } ^ { ( d ) } \right| \right| \right]
+$$
+
+We also now have that $h _ { i } ^ { ( d + 1 ) }$ is a normal random variable with variance $\leq \sigma _ { w } ^ { 2 } + \sigma _ { b } ^ { 2 }$ (as the bias is drawn from $\mathcal { N } ( 0 , \sigma _ { b } ^ { 2 } ) .$ ). So equation (b) becomes
+
+$$
+\mathbb { P } ( | h _ { i } ^ { ( d + 1 ) } | < 1 ) \ge \frac { 1 } { \sqrt { ( \sigma _ { w } ^ { 2 } + \sigma _ { b } ^ { 2 } ) } \sqrt { 2 \pi } }
+$$
+
+This gives Theorem 1
+
+$$
+\mathbb { E } [ | | \delta z ^ { ( d + 1 ) } | | ] \geq O ( \frac { \sigma _ { w } } { ( \sigma _ { w } ^ { 2 } + \sigma _ { b } ^ { 2 } ) ^ { 1 / 4 } } \cdot \frac { \sqrt { k } } { \sqrt { \sqrt { \sigma _ { w } ^ { 2 } + \sigma _ { b } ^ { 2 } } + k } } ) \mathbb { E } [ | | \delta z _ { \perp } ^ { ( d ) } | ]
+$$
+
+Statement and Proof of Upper Bound for Trajectory Growth Replace hard-tanh with a linear coordinate-wise identity map, h(d+1)i = $h _ { i } ^ { ( d + 1 ) } = ( W ^ { ( d ) } z ^ { ( d ) } ) _ { i } + b _ { i }$ . This provides an upper bound on the norm. We also then recover a chi distribution with $k$ terms, each with standard deviatio n σwk 12 ,
+
+$$
+\begin{array} { r l r } {  { \mathbb { E } [ | | \delta z ^ { ( d + 1 ) } | ] ] \leq \sqrt { 2 } \frac { \Gamma ( ( k + 1 ) / 2 ) } { \Gamma ( k / 2 ) } \frac { \sigma _ { w } } { k ^ { \frac { 1 } { 2 } } } | | \delta z ^ { ( d ) } | | } } \\ & { } & { \leq \sigma _ { w } ( \frac { k + 1 } { k } ) ^ { \frac { 1 } { 2 } } | | \delta z ^ { ( d ) } | | , } \end{array}
+$$
+
+where the second step follows from (Laforgia and Natalini, 2013), and holds for $k > 1$
+
+# B PROOFS AND ADDITIONAL RESULTS FROM SECTION 2.2.2
+
+# Proof of Theorem 2
+
+Proof. For $\sigma _ { b } = 0$ :
+
+For hiddthe large layer case, $d < n$ , consideume that $v _ { 1 } ^ { ( d ) }$ . This has as input  Furthermore, as si W (d−1)z(d−1). As we are in $\sigma$ we ass $| z _ { i } ^ { ( d - 1 ) } | = 1$ $z _ { i } ^ { ( d - 1 ) }$ and $W _ { i 1 } ^ { ( d - 1 ) }$ are both completely random, we can also assume wlog that z(d−1)i = 1. For a particular input, we can define v(d) as sensitive to $v _ { i } ^ { ( d - 1 ) }$ if $v _ { i } ^ { ( d - 1 ) }$ transitioning (to wlog $- 1$ ) will induce a transition in node $v _ { 1 } ^ { ( d ) }$ . A sufficient condition for this to happen is if $\begin{array} { r } { | W _ { i 1 } | \geq | \sum _ { j \neq i } W _ { j 1 } | } \end{array}$ . But $X = W _ { i 1 } \sim { \mathcal { N } } ( 0 , \sigma ^ { 2 } / k )$ and $\begin{array} { r } { \sum _ { j \neq i } W _ { j 1 } = Y ^ { \prime } \sim \mathcal { N } ( 0 , ( k - 1 ) \sigma ^ { 2 } / k ) } \end{array}$ . So we want to compute $\mathbb { P } ( | X | > | Y ^ { \prime } | )$ . For ease of computation, we instead look at $\mathbb { P } ( | X | > | Y | )$ , where $Y \sim { \mathcal { N } } ( 0 , \sigma ^ { 2 } )$ .
+
+But this is the same as computing $\mathbb { P } ( | X | / | Y | > 1 ) = \mathbb { P } ( X / Y < - 1 ) + \mathbb { P } ( X / Y > 1 )$ . But the ratio of two centered independent normals with variances $\sigma _ { 1 } ^ { 2 } , \sigma _ { 2 } ^ { 2 }$ follows a Cauchy distribution, with parameter $\sigma _ { 1 } / \sigma _ { 2 }$ , which in this case is $1 / \sqrt { k }$ . Substituting this in to the cdf of the Cauchy distribution, we get that
+
+$$
+\mathbb { P } \left( \frac { | X | } { | Y | } > 1 \right) = 1 - \frac { 2 } { \pi } \arctan ( \sqrt { k } )
+$$
+
+Finally, using the identity $\arctan ( x ) + \arctan ( 1 / x )$ and the Laurent series for arctan $( 1 / x )$ , we can evaluate the right hand side to be $O ( 1 / \sqrt { k } )$ . In particular
+
+$$
+\mathbb { P } \left( { \frac { | X | } { | Y | } } > 1 \right) \geq O \left( { \frac { 1 } { \sqrt { k } } } \right)
+$$
+
+This means that in expectation, any neuron in layer $d$ will be sensitive to the transitions of $\sqrt { k }$ neurons in the layer below. Using this, and the fact the while $v _ { i } ^ { ( d - 1 ) }$ might flip very quickly from say $- 1$ to 1, the gradation in the transition ensures that neurons in layer $d$ sensitive to $v _ { i } ^ { ( d - 1 ) }$ will transition at distinct times, we get the desired growth rate in expectation as follows:
+
+Let $T ^ { ( d ) }$ be a random variable denoting the number of transitions in layer $d$ . And let $T _ { i } ^ { ( d ) }$ be a random variable denoting the number of transitions of neuron $i$ in layer $d$ . Note that by linearity of expectation and symmetry, $\begin{array} { r } { \mathbb { E } \left[ T ^ { ( d ) } \right] = \sum _ { i } \mathbb { E } \left[ T _ { i } ^ { ( d ) } \right] = k \mathbb { E } \left[ T _ { 1 } ^ { ( d ) } \right] } \end{array}$
+
+Now, $\begin{array} { r } { \mathbb { E } \left[ T _ { 1 } ^ { ( d + 1 ) } \right] \geq \mathbb { E } \left[ \sum _ { i } 1 _ { ( 1 , i ) } T _ { i } ^ { ( d ) } \right] = k \mathbb { E } \left[ 1 _ { ( 1 , 1 ) } T _ { 1 } ^ { ( d ) } \right] } \end{array}$ where $1 _ { ( 1 , i ) }$ is the indicator function of neuron 1 in layer $d + 1$ being sensitive to neuron $i$ in layer $d$ .
+
+But by the independence of these two events, $\mathbb { E } \left[ 1 _ { ( 1 , 1 ) } T _ { 1 } ^ { ( d ) } \right] = \mathbb { E } \left[ 1 _ { ( 1 , 1 ) } \right] \cdot \mathbb { E } \left[ T _ { 1 } ^ { ( d ) } \right]$ . But the firt time on the right hand side is $O ( 1 / \sqrt { k } )$ by (c), so putting it all together, $\mathbb { E } \left[ T _ { 1 } ^ { ( d + \bar { 1 } ) } \right] \geq \sqrt { k } \mathbb { E } \left[ T _ { 1 } ^ { ( d ) } \right]$ . Written in terms of the entire layer, we have $\mathbb { E } \left[ T ^ { ( d + 1 ) } \right] \geq { \sqrt { k } } \mathbb { E } \left[ T ^ { ( d ) } \right]$ as desired.
+
+For $\sigma _ { b } > 0$ :
+
+We replace $\sqrt { k }$ with $\sqrt { k ( 1 + \sigma _ { b } ^ { 2 } / \sigma _ { w } ^ { 2 } ) }$ , by noting that $Y \sim \mathcal { N } ( 0 , \sigma _ { w } ^ { 2 } + \sigma _ { b } ^ { 2 } )$ . This results in a growth rate of form $\begin{array} { r } { O ( \sqrt { k } / \sqrt { 1 + \frac { \sigma _ { b } ^ { 2 } } { \sigma _ { w } ^ { 2 } } } ) } \end{array}$ . □
+
+# C PROOFS AND ADDITIONAL RESULTS FROM SECTION 2.3
+
+# Proof of Theorem 3
+
+Proof. We show inductively that $F _ { W }$ partitions the input space into convex polytopes via hyperplanes. Consider the image of the input space under the first hidden layer. Each neuron $v _ { i } ^ { ( 1 ) }$ defines hyperplane(s) on the input space: letting $W _ { i } ^ { ( 0 ) }$ be the ith row of $W ^ { ( 0 ) }$ , b(0) the bias, we have the hyperplane W (0)i ${ W _ { i } ^ { ( 0 ) } } x + b _ { i } = 0$ for a ReLU and hyperplanes ${ W _ { i } ^ { ( 0 ) } x + b _ { i } = \pm 1 }$ for a hard-tanh. Considering all such hyperplanes over neurons in the first layer, we get a hyperplane arrangement in the input space, each polytope corresponding to a specific activation pattern in the first hidden layer.
+
+Now, assume we have partitioned our input space into convex polytopes with hyperplanes from layers ≤ d−1. Consider v(d)i and a specific polytope $R _ { i }$ . Then the activation pattern on layers $\leq d - 1$ is constant on $R _ { i }$ , and so the input to $v _ { i } ^ { ( d ) }$ on $R _ { i }$ is a linear function of the inputs $\textstyle \sum _ { j } \lambda _ { j } x _ { j } + b$ and some constant term, comprising of the bias and the output of saturated units. Setting this expression to zero (for ReLUs) or to $\pm 1$ (for hard-tanh) again gives a hyperplane equation, but this time, the equation is only valid in $R _ { i }$ (as we get a different linear function of the inputs in a different region.) So the defined hyperplane(s) either partition $R _ { i }$ (if they intersect $R _ { i }$ ) or the output pattern of $v _ { i } ^ { ( d ) }$ is . The theorem then follows.
+
+This implies that any one dimensional trajectory $x ( t )$ , that does not ‘double back’ on itself (i.e. reenter a polytope it has previously passed through), will not repeat activation patterns. In particular, after seeing a transition (crossing a hyperplane to a different region in input space) we will never return to the region we left. A simple example of such a trajectory is a straight line:
+
+Corollary 2. Transitions and Output Patterns in an Affine Trajectory For any affine one dimensional trajectory $x ( t ) = x _ { 0 } + t ( x _ { 1 } - x _ { 0 } )$ input into a neural network $F _ { W }$ , we partition $\mathbb { R } \ni t$ into intervals every time a neuron transitions. Every interval has a unique network activation pattern on $F _ { W }$ .
+
+Generalizing from a one dimensional trajectory, we can ask how many regions are achieved over the entire input – i.e. how many distinct activation patterns are seen? We first prove a bound on the number of regions formed by $k$ hyperplanes in $\mathbb { R } ^ { m }$ (in a purely elementary fashion, unlike the proof presented in (Stanley, 2011))
+
+Theorem 6. Upper Bound on Regions in a Hyperplane Arrangement Suppose we have $k$ hyperplanes in $\mathbb { R } ^ { m }$ - i.e. $k$ equations of form $\alpha _ { i } x = \beta _ { i }$ . for $\alpha _ { i } \in \mathbb { R } ^ { m }$ , $\beta _ { i } \in \mathbb { R }$ . Let the number of regions (connected open sets bounded on some sides by the hyperplanes) be $r ( k , m )$ . Then
+
+$$
+r ( k , m ) \leq \sum _ { i = 0 } ^ { m } { \binom { k } { i } }
+$$
+
+# Proof of Theorem 6
+
+Proof. Let the hyperplane arrangement be denoted $\mathcal { H }$ , and let $H \in { \mathcal { H } }$ be one specific hyperplane. Then the number of regions in $\mathcal { H }$ is precisely the number of regions in $\mathcal { H } - H$ plus the number of
+
+regions in $\mathcal { H } \cap H$ . (This follows from the fact that $H$ subdivides into two regions exactly all of the regions in $\mathcal { H } \cap H$ , and does not affect any of the other regions.)
+
+In particular, we have the recursive formula
+
+$$
+r ( k , m ) = r ( k - 1 , m ) + r ( k - 1 , m - 1 )
+$$
+
+We now induct on $k + m$ to assert the claim. The base cases of $r ( 1 , 0 ) = r ( 0 , 1 ) = 1$ are trivial, and assuming the claim for $\leq k + m - 1$ as the induction hypothesis, we have
+
+$$
+\begin{array} { r l } { r ( k - 1 , m ) + r ( k - 1 , m - 1 ) \le \displaystyle \sum _ { i = 0 } ^ { m } { \binom { k - 1 } { i } } + \displaystyle \sum _ { i = 0 } ^ { m - 1 } { \binom { k - 1 } { i } } } & { } \\ { \le \binom { k - 1 } { 0 } + \displaystyle \sum _ { i = 0 } ^ { d - 1 } { \binom { k - 1 } { i } } + \binom { k - 1 } { i + 1 } } & { } \\ { \le \binom { k } { 0 } + \displaystyle \sum _ { i = 0 } ^ { m - 1 } { \binom { k } { i + 1 } } } & { } \end{array}
+$$
+
+where the last equality follows by the well known identity
+
+$$
+{ \binom { a } { b } } + { \binom { a } { b + 1 } } = { \binom { a + 1 } { b + 1 } }
+$$
+
+This concludes the proof.
+
+With this result, we can easily prove Theorem 4 as follows:
+
+Proof. First consider the ReLU case. Each neuron has one hyperplane associated with it, and so by Theorem 6, the first hidden layer divides up the inputs space into $r ( k , m )$ regions, with $r ( k , m ) \leq$ $O ( k ^ { m } )$ .
+
+Now consider the second hidden layer. For every region in the first hidden layer, there is a different activation pattern in the first layer, and so (as described in the proof of Theorem 3) a different hyperplane arrangement of $k$ hyperplanes in an $m$ dimensional space, contributing at most $r ( k , m )$ regions.
+
+In particular, the total number of regions in input space as a result of the first and second hidden layers is $\leq r ( k , m ) * r ( k , m ) \leq O ( \bar { k } ^ { 2 } m )$ . Continuing in this way for each of the $n$ hidden layers gives the $O ( k ^ { m } n )$ bound.
+
+A very similar method works for hard tanh, but here each neuron produces two hyperplanes, resulting in a bound of $O ( ( 2 k ) ^ { m n } )$ .
+
+# D PROOFS AND ADDITIONAL RESULTS FROM SECTION 2.4
+
+# D.1 UPPER BOUND FOR DICHOTOMIES
+
+The Vapnik-Chervonenkis (VC) dimension of a function class is the cardinality of the largest set of points that it can shatter. The VC dimension provides an upper (worst case) bound on the generalization error for a function class (Vapnik and Vapnik, 1998). Motivated by generalization error, VC dimension has been studied for neural networks (Sontag, 1998; Bartlett and Maass, 2003). In (Bartlett et al., 1998) an upper bound on the VC dimension $v$ of a neural network with piecewise polynomial activation function and binary output is derived. For hard-tanh units, this bound is
+
+$$
+v = 2 \left| W \right| n \log \left( 4 e \left| W \right| n k \right) + 2 \left| W \right| n ^ { 2 } \log 2 + 2 n ,
+$$
+
+where $| W |$ is the total number of weights, $n$ is the depth, and $k$ is the width of the network. The VC dimension provides an upper bound on the number of achievable dichotomies $| \mathcal F |$ by way of the
+
+![](images/6192e61c35ecf0f7a6277938370bb38b56e24e212f9ba4791a6dcac9ffc0fc64.jpg)  
+Figure 10: Here we plot the number of unique dichotomies that have been observed as a function of the number of transitions the network has undergone. Each datapoint corresponds to the number of transitions and dichotomies for a hard-tanh network of a different depth, with the weights in the first layer undergoing interpolation along a great circle trajectory $W ^ { ( 0 ) } ( \bar { t } )$ . We compare these plots to a random walk simulation, where at each transition a single class label is flipped uniformly at random. Dichotomies are measured over a dataset consisting of $s = 1 5$ random samples, and all networks had weight variance $\sigma _ { w } ^ { 2 } = 1 6$ . The blue dashed line indicates all $2 ^ { s }$ possible dichotomies.
+
+Sauer–Shelah lemma (Sauer, 1972),
+
+$$
+| \mathcal { F } | \leq \left( \frac { e | S | } { v } \right) ^ { v } .
+$$
+
+By combining Equations 4 and 5 an upper bound on the number of dichotomies is found, with a growth rate which is exponential in a low order polynomial of the network size.
+
+Our results further suggest the following conjectures:
+
+Conjecture 1. As network width $k$ increases, the exploration of the space of dichotomies increasingly resembles a simple random walk on a hypercube with dimension equal to the number of inputs $| S |$ .
+
+This conjecture is supported by Figure 10, which compares the number of unique dichotomies achieved by networks of various widths to the number of unique dichotomies achieved by a random walk. This is further supported by an exponential decrease in autocorrelation length in function space, derived in our prior work (Poole et al., 2016).
+
+Conjecture 2. The expressive power of a single weight $W _ { i j } ^ { ( d ) }$ at layer $d$ in a random network $F$ , and for a set of random inputs $S$ , is exponential in the remaining network depth $d _ { r } = ( n - d )$ . Here expressive power is the number of dichotomies achievable by adjusting only that weight.
+
+That is, the expressive power of weights in early layers in a deep hard-tanh network is exponentially greater than the expressive power of weights in later layers. This is supported by the invariance to layer number in the recurrence relations used in all proofs directly involving depth. It is also directly supported by simulation, as illustrated in Figure 5, and by experiments on MNIST and CIFAR10 as illustrated in Figures 6, 7.
+
+# E FURTHER RESULTS AND IMPLEMENTATION DETAILS FROM SECTION 3
+
+We implemented the random network architecture described in Section 2.1. In separate experiments we then swept an input vector along a great circle trajectory (a rotation) for fixed weights, and swept weights along a great circle trajectory for a fixed set of inputs, as described in Section 2.4. In both cases, the trajectory was subdivided into $1 0 ^ { 6 }$ segments. We repeated this for a grid of network widths $k$ , weight variances $\sigma _ { w } ^ { 2 }$ , and number of inputs $s$ . Unless otherwise noted, $\sigma _ { b } = 0$ for all experiments. We repeated each experiment 10 times and averaged over the results. The simulation results are discussed and plotted throughout the text.
+
+The networks trained on MNIST and CIFAR-10 were implemented using Keras and Tensorflow, and trained for a fixed number of epochs with the ADAM optimizer.
+
+![](images/baa3de5a4a17cf91ab849afbbccdc7ce819cc80e05dc786a91e745d83bd94c72.jpg)  
+Figure 11: An identical plot to Figure 8 but for transition count.
+
+![](images/b480816386068e1373f6c6d1d2105e87978aeea710d002bb3cbc0fd0624b9d17.jpg)  
+Figure 12: We repeat the experiment in Figure 6 for a convolutional network trained on CIFAR10. The network has eight convolutional hidden layers, with three by three filters and 64 filters in each layer, all with ReLU activations. The final layer is a fully connected softmax, and is trained in addition to the single convolutional layer being trained. The results again support greater expressive power with remaining depth. Note the final three convolutional layers failed to effectively train, and performed at chance level.
+
+We also have preliminary experimental results on Convolutional Networks. To try and make the comparisons fair, we implemented a fully convolutional network (no fully connected layers except for the last layer).
+
+We also include the plot showing the effect of training on number of transitions for interpolated MNIST and interpolated random points.

parse/train/B1gabhRcYX/B1gabhRcYX.md

@@ -0,0 +1,391 @@
+# BA-NET: DENSE BUNDLE ADJUSTMENT NETWORKS
+
+Chengzhou Tang School of Computer Science Simon Fraser University chengzhou_tang@sfu.ca
+
+Ping Tan School of Computer Science Simon Fraser University pingtan@sfu.ca
+
+# ABSTRACT
+
+This paper introduces a network architecture to solve the structure-from-motion (SfM) problem via feature-metric bundle adjustment (BA), which explicitly enforces multi-view geometry constraints in the form of feature-metric error. The whole pipeline is differentiable, so that the network can learn suitable features that make the BA problem more tractable. Furthermore, this work introduces a novel depth parameterization to recover dense per-pixel depth. The network first generates several basis depth maps according to the input image, and optimizes the final depth as a linear combination of these basis depth maps via feature-metric BA. The basis depth maps generator is also learned via end-to-end training. The whole system nicely combines domain knowledge (i.e. hard-coded multi-view geometry constraints) and deep learning (i.e. feature learning and basis depth maps learning) to address the challenging dense SfM problem. Experiments on large scale real data prove the success of the proposed method.
+
+# 1 INTRODUCTION
+
+The Structure-from-Motion (SfM) problem has been extensively studied in the past a few decades. Almost all conventional SfM algorithms (Agarwal et al., 2011; Wu et al., 2011; Schönberger & Frahm, 2016; Engel et al., 2018; Delaunoy & Pollefeys, 2014) jointly optimize scene structures and camera motion via the Bundle-Adjustment (BA) algorithm (Triggs et al., 2000; Agarwal et al., 2010), which minimizes the geometric (Agarwal et al., 2011; Wu et al., 2011; Schönberger & Frahm, 2016) or photometric (Engel et al., 2014; 2018; Delaunoy & Pollefeys, 2014) error through the Levenberg-Marquardt (LM) algorithm (Nocedal & Wright, 2006). Some recent works (Ummenhofer et al., 2017; Zhou et al., 2017; Wang et al., 2018) attempt to solve SfM using deep learning techniques, but most of them do not enforce the geometric constraints between 3D structures and camera motion in their networks. For example, in the recent work DeMoN (Ummenhofer et al., 2017), the scene depths and the camera motion are estimated by two individual sub-network branches.
+
+This paper formulates BA as a differentiable layer, the BA-Layer, to bridge the gap between classic methods and recent deep learning based approaches. To this end, we learn a feed-forward multilayer perceptron (MLP) to predict the damping factor in the LM algorithm, which makes all involved computation differentiable. Furthermore, unlike conventional BA that minimizes geometric or photometric error, our BA-layer minimizes the distance between aligned CNN feature maps. Our novel feature-metric BA takes CNN features of multiple images as inputs and optimizes for the scene structures and camera motion. This feature-metric BA is desirable, because it has been observed by Engel et al. (2014; 2018) that the geometric BA does not exploit all image information, while the photometric BA is sensitive to moving objects, exposure or white balance changes, etc. Most importantly, our BA-Layer can back-propagate loss from scene structures and camera motion to learn appropriate features that are most suitable for structure-from-motion and bundle adjustment. In this way, our network hard-codes the multi-view geometry constraints in the BA-Layer and learns suitable feature representations from training data.
+
+We strive to estimate a dense per-pixel depth, because dense depth is critical for many tasks such as object detection and robot navigation. A major challenge in solving dense per-pixel depth is to find a compact parameterization. Direct per-pixel depth is computational expensive, which makes the network training intractable. So we train a network to generate a set of basis depth maps for an arbitrary input image and represent the result depth map as a linear combination of these basis depth maps. The combination coefficients will be optimized in the BA-Layer together with camera motion. This novel parameterization guarantees a smooth depth map with good consistency with object boundaries. It also reduces the number of unknowns and makes dense BA possible in networks.
+
+Similar depth parameterization is introduced in a recent work, CodeSLAM (Bloesch et al., 2018). The major difference is that our method learns the basis depth map generator through the gradients back-propagated from the BA-Layer, while CodeSLAM learns the generator separately and uses its results for a standalone optimization component. Thus, our basis depth map generator has the chance to be better trained for the SfM problem. Furthermore, we use a different network structure to generate basis depth maps. CodeSLAM employs a variational auto-encoder (VAE), while we use a standard encoder-decoder. This design enables us to use the same backbone network for both feature learning and basis depth map learning, making joint training of the whole network possible.
+
+To demonstrate the effectiveness of our method, we evaluate on the ScanNet (Dai et al., 2017a) and KITTI (Geiger et al., 2012) dataset. Our method outperforms DeMoN (Ummenhofer et al., 2017), LS-Net (Clark et al., 2018), as well as several conventional baselines. Due to page limit, we move the ablation studies, evaluation on DeMoN’s dataset, multi-view SfM (up to 5 views), and comparison with CodeSLAM on the EuroC dataset (Burri et al., 2016) to the appendix.
+
+# 2 RELATED WORK
+
+Monocular Depth Estimation Networks Estimating depth from a monocular image is an ill-posed problem because an infinite number of possible scenes may have produced the same image. Before the raise of deep learning based methods, some works predict depth from a single image based on MRF (Saxena et al., 2005; 2009), semantic segmentation (Ladický et al., 2014), or manually designed features (Hoiem et al., 2005). Eigen et al. (2014) propose a multi-scale approach for depth prediction with two CNNs, where a coarse-scale network first predicts the scene depth at the global level and then a fine-scale network will refine the local regions. This approach was extended in Eigen & Fergus (2015) to handle semantic segmentation and surface normal estimation as well. Recently, Laina et al. (2016) propose to use ResNet (He et al., 2016) based structure to predict depth, and Xu et al. (2017) construct multi-scale CRFs for depth prediction. In comparison, we exploit monocular image depth estimation network for depth parameterization, which only produces a set of basis depth maps and the final result will be further improved through optimization.
+
+Structure-from-Motion Networks Recently, some works exploit CNNs to resolve the SfM problem. Handa et al. (2016) solve the camera motion by a network from a pair of images with known depth. Zhou et al. (2017) employ two CNNs for depth and camera motion estimation respectively, where both CNNs are trained jointly by minimizing the photometric loss in an unsupervised manner. Wang et al. (2018) implement the direct method (Steinbruecker et al., 2011) as a differentiable component to compute camera motion after scene depth is estimated by the method in Zhou et al. (2017). In Ummenhofer et al. (2017), the scene depth and the camera motion are predicted from optical flow features, which help to make it generalizing better to unseen data. However, the scene depth and the camera motion are solved by two separate network branches, multi-view geometry constraints between depth and motion are not enforced. Recently, Clark et al. (2018) propose to solve nonlinear least squares in two-view SfM using a LSTM-RNN (Hochreiter et al., 2001) as the optimizer.
+
+Our method belongs to this category. Unlike all previous works, we propose the BA-Layer to simultaneously predict the scene depth and the camera motion from CNN features, which explicitly enforces multi-view geometry constraints. The hard-coded multi-view geometry constraints enable our method to reconstruct more than two images, while most deep learning methods can only handle two images. Furthermore, we propose to minimize a feature-metric error instead of the photometric error in (Zhou et al., 2017; Wang et al., 2018; Clark et al., 2018) to enhance robustness.
+
+# 3 BUNDLE ADJUSTMENT REVISITED
+
+Before introducing our BA-Net architecture, we revisit the classic BA to have a better understanding about where the difficulties are and why feature-metric BA and feature learning are desirable. We only introduce the most relevant content and refer the readers to Triggs et al. (2000) and Agarwal et al. (2010) for a comprehensive introduction. Given images $\mathbb { I } = \{ I _ { i } | i = 1 \cdot \cdot \cdot N _ { i } \}$ , the geometric
+
+BA (Triggs et al., 2000; Agarwal et al., 2010) jointly optimizes camera poses $\mathbb { T } = \{ \pmb { T } _ { i } | i = 1 \cdots N _ { i } \}$ and 3D scene point coordinates $\mathbb { P } = \{ \pmb { p } _ { j } | j = 1 \cdots N _ { j } \}$ by minimizing the re-projection error:
+
+$$
+\mathcal { X } = \mathop { \mathrm { a r g m i n } } \sum _ { i = 1 } ^ { N _ { i } } \sum _ { j = 1 } ^ { N _ { j } } \| e _ { i , j } ^ { g } ( \mathcal { X } ) \| ,
+$$
+
+where the geometric distance
+
+$$
+e _ { i , j } ^ { g } ( \mathcal { X } ) = \pi ( \pmb { T } _ { i } , \pmb { p } _ { j } ) - \pmb { q } _ { i , j }
+$$
+
+measures the difference between a projected scene point and its corresponding feature point. The function $\pi$ projects scene points to image space, $\mathbf { \mathscr { q } } _ { i , j } = [ x _ { i , j } , y _ { i , j } , 1 ]$ is the normalized homogeneous pixel coordinate, and $\mathcal { X } = [ \pmb { T } _ { 1 } , \pmb { T } _ { 2 } \cdot \cdot \cdot \pmb { T } _ { N _ { i } } , \pmb { p } _ { 1 } , \pmb { p } _ { 2 } \cdot \cdot \cdot \pmb { p } _ { N _ { j } } ] ^ { \top }$ contains all the points’ and the cameras’ parameters. The general strategy to minimize Equation (1) is the Levenberg-Marquardt (LM) (Nocedal & Wright, 2006; Lourakis & Argyros, 2005) algorithm. At each iteration, the LM algorithm solves for an optimal update $\Delta \mathcal { X } ^ { * }$ to the solution by minimizing:
+
+$$
+\Delta \mathcal { X } ^ { * } = \mathrm { a r g m i n } \| J ( \mathcal { X } ) \Delta \mathcal { X } + E ( \mathcal { X } ) \| + \lambda \| D ( \mathcal { X } ) \Delta \mathcal { X } \| .
+$$
+
+Here, $E ( \mathcal { X } ) = [ e _ { 1 , 1 } ^ { g } ( \mathcal { X } ) , e _ { 1 , 2 } ^ { g } ( \mathcal { X } ) \cdot \cdot \cdot e _ { N _ { i } , N _ { j } } ^ { g } ( \mathcal { X } ) ]$ , and $J ( \mathcal { X } )$ is the Jacobian matrix of $E ( \mathcal { X } )$ respect to $\mathcal { X }$ , $D ( \mathcal { X } )$ is a non-negative diagonal matrix, typically the square root of the diagonal of the approximated Hessian $J ( \mathcal { X } ) ^ { \top } J ( \mathcal { X } )$ . The non-negative value $\lambda$ controls the regularization strength. The special structure of $J ( \mathcal { X } ) ^ { \top } J ( \mathcal { X } )$ motivates the use of Schur-Complement (Brown, 1958).
+
+This geometric BA with re-projection error is the golden standard for structure-from-motion in the last two decades, but with two main drawbacks:
+
+• Only image information conforming to the respective feature types, typically image corners, blobs, or line segments, is utilized. Features have to be matched to each other, which often result in a lot of outliers. Outlier rejection like RANSAC is necessary, which still cannot guarantee correct result.
+
+These two difficulties motivate the recent development of direct methods (Engel et al., 2014; 2018; Delaunoy & Pollefeys, 2014) which propose the photometric BA algorithm to eliminate feature matching and directly minimizes the photometric error (pixel intensity difference) of aligned pixels. The photometric error is defined as:
+
+$$
+e _ { i , j } ^ { p } ( \mathcal { X } ) = I _ { i } ( \pi ( T _ { i } , d _ { j } \cdot \pmb { q } _ { j } ) ) - I _ { 1 } ( \pmb { q } _ { j } ) ,
+$$
+
+where $d _ { j } ~ \in ~ \mathbb { D } ~ = ~ \{ d _ { j } | j ~ = ~ 1 \cdot \cdot \cdot N _ { j } \}$ is the depth of a pixel $\mathbf { \Delta } \mathbf { q } _ { j }$ at the image $I _ { 1 }$ , and $d _ { j } \cdot \mathbf { \vec { q } } _ { j }$ upgrade the pixel $\mathbf { \Delta } \mathbf { q } _ { j }$ to its 3D coordinate. Thus, the optimization parameter is $\mathcal { X } =$ $[ { \pmb T } _ { 1 } , { \pmb T } _ { 2 } \cdot \cdot \cdot { \pmb T } _ { N _ { i } } , d _ { 1 } , d _ { 2 } \cdot \cdot \cdot d _ { N _ { j } } ] ^ { \top }$ . The direct methods have the advantages of using all pixels with sufficient gradient magnitude. They have demonstrated superior performance, especially at less textured scenes. However, these methods also have some drawbacks:
+
+• They are sensitive to initialization as demonstrated in (Mur-Artal et al., 2015) and (Tang et al., 2017) because the photometric error increases the non-convexity (Engel et al., 2018).   
+• They are sensitive to camera exposure and white balance changes. An automatic photometric calibration is required (Engel et al., 2018; 2016).   
+• They are more sensitive to outliers such as moving objects.
+
+# 4 THE BA-NET ARCHITECTURE
+
+To deal with the above challenges, we propose a feature-metric BA algorithm which estimates the same scene depth and camera motion parameters $\mathcal { X }$ as in photometric BA, but minimizes the feature-metric difference of aligned pixels:
+
+$$
+e _ { i , j } ^ { f } ( \mathcal { X } ) = F _ { i } ( \pi ( \boldsymbol { T } _ { i } , d _ { j } \cdot \boldsymbol { q } _ { j } ) ) - F _ { 1 } ( \boldsymbol { q } _ { j } ) ,
+$$
+
+where $\mathbb { F } = \{ F _ { i } | i = 1 \cdot \cdot \cdot N _ { i } \}$ are feature pyramids of images $\mathbb { I } = \{ I _ { i } | i = 1 \cdot \cdot \cdot N _ { i } \}$ . Similar to the photometric BA, our feature-metric BA considers more pixels than corners or blobs. It has the potential to learn more suitable features for SfM to deal with exposure changes, moving objects, etc.
+
+![](images/51445b39ea111c0debdf69e8cec249cfec148f0981237baaf39f3d3499de2fef.jpg)  
+Figure 1: Overview of our BA-Net structure, which consists of a DRN-54 (Yu et al., 2017) as the backbone network, a Basis Depth Maps Generator that generates a set of basis depth maps, a Feature Pyramid Constructor that constructs multi-scale feature maps, and a BA-Layer that optimizes both the depth map and the camera poses through a novel differentiable LM algorithm.
+
+We learn features suitable for SfM via back-propagation, instead of using pre-trained CNN features for image classification (Czarnowski et al., 2017). Therefore, it is crucial to design a differentiable optimization layer, our BA-Layer, to solve the optimization problem, so that the loss information can be back-propagated. The BA-Layer predicts the camera poses $\mathbb { T }$ and the dense depth map $\mathbb { D }$ during forward pass and back-propagates the loss from $\mathbb { T }$ and $\mathbb { D }$ to the feature pyramids $\mathbb { F }$ for training.
+
+# 4.1 OVERVIEW
+
+As illustrated in Figure 1, our BA-Net receives multiple images and then feed them to the backbone DRN-54. We use DRN-54 (Yu et al., 2017) because it replaces max-pooling with convolution layers and generates smoother feature maps, which is desirable for BA optimization. Note the original DRN is memory inefficient due to the high resolution feature maps after dilation convolutions. We replace the dilation convolution with ordinary convolution with strides to address this issue. After DRN-54, a feature pyramid is then constructed for each input image, which are the inputs for the BA-Layer.
+
+At the same time, the basis depth maps generator generates multiple basis depth maps for the image $I _ { 1 }$ , and the final depth map is represented as a linear combination of these basis depth maps.
+
+Finally, the BA-Layer optimizes for the camera poses and the dense depth map jointly by minimizing the feature-metric error defined in Equation (4), which makes the whole pipeline end-to-end trainable.
+
+# 4.2 FEATURE PYRAMID
+
+The feature pyramid learns suitable features for the BA-Layer. Similar to the feature pyramid networks (FPN) for object detection (Lin et al., 2017), we exploit the inherent multi-scale hierarchy of deep convolutional networks to construct feature pyramids. A top-down architecture with lateral connections is applied to propagate richer context information from coarser scales to finer scales. Thus, our feature-metric BA will have a larger convergence radius.
+
+As shown in Figure 2(a), we construct a feature pyramid from the backbone DRN-54. We denote the last residual blocks of conv1, conv2, conv3, conv4 in DRN-54 as $\{ C ^ { 1 } , C ^ { 2 } , C ^ { 3 } , C ^ { 4 } \}$ , with strides $\{ 1 , 2 , 4 , 8 \}$ respectively. We upsample a feature map $C ^ { k + 1 }$ by a factor of 2 with bilinear interpolation and concatenate the upsampled feature map with $C ^ { \bar { k } }$ in the next level. This procedure is iterated until the finest level. Finally, we apply a $3 \times 3$ convolution on the concatenated feature maps to reduce its dimensionality to 128 to balance the expressiveness and computational complexity, which leads to the final feature pyramid $F _ { i } = [ F _ { i } ^ { 1 } , F _ { i } ^ { 2 } , F _ { i } ^ { 3 } ]$ for image $I _ { i }$ .
+
+We visualize some typical channels from the raw image $I$ (i.e. the RGB channels), the pre-trained DRN-54 $C ^ { 3 }$ and our learned $F ^ { 3 }$ in Figure 2(b). It is evident that, after training with our BA-Layer, the feature pyramid becomes smoother and each channel correspondences to different regions in the image. Note that our feature pyramids have higher resolution than FPN to facilitate precise alignment.
+
+To have a better intuition about how much the BA optimization benefits from our learned features, we visualize different distances in Figure 3. We evaluate the distance between a pixel marked by a yellow cross in the top image in Figure 3 (a) and all pixels in a neighbourhood of its corresponding point in the bottom image of Figure 3 (a). The distances evaluated from raw RGB values, pretrained feature $C ^ { 3 }$ , and our learned feature $F ^ { 3 }$ are visualized in (b), (c), and (d) respectively. All distances are normalized to $[ 0 , 1 ]$ and visualized as heat maps. The $x$ -axis and $y$ -axis are the offsets to the ground-truth corresponding point. The RGB distance in (b) (i.e. $e ^ { p }$ in Equation (3)) has no clear global minimum, which makes the photometric BA sensitive to initialization (Engel et al., 2014; 2018). The distance measured by the pretrained feature $C ^ { 3 }$ has both global and local minimums. Finally, the distance measured by our learned feature $F ^ { 3 }$ has a clear global minimum and smooth basin, which is helpful in gradient based optimization such as the LM algorithm.
+
+# 4.3 BUNDLE ADJUSTMENT LAYER
+
+After building feature pyramids for all images, we optimize camera poses and a dense depth map by minimizing the feature-metric error in Equation (4). Following the conventional Bundle Adjustment principle, we optimize Equation (4) using the Levenberg-Marquardt (LM) algorithm. However, the original LM algorithm is non-differentiable because of two difficulties:
+
+• The iterative computation terminates when a specified convergence threshold is reached. This if-else based termination strategy makes the output solution $\mathcal { X }$ non-differentiable with respect to the input $\mathbb { F }$ (Domke, 2012). In each iteration, it updates the damping factor $\lambda$ based on the current value of the objective function. It raises $\lambda$ if a step fails to reduce the objective; otherwise it reduces $\lambda$ . This if-else decision also makes $\mathcal { X }$ non-differentiable with respect to $\mathbb { F }$ .
+
+When the solution $\mathcal { X }$ is non-differentiable with respect to $\mathbb { F }$ , feature learning by back-propagation becomes impossible. The first difficulty has been studied in Domke (2012) and the author proposes to fix the number of iterations, which is refered as ‘incomplete optimization’. Besides making the optimization differentiable, this ‘incomplete optimization’ technique also reduces memory consumption because the number of iterations is usually fixed at a small value.
+
+The second difficulty has never been studied. Previous works mainly focus on gradient descent (Domke, 2012) or quadratic minimization (Amos & Kolter, 2017; Schmidt & Roth, 2014). In this section, we propose a simple yet effective approach to soften the if-else decision and yields a differentiable LM algorithm. We send the current objective value to a MLP network to predict $\lambda$ . This technique not only makes the optimization differentiable, but also learns to predict a better damping factor $\lambda$ , which helps the optimization to reach a better solution within limited iterations.
+
+To start with, we illustrate a single iteration of the LM optimization as a diagram in Figure 4 by interpreting intermediate variables as network nodes. During the forward pass, we compute the solution update $\Delta \mathcal { X }$ from feature pyramids $\mathbb { F }$ and current solution $\mathcal { X }$ as the following steps:
+
+• We compute the feature-metric error $E ( \mathcal { X } ) = [ e _ { 1 , 1 } ^ { f } ( \mathcal { X } ) , e _ { 1 , 2 } ^ { f } ( \mathcal { X } ) \cdots e _ { N _ { i } , N _ { j } } ^ { f } ( \mathcal { X } ) ]$ with Equation (4) on all $N _ { i }$ images and $N _ { j }$ pixels, where $\mathcal { X }$ is the solution from the previous iteration;
+
+![](images/9ce7ecb0fb83ecb8cf36b97f25a52a284e907bea6b12aeddf555810fd438bf08.jpg)  
+Figure 2: A feature pyramid and some typical channels from different feature maps.
+
+![](images/48453c521d73985079a9080b5b287ceadaf6a72242a2d461cc5cbe8fc1db9897.jpg)
+
+Figure 3: Feature distance maps defined over raw RGB values, pretrained CNN features $C ^ { 3 }$ , or our learned features $F ^ { 3 }$ . Our features produce smoother objective function to facilitate optimization.
+
+![](images/a458f2275f9f354daff70a3dc2b34e82b0d61aeb9488bd43a7479be3730985de.jpg)  
+Figure 4: A single iteration of the differentiable LM.
+
+• We then compute the Jacobian matrix $J ( \mathcal { X } )$ , the Hessian matrix $J ( \mathcal { X } ) ^ { \top } J ( \mathcal { X } )$ and its diagonal matrix $D ( \mathcal { X } )$ ;
+
+• To predict the damping factor $\lambda$ , we use global average pooling to aggregate the aboslute value of $E ( \mathcal { X } )$ over all pixels for each feature channel, and get a 128D feature vector. We then send it to a MLP sub-network to predict $\lambda$ ;
+
+• Finally, the update $\Delta \mathcal { X }$ to the current solution is computed as a standard LM step:
+
+$$
+\Delta \mathcal { X } = ( J ( \mathcal { X } ) ^ { \top } J ( \mathcal { X } ) + \lambda D ( \mathcal { X } ) ) ^ { - 1 } J ( \mathcal { X } ) ^ { \top } E ( \mathcal { X } ) .
+$$
+
+In this way, we can consider $\lambda$ as an intermediate variable and denote each LM step as a function $g$ about features pyramids $\mathbb { F }$ and the solution $\mathcal { X }$ from the previous iteration. In other words, $\Delta \mathcal { X } =$ $g ( \mathcal { X } ; \mathbb { F } )$ . Therefore, the solution after the $k$ -th iteration is:
+
+$$
+\begin{array} { r } { \mathcal { X } _ { k } = g ( \mathcal { X } _ { k - 1 } ; \mathbb { F } ) \circ \mathcal { X } _ { k - 1 } . } \end{array}
+$$
+
+Here, $\circ$ denotes parameters updating, which is addition for depth and SE(3) exponential mapping for camera poses. Equation (6) is differentiable with respect to the feature pyramids $\mathbb { F }$ , which makes back-propagation possible through the whole pipeline for feature learning. The MLP that predicts $\lambda$ is also shown in Figure 4. We stack four fully-connected layers to predict $\lambda$ from the input 128D vector. We use ReLU as the activation function to guarantee $\lambda$ is non-negative. Following the photometric BA (Engel et al., 2014; 2018), we solve our feature-metric BA using a coarse-to-fine strategy with feature map warping at each iteration. We apply the differentiable LM algorithm for 5 iterations at each pyramid level, leading to 15 iterations in total. All the camera poses are initialized with identity rotation and zero translation, and the initialization of depth map will be introduced in Section 4.4.
+
+# 4.4 BASIS DEPTH MAPS GENERATION
+
+Parameterizing a dense depth map by a per-pixel depth value is impractical under our formulation. Firstly, it introduces too many parameters for optimization. For example, an image of $3 2 0 \times 2 4 0$ pixels results in $7 6 . 8 \mathrm { k }$ parameters. Secondly, in the beginning of training, many pixels will become invisible in the other views because of the poorly predicted depth or motion. So little information can be back-propagated to improve the network, which makes training difficult.
+
+To deal with these problems, we use the convolutional network for monocular image depth estimation as a compact parameterization, rather than using it as an initialization as in Tateno et al. (2017) and Yang et al. (2018). We use a standard encoder-decoder architecture for monocular depth learning as in Laina et al. (2016). We use DRN-54 as the encoder to share the same backbone features with our feature pyramids. For the decoder, we modify the last convolutional feature maps of Laina et al. (2016) to 128 channels and use these feature maps as the basis depth maps for optimization. The final depth map is generated as the linear combination of these basis depth maps, which is:
+
+$$
+\mathbb { D } = \operatorname { R e L U } ( \pmb { w } ^ { \top } \pmb { B } ) .
+$$
+
+Here, $\mathbb { D }$ is the $h \cdot w$ depth map that contains depth values for all pixels, $\textbf {  { B } }$ is a $1 2 8 \times h \cdot w$ matrix, representing 128 basis depth maps generated from network, $\pmb { w }$ is the linear combination weights of these basis depth maps. The $\pmb { w }$ will be optimized in our BA-Layer. The ReLU activation function guarantees the final depth is non-negative. Once $\textbf {  { B } }$ is generated from the network, we fix $\textbf {  { B } }$ and use $\pmb { w }$ as a compact depth parameterization in BA optimization, and the feature-metric distance becomes:
+
+$$
+\begin{array} { r } { e _ { i , j } ^ { f } ( \mathcal { X } ) = F _ { i } ( \pi ( \pmb { T } _ { i } , \mathrm { R e L U } ( \pmb { w } ^ { \top } \pmb { B } [ j ] ) \cdot \pmb { q } _ { j } ) ) - F _ { 1 } ( \pmb { q } _ { j } ) , } \end{array}
+$$
+
+where $B [ j ]$ is the $j$ -th column of $\textbf {  { B } }$ , and $\mathrm { R e L U } ( { \pmb w } ^ { \top } { \pmb B } [ j ] )$ is the corresponding depth of $\pmb q _ { j }$ . To further speedup convergence, we learn the initial weight $\pmb { w } _ { 0 }$ as a 1D convolution filter for an arbitrary image, i.e. $\begin{array} { r } { \mathbb { D } _ { 0 } = \mathrm { R e L } \check { \mathrm { U } } ( { \pmb w } _ { 0 } ^ { \top } B ) } \end{array}$ . The $\textbf {  { B } }$ of various images are visualized in the appendix.
+
+# 4.5 TRAINING
+
+The BA-Net learns the feature pyramid, the damping factor predictor, and the basis depth maps generator in a supervised manner. We apply the following commonly used loss for training, though more sophisticated ones might be designed.
+
+Camera Pose Loss The camera rotation loss is the distance between rotation quaternion vectors $\mathcal { L } _ { r o t a t i o n } = \| \pmb { q } - \pmb { q } ^ { * } \|$ . Similarly, translation loss is the Euclidean distance between prediction and groundtruth in metric scale, $\mathcal { L } _ { t r a n s l a t i o n } = \| \pmb { t - t ^ { * } } \|$ .
+
+Depth Map Loss For each dense depth map we applies the berHu Loss (Zwald & Lambert-Lacroix, 2012) as in Laina et al. (2016).
+
+We initialize the back-bone network from DRN-54 (Yu et al., 2017), and the other components are trained with ADAM (Kingma & Ba, 2015) from scratch with initial learning rate 0.001, and the learning rate is divided by two when we observe plateaus from the Tensorboard interface.
+
+# 5 EVALUATION
+
+# 5.1 DATASET
+
+ScanNet ScanNet (Dai et al., 2017a) is a large-scale indoor dataset with 1,513 sequences in 706 different scenes. Camera poses and depth maps are not perfect, because they are estimated via BundleFusion (Dai et al., 2017b). The metric scale is known in all data from ScanNet, because the data are recorded with a depth camera which returns absolute depth values.
+
+To sample image pairs for training, we apply a simple filtering process. We first filter out pairs with a large photo-consistency error, to avoid image pairs with large pose or depth error. We also filter out image pairs, if less than $50 \%$ of the pixels from one image are visible in the other image. In addition, we also discard a pair if their roundness score (Beder & Steffen, 2006) is less than 0.001, which avoids pairs with too narrow baselines.
+
+We split the whole dataset into the training and the testing sets. The training set contains the first 1,413 sequences and the testing set contains the rest 100 sequences. We sample 547,991 training pairs and 2,000 testing pairs from the training and testing sequences respectively.
+
+KITTI KITTI (Geiger et al., 2012) is a widely used benchmark dataset collected by car-mounted cameras and a LIDAR sensor on streets. It contains 61 scenes belonging to the "city", "residential", or "road" categories. Eigen et al. (2014) select 28 scenes for testing and 28 scenes from the remaining for training. We use the same data split, to make a fair comparison with previous methods. Since ground truth pose is unavailable from the raw KITTI dataset, we compute camera poses by LibVISO2 (Geiger et al., 2011) and take them as ground truth after discarding poses with large errors.
+
+# 5.2 COMPARISONS WITH OTHER METHODS
+
+ScanNet To evaluate the results’ quality, we use the depth error metrics suggested in Eigen & Fergus (2015), where RMSE (linear, log, and log, scale inv.) measure the RMSE of the raw, the logarithmical, and aligned logarithmical depth values, while the other two metrics measure the mean of the ratios that divide the absolute and square error by groundtruth depth.. The errors in camera poses are measured by the rotation error (the angle between the ground truth and the estimated camera rotations), the translation direction error (the angle between the ground truth and estimated camera translation directions) and the absolute position error (the distance between the ground truth and the estimated camera translation vectors).
+
+<table><tr><td></td><td>Ours</td><td>Ours*</td><td>DeMoN*</td><td>Photometric BA</td><td>GeometricBA</td></tr><tr><td rowspan="3">Rotation (degree) Translation (cm) Translation (degree)</td><td>1.018</td><td>1.587</td><td>3.791</td><td>4.409</td><td>8.56</td></tr><tr><td>3.39</td><td>10.81</td><td>15.5</td><td>21.40</td><td>36.995</td></tr><tr><td>20.577</td><td>31.005</td><td>31.626</td><td>34.36</td><td>39.392</td></tr><tr><td rowspan="3">absrelative difference sqr relative difference</td><td>0.161</td><td>0.238</td><td>0.231</td><td>0.268</td><td>0.382</td></tr><tr><td>0.092</td><td>0.176</td><td>0.520</td><td>0.427</td><td>1.163</td></tr><tr><td>0.346</td><td>0.488</td><td>0.761</td><td>0.788</td><td>0.876</td></tr><tr><td>RMSE (linear) RMSE (log)</td><td>0.214</td><td>0.279</td><td>0.289</td><td>0.330</td><td>0.366</td></tr><tr><td>RMSE (log, scale inv.)</td><td>0.184</td><td>0.276</td><td>0.284</td><td>0.323</td><td>0.357</td></tr></table>
+
+Table 1: Quantitative comparisons with DeMoN and classic BA. The superindex ∗ denotes that the model is trained on the trainning set described in Ummenhofer et al. (2017).
+
+In Table 1, we compare our method with DeMoN (Ummenhofer et al., 2017) and the conventional photometric and geometric BA. Note that we cannot get DeMoN trained on the ScanNet. For fair comparison, we train our network on the same training data as DeMoN and test both networks on our testing data1. We also show the results of our network trained on ScanNet. Our BA-Net consistently performs better than DeMoN no matter which training data is used. Since DeMoN does not recover the absolute scale, we align its depth map with the groundtruth to recover its metric scale for evaluation. We further compare with conventional geometric (Nister, 2004; Agarwal et al.) and photometric (Engel et al., 2014) BA. Again, our method produces better results. The geometric BA works poorly here, because feature matching is difficult in indoor scenes. Even the RANSAC process cannot get rid of all outliers. While for photometirc BA, the highly non-convex objective function is difficult to optimize as described in Section 3.
+
+KITTI We use the same metrics as the comparisons on ScanNet for depth evaluation. To evaluate the camera poses, we follow (Zhou et al., 2017; Wang et al., 2018) to use the Absolute Trajectory Error (ATE), which measures the Euclidean differences between two trajectories (Steinbruecker et al., 2011), on the 9th and 10th sequences from the KITTI odometry data. In this experiment, we create short sequences of 5 frames by first computing 5 two-view reconstructions from our BA-Net and then align the two-view reconstructions in the coordinate system anchored at the first frame. minimize the photometric error.
+
+<table><tr><td></td><td>Ours</td><td></td><td></td><td>Wang et al. (2018) Zhou et al. (2017) Godard et al. (2017)</td><td>Eigen et al. (2014)</td></tr><tr><td>ATE(km)</td><td>0.019</td><td>0.045</td><td>0.021</td><td>N/A</td><td>N/A</td></tr><tr><td>absrel</td><td>0.083</td><td>0.151</td><td>0.208</td><td>0.148</td><td>0.203</td></tr><tr><td>sqr rel</td><td>0.025</td><td>1.257</td><td>1.768</td><td>1.344</td><td>1.548</td></tr><tr><td>RMSE(linear)</td><td>3.640</td><td>5.583</td><td>6.856</td><td>5.927</td><td>6.307</td></tr><tr><td>RMSE(log)</td><td>0.134</td><td>0.228</td><td>0.283</td><td>0.247</td><td>0.282</td></tr></table>
+
+Table 2: Quantitative comparisons on KITTI with supervised (Eigen et al., 2014) and unsupervised (Wang et al., 2018; Zhou et al., 2017; Godard et al., 2017) methods.
+
+Table 2 summarizes our results on KITTI. Our method outperforms the supervised methods (Eigen et al., 2014) as well as recent unsupervised methods (Zhou et al., 2017; Wang et al., 2018; Godard et al., 2017). Our method also achieves more accurate camera trajectories than Zhou et al. (2017) and Wang et al. (2018). We believe this is due to our feature-metric BA with features learned specifically for SfM problem, which makes the objective function closer to convex and easier to optimize as discussed in Section 4.2. In comparison, Zhou et al. (2017) and Wang et al. (2018) minimize the photometric error.
+
+More comparison with DeMoN, ablation studies, and multi-view SfM (up to 5 views) are reported in the appendix due to page limit.
+
+# 6 CONCLUSIONS AND FUTURE WORKS
+
+This paper presents the BA-Net, a network that explicitly enforces multi-view geometry constraints in terms of feature-metric error. It optimizes scene depths and camera motion jointly via feature-metric bundle adjustment. The whole pipeline is differentiable and thus end-to-end trainable, such that the features are learned from data to facilitate structure-from-motion. The dense depth is parameterized as a linear combination of several basis depth maps generated from the network. Our BA-Net nicely combines domain knowledge (hard-coded multi-view geometry constraint) with deep learning (learned feature representation and basis depth maps generator). It outperforms conventional BA and recent deep learning based methods.
+
+Acknowledgement This work is supported by the NSERC discovery grant 611664 and a project funding from Alibaba.
+
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+Ashutosh Saxena, Sung H. Chung, and Andrew Y. Ng. Learning depth from single monocular images. In International Conference on Neural Information Processing Systems (NIPS), pp. 1161–1168, 2005.   
+U. Schmidt and S. Roth. Shrinkage fields for effective image restoration. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 2774–2781, 2014.   
+Johannes Lutz Schönberger and Jan-Michael Frahm. Structure-from-motion revisited. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 4104–4113, 2016.   
+F. Steinbruecker, J. Sturm, and D. Cremers. Real-time visual odometry from dense rgb-d images. In International Conference on Computer Vision Workshop on Live Dense Reconstruction with Moving Cameras(ICCVW), 2011.   
+C. Tang, O. Wang, and P. Tan. Gslam: Initialization-robust monocular visual slam via global structure-from-motion. In International Conference on 3D Vision (3DV), pp. 239–248, 2017.   
+K. Tateno, F. Tombari, I. Laina, and N. Navab. Cnn-slam: Real-time dense monocular slam with learned depth prediction. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 6565–6574, 2017.   
+Bill Triggs, Philip F. McLauchlan, Richard I. Hartley, and Andrew W. Fitzgibbon. Bundle adjustment - a modern synthesis. In Vision Algorithms: Theory and Practice, pp. 298–372, 2000.   
+Benjamin Ummenhofer, Huizhong Zhou, Jonas Uhrig, Nikolaus Mayer, Eddy Ilg, Alexey Dosovitskiy, and Thomas Brox. Demon: Depth and motion network for learning monocular stereo. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 5622–5631, 2017.   
+Chaoyang Wang, Buenaposada, Miguel Jose, Rui Zhu, , and Simon Lucey. Learning depth from monocular videos using direct methods. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 851–859, 2018.   
+C. Wu, S. Agarwal, B. Curless, and S. M. Seitz. Multicore bundle adjustment. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 3057–3064, 2011.   
+D. Xu, E. Ricci, W. Ouyang, X. Wang, and N. Sebe. Multi-scale continuous crfs as sequential deep networks for monocular depth estimation. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 161–169, 2017.   
+Nan Yang, Rui Wang, Jorg Stuckler, and Daniel Cremers. Deep virtual stereo odometry: Leveraging deep depth prediction for monocular direct sparse odometry. In European Conference on Computer Vision (ECCV), 2018.   
+F. Yu, V. Koltun, and T. Funkhouser. Dilated residual networks. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 636–644, 2017.   
+Tinghui Zhou, Matthew Brown, Noah Snavely, and David G. Lowe. Unsupervised learning of depth and ego-motion from video. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 6612–6619, 2017.   
+L. Zwald and S. Lambert-Lacroix. The berhu penalty and the grouped effect. CoRR, abs/1207.6868, 2012.
+
+![](images/13df0af9a2a1f698d62a684f351663ff999163586b209241ce52dcdd498bd25f.jpg)  
+Figure 5: Network details for the (a) the DRN-54 backbone and (b) the basis depth generator.
+
+Network Architecture Details Figure 5 illustrates the detailed network architectures for the backbone DRN-54 and the depth basis generator. The architecture of the feature pyramid has been provided in Figure 2(a). We modify the dilated convolution of the original DRN-54 to convolution with strides and discard the conv7 and conv8 as shown in Figure 5(a). $C ^ { 1 }$ to $C ^ { 6 }$ are layers with {1,2,4,8,16,32} strides and {16,32,256,512,1024,2048} channels, where $C ^ { 1 }$ and $C ^ { 2 }$ are basic convolution layers, while $C ^ { 3 }$ to $C ^ { 6 }$ are standerd bottleneck blocks as in ResNet (He et al., 2016).
+
+Figure 5(b) visualizes our depth basis generator which adopts the up-projection structure proposed in Laina et al. (2016). The depth basis generator is a stander decoder that takes the output of $\dot { C } ^ { 6 }$ as input and stacks five up-projection blocks to generate 128 basis depth maps, and each of the basis depth maps is half the resolution of the input image. The up-projection block is shown on the right of Figure 5(b) which upsample the input by $2 \times$ and then apply convolutions with projection connection.
+
+Evaluation Time To evaluate the running time of our method, we use the Tensorflow profiler tool to retrieve the time in ms for all network nodes and then summarize the results corresponding to each component in our pipeline. As shown in Table 3, our method takes $9 5 . 2 1 \ \mathrm { m s }$ to reconstruct two $3 2 0 \times 2 4 0$ images, which is slightly faster than DeMoN that takes $1 1 0 \mathrm { m s }$ for two $2 5 6 \times 1 9 2$ images.
+
+The current computation bottleneck is the BA-Layer which contains a large amount of matrix operations and can be further speeded up by direct CUDA implementation. Since we explicitly hard-code the multi-view geometry constraints in the BA-Layer, it is possible to share the backbone DRN-54 with other high-level vision tasks, such as semantic segmentation and object detection, to maximize reuse of network structures and minimize extra computation cost.
+
+<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Backbone(DRN-54)</td><td rowspan=1 colspan=1>FeaturePyramid</td><td rowspan=1 colspan=1>Basis DepthGenerator</td><td rowspan=1 colspan=1>BA-LayerOptimization</td><td rowspan=1 colspan=1>Total</td></tr><tr><td rowspan=1 colspan=1>Time (ms)</td><td rowspan=1 colspan=1>15.04</td><td rowspan=1 colspan=1>5.87</td><td rowspan=1 colspan=1>9.81</td><td rowspan=1 colspan=1>67.22</td><td rowspan=1 colspan=1>95.21</td></tr></table>
+
+Table 3: Evaluation time for each component, which is summarized using Tensorflow profiler.
+
+# APPENDIX B: ABLATION STUDIES
+
+Learned Features vs Pre-trained Features Our learned feature pyramid improves the convexity of the objective function to facilitate the optimization. We compare our learned features with features
+
+Table 4: Ablation Study Comparisons by Disabling Different Components of BA-Net   
+
+<table><tr><td></td><td>Ours (Full)</td><td>w/o Feature Learning</td><td>w/o Joint Optimization</td><td>w/o入</td></tr><tr><td>Rotation (degree)</td><td>1.018</td><td>2.667</td><td>1.036</td><td>7.202</td></tr><tr><td>Translation (cm)</td><td>3.39</td><td>10.8</td><td>3.91</td><td>22.38</td></tr><tr><td>Translation (degree)</td><td>20.577</td><td>31.493</td><td>26.779</td><td>59.81</td></tr><tr><td>absrelative difference</td><td>0.161</td><td>0.267</td><td>0.217</td><td>0.630</td></tr><tr><td>sqr relative difference</td><td>0.092</td><td>0.242</td><td>0.145</td><td>0.549</td></tr><tr><td>RMSE (linear)</td><td>0.346</td><td>0.481</td><td>0.428</td><td>0.763</td></tr><tr><td>RMSE (log)</td><td>0.214</td><td>0.303</td><td>0.270</td><td>0.513</td></tr><tr><td>RMSE (log, scale inv.)</td><td>0.184</td><td>0.226</td><td>0.205</td><td>0.437</td></tr></table>
+
+pre-trained on ImageNet for classification tasks. As shown in Table 4, the pre-trained features (i.e.   
+w/o Feature Learning) produce larger error. This proves the discussion in Section 4.2.
+
+Bundle Adjustment Optimization vs SE(3) Pose Estimation Our BA-Layer optimizes depth and camera poses jointly. We compare it to the SE(3) camera pose estimation with fixed depth map (e.g. the initialized depth $\mathbb { D } _ { 0 }$ in Section 4.4), and similar strategy is adopted in Wang et al. (2018). To make a fair comparison, we also use our learned feature pyramids for the SE(3) camera pose estimation. As shown in Table 4, without BA optimization (i.e. w/o Joint Optimization), both the depth maps and camera poses are worse, because the errors in the depth estimation will degrades the camera pose estimation.
+
+Differentiable Levenberg-Marquardt vs Gauss-Newton To make the whole pipeline end-to-end trainable, we makes the Levenberg-Marquardt algorithm differentiable by learning the damping factor from the network. We first compare our method against vanilla Gauss-Newton without damping factor $\lambda$ (i.e. $\lambda = 0$ ). Since the objective function of feature-metric BA is non-convex, the Hessian matrix $J ( \mathcal { X } ) ^ { \top } J ( \mathcal { X } )$ might not be positive definite, which makes the matrix inversion by Cholesky decomposition fail.
+
+To deal with this problem, we use QR decomposition instead for training with Gauss-Newton. As shown in Table 4, the Gauss-Newton algorithm (i.e. w/o $\lambda$ ) generates much larger error, because the BA optimization is non-convex and the Gauss-Newton algorithm has no guaranteed convergence unless the initial solution is sufficiently close to the optimal (Nocedal & Wright, 2006). This comparison reveals that, similar to conventional BA, our differnetiable Levenberg-Marquardt algorithm is superior than the Gauss-Newton algorithm for feature-metric BA.
+
+![](images/9115916d7d58f5641ac10d6fc46f29dae3da88fc63917d5a8cc31bf6d3c24ee9.jpg)  
+Figure 6: The camera pose and the depth errors correspond to different constant $\lambda$ values.
+
+Predicted vs Constant $\lambda$ Another way to make the Levenberg-Marquardt algorithm differentiable is to fix the $\lambda$ during the iterations. We compare with this strategy. As shown in Figure 6(a), increasing $\lambda$ makes the both rotation and translation error decreases, until $\lambda = 0 . 5$ , and then increases. The reason is that a small $\lambda$ makes the algorithm close to the Gauss-Newton algorithm, which has convergence issues. A large $\lambda$ leads to a small update at each iteration, which makes it difficult to reach a good solution within limited iterations.
+
+While in Figure 6(b), increasing $\lambda$ always makes depth errors decrease, probably because a larger $\lambda$ leads to a small update and makes the final depth close to the initialed depth, which is better than the optimized one with small constant $\lambda$ .
+
+Using constant $\lambda$ value consistently generates worse results than using a predicted $\lambda$ from the MLP network, because there is no optimal $\lambda$ for all data and it should be adapted to different data and different iterations. We draw the errors of our method in Figure 6(a) and Figure 6(b) as the flat dash lines for a reference.
+
+# APPENDIX C: EVALUATION ON DEMON DATASET
+
+Table 5 summarizes our results on the DeMoN dataset. For a comparison, we also cite the results from DeMoN (Ummenhofer et al., 2017) and the most recent work LS-Net (Clark et al., 2018). We further cite the results from some conventional approaches as reported in DeMoN, indicated as Oracle, SIFT, FF, and Matlab respectively. Here, Oracle uses ground truth camera poses to solve the multi-view stereo by SGM (Hirschmuller, 2005), while SIFT, FF, and Matlab further use sparse features, optical flow, and KLT tracking respectively for feature correspondence to solve camera poses by the 8-pt algorithm (Hartley, 1997).
+
+Table 5: Quantitative comparisons on the DeMoN dataset.   
+
+<table><tr><td></td><td></td><td></td><td>Depth</td><td></td><td>Motion</td><td></td><td></td><td></td><td></td><td>Depth</td><td></td><td>Motion</td><td></td></tr><tr><td></td><td>Method</td><td>L1-inv</td><td>sc-inv</td><td>L1-rel</td><td>Rotation</td><td>Translation</td><td></td><td>Method</td><td>L1-inv</td><td>sc-inv</td><td>L1-rel</td><td>Rotation</td><td>Translation</td></tr><tr><td rowspan="8"></td><td>Oracle</td><td>0.019</td><td>0.197</td><td>0.105</td><td>0</td><td>0</td><td></td><td>Oracle</td><td>0.023</td><td>0.618</td><td>0.349</td><td>0</td><td>0</td></tr><tr><td>SIFT</td><td>0.056</td><td>0.309</td><td>0.361</td><td>21.180</td><td>60.516</td><td></td><td>SIFT</td><td>0.051</td><td>0.900</td><td>1.027</td><td>6.179</td><td>56.650</td></tr><tr><td>FF</td><td>0.055</td><td>0.308</td><td>0.322</td><td>4.834</td><td>17.252</td><td></td><td>FF</td><td>0.038</td><td>0.793</td><td>0.776</td><td>1.309</td><td>19.425</td></tr><tr><td>Matlab</td><td></td><td></td><td></td><td>10.843</td><td>32.736</td><td>Sseeeesr</td><td>Matlab</td><td></td><td></td><td></td><td>0.917</td><td>14.639</td></tr><tr><td>DeMoN</td><td>0.047</td><td>0.202</td><td>0.305</td><td>5.156</td><td>14.447</td><td></td><td>DeMoN</td><td>0.019</td><td>0.315</td><td>0.248</td><td>0.809</td><td>8.918</td></tr><tr><td>LS-Net</td><td>0.051</td><td>0.221</td><td>0.311</td><td>4.653</td><td>11.221</td><td></td><td>LS-Net</td><td>0.010</td><td>0.410</td><td>0.210</td><td>0.910</td><td>8.21</td></tr><tr><td>Ours</td><td>0.03</td><td>0.15</td><td>0.08</td><td>3.499</td><td>11.238</td><td></td><td>Ours</td><td>0.08</td><td>0.21</td><td>0.13</td><td>1.298</td><td>10.37</td></tr><tr><td colspan="2"></td><td colspan="2">Depth</td><td colspan="2">Motion</td><td></td><td></td><td></td><td colspan="2">Depth</td><td colspan="2">Motion</td></tr><tr><td></td><td>Method</td><td>L1-inv</td><td>sc-inv</td><td>L1-rel</td><td>Rotation</td><td>Translation</td><td></td><td>Method</td><td>L1-inv</td><td>sc-inv</td><td>L1-rel</td><td>Rotation</td><td>Translation</td></tr><tr><td rowspan="8">RRRPR</td><td>Oracle</td><td>0.026</td><td>0.398</td><td>0.336</td><td>0</td><td>0</td><td></td><td>Oracle</td><td>0.020</td><td>0.241</td><td>0.220</td><td>0</td><td>0</td></tr><tr><td>SIFT</td><td>0.050</td><td>0.577</td><td>0.703</td><td>12.010</td><td>56.021</td><td></td><td>SIFT</td><td>0.029</td><td>0.290</td><td>0.286</td><td>7.702</td><td>41.825</td></tr><tr><td>FF</td><td>0.045</td><td>0.548</td><td>0.613</td><td>4.709</td><td>46.058</td><td>gsun2</td><td>FF</td><td>0.029</td><td>0.284</td><td>0.297</td><td>3.681</td><td>33.301</td></tr><tr><td>Matlab</td><td></td><td></td><td></td><td>12.831</td><td>49.612</td><td></td><td>Matlab</td><td></td><td></td><td></td><td>5.920</td><td>32.298</td></tr><tr><td>DeMoN</td><td>0.028</td><td>0.130</td><td>0.212</td><td>2.641</td><td>20.585</td><td></td><td>DeMoN</td><td>0.019</td><td>0.114</td><td>0.172</td><td>1.801</td><td>18.811</td></tr><tr><td>LS-Net</td><td>0.019</td><td>0.09</td><td>0.301</td><td>1.01</td><td>22.1</td><td></td><td>LS-Net</td><td>0.015</td><td>0.189</td><td>0.650</td><td>1.521</td><td>14.347</td></tr><tr><td>Ours</td><td>0.008</td><td>0.087</td><td>0.05</td><td>2.459</td><td>14.90</td><td></td><td>Ours</td><td>0.015</td><td>0.11</td><td>0.06</td><td>1.729</td><td>13.26</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+Our method consistently outperforms DeMoN (Ummenhofer et al., 2017) at both camera motion and scene depth, except on the ‘Scenes11’ data, because we enforce multi-view geometry constraint in the BA-Layer. Our results are poorer on the ‘Scene11’ dataset, because the images there are synthesized with random objects from the ShapeNet (Chang et al., 2015) without physically correct scale. This setting is inconsistent with real data and makes it harder for our method to learn the basis depth map generator.
+
+When compared with LS-Net Clark et al. (2018), our method achieves similar accuracy on camera poses but better scene depth. It proves our feature-metric BA with learned feature is superior than the photometric BA in the LS-Net.
+
+# APPENDIX D: MULTI-VIEW STRUCTURE-FROM-MOTION
+
+Our method can be easily extended to reconstruct multiple images. We evaluate our method in the multi-view setting on the ScanNet (Dai et al., 2017a) dataset. To sample multi-view images for training, we randomly select two-view image pairs that shares a common image to construct $N$ -view sequences. Due to the limited GPU memory (12G), we limit $N$ to 5.
+
+As shown in the Table 6, the accuracy is consistently improved when more views are included, which demonstrates the strength of the multi-view geometry constraints. Instead, most existing deep learning approaches can only handle two views at a time, which is sub-optimal as known in structure-from-motion literature.
+
+<table><tr><td></td><td>Ours(2-views)</td><td>Ours(3-views)</td><td>Ours(5-views)</td></tr><tr><td rowspan="3">Rotation (degree) Translation (cm) Translation (degree)</td><td>1.018</td><td>1.013</td><td>1.009</td></tr><tr><td>3.391</td><td>2.852</td><td>2.365</td></tr><tr><td>20.577</td><td>16.423</td><td>14.626</td></tr><tr><td rowspan="4">absrelative difference sqr relative difference RMSE (linear) RMSE (log)</td><td>0.161</td><td>0.111</td><td>0.091</td></tr><tr><td>0.092</td><td>0.087</td><td>0.068</td></tr><tr><td>0.346</td><td>0.288</td><td>0.223</td></tr><tr><td>0.214</td><td>0.179</td><td>0.147</td></tr><tr><td>RMSE (log, scale inv.)</td><td>0.184</td><td>0.168</td><td>0.137</td></tr></table>
+
+Table 6: Quantitative comparisons on multi-view reconstruction on ScanNet.
+
+# APPENDIX E: QUANTITATIVE COMPARISONS WITH CODESLAM
+
+We compare our method with CodeSLAM (Bloesch et al., 2018) which adopts similar idea for depth parameterization. But the difference is that CodeSLAM learns the conditioned depth auto-encoder separately and uses the depth codes in a standalone photometric BA component, while our method learns the feature pyramid and basis depth maps generator through feature-metric BA end-to-end. Since there is no public code for CodeSLAM, we directly cite the results from their paper.2 To get the trajectory on the EuroC MH02 sequence of our method, we select one frame every four frames and concatenate the reconstructed groups that contains every five selected frames. Then we use the same evaluation metrics as in CodeSLAM, which measures the translation errors correspond to different traveled distances.
+
+![](images/da802cb4214926197154a330f9eeb5f1719fd306ae5857fe5460701cf562f196.jpg)  
+Figure 7: Quantitative Comparisons with CodeSLAM (Bloesch et al., 2018) on EuroC MH02. The 0.50.5error bars represent the maximum and the minimum errors. The orange and the blue boxes represent the median errors for CodeSLAM and our method.
+
+As shown in Figure 7, our method outperforms CodeSLAM. Our median error is less than the half of CodeSLAM’s error, i.e. CodeSLAM exhibits an error of roughly $1 \textrm { m }$ for a traveled distance of $9 \mathrm { m }$ , while our method’s error is about $0 . 4 \mathrm { m }$ . This comparison demonstrates the superiority of end-to-end learning with feature pyramid and feature-metric BA over learning depth parameterization only.
+
+# APPENDIX F: VISUALIZATION OF BASIS DEPTH MAPS
+
+In Figure 8, we visualize four typical basis depth maps as heat maps for each of the four images. An interesting observation is that one basis depth map has higher responses on close objects while another oppositely has higher responses to the far background. Some other basis depth maps have smoothly varying responses and correspond to the layouts of scenes. This observation reveals that our learned basis depth maps have captured the latent structures of scenes.
+
+![](images/0ca718d2adfbdb13f4a7196bbbedb0fd4b939b2adfb383d14592868a15728473.jpg)  
+Figure 8: Visualization of different basis depth maps.
+
+# APPENDIX G: QUALITATIVE COMPARISONS WITH OTHER METHODS
+
+Finally, we show some qualitative comparison with the previous methods. Figure 9 shows the recovered depth map by our method and DeMoN Ummenhofer et al. (2017) on the ScanNet data. As we can see from the regions highlighted with a red circle, our method recovers more shape details. This is consistent with the quantitative results in Table 1. Figure 11 shows the recovered depth maps by our method, Wang et al. (2018), and Godard et al. (2017) respectively. Similarly, we observe more shape details in our results, as reflected in the quantitative results in Table 2.
+
+![](images/b780d371c7ded7238a9b56c8ef9685c37eb1bb3b0a22bb9f09302f2916aea4cd.jpg)  
+Figure 9: Qualitative Comparisons with DeMoN (Ummenhofer et al., 2017) on ScanNet.
+
+![](images/bd53ca282836205f3f5e4d0394e1f831c885a8970627f8da70ae49c269da1f6b.jpg)  
+Figure 10: Qualitative Comparisons with DeMoN (Ummenhofer et al., 2017) on its dataset.
+
+![](images/4ea8a34be6470d158b277e4510a8d80e1b0a71f53c537acb646ba983c9ba2d9e.jpg)  
+Figure 11: Qualitative Comparisons with Wang et al. (2018) and Godard et al. (2017).

parse/train/B1gabhRcYX/B1gabhRcYX_content_list.json

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+[
+    {
+        "type": "text",
+        "text": "BA-NET: DENSE BUNDLE ADJUSTMENT NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            169,
+            99,
+            808,
+            121
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Chengzhou Tang School of Computer Science Simon Fraser University chengzhou_tang@sfu.ca ",
+        "bbox": [
+            184,
+            145,
+            390,
+            200
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Ping Tan School of Computer Science Simon Fraser University pingtan@sfu.ca ",
+        "bbox": [
+            516,
+            145,
+            705,
+            200
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            238,
+            544,
+            252
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "This paper introduces a network architecture to solve the structure-from-motion (SfM) problem via feature-metric bundle adjustment (BA), which explicitly enforces multi-view geometry constraints in the form of feature-metric error. The whole pipeline is differentiable, so that the network can learn suitable features that make the BA problem more tractable. Furthermore, this work introduces a novel depth parameterization to recover dense per-pixel depth. The network first generates several basis depth maps according to the input image, and optimizes the final depth as a linear combination of these basis depth maps via feature-metric BA. The basis depth maps generator is also learned via end-to-end training. The whole system nicely combines domain knowledge (i.e. hard-coded multi-view geometry constraints) and deep learning (i.e. feature learning and basis depth maps learning) to address the challenging dense SfM problem. Experiments on large scale real data prove the success of the proposed method. ",
+        "bbox": [
+            233,
+            268,
+            766,
+            449
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            474,
+            334,
+            491
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The Structure-from-Motion (SfM) problem has been extensively studied in the past a few decades. Almost all conventional SfM algorithms (Agarwal et al., 2011; Wu et al., 2011; Schönberger & Frahm, 2016; Engel et al., 2018; Delaunoy & Pollefeys, 2014) jointly optimize scene structures and camera motion via the Bundle-Adjustment (BA) algorithm (Triggs et al., 2000; Agarwal et al., 2010), which minimizes the geometric (Agarwal et al., 2011; Wu et al., 2011; Schönberger & Frahm, 2016) or photometric (Engel et al., 2014; 2018; Delaunoy & Pollefeys, 2014) error through the Levenberg-Marquardt (LM) algorithm (Nocedal & Wright, 2006). Some recent works (Ummenhofer et al., 2017; Zhou et al., 2017; Wang et al., 2018) attempt to solve SfM using deep learning techniques, but most of them do not enforce the geometric constraints between 3D structures and camera motion in their networks. For example, in the recent work DeMoN (Ummenhofer et al., 2017), the scene depths and the camera motion are estimated by two individual sub-network branches. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "This paper formulates BA as a differentiable layer, the BA-Layer, to bridge the gap between classic methods and recent deep learning based approaches. To this end, we learn a feed-forward multilayer perceptron (MLP) to predict the damping factor in the LM algorithm, which makes all involved computation differentiable. Furthermore, unlike conventional BA that minimizes geometric or photometric error, our BA-layer minimizes the distance between aligned CNN feature maps. Our novel feature-metric BA takes CNN features of multiple images as inputs and optimizes for the scene structures and camera motion. This feature-metric BA is desirable, because it has been observed by Engel et al. (2014; 2018) that the geometric BA does not exploit all image information, while the photometric BA is sensitive to moving objects, exposure or white balance changes, etc. Most importantly, our BA-Layer can back-propagate loss from scene structures and camera motion to learn appropriate features that are most suitable for structure-from-motion and bundle adjustment. In this way, our network hard-codes the multi-view geometry constraints in the BA-Layer and learns suitable feature representations from training data. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We strive to estimate a dense per-pixel depth, because dense depth is critical for many tasks such as object detection and robot navigation. A major challenge in solving dense per-pixel depth is to find a compact parameterization. Direct per-pixel depth is computational expensive, which makes the network training intractable. So we train a network to generate a set of basis depth maps for an arbitrary input image and represent the result depth map as a linear combination of these basis depth maps. The combination coefficients will be optimized in the BA-Layer together with camera motion. This novel parameterization guarantees a smooth depth map with good consistency with object boundaries. It also reduces the number of unknowns and makes dense BA possible in networks. ",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Similar depth parameterization is introduced in a recent work, CodeSLAM (Bloesch et al., 2018). The major difference is that our method learns the basis depth map generator through the gradients back-propagated from the BA-Layer, while CodeSLAM learns the generator separately and uses its results for a standalone optimization component. Thus, our basis depth map generator has the chance to be better trained for the SfM problem. Furthermore, we use a different network structure to generate basis depth maps. CodeSLAM employs a variational auto-encoder (VAE), while we use a standard encoder-decoder. This design enables us to use the same backbone network for both feature learning and basis depth map learning, making joint training of the whole network possible. ",
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To demonstrate the effectiveness of our method, we evaluate on the ScanNet (Dai et al., 2017a) and KITTI (Geiger et al., 2012) dataset. Our method outperforms DeMoN (Ummenhofer et al., 2017), LS-Net (Clark et al., 2018), as well as several conventional baselines. Due to page limit, we move the ablation studies, evaluation on DeMoN’s dataset, multi-view SfM (up to 5 views), and comparison with CodeSLAM on the EuroC dataset (Burri et al., 2016) to the appendix. ",
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Monocular Depth Estimation Networks Estimating depth from a monocular image is an ill-posed problem because an infinite number of possible scenes may have produced the same image. Before the raise of deep learning based methods, some works predict depth from a single image based on MRF (Saxena et al., 2005; 2009), semantic segmentation (Ladický et al., 2014), or manually designed features (Hoiem et al., 2005). Eigen et al. (2014) propose a multi-scale approach for depth prediction with two CNNs, where a coarse-scale network first predicts the scene depth at the global level and then a fine-scale network will refine the local regions. This approach was extended in Eigen & Fergus (2015) to handle semantic segmentation and surface normal estimation as well. Recently, Laina et al. (2016) propose to use ResNet (He et al., 2016) based structure to predict depth, and Xu et al. (2017) construct multi-scale CRFs for depth prediction. In comparison, we exploit monocular image depth estimation network for depth parameterization, which only produces a set of basis depth maps and the final result will be further improved through optimization. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Structure-from-Motion Networks Recently, some works exploit CNNs to resolve the SfM problem. Handa et al. (2016) solve the camera motion by a network from a pair of images with known depth. Zhou et al. (2017) employ two CNNs for depth and camera motion estimation respectively, where both CNNs are trained jointly by minimizing the photometric loss in an unsupervised manner. Wang et al. (2018) implement the direct method (Steinbruecker et al., 2011) as a differentiable component to compute camera motion after scene depth is estimated by the method in Zhou et al. (2017). In Ummenhofer et al. (2017), the scene depth and the camera motion are predicted from optical flow features, which help to make it generalizing better to unseen data. However, the scene depth and the camera motion are solved by two separate network branches, multi-view geometry constraints between depth and motion are not enforced. Recently, Clark et al. (2018) propose to solve nonlinear least squares in two-view SfM using a LSTM-RNN (Hochreiter et al., 2001) as the optimizer. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our method belongs to this category. Unlike all previous works, we propose the BA-Layer to simultaneously predict the scene depth and the camera motion from CNN features, which explicitly enforces multi-view geometry constraints. The hard-coded multi-view geometry constraints enable our method to reconstruct more than two images, while most deep learning methods can only handle two images. Furthermore, we propose to minimize a feature-metric error instead of the photometric error in (Zhou et al., 2017; Wang et al., 2018; Clark et al., 2018) to enhance robustness. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3 BUNDLE ADJUSTMENT REVISITED ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Before introducing our BA-Net architecture, we revisit the classic BA to have a better understanding about where the difficulties are and why feature-metric BA and feature learning are desirable. We only introduce the most relevant content and refer the readers to Triggs et al. (2000) and Agarwal et al. (2010) for a comprehensive introduction. Given images $\\mathbb { I } = \\{ I _ { i } | i = 1 \\cdot \\cdot \\cdot N _ { i } \\}$ , the geometric ",
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+    },
+    {
+        "type": "text",
+        "text": "BA (Triggs et al., 2000; Agarwal et al., 2010) jointly optimizes camera poses $\\mathbb { T } = \\{ \\pmb { T } _ { i } | i = 1 \\cdots N _ { i } \\}$ and 3D scene point coordinates $\\mathbb { P } = \\{ \\pmb { p } _ { j } | j = 1 \\cdots N _ { j } \\}$ by minimizing the re-projection error: ",
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8576ad06f45d655615ff9e32a7f71150c3f42abde8278c9bb133b200d8dad93a.jpg",
+        "text": "$$\n\\mathcal { X } = \\mathop { \\mathrm { a r g m i n } } \\sum _ { i = 1 } ^ { N _ { i } } \\sum _ { j = 1 } ^ { N _ { j } } \\| e _ { i , j } ^ { g } ( \\mathcal { X } ) \\| ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where the geometric distance ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fbc769eea230f223aad690cb26a22fc0fd964beb1e6af33a27d01e295dd91732.jpg",
+        "text": "$$\ne _ { i , j } ^ { g } ( \\mathcal { X } ) = \\pi ( \\pmb { T } _ { i } , \\pmb { p } _ { j } ) - \\pmb { q } _ { i , j }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "measures the difference between a projected scene point and its corresponding feature point. The function $\\pi$ projects scene points to image space, $\\mathbf { \\mathscr { q } } _ { i , j } = [ x _ { i , j } , y _ { i , j } , 1 ]$ is the normalized homogeneous pixel coordinate, and $\\mathcal { X } = [ \\pmb { T } _ { 1 } , \\pmb { T } _ { 2 } \\cdot \\cdot \\cdot \\pmb { T } _ { N _ { i } } , \\pmb { p } _ { 1 } , \\pmb { p } _ { 2 } \\cdot \\cdot \\cdot \\pmb { p } _ { N _ { j } } ] ^ { \\top }$ contains all the points’ and the cameras’ parameters. The general strategy to minimize Equation (1) is the Levenberg-Marquardt (LM) (Nocedal & Wright, 2006; Lourakis & Argyros, 2005) algorithm. At each iteration, the LM algorithm solves for an optimal update $\\Delta \\mathcal { X } ^ { * }$ to the solution by minimizing: ",
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/eb99d86f6d4ca8d2082e76c3a8df22ff93af305751d7a8622dcab4898bd6d916.jpg",
+        "text": "$$\n\\Delta \\mathcal { X } ^ { * } = \\mathrm { a r g m i n } \\| J ( \\mathcal { X } ) \\Delta \\mathcal { X } + E ( \\mathcal { X } ) \\| + \\lambda \\| D ( \\mathcal { X } ) \\Delta \\mathcal { X } \\| .\n$$",
+        "text_format": "latex",
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+    },
+    {
+        "type": "text",
+        "text": "Here, $E ( \\mathcal { X } ) = [ e _ { 1 , 1 } ^ { g } ( \\mathcal { X } ) , e _ { 1 , 2 } ^ { g } ( \\mathcal { X } ) \\cdot \\cdot \\cdot e _ { N _ { i } , N _ { j } } ^ { g } ( \\mathcal { X } ) ]$ , and $J ( \\mathcal { X } )$ is the Jacobian matrix of $E ( \\mathcal { X } )$ respect to $\\mathcal { X }$ , $D ( \\mathcal { X } )$ is a non-negative diagonal matrix, typically the square root of the diagonal of the approximated Hessian $J ( \\mathcal { X } ) ^ { \\top } J ( \\mathcal { X } )$ . The non-negative value $\\lambda$ controls the regularization strength. The special structure of $J ( \\mathcal { X } ) ^ { \\top } J ( \\mathcal { X } )$ motivates the use of Schur-Complement (Brown, 1958). ",
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "This geometric BA with re-projection error is the golden standard for structure-from-motion in the last two decades, but with two main drawbacks: ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "• Only image information conforming to the respective feature types, typically image corners, blobs, or line segments, is utilized. Features have to be matched to each other, which often result in a lot of outliers. Outlier rejection like RANSAC is necessary, which still cannot guarantee correct result. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "These two difficulties motivate the recent development of direct methods (Engel et al., 2014; 2018; Delaunoy & Pollefeys, 2014) which propose the photometric BA algorithm to eliminate feature matching and directly minimizes the photometric error (pixel intensity difference) of aligned pixels. The photometric error is defined as: ",
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+    {
+        "type": "equation",
+        "img_path": "images/56a90f148ceda20fa5264f2a6f25c9098f3e641facdcda2aa328cb806ce017ab.jpg",
+        "text": "$$\ne _ { i , j } ^ { p } ( \\mathcal { X } ) = I _ { i } ( \\pi ( T _ { i } , d _ { j } \\cdot \\pmb { q } _ { j } ) ) - I _ { 1 } ( \\pmb { q } _ { j } ) ,\n$$",
+        "text_format": "latex",
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+    },
+    {
+        "type": "text",
+        "text": "where $d _ { j } ~ \\in ~ \\mathbb { D } ~ = ~ \\{ d _ { j } | j ~ = ~ 1 \\cdot \\cdot \\cdot N _ { j } \\}$ is the depth of a pixel $\\mathbf { \\Delta } \\mathbf { q } _ { j }$ at the image $I _ { 1 }$ , and $d _ { j } \\cdot \\mathbf { \\vec { q } } _ { j }$ upgrade the pixel $\\mathbf { \\Delta } \\mathbf { q } _ { j }$ to its 3D coordinate. Thus, the optimization parameter is $\\mathcal { X } =$ $[ { \\pmb T } _ { 1 } , { \\pmb T } _ { 2 } \\cdot \\cdot \\cdot { \\pmb T } _ { N _ { i } } , d _ { 1 } , d _ { 2 } \\cdot \\cdot \\cdot d _ { N _ { j } } ] ^ { \\top }$ . The direct methods have the advantages of using all pixels with sufficient gradient magnitude. They have demonstrated superior performance, especially at less textured scenes. However, these methods also have some drawbacks: ",
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+        "type": "text",
+        "text": "• They are sensitive to initialization as demonstrated in (Mur-Artal et al., 2015) and (Tang et al., 2017) because the photometric error increases the non-convexity (Engel et al., 2018).   \n• They are sensitive to camera exposure and white balance changes. An automatic photometric calibration is required (Engel et al., 2018; 2016).   \n• They are more sensitive to outliers such as moving objects. ",
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "4 THE BA-NET ARCHITECTURE ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To deal with the above challenges, we propose a feature-metric BA algorithm which estimates the same scene depth and camera motion parameters $\\mathcal { X }$ as in photometric BA, but minimizes the feature-metric difference of aligned pixels: ",
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+        "page_idx": 2
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+    {
+        "type": "equation",
+        "img_path": "images/4374922f5e6f09f5867a8627e10d8da16587de772a504f82d90a2acaee10eb56.jpg",
+        "text": "$$\ne _ { i , j } ^ { f } ( \\mathcal { X } ) = F _ { i } ( \\pi ( \\boldsymbol { T } _ { i } , d _ { j } \\cdot \\boldsymbol { q } _ { j } ) ) - F _ { 1 } ( \\boldsymbol { q } _ { j } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathbb { F } = \\{ F _ { i } | i = 1 \\cdot \\cdot \\cdot N _ { i } \\}$ are feature pyramids of images $\\mathbb { I } = \\{ I _ { i } | i = 1 \\cdot \\cdot \\cdot N _ { i } \\}$ . Similar to the photometric BA, our feature-metric BA considers more pixels than corners or blobs. It has the potential to learn more suitable features for SfM to deal with exposure changes, moving objects, etc. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/51445b39ea111c0debdf69e8cec249cfec148f0981237baaf39f3d3499de2fef.jpg",
+        "image_caption": [
+            "Figure 1: Overview of our BA-Net structure, which consists of a DRN-54 (Yu et al., 2017) as the backbone network, a Basis Depth Maps Generator that generates a set of basis depth maps, a Feature Pyramid Constructor that constructs multi-scale feature maps, and a BA-Layer that optimizes both the depth map and the camera poses through a novel differentiable LM algorithm. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We learn features suitable for SfM via back-propagation, instead of using pre-trained CNN features for image classification (Czarnowski et al., 2017). Therefore, it is crucial to design a differentiable optimization layer, our BA-Layer, to solve the optimization problem, so that the loss information can be back-propagated. The BA-Layer predicts the camera poses $\\mathbb { T }$ and the dense depth map $\\mathbb { D }$ during forward pass and back-propagates the loss from $\\mathbb { T }$ and $\\mathbb { D }$ to the feature pyramids $\\mathbb { F }$ for training. ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.1 OVERVIEW ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As illustrated in Figure 1, our BA-Net receives multiple images and then feed them to the backbone DRN-54. We use DRN-54 (Yu et al., 2017) because it replaces max-pooling with convolution layers and generates smoother feature maps, which is desirable for BA optimization. Note the original DRN is memory inefficient due to the high resolution feature maps after dilation convolutions. We replace the dilation convolution with ordinary convolution with strides to address this issue. After DRN-54, a feature pyramid is then constructed for each input image, which are the inputs for the BA-Layer. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "At the same time, the basis depth maps generator generates multiple basis depth maps for the image $I _ { 1 }$ , and the final depth map is represented as a linear combination of these basis depth maps. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Finally, the BA-Layer optimizes for the camera poses and the dense depth map jointly by minimizing the feature-metric error defined in Equation (4), which makes the whole pipeline end-to-end trainable. ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.2 FEATURE PYRAMID ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The feature pyramid learns suitable features for the BA-Layer. Similar to the feature pyramid networks (FPN) for object detection (Lin et al., 2017), we exploit the inherent multi-scale hierarchy of deep convolutional networks to construct feature pyramids. A top-down architecture with lateral connections is applied to propagate richer context information from coarser scales to finer scales. Thus, our feature-metric BA will have a larger convergence radius. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As shown in Figure 2(a), we construct a feature pyramid from the backbone DRN-54. We denote the last residual blocks of conv1, conv2, conv3, conv4 in DRN-54 as $\\{ C ^ { 1 } , C ^ { 2 } , C ^ { 3 } , C ^ { 4 } \\}$ , with strides $\\{ 1 , 2 , 4 , 8 \\}$ respectively. We upsample a feature map $C ^ { k + 1 }$ by a factor of 2 with bilinear interpolation and concatenate the upsampled feature map with $C ^ { \\bar { k } }$ in the next level. This procedure is iterated until the finest level. Finally, we apply a $3 \\times 3$ convolution on the concatenated feature maps to reduce its dimensionality to 128 to balance the expressiveness and computational complexity, which leads to the final feature pyramid $F _ { i } = [ F _ { i } ^ { 1 } , F _ { i } ^ { 2 } , F _ { i } ^ { 3 } ]$ for image $I _ { i }$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We visualize some typical channels from the raw image $I$ (i.e. the RGB channels), the pre-trained DRN-54 $C ^ { 3 }$ and our learned $F ^ { 3 }$ in Figure 2(b). It is evident that, after training with our BA-Layer, the feature pyramid becomes smoother and each channel correspondences to different regions in the image. Note that our feature pyramids have higher resolution than FPN to facilitate precise alignment. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To have a better intuition about how much the BA optimization benefits from our learned features, we visualize different distances in Figure 3. We evaluate the distance between a pixel marked by a yellow cross in the top image in Figure 3 (a) and all pixels in a neighbourhood of its corresponding point in the bottom image of Figure 3 (a). The distances evaluated from raw RGB values, pretrained feature $C ^ { 3 }$ , and our learned feature $F ^ { 3 }$ are visualized in (b), (c), and (d) respectively. All distances are normalized to $[ 0 , 1 ]$ and visualized as heat maps. The $x$ -axis and $y$ -axis are the offsets to the ground-truth corresponding point. The RGB distance in (b) (i.e. $e ^ { p }$ in Equation (3)) has no clear global minimum, which makes the photometric BA sensitive to initialization (Engel et al., 2014; 2018). The distance measured by the pretrained feature $C ^ { 3 }$ has both global and local minimums. Finally, the distance measured by our learned feature $F ^ { 3 }$ has a clear global minimum and smooth basin, which is helpful in gradient based optimization such as the LM algorithm. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.3 BUNDLE ADJUSTMENT LAYER ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "After building feature pyramids for all images, we optimize camera poses and a dense depth map by minimizing the feature-metric error in Equation (4). Following the conventional Bundle Adjustment principle, we optimize Equation (4) using the Levenberg-Marquardt (LM) algorithm. However, the original LM algorithm is non-differentiable because of two difficulties: ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "• The iterative computation terminates when a specified convergence threshold is reached. This if-else based termination strategy makes the output solution $\\mathcal { X }$ non-differentiable with respect to the input $\\mathbb { F }$ (Domke, 2012). In each iteration, it updates the damping factor $\\lambda$ based on the current value of the objective function. It raises $\\lambda$ if a step fails to reduce the objective; otherwise it reduces $\\lambda$ . This if-else decision also makes $\\mathcal { X }$ non-differentiable with respect to $\\mathbb { F }$ . ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "When the solution $\\mathcal { X }$ is non-differentiable with respect to $\\mathbb { F }$ , feature learning by back-propagation becomes impossible. The first difficulty has been studied in Domke (2012) and the author proposes to fix the number of iterations, which is refered as ‘incomplete optimization’. Besides making the optimization differentiable, this ‘incomplete optimization’ technique also reduces memory consumption because the number of iterations is usually fixed at a small value. ",
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+    {
+        "type": "text",
+        "text": "The second difficulty has never been studied. Previous works mainly focus on gradient descent (Domke, 2012) or quadratic minimization (Amos & Kolter, 2017; Schmidt & Roth, 2014). In this section, we propose a simple yet effective approach to soften the if-else decision and yields a differentiable LM algorithm. We send the current objective value to a MLP network to predict $\\lambda$ . This technique not only makes the optimization differentiable, but also learns to predict a better damping factor $\\lambda$ , which helps the optimization to reach a better solution within limited iterations. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "To start with, we illustrate a single iteration of the LM optimization as a diagram in Figure 4 by interpreting intermediate variables as network nodes. During the forward pass, we compute the solution update $\\Delta \\mathcal { X }$ from feature pyramids $\\mathbb { F }$ and current solution $\\mathcal { X }$ as the following steps: ",
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+    {
+        "type": "text",
+        "text": "• We compute the feature-metric error $E ( \\mathcal { X } ) = [ e _ { 1 , 1 } ^ { f } ( \\mathcal { X } ) , e _ { 1 , 2 } ^ { f } ( \\mathcal { X } ) \\cdots e _ { N _ { i } , N _ { j } } ^ { f } ( \\mathcal { X } ) ]$ with Equation (4) on all $N _ { i }$ images and $N _ { j }$ pixels, where $\\mathcal { X }$ is the solution from the previous iteration; ",
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+    {
+        "type": "image",
+        "img_path": "images/9ce7ecb0fb83ecb8cf36b97f25a52a284e907bea6b12aeddf555810fd438bf08.jpg",
+        "image_caption": [
+            "Figure 2: A feature pyramid and some typical channels from different feature maps. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "image",
+        "img_path": "images/48453c521d73985079a9080b5b287ceadaf6a72242a2d461cc5cbe8fc1db9897.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Figure 3: Feature distance maps defined over raw RGB values, pretrained CNN features $C ^ { 3 }$ , or our learned features $F ^ { 3 }$ . Our features produce smoother objective function to facilitate optimization. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/a458f2275f9f354daff70a3dc2b34e82b0d61aeb9488bd43a7479be3730985de.jpg",
+        "image_caption": [
+            "Figure 4: A single iteration of the differentiable LM. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "• We then compute the Jacobian matrix $J ( \\mathcal { X } )$ , the Hessian matrix $J ( \\mathcal { X } ) ^ { \\top } J ( \\mathcal { X } )$ and its diagonal matrix $D ( \\mathcal { X } )$ ; ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "• To predict the damping factor $\\lambda$ , we use global average pooling to aggregate the aboslute value of $E ( \\mathcal { X } )$ over all pixels for each feature channel, and get a 128D feature vector. We then send it to a MLP sub-network to predict $\\lambda$ ; ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "• Finally, the update $\\Delta \\mathcal { X }$ to the current solution is computed as a standard LM step: ",
+        "bbox": [
+            215,
+            460,
+            769,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/90e737d7df9f4d9e82d5e80cb6f86a47d6ad5d6eb7ce11fdd5141c4eebc7c059.jpg",
+        "text": "$$\n\\Delta \\mathcal { X } = ( J ( \\mathcal { X } ) ^ { \\top } J ( \\mathcal { X } ) + \\lambda D ( \\mathcal { X } ) ) ^ { - 1 } J ( \\mathcal { X } ) ^ { \\top } E ( \\mathcal { X } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            362,
+            477,
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+            496
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In this way, we can consider $\\lambda$ as an intermediate variable and denote each LM step as a function $g$ about features pyramids $\\mathbb { F }$ and the solution $\\mathcal { X }$ from the previous iteration. In other words, $\\Delta \\mathcal { X } =$ $g ( \\mathcal { X } ; \\mathbb { F } )$ . Therefore, the solution after the $k$ -th iteration is: ",
+        "bbox": [
+            176,
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/548397e39e4540df62b16c1d54e586c12884616fbd23f18297c525b3d96e6aff.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathcal { X } _ { k } = g ( \\mathcal { X } _ { k - 1 } ; \\mathbb { F } ) \\circ \\mathcal { X } _ { k - 1 } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            411,
+            542,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Here, $\\circ$ denotes parameters updating, which is addition for depth and SE(3) exponential mapping for camera poses. Equation (6) is differentiable with respect to the feature pyramids $\\mathbb { F }$ , which makes back-propagation possible through the whole pipeline for feature learning. The MLP that predicts $\\lambda$ is also shown in Figure 4. We stack four fully-connected layers to predict $\\lambda$ from the input 128D vector. We use ReLU as the activation function to guarantee $\\lambda$ is non-negative. Following the photometric BA (Engel et al., 2014; 2018), we solve our feature-metric BA using a coarse-to-fine strategy with feature map warping at each iteration. We apply the differentiable LM algorithm for 5 iterations at each pyramid level, leading to 15 iterations in total. All the camera poses are initialized with identity rotation and zero translation, and the initialization of depth map will be introduced in Section 4.4. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.4 BASIS DEPTH MAPS GENERATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            703,
+            452,
+            718
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Parameterizing a dense depth map by a per-pixel depth value is impractical under our formulation. Firstly, it introduces too many parameters for optimization. For example, an image of $3 2 0 \\times 2 4 0$ pixels results in $7 6 . 8 \\mathrm { k }$ parameters. Secondly, in the beginning of training, many pixels will become invisible in the other views because of the poorly predicted depth or motion. So little information can be back-propagated to improve the network, which makes training difficult. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To deal with these problems, we use the convolutional network for monocular image depth estimation as a compact parameterization, rather than using it as an initialization as in Tateno et al. (2017) and Yang et al. (2018). We use a standard encoder-decoder architecture for monocular depth learning as in Laina et al. (2016). We use DRN-54 as the encoder to share the same backbone features with our feature pyramids. For the decoder, we modify the last convolutional feature maps of Laina et al. (2016) to 128 channels and use these feature maps as the basis depth maps for optimization. The final depth map is generated as the linear combination of these basis depth maps, which is: ",
+        "bbox": [
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+            806,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5b3e7a7e68328e741dbffe24160be17c8ae629748826a945c957fa7ba213ca94.jpg",
+        "text": "$$\n\\mathbb { D } = \\operatorname { R e L U } ( \\pmb { w } ^ { \\top } \\pmb { B } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            906,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Here, $\\mathbb { D }$ is the $h \\cdot w$ depth map that contains depth values for all pixels, $\\textbf {  { B } }$ is a $1 2 8 \\times h \\cdot w$ matrix, representing 128 basis depth maps generated from network, $\\pmb { w }$ is the linear combination weights of these basis depth maps. The $\\pmb { w }$ will be optimized in our BA-Layer. The ReLU activation function guarantees the final depth is non-negative. Once $\\textbf {  { B } }$ is generated from the network, we fix $\\textbf {  { B } }$ and use $\\pmb { w }$ as a compact depth parameterization in BA optimization, and the feature-metric distance becomes: ",
+        "bbox": [
+            173,
+            103,
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+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1b32ef418f6d9e0296dc8ce098ca2af457746c1011756c39ef2aeaaf9aefe64f.jpg",
+        "text": "$$\n\\begin{array} { r } { e _ { i , j } ^ { f } ( \\mathcal { X } ) = F _ { i } ( \\pi ( \\pmb { T } _ { i } , \\mathrm { R e L U } ( \\pmb { w } ^ { \\top } \\pmb { B } [ j ] ) \\cdot \\pmb { q } _ { j } ) ) - F _ { 1 } ( \\pmb { q } _ { j } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            320,
+            180,
+            676,
+            203
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where $B [ j ]$ is the $j$ -th column of $\\textbf {  { B } }$ , and $\\mathrm { R e L U } ( { \\pmb w } ^ { \\top } { \\pmb B } [ j ] )$ is the corresponding depth of $\\pmb q _ { j }$ . To further speedup convergence, we learn the initial weight $\\pmb { w } _ { 0 }$ as a 1D convolution filter for an arbitrary image, i.e. $\\begin{array} { r } { \\mathbb { D } _ { 0 } = \\mathrm { R e L } \\check { \\mathrm { U } } ( { \\pmb w } _ { 0 } ^ { \\top } B ) } \\end{array}$ . The $\\textbf {  { B } }$ of various images are visualized in the appendix. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.5 TRAINING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            270,
+            287,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The BA-Net learns the feature pyramid, the damping factor predictor, and the basis depth maps generator in a supervised manner. We apply the following commonly used loss for training, though more sophisticated ones might be designed. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Camera Pose Loss The camera rotation loss is the distance between rotation quaternion vectors $\\mathcal { L } _ { r o t a t i o n } = \\| \\pmb { q } - \\pmb { q } ^ { * } \\|$ . Similarly, translation loss is the Euclidean distance between prediction and groundtruth in metric scale, $\\mathcal { L } _ { t r a n s l a t i o n } = \\| \\pmb { t - t ^ { * } } \\|$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Depth Map Loss For each dense depth map we applies the berHu Loss (Zwald & Lambert-Lacroix, 2012) as in Laina et al. (2016). ",
+        "bbox": [
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+            395,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We initialize the back-bone network from DRN-54 (Yu et al., 2017), and the other components are trained with ADAM (Kingma & Ba, 2015) from scratch with initial learning rate 0.001, and the learning rate is divided by two when we observe plateaus from the Tensorboard interface. ",
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+    },
+    {
+        "type": "text",
+        "text": "5 EVALUATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.1 DATASET ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            525,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "ScanNet ScanNet (Dai et al., 2017a) is a large-scale indoor dataset with 1,513 sequences in 706 different scenes. Camera poses and depth maps are not perfect, because they are estimated via BundleFusion (Dai et al., 2017b). The metric scale is known in all data from ScanNet, because the data are recorded with a depth camera which returns absolute depth values. ",
+        "bbox": [
+            174,
+            551,
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+            607
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To sample image pairs for training, we apply a simple filtering process. We first filter out pairs with a large photo-consistency error, to avoid image pairs with large pose or depth error. We also filter out image pairs, if less than $50 \\%$ of the pixels from one image are visible in the other image. In addition, we also discard a pair if their roundness score (Beder & Steffen, 2006) is less than 0.001, which avoids pairs with too narrow baselines. ",
+        "bbox": [
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+            614,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We split the whole dataset into the training and the testing sets. The training set contains the first 1,413 sequences and the testing set contains the rest 100 sequences. We sample 547,991 training pairs and 2,000 testing pairs from the training and testing sequences respectively. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "KITTI KITTI (Geiger et al., 2012) is a widely used benchmark dataset collected by car-mounted cameras and a LIDAR sensor on streets. It contains 61 scenes belonging to the \"city\", \"residential\", or \"road\" categories. Eigen et al. (2014) select 28 scenes for testing and 28 scenes from the remaining for training. We use the same data split, to make a fair comparison with previous methods. Since ground truth pose is unavailable from the raw KITTI dataset, we compute camera poses by LibVISO2 (Geiger et al., 2011) and take them as ground truth after discarding poses with large errors. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5.2 COMPARISONS WITH OTHER METHODS ",
+        "text_level": 1,
+        "bbox": [
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+            842,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "ScanNet To evaluate the results’ quality, we use the depth error metrics suggested in Eigen & Fergus (2015), where RMSE (linear, log, and log, scale inv.) measure the RMSE of the raw, the logarithmical, and aligned logarithmical depth values, while the other two metrics measure the mean of the ratios that divide the absolute and square error by groundtruth depth.. The errors in camera poses are measured by the rotation error (the angle between the ground truth and the estimated camera rotations), the translation direction error (the angle between the ground truth and estimated camera translation directions) and the absolute position error (the distance between the ground truth and the estimated camera translation vectors). ",
+        "bbox": [
+            174,
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+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/7e5cb401e4f03073faa86ec6b9c964ed76ffc794831a17bddeb20f28f8b1b194.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 1: Quantitative comparisons with DeMoN and classic BA. The superindex ∗ denotes that the model is trained on the trainning set described in Ummenhofer et al. (2017). "
+        ],
+        "table_body": "<table><tr><td></td><td>Ours</td><td>Ours*</td><td>DeMoN*</td><td>Photometric BA</td><td>GeometricBA</td></tr><tr><td rowspan=\"3\">Rotation (degree) Translation (cm) Translation (degree)</td><td>1.018</td><td>1.587</td><td>3.791</td><td>4.409</td><td>8.56</td></tr><tr><td>3.39</td><td>10.81</td><td>15.5</td><td>21.40</td><td>36.995</td></tr><tr><td>20.577</td><td>31.005</td><td>31.626</td><td>34.36</td><td>39.392</td></tr><tr><td rowspan=\"3\">absrelative difference sqr relative difference</td><td>0.161</td><td>0.238</td><td>0.231</td><td>0.268</td><td>0.382</td></tr><tr><td>0.092</td><td>0.176</td><td>0.520</td><td>0.427</td><td>1.163</td></tr><tr><td>0.346</td><td>0.488</td><td>0.761</td><td>0.788</td><td>0.876</td></tr><tr><td>RMSE (linear) RMSE (log)</td><td>0.214</td><td>0.279</td><td>0.289</td><td>0.330</td><td>0.366</td></tr><tr><td>RMSE (log, scale inv.)</td><td>0.184</td><td>0.276</td><td>0.284</td><td>0.323</td><td>0.357</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In Table 1, we compare our method with DeMoN (Ummenhofer et al., 2017) and the conventional photometric and geometric BA. Note that we cannot get DeMoN trained on the ScanNet. For fair comparison, we train our network on the same training data as DeMoN and test both networks on our testing data1. We also show the results of our network trained on ScanNet. Our BA-Net consistently performs better than DeMoN no matter which training data is used. Since DeMoN does not recover the absolute scale, we align its depth map with the groundtruth to recover its metric scale for evaluation. We further compare with conventional geometric (Nister, 2004; Agarwal et al.) and photometric (Engel et al., 2014) BA. Again, our method produces better results. The geometric BA works poorly here, because feature matching is difficult in indoor scenes. Even the RANSAC process cannot get rid of all outliers. While for photometirc BA, the highly non-convex objective function is difficult to optimize as described in Section 3. ",
+        "bbox": [
+            173,
+            338,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "KITTI We use the same metrics as the comparisons on ScanNet for depth evaluation. To evaluate the camera poses, we follow (Zhou et al., 2017; Wang et al., 2018) to use the Absolute Trajectory Error (ATE), which measures the Euclidean differences between two trajectories (Steinbruecker et al., 2011), on the 9th and 10th sequences from the KITTI odometry data. In this experiment, we create short sequences of 5 frames by first computing 5 two-view reconstructions from our BA-Net and then align the two-view reconstructions in the coordinate system anchored at the first frame. minimize the photometric error. ",
+        "bbox": [
+            173,
+            498,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/5458c7f3b5e25f0f6f66519716e023d5741f0b1810dde5baec8530c81795d582.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 2: Quantitative comparisons on KITTI with supervised (Eigen et al., 2014) and unsupervised (Wang et al., 2018; Zhou et al., 2017; Godard et al., 2017) methods. "
+        ],
+        "table_body": "<table><tr><td></td><td>Ours</td><td></td><td></td><td>Wang et al. (2018) Zhou et al. (2017) Godard et al. (2017)</td><td>Eigen et al. (2014)</td></tr><tr><td>ATE(km)</td><td>0.019</td><td>0.045</td><td>0.021</td><td>N/A</td><td>N/A</td></tr><tr><td>absrel</td><td>0.083</td><td>0.151</td><td>0.208</td><td>0.148</td><td>0.203</td></tr><tr><td>sqr rel</td><td>0.025</td><td>1.257</td><td>1.768</td><td>1.344</td><td>1.548</td></tr><tr><td>RMSE(linear)</td><td>3.640</td><td>5.583</td><td>6.856</td><td>5.927</td><td>6.307</td></tr><tr><td>RMSE(log)</td><td>0.134</td><td>0.228</td><td>0.283</td><td>0.247</td><td>0.282</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Table 2 summarizes our results on KITTI. Our method outperforms the supervised methods (Eigen et al., 2014) as well as recent unsupervised methods (Zhou et al., 2017; Wang et al., 2018; Godard et al., 2017). Our method also achieves more accurate camera trajectories than Zhou et al. (2017) and Wang et al. (2018). We believe this is due to our feature-metric BA with features learned specifically for SfM problem, which makes the objective function closer to convex and easier to optimize as discussed in Section 4.2. In comparison, Zhou et al. (2017) and Wang et al. (2018) minimize the photometric error. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "More comparison with DeMoN, ablation studies, and multi-view SfM (up to 5 views) are reported in the appendix due to page limit. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSIONS AND FUTURE WORKS ",
+        "text_level": 1,
+        "bbox": [
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+            102,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "This paper presents the BA-Net, a network that explicitly enforces multi-view geometry constraints in terms of feature-metric error. It optimizes scene depths and camera motion jointly via feature-metric bundle adjustment. The whole pipeline is differentiable and thus end-to-end trainable, such that the features are learned from data to facilitate structure-from-motion. The dense depth is parameterized as a linear combination of several basis depth maps generated from the network. Our BA-Net nicely combines domain knowledge (hard-coded multi-view geometry constraint) with deep learning (learned feature representation and basis depth maps generator). It outperforms conventional BA and recent deep learning based methods. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgement This work is supported by the NSERC discovery grant 611664 and a project funding from Alibaba. ",
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+            280
+        ],
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+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/13df0af9a2a1f698d62a684f351663ff999163586b209241ce52dcdd498bd25f.jpg",
+        "image_caption": [
+            "Figure 5: Network details for the (a) the DRN-54 backbone and (b) the basis depth generator. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            135,
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+    {
+        "type": "text",
+        "text": "Network Architecture Details Figure 5 illustrates the detailed network architectures for the backbone DRN-54 and the depth basis generator. The architecture of the feature pyramid has been provided in Figure 2(a). We modify the dilated convolution of the original DRN-54 to convolution with strides and discard the conv7 and conv8 as shown in Figure 5(a). $C ^ { 1 }$ to $C ^ { 6 }$ are layers with {1,2,4,8,16,32} strides and {16,32,256,512,1024,2048} channels, where $C ^ { 1 }$ and $C ^ { 2 }$ are basic convolution layers, while $C ^ { 3 }$ to $C ^ { 6 }$ are standerd bottleneck blocks as in ResNet (He et al., 2016). ",
+        "bbox": [
+            173,
+            452,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Figure 5(b) visualizes our depth basis generator which adopts the up-projection structure proposed in Laina et al. (2016). The depth basis generator is a stander decoder that takes the output of $\\dot { C } ^ { 6 }$ as input and stacks five up-projection blocks to generate 128 basis depth maps, and each of the basis depth maps is half the resolution of the input image. The up-projection block is shown on the right of Figure 5(b) which upsample the input by $2 \\times$ and then apply convolutions with projection connection. ",
+        "bbox": [
+            173,
+            542,
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+            613
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Evaluation Time To evaluate the running time of our method, we use the Tensorflow profiler tool to retrieve the time in ms for all network nodes and then summarize the results corresponding to each component in our pipeline. As shown in Table 3, our method takes $9 5 . 2 1 \\ \\mathrm { m s }$ to reconstruct two $3 2 0 \\times 2 4 0$ images, which is slightly faster than DeMoN that takes $1 1 0 \\mathrm { m s }$ for two $2 5 6 \\times 1 9 2$ images. ",
+        "bbox": [
+            173,
+            619,
+            825,
+            676
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "The current computation bottleneck is the BA-Layer which contains a large amount of matrix operations and can be further speeded up by direct CUDA implementation. Since we explicitly hard-code the multi-view geometry constraints in the BA-Layer, it is possible to share the backbone DRN-54 with other high-level vision tasks, such as semantic segmentation and object detection, to maximize reuse of network structures and minimize extra computation cost. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/6f0bf70cb8f7db030f9818830ebf959c7c79e7bcf100bc28a2d18d260bc6e76a.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Backbone(DRN-54)</td><td rowspan=1 colspan=1>FeaturePyramid</td><td rowspan=1 colspan=1>Basis DepthGenerator</td><td rowspan=1 colspan=1>BA-LayerOptimization</td><td rowspan=1 colspan=1>Total</td></tr><tr><td rowspan=1 colspan=1>Time (ms)</td><td rowspan=1 colspan=1>15.04</td><td rowspan=1 colspan=1>5.87</td><td rowspan=1 colspan=1>9.81</td><td rowspan=1 colspan=1>67.22</td><td rowspan=1 colspan=1>95.21</td></tr></table>",
+        "bbox": [
+            238,
+            765,
+            754,
+            810
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Table 3: Evaluation time for each component, which is summarized using Tensorflow profiler. ",
+        "bbox": [
+            187,
+            820,
+            803,
+            837
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "APPENDIX B: ABLATION STUDIES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            863,
+            460,
+            881
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Learned Features vs Pre-trained Features Our learned feature pyramid improves the convexity of the objective function to facilitate the optimization. We compare our learned features with features ",
+        "bbox": [
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/e282dd710914d1091b4a01b72a7752f7e82b9b4b543201b964a61e8bf0dd9e99.jpg",
+        "table_caption": [
+            "Table 4: Ablation Study Comparisons by Disabling Different Components of BA-Net "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Ours (Full)</td><td>w/o Feature Learning</td><td>w/o Joint Optimization</td><td>w/o入</td></tr><tr><td>Rotation (degree)</td><td>1.018</td><td>2.667</td><td>1.036</td><td>7.202</td></tr><tr><td>Translation (cm)</td><td>3.39</td><td>10.8</td><td>3.91</td><td>22.38</td></tr><tr><td>Translation (degree)</td><td>20.577</td><td>31.493</td><td>26.779</td><td>59.81</td></tr><tr><td>absrelative difference</td><td>0.161</td><td>0.267</td><td>0.217</td><td>0.630</td></tr><tr><td>sqr relative difference</td><td>0.092</td><td>0.242</td><td>0.145</td><td>0.549</td></tr><tr><td>RMSE (linear)</td><td>0.346</td><td>0.481</td><td>0.428</td><td>0.763</td></tr><tr><td>RMSE (log)</td><td>0.214</td><td>0.303</td><td>0.270</td><td>0.513</td></tr><tr><td>RMSE (log, scale inv.)</td><td>0.184</td><td>0.226</td><td>0.205</td><td>0.437</td></tr></table>",
+        "bbox": [
+            204,
+            99,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "pre-trained on ImageNet for classification tasks. As shown in Table 4, the pre-trained features (i.e.   \nw/o Feature Learning) produce larger error. This proves the discussion in Section 4.2. ",
+        "bbox": [
+            174,
+            284,
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+            313
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Bundle Adjustment Optimization vs SE(3) Pose Estimation Our BA-Layer optimizes depth and camera poses jointly. We compare it to the SE(3) camera pose estimation with fixed depth map (e.g. the initialized depth $\\mathbb { D } _ { 0 }$ in Section 4.4), and similar strategy is adopted in Wang et al. (2018). To make a fair comparison, we also use our learned feature pyramids for the SE(3) camera pose estimation. As shown in Table 4, without BA optimization (i.e. w/o Joint Optimization), both the depth maps and camera poses are worse, because the errors in the depth estimation will degrades the camera pose estimation. ",
+        "bbox": [
+            173,
+            319,
+            826,
+            416
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Differentiable Levenberg-Marquardt vs Gauss-Newton To make the whole pipeline end-to-end trainable, we makes the Levenberg-Marquardt algorithm differentiable by learning the damping factor from the network. We first compare our method against vanilla Gauss-Newton without damping factor $\\lambda$ (i.e. $\\lambda = 0$ ). Since the objective function of feature-metric BA is non-convex, the Hessian matrix $J ( \\mathcal { X } ) ^ { \\top } J ( \\mathcal { X } )$ might not be positive definite, which makes the matrix inversion by Cholesky decomposition fail. ",
+        "bbox": [
+            173,
+            424,
+            825,
+            508
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "To deal with this problem, we use QR decomposition instead for training with Gauss-Newton. As shown in Table 4, the Gauss-Newton algorithm (i.e. w/o $\\lambda$ ) generates much larger error, because the BA optimization is non-convex and the Gauss-Newton algorithm has no guaranteed convergence unless the initial solution is sufficiently close to the optimal (Nocedal & Wright, 2006). This comparison reveals that, similar to conventional BA, our differnetiable Levenberg-Marquardt algorithm is superior than the Gauss-Newton algorithm for feature-metric BA. ",
+        "bbox": [
+            173,
+            515,
+            825,
+            599
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/9115916d7d58f5641ac10d6fc46f29dae3da88fc63917d5a8cc31bf6d3c24ee9.jpg",
+        "image_caption": [
+            "Figure 6: The camera pose and the depth errors correspond to different constant $\\lambda$ values. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            200,
+            626,
+            803,
+            792
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Predicted vs Constant $\\lambda$ Another way to make the Levenberg-Marquardt algorithm differentiable is to fix the $\\lambda$ during the iterations. We compare with this strategy. As shown in Figure 6(a), increasing $\\lambda$ makes the both rotation and translation error decreases, until $\\lambda = 0 . 5$ , and then increases. The reason is that a small $\\lambda$ makes the algorithm close to the Gauss-Newton algorithm, which has convergence issues. A large $\\lambda$ leads to a small update at each iteration, which makes it difficult to reach a good solution within limited iterations. ",
+        "bbox": [
+            173,
+            840,
+            825,
+            922
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "While in Figure 6(b), increasing $\\lambda$ always makes depth errors decrease, probably because a larger $\\lambda$ leads to a small update and makes the final depth close to the initialed depth, which is better than the optimized one with small constant $\\lambda$ . ",
+        "bbox": [
+            174,
+            103,
+            821,
+            145
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Using constant $\\lambda$ value consistently generates worse results than using a predicted $\\lambda$ from the MLP network, because there is no optimal $\\lambda$ for all data and it should be adapted to different data and different iterations. We draw the errors of our method in Figure 6(a) and Figure 6(b) as the flat dash lines for a reference. ",
+        "bbox": [
+            174,
+            152,
+            825,
+            208
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "APPENDIX C: EVALUATION ON DEMON DATASET ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            233,
+            586,
+            250
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Table 5 summarizes our results on the DeMoN dataset. For a comparison, we also cite the results from DeMoN (Ummenhofer et al., 2017) and the most recent work LS-Net (Clark et al., 2018). We further cite the results from some conventional approaches as reported in DeMoN, indicated as Oracle, SIFT, FF, and Matlab respectively. Here, Oracle uses ground truth camera poses to solve the multi-view stereo by SGM (Hirschmuller, 2005), while SIFT, FF, and Matlab further use sparse features, optical flow, and KLT tracking respectively for feature correspondence to solve camera poses by the 8-pt algorithm (Hartley, 1997). ",
+        "bbox": [
+            173,
+            267,
+            825,
+            366
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/642ee6ac4a50d545d1a10dcedc16e1db12582ca96435cb029c1483c15146e5e6.jpg",
+        "table_caption": [
+            "Table 5: Quantitative comparisons on the DeMoN dataset. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td></td><td></td><td>Depth</td><td></td><td>Motion</td><td></td><td></td><td></td><td></td><td>Depth</td><td></td><td>Motion</td><td></td></tr><tr><td></td><td>Method</td><td>L1-inv</td><td>sc-inv</td><td>L1-rel</td><td>Rotation</td><td>Translation</td><td></td><td>Method</td><td>L1-inv</td><td>sc-inv</td><td>L1-rel</td><td>Rotation</td><td>Translation</td></tr><tr><td rowspan=\"8\"></td><td>Oracle</td><td>0.019</td><td>0.197</td><td>0.105</td><td>0</td><td>0</td><td></td><td>Oracle</td><td>0.023</td><td>0.618</td><td>0.349</td><td>0</td><td>0</td></tr><tr><td>SIFT</td><td>0.056</td><td>0.309</td><td>0.361</td><td>21.180</td><td>60.516</td><td></td><td>SIFT</td><td>0.051</td><td>0.900</td><td>1.027</td><td>6.179</td><td>56.650</td></tr><tr><td>FF</td><td>0.055</td><td>0.308</td><td>0.322</td><td>4.834</td><td>17.252</td><td></td><td>FF</td><td>0.038</td><td>0.793</td><td>0.776</td><td>1.309</td><td>19.425</td></tr><tr><td>Matlab</td><td></td><td></td><td></td><td>10.843</td><td>32.736</td><td>Sseeeesr</td><td>Matlab</td><td></td><td></td><td></td><td>0.917</td><td>14.639</td></tr><tr><td>DeMoN</td><td>0.047</td><td>0.202</td><td>0.305</td><td>5.156</td><td>14.447</td><td></td><td>DeMoN</td><td>0.019</td><td>0.315</td><td>0.248</td><td>0.809</td><td>8.918</td></tr><tr><td>LS-Net</td><td>0.051</td><td>0.221</td><td>0.311</td><td>4.653</td><td>11.221</td><td></td><td>LS-Net</td><td>0.010</td><td>0.410</td><td>0.210</td><td>0.910</td><td>8.21</td></tr><tr><td>Ours</td><td>0.03</td><td>0.15</td><td>0.08</td><td>3.499</td><td>11.238</td><td></td><td>Ours</td><td>0.08</td><td>0.21</td><td>0.13</td><td>1.298</td><td>10.37</td></tr><tr><td colspan=\"2\"></td><td colspan=\"2\">Depth</td><td colspan=\"2\">Motion</td><td></td><td></td><td></td><td colspan=\"2\">Depth</td><td colspan=\"2\">Motion</td></tr><tr><td></td><td>Method</td><td>L1-inv</td><td>sc-inv</td><td>L1-rel</td><td>Rotation</td><td>Translation</td><td></td><td>Method</td><td>L1-inv</td><td>sc-inv</td><td>L1-rel</td><td>Rotation</td><td>Translation</td></tr><tr><td rowspan=\"8\">RRRPR</td><td>Oracle</td><td>0.026</td><td>0.398</td><td>0.336</td><td>0</td><td>0</td><td></td><td>Oracle</td><td>0.020</td><td>0.241</td><td>0.220</td><td>0</td><td>0</td></tr><tr><td>SIFT</td><td>0.050</td><td>0.577</td><td>0.703</td><td>12.010</td><td>56.021</td><td></td><td>SIFT</td><td>0.029</td><td>0.290</td><td>0.286</td><td>7.702</td><td>41.825</td></tr><tr><td>FF</td><td>0.045</td><td>0.548</td><td>0.613</td><td>4.709</td><td>46.058</td><td>gsun2</td><td>FF</td><td>0.029</td><td>0.284</td><td>0.297</td><td>3.681</td><td>33.301</td></tr><tr><td>Matlab</td><td></td><td></td><td></td><td>12.831</td><td>49.612</td><td></td><td>Matlab</td><td></td><td></td><td></td><td>5.920</td><td>32.298</td></tr><tr><td>DeMoN</td><td>0.028</td><td>0.130</td><td>0.212</td><td>2.641</td><td>20.585</td><td></td><td>DeMoN</td><td>0.019</td><td>0.114</td><td>0.172</td><td>1.801</td><td>18.811</td></tr><tr><td>LS-Net</td><td>0.019</td><td>0.09</td><td>0.301</td><td>1.01</td><td>22.1</td><td></td><td>LS-Net</td><td>0.015</td><td>0.189</td><td>0.650</td><td>1.521</td><td>14.347</td></tr><tr><td>Ours</td><td>0.008</td><td>0.087</td><td>0.05</td><td>2.459</td><td>14.90</td><td></td><td>Ours</td><td>0.015</td><td>0.11</td><td>0.06</td><td>1.729</td><td>13.26</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>",
+        "bbox": [
+            176,
+            383,
+            821,
+            568
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Our method consistently outperforms DeMoN (Ummenhofer et al., 2017) at both camera motion and scene depth, except on the ‘Scenes11’ data, because we enforce multi-view geometry constraint in the BA-Layer. Our results are poorer on the ‘Scene11’ dataset, because the images there are synthesized with random objects from the ShapeNet (Chang et al., 2015) without physically correct scale. This setting is inconsistent with real data and makes it harder for our method to learn the basis depth map generator. ",
+        "bbox": [
+            174,
+            612,
+            825,
+            695
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "When compared with LS-Net Clark et al. (2018), our method achieves similar accuracy on camera poses but better scene depth. It proves our feature-metric BA with learned feature is superior than the photometric BA in the LS-Net. ",
+        "bbox": [
+            176,
+            703,
+            821,
+            744
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "APPENDIX D: MULTI-VIEW STRUCTURE-FROM-MOTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            770,
+            645,
+            786
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Our method can be easily extended to reconstruct multiple images. We evaluate our method in the multi-view setting on the ScanNet (Dai et al., 2017a) dataset. To sample multi-view images for training, we randomly select two-view image pairs that shares a common image to construct $N$ -view sequences. Due to the limited GPU memory (12G), we limit $N$ to 5. ",
+        "bbox": [
+            174,
+            804,
+            825,
+            861
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "As shown in the Table 6, the accuracy is consistently improved when more views are included, which demonstrates the strength of the multi-view geometry constraints. Instead, most existing deep learning approaches can only handle two views at a time, which is sub-optimal as known in structure-from-motion literature. ",
+        "bbox": [
+            176,
+            867,
+            825,
+            922
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/ab4a902cd831ab680e7fa071eb69b53f0684749ab6db232c546f5b5b45a8803b.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 6: Quantitative comparisons on multi-view reconstruction on ScanNet. "
+        ],
+        "table_body": "<table><tr><td></td><td>Ours(2-views)</td><td>Ours(3-views)</td><td>Ours(5-views)</td></tr><tr><td rowspan=\"3\">Rotation (degree) Translation (cm) Translation (degree)</td><td>1.018</td><td>1.013</td><td>1.009</td></tr><tr><td>3.391</td><td>2.852</td><td>2.365</td></tr><tr><td>20.577</td><td>16.423</td><td>14.626</td></tr><tr><td rowspan=\"4\">absrelative difference sqr relative difference RMSE (linear) RMSE (log)</td><td>0.161</td><td>0.111</td><td>0.091</td></tr><tr><td>0.092</td><td>0.087</td><td>0.068</td></tr><tr><td>0.346</td><td>0.288</td><td>0.223</td></tr><tr><td>0.214</td><td>0.179</td><td>0.147</td></tr><tr><td>RMSE (log, scale inv.)</td><td>0.184</td><td>0.168</td><td>0.137</td></tr></table>",
+        "bbox": [
+            241,
+            101,
+            754,
+            229
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "APPENDIX E: QUANTITATIVE COMPARISONS WITH CODESLAM ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            284,
+            707,
+            300
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "We compare our method with CodeSLAM (Bloesch et al., 2018) which adopts similar idea for depth parameterization. But the difference is that CodeSLAM learns the conditioned depth auto-encoder separately and uses the depth codes in a standalone photometric BA component, while our method learns the feature pyramid and basis depth maps generator through feature-metric BA end-to-end. Since there is no public code for CodeSLAM, we directly cite the results from their paper.2 To get the trajectory on the EuroC MH02 sequence of our method, we select one frame every four frames and concatenate the reconstructed groups that contains every five selected frames. Then we use the same evaluation metrics as in CodeSLAM, which measures the translation errors correspond to different traveled distances. ",
+        "bbox": [
+            173,
+            318,
+            826,
+            444
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/da802cb4214926197154a330f9eeb5f1719fd306ae5857fe5460701cf562f196.jpg",
+        "image_caption": [
+            "Figure 7: Quantitative Comparisons with CodeSLAM (Bloesch et al., 2018) on EuroC MH02. The 0.50.5error bars represent the maximum and the minimum errors. The orange and the blue boxes represent the median errors for CodeSLAM and our method. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            210,
+            462,
+            787,
+            618
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "As shown in Figure 7, our method outperforms CodeSLAM. Our median error is less than the half of CodeSLAM’s error, i.e. CodeSLAM exhibits an error of roughly $1 \\textrm { m }$ for a traveled distance of $9 \\mathrm { m }$ , while our method’s error is about $0 . 4 \\mathrm { m }$ . This comparison demonstrates the superiority of end-to-end learning with feature pyramid and feature-metric BA over learning depth parameterization only. ",
+        "bbox": [
+            174,
+            703,
+            825,
+            760
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "APPENDIX F: VISUALIZATION OF BASIS DEPTH MAPS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            785,
+            625,
+            801
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "In Figure 8, we visualize four typical basis depth maps as heat maps for each of the four images. An interesting observation is that one basis depth map has higher responses on close objects while another oppositely has higher responses to the far background. Some other basis depth maps have smoothly varying responses and correspond to the layouts of scenes. This observation reveals that our learned basis depth maps have captured the latent structures of scenes. ",
+        "bbox": [
+            174,
+            820,
+            825,
+            890
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/0ca718d2adfbdb13f4a7196bbbedb0fd4b939b2adfb383d14592868a15728473.jpg",
+        "image_caption": [
+            "Figure 8: Visualization of different basis depth maps. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            238,
+            101,
+            758,
+            353
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "APPENDIX G: QUALITATIVE COMPARISONS WITH OTHER METHODS ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            404,
+            735,
+            420
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Finally, we show some qualitative comparison with the previous methods. Figure 9 shows the recovered depth map by our method and DeMoN Ummenhofer et al. (2017) on the ScanNet data. As we can see from the regions highlighted with a red circle, our method recovers more shape details. This is consistent with the quantitative results in Table 1. Figure 11 shows the recovered depth maps by our method, Wang et al. (2018), and Godard et al. (2017) respectively. Similarly, we observe more shape details in our results, as reflected in the quantitative results in Table 2. ",
+        "bbox": [
+            173,
+            434,
+            826,
+            518
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/b780d371c7ded7238a9b56c8ef9685c37eb1bb3b0a22bb9f09302f2916aea4cd.jpg",
+        "image_caption": [
+            "Figure 9: Qualitative Comparisons with DeMoN (Ummenhofer et al., 2017) on ScanNet. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            140,
+            531,
+            781,
+            780
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/bd53ca282836205f3f5e4d0394e1f831c885a8970627f8da70ae49c269da1f6b.jpg",
+        "image_caption": [
+            "Figure 10: Qualitative Comparisons with DeMoN (Ummenhofer et al., 2017) on its dataset. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            169,
+            126,
+            766,
+            377
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/4ea8a34be6470d158b277e4510a8d80e1b0a71f53c537acb646ba983c9ba2d9e.jpg",
+        "image_caption": [
+            "Figure 11: Qualitative Comparisons with Wang et al. (2018) and Godard et al. (2017). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            512,
+            848,
+            691
+        ],
+        "page_idx": 16
+    }
+]

parse/train/Bkeeca4Kvr/Bkeeca4Kvr.md

@@ -0,0 +1,398 @@
+# FEW-SHOT LEARNING ON GRAPHS VIA SUPERCLASSES BASED ON GRAPH SPECTRAL MEASURES
+
+Jatin Chauhan, Deepak Nathani, Manohar Kaul
+
+Department of Computer Science   
+Indian Institute of Technology Hyderabad   
+{chauhanjatin100,deepakn1019,manohar.kaul}@gmail.com
+
+# ABSTRACT
+
+We propose to study the problem of few-shot graph classification in graph neural networks (GNNs) to recognize unseen classes, given limited labeled graph examples. Despite several interesting GNN variants being proposed recently for node and graph classification tasks, when faced with scarce labeled examples in the few-shot setting, these GNNs exhibit significant loss in classification performance. Here, we present an approach where a probability measure is assigned to each graph based on the spectrum of the graph’s normalized Laplacian. This enables us to accordingly cluster the graph base-labels associated with each graph into super-classes, where the $L ^ { p }$ Wasserstein distance serves as our underlying distance metric. Subsequently, a super-graph constructed based on the super-classes is then fed to our proposed GNN framework which exploits the latent inter-class relationships made explicit by the super-graph to achieve better class label separation among the graphs. We conduct exhaustive empirical evaluations of our proposed method and show that it outperforms both the adaptation of state-ofthe-art graph classification methods to few-shot scenario and our naive baseline GNNs. Additionally, we also extend and study the behavior of our method to semi-supervised and active learning scenarios.
+
+# 1 INTRODUCTION
+
+The need to analyze graph structured data coupled with the ubiquitous nature of graphs (Borgwardt et al., 2005; Duvenaud et al., 2015; Backstrom & Leskovec, 2010; Chau et al., 2011), has given greater impetus to research interest in developing graph neural networks (GNNs) (Defferrard et al., 2016; Kipf & Welling, 2016; Hamilton et al., 2017; Velikovi et al., 2018) for learning tasks on such graphs. The overarching theme in GNNs is for each node’s feature vector to be generated by passing, transforming, and recursively aggregating feature information from a given $k$ -hop neighborhood surrounding the node. However, GNNs still fall short in the ”few-shot” learning setting, where the classifier must generalize well after seeing abundant base-class samples (while training) and very few (or even zero) samples from a novel class (while testing). Given the scarcity and difficulty involved with generation of labeled graph samples, it becomes all the more important to solve the problem of graph classification in the few-shot setting.
+
+Limitations and challenges: Recent work by Xu et. al. (Xu et al., 2019) indicated that most recently proposed GNNs were designed based on empirical intuition and heuristic approaches. They studied the representational power of these GNNs and identified that most neighborhood aggregation and graph-pooling schemes had diminished discriminative power. They rectified this problem with the introduction of a novel injective neighborhood aggregation scheme, making it as strong as the Weisfeiler-Lehman (WL) graph isomorphism test (Weisfeiler & Leman, 1968).
+
+Nevertheless, the problem posed by extremely scarce novel-class samples in the few-shot setting remains to persist as a formidable challenge, as it requires more rounds of aggregation to affect larger neighborhoods and hence necessitate greater depth in the GNN. However, when it comes to GNNs, experimental studies have shown that an increase in the number of layers results in dramatic performance drops in GNNs (Wu et al., 2019; Li et al., 2018b).
+
+Our work: Motivated by the aforementioned observations and challenges, our method does the following. We begin with a once-off preprocessing step. We assign a probability measure to each graph, which we refer to as a graph spectral measure (similar to (Gu et al., 2015)), based on the spectrum of the graph’s normalized Laplacian matrix representation. Given this metric space of graph spectral measures and the underlying distance as the $L ^ { p }$ Wasserstein distance, we compute Wasserstein barycenters (Agueh & Carlier, 2011) for each set of graphs specific to a base class and term these barycenters as prototype graphs. With this set of prototype graphs for each base class label, we cluster the spectral measures associated with each prototype graph in Wasserstein space to create a super-class label.
+
+Utilizing this super-class information, we then build a graph of graphs called a super-graph. The intuition behind this is to exploit the non-explicit and latent inter-class relationships between graphs via their spectral measures and use a GNN on this to also introduce a relational inductive bias (Battaglia et al., 2018), which in turn affords us an improved sample complexity and hence better combinatorial generalization given such few samples to begin with.
+
+Given, the super-classes and the super-graph, we train our proposed GNN model for few-shot learning on graphs. Our GNN consists of a graph isomorphism network (GIN) Xu et al. (2019) as a feature extractor $F _ { \theta } ( . )$ to generate graph embeddings; on which subsequently acts our classifier $C ( . )$ comprising of two components: (i) $C ^ { s u p }$ : a MLP layer to learn and predict the super class associated to a graph, and (ii) $\Dot { C } ^ { G A T }$ : a graph attention network (GAT) to predict the actual class label of a graph. The overall loss function is a sum of the cross-entropy losses associated with $C ^ { s u p }$ and $C ^ { G A \breve { T } }$ . We follow initialization based strategy (Chen et al., 2019), with a training and fine-tuning phase, so that in the fine-tuning phase, the pre-trained parameters associated with $F _ { \theta } ( . )$ and $C ^ { s u p }$ are frozen, and the few novel labeled graph samples are used to update the weights and attention learned by CGAT .
+
+Our contributions: To the best of our knowledge, we are the first to introduce few shot learning on graphs for graph classification. Next, we propose an architecture that makes use of the graph’s spectral measures to generate a set of super-classes and a super-graph to better model the latent relations between classes, followed by our GNN trained using an initialization method. Finally, we conduct extensive experiments to gain insight into our method. For example, in the 20-shot setting on the TRIANGLES dataset, our method shows a substantial improvement of nearly $7 \%$ and $2 0 \%$ over DL-based and unsupervised baselines, respectively.
+
+# 2 RELATED WORK
+
+Few-shot learning in the computer vision community was first introduced by (Fei-Fei et al., 2006) with the intuition that learning the underlying properties of the base classes given abundant samples can help generalize better to unseen classes with few-labeled samples available. Various learning algorithms have been proposed in the image domain, among which a broad category of initialization based methods aim to learn transferable knowledge from training classes, so that the model can be adapted to unseen classes with limited labeled examples (Finn et al., 2017); (Rusu et al., 2018); (Nichol et al., 2018). Recently proposed and widely accepted Initialization based methods can broadly be classified into: (i) methods that learn good model parameters with limited labeled examples and a small number of gradient update steps (Finn et al., 2017) and (ii) methods that learn an optimizer (Ravi & Larochelle, 2017). We refer the interested reader to Chen et. al. (Chen et al., 2019) for more examples of few-shot learning methods in vision.
+
+Graph neural networks (GNNs) were first introduced in (Gori et al., 2005); (Scarselli et al., 2009) as recurrent message passing algorithms. Subsequent work (Bruna et al., 2014); (Henaff et al., 2015) proposed to learn smooth spectral multipliers of the graph Laplacian, but incurred higher computational cost. This computational bottleneck was later resolved (Defferrard et al., 2016); (Kipf & Welling, 2016) by learning polynomials of the graph Laplacian. GNNs are a natural extension to Convolutional neural networks (CNNs) on non-Euclidean data. Recent work (Velikovi et al., 2018) introduced the concept of self-attention in GNNs, which allows each node to provide attention to the enclosing neighborhood resulting in improved learning. We refer the reader to (Bronstein et al., 2016) for detailed information on GNNs.
+
+Despite all the success of GNNs, few-shot classification remains an under-addressed problem. Some recent attempts have focused on solving the few-shot learning on graph data where GNNs are either trained via co-training and self-training (Li et al., 2018a), or extended by stacking transposed graph convolutional layers imputing a structural regularizer (Zhang et al., 2019) - however, both these works focus only on the node classification task.
+
+To the best of our knowledge, there does not exist any work pertaining few-shot learning on graphs focusing on the graph classification task, thus providing the motivation for this work.
+
+Comparison to few-shot learning on images: Few shot learning (FSL) has gained wide-spread traction in the image domain in recent years. However the success of FSL in images is not easily translated to the graph domain for the following reasons: (a) Images are typically represented in Euclidean space and thus can easily be manipulated and handled using well-known metrics like cosine similarity, $L _ { p }$ norms etc. However, graphs come from non-Euclidean domains and exhibit much more complex relationships and interdependency between objects. Furthermore, the notion of a distance between graphs is also not straightforward and requires construction of graph kernels or the use of standard metrics on graph embeddings (Kriege et al., 2019). Additionally, such graph kernels dont capture higher order relations very well. (b) In the FSL setting on images, the number of training samples from various classes is also abundantly more than what is available for graph datasets. The image domain allows training generative models to learn the task distribution and can further be used to generate samples for data augmentation, which act as very good priors. In contrast, graph generative models are still in their infancy and work in very restricted settings. Furthermore, methods like cropping and rotation to improve the models can’t be used for graphs given the permutation invariant nature of graphs. Additionally, removal of any component from the graph can adversely affect its structural properties, such as in biological datasets. (c) The image domain has very well-known regularization methods (e.g. Tikhonov, Lasso) that help generalize much better to novel datasets. Although, they dont bring any extra supervised information and hence cannot fully address the problem of FSL in the image domain. To the best of our knowledge, this is still an open research problem in the image domain. On the other hand, in the graph domain, our work would be a first step towards graph classification in an FSL setting, which would then hopefully pave the path for better FSL graph regularizers. (d) Transfer learning has led to substantial improvements on various image related tasks due to the high degree of transferability of feature extractors. Thus, downstream tasks like few-shot learning can be performed well with high quality feature extractor models, such as Resnet variants trained on Imagenet. Transfer learning, or for that matter even good feature extractors, remains a daunting challenge in the graph domain. For graphs, there neither exists a dataset which can serve as a pivot for high quality feature learning, nor does there exist a Graph NN which can capture the higher order relations between various categories of graphs, thus making this a highly challenging problem.
+
+# 3 PRELIMINARIES
+
+In this section, we introduce our notation and provide the necessary background for our few-shot learning setup on graphs. We begin by describing the various data sample types, followed by our learning procedure, in order to formally define few-shot learning on graphs. Finally, we define the graph spectral distance between a pair of graphs.
+
+Data sample sets: Let $\mathcal { G }$ denote a set of undirected unweighted graphs and $\mathcal { V }$ be the set of associated class labels. We consider two disjoint populations of labeled graphs consisting of i.i.d. graph samples, the set of base class labeled graphs $\mathbf { \bar { { G } } } _ { B } = \{ ( g _ { i } ^ { ( B ) } , y _ { i } ^ { ( B ) } ) \bar  \} _ { i = 1 } ^ { n } \}$ and the set of novel class labeled graphs $G _ { N } = \{ ( g _ { i } ^ { ( N ) } , y _ { i } ^ { ( N ) } ) \} _ { i = 1 } ^ { m }$ y(N )i )}mi=1, where g(Bi $g _ { i } ^ { ( B ) } , g _ { i } ^ { ( N ) } \in \mathcal { G } , y _ { i } ^ { ( B ) } \in \mathcal { V } ^ { ( B ) }$ y(B)i ∈ Y (B), and y(Ni $y _ { i } ^ { ( N ) } \in \mathcal { y } ^ { ( N ) }$ . Here, the set of base and novel class labels are denoted by ${ \mathcal { V } } ^ { ( B ) } = \{ 1 , \ldots , K \}$ and $\mathcal { V } ^ { ( N ) } = \{ K + 1 , \ldots , K ^ { \prime } \}$ , respectively, where $K ^ { \prime } > K$ . Both $\mathcal { V } ^ { ( B ) }$ and $\mathcal { V } ^ { ( N ) }$ are disjoint subsets of $\mathcal { V }$ , so, $\mathcal { V } ^ { ( B ) } \cap \mathcal { V } ^ { ( N ) } = \emptyset$ .
+
+Note that $m \ll n$ , i.e., there are far fewer novel class labeled graphs compared to the base class labeled ones. Besides $G _ { B }$ and $G _ { N }$ , we consider a set of $t$ unlabeled unseen graphs $\begin{array} { l l } { G _ { U } } & { : = } \end{array}$ $\{ g _ { 1 } ^ { ( U ) } , \ldots , g _ { t } ^ { ( U ) } \mid g _ { i } ^ { ( U ) } \in \pi _ { 1 } ( G _ { N } ) , i = 1 \ldots t \}$ , for testing1.
+
+Learning procedure: Inspired by the initialization based methods, we similarly follow a two-stage approach of training followed by fine-tuning.
+
+During training, we train a graph feature extractor $F _ { \theta } ( G _ { B } )$ with network parameters $\theta$ followed by a classifier $C ( G _ { B } )$ on graphs from $G _ { B }$ , where the loss function is the standard cross-entropy loss $\mathcal { L } _ { c }$ . In order to better recognize and generalize well on samples from novel classes, in the finetuning phase, the pre-trained feature extractor $F _ { \theta } ( . )$ along with its trained parameters is fixed and the classifier $C ( G _ { N } )$ is trained on the novel class labeled graph samples from $G _ { N }$ , with the same loss $\mathcal { L } _ { c }$ .
+
+Now, given the classification of data samples and the two-stage learning method, our problem of few-shot classification on graphs can be defined as follows.
+
+Problem definition: Given $n$ base-class labeled graphs from $G _ { B }$ during the training phase and $m$ novel-class labeled graphs from $G _ { N }$ during the fine-tuning phase, where $m \ll n$ , the objective of few-shot graph classification is to classify $t$ unseen test graph samples from $G _ { U }$ . Moreover, if $m = q T$ , where $T = K ^ { \prime } - K$ , i.e., each novel class label appears exactly $q$ times in $G _ { N }$ , then this setting is referred to as the $q$ -shot, $T$ -way learning.
+
+Graph spectral distance: Let us consider the graphs in $\mathcal { G }$ . The normalized Laplacian of a graph $g \in { \mathcal { G } }$ is defined as $\Delta _ { g } = I - D ^ { - 1 / 2 } A D ^ { 1 / 2 }$ , where $A$ and $D$ are the adjacency and the degree matrices of graph $g$ , respectively. The set of eigenvalues of $\Delta _ { g }$ given by $\{ \lambda _ { i } \} _ { i = 1 } ^ { | V | }$ is called the spectrum of and is denoted by . It is well known that the spectrum $\sigma ( g )$ of a normalized Laplacian matrix is contained in interval $[ 0 , 2 ]$ . We assign a Dirac mass $\delta _ { \lambda _ { i } }$ concentrated on each $\lambda _ { i } \ \in \ \sigma ( g )$ , thus associating a probability measure to $\sigma ( g )$ supported on $[ 0 , 2 ]$ , called the graph spectral measure $\mu _ { \sigma ( g ) }$ . Furthermore, let $P ( [ 0 , 2 ] )$ be the set of probability measures on interval $[ 0 , 2 ]$ .
+
+We now define the $p$ -th Wasserstein distance between probability measures, which we later use to define the spectral distance between a pair of graphs.
+
+Definition 1 Let $p \in [ 1 , \infty )$ and let $c : [ 0 , 2 ] \times [ 0 , 2 ]  [ 0 , + \infty ]$ be the cost function between the probability measures $\mu , \nu \in P ( [ 0 , 2 ] )$ . Then the $p$ -th Wasserstein distance between measures $\mu$ and $\nu$ is given by
+
+$$
+W _ { p } ( \mu , \nu ) = \left( \operatorname* { i n f } _ { \gamma } \int _ { [ 0 , 2 ] \times [ 0 , 2 ] } c ( x , y ) ^ { p } d \gamma \mid \gamma \in \Pi ( \mu , \nu ) \right) ^ { \frac { 1 } { p } }
+$$
+
+where $\Pi ( \mu , \nu )$ is the set of transport plans, i.e., the collection of all measures on $[ 0 , 2 ] \times [ 0 , 2 ]$ with marginals $\mu$ and $\nu$ .
+
+Given the general definition of the $p$ -th Wasserstein distance between probability measures and the graph spectral measure, we can now define the spectral distance between a pair of graphs in $\mathcal { G }$ .
+
+Definition 2 Given two graphs $g , g ^ { \prime } \in \mathcal { G }$ , the spectral distance between them is defined as
+
+$$
+W ^ { p } ( g , g ^ { \prime } ) : = W _ { p } \left( \mu _ { \sigma ( g ) } , \mu _ { \sigma ( g ^ { \prime } ) } \right)
+$$
+
+In words, $W ^ { p } ( g , g ^ { \prime } )$ is the optimal cost of moving mass from the graph spectral measure of graph $g$ to that of graph $g ^ { \prime }$ , where the cost of moving unit mass is proportional to the $p$ -th power of the difference of real-eigenvalues in interval $[ 0 , 2 ] ^ { 2 }$ .
+
+# 4 OUR METHOD
+
+We present our proposed approach here. First, given abundant base-class labels, we cluster them into super-classes by computing prototype graphs from each class, followed by clustering the prototype graphs based on their spectral properties. This clustering of prototype graphs induces a natural clustering on their corresponding class labels, resulting in super-classes (as outlined in Section 4.1).
+
+![](images/393d3119a11d19636ddb73ae543b4196b5496b3b1defea32cbff310a102d2d91.jpg)  
+Figure 1: The training (left) and fine-tuning (right) stages of our GNN.
+
+These super-classes are then used in the creation of a super-graph used further down by our GNN. Note that the creation of super-classes, followed by building a super-graph are a once-off process. The prototype graphs as well as the super-classes for the base classes can be stored in memory for further use.
+
+Next, we explain our graph neural network’s architecture which comprises of a feature extractor $F _ { \theta } ( . )$ and a classifier $C ( . )$ , described in Section 4.2. The classifier $C ( . )$ is further subdivided into a classifier $C ^ { s u p }$ that predicts the superclass of a graph feature vector and a graph attention network (GAT) $C ^ { G A T }$ to predict the graph’s class label. Figure 1 illustrates the training and fine-tuning phases of our GNN.
+
+# 4.1 COMPUTING SUPER CLASSES
+
+In order to exploit inter-class relationships between base-class labels, we cluster them in the following manner. First, we partition the set $G _ { B }$ into class-specific sets $G ^ { ( i ) }$ , for $i = 1 \dots K$ , where $G ^ { ( i ) }$ is the set of graphs with base-class label $i$ . Thus, $\begin{array} { r } { G _ { B } = \bigsqcup _ { i = 1 } ^ { K } G ^ { ( i ) } } \end{array}$ .
+
+Then, we compute class prototype graphs for each class-specific set. The class prototype graph for class $i$ represented by $p _ { i }$ is given by
+
+$$
+p _ { i } = \operatorname * { a r g m i n } _ { g _ { i } \in \pi _ { 1 } ( G ^ { ( i ) } ) } \frac { 1 } { | G ^ { ( i ) } | } \sum _ { j = 1 } ^ { | G ^ { ( i ) } | } W ^ { p } ( g _ { i } , g _ { j } )
+$$
+
+Essentially, the class prototype graph $p _ { i }$ for the $i$ -th class is the graph with the least average spectral distance to the rest of the graphs in the same class. Given these $K$ prototypes, we cluster them using Lloyd’s method (also known as $k$ -means)3.
+
+Clustering prototype graphs: Given $K$ unlabeled prototypes $p _ { 1 } , \dotsc , p _ { K } \in \pi _ { 1 } ( G _ { B } )$ and their associated spectral measures $ { \mu } _ { \sigma ( p _ { 1 } ) } , \ldots ,  { \mu } _ { \sigma ( p _ { K } ) } \in P ( [ 0 , 2 ] )$ . We rename the spectral measures as $s _ { 1 } , \ldots , s _ { K }$ to ease notation. Thus, our goal is to associate these spectral measures to at most $k$ clusters, where $k \geq 1$ is a user defined parameter.
+
+The $k$ -means problem finds a $k$ -partition $C = \{ C _ { 1 } , \ldots , C _ { k } \}$ that minimizes the following objective that represents the overall distortion error of the clustering
+
+$$
+\underset { C } { \operatorname { a r g m i n } } \sum _ { i = 1 } ^ { k } \sum _ { s _ { i } \in C _ { i } } W _ { p } ( s _ { i } , B ( C _ { i } ) )
+$$
+
+where $s _ { i }$ is a prototype graph in cluster $C _ { i }$ and $B ( C _ { i } )$ is the Wasserstein barycenter of the cluster $C _ { i }$ . The barycenter is computed as
+
+$$
+B ( C _ { i } ) = \operatorname * { a r g m i n } _ { p \in P ( [ 0 , 2 ] ) } \sum _ { j = 1 } ^ { | C _ { i } | } W _ { p } ( p , s ( i , j ) )
+$$
+
+![](images/0908b6d8353222c6cc3822778fc0d6149a35077983a3938dc8d0e1af7f456d8a.jpg)  
+Figure 2: An illustration of our proposed Wasserstein super-class clustering algorithm.
+
+where $s ( i , j )$ denotes the $j$ -th spectral measure in the $i$ -th cluster $C _ { i }$ .
+
+Lloyd’s algorithm: Given an initial set of Wasserstein barycenters $B ^ { ( 1 ) } ( C _ { 1 } ) , \ldots , B ^ { ( 1 ) } ( C _ { k } )$ of spectral measures at step $t = 1$ , one uses the standard Lloyd’s algorithm to find the solution by alternating between the assignment (Equation 4) and update (Equation 5) steps
+
+$$
+\begin{array} { r l } & { \quad C _ { i } ^ { ( t ) } = \left\{ s _ { p } : W _ { p } ( s _ { p } , B ^ { ( t ) } ( C _ { i } ) ) \leq W _ { p } ( s _ { p } , B ^ { ( t ) } ( C _ { j } ) ) , \forall j , 1 \leq j \leq k , 1 \leq p \leq K \right\} } \\ & { \quad C _ { i } ^ { ( t + 1 ) } = B ( C _ { i } ^ { ( t ) } ) } \end{array}
+$$
+
+Lloyd’s algorithm is known to converge to a local minimum (except in pathological cases, where it can oscillate between equivalent solutions). The final output is a grouping of the prototype graphs into $k$ groups, which also induces a grouping of the corresponding base classes. We denote these class groups as super-classes and denote the set of super-classes as $y ^ { s u p }$ .
+
+# 4.2 OUR GRAPH NEURAL NETWORK
+
+Feature extractor: To apply standard neural network architectures for downstream tasks we must embed the graphs in a finite dimensional vector space. We consider graph neural networks (GNNs) that employ the following message-passing architecture
+
+$$
+H ^ { ( j ) } = M ( A , H ^ { ( j - 1 ) } , \theta ^ { ( j ) } )
+$$
+
+where $H ^ { ( j ) } \in \mathbb { R } ^ { | V | \times d }$ are the node embeddings (i.e., messages) computed after $j$ steps of the GNN and $M$ is the message propagation function which depends on the adjacency matrix of the graph $A$ , the trainable parameters of the $j ^ { t h }$ layer $\theta ^ { ( j ) }$ , and node embeddings $H ^ { ( j - 1 ) }$ generated from the previous step.
+
+A recently proposed GNN called the graph isomorphism network (GIN) by $\mathrm { X u }$ et al. (2019) was shown to be stronger than several popular GNN variants like GCN Kipf & Welling (2016) and GraphSAGE Hamilton et al. (2017). What makes GIN so powerful and sets it apart from the other GNN variants is its injective neighborhood aggregation scheme which allows it to be as powerful as the Weisfeiler-Lehman (WL) graph isomorphism test. Motivated by this finding, we chose GIN as our graph feature extractor. The message propagation scheme in GIN is given by
+
+$$
+H ^ { ( j ) } = M L P ( ( 1 + \epsilon ) ^ { j } \odot H ^ { ( j - 1 ) } + A ^ { T } H ^ { ( j - 1 ) } )
+$$
+
+Here, $\epsilon$ is a layer-wise learnable scalar parameter and $M L P$ represents a multi-layer perceptron with layer-wise non-linearities for more expressive representations. The full GIN model run $R$ iterations of Equation 6 to generate final node embeddings which we represent by $H ^ { ( R ) }$ . As features from earlier iterations can also be helpful in achieving higher discriminative power, embeddings $H ^ { ( j ) }$
+
+from all $R$ iterations are concatenated as
+
+$$
+H _ { g } = \left\| _ { j = 1 } ^ { R } H ^ { ( j ) } , \right\|
+$$
+
+Here, $\begin{array} { r } { H ^ { ( j ) } = \sum _ { v \in V } H _ { v } ^ { ( j ) } } \end{array}$ , where $H _ { i } ^ { ( j ) }$ represents the $i$ -th node’s embedding in the $j$ -th iteration and $\parallel$ denotes a concatenation operator. $H _ { g }$ now contains the graph embedding of a graph $g$ and is passed on to the classifier.
+
+Classifier: Here, our objective is to improve the class separation produced by the graph embeddings of the feature extractor $F _ { \theta } ( . )$ and we do this by building a “graph of graph embeddings”, called a super-graph $g ^ { s u p }$ , where each node is a graph feature vector. We then employ our classifier $C ( . )$ on this super-graph to achieve better separation among the graph classes in the embedding space.
+
+During training, we first build the super-graph $g ^ { s u p }$ on a batch of base-labeled graphs as a collection of $k$ -NN graphs, where each constituent $k$ -NN graph is built on the graphs belonging to the same super-class. $g ^ { s u p }$ is then passed through a multi-layered graph attention network $C ^ { \forall G A \mathbf { \breve { T } } }$ to learn the associated class probabilities. The features extracted from $F _ { \theta } ( . )$ are passed into the MLP network $C ^ { s u p }$ to learn the associated super-class labels. $C ^ { s u p }$ and $C ^ { \widecheck { G } A T }$ combine to form our classifier $C ( . )$ . The cross-entropy losses associated with $C ^ { s u p }$ and $C ^ { G A T }$ are added to give the overall loss for $C ( . )$ . The intuition behind the construction of $g ^ { s u p }$ to train $C ^ { G A T }$ on was to further improve the existing cluster separation based on graph spectral measures by introducing a relational inductive bias (Battaglia et al., 2018) that is inherent to the GNN CGAT .
+
+Recall that we adopt an initialization method (described in 3). In our fine-tuning stage, novel class labeled graphs from $G _ { N }$ are input to the network. The pre-trained parameters learned by the feature extractor $F _ { \theta } ( . )$ are fixed and $C ^ { s u p }$ is used to infer the novel graph’s super-class label, followed by creation of super-graph on the novel graph samples and finally updating the parameters in CGAT through the loss. Finally the evaluation is performed on the samples from the unseen test set $G _ { U }$ .
+
+Discussion: We make the assumption that the novel test classes belong to the same set of super-classes from the training graphs. The reason being that the novel class labeled samples are so much fewer than the base class labeled samples, that the resulting super-graph ends up being extremely sparse and deviates a lot from the shape of the super-graph from the base classes; therefore it severely hinders $C ^ { G A T } \mathrm { s }$ ability to effectively aggregate information from the embeddings of the novel class labeled graphs. Instead, we pass the novel graph samples through our trained $C ^ { s u p }$ and infer its super-class label and this works very effectively for us, as is evidenced by our empirical results.
+
+# 5 EXPERIMENTAL RESULTS
+
+# 5.1 BASELINES AND DATASETS
+
+The standard graph classification datasets do not adequately satisfy the requirements for few-shot learning due to the dearth of unique class labels. Hence, we pick four new classification datasets, namely, Letter-High, TRIANGLES, Reddit-12K, and ENZYMES. The details and statistics for these datasets are given in Appendix A.1. As there do not exist any standard state-of-the-art methods for few-shot graph classification, we chose existing baselines for standard graph classification from both supervised and unsupervised methods.
+
+For supervised deep learning baselines, we chose - GIN (Xu et al. (2019)), CapsGNN (Xinyi & Chen (2019)), and Diffpool (Lee et al. (2019)). We ran these methods with similar settings as ours, i.e., by partitioning the main model into feature extraction and classifier sub-models to compare them in a fair and informative manner. From the unsupervised category, we consider 4 powerful SOTA methods - AWE (Ivanov & Burnaev (2018)), Graph2Vec (Narayanan et al. (2017)), WeisfeilerLehman subtree Kernel (Shervashidze et al. (2011)), and Graphlet count kernel (Shervashidze et al. (2009)). Since we want to analyze the few-shot classification abilities of these models, we essentially want to find out how well these algorithms can achieve class separation. We use $k$ -NN search on the output embeddings of these algorithms.
+
+Table 1: Results for various few-shot scenarios on Letter-High and TRIANGLES datasets. The best results are highlighted in bold while the second best results are underlined.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">Letter-High</td><td colspan="3">TRIANGLES</td></tr><tr><td>5-shot</td><td>10-shot</td><td>20-shot</td><td>5-shot</td><td>10-shot</td><td>20-shot</td></tr><tr><td>WL</td><td>65.27 ± 7.67</td><td>68.39± 4.69</td><td>72.69±3.02</td><td>51.25± 4.02</td><td>53.26 ± 2.95</td><td>57.74 ± 2.88</td></tr><tr><td>Graphlet</td><td>33.76 ± 6.94</td><td>37.59 ± 4.60</td><td>41.11 ± 3.71</td><td>40.17 ± 3.18</td><td>43.76 ± 3.09</td><td>45.90 ±2.65</td></tr><tr><td>AWE</td><td>40.60 ± 3.91</td><td>42.20 ±2.87</td><td>43.12 ± 1.00</td><td>39.36± 3.85</td><td>42.58 ± 3.11</td><td>44.98 ± 1.54</td></tr><tr><td>Graph2Vec</td><td>66.12 ± 5.21</td><td>68.17 ± 4.26</td><td>70.28 ± 2.81</td><td>48.38 ± 3.85</td><td>50.16 ± 4.15</td><td>54.90 ± 3.01</td></tr><tr><td>Diffpool</td><td>58.69 ± 6.39</td><td>61.59 ± 5.21</td><td>64.67 ± 3.21</td><td>64.17 ± 5.87</td><td>67.12 ± 4.29</td><td>73.27 ± 3.29</td></tr><tr><td>CapsGNN</td><td>56.60 ± 7.86</td><td>60.67 ± 5.24</td><td>63.97 ± 3.69</td><td>65.40 ± 6.13</td><td>68.37 ± 3.67</td><td>73.06 ± 3.64</td></tr><tr><td>GIN</td><td>65.83 ± 7.17</td><td>69.16 ± 5.14</td><td>73.28 ± 2.17</td><td>63.80 ± 5.61</td><td>67.30 ± 4.35</td><td>72.55 ± 1.97</td></tr><tr><td>GIN-k-NN</td><td>63.52 ± 7.27</td><td>65.66± 8.69</td><td>67.45± 8.76</td><td>58.34 ± 3.91</td><td>61.55± 3.19</td><td>63.45± 2.76</td></tr><tr><td>OurMethod-GCN</td><td>68.69 ± 6.50</td><td>72.80 ± 4.12</td><td>75.17 ± 3.11</td><td>69.37 ± 4.92</td><td>73.11 ± 3.94</td><td>77.86 ± 2.84</td></tr><tr><td>OurMethod-GAT</td><td>69.91 ± 5.90</td><td>73.28± 3.46</td><td>77.38 ± 1.58</td><td>71.40 ± 4.34</td><td>75.60± 3.67</td><td>80.04 ± 2.20</td></tr></table>
+
+Further configuration and implementation details for the baselines can be found in Appendix A.2. We also emphasize the benefit of using a GNN as a classifier by showing the adaptation of our model to semi-supervised fine-tuning (in Appendix A.5) and active learning (in Appendix A.6) settings.
+
+# 5.2 FEW-SHOT RESULTS
+
+We consider two variants of our model as naive baselines. In the first variant, we replace our GAT classifier with GCN Kipf & Welling (2016). We call this model OurMethod-GCN. This variant is used to justify the choice of GAT over GCN.
+
+In the second variant, we replace the entire classifier with the $k$ -NN algorithm over the features extracted from various layers of the feature extractor. We call this variant GIN- $k$ -NN and this is introduced to emphasize the significance of building a super-graph and using a GAT on it as a classifier to exploit the relational inductive bias.
+
+The results for all the datasets in various $q$ -shot scenarios, where $q \in \{ 5 , 1 0 , 2 0 \}$ are given in Table 1. We run each model 50 times and report averaged results. In every run, we select a different novel labeled subset $G _ { N }$ for fine-tuning the classifiers of the models. The evaluation for all models is done by randomly selecting a subset of 500 samples from the testing set $G _ { U }$ for Letter-High and TRIANGLES, whereas over 150 for ENZYMES and 300 for Reddit dataset and averaging over 10 such random selections.
+
+Table 2: Results for various few-shot scenarios on Reddit- $I 2 K$ and ENZYMES datasets. The best results are highlighted in bold while the second best results are underlined.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">Reddit-12K</td><td colspan="3">ENZYMES</td></tr><tr><td>5-shot</td><td>10-shot</td><td>20-shot</td><td>5-shot</td><td>10-shot</td><td>20-shot</td></tr><tr><td>WL</td><td>40.26± 5.17</td><td>42.57±3.69</td><td>44.41±3.43</td><td>55.78± 4.72</td><td>58.47±3.84</td><td>60.1±3.18</td></tr><tr><td>Graphlet</td><td>33.76 ± 6.94</td><td>37.59 ± 4.60</td><td>41.11 ± 3.71</td><td>53.17 ± 5.92</td><td>55.30 ±3.78</td><td>56.90 ±3.79</td></tr><tr><td>AWE</td><td>30.24± 2.34</td><td>33.44 ±2.04</td><td>36.13 ± 1.89</td><td>43.75 ±1.85</td><td>45.58 ± 2.11</td><td>49.98 ± 1.54</td></tr><tr><td>Graph2Vec</td><td>27.85 ± 4.21</td><td>29.97 ± 3.17</td><td>32.75 ± 2.02</td><td>55.88 ± 4.86</td><td>58.22 ± 4.30</td><td>62.28 ± 4.14</td></tr><tr><td>Diffpool</td><td>35.24 ± 5.69</td><td>37.43 ± 3.94</td><td>39.11 ± 3.52</td><td>45.64 ± 4.56</td><td>49.64 ± 4.23</td><td>54.27 ± 3.94</td></tr><tr><td>CapsGNN</td><td>36.58 ±4.28</td><td>39.16 ± 3.73</td><td>41.27 ± 3.12</td><td>52.67 ± 5.51</td><td>55.31 ± 4.23</td><td>59.34 ± 4.02</td></tr><tr><td>GIN</td><td>40.36 ± 4.69</td><td>43.70 ± 3.98</td><td>46.28 ± 3.49</td><td>55.73 ± 5.80</td><td>58.83 ±5.32</td><td>61.12 ± 4.64</td></tr><tr><td>GIN-k-NN</td><td>41.31± 2.84</td><td>43.58±2.80</td><td>45.12 ± 2.19</td><td>57.24 ± 7.06</td><td>59.34± 5.24</td><td>60.49±3.48</td></tr><tr><td>OurMethod-GCN</td><td>40.77 ± 4.32</td><td>44.28 ± 3.86</td><td>48.67 ± 4.22</td><td>54.34 ± 5.64</td><td>58.16 ± 4.39</td><td>60.86 ± 3.74</td></tr><tr><td>OurMethod-GAT</td><td>41.59 ± 4.12</td><td>45.67± 3.68</td><td>50.34± 2.71</td><td>55.42 ± 5.74</td><td>60.64 ± 3.84</td><td>62.81 ± 3.56</td></tr></table>
+
+The results clearly show that our proposed method and its GCN variant (i.e., OurMethod-GCN) outperform the baselines. GIN- $k$ -NN shows significant degradation in results on nearly all the three paradigms for all datasets with exceptions on 5-shot and 10-shot on ENZYMES dataset, thus strongly indicating that the improvements of our method can primarily be attributed to our GNN classifier fed with the super-graph constructed from our proposed method. The improvements in results are higher on TRIANGLES and Reddit datasets in contrast to Letter-High and ENZYMES, which can be attributed to the smaller size of the graphs in Letter-High making it difficult to distinguish based on graph spectra alone, whereas the complex and highly inter-related structure of enzymes makes it difficult for the DL based feature extractors as well as the graph kernel methods to segregate the classes in the feature and graph space respectively. GIN and WL show much better results as compared to other baselines for all the $q$ -shot scenarios, whereas AWE and Graphlet Kernel show significantly low results, unable to capture the properties of the graphs well. The DL baselines apart from GIN on the other hand show improvements on the TRIANGLES dataset performing close to GIN, where the unsupervised methods fails to capture the local node properties, however still perform poorly on other datasets. For the 20-shot scenario on TRIANGLES, our GAT variant shows an improvement of around $7 \%$ over DL baselines and more than $2 0 \%$ when compared to unsupervised methods. The substantial improvements of around $4 \%$ on Reddit dataset shows the superiority of our model for both the variants - GAT and GCN. Furthermore, the $t$ -SNE plots in Figure 3 show a substantial and interesting separation of class labels which strongly indicate that a good feature extractor in conjunction with a GNN perform well as a combination. The t-SNE plots for ENZYMES, Reddit, and Letter-High are shown in figures 4, 5 and 6 respectively.
+
+![](images/c52ff756e9f560ae3e4f1bc969a402fd9e4832578474fe3bc86a8e21779491da.jpg)  
+Figure 3: Visualization: t-SNE plots of the computed embeddings of test graphs on 20-shot scenario from OurMethod-GAT (left), GIN (middle) and WL Kernel (right) on TRIANGLES dataset. The embeddings for both our model and GIN are taken from the final layers of the respective models.
+
+# 5.3 ABLATION STUDY ON NUMBER OF SUPER-CLASSES
+
+Here, we study the behavior of our proposed network model without the super-class classifier $C ^ { s u p }$ . In Table 3 (10 and 20-shot setting), we observe a marked increase with the addition of our classifier which uses the super-class information and the super-graph based on spectral measures to guide $C ^ { G A T }$ towards improving the class separation of the graphs during both the training and fine-tuning stages. Using super-classes help in reducing the sample complexity of the large Hypothesis space and makes tuning of the model parameters easier during fine-tuning stage with less samples and few iterations. Negligible differences are observed on ENZYMES dataset since both the number of training classes as well as test classes are low, thus, the model performs equally well on removing super-classes. This is because of the latent inter-class representations can still be captured between few classes especially during the fine-tuning phase.
+
+Table 3: Ablation Study: “No-SC” represents our classifier $C ( . )$ without $C ^ { s u p }$ and “With-SC” represents C(.) with both Csup and CGAT present.   
+
+<table><tr><td rowspan="2">Dataset</td><td colspan="2">10-shot</td><td colspan="2">20-shot</td></tr><tr><td>No-SC</td><td>With-SC</td><td>No-SC</td><td>With-SC</td></tr><tr><td>Letter-High</td><td>71.13 ± 3.64</td><td>73.61 ± 3.19</td><td>75.23 ± 2.48</td><td>77.42 ± 1.47</td></tr><tr><td>TRIANGLES</td><td>74.03 ± 3.89</td><td>76.49 ± 3.26</td><td>76.89 ± 2.63</td><td>80.14 ± 1.88</td></tr><tr><td>Reddit-12K</td><td>43.76 ± 4.34</td><td>45.35 ± 4.06</td><td>48.19 ± 4.01</td><td>50.36 ± 3.04</td></tr><tr><td>ENZYMES</td><td>59.97 ± 3.98</td><td>59.58 ± 4.32</td><td>62.7 ± 3.63</td><td>62.39 ± 3.48</td></tr></table>
+
+# 5.4 SENSITIVITY ANALYSIS OF VARIOUS ATTRIBUTES
+
+Our proposed method contains two crucial attributes. We analyze our model by varying: (i) the number of super-classes and (ii) the $k$ -value in super-graph construction. The effect of varying these attributes on model accuracy are shown in Tables 4 and 5, respectively. For TRIANGLES and Letter
+
+Table 4: Model analysis over number of super-classes in 20-shot scenario. There is no evaluation for 5 super-classes on ENZYMES since the number of training classes is 4. Default value of parameter $k$ is fixed at 2.   
+Dataset   
+20-shot   
+Table 5: Model analysis over number of neighbors $( k )$ in super-graph for 20-shot scenario. Default value for the number of super-classes is fixed at 3.   
+
+<table><tr><td></td><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td></tr><tr><td>Letter-High</td><td>74.43± 2.61</td><td>76.61 ± 1.67</td><td>77.51 ± 1.49</td><td>76.31 ± 1.98</td><td>75.05 ± 2.29</td></tr><tr><td>TRIANGLES</td><td>76.43 ± 2.87</td><td>79.55 ± 1.91</td><td>80.51 ± 1.72</td><td>78.91 ± 2.09</td><td>78.25 ± 2.40</td></tr><tr><td>Reddit-12K</td><td>48.32 ± 4.09</td><td>50.67 ± 2.94</td><td>50.10 ± 3.02</td><td>49.52 ± 4.02</td><td>48.33 ± 4.08</td></tr><tr><td>ENZYMES</td><td>62.34 ± 4.11</td><td>62.13 ± 4.01</td><td>60.16 ± 3.81</td><td>59.34 ± 3.98</td><td></td></tr></table>
+
+Dataset   
+20-shot   
+
+<table><tr><td></td><td>2</td><td></td><td>4</td><td>6</td><td>8</td><td>Heuristic</td></tr><tr><td>Letter-High</td><td>77.33 ± 1.71</td><td></td><td>76.61 ± 1.67</td><td>75.63 ± 2.49</td><td>74.66 ± 2.61</td><td>74.35 ± 2.48</td></tr><tr><td>TRIANGLES</td><td>80.77 ± 1.57</td><td></td><td>79.85 ± 1.59</td><td>79.45 ± 1.97</td><td>78.93 ± 2.04</td><td>79.42 ± 3.16</td></tr><tr><td>Reddit-12K</td><td>50.48 ± 3.02</td><td></td><td>46.37 ± 3.03</td><td>44.12 ± 2.98</td><td>43.88 ± 3.24</td><td>44.82 ± 2.83</td></tr><tr><td>ENZYMES</td><td>62.34 ± 4.11</td><td></td><td>61.42 ± 4.42</td><td>60.23 ± 5.10</td><td>59.67 ± 4.77</td><td>61.07 ± 4.68</td></tr></table>
+
+High datasets, as we increase the number of super-classes, we observe the accuracy improving steadily up to 3 super-classes and then dropping from there onwards. For super-classes less than 3, we observe that the $k$ -NN graph does not respect the class boundaries that are already imposed by the graph spectral measures, thus connecting more arbitrary classes. For Reddit we observe the performance is slightly better on using 2 super-classes and for ENZYMES similar performances are observed for both 1 and 2 super-classes as described in the ablation study. On the other hand, increasing the number of super-classes past 3, makes each super-class cluster very sparse with few graph classes within, leading to an underflow of information between the graph classes. The same effect is observed for all datasets.
+
+The $k$ -value or the number of neighbors of each node belonging to the same connected component in the super-graph (i.e., belonging to the same super-class) is another salient parameter upon which hinges the information flow (via message passing) between the graphs of the same super-class. We analyze our model with $k$ values in the set $\{ 2 , 4 , 6 , 8 \}$ and a commonly used heuristic method, whereby each graph is connected to $\sqrt { b _ { s } }$ nearest neighboring graphs based on the Euclidean similarity of their feature representations, where $b _ { s }$ is the number of samples in the mini-batch corresponding to super-classes $s$ . We achieve best results with 2-NN graphs per super-class and increasing $k$ beyond it leads to denser graphs with unnecessary connections between classes belonging to the same super-class.
+
+# 6 CONCLUSION
+
+In this paper, we investigated the problem of few-shot learning on graphs for the graph classification task. We explicitly created a super-graph on the base-labeled graphs and then grouped / clustered their associated class labels into super-classes, based on the graph spectral measures attributed to each graph and the $L ^ { p }$ -Wasserstein distances between them. We found that training our GNN on the super-graph along with the auxiliary super-classes resulted in a marked improvement over stateof-the-art GNNs. A promising future work is to propose new GNN models that break away from current neighborhood aggregation schemes to specifically overcome the obstacle posed by few-shot learning on graphs. Our source-code and dataset splits have been made public in an attempt to attract more attention to the context of few-shot learning on graphs.
+
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+
+# A APPENDIX
+
+# A.1 DATASET DETAILS
+
+We use 4 different datasets namely - Reddit-12K, ENZYMES, Letter-High and TRIANGLES to perform exhaustive empirical evaluation of our model on various real-world datasets varying from small average graph size on Letter-High to large graphs like Reddit-12K. These datasets can be downloaded here 4. The dataset statistics are provided in Table 6, while the split statistics are provided in Table 7
+
+Table 6: Dataset Statistics   
+
+<table><tr><td>DatasetName</td><td># Classes</td><td># Graphs</td><td>Avg # Nodes</td><td>Avg #Edges</td></tr><tr><td>Reddit-12K</td><td></td><td>11929</td><td>391.41</td><td>456.89</td></tr><tr><td>ENZYMES</td><td>11 6</td><td>600</td><td>32.63</td><td>62.14</td></tr><tr><td>Letter-High</td><td>15</td><td>2250</td><td>4.67</td><td>4.50</td></tr><tr><td>TRIANGLES</td><td>10</td><td>45000</td><td>20.85</td><td>35.50</td></tr></table>
+
+Dataset Description: Reddit-12K datasets contains 11929 graphs where each graph corresponds to a thread in which each node represents a user and each edge represents that one user has responded to a comment from some other user. There are 11 different types of discussion forums corresponding to each of the 11 classes.
+
+ENZYMES is a dataset of protein tertiary structures consisting of 600 enzymes from the BRENDA enzyme database. The dataset contains 6 different graph categories corresponding to each different top-level EC enzyme.
+
+TRIANGLES dataset contain 10 different classes where the classes are numbered from 1 to 10 corresponding to the number of triangles/3-cliques in each graph of the dataset.
+
+Letter-High dataset contains graphs which represent distorted letter drawings from the english alphabets - $A , E , F , H , I , K , L , M , N , T , V , W , X , Y , Z$ . Each graph is a prototype manual construction of the alphabets.
+
+Table 7: Dataset Splits   
+
+<table><tr><td>Dataset Name</td><td>#Train Classes</td><td>#Test Classes</td><td># Training Graphs</td><td># Validation Graphs</td><td>#Test Graphs</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Reddit-12K</td><td>7</td><td>4</td><td>566</td><td>141</td><td>404</td></tr><tr><td>ENZYMES</td><td>4 11</td><td>2 4</td><td>320 1330</td><td>80 320</td><td>200 600</td></tr><tr><td>Letter-High TRIANGLES</td><td>7</td><td>3</td><td>1126</td><td>271</td><td>603</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+The validation graphs are used to assess model performance on training classes itself to check overfitting as well as for grid-search over hyperparameters. The actual train-testing class splits used for this paper are provided with the code. Since the TRIANGLES dataset has a large number of samples, this makes it infeasible to run many baselines including DL and non-DL methods. Hence, we sample 200 graphs from each class, making the total sample size 2000. Similarly we downsample the number of graphs from 11929 to 1111 (nearly 101 graphs per class). Downsampling is performed for Reddit-12K given extremely large graph sizes which makes the graph kernels as well as some deep learning baselines extremely slow.
+
+# A.2 BASELINE DETAILS
+
+This section details the implementation of the baseline methods. Since, DL-based methods - GIN, CapsGNN and DIFFPOOL have not been previously run on these datasets, we select the crucial hyper-parameters - such as number of layers heuristically based on the results of standard graph classification datasets on the best performing variants of these models. For these three methods we take the novel layers proposed in the corresponding papers as their feature extractors, while downstream MLP layers are chosen as the classifier. The training and evaluation strategies are similar to our model, i.e., the models are first trained in an end-to-end fashion on the training dataset $G _ { B }$ until convergence with learning rate decay on loss plateau and then the classifier layers are fine-tuned over $G _ { N }$ , keeping the parameters of the feature extractor layers fixed.
+
+For the unsupervised models - WL subtree kernel, Graphlet Count kernel, AWE and Graph2Vec, the evaluation is done using $k$ -NN search to assess the clustering quality of these models in our few-shot scenario. We refrain from using high-level classifier models such as SVM or MLPs, since training these classifiers on few-shot regime will not properly assess the abilities of these models to cluster together graphs of similar class labels. We empirically found that using high level classifiers resulted in higher deviations and lower mean accuracies. We choose the hyper-parameters for these models using grid-search, since they are significantly faster and each one of these models have few highly sensitive parameters which affect the model significantly. For these models, we perform a grid search for selection of $k$ in the $k$ -NN algorithm from the set $\{ 1 , 2 , 3 , 4 , 5 \}$ for the 5-shot scenario, of which $k = 1$ was found to perform the best. For higher shot scenario, the search was performed over the set $\{ 1 , 2 , 3 , 4 , 5 , 6 , 7 , \bar { 8 } , 9 , 1 0 \}$ , where we again found $k = 1$ to be the best. The validation set is used to check overfitting and hyper-parameter selection on the baseline methods.
+
+# A.3 OUR MODEL DETAILS
+
+This section provides the implementation details of our proposed model. Since, our feature extractor model is GIN, we maintain similar parameter settings as recommended by their paper. As mentioned in section 4.2, using embeddings from all iterations of the message passing network helps achieve better discriminative power and improved gradient flow, therefore we employ the same strategy in our feature extractor. The number of super-classes are selected from the set $\{ 1 , 2 , 3 , 4 , 5 \}$ using grid-search. The $k$ -value for construction of super-graph was selected from the set $\{ 2 , 4 , 6 , 8 \}$ . The feature extractor model uses batch-normalization between subsequent message passing layers. We use dropout of 0.5 in the $C ^ { s u p }$ layers. The $C ^ { G A T }$ layers undergo normalization of inputs between subsequent layers along with a dropout of 0.5, however, the normalization mechanism in classifier layers is different from batch-norm. We normalize each feature embedding to have Euclidean norm with value 1. Essentially,
+
+$$
+\mathbf { x } _ { i n p u t } ^ { j + 1 } = \frac { \mathbf { x } _ { o u t } ^ { j } } { | | \mathbf { x } _ { o u t } ^ { j } | | _ { 2 } }
+$$
+
+where $\mathbf { x } _ { i n p u t } ^ { j + 1 }$ is the input of $j + 1 ^ { t h }$ layer of classifier, $\mathbf { x } _ { o u t } ^ { j }$ is the output of the $j ^ { t h }$ layer. The inputs of the first layer of $C ^ { G A T }$ also undergo the same transformation over the outputs of the feature extractor model. We train our models with Adam (Kingma & Ba (2014)) with an initial learning rate of $1 0 ^ { - 3 }$ for 50 epochs. Each epoch has 10 iterations, where we randomly select a mini-batch from the training data $G _ { B }$ . The fine-tuning stage consists of 20 epochs with 10 iterations per epoch. We use a two-layer MLP over the final attention layer of $C ^ { G A T ^ { \bf 1 } }$ for classification. The attention layers use multi-head attention with 2 heads and leaky ReLU slope of 0.1 . The embeddings from both the attention heads are concatenated. For 20-shot, we set $k$ to 2, number of super-classes to 3 and batch size to 128 on the Letter-High dataset, while $k$ is set to 2 and batch size 64 on Reddit, ENZYMES and TRIANGLES datasets. The number of super-classes for Reddit are set to 2, for ENZYMES it is set to 1 and for TRIANGLES are 3. For ENZYMES, there are negligible differences on using 1 and 2 super-classes as shown in table 4. We used Python Optimal Transport (POT) library 5 for implementation of the $p$ -th Wasserstein distance.
+
+![](images/29fdd17a77c3e8e015beb1724b352f5089f6e6dae91756dba71ded23dab355b7.jpg)  
+Figure 4: Visualization: t-SNE plots of the computed embeddings of test graphs on 20-shot scenario from OurMethod-GAT (left), GIN (middle) and WL Kernel (right) on ENZYMES dataset.
+
+![](images/562f5e73ca715f0b27d1079b212c127e5869fabcb2d041d7a1dfb57df4883ff6.jpg)  
+Figure 5: Visualization: t-SNE plots of the computed embeddings of test graphs on 20-shot scenario from OurMethod-GAT (left), GIN (middle) and WL Kernel (right) on Reddit dataset.
+
+![](images/5cc4f83d20953c2ca1efcadb37047fc6947a35899155b9dcb5b918db26496726.jpg)  
+Figure 6: Visualization: t-SNE plots of the computed embeddings of test graphs on 20-shot scenario from OurMethod-GAT (left), GIN (middle) and WL Kernel (right) on Letter-High dataset.
+
+# A.4 SILHOUETTE SCORES
+
+To assess the clustering abilities of the models we analyze the silhouette scores of the test embeddings produced by the GAT variant of our method, GIN and WL Kernel. Silhouette coefficient essentially measures the ratio of intra-class versus inter-class distance. The Silhouette Coefficient is calculated using the mean intra-cluster distance (a) and the mean nearest-cluster distance (b) for each sample. The Silhouette Coefficient for a sample is given by $\frac { ( b - a ) } { m a x ( a , b ) }$ ,where $b$ is the distance between a sample and the nearest cluster that the sample is not a part of. The results for mean silhouette coefficient over the test samples averaged over multiple runs are shown in Table 8. We normalize the embeddings before calculating the silhouette coefficient. We can clearly see that our model creates better clusters with low intra-cluster distance as well as high inter-cluster distance. Note that the coefficient value for WL remains the same for all scenarios since it computes fixed embeddings attributed to absence of any DL component.
+
+Table 8: Silhouette coefficients of the test classes for the three dominant models - GAT variant of Our Method, GIN and WL. The best scores are highlighted in bold.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Reddit-12K</td><td colspan="2">ENZYMES</td><td colspan="2">Letter-High</td><td colspan="2">TRIANGLES</td></tr><tr><td>10-shot</td><td>20-shot</td><td>10-shot</td><td>20-shot</td><td>10-shot</td><td>20-shot</td><td>10-shot</td><td>20-shot</td></tr><tr><td>GIN</td><td>-0.0566</td><td>-0.0652</td><td>0.0168</td><td>0.0432</td><td>0.2157</td><td>0.2316</td><td>0.0373</td><td>0.1256</td></tr><tr><td>WLKernel</td><td>-0.0626</td><td>-0.0626</td><td>0.0366</td><td>0.0366</td><td>0.2490</td><td>0.2490</td><td>0.0186</td><td>0.0186</td></tr><tr><td>OurMethod-GAT</td><td>-0.0553</td><td>-0.0559</td><td>0.0296</td><td>0.1172</td><td>0.3494</td><td>0.3787</td><td>0.3824</td><td>0.4508</td></tr></table>
+
+Table 9: Semi-supervised fine-tuning results for various $p$ values on 10-shot and 20-shot scenarios, where “No Semi-Sup” represents the fine-tuning stage without additional labeled samples.
+
+<table><tr><td rowspan="2">Dataset</td><td colspan="3">10-shot</td><td colspan="3">20-shot</td></tr><tr><td>No Semi-Sup</td><td>25</td><td>50</td><td>No Semi-Sup</td><td>25</td><td>50</td></tr><tr><td>Letter-High</td><td>73.21± 3.19</td><td>74.18 ± 2.58</td><td>74.65 ± 2.16</td><td>76.95 ± 1.79</td><td>77.79 ± 1.52</td><td>78.31 ± 1.11</td></tr><tr><td>TRIANGLES</td><td>75.83 ± 2.97</td><td>76.36± 2.59</td><td>77.8 ± 2.04</td><td>80.09 ±1.78</td><td>81.29 ± 1.98</td><td>81.87 ± 1.45</td></tr><tr><td rowspan="2">Dataset</td><td></td><td>10-shot</td><td></td><td></td><td>20-shot</td><td></td></tr><tr><td>No Semi-Sup</td><td>10</td><td>20</td><td>No Semi-Sup</td><td>10</td><td>20</td></tr><tr><td>Reddit</td><td>45.41± 3.79</td><td>45.88±3.32</td><td>46.01± 2.99</td><td>50.34± 2.77</td><td>50.76±2.52</td><td>51.17 ± 2.21</td></tr><tr><td>ENZYMES</td><td>60.13 ± 3.98</td><td>60.87 ± 3.24</td><td>61.25 ± 3.17</td><td>62.74 ± 3.64</td><td>63.10± 3.47</td><td>63.67 ± 3.18</td></tr></table>
+
+# A.5 SEMI-SUPERVISED FINE-TUNING
+
+In many real-world learning scenarios, it is quite common to find abundant unlabelled data. Since our model uses a GNN classifier, this makes it possible to use unlabelled data while learning through message passing, where the fine tuning stage of our method is performed in semi-supervised settings.
+
+Essentially, while fine tuning the model, i.e., only training the classifier $C ^ { G A T }$ on $G _ { N }$ , we additionally use $p$ more graphs along with $G _ { N }$ , whose labels are unknown. The learning objective for fine tuning stage doesn’t change since the gradients are back-propagated from the labeled samples only. In this setting, each node in the attention classifier can aggregate information from unlabelled samples as well, thus allowing improved learning of the graphs features in $C ^ { G A T }$ . We show the results for $p$ values 25 and 50 on Letter-High and TRIANGLES datasets, whereas for $p$ values 10 and 20 on Reddit and ENZYMES datasets. The results are shown in Table 9. We observe an increase in the accuracy with increase in number of unlabeled samples during fine-tuning phase.
+
+# A.6 ADAPTATION TO ACTIVE-LEARNING
+
+In this section, we show the adaptation of our model to highly practical active learning scenario. In many real world applications, we might start with few samples per class, however as the number of samples to classify from these classes increase over time, some of these samples can be used by the model to adaptively learn and improve with very less human intervention, since the number of number of samples to be queried for theirs label can always be controlled.
+
+To perform active-learning, we first select a random subset of size 100 for Letter-High and TRIANGLES datasets as well as a random subset of size 40 for Reddit and ENZYMES datasets, which we term as $G _ { r a n d o m }$ , then fine tune the model on $G _ { N }$ and further evaluate the model on $G _ { r a n d o m }$ .
+
+Table 10: Active Learning Results. The value below each shot represents the number samples $l$ , added to $G _ { N }$ for second fine-tuning step, where “No AL” represents the model evaluation without additional labeled samples.   
+
+<table><tr><td rowspan="2">Dataset</td><td colspan="3">10-shot</td><td colspan="3">20-shot</td></tr><tr><td>No AL</td><td>15</td><td>25</td><td>No AL</td><td>15</td><td>25</td></tr><tr><td>Letter-High</td><td>73.34± 3.37</td><td>75.03 ± 3.24</td><td>76.89 ± 2.16</td><td>77.06 ± 1.73</td><td>78.44 ± 1.52</td><td>79.28 ± 1.36</td></tr><tr><td>TRIANGLES</td><td>76.02 ± 2.54</td><td>78.44 ± 1.84</td><td>79.91 ± 1.28</td><td>80.27 ±1.84</td><td>81.74 ± 2.03</td><td>82.58 ± 1.57</td></tr><tr><td rowspan="2">Dataset</td><td colspan="3">10-shot</td><td colspan="3">20-shot</td></tr><tr><td>No AL</td><td>10</td><td>20</td><td>No AL</td><td>10</td><td>20</td></tr><tr><td>Reddit</td><td>45.41± 3.79</td><td>46.88±3.14</td><td>47.91± 2.99</td><td>50.43±2.66</td><td>51.76± 2.32</td><td>53.07± 2.21</td></tr><tr><td>ENZYMES</td><td>60.13 ± 3.98</td><td>61.57 ± 3.48</td><td>62.25 ± 3.06</td><td>62.74 ± 3.64</td><td>63.60±3.30</td><td>64.97 ± 3.11</td></tr></table>
+
+Thereafter, l relatively important samples are chosen from $G _ { r a n d o m }$ and added to $G _ { N }$ for another step of fine-tuning. There can be multiple strategies for defining relative importance of a sample. For our purpose, we define a sample’s relative importance via its predicted class probability distribution. We sort these samples in increasing order of the difference between their highest and second highest predicted class probabilities and choose the first $l$ samples from this sorted ranking. We call this importance relative, since each sample is evaluated with respect to the set $G _ { N }$ and thus, there is transductive flow of information among the samples, hence defining the relative embeddings in the space. Intuitively speaking, we have chosen the samples lying closer to separation boundary with respect to $G _ { N }$ . The results for various values of $l$ are shown in Table 10. The evaluation is done as mentioned earlier on the unseen set $G _ { U }$ . We observe significant improvement for all the datasets. This shows our model is capable of selecting important samples with respect to the few existing samples and learn actively.
+
+# A.7 PERFORMANCE OF MODEL WITH 1 SUPER-CLASS
+
+From table 4 in correspondence to tables 1 and 2, one can observe that the results obtained by using only 1 super-class which is equivalent to removing the super-classes are still better in comparison with many GNN and graph kernel baselines. By removing the super-classes and thus forming the super-graph solely based on $\mathbf { k }$ -nearest neighbor heuristic the GAT still learns latent inter class connections via information flow better than the GNNs which use MLP as their classifier. The super graph constructed in such scenario will have arbitrary connections between classes in the beginning, however as the GIN feature extractor learns over time the segregation in the feature space increases leading to better inter as well intra-class connections. Despite this, the performance with superclasses is better as this inductive bias allows the model to initiate with a better alignment in the feature space. The silhouette score comparison for OurMethod-GAT with 1 super-class and with the best performing number of super-classes to GIN and WL clearly indicates the multifold benefits of using GNNs as a classifier via super-graph construction. The t-SNE plots for OurMethod-GAT with only 1 super-class, GIN and WL kernel on the datasets TRIANGLES, Reddit and Letter-High are provided in the figures 7, 8 and 9 respectively.
+
+Table 11: Silhouette coefficients of the test classes for three models - GAT variant of Our Method for 1 super-class which is equivalent to not using any super-classes vs the best performing number of super-classes as well as GIN and WL on 20-shot scenario. For GIN and WL both the sub-columns contain the same values as they don’t have any concept of super-classes.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="2">Reddit-12K</td><td colspan="2">ENZYMES</td><td colspan="2">Letter-High</td><td colspan="2">TRIANGLES</td></tr><tr><td>1-SC</td><td>2-SC</td><td>1-SC</td><td>2-SC</td><td>1-SC</td><td>3-SC</td><td>1-SC</td><td>3-SC</td></tr><tr><td>GIN</td><td>-0.0652</td><td>-0.0652</td><td>0.0432</td><td>0.0432</td><td>0.2316</td><td>0.2316</td><td>0.1256</td><td>0.1256</td></tr><tr><td>WLKernel</td><td>-0.0626</td><td>-0.0626</td><td>0.0366</td><td>0.0366</td><td>0.2490</td><td>0.2490</td><td>0.0186</td><td>0.0186</td></tr><tr><td>OurMethod-GAT</td><td>-0.0593</td><td>-0.0559</td><td>0.1172</td><td>0.0989</td><td>0.3519</td><td>0.3787</td><td>0.3975</td><td>0.4508</td></tr></table>
+
+![](images/1bafd2a3349ea4e2fe9c394cca4ab791a2cd1f65cf071a516bbd1c14891f4fbb.jpg)  
+Figure 7: Visualization: t-SNE plots of the computed embeddings of test graphs on 20-shot scenario from OurMethod-GAT with only 1 super-class (left), GIN (middle) and WL Kernel (right) on TRIANGLES dataset.
+
+![](images/2c26d5078ecb3af998cfa6a0df81aaa3205653b35f692f0d5124b1eb6194c0bd.jpg)  
+Figure 8: Visualization: t-SNE plots of the computed embeddings of test graphs on 20-shot scenario from OurMethod-GAT with only 1 super-class (left), GIN (middle) and WL Kernel (right) on Reddit dataset.
+
+![](images/6612059db44bda00c40722071d4b3b857b74cec9b39a0f864e9338819d122eed.jpg)  
+Figure 9: Visualization: t-SNE plots of the computed embeddings of test graphs on 20-shot scenario from OurMethod-GAT with only 1 super-class (left), GIN (middle) and WL Kernel (right) on LetterHigh dataset.

parse/train/HJgpugrKPS/HJgpugrKPS.md

@@ -0,0 +1,339 @@
+# SCALE-EQUIVARIANT STEERABLE NETWORKS
+
+Ivan Sosnovik∗, Michał Szmaja†, Arnold Smeulders   
+UvA-Bosch Delta Lab   
+University of Amsterdam
+
+# ABSTRACT
+
+The effectiveness of Convolutional Neural Networks (CNNs) has been substantially attributed to their built-in property of translation equivariance. However, CNNs do not have embedded mechanisms to handle other types of transformations. In this work, we pay attention to scale changes, which regularly appear in various tasks due to the changing distances between the objects and the camera. First, we introduce the general theory for building scale-equivariant convolutional networks with steerable filters. We develop scale-convolution and generalize other common blocks to be scale-equivariant. We demonstrate the computational efficiency and numerical stability of the proposed method. We compare the proposed models to the previously developed methods for scale equivariance and local scale invariance. We demonstrate state-of-the-art results on the MNIST-scale dataset and on the STL-10 dataset in the supervised learning setting.
+
+# 1 INTRODUCTION
+
+Scale transformations occur in many image and video analysis tasks. They are a natural consequence of the variable distances among objects, or between objects and the camera. Such transformations result in significant changes in the input space which are often difficult for models to handle appropriately without careful consideration. At a high level, there are two modeling paradigms which allow a model to deal with scale changes: models can be endowed with an internal notion of scale and transform their predictions accordingly, or instead, models can be designed to be specifically invariant to scale changes. In image classification, when scale changes are commonly a factor of 2, it is often sufficient to make class prediction independent of scale. However, in tasks such as image segmentation, visual tracking, or object detection, scale changes can reach factors of 10 or more. In these cases, it is intuitive that the ideal prediction should scale proportionally to the input. For example, the segmentation map of a nearby pedestrian should be easily converted to that of a distant person simply by downscaling.
+
+Convolutional Neural Networks (CNNs) demonstrate state-of-the-art performance in a wide range of tasks. Yet, despite their built-in translation equivariance, they do not have a particular mechanism for dealing with scale changes. One way to make CNNs account for scale is to train them with data augmentation Barnard & Casasent (1991). This is, however, suitable only for global transformations. As an alternative, Henriques & Vedaldi (2017) and Tai et al. (2019) use the canonical coordinates of scale transformations to reduce scaling to well-studied translations. While these approaches do allow for scale equivariance, they consequently break translation equivariance.
+
+Several attempts have thus been made to extend CNNs to both scale and translation symmetry simultaneously. Some works use input or filter resizing to account for scaling in deep layers Xu et al. (2014); Kanazawa et al. (2014). Such methods are suboptimal due to the time complexity of tensor resizing and the need for interpolation. In Ghosh & Gupta (2019) the authors pre-calculate filters defined on several scales to build scale-invariant networks, while ignoring the important case of scale equivariance. In contrast, Worrall & Welling (2019) employ the theory of semigroup equivariant networks with scale-space as an example; however, this method is only suitable for integer downscale factors and therefore limited.
+
+In this paper we develop a theory of scale-equivariant networks. We demonstrate the concept of steerable filter parametrization which allows for scaling without the need for tensor resizing. Then we derive scale-equivariant convolution and demonstrate a fast algorithm for its implementation. Furthermore, we experiment to determine to what degree the mathematical properties actually hold true. Finally, we conduct a set of experiments comparing our model with other methods for scale equivariance and local scale invariance.
+
+The proposed model has the following advantages compared to other scale-equivariant models:
+
+1. It is equivariant to scale transformations with arbitrary discrete scale factors and is not limited to either integer scales or scales tailored by the image pixel grid.   
+2. It does not rely on any image resampling techniques during training, and therefore, produces deep scale-equivariant representations free of any interpolation artifacts.   
+3. The algorithm is based on the combination of tensor expansion and 2-dimensional convolution, and demonstrates the same computation time as the general CNN with a comparable filter bank.
+
+# 2 PRELIMINARIES
+
+Before we move into scale-equivariant mappings, we discuss some aspects of equivariance, scaling transformations, symmetry groups, and the functions defined on them. For simplicity, in this section, we consider only 1-dimensional functions. The generalization to higher-dimensional cases is straightforward.
+
+Equivariance Let us consider some mapping $g$ . It is equivariant under $L _ { \theta }$ if and only if there exists $L _ { \theta } ^ { \prime }$ such that $g \circ L _ { \theta } = L _ { \theta } ^ { \prime } \circ g$ . In case $L _ { \theta } ^ { \prime }$ is the identity mapping, the function $g$ is invariant.
+
+In this paper we consider scaling transformations. In order to guarantee the equivariance of the predictions to such transformations, and to improve the performance of the model, we seek to incorporate this property directly inside CNNs.
+
+Scaling Given a function $f : \mathbb { R } \to \mathbb { R }$ , a scale transformation is defined as follows:
+
+$$
+L _ { s } [ f ] ( x ) = f ( s ^ { - 1 } x ) , \quad \forall s > 0
+$$
+
+We refer to cases with $s > 1$ as upscale and to cases with $s < 1$ as downscale. If we convolve the downscaled function with an arbitrary filter $\psi$ and perform a simple change of variables inside the integral, we get the following property:
+
+$$
+\begin{array} { l } { \displaystyle [ L _ { s } [ f ] \star \psi ] ( x ) = \int _ { \mathbb R } L _ { s } [ f ] ( x ^ { \prime } ) \psi ( x ^ { \prime } - x ) d x ^ { \prime } = \int _ { \mathbb R } f ( s ^ { - 1 } x ^ { \prime } ) \psi ( x ^ { \prime } - x ) d x ^ { \prime } } \\ { \displaystyle \qquad = s \int _ { \mathbb R } f ( s ^ { - 1 } x ^ { \prime } ) \psi ( s ( s ^ { - 1 } x ^ { \prime } - s ^ { - 1 } x ) ) d ( s ^ { - 1 } x ^ { \prime } ) = s L _ { s } [ f \star L _ { s ^ { - 1 } } [ \psi ] ] ( x ) } \end{array}
+$$
+
+In other words, convolution of the downscaled function with a filter can be expressed through a convolution of the function with the correspondingly upscaled filter where downscaling is performed afterwards. Equation 2 shows us that the standard convolution is not scale-equivariant.
+
+Steerable Filters In order to make computations simpler, we reparametrize $\psi _ { \sigma } ( x ) = \sigma ^ { - 1 } \psi ( \sigma ^ { - 1 } x )$ , which has the following property:
+
+$$
+L _ { s ^ { - 1 } } [ \psi _ { \sigma } ] ( x ) = \psi _ { \sigma } ( s x ) = s ^ { - 1 } \psi _ { s ^ { - 1 } \sigma } ( x )
+$$
+
+It gives a shorter version of Equation 2:
+
+$$
+L _ { s } [ f ] \star \psi _ { \sigma } = L _ { s } [ f \star \psi _ { s ^ { - 1 } \sigma } ]
+$$
+
+We will refer to such a parameterization of filters as Steerable Filters because the scaling of these filters is the transformation of its parameters. Note that we may construct steerable filters from any function. This has the important consequence that it does not restrict our approach. Rather it will make the analysis easier for discrete data. Moreover, note that any linear combination of steerable filters is still steerable.
+
+Scale-Translation Group All possible scales form the scaling group $S$ . Here we consider the discrete scale group, i.e. scales of the form $\dots a ^ { - 1 } , a ^ { - 1 } , 1 , a , a ^ { \tilde { 2 } } , \tilde { \dots } .$ with base $a$ as a parameter of our method. Analysis of this group by itself breaks the translation equivariance of CNNs. Thus we seek to incorporate scale and translation symmetries into CNNs, and, therefore consider the ScaleTranslation Group $H$ . It is a semidirect product of the scaling group $S$ and the group of translations $T \cong \mathbb { R }$ . In other words: $H = \{ ( s , t ) | s \in S , t \in T \}$ . For multiplication of group elements, we have $( s _ { 2 } , t _ { 2 } ) \cdot ( s _ { 1 } , t _ { 1 } ) = ( s _ { 2 } s _ { 1 } , s _ { 2 } t _ { 1 } + t _ { 2 } )$ and for the inverse $( s _ { 2 } , t _ { 2 } ) ^ { - 1 } \cdot ( s _ { 1 } , t _ { 1 } ) = ( s _ { 2 } ^ { - 1 } s _ { 1 } , s _ { 2 } ^ { - 1 } ( t _ { 1 } - t _ { 2 } ) )$ . Additionally, for the corresponding scaling and translation transformations, we have $L _ { s t } = L _ { s } L _ { t } \neq$ $L _ { t } L _ { s }$ , which means that the order of the operations matters.
+
+From now on, we will work with functions defined on groups, i.e. mappings $H \to \mathbb { R }$ . Note, that simple function $f : \mathbb { R }  \mathbb { R }$ may be considered as a function on $H$ with constant value along the $S$ axis. Therefore, Equation 4 holds true for functions on $H$ as well. One thing we should keep in mind is that when we apply $L _ { s }$ to functions on $H$ and $\mathbb { R }$ we use different notations. For example $L _ { s } [ f ] ( x ^ { \prime } ) = f ( s ^ { - 1 } x ^ { \prime } )$ and $\begin{array} { r } { \bar { L } _ { s } [ f ] ( s ^ { \prime } , t ^ { \prime } ) = f ( ( s , 0 ) ^ { - 1 } ( s ^ { \prime } , t ^ { \prime } ) ) = f ( s ^ { - 1 } s ^ { \prime } , s ^ { - 1 } t ^ { \prime } ) } \end{array}$
+
+Group-Equivariant Convolution Given group $G$ and two functions $f$ and $\psi$ defined on it, $G$ - equivariant convolution is given by
+
+$$
+[ f \star _ { G } \psi ] ( g ) = \int _ { G } f ( g ^ { \prime } ) L _ { g } [ \psi ] ( g ^ { \prime } ) d \mu ( g ^ { \prime } ) = \int _ { G } f ( g ^ { \prime } ) \psi ( g ^ { - 1 } g ^ { \prime } ) d \mu ( g ^ { \prime } )
+$$
+
+Here $\mu ( g ^ { \prime } )$ is the Haar measure also known as invariant measure Folland (2016). For $T \cong \mathbb { R }$ we have $d \mu ( g ^ { \prime } ) = d g ^ { \prime }$ . For discrete groups, the Haar measure is the counting measure, and integration becomes a discrete sum. This formula tells us that the output of the convolution evaluated at point $g$ is the inner product between the function $f$ and the transformed filter $L _ { g } [ \psi ]$ .
+
+# 3 SCALE-EQUIVARIANT MAPPINGS
+
+Now we define the main building blocks of scale-equivariant models.
+
+Scale Convolution In order to derive scale convolution, we start from group equivariant convolution with $G = H$ . We first use the property of semidirect product of groups which splits the integral, then choose the appropriate Haar measures, and finally use the properties of steerable filters. Given the function $f ( s , t )$ and a steerable filter $\psi _ { \sigma } ( s , t )$ defined on $H$ , a scale convolution is given by:
+
+$$
+\begin{array} { l } { { [ f \star _ { \cal H } \psi _ { \sigma } ] ( s , t ) = \displaystyle \int _ { \cal S } \int _ { \cal T } f ( s ^ { \prime } , t ^ { \prime } ) { \cal L } _ { s t } [ \psi _ { \sigma } ] ( s ^ { \prime } , t ^ { \prime } ) d \mu ( s ^ { \prime } ) d \mu ( t ^ { \prime } ) } } \\ { { { } } } \\ { { { } = \displaystyle \sum _ { s ^ { \prime } } \displaystyle \int _ { \cal T } f ( s ^ { \prime } , t ^ { \prime } ) \psi _ { s \sigma } ( s ^ { - 1 } s ^ { \prime } , t ^ { \prime } - t ) d t ^ { \prime } = \displaystyle \sum _ { s ^ { \prime } } [ f ( s ^ { \prime } , \cdot ) \star \psi _ { s \sigma } ( s ^ { - 1 } s ^ { \prime } , \cdot ) ] ( t ) } } \end{array}
+$$
+
+And for the case of $C _ { \mathrm { i n } }$ input and $C _ { \mathrm { o u t } }$ output channels we have:
+
+$$
+[ f \star _ { H } \psi _ { \sigma } ] _ { m } ( s , t ) = \sum _ { n = 1 } ^ { C _ { \mathrm { i n } } } \sum _ { s ^ { \prime } } [ f _ { n } ( s ^ { \prime } , \cdot ) \star \psi _ { n , m , s \sigma } ( s ^ { - 1 } s ^ { \prime } , \cdot ) ] ( t ) , \quad m = 1 \ldots C _ { \mathrm { o u t } }
+$$
+
+The proof of the equivariance of this convolution to transformations from $H$ is given in Appendix A.
+
+Kondor & Trivedi (2018) prove that a feed-forward neural network is equivariant to transformations from $G$ if and only if it is constructed from $\mathbf { G }$ -equivariant convolutional layers. Thus Equation 7 shows the most general form of scale-equivariant layers which allows for building scale-equivariant convolutional networks with such choice of $S$ . We will refer to models using scale-equivariant layers with steerable filters as Scale-Equivariant Steerable Networks, or shortly $S \bar { E } S N ^ { 1 }$
+
+Nonlinearities In order to guarantee the equivariance of the network to scale transformations, we use scale equivariant nonlinearities. We are free to use simple point-wise nonlinearities. Indeed, point-wise nonlinearities $\nu$ , like ReLU, commute with scaling transformations:
+
+$$
+\begin{array} { r } { [ \nu \circ L _ { s } [ f ] ] ( s ^ { \prime } , x ^ { \prime } ) = \nu ( L _ { s } [ f ] ( s ^ { \prime } , x ^ { \prime } ) ) = \nu ( f ( s ^ { - 1 } s ^ { \prime } , s ^ { - 1 } x ^ { \prime } ) ) } \\ { = \nu [ f ] ( s ^ { - 1 } s ^ { \prime } , s ^ { - 1 } x ^ { \prime } ) = [ L _ { s } \circ \nu [ f ] ] ( s ^ { \prime } , x ^ { \prime } ) } \end{array}
+$$
+
+Pooling Until now we did not discuss how to convert an equivariant mapping to invariant one. One way to do this is to calculate the invariant measure of the signal. In case of translation, such a measure could be the maximum value for example.
+
+First, we propose the maximum scale projection defined as $f ( s , x )  \operatorname* { m a x } _ { s } f ( s , x )$ . This transformation projects the function $f$ from $H$ to $T$ . Therefore, the representation stays equivariant to scaling, but loses all information about the scale itself.
+
+Second, we are free to use spatial max-pooling with a moving window or global max pooling. Transformation $f ( s , x ) \to \operatorname* { m a x } _ { x } { \overline { { f } } } ( s , x )$ projects the function $f$ from $H$ to $S$ . The obtained representation is invariant to scaling in spatial domain, however, it stores the information about scale.
+
+Finally, we can combine both of these pooling mechanisms in any order. The obtained transformation produces a scale invariant function. It is useful to utilize this transformation closer to the end of the network, when the deep representation must be invariant to nuisance input variations, but already has very rich semantic meaning.
+
+# 4 IMPLEMENTATION
+
+In this paragraph we discuss an efficient implementation of Scale-Equivariant Steerable Networks. We illustrate all algorithms in Figure 1. For simplicity we assume that zero padding is applied when it is needed for both the spatial axes and the scale axis.
+
+Filter Basis A direct implementation of Equation 7 is impossible due to several limitations. First, the infinite number of scales in $S$ calls for a discrete approximation. We truncate the scale group and limit ourselves to $N _ { S }$ scales and use discrete translations instead of continuous ones. Training of SESN involves searching for the optimal filter in functional space which is a problem by itself. Rather than solving it directly, we choose a complete basis of $N _ { b }$ steerable functions $\dot { \Psi } = \{ \psi _ { s ^ { - 1 } \sigma , i } \} _ { i = 1 } ^ { N _ { b } }$ and represent convolutional filter as a linear combination of basis functions with trainable parameters $w = \{ w _ { i } \} _ { i = 1 } ^ { N _ { b } }$ . In other words, we do the following substitution in Equation 7: $\begin{array} { r } { \psi _ { \sigma }  \kappa \stackrel { - } { = } \sum _ { i } w _ { i } \Psi _ { i } } \end{array}$
+
+In our experiments we use a basis of 2D Hermite polynomials with 2D Gaussian envelope, as it demonstrates good results. The basis is pre-calculated for all scales and fixed. For filters of size $V \times V$ , the basis is stored as an array of shape $[ N _ { b } , S , V , V ]$ . See Appendix C for more details.
+
+Conv $T \to H$ If the input signal is just a function on $T$ with spatial size $U \times U$ , stored as an array of shape $[ C _ { \mathrm { i n } } , U , U ]$ , then Equation 7 can be simplified. The summation over $S$ degenerates, and the final result can be written in the following form:
+
+$$
+\mathtt { c o n v T H } ( f , w , \Psi ) = \mathtt { s q u e e } z \in \left( \mathtt { c o n v } 2 \mathrm { d } ( f , \mathtt { e x p a n d } ( w \times \Psi ) ) \right)
+$$
+
+Here $w$ is an array of shape $[ C _ { \mathrm { o u t } } , C _ { \mathrm { i n } } , N _ { b } ]$ . We compute filter $w \times \Psi$ of shape $[ C _ { \mathrm { o u t } } , C _ { \mathrm { i n } } , S , V , V ]$ and expand it to shape $[ C _ { \mathrm { o u t } } , C _ { \mathrm { i n } } S , V , V ]$ . Then we use standard 2D convolution to produce the output with $C _ { \mathrm { o u t } } S$ channels and squeeze it to shape $[ C _ { \mathrm { o u t } } , S , U , U ]$ . Note that the output can be viewed as a stack of feature maps, where all the features in each spatial position are vectors of $S$ components instead of being scalars as in standard CNNs.
+
+Conv $H  H$ The function on $H$ has a scale axis and therefore there are two options for choosing weights of the convolutional filter. The filter may have just one scale and, therefore, does not capture the correlations between different scales of the input function; or, it may have a non-unitary extent $K _ { S }$ in the scale axis and capture the correlation between $K _ { S }$ neighboring scales. We refer to the second case as interscale interaction.
+
+It the first case $w$ has shape $[ C _ { \mathrm { o u t } } , C _ { \mathrm { i n } } , N _ { b } ]$ and Equation 7 degenerates in the same way as before
+
+$$
+\mathtt { c o n v H H } ( f , w , \Psi ) = \mathtt { s q u e e } z \in \left( \mathtt { c o n v } 2 \mathrm { d } ( \exp \mathsf { a n d } ( f ) , \mathtt { e x p a n d } ( w \times \Psi ) ) \right)
+$$
+
+We expand $f$ to an array of shape $[ C _ { \mathrm { i n } } S , U , U ]$ and expand $w \times \Psi$ to have shape $[ C _ { \mathrm { o u t } } S , C _ { \mathrm { i n } } S , V , V ]$ .   
+The result of the convolution is then squeezed in the same way as before.
+
+In the case of interscale interaction, $w$ has shape $[ C _ { \mathrm { o u t } } , C _ { \mathrm { i n } } , K _ { S } , N _ { b } ]$ . We iterate over all scales in interaction, shift $f$ for each scale, choose a corresponding part of $w$ , and apply convHH to them. We sum the obtained $K _ { S }$ results afterwards.
+
+![](images/3a842654dbcbf77a7b84f924b43c512bc70871657b145e3b30693fa35d2dfeda.jpg)  
+Figure 1: Left: the way steerable filters are computed using steerable filter basis. Middle and right: a representation of scale-convolution using Equation 9 and Equation 10. As an example we use input signal $f$ with 3 channels. It has 1 scale on $T$ and 4 scales on $H$ . It is convolved with filter $\kappa = w \times \Psi$ without scale interaction, which produces the output with 2 channels and 4 scales as well. Here we represent only channels of the signals and the filter. Spatial components are hidden for simplicity.
+
+# 5 RELATED WORK
+
+Various works on group-equivariant convolutional networks have been published recently. These works have considered roto-translation groups in 2D Cohen & Welling (2016a); Hoogeboom et al. (2018); Worrall et al. (2017); Weiler & Cesa (2019) and 3D Worrall & Brostow (2018); Kondor (2018); Thomas et al. (2018) and rotation equivariant networks in 3D Cohen et al. (2017); Esteves et al. (2018); Cohen et al. (2019). In Freeman & Adelson (1991) authors describe the algorithm for designing steerable filters for rotations. Rotation steerable filters are used in Cohen & Welling (2016b); Weiler et al. (2018a;b) for building equivariant networks. In Jacobsen et al. (2017) the authors build convolutional blocks locally equivariant to arbitrary $k$ -parameter Lie group by using a steerable basis. And in Murugan et al. the authors discuss the approach for learning steerable filters from data. To date, the majority of papers on group equivariant networks have considered rotations in 2D and 3D, but have not payed attention to scale symmetry. As we have argued above, it is a fundamentally different case.
+
+Many papers and even conferences have been dedicated to image scale-space — a concept where the image is analyzed together with all its downscaled versions. Initially introduced in Iijima (1959) and later developed by Witkin (1987); Perona & Malik (1990); Lindeberg (2013) scale space relies on the scale symmetry of images. The differential structure of the image Koenderink (1984) allows one to make a connection between image formation mechanisms and the space of solutions of the 2- dimensional heat equation, which significantly improved the image analysis models in the pre-deep learning era.
+
+One of the first works on scale equivariance and local scale invariance in the framework of CNNs was proposed by Xu et al. (2014) named SiCNN. The authors describe the model with siamese CNNs, where the filters of each instance are rescaled using interpolation techniques. This is the simplest case of equivariance where no interaction between different scales is done in intermediate layers. In SI-ConvNet by Kanazawa et al. (2014) the original network is modified such that, in each layer, the input is first rescaled, then convolved and rescaled back to the original size. Finally, the response with the maximum values is chosen between the scales. Thus, the model is locally scale-invariant. In
+
+Table 1: Comparing SESN to SiCNN $\mathrm { X u }$ et al. (2014), SI-ConvNet Kanazawa et al. (2014), SEVF Marcos et al. (2018), DSS Worrall & Welling (2019) and SS-CNN Ghosh & Gupta (2019). “Interscale” refers to the ability of capturing interscale interactions with kernels of non-unitary scale extent. “Grid” stands for the scales which generate images which lie exactly on the initial pixel grid.   
+
+<table><tr><td>Method</td><td>Equivariance</td><td>Admissible Scales</td><td>Approach</td><td>Interscale</td></tr><tr><td>SiCNN</td><td></td><td>Grid</td><td>Filter Rescaling</td><td></td></tr><tr><td>SI-ConvNet</td><td></td><td>Grid</td><td>Input Rescaling</td><td></td></tr><tr><td>SEVF</td><td></td><td>Grid</td><td>Input Rescaling</td><td></td></tr><tr><td>DSS</td><td>&lt;x&lt;/</td><td>Integer</td><td>Filter Dilation</td><td>xx&lt;&gt;</td></tr><tr><td>SS-CNN</td><td>X</td><td>Any</td><td>Steerable Filters</td><td>X</td></tr><tr><td>SESN, Ours</td><td>√</td><td>Any</td><td>Steerable Filters</td><td>√</td></tr></table>
+
+Marcos et al. (2018), in the SEVF model, the input of the layers is rescaled and convolved multiple times to form vector features instead of scalar ones. The length of the vector in each position is the maximum magnitude of the convolution, while the direction of the angle encodes the scale of the image which gave this response. These scale-equivariant networks rely on image rescaling which is quite slow. Worrall & Welling (2019) (DSS) generalize the concept of scale-space to deep networks. They use filter dilation to analyze the images on different scales. While this approach is as fast as the standard CNN, it is restricted only to integer downscale factors $2 , 4 , 8 \dots$ In Ghosh & Gupta (2019), while discussing SS-CNN the authors use scale-steerable filters to deal with scale changes. The paper does not discuss equivariance, which is an important aspect for scale.
+
+We summarize the information about these models in Table 1. In contrast to other scale-equivariant models, SESN uses steerable filters which allows for fast scale-convolution with no limitation of flexibility. With the framework of Scale-Equivariant Convolutional Networks we are free to build both equivariant and invariant models of different kinds.
+
+![](images/9d5081dc572954800739488564f5c9304a98127ff3d55fe8a82463473ff120dc.jpg)  
+Figure 2: Equivariance error $\Delta$ as a function of the number of layers (left), downscaling applied to the input image (middle), and as a function of number of scales in interscale interactions (right). The bars indicate standard deviation.
+
+In this section we conduct the experiments and compare various methods for working with scale variations in input data. Alongside SESN, we test local scale invariant SI-ConvNet and SS-CNN, scale equivariant SiCNN, SEVF and DSS. For SEVF, DSS and SS-CNN we use the code provided by authors, while for others we reimplement the main buildings blocks.
+
+We provide additional experimental results on time performance of all these methods in Appendix B. Due to the algorithm proposed in Section 4 SESN allows for training several times faster than other methods which rely on image rescaling.
+
+<table><tr><td>Method</td><td>(28×28)</td><td>(28 × 28) +</td><td>(56 × 56)</td><td>(56 × 56) +</td><td># Params</td></tr><tr><td>CNN</td><td>2.56 ± 0.04</td><td>1.96 ± 0.07</td><td>2.02 ± 0.07</td><td>1.60 ± 0.09</td><td>495K</td></tr><tr><td>SiCNN</td><td>2.40 ± 0.03</td><td>1.86 ± 0.10</td><td>2.02 ± 0.14</td><td>1.59 ± 0.03</td><td>497K</td></tr><tr><td>SI-ConvNet</td><td>2.40± 0.12</td><td>1.94 ± 0.07</td><td>1.82 ± 0.11</td><td>1.59 ± 0.10</td><td>495K</td></tr><tr><td>SEVF Scalar</td><td>2.30 ± 0.06</td><td>1.96 ± 0.07</td><td>1.87 ± 0.09</td><td>1.62 ± 0.07</td><td>494 K</td></tr><tr><td>SEVF Vector</td><td>2.63 ± 0.09</td><td>2.23 ± 0.09</td><td>2.12 ± 0.13</td><td>1.81 ± 0.09</td><td>475 K</td></tr><tr><td>DSS Scalar</td><td>2.53 ± 0.10</td><td>2.04± 0.08</td><td>1.92 ± 0.08</td><td>1.57 ± 0.08</td><td>494 K</td></tr><tr><td>DSS Vector</td><td>2.58 ± 0.11</td><td>1.95 ± 0.07</td><td>1.97 ± 0.08</td><td>1.57 ± 0.09</td><td>494 K</td></tr><tr><td>SS-CNN</td><td>2.32 ± 0.15</td><td>2.10 ±0.15</td><td>1.84 ± 0.10</td><td>1.76 ± 0.07</td><td>494 K</td></tr><tr><td>SESN Scalar</td><td>2.10 ±0.10</td><td>1.79 ± 0.09</td><td>1.74 ± 0.09</td><td>1.50 ± 0.07</td><td>495K</td></tr><tr><td>SESN Vector</td><td>2.08 ±0.09</td><td>1.76 ± 0.08</td><td>1.68 ± 0.06</td><td>1.42 ± 0.07</td><td>495 K</td></tr></table>
+
+Table 2: Classification error of different methods on MNIST-scale dataset, lower is better. In experiment we use image resolution of $2 8 \times 2 8$ and $5 6 \times 5 6$ . We test both the regime without data augmentation, and the regime with scaling data augmentation, denoted with $" + "$ . All results are reported as mean $\pm$ std over 6 different fixed realizations of the dataset. The best results are bold.
+
+# 6.1 EQUIVARIANCE ERROR
+
+We have presented scale-convolution which is equivariant to scale transformation and translation for continuous signals. While translation equivariance holds true even for discretized signals and filters, scale equivariance may not be exact. Therefore, before starting any experiments, we check to which degree the predicted properties of scale-convolution hold true. We do so by measuring the difference $\dot { \Delta } = \| [ L _ { s } \mathbf { \hat { \Phi } } ( f ) - \Phi \mathbf { \hat { L } } _ { s } ( f ) \| _ { 2 } ^ { 2 } / \| L _ { s } \Phi ( f ) \| _ { 2 } ^ { 2 }$ , where $\Phi$ is scale-convolution with randomly initialized weights.
+
+In case of perfect equivariance the difference is equal to zero. We calculate the error on randomly sampled images from the STL-10 dataset Coates et al. (2011). The results are represented in Figure 2. The networks on the left and on the middle plots do not have interscale interactions. The networks on the middle and on the right plots consist of just one layer. We use $N _ { S } = 5 , 1 3 , 5$ scales for the networks on the left, the middle, and the right plots respectively. While discretization introduces some error, it stays very low, and is not much higher than $6 \%$ for the networks with 50 layers. The difference, however, increases if the input image is downscaled more than 16 times. Therefore, we are free to use deep networks. However, we should pay extra attention to extreme cases where scale changes are of very big magnitude. These are quite rare but still appear in practice. Finally, we see that using SESN with interscale interaction introduces extra equivariance error due to the truncation of $S$ . We will build the networks with either no scale interaction or interaction of 2 scales.
+
+# 6.2 MNIST-SCALE
+
+Following Kanazawa et al. (2014); Marcos et al. (2018); Ghosh & Gupta (2019) we conduct experiments on the MNIST-scale dataset. We rescale the images of the MNIST dataset LeCun et al. (1998) to $0 . 3 - 1 . 0$ of the original size and pad them with zeros to retain the initial resolution. The scaling factors are sampled uniformly and independently for each image. The obtained dataset is then split into 10,000 for training, 2,000 for evaluation and 50,000 for testing. We generate 6 different realizations and fix them for all experiments.
+
+As a baseline model we use the model described in Ghosh & Gupta (2019), which currently holds the state-of-the-art result on this dataset. It consists of 3 convolutional and 2 fully-connected layers. Each layer has filters of size $7 \times 7$ . We keep the number of trainable parameters almost the same for all tested methods. This is achieved by varying the number of channels. For scale equivariant models we add scale projection at the end of the convolutional block.
+
+For SiCNN, DSS, SEVF and our model, we additionally train counterparts where after each convolution, an extra projection layer is inserted. Projection layers transform vector features in each spatial position of each channel into scalar ones. All of the layers have now scalar inputs instead of vector inputs. Therefore, we denote these models with “Scalar”. The original models are denoted as “Vector”. The exact type of projection depends on the way the vector features are constructed. For SiCNN, DSS, and SESN, we use maximum pooling along the scale dimension, while for SEVF, it is a calculation of the $L _ { 2 }$ -norm of the vector.
+
+All models are trained with the Adam optimizer Kingma & Ba (2014) for 60 epochs with a batch size of 128. Initial learning rate is set to 0.01 and divided by 10 after 20 and 40 epochs. We conduct the experiments with 4 different settings. Following the idea discussed in Ghosh & Gupta (2019), in addition to the standard setting we train the networks with input images upscaled to $5 6 \times 5 6$ using bilinear interpolation. This results in all image transformations performed by the network becoming more stable, which produces less interpolation artifacts. For both input sizes we conduct the experiments without data augmentation and with scaling augmentation, which results in 4 setups in total. We run the experiments on 6 different realizations of MNIST-scale and report mean $\pm$ std calculated over these runs.
+
+The obtained results are summarized in Table 2. The reported errors may differ a bit from the ones in the original paper because of the variations in generated datasets and slightly different training procedure. Nevertheless, we try to keep our configuration as close as possible to Ghosh & Gupta (2019) which currently demonstrated the best classification accuracy on MNIST-scale. For example, SS-CNN reports error of $1 . 9 1 \pm 0 . 0 4$ in Ghosh & Gupta (2019) while it has $1 . 8 4 \pm 0 . 1 0$ in our experiments.
+
+SESN significantly outperforms other methods in all 4 regimes. “Scalar” versions of it already outperform all previous methods, and “Vector” versions make the gain even more significant. The global architectures of all models are the same for all rows, which indicates that the way scale convolution is done plays an important role.
+
+# 6.3 STL-10
+
+In order to evaluate the role of scale equivariance in natural image classification, we conduct the experiments on STL-10 dataset Coates et al. (2011). This dataset consists of 8,000 training and 5,000 testing labeled images. Additionally, it includes 100,000 unlabeled images. The images have a resolution of $9 6 \times 9 6$ pixels and RGB channels. Labeled images belong to 10 classes such as bird, horse or car. We use only the labeled subset to demonstrate the performance of the models in the low data regime.
+
+The dataset is normalized by subtracting the per-channel mean and dividing by the per-channel standard deviation. During training, we augment the dataset by applying 12 pixel zero padding and randomly cropping the images to size $9 6 \times 9 6$ . Additionally, random horizontal flips with probability $5 0 \%$ and Cutout DeVries & Taylor (2017) with 1 hole of 32 pixels are used.
+
+As a baseline we choose WideResNet Zagoruyko & Komodakis (2016) with 16 layers and a widening factor of 8. We set dropout probability to 0.3 in all blocks. We train SESN-A with just vector features. For SESN-B we use maximum scalar projection several times in the intermediate layers, and for SESN-C we use interscale interaction.
+
+Table 3: Classification error on STL-10. The best results are bold. We additionally report the current best result achieved by Harm WRN from Ulicny et al. (2019).   
+
+<table><tr><td>Method</td><td>Error, %</td><td>#Params</td></tr><tr><td>WRN</td><td>11.48</td><td>11.0 M</td></tr><tr><td>SiCNN</td><td>11.62</td><td>11.0 M</td></tr><tr><td>SI-ConvNet</td><td>12.48</td><td>11.0 M</td></tr><tr><td>DSS</td><td>11.28</td><td>11.0 M</td></tr><tr><td>SS-CNN</td><td>25.47</td><td>10.8 M</td></tr><tr><td>SESN-A</td><td>10.83</td><td>11.0 M</td></tr><tr><td>SESN-B</td><td>8.51</td><td>11.0 M</td></tr><tr><td>SESN-C</td><td>14.08</td><td>11.0 M</td></tr><tr><td>Harm WRN</td><td>9.55</td><td>11.0 M</td></tr></table>
+
+All models are trained for 1000 epochs with a batch size of 128. We use SGD optimizer with Nesterov momentum of 0.9 and weight decay of $5 \cdot 1 0 ^ { - 4 } $ . The initial learning rate is set to 0.1 and divided by 5 after 300, 400, 600 and 800 epochs.
+
+The results are summarized in Table 3. We found SEVF training unstable and therefore do not include it in the table. Pure scale-invariant SI-ConvNet and SS-CNN demonstrate significantly worse results than the baseline. We note the importance of equivariance for deep networks. We also find that SESN-C performs significantly worse than SESN-A and SESN-B due to high equivariance error caused by interscale interaction. SESN-B significantly improves the results of both WRN and DSS due to the projection between scales. The maximum scale projection makes the weights of the next layer to have a maximum receptive field in the space of scales. This is an easy yet effective method for capturing the correlations between different scales. This experiment shows that scale-equivariance is a very useful inductive bias for natural image classification with deep neural networks.
+
+To the best of our knowledge, the proposed method achieves a new state-of-the-art result on the STL-10 dataset in the supervised learning setting. The previous lowest error is demonstrated in Ulicny et al. (2019). The authors propose Harm WRN — a network where the convolutional kernels are represented as a linear combination of Discrete Cosine Transform filters.
+
+# 7 DISCUSSION
+
+In this paper, we have presented the theory of Scale-Equivariant Steerable Networks. We started from the scaling transformation and its application to continuous functions. We have obtained the exact formula for scale-equivariant mappings and demonstrated how it can be implemented for discretized signals. We have demonstrated that this approach outperforms other methods for scale-equivariant and local scale-invariant CNNs. It demonstrated new state-of-the-art results on MNIST-scale and on the STL-10 dataset in the supervised learning setting.
+
+We suppose that the most exciting possible application of SESN is in computer vision for autonomous vehicles. Rapidly changing distances between the objects cause significant scale variations which makes this well suited for our work. We especially highlight the direction of siamese visual tracking where the equivariance to principle transformations plays an important role.
+
+# ACKNOWLEDGMENTS
+
+We thank Daniel Worrall for insightful discussion, Thomas Andy Keller, Victor Garcia, Artem Moskalev and Konrad Groh for valuable comments and feedback.
+
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+Kai Sheng Tai, Peter Bailis, and Gregory Valiant. Equivariant transformer networks. arXiv preprint arXiv:1901.11399, 2019.
+
+Nathaniel Thomas, Tess Smidt, Steven Kearnes, Lusann Yang, Li Li, Kai Kohlhoff, and Patrick Riley. Tensor field networks: Rotation-and translation-equivariant neural networks for 3d point clouds. arXiv preprint arXiv:1802.08219, 2018.
+
+Matej Ulicny, Vladimir A Krylov, and Rozenn Dahyot. Harmonic networks with limited training samples. arXiv preprint arXiv:1905.00135, 2019.
+
+Maurice Weiler and Gabriele Cesa. General e (2)-equivariant steerable cnns. In Advances in Neural Information Processing Systems, pp. 14334–14345, 2019.
+
+Maurice Weiler, Mario Geiger, Max Welling, Wouter Boomsma, and Taco Cohen. 3d steerable cnns: Learning rotationally equivariant features in volumetric data. In Advances in Neural Information Processing Systems, pp. 10381–10392, 2018a.
+
+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, 2018b.
+
+Andrew P Witkin. Scale-space filtering. In Readings in Computer Vision, pp. 329–332. Elsevier, 1987.   
+Daniel Worrall and Gabriel Brostow. Cubenet: Equivariance to 3d rotation and translation. In Proceedings of the European Conference on Computer Vision (ECCV), pp. 567–584, 2018.   
+Daniel E Worrall and Max Welling. Deep scale-spaces: Equivariance over scale. arXiv preprint arXiv:1905.11697, 2019.   
+Daniel E Worrall, Stephan J Garbin, Daniyar Turmukhambetov, and Gabriel J Brostow. Harmonic networks: Deep translation and rotation equivariance. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 5028–5037, 2017.   
+Yichong Xu, Tianjun Xiao, Jiaxing Zhang, Kuiyuan Yang, and Zheng Zhang. Scale-invariant convolutional neural networks. arXiv preprint arXiv:1411.6369, 2014.   
+Sergey Zagoruyko and Nikos Komodakis. Wide residual networks. arXiv preprint arXiv:1605.07146, 2016.
+
+# A PROOF OF EQUIVARIANCE
+
+Let us first show that scale-convolution defined in Equation 6 is equivariant to translations.
+
+$$
+\begin{array} { l } { [ L _ { \hat { t } } [ f ] \star _ { H } \psi _ { \sigma } ] ( s , t ) = \displaystyle \sum _ { s ^ { \prime } } [ L _ { \hat { t } } [ f ] ( s ^ { \prime } , \cdot ) \star \psi _ { s \sigma } ( s ^ { - 1 } s ^ { \prime } , \cdot ) ] ( t ) } \\ { = \displaystyle \sum _ { s ^ { \prime } } L _ { \hat { t } } [ f ( s ^ { \prime } , \cdot ) \star \psi _ { s \sigma } ( s ^ { - 1 } s ^ { \prime } , \cdot ) ] ( t ) } \\ { = L _ { \hat { t } } \Big \{ \displaystyle \sum _ { s ^ { \prime } } [ f ( s ^ { \prime } , \cdot ) \star \psi _ { s \sigma } ( s ^ { - 1 } s ^ { \prime } , \cdot ) ] \Big \} ( t ) } \\ { = L _ { \hat { t } } [ f \star _ { H } \psi _ { \sigma } ] ( s , t ) } \end{array}
+$$
+
+Now we show that scale convolution is equivariant to scale transformations:
+
+$$
+\begin{array} { l } { [ L _ { \delta } [ f ] \star _ { H } \psi _ { \sigma } ] ( s , t ) = \displaystyle \sum _ { s ^ { \prime } } [ L _ { \delta } [ f ] ( s ^ { \prime } , \cdot ) \star \psi _ { s \sigma } ( s ^ { - 1 } s ^ { \prime } , \cdot ) ] ( t ) } \\ { = \displaystyle \sum _ { s ^ { \prime } } L _ { \delta } [ f ( \hat { s } ^ { - 1 } s ^ { \prime } , \cdot ) \star \psi _ { \hat { s } ^ { - 1 } s \sigma } ( s ^ { - 1 } s ^ { \prime } , \cdot ) ] ( t ) } \\ { = \displaystyle \sum _ { s ^ { \prime \prime } } [ f ( s ^ { \prime \prime } , \cdot ) \star \psi _ { \hat { s } ^ { - 1 } s \sigma } ( \hat { s } s ^ { - 1 } s ^ { \prime \prime } , \cdot ) ] ( \hat { s } ^ { - 1 } t ) } \\ { = [ f \star _ { H } \psi _ { \sigma } ] ( \hat { s } ^ { - 1 } s , \hat { s } ^ { - 1 } t ) } \\ { = L _ { \delta } [ f \star _ { H } \psi _ { \sigma } ] ( s , t ) } \end{array}
+$$
+
+Finally, we can use the property of semidirect product of groups
+
+$$
+L _ { \hat { s t } } [ f ] \star _ { H } \psi _ { \sigma } = L _ { \hat { s } } L _ { \hat { t } } [ f ] \star _ { H } \psi _ { \sigma } = L _ { \hat { s } } [ L _ { \hat { t } } [ f ] \star _ { H } \psi _ { \sigma } ] = L _ { \hat { s } } L _ { \hat { t } } [ f \star _ { H } \psi _ { \sigma } ] = L _ { \hat { s } \hat { t } } [ f \star _ { H } \psi _ { \sigma } ] = L _ { \hat { s t } } [ f \star _ { H } \psi _ { \sigma } ]
+$$
+
+# B TIME PERFORMANCE
+
+We report the average time per epoch of different methods for scale equivariance and local scale invariance in Table 4. Experimental setups from Section 6.2 are used. We used 1 Nvidia GeForce GTX 1080Ti GPU for training the models.
+
+The methods relying on image rescaling techniques during training (SiCNN, SI-ConvNet, SEVF) demonstrate significantly worse time performance that the ones, using either steerable filters or filter dilation. Additionally, we see that our method outperforms SS-CNN by a wide margin. Despite the similar filter sizes and comparable number of parameters between SS-CNN and SESN Scalar, the second one demonstrates significantly better results due to the algorithm proposed in Section 4. Finally, DSS performs slightly faster in some cases than our method as each convolution involves less FLOPs. Dilated filters are sparse, while steerable filters are dense.
+
+<table><tr><td>Method</td><td>28×28,s</td><td>56 × 56,s</td></tr><tr><td>CNN</td><td>3.8</td><td>3.8</td></tr><tr><td>SiCNN Scalar</td><td>13.5</td><td>18.9</td></tr><tr><td>SiCNN Vector</td><td>15.3</td><td>22.8</td></tr><tr><td>SI-ConvNet</td><td>18.4</td><td>33.1</td></tr><tr><td>SEVF Scalar</td><td>21.0</td><td>38.4</td></tr><tr><td>SEVF Vector</td><td>25.4</td><td>46.0</td></tr><tr><td>DSS Scalar</td><td>3.9</td><td>5.0</td></tr><tr><td>DSS Vector</td><td>3.9</td><td>4.8</td></tr><tr><td>SS-CNN</td><td>14.8</td><td>16.6</td></tr><tr><td>SESN Scalar</td><td>3.8</td><td>5.1</td></tr><tr><td>SESN Vector</td><td>3.8</td><td>6.8</td></tr></table>
+
+Table 4: Average time per epoch during training on input data with resolution $2 8 \times 2 8$ and $5 6 \times 5 6$ .
+
+# C BASIS
+
+Assuming that the center of the filter is point $( 0 , 0 )$ in coordinates $( x , y )$ , we use the filters of the following form:
+
+$$
+\psi _ { \sigma } ( x , y ) = A \frac { 1 } { \sigma ^ { 2 } } H _ { n } \Big ( \frac { x } { \sigma } \Big ) H _ { m } \Big ( \frac { y } { \sigma } \Big ) \exp \Big [ - \frac { x ^ { 2 } + y ^ { 2 } } { 2 \sigma ^ { 2 } } \Big ]
+$$
+
+Here $A$ is a constant independent on $\sigma$ , $H _ { n }$ — Hermite polynomial of the $n$ -th order. We iterate over increasing pairs of $n , m$ to generate the required number of functions.
+
+# D MODEL CONFIGURATION
+
+# D.1 MNIST-SCALE
+
+Table 5: Number of channels in convolutional layers, number of units in fully-connected layers and number of scales used by different models in Section 6.2.   
+
+<table><tr><td>Method</td><td>Conv 1</td><td>Conv 2</td><td>Conv 3</td><td>FC1</td><td># Scales</td></tr><tr><td>CNN</td><td>32</td><td>63</td><td>95</td><td></td><td>1</td></tr><tr><td>SiCNN</td><td>32</td><td>63</td><td>95</td><td></td><td>7</td></tr><tr><td>SI-ConvNet</td><td>32</td><td>63</td><td>95</td><td></td><td>7</td></tr><tr><td>SEVF Scalar</td><td>32</td><td>63</td><td>95</td><td>256</td><td>8</td></tr><tr><td>SEVF Vector</td><td>23</td><td>45</td><td>68</td><td></td><td>8</td></tr><tr><td>DSS</td><td>32</td><td>63</td><td>95</td><td></td><td>4</td></tr><tr><td>SS-CNN</td><td>30</td><td>60</td><td>90</td><td></td><td>6</td></tr><tr><td>SESN</td><td>32</td><td>63</td><td>95</td><td></td><td>4</td></tr></table>
+
+D.2 STL-10   
+
+<table><tr><td>Method</td><td>Block 1</td><td>Block 2</td><td>Block 3</td><td># Scales</td></tr><tr><td>CNN</td><td>16</td><td>32</td><td>64</td><td>1</td></tr><tr><td>SiCNN</td><td>16</td><td>32</td><td>64</td><td>3</td></tr><tr><td>SI-ConvNet</td><td>16</td><td>32</td><td>64</td><td>3</td></tr><tr><td>SEVF</td><td>11</td><td>23</td><td>45</td><td>3</td></tr><tr><td>DSS</td><td>16</td><td>32</td><td>64</td><td>4</td></tr><tr><td>SS-CNN</td><td>11</td><td>22</td><td>44</td><td>3</td></tr><tr><td>SESN</td><td>16</td><td>32</td><td>64</td><td>3</td></tr></table>
+
+Table 6: Number of channels in convolutional blocks and number of scales used by different models in Section 6.2. We report the number of channels up to the widening factor.

parse/train/HJgpugrKPS/HJgpugrKPS_content_list.json

@@ -0,0 +1,1732 @@
+[
+    {
+        "type": "text",
+        "text": "SCALE-EQUIVARIANT STEERABLE NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
+            735,
+            121
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Ivan Sosnovik∗, Michał Szmaja†, Arnold Smeulders   \nUvA-Bosch Delta Lab   \nUniversity of Amsterdam ",
+        "bbox": [
+            183,
+            146,
+            553,
+            189
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            227,
+            544,
+            242
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The effectiveness of Convolutional Neural Networks (CNNs) has been substantially attributed to their built-in property of translation equivariance. However, CNNs do not have embedded mechanisms to handle other types of transformations. In this work, we pay attention to scale changes, which regularly appear in various tasks due to the changing distances between the objects and the camera. First, we introduce the general theory for building scale-equivariant convolutional networks with steerable filters. We develop scale-convolution and generalize other common blocks to be scale-equivariant. We demonstrate the computational efficiency and numerical stability of the proposed method. We compare the proposed models to the previously developed methods for scale equivariance and local scale invariance. We demonstrate state-of-the-art results on the MNIST-scale dataset and on the STL-10 dataset in the supervised learning setting. ",
+        "bbox": [
+            233,
+            258,
+            764,
+            425
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            454,
+            336,
+            469
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Scale transformations occur in many image and video analysis tasks. They are a natural consequence of the variable distances among objects, or between objects and the camera. Such transformations result in significant changes in the input space which are often difficult for models to handle appropriately without careful consideration. At a high level, there are two modeling paradigms which allow a model to deal with scale changes: models can be endowed with an internal notion of scale and transform their predictions accordingly, or instead, models can be designed to be specifically invariant to scale changes. In image classification, when scale changes are commonly a factor of 2, it is often sufficient to make class prediction independent of scale. However, in tasks such as image segmentation, visual tracking, or object detection, scale changes can reach factors of 10 or more. In these cases, it is intuitive that the ideal prediction should scale proportionally to the input. For example, the segmentation map of a nearby pedestrian should be easily converted to that of a distant person simply by downscaling. ",
+        "bbox": [
+            174,
+            486,
+            825,
+            654
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Convolutional Neural Networks (CNNs) demonstrate state-of-the-art performance in a wide range of tasks. Yet, despite their built-in translation equivariance, they do not have a particular mechanism for dealing with scale changes. One way to make CNNs account for scale is to train them with data augmentation Barnard & Casasent (1991). This is, however, suitable only for global transformations. As an alternative, Henriques & Vedaldi (2017) and Tai et al. (2019) use the canonical coordinates of scale transformations to reduce scaling to well-studied translations. While these approaches do allow for scale equivariance, they consequently break translation equivariance. ",
+        "bbox": [
+            174,
+            660,
+            825,
+            757
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Several attempts have thus been made to extend CNNs to both scale and translation symmetry simultaneously. Some works use input or filter resizing to account for scaling in deep layers Xu et al. (2014); Kanazawa et al. (2014). Such methods are suboptimal due to the time complexity of tensor resizing and the need for interpolation. In Ghosh & Gupta (2019) the authors pre-calculate filters defined on several scales to build scale-invariant networks, while ignoring the important case of scale equivariance. In contrast, Worrall & Welling (2019) employ the theory of semigroup equivariant networks with scale-space as an example; however, this method is only suitable for integer downscale factors and therefore limited. ",
+        "bbox": [
+            174,
+            763,
+            825,
+            876
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper we develop a theory of scale-equivariant networks. We demonstrate the concept of steerable filter parametrization which allows for scaling without the need for tensor resizing. Then we derive scale-equivariant convolution and demonstrate a fast algorithm for its implementation. Furthermore, we experiment to determine to what degree the mathematical properties actually hold true. Finally, we conduct a set of experiments comparing our model with other methods for scale equivariance and local scale invariance. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            188
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The proposed model has the following advantages compared to other scale-equivariant models: ",
+        "bbox": [
+            173,
+            212,
+            795,
+            227
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "1. It is equivariant to scale transformations with arbitrary discrete scale factors and is not limited to either integer scales or scales tailored by the image pixel grid.   \n2. It does not rely on any image resampling techniques during training, and therefore, produces deep scale-equivariant representations free of any interpolation artifacts.   \n3. The algorithm is based on the combination of tensor expansion and 2-dimensional convolution, and demonstrates the same computation time as the general CNN with a comparable filter bank. ",
+        "bbox": [
+            210,
+            238,
+            825,
+            347
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 PRELIMINARIES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            366,
+            339,
+            382
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Before we move into scale-equivariant mappings, we discuss some aspects of equivariance, scaling transformations, symmetry groups, and the functions defined on them. For simplicity, in this section, we consider only 1-dimensional functions. The generalization to higher-dimensional cases is straightforward. ",
+        "bbox": [
+            173,
+            397,
+            825,
+            454
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Equivariance Let us consider some mapping $g$ . It is equivariant under $L _ { \\theta }$ if and only if there exists $L _ { \\theta } ^ { \\prime }$ such that $g \\circ L _ { \\theta } = L _ { \\theta } ^ { \\prime } \\circ g$ . In case $L _ { \\theta } ^ { \\prime }$ is the identity mapping, the function $g$ is invariant. ",
+        "bbox": [
+            174,
+            460,
+            823,
+            489
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this paper we consider scaling transformations. In order to guarantee the equivariance of the predictions to such transformations, and to improve the performance of the model, we seek to incorporate this property directly inside CNNs. ",
+        "bbox": [
+            174,
+            496,
+            825,
+            537
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Scaling Given a function $f : \\mathbb { R } \\to \\mathbb { R }$ , a scale transformation is defined as follows: ",
+        "bbox": [
+            173,
+            545,
+            710,
+            560
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/13908d2afa72d06c5f750fd9545ff5fd44438e6667a9eef4ad001702836f5831.jpg",
+        "text": "$$\nL _ { s } [ f ] ( x ) = f ( s ^ { - 1 } x ) , \\quad \\forall s > 0\n$$",
+        "text_format": "latex",
+        "bbox": [
+            392,
+            565,
+            606,
+            584
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We refer to cases with $s > 1$ as upscale and to cases with $s < 1$ as downscale. If we convolve the downscaled function with an arbitrary filter $\\psi$ and perform a simple change of variables inside the integral, we get the following property: ",
+        "bbox": [
+            174,
+            598,
+            825,
+            640
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cecee878bf94f360d23a829c220e0eba59c655aa6e7d443c42052b97c09c1b7c.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle [ L _ { s } [ f ] \\star \\psi ] ( x ) = \\int _ { \\mathbb R } L _ { s } [ f ] ( x ^ { \\prime } ) \\psi ( x ^ { \\prime } - x ) d x ^ { \\prime } = \\int _ { \\mathbb R } f ( s ^ { - 1 } x ^ { \\prime } ) \\psi ( x ^ { \\prime } - x ) d x ^ { \\prime } } \\\\ { \\displaystyle \\qquad = s \\int _ { \\mathbb R } f ( s ^ { - 1 } x ^ { \\prime } ) \\psi ( s ( s ^ { - 1 } x ^ { \\prime } - s ^ { - 1 } x ) ) d ( s ^ { - 1 } x ^ { \\prime } ) = s L _ { s } [ f \\star L _ { s ^ { - 1 } } [ \\psi ] ] ( x ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            214,
+            646,
+            784,
+            715
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In other words, convolution of the downscaled function with a filter can be expressed through a convolution of the function with the correspondingly upscaled filter where downscaling is performed afterwards. Equation 2 shows us that the standard convolution is not scale-equivariant. ",
+        "bbox": [
+            176,
+            727,
+            823,
+            768
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Steerable Filters In order to make computations simpler, we reparametrize $\\psi _ { \\sigma } ( x ) = \\sigma ^ { - 1 } \\psi ( \\sigma ^ { - 1 } x )$ , which has the following property: ",
+        "bbox": [
+            169,
+            776,
+            823,
+            804
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5c5e01cd9c1c7f93935e09fa08fbca4b21f2258a0937e7a59b257eb0bc5c51cc.jpg",
+        "text": "$$\nL _ { s ^ { - 1 } } [ \\psi _ { \\sigma } ] ( x ) = \\psi _ { \\sigma } ( s x ) = s ^ { - 1 } \\psi _ { s ^ { - 1 } \\sigma } ( x )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            364,
+            810,
+            633,
+            829
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "It gives a shorter version of Equation 2: ",
+        "bbox": [
+            174,
+            842,
+            433,
+            858
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7651f206035b5725275d06c6bc6580fff017e6d183066a68fe1bc3c8467c7de9.jpg",
+        "text": "$$\nL _ { s } [ f ] \\star \\psi _ { \\sigma } = L _ { s } [ f \\star \\psi _ { s ^ { - 1 } \\sigma } ]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            403,
+            864,
+            594,
+            881
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We will refer to such a parameterization of filters as Steerable Filters because the scaling of these filters is the transformation of its parameters. Note that we may construct steerable filters from any function. This has the important consequence that it does not restrict our approach. Rather it will make the analysis easier for discrete data. Moreover, note that any linear combination of steerable filters is still steerable. ",
+        "bbox": [
+            174,
+            895,
+            825,
+            924
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            176,
+            103,
+            823,
+            145
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Scale-Translation Group All possible scales form the scaling group $S$ . Here we consider the discrete scale group, i.e. scales of the form $\\dots a ^ { - 1 } , a ^ { - 1 } , 1 , a , a ^ { \\tilde { 2 } } , \\tilde { \\dots } .$ with base $a$ as a parameter of our method. Analysis of this group by itself breaks the translation equivariance of CNNs. Thus we seek to incorporate scale and translation symmetries into CNNs, and, therefore consider the ScaleTranslation Group $H$ . It is a semidirect product of the scaling group $S$ and the group of translations $T \\cong \\mathbb { R }$ . In other words: $H = \\{ ( s , t ) | s \\in S , t \\in T \\}$ . For multiplication of group elements, we have $( s _ { 2 } , t _ { 2 } ) \\cdot ( s _ { 1 } , t _ { 1 } ) = ( s _ { 2 } s _ { 1 } , s _ { 2 } t _ { 1 } + t _ { 2 } )$ and for the inverse $( s _ { 2 } , t _ { 2 } ) ^ { - 1 } \\cdot ( s _ { 1 } , t _ { 1 } ) = ( s _ { 2 } ^ { - 1 } s _ { 1 } , s _ { 2 } ^ { - 1 } ( t _ { 1 } - t _ { 2 } ) )$ . Additionally, for the corresponding scaling and translation transformations, we have $L _ { s t } = L _ { s } L _ { t } \\neq$ $L _ { t } L _ { s }$ , which means that the order of the operations matters. ",
+        "bbox": [
+            173,
+            152,
+            825,
+            280
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "From now on, we will work with functions defined on groups, i.e. mappings $H \\to \\mathbb { R }$ . Note, that simple function $f : \\mathbb { R }  \\mathbb { R }$ may be considered as a function on $H$ with constant value along the $S$ axis. Therefore, Equation 4 holds true for functions on $H$ as well. One thing we should keep in mind is that when we apply $L _ { s }$ to functions on $H$ and $\\mathbb { R }$ we use different notations. For example $L _ { s } [ f ] ( x ^ { \\prime } ) = f ( s ^ { - 1 } x ^ { \\prime } )$ and $\\begin{array} { r } { \\bar { L } _ { s } [ f ] ( s ^ { \\prime } , t ^ { \\prime } ) = f ( ( s , 0 ) ^ { - 1 } ( s ^ { \\prime } , t ^ { \\prime } ) ) = f ( s ^ { - 1 } s ^ { \\prime } , s ^ { - 1 } t ^ { \\prime } ) } \\end{array}$ ",
+        "bbox": [
+            173,
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+            358
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Group-Equivariant Convolution Given group $G$ and two functions $f$ and $\\psi$ defined on it, $G$ - equivariant convolution is given by ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5988433815b304c1229351695466f4b8fe3fc481fadcd0844c1eef82a7d8bd97.jpg",
+        "text": "$$\n[ f \\star _ { G } \\psi ] ( g ) = \\int _ { G } f ( g ^ { \\prime } ) L _ { g } [ \\psi ] ( g ^ { \\prime } ) d \\mu ( g ^ { \\prime } ) = \\int _ { G } f ( g ^ { \\prime } ) \\psi ( g ^ { - 1 } g ^ { \\prime } ) d \\mu ( g ^ { \\prime } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Here $\\mu ( g ^ { \\prime } )$ is the Haar measure also known as invariant measure Folland (2016). For $T \\cong \\mathbb { R }$ we have $d \\mu ( g ^ { \\prime } ) = d g ^ { \\prime }$ . For discrete groups, the Haar measure is the counting measure, and integration becomes a discrete sum. This formula tells us that the output of the convolution evaluated at point $g$ is the inner product between the function $f$ and the transformed filter $L _ { g } [ \\psi ]$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 SCALE-EQUIVARIANT MAPPINGS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Now we define the main building blocks of scale-equivariant models. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Scale Convolution In order to derive scale convolution, we start from group equivariant convolution with $G = H$ . We first use the property of semidirect product of groups which splits the integral, then choose the appropriate Haar measures, and finally use the properties of steerable filters. Given the function $f ( s , t )$ and a steerable filter $\\psi _ { \\sigma } ( s , t )$ defined on $H$ , a scale convolution is given by: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8cb3ab9dc952636da09b958c251b9f4946d4bb834f2be16247217b30e4ccb17c.jpg",
+        "text": "$$\n\\begin{array} { l } { { [ f \\star _ { \\cal H } \\psi _ { \\sigma } ] ( s , t ) = \\displaystyle \\int _ { \\cal S } \\int _ { \\cal T } f ( s ^ { \\prime } , t ^ { \\prime } ) { \\cal L } _ { s t } [ \\psi _ { \\sigma } ] ( s ^ { \\prime } , t ^ { \\prime } ) d \\mu ( s ^ { \\prime } ) d \\mu ( t ^ { \\prime } ) } } \\\\ { { { } } } \\\\ { { { } = \\displaystyle \\sum _ { s ^ { \\prime } } \\displaystyle \\int _ { \\cal T } f ( s ^ { \\prime } , t ^ { \\prime } ) \\psi _ { s \\sigma } ( s ^ { - 1 } s ^ { \\prime } , t ^ { \\prime } - t ) d t ^ { \\prime } = \\displaystyle \\sum _ { s ^ { \\prime } } [ f ( s ^ { \\prime } , \\cdot ) \\star \\psi _ { s \\sigma } ( s ^ { - 1 } s ^ { \\prime } , \\cdot ) ] ( t ) } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "And for the case of $C _ { \\mathrm { i n } }$ input and $C _ { \\mathrm { o u t } }$ output channels we have: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/c5aa5fbbeeac0f26cf646a9653a08ff81f5b335b69d41fab9884696d6ccab114.jpg",
+        "text": "$$\n[ f \\star _ { H } \\psi _ { \\sigma } ] _ { m } ( s , t ) = \\sum _ { n = 1 } ^ { C _ { \\mathrm { i n } } } \\sum _ { s ^ { \\prime } } [ f _ { n } ( s ^ { \\prime } , \\cdot ) \\star \\psi _ { n , m , s \\sigma } ( s ^ { - 1 } s ^ { \\prime } , \\cdot ) ] ( t ) , \\quad m = 1 \\ldots C _ { \\mathrm { o u t } }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The proof of the equivariance of this convolution to transformations from $H$ is given in Appendix A. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Kondor & Trivedi (2018) prove that a feed-forward neural network is equivariant to transformations from $G$ if and only if it is constructed from $\\mathbf { G }$ -equivariant convolutional layers. Thus Equation 7 shows the most general form of scale-equivariant layers which allows for building scale-equivariant convolutional networks with such choice of $S$ . We will refer to models using scale-equivariant layers with steerable filters as Scale-Equivariant Steerable Networks, or shortly $S \\bar { E } S N ^ { 1 }$ ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Nonlinearities In order to guarantee the equivariance of the network to scale transformations, we use scale equivariant nonlinearities. We are free to use simple point-wise nonlinearities. Indeed, point-wise nonlinearities $\\nu$ , like ReLU, commute with scaling transformations: ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b2eeebeef0c9b259ac75e0c4773c21a6b103e96c44ddf703c1802c70cc02a5a1.jpg",
+        "text": "$$\n\\begin{array} { r } { [ \\nu \\circ L _ { s } [ f ] ] ( s ^ { \\prime } , x ^ { \\prime } ) = \\nu ( L _ { s } [ f ] ( s ^ { \\prime } , x ^ { \\prime } ) ) = \\nu ( f ( s ^ { - 1 } s ^ { \\prime } , s ^ { - 1 } x ^ { \\prime } ) ) } \\\\ { = \\nu [ f ] ( s ^ { - 1 } s ^ { \\prime } , s ^ { - 1 } x ^ { \\prime } ) = [ L _ { s } \\circ \\nu [ f ] ] ( s ^ { \\prime } , x ^ { \\prime } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Pooling Until now we did not discuss how to convert an equivariant mapping to invariant one. One way to do this is to calculate the invariant measure of the signal. In case of translation, such a measure could be the maximum value for example. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "First, we propose the maximum scale projection defined as $f ( s , x )  \\operatorname* { m a x } _ { s } f ( s , x )$ . This transformation projects the function $f$ from $H$ to $T$ . Therefore, the representation stays equivariant to scaling, but loses all information about the scale itself. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Second, we are free to use spatial max-pooling with a moving window or global max pooling. Transformation $f ( s , x ) \\to \\operatorname* { m a x } _ { x } { \\overline { { f } } } ( s , x )$ projects the function $f$ from $H$ to $S$ . The obtained representation is invariant to scaling in spatial domain, however, it stores the information about scale. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Finally, we can combine both of these pooling mechanisms in any order. The obtained transformation produces a scale invariant function. It is useful to utilize this transformation closer to the end of the network, when the deep representation must be invariant to nuisance input variations, but already has very rich semantic meaning. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4 IMPLEMENTATION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In this paragraph we discuss an efficient implementation of Scale-Equivariant Steerable Networks. We illustrate all algorithms in Figure 1. For simplicity we assume that zero padding is applied when it is needed for both the spatial axes and the scale axis. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Filter Basis A direct implementation of Equation 7 is impossible due to several limitations. First, the infinite number of scales in $S$ calls for a discrete approximation. We truncate the scale group and limit ourselves to $N _ { S }$ scales and use discrete translations instead of continuous ones. Training of SESN involves searching for the optimal filter in functional space which is a problem by itself. Rather than solving it directly, we choose a complete basis of $N _ { b }$ steerable functions $\\dot { \\Psi } = \\{ \\psi _ { s ^ { - 1 } \\sigma , i } \\} _ { i = 1 } ^ { N _ { b } }$ and represent convolutional filter as a linear combination of basis functions with trainable parameters $w = \\{ w _ { i } \\} _ { i = 1 } ^ { N _ { b } }$ . In other words, we do the following substitution in Equation 7: $\\begin{array} { r } { \\psi _ { \\sigma }  \\kappa \\stackrel { - } { = } \\sum _ { i } w _ { i } \\Psi _ { i } } \\end{array}$ ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In our experiments we use a basis of 2D Hermite polynomials with 2D Gaussian envelope, as it demonstrates good results. The basis is pre-calculated for all scales and fixed. For filters of size $V \\times V$ , the basis is stored as an array of shape $[ N _ { b } , S , V , V ]$ . See Appendix C for more details. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Conv $T \\to H$ If the input signal is just a function on $T$ with spatial size $U \\times U$ , stored as an array of shape $[ C _ { \\mathrm { i n } } , U , U ]$ , then Equation 7 can be simplified. The summation over $S$ degenerates, and the final result can be written in the following form: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/128a59139f7def3ef1c6918aed47f7ea431e6bba05ddfd59792f60d079b3c1ca.jpg",
+        "text": "$$\n\\mathtt { c o n v T H } ( f , w , \\Psi ) = \\mathtt { s q u e e } z \\in \\left( \\mathtt { c o n v } 2 \\mathrm { d } ( f , \\mathtt { e x p a n d } ( w \\times \\Psi ) ) \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Here $w$ is an array of shape $[ C _ { \\mathrm { o u t } } , C _ { \\mathrm { i n } } , N _ { b } ]$ . We compute filter $w \\times \\Psi$ of shape $[ C _ { \\mathrm { o u t } } , C _ { \\mathrm { i n } } , S , V , V ]$ and expand it to shape $[ C _ { \\mathrm { o u t } } , C _ { \\mathrm { i n } } S , V , V ]$ . Then we use standard 2D convolution to produce the output with $C _ { \\mathrm { o u t } } S$ channels and squeeze it to shape $[ C _ { \\mathrm { o u t } } , S , U , U ]$ . Note that the output can be viewed as a stack of feature maps, where all the features in each spatial position are vectors of $S$ components instead of being scalars as in standard CNNs. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Conv $H  H$ The function on $H$ has a scale axis and therefore there are two options for choosing weights of the convolutional filter. The filter may have just one scale and, therefore, does not capture the correlations between different scales of the input function; or, it may have a non-unitary extent $K _ { S }$ in the scale axis and capture the correlation between $K _ { S }$ neighboring scales. We refer to the second case as interscale interaction. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "It the first case $w$ has shape $[ C _ { \\mathrm { o u t } } , C _ { \\mathrm { i n } } , N _ { b } ]$ and Equation 7 degenerates in the same way as before ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/e65cff898868435e3aee58a331636287f343e513a85afceeb1662e2a44444fcb.jpg",
+        "text": "$$\n\\mathtt { c o n v H H } ( f , w , \\Psi ) = \\mathtt { s q u e e } z \\in \\left( \\mathtt { c o n v } 2 \\mathrm { d } ( \\exp \\mathsf { a n d } ( f ) , \\mathtt { e x p a n d } ( w \\times \\Psi ) ) \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We expand $f$ to an array of shape $[ C _ { \\mathrm { i n } } S , U , U ]$ and expand $w \\times \\Psi$ to have shape $[ C _ { \\mathrm { o u t } } S , C _ { \\mathrm { i n } } S , V , V ]$ .   \nThe result of the convolution is then squeezed in the same way as before. ",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "In the case of interscale interaction, $w$ has shape $[ C _ { \\mathrm { o u t } } , C _ { \\mathrm { i n } } , K _ { S } , N _ { b } ]$ . We iterate over all scales in interaction, shift $f$ for each scale, choose a corresponding part of $w$ , and apply convHH to them. We sum the obtained $K _ { S }$ results afterwards. ",
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/3a842654dbcbf77a7b84f924b43c512bc70871657b145e3b30693fa35d2dfeda.jpg",
+        "image_caption": [
+            "Figure 1: Left: the way steerable filters are computed using steerable filter basis. Middle and right: a representation of scale-convolution using Equation 9 and Equation 10. As an example we use input signal $f$ with 3 channels. It has 1 scale on $T$ and 4 scales on $H$ . It is convolved with filter $\\kappa = w \\times \\Psi$ without scale interaction, which produces the output with 2 channels and 4 scales as well. Here we represent only channels of the signals and the filter. Spatial components are hidden for simplicity. "
+        ],
+        "image_footnote": [],
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+    },
+    {
+        "type": "text",
+        "text": "5 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Various works on group-equivariant convolutional networks have been published recently. These works have considered roto-translation groups in 2D Cohen & Welling (2016a); Hoogeboom et al. (2018); Worrall et al. (2017); Weiler & Cesa (2019) and 3D Worrall & Brostow (2018); Kondor (2018); Thomas et al. (2018) and rotation equivariant networks in 3D Cohen et al. (2017); Esteves et al. (2018); Cohen et al. (2019). In Freeman & Adelson (1991) authors describe the algorithm for designing steerable filters for rotations. Rotation steerable filters are used in Cohen & Welling (2016b); Weiler et al. (2018a;b) for building equivariant networks. In Jacobsen et al. (2017) the authors build convolutional blocks locally equivariant to arbitrary $k$ -parameter Lie group by using a steerable basis. And in Murugan et al. the authors discuss the approach for learning steerable filters from data. To date, the majority of papers on group equivariant networks have considered rotations in 2D and 3D, but have not payed attention to scale symmetry. As we have argued above, it is a fundamentally different case. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Many papers and even conferences have been dedicated to image scale-space — a concept where the image is analyzed together with all its downscaled versions. Initially introduced in Iijima (1959) and later developed by Witkin (1987); Perona & Malik (1990); Lindeberg (2013) scale space relies on the scale symmetry of images. The differential structure of the image Koenderink (1984) allows one to make a connection between image formation mechanisms and the space of solutions of the 2- dimensional heat equation, which significantly improved the image analysis models in the pre-deep learning era. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "One of the first works on scale equivariance and local scale invariance in the framework of CNNs was proposed by Xu et al. (2014) named SiCNN. The authors describe the model with siamese CNNs, where the filters of each instance are rescaled using interpolation techniques. This is the simplest case of equivariance where no interaction between different scales is done in intermediate layers. In SI-ConvNet by Kanazawa et al. (2014) the original network is modified such that, in each layer, the input is first rescaled, then convolved and rescaled back to the original size. Finally, the response with the maximum values is chosen between the scales. Thus, the model is locally scale-invariant. In ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "table",
+        "img_path": "images/034542743359fb7a7b899fde9ba50daf8976a4bd6ccd05c94a4e0a5a4ca09f77.jpg",
+        "table_caption": [
+            "Table 1: Comparing SESN to SiCNN $\\mathrm { X u }$ et al. (2014), SI-ConvNet Kanazawa et al. (2014), SEVF Marcos et al. (2018), DSS Worrall & Welling (2019) and SS-CNN Ghosh & Gupta (2019). “Interscale” refers to the ability of capturing interscale interactions with kernels of non-unitary scale extent. “Grid” stands for the scales which generate images which lie exactly on the initial pixel grid. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Equivariance</td><td>Admissible Scales</td><td>Approach</td><td>Interscale</td></tr><tr><td>SiCNN</td><td></td><td>Grid</td><td>Filter Rescaling</td><td></td></tr><tr><td>SI-ConvNet</td><td></td><td>Grid</td><td>Input Rescaling</td><td></td></tr><tr><td>SEVF</td><td></td><td>Grid</td><td>Input Rescaling</td><td></td></tr><tr><td>DSS</td><td>&lt;x&lt;/</td><td>Integer</td><td>Filter Dilation</td><td>xx&lt;&gt;</td></tr><tr><td>SS-CNN</td><td>X</td><td>Any</td><td>Steerable Filters</td><td>X</td></tr><tr><td>SESN, Ours</td><td>√</td><td>Any</td><td>Steerable Filters</td><td>√</td></tr></table>",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Marcos et al. (2018), in the SEVF model, the input of the layers is rescaled and convolved multiple times to form vector features instead of scalar ones. The length of the vector in each position is the maximum magnitude of the convolution, while the direction of the angle encodes the scale of the image which gave this response. These scale-equivariant networks rely on image rescaling which is quite slow. Worrall & Welling (2019) (DSS) generalize the concept of scale-space to deep networks. They use filter dilation to analyze the images on different scales. While this approach is as fast as the standard CNN, it is restricted only to integer downscale factors $2 , 4 , 8 \\dots$ In Ghosh & Gupta (2019), while discussing SS-CNN the authors use scale-steerable filters to deal with scale changes. The paper does not discuss equivariance, which is an important aspect for scale. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We summarize the information about these models in Table 1. In contrast to other scale-equivariant models, SESN uses steerable filters which allows for fast scale-convolution with no limitation of flexibility. With the framework of Scale-Equivariant Convolutional Networks we are free to build both equivariant and invariant models of different kinds. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/9d5081dc572954800739488564f5c9304a98127ff3d55fe8a82463473ff120dc.jpg",
+        "image_caption": [
+            "Figure 2: Equivariance error $\\Delta$ as a function of the number of layers (left), downscaling applied to the input image (middle), and as a function of number of scales in interscale interactions (right). The bars indicate standard deviation. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In this section we conduct the experiments and compare various methods for working with scale variations in input data. Alongside SESN, we test local scale invariant SI-ConvNet and SS-CNN, scale equivariant SiCNN, SEVF and DSS. For SEVF, DSS and SS-CNN we use the code provided by authors, while for others we reimplement the main buildings blocks. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We provide additional experimental results on time performance of all these methods in Appendix B. Due to the algorithm proposed in Section 4 SESN allows for training several times faster than other methods which rely on image rescaling. ",
+        "bbox": [
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+    {
+        "type": "table",
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+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>(28×28)</td><td>(28 × 28) +</td><td>(56 × 56)</td><td>(56 × 56) +</td><td># Params</td></tr><tr><td>CNN</td><td>2.56 ± 0.04</td><td>1.96 ± 0.07</td><td>2.02 ± 0.07</td><td>1.60 ± 0.09</td><td>495K</td></tr><tr><td>SiCNN</td><td>2.40 ± 0.03</td><td>1.86 ± 0.10</td><td>2.02 ± 0.14</td><td>1.59 ± 0.03</td><td>497K</td></tr><tr><td>SI-ConvNet</td><td>2.40± 0.12</td><td>1.94 ± 0.07</td><td>1.82 ± 0.11</td><td>1.59 ± 0.10</td><td>495K</td></tr><tr><td>SEVF Scalar</td><td>2.30 ± 0.06</td><td>1.96 ± 0.07</td><td>1.87 ± 0.09</td><td>1.62 ± 0.07</td><td>494 K</td></tr><tr><td>SEVF Vector</td><td>2.63 ± 0.09</td><td>2.23 ± 0.09</td><td>2.12 ± 0.13</td><td>1.81 ± 0.09</td><td>475 K</td></tr><tr><td>DSS Scalar</td><td>2.53 ± 0.10</td><td>2.04± 0.08</td><td>1.92 ± 0.08</td><td>1.57 ± 0.08</td><td>494 K</td></tr><tr><td>DSS Vector</td><td>2.58 ± 0.11</td><td>1.95 ± 0.07</td><td>1.97 ± 0.08</td><td>1.57 ± 0.09</td><td>494 K</td></tr><tr><td>SS-CNN</td><td>2.32 ± 0.15</td><td>2.10 ±0.15</td><td>1.84 ± 0.10</td><td>1.76 ± 0.07</td><td>494 K</td></tr><tr><td>SESN Scalar</td><td>2.10 ±0.10</td><td>1.79 ± 0.09</td><td>1.74 ± 0.09</td><td>1.50 ± 0.07</td><td>495K</td></tr><tr><td>SESN Vector</td><td>2.08 ±0.09</td><td>1.76 ± 0.08</td><td>1.68 ± 0.06</td><td>1.42 ± 0.07</td><td>495 K</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "Table 2: Classification error of different methods on MNIST-scale dataset, lower is better. In experiment we use image resolution of $2 8 \\times 2 8$ and $5 6 \\times 5 6$ . We test both the regime without data augmentation, and the regime with scaling data augmentation, denoted with $\" + \"$ . All results are reported as mean $\\pm$ std over 6 different fixed realizations of the dataset. The best results are bold. ",
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+    {
+        "type": "text",
+        "text": "6.1 EQUIVARIANCE ERROR ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We have presented scale-convolution which is equivariant to scale transformation and translation for continuous signals. While translation equivariance holds true even for discretized signals and filters, scale equivariance may not be exact. Therefore, before starting any experiments, we check to which degree the predicted properties of scale-convolution hold true. We do so by measuring the difference $\\dot { \\Delta } = \\| [ L _ { s } \\mathbf { \\hat { \\Phi } } ( f ) - \\Phi \\mathbf { \\hat { L } } _ { s } ( f ) \\| _ { 2 } ^ { 2 } / \\| L _ { s } \\Phi ( f ) \\| _ { 2 } ^ { 2 }$ , where $\\Phi$ is scale-convolution with randomly initialized weights. ",
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+    {
+        "type": "text",
+        "text": "In case of perfect equivariance the difference is equal to zero. We calculate the error on randomly sampled images from the STL-10 dataset Coates et al. (2011). The results are represented in Figure 2. The networks on the left and on the middle plots do not have interscale interactions. The networks on the middle and on the right plots consist of just one layer. We use $N _ { S } = 5 , 1 3 , 5$ scales for the networks on the left, the middle, and the right plots respectively. While discretization introduces some error, it stays very low, and is not much higher than $6 \\%$ for the networks with 50 layers. The difference, however, increases if the input image is downscaled more than 16 times. Therefore, we are free to use deep networks. However, we should pay extra attention to extreme cases where scale changes are of very big magnitude. These are quite rare but still appear in practice. Finally, we see that using SESN with interscale interaction introduces extra equivariance error due to the truncation of $S$ . We will build the networks with either no scale interaction or interaction of 2 scales. ",
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+        "type": "text",
+        "text": "6.2 MNIST-SCALE ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Following Kanazawa et al. (2014); Marcos et al. (2018); Ghosh & Gupta (2019) we conduct experiments on the MNIST-scale dataset. We rescale the images of the MNIST dataset LeCun et al. (1998) to $0 . 3 - 1 . 0$ of the original size and pad them with zeros to retain the initial resolution. The scaling factors are sampled uniformly and independently for each image. The obtained dataset is then split into 10,000 for training, 2,000 for evaluation and 50,000 for testing. We generate 6 different realizations and fix them for all experiments. ",
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+        "type": "text",
+        "text": "As a baseline model we use the model described in Ghosh & Gupta (2019), which currently holds the state-of-the-art result on this dataset. It consists of 3 convolutional and 2 fully-connected layers. Each layer has filters of size $7 \\times 7$ . We keep the number of trainable parameters almost the same for all tested methods. This is achieved by varying the number of channels. For scale equivariant models we add scale projection at the end of the convolutional block. ",
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+        "type": "text",
+        "text": "For SiCNN, DSS, SEVF and our model, we additionally train counterparts where after each convolution, an extra projection layer is inserted. Projection layers transform vector features in each spatial position of each channel into scalar ones. All of the layers have now scalar inputs instead of vector inputs. Therefore, we denote these models with “Scalar”. The original models are denoted as “Vector”. The exact type of projection depends on the way the vector features are constructed. For SiCNN, DSS, and SESN, we use maximum pooling along the scale dimension, while for SEVF, it is a calculation of the $L _ { 2 }$ -norm of the vector. ",
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "All models are trained with the Adam optimizer Kingma & Ba (2014) for 60 epochs with a batch size of 128. Initial learning rate is set to 0.01 and divided by 10 after 20 and 40 epochs. We conduct the experiments with 4 different settings. Following the idea discussed in Ghosh & Gupta (2019), in addition to the standard setting we train the networks with input images upscaled to $5 6 \\times 5 6$ using bilinear interpolation. This results in all image transformations performed by the network becoming more stable, which produces less interpolation artifacts. For both input sizes we conduct the experiments without data augmentation and with scaling augmentation, which results in 4 setups in total. We run the experiments on 6 different realizations of MNIST-scale and report mean $\\pm$ std calculated over these runs. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The obtained results are summarized in Table 2. The reported errors may differ a bit from the ones in the original paper because of the variations in generated datasets and slightly different training procedure. Nevertheless, we try to keep our configuration as close as possible to Ghosh & Gupta (2019) which currently demonstrated the best classification accuracy on MNIST-scale. For example, SS-CNN reports error of $1 . 9 1 \\pm 0 . 0 4$ in Ghosh & Gupta (2019) while it has $1 . 8 4 \\pm 0 . 1 0$ in our experiments. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "SESN significantly outperforms other methods in all 4 regimes. “Scalar” versions of it already outperform all previous methods, and “Vector” versions make the gain even more significant. The global architectures of all models are the same for all rows, which indicates that the way scale convolution is done plays an important role. ",
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+    {
+        "type": "text",
+        "text": "6.3 STL-10 ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In order to evaluate the role of scale equivariance in natural image classification, we conduct the experiments on STL-10 dataset Coates et al. (2011). This dataset consists of 8,000 training and 5,000 testing labeled images. Additionally, it includes 100,000 unlabeled images. The images have a resolution of $9 6 \\times 9 6$ pixels and RGB channels. Labeled images belong to 10 classes such as bird, horse or car. We use only the labeled subset to demonstrate the performance of the models in the low data regime. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The dataset is normalized by subtracting the per-channel mean and dividing by the per-channel standard deviation. During training, we augment the dataset by applying 12 pixel zero padding and randomly cropping the images to size $9 6 \\times 9 6$ . Additionally, random horizontal flips with probability $5 0 \\%$ and Cutout DeVries & Taylor (2017) with 1 hole of 32 pixels are used. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "As a baseline we choose WideResNet Zagoruyko & Komodakis (2016) with 16 layers and a widening factor of 8. We set dropout probability to 0.3 in all blocks. We train SESN-A with just vector features. For SESN-B we use maximum scalar projection several times in the intermediate layers, and for SESN-C we use interscale interaction. ",
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+    {
+        "type": "table",
+        "img_path": "images/4d32f05f49eec5a1aa266914ff213cdd1b415b4908692d90c08c7b8e0fe5a749.jpg",
+        "table_caption": [
+            "Table 3: Classification error on STL-10. The best results are bold. We additionally report the current best result achieved by Harm WRN from Ulicny et al. (2019). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Error, %</td><td>#Params</td></tr><tr><td>WRN</td><td>11.48</td><td>11.0 M</td></tr><tr><td>SiCNN</td><td>11.62</td><td>11.0 M</td></tr><tr><td>SI-ConvNet</td><td>12.48</td><td>11.0 M</td></tr><tr><td>DSS</td><td>11.28</td><td>11.0 M</td></tr><tr><td>SS-CNN</td><td>25.47</td><td>10.8 M</td></tr><tr><td>SESN-A</td><td>10.83</td><td>11.0 M</td></tr><tr><td>SESN-B</td><td>8.51</td><td>11.0 M</td></tr><tr><td>SESN-C</td><td>14.08</td><td>11.0 M</td></tr><tr><td>Harm WRN</td><td>9.55</td><td>11.0 M</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "All models are trained for 1000 epochs with a batch size of 128. We use SGD optimizer with Nesterov momentum of 0.9 and weight decay of $5 \\cdot 1 0 ^ { - 4 } $ . The initial learning rate is set to 0.1 and divided by 5 after 300, 400, 600 and 800 epochs. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The results are summarized in Table 3. We found SEVF training unstable and therefore do not include it in the table. Pure scale-invariant SI-ConvNet and SS-CNN demonstrate significantly worse results than the baseline. We note the importance of equivariance for deep networks. We also find that SESN-C performs significantly worse than SESN-A and SESN-B due to high equivariance error caused by interscale interaction. SESN-B significantly improves the results of both WRN and DSS due to the projection between scales. The maximum scale projection makes the weights of the next layer to have a maximum receptive field in the space of scales. This is an easy yet effective method for capturing the correlations between different scales. This experiment shows that scale-equivariance is a very useful inductive bias for natural image classification with deep neural networks. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "To the best of our knowledge, the proposed method achieves a new state-of-the-art result on the STL-10 dataset in the supervised learning setting. The previous lowest error is demonstrated in Ulicny et al. (2019). The authors propose Harm WRN — a network where the convolutional kernels are represented as a linear combination of Discrete Cosine Transform filters. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "7 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we have presented the theory of Scale-Equivariant Steerable Networks. We started from the scaling transformation and its application to continuous functions. We have obtained the exact formula for scale-equivariant mappings and demonstrated how it can be implemented for discretized signals. We have demonstrated that this approach outperforms other methods for scale-equivariant and local scale-invariant CNNs. It demonstrated new state-of-the-art results on MNIST-scale and on the STL-10 dataset in the supervised learning setting. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We suppose that the most exciting possible application of SESN is in computer vision for autonomous vehicles. Rapidly changing distances between the objects cause significant scale variations which makes this well suited for our work. We especially highlight the direction of siamese visual tracking where the equivariance to principle transformations plays an important role. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
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+        "type": "text",
+        "text": "We thank Daniel Worrall for insightful discussion, Thomas Andy Keller, Victor Garcia, Artem Moskalev and Konrad Groh for valuable comments and feedback. ",
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+        "text": "Nathaniel Thomas, Tess Smidt, Steven Kearnes, Lusann Yang, Li Li, Kai Kohlhoff, and Patrick Riley. Tensor field networks: Rotation-and translation-equivariant neural networks for 3d point clouds. arXiv preprint arXiv:1802.08219, 2018. ",
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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, 2018b. ",
+        "bbox": [
+            174,
+            882,
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+            924
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Andrew P Witkin. Scale-space filtering. In Readings in Computer Vision, pp. 329–332. Elsevier, 1987.   \nDaniel Worrall and Gabriel Brostow. Cubenet: Equivariance to 3d rotation and translation. In Proceedings of the European Conference on Computer Vision (ECCV), pp. 567–584, 2018.   \nDaniel E Worrall and Max Welling. Deep scale-spaces: Equivariance over scale. arXiv preprint arXiv:1905.11697, 2019.   \nDaniel E Worrall, Stephan J Garbin, Daniyar Turmukhambetov, and Gabriel J Brostow. Harmonic networks: Deep translation and rotation equivariance. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 5028–5037, 2017.   \nYichong Xu, Tianjun Xiao, Jiaxing Zhang, Kuiyuan Yang, and Zheng Zhang. Scale-invariant convolutional neural networks. arXiv preprint arXiv:1411.6369, 2014.   \nSergey Zagoruyko and Nikos Komodakis. Wide residual networks. arXiv preprint arXiv:1605.07146, 2016. ",
+        "bbox": [
+            171,
+            102,
+            826,
+            335
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A PROOF OF EQUIVARIANCE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            429,
+            118
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Let us first show that scale-convolution defined in Equation 6 is equivariant to translations. ",
+        "bbox": [
+            168,
+            132,
+            766,
+            150
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a6d67739f8ff7e9b634adeaec82fac0aae10937b270cb5821f7c2f2af3c98af3.jpg",
+        "text": "$$\n\\begin{array} { l } { [ L _ { \\hat { t } } [ f ] \\star _ { H } \\psi _ { \\sigma } ] ( s , t ) = \\displaystyle \\sum _ { s ^ { \\prime } } [ L _ { \\hat { t } } [ f ] ( s ^ { \\prime } , \\cdot ) \\star \\psi _ { s \\sigma } ( s ^ { - 1 } s ^ { \\prime } , \\cdot ) ] ( t ) } \\\\ { = \\displaystyle \\sum _ { s ^ { \\prime } } L _ { \\hat { t } } [ f ( s ^ { \\prime } , \\cdot ) \\star \\psi _ { s \\sigma } ( s ^ { - 1 } s ^ { \\prime } , \\cdot ) ] ( t ) } \\\\ { = L _ { \\hat { t } } \\Big \\{ \\displaystyle \\sum _ { s ^ { \\prime } } [ f ( s ^ { \\prime } , \\cdot ) \\star \\psi _ { s \\sigma } ( s ^ { - 1 } s ^ { \\prime } , \\cdot ) ] \\Big \\} ( t ) } \\\\ { = L _ { \\hat { t } } [ f \\star _ { H } \\psi _ { \\sigma } ] ( s , t ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            302,
+            172,
+            694,
+            299
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Now we show that scale convolution is equivariant to scale transformations: ",
+        "bbox": [
+            173,
+            309,
+            671,
+            325
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a10c6b05a353e59e312fdd7276c15886a1566d7cd04c9cade1a944a14b3949e3.jpg",
+        "text": "$$\n\\begin{array} { l } { [ L _ { \\delta } [ f ] \\star _ { H } \\psi _ { \\sigma } ] ( s , t ) = \\displaystyle \\sum _ { s ^ { \\prime } } [ L _ { \\delta } [ f ] ( s ^ { \\prime } , \\cdot ) \\star \\psi _ { s \\sigma } ( s ^ { - 1 } s ^ { \\prime } , \\cdot ) ] ( t ) } \\\\ { = \\displaystyle \\sum _ { s ^ { \\prime } } L _ { \\delta } [ f ( \\hat { s } ^ { - 1 } s ^ { \\prime } , \\cdot ) \\star \\psi _ { \\hat { s } ^ { - 1 } s \\sigma } ( s ^ { - 1 } s ^ { \\prime } , \\cdot ) ] ( t ) } \\\\ { = \\displaystyle \\sum _ { s ^ { \\prime \\prime } } [ f ( s ^ { \\prime \\prime } , \\cdot ) \\star \\psi _ { \\hat { s } ^ { - 1 } s \\sigma } ( \\hat { s } s ^ { - 1 } s ^ { \\prime \\prime } , \\cdot ) ] ( \\hat { s } ^ { - 1 } t ) } \\\\ { = [ f \\star _ { H } \\psi _ { \\sigma } ] ( \\hat { s } ^ { - 1 } s , \\hat { s } ^ { - 1 } t ) } \\\\ { = L _ { \\delta } [ f \\star _ { H } \\psi _ { \\sigma } ] ( s , t ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            287,
+            332,
+            709,
+            472
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Finally, we can use the property of semidirect product of groups ",
+        "bbox": [
+            171,
+            484,
+            596,
+            501
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/209a5ce9ce9f2fb60134f9570f24b4b4f44e3552f47a5c97a83498b4435f0da6.jpg",
+        "text": "$$\nL _ { \\hat { s t } } [ f ] \\star _ { H } \\psi _ { \\sigma } = L _ { \\hat { s } } L _ { \\hat { t } } [ f ] \\star _ { H } \\psi _ { \\sigma } = L _ { \\hat { s } } [ L _ { \\hat { t } } [ f ] \\star _ { H } \\psi _ { \\sigma } ] = L _ { \\hat { s } } L _ { \\hat { t } } [ f \\star _ { H } \\psi _ { \\sigma } ] = L _ { \\hat { s } \\hat { t } } [ f \\star _ { H } \\psi _ { \\sigma } ] = L _ { \\hat { s t } } [ f \\star _ { H } \\psi _ { \\sigma } ]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            194,
+            507,
+            776,
+            525
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "B TIME PERFORMANCE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            387,
+            118
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We report the average time per epoch of different methods for scale equivariance and local scale invariance in Table 4. Experimental setups from Section 6.2 are used. We used 1 Nvidia GeForce GTX 1080Ti GPU for training the models. ",
+        "bbox": [
+            174,
+            133,
+            825,
+            176
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The methods relying on image rescaling techniques during training (SiCNN, SI-ConvNet, SEVF) demonstrate significantly worse time performance that the ones, using either steerable filters or filter dilation. Additionally, we see that our method outperforms SS-CNN by a wide margin. Despite the similar filter sizes and comparable number of parameters between SS-CNN and SESN Scalar, the second one demonstrates significantly better results due to the algorithm proposed in Section 4. Finally, DSS performs slightly faster in some cases than our method as each convolution involves less FLOPs. Dilated filters are sparse, while steerable filters are dense. ",
+        "bbox": [
+            173,
+            183,
+            825,
+            280
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/93965c7a801a3b764fa0fd4c1064ce418f0949d19c220c38a8d01f4dd5578295.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>28×28,s</td><td>56 × 56,s</td></tr><tr><td>CNN</td><td>3.8</td><td>3.8</td></tr><tr><td>SiCNN Scalar</td><td>13.5</td><td>18.9</td></tr><tr><td>SiCNN Vector</td><td>15.3</td><td>22.8</td></tr><tr><td>SI-ConvNet</td><td>18.4</td><td>33.1</td></tr><tr><td>SEVF Scalar</td><td>21.0</td><td>38.4</td></tr><tr><td>SEVF Vector</td><td>25.4</td><td>46.0</td></tr><tr><td>DSS Scalar</td><td>3.9</td><td>5.0</td></tr><tr><td>DSS Vector</td><td>3.9</td><td>4.8</td></tr><tr><td>SS-CNN</td><td>14.8</td><td>16.6</td></tr><tr><td>SESN Scalar</td><td>3.8</td><td>5.1</td></tr><tr><td>SESN Vector</td><td>3.8</td><td>6.8</td></tr></table>",
+        "bbox": [
+            352,
+            292,
+            643,
+            482
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Table 4: Average time per epoch during training on input data with resolution $2 8 \\times 2 8$ and $5 6 \\times 5 6$ . ",
+        "bbox": [
+            169,
+            497,
+            823,
+            513
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C BASIS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            541,
+            261,
+            558
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Assuming that the center of the filter is point $( 0 , 0 )$ in coordinates $( x , y )$ , we use the filters of the following form: ",
+        "bbox": [
+            174,
+            573,
+            826,
+            602
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/12e4e5c2514c596452b622d6ef91e349c9a2b5187c69a2fd93298c8308b935de.jpg",
+        "text": "$$\n\\psi _ { \\sigma } ( x , y ) = A \\frac { 1 } { \\sigma ^ { 2 } } H _ { n } \\Big ( \\frac { x } { \\sigma } \\Big ) H _ { m } \\Big ( \\frac { y } { \\sigma } \\Big ) \\exp \\Big [ - \\frac { x ^ { 2 } + y ^ { 2 } } { 2 \\sigma ^ { 2 } } \\Big ]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            321,
+            599,
+            674,
+            632
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Here $A$ is a constant independent on $\\sigma$ , $H _ { n }$ — Hermite polynomial of the $n$ -th order. We iterate over increasing pairs of $n , m$ to generate the required number of functions. ",
+        "bbox": [
+            171,
+            641,
+            823,
+            670
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "D MODEL CONFIGURATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            419,
+            118
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "D.1 MNIST-SCALE ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            133,
+            325,
+            148
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/d9d73c12207c1ef8b3a5e527b54d63142e45ebcd045aa8c1607b5bd141358528.jpg",
+        "table_caption": [
+            "Table 5: Number of channels in convolutional layers, number of units in fully-connected layers and number of scales used by different models in Section 6.2. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Conv 1</td><td>Conv 2</td><td>Conv 3</td><td>FC1</td><td># Scales</td></tr><tr><td>CNN</td><td>32</td><td>63</td><td>95</td><td></td><td>1</td></tr><tr><td>SiCNN</td><td>32</td><td>63</td><td>95</td><td></td><td>7</td></tr><tr><td>SI-ConvNet</td><td>32</td><td>63</td><td>95</td><td></td><td>7</td></tr><tr><td>SEVF Scalar</td><td>32</td><td>63</td><td>95</td><td>256</td><td>8</td></tr><tr><td>SEVF Vector</td><td>23</td><td>45</td><td>68</td><td></td><td>8</td></tr><tr><td>DSS</td><td>32</td><td>63</td><td>95</td><td></td><td>4</td></tr><tr><td>SS-CNN</td><td>30</td><td>60</td><td>90</td><td></td><td>6</td></tr><tr><td>SESN</td><td>32</td><td>63</td><td>95</td><td></td><td>4</td></tr></table>",
+        "bbox": [
+            281,
+            162,
+            715,
+            304
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/f8fdcdff123c35d84fbfe62d7502b0295003d46a9c1cd4e515b00e4296be05d0.jpg",
+        "table_caption": [
+            "D.2 STL-10 "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Block 1</td><td>Block 2</td><td>Block 3</td><td># Scales</td></tr><tr><td>CNN</td><td>16</td><td>32</td><td>64</td><td>1</td></tr><tr><td>SiCNN</td><td>16</td><td>32</td><td>64</td><td>3</td></tr><tr><td>SI-ConvNet</td><td>16</td><td>32</td><td>64</td><td>3</td></tr><tr><td>SEVF</td><td>11</td><td>23</td><td>45</td><td>3</td></tr><tr><td>DSS</td><td>16</td><td>32</td><td>64</td><td>4</td></tr><tr><td>SS-CNN</td><td>11</td><td>22</td><td>44</td><td>3</td></tr><tr><td>SESN</td><td>16</td><td>32</td><td>64</td><td>3</td></tr></table>",
+        "bbox": [
+            305,
+            407,
+            691,
+            534
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Table 6: Number of channels in convolutional blocks and number of scales used by different models in Section 6.2. We report the number of channels up to the widening factor. ",
+        "bbox": [
+            173,
+            550,
+            825,
+            579
+        ],
+        "page_idx": 13
+    }
+]

parse/train/HJr4QJ26W/HJr4QJ26W.md

@@ -0,0 +1,291 @@
+# IMPROVING IMAGE GENERATIVE MODELS WITH HUMAN INTERACTIONS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+GANs provide a framework for training generative models which mimic a data distribution. However, in many cases we wish to train a generative model to optimize some auxiliary objective function within the data it generates, such as making more aesthetically pleasing images. In some cases, these objective functions are difficult to evaluate, e.g. they may require human interaction. Here, we develop a system for efficiently training a GAN to increase a generic rate of positive user interactions, for example aesthetic ratings. To do this, we build a model of human behavior in the targeted domain from a relatively small set of interactions, and then use this behavioral model as an auxiliary loss function to improve the generative model. As a proof of concept, we demonstrate that this system is successful at improving positive interaction rates simulated from a variety of objectives, and characterize some factors that affect its performance.
+
+# 1 INTRODUCTION
+
+Generative image models have improved rapidly in the past few years, in part because of the success of Generative Adversarial Networks, or GANs (Goodfellow et al., 2014). GANs attempt to train a “generator” to create images which mimic real images, by training it to fool an adversarial “discriminator,” which attempts to discern whether images are real or fake. This is one solution to the difficult problem of learning when we don’t know how to write down an objective function for image quality: take an empirical distribution of “good” images, and try to match it.
+
+Often, we want to impose additional constraints on our goal distribution besides simply matching empirical data. If we can write down an objective which reflects our goals (even approximately), we can often simply incorporate this into the loss function to achieve our goals. For example, when trying to generate art, we would like our network to be creative and innovative rather than just imitating previous styles, and including a penalty in the loss for producing recognized styles appears to make GANs more creative (Elgammal et al., 2017). Conditioning on image content class, training the discriminator to classify image content as well as making real/fake judgements, and including a loss term for fooling the discriminator on class both allows for targeted image generation and improves overall performance (Odena et al., 2016).
+
+However, sometimes it is not easy to write an explicit objective that reflects our goals. Often the only effective way to evaluate machine learning systems on complex tasks is by asking humans to determine the quality of their results (Christiano et al., 2017, e.g.) or by actually trying them out in the real world. Can we incorporate this kind of feedback to efficiently guide a generative model toward producing better results? Can we do so without a prohibitively expensive and slow amount of data collection? In this paper, we tackle a specific problem of this kind: generating images that cause more positive user interactions. We imagine interactions are measured by a generic Positive Interaction Rate (PIR), which could come from a wide variety of sources.
+
+For example, users might be asked to rate how aesthetically pleasing an image is from 1 to 5 stars. The PIR could be computed as a weighted sum of how frequently different ratings were chosen. Alternatively, these images could be used in the background of web pages. We can assess user interactions with a webpage in a variety of ways (time on page, clicks, shares, etc.), and summarize these interactions as the PIR. In both of these tasks, we don’t know exactly what features will affect the PIR, and we certainly don’t know how to explicitly compute the PIR for an image. However, we can empirically determine the quality of an image by actually showing it to users, and in this paper we show how to use a small amount of this data (results on 1000 images) to efficiently tune a generative model to produce images which increase PIR. In this work we focus on simulated PIR values as a proof of concept, but in future work we will investigate PIR values from real interactions.
+
+![](images/ec679fbe76f983e591cddbdeceb8798f656c1b196c970471f04538046cd10173.jpg)  
+Figure 1: Diagram of our system
+
+# 2 APPROACH
+
+The most straight-forward way to improve an image GAN might be to evaluate the images the model produces with real users at each training step. However, this process is far too slow. Instead, we want to be able to collect a batch of PIR data on a batch of images, and then use this batch of data to improve the generative model for many gradient steps; we want to do this despite the fact that the images the generator is producing may evolve to be very different from the original images we collected PIR data on. In order to do this, we use the batch of image and PIR data to train a “PIR Estimator Model” which predicts PIRs on images. We then use these estimated PIRs at each step as a loss.
+
+Our approach is inspired by the work of Christiano and colleagues (Christiano et al., 2017), who integrated human preference ratings between action sequences into training of a reinforcement learning model by using the preference data to estimate a reward function. However, our problem and approach differ in several key ways. First, we are optimizing a generative image model rather than a RL model. This is more difficult in some ways, since the output space is much higher-dimensional than typical RL problems, which means that scalar feedback (like a PIR) may be harder for the system to learn from. This difficulty is partially offset by the fact that we assume we get “reward” (PIR) information for an image when we evaluate, instead of just getting preferences which we have to map to rewards. Perhaps most importantly, we use our PIR estimation model as a fully-differentiable loss function for training, instead of just using its estimated rewards. This allows us to more effectively exploit its knowledge of the objective function (but risks overfitting).
+
+Our system consists of three components: A generative image model, users who interact with the generated images in some way, and a PIR estimator that models user interactions given an image. See Fig. 1 for a diagram of the system’s general operation. The generative model produces images, which are served to users. Using interaction data from these users, we train the PIR estimator model, which predicts PIRs given a background image, and then incorporate this estimated PIR into the loss of the generative model to tune it to produce higher quality images. Below, we discuss each of these components in more detail.
+
+# 2.1 GENERATIVE MODEL
+
+We begin with a $\mathrm { G A N ^ { 1 } }$ (Goodfellow et al., 2014) which we pre-trained to produce images from a target distribution (specifically, landscapes of mountains and coasts). Let $D _ { \mathrm { s o u r c e } }$ be the source estimated by the discriminator, $G$ the generator, and $z$ a noise input to the generator sampled from a multivariate standard normal distribution $\mathcal { N } ( 0 , I )$ , and $\mathcal { T }$ be the set of real images shown to the discriminator. Define:
+
+$$
+\begin{array} { r l r } { L _ { \mathrm { f a k e i m a g e } } = } & { } & { E _ { z \sim \mathcal { N } ( 0 , I ) } \left[ \log P ( D _ { \mathrm { s o u r c e } } ( G ( z ) ) = \mathrm { f a k e } ) \right] } \\ { L _ { \mathrm { f a k e i m a g e f o o l s } } = } & { } & { E _ { z \sim \mathcal { N } ( 0 , I ) } \left[ \log P ( D _ { \mathrm { s o u r c e } } ( G ( z ) ) = \mathrm { r e a l } ) \right] } \\ { L _ { \mathrm { r e a l i m a g e } } = } & { } & { E _ { i \sim \mathcal { Z } } \left[ \log P ( D _ { \mathrm { s o u r c e } } ( i ) = \mathrm { r e a l } ) \right] } \end{array}
+$$
+
+Then the discriminator and generator are trained to maximize the following losses (respectively), where the $w _ { * }$ are weights set as hyperparameters:
+
+$$
+\begin{array} { r l r } { L _ { \mathrm { d i s c r i m i n a t o r } } = } & { { } } & { L _ { \mathrm { f a k e i m a g e } } + L _ { \mathrm { r e a l i m a g e } } } \\ { L _ { \mathrm { G e n e r a t o r } } = } & { { } } & { L _ { \mathrm { f a k e i m a g e f o o l s } } } \end{array}
+$$
+
+Note that there is a difference between these losses and the standard GAN formulation given in (Goodfellow et al., 2014) – we maximize $L _ { \mathrm { f a k e } }$ image fools $= \ \log \ P$ (classified real) rather than minimizing $\log { ( 1 - P }$ (classified real)). This seems to result in the generation of slightly better images in practice.
+
+This GAN was trained on a dataset consisting of landscape images of mountains and coastlines (see Appendix C.2 for details of the architecture and training). It is worth noting that this generative model is not photorealistic (see Fig. 3a for some samples). Its expressive capacity is limited, and it has clear output modes with limited intra-mode variability. However, for our purposes this may not matter. Indeed, it is in some ways more interesting if we can tweak this model to optimize for many objective functions, since its limited expressive capacity will make it more difficult for us to estimate and pursue the real objective – a limited set of images will effectively give us fewer points to estimate the PIR function from, and will reduce the space in which the model can easily produce images, thus reducing the possibility of getting very optimal images from the model. For example, a model which produces images of birds may not produce data points which provide good estimates of a PIR based on how much the image looks like a car, and even if it could, it may not be able to produce images which are “more car-like.” If we are able to succeed in improving PIRs with this generative model, it is likely that a better generative model would yield even better results.
+
+# 2.2 USER INTERACTIONS
+
+We will show these images to users in a variety of ways, depending on our target domain. For the purposes of this paper, however, we will use simulated interaction data (see Section 3 for details). Of course, since showing images to users is an expensive prospect, we wanted to limit the size of the datasets we used to train the model. Typical datasets used to train vision models are on the order of millions of images, (e.g. ImageNet (Russakovsky et al., 2015)), but it is completely infeasible to collect user data on this number of images. We estimated that we could show 1000 images each 1000 times to generate our datasets. We used these dataset sizes and number of impressions for all experiments discussed here, and added noise to the PIRs that was binomially distributed according to the number of times each image was shown and the “true” PIR simulate from the objective.
+
+# 2.3 PIR ESTIMATOR MODEL
+
+The final component of our system is the PIR estimator model, which learns to predict PIR from a background image. We denote this model by $R : { \mathrm { i m a g e } }  [ 0 , 1 ]$ . We parameterize this model as a deep neural network. Specifically, we take the Inception v2 architecture (Szegedy et al., 2016), remove the output layer, and replace it with a fully-connected layer to PIR estimates. We initialize the Inception v2 parameters from a version of the model trained on [dataset redacted for blind review]. See Appendix C.3 for more details.
+
+Why did we not make estimated PIR simply another auxiliary output from the discriminator, like class in the ACGAN (Appendix C.1)? Because the PIR estimator needs to be held constant in order to provide an accurate training objective. If the PIR estimates were produced by the discriminator, then as the discriminator changed to accurately discriminate the evolving generator images, the PIR estimates would tend to drift without a ground-truth to train them on. Separating the discriminator and the PIR estimator allows us to freeze the PIR estimator while still letting the discriminator adapt.
+
+# 2.4 INTEGRATION
+
+Once we have trained a PIR estimator model, we have to use it to improve the GAN. We do this as follows. Let $R$ denote the PIR estimator model, as above. Define $L _ { \mathrm { P I R } }$ to be the expectation of the estimated PIR produced over images sampled from the generator:
+
+$$
+{ \cal L } _ { \mathrm { P I R } } = { \cal E } _ { z \sim { \cal N } ( 0 , I ) , c \sim { \cal C } } \left[ R ( G ( z ) ) \right]
+$$
+
+Then we simply supplement the generator loss by adding this term times a weight $w _ { P I R }$ , set as a hyperparameter:
+
+$$
+L _ { \mathrm { G e n e r a t o r } } = L _ { \mathrm { f a k e i m a g e f o o l s } } + w _ { P I R } L _ { P I R }
+$$
+
+We set $w _ { P I R } = 1 0 0 0$ as this made the magnitude of the PIR loss and the other loss terms roughly comparable. Otherwise we used the same parameters as in the GAN training above, except that we reduced the learning rate to $1 0 ^ { - 6 }$ to allow the system to adapt more smoothly to the multiple objectives, and we trained for 50,000 steps.
+
+# 3 DATA
+
+In this paper, we use simulated interaction data as proof of concept. This raises an issue: what functions should we use to simulate user interactions? Human behavior is complex, and if we already knew precisely what guided user interactions, there would be no need to actually collect human behavioral data at all. Since we don’t know what features will guide human behavior, the next best thing we can do is to ensure that our system is able to alter the image generation model in a broad variety of ways, ranging from low level features (like making the images more colorful) to altering complex semantic features (such as including more plants in outdoor scenery). We also want to avoid hand-engineering tasks to the greatest extent possible. We present an overview of our approaches to simulating PIR data below, see Appendix C.4 for more details.
+
+# 3.1 VGG FEATURES
+
+The first approach we took to evaluating our system’s ability to train for different features was to use activity from hidden layers of a computer vision model, specifically VGG 16 (Simonyan & Zisserman, 2014) trained on ImageNet (Russakovsky et al., 2015). In particular, we took the activity of a single filter in a layer of VGG relative to the overall activity of that layer. This approach to simulating PIRs has several benefits. First, it gives a wide variety of complex objectives that can nevertheless be easily computed to simulate data. Second, models like VGG exhibit hierarchical organization, where lower levels generally respond to lower-level features such as edges and colors, while higher levels respond to higher-level semantic features such as faces (Zeiler & Fergus, 2014), and the represented features relate to those in human and macaque visual cortex (Yamins et al., 2014). Thus VGG features give a wide range of objectives which we may relate to the human perception we wish to target.
+
+There are some caveats to this approach, however. First, although the higher layers of CNNs are somewhat selective for “abstract” object categories, they are also fooled by adversarial images that humans would not be, and directly optimizing inputs for these high level features does not actually produce semantically meaningful images (Nguyen et al., 2014). Thus, even if our system succeeds in increasing activity in a targeted layer which is semantically selective, it will likely do so by adversarially exploiting particulars of VGG 16’s parameterization of the classification problem (although the fact that we are not backpropagating through the true objective will make this harder). It is not necessarily a failure of the system if it exploits simple features of the objective it is given to increase PIRs – indeed, it should be seen as a success, as long as it is generalizes to novel images. However, success on this task does not necessarily guarantee success on modifying semantic content when interacting with actual humans. It may be easier for the PIR estimator model (which is based on a CNN) to learn objectives which come from another CNN than more general possible objectives. The fact that adversarial examples can sometimes transfer between networks with different architectures (Liu et al., 2016) suggests that the computations being performed by these networks are somewhat architecture invariant. Thus CNN objectives may be easier for our estimator than human ones.
+
+We have tried to minimize these problems to the greatest extent possible by using different network architectures (Inception V2 and VGG 16, respectively) trained on different datasets ([hidden] and ImageNet (Russakovsky et al., 2015), respectively) for the estimator and the objective. However, we cannot be certain that the network is not “cheating” in some way on the VGG 16 tasks, so our results must be considered with this qualification in mind. Despite this, we think that evaluating our system’s ability to optimize for objectives generated from various layers of VGG will show its ability to optimize for a variety of complex objectives, and thus will serve as a useful indicator of its potential to improve PIRs from real users.
+
+# 3.2 MULTIPLE FILTERS
+
+After using our system on the tasks above, we noted that its performance was quite poor at layers 5, 6, and 7 of VGG compared to other tasks (see Fig. 2). This could suggest that our system was unable to capture the complex features represented at the higher levels of VGG. However, we also noticed that the feature representations at these layers tended to be quite sparse, so many of the simulated PIRs we generated were actually zero to within the bin width of our PIR estimator (see Appendix B Fig. 8 for a plot of how this affected learning). In order to evaluate whether the poor performance of our system at the higher layers of VGG was due to the number of zeros or to the complexity of the features, we created less sparse features from these layers by simply targeting a set of $k$ filters sampled without replacement from the layer, rather than a single filter. The single filter cases above can be thought of as a special case of this, where $k = 1$ . To complement these, we also tried $k = 2 0$ .
+
+This can also be thought of as perhaps a more realistic simulation of human behavior, in the sense that it is highly unlikely that there is a single feature which influences human PIRs. Rather, there are probably many related features which influence PIR in various ways. Thus it is important to evaluate our system’s ability to target these types of features as well.
+
+# 3.3 COLORS
+
+Finally, we also considered some simpler objectives based on targeting specific colors in the output images, or targeting vertical bands of two different colors, one in each half of the image, or three colors, one in each third of the image. These objectives provide a useful complement to the VGG objectives above. Although the single color objectives may be relevant to the classification task VGG 16 performs, the split color tasks are less likely to be relevant to classification. Note that it is important that we split the images along the width instead of the height dimension, as there may well be semantically relevant features corresponding to color divisions along the height dimension, e.g. a blue upper half and green lower half likely correlates with outdoor images, which would provide useful class information. By contrast, it is harder to imagine circumstances where different colors on the left and right halves of the image are semantically predictive, especially since flipping left to right is usually included in the data augmentation for computer vision systems. Thus success on optimizing for these objectives would increase our confidence in the generality of our system.
+
+# 4 RESULTS
+
+We present our results in terms of the change in mean PIR from 1000 images produced by the GAN before tuning to 1000 images produced after tuning, or in terms of the effect size of this change (Cohen’s $d$ , i.e. the change in mean PIR standardized by the standard deviation of the PIRs in the pre- and post-tuning image sets). We assess whether these changes are significant by performing a Welch’s t-test (with a significance threshold of $\alpha = 0 . 0 0 1$ ) between the pre- and post-tuning PIRs.
+
+Overall, our system was quite successful at improving PIRs across a range of simulated objective functions (see Fig. 2). Below, we discuss these results in more detail.
+
+![](images/452a4c3d84260f277ccf32fa578a0b0396ce5df970f4fe40eb77375a745ada9e.jpg)  
+Figure 2: Effect size (number of standard deviations change in mean PIR) on a variety of tasks. Values greater than zero indicate improvement. Each panel represents a set of tasks (such as single filters from VGG), each color/column represents a subset (such as a single layer within VGG), and each point represents one objective (such as a single filter from that layer) for which we optimized the generative model. Points for which the change in mean PIR is not significant by a Welch’s $t$ -test with a threshold of $\alpha = 0 . 0 0 1$ are partially transparent.
+
+# 4.1 VGG OBJECTIVES
+
+Our system largely succeed at increasing PIRs on a variety of VGG objectives (see Fig. 2). However, there are several interesting patterns to note. First, the system is not particularly successful at targeting single filters from the pool5, fc6, and fc7 layers. However, we believe this is due to the fact that filters in these layers produce relatively sparse activation, see section 3.2. Indeed, the performance of the system seems much more consistent when it is optimizing for sets of 20 filters than for single filters.
+
+Even when using 20 filters, however, there is a noticeable decline in the effect size of the improvement the system is able to make at higher layers of VGG $\beta = - 0 . 1 3$ per layer, $t = - 7 . 8$ , $\dot { p } < 1 0 ^ { - 1 0 }$ in a linear model controlling for initial standard deviation and percent zeros). This suggests that as the objectives grow more complex, the system may be finding less accurate approximations to them. However, the system’s continuing (if diminished) success at the higher layers of VGG suggests that our model is capable of at least partially capturing complex objective functions.
+
+# 4.2 COLOR OBJECTIVES
+
+Overall, the system performed quite well at optimizing for the color objectives, particularly the single and two-color results (see Fig. 2). It had more difficulty optimizing for the three-color results, and indeed had produced only very small improvements after the usual 50,000 tuning steps for the generative model, but after 500,000 steps it was able to produce significant improvements for two out of the three objectives (these longer training results are the ones included here).
+
+Because the color objectives are easiest to assess visually, we have included results for a variety of these objectives in Fig. 3. For the single color objectives, the improvement is quite clear, for example the images in Fig. 3d appear much more blue than the pre-training ones. For the two color objectives, it appears that the system found the “trick” of reducing the third color, for example the red-green split images in Fig. 3e appear much less blue than the pre-training images. Even on the three-color images where the system struggled, there are some visible signs of improvement, for example on the green-blue-red task the system has started producing a number of images with a blue streak in the middle.
+
+![](images/a677d9bfd60d90b339e6056f4eedc86e60ca0e0077525f722cfbef7db0088cd1.jpg)  
+Figure 3: Color objective sample images. Samples are randomly drawn, not cherry-picked.
+
+# 4.3 SUPPLEMENTAL ANALYSES
+
+We also conducted several supplemental analyses which can be found in detail Appendix A. In summary, the initial variability in the PIR of the images used to train the system is strongly correlated with the amount of improvement the system makes in the PIR, the system fairly consistently underestimates the performance it achieves (because of a detail of training procedure, see the Appendix), and iterating the process of improving PIRs yields better results for objectives from a lower layer of VGG but not a higher.
+
+# 5 DISCUSSION
+
+Overall, our system appears to be relatively successful. It can optimize a generative model to produce images which target a wide variety of objectives, ranging from low-level visual features such as colors and early features of VGG to features computed at the top layers of VGG. This success across a wide variety of objective functions allows us to be somewhat confident that our system will be able to achieve success in optimizing for real human interactions.
+
+Furthermore, the system did not require an inordinate amount of training data. In fact, we were able to successfully estimate many different objective functions from only 1000 images, several orders of magnitude fewer than is typically used to train CNNs for vision tasks. Furthermore, these images came from a very biased and narrow distribution (samples from our generative model) which is reflective of neither the images that were used to pre-train the Inception model in the PIR estimator, nor the images the VGG model (which produced the simulated objectives) was trained on. Our success from this small amount of data suggests that not only will our system be able to optimize for real human interactions, it will be able to do so from a feasible number of training points.
+
+These results are exciting – the model is able to approximate apparently complex objective functions from a small amount of data, even though this data comes from a very biased distribution that is unrelated to most the objectives in question. But what is really being learned? In the case of the color images, it’s clear that the model is doing something close to correct. However, for the objectives derived from VGG we have no way to really assess whether the model is making the images better or just more adversarial. For instance, when we are optimizing for the logit for “magpie,” it’s almost certainly the case that the result of this optimization will not look more like a magpie to a human, even if VGG does rate the images as more “magpie-like.” On the other hand, this is not necessarily a failure of the system – it is accurately capturing the objective function it is given. What remains to be seen is whether it can capture how background images influence human behavior as well as it can capture the vagaries of deep vision architectures.
+
+We believe there are many domains where a system similar to ours could be useful. We mentioned producing better webpage backgrounds and making more aesthetic images above, but there are many potential applications for improving GANs with a limited amount of human feedback. For example, a model could be trained to produce better music (e.g. song skip rates on streaming generated music could be treated as inverse PIRs).
+
+# 5.1 TRADING IMAGE DIVERSITY FOR PIR
+
+When tuning the GAN, the decrease in the PIR loss is usually accompanied by an increase in the generator loss, and often by a partial collapse of the generator output (for example, the optimized images generally seem to have fewer output modes than the pre-training images in Fig. 3). This is not especially surprising – because we weighted the PIR loss very highly, the model is rewarded for trading some image diversity for image optimality. Depending on the desired application, the weight on the PIR loss could be adjusted as necessary to trade off between producing images close to the data distribution and optimizing PIR. At its most extreme, one could down-weight the generator loss entirely, and train until the model just produces a single optimal image. However, the generator likely provides some regularization by constraining the images to be somewhat close to the real images, which will reduce overfitting to an imperfect estimate of the PIR function. Furthermore, in many settings we will want to generate a variety of images (e.g. backgrounds for different websites). For these reasons, we chose to keep the generator loss when tuning the GAN.
+
+# 5.2 FUTURE DIRECTIONS
+
+There are a number of future directions suggested by this work. A number of possible improvements are discussed in Appendix C.5. However, we also think this work has potential applications from the perspective of distillation or imitation approaches, which attempt to train one network to emulate another (Hinton et al., 2015; Parisotto et al., 2015, e.g), as well as from the perspective of understanding the computations that these vision architectures perform. As far as we are aware, these results are the first to show that a deep vision model can be tuned rapidly from relatively little data to produce outputs which accurately emulate the behavior of hidden layers of another deep vision architecture trained on a different dataset. This suggests both that the inductive biases shared among these architectures are causing them to find similar solutions (which is also supported by work on transferable adversarial examples (Liu et al., 2016, e.g.)), and that these networks final layers represent the computations of earlier hidden layers in a way that is somewhat accessible. It’s possible that using our system with objectives from CNN layers as we did here might help to understand the features those layers are attending to, by analyzing the distribution of images that are produced. In this sense, our system can be thought of as offering a new approach to multifaceted feature visualization (Nguyen et al., 2016), because our system attempts to optimize a distribution of images for an objective and encourages diversity in the distribution produced, rather than just optimizing a single image.
+
+# 6 CONCLUSIONS
+
+We have described a system for efficiently tuning a generative image model according to a slow-toevaluate objective function. We have demonstrated the success of this system at targeting a variety of objective functions simulated from different layers of a deep vision model, as well as from low-level visual features of the images, and have shown that it can do so from a small amount of data. We have quantified some of the features that affect its performance, including the variability of the training PIR data and the number of zeros it contains. Our system’s success on a wide variety of objectives suggests that it will be able to improve real user interactions, or other objectives which are slow and expensive to evaluate. This may have many exciting applications, such as improving machine-generated images, music, or art.
+
+# REFERENCES
+
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+
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+
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+
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+
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+
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+
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+
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+
+Emilio Parisotto, Jimmy Lei Ba, and Ruslan Salakhutdinov. Actor-Mimic: Deep Multitask and Transfer Reinforcement Learning. arXiv, pp. 1–16, 2015.
+
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+
+Christian Szegedy, Vincent Vanhoucke, Sergey Ioffe, Jonathon Shlens, and Zbigniew Wojna. Rethinking the Inception Architecture for Computer Vision. Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR), pp. 2818–2826, 2016. ISSN 08866236. doi: 10.1002/2014GB005021.
+
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+
+Matthew D Zeiler and Rob Fergus. Visualizing and Understanding Convolutional Networks arXiv:1311.2901v3 [cs.CV] 28 Nov 2013. Computer Vision–ECCV 2014, 8689:818–833, 2014. ISSN 978-3-319-10589-5. doi: 10.1007/978-3-319-10590-1 53.
+
+![](images/98240ce4438c7e3b31bfe707d1ab0bc9c597be9cf48d72834b5da56731a9b559.jpg)  
+Figure 4: Change in mean PIR vs. estimated change in mean PIR. Points for which the change in mean PIR is not significant by a Welch’s $t$ -test with a threshold of $\alpha = 0 . 0 0 1$ are partially transparent.
+
+![](images/4b39472a7f5980f4b2ddf37b5e22ac2583f56e4f0262421e27809b90bbe04416.jpg)  
+Figure 5: Change in mean PIR vs. initial variability. Points for which the change in mean PIR is not significant by a Welch’s $t$ -test with a threshold of $\alpha = 0 . 0 0 1$ are partially transparent.
+
+# A OTHER ANALYSES
+
+# A.1 INTROSPECTION
+
+Because $L _ { P I R }$ is just the expected value of the PIR, by looking at $L _ { P I R }$ before and after tuning the generative model, we can tell how well the system thinks it is doing, i.e. how much it estimates that it improved PIR. This comparison reveals the interesting pattern that the system is overly pessimistic about its performance. In fact, it tends to underestimate its performance by a factor of more than 1.5 $\beta = 1 . 6 7$ when regressing change in mean PIR on predicted change in mean PIR, see Fig. 4). However, it does so fairly consistently. This effect appears to be driven by the system consistently underestimating the (absolute) PIRs, which is probably caused by our change in the softmax temperature between training the PIR estimator and tuning the generative model (which we empirically found improves performance, as noted above).
+
+This is in contrast to the possible a priori expectation that the model would systematically overestimate its performance, because it is overfitting to an imperfectly estimated objective function. Although decreasing the softmax temperature between training and using the PIR obscures this effect, we do see some evidence of this; the more complex objectives (which the system produced lower effect sizes on) seem to both have lower estimated changes in mean PIR and true changes in PIR which are even lower than the estimated ones (see Fig. 4). Thus although the system is somewhat aware of its reduced effectiveness with these objectives (as evidenced by the lower estimates of change in mean PIR), it is not reducing its estimates sufficiently to account for the true difficulty of the objectives (as evidenced by the fact that the true change in PIR is even lower than the estimates). However, the system was generally still able to obtain positive results on these objectives (see Fig. 2).
+
+# A.2 INITIAL VARIABILITY
+
+There is a general trend (see Fig. 5) that the variability in PIR in the initial dataset is strongly positively related with the change in PIR the system is able to produce $\beta = 0 . 9 9$ , $t = 1 0 . 6$ , $p \bar { < } \bar { 1 0 } ^ { - 1 0 }$ , in a linear model controlling for initial mean PIR and initial percent of values that are zero). In fact, initial standard deviation explains about $50 \%$ of the variance in the change in mean PIR. This is perhaps not too surprising – more variability means that the generative model has capacity to produce higher PIR images without too much tweaking, and that the PIR estimator model gets a wider range of values to learn from. Still, when attempting to use this system in practice, it is important to keep in mind that starting with a sufficiently expressive generative model will more likely produce better results than starting with a more limited model.
+
+![](images/881a640f7021a2d81e1a10fbb7a3eae014fc91ad5806b8582eceaadd61ac6fd7.jpg)  
+Figure 6: Effect size evolution over two iterations
+
+# A.3 ITERATION
+
+Given that our model improves PIRs, an obvious question is whether we can iterate the process. Once we have increased PIRs, can we train a new PIR estimator on samples from our new generative model, and use that to increase PIRs again? If we could iterate this many times, we might be able to create much larger improvements in PIR than we can on a single step. On the other hand, it is possible that after a single step of optimization we will effectively have saturated the easily achievable improvement in the model, and further steps will not result in much improvement.
+
+To evaluate this, we took the subset of models trained on single filters from VGG layers pool2 and fc8, and used the set of images generated from the post-tuning model along with the original set of images to tune their PIR estimators for another 25,000 steps, and then tuned the generative model using this updated PIR estimator for another 50,000 steps. We then evaluated them as before, see Fig. 6 for the results. The second iteration results were mixed, while the pool2 models all improved from the first step to the second, none of them improved as much as they had on the first step, and many of the fc8 models actually performed worse after the second step. However, it is possible that further hyperparameter tuning could improve these results, and it is certainly the case that running multiple steps of iteration and selecting the best model by experimentation could yield better results, so this is worth investigating further.
+
+![](images/bb5568a77c88b87dcfddadac91fed93fcab073ec0eaf412bb76545bbf1d6be46.jpg)  
+Figure 7: Change in mean PIR on a variety of tasks. Each panel represents a set of tasks (such as single filters from VGG), each color/column represents a subset (such as a single layer within VGG), and each point represents one objective (such as a single filter from that layer) for which we optimized the generative model. Points for which the change in mean PIR is not significant by a Welch’s $t$ -test with a threshold of $\alpha = 0 . 0 0 1$ are partially transparent.
+
+# C DETAILED METHODS
+
+# C.1 ACGAN
+
+We describe our results using the standard GAN framework for clarity, but we actually used an ACGAN (Odena et al., 2016), which allows for conditioning for various user-specific features. This requires the following adjustments. Defining $\mathcal { C }$ to be the set of possible classes and $\mathcal { T } _ { c }$ to be the set of real images corresponding to a class $c \in { \mathcal { C } }$ :
+
+$$
+\begin{array} { r l r } { L _ { \mathrm { f a k e i m a g e } } = } & { } & { E _ { z \sim \mathcal { N } ( 0 , I ) , c \sim \mathcal { C } } \left[ \log P ( D _ { \mathrm { s o u r c e } } ( G ( z , c ) ) = \mathrm { f a k e } ) \right] } \\ { L _ { \mathrm { f a k e i m a g e ~ f o o l s } } = } & { } & { E _ { z \sim \mathcal { N } ( 0 , I ) , c \sim \mathcal { C } } \left[ \log P ( D _ { \mathrm { s o u r c e } } ( G ( z , c ) ) = \mathrm { r e a l } ) \right] } \\ { L _ { \mathrm { r e a l ~ i m a g e } } = } & { } & { E _ { c \sim \mathcal { C } , i \sim \mathcal { L } _ { c } } \left[ \log P ( D _ { \mathrm { s o u r c e } } ( i ) = \mathrm { r e a l } ) \right] } \\ { L _ { \mathrm { f a k e i m a g e c l a s s } } = } & { } & { E _ { z \sim \mathcal { N } ( 0 , I ) , c \sim \mathcal { C } } \left[ \log P ( D _ { \mathrm { c l a s s } } ( G ( z , c ) ) = c ) \right] } \\ { L _ { \mathrm { r e a l ~ i m a g e c l a s s } } = } & { } & { E _ { c \sim \mathcal { C } , i \sim \mathcal { Z } _ { c } } \left[ \log P ( D _ { \mathrm { c l a s s } } ( i ) = c ) \right] } \end{array}
+$$
+
+Then the discriminator and generator losses are modified as follows (letting $w _ { f a k e c l a s s } = 1 . 5$ and $w _ { r e a l c l a s s } = 1 . 0 \AA$ :
+
+$$
+\begin{array} { r l r } { L _ { \mathrm { d i s c r i m i n a t o r } } = } & { { } } & { L _ { \mathrm { f a k e ~ i m a g e } } + L _ { \mathrm { r e a l ~ i m a g e } } + w _ { r e a l c l a s s } L _ { \mathrm { r e a l ~ c l a s s } } } \\ { L _ { \mathrm { G e n e r a t o r } } = } & { { } } & { L _ { \mathrm { f a k e ~ i m a g e ~ f o o l s } } + w _ { f a k e c l a s s } L _ { \mathrm { f a k e ~ c l a s s } } } \end{array}
+$$
+
+Note that unlike the standard ACGAN formulation given in (Odena et al., 2016), we do not include $L _ { \mathrm { f a k e \ c l a s s } }$ in the discriminator’s loss, to keep the discriminator from cooperating with the generator on the classification task.
+
+We modified the generator network by adding one-hot class inputs, and the discriminator by adding class outputs alongside the source output, as in (Odena et al., 2016).
+
+# C.2 GENERATOR & DISCRIMINATOR
+
+We parameterized the generator as a deep neural network, which begins with a fully-connected mapping from the latent (noise) space to a $4 \times 4 \times 5 1 2$ dimensional image, and then successively upsampled (a factor of 2 by nearest neighbor), padded and applied a convolution ( $3 \times 3$ kernel, stride of 1) and a leaky ReLU $\alpha = 0 . 2$ ) nonlinearity repeatedly. We repeated this process 5 times (except with no upsampling on the first step, and a tanh nonlinearity on the last), while stepping the image depth down as follows: 512, 512, 256, 128, 64, and finally 3 (RGB) for the output image. This means that the final output images were $6 4 \times 6 4$ . We parameterized the discriminator as a convolutional network with 7 layers, 6 convolutions (kernels all $3 \times 3$ ; strides 2, 1, 2, 1, 2, 1; dropout after the 1st, 3rd, and 5th layers; filter depth 16, 32, 64, 128, 256, 512; batch normalization after each layer) and a fully connected layer to a single output for real/fake. We used a leaky ReLU $\alpha = 0 . 2$ ) nonlinearity after each layer, except the final layer, where we used a tanh.
+
+This GAN was trained on a dataset consisting of landscape images of mountains and coastlines obtained from the web. The generator was trained with the Adam optimizer (Kingma & Ba, 2015), and the discriminator with RMSProp. The learning rates for both were set to $1 0 ^ { - 5 }$ , and for Adam we set $\beta _ { 1 } = 0 . 5$ . We used a latent size of 64 units. The model was trained for $1 . 1 \times 1 0 ^ { 6 }$ gradient steps, when the generated images appeared to stop improving.
+
+# C.3 PIR ESTIMATOR
+
+Instead of predicting PIR as a scalar directly, we predict it by classifying into 100 bins via a softmax, which performs better empirically. This choice was motivated by noting that the scalar version was having trouble fitting some highly multi-modal distributions that appear in the data. We trained the PIR estimator with the Adam optimizer (learning rate $5 \cdot 1 0 ^ { - 4 }$ ). When evaluating and when using this model for improving the GAN we froze the weights of the PIR estimator. We also reduced the output softmax’s temperature to 0.01, so it was behaving almost like a max, which empirically improved results. Intuitively, a low softmax temperature in training allows the system to rapidly get gradients from many different output bins and adjust the distribution appropriately, whereas when actually using the system we want to be conservative with our estimates and not be too biased by low probability bins far from the modal estimate.
+
+# C.4 SIMULATED DATA
+
+# C.4.1 VGG FEATURES
+
+The first approach we took to evaluating our system’s ability to train for different features was to use activity from hidden layers of a computer vision model, specifically VGG 16 (Simonyan & Zisserman, 2014) trained on ImageNet (Russakovsky et al., 2015). In particular, we took the $\ell _ { 2 }$ norm of the activity of one filter within a layer, and normalized it by the $\ell _ { 2 }$ norm of the total layer’s activity, i.e. letting $\mathrm { v G G } _ { l , f } ( i )$ be the vector of unit activations in filter $f$ of layer $l$ of VGG 16 on image $i$ , we computed PIR for that image and a given layer and filter $l ^ { * } , f ^ { * }$ as:
+
+$$
+\begin{array} { r } { \mathrm { P I R } _ { l ^ { * } , f ^ { * } } ( i ) = \sqrt { \frac { \left| \mathrm { V G G } _ { l ^ { * } , f ^ { * } } ( i ) \right| _ { 2 } ^ { 2 } } { \sum _ { f \in l ^ { * } } \left| \mathrm { V G G } _ { l ^ { * } , f } ( i ) \right| _ { 2 } ^ { 2 } } } } \end{array}
+$$
+
+(Note that if we did not normalize by the activity in the whole layer, the system might be able to “cheat” to improve the PIR by just increasing the contrast of the images, which will likely increase overall network activity.) As noted above, we also added binomially distributed noise to these PIRs.
+
+# C.4.2 MULTIPLE FILTERS
+
+After using our system on the tasks above, we noted that its performance was quite poor at layers 5, 6, and 7 of VGG compared to other tasks (see Fig. 2). This could suggest that our system was unable to capture the complex features represented at the higher levels of VGG. However, we also noticed that the feature representations at these layers tended to be quite sparse, so many of the simulated PIRs we generated were actually zero to within the bin width of our PIR estimator. Respectively, these layers had only $20 \%$ , $2 5 \%$ , and $24 \%$ non-zero PIRs (collapsing across filters), and around half the filters in each (resp. 6, 4, and 5) were producing $> 9 0 \%$ zero PIRs. (By contrast, the layer with the next greatest number of zero PIRs, fc8, still had $69 \%$ nonzero PIRs overall, and had no filters in which $90 \%$ or more of the PIRs were zero.) In a few cases on layers 5, 6, and 7, all of the generated PIRs were zero. This clearly makes learning infeasible, and indeed we noted that there was a strong relationship between number of non-zero simulated PIRs in the training dataset and the ability of our system to improve PIR (see Fig. 8). This is somewhat troubling, since probably most images in the real world will not produce a PIR that is truly zero.
+
+![](images/1cc28adcf4d57ca6b2d09c22ef5c94c72c0d24e85a98baf26a61262dbd9e4b51.jpg)  
+Figure 8: Percent non-zero PIRs vs. effect size (Single-filter VGG tasks and color tasks)
+
+in order to evaluate whether the poor performance of our system at the higher layers of VGG was due to the number of zeros or to the complexity of the features, we created less sparse features from these layers by simply targeting a set of $k$ filters sampled without replacement from the layer, rather than a single filter. We did this by taking the norm across the $k$ target filters, or equivalently by summing the squared norms of the $k$ filters before taking the square root, and then normalizing by the activity in the layer as before. Formally, letting $a _ { 1 } , . . . , a _ { k }$ be a set of $k$ filter indices sampled without replacement from $\{ 0 , . . . ,$ number of filters in layer $\}$ , we computed the PIR for an image as:
+
+$$
+\mathrm { P I R } _ { l ^ { * } , f ^ { * } , k } ( i ) = \sqrt { \frac { \sum _ { j = 1 } ^ { j = k } \left| \mathrm { V G G } _ { l ^ { * } , a _ { j } } ( i ) \right| _ { 2 } ^ { 2 } } { \sum _ { f \in l ^ { * } } \left| \mathrm { V G G } _ { l ^ { * } , f } ( i ) \right| _ { 2 } ^ { 2 } } }
+$$
+
+The single filter cases above can be thought of as a special case of this, where $k = 1$ . To complement these, we also tried $k = 2 0$ . As above, we also added binomially distributed noise to these PIRs.
+
+This can also be thought of as perhaps a more realistic simulation of human behavior, in the sense that it is highly unlikely that there is a single feature which influences human PIRs. Rather, there are probably many related features which influence PIR in various ways. Thus it is important to evaluate our system’s ability to target these types of features as well.
+
+# C.4.3 COLORS
+
+Finally, we also considered some simpler objectives based on targeting specific colors in the output images. Analogously to the VGG features, we computed the PIRs from the vector norm of a given image in the targeted color, normalized by the total image value. We considered several objectives of this type:
+
+Single color: Optimizing for a single color of output image, e.g., for red the objective would be.
+
+$$
+\mathrm { P I R } _ { \mathrm { r e d } } ( i ) = \sqrt { \frac { | i ( : , : , \mathrm { r e d } ) | _ { 2 } ^ { 2 } } { | \mathbf { i } ( : , : , : ) | _ { 2 } ^ { 2 } } }
+$$
+
+Two color: We split the image horizontally into a left and right half, and then computed PIR from one color in the left half and a different color in the right half.
+
+$$
+\mathsf { P I R } _ { \mathrm { r e d \ b l u e } } ( i ) = \sqrt { \frac { \left| i ( : , : \frac { \mathrm { w i d t h } } { 2 } , \mathrm { r e d } ) \right| _ { 2 } ^ { 2 } + \left| i ( : , \frac { \mathrm { w i d t h } } { 2 } : , \mathsf { b l u e } ) \right| _ { 2 } ^ { 2 } } { | \dot { \mathbf { \mathbf { \mathbf { \mathbf { \Phi } } } } } ( : , : , : ) | _ { 2 } ^ { 2 } } }
+$$
+
+Three color: Similar to two color, but split the image into thirds, and computed PIR from a different color in each third.
+
+(As above, we also added binomially distributed noise to these PIRs.) These objectives provide a useful complement to the VGG objectives discussed in section 3.1. Although the single color objectives may be relevant to the classification task VGG 16 performs, the split color tasks are less likely to be relevant to classification. Note that it is important that we split the images along the width instead of the height dimension, as there may well be semantically relevant features corresponding to color divisions along the height dimension, e.g. a blue upper half and green lower half likely correlates with outdoor images, which would provide useful class information. By contrast, it is harder to imagine circumstances where different colors on the left and right halves of the image are semantically predictive, especially since flipping left to right is usually included in the data augmentation for computer vision systems. Thus success on optimizing for these objectives would increase our confidence in the generality of our system.
+
+# C.5 POSSIBLE IMPROVEMENTS
+
+There are a number of techniques that could be explored to improve our system. As we mentioned above, iterating for multiple steps of PIR collection and generative model tuning is worth exploring further. Also, some form of data scaling might allow the system to perform better on tasks with low variance. We briefly tried normalizing all data for an objective to have mean 0.5 and standard deviation 0.25, but did not achieve particularly good results from this, possibly because there were many outliers getting clipped to 0 or 1. Still, there are many other possibilities for scaling data that could potentially result in some improvement in performance. Also, one alternative approach to training a GAN to produce high-PIR images would be to use the PIR estimator objective in the Plug & Play Generative Networks framework (Nguyen et al., 2017) instead of using it to tune the GAN. This could be an interesting direction to explore, but its success would probably depend on expressiveness of the initial generative model. With the mediocre model we started with, it’s probably better to actually tune the model itself, which may allow it to explore parts of image space which it had not previously.

parse/train/HJr4QJ26W/HJr4QJ26W_content_list.json

@@ -0,0 +1,1606 @@
+[
+    {
+        "type": "text",
+        "text": "IMPROVING IMAGE GENERATIVE MODELS WITH HUMAN INTERACTIONS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            101,
+            826,
+            145
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            183,
+            170,
+            398,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            234,
+            544,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "GANs provide a framework for training generative models which mimic a data distribution. However, in many cases we wish to train a generative model to optimize some auxiliary objective function within the data it generates, such as making more aesthetically pleasing images. In some cases, these objective functions are difficult to evaluate, e.g. they may require human interaction. Here, we develop a system for efficiently training a GAN to increase a generic rate of positive user interactions, for example aesthetic ratings. To do this, we build a model of human behavior in the targeted domain from a relatively small set of interactions, and then use this behavioral model as an auxiliary loss function to improve the generative model. As a proof of concept, we demonstrate that this system is successful at improving positive interaction rates simulated from a variety of objectives, and characterize some factors that affect its performance. ",
+        "bbox": [
+            233,
+            263,
+            766,
+            430
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            454,
+            336,
+            470
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Generative image models have improved rapidly in the past few years, in part because of the success of Generative Adversarial Networks, or GANs (Goodfellow et al., 2014). GANs attempt to train a “generator” to create images which mimic real images, by training it to fool an adversarial “discriminator,” which attempts to discern whether images are real or fake. This is one solution to the difficult problem of learning when we don’t know how to write down an objective function for image quality: take an empirical distribution of “good” images, and try to match it. ",
+        "bbox": [
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+            484,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Often, we want to impose additional constraints on our goal distribution besides simply matching empirical data. If we can write down an objective which reflects our goals (even approximately), we can often simply incorporate this into the loss function to achieve our goals. For example, when trying to generate art, we would like our network to be creative and innovative rather than just imitating previous styles, and including a penalty in the loss for producing recognized styles appears to make GANs more creative (Elgammal et al., 2017). Conditioning on image content class, training the discriminator to classify image content as well as making real/fake judgements, and including a loss term for fooling the discriminator on class both allows for targeted image generation and improves overall performance (Odena et al., 2016). ",
+        "bbox": [
+            174,
+            575,
+            825,
+            700
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "However, sometimes it is not easy to write an explicit objective that reflects our goals. Often the only effective way to evaluate machine learning systems on complex tasks is by asking humans to determine the quality of their results (Christiano et al., 2017, e.g.) or by actually trying them out in the real world. Can we incorporate this kind of feedback to efficiently guide a generative model toward producing better results? Can we do so without a prohibitively expensive and slow amount of data collection? In this paper, we tackle a specific problem of this kind: generating images that cause more positive user interactions. We imagine interactions are measured by a generic Positive Interaction Rate (PIR), which could come from a wide variety of sources. ",
+        "bbox": [
+            174,
+            708,
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+            819
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "For example, users might be asked to rate how aesthetically pleasing an image is from 1 to 5 stars. The PIR could be computed as a weighted sum of how frequently different ratings were chosen. Alternatively, these images could be used in the background of web pages. We can assess user interactions with a webpage in a variety of ways (time on page, clicks, shares, etc.), and summarize these interactions as the PIR. In both of these tasks, we don’t know exactly what features will affect the PIR, and we certainly don’t know how to explicitly compute the PIR for an image. However, we can empirically determine the quality of an image by actually showing it to users, and in this paper we show how to use a small amount of this data (results on 1000 images) to efficiently tune a generative model to produce images which increase PIR. In this work we focus on simulated PIR values as a proof of concept, but in future work we will investigate PIR values from real interactions. ",
+        "bbox": [
+            174,
+            827,
+            825,
+            924
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/ec679fbe76f983e591cddbdeceb8798f656c1b196c970471f04538046cd10173.jpg",
+        "image_caption": [
+            "Figure 1: Diagram of our system "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            217,
+            99,
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+            342
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 APPROACH",
+        "text_level": 1,
+        "bbox": [
+            174,
+            502,
+            297,
+            517
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The most straight-forward way to improve an image GAN might be to evaluate the images the model produces with real users at each training step. However, this process is far too slow. Instead, we want to be able to collect a batch of PIR data on a batch of images, and then use this batch of data to improve the generative model for many gradient steps; we want to do this despite the fact that the images the generator is producing may evolve to be very different from the original images we collected PIR data on. In order to do this, we use the batch of image and PIR data to train a “PIR Estimator Model” which predicts PIRs on images. We then use these estimated PIRs at each step as a loss. ",
+        "bbox": [
+            173,
+            547,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our approach is inspired by the work of Christiano and colleagues (Christiano et al., 2017), who integrated human preference ratings between action sequences into training of a reinforcement learning model by using the preference data to estimate a reward function. However, our problem and approach differ in several key ways. First, we are optimizing a generative image model rather than a RL model. This is more difficult in some ways, since the output space is much higher-dimensional than typical RL problems, which means that scalar feedback (like a PIR) may be harder for the system to learn from. This difficulty is partially offset by the fact that we assume we get “reward” (PIR) information for an image when we evaluate, instead of just getting preferences which we have to map to rewards. Perhaps most importantly, we use our PIR estimation model as a fully-differentiable loss function for training, instead of just using its estimated rewards. This allows us to more effectively exploit its knowledge of the objective function (but risks overfitting). ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our system consists of three components: A generative image model, users who interact with the generated images in some way, and a PIR estimator that models user interactions given an image. See Fig. 1 for a diagram of the system’s general operation. The generative model produces images, which are served to users. Using interaction data from these users, we train the PIR estimator model, which predicts PIRs given a background image, and then incorporate this estimated PIR into the loss of the generative model to tune it to produce higher quality images. Below, we discuss each of these components in more detail. ",
+        "bbox": [
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+            827,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 GENERATIVE MODEL ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We begin with a $\\mathrm { G A N ^ { 1 } }$ (Goodfellow et al., 2014) which we pre-trained to produce images from a target distribution (specifically, landscapes of mountains and coasts). Let $D _ { \\mathrm { s o u r c e } }$ be the source estimated by the discriminator, $G$ the generator, and $z$ a noise input to the generator sampled from a multivariate standard normal distribution $\\mathcal { N } ( 0 , I )$ , and $\\mathcal { T }$ be the set of real images shown to the discriminator. Define: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/0088b43259ac008782f8d88cec6de49629fc3381a9b69629b272e6a48dd3be36.jpg",
+        "text": "$$\n\\begin{array} { r l r } { L _ { \\mathrm { f a k e i m a g e } } = } & { } & { E _ { z \\sim \\mathcal { N } ( 0 , I ) } \\left[ \\log P ( D _ { \\mathrm { s o u r c e } } ( G ( z ) ) = \\mathrm { f a k e } ) \\right] } \\\\ { L _ { \\mathrm { f a k e i m a g e f o o l s } } = } & { } & { E _ { z \\sim \\mathcal { N } ( 0 , I ) } \\left[ \\log P ( D _ { \\mathrm { s o u r c e } } ( G ( z ) ) = \\mathrm { r e a l } ) \\right] } \\\\ { L _ { \\mathrm { r e a l i m a g e } } = } & { } & { E _ { i \\sim \\mathcal { Z } } \\left[ \\log P ( D _ { \\mathrm { s o u r c e } } ( i ) = \\mathrm { r e a l } ) \\right] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Then the discriminator and generator are trained to maximize the following losses (respectively), where the $w _ { * }$ are weights set as hyperparameters: ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6cd424a9880748934e609233a2c271a2dfa0d7a3a7cf644a9761608799819961.jpg",
+        "text": "$$\n\\begin{array} { r l r } { L _ { \\mathrm { d i s c r i m i n a t o r } } = } & { { } } & { L _ { \\mathrm { f a k e i m a g e } } + L _ { \\mathrm { r e a l i m a g e } } } \\\\ { L _ { \\mathrm { G e n e r a t o r } } = } & { { } } & { L _ { \\mathrm { f a k e i m a g e f o o l s } } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            310,
+            311,
+            687,
+            348
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Note that there is a difference between these losses and the standard GAN formulation given in (Goodfellow et al., 2014) – we maximize $L _ { \\mathrm { f a k e } }$ image fools $= \\ \\log \\ P$ (classified real) rather than minimizing $\\log { ( 1 - P }$ (classified real)). This seems to result in the generation of slightly better images in practice. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "This GAN was trained on a dataset consisting of landscape images of mountains and coastlines (see Appendix C.2 for details of the architecture and training). It is worth noting that this generative model is not photorealistic (see Fig. 3a for some samples). Its expressive capacity is limited, and it has clear output modes with limited intra-mode variability. However, for our purposes this may not matter. Indeed, it is in some ways more interesting if we can tweak this model to optimize for many objective functions, since its limited expressive capacity will make it more difficult for us to estimate and pursue the real objective – a limited set of images will effectively give us fewer points to estimate the PIR function from, and will reduce the space in which the model can easily produce images, thus reducing the possibility of getting very optimal images from the model. For example, a model which produces images of birds may not produce data points which provide good estimates of a PIR based on how much the image looks like a car, and even if it could, it may not be able to produce images which are “more car-like.” If we are able to succeed in improving PIRs with this generative model, it is likely that a better generative model would yield even better results. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 USER INTERACTIONS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We will show these images to users in a variety of ways, depending on our target domain. For the purposes of this paper, however, we will use simulated interaction data (see Section 3 for details). Of course, since showing images to users is an expensive prospect, we wanted to limit the size of the datasets we used to train the model. Typical datasets used to train vision models are on the order of millions of images, (e.g. ImageNet (Russakovsky et al., 2015)), but it is completely infeasible to collect user data on this number of images. We estimated that we could show 1000 images each 1000 times to generate our datasets. We used these dataset sizes and number of impressions for all experiments discussed here, and added noise to the PIRs that was binomially distributed according to the number of times each image was shown and the “true” PIR simulate from the objective. ",
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+    {
+        "type": "text",
+        "text": "2.3 PIR ESTIMATOR MODEL ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The final component of our system is the PIR estimator model, which learns to predict PIR from a background image. We denote this model by $R : { \\mathrm { i m a g e } }  [ 0 , 1 ]$ . We parameterize this model as a deep neural network. Specifically, we take the Inception v2 architecture (Szegedy et al., 2016), remove the output layer, and replace it with a fully-connected layer to PIR estimates. We initialize the Inception v2 parameters from a version of the model trained on [dataset redacted for blind review]. See Appendix C.3 for more details. ",
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+    {
+        "type": "text",
+        "text": "Why did we not make estimated PIR simply another auxiliary output from the discriminator, like class in the ACGAN (Appendix C.1)? Because the PIR estimator needs to be held constant in order to provide an accurate training objective. If the PIR estimates were produced by the discriminator, then as the discriminator changed to accurately discriminate the evolving generator images, the PIR estimates would tend to drift without a ground-truth to train them on. Separating the discriminator and the PIR estimator allows us to freeze the PIR estimator while still letting the discriminator adapt. ",
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+    {
+        "type": "text",
+        "text": "2.4 INTEGRATION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Once we have trained a PIR estimator model, we have to use it to improve the GAN. We do this as follows. Let $R$ denote the PIR estimator model, as above. Define $L _ { \\mathrm { P I R } }$ to be the expectation of the estimated PIR produced over images sampled from the generator: ",
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+    {
+        "type": "equation",
+        "img_path": "images/31e98b68a936132de4225252446cd8cbd2a29d8084557596bd57aef4e9dd4d44.jpg",
+        "text": "$$\n{ \\cal L } _ { \\mathrm { P I R } } = { \\cal E } _ { z \\sim { \\cal N } ( 0 , I ) , c \\sim { \\cal C } } \\left[ R ( G ( z ) ) \\right]\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Then we simply supplement the generator loss by adding this term times a weight $w _ { P I R }$ , set as a hyperparameter: ",
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+    {
+        "type": "equation",
+        "img_path": "images/f70a2347392d7ceed6a2bbc190de7dcbf0d19af1844a2d8217ec78d2b91149cf.jpg",
+        "text": "$$\nL _ { \\mathrm { G e n e r a t o r } } = L _ { \\mathrm { f a k e i m a g e f o o l s } } + w _ { P I R } L _ { P I R }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We set $w _ { P I R } = 1 0 0 0$ as this made the magnitude of the PIR loss and the other loss terms roughly comparable. Otherwise we used the same parameters as in the GAN training above, except that we reduced the learning rate to $1 0 ^ { - 6 }$ to allow the system to adapt more smoothly to the multiple objectives, and we trained for 50,000 steps. ",
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+    {
+        "type": "text",
+        "text": "3 DATA ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we use simulated interaction data as proof of concept. This raises an issue: what functions should we use to simulate user interactions? Human behavior is complex, and if we already knew precisely what guided user interactions, there would be no need to actually collect human behavioral data at all. Since we don’t know what features will guide human behavior, the next best thing we can do is to ensure that our system is able to alter the image generation model in a broad variety of ways, ranging from low level features (like making the images more colorful) to altering complex semantic features (such as including more plants in outdoor scenery). We also want to avoid hand-engineering tasks to the greatest extent possible. We present an overview of our approaches to simulating PIR data below, see Appendix C.4 for more details. ",
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+    {
+        "type": "text",
+        "text": "3.1 VGG FEATURES ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The first approach we took to evaluating our system’s ability to train for different features was to use activity from hidden layers of a computer vision model, specifically VGG 16 (Simonyan & Zisserman, 2014) trained on ImageNet (Russakovsky et al., 2015). In particular, we took the activity of a single filter in a layer of VGG relative to the overall activity of that layer. This approach to simulating PIRs has several benefits. First, it gives a wide variety of complex objectives that can nevertheless be easily computed to simulate data. Second, models like VGG exhibit hierarchical organization, where lower levels generally respond to lower-level features such as edges and colors, while higher levels respond to higher-level semantic features such as faces (Zeiler & Fergus, 2014), and the represented features relate to those in human and macaque visual cortex (Yamins et al., 2014). Thus VGG features give a wide range of objectives which we may relate to the human perception we wish to target. ",
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+    {
+        "type": "text",
+        "text": "There are some caveats to this approach, however. First, although the higher layers of CNNs are somewhat selective for “abstract” object categories, they are also fooled by adversarial images that humans would not be, and directly optimizing inputs for these high level features does not actually produce semantically meaningful images (Nguyen et al., 2014). Thus, even if our system succeeds in increasing activity in a targeted layer which is semantically selective, it will likely do so by adversarially exploiting particulars of VGG 16’s parameterization of the classification problem (although the fact that we are not backpropagating through the true objective will make this harder). It is not necessarily a failure of the system if it exploits simple features of the objective it is given to increase PIRs – indeed, it should be seen as a success, as long as it is generalizes to novel images. However, success on this task does not necessarily guarantee success on modifying semantic content when interacting with actual humans. It may be easier for the PIR estimator model (which is based on a CNN) to learn objectives which come from another CNN than more general possible objectives. The fact that adversarial examples can sometimes transfer between networks with different architectures (Liu et al., 2016) suggests that the computations being performed by these networks are somewhat architecture invariant. Thus CNN objectives may be easier for our estimator than human ones. ",
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We have tried to minimize these problems to the greatest extent possible by using different network architectures (Inception V2 and VGG 16, respectively) trained on different datasets ([hidden] and ImageNet (Russakovsky et al., 2015), respectively) for the estimator and the objective. However, we cannot be certain that the network is not “cheating” in some way on the VGG 16 tasks, so our results must be considered with this qualification in mind. Despite this, we think that evaluating our system’s ability to optimize for objectives generated from various layers of VGG will show its ability to optimize for a variety of complex objectives, and thus will serve as a useful indicator of its potential to improve PIRs from real users. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.2 MULTIPLE FILTERS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "After using our system on the tasks above, we noted that its performance was quite poor at layers 5, 6, and 7 of VGG compared to other tasks (see Fig. 2). This could suggest that our system was unable to capture the complex features represented at the higher levels of VGG. However, we also noticed that the feature representations at these layers tended to be quite sparse, so many of the simulated PIRs we generated were actually zero to within the bin width of our PIR estimator (see Appendix B Fig. 8 for a plot of how this affected learning). In order to evaluate whether the poor performance of our system at the higher layers of VGG was due to the number of zeros or to the complexity of the features, we created less sparse features from these layers by simply targeting a set of $k$ filters sampled without replacement from the layer, rather than a single filter. The single filter cases above can be thought of as a special case of this, where $k = 1$ . To complement these, we also tried $k = 2 0$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "This can also be thought of as perhaps a more realistic simulation of human behavior, in the sense that it is highly unlikely that there is a single feature which influences human PIRs. Rather, there are probably many related features which influence PIR in various ways. Thus it is important to evaluate our system’s ability to target these types of features as well. ",
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+    {
+        "type": "text",
+        "text": "3.3 COLORS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Finally, we also considered some simpler objectives based on targeting specific colors in the output images, or targeting vertical bands of two different colors, one in each half of the image, or three colors, one in each third of the image. These objectives provide a useful complement to the VGG objectives above. Although the single color objectives may be relevant to the classification task VGG 16 performs, the split color tasks are less likely to be relevant to classification. Note that it is important that we split the images along the width instead of the height dimension, as there may well be semantically relevant features corresponding to color divisions along the height dimension, e.g. a blue upper half and green lower half likely correlates with outdoor images, which would provide useful class information. By contrast, it is harder to imagine circumstances where different colors on the left and right halves of the image are semantically predictive, especially since flipping left to right is usually included in the data augmentation for computer vision systems. Thus success on optimizing for these objectives would increase our confidence in the generality of our system. ",
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+    {
+        "type": "text",
+        "text": "4 RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We present our results in terms of the change in mean PIR from 1000 images produced by the GAN before tuning to 1000 images produced after tuning, or in terms of the effect size of this change (Cohen’s $d$ , i.e. the change in mean PIR standardized by the standard deviation of the PIRs in the pre- and post-tuning image sets). We assess whether these changes are significant by performing a Welch’s t-test (with a significance threshold of $\\alpha = 0 . 0 0 1$ ) between the pre- and post-tuning PIRs. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Overall, our system was quite successful at improving PIRs across a range of simulated objective functions (see Fig. 2). Below, we discuss these results in more detail. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/452a4c3d84260f277ccf32fa578a0b0396ce5df970f4fe40eb77375a745ada9e.jpg",
+        "image_caption": [
+            "Figure 2: Effect size (number of standard deviations change in mean PIR) on a variety of tasks. Values greater than zero indicate improvement. Each panel represents a set of tasks (such as single filters from VGG), each color/column represents a subset (such as a single layer within VGG), and each point represents one objective (such as a single filter from that layer) for which we optimized the generative model. Points for which the change in mean PIR is not significant by a Welch’s $t$ -test with a threshold of $\\alpha = 0 . 0 0 1$ are partially transparent. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.1 VGG OBJECTIVES ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "Our system largely succeed at increasing PIRs on a variety of VGG objectives (see Fig. 2). However, there are several interesting patterns to note. First, the system is not particularly successful at targeting single filters from the pool5, fc6, and fc7 layers. However, we believe this is due to the fact that filters in these layers produce relatively sparse activation, see section 3.2. Indeed, the performance of the system seems much more consistent when it is optimizing for sets of 20 filters than for single filters. ",
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+    {
+        "type": "text",
+        "text": "Even when using 20 filters, however, there is a noticeable decline in the effect size of the improvement the system is able to make at higher layers of VGG $\\beta = - 0 . 1 3$ per layer, $t = - 7 . 8$ , $\\dot { p } < 1 0 ^ { - 1 0 }$ in a linear model controlling for initial standard deviation and percent zeros). This suggests that as the objectives grow more complex, the system may be finding less accurate approximations to them. However, the system’s continuing (if diminished) success at the higher layers of VGG suggests that our model is capable of at least partially capturing complex objective functions. ",
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+    },
+    {
+        "type": "text",
+        "text": "4.2 COLOR OBJECTIVES ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Overall, the system performed quite well at optimizing for the color objectives, particularly the single and two-color results (see Fig. 2). It had more difficulty optimizing for the three-color results, and indeed had produced only very small improvements after the usual 50,000 tuning steps for the generative model, but after 500,000 steps it was able to produce significant improvements for two out of the three objectives (these longer training results are the ones included here). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Because the color objectives are easiest to assess visually, we have included results for a variety of these objectives in Fig. 3. For the single color objectives, the improvement is quite clear, for example the images in Fig. 3d appear much more blue than the pre-training ones. For the two color objectives, it appears that the system found the “trick” of reducing the third color, for example the red-green split images in Fig. 3e appear much less blue than the pre-training images. Even on the three-color images where the system struggled, there are some visible signs of improvement, for example on the green-blue-red task the system has started producing a number of images with a blue streak in the middle. ",
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+    {
+        "type": "image",
+        "img_path": "images/a677d9bfd60d90b339e6056f4eedc86e60ca0e0077525f722cfbef7db0088cd1.jpg",
+        "image_caption": [
+            "Figure 3: Color objective sample images. Samples are randomly drawn, not cherry-picked. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.3 SUPPLEMENTAL ANALYSES ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We also conducted several supplemental analyses which can be found in detail Appendix A. In summary, the initial variability in the PIR of the images used to train the system is strongly correlated with the amount of improvement the system makes in the PIR, the system fairly consistently underestimates the performance it achieves (because of a detail of training procedure, see the Appendix), and iterating the process of improving PIRs yields better results for objectives from a lower layer of VGG but not a higher. ",
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "5 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Overall, our system appears to be relatively successful. It can optimize a generative model to produce images which target a wide variety of objectives, ranging from low-level visual features such as colors and early features of VGG to features computed at the top layers of VGG. This success across a wide variety of objective functions allows us to be somewhat confident that our system will be able to achieve success in optimizing for real human interactions. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Furthermore, the system did not require an inordinate amount of training data. In fact, we were able to successfully estimate many different objective functions from only 1000 images, several orders of magnitude fewer than is typically used to train CNNs for vision tasks. Furthermore, these images came from a very biased and narrow distribution (samples from our generative model) which is reflective of neither the images that were used to pre-train the Inception model in the PIR estimator, nor the images the VGG model (which produced the simulated objectives) was trained on. Our success from this small amount of data suggests that not only will our system be able to optimize for real human interactions, it will be able to do so from a feasible number of training points. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "These results are exciting – the model is able to approximate apparently complex objective functions from a small amount of data, even though this data comes from a very biased distribution that is unrelated to most the objectives in question. But what is really being learned? In the case of the color images, it’s clear that the model is doing something close to correct. However, for the objectives derived from VGG we have no way to really assess whether the model is making the images better or just more adversarial. For instance, when we are optimizing for the logit for “magpie,” it’s almost certainly the case that the result of this optimization will not look more like a magpie to a human, even if VGG does rate the images as more “magpie-like.” On the other hand, this is not necessarily a failure of the system – it is accurately capturing the objective function it is given. What remains to be seen is whether it can capture how background images influence human behavior as well as it can capture the vagaries of deep vision architectures. ",
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We believe there are many domains where a system similar to ours could be useful. We mentioned producing better webpage backgrounds and making more aesthetic images above, but there are many potential applications for improving GANs with a limited amount of human feedback. For example, a model could be trained to produce better music (e.g. song skip rates on streaming generated music could be treated as inverse PIRs). ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.1 TRADING IMAGE DIVERSITY FOR PIR ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            282,
+            473,
+            296
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "When tuning the GAN, the decrease in the PIR loss is usually accompanied by an increase in the generator loss, and often by a partial collapse of the generator output (for example, the optimized images generally seem to have fewer output modes than the pre-training images in Fig. 3). This is not especially surprising – because we weighted the PIR loss very highly, the model is rewarded for trading some image diversity for image optimality. Depending on the desired application, the weight on the PIR loss could be adjusted as necessary to trade off between producing images close to the data distribution and optimizing PIR. At its most extreme, one could down-weight the generator loss entirely, and train until the model just produces a single optimal image. However, the generator likely provides some regularization by constraining the images to be somewhat close to the real images, which will reduce overfitting to an imperfect estimate of the PIR function. Furthermore, in many settings we will want to generate a variety of images (e.g. backgrounds for different websites). For these reasons, we chose to keep the generator loss when tuning the GAN. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.2 FUTURE DIRECTIONS ",
+        "text_level": 1,
+        "bbox": [
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+            494,
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "There are a number of future directions suggested by this work. A number of possible improvements are discussed in Appendix C.5. However, we also think this work has potential applications from the perspective of distillation or imitation approaches, which attempt to train one network to emulate another (Hinton et al., 2015; Parisotto et al., 2015, e.g), as well as from the perspective of understanding the computations that these vision architectures perform. As far as we are aware, these results are the first to show that a deep vision model can be tuned rapidly from relatively little data to produce outputs which accurately emulate the behavior of hidden layers of another deep vision architecture trained on a different dataset. This suggests both that the inductive biases shared among these architectures are causing them to find similar solutions (which is also supported by work on transferable adversarial examples (Liu et al., 2016, e.g.)), and that these networks final layers represent the computations of earlier hidden layers in a way that is somewhat accessible. It’s possible that using our system with objectives from CNN layers as we did here might help to understand the features those layers are attending to, by analyzing the distribution of images that are produced. In this sense, our system can be thought of as offering a new approach to multifaceted feature visualization (Nguyen et al., 2016), because our system attempts to optimize a distribution of images for an objective and encourages diversity in the distribution produced, rather than just optimizing a single image. ",
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+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSIONS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We have described a system for efficiently tuning a generative image model according to a slow-toevaluate objective function. We have demonstrated the success of this system at targeting a variety of objective functions simulated from different layers of a deep vision model, as well as from low-level visual features of the images, and have shown that it can do so from a small amount of data. We have quantified some of the features that affect its performance, including the variability of the training PIR data and the number of zeros it contains. Our system’s success on a wide variety of objectives suggests that it will be able to improve real user interactions, or other objectives which are slow and expensive to evaluate. This may have many exciting applications, such as improving machine-generated images, music, or art. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            287,
+            118
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Paul Christiano, Jan Leike, Tom B Brown, Miljan Martic, Shane Legg, and Dario Amodei. Deep reinforcement learning from human preferences. arXi, 2017. ",
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+        "text": "Anh Nguyen, Jason Yosinski, and Jeff Clune. Multifaceted Feature Visualization: Uncovering the Different Types of Features Learned By Each Neuron in Deep Neural Networks. arXiv, 2016. ",
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+    {
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+        "text": "Anh Nguyen, Jason Yosinski, Yoshua Bengio, Alexey Dosovitskiy, and Jeff Clune. Plug $\\{ \\& \\}$ Play Generative Networks: Conditional Iterative Generation of Images in Latent Space. ICCV, (3):33, 2017. doi: 10.1109/CVPR.2017.374. ",
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+        "text": "Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, Alexander C Berg, and Li Fei-Fei. ImageNet Large Scale Visual Recognition Challenge. International Journal of Computer Vision, 115(3): 211–252, 2015. ISSN 15731405. doi: 10.1007/s11263-015-0816-y. ",
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+        "text": "Christian Szegedy, Vincent Vanhoucke, Sergey Ioffe, Jonathon Shlens, and Zbigniew Wojna. Rethinking the Inception Architecture for Computer Vision. Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR), pp. 2818–2826, 2016. ISSN 08866236. doi: 10.1002/2014GB005021. ",
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+        ],
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+    {
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+        "bbox": [
+            173,
+            804,
+            826,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Matthew D Zeiler and Rob Fergus. Visualizing and Understanding Convolutional Networks arXiv:1311.2901v3 [cs.CV] 28 Nov 2013. Computer Vision–ECCV 2014, 8689:818–833, 2014. ISSN 978-3-319-10589-5. doi: 10.1007/978-3-319-10590-1 53. ",
+        "bbox": [
+            173,
+            875,
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+            915
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/98240ce4438c7e3b31bfe707d1ab0bc9c597be9cf48d72834b5da56731a9b559.jpg",
+        "image_caption": [
+            "Figure 4: Change in mean PIR vs. estimated change in mean PIR. Points for which the change in mean PIR is not significant by a Welch’s $t$ -test with a threshold of $\\alpha = 0 . 0 0 1$ are partially transparent. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            305,
+            303,
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+            680
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "image",
+        "img_path": "images/4b39472a7f5980f4b2ddf37b5e22ac2583f56e4f0262421e27809b90bbe04416.jpg",
+        "image_caption": [
+            "Figure 5: Change in mean PIR vs. initial variability. Points for which the change in mean PIR is not significant by a Welch’s $t$ -test with a threshold of $\\alpha = 0 . 0 0 1$ are partially transparent. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            258,
+            103,
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+            393
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A OTHER ANALYSES ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            464,
+            361,
+            479
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.1 INTROSPECTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            496,
+            331,
+            511
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Because $L _ { P I R }$ is just the expected value of the PIR, by looking at $L _ { P I R }$ before and after tuning the generative model, we can tell how well the system thinks it is doing, i.e. how much it estimates that it improved PIR. This comparison reveals the interesting pattern that the system is overly pessimistic about its performance. In fact, it tends to underestimate its performance by a factor of more than 1.5 $\\beta = 1 . 6 7$ when regressing change in mean PIR on predicted change in mean PIR, see Fig. 4). However, it does so fairly consistently. This effect appears to be driven by the system consistently underestimating the (absolute) PIRs, which is probably caused by our change in the softmax temperature between training the PIR estimator and tuning the generative model (which we empirically found improves performance, as noted above). ",
+        "bbox": [
+            174,
+            522,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "This is in contrast to the possible a priori expectation that the model would systematically overestimate its performance, because it is overfitting to an imperfectly estimated objective function. Although decreasing the softmax temperature between training and using the PIR obscures this effect, we do see some evidence of this; the more complex objectives (which the system produced lower effect sizes on) seem to both have lower estimated changes in mean PIR and true changes in PIR which are even lower than the estimated ones (see Fig. 4). Thus although the system is somewhat aware of its reduced effectiveness with these objectives (as evidenced by the lower estimates of change in mean PIR), it is not reducing its estimates sufficiently to account for the true difficulty of the objectives (as evidenced by the fact that the true change in PIR is even lower than the estimates). However, the system was generally still able to obtain positive results on these objectives (see Fig. 2). ",
+        "bbox": [
+            174,
+            655,
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+            795
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A.2 INITIAL VARIABILITY ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            813,
+            367,
+            827
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "There is a general trend (see Fig. 5) that the variability in PIR in the initial dataset is strongly positively related with the change in PIR the system is able to produce $\\beta = 0 . 9 9$ , $t = 1 0 . 6$ , $p \\bar { < } \\bar { 1 0 } ^ { - 1 0 }$ , in a linear model controlling for initial mean PIR and initial percent of values that are zero). In fact, initial standard deviation explains about $50 \\%$ of the variance in the change in mean PIR. This is perhaps not too surprising – more variability means that the generative model has capacity to produce higher PIR images without too much tweaking, and that the PIR estimator model gets a wider range of values to learn from. Still, when attempting to use this system in practice, it is important to keep in mind that starting with a sufficiently expressive generative model will more likely produce better results than starting with a more limited model. ",
+        "bbox": [
+            174,
+            839,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/881a640f7021a2d81e1a10fbb7a3eae014fc91ad5806b8582eceaadd61ac6fd7.jpg",
+        "image_caption": [
+            "Figure 6: Effect size evolution over two iterations "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            305,
+            103,
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+            398
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            515,
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+            558
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.3 ITERATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            641,
+            294,
+            655
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Given that our model improves PIRs, an obvious question is whether we can iterate the process. Once we have increased PIRs, can we train a new PIR estimator on samples from our new generative model, and use that to increase PIRs again? If we could iterate this many times, we might be able to create much larger improvements in PIR than we can on a single step. On the other hand, it is possible that after a single step of optimization we will effectively have saturated the easily achievable improvement in the model, and further steps will not result in much improvement. ",
+        "bbox": [
+            174,
+            694,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "To evaluate this, we took the subset of models trained on single filters from VGG layers pool2 and fc8, and used the set of images generated from the post-tuning model along with the original set of images to tune their PIR estimators for another 25,000 steps, and then tuned the generative model using this updated PIR estimator for another 50,000 steps. We then evaluated them as before, see Fig. 6 for the results. The second iteration results were mixed, while the pool2 models all improved from the first step to the second, none of them improved as much as they had on the first step, and many of the fc8 models actually performed worse after the second step. However, it is possible that further hyperparameter tuning could improve these results, and it is certainly the case that running multiple steps of iteration and selecting the best model by experimentation could yield better results, so this is worth investigating further. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/bb5568a77c88b87dcfddadac91fed93fcab073ec0eaf412bb76545bbf1d6be46.jpg",
+        "image_caption": [
+            "Figure 7: Change in mean PIR on a variety of tasks. Each panel represents a set of tasks (such as single filters from VGG), each color/column represents a subset (such as a single layer within VGG), and each point represents one objective (such as a single filter from that layer) for which we optimized the generative model. Points for which the change in mean PIR is not significant by a Welch’s $t$ -test with a threshold of $\\alpha = 0 . 0 0 1$ are partially transparent. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            142,
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+            481
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C DETAILED METHODS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            592,
+            383,
+            607
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C.1 ACGAN ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            622,
+            279,
+            637
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We describe our results using the standard GAN framework for clarity, but we actually used an ACGAN (Odena et al., 2016), which allows for conditioning for various user-specific features. This requires the following adjustments. Defining $\\mathcal { C }$ to be the set of possible classes and $\\mathcal { T } _ { c }$ to be the set of real images corresponding to a class $c \\in { \\mathcal { C } }$ : ",
+        "bbox": [
+            174,
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+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/46d61d28718b47f1447cdcde2244b97da3d2efab9461ec27d33671ddc7176ccd.jpg",
+        "text": "$$\n\\begin{array} { r l r } { L _ { \\mathrm { f a k e i m a g e } } = } & { } & { E _ { z \\sim \\mathcal { N } ( 0 , I ) , c \\sim \\mathcal { C } } \\left[ \\log P ( D _ { \\mathrm { s o u r c e } } ( G ( z , c ) ) = \\mathrm { f a k e } ) \\right] } \\\\ { L _ { \\mathrm { f a k e i m a g e ~ f o o l s } } = } & { } & { E _ { z \\sim \\mathcal { N } ( 0 , I ) , c \\sim \\mathcal { C } } \\left[ \\log P ( D _ { \\mathrm { s o u r c e } } ( G ( z , c ) ) = \\mathrm { r e a l } ) \\right] } \\\\ { L _ { \\mathrm { r e a l ~ i m a g e } } = } & { } & { E _ { c \\sim \\mathcal { C } , i \\sim \\mathcal { L } _ { c } } \\left[ \\log P ( D _ { \\mathrm { s o u r c e } } ( i ) = \\mathrm { r e a l } ) \\right] } \\\\ { L _ { \\mathrm { f a k e i m a g e c l a s s } } = } & { } & { E _ { z \\sim \\mathcal { N } ( 0 , I ) , c \\sim \\mathcal { C } } \\left[ \\log P ( D _ { \\mathrm { c l a s s } } ( G ( z , c ) ) = c ) \\right] } \\\\ { L _ { \\mathrm { r e a l ~ i m a g e c l a s s } } = } & { } & { E _ { c \\sim \\mathcal { C } , i \\sim \\mathcal { Z } _ { c } } \\left[ \\log P ( D _ { \\mathrm { c l a s s } } ( i ) = c ) \\right] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            250,
+            709,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Then the discriminator and generator losses are modified as follows (letting $w _ { f a k e c l a s s } = 1 . 5$ and $w _ { r e a l c l a s s } = 1 . 0 \\AA$ : ",
+        "bbox": [
+            173,
+            808,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/acc8f92bc4ac62832f05e8e89b7f6e8675ca13042430cbc8dea8acace320e3b8.jpg",
+        "text": "$$\n\\begin{array} { r l r } { L _ { \\mathrm { d i s c r i m i n a t o r } } = } & { { } } & { L _ { \\mathrm { f a k e ~ i m a g e } } + L _ { \\mathrm { r e a l ~ i m a g e } } + w _ { r e a l c l a s s } L _ { \\mathrm { r e a l ~ c l a s s } } } \\\\ { L _ { \\mathrm { G e n e r a t o r } } = } & { { } } & { L _ { \\mathrm { f a k e ~ i m a g e ~ f o o l s } } + w _ { f a k e c l a s s } L _ { \\mathrm { f a k e ~ c l a s s } } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            264,
+            840,
+            733,
+            877
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Note that unlike the standard ACGAN formulation given in (Odena et al., 2016), we do not include $L _ { \\mathrm { f a k e \\ c l a s s } }$ in the discriminator’s loss, to keep the discriminator from cooperating with the generator on the classification task. ",
+        "bbox": [
+            174,
+            881,
+            825,
+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We modified the generator network by adding one-hot class inputs, and the discriminator by adding class outputs alongside the source output, as in (Odena et al., 2016). ",
+        "bbox": [
+            173,
+            103,
+            823,
+            132
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C.2 GENERATOR & DISCRIMINATOR ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            148,
+            437,
+            162
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We parameterized the generator as a deep neural network, which begins with a fully-connected mapping from the latent (noise) space to a $4 \\times 4 \\times 5 1 2$ dimensional image, and then successively upsampled (a factor of 2 by nearest neighbor), padded and applied a convolution ( $3 \\times 3$ kernel, stride of 1) and a leaky ReLU $\\alpha = 0 . 2$ ) nonlinearity repeatedly. We repeated this process 5 times (except with no upsampling on the first step, and a tanh nonlinearity on the last), while stepping the image depth down as follows: 512, 512, 256, 128, 64, and finally 3 (RGB) for the output image. This means that the final output images were $6 4 \\times 6 4$ . We parameterized the discriminator as a convolutional network with 7 layers, 6 convolutions (kernels all $3 \\times 3$ ; strides 2, 1, 2, 1, 2, 1; dropout after the 1st, 3rd, and 5th layers; filter depth 16, 32, 64, 128, 256, 512; batch normalization after each layer) and a fully connected layer to a single output for real/fake. We used a leaky ReLU $\\alpha = 0 . 2$ ) nonlinearity after each layer, except the final layer, where we used a tanh. ",
+        "bbox": [
+            174,
+            174,
+            825,
+            328
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "This GAN was trained on a dataset consisting of landscape images of mountains and coastlines obtained from the web. The generator was trained with the Adam optimizer (Kingma & Ba, 2015), and the discriminator with RMSProp. The learning rates for both were set to $1 0 ^ { - 5 }$ , and for Adam we set $\\beta _ { 1 } = 0 . 5$ . We used a latent size of 64 units. The model was trained for $1 . 1 \\times 1 0 ^ { 6 }$ gradient steps, when the generated images appeared to stop improving. ",
+        "bbox": [
+            174,
+            334,
+            825,
+            405
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C.3 PIR ESTIMATOR ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            421,
+            330,
+            435
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Instead of predicting PIR as a scalar directly, we predict it by classifying into 100 bins via a softmax, which performs better empirically. This choice was motivated by noting that the scalar version was having trouble fitting some highly multi-modal distributions that appear in the data. We trained the PIR estimator with the Adam optimizer (learning rate $5 \\cdot 1 0 ^ { - 4 }$ ). When evaluating and when using this model for improving the GAN we froze the weights of the PIR estimator. We also reduced the output softmax’s temperature to 0.01, so it was behaving almost like a max, which empirically improved results. Intuitively, a low softmax temperature in training allows the system to rapidly get gradients from many different output bins and adjust the distribution appropriately, whereas when actually using the system we want to be conservative with our estimates and not be too biased by low probability bins far from the modal estimate. ",
+        "bbox": [
+            174,
+            446,
+            825,
+            587
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C.4 SIMULATED DATA ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            603,
+            339,
+            617
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C.4.1 VGG FEATURES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            628,
+            346,
+            643
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The first approach we took to evaluating our system’s ability to train for different features was to use activity from hidden layers of a computer vision model, specifically VGG 16 (Simonyan & Zisserman, 2014) trained on ImageNet (Russakovsky et al., 2015). In particular, we took the $\\ell _ { 2 }$ norm of the activity of one filter within a layer, and normalized it by the $\\ell _ { 2 }$ norm of the total layer’s activity, i.e. letting $\\mathrm { v G G } _ { l , f } ( i )$ be the vector of unit activations in filter $f$ of layer $l$ of VGG 16 on image $i$ , we computed PIR for that image and a given layer and filter $l ^ { * } , f ^ { * }$ as: ",
+        "bbox": [
+            174,
+            654,
+            826,
+            738
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a81b33135838f00b492ca1e7e4e55f5c17ef46cc5a03d415774ab2fef327b534.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathrm { P I R } _ { l ^ { * } , f ^ { * } } ( i ) = \\sqrt { \\frac { \\left| \\mathrm { V G G } _ { l ^ { * } , f ^ { * } } ( i ) \\right| _ { 2 } ^ { 2 } } { \\sum _ { f \\in l ^ { * } } \\left| \\mathrm { V G G } _ { l ^ { * } , f } ( i ) \\right| _ { 2 } ^ { 2 } } } } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            367,
+            743,
+            633,
+            795
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "(Note that if we did not normalize by the activity in the whole layer, the system might be able to “cheat” to improve the PIR by just increasing the contrast of the images, which will likely increase overall network activity.) As noted above, we also added binomially distributed noise to these PIRs. ",
+        "bbox": [
+            173,
+            799,
+            825,
+            842
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C.4.2 MULTIPLE FILTERS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            857,
+            362,
+            871
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "After using our system on the tasks above, we noted that its performance was quite poor at layers 5, 6, and 7 of VGG compared to other tasks (see Fig. 2). This could suggest that our system was unable to capture the complex features represented at the higher levels of VGG. However, we also noticed that the feature representations at these layers tended to be quite sparse, so many of the simulated PIRs we generated were actually zero to within the bin width of our PIR estimator. Respectively, these layers had only $20 \\%$ , $2 5 \\%$ , and $24 \\%$ non-zero PIRs (collapsing across filters), and around half the filters in each (resp. 6, 4, and 5) were producing $> 9 0 \\%$ zero PIRs. (By contrast, the layer with the next greatest number of zero PIRs, fc8, still had $69 \\%$ nonzero PIRs overall, and had no filters in which $90 \\%$ or more of the PIRs were zero.) In a few cases on layers 5, 6, and 7, all of the generated PIRs were zero. This clearly makes learning infeasible, and indeed we noted that there was a strong relationship between number of non-zero simulated PIRs in the training dataset and the ability of our system to improve PIR (see Fig. 8). This is somewhat troubling, since probably most images in the real world will not produce a PIR that is truly zero. ",
+        "bbox": [
+            174,
+            882,
+            825,
+            924
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/1cc28adcf4d57ca6b2d09c22ef5c94c72c0d24e85a98baf26a61262dbd9e4b51.jpg",
+        "image_caption": [
+            "Figure 8: Percent non-zero PIRs vs. effect size (Single-filter VGG tasks and color tasks) "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            287,
+            104,
+            700,
+            359
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            417,
+            825,
+            558
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "in order to evaluate whether the poor performance of our system at the higher layers of VGG was due to the number of zeros or to the complexity of the features, we created less sparse features from these layers by simply targeting a set of $k$ filters sampled without replacement from the layer, rather than a single filter. We did this by taking the norm across the $k$ target filters, or equivalently by summing the squared norms of the $k$ filters before taking the square root, and then normalizing by the activity in the layer as before. Formally, letting $a _ { 1 } , . . . , a _ { k }$ be a set of $k$ filter indices sampled without replacement from $\\{ 0 , . . . ,$ number of filters in layer $\\}$ , we computed the PIR for an image as: ",
+        "bbox": [
+            174,
+            564,
+            825,
+            662
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0d856d11f2fd4ca5070e7b117c391bd33d6cd193ae6a897db80141e6bad859d5.jpg",
+        "text": "$$\n\\mathrm { P I R } _ { l ^ { * } , f ^ { * } , k } ( i ) = \\sqrt { \\frac { \\sum _ { j = 1 } ^ { j = k } \\left| \\mathrm { V G G } _ { l ^ { * } , a _ { j } } ( i ) \\right| _ { 2 } ^ { 2 } } { \\sum _ { f \\in l ^ { * } } \\left| \\mathrm { V G G } _ { l ^ { * } , f } ( i ) \\right| _ { 2 } ^ { 2 } } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            359,
+            671,
+            637,
+            723
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "The single filter cases above can be thought of as a special case of this, where $k = 1$ . To complement these, we also tried $k = 2 0$ . As above, we also added binomially distributed noise to these PIRs. ",
+        "bbox": [
+            171,
+            731,
+            825,
+            760
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "This can also be thought of as perhaps a more realistic simulation of human behavior, in the sense that it is highly unlikely that there is a single feature which influences human PIRs. Rather, there are probably many related features which influence PIR in various ways. Thus it is important to evaluate our system’s ability to target these types of features as well. ",
+        "bbox": [
+            174,
+            766,
+            825,
+            823
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "C.4.3 COLORS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            842,
+            289,
+            856
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Finally, we also considered some simpler objectives based on targeting specific colors in the output images. Analogously to the VGG features, we computed the PIRs from the vector norm of a given image in the targeted color, normalized by the total image value. We considered several objectives of this type: ",
+        "bbox": [
+            174,
+            867,
+            823,
+            924
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Single color: Optimizing for a single color of output image, e.g., for red the objective would be. ",
+        "bbox": [
+            169,
+            102,
+            807,
+            119
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "equation",
+        "img_path": "images/49a2729840f850b1a0f27b66a0f0bf20944f3d6ea451216655f6921b402a27a7.jpg",
+        "text": "$$\n\\mathrm { P I R } _ { \\mathrm { r e d } } ( i ) = \\sqrt { \\frac { | i ( : , : , \\mathrm { r e d } ) | _ { 2 } ^ { 2 } } { | \\mathbf { i } ( : , : , : ) | _ { 2 } ^ { 2 } } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            434,
+            125,
+            620,
+            167
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Two color: We split the image horizontally into a left and right half, and then computed PIR from one color in the left half and a different color in the right half. ",
+        "bbox": [
+            173,
+            175,
+            825,
+            204
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "equation",
+        "img_path": "images/440e59433ffa75d02fe3d75cd6e9f9e9263257489294a25621957b7c08e324d4.jpg",
+        "text": "$$\n\\mathsf { P I R } _ { \\mathrm { r e d \\ b l u e } } ( i ) = \\sqrt { \\frac { \\left| i ( : , : \\frac { \\mathrm { w i d t h } } { 2 } , \\mathrm { r e d } ) \\right| _ { 2 } ^ { 2 } + \\left| i ( : , \\frac { \\mathrm { w i d t h } } { 2 } : , \\mathsf { b l u e } ) \\right| _ { 2 } ^ { 2 } } { | \\dot { \\mathbf { \\mathbf { \\mathbf { \\mathbf { \\Phi } } } } } ( : , : , : ) | _ { 2 } ^ { 2 } } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            336,
+            210,
+            720,
+            261
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Three color: Similar to two color, but split the image into thirds, and computed PIR from a different color in each third. ",
+        "bbox": [
+            173,
+            268,
+            823,
+            297
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "(As above, we also added binomially distributed noise to these PIRs.) These objectives provide a useful complement to the VGG objectives discussed in section 3.1. Although the single color objectives may be relevant to the classification task VGG 16 performs, the split color tasks are less likely to be relevant to classification. Note that it is important that we split the images along the width instead of the height dimension, as there may well be semantically relevant features corresponding to color divisions along the height dimension, e.g. a blue upper half and green lower half likely correlates with outdoor images, which would provide useful class information. By contrast, it is harder to imagine circumstances where different colors on the left and right halves of the image are semantically predictive, especially since flipping left to right is usually included in the data augmentation for computer vision systems. Thus success on optimizing for these objectives would increase our confidence in the generality of our system. ",
+        "bbox": [
+            173,
+            309,
+            825,
+            463
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "C.5 POSSIBLE IMPROVEMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            479,
+            400,
+            493
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "There are a number of techniques that could be explored to improve our system. As we mentioned above, iterating for multiple steps of PIR collection and generative model tuning is worth exploring further. Also, some form of data scaling might allow the system to perform better on tasks with low variance. We briefly tried normalizing all data for an objective to have mean 0.5 and standard deviation 0.25, but did not achieve particularly good results from this, possibly because there were many outliers getting clipped to 0 or 1. Still, there are many other possibilities for scaling data that could potentially result in some improvement in performance. Also, one alternative approach to training a GAN to produce high-PIR images would be to use the PIR estimator objective in the Plug & Play Generative Networks framework (Nguyen et al., 2017) instead of using it to tune the GAN. This could be an interesting direction to explore, but its success would probably depend on expressiveness of the initial generative model. With the mediocre model we started with, it’s probably better to actually tune the model itself, which may allow it to explore parts of image space which it had not previously. ",
+        "bbox": [
+            173,
+            505,
+            825,
+            686
+        ],
+        "page_idx": 15
+    }
+]

parse/train/HkG3e205K7/HkG3e205K7.md

@@ -0,0 +1,405 @@
+# DOUBLY REPARAMETERIZED GRADIENT ESTIMATORS FOR MONTE CARLO OBJECTIVES
+
+George Tucker Google Brain gjt@google.com
+
+Dieterich Lawson New York University jdl404@nyu.edu
+
+Shixiang Gu   
+Google Brain   
+shanegu@google.com
+
+Chris J. Maddison University of Oxford, DeepMind cmaddis@stats.ox.ac.uk
+
+# ABSTRACT
+
+Deep latent variable models have become a popular model choice due to the scalable learning algorithms introduced by (Kingma & Welling, 2013; Rezende et al., 2014). These approaches maximize a variational lower bound on the intractable log likelihood of the observed data. Burda et al. (2015) introduced a multi-sample variational bound, IWAE, that is at least as tight as the standard variational lower bound and becomes increasingly tight as the number of samples increases. Counterintuitively, the typical inference network gradient estimator for the IWAE bound performs poorly as the number of samples increases (Rainforth et al., 2018; Le et al., 2018). Roeder et al. (2017) propose an improved gradient estimator, however, are unable to show it is unbiased. We show that it is in fact biased and that the bias can be estimated efficiently with a second application of the reparameterization trick. The doubly reparameterized gradient (DReG) estimator does not suffer as the number of samples increases, resolving the previously raised issues. The same idea can be used to improve many recently introduced training techniques for latent variable models. In particular, we show that this estimator reduces the variance of the IWAE gradient, the reweighted wake-sleep update (RWS) (Bornschein & Bengio, 2014), and the jackknife variational inference (JVI) gradient (Nowozin, 2018). Finally, we show that this computationally efficient, unbiased drop-in gradient estimator translates to improved performance for all three objectives on several modeling tasks.
+
+# 1 INTRODUCTION
+
+Following the influential work by (Kingma & Welling, 2013; Rezende et al., 2014), deep generative models with latent variables have been widely used to model data such as natural images (Rezende & Mohamed, 2015; Kingma et al., 2016; Chen et al., 2016; Gulrajani et al., 2016), speech and music time-series (Chung et al., 2015; Fraccaro et al., 2016; Krishnan et al., 2015), and video (Babaeizadeh et al., 2017; Ha & Schmidhuber, 2018; Denton & Fergus, 2018). The power of these models lies in combining learned nonlinear function approximators with a principled probabilistic approach, resulting in expressive models that can capture complex distributions. Unfortunately, the nonlinearities that empower these model also make marginalizing the latent variables intractable, rendering direct maximum likelihood training inapplicable. Instead of directly maximizing the marginal likelihood, a common approach is to maximize a tractable lower bound on the likelihood such as the variational evidence lower bound (ELBO) (Jordan et al., 1999; Blei et al., 2017). The tightness of the bound is determined by the expressiveness of the variational family. For tractability, a factorized variational family is commonly used, which can cause the learned model to be overly simplistic.
+
+Burda et al. (2015) introduced a multi-sample bound, IWAE, that is at least as tight as the ELBO and becomes increasingly tight as the number of samples increases. Counterintuitively, although the bound is tighter, Rainforth et al. (2018) theoretically and empirically showed that the standard inference network gradient estimator for the IWAE bound performs poorly as the number of samples increases due to a diminishing signal-to-noise ratio (SNR). This motivates the search for novel gradient estimators.
+
+Roeder et al. (2017) proposed a lower-variance estimator of the gradient of the IWAE bound. They speculated that their estimator was unbiased, however, were unable to prove the claim. We show that it is in fact biased, but that it is possible to construct an unbiased estimator with a second application of the reparameterization trick which we call the IWAE doubly reparameterized gradient (DReG) estimator. Our estimator is an unbiased, computationally efficient drop-in replacement, and does not suffer as the number of samples increases, resolving the counterintuitive behavior from previous work (Rainforth et al., 2018). Furthermore, our insight is applicable to alternative multisample training techniques for latent variable models: reweighted wake-sleep (RWS) (Bornschein & Bengio, 2014) and jackknife variational inference (JVI) (Nowozin, 2018).
+
+In this work, we derive DReG estimators for IWAE, RWS, and JVI and demonstrate improved scaling with the number of samples on a simple example. Then, we evaluate DReG estimators on MNIST generative modeling, Omniglot generative modeling, and MNIST structured prediction tasks. In all cases, we demonstrate substantial unbiased variance reduction, which translates to improved performance over the original estimators.
+
+# 2 BACKGROUND
+
+Our goal is to learn a latent variable generative model $p _ { \theta } ( x , z ) = p _ { \theta } ( z ) p _ { \theta } ( x | z )$ where $x$ are observed data and $z$ are continuous latent variables. The marginal likelihood of the observed data, $p _ { \theta } ( x ) =$ $\textstyle \int p _ { \theta } ( x , z ) d z$ , is generally intractable. Instead, we maximize a variational lower bound on $\log p _ { \theta } ( x )$ such as the ELBO
+
+$$
+\log p _ { \theta } ( x ) = \log \mathbb { E } _ { p _ { \theta } ( z ) } [ p _ { \theta } ( x | z ) ] \geq \mathbb { E } _ { q ( z | x ) } \left[ \log \frac { p _ { \theta } ( x , z ) } { q ( z | x ) } \right] ,
+$$
+
+where $q ( z | x )$ is a variational distribution. Following the influential work by (Kingma & Welling, 2013; Rezende et al., 2014), we consider the amortized inference setting where $q _ { \phi } ( z | x )$ , referred to as the inference network, is a learnable function parameterized by $\phi$ that maps from $x$ to a distribution over $z$ . The tightness of the bound is coupled to the expressiveness of the variational family (i.e., $\{ q _ { \phi } \} _ { \phi } )$ . As a result, limited expressivity of $\{ q _ { \phi } \} _ { \phi }$ , can negatively affect the learned model.
+
+Burda et al. (2015) introduced the importance weighted autoencoder (IWAE) bound which alleviates this coupling
+
+$$
+\mathbb { E } _ { z _ { 1 : K } } \left[ \log \left( \frac { 1 } { K } \sum _ { i = 1 } ^ { K } \frac { p _ { \theta } ( x , z _ { i } ) } { q _ { \phi } ( z _ { i } | x ) } \right) \right] \le \log p _ { \theta } ( x ) ,
+$$
+
+with $z _ { 1 : K } \sim \textstyle \prod _ { i } q _ { \phi } ( z _ { i } | x )$ . The IWAE bound reduces to the ELBO when $K = 1$ , is non-decreasing as $K$ increases, and converges to $\log p _ { \theta } ( x )$ as $K  \infty$ under mild conditions (Burda et al., 2015). When $q _ { \phi }$ is reparameterizable1, the standard gradient estimator of the IWAE bound is
+
+$$
+\bigtriangledown _ { \theta , \phi } \mathbb { E } _ { z _ { 1 : K } } \left[ \log \left( \frac { 1 } { K } \sum _ { i = 1 } ^ { K } w _ { i } \right) \right] = \nabla _ { \theta , \phi } \mathbb { E } _ { \epsilon _ { 1 : K } } \left[ \log \left( \frac { 1 } { K } \sum _ { i = 1 } ^ { K } w _ { i } \right) \right] = \mathbb { E } _ { \epsilon _ { 1 : K } } \left[ \sum _ { i = 1 } ^ { K } \frac { w _ { i } } { \sum _ { j } w _ { j } } \nabla _ { \theta , \phi } \log w _ { i } \right]
+$$
+
+where $w _ { i } = p _ { \theta } ( x , z _ { i } ) / q _ { \phi } ( z _ { i } | x )$ . A single sample estimator of this expectation is typically used as the gradient estimator.
+
+As $K$ increases, the bound becomes increasingly tight, however, Rainforth et al. (2018) show that the signal-to-noise ratio (SNR) of the inference network gradient estimator goes to 0. This does not happen for the model parameters $\mathbf { \eta } ^ { ( \theta ) }$ . Following up on this work, Le et al. (2018) demonstrate that this deteriorates the performance of learned models on practical problems.
+
+Because the IWAE bound converges to $\log p _ { \theta } ( x )$ (as $K  \infty$ ) regardless of $q _ { \phi }$ , $\phi$ ’s affect on the bound must diminish as $K$ increases. It may be tempting to conclude that the SNR of the inference network gradient estimator must also decrease as $K  \infty$ . However, low SNR is a limitation of the gradient estimator, not necessarily of the bound. Although the magnitude of the gradient converges to 0, if the variance of the gradient estimator decreases more quickly, then the SNR of the gradient estimator need not degrade. This motivates the search for lower variance inference network gradient estimators.
+
+To derive improved gradient estimators for $\phi$ , it is informative to expand the total derivative2 of the IWAE bound with respect to $\phi$
+
+$$
+\mathbb { E } _ { \epsilon _ { 1 : K } } \left[ \sum _ { i = 1 } ^ { K } \frac { w _ { i } } { \sum _ { j = 1 } ^ { K } w _ { j } } \left( - \frac { \partial } { \partial \phi } \log q _ { \phi } ( z _ { i } | x ) + \frac { \partial \log w _ { i } } { \partial z _ { i } } \frac { d z _ { i } } { d \phi } \right) \right] .
+$$
+
+Previously, Roeder et al. (2017) found that the first term within the parentheses of Eq. 3 can contribute significant variance to the gradient estimator. When $K = 1$ , this term analytically vanishes in expectation, so when $K > 1$ they suggested dropping it. Below, we abbreviate this estimator as STL. As we show in Section 6.1, the STL estimator introduces bias when $K > 1$ .
+
+# 3 DOUBLY REPARAMETERIZED GRADIENT ESTIMATORS (DREGS)
+
+Our insight is that we can estimate the first term within the parentheses of Eq. 3 efficiently with a second application of the reparameterization trick. To see this, first note that
+
+$$
+\mathbb { E } _ { \epsilon _ { 1 : K } } \left[ \sum _ { i = 1 } ^ { K } \frac { w _ { i } } { \sum _ { j = 1 } ^ { K } w _ { j } } \frac { \partial } { \partial \phi } \log q ( \boldsymbol { z } _ { i } | \boldsymbol { x } ) \right] = \sum _ { i = 1 } ^ { K } \mathbb { E } _ { \epsilon _ { 1 : K } } \left[ \frac { w _ { i } } { \sum _ { j = 1 } ^ { K } w _ { j } } \frac { \partial } { \partial \phi } \log q ( \boldsymbol { z } _ { i } | \boldsymbol { x } ) \right] ,
+$$
+
+so it suffices to focus on one of the $K$ terms. Because the derivative is a partial derivativ e ∂∂ φ , it treats $z _ { i } = z ( \epsilon _ { i } , \phi )$ as a constant, so we can freely change the random variable that the expectation is over to $z _ { 1 : K }$ . Now,
+
+$$
+\mathbb { E } _ { z _ { 1 : K } } \left[ \frac { w _ { i } } { \sum _ { j } w _ { j } } \frac { \partial } { \partial \phi } \log q _ { \phi } ( z _ { i } | x ) \right] = \mathbb { E } _ { z _ { - i } } \mathbb { E } _ { z _ { i } } \left[ \frac { w _ { i } } { \sum _ { j } w _ { j } } \frac { \partial } { \partial \phi } \log q _ { \phi } ( z _ { i } | x ) \right] ,
+$$
+
+where $z _ { - i } = z _ { 1 : i - 1 , i + 1 : K }$ is the set of $z _ { 1 : K }$ without $z _ { i }$ . The inner expectation resembles a REINFORCE gradient term (Williams, 1992), where we interpret $\frac { w _ { i } } { \sum _ { j } w _ { j } }$ as the “reward”. Now, we can use the following well-known equivalence between the REINFORCE gradient and the reparameterization trick gradient (See Appendix 8.1 for a derivation)
+
+$$
+\mathbb { E } _ { q _ { \phi } ( z | x ) } \left[ f ( z ) \frac { \partial } { \partial \phi } \log q _ { \phi } ( z | x ) \right] = \mathbb { E } _ { \epsilon } \left[ \frac { \partial f ( z ) } { \partial z } \frac { \partial z ( \epsilon , \phi ) } { \partial \phi } \right] .
+$$
+
+This holds even when $f$ depends on $\phi$ . Typically, the reparameterization gradient estimator has lower variance than the REINFORCE gradient estimator because it directly takes advantage of the derivative of $f$ . Applying the identity from Eq. 5 to the right hand side of Eq. 4 gives
+
+$$
+\begin{array} { r l r } & { } & { \mathbb { E } _ { z _ { i } } \left[ \displaystyle \frac { w _ { i } } { \sum _ { j } w _ { j } } \frac { \partial } { \partial \phi } \log q _ { \phi } ( z _ { i } | x ) \right] = \mathbb { E } _ { \epsilon _ { i } } \left[ \displaystyle \frac { \partial } { \partial z _ { i } } \left( \displaystyle \frac { w _ { i } } { \sum _ { j } w _ { j } } \right) \displaystyle \frac { \partial z _ { i } } { \partial \phi } \right] } \\ & { } & { = \mathbb { E } _ { \epsilon _ { i } } \left[ \left( \displaystyle \frac { 1 } { \sum _ { j } w _ { j } } - \displaystyle \frac { w _ { i } } { ( \sum _ { j } w _ { j } ) ^ { 2 } } \right) \displaystyle \frac { \partial w _ { i } } { \partial z _ { i } } \displaystyle \frac { \partial z _ { i } } { \partial \phi } \right] = \mathbb { E } _ { \epsilon _ { i } } \left[ \left( \displaystyle \frac { w _ { i } } { \sum _ { j } w _ { j } } - \displaystyle \frac { w _ { i } ^ { 2 } } { ( \sum _ { j } w _ { j } ) ^ { 2 } } \right) \displaystyle \frac { \partial \log w _ { i } } { \partial z _ { i } } \displaystyle \frac { \partial z _ { i } } { \partial \phi } \right] . } \end{array}
+$$
+
+This last expression can be efficiently estimated with a single Monte Carlo sample. When $z _ { i }$ is not reparameterizable (e.g., the models in (Mnih & Rezende, 2016)), we can use a control variate (e.g.,
+
+$\begin{array} { r } { \frac { 1 } { K } \frac { \partial } { \partial \phi } \log q _ { \phi } ( z _ { i } | \boldsymbol { x } ) ) } \end{array}$ . In both cases, when $K = 1$ , this term vanishes exactly and we recover the estimator proposed in (Roeder et al., 2017) for the ELBO. However, when $K > 1$ , there is no reason to believe this term will analytically vanish.
+
+Substituting Eq. 6 into Eq. 3, we obtain a simplification due to cancellation of terms
+
+$$
+\nabla _ { \phi } \mathbb { E } _ { z _ { 1 : K } } \left[ \log \left( \frac { 1 } { K } \sum _ { i = 1 } ^ { K } w _ { i } \right) \right] = \mathbb { E } _ { \epsilon _ { 1 : K } } \left[ \sum _ { i = 1 } ^ { K } \left( \frac { w _ { i } } { \sum _ { j } w _ { j } } \right) ^ { 2 } \frac { \partial \log w _ { i } } { \partial z _ { i } } \frac { \partial z _ { i } } { \partial \phi } \right] .
+$$
+
+We call the algorithm that uses the single sample Monte Carlo estimator of this expression for the inference network gradient the IWAE doubly reparameterized gradient estimator (IWAE-DReG). This estimator has the property that when $q ( z | x )$ is optimal (i.e., $q ( z | x ) = p ( z | x ) )$ , the estimator vanishes exactly and has zero variance, whereas this does not hold for the standard IWAE gradient estimator. We provide an asymptotic analysis of the IWAE-DReG estimator in Appendix 8.2. The conclusion of that analysis is that, in contrast to the standard IWAE gradient estimator, the SNR of√ the IWAE-DReG estimator exhibits the same scaling behaviour of $\mathcal { O } ( \sqrt { K } )$ for both the generation and inference network gradients (i.e., improving in $K$ ).
+
+# 4 ALTERNATIVE TRAINING ALGORITHMS
+
+Now, we review alternative training algorithms for deep generative models and derive their doubly reparameterized versions.
+
+# 4.1 REWEIGHTED WAKE SLEEP (RWS)
+
+Bornschein & Bengio (2014) introduced RWS, an alternative multi-sample update for latent variable models that uses importance sampling. Computing the gradient of the log marginal likelihood
+
+$$
+\nabla _ { \theta } \log p _ { \theta } ( x ) = \frac { \nabla _ { \theta } \int _ { z } p _ { \theta } ( x , z ) d z } { p _ { \theta } ( x ) } = \frac { \int _ { z } p _ { \theta } ( x , z ) \nabla _ { \theta } \log p _ { \theta } ( x , z ) d z } { p _ { \theta } ( x ) } = \mathbb { E } _ { p _ { \theta } ( z | x ) } \left[ \nabla _ { \theta } \log p _ { \theta } ( x , z ) \right] ,
+$$
+
+requires samples from $p _ { \theta } ( z | x )$ , which is generally intractable. We can approximate the gradient with a self-normalized importance sampling estimator
+
+$$
+\mathbb { E } _ { p _ { \theta } ( z | x ) } \left[ \nabla _ { \theta } \log p _ { \theta } ( x , z ) \right] \approx \mathbb { E } _ { z _ { 1 : K } } \left[ \sum _ { i = 1 } ^ { K } \frac { w _ { i } } { \sum _ { j } w _ { j } } \nabla _ { \theta } \log p _ { \theta } ( x , z _ { i } ) \right] ,
+$$
+
+where $\begin{array} { r } { z _ { 1 : K } \sim \prod _ { i } q _ { \phi } ( z _ { i } | x ) } \end{array}$ . Interestingly, this is precisely the same as the IWAE gradient of $\theta$ , so the RWS update for $\theta$ can be interpreted as maximizing the IWAE lower bound in terms of $\theta$ . Instead of optimizing a joint objective for $p$ and $q$ , RWS optimizes a separate objective for the inference network. (Bornschein & Bengio, 2014) propose a “wake” update and a “sleep” update for the inference network. Le et al. (2018) provide empirical support for solely using the wake update for the inference network, so we focus on that update.
+
+The wake update approximately minimizes the KL divergence from $p _ { \theta } ( z | x )$ to $q _ { \phi } ( z | x )$ . The gradient of the KL term is
+
+$$
+\nabla _ { \phi } \mathbb { E } _ { p _ { \theta } ( z | x ) } \left[ \log p _ { \theta } ( z | x ) - \log q _ { \phi } ( z | x ) \right] = - \mathbb { E } _ { p _ { \theta } ( z | x ) } \left[ \frac { \partial } { \partial \phi } \log q _ { \phi } ( z | x ) \right] .
+$$
+
+The wake update of the inference network approximates the intractable expectation by selfnormalized importance sampling
+
+$$
+- \mathbb { E } _ { p _ { \theta } ( z | x ) } \left[ \frac { \partial } { \partial \phi } \log q _ { \phi } ( z | x ) \right] \approx - \mathbb { E } _ { z _ { 1 : K } } \left[ \sum _ { i = 1 } ^ { K } \frac { w _ { i } } { \sum _ { j } w _ { j } } \frac { \partial } { \partial \phi } \log q _ { \phi } ( z _ { i } | x ) \right] ,
+$$
+
+with $z _ { i } \sim q _ { \phi } ( z _ { i } | x )$ . Le et al. (2018) note that this update does not suffer from diminishing SNR as $K$ increases. However, a downside is that the updates for $p$ and $q$ are not gradients of a unified objective, so could potentially lead to instability or divergence.
+
+# DOUBLY REPARAMETERIZED REWEIGHTED WAKE UPDATE
+
+The wake update gradient for the inference network (Eq. 8) can be reparameterized
+
+$$
+- \mathbb { E } _ { z _ { 1 : K } } \left[ \sum _ { i = 1 } ^ { K } \frac { w _ { i } } { \sum _ { j } w _ { j } } \frac { \partial } { \partial \phi } \log q _ { \phi } ( z _ { i } | x ) \right] = \mathbb { E } _ { \epsilon _ { 1 : K } } \left[ \sum _ { i = 1 } ^ { K } \left( \frac { w _ { i } ^ { 2 } } { ( \sum _ { j } w _ { j } ) ^ { 2 } } - \frac { w _ { i } } { \sum _ { j } w _ { j } } \right) \frac { \partial \log w _ { i } } { \partial z _ { i } } \frac { \partial z _ { i } } { \partial \phi } \right] .
+$$
+
+We call the algorithm that uses the single sample Monte Carlo estimator of this expression as the wake update for the inference network RWS-DReG.
+
+Interestingly, the inference network gradient estimator from (Roeder et al., 2017) can be seen as the sum of the IWAE gradient estimator and the wake update of the inference network (as the wake update minimizes, we add the negative of Eq. 9). Their positive results motivate further exploration of convex combinations of IWAE-DReG and RWS-DReG
+
+$$
+\mathbb { E } _ { \epsilon _ { 1 : K } } \left[ \sum _ { i = 1 } ^ { K } \left( \alpha \frac { w _ { i } } { \sum _ { j } w _ { j } } + ( 1 - 2 \alpha ) \frac { w _ { i } ^ { 2 } } { ( \sum _ { j } w _ { j } ) ^ { 2 } } \right) \frac { \partial \log w _ { i } } { \partial z _ { i } } \frac { \partial z _ { i } } { \partial \phi } \right] .
+$$
+
+We refer to the algorithm that uses the single sample Monte Carlo estimator of this expression as $\mathrm { D R e G } ( \alpha )$ . When $\alpha = 1$ , this reduces to RWS-DReG, when $\alpha = 0$ , this reduces to IWAE-DReG and when $\alpha = 0 . 5$ , this reduces STL.
+
+# 4.2 JACKKNIFE VARIATIONAL INFERENCE (JVI)
+
+Alternatively, Nowozin (2018) reinterprets the IWAE lower bound as a biased estimator for the log marginal likelihood. He analyzes the bias and introduces a novel family of estimators, Jackknife Variational Inference (JVI), which trade off reduction in bias for increased variance. This additional flexibility comes at the cost of no longer being a stochastic lower bound on the log marginal likelihood. The first-order JVI has significantly reduced bias compared to IWAE, which empirically results in a better estimate of the log marginal likelihood with fewer samples (Nowozin, 2018). For simplicity, we focus on the first-order JVI estimator
+
+$$
+K \times \mathbb { E } _ { z _ { 1 } \cdot K } \left[ \log \left( \frac { 1 } { K } \sum _ { i = 1 } ^ { K } w _ { i } \right) \right] - \frac { K - 1 } { K } \sum _ { i = 1 } ^ { K } \mathbb { E } _ { z _ { - i } } \left[ \log \left( \frac { 1 } { K - 1 } \sum _ { j \neq i } w _ { j } \right) \right] .
+$$
+
+It is straightforward to apply our approach to higher order JVI estimators.
+
+DOUBLY REPARAMETERIZED JACKKNIFE VARIATIONAL INFERENCE (JVI)
+
+The JVI estimator is a linear combination of $K$ and $K - 1$ sample IWAE estimators, so we can use the doubly reparameterized gradient estimator (Eq. 7) for each term.
+
+# 5 RELATED WORK
+
+Mnih & Rezende (2016) introduced a generalized framework of Monte Carlo objectives (MCO). The log of an unbiased marginal likelihood estimator is a lower bound on the log marginal likelihood by Jensen’s inequality. In this view, the ELBO can be seen as the MCO corresponding to a single importance sample estimator of the marginal likelihood with $q _ { \theta }$ as the proposal distribution. Similarly, IWAE corresponds to the $K$ -sample estimator. Maddison et al. (2017) show that the tightness of an MCO is directly related to the variance of the underlying estimator of the marginal likelihood.
+
+However, Rainforth et al. (2018) point out issues with gradient estimators of multi-sample lower bounds. In particular, they show that although the IWAE bound is tighter, the standard IWAE gradient estimator’s SNR scales poorly with large numbers of samples, leading to degraded performance. Le et al. (2018) experimentally investigate this phenomenon and provide empirical evidence of this degradation across multiple tasks. They find that RWS (Bornschein & Bengio, 2014) does not suffer from this issue and find that it can outperform models trained with the IWAE bound. We conclude that it is not sufficient to just tighten the bound; it is important to understand the gradient estimators of the tighter bound as well.
+
+Wake-sleep is an alternative approach to fitting deep generative models, first introduced in (Hinton et al., 1995) as a method for training Hemholtz machines. It was extended to the multi-sample setting by (Bornschein & Bengio, 2014) and the sequential setting in (Gu et al., 2015). It has been applied to generative modeling of images (Ba et al., 2015).
+
+# 6 EXPERIMENTS
+
+To evaluate DReG estimators, we first measure variance and signal-to-noise ratio $( \mathrm { S N R } ) ^ { 3 }$ of gradient estimators on a toy example which we can carefully control. Then, we evaluate gradient variance and model learning on MNIST generative modeling, Omniglot generative modeling, and MNIST structured prediction tasks.
+
+# 6.1 TOY GAUSSIAN
+
+We reimplemented the Gaussian example from (Rainforth et al., 2018). Consider the generative model with $z \sim N ( \theta , I )$ and $x | z \sim N ( z , I )$ and inference network $\begin{array} { r } { q _ { \phi } ( z | x ) \sim N ( A x ^ { - } + b , \frac { 2 } { 3 } I ) } \end{array}$ , where $\phi = \{ A , b \}$ . As in (Rainforth et al., 2018), we sample a set of parameters for the model and inference network close to the optimal parameters (perturbed by zero-mean Gaussian noise with standard deviation 0.01), then estimate the gradient of the inference network parameters for increasing number of samples $( K )$ .
+
+In addition to signal-to-noise ratio (SNR), we plot the squared bias and variance of the gradient estimators4 in Fig. 1. The bias is computed relative to the expected value of the IWAE gradient estimator. As a result, although the average of $K$ ELBO gradient estimators is an unbiased estimator of the ELBO gradient, it is a biased gradient estimator of the IWAE objective. Importantly, SNR does not penalize estimators that are biased, so trivial constant estimators can have infinite SNR. Thus, it is important to consider additional evaluation measures as well. As $K$ increases, the SNR of the IWAE-DReG estimator increases, whereas the SNR of the standard gradient estimator of IWAE goes to 0, as previously reported. Furthermore, we can see the bias present in the STL estimator. As a check of our implementation, we verified that the observed “bias” for IWAE-DReG was statistically indistinguishable from 0 with a paired t-test. For the biased estimators (e.g., STL), we could easily reject the null hypothesis with few samples.
+
+![](images/4602530728769106fa89abfa2de18954281cbb1ea79b32fbc94d040f866f151a.jpg)  
+Figure 1: Signal-to-noise ratios (SNR), bias squared, and variance of gradient estimators with increasing $K$ over 10 random trials with 1000 measurement samples per trial (mean in bold). The observed “bias” for IWAE-DReG is not statistically significant under a paired t-test (as expected because IWAE-DReG is unbiased). IWAE-DReG is unbiased, its SNR increases with K, and it has the lowest variance of the estimators considered here.
+
+# 6.2 GENERATIVE MODELING
+
+Training generative models of the binarized MNIST digits dataset is a standard benchmark task for latent variable models. For this evaluation, we used the single latent layer architecture from (Burda et al., 2015). The generative model used 50 Gaussian latent variables with an isotropic prior and passed $z$ through two deterministic layers of 200 tanh units to parameterize factorized Bernoulli outputs. The inference network passed $x$ through two deterministic layers of 200 tanh units to parameterize a factorized Gaussian distribution over $z$ . Because our interest was in improved gradient estimators and optimization performance, we used the dynamically binarized MNIST dataset, which minimally suffers from overfitting. We used the standard split of MNIST into train, validation, and test sets.
+
+We trained models with the IWAE gradient, the RWS wake update, and with the JVI estimator. In all three cases, the doubly reparameterized gradient estimator reduced variance5 and as a result substantially improved performance (Fig. 2).
+
+![](images/df131fe5998b53bb0ac88c7a2eb163c6e64208318f090fefa87a69dbc76fe0f7.jpg)  
+Figure 2: MNIST generative modeling trained according to IWAE (left), RWS (middle), and JVI (right). The top row compares the variance of the original gradient estimator (dashed) with the variance of the doubly reparameterized gradient estimator (solid). The bottom row compares test performance. The left and middle plots show the IWAE (stochastic) lower bound on the test set. The right plot shows the JVI estimator (which is not a bound) on the test set. The bold lines are the average over three trials, and individual trials are displayed as semi-transparent). All methods used $K = 6 4$ .
+
+We found similar behavior with different numbers of samples (Fig. 3 and Appendix Fig. 8). Interestingly, the biased gradient estimators STL and RWS-DReG perform best on this task with RWSDReG slightly outperforming STL. As observed in (Le et al., 2018), RWS increasingly outperforms IWAE as $K$ increases. Finally, we experimented with convex combinations of IWAE-DReG and RWS-DReG (right Fig. 3). On this dataset, convex combinations that heavily weighted RWS-DReG had the best performance. However, as we show below, this is task dependent.
+
+Next, we performed the analogous experiment with the dynamically binarized Omniglot dataset using the same model architecture. Again, we found that the doubly reparameterized gradient estimator reduced variance and as a result improved test performance (Figs. 5 and 6 in the Appendix).
+
+# 6.3 STRUCTURED PREDICTION ON MNIST
+
+Structured prediction is another common benchmark task for latent variable models. In this task, our goal is to model a complex observation $x$ given a context $c$ (i.e., model the conditional distribution $p ( x | c ) )$ . We can use a conditional latent variable model $p _ { \theta } ( x , z | c ) = p _ { \theta } ( x | z , c ) p _ { \theta } ( z | c )$ , however, as before, computing the marginal likelihood is generally intractable. It is straightforward to adapt the bounds and techniques from the previous section to this problem.
+
+![](images/91eefa28a63730121c99b026911c4b3915048b1370526f3bf876517154d33a0f.jpg)  
+Figure 3: Log-likelihood lower bounds for generative modeling on MNIST. The left and middle plots compare performance with different number of samples $K = 3 2$ , 256. For clarity the legend is shared between the plots. The bold lines are the average over three trials, and individual trials are displayed as semi-transparent). The right plot compares performance as the convex combination between IWAE-DReG and RWS-DReG is varied (Eq. 10). To highlight differences, we plot the difference between the test IWAE bound and the test IWAE bound IWAE-DReG achieved at that step.
+
+To evaluate our method in this context, we use the standard task of modeling the bottom half of a binarized MNIST digit from the top half. We use a similar architecture, but now learn a conditional prior distribution $p _ { \theta } ( z | c )$ where $c$ is the top half of the MNIST digit. The conditional prior feeds $c$ to two deterministic layers of 200 tanh units to parameterize a factorized Gaussian distribution over $z$ . To model the conditional distribution $p _ { \theta } ( x | c , z )$ , we concatenate $z$ with $c$ and feed it to two deterministic layers of 200 tanh units to parameterize factorized Bernoulli outputs.
+
+As in the previous tasks, the doubly reparameterized gradient estimator improves across all three updates (IWAE, RWS, and JVI; Appendix Fig. 7). However, on this task, the biased estimators (STL and RWS) underperform unbiased IWAE gradient estimators (Fig. 4). In particular, RWS becomes unstable later in training. We suspect that this is because RWS does not directly optimize a consistent objective.
+
+![](images/57312dcd20d125385cf7e6526c7c22ebd0eaef48131ffdc8b992f9c2c7c72fc0.jpg)  
+Figure 4: Log-likelihood lower bounds for structured prediction on MNIST. The left plot uses $K =$ 64 samples and the right plot uses $K \ : = \ : 2 5 6$ samples. For clarity the legend is shared between the plots. The bold lines are the average over three trials, and individual trials are displayed as semi-transparent). The right plot compares performance as the convex combination between IWAEDReG and RWS-DReG is varied (Eq. 10). To highlight differences, we plot the difference between the test IWAE bound and the test IWAE bound IWAE-DReG achieved at that step.
+
+# 7 DISCUSSION
+
+In this work, we introduce doubly reparameterized estimators for the updates in IWAE, RWS, and JVI. We demonstrate that across tasks they provide unbiased variance reduction, which leads to improved performance. Furthermore, DReG estimators have the same computational cost as the original estimators. As a result, we recommend that DReG estimators be used instead of the typical gradient estimators.
+
+Variational Sequential Monte Carlo (Maddison et al., 2017; Naesseth et al., 2018; Le et al., 2018) and Neural Adapative Sequential Monte Carlo (Gu et al., 2015) extend IWAE and RWS to sequential latent variable models, respectively. It would be interesting to develop DReG estimators for these approaches as well.
+
+We found that a convex combination of IWAE-DReG and RWS-DReG performed best, however, the weighting was task dependent. In future work, we intend to apply ideas from (Baydin et al., 2017) to automatically adapt the weighting based on the data.
+
+Finally, the form of the IWAE-DReG estimator (Eq. 7) is surprisingly simple and suggests that there may be a more direct derivation that is applicable to general MCOs.
+
+# ACKNOWLEDGMENTS
+
+We thank Ben Poole and Diederik P. Kingma for helpful discussion and comments on drafts of this paper. We thank Sergey Levine and Jascha Sohl-Dickstein for insightful discussion.
+
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+
+![](images/7ffa93e6f5fc7abf77a65f5e1e89c9d93b2b6bc4de4b83ce2419386b77d72819.jpg)  
+Figure 5: Omniglot generative modeling trained according to IWAE (left), RWS (middle), and JVI (right). The top row compares the variance of the original gradient estimator (dashed) with the variance of the doubly reparameterized gradient estimator (solid). The bottom row compares test performance. The left and middle plots show the IWAE (stochastic) lower bound on the test set. The right plot shows the JVI estimator (which is not a bound) on the test set. The bold lines are the average over three trials, and individual trials are displayed as semi-transparent). All methods used $K = 6 4$ .
+
+![](images/d1fea5490bd6f00c24f8597bd2e4e41c3581964a9195ed30b99b3ed5eadd72de.jpg)  
+Figure 6: Log-likelihood lower bounds for structured prediction on Omniglot. The left plot uses $K = 6 4$ samples and the right plot uses $K = 2 5 6$ samples. For clarity the legend is shared between the plots. The bold lines are the average over three trials, and individual trials are displayed as semi-transparent). The right plot compares performance as the convex combination between IWAEDReG and RWS-DReG is varied. To highlight differences, we plot the difference between the test IWAE bound and the test IWAE bound IWAE-DReG achieved at that step.
+
+# 8.1 EQUIVALENCE BETWEEN REINFORCE GRADIENT AND REPARAMETERIZATION TRICK GRADIENT
+
+Given a function $f ( z , \phi )$ , we have
+
+$$
+\mathbb { E } _ { q _ { \phi } ( z ) } \left[ f ( z , \phi ) \frac { \partial \log q _ { \phi } ( z ) } { \partial \phi } \right] = \mathbb { E } _ { \epsilon } \left[ \frac { \partial f ( z , \phi ) } { \partial z } \frac { \partial z ( \epsilon , \phi ) } { \partial \phi } \right] ,
+$$
+
+![](images/f02e86f7fcf015fdcf9ac2e707a8330b4e3f45ea3fff361779e45e958a954f98.jpg)  
+Figure 7: Structured prediction on MNIST according to IWAE (left), RWS (middle), and JVI (right). The top row compares the variance of the original gradient estimator (dashed) with the variance of the doubly reparameterized gradient estimator (solid). The bottom row compares test performance. The left and middle plots show the IWAE (stochastic) lower bound on the test set. The right plot shows the JVI estimator (which is not a bound) on the test set. The bold lines are the average over three trials, and individual trials are displayed as semi-transparent). All methods used $K = 6 4$ .
+
+![](images/39fd367425ce5cdcaf0a8bab14584ea5412d42a363430f34083726be95e22c55.jpg)  
+Figure 8: Variance of the gradient estimators on the MNIST generative modeling task. We plot the trace of the variance of the doubly reparameterized gradient estimator relative to the original gradient estimator for IWAE (left), RWS (middle), and JVI (right) as the number of samples (K) is varied.
+
+for a reparameterizable distribution $q _ { \phi } ( z )$ . To see this, note that
+
+$$
+\begin{array} { r l } & { \displaystyle \frac { d } { d \phi } \int _ { z } q _ { \phi } ( z ) f ( z , \phi ) d z = \int _ { z } \frac { \partial } { \partial \phi } q _ { \phi } ( z ) f ( z , \phi ) d z = \int _ { z } f ( z , \phi ) \frac { \partial } { \partial \phi } q _ { \phi } ( z ) + q _ { \phi } ( z ) \frac { \partial } { \partial \phi } f ( z , \phi ) d z } \\ & { \quad \quad \quad \quad = \int _ { z } f ( z , \phi ) q _ { \phi } ( z ) \frac { \partial \log q _ { \phi } ( z ) } { \partial \phi } d z + \mathbb { E } _ { q _ { \phi } ( z ) } \left[ \frac { \partial f ( z , \phi ) } { \partial \phi } \right] } \\ & { \quad \quad \quad = \mathbb { E } _ { q _ { \phi } ( z ) } \left[ f ( z , \phi ) \frac { \partial \log q _ { \phi } ( z ) } { \partial \phi } \right] + \mathbb { E } _ { q _ { \phi } ( z ) } \left[ \frac { \partial f ( z , \phi ) } { \partial \phi } \right] , } \end{array}
+$$
+
+via the REINFORCE gradient. On the other hand,
+
+$$
+\begin{array} { l } { \displaystyle \frac { d } { d \phi } \int _ { z } q _ { \phi } ( z ) f ( z , \phi ) d z = \frac { d } { d \phi } \mathbb { E } _ { q _ { \phi } ( z ) } \left[ f ( z , \phi ) \right] = \frac { d } { d \phi } \mathbb { E } _ { \epsilon } \left[ f ( z ( \epsilon , \phi ) , \phi ) \right] = \mathbb { E } _ { \epsilon } \left[ \frac { d } { d \phi } f ( z ( \epsilon , \phi ) , \phi ) \right] } \\ { = \mathbb { E } _ { \epsilon } \left[ \frac { \partial f ( z , \phi ) } { \partial z } \frac { \partial z ( \epsilon , \phi ) } { \partial \phi } \right] + \mathbb { E } _ { \epsilon } \left[ \frac { \partial f ( z , \phi ) } { \partial \phi } \Big | _ { z = z ( \epsilon , \phi ) } \right] } \\ { = \mathbb { E } _ { \epsilon } \left[ \frac { \partial f ( z , \phi ) } { \partial z } \frac { \partial z ( \epsilon , \phi ) } { \partial \phi } \right] + \mathbb { E } _ { q _ { \phi } ( z ) } \left[ \frac { \partial f ( z , \phi ) } { \partial \phi } \right] , } \end{array}
+$$
+
+via the reparameterization trick. Thus, we conclude that
+
+$\Xi _ { q _ { \phi } ( z ) } \left[ f ( z , \phi ) \frac { \partial \log q _ { \phi } ( z ) } { \partial \phi } \right] + \mathbb { E } _ { q _ { \phi } ( z ) } \left[ \frac { \partial f ( z , \phi ) } { \partial \phi } \right] = \mathbb { E } _ { \epsilon } \left[ \frac { \partial f ( z , \phi ) } { \partial z } \frac { \partial z ( \epsilon , \phi ) } { \partial \phi } \right] + \mathbb { E } _ { q _ { \phi } ( z ) } \left[ \frac { \partial f ( z , \phi ) } { \partial \phi } \right] ,$ from which the identity follows.
+
+# 8.2 ASYMPTOTIC ANALYSIS
+
+At a high level, Rainforth et al. (2018) show that the expected value of the IWAE gradient of the inference network collapses to zero with rate √ $1 / K$ , while its standard deviation is only shrinking at a rate of $1 / \sqrt { K }$ . This is the essence of the problem that results in the SNR (expectation divided by standard deviation) of the inference network gradients going to zero at a rate √ $\mathcal { O } ( ( 1 / K ) / ( 1 / \sqrt { K } ) ) \stackrel { - } { = }$ $\mathcal { O } ( 1 / \sqrt { K } )$ , worsening with $K$ . In contrast, Rainforth et al. (2018) show that the generation network gradients scales like $\mathcal { O } ( \sqrt { K } )$ , improving with $K$ .
+
+Because the IWAE-DReG estimator is unbiased, we cannot hope to change the scaling of the expected value in $K$ , but we can hope to change the scaling of the variance. In particular, in this subsection, we provide an informal argument, via the delta method, that the standard deviation of IWAE-DReG scales like $K ^ { - 3 / 2 }$ , which results in an overall scaling of $\mathcal { O } ( \sqrt { K } )$ for the inference network gradient’s SNR (i.e., increasing with $K$ ). Thus, the SNR of the IWAE-DReG estimator improves similarly in $K$ for both inference and generation networks.
+
+We will appeal to the delta method on a two-variable function $g : \mathbb { R } ^ { 2 }  \mathbb { R }$ . Define the following notation for the partials of $g$ evaluated at the mean of random variables $X , Y$ ,
+
+$$
+g _ { x } ( X , Y ) = \left. { \frac { \partial g ( x , y ) } { \partial x } } \right| _ { ( x , y ) = ( \operatorname { \mathbb { E } } ( X ) , \operatorname { \mathbb { E } } ( Y ) ) }
+$$
+
+The delta method approximation of $\operatorname { V a r } ( g ( X , Y ) )$ is given by (Section 5.5 of Casella & Berger),
+
+$$
+\mathrm { V a r } ( g ( X , Y ) ) \approx g _ { x } ( X , Y ) ^ { 2 } \mathrm { V a r } ( X ) + 2 g _ { x } ( X , Y ) g _ { y } ( X , Y ) \mathrm { C o v } ( X , Y ) + g _ { y } ( X , Y ) ^ { 2 } \mathrm { V a r } ( Y )
+$$
+
+, f gen, and $\phi$ s a si. Let aramet, then $u _ { i } \ =$ $w _ { i } ^ { 2 } \frac { \partial \log { w _ { i } } } { \partial z _ { i } } \frac { \partial z _ { i } } { \partial \phi }$ $\textstyle X = \sum _ { i = 1 } ^ { K } u _ { i }$ $\textstyle Y = \sum _ { i = 1 } ^ { K } w _ { i }$ $g ( X , Y ) = X / Y ^ { 2 }$ $g ( X , Y )$ IWAE-DReG estimator whose variance we seek to understand. Letting $Z = \mathbb { E } ( w _ { i } )$ and $U = \mathbb { E } ( u _ { i } )$ we get in this case after cancellations,
+
+$$
+\mathrm { V a r } ( g ( X , Y ) ) \approx \frac { 1 } { Z ^ { 4 } } \frac { \mathrm { V a r } ( X ) } { K ^ { 4 } } - \frac { 4 U } { Z ^ { 5 } } \frac { \mathrm { C o v } ( X , Y ) } { K ^ { 4 } } + \frac { 4 U ^ { 2 } } { Z ^ { 6 } } \frac { \mathrm { V a r } ( Y ) } { K ^ { 4 } }
+$$
+
+Because $w _ { i }$ are all mutually independent, we get $\operatorname { V a r } ( Y ) = K \operatorname { V a r } ( w _ { i } )$ . Similarly for $\operatorname { V a r } ( X )$ and $u _ { i }$ . Because the $w _ { i }$ and $u _ { i }$ are identically distributed and independent for $i \neq j$ , we have $\operatorname { C o v } ( X , Y ) = K \operatorname { C o v } ( w _ { i } , u _ { i } )$ . All together we can see that $\mathrm { V a r } ( g ( \bar { X , Y } ) )$ scales like $\dot { K } ^ { - 3 }$ . Thus, the standard deviation scales like $K ^ { - 3 / 2 }$ .
+
+# 8.3 UNIFIED SURROGATE OBJECTIVES FOR ESTIMATORS
+
+In the main text, we assumed that $\theta$ and $\phi$ were disjoint, however, it can be helpful to share parameters between $p$ and $q$ (e.g., (Fraccaro et al., 2016)). With the IWAE bound, we differentiate a single objective with respect to both the $p$ and $q$ parameters. Thus it is straightforward to adapt IWAE and IWAE-DReG to the shared parameter setting. In this section, we discuss how to deal with shared parameters in RWS.
+
+Suppose that both $p$ and $q$ are parameterized by $\theta$ . If we denote the unshared parameters of $q$ by $\phi$ , then we can restrict the RWS wake update to only $\phi$ . Alternatively, with a modified RWS wake update, we can derive a single surrogate objective for each scenario such that taking the gradient with respect to $\theta$ results in the proper update. For clarity, we introduce the following modifier notation for $p _ { \theta } ( x , z _ { i } )$ , $q _ { \theta } ( z _ { i } | x )$ , and $w _ { i }$ which are functions of $\theta$ and $z _ { i } = z ( \theta , \epsilon _ { i } )$ . We use $\tilde { X }$ to mean $X$ with stopped gradients with respect to $z _ { i }$ , $\hat { X }$ to mean $X$ with stopped gradients with respect to $\theta$ (but not $\theta$ is not stopped in $z ( \theta , \epsilon _ { i } { \bar { ) } } )$ , and $\bar { X }$ to mean $X$ with stopped gradients for all variables. Then, we can use the following surrogate objectives:
+
+IWAE:
+
+$$
+L _ { I W A E } ( \theta ) = \mathbb { E } _ { \epsilon _ { 1 : K } } \left[ \sum _ { i = 1 } ^ { K } \frac { \bar { w } _ { i } } { \sum _ { j } \bar { w } _ { j } } \log w _ { i } \right]
+$$
+
+DReG IWAE:
+
+$$
+L _ { D R e G - I W A E } ( \theta ) = \mathbb { E } _ { \epsilon _ { 1 : K } } \left[ \sum _ { i = 1 } ^ { K } \frac { \bar { w } _ { i } } { \sum _ { j } \bar { w } _ { j } } \log \tilde { p } _ { \theta } ( x , z _ { i } ) + \left( \frac { \bar { w } _ { i } } { \sum _ { j } \bar { w } _ { j } } \right) ^ { 2 } \log \hat { w } _ { i } \right]
+$$
+
+RWS:
+
+$$
+L _ { R W S } ( \theta ) = \mathbb { E } _ { \epsilon _ { 1 : K } } \left[ \sum _ { i = 1 } ^ { K } \frac { \bar { w } _ { i } } { \sum _ { j } \bar { w } _ { j } } \left( \log \tilde { p } _ { \theta } ( x , z _ { i } ) + \log \tilde { q } _ { \theta } ( z _ { i } | x ) \right) \right]
+$$
+
+DReG RWS:
+
+$$
+L _ { D R e G - R W S } ( \theta ) = \mathbb { E } _ { \epsilon _ { 1 : K } } \left[ \sum _ { i = 1 } ^ { K } \frac { \bar { w } _ { i } } { \sum _ { j } \bar { w } _ { j } } \log \tilde { p } _ { \theta } ( x , z _ { i } ) + \left( \frac { \bar { w } _ { i } } { \sum _ { j } \bar { w } _ { j } } - \left( \frac { \bar { w } _ { i } } { \sum _ { j } \bar { w } _ { j } } \right) ^ { 2 } \right) \log \hat { w } _ { i } \right]
+$$
+
+STL:
+
+$$
+L _ { S T L } ( \theta ) = \mathbb { E } _ { \epsilon _ { 1 : K } } \left[ \sum _ { i = 1 } ^ { K } \frac { \bar { w } _ { i } } { \sum _ { j } \bar { w } _ { j } } \left( \log \tilde { p } _ { \theta } ( x , z _ { i } ) + \log \hat { w } _ { i } \right) \right]
+$$
+
+$\mathrm { D R e G } ( \alpha )$
+
+$$
+{ \bf \Psi } _ { ^ { \prime } D R e G ( \alpha ) } ( \theta ) = \mathbb { E } _ { \epsilon _ { 1 : K } } \left[ \sum _ { i = 1 } ^ { K } \frac { \bar { w } _ { i } } { \sum _ { j } \bar { w } _ { j } } \log \tilde { p } \theta ( x , z _ { i } ) + \left( \alpha \frac { \bar { w } _ { i } } { \sum _ { j } \bar { w } _ { j } } + ( 1 - 2 \alpha ) \left( \frac { \bar { w } _ { i } } { \sum _ { j } \bar { w } _ { j } } \right) ^ { 2 } \right) \log \hat { w } _ { i } \right]
+$$
+
+The only subtle difference is that DReG( $\alpha = 0 . 5$ ) does not correspond exactly to STL due to the scaling between terms:
+
+$$
+L _ { D R e G ( \alpha = 0 . 5 ) } ( \theta ) = \mathbb { E } _ { \epsilon _ { 1 : K } } \left[ \sum _ { i = 1 } ^ { K } \frac { \bar { w } _ { i } } { \sum _ { j } \bar { w } _ { j } } \left( \log \tilde { p } _ { \theta } ( x , z _ { i } ) + 0 . 5 \log \hat { w } _ { i } \right) \right]
+$$

parse/train/HkG3e205K7/HkG3e205K7_content_list.json

@@ -0,0 +1,2015 @@
+[
+    {
+        "type": "text",
+        "text": "DOUBLY REPARAMETERIZED GRADIENT ESTIMATORS FOR MONTE CARLO OBJECTIVES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            98,
+            823,
+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "George Tucker Google Brain gjt@google.com ",
+        "bbox": [
+            184,
+            170,
+            323,
+            212
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Dieterich Lawson New York University jdl404@nyu.edu ",
+        "bbox": [
+            408,
+            170,
+            550,
+            212
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Shixiang Gu   \nGoogle Brain   \nshanegu@google.com ",
+        "bbox": [
+            633,
+            171,
+            813,
+            212
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Chris J. Maddison University of Oxford, DeepMind cmaddis@stats.ox.ac.uk ",
+        "bbox": [
+            184,
+            233,
+            401,
+            275
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            311,
+            544,
+            327
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Deep latent variable models have become a popular model choice due to the scalable learning algorithms introduced by (Kingma & Welling, 2013; Rezende et al., 2014). These approaches maximize a variational lower bound on the intractable log likelihood of the observed data. Burda et al. (2015) introduced a multi-sample variational bound, IWAE, that is at least as tight as the standard variational lower bound and becomes increasingly tight as the number of samples increases. Counterintuitively, the typical inference network gradient estimator for the IWAE bound performs poorly as the number of samples increases (Rainforth et al., 2018; Le et al., 2018). Roeder et al. (2017) propose an improved gradient estimator, however, are unable to show it is unbiased. We show that it is in fact biased and that the bias can be estimated efficiently with a second application of the reparameterization trick. The doubly reparameterized gradient (DReG) estimator does not suffer as the number of samples increases, resolving the previously raised issues. The same idea can be used to improve many recently introduced training techniques for latent variable models. In particular, we show that this estimator reduces the variance of the IWAE gradient, the reweighted wake-sleep update (RWS) (Bornschein & Bengio, 2014), and the jackknife variational inference (JVI) gradient (Nowozin, 2018). Finally, we show that this computationally efficient, unbiased drop-in gradient estimator translates to improved performance for all three objectives on several modeling tasks. ",
+        "bbox": [
+            233,
+            340,
+            764,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            643,
+            336,
+            659
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Following the influential work by (Kingma & Welling, 2013; Rezende et al., 2014), deep generative models with latent variables have been widely used to model data such as natural images (Rezende & Mohamed, 2015; Kingma et al., 2016; Chen et al., 2016; Gulrajani et al., 2016), speech and music time-series (Chung et al., 2015; Fraccaro et al., 2016; Krishnan et al., 2015), and video (Babaeizadeh et al., 2017; Ha & Schmidhuber, 2018; Denton & Fergus, 2018). The power of these models lies in combining learned nonlinear function approximators with a principled probabilistic approach, resulting in expressive models that can capture complex distributions. Unfortunately, the nonlinearities that empower these model also make marginalizing the latent variables intractable, rendering direct maximum likelihood training inapplicable. Instead of directly maximizing the marginal likelihood, a common approach is to maximize a tractable lower bound on the likelihood such as the variational evidence lower bound (ELBO) (Jordan et al., 1999; Blei et al., 2017). The tightness of the bound is determined by the expressiveness of the variational family. For tractability, a factorized variational family is commonly used, which can cause the learned model to be overly simplistic. ",
+        "bbox": [
+            174,
+            674,
+            825,
+            854
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Burda et al. (2015) introduced a multi-sample bound, IWAE, that is at least as tight as the ELBO and becomes increasingly tight as the number of samples increases. Counterintuitively, although the bound is tighter, Rainforth et al. (2018) theoretically and empirically showed that the standard inference network gradient estimator for the IWAE bound performs poorly as the number of samples increases due to a diminishing signal-to-noise ratio (SNR). This motivates the search for novel gradient estimators. ",
+        "bbox": [
+            176,
+            862,
+            820,
+            890
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            823,
+            159
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Roeder et al. (2017) proposed a lower-variance estimator of the gradient of the IWAE bound. They speculated that their estimator was unbiased, however, were unable to prove the claim. We show that it is in fact biased, but that it is possible to construct an unbiased estimator with a second application of the reparameterization trick which we call the IWAE doubly reparameterized gradient (DReG) estimator. Our estimator is an unbiased, computationally efficient drop-in replacement, and does not suffer as the number of samples increases, resolving the counterintuitive behavior from previous work (Rainforth et al., 2018). Furthermore, our insight is applicable to alternative multisample training techniques for latent variable models: reweighted wake-sleep (RWS) (Bornschein & Bengio, 2014) and jackknife variational inference (JVI) (Nowozin, 2018). ",
+        "bbox": [
+            173,
+            166,
+            825,
+            291
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this work, we derive DReG estimators for IWAE, RWS, and JVI and demonstrate improved scaling with the number of samples on a simple example. Then, we evaluate DReG estimators on MNIST generative modeling, Omniglot generative modeling, and MNIST structured prediction tasks. In all cases, we demonstrate substantial unbiased variance reduction, which translates to improved performance over the original estimators. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 BACKGROUND ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            388,
+            326,
+            405
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Our goal is to learn a latent variable generative model $p _ { \\theta } ( x , z ) = p _ { \\theta } ( z ) p _ { \\theta } ( x | z )$ where $x$ are observed data and $z$ are continuous latent variables. The marginal likelihood of the observed data, $p _ { \\theta } ( x ) =$ $\\textstyle \\int p _ { \\theta } ( x , z ) d z$ , is generally intractable. Instead, we maximize a variational lower bound on $\\log p _ { \\theta } ( x )$ such as the ELBO ",
+        "bbox": [
+            173,
+            420,
+            825,
+            477
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b166f6561ef0208c010f1736de8e24017f1b77d580992f4aee4762363365eaaa.jpg",
+        "text": "$$\n\\log p _ { \\theta } ( x ) = \\log \\mathbb { E } _ { p _ { \\theta } ( z ) } [ p _ { \\theta } ( x | z ) ] \\geq \\mathbb { E } _ { q ( z | x ) } \\left[ \\log \\frac { p _ { \\theta } ( x , z ) } { q ( z | x ) } \\right] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            303,
+            481,
+            692,
+            515
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $q ( z | x )$ is a variational distribution. Following the influential work by (Kingma & Welling, 2013; Rezende et al., 2014), we consider the amortized inference setting where $q _ { \\phi } ( z | x )$ , referred to as the inference network, is a learnable function parameterized by $\\phi$ that maps from $x$ to a distribution over $z$ . The tightness of the bound is coupled to the expressiveness of the variational family (i.e., $\\{ q _ { \\phi } \\} _ { \\phi } )$ . As a result, limited expressivity of $\\{ q _ { \\phi } \\} _ { \\phi }$ , can negatively affect the learned model. ",
+        "bbox": [
+            173,
+            522,
+            825,
+            593
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Burda et al. (2015) introduced the importance weighted autoencoder (IWAE) bound which alleviates this coupling ",
+        "bbox": [
+            174,
+            598,
+            825,
+            627
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2ecd14f9e8dc4b95b84ed6512edcc08ee16a4dd9696dc2ef477d7d04301ec406.jpg",
+        "text": "$$\n\\mathbb { E } _ { z _ { 1 : K } } \\left[ \\log \\left( \\frac { 1 } { K } \\sum _ { i = 1 } ^ { K } \\frac { p _ { \\theta } ( x , z _ { i } ) } { q _ { \\phi } ( z _ { i } | x ) } \\right) \\right] \\le \\log p _ { \\theta } ( x ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            343,
+            626,
+            651,
+            670
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "with $z _ { 1 : K } \\sim \\textstyle \\prod _ { i } q _ { \\phi } ( z _ { i } | x )$ . The IWAE bound reduces to the ELBO when $K = 1$ , is non-decreasing as $K$ increases, and converges to $\\log p _ { \\theta } ( x )$ as $K  \\infty$ under mild conditions (Burda et al., 2015). When $q _ { \\phi }$ is reparameterizable1, the standard gradient estimator of the IWAE bound is ",
+        "bbox": [
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+            672,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4e2c5604fd5aa864453c18b3adeed7179a79c9536538e14bab98d09beb9513b9.jpg",
+        "text": "$$\n\\bigtriangledown _ { \\theta , \\phi } \\mathbb { E } _ { z _ { 1 : K } } \\left[ \\log \\left( \\frac { 1 } { K } \\sum _ { i = 1 } ^ { K } w _ { i } \\right) \\right] = \\nabla _ { \\theta , \\phi } \\mathbb { E } _ { \\epsilon _ { 1 : K } } \\left[ \\log \\left( \\frac { 1 } { K } \\sum _ { i = 1 } ^ { K } w _ { i } \\right) \\right] = \\mathbb { E } _ { \\epsilon _ { 1 : K } } \\left[ \\sum _ { i = 1 } ^ { K } \\frac { w _ { i } } { \\sum _ { j } w _ { j } } \\nabla _ { \\theta , \\phi } \\log w _ { i } \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            722,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $w _ { i } = p _ { \\theta } ( x , z _ { i } ) / q _ { \\phi } ( z _ { i } | x )$ . A single sample estimator of this expectation is typically used as the gradient estimator. ",
+        "bbox": [
+            173,
+            772,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "As $K$ increases, the bound becomes increasingly tight, however, Rainforth et al. (2018) show that the signal-to-noise ratio (SNR) of the inference network gradient estimator goes to 0. This does not happen for the model parameters $\\mathbf { \\eta } ^ { ( \\theta ) }$ . Following up on this work, Le et al. (2018) demonstrate that this deteriorates the performance of learned models on practical problems. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
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+    },
+    {
+        "type": "text",
+        "text": "Because the IWAE bound converges to $\\log p _ { \\theta } ( x )$ (as $K  \\infty$ ) regardless of $q _ { \\phi }$ , $\\phi$ ’s affect on the bound must diminish as $K$ increases. It may be tempting to conclude that the SNR of the inference network gradient estimator must also decrease as $K  \\infty$ . However, low SNR is a limitation of the gradient estimator, not necessarily of the bound. Although the magnitude of the gradient converges to 0, if the variance of the gradient estimator decreases more quickly, then the SNR of the gradient estimator need not degrade. This motivates the search for lower variance inference network gradient estimators. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To derive improved gradient estimators for $\\phi$ , it is informative to expand the total derivative2 of the IWAE bound with respect to $\\phi$ ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e5bccda9788f0586a6e725c0bed0f007e18056e561d504e8b573fd87009fd0cb.jpg",
+        "text": "$$\n\\mathbb { E } _ { \\epsilon _ { 1 : K } } \\left[ \\sum _ { i = 1 } ^ { K } \\frac { w _ { i } } { \\sum _ { j = 1 } ^ { K } w _ { j } } \\left( - \\frac { \\partial } { \\partial \\phi } \\log q _ { \\phi } ( z _ { i } | x ) + \\frac { \\partial \\log w _ { i } } { \\partial z _ { i } } \\frac { d z _ { i } } { d \\phi } \\right) \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            294,
+            277,
+            702,
+            321
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Previously, Roeder et al. (2017) found that the first term within the parentheses of Eq. 3 can contribute significant variance to the gradient estimator. When $K = 1$ , this term analytically vanishes in expectation, so when $K > 1$ they suggested dropping it. Below, we abbreviate this estimator as STL. As we show in Section 6.1, the STL estimator introduces bias when $K > 1$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 DOUBLY REPARAMETERIZED GRADIENT ESTIMATORS (DREGS) ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our insight is that we can estimate the first term within the parentheses of Eq. 3 efficiently with a second application of the reparameterization trick. To see this, first note that ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7cfed85029e5e6bed62e4133ce9fee579e20ba92b85589bec6add3bef30bac28.jpg",
+        "text": "$$\n\\mathbb { E } _ { \\epsilon _ { 1 : K } } \\left[ \\sum _ { i = 1 } ^ { K } \\frac { w _ { i } } { \\sum _ { j = 1 } ^ { K } w _ { j } } \\frac { \\partial } { \\partial \\phi } \\log q ( \\boldsymbol { z } _ { i } | \\boldsymbol { x } ) \\right] = \\sum _ { i = 1 } ^ { K } \\mathbb { E } _ { \\epsilon _ { 1 : K } } \\left[ \\frac { w _ { i } } { \\sum _ { j = 1 } ^ { K } w _ { j } } \\frac { \\partial } { \\partial \\phi } \\log q ( \\boldsymbol { z } _ { i } | \\boldsymbol { x } ) \\right] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            233,
+            467,
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+            512
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "so it suffices to focus on one of the $K$ terms. Because the derivative is a partial derivativ e ∂∂ φ , it treats $z _ { i } = z ( \\epsilon _ { i } , \\phi )$ as a constant, so we can freely change the random variable that the expectation is over to $z _ { 1 : K }$ . Now, ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/419c3abff00228fe517f63955ffcba2dea7c3582c8ea266544c277893217e0f7.jpg",
+        "text": "$$\n\\mathbb { E } _ { z _ { 1 : K } } \\left[ \\frac { w _ { i } } { \\sum _ { j } w _ { j } } \\frac { \\partial } { \\partial \\phi } \\log q _ { \\phi } ( z _ { i } | x ) \\right] = \\mathbb { E } _ { z _ { - i } } \\mathbb { E } _ { z _ { i } } \\left[ \\frac { w _ { i } } { \\sum _ { j } w _ { j } } \\frac { \\partial } { \\partial \\phi } \\log q _ { \\phi } ( z _ { i } | x ) \\right] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            259,
+            568,
+            736,
+            611
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $z _ { - i } = z _ { 1 : i - 1 , i + 1 : K }$ is the set of $z _ { 1 : K }$ without $z _ { i }$ . The inner expectation resembles a REINFORCE gradient term (Williams, 1992), where we interpret $\\frac { w _ { i } } { \\sum _ { j } w _ { j } }$ as the “reward”. Now, we can use the following well-known equivalence between the REINFORCE gradient and the reparameterization trick gradient (See Appendix 8.1 for a derivation) ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ec4c2e7bb6d92791b9f6d56d4a94db62f9ca44c0a6ef984a67c76c4d95a50d79.jpg",
+        "text": "$$\n\\mathbb { E } _ { q _ { \\phi } ( z | x ) } \\left[ f ( z ) \\frac { \\partial } { \\partial \\phi } \\log q _ { \\phi } ( z | x ) \\right] = \\mathbb { E } _ { \\epsilon } \\left[ \\frac { \\partial f ( z ) } { \\partial z } \\frac { \\partial z ( \\epsilon , \\phi ) } { \\partial \\phi } \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            308,
+            680,
+            687,
+            715
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "This holds even when $f$ depends on $\\phi$ . Typically, the reparameterization gradient estimator has lower variance than the REINFORCE gradient estimator because it directly takes advantage of the derivative of $f$ . Applying the identity from Eq. 5 to the right hand side of Eq. 4 gives ",
+        "bbox": [
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+            719,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d76fd48f87e3223c81da29946dcf5a875dccfa646cf5369fb933f40960516950.jpg",
+        "text": "$$\n\\begin{array} { r l r } & { } & { \\mathbb { E } _ { z _ { i } } \\left[ \\displaystyle \\frac { w _ { i } } { \\sum _ { j } w _ { j } } \\frac { \\partial } { \\partial \\phi } \\log q _ { \\phi } ( z _ { i } | x ) \\right] = \\mathbb { E } _ { \\epsilon _ { i } } \\left[ \\displaystyle \\frac { \\partial } { \\partial z _ { i } } \\left( \\displaystyle \\frac { w _ { i } } { \\sum _ { j } w _ { j } } \\right) \\displaystyle \\frac { \\partial z _ { i } } { \\partial \\phi } \\right] } \\\\ & { } & { = \\mathbb { E } _ { \\epsilon _ { i } } \\left[ \\left( \\displaystyle \\frac { 1 } { \\sum _ { j } w _ { j } } - \\displaystyle \\frac { w _ { i } } { ( \\sum _ { j } w _ { j } ) ^ { 2 } } \\right) \\displaystyle \\frac { \\partial w _ { i } } { \\partial z _ { i } } \\displaystyle \\frac { \\partial z _ { i } } { \\partial \\phi } \\right] = \\mathbb { E } _ { \\epsilon _ { i } } \\left[ \\left( \\displaystyle \\frac { w _ { i } } { \\sum _ { j } w _ { j } } - \\displaystyle \\frac { w _ { i } ^ { 2 } } { ( \\sum _ { j } w _ { j } ) ^ { 2 } } \\right) \\displaystyle \\frac { \\partial \\log w _ { i } } { \\partial z _ { i } } \\displaystyle \\frac { \\partial z _ { i } } { \\partial \\phi } \\right] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            768,
+            795,
+            856
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "This last expression can be efficiently estimated with a single Monte Carlo sample. When $z _ { i }$ is not reparameterizable (e.g., the models in (Mnih & Rezende, 2016)), we can use a control variate (e.g., ",
+        "bbox": [
+            174,
+            857,
+            826,
+            887
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "$\\begin{array} { r } { \\frac { 1 } { K } \\frac { \\partial } { \\partial \\phi } \\log q _ { \\phi } ( z _ { i } | \\boldsymbol { x } ) ) } \\end{array}$ . In both cases, when $K = 1$ , this term vanishes exactly and we recover the estimator proposed in (Roeder et al., 2017) for the ELBO. However, when $K > 1$ , there is no reason to believe this term will analytically vanish. ",
+        "bbox": [
+            174,
+            102,
+            826,
+            148
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Substituting Eq. 6 into Eq. 3, we obtain a simplification due to cancellation of terms ",
+        "bbox": [
+            174,
+            154,
+            723,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ec1bc79fa341f57839e0d4b87743c681c198c46ce9c970aa533b0c383fcf2f1a.jpg",
+        "text": "$$\n\\nabla _ { \\phi } \\mathbb { E } _ { z _ { 1 : K } } \\left[ \\log \\left( \\frac { 1 } { K } \\sum _ { i = 1 } ^ { K } w _ { i } \\right) \\right] = \\mathbb { E } _ { \\epsilon _ { 1 : K } } \\left[ \\sum _ { i = 1 } ^ { K } \\left( \\frac { w _ { i } } { \\sum _ { j } w _ { j } } \\right) ^ { 2 } \\frac { \\partial \\log w _ { i } } { \\partial z _ { i } } \\frac { \\partial z _ { i } } { \\partial \\phi } \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            254,
+            175,
+            741,
+            227
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We call the algorithm that uses the single sample Monte Carlo estimator of this expression for the inference network gradient the IWAE doubly reparameterized gradient estimator (IWAE-DReG). This estimator has the property that when $q ( z | x )$ is optimal (i.e., $q ( z | x ) = p ( z | x ) )$ , the estimator vanishes exactly and has zero variance, whereas this does not hold for the standard IWAE gradient estimator. We provide an asymptotic analysis of the IWAE-DReG estimator in Appendix 8.2. The conclusion of that analysis is that, in contrast to the standard IWAE gradient estimator, the SNR of√ the IWAE-DReG estimator exhibits the same scaling behaviour of $\\mathcal { O } ( \\sqrt { K } )$ for both the generation and inference network gradients (i.e., improving in $K$ ). ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4 ALTERNATIVE TRAINING ALGORITHMS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            366,
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+            382
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Now, we review alternative training algorithms for deep generative models and derive their doubly reparameterized versions. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.1 REWEIGHTED WAKE SLEEP (RWS) ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            441,
+            460,
+            457
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Bornschein & Bengio (2014) introduced RWS, an alternative multi-sample update for latent variable models that uses importance sampling. Computing the gradient of the log marginal likelihood ",
+        "bbox": [
+            174,
+            468,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/394d28a7dd6ec876ffa2ac39a9191b46dd55b10e0a2964d8aaa4e1f7cd17314d.jpg",
+        "text": "$$\n\\nabla _ { \\theta } \\log p _ { \\theta } ( x ) = \\frac { \\nabla _ { \\theta } \\int _ { z } p _ { \\theta } ( x , z ) d z } { p _ { \\theta } ( x ) } = \\frac { \\int _ { z } p _ { \\theta } ( x , z ) \\nabla _ { \\theta } \\log p _ { \\theta } ( x , z ) d z } { p _ { \\theta } ( x ) } = \\mathbb { E } _ { p _ { \\theta } ( z | x ) } \\left[ \\nabla _ { \\theta } \\log p _ { \\theta } ( x , z ) \\right] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
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+            540
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "requires samples from $p _ { \\theta } ( z | x )$ , which is generally intractable. We can approximate the gradient with a self-normalized importance sampling estimator ",
+        "bbox": [
+            174,
+            546,
+            821,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/128e9baf4b4a0c9d8de829e1722a45678f0f66785a520fa9c76dc5f01de0f750.jpg",
+        "text": "$$\n\\mathbb { E } _ { p _ { \\theta } ( z | x ) } \\left[ \\nabla _ { \\theta } \\log p _ { \\theta } ( x , z ) \\right] \\approx \\mathbb { E } _ { z _ { 1 : K } } \\left[ \\sum _ { i = 1 } ^ { K } \\frac { w _ { i } } { \\sum _ { j } w _ { j } } \\nabla _ { \\theta } \\log p _ { \\theta } ( x , z _ { i } ) \\right] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\begin{array} { r } { z _ { 1 : K } \\sim \\prod _ { i } q _ { \\phi } ( z _ { i } | x ) } \\end{array}$ . Interestingly, this is precisely the same as the IWAE gradient of $\\theta$ , so the RWS update for $\\theta$ can be interpreted as maximizing the IWAE lower bound in terms of $\\theta$ . Instead of optimizing a joint objective for $p$ and $q$ , RWS optimizes a separate objective for the inference network. (Bornschein & Bengio, 2014) propose a “wake” update and a “sleep” update for the inference network. Le et al. (2018) provide empirical support for solely using the wake update for the inference network, so we focus on that update. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The wake update approximately minimizes the KL divergence from $p _ { \\theta } ( z | x )$ to $q _ { \\phi } ( z | x )$ . The gradient of the KL term is ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b963f5008409444ac0ee538ae66c6072455128f643006e6f273eb0c3c30935de.jpg",
+        "text": "$$\n\\nabla _ { \\phi } \\mathbb { E } _ { p _ { \\theta } ( z | x ) } \\left[ \\log p _ { \\theta } ( z | x ) - \\log q _ { \\phi } ( z | x ) \\right] = - \\mathbb { E } _ { p _ { \\theta } ( z | x ) } \\left[ \\frac { \\partial } { \\partial \\phi } \\log q _ { \\phi } ( z | x ) \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The wake update of the inference network approximates the intractable expectation by selfnormalized importance sampling ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/41d34214020c1d54fe68a62af98587a09c357af397406ce36063115240ccf713.jpg",
+        "text": "$$\n- \\mathbb { E } _ { p _ { \\theta } ( z | x ) } \\left[ \\frac { \\partial } { \\partial \\phi } \\log q _ { \\phi } ( z | x ) \\right] \\approx - \\mathbb { E } _ { z _ { 1 : K } } \\left[ \\sum _ { i = 1 } ^ { K } \\frac { w _ { i } } { \\sum _ { j } w _ { j } } \\frac { \\partial } { \\partial \\phi } \\log q _ { \\phi } ( z _ { i } | x ) \\right] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            259,
+            830,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "with $z _ { i } \\sim q _ { \\phi } ( z _ { i } | x )$ . Le et al. (2018) note that this update does not suffer from diminishing SNR as $K$ increases. However, a downside is that the updates for $p$ and $q$ are not gradients of a unified objective, so could potentially lead to instability or divergence. ",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "DOUBLY REPARAMETERIZED REWEIGHTED WAKE UPDATE ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            103,
+            584,
+            118
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The wake update gradient for the inference network (Eq. 8) can be reparameterized ",
+        "bbox": [
+            176,
+            128,
+            720,
+            145
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ca69f97770ef62597f819b8dcb3822bb4a8c561547768c46c29a2a98f6718c19.jpg",
+        "text": "$$\n- \\mathbb { E } _ { z _ { 1 : K } } \\left[ \\sum _ { i = 1 } ^ { K } \\frac { w _ { i } } { \\sum _ { j } w _ { j } } \\frac { \\partial } { \\partial \\phi } \\log q _ { \\phi } ( z _ { i } | x ) \\right] = \\mathbb { E } _ { \\epsilon _ { 1 : K } } \\left[ \\sum _ { i = 1 } ^ { K } \\left( \\frac { w _ { i } ^ { 2 } } { ( \\sum _ { j } w _ { j } ) ^ { 2 } } - \\frac { w _ { i } } { \\sum _ { j } w _ { j } } \\right) \\frac { \\partial \\log w _ { i } } { \\partial z _ { i } } \\frac { \\partial z _ { i } } { \\partial \\phi } \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            186,
+            148,
+            812,
+            193
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We call the algorithm that uses the single sample Monte Carlo estimator of this expression as the wake update for the inference network RWS-DReG. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Interestingly, the inference network gradient estimator from (Roeder et al., 2017) can be seen as the sum of the IWAE gradient estimator and the wake update of the inference network (as the wake update minimizes, we add the negative of Eq. 9). Their positive results motivate further exploration of convex combinations of IWAE-DReG and RWS-DReG ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c354b914bd827b1d14f8e9a923ec9bbfb221c7580e7c4784f99e15a4744e457b.jpg",
+        "text": "$$\n\\mathbb { E } _ { \\epsilon _ { 1 : K } } \\left[ \\sum _ { i = 1 } ^ { K } \\left( \\alpha \\frac { w _ { i } } { \\sum _ { j } w _ { j } } + ( 1 - 2 \\alpha ) \\frac { w _ { i } ^ { 2 } } { ( \\sum _ { j } w _ { j } ) ^ { 2 } } \\right) \\frac { \\partial \\log w _ { i } } { \\partial z _ { i } } \\frac { \\partial z _ { i } } { \\partial \\phi } \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            289,
+            297,
+            707,
+            340
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We refer to the algorithm that uses the single sample Monte Carlo estimator of this expression as $\\mathrm { D R e G } ( \\alpha )$ . When $\\alpha = 1$ , this reduces to RWS-DReG, when $\\alpha = 0$ , this reduces to IWAE-DReG and when $\\alpha = 0 . 5$ , this reduces STL. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4.2 JACKKNIFE VARIATIONAL INFERENCE (JVI) ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            402,
+            521,
+            417
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Alternatively, Nowozin (2018) reinterprets the IWAE lower bound as a biased estimator for the log marginal likelihood. He analyzes the bias and introduces a novel family of estimators, Jackknife Variational Inference (JVI), which trade off reduction in bias for increased variance. This additional flexibility comes at the cost of no longer being a stochastic lower bound on the log marginal likelihood. The first-order JVI has significantly reduced bias compared to IWAE, which empirically results in a better estimate of the log marginal likelihood with fewer samples (Nowozin, 2018). For simplicity, we focus on the first-order JVI estimator ",
+        "bbox": [
+            173,
+            428,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/57da664026def4d2b565f31da4604d58d912ce6bfbde6af1ca1c694ee74b7955.jpg",
+        "text": "$$\nK \\times \\mathbb { E } _ { z _ { 1 } \\cdot K } \\left[ \\log \\left( \\frac { 1 } { K } \\sum _ { i = 1 } ^ { K } w _ { i } \\right) \\right] - \\frac { K - 1 } { K } \\sum _ { i = 1 } ^ { K } \\mathbb { E } _ { z _ { - i } } \\left[ \\log \\left( \\frac { 1 } { K - 1 } \\sum _ { j \\neq i } w _ { j } \\right) \\right] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            236,
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "It is straightforward to apply our approach to higher order JVI estimators. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "DOUBLY REPARAMETERIZED JACKKNIFE VARIATIONAL INFERENCE (JVI) ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The JVI estimator is a linear combination of $K$ and $K - 1$ sample IWAE estimators, so we can use the doubly reparameterized gradient estimator (Eq. 7) for each term. ",
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+    {
+        "type": "text",
+        "text": "5 RELATED WORK ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Mnih & Rezende (2016) introduced a generalized framework of Monte Carlo objectives (MCO). The log of an unbiased marginal likelihood estimator is a lower bound on the log marginal likelihood by Jensen’s inequality. In this view, the ELBO can be seen as the MCO corresponding to a single importance sample estimator of the marginal likelihood with $q _ { \\theta }$ as the proposal distribution. Similarly, IWAE corresponds to the $K$ -sample estimator. Maddison et al. (2017) show that the tightness of an MCO is directly related to the variance of the underlying estimator of the marginal likelihood. ",
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+    {
+        "type": "text",
+        "text": "However, Rainforth et al. (2018) point out issues with gradient estimators of multi-sample lower bounds. In particular, they show that although the IWAE bound is tighter, the standard IWAE gradient estimator’s SNR scales poorly with large numbers of samples, leading to degraded performance. Le et al. (2018) experimentally investigate this phenomenon and provide empirical evidence of this degradation across multiple tasks. They find that RWS (Bornschein & Bengio, 2014) does not suffer from this issue and find that it can outperform models trained with the IWAE bound. We conclude that it is not sufficient to just tighten the bound; it is important to understand the gradient estimators of the tighter bound as well. ",
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+    },
+    {
+        "type": "text",
+        "text": "Wake-sleep is an alternative approach to fitting deep generative models, first introduced in (Hinton et al., 1995) as a method for training Hemholtz machines. It was extended to the multi-sample setting by (Bornschein & Bengio, 2014) and the sequential setting in (Gu et al., 2015). It has been applied to generative modeling of images (Ba et al., 2015). ",
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+    {
+        "type": "text",
+        "text": "6 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To evaluate DReG estimators, we first measure variance and signal-to-noise ratio $( \\mathrm { S N R } ) ^ { 3 }$ of gradient estimators on a toy example which we can carefully control. Then, we evaluate gradient variance and model learning on MNIST generative modeling, Omniglot generative modeling, and MNIST structured prediction tasks. ",
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+    },
+    {
+        "type": "text",
+        "text": "6.1 TOY GAUSSIAN ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We reimplemented the Gaussian example from (Rainforth et al., 2018). Consider the generative model with $z \\sim N ( \\theta , I )$ and $x | z \\sim N ( z , I )$ and inference network $\\begin{array} { r } { q _ { \\phi } ( z | x ) \\sim N ( A x ^ { - } + b , \\frac { 2 } { 3 } I ) } \\end{array}$ , where $\\phi = \\{ A , b \\}$ . As in (Rainforth et al., 2018), we sample a set of parameters for the model and inference network close to the optimal parameters (perturbed by zero-mean Gaussian noise with standard deviation 0.01), then estimate the gradient of the inference network parameters for increasing number of samples $( K )$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In addition to signal-to-noise ratio (SNR), we plot the squared bias and variance of the gradient estimators4 in Fig. 1. The bias is computed relative to the expected value of the IWAE gradient estimator. As a result, although the average of $K$ ELBO gradient estimators is an unbiased estimator of the ELBO gradient, it is a biased gradient estimator of the IWAE objective. Importantly, SNR does not penalize estimators that are biased, so trivial constant estimators can have infinite SNR. Thus, it is important to consider additional evaluation measures as well. As $K$ increases, the SNR of the IWAE-DReG estimator increases, whereas the SNR of the standard gradient estimator of IWAE goes to 0, as previously reported. Furthermore, we can see the bias present in the STL estimator. As a check of our implementation, we verified that the observed “bias” for IWAE-DReG was statistically indistinguishable from 0 with a paired t-test. For the biased estimators (e.g., STL), we could easily reject the null hypothesis with few samples. ",
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+    {
+        "type": "image",
+        "img_path": "images/4602530728769106fa89abfa2de18954281cbb1ea79b32fbc94d040f866f151a.jpg",
+        "image_caption": [
+            "Figure 1: Signal-to-noise ratios (SNR), bias squared, and variance of gradient estimators with increasing $K$ over 10 random trials with 1000 measurement samples per trial (mean in bold). The observed “bias” for IWAE-DReG is not statistically significant under a paired t-test (as expected because IWAE-DReG is unbiased). IWAE-DReG is unbiased, its SNR increases with K, and it has the lowest variance of the estimators considered here. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "6.2 GENERATIVE MODELING ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Training generative models of the binarized MNIST digits dataset is a standard benchmark task for latent variable models. For this evaluation, we used the single latent layer architecture from (Burda et al., 2015). The generative model used 50 Gaussian latent variables with an isotropic prior and passed $z$ through two deterministic layers of 200 tanh units to parameterize factorized Bernoulli outputs. The inference network passed $x$ through two deterministic layers of 200 tanh units to parameterize a factorized Gaussian distribution over $z$ . Because our interest was in improved gradient estimators and optimization performance, we used the dynamically binarized MNIST dataset, which minimally suffers from overfitting. We used the standard split of MNIST into train, validation, and test sets. ",
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+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We trained models with the IWAE gradient, the RWS wake update, and with the JVI estimator. In all three cases, the doubly reparameterized gradient estimator reduced variance5 and as a result substantially improved performance (Fig. 2). ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/df131fe5998b53bb0ac88c7a2eb163c6e64208318f090fefa87a69dbc76fe0f7.jpg",
+        "image_caption": [
+            "Figure 2: MNIST generative modeling trained according to IWAE (left), RWS (middle), and JVI (right). The top row compares the variance of the original gradient estimator (dashed) with the variance of the doubly reparameterized gradient estimator (solid). The bottom row compares test performance. The left and middle plots show the IWAE (stochastic) lower bound on the test set. The right plot shows the JVI estimator (which is not a bound) on the test set. The bold lines are the average over three trials, and individual trials are displayed as semi-transparent). All methods used $K = 6 4$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We found similar behavior with different numbers of samples (Fig. 3 and Appendix Fig. 8). Interestingly, the biased gradient estimators STL and RWS-DReG perform best on this task with RWSDReG slightly outperforming STL. As observed in (Le et al., 2018), RWS increasingly outperforms IWAE as $K$ increases. Finally, we experimented with convex combinations of IWAE-DReG and RWS-DReG (right Fig. 3). On this dataset, convex combinations that heavily weighted RWS-DReG had the best performance. However, as we show below, this is task dependent. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Next, we performed the analogous experiment with the dynamically binarized Omniglot dataset using the same model architecture. Again, we found that the doubly reparameterized gradient estimator reduced variance and as a result improved test performance (Figs. 5 and 6 in the Appendix). ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "6.3 STRUCTURED PREDICTION ON MNIST ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Structured prediction is another common benchmark task for latent variable models. In this task, our goal is to model a complex observation $x$ given a context $c$ (i.e., model the conditional distribution $p ( x | c ) )$ . We can use a conditional latent variable model $p _ { \\theta } ( x , z | c ) = p _ { \\theta } ( x | z , c ) p _ { \\theta } ( z | c )$ , however, as before, computing the marginal likelihood is generally intractable. It is straightforward to adapt the bounds and techniques from the previous section to this problem. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/91eefa28a63730121c99b026911c4b3915048b1370526f3bf876517154d33a0f.jpg",
+        "image_caption": [
+            "Figure 3: Log-likelihood lower bounds for generative modeling on MNIST. The left and middle plots compare performance with different number of samples $K = 3 2$ , 256. For clarity the legend is shared between the plots. The bold lines are the average over three trials, and individual trials are displayed as semi-transparent). The right plot compares performance as the convex combination between IWAE-DReG and RWS-DReG is varied (Eq. 10). To highlight differences, we plot the difference between the test IWAE bound and the test IWAE bound IWAE-DReG achieved at that step. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "To evaluate our method in this context, we use the standard task of modeling the bottom half of a binarized MNIST digit from the top half. We use a similar architecture, but now learn a conditional prior distribution $p _ { \\theta } ( z | c )$ where $c$ is the top half of the MNIST digit. The conditional prior feeds $c$ to two deterministic layers of 200 tanh units to parameterize a factorized Gaussian distribution over $z$ . To model the conditional distribution $p _ { \\theta } ( x | c , z )$ , we concatenate $z$ with $c$ and feed it to two deterministic layers of 200 tanh units to parameterize factorized Bernoulli outputs. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "As in the previous tasks, the doubly reparameterized gradient estimator improves across all three updates (IWAE, RWS, and JVI; Appendix Fig. 7). However, on this task, the biased estimators (STL and RWS) underperform unbiased IWAE gradient estimators (Fig. 4). In particular, RWS becomes unstable later in training. We suspect that this is because RWS does not directly optimize a consistent objective. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/57312dcd20d125385cf7e6526c7c22ebd0eaef48131ffdc8b992f9c2c7c72fc0.jpg",
+        "image_caption": [
+            "Figure 4: Log-likelihood lower bounds for structured prediction on MNIST. The left plot uses $K =$ 64 samples and the right plot uses $K \\ : = \\ : 2 5 6$ samples. For clarity the legend is shared between the plots. The bold lines are the average over three trials, and individual trials are displayed as semi-transparent). The right plot compares performance as the convex combination between IWAEDReG and RWS-DReG is varied (Eq. 10). To highlight differences, we plot the difference between the test IWAE bound and the test IWAE bound IWAE-DReG achieved at that step. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "7 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this work, we introduce doubly reparameterized estimators for the updates in IWAE, RWS, and JVI. We demonstrate that across tasks they provide unbiased variance reduction, which leads to improved performance. Furthermore, DReG estimators have the same computational cost as the original estimators. As a result, we recommend that DReG estimators be used instead of the typical gradient estimators. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Variational Sequential Monte Carlo (Maddison et al., 2017; Naesseth et al., 2018; Le et al., 2018) and Neural Adapative Sequential Monte Carlo (Gu et al., 2015) extend IWAE and RWS to sequential latent variable models, respectively. It would be interesting to develop DReG estimators for these approaches as well. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We found that a convex combination of IWAE-DReG and RWS-DReG performed best, however, the weighting was task dependent. In future work, we intend to apply ideas from (Baydin et al., 2017) to automatically adapt the weighting based on the data. ",
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Finally, the form of the IWAE-DReG estimator (Eq. 7) is surprisingly simple and suggests that there may be a more direct derivation that is applicable to general MCOs. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
+        "bbox": [
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+            237,
+            356,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We thank Ben Poole and Diederik P. Kingma for helpful discussion and comments on drafts of this paper. We thank Sergey Levine and Jascha Sohl-Dickstein for insightful discussion. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
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+        "text": "Danilo Jimenez Rezende, Shakir Mohamed, and Daan Wierstra. Stochastic backpropagation and approximate inference in deep generative models. In International Conference on Machine Learning, pp. 1278–1286, 2014. ",
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+    {
+        "type": "text",
+        "text": "Ronald J Williams. Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine learning, 8(3-4):229–256, 1992. ",
+        "bbox": [
+            171,
+            880,
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+        "page_idx": 9
+    },
+    {
+        "type": "image",
+        "img_path": "images/7ffa93e6f5fc7abf77a65f5e1e89c9d93b2b6bc4de4b83ce2419386b77d72819.jpg",
+        "image_caption": [
+            "Figure 5: Omniglot generative modeling trained according to IWAE (left), RWS (middle), and JVI (right). The top row compares the variance of the original gradient estimator (dashed) with the variance of the doubly reparameterized gradient estimator (solid). The bottom row compares test performance. The left and middle plots show the IWAE (stochastic) lower bound on the test set. The right plot shows the JVI estimator (which is not a bound) on the test set. The bold lines are the average over three trials, and individual trials are displayed as semi-transparent). All methods used $K = 6 4$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            204,
+            160,
+            794,
+            383
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/d1fea5490bd6f00c24f8597bd2e4e41c3581964a9195ed30b99b3ed5eadd72de.jpg",
+        "image_caption": [
+            "Figure 6: Log-likelihood lower bounds for structured prediction on Omniglot. The left plot uses $K = 6 4$ samples and the right plot uses $K = 2 5 6$ samples. For clarity the legend is shared between the plots. The bold lines are the average over three trials, and individual trials are displayed as semi-transparent). The right plot compares performance as the convex combination between IWAEDReG and RWS-DReG is varied. To highlight differences, we plot the difference between the test IWAE bound and the test IWAE bound IWAE-DReG achieved at that step. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            205,
+            549,
+            790,
+            645
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "8.1 EQUIVALENCE BETWEEN REINFORCE GRADIENT AND REPARAMETERIZATION TRICK GRADIENT ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            805,
+            813,
+            835
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Given a function $f ( z , \\phi )$ , we have ",
+        "bbox": [
+            174,
+            852,
+            400,
+            867
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e3f84ce9fd6e9238c700c420ef29c0dfb0bedc7fa44c79a5c17c5b061c5b50cc.jpg",
+        "text": "$$\n\\mathbb { E } _ { q _ { \\phi } ( z ) } \\left[ f ( z , \\phi ) \\frac { \\partial \\log q _ { \\phi } ( z ) } { \\partial \\phi } \\right] = \\mathbb { E } _ { \\epsilon } \\left[ \\frac { \\partial f ( z , \\phi ) } { \\partial z } \\frac { \\partial z ( \\epsilon , \\phi ) } { \\partial \\phi } \\right] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            310,
+            887,
+            684,
+            922
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/f02e86f7fcf015fdcf9ac2e707a8330b4e3f45ea3fff361779e45e958a954f98.jpg",
+        "image_caption": [
+            "Figure 7: Structured prediction on MNIST according to IWAE (left), RWS (middle), and JVI (right). The top row compares the variance of the original gradient estimator (dashed) with the variance of the doubly reparameterized gradient estimator (solid). The bottom row compares test performance. The left and middle plots show the IWAE (stochastic) lower bound on the test set. The right plot shows the JVI estimator (which is not a bound) on the test set. The bold lines are the average over three trials, and individual trials are displayed as semi-transparent). All methods used $K = 6 4$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            205,
+            99,
+            794,
+            325
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/39fd367425ce5cdcaf0a8bab14584ea5412d42a363430f34083726be95e22c55.jpg",
+        "image_caption": [
+            "Figure 8: Variance of the gradient estimators on the MNIST generative modeling task. We plot the trace of the variance of the doubly reparameterized gradient estimator relative to the original gradient estimator for IWAE (left), RWS (middle), and JVI (right) as the number of samples (K) is varied. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            181,
+            446,
+            813,
+            556
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "for a reparameterizable distribution $q _ { \\phi } ( z )$ . To see this, note that ",
+        "bbox": [
+            173,
+            659,
+            591,
+            675
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4c423f2d20ec4f8ea693be4851a2c983319dabb689e10698500a4f325712605d.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\displaystyle \\frac { d } { d \\phi } \\int _ { z } q _ { \\phi } ( z ) f ( z , \\phi ) d z = \\int _ { z } \\frac { \\partial } { \\partial \\phi } q _ { \\phi } ( z ) f ( z , \\phi ) d z = \\int _ { z } f ( z , \\phi ) \\frac { \\partial } { \\partial \\phi } q _ { \\phi } ( z ) + q _ { \\phi } ( z ) \\frac { \\partial } { \\partial \\phi } f ( z , \\phi ) d z } \\\\ & { \\quad \\quad \\quad \\quad = \\int _ { z } f ( z , \\phi ) q _ { \\phi } ( z ) \\frac { \\partial \\log q _ { \\phi } ( z ) } { \\partial \\phi } d z + \\mathbb { E } _ { q _ { \\phi } ( z ) } \\left[ \\frac { \\partial f ( z , \\phi ) } { \\partial \\phi } \\right] } \\\\ & { \\quad \\quad \\quad = \\mathbb { E } _ { q _ { \\phi } ( z ) } \\left[ f ( z , \\phi ) \\frac { \\partial \\log q _ { \\phi } ( z ) } { \\partial \\phi } \\right] + \\mathbb { E } _ { q _ { \\phi } ( z ) } \\left[ \\frac { \\partial f ( z , \\phi ) } { \\partial \\phi } \\right] , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            187,
+            683,
+            810,
+            790
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "via the REINFORCE gradient. On the other hand, ",
+        "bbox": [
+            173,
+            794,
+            503,
+            809
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cb6984b844cb2ed51b88b049e3ad00081da251901be497873530b869b80bb83b.jpg",
+        "text": "$$\n\\begin{array} { l } { \\displaystyle \\frac { d } { d \\phi } \\int _ { z } q _ { \\phi } ( z ) f ( z , \\phi ) d z = \\frac { d } { d \\phi } \\mathbb { E } _ { q _ { \\phi } ( z ) } \\left[ f ( z , \\phi ) \\right] = \\frac { d } { d \\phi } \\mathbb { E } _ { \\epsilon } \\left[ f ( z ( \\epsilon , \\phi ) , \\phi ) \\right] = \\mathbb { E } _ { \\epsilon } \\left[ \\frac { d } { d \\phi } f ( z ( \\epsilon , \\phi ) , \\phi ) \\right] } \\\\ { = \\mathbb { E } _ { \\epsilon } \\left[ \\frac { \\partial f ( z , \\phi ) } { \\partial z } \\frac { \\partial z ( \\epsilon , \\phi ) } { \\partial \\phi } \\right] + \\mathbb { E } _ { \\epsilon } \\left[ \\frac { \\partial f ( z , \\phi ) } { \\partial \\phi } \\Big | _ { z = z ( \\epsilon , \\phi ) } \\right] } \\\\ { = \\mathbb { E } _ { \\epsilon } \\left[ \\frac { \\partial f ( z , \\phi ) } { \\partial z } \\frac { \\partial z ( \\epsilon , \\phi ) } { \\partial \\phi } \\right] + \\mathbb { E } _ { q _ { \\phi } ( z ) } \\left[ \\frac { \\partial f ( z , \\phi ) } { \\partial \\phi } \\right] , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            189,
+            816,
+            810,
+            926
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "via the reparameterization trick. Thus, we conclude that ",
+        "bbox": [
+            173,
+            103,
+            542,
+            118
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "$\\Xi _ { q _ { \\phi } ( z ) } \\left[ f ( z , \\phi ) \\frac { \\partial \\log q _ { \\phi } ( z ) } { \\partial \\phi } \\right] + \\mathbb { E } _ { q _ { \\phi } ( z ) } \\left[ \\frac { \\partial f ( z , \\phi ) } { \\partial \\phi } \\right] = \\mathbb { E } _ { \\epsilon } \\left[ \\frac { \\partial f ( z , \\phi ) } { \\partial z } \\frac { \\partial z ( \\epsilon , \\phi ) } { \\partial \\phi } \\right] + \\mathbb { E } _ { q _ { \\phi } ( z ) } \\left[ \\frac { \\partial f ( z , \\phi ) } { \\partial \\phi } \\right] ,$ from which the identity follows. ",
+        "bbox": [
+            173,
+            122,
+            825,
+            176
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "8.2 ASYMPTOTIC ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            193,
+            382,
+            207
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "At a high level, Rainforth et al. (2018) show that the expected value of the IWAE gradient of the inference network collapses to zero with rate √ $1 / K$ , while its standard deviation is only shrinking at a rate of $1 / \\sqrt { K }$ . This is the essence of the problem that results in the SNR (expectation divided by standard deviation) of the inference network gradients going to zero at a rate √ $\\mathcal { O } ( ( 1 / K ) / ( 1 / \\sqrt { K } ) ) \\stackrel { - } { = }$ $\\mathcal { O } ( 1 / \\sqrt { K } )$ , worsening with $K$ . In contrast, Rainforth et al. (2018) show that the generation network gradients scales like $\\mathcal { O } ( \\sqrt { K } )$ , improving with $K$ . ",
+        "bbox": [
+            173,
+            218,
+            825,
+            313
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Because the IWAE-DReG estimator is unbiased, we cannot hope to change the scaling of the expected value in $K$ , but we can hope to change the scaling of the variance. In particular, in this subsection, we provide an informal argument, via the delta method, that the standard deviation of IWAE-DReG scales like $K ^ { - 3 / 2 }$ , which results in an overall scaling of $\\mathcal { O } ( \\sqrt { K } )$ for the inference network gradient’s SNR (i.e., increasing with $K$ ). Thus, the SNR of the IWAE-DReG estimator improves similarly in $K$ for both inference and generation networks. ",
+        "bbox": [
+            173,
+            318,
+            825,
+            404
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We will appeal to the delta method on a two-variable function $g : \\mathbb { R } ^ { 2 }  \\mathbb { R }$ . Define the following notation for the partials of $g$ evaluated at the mean of random variables $X , Y$ , ",
+        "bbox": [
+            174,
+            410,
+            820,
+            439
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0550594ef77e41cdf0fe02a2709e60ecc6c162e3e5214baacdf2e5d5f524f94f.jpg",
+        "text": "$$\ng _ { x } ( X , Y ) = \\left. { \\frac { \\partial g ( x , y ) } { \\partial x } } \\right| _ { ( x , y ) = ( \\operatorname { \\mathbb { E } } ( X ) , \\operatorname { \\mathbb { E } } ( Y ) ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            366,
+            443,
+            632,
+            481
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The delta method approximation of $\\operatorname { V a r } ( g ( X , Y ) )$ is given by (Section 5.5 of Casella & Berger), ",
+        "bbox": [
+            176,
+            484,
+            802,
+            501
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b3b10e38c9369fa68d8f5f1c312fea2ad3f4e359acac9ae5c882918c57faa511.jpg",
+        "text": "$$\n\\mathrm { V a r } ( g ( X , Y ) ) \\approx g _ { x } ( X , Y ) ^ { 2 } \\mathrm { V a r } ( X ) + 2 g _ { x } ( X , Y ) g _ { y } ( X , Y ) \\mathrm { C o v } ( X , Y ) + g _ { y } ( X , Y ) ^ { 2 } \\mathrm { V a r } ( Y )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            192,
+            506,
+            805,
+            523
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": ", f gen, and $\\phi$ s a si. Let aramet, then $u _ { i } \\ =$ $w _ { i } ^ { 2 } \\frac { \\partial \\log { w _ { i } } } { \\partial z _ { i } } \\frac { \\partial z _ { i } } { \\partial \\phi }$ $\\textstyle X = \\sum _ { i = 1 } ^ { K } u _ { i }$ $\\textstyle Y = \\sum _ { i = 1 } ^ { K } w _ { i }$ $g ( X , Y ) = X / Y ^ { 2 }$ $g ( X , Y )$ IWAE-DReG estimator whose variance we seek to understand. Letting $Z = \\mathbb { E } ( w _ { i } )$ and $U = \\mathbb { E } ( u _ { i } )$ we get in this case after cancellations, ",
+        "bbox": [
+            174,
+            529,
+            823,
+            590
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5c80068d419799d37733f6db8ae9022d943350366082e7dad348c6623c447a29.jpg",
+        "text": "$$\n\\mathrm { V a r } ( g ( X , Y ) ) \\approx \\frac { 1 } { Z ^ { 4 } } \\frac { \\mathrm { V a r } ( X ) } { K ^ { 4 } } - \\frac { 4 U } { Z ^ { 5 } } \\frac { \\mathrm { C o v } ( X , Y ) } { K ^ { 4 } } + \\frac { 4 U ^ { 2 } } { Z ^ { 6 } } \\frac { \\mathrm { V a r } ( Y ) } { K ^ { 4 } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            282,
+            594,
+            714,
+            627
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Because $w _ { i }$ are all mutually independent, we get $\\operatorname { V a r } ( Y ) = K \\operatorname { V a r } ( w _ { i } )$ . Similarly for $\\operatorname { V a r } ( X )$ and $u _ { i }$ . Because the $w _ { i }$ and $u _ { i }$ are identically distributed and independent for $i \\neq j$ , we have $\\operatorname { C o v } ( X , Y ) = K \\operatorname { C o v } ( w _ { i } , u _ { i } )$ . All together we can see that $\\mathrm { V a r } ( g ( \\bar { X , Y } ) )$ scales like $\\dot { K } ^ { - 3 }$ . Thus, the standard deviation scales like $K ^ { - 3 / 2 }$ . ",
+        "bbox": [
+            173,
+            631,
+            825,
+            689
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "8.3 UNIFIED SURROGATE OBJECTIVES FOR ESTIMATORS",
+        "text_level": 1,
+        "bbox": [
+            173,
+            704,
+            581,
+            720
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In the main text, we assumed that $\\theta$ and $\\phi$ were disjoint, however, it can be helpful to share parameters between $p$ and $q$ (e.g., (Fraccaro et al., 2016)). With the IWAE bound, we differentiate a single objective with respect to both the $p$ and $q$ parameters. Thus it is straightforward to adapt IWAE and IWAE-DReG to the shared parameter setting. In this section, we discuss how to deal with shared parameters in RWS. ",
+        "bbox": [
+            173,
+            731,
+            825,
+            801
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Suppose that both $p$ and $q$ are parameterized by $\\theta$ . If we denote the unshared parameters of $q$ by $\\phi$ , then we can restrict the RWS wake update to only $\\phi$ . Alternatively, with a modified RWS wake update, we can derive a single surrogate objective for each scenario such that taking the gradient with respect to $\\theta$ results in the proper update. For clarity, we introduce the following modifier notation for $p _ { \\theta } ( x , z _ { i } )$ , $q _ { \\theta } ( z _ { i } | x )$ , and $w _ { i }$ which are functions of $\\theta$ and $z _ { i } = z ( \\theta , \\epsilon _ { i } )$ . We use $\\tilde { X }$ to mean $X$ with stopped gradients with respect to $z _ { i }$ , $\\hat { X }$ to mean $X$ with stopped gradients with respect to $\\theta$ (but not $\\theta$ is not stopped in $z ( \\theta , \\epsilon _ { i } { \\bar { ) } } )$ , and $\\bar { X }$ to mean $X$ with stopped gradients for all variables. Then, we can use the following surrogate objectives: ",
+        "bbox": [
+            173,
+            808,
+            825,
+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "IWAE: ",
+        "bbox": [
+            173,
+            103,
+            222,
+            118
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0807e5ac3d31f48ca8f39a5cb77c693d640f0f95ae0b3b88550b382043966601.jpg",
+        "text": "$$\nL _ { I W A E } ( \\theta ) = \\mathbb { E } _ { \\epsilon _ { 1 : K } } \\left[ \\sum _ { i = 1 } ^ { K } \\frac { \\bar { w } _ { i } } { \\sum _ { j } \\bar { w } _ { j } } \\log w _ { i } \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            361,
+            122,
+            635,
+            166
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "DReG IWAE: ",
+        "bbox": [
+            173,
+            178,
+            267,
+            194
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/86696c0e6cef1780922ae1e9b0fc1837df9f5e9e34a93fdf097a2d6a3c370d34.jpg",
+        "text": "$$\nL _ { D R e G - I W A E } ( \\theta ) = \\mathbb { E } _ { \\epsilon _ { 1 : K } } \\left[ \\sum _ { i = 1 } ^ { K } \\frac { \\bar { w } _ { i } } { \\sum _ { j } \\bar { w } _ { j } } \\log \\tilde { p } _ { \\theta } ( x , z _ { i } ) + \\left( \\frac { \\bar { w } _ { i } } { \\sum _ { j } \\bar { w } _ { j } } \\right) ^ { 2 } \\log \\hat { w } _ { i } \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            241,
+            199,
+            754,
+            248
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "RWS: ",
+        "bbox": [
+            173,
+            260,
+            215,
+            275
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3b1bd612ec698f7eacc35d77fb738f74d564baa33e2f25cbd9b7f33a4c727df2.jpg",
+        "text": "$$\nL _ { R W S } ( \\theta ) = \\mathbb { E } _ { \\epsilon _ { 1 : K } } \\left[ \\sum _ { i = 1 } ^ { K } \\frac { \\bar { w } _ { i } } { \\sum _ { j } \\bar { w } _ { j } } \\left( \\log \\tilde { p } _ { \\theta } ( x , z _ { i } ) + \\log \\tilde { q } _ { \\theta } ( z _ { i } | x ) \\right) \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            289,
+            280,
+            709,
+            324
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "DReG RWS: ",
+        "bbox": [
+            173,
+            335,
+            261,
+            351
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b289619d07c92dfb5f1c74cf355e14227afa9373163de7f01236cd042baf0322.jpg",
+        "text": "$$\nL _ { D R e G - R W S } ( \\theta ) = \\mathbb { E } _ { \\epsilon _ { 1 : K } } \\left[ \\sum _ { i = 1 } ^ { K } \\frac { \\bar { w } _ { i } } { \\sum _ { j } \\bar { w } _ { j } } \\log \\tilde { p } _ { \\theta } ( x , z _ { i } ) + \\left( \\frac { \\bar { w } _ { i } } { \\sum _ { j } \\bar { w } _ { j } } - \\left( \\frac { \\bar { w } _ { i } } { \\sum _ { j } \\bar { w } _ { j } } \\right) ^ { 2 } \\right) \\log \\hat { w } _ { i } \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            184,
+            356,
+            784,
+            405
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "STL: ",
+        "bbox": [
+            173,
+            417,
+            210,
+            433
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/28687d841ed056bdbeb467e88bce1b5b5e221f256330a92999f483f456aa8e18.jpg",
+        "text": "$$\nL _ { S T L } ( \\theta ) = \\mathbb { E } _ { \\epsilon _ { 1 : K } } \\left[ \\sum _ { i = 1 } ^ { K } \\frac { \\bar { w } _ { i } } { \\sum _ { j } \\bar { w } _ { j } } \\left( \\log \\tilde { p } _ { \\theta } ( x , z _ { i } ) + \\log \\hat { w } _ { i } \\right) \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            312,
+            438,
+            686,
+            481
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "$\\mathrm { D R e G } ( \\alpha )$ ",
+        "bbox": [
+            173,
+            486,
+            243,
+            501
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9d8a2b2809c28c15fcc5d0690e8085d90cc042c653b13df20bfbafaf86245c7d.jpg",
+        "text": "$$\n{ \\bf \\Psi } _ { ^ { \\prime } D R e G ( \\alpha ) } ( \\theta ) = \\mathbb { E } _ { \\epsilon _ { 1 : K } } \\left[ \\sum _ { i = 1 } ^ { K } \\frac { \\bar { w } _ { i } } { \\sum _ { j } \\bar { w } _ { j } } \\log \\tilde { p } \\theta ( x , z _ { i } ) + \\left( \\alpha \\frac { \\bar { w } _ { i } } { \\sum _ { j } \\bar { w } _ { j } } + ( 1 - 2 \\alpha ) \\left( \\frac { \\bar { w } _ { i } } { \\sum _ { j } \\bar { w } _ { j } } \\right) ^ { 2 } \\right) \\log \\hat { w } _ { i } \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
+            506,
+            825,
+            558
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The only subtle difference is that DReG( $\\alpha = 0 . 5$ ) does not correspond exactly to STL due to the scaling between terms: ",
+        "bbox": [
+            173,
+            577,
+            825,
+            606
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8a9d97d3f1527c258b339b2272b84ddf7dc05f181b1078e73d858a4a8a953f38.jpg",
+        "text": "$$\nL _ { D R e G ( \\alpha = 0 . 5 ) } ( \\theta ) = \\mathbb { E } _ { \\epsilon _ { 1 : K } } \\left[ \\sum _ { i = 1 } ^ { K } \\frac { \\bar { w } _ { i } } { \\sum _ { j } \\bar { w } _ { j } } \\left( \\log \\tilde { p } _ { \\theta } ( x , z _ { i } ) + 0 . 5 \\log \\hat { w } _ { i } \\right) \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
+            269,
+            612,
+            725,
+            656
+        ],
+        "page_idx": 13
+    }
+]

parse/train/HyPpD0g0Z/HyPpD0g0Z.md

@@ -0,0 +1,518 @@
+# GROUPING-BY-ID: GUARDING AGAINST ADVERSARIAL DOMAIN SHIFTS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+When training a deep neural network for supervised image classification, one can broadly distinguish between two types of latent features of images that will drive the classification of class $Y$ . Following the notation of Gong et al. (2016), we can divide features broadly into the classes of (i) ‘core’ or ‘conditionally invariant’ features $X ^ { c i }$ whose distribution $P ( X ^ { c i } | Y )$ does not change substantially across domains and (ii) ‘style’ or ‘orthogonal’ features $X ^ { \perp }$ whose distribution $P \bar { ( } X ^ { \bot } | Y )$ can change substantially across domains. These latter orthogonal features would generally include features such as position, rotation, image quality or brightness but also more complex ones like hair color or posture for images of persons. We try to guard against future adversarial domain shifts by ideally just using the ‘conditionally invariant’ features for classification. In contrast to previous work, we assume that the domain itself is not observed and hence a latent variable. We can hence not directly see the distributional change of features across different domains.
+
+We do assume, however, that we can sometimes observe a so-called identifier or ID variable. We might know, for example, that two images show the same person, with ID referring to the identity of the person. In data augmentation, we generate several images from the same original image, with ID referring to the relevant original image. The method requires only a small fraction of images to have an ID variable.
+
+We provide a causal framework for the problem by adding the ID variable to the model of Gong et al. (2016). However, we are interested in settings where we cannot observe the domain directly and we treat domain as a latent variable. If two or more samples share the same class and identifier, $( Y , \mathrm { I D } ) = ( y , \mathrm { i d } )$ , then we treat those samples as counterfactuals under different style interventions on the orthogonal or style features. Using this grouping-by-ID approach, we regularize the network to provide near constant output across samples that share the same ID by penalizing with an appropriate graph Laplacian. This is shown to substantially improve performance in settings where domains change in terms of image quality, brightness, color changes, and more complex changes such as changes in movement and posture. We show links to questions of interpretability, fairness and transfer learning.
+
+# 1 INTRODUCTION
+
+Deep neural networks (DNNs) have achieved outstanding performance on prediction tasks like visual object and speech recognition (Krizhevsky et al., 2012; Szegedy et al., 2015; He et al., 2015). Issues can arise when the learned representations rely on dependencies that vanish in test distributions (e.g. see Csurka (2017) and references therein). Such domain shifts can be caused by changing conditions, e.g. color, background or location changes arising when deploying the machine learning (ML) system in production. Predictive performance is then likely to degrade. For instance, the “Russian tank legend” is an example where the training data was subject to sampling biases that were not replicated in the real world. Concretely, the story relates how a machine learning system was trained to distinguish between Russian and American tanks from photos. The accuracy was very high but only due to the fact that all images of Russian tanks were of bad quality while the photos of
+
+American tanks were not. The system learned to discriminate between images of different qualities but would have failed badly in practice (Emspak, 2016)1.
+
+Hidden confounding factors like in the example above between image quality and the origin of the tank give rise to indirect associations. These are arguably one reason why deep learning requires large sample sizes as large sample sizes tend to ensure that the effect of the confounding factors averages out (although a large sample size is clearly not per se a guarantee that the confounding effect will become weaker). A large sample size is also required if one is trying to achieve invariance to known factors like translation, point of view, and rotation by using data augmentation. Another related example where human and artificial cognition deviate strongly are adversarial examples— imperceptibly but intentionally perturbed inputs that are misclassified by a ML model (Szegedy et al., 2014; Goodfellow et al., 2015). Adversarial examples do not fool humans and in general we only need to see one rotated example of the same object to achieve invariance to rotations in our perception. Our starting point is the question whether we can in a simple way mimic the human ability to learn desired invariances from a few instances of the same object and whether we can better align the features DNNs exploit with human cognition.
+
+Considerations of fairness and discrimination might be another reason why we are interested in controlling that certain characteristics of the input data are not included in the learned representations and thus have no impact on the resulting decisions (Barocas & Selbst, 2016; Kilbertus et al., 2017). Unfortunately, existing biases in datasets used for training ML algorithms tend to be replicated in the estimated models (Bolukbasi et al., 2016). For instance, in June 2015 Google’s photo app tagged two non-white people as “gorillas”—most likely because the training examples for “people” were mainly photos of white persons, making “color” predictive for the class label (Crawford, 2016; Emspak, 2016). A human would not make the same mistake after only seeing one instance of a non-white person.
+
+Addressing the issues outlined above, we propose counterfactual regularization (CORE) to control what latent features an estimator extracts from the input data. Conceptually, we take a causal view of the data generating process and categorize the latent data generating factors into ‘conditionally invariant’ (core) and ‘orthogonal’ (style) features, as in (Gong et al., 2016). It is desirable that a classifier uses only the core features as they pertain to the target of interest in a stable and coherent fashion. CORE yields an estimator which is invariant to factors of variation corresponding to style features. Consequently, it is robust with respect to adversarial domain shifts, arising through arbitrarily strong interventions on the style features. CORE relies on the fact that for certain datasets we can observe “counterfactuals” in the sense that we observe the same object under different conditions. Rather than pooling over all examples, CORE exploits knowledge about this grouping, i.e. that a number of instances relate to the same object.
+
+The remainder of this manuscript is structured as follows: $\ S 2$ starts with two motivating examples, showing how CORE can reduce the need for data augmentation and help predictive performance in small sample size settings. In $\ S 3$ we review related work and in $\ S 4$ we formally introduce counterfactual regularization, along with the CORE estimator and theoretical insights for the logistic regression setting. In §5 we further evaluate the performance of CORE in a variety of experiments.
+
+# 2 TWO MOTIVATING EXAMPLES
+
+2.1 GROUPING PHOTOS OF THE SAME PERSON: BETTER PREDICTIVE PERFORMANCE
+
+The CelebA dataset (Liu et al., 2015) contains face images of celebrities. We consider the task of classifying whether a person wears glasses. Several photos of the same person are available. We use this grouping information and constrain the classification to yield the same prediction for all images belonging to the same person and sharing the same class label. We call the additional instances of the same person counterfactual (CF) observations. Figure 1a shows examples from the training set. The standard approach would be to pool all examples. The only additional information we exploit is that some observations can be grouped. We include $n = 1 0$ identities in the training set, resulting in a total sample size $m = 3 2 1$ as there are approximately 30 images of each person2.
+
+(a) Grouping-by-ID with ID=identity.
+
+(b) Grouping-by-ID with ID $=$ original image.
+
+![](images/ba66922606a44436f08eff197b11fa3c0dbae95f221a2936a7176b3141d60bbd.jpg)  
+Figure 1: Examples from a) the subsampled CelebA dataset and b) the augmented MNIST dataset. Connected images are counterfactual examples as they share the same realization of the ID which is the identity of the person in a) and the original image used for data augmentation in b). The comparison is a training of exactly the same network architecture that does not make use of the grouping information but using a standard ridge penalty. In a) exploiting the grouping information reduces the test error by $32 \%$ compared to pooling over all samples. In b) the test error on rotated digits is reduced by $50 \%$ .
+
+Exploiting the group structure reduces the average test error from $2 4 . 7 6 \%$ to $1 6 . 8 9 \%$ , i.e. by approx. $32 \%$ , compared to the estimator which just pools all images and uses a standard ridge penalty for the cofficients3.
+
+# 2.2 GROUPING AUGMENTED IMAGES BY ORIGINAL: MORE SAMPLE EFFICIENT
+
+A different use case of CORE is to make data augmentation more efficient in terms of the required samples. In data augmentation, one creates additional samples by modifying the original inputs, e.g. by rotating, translating, or flipping the images (Scholkopf et al., 1996). In other words, additional ¨ samples are generated by interventions on style features. Using this augmented data set for training results in invariance of the estimator with respect to the transformations (style features) of interest. For CORE we can use the grouping information that the original and the augmented samples belong to the same object. This enforces the invariance with respect to the style features more strongly compared to normal data augmentation which just pools all samples. We assess this for the style feature “rotation” on MNIST (LeCun & Cortes, 2010) and only include $c = 1 0 0$ augmented training examples for $n = 1 0 0 0 0$ original samples, resulting in a total sample size of $m = 1 0 1 0 0$ . The degree of the rotations is sampled uniformly at random from [35, 70]. Figure 1b shows examples from the training set. By using CORE the average test error on rotated examples is reduced from $3 2 . 8 6 \%$ to $1 6 . 3 3 \%$ , around half its original value4.
+
+# 3 RELATED WORK
+
+Perhaps most similar to this work in terms of their goals are the work of Gong et al. (2016) and Domain-Adversarial Neural Networks (DANN) proposed in Ganin et al. (2016), an approach motivated by the work of Ben-David et al. (2007). While our approach requires grouped observations, both of these works rely on unlabeled data from the target task being available.
+
+The main idea of Ganin et al. (2016) is to learn a representation that contains no discriminative information about the origin of the input (source or target domain). This is achieved by an adversarial training procedure: the loss on domain classification is maximized while the loss of the target prediction task is minimized simultaneously. In contrast, we do not assume that we have data from different domains but just different realizations of the same object under different interventions.
+
+The data generating process assumed in Gong et al. (2016) is similar to our model, introduced in $\ S 4 . 2$ where we detail the similarities and differences between the models (cf. Figure 2). Gong et al. (2016) identify the conditionally independent features by adjusting a transformation of the variables to minimize the squared MMD distance between distributions in different domains5. The fundamental difference to our approach is that we use a different data basis. The domain identifier is explicitly observable in Gong et al. (2016), while it is latent in our approach. In contrast, we exploit presence of an identifier variable ID to penalize the classifier using any latent features outside the set of conditionally independent features.
+
+Causal modeling has related aims to the setting of transfer learning and guarding against adversarial domain shifts. Specifically, causal models have the defining advantage that the predictions will be valid even under arbitrarily large interventions on all predictor variables (Haavelmo, 1944; Aldrich, 1989; Pearl, 2009; Scholkopf et al., 2012; Peters et al., 2016; Zhang et al., 2013; 2015; X. Yu, 2017; ¨ M. Rojas-Carulla, 2017; Magliacane et al., 2017). There are two difficulties in transferring these results to the setting of adversarial domain changes in image classification. The first hurdle is that the classification task is typically anti-causal since the image we use as a predictor is a descendant of the true class of the object we are interested in rather than the other way around. The second challenge is that we do not want to guard against arbitrary interventions on any or all variables but only would like to guard against a shift of the style features. It is hence not immediately obvious how standard causal inference can be used to guard against large domain shifts.
+
+Recently, various approaches have been proposed that leverage causal motivations for deep learning or use deep learning for causal inference. In all of the following methods, the goals and the settings are different from ours. Specifically, the setting of anti-causal prediction and non-ancestral interventions on style variables is not considered. Various approaches focus on cause-effect inference where the goal is to find the causal relation between two random variables, $X$ and $Y$ (Lopez-Paz et al., 2017; Lopez-Paz & Oquab, 2017; Goudet et al., 2017). Lopez-Paz et al. (2017) propose the Neural Causation Coefficient (NCC) to estimate the probability of $X$ causing $Y$ and apply it to finding the causal relations between image features. Specifically, the NCC is used to distinguish between features of objects and features of the objects’ contexts. Lopez-Paz & Oquab (2017) note the similarity between structural equation modeling and CGANs (Mirza & Osindero, 2014). One CGAN is fitted in the direction $X  Y$ and another one is fitted for $Y  X$ . Based on a two-sample test statistic, the estimated causal direction is returned. Goudet et al. (2017) use generative neural networks for cause-effect inference, to identify $v$ -structures and to orient the edges of a given graph skeleton. Bahadori et al. (2017) devise a regularizer that combines an $\ell _ { 1 }$ penalty with weights corresponding to the estimated probability of the respective feature being causal for the target. The latter estimates are obtained by causality detection networks or scores such as estimated by the NCC. Besserve et al. (2017) draw connections between GANs and causal generative models, using a group theoretic framework. Kocaoglu et al. (2017) propose causal implicit generative models to sample from conditional as well as interventional distributions, using a conditional GAN architecture (CausalGAN). The generator structure needs to inherit its neural connections from the causal graph, i.e. the causal graph structure must be known. Louizos et al. (2017) propose the use of deep latent variable models and proxy variables to estimate individual treatment effects.
+
+Kilbertus et al. (2017) exploit causal reasoning to characterize fairness considerations in machine learning. Distinguishing between the protected attribute and its proxies, they derive causal nondiscrimination criteria. The resulting algorithms avoiding proxy discrimination require classifiers to be constant as a function of the proxy variables in the causal graph, thereby bearing some structural similarity to our style features.
+
+Distinguishing between core and style features can be seen as some form of disentangling factors of variation. Estimating disentangled factors of variation has gathered a lot of interested in the context of generative modeling (Higgins et al., 2017; Chen et al., 2016; Bouchacourt et al., 2017). For example, Matsuo et al. (2017) propose a “Transform Invariant Autoencoder” where the goal is to reduce the dependence of the latent representation on a specified transform of the object in the original image. Specifically, Matsuo et al. (2017) predefine location as the orthogonal style feature $X ^ { \perp }$ and the goal is to learn a latent representation that does not include $X ^ { \perp }$ . Here, we do not predefine which features are in $X ^ { \perp }$ . It could be location but also image quality, posture, brightness, background and contextual information. Additionally, the approach in Matsuo et al. (2017) cannot effectively deal with a confounding situation where the distribution of the style features differs conditional on the class (this is a natural restriction as the class label is not even observed in the autoencoder setting). As in CORE, Bouchacourt et al. (2017) exploit grouped observations. In a variational autoencoder framework, they aim to separate style and content—they assume that samples within a group share a common but unknown value for one of the factors of variation while the style can differ. Here we try to solve a classification task directly without estimating the latent factors explicitly as in a generative framework.
+
+![](images/092251423c06ea419ac63998157d4cdc64f2db82cb26ebd1209ccd2500749c25.jpg)  
+Figure 2: Left: data generating process for the considered model as in Gong et al. (2016), where the effect of the domain on the orthogonal features $X ^ { \perp }$ is mediated via unobserved noise $\Delta$ . Right: our setting. The domain itself is unobserved but we can now observe the ID variable we use for grouping.
+
+# 4 COUNTERFACTUAL REGULARIZATION
+
+We first describe the standard notation for classification before developing a causal graph that allows us to compare the setting of adversarial domain shifts to transfer learning, domain adaptation and adversarial examples.
+
+# 4.1 NOTATION FOR STANDARD CLASSIFICATION
+
+Let $Y \in \mathcal { V }$ be a target of interest. Typically $\mathcal { V } = \mathbb { R }$ for regression or $\mathcal { V } = \{ 1 , \ldots , K \}$ in classification with $K$ classes. Let $X \in \mathbb { R } ^ { p }$ be a predictor, for example the $p$ pixels of an image. The prediction $\hat { y }$ for $y$ , given $X = x$ , is of the form $f _ { \boldsymbol { \theta } } ( \boldsymbol { x } )$ for a suitable function $f _ { \theta }$ with parameters $\theta \in  { \mathbb { R } } ^ { d }$ , where the parameters $\theta$ correspond to the weights in a DNN. For regression, $f _ { \theta } ( x ) \in \mathbb { R }$ , whereas for classification $f _ { \theta } ( x )$ corresponds to the conditional probability distribution of $Y \in \{ 1 , \ldots , K \}$ . Let $\ell$ be a suitable loss that maps $y$ and ${ \hat { y } } = f _ { \theta } ( x )$ to $\mathbb { R } ^ { + }$ . A standard goal is to minimize the expected loss or risk
+
+$$
+L ( \theta ) \ = \ E \Big [ \ell ( Y , f _ { \theta } ( X ) ) \Big ] .
+$$
+
+Let $( x _ { i } , y _ { i } )$ for $i = 1 , \ldots , n$ be the samples that constitute the training data and $\hat { y } _ { i } = f _ { \theta } ( x _ { i } )$ the prediction for $y _ { i }$ . A standard approach to parameter estimation is penalized empirical risk minimization, where we choose the weights or parameters as $\hat { \theta } = \mathrm { a r g m i n } _ { \theta } \ L _ { n } ( \theta )$ , with the empirical loss given by $\begin{array} { r l r } { L _ { n } ( \theta ) } & { { } = } & { { \frac { 1 } { n } } \sum _ { i = 1 } ^ { n } \ell ( y _ { i } , f _ { \theta } ( x _ { i } ) ) + \lambda \cdot \mathrm { p e n } ( \theta ) } \end{array}$ , where the penalty $\mathrm { p e n } ( \theta )$ could be a ridge penalty or penalties that exploit underlying geometries such as the Laplacian regularized least squares (Belkin et al., 2006).
+
+# 4.2 CAUSAL GRAPH
+
+The full structural model for all variables is shown in the right panel of Figure 2. The domain variable $D$ is latent, in contrast to Gong et al. (2016). We add the ID variable (identity of a person, for example), whose distribution can change conditional on class $Y$ . The ID variable is used to
+
+group observations, see Section 4.4, and can be assumed to be latent in the setting of Gong et al.   
+(2016).
+
+The rest of the graph is in analogy to Gong et al. (2016). The prediction is anti-causal, that is the predictors $X$ that we use for $\hat { Y }$ are non-ancestral to $Y$ . In other words, the class label is causal for the image and not the other way around. The causal effect from the class label $Y$ on the image $X$ is mediated via two types of latent variables: the so-called core or ‘conditionally invariant’ features $X ^ { c i }$ and the orthogonal or style features $X ^ { \perp }$ . The distinguishing factor between the two is that external interventions $\Delta$ are possible on the style features but not on the core features. If the interventions $\Delta$ have different distributions in different domains, then the distribution $P ( X ^ { c i } | Y )$ is constant across domains while $P ( X ^ { \bot } | Y )$ can change across domains. The style features $X ^ { \perp }$ and $Y$ are confounded, in other words, by the latent domain $D$ . In contrast, the core or ‘conditionally invariant’ features satisfy $X ^ { c i } \perp \perp \mathsf { D } | Y$ . The dimension of $X ^ { c i }$ is chosen maximally large such that this conditional independence is still true. The style variable can include point of view, image quality, resolution, rotations, color changes, body posture, movement etc. and will in general be context-dependent6. The style intervention variable $\Delta$ influences both the latent style $\bar { X ^ { \bot } }$ , and hence also the image $X$ . In potential outcome notation, we let $X ^ { \bot } ( \Delta = \delta )$ be the style under intervention $\Delta = \delta$ , $X ( Y , \mathrm { I D } , \Delta = \delta )$ the image for class $Y$ , identity ID and style intervention $\Delta$ and this sometimes abbreviated as $X ( \Delta = \delta )$ for notational simplicity. Finally, $f _ { \theta } ( X ( \Delta = \delta ) )$ ) is the prediction under the style intervention $\Delta = \delta$ . For a formal justification of using a causal graph and potential outcome notation simultaneously see Richardson $\&$ Robins (2013).
+
+# 4.3 DOMAIN ADAPTATION, ADVERSARIAL EXAMPLES AND ADVERSARIAL DOMAIN SHIFTS
+
+In this work, we are interested in guarding against adversarial domain shifts. We use the causal graph to explain the related but not identical goals of domain adaptation, transfer learning and guarding against adversarial examples.
+
+(i) Domain adaptation and transfer learning. Assume we have $J$ different domains, each with a new distribution $F _ { j }$ for the interventions $\Delta$ (or more generally of the joint distribution of $( Y , \Delta ) )$ . The shift of $F _ { j }$ for different domains $j = 1 , \dots , J$ causes a shift in both the distribution of $X$ and in the conditional distribution $Y | X$ . If we consider domain adaptation and transfer learning together, their goal is generally to give the best possible prediction $\hat { Y } _ { j } ( x )$ in each domain $j = 1 , \dots , J$ .
+
+(ii) Standard adversarial examples. The setting of adversarial examples in the sense of Szegedy et al. (2014) and Goodfellow et al. (2015) can also be described by the causal graph above by using $X ^ { \perp } ( \Delta ) = \Delta$ and identifying $X ^ { \perp }$ with pixel-by-pixel additive effects. The magnitude of the intervention $\Delta$ is then typically assumed to be within an $\epsilon$ -ball in $\ell _ { q }$ -norm around the origin, with $q = \infty$ or $q = 2$ for example. If the input dimension is large many imperceptible changes in the coordinates of $X$ can cause a large change in the output, leading to a misclassification of the sample. The goal is to devise a classification in this graph that minimizes the adversarial loss
+
+$$
+E \Big [ \operatorname* { m a x } _ { { \Delta \in \mathbb { R } ^ { q } } \colon \| { \Delta \| _ { q } } \leq \epsilon } \ell \Big ( Y , f _ { \theta } \big ( X ( { \Delta } ) \big ) \Big ) \Big ] ,
+$$
+
+where $X ( \Delta )$ is the image under the intervention $\Delta$ and $\hat { Y } ~ = ~ f _ { \theta } ( X ( \Delta ) )$ is the estimated conditional distribution of $Y$ , given the image under the chosen interventions.
+
+(iii) Adversarial domain shifts. Here we are interested in arbitrarily strong interventions $\Delta \in \mathbb { R } ^ { q }$ on the style features $X ^ { \perp }$ , which are not known explicitly in general. Analogously to (1), the adversarial loss under arbitrarily large style interventions is
+
+$$
+L _ { a d v } ( \theta ) = E \Big [ \operatorname* { m a x } _ { \Delta \in \mathbb { R } ^ { q } } \ell \Big ( Y , f _ { \theta } \big ( X ( \Delta ) \big ) \Big ) \Big ] .
+$$
+
+In contrast to (1) the interventions can be arbitrarily strong but we assume that the style features $X ^ { \perp }$ can only change certain aspects of the image, while other aspects of the image (mediated by the core features) cannot be changed. In contrast to Ganin et al. (2016), we use the term “adversarial” to refer to adversarial interventions on the style features, while the notion of “adversarial” in domain adversarial neural networks describes the training procedure. Nevertheless, the motivation of Ganin et al. (2016) is equivalent to ours—that is, to protect against shifts in the distribution(s) of test data which we characterize by distinguishing between core and style features.
+
+# 4.4 COUNTERFACTUAL OBSERVATIONS / GROUPING
+
+The classical problem of causal inference is that we can never observe a counterfactual. For instance, we can only see the health outcome $Z$ if we take a medicine, $T = 1$ , or not, $T = 0$ , but we can never see both health outcomes simultaneously. The counterfactual in this context would be an observation where we change the treatment but hold all observed and unobserved confounders constant. If the treatment $T$ changes while all other variables are kept constant, we could just read off the treatment effect as $Z ( T = \bar { 1 } ) - Z ( T = 0 )$ if $Z$ is the health outcome of interest. Observing such counterfactuals is in general impossible as we can either observe the outcome under treatment or under no treatment but not both.
+
+Here, we use the term counterfactual for a situation where we keep class label $Y$ and ID constant but allow the value of the style intervention $\Delta$ to change. The new value of $\Delta$ could be a do-intervention (as when explicitly rotating an image in data augmentation) or it could be a noise-intervention by sampling a new realization of $\Delta$ . The style intervention $\Delta$ takes the same role as the treatment $T$ in the previous medical example. In contrast to the medical example, however, counterfactuals are conceivable for image analysis as we can see the same object $( Y , \mathrm { I D } )$ under different conditions (‘treatments’) $\Delta$ .
+
+As an example, if $Y$ is the binary variable whether a person wears glasses and ID is the identity of a person, then $\Delta$ corresponds to all other variables that determine the different images of the same person (either consistently wearing glasses or not) and includes background, posture, viewing angle, image quality, etc.
+
+In further contrast to the medical setting, we are not interested primarily in the ‘treatment effect’ of the style intervention $\Delta$ but we merely use it to implicitly rule out parts of the feature space for classification. We know that any ‘treatment effect’ of $\Delta$ occurs in the space of the style or orthogonal features $X ^ { \perp }$ and not in the ‘conditionally invariant’ space $X ^ { c i }$ and we would thus like to penalize any change in the classification under different style interventions $\Delta$ but constant class and identity $( Y , \mathrm { I D } )$ .
+
+Notationally, we have for sample $i \in \{ 1 , \ldots , n \}$ with class label and identifier $\left( Y , \mathrm { I D } \right) = \left( y _ { i } , \mathrm { i d } _ { i } \right)$ , $m _ { i }$ different images $x ( y _ { i } , \mathrm { i d } _ { i } , \Delta _ { i , j } )$ for $j ~ = ~ 1 , \dots , m _ { i }$ under different (unobserved) values of $\Delta _ { i , 1 } , \ldots , \Delta _ { i , m _ { i } }$ . Let $\begin{array} { r } { m = \sum _ { i } m _ { i } } \end{array}$ denote the total number of samples and $c = m - n$ , the number of counterfactual observations. Denote the $j$ -th observation of sample $i$ , by $x _ { i , j } \in \mathbb { R } ^ { p }$ . Typically $m _ { i } = 1$ for most samples and occasionally $m _ { i } \geq 2$ .
+
+# 4.4.1 STANDARD APPROACH: POOLED ESTIMATOR
+
+The standard approach is to simply pool over all available observations, ignoring any grouping information that might be available. The pooled estimator thus treats all examples identically by summing over the loss as
+
+$$
+{ \hat { \theta } } ^ { p o o l } = \operatorname { a r g m i n } _ { \theta } { \frac { 1 } { m } } \sum _ { i = 1 } ^ { n } \sum _ { j = 1 } ^ { m _ { i } } \left[ \ell { \bigl ( } y _ { i } , f _ { \theta } ( x _ { i , j } ) { \bigr ) } \right] + \lambda \cdot \operatorname { p e n } ( \theta ) ,
+$$
+
+where $\mathrm { p e n } ( \theta )$ could be a ridge penalty. The pooled estimator in all examples is always the ridge estimator with a cross-validated choice of the penalty parameter. The adversarial loss of the pooled estimator will in general be infinite; see $\ S 4 . 6$ for a concrete example. Using Figure 2, one can show that the pooled estimator will work well in terms of the adversarial loss $L _ { a d v }$ if both (i) $Y$ ⊥⊥ $X | X ^ { c i }$ and (ii) $\mathrm { ~ \bar { \it Y } ~ } \mathcal { Y } \mathrm { ~ \not \ = ~ } X ^ { c i } | X ^ { \bot }$ . The first condition (i) implies that if the estimator learns to extract $X ^ { c i }$ from the image $X$ , there is no further information in $X$ that explains $Y$ and, therefore, the direction corresponding to $X ^ { \perp }$ is not required for predicting $Y$ . The second condition (ii) is fulfilled if the relations between $Y$ , $X ^ { c i }$ , and $X ^ { \perp }$ are not deterministic. Intuitively, it ensures that $X ^ { \perp }$ cannot replace $X ^ { c i }$ in the first condition. From (i) and (ii), we see that the pooled estimator will work well in terms of the adversarial loss $\boldsymbol { L _ { a d v } }$ if (a) the edge from $X ^ { \perp }$ to $X$ is absent or if (b) both the edge from $D$ to $X ^ { \perp }$ and the edge from $Y$ to $X ^ { \perp }$ are absent (cf. Figure 2).
+
+# 4.5 CORE ESTIMATOR
+
+In order to minimize the adversarial loss (2) we have to ensure $f _ { \theta } ( x ( \Delta ) )$ is as constant as possible as a function of $\Delta$ for all $x \in \mathbb { R } ^ { p }$ . Let $I$ be the invariant parameter space
+
+For all $\theta \in I$ , the adversarial loss (2) is identical to the loss under no interventions at all. More precisely, let $X$ be a shorthand notation for $X ( \Delta = 0 )$ , the images in absence of external interventions:
+
+$$
+{ \mathrm { i f } } \theta \in I , { \mathrm { ~ t h e n } } \qquad E { \Bigl [ } \operatorname* { m a x } _ { \Delta \in \mathbb { R } ^ { q } } \ell { \Bigl ( } Y , f _ { \theta } { \bigl ( } X ( \Delta ) { \bigr ) } { \Bigr ) } { \Bigr ] } \ = \ E { \Bigl [ } \ell { \Bigl ( } Y , f _ { \theta } { \bigl ( } X { \bigl ) } { \Bigr ) } { \Bigr ] } .
+$$
+
+The optimal predictor in the invariant space $I$ is
+
+$$
+\theta ^ { * } \ = \ \operatorname { a r g m i n } _ { \theta } { \cal E } \Big [ \ell ( Y , f _ { \theta } ( X ) ) \Big ] \ \mathrm { s u c h ~ t h a t } \theta \in I .
+$$
+
+If $f _ { \theta }$ is only a function of the core features $X ^ { c i }$ , then $\theta \in I$ . The challenge is that the core features are not directly observable and we have to infer the invariant space $I$ from data. To get an approximation to the optimal invariant parameter vector (3), we use empirical risk minimization:
+
+$$
+{ \hat { \theta } } ^ { c o r e } = \operatorname * { a r g m i n } _ { \theta } { \frac { 1 } { n } } \sum _ { i = 1 } ^ { n } \sum _ { j = 1 } ^ { m _ { i } } \ell \bigl ( y _ { i } , f _ { \theta } ( x _ { i , j } ) \bigr ) \ \mathrm { s u c h \ t h a t } \ \theta \in I _ { n } ,
+$$
+
+where the first part is the empirical version of the expectation in (3). The unknown invariant parameters space $I$ is approximated by an empirically invariant space $I _ { n }$ , defined as
+
+$$
+I _ { n } : = \{ \theta : \sum _ { i = 1 } ^ { n } \sigma _ { i } ^ { 2 } ( \theta ) \leq \tau \} ,
+$$
+
+where $\sigma _ { i } ^ { 2 } ( \theta )$ is the variance of $f _ { \theta } ( x _ { i , j } )$ when varying $j = 1 , \dots , m _ { i }$ for a fixed value of $i$ and $\tau \geq 0$ is a regularization constant. Setting $\tau = 0$ is equivalent to demanding that the estimated predictions for the class labels are identical across all $m _ { i }$ counterfactuals of image $i$ , while slightly larger values of $\tau$ allow for some small degree of variations. For all values $\tau \geq 0$ the true invariant space $I$ is a subset of the empirically invariant subspace $I _ { n }$ , that is $I \subseteq I _ { n }$ . Under the right assumptions we get $I _ { n } = I$ for $n \to \infty$ . We return to this question in $\ S 4 . 6$ . One can equally use the Lagrangian form of the constrained optimization in (4), with a penalty parameter $\lambda$ instead of a constraint $\tau$ to get
+
+$$
+{ \hat { \theta } } ^ { c o r e } = \operatorname { \arg \operatorname* { m i n } } _ { \theta } { \frac { 1 } { n } } \sum _ { i = 1 } ^ { n } \sum _ { j = 1 } ^ { m _ { i } } \ell { \big ( } y _ { i } , f _ { \theta } ( x _ { i , j } ) { \big ) } + \lambda \cdot \operatorname { p e n } _ { \mathrm { I D } } ( \theta ) ,
+$$
+
+where $\mathrm { p e n } _ { \mathrm { I D } } ( \theta ) = \tilde { f } _ { \theta } ^ { t } L _ { \mathrm { I D } } \tilde { f } _ { \theta }$ , and $\tilde { f } _ { \theta } \in \mathbb { R } ^ { m }$ is the value of $f _ { \theta } ( x _ { i , j } )$ at all $\begin{array} { r } { m = \sum _ { i = 1 } ^ { n } m _ { i } } \end{array}$ observations. The matrix $L _ { \mathrm { I D } }$ is a graph Laplacian (Belkin et al., 2006), where the underlying graph has $n$ connectivity components as all samples that have the same ID are connected by an edge and form fully connected connectivity components. The graph Laplacian regularization is identical to penalizing the sum over the variances ${ \bar { \sigma _ { i } ^ { 2 } } } ( \theta )$ . The graph for the underlying regularization is formed in the sample space and induced by the identifier variable ID, in contrast to graphs formed in feature space as in Sandler et al. (2009), where prior knowledge is used to form the graph by connecting features that share similar characteristics.
+
+We show in $\mathrm { \ S C . 1 }$ that the outcome does not depend strongly on the chosen value of the penalty $\lambda$ and the experiments show that it is crucial to define the graph in terms of the identifier variable ID. Other regularizations do not perform nearly as well when trying to guard against adversarial domain shifts.
+
+![](images/424042f58e8c291575d11526e46731d963f19bd9ce10f72cb017452c554bca19.jpg)  
+Figure 3: a) Examples from the stickmen training set. The first three images from the left have $y \equiv c h i l d$ ; the remaining three images have $y \equiv a d u l t$ . Connected images are counterfactual examples. b) Misclassified observations from test set 2. c) Misclassification rates for $c = 5 0$ . Results for $c \in \{ \bar { 2 0 } , 5 0 0 , 2 0 0 0 \}$ can be found in Figure C.10.
+
+# 4.6 THEORETICAL RESULTS
+
+In $\ S \mathbf { A }$ we analyze the adversarial loss, defined in Eq. (2), for the pooled and the CORE estimator in a one-layer network for binary classification (logistic regression). Here, we briefly sketch the result while all details are given in $\ S \mathbf { A }$ . Assume the structural equation for the image $X \in \mathbb { R } ^ { p }$ is linear in the style features $\mathbf { \bar { A } } ^ { \perp } \in \mathbb { R } ^ { q }$ (with generally $p \gg q ,$ ), the interventions are additive and we use logistic regression to predict a class label $Y \in \{ - 1 , 1 \}$ . Under suitable assumptions (cf. Assumption 1), the pooled estimator has infinite adversarial loss while the adversarial loss of the CORE estimator converges to the optimal adversarial loss as $n \to \infty$ .
+
+# 5 EXPERIMENTS
+
+We perform an array of different experiments: in $\ S 5 . 1$ and $\ S 5 . 2$ we study how CORE can handle confounded training data sets and changing style features in test distributions. For the assessment we explicitly control the level of confounding. In $\ S 5 . 3$ , we consider classifying elephants and horses where $X ^ { \perp } \equiv c o l o r$ . In $\ S \mathbf { B }$ , we include two additional experiments: in the first one, $Y \equiv g e n d e r$ and $\begin{array} { r l } { X ^ { \bot } } & { { } \equiv } \end{array}$ wearing glasses; in the second one, $Y \equiv$ wearing glasses and $\begin{array} { r l } { X ^ { \bot } } & { { } \equiv } \end{array}$ brightness. Additional experimental results for the settings introduced in $\ S 2$ can be found in $\mathrm { \displaystyle \ S C } . 2$ and $\ S { \bf C } . 3$ . A TensorFlow (Abadi et al., 2015) implementation of CORE will be made available as well as further code necessary to reproduce the experiments. In addition to the details provided below, information on the employed architectures can be found in $\mathrm { \{ \ - \hbar \mathcal { C . 7 } } $ . An open question is how to set the value of the tuning parameter $\tau$ or the penalty $\lambda$ in Lagrangian form. We show in $\mathrm { \ S C . 1 }$ that performance is typically not very sensitive to the choice of $\lambda$ .
+
+# 5.1 STICKMEN IMAGE-BASED AGE CLASSIFICATION
+
+In this example we consider synthetically generated stickmen images (cf. Figure 3a). The target of interest is $Y \in \{ a d u l t , c h i \bar { l } d \}$ and $X ^ { \dot { c i } } \equiv h e i g h t$ . The class $Y$ is causal for height and height cannot be easily intervened on, so we consider it to be a core feature—it is a robust predictor for differentiating between children and adults. Additionally, there is a dependence between age and $\begin{array} { r l } { X ^ { \bot } } & { { } \equiv } \end{array}$ movement in the training dataset which arises through the hidden common cause $D \equiv$ place of observation. The data generating process is illustrated in Figure C.9. For instance, the images of children might mostly show children playing while the images of adults typically show them in more “static” postures. If the learned model exploits this dependence for predicting $Y$ , it will fail when presented images of, say, dancing adults.
+
+Figure 3a shows examples from the training set where large movements are associated with children and small movements are associated with adults. Test set 1 follows the same distribution. In test sets 2 and 3 $X ^ { \perp }$ is intervened on such that the edge from $D$ to $X ^ { \perp }$ is removed and the dependence between $Y$ and $X ^ { \perp }$ vanishes. In test sets 2 and 3 large movements are associated with both children and adults, while the movements are heavier in test set 3 than in test set 2. Figure C.10 shows examples from all test sets. Figure 3c shows misclassification rates for CORE and the pooled estimator for $c = 5 0$ with a total sample size of $m = 2 0 0 0 0$ . For as few as 50 counterfactual observations, CORE succeeds in achieving good predictive performance on test sets 2 and 3 where the pooled estimator fails (test errors $> 4 0 \%$ ). These results suggest that the learned representation of the pooled estimator uses movement as a predictor for age while CORE does not use this feature due to the counterfactual regularization. Importantly, including more counterfactual examples would not improve the performance of the pooled estimator as these would be subject to the same bias and hence also predominantly have examples of heavily moving children and “static” adults (also see Figure C.10 which shows results for $\bar { c ^ { \cdot } } \in \{ 2 0 , 5 0 0 , 2 0 0 0 \} )$ .
+
+![](images/c87df6c0af6f6f7d2710d03791aea64ae21fedcd4df62854e2ff4a137fdb7eb2.jpg)  
+Figure 4: a) Examples from the CelebA image quality dataset. The first three images from the left have $y \equiv n o$ glasses; the remaining three images have $y \equiv g l a s s e s$ . Connected images are counterfactual examples. b) Misclassified examples from the test sets. c) Misclassification rates for $\mu = 3 0$ and $c = 5 0 0 0$ . Results for different counterfactual settings and $\mu \in \{ 3 0 , 4 0 , 5 0 \}$ can be found in Figure C.12.
+
+# 5.2 EYEGLASSES DETECTION: IMAGE QUALITY INTERVENTION
+
+As in $\ S 2 . 1$ , we use the CelebA dataset and consider the problem of classifying whether the person in the image is wearing eyeglasses. Here, $X ^ { \perp }$ is the quality of the image which differs conditional on $Y ^ { 7 }$ —if the image shows a person wearing glasses, the image quality tends to be lower. This setting mimics the confounding that occurred in the Russian tank legend (cf. $\ S 1$ ). The strength of the image quality intervention is governed by sampling the new image quality as a percentage of the original image’s quality from a Gaussian distribution $\mathcal { N } ( \mu = 3 0 , \sigma = \bar { 1 } 0 )$ . Images of people without glasses are not changed. Thus, we only have counterfactual observations for $Y \equiv g l a s s e s$ . Figure 4a shows examples from the training set. Here, we use as the counterfactual observation the same image but with a newly sampled image quality value from $\mathcal { N } ( 3 0 , 1 0 )$ . We call using the same image as a counterfactual “CF setting $1 ^ { \circ }$ . Two alternatives for constructing counterfactual observations for this setting are discussed in $\ S \mathbf { B } . 2 . 1$ . Here, $c = 5 0 0 0$ and $m = 2 0 0 0 0$ .
+
+Figure 4c shows misclassification rates for CORE and the pooled estimator on different test sets. Examples from all test sets can be found in Figure C.11. Test set 1 follows the same distribution as the training set. In test set 2 the class of the quality intervention is reversed, i.e. the quality of images showing people without glasses tends to be lower. In test set 3 all images are left unchanged and in test set 4 the quality of all images is decreased. First, we notice that the pooled estimator performs better than CORE on test set 1. This can be explained by the fact that it can exploit the predictive information contained in an image’s quality while CORE is restricted not to do so. Second, we observe that the pooled estimator does not perform well on test sets 2–4 as its learned representation seems to use the image’s quality as a predictor for the target. In contrast, the predictive performance of CORE is hardly affected by the changing image quality distributions. More experimental details are provided in $\ S C . 5$ . Results for quality interventions of different strengths $( \mu \in \{ 3 0 , 4 0 , 5 0 \} )$ are shown in Figure C.12.
+
+![](images/f3f623c9f6127ecb1309a0e60502a320fb01b75351daef0b88f4792639b2f6cf.jpg)  
+Figure 5: a) Examples from the subsampled and augmented AwA2 dataset. The first three images from the left shows horses, the remaining three images show elephants. Connected images are counterfactual examples. b) Misclassified examples from the test sets. c) Misclassification rates.
+
+# 5.3 ELMER THE ELEPHANT
+
+In this example, we want to assess whether invariance with respect to $X ^ { \perp } \equiv c o l o r$ can be achieved. In the children’s book “Elmer the elephant”8 one instance of a colored elephant suffices to recognize it as being an elephant, making the color “gray” no longer an integral part of the object “elephant”. Motivated by this process of concept formation, we would like to assess whether CORE can exclude “color” from its learned representation by including a few counterfactuals of different color.
+
+We work with the “Animals with attributes $2 ^ { \circ }$ (AwA2) dataset (Xian et al., 2017) and consider classifying images of horses and elephants. The data generating process is illustrated in Figure C.14. We include counterfactual examples by adding grayscale images for $c = 2 5 0$ images of elephants, i.e. counterfactuals are only available for one class and the shift in color is quite subtle. The total sample size is 1850.
+
+Figure 5a shows examples from the training set and Figure 5c shows misclassification rates for CORE and the pooled estimator on different test sets. Examples from all test sets can be found in Figure C.13. Test set 1 contains original, colored images only. In test set 2 images of horses are in grayscale and the colorspace of elephant images is modified, effectively changing the color gray to red-brown. Test set 3 contains grayscale images only and in test set 4 the colorspace of all images is shifted towards red. The details are given in $\mathrm { \ S C } . 6$ . We observe that the pooled estimator does not perform well on test sets 2 and 3 as its learned representation seems to exploit the fact that “gray” is predictive for the target in the training set. Using this information helps its predictive accuracy on test set 1. In contrast, the predictive performance of CORE is hardly affected by the changing color distributions.
+
+It is noteworthy that a colored elephant can be recognized as an elephant by adding a few examples of a grayscale elephant to the very lightly colored pictures of natural elephants. If we just pool over these examples, there is still a strong bias that elephants are gray. The CORE estimator, in contrast, demands invariance of the prediction for instances of the same elephant and we can learn color invariance with a few added grayscale images.
+
+While a thorough analysis in terms of fairness considerations is beyond the scope of this work, we would like to draw the following connection. If “color” was a protected attribute or a proxy for one, CORE would satisfy fairness in the sense that it would not include it in its learned representation. In contrast, there is no way to avoid that the pooled estimator extracts and uses “color” for its decisions.
+
+# 6 CONCLUSION
+
+Distinguishing the latent features in an image into core and style features, we have proposed counterfactual regularization (CORE) to achieve robustness with respect to arbitrarily large interventions on the style or conditionally invariant features. The main idea of the CORE estimator is to exploit the fact that we often have instances of the same object in the training data. By demanding invariance of the classifier amongst a group of instances that relate to the same object, we can achieve invariance of the classification performance with respect to adversarial interventions on style features such as image quality, fashion type, color, or body posture. The training also works despite sampling biases in the data.
+
+There are two main applications areas. If the style features are known explicitly, we can achieve the same classification performance as standard data augmentation approaches but using fewer instances which, on top, do not have to be carefully balanced in the training data. Perhaps more interestingly, if the style features are unknown, the regularization of CORE avoids usage of them automatically by penalizing features that vary strongly between different instances of the same object in the training data.
+
+An interesting line of work would be to use larger models such as Inception or large ResNet architectures (Szegedy et al., 2015; He et al., 2016). These models have been trained to be invariant to an array of explicitly defined style features. In $\mathrm { \ S B . l }$ we include results which show that using Inception V3 features does not guard against interventions on more implicit style features. We would thus like to assess what benefits CORE can bring for training Inception-style models end-to-end, both in terms of sample efficiency and in terms of generalization performance.
+
+While we showed some examples where the necessary grouping information is available, an interesting possible future direction would be to use video data since objects display temporal constancy and the temporal information can hence be used for grouping and counterfactual regularization. Potentially an analogous approach could also help to debias word embeddings.
+
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+Ted Sandler, John Blitzer, Partha P Talukdar, and Lyle H Ungar. Regularized learning with networks of features. In Advances in neural information processing systems, pp. 1401–1408, 2009.   
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+
+# SUPPLEMENTARY MATERIAL
+
+# A LOGISTIC REGRESSION
+
+Assume the structural equation for the image $X \in \mathbb { R } ^ { p }$ is linear in the style features $X ^ { \bot } \in \mathbb { R } ^ { q }$ (with generally $p \gg q$ ) and we use logistic regression to predict a class label $Y \in \{ - 1 , 1 \}$ . Let the interventions $\Delta \in \mathbb { R } ^ { q }$ act additively on the style features $X ^ { \perp }$ (this is only for notational convenience) and let the style features $X ^ { \perp }$ act in a linear way on the image $X$ via a matrix $W \in \mathbb { R } ^ { p \times q }$ (this is an important assumption without which results are more involved). The core or ‘conditionally invariant’ features are $\bar { \boldsymbol X } ^ { c i } \in \mathbb { R } ^ { r }$ , where in general $r \le p$ but this is not important for the following. For independent $\varepsilon _ { Y } , \varepsilon _ { \mathrm { I D } } , \varepsilon _ { X ^ { \perp } } , \varepsilon _ { X }$ in $\mathbb { R } , \mathbb { R } ^ { q } , \mathbb { R } ^ { r } , \mathbb { R } ^ { p }$ respectively with positive density on their support and continuously differentiable functions $k _ { y } , k _ { \mathrm { I D } } , k _ { X ^ { \perp } } , k _ { X ^ { c i } } , k _ { x }$ ,
+
+Of these, $Y , X$ and ID are observed whereas $D , X ^ { c i } , \Delta , X ^ { \perp }$ and the noise variables are latent.
+
+We assume a logistic regression as a prediction of $Y$ from the image data $X$ :
+
+$$
+f _ { \theta } ( x ) : = \frac { \exp ( x ^ { t } \theta ) } { 1 + \exp ( x ^ { t } \theta ) } .
+$$
+
+Given training data with $m$ samples, we estimate $\theta$ with $\hat { \theta }$ and use here a logistic loss $\ell _ { \theta } ( y _ { i } , x _ { i } ) =$ $\log ( 1 + \exp ( - y _ { i } ( x _ { i } ^ { t } \theta ) ) )$ for training and testing. Some interesting expected losses on test data include
+
+$$
+\begin{array} { r l } & { \qquad L ( \theta ) = E \Bigl [ \ell \bigl ( Y , f _ { \theta } ( X ) ) \bigr ) \Bigr ] } \\ & { \qquad L _ { a d v } ( \theta ) = E \Bigl [ \underset { \Delta \in \mathbb { R } ^ { q } } { \operatorname* { m a x } } \ell \bigl ( Y , f _ { \theta } ( X ( \Delta ) ) \bigr ) \Bigr ] , } \end{array}
+$$
+
+where the $X$ in the first loss is a shorthand notation for $X ( \Delta = 0 )$ , that is the images in absence of interventions on the style variables. The first loss is thus a standard logistic loss in absence of adversarial interventions. The second loss is the loss under adversarial style or domain interventions as we allow arbitrarily large interventions on $X ^ { \perp }$ here. The corresponding benchmarks are
+
+$$
+L ^ { * } = \operatorname* { m i n } _ { \theta } L ( \theta ) , { \mathrm { ~ a n d ~ } } L _ { a d v } ^ { * } = \operatorname* { m i n } _ { \theta } L _ { a d v } ( \theta ) .
+$$
+
+The formulation of Theorem 1 relies on the following assumptions.
+
+Assumption 1. We require the following conditions to hold:
+
+(A1) Assume $\Delta$ is sampled from a distribution for training data in $\mathbb { R } ^ { q }$ with positive density on an $\epsilon$ -ball in $\ell _ { 2 }$ -norm around the origin for some $\epsilon > 0$ .   
+(A2) Assume the matrix $W$ has full rank $q$ .   
+(A3) Assume $c \geq q$ , that is the number $c = m - n$ of counterfactual examples in the samples is at least as large as the dimension of the style variables.
+
+Regarding (A3): the sampling process is as follows. We collect $n$ independent samples $\left( y _ { i } , \mathrm { i d } _ { i } , \delta _ { i , 1 } \right)$ from a distribution of $( Y , \mathrm { I D } , \Delta )$ that satisfies the constraints above. Then, for $c = m - n$ of the samples we select each time $i \in \{ 1 , \ldots , n \}$ at random, keep $( y _ { i } , \mathrm { i d } _ { i } )$ fixed (and hence also the realization of $X ^ { \perp }$ is fixed) and redraw a new value of $\Delta$ as $\delta _ { i , u _ { i } + 1 }$ if $u _ { i }$ is the current number of counterfactual examples for sample $i$ . This leads to $m$ samples in total with in general $n$ distinct values of $( y _ { i } , \mathrm { i d } _ { i } )$ and $m _ { i }$ counterfactuals at each sample with corresponding $x _ { i , j }$ with $i \in \{ 1 , \ldots , n \}$ and $j \in \{ 1 , \dots , m _ { i } \}$ .
+
+Theorem 1. Under Assumption $^ { l }$ , with probability $^ { l }$ with respect to the training data, the pooled estimator has infinite adversarial loss
+
+$$
+L _ { a d v } ( \hat { \theta } ^ { p o o l } ) = \infty .
+$$
+
+For the CORE estimator, for $n \to \infty$ ,
+
+$$
+L _ { a d v } ( \hat { \theta } ^ { c o r e } )  _ { p } L _ { a d v } ^ { * } .
+$$
+
+An equivalent results can be derived for misclassification loss instead of logistic loss (with infinity replaced by 1).
+
+Proof. First part. To show the first part, namely that with probability 1,
+
+$$
+L _ { a d v } ( \hat { \theta } ^ { p o o l } ) = \infty ,
+$$
+
+we need to show that $W ^ { t } \hat { \theta } ^ { p o o l } \neq 0$ with probability 1. The reason this is sufficient is as follows: if $W ^ { t } \theta \neq 0$ , then $L _ { a d v } ( \theta ) = \infty$ as we can then find a $v \in \mathbb { R } ^ { q }$ such that $\gamma : = \theta ^ { t } W v \neq 0$ . Setting $\Delta _ { \kappa } = \kappa v$ for $\kappa \in \mathbb { R }$ , we get $x ( \Delta _ { \kappa } ) ^ { t } \theta = x ( \Delta = 0 ) ^ { t } \theta + \kappa \gamma$ . Hence $\log ( 1 + \exp ( - x ( \Delta _ { \kappa } ) ^ { t } \theta ) ) \to \infty$ for either $\kappa  \infty$ or $\kappa  - \infty$ .
+
+To show that $W ^ { t } \hat { \theta } ^ { p o o l } \neq 0$ with probability 1, let ${ \hat { \theta } } ^ { * }$ be the oracle estimator that is constrained to be orthogonal to the column space of $W$ :
+
+$$
+\hat { \theta } ^ { * } = \mathrm { a r g m i n } _ { \theta : W ^ { t } \theta = 0 } L _ { n } ( \theta ) \mathrm { w i t h } L _ { n } ( \theta ) : = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \ell ( y _ { i } , f _ { \theta } ( x _ { i } ( \Delta _ { i } ) ) ) .
+$$
+
+We show $W ^ { t } \hat { \theta } ^ { p o o l } \neq 0$ by contradiction. Assume hence that $W ^ { t } \hat { \theta } ^ { p o o l } = 0$ . If this is indeed the case, then the constraint $W ^ { t } \theta = 0$ in (6) becomes non-active and we have $\hat { \theta } ^ { p o o l } = \hat { \theta } ^ { * }$ . This would imply that taking the directional derivative of the training loss with respect to any $\delta \in \mathbb { R } ^ { p }$ in the column space of $W$ should vanish at the solution ${ \hat { \theta } } ^ { * }$ . Define $r _ { i } ( \theta ) : = ( y _ { i } + 1 ) / 2 - f _ { \hat { \theta } ^ { * } }$ . For all $i = 1 , \ldots , n$ we have $r _ { i } \neq 0$ . The derivative $g ( \delta )$ of $L _ { n } ( \theta )$ in direction of $\delta$ is proportional to
+
+$$
+g ( \delta ) = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } r _ { i } ( \hat { \theta } ^ { * } ) \sum _ { j = 1 } ^ { m _ { i } } x _ { i , j } ^ { t } \delta ,
+$$
+
+$x _ { i , j } \in \mathbb { R } ^ { p }$ is the $j$ -th counterfactual for training sample $i$ (with $j \in \{ 1 , \dots , m _ { i } \} )$ . Let $x _ { i , j } ( 0 ) =$ $x _ { i , 1 } ( 0 )$ for $i = 1 , \ldots , n$ be the counterfactual training data in absence of any interventions $( \Delta _ { i , j } =$ 0). Since the interventions only have an effect on the column space of $W$ in $X$ , the oracle estimator ${ \hat { \theta } } ^ { * }$ is identical under the true training data and the counterfactual training data $x ( 0 )$ . Hence, for any $\delta$ in $\mathbb { R } ^ { p }$ , the derivative $g ( \delta )$ in (7) can also be written as
+
+$$
+g ( \delta ) = \frac { 1 } { n } \sum _ { i = 1 } ^ { n } r _ { i } ( \hat { \theta } ^ { * } ) \sum _ { j = 1 } ^ { m _ { i } } x _ { i , k } ( 0 ) ^ { t } \delta .
+$$
+
+Taking the difference between (7) and (8),
+
+$$
+{ \frac { 1 } { n } } \sum _ { i = 1 } ^ { n } r _ { i } ( { \hat { \theta } } ^ { * } ) ( \sum _ { j = 1 } ^ { m _ { i } } ( x _ { i , j } - x _ { i , j } ( 0 ) ) ^ { t } \delta ) = 0 .
+$$
+
+Now, by the model assumptions, $x _ { i , j } - x _ { i , j } ( 0 ) = W \Delta _ { i , j }$ . Since $\delta$ is in the column-space of $W$ , there exists $u \in \mathbb { R } ^ { q }$ such that $\delta = W u$ . then (9) can be written as
+
+$$
+\frac { 1 } { n } \sum _ { i = 1 } ^ { n } r _ { i } ( \hat { \theta } ^ { * } ) \sum _ { j = 1 } ^ { m _ { i } } \Delta _ { i , j } ^ { t } W ^ { t } W u = 0
+$$
+
+From (A2) we have that the eigenvalues of $W ^ { t } W$ are all positive. Also $r _ { i } (  { \hat { \theta } } ^ { * } )$ is not a function of the interventions $\Delta _ { i , j }$ since, as already argued above, the estimator ${ \hat { \theta } } ^ { * }$ is identical whether trained on the original data $x _ { i , j }$ or on the counterfactual data $x _ { i , j } ( 0 )$ . If we condition on $( x _ { i } ( 0 ) , y _ { i } )$ for $i = 1 , \ldots , n$ (that is everything except for the random $\Delta _ { i , j }$ , $i = 1 , \ldots , n )$ , then the interventions $\Delta _ { i , j }$ are by (A1) drawn from a continuous distribution. Hence the left hand side of (10) has a continuous distribution, and the probability of the left hand side of (10) being not identically 0 is 1. This completes the proof of the first part by contradiction.
+
+Second part. For the second part, we first show that with probability 1, $\hat { \theta } ^ { c o r e } = \hat { \theta } ^ { * }$ with ${ \hat { \theta } } ^ { * }$ defined as in (6). Note that the invariant space is for this model the linear subspace $I = \{ \theta : W ^ { t } \theta = 0 \}$ . Note that by their respective definitions,
+
+$$
+\begin{array} { r l r } & { } & { \hat { \theta } ^ { * } = \operatorname * { a r g m i n } _ { \theta } \ \frac { 1 } { m } \displaystyle \sum _ { i = 1 } ^ { n } \sum _ { j = 1 } ^ { m _ { i } } \ell ( y _ { i } , f _ { \theta } ( x _ { i , j } ) ) \ \mathrm { s u c h \ t h a t } \ \theta \in I , } \\ & { } & { \hat { \theta } ^ { c o r e } = \operatorname * { a r g m i n } _ { \theta } \ \frac { 1 } { m } \displaystyle \sum _ { i = 1 } ^ { n } \sum _ { j = 1 } ^ { m _ { i } } \ell ( y _ { i } , f _ { \theta } ( x _ { i , j } ) ) \ \mathrm { s u c h \ t h a t } \ \theta \in I _ { n } . } \end{array}
+$$
+
+By (A2) and (A3), with probability 1, $I _ { n } = \{ \theta : W ^ { t } \theta = 0 \}$ since the number of counterfactuals examples is equal to or exceeds the rank $q$ of $W$ and $X ^ { \perp }$ has a linear influence on $X$ . Hence with probability 1, we have $I = I _ { n }$ and hence $\hat { \theta } ^ { c o r e } = \hat { \theta } ^ { * }$ . We thus need to show that
+
+$$
+L _ { a d v } ( \hat { \theta } ^ { * } )  _ { p } L _ { a d v } ^ { * } .
+$$
+
+Since ${ \hat { \theta } } ^ { * }$ is in $I$ , we have $\ell ( y , x ( \Delta ) ) = \ell ( y , x ( 0 ) )$ , where $x ( 0 )$ are the previously discussed counterfactual data in the absence of interventions. Hence
+
+$$
+\hat { \theta } ^ { * } = \operatorname * { a r g m i n } _ { \theta } \ \frac { 1 } { m } \sum _ { i = 1 } ^ { n } \sum _ { j = 1 } ^ { m _ { i } } \ell ( y _ { i } , f _ { \theta } ( x _ { i , j } ( 0 ) ) ) \ \mathrm { s u c h ~ t h a t } \theta \in I ,
+$$
+
+that is the estimator is unchanged if we use the data without interventions $\Delta _ { i } = 0$ ) as training data. Define the population-optimal vector as
+
+$$
+\theta ^ { * } = \operatorname { a r g m i n } _ { \theta } E \big [ \operatorname* { m a x } _ { \Delta } \ell ( Y , f _ { \theta } ( X ( \Delta ) ) ) \big ] \ \mathrm { s u c h \ t h a t } \ \theta \in I ,
+$$
+
+which can for the same reason be written as
+
+$$
+\theta ^ { * } = \operatorname { a r g m i n } _ { \theta } E \left[ \ell ( Y , f _ { \theta } ( X ( \Delta = 0 ) ) ) \right] \mathrm { { s u c h t h a t } } \theta \in I .
+$$
+
+Hence (12) and (13) can be written as
+
+$$
+\begin{array} { r l } & { \hat { \theta } ^ { * } = \operatorname * { a r g m i n } _ { \theta \colon \theta \in I } L _ { n } ^ { ( 0 ) } ( \theta ) \mathrm { ~ w h e r e ~ } L _ { n } ^ { ( 0 ) } ( \theta ) : = \displaystyle \frac { 1 } { m } \sum _ { i = 1 } ^ { n } \sum _ { j = 1 } ^ { m _ { i } } \ell ( y _ { i } , f _ { \theta } ( x _ { i , j } ( 0 ) ) ) } \\ & { \theta ^ { * } = \operatorname * { a r g m i n } _ { \theta \colon \theta \in I } L ^ { ( 0 ) } ( \theta ) \mathrm { ~ w h e r e ~ } L ^ { ( 0 ) } ( \theta ) : = E [ \ell ( Y , f _ { \theta } ( X ( \Delta = 0 ) ) ) ] . } \end{array}
+$$
+
+Comparing (12) and (13), by uniform convergence of $L _ { n } ^ { ( 0 ) }$ to the population loss $L ^ { ( 0 ) }$ under the assumed sampling where $n$ samples of $( Y , \mathrm { I D } )$ are drawn independently then $c = m - n$ samples are redrawn from this empirical sample at random, we have $L ^ { ( 0 ) } ( { \hat { \theta } } ^ { * } ) \to _ { p } L ^ { ( 0 ) } ( \theta ^ { * } )$ .
+
+By definition of $I$ and $\theta ^ { * }$ we have $L _ { a d v } ^ { * } = L _ { a d v } ( \theta ^ { * } ) = L ^ { ( 0 ) } ( \theta ^ { * } )$ . As ${ \hat { \theta } } ^ { * }$ is in $I$ , we also have ${ \cal L } _ { a d v } ( \hat { \theta } ^ { * } ) = { \cal L } ^ { ( 0 ) } ( \hat { \theta } ^ { * } )$ . Since, from above, $L ^ { ( 0 ) } ( { \hat { \theta } } ^ { * } ) \to _ { p } L ^ { ( 0 ) } ( \theta ^ { * } )$ , this also implies $L _ { a d v } ( { \hat { \theta } } ^ { * } ) \to _ { p }$ $L _ { a d v } ( \theta ^ { * } ) = L _ { a d v } ^ { * }$ . This completes the proof, using the previous fact that $\hat { \theta } ^ { c o r e } = \hat { \theta } ^ { * }$ with probability 1 under (A3).
+
+# B ADDITIONAL EXPERIMENTS
+
+# B.1 GENDER CLASSIFICATION
+
+We work with the CelebA dataset (Liu et al., 2015) and consider the problem of classifying whether the person in the image is male or female. We create a confounding by including mostly images of men wearing glasses while the images of women do not include photos of women with glasses. As counterfactuals, we use an image of the same person without glasses if the person is male and with glasses if the person is female. We call using an image of the same person as counterfactual “CF setting 2”. Examples from the training and test sets are shown in Figure B.2. Test set 1 follows the same distribution as the training set. In test set 2 the association between gender and glasses is flipped: women always wear glasses while men never wear glasses.
+
+![](images/a13f32367c0487d116c0ffc11abaf9d92b2c009025197f57641f07fa81a22834.jpg)  
+Figure B.1: Examples from the CelebA gender dataset.
+
+In this example, we would like to assess whether the results will differ when (a) training a fourlayer CNN (as detailed in Table C.1) end-to-end versus (b) using Inception V3 features and merely retraining the softmax layer. Figure B.2 shows the results for varying numbers of $m$ and $c$ —in the left column for training a four-layer CNN; in the right column for using Inception V3 features. Overall, we see the same trends: As $c$ increases, the performance difference between CORE and the pooled estimator becomes smaller. This is due to the fact that $X ^ { \perp }$ is binary in this example and, therefore, including counterfactual examples corresponds to data augmentation. Interestingly, the pooled estimator performs worse on test set 2 as $m$ becomes larger. It thus seems to exploit $\dot { X } ^ { \perp }$ to a larger extent as $m$ grows.
+
+# B.2 EYEGLASSES DETECTION: BRIGHTNESS INTERVENTION
+
+As in $\ S 5 . 2$ we work with the CelebA dataset and consider the problem of classifying whether the person in the image is wearing eyeglasses. Here we analyze a confounded setting that could arise as follows. Say the hidden common cause of $Y$ and $X ^ { \perp }$ , $D$ indicates whether the image was taken outdoors or indoors. If it was taken outdoors, then the person wears glasses and the image tends to be brighter. If the image was taken indoors, then the person does not wear glasses and the image tends to be darker. In other words, $X ^ { \perp } \equiv .$ brightness and the structure of the data generating process is equivalent to the one shown in Figure C.9. Figure B.3a shows examples from the training set. Here, we use as the counterfactual observation the same image (CF setting 1) but with a different brightness. Two alternatives for constructing counterfactual observations in this setting are discussed in $\ S \mathbf { B } . 2 . 1$ . We use $c = 2 0 0 0$ and $m = 2 0 0 0 0$ .
+
+For the brightness intervention, we sample the value for the magnitude of the brightness increase resp. decrease from an exponential distribution with mean $\beta = 2 0$ . Specifically, we use ImageMag$\mathrm { i c k } ^ { \mathrm { \scriptsize 9 } }$ to modify the brightness of each image. In the training set and test set 1, we sample the brightness value as $b _ { i , j } = 1 0 0 + y _ { i } e _ { i , j }$ where $e _ { i , j } \sim E x p ( \beta ^ { - \bar { 1 } } )$ and $y _ { i } \in \{ - 1 , 1 \}$ . $y _ { i } = 1$ corresponds to $y _ { i } \equiv g l a s s e s$ . We then apply the command convert -modulate b ij, $^ { 1 0 0 , 1 0 0 }$ input.jpg output.jpg to the image. Importantly, since we sample from an exponential distribution, the brightness interventions are quite subtle in many cases as can be seen in Figure B.3a.
+
+Figure B.3c shows misclassification rates for CORE and the pooled estimator on different test sets. Examples from all test sets can be found in Figure B.4. Test set 1 follows the same distribution as the training set. In test set 2 the sign of the brightness intervention is reversed, i.e. images of people with glasses tend to be darker; images of people without glasses tend to be brighter. In test set 3 all images are left unchanged and in test set 4 the brightness of all images is increased. First, we notice that the pooled estimator performs better than CORE on test set 1. This can be explained by the fact that it can exploit the predictive information contained in the brightness of an image while CORE is restricted not to do so. Second, we observe that the pooled estimator does not perform well on test sets 2 and 4 as its learned representation seems to use the image’s brightness as a predictor for the response which fails when the brightness distribution in the test set differs significantly from the training set. In contrast, the predictive performance of CORE is hardly affected by the changing brightness distributions. Results for $\beta \in \{ 5 , 1 0 , 2 0 \}$ and $c \in \{ 2 0 0 , 5 0 0 0 \}$ can be found in Figure B.5.
+
+![](images/7a62ea5acb75aa24137a62fbeed37566820efbb3b793133489f0443118d7d94e.jpg)  
+Figure B.2: Misclassification rates for the CelebA gender datasets with varying numbers for $m$ and $c$ . The left column shows results for training a four-layer CNN (cf. Table C.1) end-to-end, the right column shows results for using Inception V3 features and retraining the softmax layer.
+
+![](images/d16e6f39caf4bbec0a6a5f5d4cc670f751d20a0a246893e2e34dc69efaf115c2.jpg)  
+Figure B.3: a) Examples from the CelebA brightness dataset. The first three images from the left have $y \equiv n o$ glasses; the remaining three images have $y \equiv g l a s s e s$ . Connected images are counterfactual examples. b) Misclassified examples from the test sets. c) Misclassification rates for $\beta = 2 0$ and $c = 2 0 0 0$ . Results for different counterfactual settings, $\beta \in \{ 5 , 1 0 , 2 0 \}$ and $c \in \{ 2 0 0 , 5 0 0 0 \}$ can be found in Figure B.5.
+
+# B.2.1 COUNTERFACTUAL SETTINGS 2 AND 3
+
+Above we used the same image to create a counterfactual observation by sampling a different value for the brightness intervention. A plausible alternative is to use a different image of the same person as counterfactual. We call this “CF setting $2 ^ { \circ }$ . For comparison, we also evaluate using an image of a different person as counterfactual as a baseline (“CF setting $3 ^ { \circ }$ ). Examples from the training sets using CF setting 2 and 3 can be found in Figure B.4.
+
+Results for all counterfactual settings, $\beta \in \{ 5 , 1 0 , 2 0 \}$ and $c \in \{ 2 0 0 , 5 0 0 0 \}$ can be found in Figure B.5. We see that using counterfactual setting 1 works best since we could explicitly control that only $X ^ { \perp } \equiv$ brightness varies between counterfactual examples. In counterfactual setting 2, different images of the same person can vary in many factors, making it more challenging to isolate brightness as the factor to be invariant against. Lastly, we see that even grouping images of different persons can still help predictive performance to some degree.
+
+# C EXPERIMENTAL DETAILS AND ADDITIONAL RESULTS FOR EXPERIMENTS INTRODUCED IN §2 AND §5
+
+# C.1 CHOOSING THE TUNING PARAMETER $\lambda$
+
+An open question is how to set the value of the tuning parameter $\tau$ in Eq. (4) or the penalty $\lambda$ in the Lagrangian form. Figure C.6 shows the misclassification rates of CORE on the subsampled and augmented AwA2 dataset as a function of the penalty $\lambda$ . We see that performance is not very sensitive to the choice of $\lambda$ .
+
+# .2 GROUPING PHOTOS OF THE SAME PERSON: BETTER PREDICTIVE PERFORMAN
+
+Here, we show further results for the experiment introduced in $\ S 2 . 1$ . We vary the number of identities included in the training data set $n \in \{ 1 0 , 2 0 , 4 0 , 8 0 , 1 6 0 \}$ . This results in total sample sizes $m$ ranging from 321 for $n = 1 0$ to 4386 for $n = 1 6 0$ , implying that the average number of counterfactual observations per person varies between 27 and 32. Figure ${ \mathrm { C . 7 b } }$ shows the misclassification rates for the test set which consists of 5000 examples. We see that CORE helps predictive performance compared to the estimator which just pools all images, notably when $n$ is very small. It thus successfully mitigates the effect of potential confounders arising due to small sample sizes. As $n$ and $m$ increase the performance of CORE and the pooled estimator become comparable—the larger sample sizes ensure that fewer confounding factors are present in the training data and exploited by the pooled estimator.
+
+![](images/6fb23cf7597ce0bfd497baca22426a641baef6f9bc618d2b2178e4f022941491.jpg)  
+Figure B.4: Examples from the CelebA brightness datasets, counterfactual settings 1–3 with $\beta \in \{ 5 , 1 0 , 2 0 \}$ . In all rows, the first three images from the left have $y \equiv n o$ glasses; the remaining three images have $y \equiv$ glasses. Connected images are counterfactual examples. In panels (a)–(c), row 1 shows examples from the training set, rows 2–4 contain examples from test sets 2–4, respectively. Panels (d)–(i) show examples from the respective training sets.
+
+![](images/46e82328bf8065a71777f633a5c0c0720b996241c71a6addc6d310cf0f853153.jpg)  
+Figure B.5: Misclassification rates for the CelebA brightness datasets, counterfactual settings 1–3 with $c \in$ $\{ 2 0 0 , 2 0 0 0 , 5 0 0 0 \}$ and the mean of the exponential distribution $\beta \in \{ 5 , 1 0 , 2 0 \}$ .
+
+![](images/6b32a20d90e1783ad0b6b30717558e2450097912c52083e4a7a41f6a76d45528.jpg)  
+Figure C.6: Misclassification rates of CORE on the subsampled and augmented AwA2 dataset as a function of the penalty $\lambda$ . The outcome does not depend strongly on the chosen value.
+
+(a) Training examples, grouped by identity.
+
+![](images/ee4ac5751f587d50c9f8315fb61544cc4bde63f0bd97fa249da7deecdf0c7564.jpg)  
+Figure C.7: a) Examples from the subsampled CelebA dataset. In each row, the first three images from the left have $y \equiv n o$ glasses; the remaining three images have $y \equiv g l a s s e s$ . Connected images are counterfactual examples. b) Misclassification rates for different numbers of identities, included in the training data.
+
+![](images/4ef8e29ff893328954129682c5011e461c940666e8fd2abc1afc62a82f128e8b.jpg)  
+Figure C.8: Data augmentation setting: Misclassification rates for MNIST and $S \equiv r o t a t i o n$ . In test set 1 all digits are rotated by a degree randomly sampled from [35, 70]. Test set 2 is the usual MNIST test set.
+
+![](images/d17d5ef3f7e0a2b80252731d113ac9911a85a5c325b0faf2ae4dd6d19f3ba48c.jpg)  
+Figure C.10: a) Examples from the stickmen test set 1 (row 1), test set 2 (row 2) and test sets 3 (row 3). In each row, the first three images from the left have $y \equiv c h i l d$ ; the remaining three images have $y \equiv a d u l t$ . Connected images are counterfactual examples. b) Misclassification rates for different numbers of counterfactual examples.
+
+# C.4 STICKMEN IMAGE-BASED AGE CLASSIFICATION
+
+![](images/4c5fa35fe40db3bacec300cc709251e0a4e4a044f907b839bf20a83e1e3f815b.jpg)  
+Figure C.9: Data generating process for the stickmen example.
+
+Here, we show further results for the experiment introduced in $\ S 5 . 1$ . Figure C.10b shows results for different numbers of counterfactual examples. For $c = 2 0$ the misclassification rate of CORE estimator has a large variance. For $c \in \{ 5 0 , 5 0 0 , 2 0 0 0 \}$ , the CORE estimator shows similar results. Its performance is thus not sensitive to the number of counterfactual examples, once there are sufficiently many counterfactual observations in the training set. The pooled estimator fails to achieve good predictive performance on test sets 2 and 3 as it seems to use “movement” as a predictor for “age”.
+
+# C.5 EYEGLASSES DETECTION: IMAGE QUALITY INTERVENTION
+
+Here, we show further results for the experiment introduced in $\ S 5 . 2$ . Specifically, we consider interventions of different strengths by varying the mean of the quality intervention in $\mu \in \{ 3 0 , 4 0 , 5 0 \}$ . As in $\mathrm { \ S B } . 2$ , we use ImageMagick, this time to modify the image quality. In the training set and in test set 1, we sample the image quality value as $q _ { i , j } \sim \mathcal { N } ( \mu , \overset { \cdot } { \sigma } = 1 0 \overset { \cdot } { ) }$ and apply the command convert -quality q ij input.jpg output.jpg if $y _ { i } \equiv g l a s s e s$ . If $y _ { i } \equiv n o$ glasses, the image is not modified. In test set 2, the above command is applied if $y _ { i } \equiv n o$ glasses while images with $y _ { i } \equiv g l a s s e s$ are not changed. In test set 3 all images are left unchanged and in test set 4 the command is applied to all images, i.e. the quality of all images is reduced.
+
+We run experiments for counterfactual settings 1–3 and for $c = 5 0 0 0$ . Figure C.11 shows examples from the respective training and test sets and Figure C.12 shows the corresponding misclassification rates. Again, we observe that counterfactual setting 1 works best while there are only small differences in predictive performance between counterfactual settings 2 and 3. Interestingly, there is a large performance difference between $\mu = 4 0$ and $\mu = 5 0$ for the pooled estimator. Possibly, with $\mu = 5 0$ the image quality is not sufficiently predictive for the target.
+
+![](images/5f2946bdfa2e233a04f7b8e9bee68362bd4382f711992275bfb61f9674f1965f.jpg)  
+Figure C.11: Examples from the CelebA image quality datasets, counterfactual settings 1–3 with $\mu ~ \in$ $\{ 3 0 , 4 0 , 5 0 \}$ . In all rows, the first three images from the left have $y \equiv n o$ glasses; the remaining three images have $y \equiv g l a s s e s$ . Connected images are counterfactual examples. In panels (a)–(c), row 1 shows examples from the training set, rows 2–4 contain examples from test sets 2–4, respectively. Panels (d)–(i) show examples from the respective training sets.
+
+![](images/6d8d3765314d0a31ea31ac87ae0d7876cc86ea63abc4e7c10386bd3c0a8f807b.jpg)  
+Figure C.12: Misclassification rates for the CelebA image quality datasets, counterfactual settings 1–3 with $c = 5 0 0 0$ and the mean of the Gaussian distribution $\mu \in \{ 3 0 , 4 0 , 5 0 \}$ .
+
+# C.6 ELMER THE ELEPHANT
+
+The color interventions for the experiment introduced in $\ S 5 . 3$ are created as follows. In the training set, if $y _ { i } \equiv$ elephant we apply the following ImageMagick command only for the counterfactual examples convert -modulate $1 0 0 , 0 , 1 0 0$ input.jpg output.jpg, producing a grayscale image. In test set 1, all images are left unchanged. In test set 2, the above command is applied if $y _ { i } \equiv h o r s e$ ; if $y _ { i } \equiv$ elephant we sample $c _ { i , j } \sim \mathrm { \bar { \mathcal { N } } } ( \mu = 2 0 , \sigma = 1 )$ and apply convert -modulate $^ { 1 0 0 , 1 0 0 , 1 0 0 - \mathtt { C } _ { - } \mathtt { i } \mathtt { j } }$ input.jpg output.jpg to the image. In test set 4, the latter command is applied to all images. It rotates the colors of the image, in a cyclic manner10. In test set 3, all images are changed to grayscale.
+
+![](images/a3ed161efa8c79e208b50a7a239a3dc2954fcc1872197bb65b2f2b9c10c0cbd0.jpg)  
+Figure C.13: Examples from the subsampled and augmented AwA2 dataset. Row 1 shows examples from the training set, rows 2–5 show examples from test sets 1–4, respectively.
+
+![](images/c2ddfa99cc55938f7a9817fc6b4e78250561212fc699556819813dac2971cdbd.jpg)  
+Figure C.14: Data generating process for the Elmer the elephant example.
+
+# C.7 NETWORK ARCHITECTURES
+
+We implemented the considered models in TensorFlow (Abadi et al., 2015). The model architectures used are detailed in Table C.1. CORE and the pooled estimator thus use the same network architecture and training procedure; merely the loss function differs by the counterfactual regularization term. In all experiments we use the Adam optimizer (Kingma & Ba, 2015).
+
+All experimental results are based on training the respective model five times (using the same data) to assess the variance due to the randomness in the training procedure.
+
+In each epoch of the training, the training data $x _ { i , \cdot } , i = 1 , \ldots , n$ is randomly shuffled, keeping the counterfactual observations $x _ { i , j } , j = 1 , \dotsc , m _ { i }$ together to ensure that mini batches will contain counterfactual observations. In all experiments the mini batch size is set to 120. For small $c$ this implies that not all mini batches contain counterfactual observations, making the optimization more challenging.
+
+Table C.1: Details of the model architectures used.   
+
+<table><tr><td>Dataset</td><td>Optimizer</td><td>Architecture</td><td></td></tr><tr><td>MNIST</td><td>Adam</td><td>Input CNN</td><td>28×28×1 Conv5×5×16,5×5×32</td></tr><tr><td>Stickmen</td><td>Adam</td><td>Input CNN</td><td>(same padding,strides= 2,ReLu activation), fully connected, softmax layer 64×64×1 Conv5×5×16,5×5×32,5×5×64,5×5×128</td></tr><tr><td>CelebA (all experiments</td><td>Adam</td><td>Input CNN</td><td>(same padding,strides = 2, leaky ReLu activation), fully connected, softmax layer 64×48×3 Conv5×5×16,5×5×32,5×5×64,5×5×128</td></tr><tr><td>using CelebA) AwA2</td><td>Adam</td><td>Input</td><td>(same padding,strides = 2,leaky ReLu activation), fully connected, softmax layer</td></tr><tr><td></td><td></td><td>CNN</td><td>32 ×32×3 Conv5×5×16,5×5×32,5×5×64,5×5×128 (same padding,strides = 2,leaky ReLu activation), fully connected, softmax layer</td></tr></table>

parse/train/Hygxb2CqKm/Hygxb2CqKm.md

@@ -0,0 +1,667 @@
+# STABLE RECURRENT MODELS
+
+John Miller & Moritz Hardt University of California, Berkeley {miller john,hardt}@berkeley.edu
+
+# ABSTRACT
+
+Stability is a fundamental property of dynamical systems, yet to this date it has had little bearing on the practice of recurrent neural networks. In this work, we conduct a thorough investigation of stable recurrent models. Theoretically, we prove stable recurrent neural networks are well approximated by feed-forward networks for the purpose of both inference and training by gradient descent. Empirically, we demonstrate stable recurrent models often perform as well as their unstable counterparts on benchmark sequence tasks. Taken together, these findings shed light on the effective power of recurrent networks and suggest much of sequence learning happens, or can be made to happen, in the stable regime. Moreover, our results help to explain why in many cases practitioners succeed in replacing recurrent models by feed-forward models.
+
+# 1 INTRODUCTION
+
+Recurrent neural networks are a popular modeling choice for solving sequence learning problems arising in domains such as speech recognition and natural language processing. At the outset, recurrent neural networks are non-linear dynamical systems commonly trained to fit sequence data via some variant of gradient descent.
+
+Stability is of fundamental importance in the study of dynamical system. Surprisingly, however, stability has had little impact on the practice of recurrent neural networks. Recurrent models trained in practice do not satisfy stability in an obvious manner, suggesting that perhaps training happens in a chaotic regime. The difficulty of training recurrent models has compelled practitioners to successfully replace recurrent models with non-recurrent, feed-forward architectures.
+
+This state of affairs raises important unresolved questions. Is sequence modeling in practice inherently unstable? When and why are recurrent models really needed?
+
+In this work, we shed light on both of these questions through a theoretical and empirical investigation of stability in recurrent models.
+
+We first prove stable recurrent models can be approximated by feed-forward networks. In particular, not only are the models equivalent for inference, they are also equivalent for training via gradient descent. While it is easy to contrive non-linear recurrent models that on some input sequence cannot be approximated by feed-forward models, our result implies such models are inevitably unstable. This means in particular they must have exploding gradients, which is in general an impediment to learnibility via gradient descent.
+
+Second, across a variety of different sequence tasks, we show how recurrent models can often be made stable without loss in performance. We also show models that are nominally unstable often operate in the stable regime on the data distribution. Combined with our first result, these observation helps to explain why an increasingly large body of empirical research succeeds in replacing recurrent models with feed-forward models in important applications, including translation (Vaswani et al., 2017; Gehring et al., 2017), speech synthesis (Van Den Oord et al., 2016), and language modeling (Dauphin et al., 2017). While stability does not always hold in practice to begin with, it is often possible to generate a high-performing stable model by imposing stability during training.
+
+Our results also shed light on the effective representational properties of recurrent networks trained in practice. In particular, stable models cannot have long-term memory. Therefore, when stable and unstable models achieve similar results, either the task does not require long-term memory, or the unstable model does not have it.
+
+# 1.1 CONTRIBUTIONS
+
+In this work, we make the following contributions.
+
+1. We present a generic definition of stable recurrent models in terms of non-linear dynamical systems and show how to ensure stability of several commonly used models. Previous work establishes stability for vanilla recurrent neural networks. We give new sufficient conditions for stability of long short-term memory (LSTM) networks. These sufficient conditions come with an efficient projection operator that can be used at training time to enforce stability.   
+2. We prove, under the stability assumption, feed-forward networks can approximate recurrent networks for purposes of both inference and training by gradient descent. While simple in the case of inference, the training result relies on non-trivial stability properties of gradient descent.   
+3. We conduct extensive experimentation on a variety of sequence benchmarks, show stable models often have comparable performance with their unstable counterparts, and discuss when, if ever, there is an intrinsic performance price to using stable models.
+
+# 2 STABLE RECURRENT MODELS
+
+In this section, we define stable recurrent models and illustrate the concept for various popular model classes. From a pragmatic perspective, stability roughly corresponds to the criterion that the gradients of the training objective do not explode over time. Common recurrent models can operate in both the stable and unstable regimes, depending on their parameters. To study stable variants of common architectures, we give sufficient conditions to ensure stability and describe how to efficiently enforce these conditions during training.
+
+# 2.1 DEFINING STABLE RECURRENT MODELS
+
+A recurrent model is a non-linear dynamical system given by a differentiable state-transition map $\phi _ { w } : { \mathbf { R } } ^ { n } \times { \mathbf { R } } ^ { d } \to { \mathbf { R } } ^ { n }$ , parameterized by $w \in { \mathbf { R } } ^ { m }$ . The hidden state $h _ { t } \in \mathbf { R } ^ { n }$ evolves in discrete time steps according to the update rule
+
+$$
+h _ { t } = \phi _ { w } ( h _ { t - 1 } , x _ { t } ) ,
+$$
+
+where the vector $\boldsymbol { x } _ { t } \in \mathbb { R } ^ { d }$ is an arbitrary input provided to the system at time $t$ . This general formulation allows us to unify many examples of interest. For instance, for a recurrent neural network, given weight matrices $W$ and $U$ , the state evolves according to
+
+$$
+h _ { t } = \phi _ { W , U } \big ( h _ { t - 1 } , x _ { t } \big ) = \operatorname { t a n h } \big ( W h _ { t - 1 } + U x _ { t } \big ) .
+$$
+
+Recurrent models are typically trained using some variant of gradient descent. One natural—even if not strictly necessary—requirement for gradient descent to work is that the gradients of the training objective do not explode over time. Stable recurrent models are precisely the class of models where the gradients cannot explode. They thus constitute a natural class of models where gradient descent can be expected to work. In general, we define a stable recurrent model as follows.
+
+Definition 1. A recurrent model $\phi _ { w }$ is stable if there exists some $\lambda < 1$ such that, for any weights $w \in { \mathbf { R } } ^ { m }$ , states $h , h ^ { \prime } \in \mathbf { R } ^ { n }$ , and input $\boldsymbol { x } \in \mathbf { R } ^ { d }$ ,
+
+$$
+\begin{array} { r } { \| \phi _ { w } ( h , x ) - \phi _ { w } ( h ^ { \prime } , x ) \| \leq \lambda \| h - h ^ { \prime } \| . } \end{array}
+$$
+
+Equivalently, a recurrent model is stable if the map $\phi _ { w }$ is $\lambda$ -contractive in $h$ . If $\phi _ { w }$ is $\lambda$ -stable, then $\| \nabla _ { h } \phi _ { w } ( h , x ) \| < \lambda$ , and for Lipschitz loss $p$ , $\| \nabla _ { w } p \|$ is always bounded (Pascanu et al., 2013).
+
+Stable models are particularly well-behaved and well-justified from a theoretical perspective. For instance, at present, only stable linear dynamical systems are known to be learnable via gradient
+
+descent (Hardt et al., 2018). In unstable models, the gradients of the objective can explode, and it is a delicate matter to even show that gradient descent converges to a stationary point. The following proposition offers one such example. The proof is provided in the appendix.
+
+Proposition 1. There exists an unstable system $\phi _ { w }$ where gradient descent does not converge to a stationary point, and $\| \nabla _ { w } p \| \to \infty$ as the number of iterations $N \to \infty$ .
+
+# 2.2 EXAMPLES OF STABLE RECURRENT MODELS
+
+In this section, we provide sufficient conditions to ensure stability for several common recurrent models. These conditions offer a way to require learning happens in the stable regime– after each iteration of gradient descent, one imposes the corresponding stability condition via projection.
+
+Linear dynamical systems and recurrent neural networks. Given a Lipschitz, point-wise nonlinearity $\rho$ and matrices $W \in \mathbf { R } ^ { n \times n }$ and $U \in \mathbf { R } ^ { n \times d }$ , the state-transition map for a recurrent neural network (RNN) is
+
+$$
+h _ { t } = \rho ( W h _ { t - 1 } + U x _ { t } ) .
+$$
+
+If $\rho$ is the identity, then the system is a linear dynamical system. Jin et al. (1994) show if $\rho$ is $L _ { \rho }$ -Lipschitz, then the model is stable provided $\begin{array} { r } { \| \dot { W } \| < \frac { 1 } { L _ { \rho } } } \end{array}$ . Indeed, for any states $h , h ^ { \prime }$ , and any $x$ ,
+
+$$
+\left\| \rho ( W h + U x ) - \rho ( W h ^ { \prime } + U x ) \right\| \leq L _ { \rho } \left\| W h + U x - W h ^ { \prime } - U x \right\| \leq L _ { \rho } \left\| W \right\| \left\| h - h ^ { \prime } \right\| .
+$$
+
+In the case of a linear dynamical system, the model is stable provided $\| W \| < 1$ . Similarly, for the 1-Lipschitz tanh-nonlinearity, stability obtains provided $\| W \| < 1$ . In the appendix, we verify the assumptions required by the theorems given in the next section for this example. Imposing this condition during training corresponds to projecting onto the spectral norm ball.
+
+Long short-term memory networks. Long Short-Term Memory (LSTM) networks are another commonly used class of sequence models (Hochreiter & Schmidhuber, 1997). The state is a pair of vectors $s = ( c , h ) \in \mathbf { R } ^ { 2 \bar { d } }$ , and the model is parameterized by eight matrices, $W _ { \perp } \in \mathbf { R } ^ { d \times d }$ and $U _ { \bigstar } \in \mathbf { R } ^ { d \times n }$ , for $\boldsymbol { \bigtriangledown } \in \{ i , f , o , z \}$ . The state-transition map $\phi _ { \mathrm { L S T M } }$ is given by
+
+$$
+\begin{array} { r l } & { f _ { t } = \sigma ( W _ { f } h _ { t - 1 } + U _ { f } x _ { t } ) } \\ & { i _ { t } = \sigma ( W _ { i } h _ { t - 1 } + U _ { i } x _ { t } ) } \\ & { o _ { t } = \sigma ( W _ { o } h _ { t - 1 } + U _ { o } x _ { t } ) } \\ & { z _ { t } = \operatorname { t a n h } ( W _ { z } h _ { t - 1 } + U _ { z } x _ { t } ) } \\ & { c _ { t } = i _ { t } \circ z _ { t } + f _ { t } \circ c _ { t - 1 } } \\ & { h _ { t } = o _ { t } \cdot \operatorname { t a n h } ( c _ { t } ) , } \end{array}
+$$
+
+where $\circ$ denotes elementwise multiplication, and $\sigma$ is the logistic function.
+
+We provide conditions under which the iterated system $\phi _ { \mathrm { L S T M } } ^ { r } = \phi _ { \mathrm { L S T M } } \circ \cdot \cdot \cdot \circ \phi _ { \mathrm { L S T M } }$ is stable. Let $\left\| f \right\| _ { \infty } = \operatorname* { s u p } _ { t } \left\| f _ { t } \right\| _ { \infty }$ . If the weights $W _ { f } , U _ { f }$ and inputs $x _ { t }$ are bounded, then $\| f \| _ { \infty } < 1$ since $| \sigma | < 1$ for any finite input. This means the next state $c _ { t }$ must “forget” a non-trivial portion of $c _ { t - 1 }$ . We leverage this phenomenon to give sufficient conditions for $\phi _ { \mathrm { L S T M } }$ to be contractive in the $\ell _ { \infty }$ norm, which in turn implies the iterated system $\phi _ { \mathrm { L S T M } } ^ { r }$ is contractive in the $\ell _ { 2 }$ norm for $r = O ( \log ( d ) )$ . Let $\| W \| _ { \infty }$ denote the induced $\ell _ { \infty }$ matrix norm, which corresponds to the maximum absolute row sum $\operatorname* { m a x } _ { i } \sum _ { j } ^ { \sim } | W _ { i j } |$ .
+
+Proposition 2. If $\left\| \boldsymbol { W } _ { i } \right\| _ { \infty } , \left\| \boldsymbol { W } _ { o } \right\| _ { \infty } < ( 1 - \left\| \boldsymbol { f } \right\| _ { \infty } )$ , $\| W _ { z } \| _ { \infty } \leq ( 1 / 4 ) ( 1 - \| f \| _ { \infty } )$ , $\| W _ { f } \| _ { \infty } <$ $( 1 - \| f \| _ { \infty } ) ^ { 2 }$ , and $r = O ( \log ( d ) )$ , then the iterated system $\phi _ { \mathrm { L S T M } } ^ { r }$ is stable.
+
+The proof is given in the appendix. The conditions given in Proposition 2 are fairly restrictive. Somewhat surprisingly we show in the experiments models satisfying these stability conditions still achieve good performance on a number of tasks. We leave it as an open problem to find different parameter regimes where the system is stable, as well as resolve whether the original system $\phi _ { \mathrm { L S T M } }$ is stable. Imposing these conditions during training and corresponds to simple row-wise normalization of the weight matrices and inputs. More details are provided in Section 4 and the appendix.
+
+# 3 STABLE RECURRENT MODELS HAVE FEED-FORWARD APPROXIMATIONS
+
+In this section, we prove stable recurrent models can be well-approximated by feed-forward networks for the purposes of both inference and training by gradient descent. From a memory perspective, stable recurrent models are equivalent to feed-forward networks—both models use the same amount of context to make predictions. This equivalence has important consequences for sequence modeling in practice. When a stable recurrent model achieves satisfactory performance on some task, a feed-forward network can achieve similar performance. Consequently, if sequence learning in practice is inherently stable, then recurrent models may not be necessary. Conversely, if feed-forward models cannot match the performance of recurrent models, then sequence learning in practice is in the unstable regime.
+
+# 3.1 TRUNCATED RECURRENT MODELS
+
+For our purposes, the salient distinction between a recurrent and feed-forward model is the latter has finite-context. Therefore, we say a model is feed-forward if the prediction made by the model at step $t$ is a function only of the inputs $x _ { t - k } , \ldots , x _ { t }$ for some finite $k$ .
+
+While there are many choices for a feed-forward approximation, we consider the simplest one— truncation of the system to some finite context $k$ . In other words, the feed-forward approximation moves over the input sequence with a sliding window of length $k$ producing an output every time the sliding window advances by one step. Formally, for context length $k$ chosen in advance, we define the truncated model via the update rule
+
+$$
+h _ { t } ^ { k } = \phi _ { w } \big ( h _ { t - 1 } ^ { k } , x _ { t } \big ) , \quad h _ { t - k } ^ { k } = 0 .
+$$
+
+Note that $h _ { t } ^ { k }$ is a function only of the previous $k$ inputs $x _ { t - k } , \ldots , x _ { t }$ . While this definition is perhaps an abuse of the term “feed-forward”, the truncated model can be implemented as a standard autoregressive, depth- $k$ feed-forward network, albeit with significant weight sharing.
+
+Let $f$ denote a prediction function that maps a state $h _ { t }$ to outputs $f ( h _ { t } ) = y _ { t }$ . Let $y _ { t } ^ { k }$ denote the predictions from the truncated model. To simplify the presentation, the prediction function $f$ is not parameterized. This is without loss of generality because it is always possible to fold the parameters into the system $\phi _ { w }$ itself. In the sequel, we study $\left\| y _ { t } - y _ { t } ^ { k } \right\|$ both during and after training.
+
+# 3.2 APPROXIMATION DURING INFERENCE
+
+Suppose we train a full recurrent model $\phi _ { w }$ and obtain a prediction $y _ { t }$ . For an appropriate choice of context $k$ , the truncated model makes essentially the same prediction $y _ { t } ^ { k }$ as the full recurrent model. To show this result, we first control the difference between the hidden states of both models.
+
+Lemma 1. Assume $\phi _ { w }$ is $\lambda$ -contractive in $h$ and $L _ { x }$ -Lipschitz in $x$ . Assume the input sequence $\| x _ { t } \| \leq B _ { x }$ for all $t$ . If the truncation length $\begin{array} { r } { k \ge \log _ { 1 / \lambda } \left( \frac { L _ { x } B _ { x } } { ( 1 - \lambda ) \varepsilon } \right) } \end{array}$ , then the difference in hidden states $\left\| h _ { t } - h _ { t } ^ { k } \right\| \leq \varepsilon$ .
+
+Lemma 1 effectively says stable models do not have long-term memory– distant inputs do not change the states of the system. A proof is given in the appendix. If the prediction function is Lipschitz, Lemma 1 immediately implies the recurrent and truncated model make nearly identical predictions.
+
+Proposition 3. If $\phi _ { w }$ is a $L _ { x }$ -Lipschitz and $\lambda$ -contractive map, and $f$ is $L _ { f }$ Lipschitz, and the truncation length $\begin{array} { r } { k \ge \log _ { 1 / \lambda } \left( \frac { L _ { f } L _ { x } B _ { x } } { ( 1 - \lambda ) \varepsilon } \right) } \end{array}$ , then $\left\| y _ { t } - y _ { t } ^ { k } \right\| \leq \varepsilon$ .
+
+# 3.3 APPROXIMATION DURING TRAINING VIA GRADIENT DESCENT
+
+Equipped with our inference result, we turn towards optimization. We show gradient descent for stable recurrent models finds essentially the same solutions as gradient descent for truncated models. Consequently, both the recurrent and truncated models found by gradient descent make essentially the same predictions.
+
+Our proof technique is to initialize both the recurrent and truncated models at the same point and track the divergence in weights throughout the course of gradient descent. Roughly, we show if $k \approx { \cal O } ( \log ( N / \varepsilon ) )$ , then after $N$ steps of gradient descent, the difference in the weights between the recurrent and truncated models is at most $\varepsilon$ . Even if the gradients are similar for both models at the same point, it is a priori possible that slight differences in the gradients accumulate over time and lead to divergent weights where no meaningful comparison is possible. Building on similar techniques as Hardt et al. (2016), we show that gradient descent itself is stable, and this type of divergence cannot occur.
+
+Our gradient descent result requires two essential lemmas. The first bounds the difference in gradient between the full and the truncated model. The second establishes the gradient map of both the full and truncated models is Lipschitz. We defer proofs of both lemmas to the appendix.
+
+Let $p _ { T }$ denote the loss function evaluated on recurrent model after $T$ time steps, and define $p _ { T } ^ { k }$ similarly for the truncated model. Assume there some compact, convex domain $\boldsymbol { \Theta } \subset \mathbf { R } ^ { m }$ so that the map $\phi _ { w }$ is stable for all choices of parameters $w \in \Theta$ .
+
+Lemma 2. Assume $p$ (and therefore $p ^ { k }$ ) is Lipschitz and smooth. Assume $\phi _ { w }$ is smooth, $\lambda$ - contractive, and Lipschitz in $x$ and $w$ . Assume the inputs satisfy $\| x _ { t } \| \leq B _ { x }$ , then
+
+$$
+\left\| \nabla _ { w } p _ { T } - \nabla _ { w } p _ { T } ^ { k } \right\| = \gamma k \lambda ^ { k } ,
+$$
+
+where $\gamma = O \left( B _ { x } ( 1 - \lambda ) ^ { - 2 } \right)$ , suppressing dependence on the Lipschitz and smoothness parameters.
+
+Lemma 3. For any $w , w ^ { \prime } \in \Theta$ , suppose $\phi _ { w }$ is smooth, $\lambda$ -contractive, and Lipschitz in $w$ . If $p$ is Lipschitz and smooth, then
+
+$$
+\begin{array} { r } { \| \nabla _ { w } p _ { T } ( w ) - \nabla _ { w } p _ { T } ( w ^ { \prime } ) \| \leq \beta \| w - w ^ { \prime } \| , } \end{array}
+$$
+
+where $\beta = O \left( ( 1 - \lambda ) ^ { - 3 } \right)$ , suppressing dependence on the Lipschitz and smoothness parameters.
+
+cate Let wirecurr be the weights of tl. At initialization, l on s. For p s $i$ and define fficiently la $w _ { \mathrm { t r u n c } } ^ { i }$ similarly for the trun-mma 2 guarantees the $w _ { \mathrm { r e c u r r } } ^ { 0 } = w _ { \mathrm { t r u n c } } ^ { 0 }$ $k$ difference between the gradient of the recurrent and truncated models is negligible. Therefore, after a gradient update, $\lVert w _ { \mathrm { r e c u r r } } ^ { 1 } - w _ { \mathrm { t r u n c } } ^ { 1 } \rVert$ is small. Lemma 3 then guarantees that this small difference in weights does not lead to large differences in the gradient on the subsequent time step. For an appropriate choice of learning rate, formalizing this argument leads to the following proposition.
+
+Proposition 4. Under the assumptions oprojected gradient descent with step size $\Theta$ $N$ steps of $\alpha _ { t } = \alpha / t$ $| t , \big | \big | w _ { \mathrm { r e c u r r } } ^ { N } - w _ { \mathrm { t r u n c } } ^ { N ^ { \ast } } \big | \big | \leq \alpha \gamma k \lambda ^ { k } N ^ { \tilde { \alpha } \beta + 1 }$
+
+The decaying step size in our theorem is consistent with the regime in which gradient descent is known to be stable for non-convex training objectives (Hardt et al., 2016). While the decay is faster than many learning rates encountered in practice, classical results nonetheless show that with this learning rate gradient descent still converges to a stationary point; see p. 119 in Bertsekas (1999) and references there. In the appendix, we give empirical evidence the $O ( 1 / t )$ rate is necessary for our theorem and show examples of stable systems trained with constant or $O ( 1 / \sqrt { t } )$ rates that do not satisfy our bound.
+
+Critically, the bound in Proposition 4 goes to 0 as $k  \infty$ . In particular, if we take $\alpha = 1$ and $k \geq \Omega ( \log ( \gamma N ^ { \beta } / \varepsilon ) )$ , then after $N$ → ∞steps of projected gradient descent, $\lVert w _ { \mathrm { r e c u r r } } ^ { N } - w _ { \mathrm { t r u n c } } ^ { N } \rVert \leq \varepsilon$ . For this choice of $k$ , we obtain the main theorem. The proof is left to the appendix.
+
+Theorem 1. Let $p$ be Lipschitz and smooth. Assume $\phi _ { w }$ is smooth, $\lambda$ -contractive, Lipschitz in $x$ and $w$ . Assume the inputs are bounded, and the prediction function $f$ is $L _ { f }$ -Lipschitz. If $k \geq \Omega ( \log ( \gamma N ^ { \beta } / \varepsilon ) )$ , then after $N$ steps of projected gradient descent with step size $\alpha _ { t } = 1 / t$ , $\left\| y _ { T } - y _ { T } ^ { k } \right\| \leq \varepsilon$ .
+
+# 4 EXPERIMENTS
+
+In the experiments, we show stable recurrent models can achieve solid performance on several benchmark sequence tasks. Namely, we show unstable recurrent models can often be made stable without a loss in performance. In some cases, there is a small gap between the performance between unstable and stable models. We analyze whether this gap is indicative of a “price of stability” and show the unstable models involved are stable in a data-dependent sense.
+
+# 4.1 TASKS
+
+We consider four benchmark sequence problems–word-level language modeling, character-level language modeling, polyphonic music modeling, and slot-filling.
+
+Language modeling. In language modeling, given a sequence of words or characters, the model must predict the next word or character. For character-level language modeling, we train and evaluate models on Penn Treebank (Marcus et al., 1993). To increase the coverage of our experiments, we train and evaluate the word-level language models on the Wikitext-2 dataset, which is twice as large as Penn Treebank and features a larger vocabulary (Merity et al., 2017). Performance is reported using bits-per-character for character-level models and perplexity for word-level models.
+
+Polyphonic music modeling. In polyphonic music modeling, a piece is represented as a sequence of 88-bit binary codes corresponding to the 88 keys on a piano, with a 1 indicating a key that is pressed at a given time. Given a sequence of codes, the task is to predict the next code. We evaluate our models on JSB Chorales, a polyphonic music dataset consisting of 382 harmonized chorales by J.S. Bach (Allan & Williams, 2005). Performance is measured using negative log-likelihood.
+
+Slot-filling. In slot filling, the model takes as input a query like “I want to Boston on Monday” and outputs a class label for each word in the input, e.g. Boston maps to Departure City and Monday maps to Departure Time. We use the Airline Travel Information Systems (ATIS) benchmark and report the F1 score for each model (Price, 1990).
+
+# 4.2 COMPARING STABLE AND UNSTABLE MODELS
+
+For each task, we first train an unconstrained RNN and an unconstrained LSTM. All the hyperparameters are chosen via grid-search to maximize the performance of the unconstrained model. For consistency with our theoretical results in Section 3 and stability conditions in Section 2.2, both models have a single recurrent layer and are trained using plain SGD. In each case, the resulting model is unstable. However, we then retrain the best models using projected gradient descent to enforce stability without retuning the hyperparameters. In the RNN case, we constrain $\| W \| < 1$ . After each gradient update, we project the $W$ onto the spectral norm ball by computing the SVD and thresholding the singular values to lie in $[ 0 , 1 )$ . In the LSTM case, after each gradient update, we normalize each row of the weight matrices to satisfy the sufficient conditions for stability given in Section 2.2. Further details are given in the appendix.
+
+Stable and unstable models achieve similar performance. Table 1 gives a comparison of the performance between stable and unstable RNNs and LSTMs on each of the different tasks. Each of the reported metrics is computed on the held-out test set. We also show a representative comparison of learning curves for word-level language modeling and polyphonic music modeling in Figures 1(a) and 1(b).
+
+![](images/1628a1c276de18cff1c5c32e660b499e8c36a246c3042a510e4c43b7e79686a3.jpg)  
+Figure 1: Stable and unstable variants of common recurrent architectures achieve similar performance across a range of different sequence tasks.
+
+Table 1: Comparison of stable and unstable models on a variety of sequence modeling tasks. For all the tasks, stable and unstable RNNs achieve the same performance. For polyphonic music and slot-filling, stable and unstable LSTMs achieve the same results. On language modeling, there is a small gap between stable and unstable LSTMs. We discuss this in Section 4.3. Performance is evaluated on the held-out test set. For negative log-likelihood (nll), bits per character (bpc), and perplexity, lower is better. For F1 score, higher is better.
+
+Model   
+
+<table><tr><td rowspan="2">Sequence Task</td><td rowspan="2">Dataset (measure)</td><td colspan="2">RNN</td><td colspan="2">LSTM</td></tr><tr><td>Unstable</td><td>Stable</td><td>Unstable</td><td>Stable</td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Polyphonic Music</td><td>JSB Chorales (nll)</td><td>8.9</td><td>8.9</td><td>8.5</td><td>8.5</td></tr><tr><td>Slot-Filling</td><td>Atis (F1 score)</td><td>94.7</td><td>94.7</td><td>95.1</td><td>94.6</td></tr><tr><td>Word-level LM</td><td>Wikitext-2 (perplexity)</td><td>146.7</td><td>143.5</td><td>95.7</td><td>113.2</td></tr><tr><td>Character-level LM</td><td>Penn Treebank (bpc)</td><td>1.8</td><td>1.9</td><td>1.4</td><td>1.9</td></tr></table>
+
+Across all the tasks we considered, stable and unstable RNNs have roughly the same performance. Stable RNNs and LSTMs achieve results comparable to published baselines on slot-filling (Mesnil et al., 2015) and polyphonic music modeling (Bai et al., 2018). On word and character level language modeling, both stable and unstable RNNs achieve comparable results to (Bai et al., 2018).
+
+On the language modeling tasks, however, there is a gap between stable and unstable LSTM models. Given the restrictive conditions we place on the LSTM to ensure stability, it is surprising they work as well as they do. Weaker conditions ensuring stability of the LSTM could reduce this gap. It is also possible imposing stability comes at a cost in representational capacity required for some tasks.
+
+4.3 WHAT IS THE “PRICE OF STABILITY” IN SEQUENCE MODELING?
+
+The gap between stable and unstable LSTMs on language modeling raises the question of whether there is an intrinsic performance cost for using stable models on some tasks. If we measure stability in a data-dependent fashion, then the unstable LSTM language models are stable, indicating this gap is illusory. However, in some cases with short sequences, instability can offer modeling benefits.
+
+LSTM language models are stable in a “data-dependent” way. Our notion of stability is conservative and requires stability to hold for every input and pair of hidden states. If we instead consider a weaker, data-dependent notion of stability, the word and character-level LSTM models are stable (in the iterated sense of Proposition 2). In particular, we compute the stability parameter only using input sequences from the data. Furthermore, we only evaluate stability on hidden states reachable via gradient descent. More precisely, to estimate $\lambda$ , we run gradient ascent to find worst-case hidden states $h , h ^ { \prime }$ to maximize kφw(h,x)−φw(h0,x)kk − 0k . More details are provided in the appendix.
+
+The data-dependent definition given above is a useful diagnostic— when the sufficient stability conditions fail to hold, the data-dependent condition addresses whether the model is still operating in the stable regime. Moreover, when the input representation is fixed during training, our theoretical results go through without modification when using the data-dependent definition.
+
+Using the data-dependent measure, in Figure 2(a), we show the iterated character-level LSTM, $\phi _ { \mathrm { L S T M } } ^ { r }$ , is stable for $r \approx 8 0$ iterations. A similar result holds for the word-level language model for $r \approx 1 0 0$ . These findings are consistent with experiments in Laurent & von Brecht (2017) which find LSTM trajectories converge after approximately 70 steps only when evaluated on sequences from the data. For language models, the “price of stability” is therefore much smaller than the gap in Table 1 suggests– even the “unstable” models are operating in the stable regime on the data distribution.
+
+Unstable systems can offer performance improvements for short-time horizons. When sequences are short, training unstable models is less difficult because exploding gradients are less of an issue. In these case, unstable models can offer performance gains. To demonstrate this, we train truncated unstable models on the polyphonic music task for various values of the truncation parameter $k$ . In Figure 2(b), we simultaneously plot the performance of the unstable model and the stability parameter $\lambda$ for the converged model for each $k$ . For short-sequences, the final model is more unstable $( \lambda \approx 3 . 5 )$ and achieves a better test-likelihood. For longer sequence lengths, $\lambda$ decreases closer to the stable regime $\left( \lambda \approx 1 . 5 \right)$ ), and this improved test-likelihood performance disappears.
+
+![](images/527d5e269348aa2db67694da6730ff856052b523d9bea2fda4f42ef8222a5d43.jpg)
+
+(a) Data-dependent stability of character-level language models. The iterated-LSTM refers to the iteration system $\phi _ { \mathrm { L S T M } } ^ { r } = \phi _ { \mathrm { L S T M } } \circ \cdot \cdot \cdot \circ \phi _ { \mathrm { L S T M } }$ .
+
+![](images/f282c4adb4fa436abe557a08aac14b09601a6865f3b718a3c23d995e0102b57e.jpg)  
+(b) Unstable models can boost performance for short sequences.
+
+Figure 2: What is the intrinsic “price of stability”? For language modeling, we show the unstable LSTMs are actually stable in weaker, data-dependent sense. On the other hand, for polyphonic music modeling with short sequences, instability can improve model performance.
+
+# 4.4 UNSTABLE MODELS OPERATE IN THE STABLE REGIME
+
+In the previous section, we showed nominally unstable models often satisfy a data-dependent notion of stability. In this section, we offer further evidence unstable models are operating in the stable regime. These results further help explain why stable and unstable models perform comparably in experiments.
+
+Vanishing gradients. Stable models necessarily have vanishing gradients, and indeed this ingredient is a key ingredient in the proof of our training-time approximation result. For both word and character-level language models, we find both unstable RNNs and LSTMs also exhibit vanishing gradients. In Figures 3(a) and 3(b), we plot the average gradient of the loss at time $t + i$ with respect to the input at time $t$ , $\| \nabla _ { x _ { t } } p _ { t + i } \|$ as $t$ ranges over the training set. For either language modeling task, the LSTM and the RNN suffer from limited sensitivity to distant inputs at initialization and throughout training. The gradients of the LSTM vanish more slowly than those of the RNN, but both models exhibit the same qualitative behavior.
+
+![](images/562b80e4783bef0c16d64d02a308f7983e074a824796ca50596c255ca3d40b17.jpg)  
+Figure 3: Unstable word and character-level language models exhibit vanishing gradients. We plot the norm of the gradient with respect to inputs, $\| \nabla _ { x _ { t } } p _ { t + i } \|$ , as the distance between the input and the loss grows, averaged over the entire training set. The gradient vanishes for moderate values of $i$ for both RNNs and LSTMs, though the decay is slower for LSTMs.
+
+Truncating Unstable Models. The results in Section 3 show stable models can be truncated without loss of performance. In practice, unstable models can also be truncated without performance loss. In Figures 4(a) and 4(b), we show the performance of both LSTMs and RNNs for various values of the truncation parameter $k$ on word-level language modeling and polyphonic music modeling. Initially, increasing $k$ increases performance because the model can use more context to make predictions. However, in both cases, there is diminishing returns to larger values of the truncation parameter $k$ . LSTMs are unaffected by longer truncation lengths, whereas the performance of RNNs slightly degrades as $k$ becomes very large, possibly due to training instability. In either case, diminishing returns to performance for large values of $k$ means truncation and therefore feed-forward approximation is possible even for these unstable models.
+
+![](images/dc44c6050af8f079adf13da995f8e80349dcbf7d9d878975267f82b08c9a76f1.jpg)  
+Figure 4: Effect of truncating unstable models. On both language and music modeling, RNNs and LSTMs exhibit diminishing returns for large values of the truncation parameter $k$ . In LSTMs, larger $k$ doesn’t affect performance, whereas for unstable RNNs, large $k$ slightly decreases performance
+
+Proposition (4) holds for unstable models. In stable models, Proposition (4) in Section 3 ensures the distance between the weight matrices $\lVert w _ { \mathrm { r e c u r r } } - w _ { \mathrm { t r u n c } } \rVert$ grows slowly as training progresses, and this rate decreases as $k$ becomes large. In Figures 5(a) and 5(b), we show a similar result holds empirically for unstable word-level language models. All the models are initialized at the same point, and we track the distance between the hidden-to-hidden matrices $W$ as training progresses. Training the full recurrent model is impractical, and we assume $k = 6 5$ well captures the fullrecurrent model. In Figures 5(a) and 5(b), we plot $\| W _ { k } - W _ { 6 5 } \|$ for $k \in \{ 5 , 1 0 , 1 5 , 2 5 , 3 5 , 5 0 , 6 4 \}$ throughout training. As suggested by Proposition (4), after an initial rapid increase in distance, $\| W _ { k } - W _ { 6 5 } \|$ grows slowly, as suggested by Proposition 4. Moreover, there is a diminishing return to choosing larger values of the truncation parameter $k$ in terms of the accuracy of the approximation.
+
+![](images/e62eb2ca9d12bb93404441676bbbe47f26c133a3a792ca5cbcad142eb241d7dc.jpg)  
+Figure 5: Qualitative version of Proposition 4 for unstable, word-level language models. We assume $k = 6 5$ well-captures the full-recurrent model and plot $\left. w _ { \mathrm { t r u n c } } - w _ { \mathrm { r e c u r r } } \right. = \left. W _ { k } - W _ { 6 5 } \right.$ as training proceeds, where $W$ denotes the recurrent weights. As Proposition 4 suggests, this quantity grows slowly as training proceeds, and the rate of growth decreases as $k$ increases.
+
+# 5 ARE RECURRENT MODELS TRULY NECESSARY?
+
+Our experiments show recurrent models trained in practice operate in the stable regime, and our theoretical results show stable recurrent models are approximable by feed-forward networks, As a consequence, we conjecture recurrent networks trained in practice are always approximable by feed-forward networks. Even with this conjecture, we cannot yet conclude recurrent models as commonly conceived are unnecessary. First, our present proof techniques rely on truncated versions of recurrent models, and truncated recurrent architectures like LSTMs may provide useful inductive bias on some problems. Moreover, implementing the truncated approximation as a feed-forward network increases the number of weights by a factor of $k$ over the original recurrent model. Declaring recurrent models truly superfluous would require both finding more parsimonious feed-forward approximations and proving natural feed-forward models, e.g. fully connected networks or CNNs, can approximate stable recurrent models during training. This remains an important question for future work.
+
+# 6 RELATED WORK
+
+Learning dynamical systems with gradient descent has been a recent topic of interest in the machine learning community. Hardt et al. (2018) show gradient descent can efficiently learn a class of stable, linear dynamical systems, Oymak (2018) shows gradient descent learns a class of stable, non-linear dynamical systems. Work by Sedghi & Anandkumar (2016) gives a moment-based approach for learning some classes of stable non-linear recurrent neural networks. Our work explores the theoretical and empirical consequences of the stability assumption made in these works. In particular, our empirical results show models trained in practice can be made closer to those currently being analyzed theoretically without large performance penalties.
+
+For linear dynamical systems, Tu et al. (2017) exploit the connection between stability and truncation to learn a truncated approximation to the full stable system. Their approximation result is the same as our inference result for linear dynamical systems, and we extend this result to the non-linear setting. We also analyze the impact of truncation on training with gradient descent. Our training time analysis builds on the stability analysis of gradient descent in Hardt et al. (2016), but interestingly uses it for an entirely different purpose. Results of this kind are completely new to our knowledge.
+
+For RNNs, the link between vanishing and exploding gradients and $\| W \|$ was identified in Pascanu et al. (2013). For 1-layer RNNs, Jin et al. (1994) give sufficient conditions for stability in terms of the norm $\| W \|$ and the Lipschitz constant of the non-linearity. Our work additionally considers LSTMs and provides new sufficient conditions for stability. Moreover, we study the consequences of stability in terms of feed-forward approximation.
+
+A number of recent works have sought to avoid vanishing and exploding gradients by ensuring the system is an isometry, i.e. $\lambda = 1$ . In the RNN case, this amounts to constraining $\lvert | W \rvert | = \bar { 1 }$ (Arjovsky et al., 2016; Wisdom et al., 2016; Jing et al., 2017; Mhammedi et al., 2017; Jose et al., 2018). Vorontsov et al. (2017) observes strictly requiring $\| W \| = 1$ reduces performance on several tasks, and instead proposes maintaining $\| W \| \in [ 1 - \varepsilon , 1 + \varepsilon ]$ . Zhang et al. (2018) maintains this “soft-isometry” constraint using a parameterization based on the SVD that obviates the need for the projection step used in our stable-RNN experiments. Kusupati et al. (2018) sidestep these issues and stabilizes training using a residual parameterization of the model. At present, these unitary models have not yet seen widespread use, and our work shows much of the sequence learning in practice, even with nominally unstable models, actually occurs in the stable regime.
+
+From an empirical perspective, Laurent $\&$ von Brecht (2017) introduce a non-chaotic recurrent architecture and demonstrate it can perform as well more complex models like LSTMs. Bai et al. (2018) conduct a detailed evaluation of recurrent and convolutional, feed-forward models on a variety of sequence modeling tasks. In diverse settings, they find feed-forward models outperform their recurrent counterparts. Their experiments are complimentary to ours; we find recurrent models can often be replaced with stable recurrent models, which we show are equivalent to feed-forward networks.
+
+# ACKNOWLEDGEMENTS
+
+This material is based upon work supported by the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE 1752814 and a generous grant from the AWS Cloud Credits for Research program.
+
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+
+# A PROOFS FROM SECTION 2
+
+# A.1 GRADIENT DESCENT ON UNSTABLE SYSTEMS NEED NOT CONVERGE
+
+Proof of Proposition $^ { l }$ . Consider a scalar linear dynamical system
+
+$$
+\begin{array} { l } { { h _ { t } = a h _ { t - 1 } + b x _ { t } } } \\ { { \hat { y } _ { t } = h _ { t } , } } \end{array}
+$$
+
+where $h _ { 0 } = 0$ , $a , b \in \mathbf { R }$ are parameters, and $x _ { t } , y _ { t } \in \textbf { R }$ are elements the input-output sequence $\left\{ \left( x _ { t } , y _ { t } \right) \right\} _ { t = 1 } ^ { T }$ , where $L$ is the sequence length, and $\hat { y } _ { t }$ is the prediction at time $t$ . Stability of the above system corresponds to $| a | \overset { \vartriangle } { < } 1$ .
+
+Suppose $( x _ { t } , y _ { t } ) \ : = \ : ( 1 , 1 )$ for $t = 1 , \dots , L$ . Then the desired system (4) simply computes the identity mapping. Suppose we use the squared-loss $\ell ( y _ { t } , \hat { y } _ { t } ) = ( 1 / \overset { \cdot } { 2 } ) ( y _ { t } - \hat { y } _ { t } ) ^ { 2 }$ , and suppose further $b = 1$ , so the problem reduces to learning $a = 0$ . We first compute the gradient. Compactly write
+
+$$
+h _ { t } = \sum _ { i = 0 } ^ { t - 1 } a ^ { t } b = \left( \frac { 1 - a ^ { t } } { 1 - a } \right) .
+$$
+
+Let $\delta _ { t } = \left( \hat { y } _ { t } - y _ { t } \right)$ . The gradient for step $T$ is then
+
+$$
+\begin{array} { r l } { \displaystyle \frac { d } { d a } \ell ( y _ { T } , \hat { y } _ { T } ) = \delta _ { T } \frac { d } { d a } = \delta _ { T } \sum _ { t = 0 } ^ { T - 1 } a ^ { T - 1 - t } h _ { t } } & { } \\ { = \delta _ { T } \sum _ { t = 0 } ^ { T - 1 } a ^ { T - 1 - t } \left( \frac { 1 - a ^ { t } } { 1 - a } \right) } & { } \\ { = \delta _ { T } \left[ \displaystyle \frac { 1 } { ( 1 - a ) } \sum _ { t = 0 } ^ { T - 1 } a ^ { t } - \frac { T a ^ { T - 1 } } { ( 1 - a ) } \right] } & { } \\ { = \delta _ { T } \left[ \displaystyle \frac { ( 1 - a ^ { T } ) } { ( 1 - a ) ^ { 2 } } - \frac { T a ^ { T - 1 } } { ( 1 - a ) } \right] . } \end{array}
+$$
+
+Plugging in $y _ { t } = 1$ , this becomes
+
+$$
+\frac { d } { d a } \ell ( y _ { T } , \hat { y } _ { T } ) = \left( \frac { ( 1 - a ^ { T } ) } { ( 1 - a ) } - 1 \right) \left[ \frac { ( 1 - a ^ { T } ) } { ( 1 - a ) ^ { 2 } } - \frac { T a ^ { T - 1 } } { ( 1 - a ) } \right] .
+$$
+
+For large $T$ , if $| a | > 1$ , then $a ^ { L }$ grows exponentially with $T$ and the gradient is approximately
+
+$$
+\frac { d } { d a } \ell ( y _ { T } , \hat { y } _ { T } ) \approx \left( a ^ { T - 1 } - 1 \right) T a ^ { T - 2 } \approx T a ^ { 2 T - 3 }
+$$
+
+Therefore, if $a ^ { 0 }$ is initialized outside of $[ - 1 , 1 ]$ , the iterates $a ^ { i }$ from gradient descent with step size $\alpha _ { i } = ( 1 / i )$ diverge, i.e. $a ^ { i }  \infty$ , and from equation (6), it is clear that such $a ^ { i }$ are not stationary points. □
+
+# A.2 PROOFS FROM SECTION 2.2
+
+# A.2.1 RECURRENT NEURAL NETWORKS
+
+Assume $\| W \| \le \lambda < 1$ and $\| U \| \le B _ { U }$ . Notice $\operatorname { t a n h } ^ { \prime } ( x ) = 1 - \operatorname { t a n h } ( x ) ^ { 2 }$ , so since $\operatorname { t a n h } ( x ) \in$ $[ - 1 , 1 ]$ , $\operatorname { t a n h } ( x )$ is 1-Lipschitz and 2-smooth. We previously showed the system is stable since, for any states $h , h ^ { \prime }$ ,
+
+$$
+\begin{array} { r l } & { \| \mathrm { t a n h } ( W h + U x ) - \operatorname { t a n h } ( W h ^ { \prime } + U x ) \| } \\ & { \leq \| W h + U x - W h ^ { \prime } - U x \| } \\ & { \leq \| W \| \| h - h ^ { \prime } \| . } \end{array}
+$$
+
+Using Lemma 1 with $k = 0$ , $\begin{array} { r } { \| h _ { t } \| \le \frac { B _ { U } B _ { x } } { ( 1 - \lambda ) } } \end{array}$ for all $t$ . Therefore, for any $W , W ^ { \prime } , U , U ^ { \prime }$ ,
+
+$$
+\begin{array} { r l } & { \| \mathrm { t a n h } ( W h _ { t } + U x ) - \mathrm { t a n h } ( W ^ { \prime } h _ { t } + U ^ { \prime } x ) \| } \\ & { \leq \| W h _ { t } + U x - W ^ { \prime } h _ { t } - U ^ { \prime } x \| } \\ & { \leq \underset { t } { \operatorname* { s u p } } \left\| h _ { t } \right\| \| W - W ^ { \prime } \| + B _ { x } \left\| U - U ^ { \prime } \right\| . } \\ & { \leq \frac { B _ { U } B _ { x } } { ( 1 - \lambda ) } \left\| W - W ^ { \prime } \right\| + B _ { x } \left\| U - U ^ { \prime } \right\| , } \end{array}
+$$
+
+so the model is Lipschitz in $U , W$ . We can similarly argue the model is $B _ { U }$ Lipschitz in $x$ . For smoothness, the partial derivative with respect to $h$ is
+
+$$
+\frac { \partial \phi _ { w } ( h , x ) } { \partial h } = { \bf d i a g } ( \operatorname { t a n h } ^ { \prime } ( W h + U x ) ) W ,
+$$
+
+so for any $h , h ^ { \prime }$ , bounding the $\ell _ { \infty }$ norm with the $\ell _ { 2 }$ norm,
+
+$$
+\begin{array} { r l } & { \left\| \frac { \partial \phi _ { w } ( h , x ) } { \partial h } - \frac { \partial \phi _ { w } ( h ^ { \prime } , x ) } { \partial h } \right\| = \left\| \mathbf { d i a g } ( \operatorname { t a n h } ^ { \prime } ( W h + U x ) ) W - \mathbf { d i a g } ( \operatorname { t a n h } ^ { \prime } ( W h ^ { \prime } + U x ) ) W \right\| } \\ & { \qquad \leq \| W \| \left\| \mathbf { d i a g } ( \operatorname { t a n h } ^ { \prime } ( W h + U x ) - \operatorname { t a n h } ^ { \prime } ( W h ^ { \prime } + U x ) ) \right\| } \\ & { \qquad \leq 2 \left\| W \right\| \left\| W h + U x - W h ^ { \prime } - U x \right\| _ { \infty } } \\ & { \qquad \leq 2 \lambda ^ { 2 } \left\| h - h ^ { \prime } \right\| . } \end{array}
+$$
+
+For any $W , W ^ { \prime } , U , U ^ { \prime }$ satisfying our assumptions,
+
+$$
+\begin{array} { r l } { \| \frac { \partial \phi _ { w } ( h , x ) } { \partial h } - \frac { \partial \phi _ { w ^ { \prime } } ( h , x ) } { \partial h } \| = \| \mathrm { d i a g } ( \operatorname { t a n h } ^ { \prime } ( W h + U x ) ) W - \mathrm { d i a g } ( \operatorname { t a n h } ^ { \prime } ( W ^ { \prime } h + U ^ { \prime } x ) ) W ^ { \prime } \| } & { } \\ & { \leq \| \mathrm { d i a g } ( \operatorname { t a n h } ^ { \prime } ( W h + U x ) - \operatorname { t a n h } ^ { \prime } ( W ^ { \prime } h + U ^ { \prime } x ) ) \| \| W \| } \\ & { \quad + \| \mathrm { d i a g } ( \operatorname { t a n h } ^ { \prime } ( W ^ { \prime } h + U ^ { \prime } x ) ) \| \| \| W - W ^ { \prime } \| } \\ & { \leq 2 \lambda \| ( W - W ^ { \prime } ) h + ( U - U ^ { \prime } ) x \| _ { \infty } + \| W - W ^ { \prime } \| } \\ & { \leq 2 \lambda \| ( W - W ^ { \prime } ) \| \| h \| + 2 \lambda \| U - U ^ { \prime } \| \| x \| + \| W - W ^ { \prime } \| } \\ & { \leq \frac { 2 \lambda B _ { U } B _ { x } + ( 1 - \lambda ) } { ( 1 - \lambda ) } \| W - W ^ { \prime } \| + 2 \lambda B _ { x } \| U - U ^ { \prime } \| . } \end{array}
+$$
+
+Similar manipulations establish $\frac { \partial \phi _ { w } ( h , x ) } { \partial w }$ is Lipschitz in $h$ and $w$ .
+
+# A.2.2 LSTMS
+
+Similar to the previous sections, we assume $s _ { 0 } = 0$ .
+
+The state-transition map is not Lipschitz in $s$ , much less stable, unless $\| c \|$ is bounded. However, assuming the weights are bounded, we first prove this is always the case.
+
+Lemma 4. Let $\left\| f \right\| _ { \infty } = \operatorname* { s u p } _ { t } \left\| f _ { t } \right\| _ { \infty }$ . If $\| W _ { f } \| _ { \infty } < \infty$ , $\| U _ { f } \| _ { \infty } < \infty$ , and $\| x _ { t } \| _ { \infty } \leq B _ { x }$ , then $\| f \| _ { \infty } < 1$ and $\begin{array} { r } { \left\| c _ { t } \right\| _ { \infty } \le \frac { 1 } { ( 1 - \left\| f \right\| _ { \infty } ) } } \end{array}$ for all $t$ .
+
+Proof of Lemma 4. Note $| \mathrm { t a n h } ( x ) | , | \sigma ( x ) | \le 1$ for all $x$ . Therefore, for any $t$ , $\| h _ { t } \| _ { \infty } \ =$ $\left\| o _ { t } \circ \operatorname { t a n h } ( c _ { t } ) \right\| _ { \infty } \leq 1$ . Since $\sigma ( x ) < 1$ for $x < \infty$ and $\sigma$ is monotonically increasing
+
+$$
+\begin{array} { r l } & { \left\| f _ { t } \right\| _ { \infty } \leq \sigma \left( \left\| W _ { f } h _ { t - 1 } + U _ { f } x _ { t } \right\| _ { \infty } \right) } \\ & { \qquad \leq \sigma \left( \left\| W _ { f } \right\| _ { \infty } \left\| h _ { t - 1 } \right\| _ { \infty } + \left\| U _ { f } \right\| _ { \infty } \left\| x _ { t } \right\| _ { \infty } \right) } \\ & { \qquad \leq \sigma \left( B _ { W } + B _ { u } B _ { x } \right) } \\ & { \qquad < 1 . } \end{array}
+$$
+
+Using the trivial bound, $\| i _ { t } \| _ { \infty } \leq 1$ and $\begin{array} { r } { \| z _ { t } \| _ { \infty } \leq 1 } \end{array}$ , so
+
+$$
+\left\| { c _ { t + 1 } } \right\| _ { \infty } = \left\| { i _ { t } \circ z _ { t } + f _ { t } \circ c _ { t } } \right\| _ { \infty } \leq 1 + \left\| { f _ { t } } \right\| _ { \infty } \left\| { c _ { t } } \right\| _ { \infty } .
+$$
+
+Unrolling this recursion, we obtain a geometric series
+
+$$
+\left\| { c _ { t + 1 } } \right\| _ { \infty } \leq \sum _ { i = 0 } ^ { t } \left\| { f _ { t } } \right\| _ { \infty } ^ { i } \leq \frac { 1 } { \left( 1 - \left\| { f } \right\| _ { \infty } \right) } .
+$$
+
+Proof of Proposition 2. We show $\phi _ { \mathrm { L S T M } }$ is $\lambda$ -contractive in the $\ell _ { \infty }$ -norm for some $\lambda < 1$ . For $r \geq \log _ { 1 / \lambda } ( \sqrt { d } )$ , this in turn implies the iterated system $\phi _ { \mathrm { L S T M } } ^ { r }$ is contractive is the $\ell _ { 2 }$ -norm.
+
+Consider the pair of reachable hidden states $s = ( c , h ) , s ^ { \prime } = ( c ^ { \prime } , h ^ { \prime } )$ . By Lemma 4, $c , c ^ { \prime }$ are bounded. Analogous to the recurrent network case above, since $\sigma$ is $( 1 / 4 )$ -Lipschitz and tanh is 1-Lipschitz,
+
+$$
+\begin{array} { l } { \displaystyle \| i - i ^ { \prime } \| \leq \frac 1 4 \| W _ { i } \| _ { \infty } \| h - h ^ { \prime } \| _ { \infty } } \\ { \displaystyle \| f - f ^ { \prime } \| \leq \frac 1 4 \| W _ { f } \| _ { \infty } \| h - h ^ { \prime } \| _ { \infty } } \\ { \displaystyle \| o - o ^ { \prime } \| \leq \frac 1 4 \| W _ { o } \| _ { \infty } \| h - h ^ { \prime } \| _ { \infty } } \\ { \displaystyle \| z - z ^ { \prime } \| \leq \| W _ { z } \| _ { \infty } \| h - h ^ { \prime } \| _ { \infty } . } \end{array}
+$$
+
+Both $\| z \| _ { \infty } , \| i \| _ { \infty } \leq 1$ since they’re the output of a sigmoid. Letting $c _ { + }$ and $c _ { + } ^ { \prime }$ denote the state on the next time step, applying the triangle inequality,
+
+$$
+\begin{array} { r l } & { \left\| c _ { + } - c _ { + } ^ { \prime } \right\| _ { \infty } \leq \left\| i \circ z - i ^ { \prime } \circ z ^ { \prime } \right\| _ { \infty } + \left\| f \circ c - f ^ { \prime } \circ c ^ { \prime } \right\| _ { \infty } } \\ & { \qquad \leq \left\| \left( i - i ^ { \prime } \right) \circ z \right\| _ { \infty } + \left\| i ^ { \prime } \circ \left( z - z ^ { \prime } \right) \right\| _ { \infty } + \left\| f \circ \left( c - c ^ { \prime } \right) \right\| _ { \infty } + \left\| c \circ \left( f - f ^ { \prime } \right) \right\| _ { \infty } } \\ & { \qquad \leq \left\| i - i ^ { \prime } \right\| _ { \infty } \left\| z \right\| _ { \infty } + \left\| z - z ^ { \prime } \right\| _ { \infty } \left\| i ^ { \prime } \right\| _ { \infty } + \left\| c - c ^ { \prime } \right\| _ { \infty } \left\| f \right\| _ { \infty } + \left\| f - f ^ { \prime } \right\| _ { \infty } \left\| c \right\| _ { \infty } } \\ & { \qquad \leq \left( \frac { \left\| W _ { i } \right\| _ { \infty } + \left\| c \right\| _ { \infty } \left\| W _ { f } \right\| _ { \infty } } { 4 } + \left\| W _ { z } \right\| _ { \infty } \right) \left\| h - h ^ { \prime } \right\| _ { \infty } + \left\| f \right\| _ { \infty } \left\| c - c ^ { \prime } \right\| _ { \infty } . } \end{array}
+$$
+
+A similar argument shows
+
+$$
+\left\| h _ { + } - h _ { + } ^ { \prime } \right\| _ { \infty } \leq \left\| o - o ^ { \prime } \right\| _ { \infty } + \left\| c _ { + } - c _ { + } ^ { \prime } \right\| _ { \infty } \leq \frac { \| W _ { o } \| _ { \infty } } { 4 } \left\| h - h ^ { \prime } \right\| _ { \infty } + \left\| c _ { + } - c _ { + } ^ { \prime } \right\| _ { \infty } .
+$$
+
+By assumption,
+
+$$
+\left( \frac { \| W _ { i } \| _ { \infty } + \| c \| _ { \infty } \| W _ { f } \| _ { \infty } + \| W _ { o } \| _ { \infty } } { 4 } + \| W _ { z } \| _ { \infty } \right) < 1 - \| f \| _ { \infty } ,
+$$
+
+and so
+
+$$
+\begin{array} { r } { \left\| h _ { + } - h _ { + } ^ { \prime } \right\| _ { \infty } < \left( 1 - \left\| f \right\| _ { \infty } \right) \left\| h - h ^ { \prime } \right\| _ { \infty } + \left\| f \right\| _ { \infty } \left\| c - c ^ { \prime } \right\| _ { \infty } \leq \left\| s - s ^ { \prime } \right\| _ { \infty } , } \end{array}
+$$
+
+as well as
+
+$$
+\left\| c _ { + } - c _ { + } ^ { \prime } \right\| _ { \infty } < ( 1 - \| f \| _ { \infty } ) \left\| h - h ^ { \prime } \right\| _ { \infty } + \left\| f \right\| _ { \infty } \left\| c - c ^ { \prime } \right\| _ { \infty } \leq \left\| s - s ^ { \prime } \right\| _ { \infty } ,
+$$
+
+which together imply
+
+$$
+\begin{array} { r } { \left\| s _ { + } - s _ { + } ^ { \prime } \right\| _ { \infty } < \| s - s ^ { \prime } \| _ { \infty } , } \end{array}
+$$
+
+establishing $\phi _ { \mathrm { L S T M } }$ is contractive in the $\ell _ { \infty }$ norm.
+
+# B PROOFS FROM SECTION 3
+
+Throughout this section, we assume the initial state $h _ { 0 } = 0$ . Without loss of generality, we also assume $\phi _ { w } ( 0 , 0 ) = 0$ for all $w$ . Otherwise, we can reparameterize $\phi _ { w } ( h , x ) \mapsto \phi _ { w } ( h , x ) - \phi _ { w } ( 0 , 0 )$ without affecting expressivity of $\phi _ { w }$ . For stable models, we also assume there some compact, convex domain $\boldsymbol { \Theta } \subset \mathbf { R } ^ { m }$ so that the map $\phi _ { w }$ is stable for all choices of parameters $w \in \Theta$ .
+
+Proof of Lemma $^ { l }$ . For any $t \geq 1$ , by triangle inequality,
+
+$$
+\begin{array} { r } { \| h _ { t } \| = \left\| \phi _ { w } ( h _ { t - 1 } , x _ { t } ) - \phi _ { w } ( 0 , 0 ) \right\| \leq \left\| \phi _ { w } ( h _ { t - 1 } , x _ { t } ) - \phi _ { w } ( 0 , x _ { t } ) \right\| + \left\| \phi _ { w } ( 0 , x _ { t } ) - \phi _ { w } ( 0 , 0 ) \right\| . } \end{array}
+$$
+
+Applying the stability and Lipschitz assumptions and then summing a geometric series,
+
+$$
+\left\| h _ { t } \right\| \leq \lambda \left\| h _ { t - 1 } \right\| + L _ { x } \left\| x _ { t } \right\| \leq \sum _ { i = 0 } ^ { t } \lambda ^ { i } L _ { x } B _ { x } \leq { \frac { L _ { x } B _ { x } } { ( 1 - \lambda ) } } .
+$$
+
+Now, consider the difference between hidden states at time step $t$ . Unrolling the iterates $k$ steps and then using the previous display yields
+
+$$
+\left\| h _ { t } - h _ { t } ^ { k } \right\| = \left\| \phi _ { w } ( h _ { t - 1 } , x _ { t } ) - \phi _ { w } ( h _ { t - 1 } ^ { k } , x _ { t } ) \right\| \leq \lambda \left\| h _ { t - 1 } - h _ { t - 1 } ^ { k } \right\| \leq \lambda ^ { k } \left\| h _ { t - k } \right\| \leq { \frac { \lambda ^ { k } L _ { x } B _ { x } } { ( 1 - \lambda ) } } ,
+$$
+
+and solving for $k$ gives the result.
+
+# B.1 PROOFS FROM SECTION 3.3
+
+Before proceeding, we introduce notation for our smoothness assumption. We assume the map $\phi _ { w }$ satisfies four smoothness conditions: for any reachable states $h , h ^ { \prime }$ , and any weights $w , w ^ { \prime } \in \Theta$ , there are some scalars $\beta _ { w w } , \beta _ { w h } , \beta _ { h w } , \beta _ { h h }$ such that
+
+$$
+\begin{array} { r l } & { \left| \frac { \partial \phi _ { w } ( h , x ) } { \partial w } - \frac { \partial \phi _ { w ^ { \prime } } ( h , x ) } { \partial w } \right| \leq \beta _ { w w } \left\| w - w ^ { \prime } \right\| . } \\ & { \left\| \frac { \partial \phi _ { w } ( h , x ) } { \partial w } - \frac { \partial \phi _ { w } ( h ^ { \prime } , x ) } { \partial w } \right\| \leq \beta _ { w h } \left\| h - h ^ { \prime } \right\| . } \\ & { \left\| \frac { \partial \phi _ { w } ( h , x ) } { \partial h } - \frac { \partial \phi _ { w ^ { \prime } } ( h , x ) } { \partial h } \right\| \leq \beta _ { h w } \left\| w - w ^ { \prime } \right\| . } \\ & { \left\| \frac { \partial \phi _ { w } ( h , x ) } { \partial h } - \frac { \partial \phi _ { w } ( h ^ { \prime } , x ) } { \partial h } \right\| \leq \beta _ { h h } \left\| h - h ^ { \prime } \right\| . } \end{array}
+$$
+
+# B.1.1 GRADIENT DIFFERENCE DUE TO TRUNCATION IS NEGLIGIBLE
+
+In the section, we argue the difference in gradient with respect to the weights between the recurrent and truncated models is $O ( k \lambda ^ { k } )$ . For sufficiently large $k$ (independent of the sequence length), the impact of truncation is therefore negligible. The proof leverages the “vanishing-gradient” phenomenon– the long-term components of the gradient of the full recurrent model quickly vanish. The remaining challenge is to show the short-term components of the gradient are similar for the full and recurrent models.
+
+Proof of Lemma 2. The Jacobian of the loss with respect to the weights is
+
+$$
+\frac { \partial p _ { T } } { \partial w } = \frac { \partial p _ { T } } { \partial h _ { T } } \left( \sum _ { t = 0 } ^ { T } \frac { \partial h _ { T } } { \partial h _ { t } } \frac { \partial h _ { t } } { \partial w } \right) ,
+$$
+
+where ∂ht∂w is the partial derivative of ht with respect to w, assuming ht−1 is constant with respect to $w$ . Expanding the expression for the gradient, we wish to bound
+
+$$
+\begin{array} { r l } {  { \| \nabla _ { w } p _ { T } ( w ) - \nabla _ { w } p _ { T } ^ { k } ( w ) \| = \| \sum _ { t = 1 } ^ { T } ( \frac { \partial h _ { T } } { \partial h _ { t } } \frac { \partial h _ { t } } { \partial w } ) ^ { \top } \nabla _ { h _ { T } } p _ { T } - \sum _ { t = T - k + 1 } ^ { T } ( \frac { \partial h _ { T } ^ { k } } { \partial h _ { t } ^ { k } } \frac { \partial h _ { t } ^ { k } } { \partial w } ) ^ { \top } \nabla _ { h _ { T } ^ { k } } p _ { T } ^ { k } \| } } \\ & { \leq \| \sum _ { t = 1 } ^ { T - k } ( \frac { \partial h _ { T } } { \partial h _ { t } } \frac { \partial h _ { t } } { \partial w } ) ^ { \top } \nabla _ { h _ { T } } p _ { T } \| } \\ & { +  \sum _ { t = T - k + 1 } ^ { T } \| ( \frac { \partial h _ { T } } { \partial h _ { t } } \frac { \partial h _ { t } } { \partial w } ) ^ { \top } \nabla _ { h _ { T } } p _ { T } - ( \frac { \partial h _ { T } ^ { k } } { \partial h _ { t } ^ { k } } \frac { \partial h _ { t } ^ { k } } { \partial w } ) ^ { \top } \nabla _ { h _ { T } ^ { k } } p _ { T } \| . } \end{array}
+$$
+
+The first term consists of the “long-term components” of the gradient for the recurrent model. The second term is the difference in the “short-term components” of the gradients between the recurrent and truncated models. We bound each of these terms separately.
+
+For the first term, by the Lipschitz assumptions, $\| \nabla _ { h _ { T } } p _ { T } \| \le L _ { p }$ and $\| \nabla _ { w } h _ { t } \| \le L _ { w }$ . Since $\phi _ { w }$ is $\lambda$ -contractive, so $\left\| { \frac { \partial h _ { t } } { \partial h _ { t - 1 } } } \right\| \leq \lambda$ . Using submultiplicavity of the spectral norm,
+
+$$
+\left\| \frac { \partial p _ { T } } { \partial h _ { T } } \sum _ { t = 0 } ^ { T - k } \frac { \partial p _ { T } } { \partial h _ { t } } \frac { \partial h _ { t } } { \partial w } \right\| \leq \| \nabla _ { h _ { T } } p _ { T } \| \sum _ { t = 0 } ^ { T - k } \left\| \prod _ { i = t } ^ { T } \frac { \partial h _ { i } } { \partial h _ { i - 1 } } \right\| \| \nabla _ { w } h _ { t } \| \leq L _ { p } L _ { w } \sum _ { t = 0 } ^ { T - k } \lambda ^ { T - t } \leq \lambda ^ { k } \frac { L _ { p } L _ { w } } { ( 1 - \lambda ) } .
+$$
+
+Focusing on the second term, by triangle inequality and smoothness,
+
+$$
+\begin{array} { r l } & { \underset { t = T - k + 1 } { \overset { T } { \sum } } \left. \left( \frac { \partial h _ { T } } { \partial h _ { t } } \frac { \partial h _ { t } } { \partial w } \right) ^ { \top } \nabla _ { h _ { T } } p _ { T } - \left( \frac { \partial h _ { T } ^ { k } } { \partial h _ { t } ^ { k } } \frac { \partial h _ { t } ^ { k } } { \partial w } \right) ^ { \top } \nabla _ { h _ { T } ^ { k } } p _ { T } \right. } \\ & { \leq \underset { t = T - k + 1 } { \overset { T } { \sum } } \left. \nabla _ { h _ { T } } p _ { T } - \nabla _ { h _ { T } ^ { k } } p _ { T } ^ { k } \right. \left. \frac { \partial h _ { T } ^ { k } } { \partial h _ { t } ^ { k } } \frac { \partial h _ { t } ^ { k } } { \partial w } \right. + \left. \nabla _ { h _ { T } } p _ { T } \right. \left. \frac { \partial h _ { T } } { \partial h _ { t } } \frac { \partial h _ { t } } { \partial w } - \frac { \partial h _ { T } ^ { k } } { \partial h _ { t } ^ { k } } \frac { \partial h _ { t } ^ { k } } { \partial w } \right. } \\ & { \leq \underset { t = T - k + 1 } { \overset { T } { \sum } } \underset { ( \alpha ) } { \overset { \beta _ { p } } { \sum } } \left. h _ { T } - h _ { T } ^ { k } \right. \lambda ^ { T - t } L _ { w } + \underset { ( \alpha ) } { \overset { \beta _ { p } } { \sum } } \left. \frac { \partial h _ { T } } { \partial h _ { t } } \frac { \partial h _ { t } } { \partial w } - \frac { \partial h _ { T } ^ { k } } { \partial h _ { t } ^ { k } } \frac { \partial h _ { t } ^ { k } } { \partial w } \right. . } \end{array}
+$$
+
+Using Lemma 1 to upper bound (a),
+
+$$
+\sum _ { t = T - k } ^ { T } \beta _ { p } \left. h _ { T } - h _ { T } ^ { k } \right. \lambda ^ { T - t } L _ { w } \leq \sum _ { t = T - k } ^ { T } \lambda ^ { T - t } \frac { \lambda ^ { k } \beta _ { p } L _ { w } L _ { x } B _ { x } } { ( 1 - \lambda ) } \leq \frac { \lambda ^ { k } \beta _ { p } L _ { w } L _ { x } B _ { x } } { ( 1 - \lambda ) ^ { 2 } } .
+$$
+
+Using the triangle inequality, Lipschitz and smoothness, (b) is bounded by
+
+$$
+\begin{array} { r l } & { \quad \displaystyle { \sum _ { t = T - k + 1 } ^ { T } } L _ { p } \left\| \frac { \partial h _ { T } } { \partial h _ { t } } \frac { \partial h _ { t } } { \partial w } - \frac { \partial h _ { t } ^ { k } } { \partial h _ { t } ^ { k } } \frac { \partial h _ { t } ^ { k } } { \partial w } \right\| } \\ & { \leq \displaystyle { \sum _ { t = T - k + 1 } ^ { T } } L _ { p } \left\| \frac { \partial h _ { T } } { \partial h _ { t } } \right\| \left\| \frac { \partial h _ { t } } { \partial w } - \frac { \partial h _ { t } ^ { k } } { \partial w } \right\| + L _ { p } \left\| \frac { \partial h _ { t } ^ { k } } { \partial w } \right\| \left\| \frac { \partial h _ { T } } { \partial h _ { t } } - \frac { \partial h _ { T } ^ { k } } { \partial h _ { t } ^ { k } } \right\| } \\ & { \leq \displaystyle { \sum _ { t = T - k + 1 } ^ { T } } L _ { p } \lambda ^ { T - t } \beta _ { w h } \left\| h _ { t } - h _ { t } ^ { k } \right\| + L _ { p } L _ { w } \left\| \frac { \partial h _ { T } } { \partial h _ { t } } - \frac { \partial h _ { T } ^ { k } } { \partial h _ { t } ^ { k } } \right\| } \\ & { \leq k \lambda ^ { k } \frac { L _ { p } \beta _ { w h } L _ { x } B _ { x } } { ( 1 - \lambda ) } + \underbrace { L _ { p } L _ { w } } _ { \mathrm { r e r - } k + 1 } \left\| \frac { \partial h _ { T } } { \partial h _ { t } } - \frac { \partial h _ { T } ^ { k } } { \partial h _ { t } ^ { k } } \right\| , } \end{array}
+$$
+
+where the last line used $\begin{array} { r } { \left\| h _ { t } - h _ { t } ^ { k } \right\| \leq \lambda ^ { t - ( T - k ) } \frac { L _ { x } B _ { x } } { ( 1 - \lambda ) } } \end{array}$ for $t \geq T - k$ . It remains to bound (c), the difference of the hidden-to-hidden Jacobians. Peeling off one term at a time and applying triangle inequality, for any $t \geq T - k + 1$ ,
+
+$$
+\begin{array} { l } { \displaystyle \left| \left| \frac { \partial h _ { T } } { \partial h _ { t } } - \frac { \partial h _ { T } ^ { k } } { \partial h _ { t } ^ { k } } \right| \right| \leq \left\| \frac { \partial h _ { T } } { \partial h _ { T - 1 } } - \frac { \partial h _ { T } ^ { k } } { \partial h _ { T - 1 } ^ { k } } \right\| \left\| \frac { \partial h _ { T - 1 } } { \partial h _ { t } } \right\| + \left\| \frac { \partial h _ { T } ^ { k } } { \partial h _ { T - 1 } ^ { k } } \right\| \left\| \frac { \partial h _ { T - 1 } } { \partial h _ { t } } - \frac { \partial h _ { T - 1 } ^ { k } } { \partial h _ { t } ^ { k } } \right\| } \\ { \leq \beta _ { h 1 } \| h _ { T - 1 } - h _ { T - 1 } \| \boldsymbol \lambda ^ { T - t - 1 } + \lambda \left\| \frac { \partial h _ { T - 1 } } { \partial h _ { t } } - \frac { \partial h _ { T - 1 } ^ { k } } { \partial h _ { t } ^ { k } } \right\| } \\ { \leq \displaystyle \sum _ { i = t } ^ { T - 1 } \beta _ { h k } \boldsymbol \lambda ^ { T - t - 1 } \left\| h _ { i } - h _ { i } ^ { k } \right\| } \\ { \leq \boldsymbol \lambda ^ { k } \frac { \beta _ { h k } L _ { x } B _ { z } } { ( 1 - \boldsymbol \lambda ) } \displaystyle \sum _ { i = t } ^ { T - 1 } \boldsymbol \lambda ^ { i - t } } \\ { \leq \boldsymbol \lambda ^ { k } \frac { \beta _ { h k } L _ { x } B _ { z } } { ( 1 - \boldsymbol \lambda ) ^ { 2 } } , } \end{array}
+$$
+
+so (c) is bounded by $\begin{array} { r } { k \lambda ^ { k } \frac { L _ { p } L _ { w } \beta _ { h h } L _ { x } B _ { x } } { ( 1 - \lambda ) ^ { 2 } } } \end{array}$ . Ignoring Lipschitz and smoothness constants, we’ve shown the entire sum is $\begin{array} { r } { O \left( \frac { k \lambda ^ { k } } { ( 1 - \lambda ) ^ { 2 } } \right) } \end{array}$
+
+In this section, we prove that the gradient map $\nabla _ { w } p _ { T }$ is Lipschitz. First, we show on the forward pass, the difference between hidden states $h _ { t } ( w )$ and $h _ { t } ^ { \prime } ( w ^ { \prime } )$ obtained by running the model with weights $w$ and $w ^ { \prime }$ , respectively, is bounded in terms of $\| w - w ^ { \prime } \|$ . Using smoothness of $\phi$ , the difference in gradients can be written in terms of $\| h _ { t } ( w ) - h _ { t } ^ { \prime } ( w ^ { \prime } ) \|$ , which in turn can be bounded in terms of $\| w - w ^ { \prime } \|$ . We repeatedly leverage this fact to conclude the total difference in gradients must be similarly bounded.
+
+We first show small differences in weights don’t significantly change the trajectory of the recurrent model.
+
+Lemma 5. For some $w , w ^ { \prime }$ , suppose $\phi _ { w } , \phi _ { w ^ { \prime } }$ are $\lambda$ -contractive and $L _ { w }$ Lipschitz in $w$ . Let $h _ { t } ( w ) , h _ { t } ( w ^ { \prime } )$ be the hidden state at time $t$ obtain from running the model with weights $w , w ^ { \prime }$ on common inputs $\{ x _ { t } \}$ . If $h _ { 0 } ( w ) = h _ { 0 } ( w ^ { \prime } )$ , then
+
+$$
+\| h _ { t } ( w ) - h _ { t } ( w ^ { \prime } ) \| \leq \frac { L _ { w } \| w - w ^ { \prime } \| } { ( 1 - \lambda ) } .
+$$
+
+Proof. By triangle inequality, followed by the Lipschitz and contractivity assumptions,
+
+$$
+\begin{array} { r l } & { \left\| h _ { t } ( w ) - h _ { t } ( w ^ { \prime } ) \right\| } \\ & { = \left\| \phi _ { w } ( h _ { t - 1 } ( w ) , x _ { t } ) - \phi _ { w ^ { \prime } } ( h _ { t - 1 } ( w ^ { \prime } ) , x _ { t } ) \right\| } \\ & { \leq \left\| \phi _ { w } ( h _ { t - 1 } ( w ) , x _ { t } ) - \phi _ { w ^ { \prime } } ( h _ { t - 1 } ( w ) , x _ { t } ) \right\| + \left\| \phi _ { w ^ { \prime } } ( h _ { t - 1 } ( w ) , x _ { t } ) - \phi _ { w ^ { \prime } } ( h _ { t - 1 } ( w ^ { \prime } ) , x _ { t } ) \right\| } \\ & { \leq L _ { w } \left\| w - w ^ { \prime } \right\| + \lambda \left\| h _ { t - 1 } ( w ) - h _ { t - 1 } ( w ^ { \prime } ) \right\| . } \end{array}
+$$
+
+Iterating this argument and then using $h _ { 0 } ( w ) = h _ { 0 } ( w ^ { \prime } )$ , we obtain a geometric series in $\lambda$ .
+
+$$
+\begin{array} { r l } & { \| h _ { t } ( w ) - h _ { t } ( w ^ { \prime } ) \| \leq L _ { w } \| w - w ^ { \prime } \| + \lambda \| h _ { t - 1 } ( w ) - h _ { t - 1 } ( w ^ { \prime } ) \| } \\ & { \qquad \leq \displaystyle \sum _ { i = 0 } ^ { t } L _ { w } \| w - w ^ { \prime } \| \lambda ^ { i } } \\ & { \qquad \leq \displaystyle \frac { L _ { w } \| w - w ^ { \prime } \| } { ( 1 - \lambda ) } . } \end{array}
+$$
+
+The proof of Lemma 3 is similar in structure to Lemma 2, and follows from repeatedly using smoothness of $\phi$ and Lemma 5.
+
+Proof of Lemma 3. Let $h _ { t } ^ { \prime } ~ = ~ h _ { t } ( w ^ { \prime } )$ . Expanding the gradients and using $\lVert h _ { t } ( w ) - h _ { t } ( w ^ { \prime } ) \rVert ~ \leq$ $\frac { L _ { w } \left| \left| w - w ^ { \prime } \right| \right| } { ( 1 - \lambda ) }$ from Lemma 5.
+
+$$
+\begin{array} { r l } & { \| \nabla _ { x } p _ { T } ( w ) - \nabla _ { w } p _ { T } ( w ^ { ' } ) \| } \\ & { \le \displaystyle \sum _ { t = 1 } ^ { T } \left\| \left( \frac { \partial h _ { T } } { \partial h _ { t } } \frac { \partial h _ { t } } { \partial w } \right) ^ { \top } \nabla _ { h } p _ { T } - \left( \frac { \partial h _ { T } ^ { t } } { \partial h _ { t } ^ { t } } \frac { \partial h _ { t } ^ { t } } { \partial w } \right) ^ { \top } \nabla _ { h _ { T } , p } \eta \right\| } \\ & { \le \displaystyle \sum _ { t = 1 } ^ { T } \left\| \nabla _ { h _ { T } } p _ { T } - \nabla _ { h _ { T } ^ { t } } p _ { T } \right\| \frac { \partial h _ { T } ^ { t } } { \partial h _ { t } ^ { t } } \frac { \partial h _ { t } ^ { t } } { \partial w } \bigg \| + \| \nabla _ { t _ { 2 } } p _ { T } \| \left\| \frac { \partial h _ { T } } { \partial h _ { t } } \frac { \partial h _ { t } } { \partial w } - \frac { \partial h _ { T } ^ { t } } { \partial h _ { t } ^ { t } } \frac { \partial h _ { t } ^ { t } } { \partial w } \right\| } \\ & { \le \displaystyle \sum _ { t = 1 } ^ { T } \beta _ { \eta } \| h _ { T } - h _ { T } ^ { t } \| \boldsymbol { \lambda } ^ { T - t } L _ { w } + L _ { p } \left\| \frac { \partial h _ { T } } { \partial h _ { t } } \frac { \partial h _ { t } } { \partial w } - \frac { \partial h _ { T } ^ { t } } { \partial w } \frac { \partial h _ { t } ^ { t } } { \partial w } \right\| } \\ & { \le \frac { \beta _ { \eta } L _ { \eta } ^ { 2 } \left\| w - w ^ { \prime } \right\| } { ( 1 - \lambda ) ^ { 2 } } + L _ { p } \displaystyle \sum _ { t = 1 } ^ { T } \left\| \frac { \partial h _ { T } } { \partial h _ { t } } \frac { \partial h _ { t } } { \partial w } - \frac { \partial h _ { T } ^ { t } } { \partial h _ { t } ^ { t } } \frac { \partial h _ { t } ^ { t } } { \partial w } \right\| . } \end{array}
+$$
+
+Focusing on term (a),
+
+$$
+\begin{array} { r l } & { \displaystyle L _ { p } \sum _ { \ell = 1 } ^ { T } \left\| \frac { \partial h _ { T } } { \partial h _ { t } } \frac { \partial h _ { t } } { \partial w } - \frac { \partial h _ { T } ^ { \prime } } { \partial h _ { t } ^ { \prime } } \frac { \partial h _ { t } ^ { \prime } } { \partial w } \right\| } \\ & { \displaystyle \leq L _ { p } \sum _ { \ell = 1 } ^ { T } \left\| \frac { \partial h _ { T } } { \partial h _ { t } } - \frac { \partial h _ { T } ^ { \prime } } { \partial h _ { t } ^ { \prime } } \right\| \left\| \frac { \partial h _ { t } } { \partial w } \right\| + L _ { p } \left\| \frac { \partial h _ { T } ^ { \prime } } { \partial h _ { t } ^ { \prime } } \right\| \left\| \frac { \partial h _ { t } } { \partial w } - \frac { \partial h _ { t } ^ { \prime } } { \partial w } \right\| } \\ & { \displaystyle \leq L _ { p } L _ { w } \sum _ { \ell = 1 } ^ { T } \left\| \frac { \partial h _ { T } } { \partial h _ { t } } - \frac { \partial h _ { T } ^ { \prime } } { \partial h _ { t } ^ { \prime } } \right\| + L _ { p } \sum _ { \ell = 1 } ^ { T } \lambda ^ { T - t } ( \beta _ { w h } \| h _ { t } - h _ { t } ^ { \prime } \| + \beta _ { w w } \| w - w ^ { \prime } \| ) } \\ & { \displaystyle \leq L _ { p } L _ { w } \sum _ { \ell = 1 } ^ { T } \left\| \frac { \partial h _ { T } } { \partial h _ { t } } - \frac { \partial h _ { T } ^ { \prime } } { \partial h _ { t } ^ { \prime } } \right\| + \frac { L _ { p } \beta _ { w h } L _ { w } \| w - w ^ { \prime } \| } { ( 1 - \lambda ) ^ { 2 } } + \frac { L _ { p } \beta _ { w w } \| w - w ^ { \prime } \| } { ( 1 - \lambda ) } , } \end{array}
+$$
+
+where the penultimate line used,
+
+$$
+\begin{array} { r l } {  { \| \frac { \partial h _ { t } } { \partial w } - \frac { \partial h _ { t } ^ { \prime } } { \partial w } \| \leq \| \frac { \partial \phi _ { w } ( h _ { t - 1 } , x _ { t } ) } { \partial w } - \frac { \partial \phi _ { w } ( h _ { t - 1 } ^ { \prime } , x _ { t } ) } { \partial w } \| + \| \frac { \partial \phi _ { w } ( h _ { t - 1 } ^ { \prime } , x _ { t } ) } { \partial w } - \frac { \partial \phi _ { w ^ { \prime } } ( h _ { t - 1 } ^ { \prime } , x _ { t } ) } { \partial w } \| } } \\ & { \leq \beta _ { w h } \| h - h ^ { \prime } \| + \beta _ { w w } \| w - w ^ { \prime } \| . } \end{array}
+$$
+
+To bound (b), we peel off terms one by one using the triangle inequality,
+
+$$
+\begin{array} { r l } &  \begin{array} { r l } & { L _ { P r } L _ { \infty } \displaystyle \sum _ { i = 1 } ^ { N } \left\| \frac { \partial L _ { T } } { \partial t _ { i } } - \frac { \partial N _ { T } ^ { * } } { \partial t _ { i } ^ { * } } \right\| } \\ & { \leq L _ { P r } L _ { \infty } \displaystyle \sum _ { i = 1 } ^ { N } \left\| \frac { \partial L _ { T } } { \partial t _ { i - 1 } } - \frac { \partial N _ { T } ^ { * } } { \partial t _ { i - 1 } ^ { * } } \right\| \left\| \frac { \partial B _ { P r - 1 } } { \partial t _ { i + 1 } } \right\| _ { t } \left\| \frac { \partial N _ { T } ^ { * } } { \partial t _ { i - 1 } ^ { * } } \right\| \left\| \frac { \partial B _ { T - 1 } } { \partial t _ { i } } - \frac { \partial N _ { T - 1 } ^ { * } } { \partial t _ { i } ^ { * } } \right\| } \\ & { \leq L _ { P r } L _ { \infty } \displaystyle \sum _ { i = 1 } ^ { N } \left[ \left( \partial _ { t h } \left\| b _ { T - 1 } - h _ { T - 1 } ^ { * } \right\| \right\| , \ \left| \ \partial _ { t h } \left\| | \sigma _ { i } - \sigma ^ { * } \right\| \right) \right] ^ { 1 / 2 - \epsilon - 1 } \mathrm { ~ t ~ } \lambda _ { t } \left\| \frac { \partial N _ { T - 1 } } { \partial t _ { t } } - \frac { \partial N _ { T - 1 } ^ { * } } { \partial t _ { i } ^ { * } } \right\| } \\ & { \leq L _ { P r } L _ { \infty } \displaystyle \sum _ { i = 1 } ^ { N } \left[ \beta _ { h \infty } ( T - t ) \chi ^ { * - t _ { i - 1 } } \left\| \mathrm { ~ t ~ } - \sigma ^ { * } \right\| _ { t } \left\| b _ { T - 1 } \right\| _ { t } \left\| b _ { T - 2 } - b _ { T - 1 } ^ { * } \right\| \chi ^ { * - t _ { i } } \right\| \right] } \\ &  \leq L _ { P r } L _ { \infty } \displaystyle \sum _ { i = 1 } ^ { N } \left[ \beta _ { h \infty } ( T - t ) \chi ^ { * - t _ { i } } \right\| \| x - \sigma ^ { * } \| + \beta _ { h \infty } \displaystyle \sum _ { i = 1 } ^ { T - t _ { i } } \left\| b _ { T - i } - \frac { \partial L _ { T - 1 } }  \end{array} \end{array}
+$$
+
+Supressing Lipschitz and smoothness constants, we’ve shown the entire sum is ${ \cal O } ( 1 / ( 1 - \lambda ) ^ { 3 } )$ , as required. □
+
+# B.1.3 GRADIENT DESCENT ANALYSIS
+
+Equipped with the smoothness and truncation lemmas (Lemmas 2 and 3), we turn towards proving the main gradient descent result.
+
+Proof of Proposition 4. Let $\Pi _ { \Theta }$ denote the Euclidean projection onto $\Theta$ , and let $\begin{array} { r l } { \delta _ { i } } & { { } = } \end{array}$ $\lVert \boldsymbol { w } _ { \mathrm { r e c u r r } } ^ { i } - \boldsymbol { \dot { w } } _ { \mathrm { t r u n c } } ^ { i } \rVert$ . Initially $\delta _ { 0 } = 0$ , and on step $i + 1$ , we have the following recurrence rela
+
+tion for $\delta _ { i + 1 }$
+
+$$
+\begin{array} { r l } { \delta _ { i + 1 } = \left\| w _ { \mathrm { r e c u r r } } ^ { i + 1 } - w _ { \mathrm { t r u n c } } ^ { i + 1 } \right\| } & { } \\ { = \left\| \Pi _ { \Theta } ( w _ { \mathrm { r e c u r r } } ^ { i } - \alpha _ { i } \nabla p _ { T } ( w ^ { i } ) ) - \Pi _ { \Theta } ( w _ { \mathrm { t r u n c } } ^ { i } - \alpha _ { i } \nabla p _ { T } ^ { k } ( w _ { \mathrm { t r u n c } } ^ { i } ) ) \right\| } & { } \\ { \leq \left\| w _ { \mathrm { r e c u r r } } ^ { i } - \alpha _ { i } \nabla p _ { T } ( w ^ { i } ) ) - w _ { \mathrm { t r u n c } } ^ { i } - \alpha _ { i } \nabla p _ { T } ^ { k } ( w _ { \mathrm { t r u n c } } ^ { i } ) \right\| } & { } \\ { \leq \left\| w _ { \mathrm { r e c u r r } } ^ { i } - w _ { \mathrm { t r u n c } } ^ { i } \right\| + \alpha _ { i } \left\| \nabla p _ { T } ( w _ { \mathrm { r e c u r r } } ^ { i } ) - \nabla p _ { T } ^ { k } ( w _ { \mathrm { t r u n c } } ^ { i } ) \right\| } & { } \\ { \leq \delta _ { i } + \alpha _ { i } \left\| \nabla p _ { T } ( w _ { \mathrm { r e c u r r } } ^ { i } ) - \nabla p _ { T } ( w _ { \mathrm { t r u n c } } ^ { i } ) \right\| + \alpha _ { i } \left\| \nabla p _ { T } ( w _ { \mathrm { t r u n c } } ^ { i } ) - \nabla p _ { T } ^ { k } ( w _ { \mathrm { t r u n c } } ^ { i } ) \right\| } & { } \\ { \leq \delta _ { i } + \alpha _ { i } \left( \beta \delta _ { i } + \gamma k \lambda ^ { k } \right) } & { } \\ { \leq \exp \left( \alpha _ { i } \beta \right) \delta _ { i } + \alpha _ { i } \gamma k \lambda ^ { k } , } \end{array}
+$$
+
+the penultimate line applied lemmas 2 and 3, and the last line used $1 + x \leq e ^ { x }$ for all $x$ . Unwinding the recurrence relation at step $N$ ,
+
+$$
+\begin{array} { l } { \displaystyle \delta _ { N } \leq \sum _ { i = 1 } ^ { N } \left\{ \prod _ { j = i + 1 } ^ { N } \exp ( \alpha _ { j } \beta ) \right\} \alpha _ { i } \gamma k \lambda ^ { k } } \\ { \displaystyle \quad \leq \sum _ { i = 1 } ^ { N } \left\{ \prod _ { j = i + 1 } ^ { N } \exp \left( \frac { \alpha \beta } { j } \right) \right\} \frac { \alpha \gamma k \lambda ^ { k } } { i } } \\ { \displaystyle \quad = \sum _ { i = 1 } ^ { N } \left\{ \exp \left( \alpha \beta \sum _ { j = i + 1 } ^ { N } \frac { 1 } { j } \right) \right\} \frac { \alpha \gamma k \lambda ^ { k } } { i } . } \end{array}
+$$
+
+Bounding the inner summation via an integral, $\begin{array} { r } { \sum _ { j = i + 1 } ^ { N } \frac { 1 } { j } \le \log ( N / i ) } \end{array}$ and simplifying the resulting expression,
+
+$$
+\begin{array} { c } { { \delta _ { N } \leq \displaystyle \sum _ { i = 1 } ^ { N } \exp ( \alpha \beta \log ( N / i ) ) \frac { \alpha \gamma k \lambda ^ { k } } { i } } } \\ { { = \alpha \gamma k \lambda ^ { k } N ^ { \alpha \beta } \displaystyle \sum _ { i = 1 } ^ { N } \frac { 1 } { i ^ { \alpha \beta + 1 } } } } \\ { { \leq \alpha \gamma k \lambda ^ { k } N ^ { \alpha \beta + 1 } . } } \end{array}
+$$
+
+# B.1.4 PROOF OF THEOREM 1
+
+Proof of Theorem $^ { l }$ . Using $f$ is $L _ { f }$ -Lipschitz and the triangle inequality,
+
+$$
+\begin{array} { r l } & { \left\| y _ { T } - y _ { T } ^ { k } \right\| \leq L _ { f } \left\| h _ { T } ( w _ { \mathrm { r e c u r r } } ^ { N } ) - h _ { T } ^ { k } ( w _ { \mathrm { t r u n c } } ^ { N } ) \right\| } \\ & { \qquad \leq L _ { f } \left\| h _ { T } ( w _ { \mathrm { r e c u r r } } ^ { N } ) - h _ { T } ( w _ { \mathrm { t r u n c } } ^ { N } ) \right\| + L _ { f } \left\| h _ { T } ( w _ { \mathrm { t r u n c } } ^ { N } ) - h _ { T } ^ { k } ( w _ { \mathrm { t r u n c } } ^ { N } ) \right\| . } \end{array}
+$$
+
+By Lemma 5, the first term is bounded by $\frac { L _ { w } \left| \left| w _ { \mathrm { r e c u r r } } ^ { N } - w _ { \mathrm { t r u n c } } ^ { N } \right| \right| } { ( 1 - \lambda ) }$ , and by Lemma 1, the second term is bounded by λk LxBx(1−λ) . Using Proposition 4, after $N$ steps of gradient descent, we have
+
+$$
+\begin{array} { r l r } {  { \| y _ { T } - y _ { T } ^ { k } \| \leq \frac { L _ { f } L _ { w } \| w _ { \mathrm { r e c u r r } } ^ { N } - w _ { \mathrm { t r u n c } } ^ { N } \| } { ( 1 - \lambda ) } + \lambda ^ { k } \frac { L _ { f } L _ { x } B _ { x } } { ( 1 - \lambda ) } } } \\ & { } & { \leq k \lambda ^ { k } \frac { \alpha L _ { f } L _ { w } N ^ { \alpha \beta + 1 } } { ( 1 - \lambda ) } + \lambda ^ { k } \frac { L _ { f } L _ { x } B _ { x } } { ( 1 - \lambda ) } , ~ } \end{array}
+$$
+
+and solving for $k$ such that both terms are less than $\varepsilon / 2$ gives the result.
+
+![](images/06878230c5636e61275c6ed52c1e2349ba3b84d29b5d7597346a85e401456e43.jpg)  
+Figure 6: Empirical validation Proposition 4 on random Gaussian instances. Without the $1 / t$ rate, the gradient descent bound no longer appears qualitatively correct, suggesting the $O ( 1 / t )$ rate is necessary.
+
+# C EXPERIMENTS
+
+The $O ( 1 / t )$ rate may be necessary. The key result underlying Theorem 1 is the bound on the parameter difference $\lVert w _ { \mathrm { t r u n c } } - w _ { \mathrm { r e c u r r } } \rVert$ while running gradient descent obtained in Proposition 4. We show this bound has the correct qualitative scaling using random instances and training randomly initialized, stable linear dynamical systems and tanh-RNNs. In Figure 6, we plot the parameter error $\lVert w _ { \mathrm { t r u n c } } ^ { t } - w _ { \mathrm { r e c u r r } } ^ { t } \rVert$ as training progresses for both models (averaged over 10 runs). The error scales comparably with the bound given in Proposition 4. We also find for larger step-sizes like $\alpha / \sqrt { t }$ or constant $\alpha$ , the bound fails to hold, suggesting the $O ( 1 / t )$ condition is necessary.
+
+Concretely, we generate random problem instance by fixing a sequence length $T = 2 0 0$ , sampling input data $x _ { t } \stackrel { \mathrm { i . i . d . } } { \sim } \mathcal { N } \left( 0 , 4 \cdot I _ { 3 2 } \right)$ , and sampling $y _ { T } \ \sim \ \mathrm { U n i f [ - 2 , 2 ] }$ . Next, we set $\lambda = 0 . 7 5$ and randomly initialize a stable linear dynamical system or RNN with tanh non-linearity by sampling $U _ { i j } , W _ { i j } \stackrel { \mathrm { i . i . d . } } { \sim } \ N ( 0 , 0 . 5 )$ and thresholding the singular values of $W$ so $\| W \| \leq \lambda$ . We use the squared loss and prediction function $f ( h _ { t } , x _ { t } ) = C h _ { t } + D x _ { t }$ , where $C , D \stackrel { \mathrm { i . i . d . } } { \sim } \mathcal { N } \left( 0 , I _ { 3 2 } \right)$ . We fix the truncation length to $k = 3 5$ , set the learning rate to $\alpha _ { t } = \alpha / t$ for $\alpha = 0 . 0 1$ , and take $N = 2 0 0$ gradient steps. These parameters are chosen so that the $\gamma k \lambda ^ { k } N ^ { \dot { \alpha } \beta + 1 }$ bound from Proposition 4 does not become vacuous – by triangle inequality, we always have $\lVert w _ { \mathrm { t r u n c } } - w _ { \mathrm { r e c u r r } } \rVert \leq 2 \lambda$ .
+
+Stable vs. unstable models. The word and character level language modeling experiments are based on publically available code from Merity et al. (2018). The polyphonic music modeling code is based on the code in Bai et al. (2018), and the slot-filling model is a reimplementation of Mesnil et al. (2015) 1
+
+Since the sufficient conditions for stability derived in Section 2.2 only apply for networks with a single layer, we use a single layer RNN or LSTM for all experiments. Further, our theoretical results are only applicable for vanilla SGD, and not adaptive gradient methods, so all models are trained with SGD. Table 2 contains a summary of all the hyperparameters for each experiment.
+
+All hyperparameters are shared between the stable and unstable variants of both models. In the RNN case, enforcing stability is conceptually simple, though computationally expensive. Since tanh is 1- Lipschitz, the RNN is stable as long as $\lVert W \rVert < 1$ . Therefore, after each gradient update, we project $W$ onto the spectral norm ball by taking the SVD and thresholding the singular values to lie in $[ 0 , 1 )$ . In the LSTM case, enforcing stability is conceptually more difficult, but computationally simple. To ensure the LSTM is stable, we appeal to Proposition 2. We enforce the following inequalities after each gradient update
+
+Table 2: Hyperparameters for all experiments   
+
+<table><tr><td rowspan="2"></td><td colspan="2">Model</td></tr><tr><td></td><td>RNNLSTM</td></tr><tr><td colspan="3">Word LM</td></tr><tr><td rowspan="10"></td><td>Number layers</td><td>1</td><td>1</td></tr><tr><td>Hidden units</td><td>256</td><td>1024</td></tr><tr><td>Embedding size</td><td>1024</td><td>512</td></tr><tr><td>Dropout</td><td>0.25</td><td>0.65</td></tr><tr><td>Batch size</td><td>20</td><td>20</td></tr><tr><td>Learning rate</td><td>2.0</td><td>20.</td></tr><tr><td>BPTT</td><td>35</td><td>35</td></tr><tr><td>Gradient clipping</td><td>0.25</td><td>1.0</td></tr><tr><td>Epochs</td><td>40</td><td>40</td></tr><tr><td>Number layers</td><td>1</td><td>1</td></tr><tr><td rowspan="14"></td><td>Hidden units</td><td>768</td><td>1024</td></tr><tr><td>Embedding size</td><td>400</td><td>400</td></tr><tr><td>Dropout</td><td>0.1</td><td>0.1</td></tr><tr><td>Weight decay</td><td>1e-6</td><td>1e-6</td></tr><tr><td>Batch size</td><td>80</td><td>80</td></tr><tr><td>Learning rate</td><td>2.0 150</td><td>20.0</td></tr><tr><td>BPTT</td><td></td><td>150</td></tr><tr><td>Gradient clipping</td><td>1.0</td><td>1.0</td></tr><tr><td>Epochs Number layers</td><td>300 1</td><td>300</td></tr><tr><td>Polyphonic Music</td><td></td><td>1</td><td></td></tr><tr><td>Hidden units</td><td>1024</td><td></td><td>1024</td></tr><tr><td>Dropout Batch size</td><td></td><td>0.1</td><td>0.1</td></tr><tr><td></td><td></td><td>1</td><td>1</td></tr><tr><td></td><td>Learning rate</td><td>0.05</td><td>2.0</td></tr><tr><td></td><td>Gradient clipping</td><td>5.0</td><td>5.0</td></tr><tr><td>Slot-Filling</td><td>Epochs</td><td>100</td><td>100</td></tr><tr><td></td><td>Number layers</td><td>1</td><td>1</td></tr><tr><td></td><td>Hidden units</td><td>128</td><td>128</td></tr><tr><td></td><td>Embedding size</td><td>64</td><td>64</td></tr><tr><td></td><td>Dropout</td><td>0.5</td><td>0.5</td></tr><tr><td></td><td>Weight decay</td><td>1e-4</td><td>1e-4</td></tr><tr><td></td><td>Batch size</td><td>128</td><td>128</td></tr><tr><td></td><td>Learning rate</td><td>10.0</td><td>10.0</td></tr><tr><td></td><td>Gradient clipping</td><td>1.0</td><td>1.0</td></tr><tr><td></td><td>Epochs</td><td>100</td><td>100</td></tr></table>
+
+1. The hidden-to-hidden forget gate matrix should satisfy $\| W _ { f } \| _ { \infty } < 0 . 1 2 8$ , which is enforced by normalizing the $\ell _ { 1 }$ - norm of each row to have value at most 0.128.   
+2. The input vectors $x _ { t }$ must satisfy $\| x _ { t } \| _ { \infty } \leq B _ { x } = 0 . 7 5$ , which is achieved by thresholding all values to lie in $[ - 0 . 7 5 , 0 . 7 5 ]$ .   
+3. The bias of the forget gate $b _ { f }$ , must satsify $\| b _ { f } \| _ { \infty } \leq 0 . 2 5$ , which is again achieved by thresholding all values to lie in $[ - 0 . 2 5 , 0 . 2 5 ]$ .   
+4. The input-hidden forget gate matrix $U _ { f }$ should satisfy $\| U _ { f } \| _ { \infty } \le 0 . 2 5$ . This is enforced by normalizing the $\ell _ { 1 } \cdot$ - norm of each row to have value at most 0.25.   
+5. Given 1-4, the forget gate can take value at most $f _ { \infty } < 0 . 6 4$ . Consequently, we enforce $\| W _ { i } \| _ { \infty } , \| W _ { o } \| _ { \infty } \overset { \_ } { \leq } 0 . 3 6$ , $\lVert W _ { z } \rVert \leq 0 . 0 9 1$ , and $\| \dot { W } _ { f } \| _ { \infty } < \operatorname* { m i n } \left\{ 0 . 1 \dot { 2 } 8 , ( 1 \dot { - } 0 . 6 4 ) ^ { 2 } \right\} =$ 0.128.
+
+After 1-5 are enforced, by Proposition 2, the resulting (iterated)-LSTM is stable. Although the above description is somewhat complicated, the implementation boils down to normalizing the rows of the LSTM weight matrices, which can be done very efficiently in a few lines of PyTorch.
+
+Data-dependent stability. Unlike the RNN, in an LSTM, it is not clear how to analytically compute the stability parameter $\lambda$ . Instead, we rely on a heuristic method to estimate $\lambda$ . Recall a model is stable if for all ${ \bar { \boldsymbol { x } } } , h , h ^ { \prime }$ , we have
+
+$$
+S ( h , h ^ { \prime } , x ) : = \frac { \| \phi _ { w } ( h , x ) - \phi _ { w } ( h ^ { \prime } , x ) \| } { \| h - h ^ { \prime } \| } \leq \lambda < 1 .
+$$
+
+To estimate $\begin{array} { r } { \operatorname* { s u p } _ { h , h ^ { \prime } , x } S ( h , h ^ { \prime } , x ) } \end{array}$ , we do the following. First, we take $x$ to be point in the training set. In the language modeling case, $x$ is one of the learned word-vectors. We randomly sample and fix $x$ , and then we perform gradient ascent on $S ( h , h ^ { \prime } , x )$ to find worst-case $h , h ^ { \prime }$ . In our experiments, we initialize $h , h ^ { \prime } \sim \mathcal { N } \left( 0 , 0 . 1 \cdot I \right)$ and run gradient ascent with learning rate 0.9 for 1000 steps. This procedure is repeated 20 times, and we estimate $\lambda$ as the maximum value of $S ( h , h ^ { \prime } , x )$ encounted during any iteration from any of the 20 random starting points.

parse/train/JzdYX8uzT4W/JzdYX8uzT4W.md

@@ -0,0 +1,352 @@
+# Decoupled Contrastive Learning
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 Contrastive learning (CL) is one of the most successful paradigms for self  
+2 supervised learning (SSL). Specifically, contrastive learning treats two augmented   
+3 “views” of the same sample as positive, pulling them close and treating all other   
+4 samples as negative to push them far apart. Despite the evident success of CL SSL   
+5 methods, there are several challenges in the existing methods as they may require   
+6 special structures, large batches, or huge training epochs, etc. Our motivation in   
+7 this work is to provide a simple, efficient, and yet competitive contrastive learning   
+8 baseline. Through both theoretical and empirical studies, we identified a strong   
+9 negative-positive-coupling (NPC) effect in the widely used cross-entropy loss in   
+10 CL SSL methods. We hypothesize that the NPC effect may be a major cause of the   
+11 inefficiency in many contrastive learning methods. By removing the NPC effect,   
+12 we reach a decoupled contrastive learning (DCL) objective function, which signifi  
+13 cantly improves the training efficiency. DCL can achieve competitive performance,   
+14 requiring neither large batches in SimCLR, momentum encoding in Moco, or large   
+15 epochs. We demonstrate the benefit of DCL in various benchmarks. Further, DCL   
+16 is also much less sensitive to suboptimal hyperparameters. Notably, our approach   
+17 achieves $6 6 . 9 \%$ ImageNet top-1 accuracy with 256 batch size within 200 epochs   
+18 pre-training, which outperforms its baseline SimCLR by $5 . 1 \%$ . We believe DCL   
+19 may provide a strong baseline for future contrastive learning-based SSL studies.
+
+# 20 1 Introduction
+
+21 As a fundamental task in machine learning, representation learning aims to extract features to   
+22 reconstruct the raw data fully. It has been regarded as a long-acting goal over the past decades. Recent   
+23 progress on representation learning has achieved a significant milestone over self-supervised learning   
+24 (SSL), facilitating feature learning with its competence in exploiting massive raw data without any   
+25 annotated supervision. In the early stage of SSL, representation learning has focused on exploiting   
+26 pretext tasks, which are addressed by generating pseudo-labels to the unlabeled data through different   
+27 transformations, such as solving jigsaw puzzles [1], colorization [2] and rotation prediction [3].   
+28 Though these approaches achieve some success in computer vision, there is a large gap between   
+29 these methods and supervised learning. Recently, there has been a significant advancement in using   
+30 contrastive learning [4, 5, 6, 7, 8] for self-supervised pre-training, which significantly closes the gap   
+31 between the SSL method and supervised learning. Contrastive SSL methods, e.g., SimCLR [8], in   
+32 general, try to pull different views of the same instance close and push different instances far apart in   
+33 the representation space.   
+34 Despite the evident progress of the state-of-the-art contrastive SSL methods, there have been several   
+35 challenges in future developing this direction: 1) The SOTA models [7] may require unique structures   
+36 like the momentum encoder and large memory queues, which may complicate the understanding. 2)   
+37 The contrastive SSL models [8] may depend on large batch size and huge epoch numbers to achieve   
+38 competitive performance, posing a computational challenge for academia to explore this direction.   
+39 3) They may be sensitive to hyperparameters and optimizers, introducing additional difficulty to   
+40 reproduce the results on various benchmarks.   
+41 Our motivation in this work is to provide a simple, efficient, and yet competitive contrastive learning   
+42 baseline. We choose SimCLR as our starting point, given its conceptual simplicity. By analyzing the   
+43 objective function, we identified a Negative-Positive-Coupling (NPC) multiplier $q _ { B }$ in the gradient   
+44 as shown in Proposition 1. The NPC multipliers modulate the gradient of each sample, and it   
+45 mistakenly increases the impact of both negative samples and positive samples, given either of them   
+46 is more informative. Such a coupling exacerbates when smaller batch sizes are used. By removing   
+47 the coupling term, we reach a new formulation, the decoupled contrastive learning (DCL). The   
+48 new objective function significantly improves the training efficiency, requires neither large batches,   
+49 momentum encoding, or large epochs to achieve competitive performance on various different   
+50 benchmarks. Specifically, DCL reaches $6 6 . 9 \%$ ImageNet top-1 (linear probing) accuracy with batch   
+51 size 256, SGD optimizer within 200 epochs. Even if DCL is trained for 100 epochs, it still reaches   
+52 $6 4 . 6 \%$ ImageNet top-1 accuracy with batch size 256.
+
+![](images/bc25c291918beafdd2137b380f58aba11d9866e7e06ee97d4153116558915405.jpg)  
+Figure 1: An overview of the batch size issue in the general contrastive approaches: (a) shows the NPC multiplier $q _ { B }$ in different batch sizes. As the large batch size increasing the $q _ { B }$ will approach 1 with a small coefficient of variation. (b) illustrates the distribution of $q _ { B }$ .
+
+53 In short, this work makes the following contributions:
+
+1) We provide both theoretical analysis and empirical evidence to show the negative-positive coupling in the gradient of contrastive learning;   
+2) We introduce a new, decoupled contrastive learning (DCL) objective, which casts off the coupling phenomenon between positive and negative samples in contrastive learning, and significantly improves the training efficiency; Additionally, the proposed DCL objective is less sensitive the several important hyperparameters;   
+3) We demonstrate our approach via extensive experiments and analysis on both large and small-scale vision benchmarks, with an optimal configuration for the standard SimCLR baseline to have a competitive performance within contrastive approaches.
+
+# 63 2 Related work
+
+# 2.1 Self-supervised representation learning
+
+65 Self-supervised representation learning (SSL) aims to learn a robust embedding space from data   
+66 without human annotation. Previous arts can be roughly categorized into generative and discriminative.   
+67 Generative approaches, such as autoencoders and adversarial learning, focus on reconstructing   
+68 images from latent representations [9, 10]. Conversely, recent discriminative approaches, especially   
+69 contrastive learning-based approaches, have gained the most ground and achieved state-of-the-art   
+70 standard large-scale image classification benchmarks with increasingly more compute and data   
+71 augmentations.
+
+# 72 2.2 Contrastive learning
+
+73 Contrastive learning (CL) constructs positive and negative sample pairs to extract information from   
+74 the data itself. In CL, each anchor image in a batch has only one positive sample to construct a positive   
+75 sample pair [11, 8, 7]. CPC [5] predicts the future output of sequential data by using current output   
+76 as prior knowledge, which can improve the feature representing the ability of the model. Instance   
+77 discrimination [4] proposes a non-parametric cross-entropy loss to optimize the model at the instance   
+78 level. Inv. spread [12] makes use of data augmentation invariants and the spread-out property of   
+79 instance to learn features. MoCo [7] proposes a dictionary to maintain a negative sample set, thus   
+80 increasing the number of negative sample pairs. Different from the aforementioned self-supervised   
+81 CL approaches, [13] proposes a supervised CL that considers all the same categories as positive pairs   
+82 to increase the utility of images.
+
+# 83 2.3 Collapsing issue via batch size and negative sample
+
+84 In CL, the objective is to maximize the mutual information between the positive pairs. However, to   
+85 avoid the “collapsing output”, vast quantities of negative samples are needed so that the learning   
+86 objectives obtain the maximum similarity and have the minimum similarity with negative samples.   
+87 For instance, in SimCLR [8], training requires many negative samples, leading to a large batch size   
+88 (i.e., 4096). Furthermore, to optimize such a huge batch, a specially designed optimizer LARS [14]   
+89 is used. Similarly, MoCo [7] needs a vast queue (i.e., 65536) to achieve competitive performance.   
+90 BYOL [15] does not collapse output without using any negative samples by considering all the   
+91 images are positive and to maximize the similarity of “projection” and “prediction ” features. On the   
+92 other hand, Simsam [16] leverages the Siamese network to introduce inductive biases for modeling   
+93 invariance. With the small batch size (i.e., 256), Simsam is a rival to BYOL (4096). Unlike both   
+94 approaches that achieved their success through empirical studies, this paper tackles from a theoretical   
+95 perspective, proving that an intertwined multiplier $q _ { B }$ of positive and negative is the main issue to   
+96 contrastive learning.
+
+![](images/7cac23c9fb9b0b9a2a31e147df9e235518441732db80a04de0db6a27da23015b.jpg)  
+97 3 Decouple negative and positive samples in contrastive learning   
+Figure 2: Contrastive learning and negative-positive coupling (NPC). (a) In SimCLR, each sample $\mathbf { x } _ { i }$ has two augmented views $\{ \mathbf { x } _ { i } ^ { ( 1 ) } , \mathbf { x } _ { i } ^ { ( 2 ) } \}$ . They are encoded by the same encoder $f$ and further projected to $\{ \mathbf { z } _ { i } ^ { ( 1 ) } , \mathbf { z } _ { i } ^ { ( 2 ) } \}$ by a normalized MLP. (b) According to Equation 3. For the view $\mathbf { x } _ { i } ^ { ( 1 ) }$ , the cross-entropy loss ${ L } _ { i } ^ { ( 1 ) }$ leads to a positive force $\mathbf { z } _ { i } ^ { ( 2 ) }$ , which comes from the other view $\mathbf { x } _ { i } ^ { ( 2 ) }$ of $\mathbf { x }$ and a negative force, which is a weighted average of all the negative samples, i.e. $\{ \mathbf { z } _ { j } ^ { ( l ) } | l \in \{ 1 , 2 \} , j \neq i \}$ . However, the gradient $- \nabla _ { \mathbf { z } _ { i } ^ { ( 2 ) } } L _ { i } ^ { ( 1 ) }$ is proportional to the NPC multiplier.
+
+98 We choose to start from SimCLR because of its conceptual simplicity. Given a batch of $N$ samples (e.g. images), 99 $\{ \mathbf { x } _ { 1 } , \dotsc , \mathbf { x } _ { N } \}$ , let $\mathbf { x } _ { i } ^ { ( 1 ) } , \mathbf { x } _ { i } ^ { ( 2 ) }$ be two augmented views of the sample $x _ { i }$ and $B$ be the set of all of the augmented views in the batch, i.e. 100 $B = \{ \mathbf { x } _ { i } ^ { ( k ) } | k \in \{ 1 , 2 \} , i \in \mathbb { I } ^ { 1 , N } \mathbb { I } \}$ . As
+
+101 shown by Figure 2(a), each of the views $\mathbf { x } _ { i } ^ { ( k ) }$ is sent into the same encoder network $f$ and the output   
+102 h(k)i $\mathbf { h } _ { i } ^ { ( k ) } = f ( \mathbf { x } _ { i } ^ { ( k ) } )$ is then projected by a normalized MLP projector that $\mathbf { z } _ { i } ^ { ( k ) } = { g ( \mathbf { h } _ { i } ^ { ( k ) } ) } / \| g ( \mathbf { h } _ { i } ^ { ( k ) } ) \|$ .   
+103 For each augmented view $\mathbf { x } _ { i } ^ { ( k ) }$ , SimCLR solves a classification problem by using the rest of the   
+104 105 views in creates a $B$ as targets, and assigns thoss-entropy loss function $L _ { i } ^ { ( k ) }$ y positive labelfor each view $\mathbf { x } _ { i } ^ { ( k ) }$ ${ \bf x } _ { i } ^ { ( u ) }$ , where d the o $u \ne k$ . So SimCLRoss function is   
+106 $\begin{array} { r } { L = \sum _ { k \in \{ 1 , 2 \} , i \in [ [ 1 , N ] ] } L _ { i } ^ { ( k ) } } \end{array}$
+
+$$
+L _ { i } ^ { ( k ) } = - \log \frac { \exp ( \langle \mathbf { z } _ { i } ^ { ( 1 ) } , \mathbf { z } _ { i } ^ { ( 2 ) } \rangle / \tau ) } { \exp ( \langle \mathbf { z } _ { i } ^ { ( 1 ) } , \mathbf { z } _ { i } ^ { ( 2 ) } \rangle / \tau ) + \sum _ { l \in \{ 1 , 2 \} , j \in [ 1 , N ] , j \neq i } \exp ( \langle \mathbf { z } _ { i } ^ { ( k ) } , \mathbf { z } _ { j } ^ { ( l ) } \rangle / \tau ) }
+$$
+
+Proposition 1. There exists a negative-positive coupling (NPC) multiplier 107 $q _ { B , i } ^ { ( 1 ) }$ in the gradient of 108 ${ L } _ { i } ^ { ( 1 ) }$ :
+
+$$
+\begin{array}{c} \begin{array} { r } { \left\{ - \nabla _ { \mathbf { z } _ { i } ^ { ( 1 ) } } L _ { i } ^ { ( 1 ) } = \frac { q _ { B , i } ^ { ( 1 ) } } { \tau } \left[ \mathbf { z } _ { i } ^ { ( 2 ) } - \sum _ { l \in \{ 1 , 2 \} , j \in [ 1 , N ] , j \neq i } \frac { \exp { \langle \mathbf { z } _ { i } ^ { ( 1 ) } , \mathbf { z } _ { j } ^ { ( l ) } \rangle } / \tau } { \sum _ { q \in \{ 1 , 2 \} , j \in [ 1 , N ] , j \neq i } \exp ( \langle \mathbf { z } _ { i } ^ { ( 1 ) } , \mathbf { z } _ { j } ^ { ( 0 ) } \rangle / \tau ) } \cdot \mathbf { z } _ { j } ^ { ( l ) } \right] \right.} \\ { - \nabla _ { \mathbf { z } _ { i } ^ { ( 2 ) } } L _ { i } ^ { ( 1 ) } = \frac { q _ { B , i } ^ { ( 1 ) } } { \tau } \cdot \mathbf { z } _ { i } ^ { ( 1 ) } } \\ { - \nabla _ { \mathbf { z } _ { j } ^ { ( l ) } } L _ { i } ^ { ( 1 ) } = - \frac { q _ { B , i } ^ { ( 1 ) } } { \tau } \frac { \exp { \langle \mathbf { z } _ { i } ^ { ( 1 ) } , \mathbf { z } _ { j } ^ { ( l ) } \rangle } / \tau } { \sum _ { q \in \{ 1 , 2 \} , j \in [ 1 , N ] , j \neq i } \exp ( \langle \mathbf { z } _ { i } ^ { ( 1 ) } , \mathbf { z } _ { j } ^ { ( q ) } \rangle / \tau ) } \cdot \mathbf { z } _ { i } ^ { ( 1 ) } } \end{array}   \end{array}
+$$
+
+where the NPC multiplier 109 $q _ { B , i } ^ { ( 1 ) }$ is:
+
+$$
+q _ { B , i } ^ { ( 1 ) } = 1 - \frac { \exp ( \langle \mathbf { z } _ { i } ^ { ( 1 ) } , \mathbf { z } _ { i } ^ { ( 2 ) } \rangle / \tau ) } { \sum _ { q \in \{ 1 , 2 \} , j \in \mathbb { [ 1 , N ] } , j \neq i } \exp ( \langle \mathbf { z } _ { i } ^ { ( 1 ) } , \mathbf { z } _ { j } ^ { ( q ) } \rangle / \tau ) }
+$$
+
+Due to the symmetry, a similar NPC multiplier 110 $q _ { B , i } ^ { ( k ) }$ exists in the gradient of $L _ { i } ^ { ( k ) } , k \in \{ 1 , 2 \} , i \in$ 111 $[ 1 , N ]$ .
+
+112 $q _ { B , i } ^ { ( k ) }$ we can see, all of the partial gradients in Equation 2 are modified by the commonin Equation 3. Equation 3 makes intuitive sense: 1) When a positive sample pair $\{ \mathbf { z } _ { i } ^ { ( 1 ) } , \mathbf { z } _ { i } ^ { ( 2 ) } \}$ lier are   
+114 farther, the corresponding NPC multiplier $q _ { B , i } ^ { ( 1 ) }$ is larger. This will makes the overall gradient larger.   
+115 Otherwise, the gradient is smaller. 2) When a negative sample is closer to $\mathbf { z } _ { i } ^ { ( 1 ) }$ , it makes $q _ { B , i } ^ { ( 1 ) }$ larger.   
+116 Overall, the intuition here is that a positive sample farther from the target or a negative sample closer   
+117 to the target is more informative. However, the positive samples and negative samples are strongly   
+118 coupled. An outlier positive sample also makes the gradient from the negative samples significantly   
+119 larger and vice versa.   
+120 Figure 1(b) shows the NPC multiplier $q _ { B }$ distribution shift w.r.t. different batch sizes for a pre-trained   
+121 SimCLR baseline model. While all of the shown distributions have prominent fluctuation, the smaller   
+122 batch size makes $q _ { B }$ cluster towards 0, while the larger batch size pushes the distribution towards   
+123 $\delta ( 1 )$ . Figure 1(a) shows the averaged NPC multiplier $\langle q _ { B } \rangle$ changes w.r.t. the batch size and the   
+124 relative fluctuation. The small batch sizes introduce significant NPC fluctuation. Based on this   
+125 observation, we propose to remove the NPC multipliers from the gradients, which corresponds to the   
+126 case $q _ { B , N  \infty }$ . This leads to the decoupled contrastive learning formulation.   
+127 Proposition 2. Removing the positive pair from the denominator of Equation 2 leads to a decoupled   
+128 contrastive learning loss. If we remove the NPC multiplier $q _ { B , i } ^ { ( k ) }$ from Equation 2, we reach a   
+129 decoupled contrastive learning loss LDC = Pk∈{1,2},i∈[[1,N]] , where $L _ { D C , i } ^ { ( k ) }$ is:
+
+$$
+\begin{array} { r l } & { L _ { D C , i } ^ { ( k ) } = - \log \frac { \exp ( \langle \mathbf { z } _ { i } ^ { ( 1 ) } , \mathbf { z } _ { i } ^ { ( 2 ) } \rangle / \tau ) } { \exp ( \langle \mathbf { \hat { z } } _ { i } ^ { ( 1 ) } , \mathbf { z } _ { i } ^ { ( 2 ) } \rangle / \tau ) } + \sum _ { l \in \{ 1 , 2 \} , j \in [ 1 , N ] , j \neq i } \exp ( \langle \mathbf { z } _ { i } ^ { ( k ) } , \mathbf { z } _ { j } ^ { ( l ) } \rangle / \tau ) } \\ & { \quad \quad \quad = - \langle \mathbf { z } _ { i } ^ { ( 1 ) } , \mathbf { z } _ { i } ^ { ( 2 ) } \rangle / \tau + \log \underset { l \in \{ 1 , 2 \} , j \in [ 1 , N ] , j \neq i } { \sum } \exp ( \langle \mathbf { z } _ { i } ^ { ( k ) } , \mathbf { z } _ { j } ^ { ( l ) } \rangle / \tau ) } \end{array}
+$$
+
+![](images/073f9e49f7abf8410eb3e0e37d28c2dec0fd75636d13245c1eaaec9e61fc0758.jpg)  
+Figure 3: Comparisons on ImageNet-1K with/without DCL under different numbers of (a): batch sizes for SimCLR [8] and (b): queues for MoCo [7]. Without DCL, the top-1 accuracy significantly drops when batch size (SimCLR) or queues (MoCo) becomes very small.
+
+130 The proofs of Proposition 1 and 2 are given in Appendix. Further, we can generalize the loss   
+131 function $L _ { D C }$ to $L _ { D C W }$ by introducing a weighting function for the positive pairs i.e. $L _ { D C W } =$   
+132 $\begin{array} { r } { \sum _ { k \in \{ 1 , 2 \} , i \in [ [ 1 , N ] ] } L _ { D C W , i } ^ { ( k ) } } \end{array}$ .
+
+$$
+L _ { D C W , i } ^ { ( k ) } = - w ( \mathbf { z } _ { i } ^ { ( 1 ) } , \mathbf { z } _ { i } ^ { ( 2 ) } ) ( \langle \mathbf { z } _ { i } ^ { ( 1 ) } , \mathbf { z } _ { i } ^ { ( 2 ) } \rangle / \tau ) + \log \sum _ { \substack { l \in \{ 1 , 2 \} , j \in [ 1 , N ] , j \neq i } } \exp ( \langle \mathbf { z } _ { i } ^ { ( k ) } , \mathbf { z } _ { j } ^ { ( l ) } \rangle / \tau )
+$$
+
+133 where we can intuitively choose $w$ to be a negative von Mises-Fisher weighting function that   
+134 $\begin{array} { r } { w ( \mathbf { z } _ { i } ^ { ( 1 ) } , \mathbf { z } _ { i } ^ { ( 2 ) } ) = 2 - \frac { \exp ( \langle \mathbf { z } _ { i } ^ { ( 1 ) } , \mathbf { z } _ { i } ^ { ( 2 ) } \rangle / \sigma ) } { \mathrm { E } _ { i } \left[ \exp ( \langle \mathbf { z } _ { i } ^ { ( 1 ) } , \mathbf { z } _ { i } ^ { ( 2 ) } \rangle / \sigma ) \right] } } \end{array}$ and E $[ w ] = 1$ . $L _ { D C }$ is a special case of $L _ { D C W }$ and we   
+135 can see that $\begin{array} { r } { \operatorname* { l i m } _ { \sigma \to \infty } L _ { D C W } = L _ { D C } } \end{array}$ . The intuition behind $w ( \mathbf { z } _ { i } ^ { ( 1 ) } , \mathbf { z } _ { i } ^ { ( 2 ) } )$ z(2)i ) is that there is more learning   
+136 signal when a positive pair of samples are far from each other.
+
+# 137 4 Experiments
+
+138 This section evaluates our proposed decoupled contrastive learning (DCL) empirically and compares   
+139 it to the general contrastive learning methods. We summarize our experiments and analysis as the   
+140 following: (1) our proposed work significantly outperforms the general contrastive learning on large   
+141 and small-scale vision benchmarks; (2) we show the better version of DCL: LDCW could further   
+142 improve the representation quality. (3) we further analyze our DCL with few learning epochs, which   
+143 shows fast convergence of the proposed DCL. Detailed experimental settings can be found in the   
+144 Appendix.
+
+# 145 4.1 Implementation details
+
+146 To understand the effect of the sample decoupling, we consider our proposed DCL, which is based on   
+147 the general contrastive learning, where model optimization is irrelevant to the size of batches (i.e.,   
+148 negative samples). Extensive experiments and analysis are demonstrated on large-scale benchmarks:   
+149 ImageNet-1K [19], ImageNet-100 [6], and small-scale benchmark: CIFAR [20], and STL10 [21].   
+150 Note that all of our experiments are conducted with 8 Nvidia V100 GPUs on a single machine.   
+151 ImageNet For a fair comparison on ImageNet data, we implement our proposed decoupled structure,   
+152 DCL by following SimCLR [8] with ResNet-50 [22] as the encoder backbone and use cosine annealing   
+153 schedule. We set the temperature $\tau$ to 0.1 and the latent vector dimension to 128. Following [23],   
+154 we evaluate the pre-trained models by training a linear classifier with frozen learned embedding on   
+155 ImageNet data. We further consider evaluating our approach on ImageNet-100, a selected subset of   
+156 100 classes of ImageNet-1K.   
+157 CIFAR and STL10 For CIFAR10, CIFAR100, and STL10, ResNet-18 [22] is used as the encoder   
+158 architecture. We set the temperature $\tau$ to 0.07. All models are trained for 200 epochs with SGD   
+159 optimizer and a base $l r = 0 . 0 3 * $ batchsize/256. We follow NPID [4] on using $k = 2 0 0$ nearest   
+160 neighbor (kNN) classifier. Note that on STL10, we follow [24] to use both train set and unlabeled   
+161 set for model pre-training.
+
+Table 1: Comparisons with/without DCL under different numbers of batch sizes from 32 to 512. Results show the effectness of DCL on four widely used benchmarks. The performance of DCL keeps steadier than the SimCLR baseline while the batch size is varied.   
+
+<table><tr><td>Dataset</td><td colspan="5">ImageNet-100 (linear)</td><td colspan="5">CIFAR10 (kNN)</td></tr><tr><td>Batch Size</td><td>32</td><td>64</td><td>128</td><td>256</td><td>512</td><td>32</td><td>64</td><td>128</td><td>256</td><td>512</td></tr><tr><td>SimCLR [8]</td><td>74.2</td><td>77.6</td><td>79.3</td><td>80.7</td><td>81.3</td><td>78.9</td><td>80.4</td><td>81.1</td><td>81.4</td><td>81.3</td></tr><tr><td>SimCLR w/DCL</td><td>80.8</td><td>82.0</td><td>81.9</td><td>83.1</td><td>82.8</td><td>83.7</td><td>84.4</td><td> 84.4</td><td>84.2</td><td>83.5</td></tr><tr><td>Dataset</td><td></td><td>CIFAR100 (kNN)</td><td></td><td></td><td></td><td></td><td>STL10 (kNN)</td><td></td><td></td><td></td></tr><tr><td>Batch Size</td><td>32</td><td>64</td><td>128</td><td>256</td><td>512</td><td>32</td><td>64</td><td>128</td><td>256</td><td>512</td></tr><tr><td>SimCLR [8]</td><td>49.4</td><td>50.3</td><td>51.8</td><td>52</td><td>52.4</td><td>74.1</td><td>76.2</td><td>76.9</td><td>77.3</td><td>77.6</td></tr><tr><td>SimCLR w/ DCL</td><td> 51.1</td><td> 54.3</td><td>54.6</td><td> 54.9</td><td>55</td><td>82.0</td><td>82.8</td><td> 81.8</td><td>81.2</td><td>81.0</td></tr></table>
+
+Table 2: kNN top-1 accuracy $( \% )$ comparison of SSL approaches on small-scale benchmarks: CIFAR10, CIFAR100, and STL10. Results show that DCL consistently improves its SimCLR baseline. With multi-cropping [17], our DCLW reaches competitive performance within other contrastive learning approaches [8, 7, 4, 12, 18].   
+
+<table><tr><td>kNN (top-1)</td><td>SimCLR</td><td>MoCo</td><td>MoCo +CLD</td><td>NPID</td><td>NPID + CLD</td><td>Inv. Spread</td><td>Exemplar</td><td>DCL</td><td>DCLW w/ mcrop</td></tr><tr><td>CIFAR10</td><td>81.4</td><td>82.1</td><td>87.5</td><td>80.8</td><td>86.7</td><td>83.6</td><td>76.5</td><td>84.1</td><td>87.8</td></tr><tr><td>CIFAR100</td><td>52.0</td><td>53.1</td><td>58.1</td><td>51.6</td><td>57.5</td><td>N/A</td><td>N/A</td><td>54.9</td><td>58.8</td></tr><tr><td>STL10</td><td>77.3</td><td>80.8</td><td>84.3</td><td>79.1</td><td>83.6</td><td>81.6</td><td>79.3</td><td>81.2</td><td>84.1</td></tr></table>
+
+# 4.2 Experiments and analysis
+
+DCL on ImageNet This section illustrates the effect of our DCL under different batch sizes and queues. The initial setup is to have 1024 batch size (SimCLR [8]) and 65536 queues (MoCo [7]) and gradually reduce the batch size (SimCLR) and queue (MoCo) to show the corresponding top-1 accuracy by linear evaluation. Figure 3 indicates that without DCL, the top-1 accuracy drastically drops when batch size (SimCLR) or queue (MoCo) becomes very small. While with DCL, the performance keeps steadier than baselines (SimCLR: $- 4 . 1 \%$ vs. $- 8 . \dot { 3 } \%$ , MoCo: $- 0 . 4 \%$ vs. $- 5 . 9 \%$ ).
+
+169 Specifically, Figure 3 further shows that in SimCLR, the performance with DCL improves from   
+170 $6 \bar { 1 } . 8 \%$ to $6 \mathrm { \bar { 5 } } . 9 \%$ under 256 batch size; MoCo with DCL improves from $5 4 . 7 \%$ to $6 0 . 8 \%$ under 256   
+171 queues. The comparison fully demonstrates the necessity of DCL, especially when the number of   
+172 negatives is small. Although batch size is increased to 1024, we also note that our DCL $( 6 6 . 1 \% )$ still   
+173 improves over the SimCLR baseline $( 6 5 . 1 \% )$ .   
+174 We further observe the same phenomenon on ImageNet-100 data. Table 1 shows that, while with   
+175 DCL, the performance only drops $2 . 3 \%$ compare to the SimCLR baseline of $7 . 1 \%$ .   
+176 In summary, it is worth noting that, while the batch size is small, the strength of $q _ { B , i }$ , which is used to   
+177 push the negative samples away from the positive sample, is also relatively weak. This phenomenon   
+178 tends to reduce the efficiency of learning representation. While taking advantage of DCL alleviates   
+179 the performance gap between small and large batch sizes. Hence, through the analysis, we find out   
+180 DCL can simply tackle the batch size issue in contrastive learning. With this considerable advantage   
+181 given by DCL, general SSL approaches can be implemented with fewer computational resources or   
+182 lower standard platforms.   
+183 DCL on CIFAR and STL10 In Table 1 and Table 3, it is observed that DCL also demonstrates   
+184 its effectiveness on small-scale benchmarks. In summary, DCL outperforms its baseline by $3 . 1 \%$   
+185 (CIFAR10) and $2 . 8 \%$ (CIFAR100) and keeps the performance relatively steady under batch size 256.   
+186 We also improve the kNN accuracy of the SimCLR baseline on STL10 by $3 . { \dot { 9 } } \%$ .   
+187 Decoupled objective with re-weighting DCLW We only replace $L _ { D C }$ with $L _ { D C W }$ with no pos  
+188 sible advantage from additional tricks. That is, both our approach and the baselines apply the same   
+189 training instruction of the OpenSelfSup benchmark [23] for fairness. Note that we empirically choose   
+190 $\sigma = 0 . 5$ in the experiments.   
+191 Results in Table 3 indicates that, DCLW achieves extra $5 . 1 \%$ (ImageNet-1K), $3 . 5 \%$ (ImageNet-100)   
+192 gains compared to the baseline. For CIFAR data, extra $3 . 7 \%$ (CIFAR10), $3 . 1 \%$ are gained from the   
+193 addition of DCLW. It is worth to note that, trained with 200 epochs, our DCLW reaches $6 6 . 9 \%$ with   
+194 batch size 256, surpassing the SimCLR [8] baseline: $6 6 . 2 \%$ with batch size 8192.
+
+Table 3: Comparisons between SimCLR baseline, DCL, and DCLW. Results indicate that DCL improves the performance of baseline, and DCLW further provides an extra boost. Note that results are under the batch size 256 and epoch 200. All of models are both trained and evaluated with same experimental settings.   
+
+<table><tr><td></td><td>Baseline</td><td>DCL</td><td>DCLW</td></tr><tr><td>CIFAR10</td><td>81.8</td><td>84.2 (+3.1)</td><td>84.8 (+3.7)</td></tr><tr><td>CIFAR100</td><td>51.8</td><td>54.6 (+2.8)</td><td>54.9 (+3.1)</td></tr><tr><td>ImageNet-100</td><td>79.3</td><td>81.9 (+2.6)</td><td>82.8 (+3.5)</td></tr><tr><td>ImageNet-1K</td><td>61.8</td><td>65.9 (+4.1)</td><td>66.9 (+5.1)</td></tr></table>
+
+Table 4: ImageNet-1K top-1 accuracy $( \% )$ on SimCLR and MoCo v2 with/without DCL under few training epochs. We further list results under 200 epochs for clear comparison. With DCL, the performance of SimCLR trained under 100 epochs nearly reaches its performance under 200 epochs. The MoCo v2 with DCL also reaches higher accuracy than the baseline under 100 epochs.   
+
+<table><tr><td></td><td>SimCLR[8]</td><td>SimCLR w/ DCL</td><td>MoCo v2[25]</td><td>MoCo v2 w/ DCL</td></tr><tr><td>100 epoch</td><td>57.5</td><td>64.6</td><td>63.6</td><td>64.4</td></tr><tr><td>200 epoch</td><td>61.8</td><td>65.9</td><td>67.5</td><td>67.7</td></tr></table>
+
+# 195 4.3 Small-scale benchmark results: STL10, CIFAR10, and CIFAR100
+
+196 For STL10, CIFAR10, and CIFAR100, we implement our DCL with ResNet-18 [22] as encoder   
+197 backbone by following small-scale benchmark of CLD [24]. All the models are trained for 200   
+198 epochs with 256 batch size and evaluate by using kNN accuracies $k = 2 0 0$ ).   
+199 Results in Table 2 indicates that, our DCLW with multi-cropping [17] consistently outperforms the   
+200 state-of-the-art baselines on CIFAR10, STL10, and CIFAR100. Our DCL also demonstrates its   
+201 capability while comparing against other baselines. More analysis of large-scale benchmarks can be   
+202 found in Appendix.
+
+# 4.4 Ablations
+
+We perform extensive ablations on the hyperparameters of our DCL and DCLW on both ImageNet data and other small-scale data, i.e., CIFAR10, CIFAR100, and STL10. By seeking better configurations empirically, we see that our approach gives consistent gains over the standard SimCLR baseline. In other ablations, we see that our DCL achieves more gains over both SimCLR and MoCo v2, i.e., contrastive learning baselines, also when training for 100 epochs only.
+
+![](images/b6547bd17add024c15718688b0d8b924b97090935ad8506ae57ec959de59799e.jpg)  
+Figure 4: During the SSL pre-training, DCL speeds up the model convergence and provides better performance than the baseline on CIFAR and STL10 data.
+
+09 Few learning epochs Our DCL is inspired by the traditional contrastive learning framework, which   
+10 needs a large batch size, long learning epochs to achieve higher performance. The previous state  
+11 of-the-art, SimCLR [8], heavily rely on large quantities of learning epochs to obtain high top-1   
+12 accuracy. (e.g., $6 9 . 3 \%$ with up to 1000 epochs). The purpose of our DCL is to achieve higher learning   
+13 efficiency with few learning epochs. We demonstrate the effectiveness of DCL in contrastive learning   
+14 frameworks SimCLR and MoCo v2. We choose the batch size of 256 (queue of 65536) as the baseline   
+15 and train the model with only 100 epochs instead of the normal number of 200. We make sure other   
+16 parameter settings are the same for a fair comparison. Table 4 shows the result on ImageNet-1K   
+17 using linear evaluation. With DCL, SimCLR can achieve $6 4 . 6 \%$ top-1 accuracy with only 100 epochs   
+18 compared to SimCLR baseline: $5 7 . 5 \%$ ; MoCo v2 with DCL reaches $6 4 . 4 \%$ compared to MoCo v2   
+19 baseline: $6 3 . 6 \%$ with 100 epochs pre-training.
+
+We further demonstrate that, with DCL, learning representation becomes faster during the early stage of training. The reason is that DCL successfully solves the decoupled issue between positive and negative pairs. Figure 4 (a), (b), and (c), show that our DCL improves the speed of convergence and reaches higher performance than the baseline on CIFAR and STL10 data.
+
+# 5 Conclusion
+
+In this paper, we identify the negative-positive-coupling (NPC) effect in SimCLR. By removing the NPC effect, we reach a new objective function, decoupled contrastive learning (DCL). The proposed DCL loss function requires minimal modification to the SimCLR baseline and provides efficient, reliable, and nontrivial performance improvement on various benchmarks. Given the conceptual simplicity of DCL and that it requires neither momentum encoding, large batch sizes, or long epochs to reach competitive performance, we wish that DCL can serve as a strong baseline for the contrastive-based SSL methods.
+
+# References
+
+[1] Mehdi Noroozi and Paolo Favaro. Unsupervised learning of visual representations by solving jigsaw puzzles. In European Conference on Computer Vision, pages 69–84. Springer, 2016.   
+[2] Richard Zhang, Phillip Isola, and Alexei A Efros. Colorful image colorization. In European conference on computer vision, pages 649–666. Springer, 2016.   
+[3] Spyros Gidaris, Praveer Singh, and Nikos Komodakis. Unsupervised representation learning by predicting image rotations. arXiv preprint arXiv:1803.07728, 2018.   
+[4] Zhirong Wu, Yuanjun Xiong, Stella X Yu, and Dahua Lin. Unsupervised feature learning via non-parametric instance discrimination. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 3733–3742, 2018.   
+[5] Aaron van den Oord, Yazhe Li, and Oriol Vinyals. Representation learning with contrastive predictive coding. arXiv preprint arXiv:1807.03748, 2018.   
+[6] Yonglong Tian, Dilip Krishnan, and Phillip Isola. Contrastive multiview coding. arXiv preprint arXiv:1906.05849, 2019.   
+[7] Kaiming He, Haoqi Fan, Yuxin Wu, Saining Xie, and Ross Girshick. Momentum contrast for unsupervised visual representation learning. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 9729–9738, 2020.   
+[8] Ting Chen, Simon Kornblith, Mohammad Norouzi, and Geoffrey Hinton. A simple framework for contrastive learning of visual representations. arXiv preprint arXiv:2002.05709, 2020. [9] Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Advances in neural information processing systems, pages 2672–2680, 2014.   
+[10] Alec Radford, Luke Metz, and Soumith Chintala. Unsupervised representation learning with deep convolutional generative adversarial networks. arXiv preprint arXiv:1511.06434, 2015.   
+[11] Raia Hadsell, Sumit Chopra, and Yann LeCun. Dimensionality reduction by learning an invariant mapping. In 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR 2006), 17-22 June 2006, New York, NY, USA, pages 1735–1742. IEEE Computer Society, 2006. [12] Mang Ye, Xu Zhang, Pong C Yuen, and Shih-Fu Chang. Unsupervised embedding learning via invariant and spreading instance feature. In Proceedings of the IEEE Conference on computer vision and pattern recognition, pages 6210–6219, 2019.   
+[13] Prannay Khosla, Piotr Teterwak, Chen Wang, Aaron Sarna, Yonglong Tian, Phillip Isola, Aaron Maschinot, Ce Liu, and Dilip Krishnan. Supervised contrastive learning. CoRR, abs/2004.11362, 2020.   
+[14] Yang You, Igor Gitman, and Boris Ginsburg. Large batch training of convolutional networks. arXiv preprint arXiv:1708.03888, 2017.   
+[15] Jean-Bastien Grill, Florian Strub, Florent Altché, Corentin Tallec, Pierre H. Richemond, Elena Buchatskaya, Carl Doersch, Bernardo Ávila Pires, Zhaohan Guo, Mohammad Gheshlaghi Azar, Bilal Piot, Koray Kavukcuoglu, Rémi Munos, and Michal Valko. Bootstrap your own latent - A new approach to self-supervised learning. In Hugo Larochelle, Marc’Aurelio Ranzato, Raia Hadsell, Maria-Florina Balcan, and Hsuan-Tien Lin, editors, Advances in Neural Information Processing Systems 33: Annual Conference on Neural Information Processing Systems 2020, NeurIPS 2020, December 6-12, 2020, virtual, 2020.   
+[16] Xinlei Chen and Kaiming He. Exploring simple siamese representation learning. CoRR, abs/2011.10566, 2020.   
+[17] Mathilde Caron, Ishan Misra, Julien Mairal, Priya Goyal, Piotr Bojanowski, and Armand Joulin. Unsupervised learning of visual features by contrasting cluster assignments. arXiv preprint arXiv:2006.09882, 2020.   
+[18] Alexey Dosovitskiy, Philipp Fischer, Jost Tobias Springenberg, Martin Riedmiller, and Thomas Brox. Discriminative unsupervised feature learning with exemplar convolutional neural networks. IEEE transactions on pattern analysis and machine intelligence, 38(9):1734–1747, 2015.   
+[19] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A largescale hierarchical image database. In 2009 IEEE conference on computer vision and pattern recognition, pages 248–255. Ieee, 2009.   
+[20] Alex Krizhevsky, Geoffrey Hinton, et al. Learning multiple layers of features from tiny images. 2009.   
+[21] Adam Coates, Andrew $\mathrm { N g }$ , and Honglak Lee. An analysis of single-layer networks in unsupervised feature learning. In Proceedings of the fourteenth international conference on artificial intelligence and statistics, pages 215–223. JMLR Workshop and Conference Proceedings, 2011.   
+[22] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 770–778, 2016.   
+[23] Xiaohang Zhan, Jiahao Xie, Ziwei Liu, Dahua Lin, and Chen Change Loy. OpenSelfSup: Open mmlab self-supervised learning toolbox and benchmark. 2020.   
+[24] Xudong Wang, Ziwei Liu, and X Yu Stella. Unsupervised feature learning by cross-level instance-group discrimination.
+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   
+(b) Did you describe the limitations of your work? [Yes] See supplementary.   
+(c) Did you discuss any potential negative societal impacts of your work? [No]   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] See Section 3.   
+(b) Did you include complete proofs of all theoretical results? [Yes] See Section 3.
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [N/A] It can be straightforward to reproduce. In addition, we will release our code public after the review process.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Section4.   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No] . We only report the mean value of 5 runs but without standard deviation.   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Section4.
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes]   
+(b) Did you mention the license of the assets? [Yes]   
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [Yes]   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [No]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and scu’re using/cuapplicable? [Yes]   
+(b) Did you describe any potential participant risks, with links to Inu’re using/cuview Board (IRB) approvals, if applicable? [Yes]   
+(c) Did you include the estimated hourly wage paid to participants and the tou’re using/cunt on participant compensation? [Yes]

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+        "type": "text",
+        "text": "Decoupled Contrastive Learning ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
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+        "text": "Abstract ",
+        "text_level": 1,
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+        ],
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+        "type": "text",
+        "text": "1 Contrastive learning (CL) is one of the most successful paradigms for self  \n2 supervised learning (SSL). Specifically, contrastive learning treats two augmented   \n3 “views” of the same sample as positive, pulling them close and treating all other   \n4 samples as negative to push them far apart. Despite the evident success of CL SSL   \n5 methods, there are several challenges in the existing methods as they may require   \n6 special structures, large batches, or huge training epochs, etc. Our motivation in   \n7 this work is to provide a simple, efficient, and yet competitive contrastive learning   \n8 baseline. Through both theoretical and empirical studies, we identified a strong   \n9 negative-positive-coupling (NPC) effect in the widely used cross-entropy loss in   \n10 CL SSL methods. We hypothesize that the NPC effect may be a major cause of the   \n11 inefficiency in many contrastive learning methods. By removing the NPC effect,   \n12 we reach a decoupled contrastive learning (DCL) objective function, which signifi  \n13 cantly improves the training efficiency. DCL can achieve competitive performance,   \n14 requiring neither large batches in SimCLR, momentum encoding in Moco, or large   \n15 epochs. We demonstrate the benefit of DCL in various benchmarks. Further, DCL   \n16 is also much less sensitive to suboptimal hyperparameters. Notably, our approach   \n17 achieves $6 6 . 9 \\%$ ImageNet top-1 accuracy with 256 batch size within 200 epochs   \n18 pre-training, which outperforms its baseline SimCLR by $5 . 1 \\%$ . We believe DCL   \n19 may provide a strong baseline for future contrastive learning-based SSL studies. ",
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+    {
+        "type": "text",
+        "text": "20 1 Introduction ",
+        "text_level": 1,
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+        ],
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+        "text": "21 As a fundamental task in machine learning, representation learning aims to extract features to   \n22 reconstruct the raw data fully. It has been regarded as a long-acting goal over the past decades. Recent   \n23 progress on representation learning has achieved a significant milestone over self-supervised learning   \n24 (SSL), facilitating feature learning with its competence in exploiting massive raw data without any   \n25 annotated supervision. In the early stage of SSL, representation learning has focused on exploiting   \n26 pretext tasks, which are addressed by generating pseudo-labels to the unlabeled data through different   \n27 transformations, such as solving jigsaw puzzles [1], colorization [2] and rotation prediction [3].   \n28 Though these approaches achieve some success in computer vision, there is a large gap between   \n29 these methods and supervised learning. Recently, there has been a significant advancement in using   \n30 contrastive learning [4, 5, 6, 7, 8] for self-supervised pre-training, which significantly closes the gap   \n31 between the SSL method and supervised learning. Contrastive SSL methods, e.g., SimCLR [8], in   \n32 general, try to pull different views of the same instance close and push different instances far apart in   \n33 the representation space.   \n34 Despite the evident progress of the state-of-the-art contrastive SSL methods, there have been several   \n35 challenges in future developing this direction: 1) The SOTA models [7] may require unique structures   \n36 like the momentum encoder and large memory queues, which may complicate the understanding. 2)   \n37 The contrastive SSL models [8] may depend on large batch size and huge epoch numbers to achieve   \n38 competitive performance, posing a computational challenge for academia to explore this direction.   \n39 3) They may be sensitive to hyperparameters and optimizers, introducing additional difficulty to   \n40 reproduce the results on various benchmarks.   \n41 Our motivation in this work is to provide a simple, efficient, and yet competitive contrastive learning   \n42 baseline. We choose SimCLR as our starting point, given its conceptual simplicity. By analyzing the   \n43 objective function, we identified a Negative-Positive-Coupling (NPC) multiplier $q _ { B }$ in the gradient   \n44 as shown in Proposition 1. The NPC multipliers modulate the gradient of each sample, and it   \n45 mistakenly increases the impact of both negative samples and positive samples, given either of them   \n46 is more informative. Such a coupling exacerbates when smaller batch sizes are used. By removing   \n47 the coupling term, we reach a new formulation, the decoupled contrastive learning (DCL). The   \n48 new objective function significantly improves the training efficiency, requires neither large batches,   \n49 momentum encoding, or large epochs to achieve competitive performance on various different   \n50 benchmarks. Specifically, DCL reaches $6 6 . 9 \\%$ ImageNet top-1 (linear probing) accuracy with batch   \n51 size 256, SGD optimizer within 200 epochs. Even if DCL is trained for 100 epochs, it still reaches   \n52 $6 4 . 6 \\%$ ImageNet top-1 accuracy with batch size 256. ",
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+            "Figure 1: An overview of the batch size issue in the general contrastive approaches: (a) shows the NPC multiplier $q _ { B }$ in different batch sizes. As the large batch size increasing the $q _ { B }$ will approach 1 with a small coefficient of variation. (b) illustrates the distribution of $q _ { B }$ . "
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+        "type": "text",
+        "text": "53 In short, this work makes the following contributions: ",
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+        "text": "1) We provide both theoretical analysis and empirical evidence to show the negative-positive coupling in the gradient of contrastive learning;   \n2) We introduce a new, decoupled contrastive learning (DCL) objective, which casts off the coupling phenomenon between positive and negative samples in contrastive learning, and significantly improves the training efficiency; Additionally, the proposed DCL objective is less sensitive the several important hyperparameters;   \n3) We demonstrate our approach via extensive experiments and analysis on both large and small-scale vision benchmarks, with an optimal configuration for the standard SimCLR baseline to have a competitive performance within contrastive approaches. ",
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+    {
+        "type": "text",
+        "text": "63 2 Related work ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "2.1 Self-supervised representation learning ",
+        "text_level": 1,
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+        "text": "65 Self-supervised representation learning (SSL) aims to learn a robust embedding space from data   \n66 without human annotation. Previous arts can be roughly categorized into generative and discriminative.   \n67 Generative approaches, such as autoencoders and adversarial learning, focus on reconstructing   \n68 images from latent representations [9, 10]. Conversely, recent discriminative approaches, especially   \n69 contrastive learning-based approaches, have gained the most ground and achieved state-of-the-art   \n70 standard large-scale image classification benchmarks with increasingly more compute and data   \n71 augmentations. ",
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+        "type": "text",
+        "text": "72 2.2 Contrastive learning ",
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+        "text": "73 Contrastive learning (CL) constructs positive and negative sample pairs to extract information from   \n74 the data itself. In CL, each anchor image in a batch has only one positive sample to construct a positive   \n75 sample pair [11, 8, 7]. CPC [5] predicts the future output of sequential data by using current output   \n76 as prior knowledge, which can improve the feature representing the ability of the model. Instance   \n77 discrimination [4] proposes a non-parametric cross-entropy loss to optimize the model at the instance   \n78 level. Inv. spread [12] makes use of data augmentation invariants and the spread-out property of   \n79 instance to learn features. MoCo [7] proposes a dictionary to maintain a negative sample set, thus   \n80 increasing the number of negative sample pairs. Different from the aforementioned self-supervised   \n81 CL approaches, [13] proposes a supervised CL that considers all the same categories as positive pairs   \n82 to increase the utility of images. ",
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+    {
+        "type": "text",
+        "text": "83 2.3 Collapsing issue via batch size and negative sample ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "84 In CL, the objective is to maximize the mutual information between the positive pairs. However, to   \n85 avoid the “collapsing output”, vast quantities of negative samples are needed so that the learning   \n86 objectives obtain the maximum similarity and have the minimum similarity with negative samples.   \n87 For instance, in SimCLR [8], training requires many negative samples, leading to a large batch size   \n88 (i.e., 4096). Furthermore, to optimize such a huge batch, a specially designed optimizer LARS [14]   \n89 is used. Similarly, MoCo [7] needs a vast queue (i.e., 65536) to achieve competitive performance.   \n90 BYOL [15] does not collapse output without using any negative samples by considering all the   \n91 images are positive and to maximize the similarity of “projection” and “prediction ” features. On the   \n92 other hand, Simsam [16] leverages the Siamese network to introduce inductive biases for modeling   \n93 invariance. With the small batch size (i.e., 256), Simsam is a rival to BYOL (4096). Unlike both   \n94 approaches that achieved their success through empirical studies, this paper tackles from a theoretical   \n95 perspective, proving that an intertwined multiplier $q _ { B }$ of positive and negative is the main issue to   \n96 contrastive learning. ",
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+        "img_path": "images/7cac23c9fb9b0b9a2a31e147df9e235518441732db80a04de0db6a27da23015b.jpg",
+        "image_caption": [
+            "97 3 Decouple negative and positive samples in contrastive learning ",
+            "Figure 2: Contrastive learning and negative-positive coupling (NPC). (a) In SimCLR, each sample $\\mathbf { x } _ { i }$ has two augmented views $\\{ \\mathbf { x } _ { i } ^ { ( 1 ) } , \\mathbf { x } _ { i } ^ { ( 2 ) } \\}$ . They are encoded by the same encoder $f$ and further projected to $\\{ \\mathbf { z } _ { i } ^ { ( 1 ) } , \\mathbf { z } _ { i } ^ { ( 2 ) } \\}$ by a normalized MLP. (b) According to Equation 3. For the view $\\mathbf { x } _ { i } ^ { ( 1 ) }$ , the cross-entropy loss ${ L } _ { i } ^ { ( 1 ) }$ leads to a positive force $\\mathbf { z } _ { i } ^ { ( 2 ) }$ , which comes from the other view $\\mathbf { x } _ { i } ^ { ( 2 ) }$ of $\\mathbf { x }$ and a negative force, which is a weighted average of all the negative samples, i.e. $\\{ \\mathbf { z } _ { j } ^ { ( l ) } | l \\in \\{ 1 , 2 \\} , j \\neq i \\}$ . However, the gradient $- \\nabla _ { \\mathbf { z } _ { i } ^ { ( 2 ) } } L _ { i } ^ { ( 1 ) }$ is proportional to the NPC multiplier. "
+        ],
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+        "text": "98 We choose to start from SimCLR because of its conceptual simplicity. Given a batch of $N$ samples (e.g. images), 99 $\\{ \\mathbf { x } _ { 1 } , \\dotsc , \\mathbf { x } _ { N } \\}$ , let $\\mathbf { x } _ { i } ^ { ( 1 ) } , \\mathbf { x } _ { i } ^ { ( 2 ) }$ be two augmented views of the sample $x _ { i }$ and $B$ be the set of all of the augmented views in the batch, i.e. 100 $B = \\{ \\mathbf { x } _ { i } ^ { ( k ) } | k \\in \\{ 1 , 2 \\} , i \\in \\mathbb { I } ^ { 1 , N } \\mathbb { I } \\}$ . As ",
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+        "type": "text",
+        "text": "101 shown by Figure 2(a), each of the views $\\mathbf { x } _ { i } ^ { ( k ) }$ is sent into the same encoder network $f$ and the output   \n102 h(k)i $\\mathbf { h } _ { i } ^ { ( k ) } = f ( \\mathbf { x } _ { i } ^ { ( k ) } )$ is then projected by a normalized MLP projector that $\\mathbf { z } _ { i } ^ { ( k ) } = { g ( \\mathbf { h } _ { i } ^ { ( k ) } ) } / \\| g ( \\mathbf { h } _ { i } ^ { ( k ) } ) \\|$ .   \n103 For each augmented view $\\mathbf { x } _ { i } ^ { ( k ) }$ , SimCLR solves a classification problem by using the rest of the   \n104 105 views in creates a $B$ as targets, and assigns thoss-entropy loss function $L _ { i } ^ { ( k ) }$ y positive labelfor each view $\\mathbf { x } _ { i } ^ { ( k ) }$ ${ \\bf x } _ { i } ^ { ( u ) }$ , where d the o $u \\ne k$ . So SimCLRoss function is   \n106 $\\begin{array} { r } { L = \\sum _ { k \\in \\{ 1 , 2 \\} , i \\in [ [ 1 , N ] ] } L _ { i } ^ { ( k ) } } \\end{array}$ ",
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+        "img_path": "images/c84a0e7fdd7aa55b808166593a7f4a683c57394233c6453eaa9c8a08fbf453da.jpg",
+        "text": "$$\nL _ { i } ^ { ( k ) } = - \\log \\frac { \\exp ( \\langle \\mathbf { z } _ { i } ^ { ( 1 ) } , \\mathbf { z } _ { i } ^ { ( 2 ) } \\rangle / \\tau ) } { \\exp ( \\langle \\mathbf { z } _ { i } ^ { ( 1 ) } , \\mathbf { z } _ { i } ^ { ( 2 ) } \\rangle / \\tau ) + \\sum _ { l \\in \\{ 1 , 2 \\} , j \\in [ 1 , N ] , j \\neq i } \\exp ( \\langle \\mathbf { z } _ { i } ^ { ( k ) } , \\mathbf { z } _ { j } ^ { ( l ) } \\rangle / \\tau ) }\n$$",
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+        "type": "text",
+        "text": "Proposition 1. There exists a negative-positive coupling (NPC) multiplier 107 $q _ { B , i } ^ { ( 1 ) }$ in the gradient of 108 ${ L } _ { i } ^ { ( 1 ) }$ : ",
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+        "type": "equation",
+        "img_path": "images/3d09e27f9a9a2a4cb8e39673fde432b4e72ae1a66060b70cb3444f12bebcd3ea.jpg",
+        "text": "$$\n\\begin{array}{c} \\begin{array} { r } { \\left\\{ - \\nabla _ { \\mathbf { z } _ { i } ^ { ( 1 ) } } L _ { i } ^ { ( 1 ) } = \\frac { q _ { B , i } ^ { ( 1 ) } } { \\tau } \\left[ \\mathbf { z } _ { i } ^ { ( 2 ) } - \\sum _ { l \\in \\{ 1 , 2 \\} , j \\in [ 1 , N ] , j \\neq i } \\frac { \\exp { \\langle \\mathbf { z } _ { i } ^ { ( 1 ) } , \\mathbf { z } _ { j } ^ { ( l ) } \\rangle } / \\tau } { \\sum _ { q \\in \\{ 1 , 2 \\} , j \\in [ 1 , N ] , j \\neq i } \\exp ( \\langle \\mathbf { z } _ { i } ^ { ( 1 ) } , \\mathbf { z } _ { j } ^ { ( 0 ) } \\rangle / \\tau ) } \\cdot \\mathbf { z } _ { j } ^ { ( l ) } \\right] \\right.} \\\\ { - \\nabla _ { \\mathbf { z } _ { i } ^ { ( 2 ) } } L _ { i } ^ { ( 1 ) } = \\frac { q _ { B , i } ^ { ( 1 ) } } { \\tau } \\cdot \\mathbf { z } _ { i } ^ { ( 1 ) } } \\\\ { - \\nabla _ { \\mathbf { z } _ { j } ^ { ( l ) } } L _ { i } ^ { ( 1 ) } = - \\frac { q _ { B , i } ^ { ( 1 ) } } { \\tau } \\frac { \\exp { \\langle \\mathbf { z } _ { i } ^ { ( 1 ) } , \\mathbf { z } _ { j } ^ { ( l ) } \\rangle } / \\tau } { \\sum _ { q \\in \\{ 1 , 2 \\} , j \\in [ 1 , N ] , j \\neq i } \\exp ( \\langle \\mathbf { z } _ { i } ^ { ( 1 ) } , \\mathbf { z } _ { j } ^ { ( q ) } \\rangle / \\tau ) } \\cdot \\mathbf { z } _ { i } ^ { ( 1 ) } } \\end{array}   \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where the NPC multiplier 109 $q _ { B , i } ^ { ( 1 ) }$ is: ",
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+        "img_path": "images/a193f9ea65313fe38640fa6955f1946f823e28106e7e2852b8e72654d0105227.jpg",
+        "text": "$$\nq _ { B , i } ^ { ( 1 ) } = 1 - \\frac { \\exp ( \\langle \\mathbf { z } _ { i } ^ { ( 1 ) } , \\mathbf { z } _ { i } ^ { ( 2 ) } \\rangle / \\tau ) } { \\sum _ { q \\in \\{ 1 , 2 \\} , j \\in \\mathbb { [ 1 , N ] } , j \\neq i } \\exp ( \\langle \\mathbf { z } _ { i } ^ { ( 1 ) } , \\mathbf { z } _ { j } ^ { ( q ) } \\rangle / \\tau ) }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Due to the symmetry, a similar NPC multiplier 110 $q _ { B , i } ^ { ( k ) }$ exists in the gradient of $L _ { i } ^ { ( k ) } , k \\in \\{ 1 , 2 \\} , i \\in$ 111 $[ 1 , N ]$ . ",
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+        "type": "text",
+        "text": "112 $q _ { B , i } ^ { ( k ) }$ we can see, all of the partial gradients in Equation 2 are modified by the commonin Equation 3. Equation 3 makes intuitive sense: 1) When a positive sample pair $\\{ \\mathbf { z } _ { i } ^ { ( 1 ) } , \\mathbf { z } _ { i } ^ { ( 2 ) } \\}$ lier are   \n114 farther, the corresponding NPC multiplier $q _ { B , i } ^ { ( 1 ) }$ is larger. This will makes the overall gradient larger.   \n115 Otherwise, the gradient is smaller. 2) When a negative sample is closer to $\\mathbf { z } _ { i } ^ { ( 1 ) }$ , it makes $q _ { B , i } ^ { ( 1 ) }$ larger.   \n116 Overall, the intuition here is that a positive sample farther from the target or a negative sample closer   \n117 to the target is more informative. However, the positive samples and negative samples are strongly   \n118 coupled. An outlier positive sample also makes the gradient from the negative samples significantly   \n119 larger and vice versa.   \n120 Figure 1(b) shows the NPC multiplier $q _ { B }$ distribution shift w.r.t. different batch sizes for a pre-trained   \n121 SimCLR baseline model. While all of the shown distributions have prominent fluctuation, the smaller   \n122 batch size makes $q _ { B }$ cluster towards 0, while the larger batch size pushes the distribution towards   \n123 $\\delta ( 1 )$ . Figure 1(a) shows the averaged NPC multiplier $\\langle q _ { B } \\rangle$ changes w.r.t. the batch size and the   \n124 relative fluctuation. The small batch sizes introduce significant NPC fluctuation. Based on this   \n125 observation, we propose to remove the NPC multipliers from the gradients, which corresponds to the   \n126 case $q _ { B , N  \\infty }$ . This leads to the decoupled contrastive learning formulation.   \n127 Proposition 2. Removing the positive pair from the denominator of Equation 2 leads to a decoupled   \n128 contrastive learning loss. If we remove the NPC multiplier $q _ { B , i } ^ { ( k ) }$ from Equation 2, we reach a   \n129 decoupled contrastive learning loss LDC = Pk∈{1,2},i∈[[1,N]] , where $L _ { D C , i } ^ { ( k ) }$ is: ",
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+        "text": "$$\n\\begin{array} { r l } & { L _ { D C , i } ^ { ( k ) } = - \\log \\frac { \\exp ( \\langle \\mathbf { z } _ { i } ^ { ( 1 ) } , \\mathbf { z } _ { i } ^ { ( 2 ) } \\rangle / \\tau ) } { \\exp ( \\langle \\mathbf { \\hat { z } } _ { i } ^ { ( 1 ) } , \\mathbf { z } _ { i } ^ { ( 2 ) } \\rangle / \\tau ) } + \\sum _ { l \\in \\{ 1 , 2 \\} , j \\in [ 1 , N ] , j \\neq i } \\exp ( \\langle \\mathbf { z } _ { i } ^ { ( k ) } , \\mathbf { z } _ { j } ^ { ( l ) } \\rangle / \\tau ) } \\\\ & { \\quad \\quad \\quad = - \\langle \\mathbf { z } _ { i } ^ { ( 1 ) } , \\mathbf { z } _ { i } ^ { ( 2 ) } \\rangle / \\tau + \\log \\underset { l \\in \\{ 1 , 2 \\} , j \\in [ 1 , N ] , j \\neq i } { \\sum } \\exp ( \\langle \\mathbf { z } _ { i } ^ { ( k ) } , \\mathbf { z } _ { j } ^ { ( l ) } \\rangle / \\tau ) } \\end{array}\n$$",
+        "text_format": "latex",
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+        "image_caption": [
+            "Figure 3: Comparisons on ImageNet-1K with/without DCL under different numbers of (a): batch sizes for SimCLR [8] and (b): queues for MoCo [7]. Without DCL, the top-1 accuracy significantly drops when batch size (SimCLR) or queues (MoCo) becomes very small. "
+        ],
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+        "text": "130 The proofs of Proposition 1 and 2 are given in Appendix. Further, we can generalize the loss   \n131 function $L _ { D C }$ to $L _ { D C W }$ by introducing a weighting function for the positive pairs i.e. $L _ { D C W } =$   \n132 $\\begin{array} { r } { \\sum _ { k \\in \\{ 1 , 2 \\} , i \\in [ [ 1 , N ] ] } L _ { D C W , i } ^ { ( k ) } } \\end{array}$ . ",
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+        "img_path": "images/35cb19aad728c41699e7a8b2a06d47dc92f6bc5fd71f660f746a7ee12f79c412.jpg",
+        "text": "$$\nL _ { D C W , i } ^ { ( k ) } = - w ( \\mathbf { z } _ { i } ^ { ( 1 ) } , \\mathbf { z } _ { i } ^ { ( 2 ) } ) ( \\langle \\mathbf { z } _ { i } ^ { ( 1 ) } , \\mathbf { z } _ { i } ^ { ( 2 ) } \\rangle / \\tau ) + \\log \\sum _ { \\substack { l \\in \\{ 1 , 2 \\} , j \\in [ 1 , N ] , j \\neq i } } \\exp ( \\langle \\mathbf { z } _ { i } ^ { ( k ) } , \\mathbf { z } _ { j } ^ { ( l ) } \\rangle / \\tau )\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "133 where we can intuitively choose $w$ to be a negative von Mises-Fisher weighting function that   \n134 $\\begin{array} { r } { w ( \\mathbf { z } _ { i } ^ { ( 1 ) } , \\mathbf { z } _ { i } ^ { ( 2 ) } ) = 2 - \\frac { \\exp ( \\langle \\mathbf { z } _ { i } ^ { ( 1 ) } , \\mathbf { z } _ { i } ^ { ( 2 ) } \\rangle / \\sigma ) } { \\mathrm { E } _ { i } \\left[ \\exp ( \\langle \\mathbf { z } _ { i } ^ { ( 1 ) } , \\mathbf { z } _ { i } ^ { ( 2 ) } \\rangle / \\sigma ) \\right] } } \\end{array}$ and E $[ w ] = 1$ . $L _ { D C }$ is a special case of $L _ { D C W }$ and we   \n135 can see that $\\begin{array} { r } { \\operatorname* { l i m } _ { \\sigma \\to \\infty } L _ { D C W } = L _ { D C } } \\end{array}$ . The intuition behind $w ( \\mathbf { z } _ { i } ^ { ( 1 ) } , \\mathbf { z } _ { i } ^ { ( 2 ) } )$ z(2)i ) is that there is more learning   \n136 signal when a positive pair of samples are far from each other. ",
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+    {
+        "type": "text",
+        "text": "137 4 Experiments ",
+        "text_level": 1,
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+        "text": "138 This section evaluates our proposed decoupled contrastive learning (DCL) empirically and compares   \n139 it to the general contrastive learning methods. We summarize our experiments and analysis as the   \n140 following: (1) our proposed work significantly outperforms the general contrastive learning on large   \n141 and small-scale vision benchmarks; (2) we show the better version of DCL: LDCW could further   \n142 improve the representation quality. (3) we further analyze our DCL with few learning epochs, which   \n143 shows fast convergence of the proposed DCL. Detailed experimental settings can be found in the   \n144 Appendix. ",
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+    {
+        "type": "text",
+        "text": "145 4.1 Implementation details ",
+        "text_level": 1,
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+        "text": "146 To understand the effect of the sample decoupling, we consider our proposed DCL, which is based on   \n147 the general contrastive learning, where model optimization is irrelevant to the size of batches (i.e.,   \n148 negative samples). Extensive experiments and analysis are demonstrated on large-scale benchmarks:   \n149 ImageNet-1K [19], ImageNet-100 [6], and small-scale benchmark: CIFAR [20], and STL10 [21].   \n150 Note that all of our experiments are conducted with 8 Nvidia V100 GPUs on a single machine.   \n151 ImageNet For a fair comparison on ImageNet data, we implement our proposed decoupled structure,   \n152 DCL by following SimCLR [8] with ResNet-50 [22] as the encoder backbone and use cosine annealing   \n153 schedule. We set the temperature $\\tau$ to 0.1 and the latent vector dimension to 128. Following [23],   \n154 we evaluate the pre-trained models by training a linear classifier with frozen learned embedding on   \n155 ImageNet data. We further consider evaluating our approach on ImageNet-100, a selected subset of   \n156 100 classes of ImageNet-1K.   \n157 CIFAR and STL10 For CIFAR10, CIFAR100, and STL10, ResNet-18 [22] is used as the encoder   \n158 architecture. We set the temperature $\\tau$ to 0.07. All models are trained for 200 epochs with SGD   \n159 optimizer and a base $l r = 0 . 0 3 * $ batchsize/256. We follow NPID [4] on using $k = 2 0 0$ nearest   \n160 neighbor (kNN) classifier. Note that on STL10, we follow [24] to use both train set and unlabeled   \n161 set for model pre-training. ",
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+        "table_caption": [
+            "Table 1: Comparisons with/without DCL under different numbers of batch sizes from 32 to 512. Results show the effectness of DCL on four widely used benchmarks. The performance of DCL keeps steadier than the SimCLR baseline while the batch size is varied. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td colspan=\"5\">ImageNet-100 (linear)</td><td colspan=\"5\">CIFAR10 (kNN)</td></tr><tr><td>Batch Size</td><td>32</td><td>64</td><td>128</td><td>256</td><td>512</td><td>32</td><td>64</td><td>128</td><td>256</td><td>512</td></tr><tr><td>SimCLR [8]</td><td>74.2</td><td>77.6</td><td>79.3</td><td>80.7</td><td>81.3</td><td>78.9</td><td>80.4</td><td>81.1</td><td>81.4</td><td>81.3</td></tr><tr><td>SimCLR w/DCL</td><td>80.8</td><td>82.0</td><td>81.9</td><td>83.1</td><td>82.8</td><td>83.7</td><td>84.4</td><td> 84.4</td><td>84.2</td><td>83.5</td></tr><tr><td>Dataset</td><td></td><td>CIFAR100 (kNN)</td><td></td><td></td><td></td><td></td><td>STL10 (kNN)</td><td></td><td></td><td></td></tr><tr><td>Batch Size</td><td>32</td><td>64</td><td>128</td><td>256</td><td>512</td><td>32</td><td>64</td><td>128</td><td>256</td><td>512</td></tr><tr><td>SimCLR [8]</td><td>49.4</td><td>50.3</td><td>51.8</td><td>52</td><td>52.4</td><td>74.1</td><td>76.2</td><td>76.9</td><td>77.3</td><td>77.6</td></tr><tr><td>SimCLR w/ DCL</td><td> 51.1</td><td> 54.3</td><td>54.6</td><td> 54.9</td><td>55</td><td>82.0</td><td>82.8</td><td> 81.8</td><td>81.2</td><td>81.0</td></tr></table>",
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+            "Table 2: kNN top-1 accuracy $( \\% )$ comparison of SSL approaches on small-scale benchmarks: CIFAR10, CIFAR100, and STL10. Results show that DCL consistently improves its SimCLR baseline. With multi-cropping [17], our DCLW reaches competitive performance within other contrastive learning approaches [8, 7, 4, 12, 18]. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>kNN (top-1)</td><td>SimCLR</td><td>MoCo</td><td>MoCo +CLD</td><td>NPID</td><td>NPID + CLD</td><td>Inv. Spread</td><td>Exemplar</td><td>DCL</td><td>DCLW w/ mcrop</td></tr><tr><td>CIFAR10</td><td>81.4</td><td>82.1</td><td>87.5</td><td>80.8</td><td>86.7</td><td>83.6</td><td>76.5</td><td>84.1</td><td>87.8</td></tr><tr><td>CIFAR100</td><td>52.0</td><td>53.1</td><td>58.1</td><td>51.6</td><td>57.5</td><td>N/A</td><td>N/A</td><td>54.9</td><td>58.8</td></tr><tr><td>STL10</td><td>77.3</td><td>80.8</td><td>84.3</td><td>79.1</td><td>83.6</td><td>81.6</td><td>79.3</td><td>81.2</td><td>84.1</td></tr></table>",
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+        "text": "4.2 Experiments and analysis ",
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+        "text": "DCL on ImageNet This section illustrates the effect of our DCL under different batch sizes and queues. The initial setup is to have 1024 batch size (SimCLR [8]) and 65536 queues (MoCo [7]) and gradually reduce the batch size (SimCLR) and queue (MoCo) to show the corresponding top-1 accuracy by linear evaluation. Figure 3 indicates that without DCL, the top-1 accuracy drastically drops when batch size (SimCLR) or queue (MoCo) becomes very small. While with DCL, the performance keeps steadier than baselines (SimCLR: $- 4 . 1 \\%$ vs. $- 8 . \\dot { 3 } \\%$ , MoCo: $- 0 . 4 \\%$ vs. $- 5 . 9 \\%$ ). ",
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+        "text": "169 Specifically, Figure 3 further shows that in SimCLR, the performance with DCL improves from   \n170 $6 \\bar { 1 } . 8 \\%$ to $6 \\mathrm { \\bar { 5 } } . 9 \\%$ under 256 batch size; MoCo with DCL improves from $5 4 . 7 \\%$ to $6 0 . 8 \\%$ under 256   \n171 queues. The comparison fully demonstrates the necessity of DCL, especially when the number of   \n172 negatives is small. Although batch size is increased to 1024, we also note that our DCL $( 6 6 . 1 \\% )$ still   \n173 improves over the SimCLR baseline $( 6 5 . 1 \\% )$ .   \n174 We further observe the same phenomenon on ImageNet-100 data. Table 1 shows that, while with   \n175 DCL, the performance only drops $2 . 3 \\%$ compare to the SimCLR baseline of $7 . 1 \\%$ .   \n176 In summary, it is worth noting that, while the batch size is small, the strength of $q _ { B , i }$ , which is used to   \n177 push the negative samples away from the positive sample, is also relatively weak. This phenomenon   \n178 tends to reduce the efficiency of learning representation. While taking advantage of DCL alleviates   \n179 the performance gap between small and large batch sizes. Hence, through the analysis, we find out   \n180 DCL can simply tackle the batch size issue in contrastive learning. With this considerable advantage   \n181 given by DCL, general SSL approaches can be implemented with fewer computational resources or   \n182 lower standard platforms.   \n183 DCL on CIFAR and STL10 In Table 1 and Table 3, it is observed that DCL also demonstrates   \n184 its effectiveness on small-scale benchmarks. In summary, DCL outperforms its baseline by $3 . 1 \\%$   \n185 (CIFAR10) and $2 . 8 \\%$ (CIFAR100) and keeps the performance relatively steady under batch size 256.   \n186 We also improve the kNN accuracy of the SimCLR baseline on STL10 by $3 . { \\dot { 9 } } \\%$ .   \n187 Decoupled objective with re-weighting DCLW We only replace $L _ { D C }$ with $L _ { D C W }$ with no pos  \n188 sible advantage from additional tricks. That is, both our approach and the baselines apply the same   \n189 training instruction of the OpenSelfSup benchmark [23] for fairness. Note that we empirically choose   \n190 $\\sigma = 0 . 5$ in the experiments.   \n191 Results in Table 3 indicates that, DCLW achieves extra $5 . 1 \\%$ (ImageNet-1K), $3 . 5 \\%$ (ImageNet-100)   \n192 gains compared to the baseline. For CIFAR data, extra $3 . 7 \\%$ (CIFAR10), $3 . 1 \\%$ are gained from the   \n193 addition of DCLW. It is worth to note that, trained with 200 epochs, our DCLW reaches $6 6 . 9 \\%$ with   \n194 batch size 256, surpassing the SimCLR [8] baseline: $6 6 . 2 \\%$ with batch size 8192. ",
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+        "table_caption": [
+            "Table 3: Comparisons between SimCLR baseline, DCL, and DCLW. Results indicate that DCL improves the performance of baseline, and DCLW further provides an extra boost. Note that results are under the batch size 256 and epoch 200. All of models are both trained and evaluated with same experimental settings. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Baseline</td><td>DCL</td><td>DCLW</td></tr><tr><td>CIFAR10</td><td>81.8</td><td>84.2 (+3.1)</td><td>84.8 (+3.7)</td></tr><tr><td>CIFAR100</td><td>51.8</td><td>54.6 (+2.8)</td><td>54.9 (+3.1)</td></tr><tr><td>ImageNet-100</td><td>79.3</td><td>81.9 (+2.6)</td><td>82.8 (+3.5)</td></tr><tr><td>ImageNet-1K</td><td>61.8</td><td>65.9 (+4.1)</td><td>66.9 (+5.1)</td></tr></table>",
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+        "table_caption": [
+            "Table 4: ImageNet-1K top-1 accuracy $( \\% )$ on SimCLR and MoCo v2 with/without DCL under few training epochs. We further list results under 200 epochs for clear comparison. With DCL, the performance of SimCLR trained under 100 epochs nearly reaches its performance under 200 epochs. The MoCo v2 with DCL also reaches higher accuracy than the baseline under 100 epochs. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>SimCLR[8]</td><td>SimCLR w/ DCL</td><td>MoCo v2[25]</td><td>MoCo v2 w/ DCL</td></tr><tr><td>100 epoch</td><td>57.5</td><td>64.6</td><td>63.6</td><td>64.4</td></tr><tr><td>200 epoch</td><td>61.8</td><td>65.9</td><td>67.5</td><td>67.7</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "195 4.3 Small-scale benchmark results: STL10, CIFAR10, and CIFAR100 ",
+        "text_level": 1,
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+        "text": "196 For STL10, CIFAR10, and CIFAR100, we implement our DCL with ResNet-18 [22] as encoder   \n197 backbone by following small-scale benchmark of CLD [24]. All the models are trained for 200   \n198 epochs with 256 batch size and evaluate by using kNN accuracies $k = 2 0 0$ ).   \n199 Results in Table 2 indicates that, our DCLW with multi-cropping [17] consistently outperforms the   \n200 state-of-the-art baselines on CIFAR10, STL10, and CIFAR100. Our DCL also demonstrates its   \n201 capability while comparing against other baselines. More analysis of large-scale benchmarks can be   \n202 found in Appendix. ",
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+    {
+        "type": "text",
+        "text": "4.4 Ablations ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We perform extensive ablations on the hyperparameters of our DCL and DCLW on both ImageNet data and other small-scale data, i.e., CIFAR10, CIFAR100, and STL10. By seeking better configurations empirically, we see that our approach gives consistent gains over the standard SimCLR baseline. In other ablations, we see that our DCL achieves more gains over both SimCLR and MoCo v2, i.e., contrastive learning baselines, also when training for 100 epochs only. ",
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+        "image_caption": [
+            "Figure 4: During the SSL pre-training, DCL speeds up the model convergence and provides better performance than the baseline on CIFAR and STL10 data. "
+        ],
+        "image_footnote": [],
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+        "text": "09 Few learning epochs Our DCL is inspired by the traditional contrastive learning framework, which   \n10 needs a large batch size, long learning epochs to achieve higher performance. The previous state  \n11 of-the-art, SimCLR [8], heavily rely on large quantities of learning epochs to obtain high top-1   \n12 accuracy. (e.g., $6 9 . 3 \\%$ with up to 1000 epochs). The purpose of our DCL is to achieve higher learning   \n13 efficiency with few learning epochs. We demonstrate the effectiveness of DCL in contrastive learning   \n14 frameworks SimCLR and MoCo v2. We choose the batch size of 256 (queue of 65536) as the baseline   \n15 and train the model with only 100 epochs instead of the normal number of 200. We make sure other   \n16 parameter settings are the same for a fair comparison. Table 4 shows the result on ImageNet-1K   \n17 using linear evaluation. With DCL, SimCLR can achieve $6 4 . 6 \\%$ top-1 accuracy with only 100 epochs   \n18 compared to SimCLR baseline: $5 7 . 5 \\%$ ; MoCo v2 with DCL reaches $6 4 . 4 \\%$ compared to MoCo v2   \n19 baseline: $6 3 . 6 \\%$ with 100 epochs pre-training. ",
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+        "text": "We further demonstrate that, with DCL, learning representation becomes faster during the early stage of training. The reason is that DCL successfully solves the decoupled issue between positive and negative pairs. Figure 4 (a), (b), and (c), show that our DCL improves the speed of convergence and reaches higher performance than the baseline on CIFAR and STL10 data. ",
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+        "type": "text",
+        "text": "5 Conclusion ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this paper, we identify the negative-positive-coupling (NPC) effect in SimCLR. By removing the NPC effect, we reach a new objective function, decoupled contrastive learning (DCL). The proposed DCL loss function requires minimal modification to the SimCLR baseline and provides efficient, reliable, and nontrivial performance improvement on various benchmarks. Given the conceptual simplicity of DCL and that it requires neither momentum encoding, large batch sizes, or long epochs to reach competitive performance, we wish that DCL can serve as a strong baseline for the contrastive-based SSL methods. ",
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+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
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+        "text": "[1] Mehdi Noroozi and Paolo Favaro. Unsupervised learning of visual representations by solving jigsaw puzzles. In European Conference on Computer Vision, pages 69–84. Springer, 2016.   \n[2] Richard Zhang, Phillip Isola, and Alexei A Efros. Colorful image colorization. In European conference on computer vision, pages 649–666. Springer, 2016.   \n[3] Spyros Gidaris, Praveer Singh, and Nikos Komodakis. Unsupervised representation learning by predicting image rotations. arXiv preprint arXiv:1803.07728, 2018.   \n[4] Zhirong Wu, Yuanjun Xiong, Stella X Yu, and Dahua Lin. Unsupervised feature learning via non-parametric instance discrimination. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 3733–3742, 2018.   \n[5] Aaron van den Oord, Yazhe Li, and Oriol Vinyals. Representation learning with contrastive predictive coding. arXiv preprint arXiv:1807.03748, 2018.   \n[6] Yonglong Tian, Dilip Krishnan, and Phillip Isola. Contrastive multiview coding. arXiv preprint arXiv:1906.05849, 2019.   \n[7] Kaiming He, Haoqi Fan, Yuxin Wu, Saining Xie, and Ross Girshick. Momentum contrast for unsupervised visual representation learning. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 9729–9738, 2020.   \n[8] Ting Chen, Simon Kornblith, Mohammad Norouzi, and Geoffrey Hinton. A simple framework for contrastive learning of visual representations. arXiv preprint arXiv:2002.05709, 2020. [9] Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Advances in neural information processing systems, pages 2672–2680, 2014.   \n[10] Alec Radford, Luke Metz, and Soumith Chintala. Unsupervised representation learning with deep convolutional generative adversarial networks. arXiv preprint arXiv:1511.06434, 2015.   \n[11] Raia Hadsell, Sumit Chopra, and Yann LeCun. Dimensionality reduction by learning an invariant mapping. In 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR 2006), 17-22 June 2006, New York, NY, USA, pages 1735–1742. IEEE Computer Society, 2006. [12] Mang Ye, Xu Zhang, Pong C Yuen, and Shih-Fu Chang. Unsupervised embedding learning via invariant and spreading instance feature. In Proceedings of the IEEE Conference on computer vision and pattern recognition, pages 6210–6219, 2019.   \n[13] Prannay Khosla, Piotr Teterwak, Chen Wang, Aaron Sarna, Yonglong Tian, Phillip Isola, Aaron Maschinot, Ce Liu, and Dilip Krishnan. Supervised contrastive learning. CoRR, abs/2004.11362, 2020.   \n[14] Yang You, Igor Gitman, and Boris Ginsburg. Large batch training of convolutional networks. arXiv preprint arXiv:1708.03888, 2017.   \n[15] Jean-Bastien Grill, Florian Strub, Florent Altché, Corentin Tallec, Pierre H. 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Discriminative unsupervised feature learning with exemplar convolutional neural networks. IEEE transactions on pattern analysis and machine intelligence, 38(9):1734–1747, 2015.   \n[19] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A largescale hierarchical image database. In 2009 IEEE conference on computer vision and pattern recognition, pages 248–255. Ieee, 2009.   \n[20] Alex Krizhevsky, Geoffrey Hinton, et al. Learning multiple layers of features from tiny images. 2009.   \n[21] Adam Coates, Andrew $\\mathrm { N g }$ , and Honglak Lee. An analysis of single-layer networks in unsupervised feature learning. In Proceedings of the fourteenth international conference on artificial intelligence and statistics, pages 215–223. JMLR Workshop and Conference Proceedings, 2011.   \n[22] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 770–778, 2016.   \n[23] Xiaohang Zhan, Jiahao Xie, Ziwei Liu, Dahua Lin, and Chen Change Loy. OpenSelfSup: Open mmlab self-supervised learning toolbox and benchmark. 2020.   \n[24] Xudong Wang, Ziwei Liu, and X Yu Stella. Unsupervised feature learning by cross-level instance-group discrimination. ",
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+        "text": "Checklist ",
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+        "type": "text",
+        "text": "1. For all authors... ",
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+    {
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+        "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   \n(b) Did you describe the limitations of your work? [Yes] See supplementary.   \n(c) Did you discuss any potential negative societal impacts of your work? [No]   \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ",
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+        "type": "text",
+        "text": "2. If you are including theoretical results... ",
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+    {
+        "type": "text",
+        "text": "(a) Did you state the full set of assumptions of all theoretical results? [Yes] See Section 3.   \n(b) Did you include complete proofs of all theoretical results? [Yes] See Section 3. ",
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+        "text": "3. If you ran experiments... ",
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+        "text": "(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [N/A] It can be straightforward to reproduce. In addition, we will release our code public after the review process.   \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Section4.   \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No] . We only report the mean value of 5 runs but without standard deviation.   \n(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Section4. ",
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parse/train/S1EwLkW0W/S1EwLkW0W.md

@@ -0,0 +1,593 @@
+# DISSECTING ADAM: THE SIGN, MAGNITUDE ANDVARIANCE OF STOCHASTIC GRADIENTS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+The ADAM optimizer is exceedingly popular in the deep learning community. Often it works very well, sometimes it doesn’t. Why? We interpret ADAM as a combination of two aspects: for each weight, the update direction is determined by the sign of the stochastic gradient, whereas the update magnitude is solely determined by an estimate of its relative variance. We disentangle these two aspects and analyze them in isolation, shedding light on ADAM’s inner workings. Transferring the “variance adaptation” to momentum-SGD gives rise to a novel method, completing the practitioner’s toolbox for problems where ADAM fails.
+
+# 1 INTRODUCTION
+
+Many prominent machine learning models pose empirical risk minimization problems of the form
+
+$$
+\operatorname* { m i n } _ { \theta \in \mathbb { R } ^ { d } } \mathcal { L } ( \theta ) = \frac { 1 } { M } \sum _ { k = 1 } ^ { M } \ell ( \theta ; x _ { k } ) , \quad \mathrm { w i t h ~ g r a d i e n t } \quad \nabla \mathcal { L } ( \theta ) = \frac { 1 } { M } \sum _ { k = 1 } ^ { M } \nabla \ell ( \theta ; x _ { k } ) ,
+$$
+
+where $\theta \in \mathbb { R } ^ { d }$ is a vector of parameters, $\{ x _ { 1 } , \ldots , x _ { M } \}$ a training set, and $\ell ( \theta ; x )$ is a loss quantifying the performance of parameter vector $\theta$ on example $x$ . Computing the exact gradient in each step of an iterative optimization algorithm becomes inefficient for large $M$ . Instead, we construct a minibatch $B \subset \{ 1 , \ldots , M \}$ of $| B | \ll M$ data points sampled uniformly and independently from the training set and compute an approximate stochastic gradient
+
+$$
+g ( \theta ) = \frac { 1 } { | \boldsymbol { \mathcal { B } } | } \sum _ { \boldsymbol { k } \in \boldsymbol { B } } \nabla \ell ( \theta ; x _ { k } ) ,
+$$
+
+which is an unbiased estimate, $\mathbf { E } [ g ( \theta ) ] ~ = ~ \nabla \mathcal { L } ( \theta )$ . We will denote by $\sigma ( \theta ) _ { i } ^ { 2 } : = { \bf v a r } [ g ( \theta ) _ { i } ]$ its element-wise variances.1
+
+The basic stochastic optimizer is stochastic gradient descent (SGD, Robbins & Monro, 1951) and its momentum variants (Polyak, 1964; Nesterov, 1983). A number of methods, widely-used in the deep learning community, choose per-element update magnitudes based on the history of stochastic gradient observations. Among these are ADAGRAD (Duchi et al., 2011), RMSPROP (Tieleman & Hinton, 2012), ADADELTA (Zeiler, 2012) and ADAM (Kingma & Ba, 2015).
+
+# 1.1 A CLOSER LOOK AT ADAM
+
+We start out from a reinterpretation of the widely-used ADAM optimizer. Some of the considerations naturally extend to ADAM’s close relatives RMSPROP and ADADELTA, but we restrict our attention to ADAM to keep the presentation concise. ADAM maintains moving averages of the observed stochas
+
+![](images/404fb4e564117211c1dc17e230c6d82c1898c0efaa509e7f6c6fab1fff3186f8.jpg)  
+Figure 1: Conceptual sketch of variance adaptation, ignoring the sign aspect of ADAM. The left panel shows the true gradient $\nabla { \mathcal { L } } = ( 2 , 1 )$ and stochastic gradients scattered around it with $( \sigma _ { 1 } , \bar { \sigma _ { 2 } } ) =$ (1, 1.5). In the right panel, we employ a variance adaptation (to be derived in $\ S 3 . 2 )$ that scales the $i$ -th coordinate by $( 1 \dot { + } \eta _ { i } ^ { 2 } ) ^ { - 1 }$ . In this example, the $\theta _ { 2 }$ -coordinate has much higher relative variance $( \eta _ { 2 } ^ { 2 } = 2 . 2 5 )$ than the $\theta _ { 1 }$ -coordinate $( \eta _ { 1 } ^ { 2 } = 0 . 2 5 )$ and is thus shortened. This reduces the variance of the update direction at the expense of biasing it away from the true gradient in expectation.
+
+tic gradients and their element-wise square2,
+
+$$
+\begin{array} { r l } & { \tilde { m } _ { t } = \beta _ { 1 } \tilde { m } _ { t - 1 } + ( 1 - \beta _ { 1 } ) g _ { t } , \qquad m _ { t } = ( 1 - \beta _ { 1 } ^ { t } ) ^ { - 1 } \tilde { m } _ { t } , } \\ & { \tilde { v } _ { t } = \beta _ { 2 } \tilde { v } _ { t - 1 } + ( 1 - \beta _ { 2 } ) g _ { t } ^ { 2 } , \qquad v _ { t } = ( 1 - \beta _ { 2 } ^ { t } ) ^ { - 1 } \tilde { v } _ { t } . } \end{array}
+$$
+
+Here, $m _ { t }$ and $v _ { t }$ are “bias-corrected” versions of the exponential moving averages to obtain convex combinations of past observed (squared) gradients. ADAM then updates
+
+$$
+\theta _ { t + 1 } = \theta _ { t } - \alpha \frac { m _ { t } } { \sqrt { v _ { t } } + \varepsilon }
+$$
+
+with a small constant $\varepsilon > 0$ guaranteeing numerical stability of this division. Ignoring $\varepsilon$ and assuming $| m _ { t , i } | > 0$ for the moment, we can rewrite the update direction as3
+
+$$
+{ \frac { m _ { t } } { \sqrt { v _ { t } } } } = { \frac { \mathrm { s i g n } ( m _ { t } ) | m _ { t } | } { \sqrt { v _ { t } } } } = { \frac { \mathrm { s i g n } ( m _ { t } ) } { \sqrt { \frac { v _ { t } } { m _ { t } ^ { 2 } } } } } = { \frac { \mathrm { s i g n } ( m _ { t } ) } { \sqrt { 1 + { \frac { v _ { t } - m _ { t } ^ { 2 } } { m _ { t } ^ { 2 } } } } } } .
+$$
+
+Since $m _ { t }$ and $v _ { t }$ approximate the first and second moment of the stochastic gradient $g _ { t }$ , respectively, $v _ { t } - m _ { t } ^ { 2 }$ can be seen as an estimate of element-wise stochastic gradient variances. The division by the non-central second moment effectively removes the magnitude of $m _ { t }$ ; it only appears in the ratio $( v _ { t } - m _ { t } ^ { 2 } ) / m _ { t } ^ { 2 }$ . Hence, ADAM can be interpreted as a combination of the two following aspects:
+
+• The update direction $( \pm )$ for the $i$ -th weight is given by the sign of $m _ { t , i }$ • The update magnitude for the $i$ -th weight is uniquely determined by the global step size $\alpha$ and an estimate of the relative variance,
+
+$$
+\hat { \eta } _ { t , i } ^ { 2 } : = \frac { v _ { t , i } - m _ { t , i } ^ { 2 } } { m _ { t , i } ^ { 2 } } \approx \frac { \sigma _ { t , i } ^ { 2 } } { \nabla { \mathcal { L } _ { t , i } ^ { 2 } } } = : \eta _ { t , i } ^ { 2 } .
+$$
+
+Specifically, the update in the $i$ -th coordinate is scaled by $\left( 1 + \hat { \eta } _ { t , i } ^ { 2 } \right) ^ { - 1 / 2 }$ , shortening steps in high-relative-variance coordinates. Fig. 1 shows a sketch of this variance adaptation.
+
+Table 1: The methods under consideration in this paper.   
+
+<table><tr><td></td><td>Sign + Magnitude</td><td>Sign</td></tr><tr><td>Not Variance-Adapted</td><td>SGD</td><td>SSD &quot;Stochastic Sign Descent&quot;</td></tr><tr><td rowspan="2">Variance-Adapted</td><td></td><td></td></tr><tr><td>SVAG “Stochastic Variance-Adapted Gradient&quot;</td><td>ADAM</td></tr></table>
+
+# 1.2 OVERVIEW
+
+Both aspects of ADAM—taking the sign and variance adaptation—are briefly mentioned in Kingma & Ba (2015), who note that “[t]he effective stepsize [...] is also invariant to the scale of the gradients” and refer to $m _ { t } / \sqrt { v _ { t } }$ as a “signal-to-noise ratio”. The purpose of this paper is to disentangle these two intertwined aspects in order to discuss and analyze them in isolation.
+
+This perspective naturally suggests two alternative methods by incorporating one of the aspects but not the other (see Table 1). Taking the sign of the stochastic gradient (or momentum term) without any further modification gives rise to “Stochastic Sign Descent” (SSD). On the other hand, “Stochastic Variance-Adapted Gradient” (SVAG) applies element-wise variance adaptation factors directly on the stochastic gradient (or momentum term) instead of on its sign. We proceed as follows: In Section 2, we investigate the sign aspect. In the simplified setting of stochastic quadratic problems, we derive conditions under which the element-wise sign of a stochastic gradient can be a better update direction than the stochastic gradient itself. Section 3 discusses the variance adaptation. We present a principled derivation of “optimal” element-wise variance adaptation factors for a stochastic gradient as well as its sign. Subsequently, we incorporate momentum and briefly discuss the practical estimation of stochastic gradient variance. Section 4 presents some experimental results.
+
+# 1.3 RELATED WORK
+
+The idea of using the sign of the gradient as the principal source of the optimizer update has already received some attention in the literature. The RPROP algorithm (Riedmiller & Braun, 1993) ignores the magnitude of the gradient and dynamically adapts the per-element magnitude of the update based on observed sign changes. With the goal of reducing communication cost in distributed training of neural networks, Seide et al. (2014) empirically investigate the use of the sign of stochastic gradients. Regarding the variance adaptation, Schaul et al. (2013) derive element-wise step sizes for stochastic gradient descent that have (among other factors) a dependency on the stochastic gradient variance.
+
+# 1.4 THE SIGN OF A STOCHASTIC GRADIENT
+
+We briefly establish a fact that will be used throughout the paper. The sign of a stochastic gradient $s ( \theta ) = \mathrm { s i g n } ( g ( \theta ) )$ estimates the sign of the true gradient. Its distribution (and thus the quality of this estimate) is fully characterized by the success probabilities $\rho _ { i } : = \mathbf { P } \left[ s ( \theta ) _ { i } = \mathrm { s i g n } ( \nabla \mathcal { L } ( \theta ) _ { i } ) \right]$ . These depend on the distribution of the stochastic gradient. If we assume $g ( \theta )$ to be Gaussian—which is strongly supported by a Central Limit Theorem argument on Eq. (2)—we have
+
+$$
+\rho _ { i } : = \mathbf { P } \left[ s ( \theta ) _ { i } = \operatorname { s i g n } ( \nabla \mathcal { L } ( \theta ) _ { i } ) \right] = \frac { 1 } { 2 } + \frac { 1 } { 2 } \operatorname { e r f } \left( \frac { | \nabla \mathcal { L } ( \theta ) _ { i } | } { \sqrt { 2 } \sigma ( \theta ) _ { i } } \right) ,
+$$
+
+see $\mathrm { \ S B } . 2$ in the supplements. Furthermore, it is $\mathbf { E } [ s ( \theta ) _ { i } ] = ( 2 \rho _ { i } - 1 ) \operatorname { s i g n } ( \nabla { \mathcal { L } } ( \theta ) _ { i } ) .$
+
+# 2 WHY THE SIGN?
+
+Can it make sense to ignore the gradient magnitude? We provide some intuition under which circumstances the element-wise sign of a stochastic gradient is a better update direction than the stochastic gradient itself. This question is difficult to tackle in general, which is why we restrict the problem
+
+class to the simple, yet insightful, case of stochastic quadratic problems, where we can investigate the effects of curvature properties and its interaction with stochastic noise.
+
+Model Problem (Stochastic Quadratic Problem (QP)). Consider the loss function $\ell ( \theta , x ) ~ =$ $0 . 5 ( \theta - x ) ^ { T } Q ( \theta - x )$ with a symmetric positive definite matrix $Q \in \mathbb { R } ^ { d }$ and “data” coming from the distribution $x \sim \dot { \mathcal { N } } ( x ^ { * } , \nu ^ { 2 } I )$ . It is
+
+$$
+\mathcal { L } ( \theta ) : = \mathbf { E } _ { x } [ \ell ( \theta , x ) ] = \frac { 1 } { 2 } ( \theta - x ^ { * } ) ^ { T } Q ( \theta - x ^ { * } ) + \frac { \nu ^ { 2 } } { 2 } \operatorname { t r } ( Q ) ,
+$$
+
+with $\nabla { \mathcal { L } } ( \theta ) = Q ( \theta - x ^ { * } )$ . Stochastic gradients are given by $g ( \theta ) = Q ( \theta - x ) \sim \mathcal { N } ( x ^ { * } , \nu ^ { 2 } I )$
+
+# 2.1 THEORETICAL COMPARISON
+
+We want to compare update directions on stochastic QPs in terms of their expected decrease in function value from a single update step. If we update from $\theta$ to $\theta + \alpha z$ , we have
+
+$$
+\mathbf { E } [ \mathcal { L } ( \theta + \alpha z ) ] = \mathcal { L } ( \theta ) + \alpha \nabla \mathcal { L } ( \theta ) ^ { T } \mathbf { E } [ z ] + \frac { \alpha ^ { 2 } } { 2 } \mathbf { E } [ z ^ { T } Q z ] .
+$$
+
+For this comparison of update directions, we allow for the optimal step size that minimizes Eq. (10), which is easily found to be $\alpha _ { * } = - \nabla \mathcal { L } ( \boldsymbol { \theta } ) ^ { T } \mathbf { E } [ \boldsymbol { z } ] / \mathbf { E } [ \boldsymbol { z } ^ { T } Q \boldsymbol { z } ]$ and yields an expected improvement of
+
+$$
+\mathcal { I } ( z ) : = \left| \mathbf { E } [ \mathcal { L } ( \theta + \alpha _ { * } z ) ] - \mathcal { L } ( \theta ) \right| = \frac { ( \nabla \mathcal { L } ( \theta ) ^ { T } \mathbf { E } [ z ] ) ^ { 2 } } { 2 \mathbf { E } [ z ^ { T } Q z ] } .
+$$
+
+We find the following expressions/bounds for the improvement of SGD and SSD:
+
+$$
+\mathcal { I } ( g ) = \frac { 1 } { 2 } \frac { ( \nabla \mathcal { L } ( \theta ) ^ { T } \nabla \mathcal { L } ( \theta ) ) ^ { 2 } } { \nabla \mathcal { L } ( \theta ) ^ { T } Q \nabla \mathcal { L } ( \theta ) + \nu ^ { 2 } \sum _ { i = 1 } ^ { d } \lambda _ { i } ^ { 3 } } , \quad \mathcal { Z } ( s ) \ge \frac { 1 } { 2 } \frac { \left( \sum _ { i = 1 } ^ { d } ( 2 \rho _ { i } - 1 ) | \nabla \mathcal { L } ( \theta ) _ { i } | \right) ^ { 2 } } { \sum _ { i , j = 1 } ^ { d } | q _ { i j } | }
+$$
+
+where the $\lambda _ { i } \in \mathbb { R } _ { + }$ are the eigenvalues of $Q$ with orthonormal eigenvectors $v _ { i } \in \mathbb { R } ^ { d }$ . Derivations can be found in $\mathrm { \ S B . l }$ of the supplements. Comparing these expressions, we make two observations.
+
+Firstly, $\mathcal { T } ( s )$ has a dependency on $\textstyle \sum _ { i , j } | q _ { i j } |$ . This quantity relates to the eigenvalues, as well as the orientation of the eigenbasis of $Q$ . By writing $Q$ in its eigendecomposition one finds that $\begin{array} { r } { \sum _ { i , j } | q _ { i j } | \leq \sum _ { i } \lambda _ { i } \| v _ { i } \| _ { 1 } ^ { 2 } } \end{array}$ . If the eigenvectors are perfectly axis-aligned (diagonal $Q$ ), their 1-norms are $\mathbf { \| } v _ { i } \| _ { 1 } = \| v _ { i } \| _ { 2 } = 1$ . It is intuitive that this is the best case for the intrinsically axis-aligned sign update. In general, the 1-norm is only bounded by $\| v _ { i } \| _ { 1 } \leq \sqrt { d } \| v _ { i } \| _ { 2 } = \sqrt { d }$ , suggesting that the sign update will have difficulties with arbitrarily oriented eigenbases. We can alternatively express this matter in terms of “diagonal dominance”. Assuming $Q$ has a percentage $c \in [ 0 , 1 ]$ of its “mass” on the diagonal, i.e., $\begin{array} { r } { \sum _ { i } | \bar { q _ { i i } } | \geq c \sum _ { i , j } | q _ { i j } | } \end{array}$ , we can write
+
+$$
+\mathcal { T } ( s ) \geq \frac { 1 } { 2 } \frac { \Big ( \sum _ { i = 1 } ^ { d } ( 2 \rho _ { i } - 1 ) | \nabla \mathcal { L } ( \theta ) _ { i } | \Big ) ^ { 2 } } { c ^ { - 1 } \sum _ { i = 1 } ^ { d } | q _ { i i } | } = \frac { 1 } { 2 } \frac { \Big ( \sum _ { i = 1 } ^ { d } ( 2 \rho _ { i } - 1 ) | \nabla \mathcal { L } ( \theta ) _ { i } | \Big ) ^ { 2 } } { c ^ { - 1 } \sum _ { i = 1 } ^ { d } \lambda _ { i } } .
+$$
+
+Becker & LeCun (1988) empirically investigated the diagonal dominance of Hessians in optimization problems arising from neural networks and found relatively high percentages of mass on the diagonals of $c = 0 . 1$ up to $c = 0 . 6$ for the problems they investigated.
+
+Secondly, hugely ob $\boldsymbol { \mathcal { T } } ( \boldsymbol { g } )$ contains the constant offset ive for ill-conditioned and no $\nu ^ { 2 } \textstyle \sum _ { i = 1 } ^ { d } \lambda _ { i } ^ { 3 }$ in ts. In nominator, which can become, on the other hand, there is no $\mathcal { T } ( s )$ such interaction between the magnitude of the noise and the eigenspectrum; the noise only manifests in the element-wise success probabilities $\rho _ { i }$ , its effect in the denominator is bounded. A recent paper (Chaudhari et al., 2016) investigated the eigenspectrum in deep learning problems and found it to be very ill-conditioned with the majority of eigenvalues close to zero and a few very large ones.
+
+In summary, we can expect the sign update to be beneficial for noisy, ill-conditioned problems with “diagonally dominant” Hessians. There is some (weak) empirical evidence that these conditions might be fulfilled in deep learning problems.
+
+![](images/19e6e17310caf2167b2a57e86acc43783b4c4f1e0c5d730fa1002011c3ff5e79.jpg)  
+Figure 2: Performance of SGD and SSD on 100-dimensional stochastic quadratic problems. Rows correspond to different $\mathrm { Q P s }$ : the eigenspectrum is shown and each is used with a randomly rotated and an axis-aligned eigenbasis. Columns correspond to different noise levels. Horizontal axis is number of steps; vertical axis is log function value and is shared per row for comparability.
+
+# 2.2 EXPERIMENTAL EVALUATION
+
+We verify the above findings on artificially generated stochastic QPs, where all relevant quantities are known analytically and controllable. We control the eigenspectrum by specifying a diagonal matrix $\Lambda$ of eigenvalues: (1) a mildly-conditioned problem with eigenvalues drawn uniformly from [0.1, 1.1] and (2) an ill-conditioned problem with a structured eigenspectrum similar to the one reported for neural networks by Chaudhari et al. (2016) by uniformly drawing $90 \%$ of the eigenvalues from $[ 0 , 1 ]$ and $10 \%$ from [30, 60]. $Q$ is then generated by (1) $Q = \Lambda$ to produce an axis-aligned problem and (2) $Q \ = \ R \mathring { \Lambda } R ^ { \hat { T } }$ with a rotation matrix $R$ drawn uniformly at random (see Diaconis & Shahshahani, 1987). This makes four different matrices, which we consider at noise levels $\nu \in \{ 0 , 0 . 1 , 4 . 0 \}$ . We compare SGD and SSD, both with the optimal step size as derived from Eq. (10), which can be computed exactly in this setting.
+
+Figure 2 shows the results, which confirm the theoretical findings. On the well-conditioned, noisefree problem, gradient descent vastly outperforms the sign-based method. Surprisingly, adding even a little noise almost evens out the difference in performance. The orientation of the eigenbasis had little effect on the performance of SSD in the well-conditioned case. On the ill-conditioned problem, the methods work roughly equally well when the eigenbasis is randomly rotated. As predicted, SSD benefits drastically from an axis-aligned eigenbasis (last row), where it clearly outperforms SGD.
+
+# 3 VARIANCE-BASED ELEMENT-WISE STEP SIZE ADAPTATION
+
+Besides the sign direction, the other defining property of ADAM are variance-based element-wise step sizes. Considering the variance adaptation in isolation from the sign aspect naturally suggests to employ it directly on the stochastic gradient, without taking the sign. In both cases, a motivation arises from the following consideration:
+
+Assume we want to update in a direction $p \in \mathbb { R } ^ { d }$ (or $\mathrm { s i g n } ( p ) )$ , but only have access to an unbiased estimate $\hat { p } \in \mathbb { R } ^ { d }$ with $\mathbf { E } [ \hat { p } ] = p$ . We allow for element-wise factors $\gamma \in \mathbb { R } ^ { d }$ , i.e., we update $\gamma \odot \hat { p }$ or $\gamma \odot \mathrm { s i g n } ( \hat { p } )$ . One way to make “optimal” use of these factors is to choose them such as to minimize the expected distance to the desired update direction. Using the squared Euclidean norm as a distance measure, we find the following result.
+
+Lemma 1. Let $\hat { p } \in \mathbb { R } ^ { d }$ be a random variable with $\mathbf { E } [ \hat { p } ] = p$ and $\mathbf { v a r } [ p _ { i } ] = \sigma _ { i } ^ { 2 }$ . Then
+
+$$
+\operatorname* { m i n } _ { \gamma \in \mathbb { R } ^ { d } } \mathbf { E } [ \| \gamma \odot \hat { p } - p \| _ { 2 } ^ { 2 } ] \quad i s ~ s o l \nu e d b y \quad \gamma _ { i } = \frac { p _ { i } ^ { 2 } } { p _ { i } ^ { 2 } + \sigma _ { i } ^ { 2 } } = \frac { 1 } { 1 + \sigma _ { i } ^ { 2 } / p _ { i } ^ { 2 } }
+$$
+
+and
+
+$$
+\operatorname* { m i n } _ { \gamma \in \mathbb { R } ^ { d } } \mathbf { E } [ \| \gamma \odot \mathrm { s i g n } ( \hat { p } ) - \mathrm { s i g n } ( p ) \| _ { 2 } ^ { 2 } ] \quad i s s o l \nu e d b y \quad \gamma _ { i } = ( 2 \rho _ { i } - 1 ) ,
+$$
+
+where $\rho _ { i } = \mathbf { P } [ \mathrm { s i g n } ( \hat { p } _ { i } ) = \mathrm { s i g n } ( p _ { i } ) ]$ .
+
+In the sign case, $\gamma _ { i }$ is proportional to the success probability with $\gamma _ { i } = 1$ if we are certain about the sign $\rho _ { i } = 1 { \ : }$ ) and $\gamma _ { i } = 0$ if we have no information about the sign at all $\rho _ { i } = . 5$ ).
+
+# 3.1 VARIANCE ADAPTATION FOR THE SIGN OF A STOCHASTIC GRADIENT
+
+Applying Eq. (15) to $\hat { p } = g$ , the optimal variance adaptation factors for the sign of a stochastic gradient are found to be $\gamma _ { i } = 2 \rho _ { i } - 1$ , where $\rho _ { i } = { \bf P } [ \mathrm { s i g n } ( g _ { i } ) = \mathrm { s i g n } ( \nabla { \mathcal { L } } _ { i } ) ]$ . Recall from Eq. (8) that, under the Gaussian assumption, the success probabilities of the sign of a stochastic gradient are $2 \rho _ { i } - 1 = \mathrm { e r f } [ ( \sqrt { 2 } \eta _ { i } ) ^ { - 1 } ]$ . ADAM uses the variance adaptation factors $( 1 + \eta _ { i } ^ { 2 } ) ^ { - 1 / 2 }$ , which turns out to be a close approximation of $\mathrm { e r f } [ ( \sqrt { 2 } \eta _ { i } ) ^ { - 1 } ]$ , as shown in Figure 5 in the supplements. Hence, ADAM can be regarded as an approximate realization of this optimal variance adaptation scheme. We experimented with both variants and found them to have identical effects. The small difference between them can be regarded as insignificant when $\eta$ itself is subject to approximation error. We thus stick to $( 1 + \eta _ { i } ^ { 2 } ) ^ { - 1 / 2 }$ for accordance with ADAM and to avoid the (more costly) error function.
+
+# 3.2 STOCHASTIC VARIANCE-ADAPTED GRADIENT (SVAG)
+
+Applying Eq. (14) to $\hat { p } = g$ , the optimal variance adaptation factors for SGD are found to be
+
+$$
+\gamma _ { i } = \frac { 1 } { 1 + \sigma _ { i } ^ { 2 } / \nabla \mathcal { L } _ { i } ^ { 2 } } = \frac { 1 } { 1 + \eta _ { i } ^ { 2 } } .
+$$
+
+This term is known from Schaul et al. (2013), where it appears together with diagonal curvature estimates in element-wise step sizes for SGD. We refer to this method (without curvature estimates) as “Stochastic Variance-Adapted Gradient” (SVAG). A momentum variant will be derived below.
+
+Intriguingly, variance adaptation of this form guarantees convergence without manually decreasing the global step size. We recover the $\mathcal { O } ( 1 / t )$ rate of SGD for smooth, strongly convex functions. We emphasize that this result considers an “idealized” version of SVAG with exact $\eta _ { i } ^ { 2 }$ . It is a motivation for this form of variance adaptation, not a statement about the performance with estimated variances.
+
+Theorem 1. Let $f$ be $\mu$ -strongly convex and $L$ -smooth. Assume we update $\theta _ { t + 1 } = \theta _ { t } - \alpha ( \gamma _ { t } \odot g _ { t } )$ , where $g _ { t }$ is a stochastic gradient with $\mathbf { E } [ g _ { t } | \theta _ { t } ] = \nabla f ( \theta _ { t } )$ , $\mathbf { v a r } [ g _ { t , i } | \theta _ { t } ] = \sigma _ { t , i } ^ { 2 } .$ , variance adaptation factors $\gamma _ { t , i } = ( 1 + \sigma _ { t , i } ^ { 2 } / \nabla f _ { t , i } ^ { 2 } ) ^ { - 1 }$ , and $\alpha = 1 / L$ . Assume $\mathbf { E } [ \| g _ { t } \| ^ { 2 } ] \leq G ^ { 2 }$ . Then
+
+$$
+\mathbf { E } [ f ( \theta _ { t } ) - f _ { * } ] \in \mathcal { O } \left( \frac { 1 } { t } \right) ,
+$$
+
+where $f _ { * }$ is the minimum value of $f$ .
+
+(Proof in $\ S B . 4 )$
+
+# 3.3 ESTIMATING GRADIENT VARIANCE
+
+In practice, the relative variance is of course not known and must be estimated. As noted in the $\sigma _ { t , i } ^ { 2 } \approx \widehat { s } _ { t , i } = v _ { t , i } - m _ { t , i } ^ { 2 }$ ains an estimate of the stochastic gradient variance from moving averages,. The underlying assumption is that the function does not change drastically horizon” of the moving average, such that the recent gradients can approximately be considered to be iid draws from the stochastic gradient distribution. An estimate of the relative variance can then be obtained by $( v _ { t } - m _ { t } ^ { 2 } ) / ( m _ { t } ^ { 2 } )$ , as in ADAM.
+
+Unlike ADAM we do not use different moving average constants for $m _ { t }$ and $v _ { t }$ . The constant for the moving average should define a time horizon over which the gradients can approximately be considered to come from the same distribution. From this perspective, it is hardly justifiable to use different horizons for the gradient and its square. Furthermore, we found individual moving average constants for $m _ { t }$ and $v _ { t }$ to have only minor effect on the performance of our methods.
+
+An alternative variance estimate can be computed locally “within” a single mini-batch. A more detailed discussion of both estimators can be found in $\mathrm { \ S C }$ of the supplements. We have experimented with both estimators and found them to work equally well for our purpose of variance adaptation. We thus stick to moving average-based estimates for the main paper. Appendix D provides details and experimental results for the mini-batch variant.
+
+# 3.4 INCORPORATING MOMENTUM
+
+When we add momentum—i.e., we want to update in the direction $r _ { t }$ or $\mathrm { s i g n } ( r _ { t } )$ with a momentum term $\begin{array} { r } { r _ { t } = \mu r _ { t - 1 } + g _ { t } = \sum _ { s = 0 } ^ { t } \mu ^ { s } g _ { t - s } . } \end{array}$ —the variance adaptation factors should be determined by the relative variance of $r _ { t }$ , according to Lemma 1. It is
+
+$$
+\mathbf { E } [ r _ { t } ] = \sum _ { s = 0 } ^ { t } \mu ^ { s } \nabla { \mathcal { L } } _ { t - s } , \quad \mathbf { v a r } [ r _ { t , i } ] = \sum _ { s = 0 } ^ { t } ( \mu ^ { s } ) ^ { 2 } \mathbf { v a r } [ g _ { t - s , i } ] = \sum _ { s = 0 } ^ { t } \mu ^ { 2 s } \sigma _ { t - s , i } ^ { 2 } .
+$$
+
+Replacing $\mathbf { E } [ g _ { t - s } ] \approx m _ { t - s }$ and $\mathbf { v a r } [ g _ { t - s } ] \approx v _ { t - s } - m _ { t - s } ^ { 2 }$ we could compute these quantities. However, this would require two additional moving averages and can thus be discarded as impractical. Fortunately, we can motivate an approximation that does not require any additional memory requirements (see $\mathrm { \ S C ) }$ :
+
+$$
+\frac { \mathbf { v a r } [ r _ { t } ] } { \mathbf { E } [ r _ { t } ] ^ { 2 } } \approx \kappa ( \mu , t ) \frac { v _ { t } - m _ { t } ^ { 2 } } { m _ { t } ^ { 2 } } ~ \mathrm { w i t h } ~ \kappa ( \mu , t ) : = \frac { ( 1 - \mu ^ { 2 t } ) ( 1 - \mu ) ^ { 2 } } { ( 1 - \mu ^ { 2 } ) ( 1 - \mu ^ { t } ) ^ { 2 } } .
+$$
+
+Note that the correction factor $\kappa ( \mu , t )$ does not appear in ADAM, which updates in the direction $\mathrm { s i g n } ( m _ { t } ) = \mathrm { s i g n } ( r _ { t } )$ but performs variance adaptation based on $( v _ { t } - m _ { t } ^ { 2 } ) \bar { / } m _ { t } ^ { 2 }$ . The supplements contain experiments with a variant of ADAM that includes this correction factor.
+
+# 4 EXPERIMENTS
+
+We compare momentum-SGD (M-SGD) and ADAM to two new methods: First, we consider M-SSD: stochastic sign descent using a momentum term. The second method is M-SVAG, i.e., SGD with momentum and variance adaptation of the form $( 1 + \eta ^ { 2 } ) ^ { - 1 }$ , where the relative variance of the momentum term is estimated from moving averages according to Eq. (19). These four methods are the four possible recombinations of the sign aspect and the variance adaptation aspect of ADAM, as laid out in Table 1. Algorithms 1 and 2 provide pseudo-code for M-SSD and M-SVAG. For all experiments, we use $\mu = 0 . 9$ for M-SGD, M-SSD and M-SVAG and default parameters $\beta _ { 1 } =$ $0 . 9 , \mathring { \beta _ { 2 } } = 0 . 9 9 9 , \varepsilon = 1 0 ^ { - 8 } ,$ for ADAM. Note that M-SVAG does not use an $\varepsilon$ -parameter, see Alg. 2.
+
+# Algorithm 1 M-SSD (Stochastic Sign Descent with Momentum)
+
+<table><tr><td colspan="2">Require: initial value 0o,step size α, momentum parameter μ ∈ [0,1], number of steps T</td></tr><tr><td colspan="2">1:Initialize m= O,v = 0</td></tr><tr><td>2: for t =1,...,T do</td><td></td></tr><tr><td>3: 4:</td><td>Compute stochastic gradient g = g(0)</td></tr><tr><td>5:</td><td>Update moving average m ← μm + g</td></tr><tr><td>6: end for</td><td>Updateθ ←θ-α sign(m)</td></tr><tr><td colspan="2"></td></tr></table>
+
+# Algorithm 2 M-SVAG (Stochastic Variance-Adapted Gradient with Momentum)
+
+Require: initial value $\theta _ { 0 }$ , step size $\alpha$ , momentum parameter $\mu \in [ 0 , 1 ]$ , number of steps $T$
+
+1: Initialize $\tilde { m } = 0$ , $\tilde { v } = 0$   
+2: for $t = 1 , \dots , T$ do   
+3: Compute stochastic gradient $g = g ( \theta )$   
+4: Update moving averages $\tilde { m }  \mu \tilde { m } + ( 1 - \mu ) g , \quad \tilde { v }  \mu \tilde { v } + ( 1 - \mu ) g ^ { 2 }$   
+5: Bias-correct $\bar { m } = ( 1 \bar { - } \mu ^ { t } ) ^ { - 1 } \tilde { m }$ , $v = ( 1 - \mu ^ { t } ) ^ { - 1 } \tilde { v }$   
+6: Compute relative variance estimate η2 = κ(µ, t) v−m2m2   
+7: Compute variance adaptation factors $\gamma = ( 1 + \eta ^ { 2 } ) ^ { - 1 }$   
+8: Update $\theta  \theta - \alpha ( \gamma \overset { \cdot } { \odot } m )$
+
+. Eq. (19)
+
+# 9: end for
+
+We do not use an $\varepsilon$ -parameter as in ADAM. In the (rare) case that $m _ { i } = 0$ for coordinate $i$ , the division by zero in line 6 is caught and the update magnitude will be set to zero in line 8.
+
+# 4.1 EXPERIMENTAL SET-UP
+
+We tested all methods on three problems: a simple fully-connected neural network on the MNIST data set (LeCun et al., 1998), as well as convolutional neural networks (CNNs) on the CIFAR-10 and CIFAR-100 data sets (Krizhevsky, 2009). On CIFAR-10, we used a simple CNN with three convolutional layers, interspersed with max-pooling, and three fully-connected layers. On CIFAR100 we used the AllCNN architecture of Springenberg et al. (2014) with a total of nine convolutional layers. A complete description of all network architectures has been moved to $\ S \mathbf { A }$ . While MNIST and CIFAR-10 are trained with a constant global step size $( \alpha )$ , we used a fixed decreasing schedule for CIFAR-100, dividing by 10 after $4 0 \mathrm { k }$ and $5 0 \mathrm { k }$ steps (adopted from Springenberg et al., 2014). We used a batch size of 128 on MNIST and 256 on the two CIFAR data sets.
+
+Step sizes (initial step sizes in the case of CIFAR-100) were tuned for each method individually by first finding the maximal stable step size by trial and error, then searching downwards over two orders of magnitude (details in $\ S \mathbf { A }$ ). We selected the one that yielded maximal overall test accuracy within the fixed number of training steps. Experiments with the best step size have been replicated ten times with different random seeds and all performance indicators are reported as mean plus/minus one standard deviation.
+
+# 4.2 RESULTS
+
+Results are shown in Figure 3. On MNIST, ADAM clearly outperforms M-SGD. Interestingly, there is only a very small difference in performance between the two sign-based methods, M-SSD and ADAM. Apparently, the advantage of ADAM over M-SGD on this problem is primarily due to the sign aspect. Going from M-SGD to M-SVAG, gives a considerable boost in performance, but MSVAG is still outperformed by the two sign-based methods.
+
+On CIFAR-10, the sign-based methods again have superior performance. Neither M-SSD nor M-SGD can benefit significantly from adding variance adaptation.
+
+Finally, the situation is reversed on CIFAR-100, where M-SGD outperforms ADAM. It attains lower minimal loss values (both training and test) and converges faster. This is also reflected in the test accuracies, where M-SGD beats ADAM by almost 10 percentage points. Furthermore, ADAM is much less stable with significantly larger variance in performance. On this problem, variance adaptation has a small but significant positive effect for the sign-based methods as well as for M-SGD. When going from M-SGD to M-SVAG we gain some speed in the initial phase. The difference is later evened out by the manual learning rate decrease (which was necessary, for all methods, to train this architecture to satisfying performance).
+
+# 5 DISCUSSION AND CONCLUSION
+
+We have argued that ADAM combines two aspects: taking signs and variance adaptation. Our separate analysis of both aspects provides some insight into the inner workings of this method.
+
+![](images/da6602d5d1724210da9fbd5b18e0ee47ec020f1c161f9153c94b4ffc4d6341ed.jpg)  
+Figure 3: Experimental results on the three test problems. Plots display training and test loss over the number of steps. Curves for the different optimization methods are color-coded. The shaded area spans plus/minus one standard deviation, obtained from ten replications. The table below contains test accuracies evaluated after the last iteration.
+
+Taking the sign can be beneficial, but does not need to be. Our theoretical analysis suggests that it depends on the interplay of stochasticity, the conditioning of the problem, and its “axis-alignment”. Our experiments confirm that sign-based methods work well on some, but not all problems.
+
+Variance adaptation can be applied to any stochastic update direction. In our experiments it was beneficial in all cases, but its effect can sometimes be minuscule. M-SVAG, a variance-adapted variant of momentum-SGD, is a useful addition to the practitioner’s toolbox for problems where sign-based methods like ADAM fail. Its memory and computation cost are identical to ADAM and it has two hyper-parameters, the momentum constant $\mu$ and the global step size $\alpha$ . Our TensorFlow (Abadi et al., 2015) implementation of this method will be made available upon publication.
+
+# ACKNOWLEDGMENTS
+
+We want to thank [names removed] for many helpful discussions.
+
+# REFERENCES
+
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+
+SUPPLEMENTARY MATERIAL
+
+A DESCRIPTION OF EXPERIMENTS
+
+# A.1 NETWORK ARCHITECTURES
+
+MNIST We train a simple fully-connected neural network with three hidden layers of 1000, 500 and 100 units with ReLU activation. The output layer has 10 units with softmax activation. We use the cross-entropy loss function and apply $L _ { 2 }$ -regularization on all weights, but not the biases. We use a batch size of 128. The global learning rate $\alpha$ stays constant.
+
+CIFAR-10 The CIFAR-10 data set consists of $3 2 \times 3 2 \mathrm { p x }$ RGB images with one of ten categorical labels. We train a convolutional neural network (CNN) with three convolutional layers (64 filters of size $5 \times 5$ , 96 filters of size $3 \times 3$ , and 128 filters of size $3 \times 3$ ) interspersed with max-pooling over $3 \times 3$ areas with stride 2. Two fully-connected layers with 512 and 256 units follow. We use ReLU activation function for all layers. The output layer has 10 units for the 10 classes of CIFAR-10 with softmax activation. We use the cross-entropy loss function and apply $L _ { 2 }$ -regularization on all weights, but not the biases. During training we perform some standard data augmentation operations (random cropping of sub-images, left-right mirroring, color distortion) on the input images. We use a batch size of 256. The global learning rate $\alpha$ stays constant.
+
+CIFAR-100 We use the AllCNN architecture of Springenberg et al. (2014). It consists of seven convolutional layers, some of them with stride, and no pooling layers. The fully-connected layers are replaced with two layers of $1 \times 1$ convolutions with global spatial averaging in the end. ReLU activation function is used in all layers. Details can be found in the original paper. We use the cross-entropy loss function and apply $L _ { 2 }$ -regularization on all weights, but not the biases. We used the same data augmentation operations as for CIFAR-10 and a batch size of 256. The global learning rate $\alpha$ is decreased by a factor of 10 after $4 0 \mathrm { k }$ and $5 0 \mathrm { k }$ steps.
+
+# A.2 LEARNING RATE TUNING
+
+Learning rates for each optimizer have been tuned by first finding the maximal stable learning rate by trial and error and then searching downwards over two orders of magnitude with learning rates $6 \cdot 1 0 ^ { m }$ , $3 \cdot 1 0 ^ { m }$ , and $1 \cdot 1 0 ^ { m }$ for order of magnitude $m$ . We evaluated loss and accuracy on the full test set at a constant interval and selected the best-performing learning rate for each method in terms of maximally reached test accuracy. Using the best learning rate, we replicated the experiment ten times with different random seeds.
+
+# B MATHEMATICAL DETAILS
+
+# B.1 DETAILS OF THE ANALYSIS ON STOCHASTIC QPS
+
+We derive the expressions for $\mathcal { T } ( s )$ and $\boldsymbol { \mathcal { T } } ( \boldsymbol { g } )$ in Eq. (12). We drop the fixed $\theta$ from the notation for readability. For SGD, we have $\mathbf { E } [ g ] = \nabla \mathcal { L }$ and $\mathbf { E } [ g ^ { T } Q g ] = \nabla \mathcal { L } ^ { \hat { T } } Q \nabla \mathcal { L } + \mathrm { t r } ( Q \mathbf { c o v } [ g ] )$ , which is a general fact for quadratic forms of random variables. For the stochastic QP the gradient covariance is $\mathbf { \bar { c o v } } [ g ] = \nu ^ { 2 } Q \dot { Q }$ , thus $\begin{array} { r } { \mathrm { t r } ( Q \mathbf { c o v } [ g ] ) = \nu ^ { 2 } \mathrm { t r } ( Q Q Q ) = \nu ^ { 2 } \sum _ { i } \lambda _ { i } ^ { 3 } } \end{array}$ . Plugging everything into Eq. (11) yields
+
+$$
+\mathcal { T } ( g ) = \frac { ( \nabla \mathcal { L } ^ { T } \nabla \mathcal { L } ) ^ { 2 } } { \nabla \mathcal { L } ^ { T } Q \nabla \mathcal { L } + \nu ^ { 2 } \sum _ { i = 1 } ^ { d } \lambda _ { i } ^ { 3 } } .
+$$
+
+For stochastic sign descent, we have $\mathbf { E } [ s _ { i } ] ~ = ~ ( 2 \rho _ { i } ~ - ~ 1 ) \mathrm { s i g n } ( \nabla { \mathcal { L } } _ { i } )$ and thus $\nabla \mathcal { L } ^ { T } \mathbf { E } [ s ] ~ =$ $\begin{array} { r } { \sum _ { i = 1 } ^ { d } \nabla \mathcal { L } _ { i } \mathbf { E } [ s _ { i } ] = \sum _ { i } ( 2 \rho _ { i } - 1 ) | \nabla \mathcal { L } _ { i } | } \end{array}$ . Regarding the denominator, it is
+
+$$
+0 \leq s ^ { T } H s = | s ^ { T } Q s | = \left| \sum _ { i = 1 } ^ { d } q _ { i j } s _ { i } s _ { j } \right| \leq \sum _ { i = 1 } ^ { d } | q _ { i j } | | s _ { i } | | s _ { j } | = \sum _ { i = 1 } ^ { d } | q _ { i j } | .
+$$
+
+![](images/50cd5e3eb94599e4df6f4c2182fdf84af844b55df0e22a53f83f911479f0cbaa.jpg)  
+Figure 4: Probability density functions (pdf) of three Gaussian distributions, all with $\mu = 1$ , but different variances $\sigma ^ { 2 } = 0 . 5$ (left), $\sigma ^ { 2 } = \mathrm { { 1 . 0 } }$ (middle), $\sigma ^ { 2 } = 4 . 0$ (right). The shaded area under the curve corresponds to the probability that a sample from the distribution has the opposite sign than its mean. For the Gaussian distribution, this probability is uniquely determined by the fraction $\sigma / | \mu |$ , as shown in Lemma 2.
+
+Plugging everything into Eq. (11) yields
+
+$$
+\mathcal { T } ( s ) \geq \frac { \left( \sum _ { i = 1 } ^ { d } ( 2 \rho _ { i } - 1 ) | \nabla \mathcal { L } _ { i } | \right) ^ { 2 } } { \sum _ { i = 1 } ^ { d } | q _ { i j } | } .
+$$
+
+B.2 SUCCESS PROBABILITIES OF THE SIGN OF A STOCHASTIC GRADIENT
+
+We have stated in the main text that the sign of a stochastic gradient, $s ( \theta ) = \mathrm { s i g n } ( g ( \theta ) )$ , has success probabilities
+
+$$
+\rho _ { i } = \mathbf { P } [ s ( \theta ) _ { i } = \operatorname { s i g n } ( \nabla { \mathcal { L } } ( \theta ) _ { i } ) ] = { \frac { 1 } { 2 } } + { \frac { 1 } { 2 } } \operatorname { e r f } \left( { \frac { | \nabla { \mathcal { L } } ( \theta ) _ { i } | } { \sqrt { 2 } \sigma ( \theta ) _ { i } } } \right)
+$$
+
+under the assumption that $g \sim \mathcal { N } ( \nabla \mathcal { L } , \Sigma )$ . The following Lemma formally proves this statement and Figure 4 provides a pictorial illustration.
+
+Lemma 2. If $X \sim { \mathcal { N } } ( \mu , \sigma ^ { 2 } )$ then
+
+$$
+\rho = \mathbf { P } [ \operatorname { s i g n } ( X ) = \operatorname { s i g n } ( \mu ) ] = { \frac { 1 } { 2 } } \left( 1 + \operatorname { e r f } \left( { \frac { | \mu | } { \sqrt { 2 } \sigma } } \right) \right) .
+$$
+
+Proof. The cumulative density function (cdf) of $X \sim \mathcal { N } ( \mu , \sigma ^ { 2 } )$ is $\mathbf { P } [ X \leq x ] = \Phi ( ( x - \mu ) / \sigma )$ , where $\Phi ( z ) = 0 . 5 ( 1 + \mathrm { e r f } ( z / \sqrt { 2 } ) )$ is the cdf of the standard normal distribution. If $\mu < 0$ , then
+
+$$
+\rho = \mathbf { P } [ X < 0 ] = \Phi \left( \frac { 0 - \mu } { \sigma } \right) = \frac { 1 } { 2 } \left( 1 + \operatorname { e r f } \left( \frac { - \mu } { \sqrt { 2 } \sigma } \right) \right) .
+$$
+
+If $\mu > 0$ , then
+
+$$
+\begin{array} { l } { \displaystyle \rho = \mathbf { P } [ X > 0 ] = 1 - \mathbf { P } [ X \leq 0 ] = 1 - \Phi \left( \frac { 0 - \mu } { \sigma } \right) } \\ { \displaystyle = 1 - \frac { 1 } { 2 } \left( 1 + \mathrm { e r f } \left( \frac { - \mu } { \sqrt { 2 } \sigma } \right) \right) = \frac { 1 } { 2 } \left( 1 + \mathrm { e r f } \left( \frac { \mu } { \sqrt { 2 } \sigma } \right) \right) , } \end{array}
+$$
+
+where the last step used the anti-symmetry of the error function.
+
+![](images/c08f2f0fa8bc643522d8443b13fdf2ac0a7fd1d4e227a8dbde42544aa741ee41.jpg)  
+Figure 5: Variance adaptation factors as functions of the relative standard deviation $\eta$ . $( 1 + \eta ^ { 2 } ) ^ { - 1 }$ is the optimal variance adaptation factor for SGD (Eq. 16). The optimal factor for the sign of a stochastic gradient is $\mathrm { e r f } ( ( \sqrt { 2 } \eta ) ^ { - 1 } )$ under the Gaussian assumption (Eq. 15). It is closely approximated by $( 1 + \eta ^ { 2 } ) ^ { - 1 / 2 }$ , which is the factor implicitly employed by ADAM (Eq. 6).
+
+B.3 DETAILS ON VARIANCE ADAPTATION FACTORS
+
+Proof of Lemma $^ { l }$ . Using $\mathbf { E } [ \hat { p } _ { i } ] = p _ { i }$ and ${ \bf E } [ \hat { p } _ { i } ^ { 2 } ] = p _ { i } ^ { 2 } + \sigma _ { i } ^ { 2 }$ , we get
+
+$$
+\begin{array} { l } { { \displaystyle { \bf E } [ \| \gamma \odot \hat { p } - p \| _ { 2 } ^ { 2 } ] = \sum _ { i = 1 } ^ { d } { \bf E } [ ( \gamma _ { i } \hat { p } _ { i } - p _ { i } ) ^ { 2 } ] = \sum _ { i = 1 } ^ { d } \gamma _ { i } ^ { 2 } { \bf E } [ \hat { p } _ { i } ^ { 2 } ] - 2 \gamma _ { i } p _ { i } { \bf E } [ \hat { p } _ { i } ] + p _ { i } ^ { 2 } } } \\ { { \displaystyle \quad \quad = \sum _ { i = 1 } ^ { d } \gamma _ { i } ^ { 2 } ( p _ { i } ^ { 2 } + \sigma _ { i } ^ { 2 } ) - 2 \gamma _ { i } p _ { i } ^ { 2 } + p _ { i } ^ { 2 } . } } \end{array}
+$$
+
+Setting the derivative w.r.t. $\gamma _ { i }$ to zero, we find the optimal choice
+
+$$
+\gamma _ { i } = \frac { p _ { i } ^ { 2 } } { p _ { i } ^ { 2 } + \sigma _ { i } ^ { 2 } } .
+$$
+
+Using $\mathbf { E } [ \mathrm { s i g n } ( \hat { p } _ { i } ) ] = ( 2 \rho _ { i } - 1 ) \mathrm { s i g n } ( p _ { i } )$ and $\mathrm { s i g n } ( \cdot ) ^ { 2 } = 1$ , we get
+
+$$
+\begin{array} { l l l } { \displaystyle \mathbf { E } [ \| \gamma \odot \mathrm { s i g n } ( \hat { p } ) - \mathrm { s i g n } ( p ) \| _ { 2 } ^ { 2 } ] = \sum _ { i = 1 } ^ { d } \gamma _ { i } ^ { 2 } \mathbf { E } [ \mathrm { s i g n } ( \hat { p } _ { i } ) ^ { 2 } ] - 2 \gamma _ { i } \mathrm { s i g n } ( p _ { i } ) \mathbf { E } [ \mathrm { s i g n } ( \hat { p } _ { i } ) ] + \mathrm { s i g n } ( p _ { i } ) ^ { 2 } } \\ { \displaystyle \qquad = \gamma _ { i } ^ { 2 } - 2 \gamma _ { i } ( 2 \rho _ { i } - 1 ) + 1 } \end{array}
+$$
+
+and easily find the optimal choice
+
+$$
+\gamma _ { i } = 2 \rho _ { i } - 1 .
+$$
+
+by setting the derivative to zero.
+
+See Figure 5 for a plot of the variance adaptation factors considered in this paper.
+
+# B.4 CONVERGENCE OF IDEALIZED STOCHASTIC VARIANCE-ADAPTED GRADIENT
+
+We proof the convergence results for idealized variance-adapted stochastic gradient descent. We have to clarify an aspect that we have glossed over in the main text. A stochastic optimizer generates a discrete stochastic process $\{ \boldsymbol { \theta } _ { t } \} _ { t \in  { \mathbb { N } } _ { 0 } }$ . We denote as $\mathbf { E } _ { t } [ \cdot ] = \mathbf { E } [ \cdot | \theta _ { 0 } , \ldots , \theta _ { t } ]$ the conditional expectation given a realization of that process up to time step $t$ . Recall that $\mathbf { E } [ \mathbf { E } _ { t } [ \cdot ] ] = \mathbf { E } [ \cdot ]$ .
+
+Proof of Theorem $^ { l }$ . Using the Lipschitz continuity of $\nabla f$ , we can bound $f ( \theta + \Delta \theta ) \leq f ( \theta ) +$ $\begin{array} { r } { \nabla f ( \theta ) ^ { T } \Delta \theta + \frac { L } { 2 } \| \Delta \theta \| ^ { 2 } } \end{array}$ . Hence,
+
+$$
+\begin{array} { r l } & { \mathbf { E } _ { t } \big [ f _ { t + 1 } \big ] \leq f _ { t } - \alpha \mathbf { E } _ { t } \big [ \nabla f _ { t } ^ { T } ( \gamma _ { t } \odot g _ { t } ) \big ] + \displaystyle \frac { L \alpha ^ { 2 } } { 2 } \mathbf { E } _ { t } \big [ \| \gamma _ { t } \odot g _ { t } \| ^ { 2 } \big ] } \\ & { \qquad = f _ { t } - \displaystyle \frac { 1 } { L } \sum _ { i = 1 } ^ { d } \gamma _ { t , i } \nabla f _ { t , i } \mathbf { E } [ g _ { t , i } ] + \displaystyle \frac { 1 } { 2 L } \sum _ { i = 1 } ^ { d } \gamma _ { t , i } ^ { 2 } \mathbf { E } _ { t } [ g _ { t , i } ^ { 2 } ] } \\ & { \qquad = f _ { t } - \displaystyle \frac { 1 } { L } \sum _ { i = 1 } ^ { d } \gamma _ { t , i } \nabla f _ { t , i } ^ { 2 } + \displaystyle \frac { 1 } { 2 L } \sum _ { i = 1 } ^ { d } \gamma _ { t , i } ^ { 2 } ( \nabla f _ { t , i } ^ { 2 } + \sigma _ { t , i } ^ { 2 } ) . } \end{array}
+$$
+
+Plugging in the definition
+
+$$
+\gamma _ { t , i } = \frac { \nabla f _ { t , i } ^ { 2 } } { \nabla f _ { t , i } ^ { 2 } + \sigma _ { t , i } ^ { 2 } }
+$$
+
+and simplifying, we get
+
+$$
+\mathbf { E } _ { t } [ f _ { t + 1 } ] \leq f _ { t } - \frac { 1 } { 2 L } \sum _ { i = 1 } ^ { d } \frac { \nabla f _ { t , i } ^ { 2 } } { \nabla f _ { t , i } ^ { 2 } + \sigma _ { t , i } ^ { 2 } } \nabla f _ { t , i } ^ { 2 } .
+$$
+
+Using Jensen’s inequality4
+
+$$
+\begin{array} { r l } & { \displaystyle \sum _ { i = 1 } ^ { d } \frac { \nabla f _ { t , i } ^ { 2 } } { \nabla f _ { t , i } ^ { 2 } + \sigma _ { t , i } ^ { 2 } } \nabla f _ { t , i } ^ { 2 } = \| \nabla f _ { t } \| ^ { 2 } \sum _ { i = 1 } ^ { d } \frac { \nabla f _ { t , i } ^ { 2 } } { \| \nabla f _ { t } \| ^ { 2 } } \left( \frac { \nabla f _ { t , i } ^ { 2 } + \sigma _ { t , i } ^ { 2 } } { \nabla f _ { t , i } ^ { 2 } } \right) ^ { - 1 } } \\ & { \qquad \geq \| \nabla f _ { t } \| ^ { 2 } \left( \displaystyle \sum _ { i = 1 } ^ { d } \frac { \nabla f _ { t , i } ^ { 2 } } { \| \nabla f _ { t } \| ^ { 2 } } \frac { \nabla f _ { t , i } ^ { 2 } + \sigma _ { t , i } ^ { 2 } } { \nabla f _ { t , i } ^ { 2 } } \right) ^ { - 1 } } \\ & { \qquad = \frac { \| \nabla f _ { t } \| ^ { 4 } } { \sum _ { i = 1 } ^ { d } ( \nabla f _ { t , i } ^ { 2 } + \sigma _ { t , i } ^ { 2 } ) } \geq \frac { \| \nabla f _ { t } \| ^ { 4 } } { G ^ { 2 } } . } \end{array}
+$$
+
+Due to strong convexity, we have $\| \nabla f _ { t } \| ^ { 2 } \geq 2 \mu ( f _ { t } - f _ { * } )$ and can further bound
+
+$$
+\sum _ { i = 1 } ^ { d } \frac { \nabla f _ { t , i } ^ { 2 } } { \nabla f _ { t , i } ^ { 2 } + \sigma _ { t , i } ^ { 2 } } \nabla f _ { t , i } ^ { 2 } \geq \frac { 4 \mu ^ { 2 } ( f _ { t } - f _ { * } ) ^ { 2 } } { G ^ { 2 } } .
+$$
+
+Inserting this in (33) and subtracting $f _ { * }$ , we get
+
+$$
+\mathbf { E } _ { t } [ f _ { t + 1 } ] - f _ { * } \leq f _ { t } - f _ { * } - \frac { 2 \mu ^ { 2 } } { L G ^ { 2 } } ( f _ { t } - f _ { * } ) ^ { 2 } ,
+$$
+
+and, consequently, by total expectation
+
+$$
+\begin{array} { l } { { \displaystyle { \bf E } [ f _ { t + 1 } - f _ { * } ] = { \bf E } \left[ { \bf E } _ { t } [ f _ { t + 1 } ] - f _ { * } \right] \leq { \bf E } [ f _ { t } - f _ { * } ] - \frac { 2 \mu ^ { 2 } } { L G ^ { 2 } } { \bf E } [ ( f _ { t } - f _ { * } ) ^ { 2 } ] } \ ~ } \\ { { \displaystyle \phantom { \frac { 1 } { 1 } } \leq { \bf E } [ f _ { t } - f _ { * } ] - \frac { 2 \mu ^ { 2 } } { L G ^ { 2 } } { \bf E } [ f _ { t } - f _ { * } ] ^ { 2 } } , } \end{array}
+$$
+
+which we rewrite, using the shorthand $e _ { t } : = \mathbf { E } [ f _ { t } - f _ { * } ]$ , as
+
+$$
+0 \leq e _ { t + 1 } \leq e _ { t } ( 1 - c e _ { t } ) , \quad c = \frac { 2 \mu ^ { 2 } } { L G ^ { 2 } } .
+$$
+
+To conclude the proof, we will show that this implies $\textstyle e _ { t } \in { \mathcal { O } } { \bigl ( } { \frac { 1 } { t } } { \bigr ) }$ . Without loss of generality, we assume $e _ { t + 1 } > 0$ and get
+
+$$
+e _ { t + 1 } ^ { - 1 } \geq e _ { t } ^ { - 1 } ( 1 - c e _ { t } ) ^ { - 1 } \geq e _ { t } ^ { - 1 } ( 1 + c e _ { t } ) = e _ { t } ^ { - 1 } + c ,
+$$
+
+where the second step is due to the simple fact that $( 1 - x ) ^ { - 1 } \geq ( 1 + x )$ for any $x \in [ 0 , 1 )$ . Summing this inequality over $t = 0 , \ldots , T - 1$ yields $e _ { T } ^ { - 1 } \geq e _ { 0 } ^ { - 1 } + T c$ and, thus,
+
+$$
+T e _ { T } \le \left( \frac { 1 } { T e _ { 0 } } + c \right) ^ { - 1 } \stackrel { T \to \infty } { \longrightarrow } \frac { 1 } { c } < \infty ,
+$$
+
+which shows that $\textstyle e _ { t } \in { \mathcal { O } } { \bigl ( } { \frac { 1 } { t } } { \bigr ) }$ .
+
+# C MORE ON GRADIENT VARIANCE ESTIMATION
+
+# C.1 ESTIMATES FROM MOVING AVERAGES
+
+Iterating the recursive formula for $\tilde { m } _ { t }$ backwards, we get
+
+$$
+m _ { t } = \frac { \tilde { m } _ { t } } { 1 - \beta _ { 1 } ^ { t } } = \frac { 1 } { 1 - \beta _ { 1 } ^ { t } } \left( \beta _ { 1 } m _ { t - 1 } + ( 1 - \beta _ { 1 } ) g _ { t } \right) = . . . = \frac { 1 - \beta _ { 1 } } { 1 - \beta _ { 1 } ^ { t } } \sum _ { s = 0 } ^ { t - 1 } \beta _ { 1 } ^ { s } g _ { t - s } .
+$$
+
+Hence, $\beta _ { 1 } ) / ( 1 - \beta _ { 1 } ^ { t } )$ $m _ { t }$ is a weighted average of past observed gradients with coefficients , which sum to one, sinus statement holds for $\begin{array} { r } { \sum _ { s = 0 } ^ { t - 1 } \beta _ { 1 } ^ { s } = ( 1 - \beta _ { 1 } ^ { t } ) / ( 1 - \beta _ { 1 } ) } \end{array}$ by the geometric sum formula.itates a variance estimate from $c ( \beta _ { 1 } , t , s ) : = \beta _ { 1 } ^ { s } ( 1 -$ $v _ { t }$   
+past gradient observation is to assume that the true gradient does not change drastically over the effective time horizon of the exponential moving average. For mathematical simplicity, we can translate this assumption to mean that, at the $t$ -th step, we treat all $\{ g _ { t - s , i } \mid s = 0 , \ldots , t - 1 \}$ as iid with mean $\nabla { \mathcal { L } } _ { t , i }$ and variance $\sigma _ { t , i } ^ { 2 }$ . This will of course be utterly wrong for gradient observations that are far in the past, but since $c ( \mu , t , s )$ is very small for large $t - s$ , these won’t contribute significantly to the moving average. The moving average constant defines the effective time horizon, for which we implicitly make this assumption.
+
+Under this peculiar assumption, $m _ { t }$ and $v _ { t }$ are unbiased estimates of the first and second moment of $g _ { t }$ , respectively:
+
+$$
+\begin{array} { r l r } {  { \mathbf { E } [ m _ { t , i } ] = \sum _ { s = 0 } ^ { t - 1 } c ( \mu , t , s ) \mathbf { E } [ g _ { t - s , i } ] = \nabla \mathcal { L } _ { t , i } \sum _ { s = 0 } ^ { t - 1 } c ( \mu , t , s ) = \nabla \mathcal { L } _ { t , i } , } } \\ & { } & { \quad \mathbf { E } [ v _ { t , i } ] = \sum _ { s = 0 } ^ { t - 1 } c ( \mu , t , s ) \mathbf { E } [ g _ { t - s , i } ^ { 2 } ] = ( \nabla \mathcal { L } _ { t , i } ^ { 2 } + \sigma _ { t , i } ^ { 2 } ) \sum _ { s = 0 } ^ { t - 1 } c ( \mu , t , s ) = \nabla \mathcal { L } _ { t , i } ^ { 2 } + \sigma _ { t , i } ^ { 2 } , } \end{array}
+$$
+
+motivating $v _ { t } - m _ { t } ^ { 2 }$ as a gradient variance estimate. However, $v _ { \frac { t } { r } } - m _ { t } ^ { 2 }$ is not an unbiased variance estimate due to the fact $\bar { m } _ { t } ^ { 2 }$ is not an unbiased estimate of $\nabla { \mathcal { L } } _ { t } ^ { 2 }$ . The error arising from this bias should generally be dominated by other error sources and will thus be ignored.
+
+# C.2 MINI-BATCH ESTIMATES
+
+An alternative gradient variance estimate can be obtained locally, within a single mini-batch. The individual gradients $\nabla \ell ( \theta , x _ { k } )$ in a mini-batch are iid random variables and, as noted in the introduction, $\mathbf { v a r } [ g ( \theta ) ] = | \vartheta | ^ { - 1 } \mathbf { v a r } [ \nabla \ell ( \theta , x _ { k } ) ]$ . We can thus estimate $g ( \theta )$ ’s variances by computing the sample variance of the $\{ \nabla \ell ( \theta , x _ { k } ) \} _ { k \in B }$ , then scaling by $| B | ^ { - 1 }$ ,
+
+$$
+\hat { s } ( \theta ) = \frac { 1 } { | \mathcal { B } | } \left( \frac { 1 } { | \mathcal { B } | - 1 } \sum _ { k \in \mathcal { B } } \nabla \ell ( \theta , x _ { k } ) ^ { 2 } - g ( \theta ) ^ { 2 } \right) .
+$$
+
+Several recent papers (Mahsereci & Hennig, 2015; Balles et al., 2017b; Mahsereci et al., 2017) have used this variance estimate for other aspects of stochastic optimizers. In contrast to $v _ { t } - m _ { t } ^ { 2 }$ , this is an unbiased estimate of the local gradient variance. The (non-trivial) implementation of this estimator for neural networks is described in Balles et al. (2017a).
+
+# C.3 RELATIVE VARIANCE OF A MOMENTUM TERM (DERIVATION OF EQ. 19)
+
+When estimating the variance with moving averages, we assume that $\mathbf { E } [ g _ { t } ] = m _ { t }$ and $\mathbf { v a r } [ g _ { t } ] =$ $v _ { t } - m _ { t } ^ { 2 }$ . Plugging this into Eq. (18) we can approximate the mean and variance of the momentum term by
+
+$$
+\mathbf { E } [ r _ { t } ] ^ { 2 } \approx \left( \sum _ { s = 0 } ^ { t } \mu ^ { s } m _ { t - s } \right) ^ { 2 } , \quad \mathbf { v a r } [ r _ { t } ] \approx \sum _ { s = 0 } ^ { t } \mu ^ { 2 s } ( v _ { t - s } - m _ { t - s } ^ { 2 } ) .
+$$
+
+Computing these two expressions would require two more moving averages in addition to $m _ { t }$ and $v _ { t }$ . However, $m _ { t }$ and $v _ { t }$ will change slowly over time and, by using $v _ { t } - m _ { t } ^ { 2 }$ as the variance estimate for
+
+$g _ { t }$ , we anyways make the assumption that all gradients in the effective time horizon of the moving average have the same mean and variance. We thus further approximate by replacing $m _ { t - s }$ with $m _ { t }$ and get
+
+$$
+\begin{array} { c } { { \displaystyle { \bf E } [ r _ { t } ] ^ { 2 } \approx m _ { t } ^ { 2 } \left( \sum _ { s = 0 } ^ { t } \mu ^ { s } \right) ^ { 2 } = m _ { t } ^ { 2 } \left( \frac { 1 - \mu ^ { t } } { 1 - \mu } \right) ^ { 2 } , } } \\ { { \displaystyle { \bf v a r } [ r _ { t } ] \approx ( v _ { t } - m _ { t } ^ { 2 } ) \sum _ { s = 0 } ^ { t } \mu ^ { 2 s } = ( v _ { t } - m _ { t } ^ { 2 } ) \frac { 1 - \mu ^ { 2 t } } { 1 - \mu ^ { 2 } } . } } \end{array}
+$$
+
+The two scalar factors lead to the correction term $\kappa ( \mu , t )$ in Eq. (19).
+
+When estimating the gradient variance from the mini-batch (Eq. 44), we can obtain an unbiased estimate of $\mathbf { v a r } [ \bar { \boldsymbol { r } } _ { t } ]$ in Eq. (18) via
+
+$$
+\bar { s } _ { t } = \mu ^ { 2 } \bar { s } _ { t - 1 } + \hat { s } _ { t } ,
+$$
+
+where $\hat { s } _ { t }$ is given by Eq. (44).
+
+# D VARIATIONS OF VARIANCE-ADAPTED METHODS
+
+Based on the considerations in Section 3, we examined three more variance-adapted methods. The first is a variation of M-SVAG which estimates stochastic gradient variances locally within the minibatch, as explained in $\mathrm { \displaystyle \ S C } . 2$ . Pseudo-code can be found in Alg. 5. Furthermore, we tested a variant of ADAM that applies the correction factor from Eq. (19) to the estimate of the relative variance of the momentum term. We refer to this method as ADAM\*. Two variants of ADAM\* with the two variance estimates can be found in Algorithms 4 and 5.
+
+Require: initial value $\theta _ { 0 }$ , step size $\alpha$ , momentum parameter $\mu \in [ 0 , 1 ]$ , number of steps T   
+1: Initialize $m = 0$ , $\bar { s } = 0$   
+2: for $t = 1 , \dots , T$ do   
+3: Compute stochastic gradient $g ( \theta )$ and variance estimate $\hat { s } ( \theta )$ . Eq. (44)   
+4: Update aggregators $\bar { m }  \mu \bar { m } + g ( \theta ) , \quad \bar { s }  \mu ^ { 2 } \bar { s } + \hat { s } ( \theta )$   
+5: Compute relative variance estimate $\eta ^ { 2 } = \bar { s } / m ^ { 2 }$   
+6: Compute variance adaptation factors $\gamma = ( 1 + \eta ^ { 2 } ) ^ { - 1 }$   
+7: Update $\theta  \theta - \alpha ( \gamma \overset { \cdot } { \odot } m )$   
+8: end for
+
+Algorithm 4 ADAM\* (with exp. moving average variance estimates)
+
+Require: initial value $\theta _ { 0 }$ , step size $\alpha$ , momentum/averaging constant $\mu \in [ 0 , 1 ]$ , number of step
+
+1: Initialize $m = 0$ , $v = 0$   
+2: for $t = 1 , \dots , T$ do   
+3: Compute stochastic gradient $g = g ( \theta )$   
+4: Update moving averages $m  \mu m + ( 1 - \mu ) g , \quad v  \mu v + ( 1 - \mu ) g ^ { 2 }$   
+5: Bias-correct $m = ( 1 - \mu ^ { t } ) ^ { - 1 } \tilde { m } , \quad v = ( 1 - \mu ^ { t } ) ^ { - 1 } \tilde { v }$   
+6: Compute relative variance estimate η2 = κ(µ, t) v−m2m2 . Eq. (19)   
+7: Compute variance adaptation factors $\gamma = ( 1 + \eta ^ { 2 } ) ^ { - 1 / 2 }$   
+8: Update $\theta  \theta - \alpha ( \gamma \odot \mathrm { s i g n } ( m ) )$
+
+# 9: end for
+
+This is ADAM $( \beta _ { 1 } = \beta _ { 2 } = \mu , \varepsilon = 0 )$ , expect for the correction factor $\kappa ( \mu , t )$ for the relative variance.
+
+![](images/ea33cd7452c281ad0e1e0feae26418bcae08dc841fbde38062109358cc998549.jpg)  
+Figure 6: Comparison of the original ADAM algorithm to the variants in Algs. 4 and 5. Set-up of the plots as in Fig. 3. All three algorithms exhibit very similar performance on both problems.
+
+Algorithm 5 ADAM\*-mb (with mini-batch variance estimates)
+
+Require: initial value $\theta _ { 0 }$ , step size $\alpha$ , momentum/averaging constant $\mu \in [ 0 , 1 ]$ , number of steps T   
+1: Initialize $m = 0$ , $\bar { s } = 0$   
+2: for $t = 1 , \dots , T$ do   
+3: Compute stochastic gradient $g ( \theta )$ and variance estimate $\hat { s } ( \theta )$ . Eq. (44)   
+4: Update aggregators $\bar { m }  \mu \bar { m } + g ( \theta ) , \quad \bar { s }  \mu ^ { 2 } \bar { s } + \hat { s } ( \theta )$   
+5: Compute relative variance estimate $\dot { \eta } ^ { 2 } = \bar { s } / m ^ { 2 }$   
+6: Compute variance adaptation factors $\gamma = ( 1 + \eta ^ { 2 } ) ^ { - 1 / 2 }$   
+7: Update $\theta  \theta - \alpha ( \gamma \odot \mathrm { s i g n } ( m ) )$   
+8: end for
+
+# D.1 EXPERIMENTAL RESULTS
+
+We evaluated the variants on the two CIFAR test problems. Figure 6 shows a comparison of the two $\mathbf { A D A M } ^ { * }$ variants with the original ADAM. Figure 7 compares the mini-batch variant of M-SVAG to the one with exponential moving averages.
+
+![](images/06bf29d5a07ec1d70e7a05d48c66b1da35f79b18b7c5bf0a47be2bf43def9d71.jpg)  
+Figure 7: Comparison of the two variants of the M-SVAG algorithm. Set-up of the plots as in Fig. 3. Both variants exhibit very similar performance on both problems.

parse/train/S1xiOjC9F7/S1xiOjC9F7.md

@@ -0,0 +1,405 @@
+# GRAPH MATCHING NETWORKS FOR LEARNING THE SIMILARITY OF GRAPH STRUCTURED OBJECTS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+This paper addresses the challenging problem of retrieval and matching of graph structured objects, and makes two key contributions. First, we demonstrate how Graph Neural Networks (GNN), which have emerged as an effective model for various supervised prediction problems defined on structured data, can be trained to produce embedding of graphs in vector spaces that enables efficient similarity reasoning. Second, we propose a novel Graph Matching Network model that, given a pair of graphs as input, computes a similarity score between them by jointly reasoning on the pair through a new cross-graph attention-based matching mechanism. We demonstrate the effectiveness of our models on different domains including the challenging problem of control-flow-graph based function similarity search that plays an important role in the detection of vulnerabilities in software systems. The experimental analysis demonstrates that our models are not only able to exploit structure in the context of similarity learning but they can also outperform domain-specific baseline systems that have been carefully hand-engineered for these problems.
+
+# 1 INTRODUCTION
+
+Graphs are natural representations for encoding relational structures that are encountered in many domains. Expectedly, computations defined over graph structured data are employed in a wide variety of fields, from the analysis of molecules for computational biology and chemistry (Gilmer et al., 2017; Yan et al., 2005), to the analysis of knowledge graphs or graph structured parses for natural language understanding.
+
+In the past few years graph neural networks (GNNs) have emerged as an effective class of models for learning representations of structured data and for solving various supervised prediction problems on graphs. Such models are invariant to permutations of graph elements by design and compute graph node representations through a propagation process which iteratively aggregates local structural information (Scarselli et al., 2009; Li et al., 2015; Gilmer et al., 2017). These node representations are then used directly for node classification, or pooled into a graph vector for graph classification. Problems beyond supervised classification or regression are relatively less well-studied for GNNs.
+
+In this paper we study the problem of similarity learning for graph structured objects, which appears in many important real world applications, in particular similarity based retrieval in graph databases. One motivating application is the computer security problem of binary function similarity search, where given a binary which may or may not contain code with known vulnerabilities, we wish to check whether any control-flow-graph in this binary is sufficiently similar to a database of known-vulnerable functions. This helps identify vulnerable statically linked libraries in closed-source software, a recurring problem (CVE, 2010; 2018) for which no good solutions are currently available. Figure 1 shows one example from this application, where the binary functions are represented as control flow graphs annotated with assembly instructions. This similarity learning problem is very challenging as subtle differences can make two graphs be semantically very different, while graphs with different structures can still be similar. A successful model for this problem should therefore (1) exploit the graph structures, and (2) be able to reason about the similarity of graphs both from the graph structures as well as from learned semantics.
+
+In order to solve the graph similarity learning problem, we investigate the use of GNNs in this context, explore how they can be used to embed graphs into a vector space, and learn this embedding model to make similar graphs close in the vector space, and dissimilar graphs far apart. One important property of this model is that, it maps each graph independently to an embedding vector, and then all the similarity computation happens in the vector space. Therefore, the embeddings of graphs in a large database can be precomputed and indexed, which enables efficient retrieval with fast nearest neighbor search data structures like k-d trees (Bentley, 1975) or locality sensitive hashing (Gionis et al., 1999).
+
+We further propose an extension to GNNs which we call Graph Matching Networks (GMNs) for similarity learning. Instead of computing graph representations independently for each graph, the GMNs compute a similarity score through a cross-graph attention mechanism to associate nodes across graphs and identify differences. By making the graph representation computation dependent on the pair, this matching model is more powerful than the embedding model, providing a nice accuracy-computation trade-off.
+
+![](images/9e7225813589a8cf201b5c4b38bc3bf6d40968e6b4043da81e648958769e7427.jpg)  
+Figure 1: The binary function similarity learning problem. Checking whether two graphs are similar requires reasoning about both the structure as well as the semantics of the graphs. Here the left two control flow graphs correspond to the same function compiled with different compilers (and therefore similar), while the graph on the right corresponds to a different function.
+
+We evaluate the proposed models and baselines   
+on three tasks: a synthetic graph edit-distance learning task which captures structural similarity only, and two real world tasks - binary function similarity search and mesh retrieval, which require reasoning about both the structural and semantic similarity. On all tasks, the proposed approaches outperform established baselines and structure agnostic models; in more detailed ablation studies, we found that the Graph Matching Networks consistently outperform the graph embedding model and Siamese networks.
+
+To summarize, the contributions of this paper are: (1) we demonstrate how GNNs can be used to produce graph embeddings for similarity learning; (2) we propose the new Graph Matching Networks that computes similarity through cross-graph attention-based matching; (3) empirically we show that the proposed graph similarity learning models achieve good performance across a range of applications, outperforming structure agnostic models and established hand-engineered baselines.
+
+# 2 RELATED WORK
+
+Graph Neural Networks and Graph Representation Learning The history of graph neural networks (GNNs) goes back to at least the early work by Gori et al. (2005) and Scarselli et al. (2009), who proposed to use a propagation process to learn node representations. These models have been further developed by incorporating modern deep learning components (Li et al., 2015; Velickovi ˇ c´ et al., 2017; Bruna et al., 2013). A separate line of work focuses on generalizing convolutions to graphs (Bruna et al., 2013; Bronstein et al., 2017). Popular graph convolutional networks also compute node updates by aggregating information in local neighborhoods (Kipf & Welling, 2016), making them the same family of models as GNNs. GNNs have been successfully used in many domains (Kipf & Welling, 2016; Velickovi ˇ c et al., 2017; Battaglia et al., 2016; 2018; Niepert et al., ´ 2016; Duvenaud et al., 2015; Gilmer et al., 2017; Dai et al., 2017; Li et al., 2018; Wang et al., 2018a;b). Most of the previous work on GNNs focus on supervised prediction problems (with exceptions like (Dai et al., 2017; Li et al., 2018; Wang et al., 2018a)). The graph similarity learning problem we study in this paper and the new graph matching model can be good additions to this family of models.
+
+Graph Similarity Search and Graph Kernels Graph similarity search has been studied extensively in database and data mining communities (Yan et al., 2005; Dijkman et al., 2009). The similarity is typically defined by either exact matches (full-graph or sub-graph isomorphism) (Berretti et al., 2001; Shasha et al., 2002; Yan et al., 2004; Srinivasa & Kumar, 2003) or some measure of structural similarity, e.g. in terms of graph edit distances (Willett et al., 1998; Raymond et al., 2002). Most of the approaches proposed in this direction are not learning-based, and focus on efficiency.
+
+Graph kernels are kernels on graphs designed to capture the graph similarity, and can be used in kernel methods for e.g. graph classification (Vishwanathan et al., 2010; Shervashidze et al., 2011). Popular graph kernels include those that measure the similarity between walks or paths on graphs (Borgwardt & Kriegel, 2005; Kashima et al., 2003; Vishwanathan et al., 2010), kernels based on limited-sized substructures (Horváth et al., 2004; Shervashidze et al., 2009) and kernels based on sub-tree structures (Shervashidze & Borgwardt, 2009; Shervashidze et al., 2011). Graph kernels are usually used in models that may have learned components, but the kernels themselves are handdesigned and motivated by graph theory. They can typically be formulated as first computing the feature vectors for each graph (the kernel embedding), and then take inner product between these vectors to compute the kernel value. One exception is (Yanardag & Vishwanathan, 2015) where the co-occurrence of graph elements (substructures, walks, etc.) are learned, but the basic elements are still hand-designed. Compared to these approaches, our graph neural network based similarity learning framework learns the similarity metric end-to-end.
+
+Distance Metric Learning Learning a distance metric between data points is the key focus of the area of metric learning. Most of the early work on metric learning assumes that the data already lies in a vector space, and only a linear metric matrix is learned to properly measure the distance in this space to group similar examples together and dissimilar examples to be far apart (Xing et al., 2003; Weinberger & Saul, 2009; Davis et al., 2007). More recently the ideas of distance metric learning and representation learning have been combined in applications like face verification, where deep convolutional neural networks are learned to map similar images to similar representation vectors (Chopra et al., 2005; Hu et al., 2014; Sun et al., 2014). In this paper, we focus on representation and similarity metric learning for graphs, and our graph matching model goes one step beyond the typical representation learning methods by modeling the cross-graph matchings.
+
+Siamese Networks Siamese networks (Bromley et al., 1994; Baldi & Chauvin, 1993) are a family of neural network models for visual similarity learning. These models typically consist of two networks with shared parameters applied to two input images independently to compute representations, a small network is then used to fuse these representations and compute a similarity score. They can be thought of as learning both the representations and the similarity metric. Siamese networks have achieved great success in many visual recognition and verification tasks (Bromley et al., 1994; Baldi & Chauvin, 1993; Koch et al., 2015; Bertinetto et al., 2016; Zagoruyko & Komodakis, 2015). In the experiments we adapt Siamese networks to handle graphs, but found our graph matching networks to be more powerful as they do cross-graph computations and therefore fuse information from both graphs early in the computation process. Independent of our work, recently (Shyam et al., 2017) proposed a cross-example attention model for visual similarity as an alternative to Siamese networks based on similar motivations and achieved good results.
+
+# 3 DEEP GRAPH SIMILARITY LEARNING
+
+Given two graphs $G _ { 1 } = ( V _ { 1 } , E _ { 1 } )$ and $G _ { 2 } = ( V _ { 2 } , E _ { 2 } )$ , we want a model that produces the similarity score $s ( G _ { 1 } , G _ { 2 } )$ between them. Each graph $G = ( V , E )$ is represented as sets of nodes $V$ and edges $E$ , optionally each node $i \in V$ can be associated with a feature vector $\mathbf { x } _ { i }$ , and each edge $( i , j ) \in E$ associated with a feature vector $\mathbf { x } _ { i j }$ . These features can represent, e.g. type of a node, direction of an edge, etc. If a node or an edge does not have any associated features, we set the corresponding vector to a constant vector of 1s. We propose two models for graph similarity learning: a model based on standard GNNs for learning graph embeddings, and the new and more powerful GMNs. The two models are illustrated in Figure 2.
+
+# 3.1 GRAPH EMBEDDING MODELS
+
+Graph embedding models embed each graph into a vector, and then use a similarity metric in that vector space to measure the similarity between graphs. Our GNN embedding model comprises 3 parts: (1) an encoder, (2) propagation layers, and (3) an aggregator.
+
+![](images/65584d1e56fbd621110b6ab4af106377ea3ca09c6528543dfddad7e6ff636d03.jpg)  
+Figure 2: Illustration of the graph embedding (left) and matching models (right).
+
+Encoder The encoder maps the node and edge features to initial node and edge vectors through separate MLPs:
+
+$$
+\mathbf { h } _ { i } ^ { ( 0 ) } = \mathrm { M L P } _ { \mathrm { n o d e } } ( \mathbf { x } _ { i } ) , \quad \forall i \in V \qquad \mathbf { e } _ { i j } = \mathrm { M L P } _ { \mathrm { e d g e } } ( \mathbf { x } _ { i j } ) , \quad \forall ( i , j ) \in E .
+$$
+
+Propagation Layers A propagation layer maps a set of node representations $\{ \mathbf { h } _ { i } ^ { ( t ) } \} _ { i \in V }$ to new node representations $\{ \mathbf { h } _ { i } ^ { ( t + 1 ) } \} _ { i \in V }$ , as the following:
+
+$$
+\mathbf { m } _ { j \to i } = f _ { \mathrm { m e s s a g e } } ( \mathbf { h } _ { i } ^ { ( t ) } , \mathbf { h } _ { j } ^ { ( t ) } , \mathbf { e } _ { i j } ) , \qquad \mathbf { h } _ { i } ^ { ( t + 1 ) } = f _ { \mathrm { n o d e } } \left( \mathbf { h } _ { i } ^ { ( t ) } , \sum _ { j : ( j , i ) \in E } \mathbf { m } _ { j \to i } \right)
+$$
+
+Here $f _ { \mathrm { m e s s a g e } }$ is typically an MLP on the concatenated inputs, and $f _ { \mathrm { n o d e } }$ can be either an MLP or a recurrent neural network core, e.g. RNN, GRU or LSTM (Li et al., 2015). To aggregate the messages, we use a simple sum which may be alternatively replaced by other commutative operators such as mean, max or the attention-based weighted sum (Velickovi ˇ c et al., 2017). Through multiple layers of ´ propagation, the representation for each node will accumulate information in its local neighborhood.
+
+Aggregator After a certain number $T$ rounds of propagations, an aggregator takes the set of node representations $\{ \mathbf { h } _ { i } ^ { ( T ) } \}$ as input, and computes a graph level representation ${ \bf h } _ { G } = f _ { G } ( \{ { \bf h } _ { i } ^ { ( T ) } \} )$ , as
+
+$$
+\mathbf { h } _ { G } = \mathrm { M L P } _ { G } \left( \sum _ { i \in V } \sigma ( \mathrm { M L P } _ { \mathrm { g a t e } } ( \mathbf { h } _ { i } ^ { ( T ) } ) ) \odot \mathrm { M L P } ( \mathbf { h } _ { i } ^ { ( T ) } ) \right) ,
+$$
+
+which transforms node representations and then uses a weighted sum with gating vectors to aggregate across nodes. The weighted sum can focus only on the important nodes, it is more powerful than a simple sum and also works significantly better empirically.
+
+After the graph representations $\mathbf { h } _ { G _ { 1 } }$ and $\mathbf { h } _ { G _ { 2 } }$ are computed for the pair $( G _ { 1 } , G _ { 2 } )$ , we compute the similarity between them using a similarity metric in the vector space, for example the Euclidean, cosine or Hamming similarities.
+
+Note that without the propagation layers (or with 0 propagation steps), this model becomes an instance of the Deep Set (Zaheer et al., 2017) or PointNet (Qi et al., 2017), which does computation on the individual nodes, and then pool the node representations into a representation for the whole graph. Such a model, however, ignores the structure and only treats the data as a set of independent nodes.
+
+# 3.2 GRAPH MATCHING NETWORKS
+
+Graph matching networks take a pair of graphs as input and compute a similarity score between them. Compared to the embedding models, these matching models compute the similarity score jointly on the pair, rather than first independently mapping each graph to a vector. Therefore these models are potentially stronger than the embedding models, at the cost of some extra computation efficiency.
+
+We propose the following graph matching network, which changes the node update module in each propagation layer to take into account not only the aggregated messages on the edges for each graph as before, but also a cross-graph matching vector which measures how well a node in one graph can
+
+be matched to one or more nodes in the other:
+
+$$
+\begin{array} { r l } & { \mathbf { m } _ { j \to i } = f _ { \mathrm { m e s s a g e } } ( \mathbf { h } _ { i } ^ { ( t ) } , \mathbf { h } _ { j } ^ { ( t ) } , \mathbf { e } _ { i j } ) , \quad \forall ( i , j ) \in E _ { 1 } \cup E _ { 2 } } \\ & { \mu _ { j \to i } = f _ { \mathrm { m a t c h } } ( \mathbf { h } _ { i } ^ { ( t ) } , \mathbf { h } _ { j } ^ { ( t ) } ) , \quad \forall i \in V _ { 1 } , j \in V _ { 2 } , \mathrm { o r } i \in V _ { 2 } , j \in V _ { 1 } } \\ & { \mathbf { h } _ { i } ^ { ( t + 1 ) } = f _ { \mathrm { n o d e } } \left( \mathbf { h } _ { i } ^ { ( t ) } , \displaystyle \sum _ { j } \mathbf { m } _ { j \to i } , \displaystyle \sum _ { j ^ { \prime } } \mu _ { j ^ { \prime } \to i } \right) } \\ & { \mathbf { h } _ { G _ { 1 } } = f _ { G } ( \{ \mathbf { h } _ { i } ^ { ( T ) } \} _ { i \in V _ { 1 } } ) , \quad \mathbf { h } _ { G _ { 2 } } = f _ { G } ( \{ \mathbf { h } _ { i } ^ { ( T ) } \} _ { i \in V _ { 2 } } ) , \quad s = f _ { s } ( \mathbf { h } _ { G _ { 1 } } , \mathbf { h } _ { G _ { 2 } } ) . } \end{array}
+$$
+
+Here $f _ { s }$ is a standard vector space similarity between $\mathbf { h } _ { G _ { 1 } }$ and $\mathbf { h } _ { G _ { 2 } }$ . $f _ { \mathrm { m a t c h } }$ is a function that communicates cross-graph information, which we propose to use an attention-based module:
+
+$$
+\begin{array} { l } { { \displaystyle a _ { j \to i } = \frac { \exp \bigl ( s _ { h } ( { \bf h } _ { i } ^ { ( t ) } , { \bf h } _ { j } ^ { ( t ) } ) \bigr ) } { \sum _ { j ^ { \prime } } \exp \bigl ( s _ { h } ( { \bf h } _ { i } ^ { ( t ) } , { \bf h } _ { j ^ { \prime } } ^ { ( t ) } ) \bigr ) } , \qquad \mu _ { j \to i } = a _ { j \to i } \bigl ( { \bf h } _ { i } ^ { ( t ) } - { \bf h } _ { j } ^ { ( t ) } \bigr ) \quad \mathrm { a n d ~ t h e r e f o r e } } } \\ { { \displaystyle \sum _ { j } \mu _ { j \to i } = \sum _ { j } a _ { j \to i } \bigl ( { \bf h } _ { i } ^ { ( t ) } - { \bf h } _ { j } ^ { ( t ) } \bigr ) = { \bf h } _ { i } ^ { ( t ) } - \sum _ { j } a _ { j \to i } { \bf h } _ { j } ^ { ( t ) } } . } \end{array}
+$$
+
+$s _ { h }$ is again a vector space similarity metric, like Euclidean or cosine similarity, $a _ { j \to i }$ are the attention weights, and $\textstyle \sum _ { j } \mu _ { j \to i }$ intuitively measures the difference between $\mathbf { h } _ { i } ^ { ( t ) }$ and its closest neighbor in the other graph. Note that because of the normalization in $a _ { j \to i }$ , the function $f _ { \mathrm { m a t c h } }$ implicitly depends on the whole set of $\{ \mathbf { h } _ { j } ^ { ( t ) } \}$ , which we omitted in Eq. 8 for a cleaner notation. Since attention weights are required for every pair of nodes across two graphs, this operation has a computation cost of $O ( | V _ { 1 } | | V _ { 2 } | )$ , while for the GNN embedding model the cost for each round of propagation is $O ( | V | + | E | )$ . The extra power of the GMNs comes from utilizing the extra computation.
+
+Note By construction, the attention module has a nice property that, when the two graphs can be perfectly matched, and when the attention weights are peaked at the exact match, we have $\textstyle \sum _ { j } { \pmb { \mu } } _ { j \to i } = { \bf 0 }$ , which means the cross-graph communications will be reduced to zero vectors, and the two graphs will continue to compute identical representations in the next round of propagation. On the other hand, the differences across graphs will be captured in the cross-graph matching vector $\textstyle \sum _ { j } \mu _ { j \to i }$ , which will be amplified through the propagation process, making the matching model more sensitive to these differences.
+
+Compared to the graph embedding model, the matching model has the ability to change the representation of the graphs based on the other graph it is compared against. The model will adjust graph representations to make them become more different if they do not match.
+
+# 3.3 LEARNING
+
+The proposed graph similarity learning models can be trained on a set of example pairs or triplets. Pairwise training requires us to have a dataset of pairs labeled as positive (similar) or negative (dissimilar), while triplet training only needs relative similarity, i.e. whether $G _ { 1 }$ is closer to $G _ { 2 }$ or $G _ { 3 }$ . We describe the losses on pairs and triplets we used below, which are then optimized with gradient descent based algorithms.
+
+When using Euclidean similarity, we use the following margin-based pairwise loss:
+
+$$
+L _ { \mathrm { p a i r } } = \mathbb { E } _ { ( G _ { 1 } , G _ { 2 } , t ) } [ \operatorname* { m a x } \{ 0 , \gamma - t ( 1 - d ( G _ { 1 } , G _ { 2 } ) ) \} ] ,
+$$
+
+where $t \in \{ - 1 , 1 \}$ is the label for this pair, $\gamma > 0$ is a margin parameter, and $d ( G _ { 1 } , G _ { 2 } ) =$ $\| \mathbf { h } _ { G _ { 1 } } - \mathbf { h } _ { G _ { 2 } } \| ^ { 2 }$ is the Euclidean distance. This loss encourages $d ( G _ { 1 } , G _ { 2 } ) < 1 - \gamma$ when the pair is similar $\mathit { t } = 1$ ), and $d ( G _ { 1 } , G _ { 2 } ) > 1 + \gamma$ when $t = - 1$ . Given triplets where $G _ { 1 }$ and $G _ { 2 }$ are closer than $G _ { 1 }$ and $G _ { 3 }$ , we optimize the following margin-based triplet loss:
+
+$$
+L _ { \mathrm { t r i p l e t } } = \mathbb { E } _ { ( G _ { 1 } , G _ { 2 } , G _ { 3 } ) } [ \operatorname* { m a x } \{ 0 , d ( G _ { 1 } , G _ { 2 } ) - d ( G _ { 1 } , G _ { 3 } ) + \gamma \} ] .
+$$
+
+This loss encourages $d ( G _ { 1 } , G _ { 2 } )$ to be smaller than $d ( G _ { 1 } , G _ { 3 } )$ by at least a margin $\gamma$ .
+
+For applications where it is necessary to search through a large database of graphs with low latency, it is beneficial to have the graph representation vectors be binary, i.e. $\mathbf { h } _ { G } ^ { - } \in \{ - 1 , 1 \} ^ { H }$ , so that efficient nearest neighbor search algorithms (Gionis et al., 1999) may be applied. In such cases, we can minimize the Hamming distance of positive pairs and maximize it for negative pairs. With this restriction the graph vectors can no longer freely occupy the whole Euclidean space, but we gain the efficiency for fast retrieval and indexing. To achieve this we propose to pass the $\mathbf { h } _ { G }$ vectors through a tanh transformation, and optimize the following pair and triplet losses:
+
+$$
+\begin{array} { r l } & { \quad L _ { \mathrm { p a i r } } = \mathbb { E } _ { ( G _ { 1 } , G _ { 2 } , t ) } [ ( t - s ( G _ { 1 } , G _ { 2 } ) ) ^ { 2 } ] / 4 , \quad \mathrm { a n d } } \\ & { \quad L _ { \mathrm { t r i p l e t } } = \mathbb { E } _ { ( G _ { 1 } , G _ { 2 } , G _ { 3 } ) } [ ( s ( G _ { 1 } , G _ { 2 } ) - 1 ) ^ { 2 } + ( s ( G _ { 1 } , G _ { 3 } ) + 1 ) ^ { 2 } ] / 8 , } \end{array}
+$$
+
+where $\begin{array} { r } { s ( G _ { 1 } , G _ { 2 } ) = \frac { 1 } { H } \sum _ { i = 1 } ^ { H } \operatorname { t a n h } ( h _ { G _ { 1 } { i } } ) \cdot \operatorname { t a n h } ( h _ { G _ { 2 } { i } } ) } \end{array}$ is the approximate average Hamming similarity. Both losses are bounded in $[ 0 , 1 ]$ , and they push positive pairs to have Hamming similarity close to 1, and negative pairs to have similarity close to -1. We found these losses to be a bit more stable than margin based losses for Hamming similarity.
+
+# 4 EXPERIMENTS
+
+In this section, we evaluate the graph similarity learning (GSL) framework and the graph embedding (GNNs) and graph matching networks (GMNs) on three tasks and compare these models with other competing methods. Overall the empirical results demonstrate that the GMNs excel on graph similarity learning, consistently outperforming all other approaches.
+
+# 4.1 LEARNING GRAPH EDIT DISTANCES
+
+Problem Background Graph edit distance between graphs $G _ { 1 }$ and $G _ { 2 }$ is defined as the minimum number of edit operations needed to transform $G _ { 1 }$ to $G _ { 2 }$ . Typically the edit operations include add/remove/substitute nodes and edges. Graph edit distance is naturally a measure of similarity between graphs and has many applications in graph similarity search (Dijkman et al., 2009; Zeng et al., 2009; Gao et al., 2010). However computing the graph edit distance is NP-hard in general (Zeng et al., 2009), therefore approximations have to be used. Through this experiment we show that the GSL models can learn structural similarity between graphs on very challenging problems.
+
+Training Setup We generated training data by sampling random binomial graphs $G _ { 1 }$ with $n$ nodes and edge probability $p$ (Erdös & Rényi, 1959), and then create positive example $G _ { 2 }$ by randomly substituting $k _ { p }$ edges from $G _ { 1 }$ with new edges, and negative example $G _ { 3 }$ by substituting $k _ { n }$ edges from $G _ { 1 }$ , where $k _ { p } < k _ { n } ^ { \phantom { } 1 }$ . A model needs to predict a higher similarity score for positive pair $( G _ { 1 } , G _ { 2 } )$ than negative pair $\left( G _ { 1 } , G _ { 3 } \right)$ . Throughout the experiments we fixed the dimensionality of node vectors to 32, and the dimensionality of graph vectors to 128 without further tuning. We also tried different number of propagation steps $T$ from 1 to 5, and observed consistently better performance with increasing $T$ . The results reported in this section are all with $T = 5$ unless stated otherwise. More details are included in Appendix B.1.
+
+Baseline We compare our models with the popular Weisfeiler Lehman (WL) kernel (Shervashidze et al., 2011), which has been shown to be very competitive on graph classification tasks and the Weisfeiler Lehman algorithm behind this kernel is a strong method for checking graph isomorphism (edit distance of 0), a closely related task (Weisfeiler & Lehman, 1968; Shervashidze et al., 2011).
+
+Evaluation The performance of different models are evaluated using two metrics: (1) pair AUC - the area under the ROC curve for classifying pairs of graphs as similar or not on a fixed set of 1000 pairs and (2) triplet accuracy - the accuracy of correctly assigning higher similarity to the positive pair in a triplet than the negative pair on a fixed set of 1000 triplets.
+
+Results We trained and evaluated the GSL models on graphs of a few specific distributions with different $n , p .$ , with $k _ { p } = 1$ and $k _ { n } = 2$ fixed. The evaluation results are shown in Table 1. We can see that by learning on graphs of specific distributions, the GSL models are able to do better than generic baselines, and the GMNs consistently outperform the embedding model (GNNs).
+
+For the GMNs, we can visualize the cross-graph attention to gain further insight into how it is working. Figure 3 shows two examples of this for a matching model trained with $n$ sampled from [20, 50],
+
+<table><tr><td>Graph Distribution</td><td>WL kernel</td><td>GNN</td><td>GMN</td></tr><tr><td>n= 20,p=0.2</td><td>80.8/83.2</td><td>88.8/94.0</td><td>95.0/95.6</td></tr><tr><td>n= 20,p= 0.5</td><td>74.5 / 78.0</td><td>92.1 /93.4</td><td>96.6 /98.0</td></tr><tr><td>n= 50,p 二 0.2</td><td>93.9 /97.8</td><td>95.9 /97.2</td><td>97.4 /97.6</td></tr><tr><td>n = 50,p = 0.5</td><td>82.3 / 89.0</td><td>88.5 /91.0</td><td>93.8 / 92.6</td></tr></table>
+
+Table 1: Comparing the graph embedding (GNN) and matching (GMN) models trained on graphs from different distributions with the baseline, measuring pair AUC / triplet accuracy $( \times 1 0 0 )$ .
+
+![](images/c83ceeda3457d9b386a8e617206399f3aba89b3c9ef41cc902ee98bae62102fe.jpg)  
+Figure 3: Visualization of cross-graph attention for GMNs after 5 propagation layers. In each pair of graphs the left figure shows the attention from left graph to the right, the right figure shows the opposite.
+
+tested on graphs of 10 nodes. The cross-graph attention weights are shown in green, with the scale of the weights shown as the transparency of the green edges. We can see that the attention weights can align nodes well when the two graphs match, and tend to focus on nodes with higher degrees when they don’t. However the pattern is not as interpretable as in standard attention models.
+
+More experiments on generalization capabilities of these models (train on small graphs, test on larger graphs, train on graphs with some $k _ { p } , k _ { n }$ combinations, test on others) are included in Appendix B.1.
+
+# 4.2 CONTROL FLOW GRAPH BASED BINARY FUNCTION SIMILARITY SEARCH
+
+Problem Background Binary function similarity search is an important problem in computer security. The need to analyze and search through binaries emerges when we do not have access to the source code, for example when dealing with commercial or embedded software or suspicious executables. Combining a disassembler and a code analyzer, we can extract a control-flow graph (CFG) which contains all the information in a binary function in a structured format. See Figure 1 and Appendix B.2 for a few example CFGs. In a CFG, each node is a basic block of assembly instructions, and the edges between nodes represent the control flow, indicated by for example a jump or a return instruction used in branching, loops or function calls. In this section, we target the vulnerability search problem, where a piece of binary known to have some vulnerabilities is used as the query, and we search through a library to find similar binaries that may have the same vulnerabilities.2 Accurate identification of similar vulnerabilities enables security engineers to quickly narrow down the search space and apply patches.
+
+In the past the binary function similarity search problem has been tackled with classical graph theoretical matching algorithms (Eschweiler et al., 2016; Pewny et al., 2015), and Xu et al. (2017) and Feng et al. (2016) proposed to learn embeddings of CFGs and do similarity search in the embedding space. Xu et al. (2017) in particular proposed an embedding method based on graph neural networks, starting from some hand selected feature vectors for each node. Here we study further the performance of graph embedding and matching models, with pair and triplet training, different number of propagation steps, and learning node features from the assembly instructions.
+
+Training Setup and Baseline We train and evaluate our model on data generated by compiling the popular open source video processing software ffmpeg using different compilers gcc and clang, and different compiler optimization levels, which results in 7940 functions and roughly 8 CFGs per function. The average size of the CFGs is around 55 nodes per graph, with some larger graphs having up to a few thousand nodes (see Appendix B.2 for more detailed statistics). Different compiler optimization levels result in CFGs of very different sizes for the same function. We split the data and used $80 \%$ functions and the associated CFGs for training, $10 \%$ for validation and $10 \%$ for testing. The models were trained to learn a similarity metric on CFGs such that the CFGs for the same function have high similarity, and low similarity otherwise. Once trained, this similarity metric can be used to search through library of binaries and be invariant to compiler type and optimization levels.
+
+![](images/af414ae2d5002694a07136f9a0949cb2518203a5f57033ce039d41d5a1726232.jpg)  
+Figure 4: Performance $\times 1 0 0 $ ) of different models on the binary function similarity search task.
+
+We compare our graph embedding and matching models with Google’s open source function similarity search tool (Dullien, 2018), which has been used to successfully find vulnerabilities in binaries in the past. This tool computes representations of CFGs through a hand-engineered graph hashing process which encodes the neighborhood structure of each node by hashing the degree sequence from a traversal of a 3-hop neighborhood, and also encodes the assembly instructions for each basic block by hashing the trigrams of assembly instruction types. These features are then combined by using a SimHash-style (Charikar, 2002) algorithm with learned weights to form a 128-dimensional binary code. An LSH-based search index is then used to perform approximate nearest neighbor search using hamming distance.
+
+Following (Dullien, 2018), we also map the CFGs to 128-dimensional binary vectors, and use the Hamming similarity formulation described in Section 3 for training. We further studied two variants of the data, one that only uses the graph structure, and one that uses both the graph structure and the assembly instructions with learned node features. When assembly instructions are available, we embed each instruction type into a vector, and then sum up all the embedding vectors for instructions in a basic block as the initial representation vector (the $\mathbf { x } _ { i }$ ’s) for each node, these embeddings are learned jointly with the rest of the model.
+
+Results Figure 4 shows the performance of different models with different number of propagation steps and in different data settings. We again evaluate the performance of these models on pair AUC and triplet accuracy on fixed sets of pairs and triplets from the test set. It is clear from results that: (1) the performance of both the graph embedding and matching models consistently go up with more propagation steps, and in particular significantly outperforming the structure agnostic model special case which uses 0 propagation steps; (2) the graph embedding model is consistently better than the baselines with enough propagation steps; and (3) graph matching models outperforms the embedding models across all settings and propagation steps. Additionally, we have tried the WL kernel on this task using only the graph structure, and it achieved 0.619 AUC and $2 4 . 5 \%$ triplet accuracy. This is not surprising as the WL kernel is not designed for solving this task, while our models learn the features useful for the task of interest, and can achieve better performance than generic similarity metrics.
+
+# 4.3 MORE BASELINES AND ABLATION STUDIES
+
+In this section, we carefully examine the effects of the design decisions we made in the GMN model and compare it against a few more alternatives. In particular, we evaluate the popular Graph Convolutional Network (GCN) model by Kipf & Welling (2016) as an alternative to our GNN model, and Siamese versions of the GNN/GCN embedding models. The GCN model replaces the message passing in Eq. 2 with graph convolutions, and the Siamese model predicts a distance value by concatenating two graph vectors and then pass through a 2 layer MLP. The comparison with Siamese networks can in particular show the importance of the cross-graph attention early on in the similarity computation process, as Siamese networks fuse the representations for 2 graphs only at the very end.
+
+We focus on the function similarity search task, and also conduct experiments on an extra COILDEL mesh graph dataset (Riesen & Bunke, 2008), which contains 100 classes of mesh graphs corresponding to 100 types of objects. We treat graphs in the same class as similar, and used identical setup as the function similarity search task for training and evaluation.
+
+<table><tr><td>Model</td><td>Pair AUC</td><td>Triplet Acc</td></tr><tr><td>Baseline</td><td>96.09</td><td>96.35</td></tr><tr><td>GCN</td><td>96.67</td><td>96.57</td></tr><tr><td>Siamese-GCN</td><td>97.54</td><td>97.51</td></tr><tr><td>GNN</td><td>97.71</td><td>97.83</td></tr><tr><td>Siamese-GNN</td><td>97.76</td><td>97.58</td></tr><tr><td>GMN</td><td>99.28</td><td>99.18</td></tr></table>
+
+Function Similarity Search
+
+<table><tr><td>Model</td><td>Pair AUC</td><td>Triplet Acc</td></tr><tr><td>GCN</td><td>94.80</td><td>94.95</td></tr><tr><td>Siamese-GCN</td><td>95.90</td><td>96.10</td></tr><tr><td>GNN</td><td>98.58</td><td>98.70</td></tr><tr><td>Siamese-GNN</td><td>98.76</td><td>98.55</td></tr><tr><td>GMN</td><td>98.97</td><td>98.80</td></tr></table>
+
+COIL-DEL
+
+Table 2: More results on the function similarity search task and the extra COIL-DEL dataset.
+
+Table 2 summarizes the experiment results, which clearly show that: (1) the GNN embedding model is a competitive model (more powerful than the GCN model); (2) using Siamese network architecture to learn similarity on top of graph representations is better than using a prespecified similarity metric (Euclidean, Hamming etc.); (3) the GMNs outperform the Siamese models showing the importance of cross-graph information communication early in the computation process.
+
+# 5 CONCLUSIONS AND DISCUSSION
+
+In this paper we studied the problem of graph similarity learning using graph neural networks. Compared to standard prediction problems for graphs, similarity learning poses a unique set of challenges and potential benefits. For example, the graph embedding models can be learned through a classification setting when we do have a set of classes in the dataset, but formulating it as a similarity learning problem can handle cases where we have a very large number of classes and only very few examples for each class. The representations learned from the similarity learning setting can also easily generalize to data from classes unseen during training (zero-shot generalization).
+
+We proposed the new graph matching networks as a stronger alternative to the graph embedding models. The added power for the graph matching models comes from the fact that they are not independently mapping each graph to an embedding, but rather doing comparisons at all levels across the pair of graphs, in addition to the embedding computation. The model can then learn to properly allocate capacity toward the embedding part or the matching part. The price to pay for this expressivity is the added computation cost in two aspects: (1) since each cross-graph matching step requires the computation of the full attention matrices, which requires at least $O ( | V _ { 1 } | | V _ { 2 } | )$ time, this may be expensive for large graphs; (2) the matching models operate on pairs, and cannot directly be used for indexing and searching through large graph databases. Therefore it is best to use the graph matching networks when we (1) only care about the similarity between individual pairs, or (2) use them in a retrieval setting together with a faster filtering model like the graph embedding model or standard graph similarity search methods, to narrow down the search to a smaller candidate set, and then use the more expensive matching model to rerank the candidates to improve precision.
+
+Developing neural models for graph similarity learning is an important research direction with many applications. There are still many interesting challenges to resolve, for example to improve the efficiency of the matching models, study different matching architectures, adapt the GNN capacity to graphs of different sizes, and applying these models to new application domains. We hope our work can spur further research in this direction.
+
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+# A EXTRA DETAILS ON MODEL ARCHITECTURES
+
+In the propagation layers of the graph embedding and matching models, we used an MLP with one hidden layer as the $f _ { \mathrm { m e s s a g e } }$ module, with a ReLU nonlinearity on the hidden layer. For node state vectors (the $\mathbf { h } _ { i } ^ { ( t ) }$ vectors) of dimension $D$ , the size of the hidden layer and the output is set to $2 D$ . We found it to be beneficial to initialize the weights of this $f _ { \mathrm { m e s s a g e } }$ module to be small, which helps stablizing training. We used the standard Glorot initialization with an extra scaling factor of 0.1. When not using this small scaling factor, at the begining of training the message vectors when summed up can have huge scales, which is bad for learning.
+
+One extra thing to note about the propagation layers is that we can make all the propagation layers share the same set of parameters, which can be useful if this is a suitable inductive bias to have.
+
+We tried different $f _ { \mathrm { n o d e } }$ modules in both experiments, and found GRUs to generally work better than one-hidden layer MLPs, and all the results reported uses GRUs as $f _ { \mathrm { n o d e } }$ , with the sum over edge messages $\textstyle \sum _ { j } \mathbf { m } _ { j \to i }$ treated as the input to the GRU for the embedding model, and the concatenation of $\begin{array} { r } { \sum _ { j } \mathbf { m } _ { j  i } } \end{array}$ and $\textstyle \sum _ { j ^ { \prime } } \pmb { \mu } _ { j ^ { \prime }  i }$ as the input to the GRU for the matching model.
+
+In the aggregator module, we used a single linear layer for the node transformation MLP and the gating $\mathrm { M L P _ { g a t e } }$ in Eq.3. The output of this linear layer has a dimensionality the same as the required dimensionality for the graph vectors. $\begin{array} { r } { \sigma ( x ) = \frac { 1 } { 1 + e ^ { - x } } } \end{array}$ is the logistic sigmoid function, and $\odot$ is the element-wise product. After the weighted sum, another MLP with one hidden layers is used to further transform the graph vector. The hidden layer has the same size as the output, with a ReLU nonlinearity.
+
+For the matching model, the attention weights are computed as
+
+$$
+a _ { j  i } = \frac { \exp ( s _ { h } ( \mathbf { h } _ { i } ^ { ( t ) } , \mathbf { h } _ { j } ^ { ( t ) } ) ) } { \sum _ { j ^ { \prime } } \exp ( s _ { h } ( \mathbf { h } _ { i } ^ { ( t ) } , \mathbf { h } _ { j ^ { \prime } } ^ { ( t ) } ) ) } .
+$$
+
+We have tried the Euclidean similarity $s _ { h } ( \mathbf { h } _ { i } , \mathbf { h } _ { j } ) = - \| \mathbf { h } _ { i } - \mathbf { h } _ { j } \| ^ { 2 }$ for $s _ { h }$ , as well as the dot-product similarity $s _ { h } ( \mathbf h _ { i } , \mathbf h _ { j } ) = \mathbf h _ { i } ^ { \top } \mathbf h _ { j }$ , and they perform similarly without significant difference.
+
+# B EXTRA EXPERIMENT DETAILS
+
+We fixed the node state vector dimensionality to 32, and graph vector dimensionality to 128 throughout both the graph edit distance learning and binary function similarity search tasks. We tuned this initially on the function similarity search task, which clearly performs better than smaller models. Increasing the model size however leads to overfitting for that task. We directly used the same setting for the edit distance learning task without further tuning. Using larger models there should further improve model performance.
+
+# B.1 LEARNING GRAPH EDIT DISTANCES
+
+In this task the nodes and edges have no extra features associated with them, we therefore initialized the $\mathbf { x } _ { i }$ and $\mathbf { x } _ { i j }$ vectors as vectors of 1s, and the encoder MLP in Eq.1 is simply a linear layer for the nodes and an identity mapping for the edges.
+
+We searched through the following hyperparameters: (1) triplet vs pair training; (2) number of propagation layers; (3) share parameters on different propagation layers or not. Learning rate is fixed at 0.001 for all runs and we used the Adam optimizer Kingma & Ba (2014). Overall we found: (1) triplet and pair training performs similarly, with pair training slightly better, (2) using more propagation layers consistently helps, and increasing the number of propagation layers $T$ beyond 5 may help even more, (3) sharing parameters is useful for performance more often than not.
+
+Intuitively, the baseline WL kernel starts by labeling each node by its degree, and then iteratively updates a node’s representation as the histogram of neighbor node patterns, which is effectively also a graph propagation process. The kernel value is then computed as a dot product of graph representation vectors, which is the histogram of different node representations. When using the kernel with $T$ iterations of computation, a pair of graphs of size $| V |$ can have as large as a $2 \bar { | V | } T$
+
+<table><tr><td>Eval Graphs</td><td>WL kernel</td><td>GNN</td><td>GMN</td></tr><tr><td>n=100,p=0.2</td><td>98.5 /99.4</td><td>96.6/96.8</td><td>96.8/97.7</td></tr><tr><td>n= ：100,p=0.5</td><td>86.7 /97.0</td><td>79.8 / 81.4</td><td>83.1 / 83.6</td></tr><tr><td>n= 200,p=( ：0.2</td><td>99.9 / 100.0</td><td>88.7 /88.5</td><td>89.4 /90.0</td></tr><tr><td>n= 200,p=0.5</td><td>93.5 /99.2</td><td>72.0 / 72.3</td><td>68.3 / 70.1</td></tr></table>
+
+Table 3: Generalization performance on large graphs for the GSL models trained on small graphs with $2 0 \leq n \leq 5 0$ and $0 . 2 \leq p \leq 0 . 5$ .
+
+dimensional representation vector for each graph, and these sets of effective ‘feature’ types are different for different pairs of graphs as the node patterns can be very different. This is an advantage for WL kernel over our models as we used a fixed sized graph vector regardless of the graph size. We evaluate WL kernel for $T$ up to 5 and report results for the best $T$ on the evaluation set.
+
+In addition to the experiments presented in the main paper, we have also tested the generalization capabilities of the proposed models, and we present the extra results in the following.
+
+Train on small graphs, generalize to large graphs. In this experiment, we trained the GSL models on graphs with $n$ sampled uniformly from 20 to 50, and $p$ sampled from range [0.2, 0.5] to cover more variability in graph sizes and edge density for better generalization, and we again fix $k _ { p } = 1 , k _ { n } = 2$ . For evaluation, we tested the best embedding models and matching models on graphs with $n = 1 0 0$ , 200 and $p = 0 . 2 , 0 . 5$ , with results shown in Table 3. We can see that for this task the GSL models trained on small graphs can generalize to larger graphs than they are trained on. The performance falls off a bit on much larger graphs with much more nodes and edges. This is also partially caused by the fact that we are using a fixed sized graph vector throughout the experiments , but the WL kernel on the other hand has much more effective ‘features’ to use for computing similarity. On the other hand, as shown before, when trained on graphs from distributions we care about, the GSL models can adapt and perform much better.
+
+Train on some $k _ { p } , k _ { n }$ combinations, test on other combinations. We have also tested the model trained on graphs with $n \in [ 2 0 , 5 0 ]$ , $p \in [ 0 . 2 , 0 . 5 ]$ , $k _ { p } = 1 , k _ { n } = 2$ , on graphs with different $k _ { p }$ and $k _ { n }$ combinations. In particular, when evaluated on $\bar { k _ { p } } = 1 , k _ { n } = 4$ , the models perform much better than on $k _ { p } = 1 , k _ { n } = 2$ , reaching 1.0 AUC and $100 \%$ triplet accuracy easily, as this is considerably simpler than the $k _ { p } = 1 , k _ { n } = 2$ setting. When evaluated on graphs with $k _ { p } = 2 , k _ { n } = 3$ , the performance is workse than $k _ { p } = 1 , k _ { n } = 2$ as this is a harder setting.
+
+In addition, we have also tried training on the more difficult setting $k _ { p } = 2 , k _ { n } = 3$ , and evaluate the models on graphs with $k _ { p } = 1 , k _ { n } = 2$ and $n \in [ 2 0 , 5 0 ] , p \in [ 0 . 2 , 0 . 5 ]$ . The performance of the models on these graphs are actually be better than the models trained on this setting of $k _ { p } = 1 , k _ { n } = 2$ , which is surprising and clearly demonstrates the value of good training data. However, in terms of generalizing to larger graphs models trained on $k _ { p } = 2 , k _ { n } = 3$ does not have any significant advantages.
+
+# B.2 BINARY FUNCTION SIMILARITY SEARCH
+
+In this task the edges have no extra features so we initialize them to constant vectors of 1s, and the encoder MLP for the edges is again just an identity mapping. When using the CFG graph structure only, the nodes are also initialized to constant vectors of 1s, and the encoder MLP is a linear layer. In the case when using assembly instructions, we have a list of assembly code associated with each node. We extracted the operator type (e.g. add, mov, etc.) from each instruction, and then embeds each operator into a vector, the initial node representation is a sum of all operator embeddings.
+
+We searched through the following hyperparameters: (1) triplet or pair training, (2) learning rate in $\lbrace 1 0 ^ { - 3 } , 1 0 ^ { - 4 } \rbrace$ , (3) number of propagation layers; (4) share propagation layer parameters or not; (5) GRU vs one-layer MLP for the $f _ { \mathrm { n o d e } }$ module.
+
+Overall we found that (1) triplet training performs slightly better than pair training in this case; (2) both learning rates can work but the smaller learning rate is more stable; (3) increasing number of propagation layers generally helps; (4) using different propagation layer parameters perform better than using shared parameters; (5) GRUs are more stable than MLPs and performs overall better.
+
+![](images/593899a902ab852a2843b54ce2b3265206c74babd8a7390dfb46dede3315c18f.jpg)  
+Figure 5: Example control flow graphs for the same binary function, compiled with different compilers (clang for the leftmost one, gcc for the others) and optimization levels. Note that each node in the graphs also contains a set of assembly instructions which we also take into account when computing similarity using learned features.
+
+In addition to the results reported in the main paper, we have also tried the same models on another dataset obtained by compiling the compression software unrar with different compilers and optimization levels. Our graph similarity learning methods also perform very well on the unrar data, but this dataset is a lot smaller, with around 400 functions only, and overfitting is therefore a big problem for any learning based model, so the results on this dataset are not very reliable to draw any conclusions.
+
+A few more control-flow graph examples are shown in Figure 5. The distribution of graph sizes in the training set is shown in Figure 6.
+
+# C EXTRA ATTENTION VISUALIZATIONS
+
+A few more attention visualizations are included in Figure 7, Figure 8 and Figure 9. Here the graph matching model we used has shared parameters for all the propagation and matching layers and was trained with 5 propagation layers. Therefore we can use a number $T$ different from the number of propagation layers the model is being trained on to test the model’s performance. In both visualizations, we unrolled the propagation for up to 9 steps and the model still computes sensible attention maps even with $T > 5$ .
+
+![](images/8657a3e5631d8bd6ee1f6dc76ac824725490c4acac7510eb78f3f751399c367e.jpg)  
+Figure 6: Control flow graph size distribution in the training set. In this plot the graphs are sorted by size on the $\mathbf { X }$ axis, each point in the figure corresponds to the size of one graph.
+
+Note that the attention maps do not converge to very peaked distributions. This is partially due to the fact that we used the node state vectors both to carry information through the propagation process, as well as in the attention mechanism as is. This makes it hard for the model to have very peaked attention as the scale of these node state vectors won’t be very big. A better solution is to compute separate key, query and value vectors for each node as done in the tensor2tensor self-attention formulation Vaswani et al. (2017), which may further improve the performance of the matching model.
+
+Figure 7 shows another possibility where the attention maps do not converge to very peaked distributions because of in-graph symmetries. Such symmetries are very typical in graphs. In this case even though the attention maps are not peaked, the cross graph communication vectors $\mu$ are still zero, and the two graphs will still have identical representation vectors.
+
+![](images/f69e60bd25ae00a37a03bd4120248e00d16c02d8c27bf6cf1d1155a28faa1217.jpg)  
+Figure 7: The change of cross-graph attention over propagation layers. Here the two graphs are two isomorphic chains and there are some in-graph symmetries. Note that in the end the nodes are matched to two corresponding nodes with equal weight, except the one at the center of the chain which can only match to a single other node.
+
+![](images/5bdf0211064fb6cc7078e04b791f4eb52b50e9dd5275d0e3abac2e6f046e5e4c.jpg)  
+Figure 8: The change of cross-graph attention over propagation layers. Here the two graphs are isomorphic, with graph edit distance 0. Note that in the end a lot of the matchings concentrated on the correct match.
+
+![](images/7abeb45f8c7ff4e0d7cc0be473c7585f5ba2fb4d7fb80b63d7d6a4e0bc6c8519.jpg)  
+Figure 9: The change of cross-graph attention over propagation layers. The edit distance between these two graphs is 1.